The Finite Element Method for Solid and Structural Mechanics Sixth edition
Professor O.C. Zienkiewicz, CBE, FRS, FREng is Professor Emeritus at the Civil and Computational Engineering Centre, University of Wales Swansea and previously Director of the Institute for Numerical Methods in Engineering at the University of Wales Swansea, UK. He holds the UNESCO Chair of Numerical Methods in Engineering at the Technical University of Catalunya, Barcelona, Spain. He was the head of the Civil Engineering Department at the University of Wales Swansea between 1961 and 1989. He established that department as one of the primary centres of finite element research. In 1968 he became the Founder Editor of the International Journal for Numerical Methods in Engineering which still remains today the major journal in this field. The recipient of 27 honorary degrees and many medals, Professor Zienkiewicz is also a member of five academies - an honour he has received for his many contributions to the fundamental developments of the finite element method. In 1978, he became a Fellow of the Royal Society and the Royal Academy of Engineering. This was followed by his election as a foreign member to the US Academy of Engineering (1981), the Polish Academy of Science (1985), the Chinese Academy of Sciences (1998), and the National Academy of Science, Italy (Academia dei Lincei) (1999). He published the first edition of this book in 1967 and it remained the only book on the subject until 1971. Professor R.L. Taylor has more than 40 years' experience in the modelling and simulation of structures and solid continua including two years in industry. He is Professor in the Graduate School and the Emeritus T.Y. and Margaret Lin Professor of Engineering at the University of California at Berkeley. In 1991 he was elected to membership in the US National Academy of Engineering in recognition of his educational and research contributions to the field of computational mechanics. Professor Taylor is a Fellow of the US Association of Computational Mechanics - USACM (1996) and a Fellow of the International Association of Computational Mechanics- IACM (1998). He has received numerous awards including the Berkeley Citation, the highest honour awarded by the University of California at Berkeley, the USACM John von Neumann Medal, the IACM Gauss-Newton Congress Medal and a Dr.-Ingenieur ehrenhalber awarded by the Technical University of Hannover, Germany. Professor Taylor has written several computer programs for finite element analysis of structural and non-structural systems, one of which, FEAP, is used world-wide in education and research environments. A personal version, FEAPpv, available from the publisher's website, is incorporated into the book.
The Finite Element M ethod for Solid and Structural Mechanics Sixth edition O.C. Zienkiewicz, CBE, FRS
UNESCO Professor of Numerical Methods in Engineering International Centre for Numerical Methods in Engineering, Barcelona Previously Director of the Institute for Numerical Methods in Engineering University of Wales Swansea
R.L. Taylor
Professor in the Graduate School Department of Civil and Environmental Engineering University of California at Berkeley Berkeley, California
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Butterworth-Heinemann is an imprint of Elsevier Linacre House, Jordan Hill, Oxford OX2 8DP, UK 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA First edition published in 1967 by McGraw-Hill Fifth edition published by Butterworth-Heinemann 2000 Reprinted 2002 Sixth edition 2005 Reprinted 2006 (twice) Copyright 9 2000, 2005, O.C. Zienkiewicz and R.L. Taylor. Published by Elsevier Ltd. All rights reserved The rights of O.C. Zienkiewicz and R.L. Taylor to be identified as the authors of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988 No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier's Science & Technology Rights Department in Oxford, UK: phone: (+44) (0) 1865 843830; fax: (+44) (0) 1865 853333; email:
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British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN-13:978-0-7506-6321-2 ISBN-10:0-7506-6321-9 For information on all Butterworth-Heinemann publications visit our website at books.elsevier.com Printed and bound in Great Britain 06 07 08 09 10 10 9 8 7 6 5 4 3
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Dedication This book is dedicated to our wives Helen and Mary Lou and our families for their support and patience during the preparation of this book, and also to all of our students and colleagues who over the years have contributed to our knowledge of the finite element method. In particular we would like to mention Professor Eugenio Ofiate and his group at CIMNE for their help, encouragement and support during the preparation process.
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Contents
P refa c e
1.
2.
,
,
xiii
General problems in solid mechanics and non-linearity 1.1 Introduction 1.2 Small deformation solid mechanics problems 1.3 Variational forms for non-linear elasticity 1.4 Weak forms of governing equations 1.5 Concluding remarks References
4 12 14 15 15
Galerkin method of approximation- irreducible and mixed forms 2.1 Introduction 2.2 Finite element approximation- Galerkin method 2.3 Numerical integration - quadrature 2.4 Non-linear transient and steady-state problems 2.5 Boundary conditions: non-linear problems 2.6 Mixed or irreducible forms 2.7 Non-linear quasi-harmonic field problems 2.8 Typical examples of transient non-linear calculations Concluding remarks 2.9 References
17 17 17 22 24 28 33 37 38 43 44
Solution of non-linear algebraic equations 3.1 Introduction 3.2 Iterative techniques 3.3 General remarks - incremental and rate methods References
46 46 47 58 60
Inelastic and non-linear materials 4.1 Introduction 4.2 Viscoelasticity - history dependence of deformation 4.3 Classical time-independent plasticity theory 4.4 Computation of stress increments
62 62 63 72 80
1 1
viii
Contents
4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13
Isotropic plasticity models Generalized plasticity Some examples of plastic computation Basic formulation of creep problems Viscoplasticity- a generalization Some special problems of brittle materials Non-uniqueness and localization in elasto-plastic deformations Non-linear quasi-harmonic field problems Concluding remarks References
85 92 95 100 102 107 112 116 118 120
5.
Geometrically non-linear problems - finite deformation 5.1 Introduction 5.2 Governing equations Variational description for finite deformation 5.3 5.4 Two-dimensional forms A three-field, mixed finite deformation formulation 5.5 A mixed-enhanced finite deformation formulation 5.6 Forces dependent on deformation- pressure loads 5.7 Concluding remarks 5.8 References
127 127 128 135 143 145 150 154 155 156
6.
Material constitution for finite deformation 6.1 Introduction 6.2 Isotropic elasticity 6.3 Isotropic viscoelasticity 6.4 Plasticity models 6.5 Incremental formulations 6.6 Rate constitutive models 6.7 Numerical examples 6.8 Concluding remarks References
158 158 158 172 173 174 176 178 185 189
7.
Treatment of constraints - contact and fled interfaces 7.1 Introduction 7.2 Node-node contact: Hertzian contact 7.3 Tied interfaces 7.4 Node-surface contact 7.5 Surface-surface contact 7.6 Numerical examples 7.7 Concluding remarks References
191 191 193 197 200 218 219 224 224
Pseudo-rigid and rigid-flexible bodies 8.1 Introduction 8.2 Pseudo-rigid motions 8.3 Rigid motions
228 228 228 230
Contents
8.4 8.5 8.6 9.
Connecting a rigid body to a flexible body Multibody coupling by joints Numerical examples References
Discrete element methods 9.1 Introduction 9.2 Early DEM formulations Contact detection 9.3 9.4 Contact constraints and boundary conditions Block deformability 9.5 9.6 Time integration for discrete element methods 9.7 Associated discontinuous modelling methodologies 9.8 Unifying aspects of discrete element methods 9.9 Concluding remarks References
10. Structural mechanics problems in one dimension- rods 10.1 Introduction 10.2 Governing equations 10.3 Weak (Galerkin) forms for rods 10.4 Finite element solution: Euler-Bernoulli rods 10.5 Finite element solution: Timoshenko rods 10.6 Forms without rotation parameters 10.7 Moment resisting frames 10.8 Concluding remarks References 11. Plate bending approximation: thin (Kirchhoff) plates and C1 continuity requirements 11.1 Introduction 11.2 The plate problem: thick and thin formulations 11.3 Rectangular element with comer nodes (12 degrees of freedom) 11.4 Quadrilateral and parallelogram elements 11.5 Triangular element with comer nodes (9 degrees of freedom) 11.6 Triangular element of the simplest form (6 degrees of freedom) 11.7 The patch test- an analytical requirement 11.8 Numerical examples 11.9 General remarks 11.10 Singular shape functions for the simple triangular element 11.11 An 18 degree-of-freedom triangular element with conforming shape functions 11.12 Compatible quadrilateral elements 11.13 Quasi-conforming elements 11.14 Hermitian rectangle shape function 11.15 The 21 and 18 degree-of-freedom triangle 11.16 Mixed formulations - general remarks
234 237 240 242 245 245 247 250 256 260 267 270 271 272 273 278 278 279 285 290 305 317 319 320 320 323 323 325 336 340 340 345 346 348 357 357 360 361 362 363 364 366
ix
x
Contents
11.17 11.18 11.19 11.20 11.21
Hybrid plate elements Discrete Kirchhoff constraints Rotation-free elements Inelastic material behaviour Concluding remarks- which elements? References
368 369 371 374 376 376
12. 'Thick' Reissner-Mindlin plates- irreducible and mixed formulations 12.1 Introduction 12.2 The irreducible formulation- reduced integration 12.3 Mixed formulation for thick plates 12.4 The patch test for plate bending elements 12.5 Elements with discrete collocation constraints 12.6 Elements with rotational bubble or enhanced modes 12.7 Linked interpolation- an improvement of accuracy 12.8 Discrete 'exact' thin plate limit 12.9 Performance of various 'thick' plate elements - limitations of thin plate theory 12.10 Inelastic material behaviour 12.11 Concluding remarks - adaptive refinement References
382 382 385 390 392 397 405 408 413
13. Shells 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8
426 426 428 429 431 435 440 440 441 450
as an assembly of fiat elements Introduction Stiffness of a plane element in local coordinates Transformation to global coordinates and assembly of elements Local direction cosines 'Drilling' rotational stiffness- 6 degree-of-freedom assembly Elements with mid-side slope connections only Choice of element Practical examples References
14. Curved rods and axisymmetric shells 14.1 Introduction 14.2 Straight element 14.3 Curved elements 14.4 Independent slope-displacement interpolation with penalty functions (thick or thin shell formulations) References 15. Shells as a special case of three-dimensional analysis - Reissner-Mindlin assumptions 15.1 Introduction 15.2 Shell element with displacement and rotation parameters 15.3 Special case of axisymmetric, curved, thick shells 15.4 Special case of thick plates
415 419 420 421
454 454 454 461 468 473
475 475 475 484 487
Contents
15.5 15.6 15.7 15.8
Convergence Inelastic behaviour Some shell examples Concluding remarks References
487 488 488 493 495
16. Semi-analytical finite element processes - use of orthogonal functions
and 'finite strip' methods 16.1 Introduction 16.2 Prismatic bar 16.3 Thin membrane box structures 16.4 Plates and boxes with flexure 16.5 Axisymmetric solids with non-symmetrical load 16.6 Axisymmetric shells with non-symmetrical load 16.7 Concluding remarks References
498 498 501 504 505 507 510 514 515
17. Non-linear structural problems- large displacement and instability 17.1 Introduction 17.2 Large displacement theory of beams 17.3 Elastic stability- energy interpretation 17.4 Large displacement theory of thick plates 17.5 Large displacement theory of thin plates 17.6 Solution of large deflection problems 17.7 Shells 17.8 Concluding remarks References
517 517 517 523 526 532 534 537 542 543
18. Multiscale modelling 18.1 Introduction 18.2 Asymptotic analysis 18.3 Statement of the problem and assumptions 18.4 Formalism of the homogenization procedure 18.5 Global solution 18.6 Local approximation of the stress vector 18.7 Finite element analysis applied to the local problem 18.8 The non-linear case and bridging over several scales 18.9 Asymptotic homogenization at three levels: micro, meso and macro 18.10 Recovery of the micro description of the variables of the problem 18.11 Material characteristics and homogenization results 18.12 Multilevel procedures which use homogenization as an ingredient 18.13 General first-order and second-order procedures 18.14 Discrete-to-continuum linkage 18.15 Local analysis of a unit cell 18.16 Homogenization procedure - definition of successive yield surfaces
547 547 549 550 552 553 554 555 560 561 562 565 567 570 572 578 578
xi
xii
Contents
18.17 Numerically developed global self-consistent elastic-plastic constitutive law 18.18 Global solution and stress-recovery procedure 18.19 Concluding remarks References
580 581 586 587
19. Computer procedures for finite element analysis 19.1 Introduction 19.2 Solution of non-linear problems 19.3 Eigensolutions 19.4 Restart option 19.5 Concluding remarks References
590 590 591 592 594 595 595
Appendix A Appendix B
597 604 609 619
Author index Subject index
Isoparametric finite element approximations Invariants of second-order tensors
Preface It is thirty-eight years since the The Finite Element Method in Structural and Continuum Mechanics was first published. This book,which was the first dealing with the finite element method, provided the basis from which many further developments occurred. The expanding research and field of application of finite elements led to the second edition in 1971, the third in 1977, the fourth as two volumes in 1989 and 1991 and the fifth as three volumes in 2000. The size of each of these editions expanded geometrically (from 272 pages in 1967 to the fifth edition of 1482 pages). This was necessary to do justice to a rapidly expanding field of professional application and research. Even so, much filtering of the contents was necessary to keep these editions within reasonable bounds. In the present edition we retain the three volume format of the fifth edition but have decided not to pursue having three contiguous volumes - rather we treat the whole work as an assembly of three separate works. Each one is capable of being used without the others and each one appeals perhaps to a different audience. Though naturally we recommend the use of the whole ensemble to people wishing to devote much of their time and study to the finite element method. The first volume is renamed The Finite Element Method: Its l~asis and Fundamentals. This volume covers the topic starting from a physical approach4o.solve problems in linear elasticity. The volume then presents a mathematical framework from which general problems may be formulated and solved using variational and Galerkin methods. The general topic of shape functions is also presented for situations in which the approximating functions are C Ocontinuous. The two- and three-dimensional problems of linear elasticity are then presented in a unified manner using higher order shape functions. This is followed by consideration of quasi-harmonic problems governed by Laplace and Poisson differential equations. The patch test is introduced and used as a means to guarantee convergence of the method. We also cover in some depth solution forms using mixed methods with special consideration given to problems in which incompressibility can occur. The solution of transient problems is presented using semi-discrete formulations and finite element in time concepts. The volume concludes with a presentation of coupled problems. In this volume we consider more advanced problems in solid and structural mechanics while in a third volume we consider applications in fluid dynamics. It is our intent that the present volume can be used by investigators familiar with the finite element method N
xiv Preface
at the level presented in the first volume or any other basic textbook on the subject. However, the volume has been prepared such that it can stand alone. The volume has been reorganized from the previous edition to cover consecutively two main subject areas. In the first part we consider non-linear problems in solid mechanics and in the second part linear and non-linear problems in structural mechanics. In Chapters 1 to 9 we consider non-linear problems in solid mechanics. In these chapters the special problems of solving non-linear equation systems are addressed. We begin by restricting our attention to non-linear behaviour of materials while retaining the assumptions on small strain. This serves as a bridge to more advanced studies later in which geometric effects form large displacements and deformations are presented. Indeed, non-linear applications are of great importance today and of practical interest in most areas of engineering and physics. By starting our study first using a small strain approach we believe the reader can more easily comprehend the various aspects which need to be understood to master the subject matter. We cover in some detail formulations of material models for viscoelasticity, plasticity and viscoplasticity which should serve as a basis for applications to other material models. In our study of finite deformation problems we present a series of approaches which may be used to solve problems including extensions for treatment of constraints such as near incompressibility, rigid and multi-body motions and discrete element forms. The chapter on discrete element methods was prepared by Professor Nenad Bi6ani6 of the University of Glasgow, UK. In the second part of the volume we consider problems in structural mechanics. In this class of applications the dimension of the problem is reduced using basic kinematic assumptions. We begin the presentation in a new chapter that considers rod problems where two of the dimensions of the structure are small compared to the third. This class of problems is a combination of beam bending, axial extension and torsion. Again we begin from a small strain assumption and introduce alternative forms of approximation for the Euler-Bernoulli and the Timoshenko theory. In the former theory it is necessary now to use C 1 interpolation (i.e. continuous displacement and slope) to model the bending behaviour, whereas in the latter theory use of C O interpolation is permitted when special means are included to avoid 'locking' in the transverse shear response. Based upon the study of rods we then present a detailed study of problems in which only one dimension is small compared to the other two. Building on the results from rods we present a coverage for thin plates (Kirchhoff theory), thick plates (Reissner-Mindlin theory) and their corresponding forms for shells. We then consider the problem of large strains and present forms for buckling and large displacements. The volume includes a new chapter on multi-scale effects. This is a recent area of much research and the chapter presents a summary of some notable recent results. We are indebted to Professor Bernardo Schrefler of the University of Padova, Italy, for preparing this timely contribution. The volume concludes with a short chapter on computational methods that describes a companion computer program that can be used to solve several of the problem classes described in this volume. We emphasize here the fact that all three of our volumes stress the importance of considering the finite element method as a unique and whole basis of approach and that it contains many of the other numerical analysis methods as special cases. Thus, imagination and knowledge should be combined by the readers in their endeavours.
Preface
The authors are particularly indebted to the International Centre of Numerical Methods in Engineering (CIMNE) in Barcelona who have allowed their pre- and postprocessing code (GiD) to be accessed from the web site. This allows such difficult tasks as mesh generation and graphic output to be dealt with efficiently. The authors are also grateful to Professor Eric Kasper for his careful scrutiny of the entire text. We also acknowledge the assistance of Matt Salveson who also helped in proofreading the text.
Resources to accompany this book
Complete source code and user manual for program FEAPpv may be obtained at no cost from the publisher's web page: http://books.elsevier.com/companions/or from the author's web page: http://www.ce.berkeley.edu/~rlt OCZ and RLT
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General problems problems in solid mechanics mechanics and non-linearity 1.1 Introduction Many introductory texts on the finite element method discuss the solution for linear I 3 problems of of elasticity and field equations. 1-3 - In practical applications the limitation of linear elasticity, or more generally of linear behaviour, often precludes precludes obtaining an accurate assessment of the solution because because of the presence of 'non-linear' effects and/or because the geometry has a 'thin' dimension in one or more directions. In this book we describe extensions to the formulations introduced to solve linear problems to permit solutions to both classes of of problems. Non-linear behaviour of solids takes two forms: material non-linearity and geometric non-linearity. The simplest form of of non-linear material behaviour is that of of elasticity for which the stress is not linearly proportional to the strain. More general situations are those in which the loading and unloading response of the material is different. Typical here is the case of classical elastic-plastic behaviour. When the deformation of a solid reaches a state for which the undeformed and deformed shapes are substantially different a state offinite deformation occurs. In this strain-displacement or equilibrium equations case it is no longer possible to write linear strain-displacement on the undeformed geometry. Even before finite deformation exists it is possible to observe buckling or load bifurcations in some solids and non-linear equilibrium effects need to be considered. The classical Euler column, where the equilibrium of this class equation for buckling includes the effect of axial loading, is an example of of problem. When deformation is large the boundary conditions can also become nonlinear. Examples are pressure loading that remains normal to the deformed body and also the case where the deformed boundary interacts with another body. This latter example defines a class known as contact problems and much research is currently performed in this area. An example of a class of problems involving non-linear effects behaviour and contact is the analysis of of a rolling in deformation measures, material behaviour tyre. A typical mesh for a tyre analysis is shown in Fig. 1.1. The cross-section shown character of a tread. is able to model the layering of rubber and cords and the overall character The full mesh is generated by sweeping the cross-section around the wheel axis with a variable spacing in the area which will be in contact. A formulation in which the mesh is fixed and the material rotates is commonly used to perform the analysis.4-7 analysis. 4-7
2 General problems in solid mechanics and non-linearity
(a) Tyre cross-section.
(b) Full mesh.
Fig. 1.1 Finite element mesh for tyre analysis.
Generally the accurate solution of solid problems which have one (or more) small dimension(s) compared to the others cannot be achieved efficiently using standard two- or three-dimensional finite element formulations. Traditionally separate theories of structural mechanics are introduced to solve this class of problems. A plate is a fiat structure with one thin (small) direction which is called the thickness. A shell is a curved structure in space with one such small thickness direction. Structures with two small dimensions are called beams, frames, or rods. A primary reason why use of standard two- or three-dimensional finite element formulations do not yield accurate solutions is the numerical ill-conditioning which results in their algebraic equations. In this book we combine the traditional approaches of structural mechanics with a much stronger link to the full three-dimensional theory of solids to obtain formulations which are easily solved using standard finite element approaches. This book considers both solid and structural mechanics problems and formulations which make practical finite element solutions feasible. We divide the volume into two main parts. In the first part we consider problems in which continuum theory of solids continues to be used, whereas in the second part we focus attention on theories of structural mechanics to describe the behaviour of rods, plates and shells. In the present chapter we review the general equations for analysis of solids in which deformations remain 'small' but material behaviour includes effects of a nonlinear kind. We present the theory in both an indicial (or tensorial) form as well as in the matrix form commonly used in finite element developments. We also reformulate the equations of solids in a variational (Galerkin) form. In Chapter 2 we present a general scheme based on the Galerkin method to construct a finite element approximate solution to problems based on variational forms. In this chapter we consider both irreducible
Introduction
and mixed forms of finite element approximation and indicate where the mixed forms have distinct advantages. Here we also show how the linear problems of solids for steady state and transient behaviour become non-linear when the material constitutive model is represented in a non-linear form. Some discussion on the solution of transient non-linear finite element forms is included. Since the form of the inertial effects is generally unaffected by non-linearity, in the remainder of this volume we shall primarily confine our remarks to terms arising from non-linear material behaviour and finite deformation effects. In Chapter 3 we describe various possible methods for solving non-linear algebraic equations. This is followed in Chapter 4 by consideration of material non-linear behaviour and completes the development of a general formulation from which a finite element computation can proceed. In Chapter 5 we present a summary for the study of finite deformation of solids. Basic relations for defining deformation are presented and used to write variational (Galerkin) forms related to the undeformed configuration of the body and also to the deformed configuration. It is shown that by relating the formulation to the deformed body a result is obtained which is nearly identical to that for the small deformation problem we considered in the small deformation theory treated in the early chapters of this volume. Essential differences arise only in the constitutive equations (stress-strain laws) and the addition of a new stiffness term commonly called the geometric or initial stress stiffness. For constitutive modelling we summarize in Chapter 6 alternative forms for elastic and inelastic materials. Contact problems are discussed in Chapter 7. Here we summarize methods commonly used to model the interaction of intermittent contact between surfaces of bodies. In Chapter 8 we show that analyses of rigid and so-called pseudo-rigid bodies 8 may be developed directly from the theory of deformable solids. This permits the inclusion in programs of options for multi-body dynamic simulations which combine deformable solids with objects modelled as rigid bodies. In Chapter 9 we discuss specialization of the finite deformation problem to address situations in which a large number of small bodies interact [multi-particle or granular bodies commonly referred to as discrete element methods (DEM) or discrete deformation analysis (DDA)]. In the second part of this book we study the behaviour of problems of structural mechanics. In Chapter 10 we present a summary of the behaviour of rods (beams) modelled by linear kinematic behaviour. We consider cases where deformation effects include axial, bending and transverse shearing strains (Timoshenko beam theory 9) as well as the classical theory where transverse effects are neglected (Euler-Bernoulli theory). We then describe the solution of plate problems, considering first the problem of thin plates (Chapter 11) in which only bending deformations are included and, second, the problem in which both bending and shearing deformations are present (Chapter 12). The problem of shell behaviour adds in-plane membrane deformations and curved surface modelling. Here we split the problem into three separate parts. The first combines simple flat elements which include bending and membrane behaviour to form a faceted approximation to the curved shell surface (Chapter 13). Next we involve the addition of shearing deformation and use of curved elements to solve axisymmetric shell problems (Chapter 14). We conclude the presentation of shells with a general form using curved isoparametric element shapes which include the effects of bending,
3
4 General problems in solid mechanics and non-linearity shearing, and membrane deformations (Chapter 15). Here a very close link with the full three-dimensional analysis will be readily recognized. In Chapter 16 we address a class of problems in which the solution in one coordinate direction is expressed as a series, for example a Fourier series. Here, for linear material behaviour, very efficient solutions can be achieved for many problems. Some extensions to non-linear behaviour are also presented. In Chapter 17 we specialize the finite deformation theory to that which results in large displacements but small strains. This class of problems permits use of all the constitutive equations discussed for small deformation problems and can address classical problems of instability. It also permits the construction of non-linear extensions to plate and shell problems discussed in Chapters 11-15 of this volume. We conclude the descriptions applied to solids in Chapter 18 with a presentation of multi-scale effects in solids. In the final chapter we summarize the capabilities of a companion computer program (called FEAPpv) that is available at the publisher's web site. This program may be used to address the class of non-linear solid and structural mechanics problems described in this volume.
1.2.1 Strong form of equations- indicial notation In this general section we shall describe how the various equations of solid mechanics* can become non-linear under certain circumstances. In particular this will occur for solid mechanics problems when non-linear stress-strain relationships are used. The chapter also presents the notation and the methodology which we shall adopt throughout this book. The reader will note how simply the transition between forms for linear and non-linear problems occurs. The field equations for solid mechanics are given by equilibrium behaviour (balance of momentum), strain-displacement relations, constitutive equations, boundary conditions, and initial conditions. 1~ In the treatment given here we will use two notational forms. The first is a cartesian tensor indicial form and the second is a matrix form (see reference 1 for additional details on both approaches). In general, we shall find that both are useful to describe particular parts of formulations. For example, when we describe large strain problems the development of the so-called 'geometric' or 'initial stress' stiffness is most easily described by using an indicial form. However, in much of the remainder, we shall find that it is convenient to use a matrix form. The requirements for transformations between the two will also be indicated. In the sequel, when we use indicial notation an index appearing once in any term is called a free index and a repeated index is called a dummy index. A dummy index may only appear twice in any term and implies summation over the range of the index. * More general theories for solid mechanics problems exist that involve higher order micro-polar or couple stress effects; however, we do not consider these in this volume.
Small deformation solid mechanics problems 5 Thus if two vectors ai and bi each have three terms the form ai bi implies aibi -- a l b l -Jr a2b2 -Jr-a3b3
Note that a dummy index may be replaced by any other index without changing the meaning, accordingly aibi =~ a j b j
Coordinates and displacements
For a fixed Cartesian coordinate system we denote coordinates as x, y, z or in index form as x l, x2, x3. Thus the vector of coordinates is given by X
: Xlel -4- x2e2 -+- x3e3 : xi ei
in which ei are unit base vectors of the Cartesian system and the summation convention described above is adopted. Similarly, the displacements will be denoted as u, v, w or ul, u2, u3 and the vector of displacements by u2e2 + u3e3 -- ui ei
u -- u~e~ +
Generally, we will denote all quantifies by their components and where possible the coordinates and displacements will be denoted as xi and ui, respectively, in which the range of the index i is 1,2, 3 for three-dimensional applications (or 1,2 for twodimensional problems).
Stra in-displacemen t relations
The strains may be expressed in Cartesian tensor form as (1.1) and are valid measures provided deformations are small. By a small deformation problem we mean that I~ij[ < < 1
and
Ioj2jl < < lIE/jI[
where [-I denotes absolute value and II 9 [I a suitable norm. In the above o.)ij denotes a small rotation given by ~vij -
l/Oui
OuJ /
~
Ox i
Ox j
(1.2)
and thus the displacement gradient may be expressed as Oui Oxj
: ~ij + (a')ij
(1.3)
6 General problems in solid mechanics and non-linearity
Equilibrium equations - balance of momentum
The equilibrium equations (balance of linear momentum) are given in index form as
ffji,j
Jr
bi
--
i, j = 1, 2, 3
p/~i,
(1.4)
where aij are components of (Cauchy) stress, p is mass density, and bi are body force components. In the above, and in the sequel, we use the convention that the partial derivatives are denoted by
Of f,i-- OXi
Of
and
3~ = -~-
for coordinates and time, respectively. Similarly, moment equilibrium (balance of angular momentum) yields symmetry of stress given in indicial form as (1.5)
(Tij - - a j i
Equations (1.4) and (1.5) hold at all points xi in the domain of the problem f2.
Boundary conditions
Stress boundary conditions are given by the traction condition ti - - f f j i n j
(1.6)
- - ti
for all points which lie on the part of the boundary denoted as Ft. A quantity with a 'bar' denotes a specified function. Similarly, displacement boundary conditions are given by Ui -- Ui
(1.7)
and apply for all points which lie on the part of the boundary denoted as F~. Many additional forms of boundary conditions exist in non-linear problems. Conditions where the boundary of one part interacts with another part, so-called contact conditions, will be taken up in Chapter 7. Similarly, it is necessary to describe how loading behaves when deformations become large. Follower pressure loads are one example of this class and we consider this further in Sec. 5.7.
Initial conditions
Finally, for transient problems in which the inertia term p/~i is important, initial conditions are required. These are given for an initial time denoted as 'zero' by U i ( X j , O) ~"
di(xj)
and
l ~ i ( X j , O) "~ ~3i(Xj)
in f2
(1.8)
It is also necessary in some problems to specify the state of stress at the initial time.
Constitutive relations
All of the above equations apply to any material provided the deformations remain small. The specific behaviour of a material is described by constitutive equations which relate the stresses to imposed strains and, often, other sources which cause deformation (e.g. temperature).
Small deformation solid mechanics problems 7 The simplest material model is that of linear elasticity where qUite generally _(0) ) - ~kl
O'ij - - C i j k l ( e k l
(1.9a)
in which Cijkl are e l a s t i c m o d u l i and _~0~ kl are strains arising from sources other than displacement. For example, in thermal problems strains result from change in temperature and these may be given by C(0) kl - - a u [ r - To] (1.9b) in which akl are coefficients of linear expansion and T is temperature with To a reference temperature for which thermal strains are zero. For linear i s o t r o p i c materials these relations simplify to O'ij = /~(~ij (Ekk - ekk _(0) ) + 2 #
and
E(klO)
- - t~ijO~ [ T -
(eij
_(0) ) -- e-.ij
To]
(1.10a) (1 10b)
w h e r e A and # are Lam6 elastic parameters and a is a scalar coefficient of linear expansion. 1~ In addition, ~Sij is the Kronecker delta function given by
1; ~ij - -
O;
for i = j for i # j
Many materials are not linear nor are they elastic. The construction of appropriate constitutive models to represent experimentally observed behaviour is extremely complex. In this book we will illustrate a few classical models of behaviour and indicate how they can be included in a general solution framework. Here we only wish to indicate how a non-linear material behaviour affects our formulation. To do this we consider non-linear elastic behaviour represented by a strain-energy density function W in which stress is computed as ll OW
O'iJ - -
tgeij
(1.11)
Materials based on this form are called h y p e r e l a s t i c . When the strain-energy is given by the quadratic form 1
/-,
W -- ~EijCijklEkl
,.(0)
-- Eijt...ijklakl
(1.12)
we obtain the linear elastic model given by Eq. (1.9a). More general forms are permitted, however, including those leading to non-linear elastic behaviour.
1.2.2 Matrix notation
Ixllxll
In this book we will often use a matrix form to write the equations. In this case we denote the coordinates as x=
y Z
=
x2 X3
(1.13)
8 General problems in solid mechanics and non-linearity
lullul
and displacements as U --"
1)
--
(1.14)
U2
w
u3
For two-dimensional forms we often ignore the third component. The transformation to matrix form for stresses is given in the order
~--[~ll
~22
~33
~12
~23
~311 ~
= [~xx ~yy
~zz
~xy
~yz
~zxl ~
C22
E33
"?'12 ')'23 ")/31]T
Eyy
Ezz
")/xy
(1.15)
and strains by ~=
JEll
--[Cxx
~yz
(1.16)
~zx] T
where symmetry of the tensors is assumed and 'engineering' shear strains are introduced as
")'ij = 2cij,
i ~= j
(1.17)
to make writing of subsequent matrix relations in a concise manner. The transformation to the six independent components of stress and strain is performed by using the index order given in Table 1.1. This ordering will apply to many subsequent developments also. The order is chosen to permit reduction to twodimensional applications by merely deleting the last two entries and treating the third entry as appropriate for plane or axisymmetric applications. The strain-displacement equations are expressed in matrix form as e = ,Su
(1.18)
with the three-dimensional strain operator given by -O S T =
0
-
0
o
~2
0
o 0 0 ~3
O ~
~1
0
0
O0
~3
0
~2
0 ~1-
Table 1.1 Index relation between tensor and matrix forms Form
Index value
Matrix Tensor (1, 2, 3)
1
2
3
4
5
6
11
22
33
12 21
23 32
31 13
Cartesian (x, y, z)
xx
yy
zz
xy yx
yz zy
zx xz
Cylindrical (r, z, 0)
rr
zz
O0
rz zr
zO Oz
Or rO
Small deformation solid mechanics problems 9 The same operator may be used to write the equilibrium equations (1.4) as S r o r + b = pii
(1.19)
The boundary conditions for displacement and traction are given by u=fi
OnFu
where Gr =
and
t-Grtr=t
[i n20 00 0
nl 0
n3
onFt
(1.20)
0 n3] 0
n3 n2
nl
in which n = (n~, n2, n3) are direction cosines of the normal to the boundary r'. We note further that the non-zero structure of S and G are the same. For transient problems, initial conditions are denoted by u(x, 0) - d(x)
and
ti(x, 0) - ~(x)
in f2
(1.21)
The constitutive equations for a linear elastic material are given in matrix form by or - O(e - e0)
(1.22)
where in Eq. (1.9a) the index pairs ij and kl for Cijkl are transformed to the 6 x 6 matrix D terms using Table 1.1. For a general hyperelastic material we use or =
OW
(1.23)
Oe
1.2.3 Two-dimensional problems There are several classes of two-dimensional problems which may be considered. The simplest are plane stress in which the plane of deformation (e.g. Xl - x2) is thin and stresses 0 3 3 - - 7-13 - - 7-23 - - 0 ; and plane strain in which the plane of deformation (e.g. X 1 - - X 2 ) is one for which e 3 3 " - - ")/13 - - ")/23 - - 0. Another class is called axisymmetric where the analysis domain is a three-dimensional body of revolution defined in cylindrical coordinates (r, 0, z) but deformations and stresses are two-dimensional functions of r, z only.
Plane stress and plane strain
For plane stress and plane strain problems which have mation, the displacements are assumed in the form U =
Ul(Xl,X2, t)} u2(Xl,X2, t)
X l --
X2
as the plane of defor-
(1.24)
10 General problems in solid mechanics and non-linearity and thus the strains may be defined by: 11 o
=
Cll C22 C33
=Su+e3
o
o
=
0
0
0
0
Ox 2
Ox l
"712
0
{ul)+ o U2
C33
(1.25)
0
Here the C33 is either zero (plane strain) or determined from the material constitution by assuming 0"33is zero (plane stress). The components of stress are taken in the matrix form ( 1.26) or.T = { 0"11 0"22 0"33 712 } where 0"33 is determined from material constitution (plane strain) or taken as zero (plane stress). We note that the local 'energy' term E = crr e
(1.27)
does not involve ~33 for either plane stress or plane strain. Indeed, it is not necessary to compute the 0"33 (or c33) until after a problem solution is obtained. The traction vector for plane problems is given by t=Grcr
where
Gr
~---
~.,[n, ]
[,,
0
l'/2
0 0
n21] nl j
(1.28)
and once again we note that ,5 and G have the same non-zero structure.
Axisymmetricproblems
Ixlllrl
In an axisymmetric problem we use the cylindrical coordinate system
X --
X2
x3
=
Z
(1.29)
0
This ordering permits the two-dimensional axisymmetric and plane problems to be written in a very similar manner. The body is three dimensional but defined by a surface of revolution such that properties and boundaries are independent of the 0 coordinate. For this case the displacement field may be taken as u--
u2(xl, x2, t) u3(xl, x2, t)
and, thus, also is taken as independent of 0.
-- ] Uz(r, z, t) kuo(r, z, t)
(1.3o)
Small deformation solid mechanics problems The strains for the axisymmetric case are given by "11 -0
o 0
~00
E'll E'22 C33
%z %o ")'Or
")q2 %3 ")'31
Err Ezz __
=Su=
o
&2
1
0
{Ul}
X1
a
o
OX2
OX1
0
0
0
0
U2 U3
(1.31)
0
(o 1) 0Xl
X1
The stresses are written in the same order as 03"--{fill
0"22 0"33 TI2
(1.32)
7"23 7"31}
Similar to the three-dimensional problem the traction is given by t =Grtr
where
Gr =
[o 00 n2 0
0 0
nl 0
0 n2
01
(1.33)
nl
where we note that n3 cannot exist for a complete body of revolution. Once again we note that S and G have the same non-zero structure. We note that the strain-displacement relations between the u 1, u2 and u3 components are uncoupled. If the material constitution is also uncoupled between the first four and the last two components of strain (i.e. the first four stresses are related only to the first four strains) we may separate the axisymmetric problem into two parts: (a) a part which depends only on the first four strains which are expressed in ul, u2; and (b) a problem which depends only on the last two shear strains and u3. The first problem is sometimes referred to as torsionless and the second as a torsion problem. However, when the constitution couples the effects, as in classical elastic-plastic solution of a bar which is stretched and twisted, it is necessary to consider the general case. The torsionless axisymmetric problem is given by -0 "Crr ~
"C11
E00
C22 E33
'ffr z
")/12
Ezz
=
=Su=
o
o
1 X1
0
Ul 0
0
0
.OqX2
OX1 -
U2
(1.34)
with stresses given by Eq. (1.26) and tractions by Eq. (1.28). Thus the only difference in these two classes of problems is the presence of the Ul/Xl for the third strain in
11
12 General problems in solid mechanics and non-linearity
the axisymmetric case (of course the two differ also in the domain description of the problem as we shall point out later). iii.......ii '~::'~::~'~~::~'~'............... ~'~::~::~~:~.!i...~............~.~.i...ii....~i.!.i.i.i.i.?~.i.~.!......ii ............................................. ::'::'::::::i~::...................... :': ~ !", i"~'~ii",iiiiiiTiiii'?:'::::"ii"............................................................. ~ ii............................... ': : ~~iiiii : : : iiiii.................................... ii..i.i.iii.. i.iii.ii~.. .i~:~:i.M.i.iT~.~.~.iii!i.i~ii~' ii.~.i~.~i',.~'i~i.,iiiii!iiii!iiiiiiiiiiiiiiii!iili i.i.i.i.i.i.i 'ii, iiii!i'i'i',~ii,i',ii'~i'iii',i',,ii',iiii~,',',',' ii",",i'~iii~ i'~~,'~',",',"~"~ ',}i"~i',",~~,,'"",,' !',' ~ii,"i,i',',i~,i",",i i',~,i"~ ~i~::~,::~i~:::i.i.~i~i~::.-~::~..::~i~-.---::!ii-~v-.::::::~:::::::--~,~..........~. ", !~iil :!i:~ '~......... ::....~i~i~ .:::........, ::~ii .... i~ '~::.... :N~ ~: ...... ::........~::~:':~,~!i:ii~i~i~i' ::~ ,i':!i ~ii. '~s !', iL,,!|~ ...~...~?~g~?~:~~:~
For an elastic material as specified by Eq. (1.23), the above equations may be given in a variational form when no inertial effects are included. The simplest form is the potential energyprinciple where
l"IpE= ~ W ( , . q u ) d ~ 2 - ~ u r b d ~ 2 - f r
urtdF
(1.35)
t
The first variation yields the governing equation of the functional ~rlee =
~(su)r O,Nu d~ -
~urb d~2 -
a s 16
~uridl -' = 0
(1.36)
t
After integration by parts and collecting terms we obtain ~l-Ipe=-- ~ ~ u r ( S r t r + b )
d~2 (1.37)
+fr
~ur ( G r a -
i) d[" = 0
t
where O'--
OW 08u
When W is given by the quadratic form (1.12) we recover the linear problem given by Eq. (1.22). In this case the form becomes the principle of minimum potential energy and the displacement field which renders W an absolute minimum is an exact solution to the problem, ll We note that the potential energy principle includes the strain-displacement equations and the elastic model expressed in terms of displacement-based strains. It also requires the displacement boundary condition to be stated in addition to the theorem. It is, however, the simplest variational form and only requires knowledge of the displacement field to be valid. This form is a basis for irreducible(or displacement) methods of approximate solution. A general variational theorem, which includes all the equations and boundary conditions, is given by the Hu-Washizu variational theorem. 17 This theorem is given by II HW(U ,
0")
f
[W(e) + err (Su - e)] dr2
,If2
(1.38)
_s t
u
in which t = Gr o'. The proof that the theorem contains all the governing equations is obtained by taking the variation of Eq. (1.38) with respect to u, s and o'. Accordingly,
Variational forms for non-linear elasticity taking the variation of (1.38) and performing an integration by parts on 3(,Su) we obtain
5Fl~w = f Ssr [ ~ - ~r] d~2 + f,o'r
[Su-s]
d~-fr
,t r (u-O)dF
(1.39)
u
-f'ur(sr~r+b)
d~+fr
'ur (t-i)
dF=0
t
and it is evident that the Hu-Washizu variational theorem yields all the equations for the non-linear elastostatic problem. We may also establish a direct link between the Hu-Washizu theorem and other variational principles. If we express the strains s in terms of the stresses using the Laurant transformation U (rr) + W (e) = rr Te (1.40) we recover the Hellinger-Reissner variational principle given by 18-2~
1-IMR(U, ~r) = ~_ [crrSu -- U(~r)] dr2
- f urbdf~-fruridF-frtr(u-fi,
dF
(1.41)
In the linear elastic case we have, ignoring initial strain and stress effects, 1
U (o') -- ~ oij Sijkl O'kl
(1.42)
where Sijkl are elastic compliances. While this form is also formally valid for general elastic problems. We shall find that in the non-linear case it is not possible to find unique relationsfor the constitutive behaviour in terms of stress forms. Thus, we shall often rely on use of the Hu-Washizu functional as the basis for a mixed formulation. We may also establish a direct link to the minimum potential energy form and the Hu-Washizu theorem. If we satisfy the displacement boundary condition (1.20) apriori the integral term over 1", is eliminated from Eq. (1.38). Generally, in our finite element approximations based on the Hu-Washizu theorem (or variants of the theorem) we shall satisfy the displacement boundary conditions explicitly and thus avoid approximating the Fu term. If we then satisfy the strain-displacement relations a priori then the Hu-Washizu theorem is identical with the potential energy principle. In constructing finite element approximations, the potential energy principle is a basis for developing displacement models (also referred to as irreduciblemodels1) whereas the Hu-Washizu form is a basis for developing mixedmodels.1As we will show in Chapter 2 mixed methods have distinct advantages in constructing robust finite element formulations. However, there are also advantages in having a finite element formulation where the global problem is expressed in a displacement form. Noting how the Hu-Washizu form reduces to the potential energy principle provides a link on treating the reductions to their approximate counterparts (see Sec. 2.6).
13
14 General problems in solid mechanics and non-linearity One advantage of a variational theorem is that symmetry conditions are automatically obtained; however, a distinct disadvantage is that only elastic behaviour and static forms may be considered. In the next section we consider an alternative approach of weak forms which is valid for both elastic or inelastic material forms and directly admits the inertial effects. We shall observe that for the elastostatic problem a weak form is equivalent to the variation of a theorem.
A variational (weak) form for any set of equations is a scalar relation and may be constructed by multiplying the equation set by an appropriate arbitrary function which has the same free indices as in the set of governing equations (which then becomes a dummy index and sums over its range), integrating over the domain of the problem and setting the result to zero. 1'17
1.4.1 Weak form for equilibrium equation For example, in indicial form the equilibrium equation (1.4) has the free index i, thus to construct a weak form we multiply by an arbitrary vector with index i and integrate the result over the domain f2. Virtual work is a weak form in which the arbitrary function is a virtual displacement ~ui, accordingly using this function we obtain the form
(~I'Ieq -- f ~ t~ui [pu i - o'ji,j -- b i ] d S2 :
0
Generally stress will depend on strains which are derivatives of displacements. Thus, the above form will require computation of second derivatives of displacement to form the integrands. The need to compute second derivatives may be reduced (i.e. 'weakened') by performing an integration by parts and upon noting the symmetry of the stress we obtain
'l-leq :f
t~uipuid~Wf t~Eij(uk)ffijd~-f
'uibid~2-fr'uitid~--O (1.43)
where virtual strains are related to virtual displacements as
t~'Cij(Uk) = l (t~Ui,j -~-(~Uj,i)
(1.44)
This may be further simplified by splitting the boundary into parts where traction is specified, Ft, and parts where displacements are specified, Fu. If we enforce pointwise all the displacement boundary conditions* and impose a constraint that ~u~ vanishes on Fu, we obtain the final result
t~I'Ieq--f t~uipi~lid~"~f-f t~Eij(Uk)O'ijd~'~--f (~uibid~'~-fF (~ui-tid~'~--- 0 t
(1.45) * Alternatively, we can combine this term with another from the integration by parts of the weak form of the strain--displacement equations.
References References 15 15
in matrix form as as or in
1n
r
r
r
T T 80 ( ~ I ' Ieq e q -= L 8uT ~uT piidQ piidQ + + L 8(SU)T ~(Su)r udQ o ' d Q -- L Bu ~uT bdQ b d Q -- f r Bu ~ u Ttdr tdF = = 0
in
in
ir,
t
(1.46)
The first term is the virtual work of internal inertial forces, the second the virtual work of the internal stresses and the last two the virtual work of body and traction forces, respectively. The above weak form provides the basis from which a finite element formulation of equilibrium may be deduced for general applications. It is necessary to add appropriate expressions for the strain-displacement and constitutive equations to complete a problem formulation. Weak forms for these may be written immediately from the of the Hu-Washizu principle given in Eq. (1.39). variation ofthe We note that the form adopted to define the matrices of stress and strain permits the internal work of stress and strain to be written as Eij O'ij - - "cTo " :
(1.47)
0 "T
Similarly, the internal virtual work per unit volume may be expressed by (~W = : &ij (~"Cij aij o'ij = : 8e (~~ TT u or" 8W
(1.48)
In Chapter Chapter 4 we will discuss this in more detail and show that constructing constructing constitutive equations of six components components of of stress and strain must be treated appropriately equations in terms of in reductions reductions from the original original nine tensor components. components.
i 1.5 i ii!iConcluding ii ijii remarks iii iii!! iiii !
i
i iiliii ii!iii!iii i iiiiiiii
In this chapter chapter we we have summarized summarized the basic basic steps needed needed to formulate a general general The formulation has has been been presented presented in in a strong strong small-strain solid solid mechanics mechanics problem. problem.' The small-strain terms of of partial partial differential differential equations equations and and in a wweak terms of of integral integral fform o r m in terms e a k fform o r m in terms expressions. expressions. We We have have also also indicated indicated how how the the general general problem problem can can become become non-linear. non-linear. In the the next next chapter chapter we we describe describe the the use use of of the the finite element element method method to to construct construct In approximate solutions solutions to to weak weak forms forms for non-linear non-linear transient transient solid solid mechanics mechanics problems. problems. approximate
References
iii !~ !iiiiiii!i!iiiiiii!i!!i!iiiii!iii!i!i!iiiiiiiiiiii!iii! i!i i i!i i
ili i ilii2iiili! ill ii ii ii !i
iiiiiiiiii iiiiiiii!i]!i!!!i i iii iii
i
ill ii i
i i i ii i!ii!!ii••i•ii••••iii••iii!••i•i••ii!•••ii•ii••!•ii!••i••••••ii••]•iiii!ii••ii
1. O.C. Zienkiewicz, R.L. R.L. Taylor Taylor and J.Z. Zhu. The The Finite Finite Element Element Method: Method: Its Its Basis Basis and and FundaFundamentals. Butterworth-Heinemann, Oxford, 6th edition, 2005. mentals. Butterworth-Heinemann, 2. 2. T.J.R. T.I.R. Hughes. Hughes. The The Finite Finite Element Element Method: Method: Linear Linear Static Static and and Dynamic Dynamic Analysis. Analysis. Dover PubliPublications, New New York, York, 2000. 3. R.D. R.D. Cook, D.S. D.S. Malkus, MaIkus, M.E. M.E. Plesha Plesha and R.J. R.I. Witt. Witt. Concepts Concepts and and Applications Applications of of Finite Finite 3. Element John Wiley Wiley & Sons, New New York, York, 4th 4th edition, edition, 2001. 2001. Element Analysis. Analysis. John 4. 4. EF. de S. Lynch. Lynch. A A finite finite element element method method of of viscoelastic viscoelastic stress stress analysis analysis with with application to rolling rolling contact contact problems. problems. International International Journal Journal for for Numerical Numerical Methods Methods in in Engineering, Engineering, to 1:379-394, 1:379-394, 1969. 1969.
16 General problems in solid mechanics and non-linearity 5. J.T. Oden and T.L. Lin. On the general rolling contact problem for finite deformations of a viscoelastic cylinder. Computer Methods in Applied Mechanics and Engineering, 57:297-367, 1986. 6. P. le Tallec and C. Rahier. Numerical models of steady rolling for non-linear viscoelastic structures in finite deformation. International Journal for Numerical Methods in Engineering, 37:1159-1186, 1994. 7. S. Govindjee and P.A. Mahalic. Viscoelastic constitutive relations for the steady spinning of a cylinder. Technical Report UCB/SEMM Report 98/02, University of California at Berkeley, 1998. 8. H. Cohen and R.G. Muncaster. The Theory of Pseudo-rigid Bodies. Springer, New York, 1988. 9. S.P. Timoshenko and J.M. Gere. Theory of Elastic Stability. McGraw-Hill, New York, 1961. 10. S.P. Timoshenko and J.N. Goodier. Theory of Elasticity. McGraw-Hill, New York, 3rd edition, 1969. 11. I.S. Sokolnikoff. The Mathematical Theory of Elasticity. McGraw-Hill, New York, 2nd edition, 1956. 12. L.E. Malvern. Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, Englewood Cliffs, NJ, 1969. 13. A.P. Boresi and K.P. Chong. Elasticity in Engineering Mechanics. Elsevier, New York, 1987. 14. P.C. Chou and N.J. Pagano. Elasticity: Tensor, Dyadic and Engineering Approaches. Dover Publications, Mineola, NY, 1992. Reprinted from 1967 Van Nostrand edition. 15. I.H. Shames and EA. Cozzarelli. Elastic and Inelastic Stress Analysis. Taylor & Francis, Washington, DC, 1997. (Revised printing.) 16. F.B. Hildebrand. Methods ofApplied Mathematics. Prentice-Hall (reprinted by Dover Publishers, 1992), 2nd edition, 1965. 17. K. Washizu. Variational Methods in Elasticity and Plasticity. Pergamon Press, New York, 3rd edition, 1982. 18. E. Hellinger. Die aUgemeine Aussetze der Mechanik der Kontinua. In E Klein and C. Muller, editors, Encyclopedia der Mathematishen Wissnschafien, volume 4. Tebner, Leipzig, 1914. 19. E. Reissner. On a variational theorem in elasticity. Journal of Mathematics and Physics, 29(2): 90-95, 1950. 20. E. Reissner. A note on variational theorems in elasticity. International Journal of Solids and Structures, 1:93-95, 1965.
22
i~!iii~ii~i~ii!!!~!~i~i~ii!~!i~!~!ii!i!!!i!i~i~!~iii~iiii~i~i!i~i!i
method of approximation Galerkin method --irreducible irreducible and mixed mixed forms
iJ!ii~iii!iii!!i!iii~i~i!iiiiii~i!iiii!i~~iiiiiii~iiiiii!i!iii~iiiiiii~ii~ii~iii!~i~iiiii~i~iiiiiiiiiiiiiiii~iiiiiiiiii~iiiiiii!iiiiiiii~ii!ii~iiii~iii~i!iii!iiiiiiiii!~iiiii~!iiiiiiiii~i~i~iiiiii~iii~i~iii~iiiii~!iiii~i~i!i~iii~iiiii!~ii~i~iiii iiiii~i~d~i~~iii~iiiiiiii~2~iiii~ii~i~i~ii~i~ii!~ii~i~i~ii~i
2.1 Introduction
In the previous chapter we presented the basic equations for problems in non-linear solid mechanics in which strains remain small. We showed that the equations can be presented in a strong form as a set of partial differential equations or alternatively in terms of a variational principle or weak form expressed as an integral over the domain of interest. In the present chapter we use the weak form to construct approximate solutions based on the finite element method. This results in a Galerkin method for which general properties are well known. 1-4 1-4 Although it is assumed that the reader is familiar with finite element methods for small deformation linear problems, we present a full summary of the basic steps to construct a solution for the transient problem. We emphasize the differences between linear and non-linear effects as well as the numerical procedures used to establish the final discrete form of the equations which is the form used in computer analysis. We irreducible and mixed mixed forms of approximation. The mixed forms also consider both irreducible are introduced to overcome deficiencies arising in use of low order elements based on irreducible forms. In particular, in this chapter we consider a mixed form appropriate for use in problems in which near incompressible behaviour can occur. In the second part of this book, we consider forms for structural problems where so-called 'shear locking' can occur in bending of thin rods, plates and shells. chapter by applying the methods developed for the equations of We conclude this chapter solid mechanics to that for thermal analysis based on a non-linear form of the quasiharmonic equation.
i!ii!ii2.2 iiiiii~'ii'~'~i~iiii!i!iiiiiiiiiiiii' "'Finite ~~!~i '~'~"~i i~iiii~i~iii~ii~~' ~ii~element i~i~~:ii~i:~i!i!~ii!~ii!i!i~i!ii~iii~ii~iapproximation i~i~!iiiiii~ii~ii!i~ii~i!iiii~!~iiii~ii~i~~iiiii~!iii~~~i ii!i~i-i~i~iii~iiGalerkin .i!~.i.i~.i.i~ii~i~i~iii!~~ii~ii~iiii~ii:.:i.~i:.!i~ii!~!~i i~i!i~i~imethod ii~ii~ii~ii!~iii~i~~i~ii.~i:.i:.~iiii~i~ii~i!i!i~!ii~iii~ii~i~iii!i~i~iii~ii~ii~i~~ii~ii~iiiiiii~ii~i~ii iiii~i~iii~ii~!iii~i~i~i~i~i~iii~i!~i!i!i!i!iiii~i~i!~i~ii!iiii~i~iiii~!~i~i!~i~iiiii~iii~i~i!i!ii~iiiii~iiiiiiii!ii~i~iiiii~i!~i~iii~i~i~i~ii!i!ii~iiiiiiii~iii~i~ii~i!~i~!iii!~iiiii~iii~i~i~ii~ii!~i~iiii~i~ The finite element approximation to a problem starts by dividing the domain of interest, Q, f2, into a set of subdomains (called elements), Qe, fie, such that f2
= ~
~'2e
(2.la) (2.1a)
ee
Similarly the boundary is divided into subdomains as
L:r L:r L:rI~ue
rr~r= ,~ F - ~--~ Fee = -- ~-~ Ftete + -~- ~ e
et et
eu eu
ue
(2.tb) (2.1b)
18 Galerkinmethod of approximation -irreducible and mixed forms where Fte is a boundary segment on which tractions are specified and F/./e o n e where displacements are specified. We note that in general the domain of a finite element analysis is an approximation to the true domain which depends on the boundary shape of elements. The weak form for the governing equations is written for the problem domain (2 and also written as a sum over the element domains. Thus, the weak form given in Eq. (1.46) for the equilibrium equation becomes
r
~ (~'leq : ~-~ [ J~ 'uTpiidf2 + J~ e
e
et
= ~
6urbdf2] e
te
t~leIe -+- y ~ e
'(Su)To'df2-f~ e
t~leIt = 0
et
In the above r e a r e terms within the domain ~'~e of each element and ~l~It those which belong to traction boundary surfaces Fte. A Galerkin method of solution is obtained using approximations to the dependent variables and their virtual forms. 1'2 For an irreducible (or displacement) finite element form we only need approximations for u and ~u. In order for a variational theorem or a weak form to be split into the additive sum indicated in Eq. (2.2) the highest derivatives appearing in the functional must be at least piecewise continuous so that all the integrals exist and no contributions across interelement boundaries are present.* For a functional containing a variable with a highest derivative of order m + 1 the functions used to approximate the variable must have all derivatives up to order m continuous in the entire domain, f2 - such functions are called C m. For the weak forms considered for problems in solid mechanics we will encounter functionals which contain only first derivatives and thus will need only C O functions for the approximation. Indeed some functions in mixed forms will have no derivatives and these may be approximated by discontinuous functions in f2. Generally, one should respect the order of approximation (i.e. C m) where an exact solution can have discontinuous behaviour. For solid mechanics there are discontinuities in the displacement at material interfaces and at some singular load forms (e.g. point loads or line loads). Material interfaces are real; however, use of a point or line load is not and, when used, is an approximation to a physical action. Use of functions with added continuity over C m can be beneficial where solutions are smooth. Thus there are some forms of interpolation being introduced in recent literature that have increased smoothness. In this volume, however, we will generally present only those forms which provide C m continuity. In the first part of this volume concerning problems in solid mechanics approximations will be made with C O functions. In the second part concerning structural mechanics problems for rods, plates and shells we shall need C 1 functions for some formulations, whereas in others we can still use C Oforms. There are formulations which violate the continuity conditions leading to so-called incompatible approximation (e.g. see Wilson et al.6). Strang termed such approxima* It is possible to add interelement jump terms to a functional leading to a D i s c o n t i n u o u s see Cockburn e t al. 5 for more information.
Galerkinformulation
-
Finite element approximation-Galerkin method
tions a v a r i a t i o n a l c r i m e 7 but showed convergence could still be achieved provided certain requirements were met. Most incompatible formulations that perform well have subsequently been shown to be members of a valid mixed formulation (e.g. see Simo and Rifai8). In the forms given for problems in solid mechanics we will use interpolations which satisfy the C Orequirement; however, in the study of thin plates we will present some forms which violate the C 1 requirement. In addition to the continuity requirement it is necessary for C m functions to possess complete polynomials to order m + 1 to ensure that the derivatives up to order m + 1 can assume constant values. Both of the above requirements are covered in standard introductory texts on the finite element method (e.g. see reference 2 or 3). They remain equally valid for the study of non-linear problems - both for forms with material non-linearity as well as those with large deformations where kinematic conditions are non-linear. The patch test also remains valid in assessing the available continuity and derivatives present in any approximation (see reference 2 for a general discussion on the patch test for irreducible and mixed finite element formulations).
Displacementapproximation
A finite element approximation for displacements is given by u(x,
t) ~ fi -
~
Nb(X)fib(t) = N(x)fi(t)
(2.3)
b
where Nb are element shape functions, fib(t) are time dependent nodal displacements and the sum ranges over the number of nodes associated with an element. Alternatively, in i s o p a r a m e t r i c f o r m 2 the expressions are given by (as shown in Fig, 2.1 for a four-node two-dimensional quadrilateral) U(~, t) ~ ~t(~, t) -- ~
Nb(~)fib(t) --
N(~)fi(t);
b Nb(~')'~b -- N(~)'~
x(~) - ~
(2.4a)
b
~2
~2
1
I
I
r
( 1
._1~. f l
~,
(a) Element in ~ccoordinates.
1
:> Xl
r
(b) Element in x coordinates.
Fig. 2.1 Isoparametric map for 4-node two-dimensional quadrilateral.
19
20
Galerkin method of approximation -irreducible and mixed forms
where ~ represent nodal coordinate parameters and ( are the parametric coordinates for each element. An approximation for the virtual displacement is given by
611(~) ~ 61_1(~) -- Z
a
Na(~)lla -- N(~)u
(2.4b)
A summary of procedures used to construct shape functions for some isoparametric elements is included in Appendix A.
Derivatives
The weak forms presented in Chapter 1 all include first derivative of displacements. For the isoparametric approximation given in Eq. (2.4a) we need first derivatives of the shape functions with respect to xj. These are computed using the chain rule as:
ONa O~i or in matrix form
OXj ONa
=
ONa
(2.5)
O~i OXj
ON.
= J ~
(2.6a)
where
" 01~ ON a 0( =
ON N 01~ --~
;
ONa 0x =
ON,
Ox 1
~X1 ON ~ ON,
0~10~l Ox1 Ox2. 0~2 0~2 OXl Ox2
;
J=
~
-0~3
OX2
0~3
OX3 O~1 Ox3 0~2 Ox3
(2.6b)
0~3
in which J is the Jacobian transformation between x and ~. Using the above the shape function derivatives are given by
ONa = j-10Na
(2.6c)
In two-dimensional problems only the first two coordinates are involved, thus reducing the size of J to a 2 x 2 matrix. In the sequel we will often use the notation
ONa ONa Oxj = Na,xj and O~i = Na,~,
(2.7)
Strain-displacement equations
Using (1.18) the strain--displacement equations are given by -- S u ,~ ~--~(,-qNb)fib = ~ Bbfib -- Bfi b b
(2.8)
In a general three-dimensional problem the strain matrix at each node of an element is defined by [Nb,xl 0 0 Nb,x2 0 Nb,x3] B~-/ 0 Nb,x2 0 Nb,xl Nb,x3 0 J (2.9a) L0 0 Nb,x3 0 Na,x2 Nb,x,
Finite element approximation-Galerkin method
For the two-dimensional plane stress, plane strain and torsionless axisymmetric problem the strain matrix at a node is given by
B~-
[N~x,
0 CNb/Xl Nb,x2] Nb,x2 0 Nb,x,
(2.9b)
where c = 0 for plane stress and strain and c - 1 for the torsionless axisymmetric case. For the axisymmetric problem with torsion the strain matrix becomes
rNb,xl 0 Nb/Xl BTb-- [ 0 Nb,x: 0
0
0
Nb,x2 Nb,x,
0
0 0
0 0
]
]
(2.9c)
Nb,x2 (Nb,x, -- Nb/X~)
Weak form
Substituting the above forms for displacement and strains into the weak form of equilibrium given in Eq. (2.2) yields, for a single element,
(~I"Ieq- "e
(~1-1 T
[f~ e
N r pN dr2 fi +
f~ e
B r or dr2 -
f~ e
Nrb dr2 -
Ji te
N r t dF
(2.10)
]
Performing the sum over all elements and noting that the virtual parameters 6fi are arbitrary we obtain a semi-discrete problem given by the set of ordinary differential equations (2.11a) M a + P(or) = f where
M(e);
M = ~
e = y ~ e(e) and f = ~
e
e
f(e)
(2.1 l b)
e
with the element arrays specified by
M(e)=[NrpNdf2;
P(e)(or)-[Brordf2
J ~'~e
J ~"~e
and f ( e ) = [ N r b d f 2
+fNrtdF
J ~'~e
te
(2.11c) The term P is often referred to as the stress divergence or stressforce term. While the form for the arrays given above is valid for all problem classes the volume element differs and is given by: df2 dr2 dr2 dr2
= dXl dx2 dx3;
= dXl dx2;
-- h3 dXl dx2; = 2 7rxl dxl dxa;
General three-dimensional problems, Plane strain problems, Plane stress problems, Axisymmetric problems.
In the above we assume a unit thickness in the x3 direction for plane strain, h3 is the thickness of a plane stress slab; and the factor 2 7r in axisymmetric problems results from the integration of f dx3 = f d0 of the body of revolution.* In the sequel we will discuss the finite element form for solids in a general context using the coordinates, xi, displacements, ui, etc. Unless otherwise stated, we will also assume that the forms for B, f2e, dr2, etc. are always replaced by that appropriate for the problem class considered (i.e. plane stress, plane strain, axisymmetric, or general three dimensions).
* Someprograms,includingFEAPpvavailableat the publisher'sweb site, omitthe factor271"in axisymmetric forms.
21
22 22
Galerkin method method of of approximation approximation -irreducible - irreducible and and mixed mixed forms forms Galerkin
displacement method method 2.2.1 Irreducible displacement In In the the case case of of linear linear elasticity elasticity the the constitutive constitutive equations equations are are given given by by Eq. Eq. (1.22) (1.22) and and using Eq. Eq. (2.8) (2.8) an an irreducible irreducible displacement displacement method method results results22 with with using p(e)(o-) -= ( f P(e'(~r)
rBrDBdf2)fi = K(e)fl
ln,
BTDBdQ)fi = K(e)fi
(2.12) (2.12)
e
in in which which K(e) K(e) is is aa linear linear stiffness stiffness matrix. matrix. In In many many situations, situations, however, however, itit is is necessary necessary to to use non-linear or time-dependent stress-strain (constitutive) relations and in use non-linear or time-dependent stress-strain (constitutive) relations and in these these cases cases we l a) to l c). This we need need to to develop develop solution solution strategies strategies directly directly from from Eqs Egs (2.1 (2.11a) to (2.1 (2.11c). This will will be be considered considered further further in in detail detail in in later later chapters chapters for quite quite general general constitutive constitutive behaviour. behaviour. this stage stage we we simply simply need need to note note that that However, at this a0- == er(e) o-(e:)
(2.13) (2.13)
and that that the the functional functional relationship relationship can can be be very non-linear non-linear and occasionally occasionally non-unique. non-unique. Furthermore, Furthermore, it will will be be necessary necessary to use use a mixed mixed approach approach if if constraints, constraints, such such as near near incompressibility, incompressibility, are encountered. encountered. 22 We address this latter aspect aspect in Sec. 2.6; however, before before doing doing so we we consider consider the manner manner whereby whereby calculation calculation of of the finite element element arrays and solution solution of of the transient equations equations may be computed computed using numerical numerical methods. methods. ','i,i i',i',i'~:i!'~,i,'i'li,:,~': ,'~,~',::i',~i~,,i~::','i*',,'i,*,',i~,::~'~,'i,~i':~i i~iT'~'~"'~: 'i'~l'i~: i',ili~i,'~i',i~i'~',i',~i~,i!,i~:'~,i,i~':i~i'~i'~',i,i~,i',i,'~'~i,'i:~i iii'~~'(~ii',',:,i~'!~'i,i',:'i,~ii~,ii',':~i,i i':i,i':',:~i'~,'i,~ii ',i!'~i',i',~i',~,~',~''i,',',~',~ii!i'~i:~,',~i~,',i i'i~!i',i~ii~,i',i i'::i,~,i'i,'i,',i i',i'~',i:',i:~~',:i~',i:'~,~i',:'i,i',i'~,i'~,~,i','~i~ii',i!',i:i,'~,,',~~'i~,',~~,iiiii!','i,'i,~ii~'',i,:i~',~,,:'~i'i,'i,iii~' ','i:,',i~,','i,',i'~,',',',i',~i','i~',:,'~i,~'i',~',:':'i:f'~:,:,i,:::ii'i~!i:~',~i ,:(,i,i,':,::i~:~i:i','~,i',i!i':,i!'~',i'i,~':'i,'i,iiiii~,i~i'~,i',ii~i~,i~l i'i,~ii'~:~ii ',i',i'~,i'i:,i!i i i i!ii',',i i',:',i:iii i','i,iiiiiiiiii~~i,ii',i',i',',',i:'i~i:i,ii i'~i ili',iii~,i~' ,i'l,i',iii!'i,iii',i',iiil'~i!',i',i',iii'~,i~,ii~,'i~',il,i~,i',i',i',i~i,'i~:,i,i'~i:,ii~'i,'i:~i:i'~,i!i:',,i~:izi':i',i',iiii',i',i',i',i',':i
2.3 Numerical integration - quadrature
element arrays are most conveniently The integrations needed to compute the finite element performed numerically by quadrature. 22'3'9 ,3,9 Many forms of quadrature formulas exist; expressions is Gauss-Legendre Gauss-Legendre quadrahowever, the most accurate for polynomial expressions ture. IO 1~ Gauss-Legendre Gauss-Legendre quadrature tables are generally tabulated over the range of of - 1 < ~~ < 1 (hence our main reason for also choosing many shape funccoordinates -1 tion on this interval). Gaussian quadrature integrates a function as
(d f)
1 fl
2n
d~ 2n f ) f (~) d~d~ =-- ~ y n~n f(~j f )(~j) Wj wj + + 00 (d2,, _I f(~) den 1
1
(2.14a)
j=l
wj is a weight. Thus, an where ~j~j are the points where the function is evaluated and Wj n-point formula integrates exactly a polynomial of order 2n - 1. Table 2.1 presents the location of points and weights for the first five members of the family. Integrations over multi-dimensional domains may be performed by products of the one-dimensional formula. Thus, in two dimensions we use (with 17~7== - ~2)~2)
LLf(~, 1
1
~
f (~, n) ~7)d{ d~ dn d~7 =
t. t, j=l k=l
ff«j. (~j, n,) r/k) W tvjj w, wk
(2.14b) (2. 14b)
and and in in three dimensions (with (~ == ~3)~3)
111111 f(~, I
1
I
1
I n n
1
()d~d17d(
f(~, 17, r/, 0 d~ d~dr =--
~f; j=l
n
7~k, (/) ~l) Wj Wj Wk W k W/ 11)l ~ ff(~j' (~j, 17k'
k=l l=1
are exact exact when polynomials in in any any direction direction are are less than order order 2n. 2n. which are
(2.14c)
Numerical integration - quadrature Table 2.1 ~"~=1
Gaussian quadrature abscissae and weights for f_l I f ( ~ ) d ~
--
f (~j)wj. j
Order
~j
wj 2
n=l
1
0
n=2
1
+ 1/v/'3
1
2
-1/4~
1 5/9
n=3
n -- 4
1
+ ~/-0--.6
2
0
8/9
3
- ~/-O.-.-~
5/9
1 4-,,/((3 4- a ) / 7 )
0.5 - 1 / ( 3 a )
2
4- ~/((3 - a ) / 7 )
0.5 4- 1 / ( 3 a )
3
- ~/((3 - a ) / 7 )
0.5 4- 1 / ( 3 a )
4
- ~/((3 4- a ) / 7 )
0.5 - 1 / ( 3 a )
1
+ v/b
2
+~/-d
n = 5
3
0
4
- ~
5
- v/b
a=
((5c - 3)d/b) ((3-5b)d/c) 2-
44~.8
a -- v / l 1 2 0 b = (70 4- a ) / 1 2 6 c = (70 - a ) / 1 2 6
2(wl + / 0 2 )
5b)d/c) ((5c - 3)d/b) ((3 -
d = 1/(15(c - b))
Volume integrals
The problem remains to transform our integrals from the element region ~'2e to the gaussian range - 1 < ~ < 1. The determinant of J appearing in Eq. (2.6a) is used to transform the volume element from the cartesian coordinates to the natural coordinates as
dxldx2dx3 = d e t J d~l d~2 d~3 = j(~l, ~2, ~3) d~l d~2d~3
(2.15)
in which detJ - j must be positive to maintain a correct volume element. Using the above type of transformation, integrals of finite element arrays are given by f ( x ) d ~ = f~ f ( , ) j ( , ) d Q
(2.16)
e
where f is the function f written in terms of the parent coordinates ~, D denotes the range of parent coordinates for the dimension of problem considered and j (~) is the appropriate Jacobian transformation for the coordinate system considered. For the various problem classes these are given by: Problem type Three-dimensional: Plane strain: Plane stress: Axisymmetric:
n - domain d~l d~2 d~3 d~l d~2 d~l d~2 d~l d(2
j
-
Jacobian
J (~1' ~2' ~3) J (~1, ~2) h3 j (~l, ~2) 27rXl j (~1, ~2)
where for two-dimensional problems
I
0Xl
J (~1, ~2) ~"
det
0x2
OXl 0x2
>0
(2.17)
23
24 Galerkin Galerkinmethod of approximation approximation --irreducible mixed forms forms irreducible and mixed
The permissible is chosen so that The minimum number number of of quadrature quadrature points permissible 1. Elements Elements which have constant jacobians jacobians j are exactly integrated; or element stiffness matrix has full rank. 2. The resulting element
In either The first either case, the consistency part part of of the patch test must also be satisfied. 22 The criterion criterion may be used only for linear materials materials in small strain (such as discussed in this chapter). The non-linear problems. The second is applicable applicable to both linear and non-linear problems. Use of the quadrature than that satisfying the above is called reduced quadrature next lower order quadrature and generally should be avoided.* When integrated using 'full' order When some terms are integrated and some with 'reduced' method is referred 'reduced' order quadrature quadrature the method referred to as selective reduced 2•3 integration. 2,3 The above form of natural coordinates coordinates ~~ assumes that each finite element element is a line, The a quadrilateral quadrilateral or a hexahedron. hexahedron. For other shapes, such as a triangle or a tetrahedron, tetrahedron, For other shapes, appropriate changes changes are made coordinates and integration formula used appropriate made for the natural coordinates A).2.3.9 (see also Appendix Appendix A ) . 2'3'9
Surface integrals integrals Surface
element surfaces surfaces and this is most easily It is also necessary necessary to compute integrals integrals over element accomplished by considering considering df dF as a vector oriented in the direction normal to the accomplished surface. For three-dimensional three-dimensional problems problems we form the vector product product Ox Ox &x &x ndf d~1 n dr" = - dr dr = = -8 0s xx -8 ~ d~ 1 d~2 d~2 ~I ~2 = (VI (Vl x V2) v2) d~1 d~l d~2 d~ 2 = -- V Vn d~l d~2 d(2 = n d~1
(2.18) (2.18)
where ~I~1 and ~2~2 are parent coordinates for the surface surface element element and x denotes a vector where parent coordinates cross product. Surfaces for two-dimensional problems described in terms of ';1 ~1 only and Surfaces problems may be described Ox/O~2 by e3 (the ( t h e unit normal to the plane plane of of deformation). for this case we replace replace &x/a~2 If If necessary, the surface differential differential may be computed computed from T ) 1/2 dr = - ((VTVn)1/2 d~, d~2 d~2 df Vn Vn d~1
(2.19)
iiiiiiiMiii~iiiiiiiiii~iliiii~ii~ii~iiii ~iMi~i~i~i~ii~iiii~ii~i~i~i~i~ii~i~ii~i~i!iiiiiiiiiiiiiiiiii!i~i~i~i~iiiiiiiiiiiiiiiMii~iiiiiiiiiiiiiii~i~ii~i~i~i~iii~iiiiiiiiiiiiii~iii~iiiiiiii~iiiii~iii!iiiiiiiii~iiiiiiii~iii~iii~iiiiii~ii~i~%i!~i!iiiiiiii!~!iii~iiiii~i!iii~iiiiiiiiiiiiiiiiii!i~iiiiii iiii!ii~i~i~i iii~iiiiiii!i!iiiiiiiiiiiiiiiiiiiiii!iiiiiiiiiiTiiTiikiiii!ili !iiiiiiiiikliiikiiiii ii!iiiiiiiiiiiiiiiiiiiiiiiiiiiliiiiiiiiii!iiiii iiiiiiiiiiii!iii~':iiii! ~ iiiiiiii!iii :~ i~: ':~.......i~ i i........: ~:ii~-k~i ,,~iiii~~,,,i~ i.......~i!ill'~'~ ! ........i ,iiilJ ~i......i~..... iiiii!!!:i~ ,!': ii~ :i~: l~'~i!i !:!~ ~iii~'~i~~': iiii!i!iiii ........~i~'........i :~i !~........i........ !ik~:~ii!iiiiiii~iiiiiiiii!i!i!i!ii~iii~i!~!iii~i!ii!i~i!i!i!iii~!~!!ii!i~i!i~i!i~ii~i
2.4 Non-linear transient and steady-state problems
To obtain a set of algebraic equations for transient transient problems problems we introduce introduce a discrete approximation in time. We write the approximation approximation to the solution as U(tn+l) ~~
U(tn+l)
U U nn++,; l;
U(tn+d ~~ VVn+1 and UU(tn+l) ~~ an+1 U(tn+l) n+l (tn+l) an+l
where discrete variables variables is omitted for simplicity. Thus, the equilibrium equilibrium where the tilde on discrete equation (2.IIa) (2.1 la) at each discrete time tn+1 tn+l may be written in a residual form as 0
(2.20a)
Pn+l -- /~, B Trtrn+a d~ = - - P(un+d P(Un+l) P O'n+1 dQ n+1 ==
(2.20b)
~n+l = fn+l -- M
where where
1
a n + l -- P n + l =
a n y explicit e x p l i c i t codes c o d e s uuse s e reduced r e d u c e d qquadrature uadrature in combination w i t h so-called s o - c a l l e d hhour-glass our-glass stabilization. 4 4,11 •J I •* M Many in combination with stabilization.
Non-linear transient and steady-state problems 25 Here we have indicated that P can be expressed in terms of the displacement alone. This is correct for elastic materials but with inelastic behaviour the material model will depend on the solution variables in a more general form. In Chapter 4 we will show that most constitutive models may be given in terms of increments of u. Thus the above assumed form will need only a minor modification that does not significantly affect the following discussion. For transient problems we apply the GN22 method 2 (which is identical to the Newmark procedure ~2 except for the manner parameters are defined) to equations with second derivatives in time. The GN22 method relates the discrete displacements, velocities, and accelerations at tn+l to those at tn by the formulas Iln+ 1 =
U n
-q-- At Vn +
Yn+l = Yn + (1
1 ~(1 -/~2)At2an
-4-
1 ~/32At2an+l =
IJn+ 1 +
1 ~/32At2an+l
- / 3 l) At an +/31 At an+l - - Cgn+l + ~1 At an+l
(2.21)
in which At = tn+l -- tn is a time increment and fin+l, ~n+l are values depending only on the solution at t,. This one-step form is very desirable as it allows the At to change from one step to the next without introducing any complications (although very large changes should always be avoided). The two parameters/31 and/32 are selected to control accuracy and stability. The transient problem is now obtained for each time tn+l by solving the non-linear equation set (2.20a) and the pair of linear equations with scalar coefficients (2.21). It is possible to take as the basic unknown any one of the three variables at time tn+l (i.e., Un+l, Vn+l or an+l). Using Eq. (2.21) the non-linear equation may then be given in terms of a single unknown.
2.4.1 Explicit GN22 method A very convenient choice is to take/32 - 0 and select an+l as the primary unknown. Using Eq. (2.211) we immediately obtain ~ n + l "-- Un+l
and thus from Eq. (2.20a) M an+l - - fn+l -- P ( ~ )
may be solved directly for an+l. The velocity Yn+l is then obtained from Eq. (2.212). This leads to a so-called explicit scheme since only linear equations are solved. If the M matrix is diagonal 2 (or lumped) the solution for an+l is trivial and the problem can be considered solved since
I
1/Mll
M-l_
..
1/Mmml
where m is the total number of equations in the problem. However, explicit schemes are only conditionally stable with At < Atcrit, where Atcrit is related to the smallest
26
Galerkin method of approximation -irreducible and mixed forms time it takes for 'wave propagation' across any element or, alternatively, the highest 'frequency' in the finite element mesh. 2 Thus a solution by an explicit scheme may require many thousands of time steps to cover a specified time interval. For many transient problems, and indeed for most static (steady state) problems, it is often more efficient to deal with implicit methods for which much larger time steps may be used.
2.4.2 Implicit GN22 method In an implicit method it is convenient to use U~+l as the basic variable and to calculate Vn+l and an+l using Eq. (2.21). With this form we merely set Vn+l = an+l -- 0 to consider a quasi-static problem.* The equation system (2.20a) now can be written as 2c /32At2 M [Un+ 1 - a n + l ] -- Pn+l -- 0
~IJ(Un+l) -- fn+l
(2.22)
where c = 1 for transient problems and c = 0 for quasi-static ones. The solution to this set of equations requires an iterative process when any of the terms is non-linear. We shall discuss various non-linear calculation processes in some detail in Chapter 3; however, we note here that Newton's method t forms the basis of most practical schemes. In this method an iteration is written as ~ lI/k+ l ,~ k k n+l It/n+ 1 "t- dffffn+ 1 -- 0
(2.23a)
where, for fn+l independent of deformation, the increment of Eq. (2.22) is given by dffffkn+ 1 - - -
/32At2
M-
0un+----~n+l
The displacement increment is computed from k
A n + l dUn+ l :
k l ffff n+
(2.24a)
and the solution is updated using lak+l k n+l - - Un+l -~" dUkn+l ak+l n+l :
2 r k+l / 3 2 A t 2 [Un+l -- I]n+l]
u k+l - " ~rn+l "~- /31 A t
(2.24b)
a~++11
An initial iterate may be taken as zero or, more appropriately, as the converged solution from the last time step. Accordingly, a n1+ 1 - - a n
(2.25a)
* A quasi-static problem may be time or load path dependent; however, inertia effects are not included. t Often also called the Newton-Raphson method. See reference 13 for a discussion on the history of the method. ~t Note that an italic 'd' is used for a solution increment and an upright 'd' for a differential.
Non-linear transient and steady-state problems 27 in which a quantity without the superscript k denotes a converged value. For transient problems initial velocities and accelerations are given by 2
1 __ [U n _ i~n+l ] a n + l - - /~2 A t 2 1 1 Yn+l - - Vn+l -~-/~1 A t an+l
(2.25b)
Iteration continues until a convergence criterion of the form II kn+ 1 II _< e II~I'n+ 1 1 II
(2.26)
or similar is satisfied for some small tolerance e. A good practice when all terms of the Newton method are accurately computed is to assume the tolerance at half machine precision. Thus, if the machine can compute to about 16 digits of accuracy, selection of e = 10 -8 is appropriate. Additional discussion on selection of appropriate convergence criteria is presented in Chapter 3. A common choice of parameters is/31 =/32 = 1/2 which is also known as the 'trapezoidal integration rule'. The derivative of P appearing in Eq. (2.23b) is computed for each element from Eq. (2.1 lc2) as 0p(e) I k - - - ~ n+l
BrDkrBd~ - Kkr
(2.27)
e
We note that the above relation is similar but not identical to that of linear elasticity. Here DkT is the tangent modulus matrix for the stress-strain relation (which may or may not be unique but generally is related to deformations in a non-linear manner) and K r is the tangent stiffness matrix. Various forms of non-linear elasticity have in fact been used in the present context and here we present a simple approach in which we define a strain energy density, W, as a function of e W = W ( e ) "-" W ( e i j )
and we note that this definition gives us immediately cr =
OW 0e
(2.28)
If the nature of the function W is known the tangent modulus Dkr becomes 0trlk _ 02Wlk -Dk = ~ n+l 0 ~ 0 S n+l For problems in which path dependence is involved to compute O'n_t_1 (viz. Chapter 4) it is necessary to keep track of the total increment during the solution step from tn to tn+l and write tlk+l (2.29a) n+l - - U n + A U. nk+l +l with A U n1+ 1 -- 0 The total increment can be accumulated using the solution increments as .k+l
. k+l
k
k
m u n + 1 = Un+ 1 -- U n - - m U n + 1 "q- dun+ 1
(2.29b)
In an implicit scheme it is desirable to use the displacement from the last iteration to compute both A and ~I, - especially when inelastic material behaviour or large strains are considered.
28 Galerkin Galerkin method method of of approximation approximation -irreducible - irreducible and and mixed mixed forms forms 28
2.4.3 Generalized mid-point mid-point implicit implicit form 2.4.3 An alternative alternative form form to to that that just just discussed discussed satisfies satisfies the the balance balance of of momentum momentum equation equation An at at an an intermediate intermediate time time between between tn t n and and tn+l. tn+l' In In this this form form we we interpolate interpolate the the variables variables aas s
Un+o~ Un+a -=-
(1 (l - ce) a) Un Un + + ceaUU nn++11
Vn+a "-= (1 (1 Yn+o~
a) Vn Vn -q+ ceaVY nn++1l - ce)
(2.30) (2.30)
an+a -= (1 (l -- Ce)an a) an + + O~ aa an+a an+l n+1
and write write momentum momentum balance balance as and ~I'n+~ - fn+~ - M an+c~ - Pn+c~ "-- 0
=
(2.31) (2.31)
=
We We select select the the parameters parameters in the the GN22 GN22 algorithm algorithm as/31 as /31 =/32 /32 = / 3/3 and, and, thus, thus, obtain obtain the the simple form simple Un+ 1 -- Un +
1 At
(Vn+l -- u
--
1
~ At (v n + Yn+l)
(1 -/3)
(2.32) (2.32) an - F / ~ a n + l
Selecting now c~ equation in the form Selecting a -=/ 3f3 gives the the momentum momentum equation tTffn+o~ - - fn+o~ - - ~
1
At
M
(u
- - Vn) - -
Pn+~ - 0
(2.33)
which is the form utilized by Simo et at. al.** as part of of an energy-momentum conserving 15 14,15 a = / 3f3 - 1/2 1/2 method. 14 • For a linear elastic problem it is easy to show that choosing a If non-linear elastic forms are used will conserve energy during free motion (i.e. f = 0). If with these values, it is necessary to modify the manner by which acrn+l/2 n +l/2 is computed 16 We shall address this more in Chapters 5 and 6 to preserve the conservation property. 16 when we consider finite deformation forms and hyperelastic constitutive models. Un+l as the The solution of the above form of the balance equation may still use Un+l primary variable. The only modification is the appearance of the parameter parameter a in terms Pn+a. arising from linearizations of P n+a'
= =
tliitii!tilitiiiiiiiiiQ•it!•i!i•t•i!•i!•i!•i@•i!i•
i•i•i;i•ii•i•i•i!i•iit•it•iti!it•it!i!iti•i•i•ii;
i!i!ititi•i•iti•i!ii•ii•i•i•i!it!•i
il!iiii!i~i~i!i@i~ iiii~i~i~itiiii~i~i~i!iii~i~i~i!~i~i!iii~i~i~iii~i~i~!~i!ii!i!i~ii!ii~i~i!i~!i~i~!~
2.5 Boundary conditions: non-linear problems
In constructing a solution from a variational (weak) form, boundary conditions are classified into two categories: natural conditions that are satisfied by the variational special considerations; and essential conditions for which modifications form without special to the solution process must be made to make the variational form valid. For example, in an irreducible displacement method the traction boundary condition is a natural form condition is an an essential form and and must be imposed and the displacement boundary condition separately. separately. Simo et et ai. al. did did not not interpolate interpolate the the inertia inertia term term and, and, thus, thus, needed needed different different parameters parameters to to obtain obtain the the conservation conservation •* Simo property. property.
Boundary conditions: non-linear problems 29
2,5.1 Displacement (essential) condition The specification of a boundary condition for displacement is given by Eq. (1.7). In a finite element calculation the usual procedure to specify a displacement boundary condition merely assigns the value at a node as (~.la) i
-- Ui(Xa)
(2.34)
where (fia)i is the value at node a in the direction i, as shown for a two-dimensional case in Fig. 2.2. Here we note that the condition is imposed on the finite element approximation to the boundary Fuh and not the true boundary Pu. As the mesh is refined near the boundary the two converge, generally at a rate equal to or higher than errors from other approximations. Imposing a specified displacement condition may be implemented in several different ways. For example, consider the linear static problem given by
Igll K12] Ul < (u2}:{ h }
(2.35)
in which the condition Ul -- fil is to be imposed. 1. Impose the condition by replacing the first equation (associated with 6ui in a weak form) by the boundary condition giving
1 K0221{Ul} u2 _ (~1 which yields the desired solution. This method is not efficient if K is symmetric. However, by writing the system as
I0 K22 0 ] {Ul} { Ul } u2 = t " 2 - K z l u l
(2.36a)
the problem again becomes symmetric. 2. A second approach is to perform the modification as above and eliminate all equations for which values are known. Accordingly, we then have K22u2 = f2 - K21 ~1
Fu
[u=~]
Fig. 2.2 Boundary conditions for specified displacements.
(2.36b)
Fu
30
Galerkinmethod of approximation -irreducible and mixed forms together with the known condition/'/1 -- fi 1o This approach leads to a final set of equations with a minimum number of unknowns and is the one adopted in FEAPpv. 3. A third method uses a 'penalty' approach 2 in which the equations are given as kll K21
fk11 fi
K12] Ul K22J { u 2 } -
~
f2 1}
(2.36C)
in which kll -- c~K11 where c~ > > 1. This method is very easy to implement but requires selection of an appropriate value of c~. For simple point constraints, such as considered for u l = ill, a choice of c~ = 106 to 108 usually is adequate. When transient non-linear problems are encountered the imposition of displacement boundary conditions becomes slightly more involved. First, the boundary condition needs to be implemented on the incremental equations. This requires computation of initial values for the displacement, velocity and acceleration variables. As noted above there are two basic forms to consider: the explicit form and the implicit form.
Non-linear explicit problems
The explicit form is straightforward if velocity terms do not appear in the equilibrium equation. Here, for the GN22 algorithm,/32 = 0 and the value of the displacement at tn+l is obtained from Eq. (2.211) including those for the boundary Fu. Next we solve M an+l --- fn+l -- P(o'n+l)
(2.37)
for the new acceleration, with M given by a diagonal form. The velocity is then computed with the known acceleration using Eq. (2.212). By employing a diagonal M, there is no coupling of the acceleration between boundary and non-boundary nodes. If the velocity appears explicitly in the equilibrium equation an iterative strategy can be adopted where Vn+ 11 is taken as Vn and a 'trial' value of the acceleration is computed. Computing the velocity from the trial acceleration and performing one more iteration yields results which are adequate. Here it may be necessary to devise an expression for the boundary velocity updates to maintain high accuracy in final results.
Non-linear implicit problems
In an implicit form using the GN22 algorithm both /~1 and/32 are non-zero. If the time increment At is zero both the displacement and the velocity do not change [viz. Eq. (2.21)] and the new acceleration is determined from Eq. (2.37) - accounting only for any instantaneous change in fn+l. When At > 0, Eq. (2.24a) is used to impose the constraint Un+1 -- fin+l. In the first iteration we obtain du 1+1 = dUn+l -1 from Eqs (2.24bl) and (2.25a) such that Un+l e -- Un+l. This increment of the displacement boundary condition is employed in the incremental form
All AI2I A=I A22]
d~l } du2
"- {~12}
(2,38)
during the first iteration only. In the above the set Ul are associated with known displacements of boundary nodes and set u2 with the 'unknown' displacements. Any of the methods described above for the linear static problem may be used to obtain the solution.
Boundary conditions: non-linear problems 31
2.5.2 Traction condition The application of a traction is a 'natural' variational boundary condition and does not affect the active nodal displacements at a boundary- it only affects the applied nodal force condition. The imposition of a non-zero traction on the boundary requires an integration over the surface of each element. Thus for a typical node a as shown in Fig. 2.3 it is necessary to evaluate the integral
fa-- f e
(2.39) t
where e ranges over all elements belonging to 1-'t that include node a (e.g. for the two-dimensional case shown in Fig. 2.3 this is the element above and the element below node a). Of course if t is zero no evaluation of the integral is required.
Pressure loading
One important example is the application of a normal 'pressure' to a surface. Here the traction is given by
i - p,,n where Pn is the specified normal pressure (taken as positive when in tension) and n is a unit outward normal to the boundary rt, see Fig. 2.4. In this case Eq. (2.39) becomes
fa = Z fF Na [:?nIld r e
(2.40a)
t
Using Eq. (2.18) the computation of pressure loading is given by
fa -- ~ /O Na(~) Pn(~) (Vl X
(2.40b)
Y2) d F ]
e
where 121 = d~l d~2 and each element integral is performed on the natural coordinate system directly. For two-dimensional problems the surface shape functions are given h
rt
t=t]
Fig. 2.3 Boundaryconditions for specified traction.
Ft
fa
32
Galerkin method of approximation -irreducible and mixed forms
Fig. 2.4 Normal to surface.
by Na (~1),
1-"1--- d~ 1 and we use u ~
for plane strain for plane stress for axisymmetry
e3, h3 e3, 2"n X 1 e3,
where e3 is the unit normal vector to the plane of deformation.
2.5.3 Mixed displacement/traction condition The treatment of a mixed condition in which some displacement components are specified together with some traction components often requires a change in the nodal parameters. For example, a shaft with axis in the x3 direction and radius R that rotates inside a bearing (without friction or gaps) requires un = u r ( g ) -- 0 and to(R) = t z ( R ) = 0 (where the coordinate origin is placed at the centre of the shaft). In this case it is necessary to transform the degrees of freedom at each node on the boundary of the shaft such that
{(~l)a } [COS 0a Ua "-" (fi2)a "-" si 0 a
-
sin Oa
COS0 a 0
i]{ora} (blO)a
--
La ~la'
(2.41)
(~lz)a
This transformation is then applied to a residual as Ra, = LTRa
(2.42a)
and to the mass and stiffness as Ma'b' - L T M a b Lb; Ma,c -- LTMac; Mcb' ---- Mcb Lb Ka'b' -- L T K a b Lb; Ka,c -- L aT Kac; Kcb' -- Kcb Lb
(2.42b)
where a and b belong to transformed nodes and c to a node which retains its original orientation. It is usually convenient to perform these transformations on each individual element; however, if desired they can be applied to the assembled arrays. Once the transformation is performed, each individual displacement and traction condition may be imposed as described above.
Mixed or irreducible forms
2.6 Mixed or irreducible forms The evaluation of the stiffness given by Eq. (2.12) was cast entirely in terms of of the so-called displacement formulation which indeed is extensively used in many finite mixed finite eleelement solutions. However, on some occasions it is convenient to use mixedfinite ment forms and these are especially necessary when constraints such as (near) incompressibility arise. It has been frequently noted that certain constitutive laws, such as those of viscoelasticity and associative plasticity that we will discuss in Chapter 4, the material behaves in a nearly incompressible manner. For such problems a reformulation is necessary. On such occasions we have two choices of formulation. We can have the variables u and p (where p is the mean stress) as a two-fieldformulation two-field formulation or we can have the variables u, p and c:eov (where c:e~v is the volume change) as a three-field formulation (e.g. see reference 2 for more details). Here several alternatives are available and the matter of which we use may depend on the form of the constitutive equation employed. For situations where changes in volume affect only the pressure the two-field form can be easily used. However, for problems in which the response may become coupled between the deviatoric and mean components of stress and strain the three-field formulations lead to much simpler forms from which to develop a finite element model. To illustrate this point we present a general three-field mixed formulation and show in detail how such coupled effects can be easily included without any change change to the previous discussion on solving non-linear problems. The development also serves as a basis for the development of an extended form which permits the treatment of finite Chapter 5. deformation problems. This extension will be presented presented in Chapter
Deviatoric and components Deviatoric and mean mean stress and and strain components
The treatment of nearly incompressible materials is most easily considered by splitting the stress and strain into their deviatoric (isochoric) and mean parts. Accordingly, we define the mean stress (pressure) as P
--
1
5
1
[fill ~- 0"22 -~- 0"33] --- g O'ii
(2.43a)
and the deviator stress as (ad)ij - oij p (O'd)ij = -'- aij CTij -(~ijP
(2.43b)
(~ij is the Kronecker delta function where oij
1; (~ij --
0;
forii = j for 7~ j for i i=
Similarly, we define the mean strain (volume change) as /~v -- [ e l l -~- E'22 "~- 6"331 = eii
(2.44a)
and the deviator strain as (ed)ij
-- eij -- -~1 (~ij l?'v
(2.44b)
Note that the placement 1/3 factor appears in both, but at different locations in placement of the 1/3 the expressions.
33
34 Galerkinmethod of approximation -irreducible and mixed forms
A three-field mixed method for general constitutive models
In order to develop a mixed form for use with constitutive models in which mean and deviatoric effects can be coupled we define mean and deviatoric matrix operators given by 1 1 0 0 0] r and Id = I - - I gmm r , m-(2.45)
[1
respectively, where I is the identity matrix. The strains may now be expressed in a mixed form as e -- ld(SU)
1
+ 5 m
eo
(2.46a)
where the first term is the deviatoric part and the second the mean part. Similarly, the stresses may now be expressed in a mixed form as or = ld # + m p
(2.46b)
where 6" is the set of stresses deduced directly from the strains, incremental strains, or strain rates, depending on the particular constitutive model form. For the present we shall denote this stress by 6" = tr(e) (2.47) and we note that it is not necessary to split the model into mean and deviatoric parts. The weak form (variational Galerkin equations) for the case including transients is now given by
t
f~eo
[1mr*-
el dQ
0
(2.48)
f a 3 p [m r (Su) - eo] d a = 0 Introducing finite element approximations to the variables as u ~ fi = Nufi,
p ,~ p -
Npp
and
ev ~ ~v = Nolo
and similar approximations to virtual quantities as ~u ~ d;fi = Nu6fi,
6p ~ t~/~ -- Npt~p
and
&o ~ 3~v = No~o
the strain in an element becomes e = IdBfi+
1
~ mN~ ~
(2.49)
in which B is the standard strain--displacement matrix given in Eq. (2.9a). Similarly, the stresses in each element may be computed by using tr - Id t~ + m Np p where again ~ are stresses computed as in Eq. (2.47) in terms of the strains e.
(2.50)
Mixed or irreducible forms
Substituting the element stress and strain expressions from Eqs (2.49) and (2.50) into Eq. (2.48) we obtain the set of finite element equations oo
P+Mfi=f Pp - Kvp
p= 0
(2.51)
--K~Tg~ + Keu fi = 0
where f p - In Bror dr2' K~p
T T,, Pp - ~1 L Nom crdf2
Kpu = f~ NpTmTBdr2
f~ NrNp dr2,
(2.52)
t
If the pressure and volumetric strain approximations are taken locally in each element and N~ = Np it is possible to solve the second and third equation of (2.51) in each element individually. Noting that the array K~p is now symmetric positive definite, we may always write these as = K~Ipp ~ = K~plKeu~ = W~
(2.53)
The mixed strain in each element may now be computed as e-where
[ ; I E ldB+
mBo f i =
Id
~m
Bo fi
Bo = NoW
(2.54) (2.55)
defines a mixedform of the volumetric strain--displacement equations. From the above results it is possible to write the vector P in the alternative forms ~7'18 lnT T1J 6,dS2 = f~ [B T B T] [ l imT d] ~dfl P = f~ BTcrdfl - f~ [BTId+ ~n~m (2.56) Based on this result we observe that it is not necessary to compute the true mixed stress except when reporting final results. This is particularly important when we consider the effects of other material models in Chapter 4. The last step in the process is the computation of the tangent for the equations. This is straightforward using forms given by Eq. (2.47) where we obtain
d6"- OTdr Use of Eq. (2.54) to express the incremental mixed strains then gives Kr=
[B r
BT]
~r
Dr lid
89
(2.57a)
35
36
Galerkin method of approximation -irreducible and mixed forms
It should be noted that construction of a modified modulus term given by I)r -- / l m r L~ld
]
l)r[ld
[ Idi)rld
[ 89
89
3ldl)rm ]
[I)ll
I)12]
d 9 m r l ) r m j = [I)21 DEEJ
(2.57b)
requires very few operations because of the sparsity and form of the arrays Id and m. Consequently, the multiplication by the coefficient matrices B and B~ in this form is far more efficient than constructing a l] as 2 = IdB d- ~1 m B~
(2.58)
and computing the tangent from
KT -- ~ l~rl)rl~ d ~
(2.59)
In this form I~ has few zero terms which accounts for the difference in effort.
Example 2.1 Linear elastic tangent
As an example we consider a linear elastic material with the constitutive equation expressed as (2.60) o" = (K m m r + 2G I0) e in which
2 0 Io-~
1
0 0
0 0 0
0
0
0 0 0000
2
0
0
0 0 0
(2.61)
1 0 ! 0 1 0 0
accounts for the transformations used to define the strain. The incremental form is identical to Eq. (2.60) with o., e replaced by do', de. The above form for the mixed element is valid for use with many different linear and non-linear constitutive models. In Chapter 4 we consider stress-strain behaviour modelled by viscoelasticity, classical plasticity, and generalized plasticity formulations. Each of these forms can lead to situations in which a nearly incompressible response is required and for many examples included in this book we shall use the above mixed formulation. Here two basic forms of finite element approximations are considered: a four-node quadrilateral or an eight-node brick isoparametric element with constant interpolation in each element for one-term approximations to No and Np by unity; and a nine-node quadrilateral or a 27-node brick isoparametric element with linear interpolation for Np and No.* Accordingly, for the latter class of elements in two dimensions we use
Np--Nv---[1
~ r/]
or
[1
x
y]
* Formulationsusingthe eight-nodequadrilateraland 20-nodebrick serendipityelementsmay alsobe constructed; however, these elements do not fully satisfy the mixed patch test (see reference 2).
Non-linear quasi-harmonic field problems 37 and in three dimensions Np--N~=
[1
~ r/ ~]
or
[1
x
y
z]
The elements created by this process may be used to solve a wide range of problems in solid mechanics, as we shall illustrate in later chapters of this volume.
~~~[iii[i[iii!!~!~i[i~iii~i!i!i!iii!ii~i~i~i~i!ii!i!i[i!i}ii~i~i~i!i!i!i!ii~!~i~!~i!i!i~i!i!i!i!iii!i!~i!i~!~!iiiiii!i!i!i!i~!!ii~ii~i~i~iiiii!~iiiii!i!i!iii~!i~iii!ii~i!iiii~!i!~!~iii~i~i~i!i!iii!i~i~ii~i!i~iii~i!i~ii!
In subsequent chapters we shall touch upon non-linear problems in the context of inelastic constitutive equations for solids, plates, and shells and in geometric effects arising from finite deformation. Non-linear effects can also be considered for various fluid mechanics situations (e.g. see reference 19). However, non-linearity occurs in many other problems and in these the techniques described in this chapter are still universally applicable. An example of such situations is the quasi-harmonic equation which is encountered in many fields of engineering. Here we consider a simple quasiharmonic problem given by (e.g. heat conduction)
pc~b + V r q - a ( r
= 0
(2.62)
with suitable boundary conditions. Such a form may be used to solve problems ranging from temperature response in solids, seepage in porous media, magnetic effects in solids, and potential fluid flow. In the above, q is a flux and quite generally this can be written as q - q(r V r - - k(r V r 1 6 2 or, after linearization,
dq = - k~162- k l d ( V r
where kO =
(~qi 0r
and
klj--
Oqi Or
The source term Q(r also can introduce non-linearity. A discretization based on Galerkin procedures gives after integration by parts of the q term the problem
,H - j~ ~r p c ~pdf2 - J~ (V ~r r q df2 (2.63) -~6r162
6r q
and is still valid if q and/or Q (and indeed the boundary conditions) are dependent on r or its derivatives. Introducing the interpolations r
N~b(t)
and
~r
N~(b
(2.64)
a discretized form is given as ~I, - f((b) - CO - Pq ((b) - 0
(2.65a)
38
Galerkin method of approximation -irreducible and mixed forms where C-
~NrpcNdf~
pq __ _/,., (~rN)r q dr2
(2.65b)
q
Equation (2.65a) may be solved following similar procedures described above. For instance, just as we did with GN22, we can now use GN 11 as ~ ~n+l -- ~n + (1 - 0 ) ~ n .Ji-Off)n+ 1
(2.66)
Once again we have the choice of using ~b.+~ or ~n+~ as the primary solution variable. To this extent the process of solving transient problems follows the same lines as those described in the previous section and need not be further discussed here. We note again that the use of q~.+~ as the chosen variable will allow the solution method to be applied to static (steady state) problems in which the first term of Eq. (2.62) becomes zero.
iii~i!iii~iiiiiii~i~ii~iiii~ii~ii~iiii!i!iiiiiiiiiii~i~;i~ii~iii!i~i~!!i~iii~iii~i~!i!ii!!~J~!~ii!~!i!ii!ii~!i~i~i1iii~i~i~i~iiiiiiii~iii~iiii~i~i!ii~!i~iiii~i~!i~i!~:~!!iiiiiii!~!~i~i~i~i~i~i~!~!~:~!iiii~:~i!i~iii~i~i~ In this section we report results of some transient problems of structural mechanics as well as field problems. As we mentioned earlier, we usually will not consider transient behaviour in later parts of this book as the solution process for transients essentially follows the path described above.
Transient heat conduction
The governing equation for this set of physical problems is discussed in the previous section, with ~bbeing the temperature T now [Eq. (2.62)]. Non-linearity clearly can arise from the specific heat, c, thermal conductivity, k, and source, Q, being temperature dependent or from a radiation boundary condition aT k-- = -c~(TOn
To) n
(2.67)
with n # 1. Here c~ is a convective heat transfer coefficient and To is an ambient external temperature. We shall show two examples to illustrate the above. The first concerns the freezing of ground in which the latent heat of freezing is represented by varying the material properties with temperature in a narrow zone, as shown in Fig. 2.5. Further, in the transition from the fluid to the frozen state a variation in conductivity occurs. We now thus have a problem in which both matrices C and P [Eq. (2.65b)] are variable, and solution in Fig. 2.6 illustrates the progression of a freezing front which was derived by using the three-point (Lees) algorithm 2~ with C - Cn and P = Pn. A computational feature of some significance arises in this problem as values of the specific heat become very high in the transition zone and in time stepping can be missed
Typical examples of transient non-linear calculations !
:z: f5 CL
k(T)
pc(T)
!
---,
i
i
i
i i I
I
To
~-- 2AT
r~
Fig. 2.5 Estimation of thermophysical properties in phase change problems. The latent heat effect is approximated by a large capacity over a small temperature interval 2A T.
if the temperature step s t r a d d l e s the freezing point. To avoid this difficulty and keep the heat balance correct the concept of enthalpy is introduced, defining H =
f0Tpc d T
(2.68)
Now, whenever a change of temperature is considered, an appropriate value of pc is calculated that gives the correct change of H. The heat conduction problem involving phase change is of considerable importance in welding and casting technology. Some very useful finite element solutions of these T... 289K
~o,
,4
P~ss~ of frozen zone.
l
.,~ d I 1 :l .# /
gi
.4
.E,
=,2
,2 -6
(al
+
0
8
x(m)
Fig. 2.6 Freezing of a moist soil (sand).
20
8
~)
,(m)
10
12
14
39
40
Galerkin method of approximation -irreducible and mixed forms problems have been obtained. 22 Further elaboration of the procedure described above is given in reference. 23 The second non-linear example concerns the problem of spontaneous ignition. 24 We will discuss the steady state case of this problem in Chapter 4 and now will be concerned only with transient behaviour. Here the heat generated depends on the temperature Q = ~Ser
(2.69)
and the situation can become physically unstable with the computed temperature rising continuously to extreme values. In Fig. 2.7 we show a transient solution of a sphere at an initial temperature of T = 290 K immersed in a bath of 500 K. The solution is given for two values of the parameter ~ with k = pc = 1, and the non-linearity is now so severe that a full Newton iterative solution in each time increment is necessary. For the larger value of ~ the temperature increases to an 'infinite' value in afinite time and the time interval for the computation had to be changed continuously to account for this. The finite time for this point to be reached is known as the induction time and is shown in Fig. 2.7 for various values of ~. The question of changing the time interval during the computation has not been discussed in detail, but clearly this must be done quite frequently to avoid large changes of the unknown function which will result in inaccuracies.
Structural dynamics
Here the examples concern dynamic structural transients with material and geometric non-linearity. A highly non-linear geometrical and material non-linearity generally occurs. Neglecting damping forces, Eq. (2.11 a) can be explicitly solved in an efficient manner. If the explicit computation is pursued to the point when steady state conditions are approached, that is, until a = v ~ 0, the solution to a static non-linear problem is obtained. This type of technique is frequently efficient as an alternative to the methods described above and in Chapter 3 has been applied successfully in the context of finite differences under the name of 'dynamic relaxation' for the solution of non-linear static problems. 25 Two examples of explicit dynamic analysis will be given here. The first problem, illustrated in Plate 3, is a large three-dimensional problem and its solution was obtained with the use of an explicit dynamic scheme. In such a case implicit schemes would be totally inapplicable and indeed the explicit code provides a very efficient solution of the crash problem shown. It must, however, be recognized that such final solutions are not necessarily unique. As a second example Fig. 2.8 shows a typical crash analysis of a motor vehicle carried out by similar means.
Earthquake response of soil- structures
The interaction of the soil skeleton or matrix with the water contained in the pores is of extreme importance in earthquake engineering and here again solution of transient non-linear equations is necessary. As in the mixed problem which we referred to earlier, the variables include displacement, and the pore pressure in the fluid p.
Typical examples of transient non-linear calculations
10
~crit
8
1
2
(a) 700
a=2
700
600 v
10
20
40
8=16
600
t = 330, At = 160
L_
4
v
= 500
= 500
E
E
400
400
300
300
(b) Fig. 2.7 Reactive sphere. Transient temperature behaviour for ignition (~ -- 16) and non-ignition (~ --- 2) cases: (a)ind_uction time versus Frank-K_amenetskii parameter; temperature profiles; (b) temperature profiles for ignition (6 - 16) and non-ignition (6 - 2) transient behaviour of a reactive sphere.
We have in fact shown a comparison between some centrifuge results and computations elsewhere (viz. Chapter 18 of reference 2). These illustrate the development of the pore pressure arising from a particular form of the constitutive relation assumed. Many such examples and indeed the full theory are given in reference 26 and in Fig. 2.9 we show an example of comparison of calculations and a centrifuge model presented at a 1993 workshop known as VELACS. 27 This figure shows the displacements of a big retaining wall after the passage of an earthquake, which were measured in the centrifuge and also calculated.
41
42
Galerkin method of approximation - irreducible and mixed forms
Fig. 2.8 Crash analysis: (a) mesh at t == 0 ms; (b) mesh at t == 20 ms; (c) mesh at t == 40 ms.
Concluding Concluding remarks remarks 43
...
45.0 45.0
_...~ 5.0 5.0 .1 .1.
o0.5 . ~~li"'l·.~"'I~.t- =
17.o 17.0
't
....
~ CBI
ACCIO ACC,0
...
_ ~l ' _ _ACC12 L ~.t. ' ~ACC11 c" .i.ll. ~
~';m
PP~,6
PPT2 P"~T5 I • ACC9 PPT5
PPT3 PPT3
PPT2
~ac~,,,
....
,
= "~=1
backfill
lI!!IlI ACCS.-:l
...
z Z ,~x x
~
PPT4 PPT4 lI!!IlI
ACC3
r
it
"'r_:.1
20.0 -,,20:::·0'-
~, ~o,~
~5
lVDTl LVDT1 ~ -'=- T
I
/
model 5.0 water I wall w='m~176176 w=o, /
~~~ ~
~o 2.0
~ ~
;ACC2 ~;CCc, Acel
P
.~n 50
0.5 925.~ 0.35 0.35 0.5
I• . • ,0~o~ - .~=~
.0.20
V
1~1+
--.(------------- r-"
:0.35 • :'5' .., ............. 1~:~ ~.~o ,
Water saturated sand : Water saturate~sanOl i
Cambridge-test Cambridge-test
L~s 0.75 Water W ater
5.0
2.0 2.0
Jr
- - Initial boundary Initial boundary Boundary after after earthquate earthquate test test .- .-.-. -. .- - Boundary
0.5 0.5
Location 12.0, 7.0) = (25.0, (25.0, 12.0. 7.0) m m Location = LVDT1 LVDT1
~" 0.4 v:[0.4 r 'E r~ 0.3 0.3 E (9 Gl 0 co ~ 0.2 0.2
0.5 0.5
I Location -= (25.0, 12.0,7.0) 12.0, 7.0) m horizontal (LVDT1) horizontal
E 0.4 0.4 p
'E 8~ 0.3 0.3
L .~
|.~Gl o.e /
Q. a. r
'"0 .0.1 ~Ci 1 f 0
~ 0.2 ~. a. (n '" 0.1 F , ~ Ci ~o.1
ridge-test i
I
55
I
Time Time (s) (s)
10 10
15 15
00 I ~ 9 00
I 55
LVDT1 DT1 Prediction Prediction
Time (s)
I 10 10
15 15
Fig. 2.9 Retaining Retaining wall wall subjected subjected to earthquake earthquake excitation' excitation: comparison comparison of experiment experiment (centrifuge) (centrifuge) and 26 calculations. calculations.26 ~iii~iii~i~i~i i~i~i~i~%!~ii~%iiii~i~~i~ii!ii~!~%ii~ii~i~i~~i~i!ii~ii~ii~i~ii!i!i!~i~i~%i i~ii!!ii~ii~!ii~!iiiiiii~i:i~!i!iili~!i!i!i!ilii!~i!li!i!iiiiiiiiii!iil l ~ iiii!iliiiiiiiiiiliii~':i~:'iii!iiiiiiii!iiiiiiiiiiiiiiiiiiiiii iiiiiiiiiiiiiiiiii!iiiii!i!ii!i!iiiiiiiiiiiiiiii!iiii!iiiiiiiiiiiiiiiiiiiiii iiiiiiiiii!iiiiiiiii!iiiiiiiiii!iiiiiiiiiii!iiiiiiiiiiiiiiiiiiiiiii!iiiii!iiiil iii!iiiiiiiiiiiiiiii!!iiiiii!ii!iiiiii!ii!il!il!iii!iliiiiiiiiiiiiiiiiiiii!iiiiiiii i!ii!i!iiiii!iiiii
2.9 Concluding remarks
!ii!ii!ii!ii!iii!!iii!iiiiiii!ii!iii! i!iiiii!ii!i !i!i!ii•i!ii!!i;i•i!i!!iii!i!!!!!!i!i!!!ii!i:i!i•i!i!•!!ii!!i!iii! !!ii!i!! i•!!!•!i ii!ii!iii !•iii!i iiiiiii,!!::,,~,~!! !iiiiiiiiiiiii!!i!iiii!i!iii!ili!!!ii!i!!!iiiiiii!i!iii!!!iiii !!!i!iiiiiiiiiii!iiiiiiiiiiiiiiiiiiiiiiii!!ii!i!iiiiiiiii! ii!!!!!!!i!ii!iiii iii!iii!ii!;!ii!i;iiil;;;i;i~ii~~i!iii!!iii!il!;!;i;;;;~ili;i~i;!i:!~ili~!ii~ii;ili!i!!!;!~i;;~i!i!i!;iii!i!i!i!i!iii!i!i!;!!!i~!; !!i;i~!i!~!~!;!~i~ii;~iii;iii ili; iiiiii~~i~i~i~i;i~i;i;;;;ii;!;;iiii;iiiiiiiiiiili;i;ilili;ii!;i;
this chapter chapter we we have have summarized summarized the the basic basic steps steps needed needed to solve solve a general general smallsmallIn this strain the quasi-harmonic quasi-harmonic field problem. problem. Only Only a strain solid solid mechanics mechanics problem problem as well well as the standard standard Newton Newton solution solution method method has has been been mentioned mentioned to to solve solve the the resulting resulting non-linear non-linear algebraic algebraic problem. problem. For For problems problems which which include include non-linear non-linear behaviour behaviour there there are are many many
44 Galerkin irreducible and mixed Galerkinmethod method of approximation approximation --irreducible mixed forms forms
situations where additional situations where additional solution solution strategies are required. required. In the next chapter chapter we problems. In will consider consider some some basic schemes schemes for solving solving such such non-linear non-linear algebraic algebraic problems. subsequent chapters chapters we shall address address some some of of these context of of particular subsequent these in the the context particular problems problems classes. classes. The reader will note except in the example example solutions, solutions, we have not discussed The reader note that, except not discussed problems solution problems in which which large strains occur. We can note here, however, that the solution described above remains The parts change are associated associated with strategy described remains valid. The parts that that change with the effects of of finite deformation deformation and and the manner manner in which computing which these affect the the computing of stresses, the stress-divergence stress-divergence term and the resulting tangent moduli of stresses, term and resulting tangent moduli and and stiffness. As these aspects involve more advanced we have have deferred treatment of As involve more advanced concepts concepts we deferred the treatment of finite strain problems problems to later later chapters where we we will address basic formulations formulations and chapters where address basic applications. applications.
References Vestn. equilibrium of rods and plates. Vestn. 1. B.G. Galerkin. Series solution of some problems in elastic equilibrium 19:897-908, 1915. Inzh. Tech., Tech., 19:897-908, Zhu. The Finite Element Method: Its Basis and Funda2. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. Butterworth-Heinemann, Oxford, 6th edition, 2005. mentals. Butterworth-Heinemann, T.J.R. Hughes. The Finite Element Method: Linear Static and Dynamic Analysis. Dover Publi3. TJ.R. cations, New York, York, 2000. W.K. Liu and B. Moran. Nonlinear Finite Elements for Continua and Structures. 4. T. Belytschko, W.K. John Wiley & & Sons, Chichester, 2000. 5. B. Cockburn, G.E. Karniadakis and Chi-Wang Shu. Theory, Shu. Discontinuous Galerkin Methods: Theory, Computation and Applications. Springer-Verlag, Berlin, 2000. 6. E.L. Wilson, R.L. Taylor, w.P. Doherty and J. Ghaboussi. Incompatible displacement models. Taylor, W.P. editor, Numerical and Computer Methods in Structural Mechanics, pages S.T. Fenves et al., editor, In ST. 43-57. Academic Press, New York, York, 1973. 7. G. Strang and GJ. G.J. Fix. An Analysis of of the Finite Element Method. Prentice-Hall, Englewood Cliffs, NJ, 1973. 8. J.C. Simo and M.S. Rifai. A class of mixed assumed strain methods and the method of incompatible modes. International Journal for Numerical Methods in Engineering, 29:1595-1638, 1990. Witt. Concepts andApplications and Applications ofFinite of Finite Element 9. R.D. Cook, D.S. D .S. Malkus, M.E. Plesha and R.J. RJ. Witt. Analysis. John Wiley & & Sons, New York, 4th edition, 2001. of Mathematical Functions. Dover PubliI.A. Stegun, editors. Handbook of 10. M. Abramowitz and LA. cations, New York, York, 1965. II. Frasier. Treatment of hour glass patterns in low order finite 11. D. Kosloff and G.A. Frasier. finite element codes. International Journal for Numerical Analysis Methods in Geomechanics, 2:57-72,1978. 2:57-72, 1978. Journalfor 12. N. Newmark. A method of computation for structural dynamics. J. Engineering Mechanics, ASCE, 85:67-94, 1959. 13. N. Bicanic Bi6ani6 and K.w. K.W. Johnson. Who was 'Raphson'? International Journal for Numerical 14:148-152, 1979. Methods in Engineering, 14:148-152, 14. J.C. Simo and N. Tarnow. Tarnow. The discrete energy-momentum method. Conserving algorithm for ftlr Mathematik und Physik, 43:757-793, 1992. nonlinear elastodynamics. Zeitschrift Zeitschriftfur J.C. Simo and N. Tarnow. Tarnow. Exact energy-momentum conserving algorithms and symplectic 15. lC. schemes for nonlinear dynamics. Computer Methods in Applied Mechanics and Engineering, 100:63-116, 100:63-116, 1992.
References 45 16. J.C. Simo and O. Gonz~ilez. Recent results on the numerical integration of infinite dimensional hamiltonian systems. In Recent Developments in Finite Element Analysis. CIMNE, Barcelona, Spain, 1994. 17. T.J.R. Hughes. Generalization of selective integration procedures to anisotropic and non-linear media. International Journal for Numerical Methods in Engineering, 15:1413-1418, 1980. 18. J.C. Simo and T.J.R. Hughes. On the variational foundations of assumed strain methods. J. Appl. Mech., 53(1):51-54, 1986. 19. O.C. Zienkiewicz, R.L. Taylor and P. Nithiarasu. The Finite Element Methodfor Fluid Dynamics. Butterworth-Heinemann, Oxford, 6th edition, 2005. 20. M. Lees. A linear three level difference scheme for quasilinear parabolic equations. Maths. Comp., 20:516--622, 1966. 21. G. Comini, S. Del Guidice, R.W. Lewis and O.C. Zienkiewicz. Finite element solution of nonlinear conduction problems with special reference to phase change. International Journal for Numerical Methods in Engineering, 8:613-624, 1974. 22. H.D. Hibbitt and P.V. Marqal. Numerical thermo-mechanical model for the welding and subsequent loading of a fabricated structure. Computers and Structures, 3:1145-1174, 1973. 23. K. Morgan, R.W. Lewis and O.C. Zienkiewicz. An improved algorithm for heat convection problems with phase change. International Journal for Numerical Methods in Engineering, 12:1191-1195, 1978. 24. C.A. Anderson and O.C. Zienkiewicz. Spontaneous ignition: finite element solutions for steady and transient conditions. Trans ASME, J. Heat Transfer, pages 398-404, 1974. 25. J.R.H. Otter, E. Cassel and R.E. Hobbs. Dynamic relaxation. Proc. Inst. Civ. Eng., 35:633-656, 1966. 26. O.C. Zienkiewicz, A.H.C. Chan, M. Pastor, B.A. Schrefler and T. Shiomi. Computational Geomechanics: With Special Reference to Earthquake Engineering. John Wiley & Sons, Chichester, 1999. 27. K. Arulanandan and R.F. Scott, editors. Proceedings of VELACS Symposium, Rotterdam, 1993. Balkema.
iii iiiiii i iiiiiiiiiii i i i !i iii Solution of non-linear Solution of non-linear algebraic equations algebraic equations i!i!ii~i~i~~ii~iii i~i!~ii~i~i~iiiii~iii~ i~ii~i~'i'~i~ili!ii i!i~iiiii! iiii~!~i!i!ii!i!i!i ii~ii~ilili!i!i iiiiiiiiiiiiiiiiiiiii:':' 'iiii!iiiiiliiiiiiii!i!i!!!i iiiiiiiii~iiiiiiiiiiiiiiiiii~i~i~i i!iiiiii i~iiiiii iiliiiiiiiiiiii~iiii!i!il li iiiiiii iiiiii!iiiiiiiiiiiiil iii~i~iliiiili i!iii!i il iiiiiiiiiiiiiili!ililiiiii~iiiiiiiiiiiiiiiliiiiiil iiiiiiiii!ii i~iiif!ilii~iiiiiii i !!ii!iiiiii ili~iiiiiii! iiiiHiiii ~i~i!ii!iiiiiiiiiiiilii!iii~ii~~ii~ii~!iiliii iiiiiiii i ii!~ii~!i~!~i i ~iiiii~i!iiiliiiiii i~ili ii~iiiiii iiiii~i~i~i!i~i!iiii liiili~iiiiiiiii i iili~ii~ii~i!iiii!ii ii!iii~iiiiiiil iiii!iii~iiiiiiiil !i~~iii!~i~i i!iliiiiii!i !i~iiiiiiiii i ~!~!~i iiili!ili~iiiilil iiii ii i iii!ili!!iiiiiiiii iiiiilili!iii!~iiiiiliii ~iliiiiiiii i iiiiii i~i~i~iiii!il i!iliiiiiiiiiiil i~iiiiiiil i iiiiiiiiii ilili!i!iii~i!i~iiiiiiili i~ilHi !iiiiilii iliii
3.1 Introduction
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In the solution linear problems by a finitebyelement alwaysweneed to solve In the of solution of linear problems a finite method elementwe method always need to solve a set of simultaneous algebraic algebraic equationsequations of the form a set of simultaneous of the form Ku = f
Ku=f
(3.1)
(3.1)
Provided Provided the coefficient matrix is matrix non-singular the solution these equations is unique.is unique. the coefficient is non-singular thetosolution to these equations In the solution non-linear problemsproblems we will always obtain a set of algebraic equations;equations; of non-linear we will always obtain a set ofalgebraic In theof solution however, however, they generally will be non-linear. For example, in Chapterin2Chapter we obtained the they generally will be non-linear. For example, 2 we obtained the set (2.22)set at (2.22) each discrete tn+l.time Here, we Here, consider the genetic wewhich we at eachtime discrete tn +l. we consider theproblem generic which problem indicate as indicate as ~I/n+l "-- ~I/(Un+l) : fn+l -- =P(un+l) (3.2) W n +l = W(U f n + 1 - P(Un +l) (3.2) n +l) where u~+l is the set isofthe discretization parameters, f,,+l a vector is which independent where Un +l set of discretization parameters, f n +) which a vector is independent of the parameters and P a vector on the parameters. These equations may of the parameters and P dependent a vector dependent on the parameters. These equations may have multiple [i.e. more[i.e. thanmore one set Eq. (3.2)]. Thus, if a have solutions multiple solutions thanofu~+l one setmay OfUsatisfy +l may satisfy Eq. (3.2)]. Thus, if a n solution issolution achieved it may notitnecessarily be the solution is achieved may not necessarily be the sought. solutionPhysical sought. insight Physicalinto insight into the naturethe of nature the problem usually, incremental approaches from known of the and, problem and,small-step usually, small-step incremental approaches from known solutions solutions are essential to obtain to realistic Such increments are indeedarealways are essential obtain answers. realistic answers. Such increments indeed always required required if the problem is transient, if the constitutive law relating andstress strainand strain if the problem is transient, if the constitutive law stress relating is path dependent and/or if and/or the load-displacement path has path bifurcations or multiple is path dependent if the load-displacement has bifurcations or multiple branches branches at certainatload levels. certain load levels. The general from a nearby Theproblem general should problemalways shouldstarts always starts from asolution nearby at solution at u = u~, U=~ Un,= 0,
-- fn Wn f=0,
(3.3)
(3.3)
(3.4)
(3.4)
(3.5)
(3.5)
and oftenand arises from changes the forcing to often arises from in changes in thefunction forcing fn function f n to fn+l = fnf+ Afn+l n + 1 = fn
+ b.fn + 1
The determination of the change The determination of theAun+~ changesuch b.Uthat n +l such that Un+ 1 -- U n +
mUn+
1
will be thewill objective and generally the increments of Afn+l will ben +kept reasonably small be the objective and generally the increments of b.f 1 will be kept reasonably small so that path dependence can be followed. Further, such incremental procedures will so that path dependence can be followed. Further, such incremental procedures will
Iterative Iterative techniques techniques 47 47 f f~ PP
Softening range P=f
v
u
u
Fig. 3.1 Possibility Possibilityof multiple solutions.
be be useful in avoiding excessive numbers of of iterations iterations and in following the physically correct correct path. In Fig. 3.1 we show a typical non-uniqueness which may occur occur if the function ~I'n+l parameter u,+l un +! uniformly n +! decreases and subsequently increases as the parameter increases. It is clear that to follow the path ilf Af,+l n +! will have both positive and negative complete computation computation process. process. signs during a complete It is possible possible to obtain solutions in a single increment increment only in the case of mild nondependence), that is, with linearity (and no path dependence),
w
fn = 0,
Af,+l = f,+l
(3.6)
approaches and on particular The literature on general solution approaches particular applications is extenpossible to encompass fully all the variants which sive and, in a single chapter, it is not possible have been introduced. However, we shall attempt to give a comprehensive picture by outlining first the general general solution procedures. procedures. In later chapters we shall focus on procedures associated with rate-independent material non-linearity (plasticity), rate-dependent material non-linearity (creep and viscoplasticity), some non-linear field problems, large displacements and other special examples.
3.2 Iterative techniques 3.2.1 3.2.1 General General remarks remarks (3.2)-(3.5) cannot be approached directly The solution of the problem posed by Eqs (3.2)-(3.5) and some form of iteration will always be required. We shall concentrate here on procedures in which repeated solution of linear equations (i.e. iteration) of the form id ii ii i K K d Uunn++ 11 = --- rr nn ++ 11
(3.7) (3.7)
in which a superscript i indicates the iteration number. In these a solution increment i 1 is computed. Direct (Gaussian) elimination techniques or iterative methods can dun+ dU~+1
48
Solutionof non-linear algebraic equations be used to solve the linear equations associated with each iteration. 1-3 However, the application of an iterative solution method may prove to be more economical, and in later chapters we shall frequently refer to such possibilities although they have not been fully explored. Many of the iterative techniques currently used to solve non-linear problems originated by intuitive application of physical reasoning. However, each of such techniques has a direct association with methods in numerical analysis, and in what follows we shall use the nomenclature generally accepted in texts on this subject. 2'4-7 Although we state each algorithm for a set of non-linear algebraic equations, we shall illustrate each procedure by using a single scalar equation. This, though useful from a pedagogical viewpoint, is dangerous as convergence of problems with numerous degrees of freedom may depart from the simple pattern in a single equation.
3.2.2 Newton's method Newton's method is the most rapidly convergent process for solutions of problems in which only one evaluation of 9 is made in each iteration. Of course, this assumes that the initial solution is within the zone of attraction and, thus, divergence does not occur. Indeed, Newton's method is the only process described here in which the asymptotic rate of convergence is quadratic. The method is often called the Newton-Raphson method as it appears to have been simultaneously derived by Newton and Raphson, and an interesting history of its origins is given in reference 8. In this iterative method we note that, to the first order, Eq. (3.2) can be approximated as
(__..~) i
duin+l -- 0
ltX/~,Un+l ) ( . i+1,~ ~,~ ~ii(Uin+l) +
(3.8)
n+l
Here the iteration counter i usually starts by assuming 1
(3.9)
Un+ 1 -- a n
in which a n is a converged solution at a previous load level or time step. The jacobian matrix (or in structural terms the stiffness matrix) corresponding to a tangent direction is given by OP OffJ KT = ~ -Ou (3.10) Equation (3.8) gives immediately the iterative correction as i
i
KTdUn+ 1 -
i
~IIn+ 1
or
dUin+ 1 __ ( K ~ ) -1 ~IIn+ i 1
(3.11)
A series of successive approximations gives di+l i n+l -- Un+l -~-
i
dUn+l
i = U n q- A n n + 1
(3.12)
Iterative techniques
T
P
fn,1
i
. . . . . . . . . . . . . . . . . . lit1+ I
t.
_
~ Q I I I ! ! ! ! !
i !
'
'
:
',
i i
! i
"'
,'!
.~ ,
1
,
l! ,'L.
i
I i
AU ~
! i I
,.
I II
i ! I
4.,
,L,d§
v
u
Fig. 3.2 Newton's method.
where
i i AUn+ 1 = y~
dUnk+l
(3.13)
k=l
The process is illustrated in Fig. 3.2 and shows the very rapid convergence that can be achieved. i 1 is perhaps not obvious The need for the introduction of the total increment AUn+ here but in fact it is essential if the solution process is path dependent, as we shall see in Chapter 4 for some non-linear constitutive equations of solids. The Newton process, despite its rapid convergence, has some negative features: 1. a new KT matrix has to be computed at each iteration; 2. if direct solution for Eq. (3.11) is used the matrix needs to be factored at each iteration; 3. on some occasions the tangent matrix is symmetric at a solution state but unsymmetric otherwise (e.g. in some schemes for integrating large rotation parameters 9 or non-associated plasticity). In these cases an unsymmetric solver is needed in general. Some of these drawbacks are absent in alternative procedures, although generally then a quadratic asymptotic rate of convergence is lost.
3.2.3 Modified Newton's method This method uses essentially the same algorithm as the Newton process but replaces the variable jacobian matrix K~ by a constant approximation
K~ ~ giving in place of Eq. (3.11),
I~ T
i i dUn+l -- KT 1ffJ,+l
(3.14) (3.15)
49
50
Solution of non-linear algebraic equations f u
'n.,--'T-ill-
2 .......
|
....
-
A~
', ....
"i ,"
~
AUn~
_
ur,
P
: '
~1
J,
Ill
v
un~.~ Un2+, u3+~
U
(a) fb u
1
Q
[n+l
Atn
14/1+1
i~t11 ~-i i
1~/12', ; rkl3
AUn'~i i i
! !
|
i
i
i
',
Un
,11 , ! Un + 1 U2 + 1
.....
U
(b)
Fig. 3.3 The modified Newton method: (a) with initial tangent in increment; (b) with initial problem tangent. Many possible choices exist here. For instance, KT can be chosen as the matrix corresponding to the first iteration K~ [as shown in Fig. 3.3(a)] or may even be one corresponding to some previous time step or load increment K ~ [as shown in Fig. 3.3(b)]. In the context of solving problems in solid mechanics the method is also known as the stress transfer or initial stress method. Alternatively, the approximation can be chosen every few iterations as I~T = K~ where j < i. Obviously, the procedure generally will converge at a slower rate (generally a norm of the residual @ has linear asymptotic convergence instead of the quadratic one in the full Newton method) but some of the difficulties mentioned above for the Newton process disappear. However, some new difficulties can also arise as this method fails to converge when the tangent used has opposite 'slope' to the one at the current solution (e.g. as shown by regions with different slopes in Fig. 3.1). Frequently the 'zone of
Iterative techniques 51 attraction' for the modified process is increased and previously divergent approaches can be made to converge, albeit slowly. Many variants of this process can be used and symmetric solvers often can be employed when a symmetric form of KT is chosen.
3.2.4 Incremental-secant or quasi-Newton methods Once the first iteration of the preceding section has been established giving ] 1 dUn+ 1 --" K~ -1~I/n+ 1
(3 16)
a secant 'slope' can be found, as shown in Fig. 3.4, such that
dunl+l = (K2) -1 (~I/l+l- tI/n2+l )
(3.17)
This 'slope' can now be used to establish u,2 by using
2 -1 dUn+, 2 1 -- (Ks It/n+
(3.18)
Quite generally, one could write in place of Eq. (3.18) for i > 1, now dropping subscripts, d u i - - (K~) -1 ~I/i (3.19) where (K~) -1 is determined so that dui-1
__ (K~) -1 (~i/i-1-
ii/i)__
(K~) -I ,,y/-1
(3.20)
For the scalar system illustrated in Fig. 3.4 the determination of K~ is trivial and, as shown, the convergence is much more rapid than in the modified Newton process (generally a super-linear asymptotic convergence rate is achieved for a norm of the residual).
1 Af n
.
.
K~
~ln. 1
1, l .
.
.
~,,
' !
.
.
.
i-"
i ! ! .
,,,..-i-.~
' AUn1 ! ~
i ! .
L__
OUn i
! ~
,r,
i ! !
i
!
!
!
V
i
Un
Fig. 3.4 The secant method starting from a K~ prediction.
v
U
52 Solution of non-linear algebraic equations For systems with more than one degree of freedom the determination of K~ or its inverse is more difficult and is not unique. Many different forms of the matrix K~ can satisfy relation (3.1) and, as expected, many alternatives are used in practice. All of these use some form of updating of a previously determined matrix or of its inverse in a manner that satisfies identically Eq. (3.20). Some such updates preserve the matrix symmetry whereas others do not. Any of the methods which begin with a symmetric tangent can avoid the difficulty of non-symmetric matrix forms that arise in the Newton process and yet achieve a faster convergence than is possible in the modified Newton procedures. Such secant update methods appear to stem from ideas introduced first by Davidonl~ and developed later by others. Dennis and More 11 survey the field extensively, while Matthies and Strang 12 appear to be the first to use the procedures in the finite element context. Further work and assessment of the performance of various update procedures is available in references 13-16. The BFGS update 11 (named after Broyden, Fletcher, Goldfarb and Shanno) and the DFP update 11 (Davidon, Fletcher and Powell) preserve matrix symmetry and positive definiteness and both are widely used. We summarize below a step of the BFGS update for the inverse, which can be written as
(Ki) -1 = (I + wiu T) (Ki-I) -1 (I + SriwT)
(3.21)
where I is an identity matrix and
u
(dui-1)T"Y i-1 ] ffffi-
[1-- d (ui ) T lt~i-1 1
w i : du(i_l)T,.yi_ 1
l _
~i
(3.22)
d u i- 1
where "7 is defined by Eq. (3.20). Some algebra will readily verify that substitution of Eqs (3.21) and (3.22) into Eq. (3.20) results in an identity. Further, the form of Eq. (3.21) guarantees preservation of the symmetry of the original matrix. The nature of the update does not preserve any sparsity in the original matrix. For this reason it is convenient at every iteration to return to the original (sparse) matrix K 1, used in the first iteration and to reapply the multiplication of Eq. (3.21) through all previous iterations. This gives the algorithm in the form
i
T ~i/i
hi = 1-I (I + v i i i )
j=2
b2 = (K~)-' bl
i-2 dui : I X j=o
(3.23)
(I + Wi_jvTi_j)bE
This necessitates the storage of the vectors v j and w j for all previous iterations and their successive multiplications. Further details on the operations are described well in reference 12.
Iterative techniques
~n+l --
.............
,,,.
,
:,,
3
4
v,_. ~.v,.. ,
~
~
1,.,' ivL,
,+,1+,
17/i
i i i i i i
I Un
U
Fig. 3.5 Direct (or Picard)iteration. When the number of iterations is large (i > 15) the efficiency of the update decreases as a result of incipient instability. Various procedures are open at this stage, the most effective being the recomputation and factorization of a tangent matrix at the current solution estimate and restarting the process. Another possibility is to disregard all the previous updates and return to the original matrix K 1. Such a procedure was first suggested by Crisfield 13'17'18in the finite element context and is illustrated in Fig. 3.5. It is seen to be convergent at a slightly slower rate but avoids totally the stability difficulties previously encountered and reduces the storage and number of operations needed. Obviously any of the secant update methods can be used here. The procedure of Fig. 3.5 is identical to that generally known as direct (or Picard) iteration 4 and is particularly useful in the solution of non-linear problems which can be written as 9 (u) -- f - K(u)u = 0 (3.24) 1 In such a case un+ 1 -- un is taken and the iteration proceeds as
= [I,:min+,)] -' fn+,
(3.25)
3.2.5 Line search procedures- acceleration of convergence All the iterative methods of the preceding section have an identical structure described by Eqs (3.11)-(3.13) in which various approximations to the Newton matrix K~ are used. For all of these an iterative vector is determined and the new value of the unknowns found as iJti+l i i n+l - - U n + l "1- d U n + l
(3.26)
53
54
Solution of non-linear algebraic equations starting from 1
Un+l~Un
in which u, is the known (converged) solution at the previous time step or load level. The objective is to achieve the reduction of oi+~ =,+1 to zero, although this is not always easily achieved by any of the procedures described even in the scalar example illustrated. To get a solution approximately satisfying such a scalar non-linear problem would have been in fact easier by simply evaluating the scalar dfiri+l :": n+l for various values of U,+l and by suitable interpolation arriving at the required answer. For multi-degree-of-freedom systems such an approach is obviously not possible unless some scalar norm of the residual is considered. One possible approach is to write ui+l,j n+l
i i - - U n + l "~- T]i,jdUn+l
(3.27)
and determine the step size rli,j so that a projection of the residual on the search direction i 1 is made zero. We could define this projection as du,+
Gi,j ~ (duin+l) Tltl;i+l'j :=n+l where
l.~i + l,j n+l
~ ~I/ (Uin+l +
(3.28)
7]i,0
Tli,jdUin),
=
1
Here, of course, other norms of the residual could be used. This process is known as a line search, and ~Ti.jcan conveniently be obtained by using a regulafalsi (or secant) procedure as illustrated in Fig. 3.6. An obvious disadvantage of a line search is the need for several evaluations of ~ . However, the acceleration of the overall convergence can be remarkable when applied to modified or quasi-Newton methods. Indeed, line search is also useful in the full Newton method by making the radius of attraction larger. A compromise frequently used ]2 is to undertake the search only if + l, J Gi, 0 > 8 (gain+l) T ~jri =,,+] (3.29) where the tolerance e is set between 0.5 and 0.8. This means that if the iteration process directly resulted in a reduction of the residual to e or less of its original value a line search is not used.
GI
~
G,, L.,
E I"
(a)
x
~1
rli, 1 qi.2
"1
~ "1
I~
rli, 1 qi.2
(b)
Fig. 3.6 Regula falsi applied to line search: (a) extrapolation; (b) interpolation.
._1 "1
Iterative techniques 55
3.2.6 'Softening' behaviour and displacement control In applying the preceding to load control problems we have implicitly assumed that the iteration is associated with positive increments of the forcing vector, f, in Eq. (3.4). In some structural problems this is a set of loads that can be assumed to be proportional to each other, so that one can write Afn+l
-- A / ~ n + l f 0
(3.30)
In many problems the situation will arise that no solution exists above a certain maximum value of f and that the real solution is a 'softening' branch, as shown in Fig. 3.1. In such cases AAn+I will need to be negative unless the problem can be recast as one in which the forcing can be applied by displacement control. In a simple case of a single load it is easy to recast the general formulation to increments of a single prescribed displacement and much effort has gone into such solutions. 13'19-25 In all the successful approaches of incrementation of AAn+I the original problem of Eq. (3.2) is rewritten as the solution of ~I~n+l ~ ~ n + l f 0 -- P(u~+l) = 0
with U.+l = u. + A u . + l
(3.31)
and '~n+l "-" '~n + A / ~ n + l
being included as variables in any increment. Now an additional equation (constraint) needs tobe provided to solve for the extra variable AAn+l. This additional equation can take various forms. Riks 2~ assumes that in each increment AuT+IAun+I + AA2f~f0- A/2 (3.32) where A l i s a prescribed 'length' in the space of n + 1 dimensions. Crisfield 13'26 provides a more natural control on displacements, requiting that T 1Au,,+l = AI 2 Au~+ (3.33) These so-called arc-length and spherical path controls are but some of the possible constraints. Direct addition of the constraint Eq. (3.32) or (3.33) to the system of Eqs (3.31) is now possible and the previously described iterative methods could again be used. However, the 'tangent' equation system would always lose its symmetry so an alternative procedure is generally used. We note that for a given iteration i we can write quite generally the solution as
lt~in+1
i
i
--- )~n+lf0 __ P ( U n + l )
~I/i+l i n+l ~ ~Itn+l +
i If 0 dan+
- K Ti d U ni + l
(3.34)
The solution increment for u may now be given as dUn+l i f0] i __ (K~) -1 [~I/in+l ..1_ d,~n+l i ,,i i ,-i d U n + 1 -- d U n + 1 q- dAn+ldun+ 1
(3.35)
56
Solution of non-linear algebraic equations where dUn+ -'i 1 .__ (K~) -1 itI/ni + 1 ,,i dUn+,-
(3.36)
f0
Now an additional equation is cast using the constraint. Thus, for instance, with Eq. (3.3 3) we have (AUin~ll -~-
duin+l) T (AUin+ll
duin+l) =
+
AI 2
(3.37)
where Au~ll is defined by Eq. (3.13). On substitution of Eq. (3.35) into Eq. (3.37) a i (which quadratic equation is available for the solution of the remaining unknown dA,+l may well turn out to be negative). Additional details may be found in references 13 and 26. A procedure suggested by Bergan 22'25is somewhat different from those just described. Here a fixed load increment AAn+I is first assumed and any of the previously introi 1. Now a new duced iterative procedures are used for calculating the increment dun+ increment AAn+1 is calculated so that it minimizes a norm of the residual *
(AAn+lf0-
pi+l
T
pi+l)]
I n+l)(AAn+lf0
-- I n+l
__ At2
(3.38)
The result is thus computed from dAl 2 dAA*n+l
=0
and yields the solution AA~+I =
f0rv~+l " n+l fTfo
(3.39)
This quantity may again well be negative, requiting a load decrease, and it indeed results in a rapid residual reduction in all cases, but precise control of displacement magnitudes becomes more difficult. The interpretation of the Bergan method in a one-dimensional example, shown in Fig. 3.7, is illuminating. Here it gives the exact a n s w e r s - with a displacement control, the magnitude of which is determined by the initial AAn+~ assumed to be the slope KT used in the first iteration.
3.2.7 Convergence criteria In all the iterative processes described the numerical solution is only approximately achieved and some tolerance limits have to be set to terminate the iteration. Since finite precision arithmetic is used in all computer calculations, one can never achieve a better solution than the round-off limit of the calculations. Frequently, the criteria used involve a norm of the displacement parameter changes i [[dUn+l II or, more logically, that of the residuals II~in+~II. In the latter case the limit can often be expressed as some tolerance of the norm of forces Ilfn+~II- Thus, we may require that i II'I'n+~ II 5_ ellfn+l II (3.40)
Iterative techniques 57
| !
T
I//" IA :, i ! i i I i i i i
i
!
"
r
U
Fig. 3.7 One-dimensionalinterpretation of the Bergan procedure. where e is chosen as a small number, and
II'I'll = ('I'T~I ') 1/2
(3.41)
Other alternatives exist for choosing the comparison norm, and another option is to use the residual of the first iteration as a basis. Thus, [l~/n+l II
_< ell~I'n~+~II
(3.42)
The error due to the incomplete solution of the discrete non-linear equations is of course additive to the error of the discretization that we frequently measure in the energy n o r m . 27 It is possible therefore to use the same norm for bounding of the iteration process. We could, as a third option, require that the error in the energy norm satisfy
d E i _ (dUn,+T1~in+ 1)1/2 __< e (dul~l li/n+l ) 1 1/2
(3.43)
< edE 1 In each of the above forms, problem types exist where the fight-hand-side norm is zero. Thus a fourth form, which is quite general, is to compute the norm of the element residuals. If the problem residual is obtained as a sum over elements as
kX/n+1 = ~
~3e+l
(3.44)
e
where e denotes an individual element and ~3 e the residual from each element, we can express the convergence criterion as
IIlI/n+l i
e II II _< ell~n+l
(3.45)
where
[[~)e+l ]]- ~
11(~3e+l) i II e
(3.46)
58 Solutionof non-linear algebraic equations Once a criterion is selected the problem still remains to choose an appropriate value for e. In cases where a full Newton scheme is used (and thus asymptotic quadratic convergence should occur) the tolerance may be chosen at half the machine precision. Thus if the precision of calculations is about 16 digits one may choose e = 10 -8 since quadratic convergence assures that the next residual (in the absence of round-off) would achieve full precision. For modified or quasi-Newton schemes such asymptotic rates are not assured, necessitating more iterations to achieve high precision. In these cases it is common practice by some to use much larger tolerance values (say 0.01 to 0.001). However, for problems where large numbers of steps are taken, instability in the solution may occur if the convergence tolerance is too large. We recommend therefore that whenever practical a tolerance of half machine precision be used.
ii!i!iii!~iiii!iii!!!i!iiii ~':~'~i~iiiiiiiiiii! ~ ~i~i{i~i!~i!i~!~i!i!~i~!~!!~i~ii~i~i~ii~i~i~i!~!~i~ii!i!~i~!~i~i~i~ii!i~!i !i!i!~ii~i!!~ii!ii~!~i!~i~i~!i!i!~i!i~i!~{ii~i~i!~!i!i~i~!i~i!~!!ii~i!i!i~ii!!ii~iii~{!!~iii~ii!~!iiiiii!i~ii~{~!!i~i!~!i!~i~{!i!i!i!~i~!iiii~!iii~ii!~i!ii!~!iiii!iii~i!!ii!!i!ii!!iiiii~ii~ii~iii!~!i!i~!~i!ii~iii~i!~iii{~iii~iii~i~i~i!~!i!iiiiii~i~iii~:~!i!~!iiiiii!i~i!i!~!~ ~i~i~!!i!iiii~i!ii~!ii~ii ~iiiiiiiii~ii!~iiiiiiiii~!iiiii~i!~i!ii~i~!~!~!i!i~i!iii{iiiiiiii!i{!!i!iiiii~iiiiii!i{!iii!ii{!~ii!~iii!~iii~ii!iiii~i~iiii~:~i~!!i!!ii!~ii~!~!i~i!i~!!:!iiiii!iiiii ~ii~ii~i~i!i~ii~i~i~i~i!ii!~!~i~i~i!~iiii~i~ii~!~!i!i~i~i~i~i!~!i{~i~ii~i~ iiiiiiiiili!ii~iliiiiiiiiiiiii! .:~iiiiiili!-:iiiiii!iii!~iii~:~i ,- : i ~ : .,~,!, ~ iiiili :~....... i >. i .... .~i .::iiii!~!iii i :i: ...... i: , :i' : :~ ,~ :i iiiii~.. :i : !ilii ,~:. ~ ,:: i!iii i" :~ ~, :i........ i: i .:~:ii!!iiiiiii!ii!ii!iiiii~ii!iliiiiiiiiiiiiiiii~ili
The various iterative methods described provide an essential tool kit for the solution of non-linear problems in which finite element discretization has been used. The precise choice of the optimal methodology is problem dependent and although many comparative solution cost studies have been published 12'17'28the differences are often marginal. There is little doubt, however, that exact Newton processes (with line search) should be used when convergence is difficult to achieve. Also the advantage of symmetric update matrices in the quasi-Newton procedures frequently make these a very economical candidate. When non-symmetric tangent moduli exist it may be better to consider one of the non-symmetric updates, for example a Broyden method. 13'29 We have not discussed in the preceding direct iterative methods such as the various conjugate direction methods 3~ or dynamic relaxation methods in which an explicit dynamic transient analysis (see Chapter 2) is carried out to achieve a steady-state solution. 35'36 These forms are often characterized by" 1. a diagonal or very sparse form of the matrix used in computing trial increments du (and hence very low cost of an iteration) and 2. a significant number of total iterations and hence evaluations of the residual ~I,. These opposing trends imply that such methods offer the potential to solve large problems efficiently. However, to date such general solution procedures are effective only in certain problems. 37 One final remark concerns the size of increments Af or AA to be adopted. First, it is clear that small increments reduce the total number of iterations required per computational step, and in many applications automatic guidance on the size of the increment to preserve a (nearly) constant number of iterations is needed. Here such processes as the use of the 'current stiffness parameter' introduced by Bergan 22 can be effective. Second, if the behaviour is path dependent (e.g. as in plasticity-type constitutive laws) the use of small increments is desirable to preserve accuracy in solution changes. In this context, we have already emphasized the need for calculating such changes by i always using the accumulated AUn+ 1 change and not in adding changes arising from i each iterative dUn+1 step in an increment.
General remarks-incremental and rate methods 59
Third, if only a single Newton iteration is used in each increment of A)~ then the procedure is equivalent to the solution of a standard rate problem incrementally by direct forward integration. Here we note that if Eq. (3.2) is rewritten as P(u) = )~f0
(3.47)
we can, on differentiation with respect to )~, obtain dP du
= f0
(3.48)
= KT~f0
(3.49)
du d/k
and write this as du d--~
Incrementally, this may be written in an explicit form by using a Euler method as mUn_t_ 1 -- m / ~ K T l f 0
(3.50)
This direct integration is illustrated in Fig. 3.8 and can frequently be divergent as well as being only conditionally stable as a result of the Euler explicit method used. Obviously, other methods can be used to improve accuracy and stability. These include Euler implicit schemes and Runge-Kutta procedures.
t(;r
Possible divergence
\
sw
r
U
Fig. 3.8 Direct integration procedure.
60
Solution of non-linear algebraic equations
1. G. Strang. Linear Algebra and its Application. Academic Press, New York, 1976. 2. J. Demmel. Applied Numerical Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1997. 3. R.M. Ferencz and T.J.R. Hughes. Iterative finite element solutions in nonlinear solid mechanics. In RG. Ciarlet and J.L. Lions, editors, Handbook of Numerical Analysis, volume III, pages 3-178. Elsevier Science Publisher BV, 1999. 4. A. Ralston. A First Course in Numerical Analysis. McGraw-Hill, New York, 1965. 5. L. Collatz. The Numerical Treatment of Differential Equations. Springer, Berlin, 1966. 6. G. Dahlquist and/~. Bj6rck. Numerical Methods. Prentice-Hall, Englewood Cliffs, NJ, 1974. Reprinted by Dover, New York, 2003. 7. H.R. Schwarz. Numerical Analysis. John Wiley & Sons, Chichester, 1989. 8. N. Bidanid and K.W. Johnson. Who was 'Raphson'? International Journal for Numerical Methods in Engineering, 14:148-152, 1979. 9. J.C. Simo and L. Vu-Quoc. A three-dimensional finite strain rod model. Part II: Geometric and computational aspects. Computer Methods in Applied Mechanics and Engineering, 58:79-116, 1986. 10. W.C. Davidon. Variable metric method for minimization. Technical Report ANL-5990, Argonne National Laboratory, 1959. 11. J.E. Dennis and J. More. Quasi-Newton methods - motivation and theory. SIAM Rev., 19:46-89, 1977. 12. H. Matthies and G. Strang. The solution of nonlinear finite element equations. International Journal for Numerical Methods in Engineering, 14:1613-1626, 1979. 13. M.A. Crisfield. Non-linear Finite Element Analysis of Solids and Structures, volume 1. John Wiley & Sons, Chichester, 1991. 14. M.A. Crisfield. Non-linear Finite Element Analysis of Solids and Structures, volume 2. John Wiley & Sons, Chichester, 1997. 15. K.-J. Bathe and A.R Cimento. Some practical procedures for the solution of nonlinear finite element equations, cmame, 22:59-85, 1980. 16. M. Geradin, S. Idelsohn and M. Hogge. Computational strategies for the solution of large nonlinear problems via quasi-Newton methods. Computers and Structures, 13:73-81, 1981. 17. M.A. Crisfield. Finite element analysis for combined material and geometric nonlinearity. In W. Wunderlich, E. Stein and K.-J. Bathe, editors, Nonlinear Finite Element Analysis in Structural Mechanics. Springer-Verlag, Berlin, 1981. 18. M.A. Crisfield. A fast incremental/iterative solution procedure that handles 'snap through'. Computers and Structures, 13:55-62, 1981. 19. T.H.H. Pian and R Tong. Variational formulation of finite displacement analysis. In Syrup. on High Speed Electronic Computation of Structures, Liege, 1970. 20. O.C. Zienkiewicz. Incremental displacement in non-linear analysis. International Journal for Numerical Methods in Engineering, 3:587-592, 1971. 21. E. Riks. An incremental approach to the solution of snapping and buckling problems. International Journal of Solids and Structures, 15:529-551, 1979. 22. RG. Bergan. Solution algorithms for nonlinear structural problems. In Int. Conf. on Engineering Applications of the Finite Element Method, pages 13.1-13.39. Computas, 1979. 23. J.L. Batoz and G. Dhatt. Incremental displacement algorithms for nonlinear problems. International Journal for Numerical Methods in Engineering, 14:1261-1266, 1979. 24. E. Ramm. Strategies for tracing nonlinear response near limit points. In W. Wunderlich, E. Stein and K.-J. Bathe, editors, Nonlinear Finite Element Analysis in Structural Mechanics, pages 63-89. Springer-Verlag, Berlin, 1981.
References 61 25. E Bergan. Solution by iteration in displacement and load spaces. In W. Wundeflich, E. Stein and K.-J. Bathe, editors, Nonlinear Finite Element Analysis in Structural Mechanics. SpringerVerlag, Berlin, 1981. 26. M.A. Crisfield. Incremental/iterative solution procedures for nonlinear structural analysis. In C. Taylor, E. Hinton, D.R.J. Owen and E. Ofiate, editors, Numerical Methods for Nonlinear Problems. Pineridge Press, Swansea, 1980. 27. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann, Oxford, 6th edition, 2005. 28. A. Pica and E. Hinton. The quasi-Newton BFGS method in the large deflection analysis of plates. In C. Taylor, E. Hinton, D.R.J. Owen and E. Ofiate, editors, Numerical Methods for Nonlinear Problems. Pineridge Press, Swansea, 1980. 29. C.G. Broyden. Quasi-Newton methods and their application to function minimization. Math. Comp., 21:368-381, 1967. 30. M. Hestenes and E. Stiefel. Method of conjugate gradients for solving linear systems. J. Res. Natl. Bur. Stand., 49:409-436, 1954. 31. R. Fletcher and C.M. Reeves. Function minimization by conjugate gradients. The Computer Journal, 7:149-154, 1964. 32. E. Polak. Computational Methods in Optimization. A Unified Approach. Academic Press, London, 1971. 33. B.M. Irons and A.E Elsawaf. The conjugate Newton algorithm for solving finite element equations. In K.-J. Bathe, J.T. Oden and W. Wunderlich, editors, Proc. U.S.-German Symp. on Formulations and Algorithms in Finite Element Analysis, pages 656-672, Cambridge, Mass., 1977. MIT Press. 34. M. Papadrakakis and P. Ghionis. Conjugate gradient algorithms in nonlinear structural analysis problems. Computer Methods in Applied Mechanics and Engineering, 59:11-27, 1986. 35. J.R.H. Otter, E. Cassel and R.E. Hobbs. Dynamic relaxation. Proc. Inst. Civ. Eng., 35:633-656, 1966. 36. O.C. Zienkiewicz and R. L6hner. Accelerated relaxation or direct solution? Future prospects for FEM. International Journal for Numerical Methods in Engineering, 21:1-11, 1986. 37. M. Adams. Parallel multigrid algorithms for unstructured 3D large deformation elasticity and plasticity finite element problems. Technical Report UCB//CSD-99-1036, University of California, Berkeley, 1999.
Inelastic and non-linear materials 4.1 Introduction In Chapter 2 we presented a framework for solving general problems in solid mechanics. In this chapter we consider several classical models for describing the behaviour of engineering materials. Each model we describe is given in a strain-driven form in which a strain or strain increment obtained from each finite element solution step is used to compute the stress needed to evaluate the internal force, BTudQ, as well as a tangent modulus matrix, or its approximation, for use in constructing the tangent stiffness matrix. Quite generally in the study of small deformation and inelastic materials (and indeed in some forms applied to large deformation) the strain (or strain rate) or the stress is assumed to split into an additive sum of parts. We can write this as
in which we shall generally assume that the elastic part is given by the linear model
in which D is the matrix of elastic moduli. In the following sections we shall consider the problems of viscoelasticity, plasticity, and general creep in quite general form. By using these general types it is possible to present numerical solutions which accurately predict many physical phenomena. We begin with viscoelasticity, where we illustrate the manner in which we shall address the solution of problems given in a rate or differential form. This rate form of course assumes time dependence and all viscoelastic phenomena are indeed transient, with time playing an important part. We shall follow this section with a description of plasticity models in which time does not explicitly arise and the problems are time independent. However, we shall introduce for convenience a rate description of the behaviour. This is adopted to allow use of the same kind of algorithms for all forms discussed in this chapter.
Viscoelasticity-history dependence of deformation 63
Viscoelastic phenomena are characterized by the fact that the rate at which inelastic strains develop depends not only on the current state of stress and strain but, in general, on the full history of their development. Thus, to determine the increment of inelastic strain over a given time interval (or time step) it is necessary to know the state of stress and strain at all preceding times. In the computation process these can in fact be obtained and in principle the problem presents little theoretical difficulty. Practical limitations appear immediately, however, that each computation point must retain this history information-thus leading to very large storage demands. In the context of linear viscoelasticity, means of overcoming this limitation were introduced by Zienkiewicz et al. l and White. 2 Extensions to include thermal effects were also included in some of this early work. 3 Further considerations which extend this approach are also discussed in earlier editions of this book. 4,5
4.2.1 Linear models for viscoelasticity The representation of a constitutive equation for linear viscoelasticity may be given in the form of either a differential equation or an integral equation. 6,7 In a differential model the constitutive equation may be written as a linear elastic part with an added series of partial strains q. Accordingly, we write M
tr(t) -- Doe(t) + Z
Dmq(m)(t)
(4.4)
m=l
where for a linear model the partial strains are solutions of the first-order differential equations (l (m) "q- Tmq (m) -- ~ (4.5) with Tm a constant matrix of reciprocal relaxation times and Do, Dm constant moduli matrices. The presence of a split of stress as given by Eq. (4.2) is immediately evident in the above. Each of the forms in Eq. (4.5) represents an elastic response in series with a viscous response and is known as a Maxwell model. In terms of a springdashpot model, a representation for the Maxwell material is shown in Fig. 4.1(a) for a single stress component. Thus, the sum given by Eq. (4.4) describes a generalized
w
(a)
(b)
Fig. 4.1 Spring-dashpot models for linear viscoelasticity: (a) Maxwell element; (b) Kelvin element.
64
Inelasticand non-linear materials
Maxwell solid in which several elements are assembled in a parallel form and the Do term becomes a spring alone. In an integral form the stress-strain behaviour may be written in a convolution form as o" = D(t)e(0) +
f0 t D(t -
t')~- 7 dt'
(4.6)
where components of D(t) are relaxation moduli functions. Inverse relations may be given where the differential model is expressed as M e(t) = Jotr(t) + Z Jmr(m)(t)
(4.7)
m=l
where for a linear model the partial stresses r are solutions of r(m) "l- Vm r(m) : O"
(4.8)
in which Vm are constant reciprocal retardation time parameters and J0, Jm constant compliance ones (i.e. reciprocal moduli). Each partial stress corresponds to a solution in which a linear elastic and a viscous response are combined in parallel to describe a Kelvin model as shown in Fig. 4.1(b). The total model thus is a generalized Kelvin solid. In an integral form the strain-stress constitutive relation may be written as 00"7 dt' e -- J(t)tr(0) + f0 t J(t - t')~-
(4.9)
where J(t) are known as creep compliance functions. The parameters in the two forms of the model are related. For example, the creep compliances and relaxation moduli are related through 0D 0J7 d t ' = I J(t)D(0) + f0 t J(t - t')-0~ d t ' = D(t)J(0) + fot D(t - t')~-
(4.10)
as may easily be shown by applying, for example, Laplace transform theory to Eqs (4.6) and (4.9). The above forms hold for isotropic and anisotropic linear viscoelastic materials. Solutions may be obtained by using standard numerical techniques to solve the constant coefficient differential or integral equations. Here we will proceed to describe a solution for the isotropic case where specific numerical schemes are presented. Generalization of the methods to the anisotropic case may be constructed by using a similar approach and is left as an exercise to the reader.
4.2.2 Isotropic models To describe in more detail the ideas presented above we consider here isotropic models where we split the stress as o" -- s + m p
with
1 Ttr p - ~m
(4.11)
Viscoelasticity- history dependence of deformation where s is the stress deviator, p is the mean (pressure) stress and, for a three-dimensional state of stress, In is given in Eq. (2.45). Similarly, a split of strain is expressed as e = e + ~ Ira0
with
0 - mTe
(4.12)
where e is the strain deviator and 0 is the volume change. In the presentation given here, for simplicity we restrict the viscoelastic response to deviatoric parts and assume pressure-volume response is given by the linear elastic model p = KO (4.13) where K is an elastic bulk modulus. A generalization to include viscoelastic behaviour in this component also may be easily performed by using the method described below for deviatoric components.
Differential equation model The deviatoric part may be stated as differential equation models or in the form of integral equations as described above. In the differential equation model the constitutive equation may be written as s = 2G
#mq(m)
#0 e +
(4.14)
m=l
in which ~m are dimensionless parameters satisfying M
~m "--" 1
with
#m > 0
(4.15)
m=0
and dimensionless partial deviatoric strains q(m) are obtained by solving q(m) + l q ( m ) '~m
__ 6
(4.16)
in which )k m a r e relaxation times. This form of the representation is again a generalized Maxwell model (a set of Maxwell models in parallel). Each differential equation set may be solved numerically using a one-step time integration method [e.g., GN 11 in Sec. 2.7, Eq. (2.66)]. 8,9 To solve numerically we first define a set of discrete points, tk, at which we wish to obtain the solution. For a time tn+l we assume the solution at all previous points up to tn are known. Using a simple single-step method the solution for each partial stress is given by:
(OAt)-(m)((1-O)At) 1+
'~m J q n + l
--
1 --
Am
-(m)
tl n
+ e n + l - - en
(4.17)
in which At = t,+l - tn. We note that this form of the solution is given directly in a strain-driven form. Accordingly, given the strain from any finite element solution step we can immediately compute the stresses by using Eqs (4.13), (4.14) and (4.17) in Eqs (4.11) and (4.12).
65
66
Inelasticand non-linear materials
Inserting the above into a Newton-type solution strategy requires the computation of the tangent moduli. The tangent moduli for the viscoelastic model are deduced from
O0"n+1
OqSn._bl
OPn+l
DTIn+I ---- O~n+1 = O~n+l+ m ~O~n+1
(4.18)
The tangent part for the volumetric term is elastic and given by
Op,,+l
OPn+l OOn+1 = Kmm T 00n+l O~n+l
m ~
= m ~
O~n+l
(4.19)
Similarly, the tangent part for the deviatoric term is deduced from Eq. (4.17) as
OSn+l O~n+l
M #o -{- E
-- O~n+l Oqen+l -- 2G O~n+l
OSn+l
m=l
la
~m 1+
-~m
(4.20)
where la is defined in Eq. (2.45). Using the above, tangent moduli are expressed as M
DTI,,+I -- K m m T + 2G
#0+~( m=l
/~0/~t)
ld
(4.21)
1 ..+- --~m J
and we note that the only difference from a linear elastic material is the replacement of the elastic shear modulus by the viscoelastic term M
G --+ G m=l
l+-~m
This relation is independent of stress and strain and hence when it is used with a Newton scheme it converges in one iteration (i.e. the residual of a second iteration is numerically zero). The set of first-order differential equations (4.16) may be integrated exactly for specified strains, e. The integral for each term is given by q~m)(t) --
f
t oo
Oe exp [--(t -- t')/Am] -~7 dt'
(4.22)
An advantage to the differential equation form, however, is that it may be extended to include aging or other non-linear effects by making the parameters time or solution dependent. The exact solution to the differential equations for such a situation will then involve integrating factors, leading to more involved expressions. In the following parts of this section we consider the integral equation form and its numerical solution for linear viscoelastic behaviour. Models and their solutions for more general cases are left as an exercise for the reader.
Viscoelasticity-history
dependence
of d e f o r m a t i o n
Integral equation model
The integral equation form for the deviatoric stresses is expressed in terms of a relaxation modulus function which is defined by an idealized experiment in which, at time zero (t = 0), a specimen is subjected to suddenly applied and constant strain, e0, and the stress response, s(t), is measured. For a linear material a unique relation is obtained which is independent of the magnitude of the applied strain. This relation may be written as s(t) = 2G(t)e0 (4.23) where G(t) is defined as the shear relaxation modulus function. A typical relaxation function is shown in Fig. 4.2. The function is shown on a logarithmic time scale since typical materials have time effects which cover wide ranges in time. Using linearity and superposition for an arbitrary state of strain yields the integral equation specified as s(t) =
(4.24)
2G(t - t')-~, dt'
We note that the above form is a generalization to the Maxwell material. However, the integral equation form may be specialized to the generalized Maxwell model by assuming the shear relaxation modulus function in a Prony series form #m exp(--t/,Xm)
G(t) = G #o +
(4.25)
m--1
where the #m satisfy Eq. (4.15).
Solution to integral equation with Prony series
The solution to the viscoelastic model is performed for a set of discrete points tk. Thus, again assuming that all solutions are available up to time t~, we desire to compute the next step for time tn+l. Solution of the general form would require summation over
1.0 ,ww.,,
0.8
C
, .o ..
~ 0.6 C
o
..=,
x
0.4
i .
(b
n"
0.2
0 -5
I I -4-3-2
I
Fig. 4.2 Typical viscoelastic relaxation function.
I I I -1 0 1 log time t
I 2
I 3
I 4
5
67
68
Inelastic and non-linear materials
all previous time steps for each new time; however, by using the generalized Maxwell model we may reduce the solution to a recursion formula in which each new solution is computed by a simple update of the previous solution. We will consider a special case of the generalized Maxwell material in which the number of terms M is equal to 1 [which defines a standard linear solid, Fig. 4.3(a)]. The addition of more terms is easily performed from the one-term solution. Accordingly, we take G(t) = G [#0 + #1 exp(-t/A1)] (4.26) where #0 + #1 = 1. For the standard solid only a limited range of time can be considered, as can be observed from Fig. 4.3(b) for the model given by
G(t) = G[0.15 + 0.85 e x p ( - t ) ] To consider a wider range it is necessary to use terms in which the '~m cover the total time by using at least one term for each decade of time (a decade being one unit on the lOgl0 time scale). Substitution of Eq. (4.26) into Eq. (4.24) yields
$(t) = 2G
[~0 + ~1
exp(-(t - t')/)~l)] ~ dt'
(4.27)
which may be split and expressed as s(t) - 2G#oe(t) + 2G#l
e x p ( - ( t - t')/)~l)~-; dt'
(4.28/
oo
= 2G[#oe(t ) + ~lq(1)(t)] where we note that q(l~ is identical to the form given in Eq. (4.22). Thus use of a Prony series for G(t) is identical to solving the differential equation model exactly. In applications involving a linear viscoelastic model, it is usually assumed that the material is undisturbed until a time identified as zero. At time zero a strain may be
q~ 1.0
tO
0.8
~ 0.6 tO .~
0------
0.4
n." 0.2 0 I I -5 -4-3-2
(a)
I
I I i -1 0 1 log time t
I 2
(b)
Fig. 4.3 Standard linear viscoelastic solid: (a) model for standard solid; (b) relaxation function.
I 3
I 4
5
Viscoelasticity- history dependence of deformation suddenly applied and then varied over subsequent time. To evaluate a solution at time tn+l the integral representation for the model may be simplified by dividing the integral into t.+l
O-
tn
(.) dt' =
(.) dt' +
J --00
(.1 dt' +
t.+l
(.) dt' +
O0
(.) dt'
(4.29)
J tn
In each analysis considered here the material is assumed to be unstrained before the time denoted as zero. Thus, the first term on the right-hand side is zero, the second term includes a jump term associated with e0 at time zero, and the last two terms cover the subsequent history of strain. The result of this separation when applied to Eq. (4.27) gives the recursion 3 q(1) n+l - - e x p ( - A t / A 1 ) q ~ 1) + Aq (1) (4.30) where f
t~+l
Aq (1) --
Oe
exp[-(tn+l - t')/Az]~- 7 dt'
(4.31)
d tn
and q<01)= e0. To obtain a numerical solution, we approximate the strain rate in each time increment by a constant to obtain 1
=
ft.+l
exp[-(t,,+l - t')/Al] [en+l --e,]
dt'
(4.32)
The integral may now be evaluated directly over each time step as 3 A q n~1) +l
=
"A1 ~
[1-exp(-At/A
1 )]
(e,+l
-
en)
=
_~ q n +~1) l
(en+l --
e,)
(4.33)
This approximation is singular for zero time steps; however, the limit value at At ----0 is one. Thus, for small time steps a series expansion may be used to yield accurate values, giving A (1) qn+l
"-
1
1 (A~lt) 1 (At) 2 1 (A.~lt)3 2 + ~ -~1 - ~ +--.
(4.34)
Using a few terms for very small time increment ratios yields numerically correct answers (to computer precision). Once the time increment ratio is larger than a certain small value the representation given in Eq. (4.33) is used directly. The above form gives a recursion which is stable for small and large time steps and produces very smooth transitions under variable time steps. A numerical approximation to Eq. (4.32) in which the integrand of Eq. (4.31) is evaluated at tn+l/2 has also been used with success, l~ In the above recursion we note that a zero and infinite value of a time step produces a correct instantaneous and zero response, respectively, and thus is asymptotically accurate at both limits. The use of finite difference approximations on the differential equation form directly does not produce this property unless 0 -- 1 and for this value is much less accurate than the solution given by Eq. (4.33).
69
70
Inelasticand non-linear materials
Using the recursion formula, the constitutive equation now has the simple form Sn+l = 2G[#oen+l + #1q~.~1]
(4.35)
The process may also be extended to include effects of temperature on relaxation times for use with thermorheologically simple materials. 3 The implementation of the above viscoelastic model into a Newton-type solution process again requires the computation of a tangent tensor. Accordingly, for the deviatoric part we need to compute ~ S n + l __-- ~Sn+l Id (4.36) O~, n-t-1
C~n + l
The partial derivative with respect to the deviatoric stress follows from Eq. (4.35) as ~
= 2G #oi + ~1 .....0 e
(4.37)
Using Eq. (4.33) the derivative of the last term becomes tl 1) n+l ,, (1)(At)l Oen+ 1 -- /-Xqn+ 1
(4.38)
Thus, the tangent tensor is given by OOSn+I
O~,n+l
,, (1)(At)lid = 2G[#0 + #1 ['-~qn+l
(4.39)
Again, the only modification from a linear elastic material is the substitution of the elastic shear modulus by (1)
G --> G [#o + # 1z_xq.+1(At) ]
(4.40)
We note that for zero At the full elastic modulus is recovered, whereas for very large increments the equilibrium modulus #0 G is used. Since the material is linear, use of this tangent modulus term again leads to convergence in one iteration (the second iteration produces a n u m e r i c a l l y zero residual). The inclusion of more terms in the series reduces to evaluation of additional t..~m) ln+l integral recursions. Computer storage is needed to retain the q~m) for each solution (quadrature) point in the problem and each term in the series.
Example 4.1" A thick-walled cylinder subjected to internal pressure
To illustrate the importance of proper element selection when performing analyses in which material behaviour approaches a near incompressible situation we consider the case of internal pressure on a thick-walled cylinder. The material is considered to be isotropic and modelled by viscoelastic response in deviatoric stress-strain only. Material properties are: modulus of elasticity, E -- 1000; Poisson's ratio, u -- 0.3; #1 - - 0.99; and '~1 -'- 1. Thus, the viscoelastic relaxation function is given by 1000
G ( t ) -" ~ [ 0 . 0 1
2.6
+ 0.99 exp(-t)]
Viscoelasticity-history
dependence
of d e f o r m a t i o n
The ratio of the bulk modulus to shear modulus for instantaneous loading is given by K/G(O) = 2.167 and for long time loading by K/G(oo) = 216.7 which indicates a near incompressible behaviour for sustained loading cases (the effective Poisson ratio for infinite time is 0.498). The response for a suddenly applied internal pressure, p = 10, is computed to time 20 by using both displacement and the mixed element described in Chapter 2. Quadrilateral elements with four nodes (Q4) and nine nodes (Q9) are considered, and meshes with equivalent nodal forces are shown in Fig. 4.4. The exact solution to this problem is one dimensional and, since all radial boundary conditions are traction ones, the stress distribution should be time independent. During the early part of the solution, when the response is still in the compressible range, the solutions from the two formulations agree well with this exact solution. However, during the latter part of the solution the answers from a displacement element diverge because of near incompressibility effects, whereas those from a mixed element do not. The distribution of quadrature point radial stresses at time t = 20 is shown in Fig. 4.5 where
(a)
(b)
Fig. 4.4 Mesh and loads for internal pressure on a thick-walled cylinder: (b) nine-node quadrilaterals.
10
10 o
5-
~
0 m
(a) four-node quadrilaterals;
~
-5
-~ -10 .N
o o
O-
o
"0
-20 -25 -4.0 (a) Fig. 4.5 model.
o Q4/1 - Mixed x Q9/3 - Mixed --- Exact
I
-3.5
I
I
-2.0
-1.5
o
o
o
o
o o
-20 -
I
o
o
n- - 1 5 -
-3.0 -2.5 r-coordinate
o
-5-
--~ - 1 0 -
rr - 1 5 -
o
-25 -4.0
I
-3.5
I
o Q4-Displ x Q9-Displ ---9 Exact
I
-3.0 -2.5 r-coordinate
I
-2.0
-1.5
(b) Radial stress for internal pressure on a thick-walled cylinder: (a) mixed model; (b) displacement
71
72
Inelastic and non-linear materials
the highly oscillatory response of the displacement form is clearly evident. We note that extrapolation to 'reduced quadrature' points would avoid these oscillations; however, use of fully reduced integration would lead to singularity in the stiffness matrix and selective reduced integration is difficult to use with general non-linear material behaviour. Thus, for general applications the use of mixed elements is preferred.
4.2.3 Solution by analogies The labour of step-by-step solutions for linear viscoelastic media can, on occasion, be substantially reduced. In the case of a homogeneous structure with linear isotropic viscoelasticity and constant Poisson ratio operator, the McHenry-Alfrey analogies allow single-step elastic solutions to be used to obtain stresses and displacements at a given time by the use of equivalent loads, displacements and temperatures. ~'12 Some extensions of these analogies have been proposed by Hilton and Russell. ~3 Further, when subjected to steady loads and when strains tend to a constant value at an infinite time, it is possible to determine the final stress distribution even in cases where the above analogies are not applicable. Thus, for instance, where the viscoelastic properties are temperature dependent and the structure is subject to a system of loads and temperatures which remain constant with time, long-term 'equivalent' elastic constants can be found and the problem solved as a single, non-homogeneous elastic one. ~4 The viscoelastic problem is a particular case of a creep phenomenon to which we shall return in Sec. 4.9.3 using some other classical non-linear models to represent material behaviour.
~::i',:',:~',::,'!,i:!i',i!','~',i~::~;'J,:~'"",iii',~',~',i':,!~'~"~'~~:"'~:i','~i',',~ii'i'i,i~',~::' ',i',~'~~'~'~':~i::~::':i~~','i, i~',iii''~'~'i,i',:,','i,i',',i':~'',i,i~' !'',',!',ii'~i',ii',i~i','i,'~,'~,'!,',~ilii';:,'~,~:::' i, ',ii''~'i',~','~,'~',~::il~,~ '~'~,ii:~i' i ~,~'~''~,,~',ii'',~',i~i'',~'~i',~',ii!i',~~,i',i'~i'~','i,i~''i, ',~'~'~~'i',~',' ,i'~,i',~' ','~'?i:''~~i',~i~iiiii~' '~'~'~',~i!i~',ii~'i~iiii'i~i'~i'~,~ii',iiiiiilili',i',iilii?:'ii'~i~iii'i,iiii' ',iiiiiiii,'i',i,',~i',~ii~~iii',ii'iii,iiiii',iiiiiliiiiiii':':'~iiii',i',~i''i,iliiiiiiiiiiiiii',~i~~:i:~i~~ iiiii',i',ii',i:,i'',~,~':~:~iiii:,ii',i',!:,iii'i:,i,iii~ i:,i~:'':,i?':i "i~' : ',ii',iii:,iiiiiiii iiiii':'i:!iiii!',i~:i!i:i~' '~i~i,~ii:',i',i':i'':i~i'i,'i'i,:i:~iiii:iiii,i~i~,iiiilii~,~,','~,'i,ii':'~,i'l,i:,!',i',~',:,i'i:i::iiiiiii',',~',!',i',i',!i!i:, ',i'i:,,ii'i,iiil'i,ii~il iiiii'~'~,i',iii',:,i':~,:,i':,iii~:i' ',i',i:,':ii~:i!i' :i :iii :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
i.......~...`~.~...~.`~..~.~.~::i~::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::~::;::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::~;~..~ ~::::::::::::i:::.~::~::~:~::~i~i~i~:.~:~::~i;::::~::i:::::~::~::::::::::[?:i..```.~.~..`!z::!::~!~::::::~!~:;:::::~i~i~::~i~::~i~::~!~::!i~i~i~::~i :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ~::~!~::::~!~::ii::i ::~!~:.!::ii!:;:;ii::~i~
Classical 'plastic' behaviour of solids is characterized by a non-unique stress-strain relationship which is independent of the rate of loading but does depend on loading sequence that may be conveniently represented as a process evolving in time. Indeed, one definition of plasticity is the presence of irrecoverable strains on load removal. If uniaxial behaviour of a material is considered, as shown in Fig. 4.6(a), a non-linear relationship on loading alone does not determine whether non-linear elastic or plastic behaviour is exhibited. Unloading will immediately discover the difference, with an elastic material following the same path and a plastic material showing a historydependent different path. We have referred to non-linear elasticity already in Sec. 2.4 [see Eq. (2.28)] and will not give further attention to it here as the techniques used for plasticity problems or non-linear elasticity show great similarity. Representation of non-linear elastic behaviour for finite deformation applications is more complex as we shall show in Chapter 5. Some materials show a nearly ideal plastic behaviour in which a limiting yield stress, Y (or Cry), exists at which the strains are indeterminate. For all stresses below such yield, a linear (or non-linear) elastic relationship is assumed, Fig. 4.6(b) illustrates this. A further refinement of this model is one of a hardening~softening plastic material in which the yield stress depends on some parameter ~ (such as the accumulated plastic
Classical time-independent plasticity theory
"
Loading
~ l Loading
Unloading
Snloading
Non-linearelasti/ c
Gy
Plastic
= constant
E
(a)
s
(b)
i
EP
,"
~y = ay (EP)
E
(c) Fig. 4.6 Uniaxialbehaviourof materials: (a) non-linearelastic and plastic behaviour;(b)ideal plasticity; (c) strainhardeningplasticity. strain e p) [Fig. 4.6(c)]. It is with such kinds of plasticity that this section is concerned and for which much theory has been developed. 15'16 In a multiaxial rather than a uniaxial state of stress the concept of yield needs to be generalized. It is important to note that in the following development of results in a matrix form all nine tensor components are used instead of the six 'engineering' component form used previously. To distinguish between the two we introduce an underbar on the symbol for all nine-component forms. Thus, we shall use: 0 "n
IO'x
O'y
O"z
O'xy
O'yz
O'zx] T
O" n
[fix
O'y
(7"z
O'xy
O'yx
O'yz
"C=['Cx
"Cy
"Cz ~xy
"~yz "yzx] T
~ --- [~x
~y
CZ
~yx
~xy
~yz
O'zx
O'zy
EZy
Ezx
O'xz ] T
(4.41)
Exz] T
in which ~ij -- 2Eij. The transformations between the nine- and six-component forms needed later are obtained by using e _ - Pe
and
o " - pTo"
(4.42)
73
74
Inelastic and non-linear materials
where
p~__ 1 2
2 0 02 002 0 0 0 0 0 0
0 0 0 0 0
0 0 0 1 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 1 0
0 0 0 0 0 1
0" 0 0 0 0 1
Accordingly, we first make all computations by using the nine 'tensor' components of stress and strain and only at the end do we reduce the computations to expressions in terms of the six independent 'engineering' quantifies using P. This will permit final expressions for strain and equilibrium to be written in terms of B as in all previous developments. In addition we note that:
pTip = pTp = Io
with
Io =
1
(4.43)
4.3.1 Yield |unct.ions It is quite generally postulated, as an experimental fact, that yielding can occur only if the stress satisfies the general yield criterion .F(o-, ~, ~) -- 0
(4.44)
where tr denotes a matrix form with all nine components of stress, _~ represents kinematic hardening parameters and ~ an isotropic hardening parameter. 15We shall discuss these particular sets of parameters later but, of course, many other types of parameters also can be used to define hardening. This yield condition can be visualized as a surface in an n-dimensional space of stress with the position and size of the surface dependent on the instantaneous value of the parameters _~ and ~; (Fig. 4.7).
4.3.2 Flow rule (normality principle) Von Mises first suggested that basic behaviour defining the plastic strain increments is related to the yield surface. 17 Heuristic arguments for the validity of the relationship proposed have been given by various workers in the field 18-25 and at the present time the following hypothesis appears to be generally accepted for many materials; if pe
Classical time-independent plasticity theory
.........
l
.....M ... C'- .....'~ .......~........ r
--
H
Fig. 4.7 Yield surface and normality criterion in two-dimensional stress space.
denotes the components of the plastic strain tensor the rate of plastic strain is assumed to be given by* ~P = ~ F o .
(4.45)
OF F_~ _-- 0tr
(4.46)
where the notation
is introduced. In the above, ,~ is a proportionality constant, as yet undetermined, often referred to as the 'plastic consistency' parameter. During sustained plastic deformation we must have /~ = 0 and ~ > 0 (4.47) whereas during elastic loading/unloading ~ = 0 and /7 ~ 0 leading to a general constraint condition in Kuhn-Tucker foITn 16 F,k = 0
(4.48)
The above rule is known as the normality principle because relation (4.45) can be interpreted as requiting the plastic strain rate components to be normal to the yield surface in the space of nine stress and strain dimensions. Restrictions of the above rule can be removed by specifying separately a plasticflow
rule potential
Q = Q(_~, ~)
(4.49)
* Some authors prefer to write Eq. (4.45) in an incremental form d ~ p -- dA F.,r where d ~ p = ~Pdt, and t is some pseudo-time variable. Here we prefer the rate form to permit use of common solution algorithms in which d E will denote an increment in a Newton-type solution. (Also note the difference in notation between a small increment 'd' and a differential 'd'.)
75
76
Inelastic and non-linear materials
which defines the plastic strain rate similarly to Eq. (4.45); that is, giving this as e-"p = "~a , z ,
'~ > 0
(4.50)
The particular case of Q = F is known as associative plasticity. When this relation is not satisfied the plasticity is non-associative. In what follows this more general form will be considered initially (reductions to the associative case follow by simple substitution of Q = F). The satisfaction of the normality rule for the associative case is essential for proving so-called upper and lower bound theorems of plasticity as well as uniqueness. In the non-associative case the upper and lower bounds do not exist and indeed it is not certain that the solutions are always unique. This does not prevent the validity of nonassociated rules as it is well known that in frictional materials, for instance, uniqueness is seldom achieved but the existence of friction cannot be denied.
4.3.3 Hardening/softening rules
Isotropic hardening
The parameters ~ and ~ must also be determined from rate equations and define hardening (or softening) of the plastic behaviour of the material. The evolution of ~, governing the size of the yield surface, is commonly related to the rate of plastic work or directly to the consistency parameter. If related to the rate of plastic work ~ has dimensions of stress and a relation of the type = tr Tk p = Y (~)k p
(4.51)
is used to match behaviour to a uniaxial tension or compression result. The slope
OY
A -- ~ 0~
(4.52)
provides a modulus defining instantaneous isotropic hardening. In the second approach ~ is dimensionless (e.g. an accumulated plastic strain 16) and is related directly to the consistency parameter using = [(_~p)T~p] 1/2 = ~[aXtra,tr]~/2
(4.53)
A constitutive equation is then introduced to match uniaxial results. For example, a simple linear form is given by
O'y(/'C) - -
O'y 0 31-
Hi ol~
where Hio is a constant isotropic hardening modulus.
Kinematic hardening
A classical procedure to represent kinematic hardening was introduced by Prager 26 and modified by Ziegler. 27 Here the stress in each yield surface is replaced by a linear relation in terms of a 'back stress' ~ as m
= tr- ~
(4.54)
Classical time-independent plasticity theory 77 with the yield function now given as F ( g - ~, t~) = F (5_, t~) = 0
(4.55)
during plastic behaviour. We note that with this approach derivatives of the yield surface differ only by a sign and are given by
F,r
F_~ =-F,_~
(4.56)
m
Accordingly, the yield surface will now translate, and if isotropic hardening is present will also expand or contract, during plastic loading. A rate equation may be specified most directly by introducing a conjugate work variable 13 from which the hardening parameter ~ is deduced by using a hardening potential 7-/. This may be stated as
- -7-t,~
(4.57)
which is completely analogous to use of an elastic energy to relate r and e . A rate equation may be expressed now as /3 = j~Q,_~
(4.58)
It is immediately obvious that here also we have two possibilities. Using Q in the above expression defines a non-associative hardening, whereas replacing Q by F would give an associative hardening. Thus for a fully associative model we require that F be used to define both the plastic potential and the hardening. In such a case the relations of plasticity also may be deduced by using the principle of maximum plastic dissipation. 15'16'28"29A quadratic form for the hardening potential may be adopted and written as _ 7 - t - ~1f-.ITH _k/3 (4.59) in which H~ is assumed to be an invertible set of constant hardening parameters. Now 13 may be eliminated to give the simple rate form
~_-
-~I'Ik
OQ O~
-
-
AH Q,,, -k
_
(4.60)
Use of a linear shift in relation (4.54) simplifies this, noting Eq. (4.56), to -- ,~H k Q,r
(4.61)
In our subsequent discussion we Shall usually assume a general quadratic model for both elastic and hardening potentials. For a more general treatment the reader is referred to references 16 and 30. Another approach to kinematic hardening was introduced by Armstrong and Frederick 31 and provides a means of retaining smoother transitions from elastic to inelastic behaviour during cyclic loading. Here the hardening is given as
= A[Hk Q,~- HNL~]
(4.62)
78
Inelastic and non-linear materials
Applications of this approach are presented by Chaboche 32'33 and numerical comparisons to a simpler approach using a generalized plasticity model 34'35 are given by Auricchio and Taylor. 36 Many other approaches have been proposed to represent classical hardening behaviour and the reader is referred to the literature for additional information and discussion. 21-23'37-39 A physical procedure utilizing directly the finite element method is available to obtain both ideal plasticity and hardening. Here several ideal plasticity components, each with different yield stress, are put in series and it will be found that both hardening and softening behaviour can be obtained easily retaining the properties so far described. This approach was named by many authors as an 'overlay' model 4~ and by others is described as a 'sublayer' model. There are of course many other possibilities to define change in surfaces during the process of loading and unloading. Here frictional soils present one of the most difficult materials to model and for the non-associative case we find it convenient to use the generalized plasticity method described in Sec. 4.6.
4.3.4 Plastic stress-strain relations To construct a constitutive model for plasticity, the strains are assumed to be divisible into elastic and plastic parts given as g_ = s__e + s__p
(4.63)
For linear elastic behaviour, the elastic strains are related to stresses by a symmetric 9 • 9 matrix of constants D__. Differentiating Eq. (4.63) and incorporating the plastic relation (4.50) we obtain - - I ) - l l ~ " -t- ,~a,ff (4.64) The plastic strain (rate) will occur only if the 'elastic' stress changes &__e__ DS
(4.65)
tend to put the stress outside the yield surface, that is, is in the plastic loading direction. If, on the other hand, this stress change is such that unloading occurs then of course no plastic straining will be present, as illustrated for the one-dimensional case in Fig. 4.6. The test of the above relation is therefore crucial in differentiating between loading and unloading operations and underlines the importance of the straining path in computing stress changes. When plastic loading is occurring the stresses are on the yield surface given by Eq. (4.44). Differentiating this we can therefore write
OF
P -- ~ O ' x 00"x
OF
"JI-~ ( T y O0"y
-~-''"-t"
OF ~
~x "Jr-
OF ~
~y -t"''" "JI-
OF
~ --- 0
or
P -- F ~
+ F~-
Hi ~ = 0
(4.66)
in which we make the substitution
OF
Hi,~ = - 0----~-/~= - F,~t~ where Hi denotes an isotropic hardening modulus.
(4.67)
Classical time-independent plasticity theory 79 For the case where kinematic hardening is introduced, using Eq. (4.54) we can substitute Eq. (4.61) and modify Eq. (4.64) to De = ~_+ (D D_ + I-Ik)J~Q,r
(4.68)
Similarly, introducing Eq. (4.56) into Eq. (4.66) we obtain (4.69)
P - F T ~ - Hi~ -- 0
Equations (4.68) and (4.69) now can be written in matrix form as (D__+ H__k)Q,_~] -Hi {~}
(4.70)
The indeterminate constant ~ can now be eliminated (taking care not to multiply or divide by Hi or ~ which are zero in ideal plasticity). To accomplish the elimination we solve the first set of Eq. (4.70) for ~, giving ~ = D e - (_D_+ H__s)Q,~J~
and substitute into the second, yielding the expression FTDe - [Hi + F,T(D__+ H__~)Q,r
= 0
Equation (4.64) now results in an explicit expansion that determines the stress changes in terms of imposed strain changes. Using Eq. (4.43) this may now be reduced to a form in which only six independent components are present and expressed as* ,k
9
O" - - Dep~:
(4.71)
and 1 Dep = pTD--P - H* pTDQ'-r FTDp = D-
1
(4.72)
H.pTD__Q,sF~_DP_~
where n*--
H i -Jr-F T (I)-t--a_Hk) Q,g__
The elasto-plastic matrix De*p takes the place of the elasticity matrix D T in a continuum rate formulation. We note that in the absence of kinematic hardening it is possible to make reductions to the six-component form for all the computations at the very beginning. However, the manner in which the back stress enters the computation is not the same as that for the plastic strain and would be necessary to scale the two differently to make the general reduction. Thus, for the developments reported here we prefer to * We shall show this step in more detail below for the J2 plasticity model. In general, however, the final result involves only the usual form of the D matrix and six independent components from the derivative of the yield function.
80
Inelasticand non-linear materials
carry out all calculations using the full nine-component form (or, in the case of plane stress, to follow a four-component form) and make final reductions using Eq. (4.72). For a generalization of the above concepts to a yield surface possessing 'comers' where Q,_~is indeterminate, the reader is referred to the work of Koiter ~9or the multiple surface treatments in Simo and Hughes. 16 An alternative procedure exists here simply by smoothing the comers. We shall refer to it later in the context of the Mohr-Coulomb surface often used in geomechanics and the procedure can be applied to any form of yield surface. The continuum elasto-plastic matrix is symmetric only when plasticity is associative and when kinematic hardening is symmetric. In general, non-associative materials present stability difficulties, and special care is needed to use them effectively. Similar difficulties occur if the hardening moduli are negative which, in fact, leads to a softening behaviour. This is addressed further in Sec. 4.11 and later for large strain in Sec. 6.7.2. The elasto-plastic matrix given above is defined even for ideal plasticity when Hi and H_H_kare zero. Direct use of the continuum tangent in an incremental finite element context where the rates are approximated by
e~+~At .~ Aen+~
and
b'n+~At
~
Aorn+ 1
was first made by Yamada et al. 42 and Zienkiewicz et al. 43 However, this approach does not give quadratic convergence when used in the Newton scheme. For the associative case we can introduce a discrete time integration algorithm in order to develop an exact (numerically consistent) tangent which does produce quadratic convergence when used in the Newton iterative algorithm.
We have emphasized that with the use of iterative procedures within a particular increment of loading, it is important always to compute the stresses as k
O'n+ 1 :
(4.73)
or n + mOr~n
corresponding to the total change in displacement parameters A Uknand hence the total strain change k
AS nk--BAUkn
Auk -- ~-'~duin
(4.74)
i=O
which has accumulated in all previous iterations within the step. This point is of considerable importance as constitutive models with path dependence (viz. plasticitytype models) have different responses for loading and unloading. If a decision on loading/unloading is based on the increment dUkn erroneous results will be obtained. Such decisions must always be performed with respect to the total increment A ukn. In terms of the elasto-plastic modulus matrix given by Eq. (4.72) this means that the stresses have to be integrated as
Computation of stress increments 81 k O'n+ 1 - - Orn "{-
f
zxe,k * I)ep de
(4.75)
do
incorporating into D*ep the dependence on variables in a manner corresponding to a linear increase of Aekn (or Au~). Here, of course, all other rate equations have to be suitably integrated, though this generally presents little additional difficulty. Various procedures for integration of Eq. (4.75) have been adopted and can be classified into explicit and implicit categories.
4.4.1 Explicit methods In explicit procedures either a direct integration process is used or some form of the Runge-Kutta process is adopted, an In the former the known increment Ae nk is subdivided into m intervals and the integral of Eq. (4.75) is replaced by direct summation, writing m-1
AO~n -- % ~ I)(n+j/m)Aekn m
(4.76)
j=0
where I)(n+j/m ) denotes the tangent matrix computed for stresses and hardening parameters updated from the previous increment in the sum. This procedure, originally introduced in reference 45 and described in detail in references 46 and 47, is known as subincrementation. Its accuracy increases with the number of subincrements, m, used. In general it is difficult a priori to decide on this number, and accuracy of prediction is not easy to determine. Such integration will generally result in the stress change departing from the yield surface by some margin. In problems such as those of ideal plasticity where the yield surface forms a meaningful limit a proportional scaling of stresses (or return map) has been practised frequently to obtain stresses which are on the yield surface at all times. 47,48In this process the effects of integrating the evolution equation for hardening must also be treated. A more precise explicit procedure is provided by use of a Runge-Kutta method. Here, first an increment of Ae/2 is applied in a single-step explicit manner to obtain Aorn+l/2
--
1 . -~DnAe n
(4.77)
using the initial elasto-plastic matrix. This increment of stress (and corresponding tCn+~/2) is evaluated to compute l)n+l/2 and finally we evaluate mor n -- Dn+l/zAen
(4.78)
This process has a second-order accuracy and, in addition, can give an estimate of errors incurred as mor n - 2Aorn+l/2 (4.79) If such stress errors exceed a certain norm the size of the increment can be reduced. This approach is particularly useful for integration of non-associative models or models without yield functions where 'tangent' matrices are simply evaluated (see Sec. 4.6).
82
Inelastic and non-linear materials
4.4.2 Implicit methods: return map algorithm The integration of Eq. (4.75) can, of course, be written in an implicit form. For instance, we could write in place of Eq. (4.75), during each iteration k, that Ao~n+l = [(1 -- O)Dn+ODn~l] km~n+1
(4.80)
where here D~, denotes the value of the tangential matrix at the beginning of the time *,k step and Dn+ 1 the current estimate to the tangential matrix at the end of the step. This non-linear equation set could be solved by any of the procedures previously described; however, derivatives of the tangent matrix are quite complex and in any case a serious error is committed in the approximate form of Eq. (4.80). Further, there is no guarantee that the stresses do not depart from the yield surface.
Return map algorithm
In 1964 a very simple algorithm was introduced simultaneously by Maenchen and Sacks 49 and by Wilkins. 5~ This algorithm uses a two-step process to compute the new stress and was originally implemented in an explicit time integration form, thus requiring no explicit construction of an elasto-plasfic tangent matrix; however, later its versatility and robustness was demonstrated for implicit solutions. 5~'52 The steps of the algorithm are: 1. Perform a predictor step in which the entire increment of strain (for the present discussion we omit the iteration counter k for simplicity)
is used to compute trial stresses (denoted by superscript TR) assuming elastic behaviour. Accordingly, o',a~+, = D (~_,+1 - ~_P,) (4.81) where only an elastic modulus D_Dis _ required. 2. Evaluate the yield function in terms of the trial stress and the values of the plastic parameters at the previous time: F(cr a~ ~ , , t~) --- ' - -
< O, > 0,
elastic plastic
(4.82)
(a) For an elastic value of F set the current stress to the trial value, accordingly TR oon+I - - o o n + I ,
---.~n+l = ~ n
and
B;n+ 1 :
B;n
(b) For a plastic state solve a discretized set of plasticity rate equations (namely, using any appropriate time integration method as described, for example, in reference 8) such that the final value of Fn+l is zero. A plastic correction can be most easily developed by returning to the original Eq. (4.64) and writing the relation for stress increment as moo n = D (m~__ n - - A~__Pn)
(4.83)
Computation of stress increments 83 Now integrating the plastic strain relation (4.50) using a form similar to that in Eq. (4.80) yields Ae~ = AA[(1 - o ) a , g l n -I- o a , g l n + l ]
(4.84)
where AA represents an approximation to the change in consistency parameter over the time increment. Kinematic hardening is included by integrating Eq. (4.60) as m__.~ n =
(4.85)
-AAI-Ik[(1 - O)Q,~ln -i- OQ,~ln+l]
Finally, during the plastic solution we enforce F,,+I = 0
(4.86)
thus ensuring that final values at tn+l satisfy the yield condition exactly. The above solution process is particularly simple for 0 = 1 (backward difference or Euler implicit) and now, eliminating A_e~, we can write the above non-linear system in residual form Ricr - - m e n Rixo ri
-- I ) - l m o . i n
--Hkl
AIr n ~
_
AAa,zlin+,
A A Q , xo i
~
_]n+l
__ ~ F/+I
and seek solutions which satisfy R / - 0, Ri~ = 0 and r i - O. Any of the general iterative schemes described in Chapter 3 can now be used. In particular, the full Newton process is convenient. Noting that A ~ is treated here as a specified constant (actually, the Aekn from the current finite element solution), we can write, on linearization,
["
-1 + AA Q,,~__z~ AA Q , ~__n~ AAQ,,~ I-I~ 1 + A A Q , ~ F T ~ ,tr
F T '~
Q ,z a,,, -I~i
--
li / / / / d~ i
n+l
dro i d-~ i
R__i
=
Ri re
(4.87)
where Hi is the same hardening parameter as that obtained in Eq. (4.67). Some complexity is introduced by the presence of the second derivatives of Q in Eq. (4.87) and the term may be omitted for simplicity (although at the expense of asymptotic quadratic convergence in the Newton iteration). Analytical forms of such second derivatives are available for frequently used potential surfaces. 16'30'51-53Appendix A also presents results for second derivatives of stress invariants. It is important to note that the requirement that/7,+1 = - r i [Eq. (4.87)] ensures that the r i residual measures precisely the departure from the yield surface. This measure is not available for any of the tangential forms if Oep is adopted. For the solution it is only necessary to compute d)~ i and update as i
AA / -- ~ j=0
dA j
(4.88)
84
Inelastic and non-linear materials
This solution process can be done in precisely the same way as was done in establishing Eq. (4.72). Thus, a solution may be constructed by defining the following:
V__F=
F o. , V_.Q=
i..l k 1 + A
Q,trS
(4.89) Q ,~--~
and expressing Eq. (4.87) as d ~_ i
= _A_IR _ i -- - A1- - ~ A - I ~ T Q i [ ( ~ _ _ F i ) T A - 1 R
i - r i]
where A* -
(4.90) (4.91)
Hi + (V__Fi)TA__-IV___Qi
Immediately, we observe that at convergence R i -- 0 and r i -- 0, thus, here we obtain a zero stress increment. At this point we have computed a stress state ~n+l which satisfies the yield condition exactly. However, this stress, when substituted back into the finite element residual [e.g. Eq. (2.20a) or (2.51)], may not satisfy the equilibrium condition and it is now necessary to compute a new iteration k and obtain a new strain increment d~_kn from which the process is repeated. We note that inserting this new increment into Eq. (4.87) will again result in a non-zero value for R~, but that R~ and r remain zero until subsequent iterations. Thus, Eq. (4.90) provides directly the ~, required tangent matrix l)ep from d= ~ }_ {dcr
[A-1 - ~1 1 a *A - V__ Q ( V F__) T A -_ 13 { d g } =
[D--.ep " ] { d g }
(4.92)
Thus, we find the tangent matrix ~ep is obtained from the upper diagonal block of Eq. (4.92). We note that this development also follows exactly the procedure for computing Dep in Eq. (4.72). At this stage the terms may once again be reduced to their six-component form using P as indicated in Eq. (4.42). Some remarks on the above algorithm are in order: 1. For non-associative plasticity (namely, Q r F) the return direction is n o t normal to the yield surface. In this case no solution may exist for some strain increments (in general, arbitrary selection of F and Q forms in non-associative plasticity does not assure stability) and the iteration process will not converge. 2. For associative plasticity the normality principle is valid, requiting a convex yield surface. In this case the above iteration process always converges for a hardening material. 3. Convergence of the finite element equations may not always occur if more than one quadrature point changes from elastic to plastic or from plastic to elastic in subsequent iterations. Based on these comments it is evident that no universal method exists that can be used with the many alternatives which can occur in practice. In the next several sections we illustrate some formulations which employ the alternatives we have discussed above.
Isotropic plasticity models 85
iiiii~iiii~i1ii4iii5i~i|i1i~iiiiii~iiiii~iiiiiiiiP|~ ii,i'i~iiiiiP|~S~|~ ii~iiiiii~Iiiiiii~iiiiiiiiiiiiii~i~i~iii,i~~iiiiii,mD~i~ ~i~2~ i~i~"i'i~i~~ ~i~i~i~i~~i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~i ~~~~~~~~~~~~~~ i~i~~i~~~~i ~~~i~i~i~i~~i ~~~i~i~~ii~i~~i~~~~~~~~~~~~~~~~~ We consider here some simple cases for isotropic plasticity-type models in which both a yield function and a flow rule are used. For an isotropic material linear elastic response may be expressed by moduli defined with two parameters. Here we shall assume these to be the bulk and shear moduli, as used previously in the viscoelastic section (Sec. 4.2). Accordingly, the stress at any discrete time tn+l is computed from elastic strains in matrix form as O ' n + 1 "-" P n + l m
-k- Sn+ l - - K m m e e + l
-+-
2 G ( I - gmmlT)~n+le (4.93)
- - D-D-(gn+l - ~--Pn+l)
where the elastic modulus matrix for an isotropic material is given in the simple form D__= K m m T + 2 G ( I - 3mm v)
(4.94)
and I is the 9 x 9 identity matrix and in is the nine-component matrix m--[1
1
1 0
0
0
0
0
0] T
Using Eqs (4.42) and (4.43) immediately reduces the above to D-
K m m x + 2G(I0 - 3mm x)
(4.95)
The above relation yields the stress at the current time provided we know the current total strain and the current plastic strain values. The total strain is available from the finite element equations using the current value of nodal displacements, and the plastic strain is assumed to be computed with use of one of the algorithms given above. In the discussion to follow we consider relations for various classical yield surfaces.
4.5.1 Isotropic yield surfaces The general procedures outlined in the previous section allow determination of the tangent matrices for almost any yield surface applicable in practice. For an isotropic material all functions can be represented in terms of the three stress invariants:* I1 =
oi i ~
2J2
-- sijsji
3J3
=
SijSjkSki
raTtY
(4.96)
- - s T s - - IS[ 2 --
dets
where we can observe that definition of all the invariants is most easily performed in indicial notation. One useful form of these invariants for use in yield functions is given by 45 3Crm - -
I1
wit * Appendix B presents a summary of invariants and their derivatives.
71"
6-
~0~
71"
-6
-
-
(4.97)
86 Inelasticand non-linear materials Using these definitions the surface for several classical yield conditions can be given as"
1. Tresca: F = 2# cos 0 - Y (~) = 0
(4.98)
2. Huber-von Mises: F = ~/-2~-
Y(~;) = I_sl-
Y(~;) = 0
(4.99)
Both conditions 1 and 2 are well verified in metal plasticity. For soils, concrete and other 'frictional' materials the Mohr-Coulomb or Drucker-Prager surfaces is frequently used. 54 3. Mohr--Coulomb: F
=
om
sin ~b+ #
(
1
cos 0
,r
sin ~bsin 0
)
- c cos ~b = 0
(4.100)
where c(n) and qS(t~) are the cohesion and the angle of friction, respectively, which can depend on an isotropic strain hardening parameter ~. 4. Drucker-Prager: F = 3o/(/s;)ffm + # - K(~) = 0 (4.101) where o/=
2 sin ~b ~/~(3 - sin qS)
K=
6 cos ~b ~/3(3 - sin ~b)
and again c and 05 can depend on a strain hardening parameter. These forms lead to a convenient definition of the gradients F,z or Q,_~, irrespective of whether the surface is used as a yield condition or a flow potential. Thus we can always write F_r = F r
0~,n 0# 00 0rr + F,~ ~ + F,o 0rr
(4.102)
and upon noting that
O# Oor
O# OJ2 OJ2 0o"
1 OJ2 2~/~
oo
oo o12
oo oj3
= tan30
1 OJ3
1jz OJ2 ]
~J3-~ - g
(4.103)
-~j
Alternatively, we can always write"
O0"m OJ2 OJ3 F'O" = F'o'm OCT + F J2 ~ + F'J3 0~ r
(4.104)
which can be put into a matrix form as shown in Appendix B. The values of the three derivatives with respect to the invariants are shown in Table 4.1 for the various yield surfaces mentioned. The form of the various yield surfaces given above is shown with respect to the principal stress space in Fig. 4.8, though many more elaborate ones have been developed, particularly for soil (geomechanics) problems. 55-57
Isotropic plasticity models 87 "(73
Drucker-Prager ~ > 0 .G1= 02 = ($3
cot o__M~ses
~/3C
- - O ~ n
"~3 ~/3c cot~
Mohr-Coulomb ~ > 0 01 = G2 = ($3
/~
--02
Tr'~esca ---02
1
"~1
(a)
(b)
Fig. 4.8 Isotropic yield surfaces in principal stress space: (a) Drucker-Prager and von Mises; (b) Mohr-Coulomb and Trexa.
4.5.2 -/2 model with isotropic and kinematic hardening
(PrandtI-Reuss equations)
As noted in Table 4.1 a particularly simple form results if we assume the yield function involves only the second invariant of the deviatoric stresses J2. Here we present a more detailed discussion of results obtained by using an associated form and the return map algorithm. Since the yield function involves deviatoric quantifies only we can initially make all the calculations in terms of these. Accordingly, the elastic deviatoric stress-strain relation is given as _s = 2GeZ - 2G (g - e_p)
(4.105)
Continuum rate form
Before constructing the return map solution we first consider the form of the plasticity equations in rate form for this simple model. The plastic deviatoric strain rates are deduced from eZ
9O F
= A--~- = J~F,_s
(4.106)
Including the effects of isotropic and kinematic hardening the Huber-von Mises yield function may be expressed as F
Is- El-
~Y(~)
- 0
(4.107)
in which ~ are back stresses from kinematic hardening and t~ is an isotropic hardening parameter. We assume linear isotropic hardening given by* Y ( n ) - u + Hie;
(4.108)
* More general forms of hardening may be approximated by piecewise linear segments, thus making the present formulation quite general.
88
Inelastic and non-linear materials Table 4.1 Invariant derivatives for various yield conditions Yield condition
F.crm
~/-~F, J2
J2 F, J3
Tresca
0
2 cos 0(1 + tan 0 tan 30)
Huber-von Mises
0
4c3
4 ~ sin 0 cos 30 0
Mohr-Coulomb
89c o s 0 [ 1 + tan 0 sin 30
sin
L
"1
~/3 sin 0 + sin 4~cos 0 2 cos 30
+_!_1 sin O(tan 3 0 - tan 0)[ Drucker-Prager
3a'
1
1
0
Here a rate of ~ is computed from a norm of the plastic strains, by using Eq. (4.53), as /c - ~ , ~
(4.109)
in which the factor v/2-/3 is introduced to match uniaxial behaviour given by Eq. (4.108). On differentiation of F it will be found that OF
0s_
=
OF
=n
where
-
n-
-
s-
-
Is-_
ro
-
l
(4.110)
Using the above, the plastic strains are given by
An
eP -
(4.111)
and, when substituted into a rate-of-stress relation, yield s_"= 2G [e_"- ,~n_]
(4.112)
A rate form for the kinematic hardening is taken as 2
(4.113)
P = n__. T (s_-k__) - 52 H i ~
(4.114)
The rate of the yield function becomes
and when combined with the other rate equations gives the expression for the plastic consistency parameter as (noting that with the nine-component form nTn = 1) A = GnTe
where G*=G+:
l ( n i --I- Hk)
(4.115) (4.116)
Substitution of Eq. (4.115) into Eq. (4.112), and using Eq. (4.42) to reduce to the six-component form, gives the rate form for stress-strain deviators as s -- 2G Io - G---~nnT /~
(4.117)
Isotropic plasticity models We note that for perfect plasticity Hi = Hk = 0 leading to G~ G* = 1 and, thus, the elastic-plastic tangent for this special case is also here obtained. Use of Eq. (4.117) in the rate form of Eq. (4.93) gives the final continuum elasticplastic tangent I)ep KmmT + 2G I i 0 . l m m T 2G nn T1 * -(4.118) 3 G* This then establishes the well-known Prandtl-Reuss stress-strain relations generalized for linear isotropic and kinematic hardening.
Incremental return map form
The return map form for the equations is established by using a backward (Euler implicit) difference form as described previously (see Sec. 4.4.2). Omitting the subscript on the n + 1 quantities the plastic strain equation becomes, using Eqs (4.106) and (4.110), gP - gPn + AA_n (4.119) and the accumulated (effective) plastic strain ~; = ~;n +
AA
(4.120)
Thus, now the discrete constitutive equation is s -- 2G (e_ - _ep) the kinematic hardening is 2
(4.121)
HkAAn
(4.122)
and the yield function is F = Is_- El _ ~ / 3ny2 _ g2Hi AA
(4.123)
where Yn -- Yo -t- ,v/2/ 3 t% . The trial stress, which establishes whether plastic behaviour occurs, is given by s TR = 2G ( e - e__p)
(4.124)
which for situations where plasticity occurs permits the final stress to be given as s - s_TR - 2GAAn__
(4.125)
Using the definition of n__,we may now combine the stress and kinematic hardening relations as I s - ~ l l l - IsT R - ~ n l n T R - ( 2 G + g 2 Hk) AAn (4.126) and noting from this that we must have 111T R - -
(4.127)
n
we may solve the yield function directly for the consistency parameter as 16'36 AA -where G* is given by Eq. (4.116).
[swR -
---~n I -
2G*
~/~-/3
Yn
(4.128)
89
90 Inelasticand non-linear materials We can also easily establish the relations for the consistent tangent matrix for this J2 model. From Eqs (4.121) and (4.119) we obtain the incremental expression ds_ = 2G [ d g - _ndA- AAdn__]
(4.129)
The increment of relation (4.127) gives 16
d n = d n rR= -
-
2G
[l-nn ride
I s - _ ~ l
- -
(4.130)
-
and we have have and from from Eq. Eq. (4.128) (4.128) we d)~ = ur nrde .
~
(4.131)
m
Substitution into Eq. (4.129) gives the consistent tangent matrix ds - 2G
I(
1 --
2GAA
]sT R __ ----~n I
)I_(G
2GAA
G*
-
1-STR -- ---~n ]
)nnT]de
(4.132)
This may now be expressed in terms of the total strains, combined with the elastic volumetric term and reduced to six-components to give Dep
= Kmm T + 2G
1 --
-~mm)-
IsTR __ ----.~nI
G*
l_sTR -- -~-~n I
nnT
(4.133) We here note also that when A)~ = 0 the tangent for the return map becomes the continuum tangent, thus establishing consistency of form.
4.5.3 J2 plane stress The discussion in the previous part of this section may be applied to solve problems in plane strain, axisymmetry, and general three-dimensional behaviour. In plane strain and axisymmetric problems it is only necessary to note that some strain components are zero. For problems in plane stress, however, it is necessary to modify the algorithm to achieve an efficient solution process. In a plane stress process only the four stresses O'x, O'y, 7"xy and Ty x need be considered. When considering deviatoric components, however, there are five components, sx, Sy, Sz, Sxy a n d Syx. The deviators may be expressed in terms of the independent stresses as Sx
2
Sy s =
Sz
Sxy Syx
-
1
-~
-I
-
-1
2
0
0
0
~
- 1 0 0 0 3 0
0
0
ax Oy
= Ps_~
(4.134)
7"xy TYX
The Huber-von Mises yield function may be written as F = [(~ - ~)TPsrPs(cr - ~)] 1/2 _ ~y2 (~) < 0
(4.135)
Isotropic plasticity models 91 Expanding Eq. (4.135) gives the plane stress yield function F = [9x2 -
ffxffy
+ 9y2 + 1.5(g'xZy+ ff2x)] 1/2_ Y(tc) < 0
(4.136)
where
g'x ~ O'x -- ~x,
~y - - O'y ~ I'f,y,
~yx - " 7"yx -- I'~yx
~xy - - Txy -- I'~xy ,
(4.137)
define stresses which are shifted by the kinematic hardening back stress. Plastic strain rates may now be computed by direct differentiation of the yield function, giving kx ~Z--
~F,o"
"~
2
~y ~xy
~ -- 2~1
- 1
--i
kyx
0
0
Crx
02 03 i
O'y Txy
0
7"yx
0
.
= i_~1As~
(4.138)
where As = PsTPs. Similarly, the rate of the back stress for the kinematic hardening case is given by 1
~/~ =
As~r
nk--
(4.139)
--
The elastic components are computed by using the plane stress relation. Accordingly, for a plastic step the constitution is given by = D_D(_e _ - ~P)
(4.140)
where for isotropic behaviour
~
E 1 -u2
1
u
0
u
1 0 0
0 1
0
u 0
0 0 1-
(4.141)
with E the modulus of elasticity and u the Poisson's ratio. We note that for a J2 model the volumetric plastic strain must always be zero; consequently, we can complete the determination of plastic strains at any instant by using ezp - -exp - ePy (4.142) This may be combined with the elastic strain given by ee =
/]
E (ax + O-y)
(4.143)
to compute the total strain ez and, thus, the thickness change. The solution process now follows the procedures given for the general return mapping case. A procedure which utilizes a spectral transformation on the elastic and plastic parts is given in references 16 and 52. The process given there is more elegant but lacks the clarity of working directly with the stress and plastic strain increments.
92
Inelastic and non-linear materials
Plastic behaviour characterized by irreversibility of stress paths and the development of permanent strain changes after a stress cycle can be described in a variety of ways. One form of such description has been given in Sec. 4.3. Another general method is presented here.
4.6.1 Non-associative case- frictional materials This approach assumes a priori the existence of a rate process which may be written directly as -- D*e (4.144) in which the matrix D__*depends not only on the stress o" and the state of parameters ~, but also on the direction of the applied stress (or strain) rate ~r (or ~).58 A slightly less ambitious description arises if we accept the dependence of D* only on two directions those of loading and unloading. If in the general stress space we specify a 'loading' direction by a unit vector n_ given at every point (and also depending on the state parameters e~), as shown in Fig. 4.9, we can describe plastic loading and unloading by the sign of the projection n T&. Thus
-
nT&
~ > 0 for loading L< 0 for unloading
(4.145)
while nTb " = 0 is a neutral direction in which only elastic straining occurs. One can now write quite generally that f D___[e_" for loading = I,D~k for unloading where the matrices D__[and D__~depend only on the state described by tr and ~. 0'2
,•
(load)
6 (unload)
Fig. 4.9 Loading and unloading directions in stress space.
(4.146)
Generalized plasticity 93 The specification of D___[and I)~ must be such that in the neutral direction of the stress increment ~ the strain rates corresponding to this are equal. Thus we require ~"
~
~r
(DL)
--1
"
O" = O~l-l____~
when n T& = 0
(4.147)
A general way to achieve this end is to write 1
(D~) -1 ~ D -1 --['- ~LngL aT
1
( D u ) -1 -- D -1 -~- -~ungU aT
and
(4.148)
where D__is the elastic matrix, ng L and ng U are arbitrary unit stress vectors for loading and unloading directions, and HL and Hu are appropriate plastic moduli which in general depend on cr and ~. The value of the tangent matrices D[ and D~ can be obtained by direct inversion if HL/u ~ O, but more generally can be deduced following procedures given in Sec. 4.3.4 or can be written directly using the Sherman-Morrison-Woodbury f o r m u l a 59 as" D[ - D
H~ = HL + nXDngL
H~DngLnTD
(4.149)
This form resembles Eq. (4.72) and indeed its derivation is almost identical. We note further that (D[) -1 is now well behaved for HL zero and a form identical to that of perfect plasticity is represented. Of course, a similar process is used to obtain D__~. This simple and general description of generalized plasticity was introduced by Mr6z and Z i e n k i e w i c z . 6~ It allows: 1. the full model to be specified by a direct prescription of n, ng and H for loading and unloading at any point of the stress space; 2. existence of plasticity in both loading and unloading directions; 3. relative simplicity for description of experimental results when these are complex and when the existence of a yield surface of the kind encountered in ideal plasticity is uncertain. For the above reasons the generalized plasticity forms have proved useful in describing the complex behaviour of soils. 62'62-65 Here other descriptions using various interpolations of n and moduli form a unique yield surface, known as bounding surface plasticity models, are indeed particular forms of the above generalization and have proved to be useful. 66 Classical plasticity is indeed a special case of the generalized models. Here the yield surface may be used to define a unit normal vector as 1
n__= tr b_r, jr,-T,,_11/F_~ 2
(4.150)
and the plastic potential may be used to define 1
ng = r,t~g.~g -,T ,-,,_,1/2 J Q.z
(4.151)
where once again some care must be exercised in defining the matrix notation. Substitution of such values for the unit vectors into Eq. (4.149) will of course retrieve
94
Inelastic and non-linear materials
Experimental
100
a_
Computational model
100
v
50
50 L_
0
0 0
0
L_
"~ -50
.~ -50 >
>
a-100 O
~
0
~
I 200
250
a -100 nt~
j
v
50 0
100 ...,.
o "~ -50 "N
._~ -50 > r
L
a-1~176
I
!
I
I
-6 -4 -2
0 Axial strain (%)
250
Mean effective pressure p' (kPa)
Mean effective pressure p' (kPa) t~ 100
0
i 2
(D
I a -100 -" 0 - 8
I I I -6 -4 -2 0 Axial strain (%)
I 2
Fig. 4.10 A generalizedplasticity model describinga very complex path, and comparisonwith experimental data. Undrainedtwo-way cyclicloading of Nigata sand.67(Note that in an undrainedsoil test the fluid restrains all volumetricstrains,and pore pressuresdevelop;see reference68.)
the original form of Eq. (4.72). However, interpretation of generalized plasticity in classical terms is more difficult. The success of generalized plasticity in practical applications has allowed many complex phenomena of soil dynamics to be solved. 69'70 We shall refer to such applications later but in Fig. 4.10 we show how complex cyclic response with plastic loading and unloading can be followed. While we have specified initially the loading and unloading directions in terms of the total stress rate ~ this definition ceases to apply when strain softening occurs and the plastic modulus H becomes negative. It is therefore more convenient to check the loading or unloading direction by the elastic stress increment cre of Eq. (4.65) and to specify nT&, ~ > 0 for loading L < 0 for unloading
(4.152)
This, of course, becomes identical to the previous definition of loading and unloading in the case of hardening.
4.6.2 Associative case- J2 generalized plasticity Another modification to the classical rate-independent approach is one in which the transition from an elastic to a fully plastic solution is accomplished with a smooth
Some examples of plastic computation 95 transition. This approach is useful in improving the match with experimental data for cyclic loading. A particularly simple form applicable to the J2 model was introduced by Lubliner. 34'35 In this approach, the yield function is modified to a rate form directly and is expressed as h(F)l~-A =0 (4.153) where h(F) is given by the function
h(F) =
F ( f l - F)~ + H/3
(4.154)
in which H = Hi + Hk, and ~, 3 are two positive parameters with dimension of stress. In particular, fl is a distance between a limit plastic state and the current radius of the yield surface, and d; is a parameter controlling the approach to the limit state with increasing accumulated plastic strain. On discretization and combination with the return map algorithm a rate-independent process is evident and again only minor modifications to the algorithm presented previously are necessary. A full description of the steps involved is given by Auricchio and Taylor.36 Their paper also includes a development for the non-linear kinematic hardening model given in Eq. (4.62). In the case where the yield function is associative (i.e. F = Q) the use of the non-linear kinematic hardening model leads to an unsymmetric tangent stiffness when used with the return map algorithm. On the other hand, the generalized plasticity model is fully symmetric for this case. In the next section we present further discussion on the use of generalized plasticity to model the behaviour of frictional materials. In general, these involve use of nonassociative models where the retum map algorithm cannot be used effectively.
The finite element discretization technique in plasticity problems follows precisely the same procedures as those of corresponding elasticity problems. Any of the elements already discussed can be used for problems in plane stress; however, for plane strain, axisymmetry, and three-dimensional problems it is usually necessary to use elements which perform well in constrained situations such as encountered for near incompressibility. For this latter class of problems use of mixed elements is generally recommended, although elements and constitutive forms that permit use of reduced integration may also be used. The use of mixed elements is especially important in metal plasticity as the Hubervon Mises flow rule does not permit any volume changes. As the extent of plasticity spreads at the collapse load the deformation becomes nearly incompressible, and with conventional (fully integrated) displacement elements the system locks and a true collapse load cannot be obtained. 71'72 Finally, we should remark that the possibility of solving plastic problems is not limited to a displacement and mixed formulation alone. Equilibrium fields form a suitable vehicle, 73-75 but owing to their convenient and easy interpretation displacement and mixed forms are most commonly used.
96
Inelastic and non-linear materials
4.7.1 Perforated plate- plane stress solutions Figure 4.11 shows the configuration and the division into simple triangular and quadrilateral elements. In this example plane stress conditions are assumed and solution is obtained for both ideal plasticity and strain hardening. This problem was studied experimentally by Theocaris and Marketos 76 and was first analysed using finite element methods by Marcal and King 77 and Zienkiewicz et al. 43 (See reference 5 for discussion on these early solutions.) The von Mises criterion is used and, in the case of strain hardening, a constant slope of the uniaxial hardening curve, H, is taken. Data for the problem, from reference 76, are E = 7000 kg/mm 2, H = 225 kg/mm 2 and tZy = 24.3 kg/mm 2. Poisson's ratio is not given but is here taken as in reference 43 as v, = 0.3. To match a configuration considered in the experimental study a strip with 200 mm width and 360 mm length containing a central hole of 200 mm diameter. Using symmetry only one quadrant is discretized as shown in Fig. 4.11. Displacement boundary restraints are imposed for normal components on symmetry boundaries and the top boundary. Sliding is permitted, to impose the necessary zero tangential traction boundary condition. Loading is applied by a uniform non-zero normal displacement (a)
(b)
[/IZ~
(c)
(d)
Fig. 4.11 Perforated plane stress tension strip: mesh used and development of plastic zones at loads of 0.55, 0.66, 0.75, 0.84, 0.92, 0.98, 1.02 times O-y. (a) T3 triangles; (b) plastic zone spread; (c) Q4 quadrilaterals; (d) Q9 quadrilaterals.
Some examples of plastic computation 97
1.0 0.8
G)
E
0.6
0.4 0.2
rU 0
Ex0ermenta I
0.5
I
I
1.0
I
I
I
I
I
I
1.5 2.0 2.5 3.0 3.5 4.0 4.5 E (ey/Oy)
5.0
Fig. 4.12 Perforated plane stress tension strip: load deformation for strain hardening case (H = 225 kg/mm2). with equal increments. Displacement elements of type T3, Q4, and Q9 are used with the same nodal layout. Results for the three elements are nearly the same, with the extent of plastic zones indicated for various loads in Fig. 4.11 obtained using the Q4 element. The load--deformation characteristics of the problem are shown in Fig. 4.12 and compared to experimental results. The strain ~y is the peak value occurring at the hole boundary. This plane stress problem is relatively insensitive to element type and load increment size. Indeed, doubling the number of elements resulted in small changes of all essential quantities.
4.7.2 Perforated plate- plane strain solutions .........................................................................................................................................................
-=-
. ...........................................................................................................................................................................................................................
The problem described above is now analysed assuming a plane strain situation. Data are the same as for the plane stress case except the lateral boundaries are also restrained to create a zero normal displacement boundary condition. This increases the confinement on the mesh and shows more clearly the locking condition cited previously. In Fig. 4.13 we plot the resultant axial load for each load step in the solution. Figure 4.13(a) shows results for the displacement model using T3, Q4, and Q9 elements and it is evident that the T3 and Q4 elements result in an erroneous increasing resultant load after the fully plastic state has developed. The Q9 element shows a clear limit state and indicates that higher order elements are less prone to locking (even though we have shown that for the fully incompressible state the Q9 displacement element will lock!). Figure 4.13(b) presents the same results for the Q4/1 and Q9/3 mixed elements and both give a clear limit load after the fully plastic state is reached.
4.7.3 Steel pressure vessel This final example, for which test results obtained by Dinno and Gill TM are available, illustrates a practical application, and the objectives are twofold. First, we show that this
98
Inelastic
and non-linear
materials
problem which can really be described as a thin shell can be adequately represented by a limit number (53) of isoparametric quadratic elements. Indeed, this model simulates well both the overall behaviour and the local stress concentration effects [Fig. 4.14(a)]. Second, this problem is loaded by an internal pressure and a solution is performed up to the 'collapse' point (where, because there is no hardening, the strains increase without limit) by incrementing the pressure rather than displacement. A comparison of calculated and measured deflections in Fig. 4.14(b) shows how well the objectives are achieved.
2500
2000 "o 1500
m
O ..=,, .m X
< 1000 .............. T 3 - Displ. . . . . . . . Q 4 - Displ.
500 0
i 0
Q 9 - Displ. I
5
I
10
I
15
I
I
I
I
I
I
20 25 30 Step number
35
40
45
50
20 25 30 Step number
35
40
45
50
(a)
2500 2000 ~5oo
~ 1000 500 0
0
5
10
15
(b)
Fig. 4.13 Limit load behaviour for plane strain perforated strip: (a) displacement (displ.) formulation results; (b) mixed formulation results.
Some examples of plastic computation E = 29120000 Ib/in 2 1.60inl
v=0.3 O'y = 40540 Ib/in 2 No strain hardening
1000
I
I
I
-~1
, __~ ~_0_0.125in . __~ 2~8125 in E_~ D
900 o
760
0.25 in
9
i
,
1
I!
L
8.687in
p
J
r!
1000 1080
0.545
1400
..
1000 -a ~- 800
,., . .
,,,.,
......
"
"
"
"
" "
"
"
"
" "
Finite element analysis
t.D
=
/
Experimental results - Dinno and Gill 78
1200
r~
-i
Contours for plastic zone at different pressures (Ib/in 2)
(a)
z..
j
600
r~
n
400 200 0
(b)
0
I 10
I I I 20 30 40 Vertical deflection of point A (x 10 -3 in)
I 50
Fig. 4.14 Steel pressure vessel: (a) element subdivision and spread of plastic zones; (b) vertical deflection at point A with increasing pressure.
99
100
Inelastic and non-linear materials
The phenomenon of 'creep' is manifested by a time-dependent deformation under a constant stress. Indeed the viscoelastic behaviour described in Sec. 4.2 is a particular model for linear creep. Here we shall deal with some non-linear models. Thus, in addition to an instantaneous strain, the material develops creep strains, e c, which generally increase with duration of loading. The constitutive law of creep will usually be of a form in which the rate of creep strain is defined as some function of stresses and the total creep strains (ec), that is, kc -
0e c
Ot
=/3(or, e c)
(4.155)
If we consider the instantaneous strains are elastic (ee), the total strain can be written again in an additive form as : ~e _~_ ~c (4.156) with (4.157)
~e _. n - l o .
where we neglect any initial (thermal) strains or initial (residual) stresses. A special case of this form was considered for linear viscoelasticity in Sec. 4.2. Here we consider a more general non-linear approach commonly used in modelling behaviour of metals at elevated temperatures and in modelling creep in cementitious materials. We can again use any of the time integration schemes considered above and approximate the constitutive equations in a form similar to that used in plasticity as o'.+1 -- D (r - eC+l) c c ~n+l -- ~n -Jl" Att~n+O
(4.158)
where t~n+O is calculated as i~n+O --- (1 -- O)jO n +
0t~n+ 1
On eliminating Ae c we have simply a non-linear equation Rn+l
~
~n+l
D-10rn+ 1
--
__ ~c
__
At/3.+ 0 : 0
(4.159)
The system of equations can be solved iteratively using, say, the Newton procedure. Starting from some initial guess, say Crn+l = ~r. and an increment of strain is given by the finite element process, the general iterative/incremental solution can be written as R i+1 - - 0 -
Ri -
( D -1 +
AtCn+l)dOrin+ 1
(4.160)
where Cn+l =
~
n+O
(4.161) n+l
Solving this set of equations until the residual R is zero we obtain a set of stresses o'.+ and tangent matrix Dn+ 1 --= [ D - l + Atfn+l] -1 (4.162)
Basic formulation of creep problems 101 which may once again be used to perform any needed iterations on the finite element equilibrium equations. The iterative computation that follows is very similar to that used in plasticity, but here At is an actual time and the solution becomes rate dependent. While in plasticity we have generally used implicit (backward difference) procedures; here many simple alternatives are possible. In particular, two schemes with a single iterative step are popular.
4.8.1 Fully explicit solutions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
"Initial strain" procedure: o - 0
Here, from Eqs (4.161) and (4.162) we see that Cn+l - 0
and
Dn+ 1 = D
(4.163)
~rn+l = D[en+l - e c - At/3n]
(4.164)
Thus, from Eq. (4.159) we obtain
which may be used in Eq. (2.1 lc2) of Chapter 2 to satisfy a discretized equilibrium equation. We note that this form will lead to a standard elastic stiffness matrix. This, of course, is equivalent to evaluating the increment of creep strain from the initial stress values at each time tn and is exceedingly simple to calculate. While the process has been popular since the earliest days of finite elements 79-81 it is obviously less accurate for a finite step than other alternatives. Of course accuracy will improve if small time steps are used in such calculations. Further, if the time step is too large, unstable results will be obtained. Thus it is necessary for At _< Atcrit
(4.165)
where Atcrit is determined in a suitable manner (see, for example, Chapter 17 in reference 8). A 'rule of thumb' that proves quite effective in practice is that the increment of creep strain should not exceed one half the total elastic strain 82 A t [tOnt~n]
~_~ ~
(4.166)
Fully explicit process with modified stiffness: 1/2 <_ 0 <_ 1
Here the main difference from the first explicit process is that the matrix C is not equal to zero but within a single step is taken as a constant, that is,
C.+1
0
(4.167)
This is equivalent to a modified Newton scheme in which the tangent is held constant at its initial value in the step. Now
D: +1
=
[D-I +
AtCn+l
]-I
This process is more expensive than the simple explicit one previously mentioned, as the finite element tangent matrix has to be formed and solved for every time step. Further, such matrices can be non-symmetric, adding to computational expense.
102 Inelasticandnon-linearmaterials Neither of the simplified iteration procedures described above give any attention to errors introduced in the estimates of the creep strain. However, for accuracy the iterative process with 0 > 1/2 is recommended. Such full iterative procedures were introduced by Cyr and Teter, 83 and later by Zienkiewicz and co-workers. 84,85 We shall note that the process has much similarity with iterative solutions of plastic problems of Sec. 4.3 in the case of viscoplasticity, which we shall discuss in the next section.
4.9.1 General remarks .... = ...............
-------=:=
..................
: ......................
- ......................
- ................
- ..............................
- ......................................................................
- ...............................
--:
.....................
-
.........................................
= ......................................................................................
The purely plastic behaviour of solids postulated in Sec. 4.3 is probably a fiction as the maximum stress that can be carded is invariably associated with the rate at which this is applied. A purely elasto-plastic behaviour in a uniaxial loading is described in a model of Fig. 4.15(a) in which the plastic strain rate is zero for stresses below yield, that is, kp -- 0 if I 0 " - fly[ < 0 and Icrl > 0 and kp is indeterminate when c r - Oy --- 0. An elasto-viscoplastic material, on the other hand, can be modelled as shown in Fig. 4.15(b), where a dashpot is placed in parallel with the plastic element. Now stresses can exceed Oy for strain rates other than zero. The viscoplastic (or creep) strain rate is now given by a general expression
(4.168)
~vp--" ")/(~(O" -- O'y) ~
where the arbitrary function ~bis such that
i
(~)(O" -- O'y)/ : 0 ~(0"-
(a) 1
O'y)~ : ~(0"- O'y)
(b)
15
....
•
Ep ~ 0 15 = 15y
(at yield)
(c)
1 15
69 ; Li' I ,
if Icr-cryl <_ 0 if Icr- Cryl > 0
T
EvP
•
o
1'9
(4.169)
_L se
Evp ~ 0 15 > 15y
(at yield)
•
Fig. 4.15 (a) Elasto-plastic;(b) elasto-viscoplastic;(c) seriesof elasto-viscoplasticmodels.
Viscoplasticity- a generalization The model suggested is, in fact, of a creep-type category described in the previous sections and often is more realistic than that of classical plasticity. A viscoplastic model for a general stress state is given here and follows precisely the arguments of the plasticity section. In a three-dimensional context 05 becomes a function of the yield condition F(~r, e~, n) defined in Eq. (4.44). If this is less than zero, no 'plastic' flow will occur. To include the viscoplastic behaviour we modify Eq. (4.44) as "7 (qS(F)) - )~ = 0 (4.170) and use Eq. (4.45) to define the plastic strain. Equation (4.170) implies (qS(F))=
0 if F < 0 ~b(F) if F > 0
(4.171)
and-y is some 'viscosity' parameter. Once again associatedor non-associatedflows can be invoked, depending on whether F = Q or not. Further, any of the yield surfaces described in Sec. 4.3.1 and hardening forms described in Sec. 4.3.3 can be used to define the appropriate flow in detail. For simplicity, ~b(F) = F m where m is a positive power often used to define the viscoplastic rate effects in Eq. (4.170). 86,87 The concept of viscoplasticity in one of its earliest versions was introduced by Bingham in 192288 and a survey of such modelling is given in references 86, 89 and 90. The computational procedure for a viscoplastic model can follow any of the general methods described in Sec. 4.4. Early applications commonly used the straightforward Euler (explicit) method. 91-95 The stability requirements for this approach have been considered for several types of yield conditions by Cormeau. 96 A tangentialprocess can again be used, but unless the viscoplastic flow is associated (F = Q), non-symmetric systems of equations have to be solved at each step. Use of an explicit method will yield solution for the associative and non-associative cases and the system matrix remains symmetric. This process is thus similar to that of a modified Newton method (initial stress method) and is quite efficient. Indeed within the stability limit it has been shown that use of an overrelaxation method leads to rapid convergence.
4.9.2 Iterative solution The complete iterative solution scheme for viscoplasticity is identical to that used in plasticity except for the use of Eq. (4.170) instead of Eq. (4.44). To underline this similarity we consider the constitutive model without hardening and use the return map implicit algorithm. The linearized relations are identical except for the treatment of relation (4.175). The form becomes ID-1 + A/~a,~rcr
a,~r 1 ]i
~d)~lfd~r!)(Ri ri
(4.172)
n+l
where the discrete residual for Eq. (4.171) is given by 1
rn ---q~(F)n -t---m/~n "/At
(4.173)
103
104
Inelastic and non-linear materials
and dF Now the equations are almost identical to those of plasticity [see Eq. (4.87)], with differences appearing only in the q~' and 1/(TAt) terms. Again, a consistent tangent can be obtained by elimination of the dA i and a general iterative scheme is once more available. Indeed, as expected, 7At -- c~ will now correspond to the exact plasticity solution. This will always be reached by any solution tending to steady state. However, for transient situations this is not the case and use of finite values for ,,/At will invariably lead to some rate effects being present in the solution. The viscoplastic laws can easily be generalized to include a series of components, as shown in Fig. 4.15(c). Now we write ~V
"V "V - - S 1 -~- S 2 "q- " ' "
(4.174)
and again the standard formulation suffices. If, as shown in the last element of Fig. 4.15(c), the plastic yield is set to zero, a 'pure' creep situation arises in which flow occurs at all stress levels. If a finite value is in a term a corresponding rate equation for the associated ,~j must be used. This is similar to the Koiter treatment for multi-surface plasticity. 19'97 The use of the Duvaut and Lions 9~ approach modifies the return map algorithm for a rate-independent plasticity solution. Once this solution is available a reduction in the value of A A is computed to account for rate effects. The interested reader should consult references 16 and 30 for additional information on this approach.
4.9.3 Creep of metals If an associated form of viscoplasticity using the von Mises yield criterion of Eq. (4.99) is considered the viscoplastic strain rate can be written as =
01sl =
(4.175)
with the rate expressed again as ~ --- 'ff (0" - - O'y) I f O-y,
(4.176)
the yield stress, is set to zero we can write the above as cvp __ TO.ran
(4.177)
and we obtain the well-known Norton-Soderberg creep law. In this, generally the parameter "7 is a function of time, temperature, and the total creep strain (e.g. the analogue to the plastic strain eP). For a survey of such laws the reader can consult specialized references 98 and 99. An example initially solved using a large number of triangular elements 85 is presented in Fig. 4.16, where a much smaller number of isoparametric quadrilaterals are used in a general viscoplastic program. 95
Viscoplasticity- a generalization 200
300 400
500 9 9 mmlmmmmlmtSInltm~ 9
900 1000 1100 200 300 500 Ib/in 2 9 600 Internal pressure: ~'~ ~'~" ~ ~ ~ 7 0 0 445 Ib/in2 ~.,/'~~~_________.-~~ 900 E= 20 x 1061b/in2 ~ , / ~ 1200 v = 0.30 Von Mises L,,f/,"/~,,~l 1200 . . . . . . associative flow 700 ~ { ~ } J 300 Results of Greenbaum Power flow rule !~ 7 / / ~ 100 and Rubinstein 81 exponent: m = 3.61 coefficient: 7 21.146 x 10-16 Units: lb. inch. hour
.y. './:' 600 700 800 900 1000
01 o, / 9
I
l l'./i ~
9
Axis o! revolution
0.091 in - - - 4 ,
,~ .
0.25 in
.
.
.
0.445 in
(a) 9
9
o o'~
/
9 9
9
9
800
i i
49oo
500 900 600 700 k 9 9
800
800
700
500
750 Ib/in 2
....
500 400 4O0
900 650
350
/
600 I C E S ~ 1 7 6 650 ~-"-"-,/7/1300 700 ~ _ . ~ ; 5 0
600
800 ~
~ 9 5 0
Results of Greenbaum and Rubinstein81
250
(b)
Fig. 4.16 Creep in a pressure vessel: (a) mesh end effective stress contours at start of pressurization; (b) effective stress contours 3 h after pressurization.
4.9.4 Soil mechanics applications As we have already mentioned, the viscoplastic model provides a simple and effective tool for the solution of plasticity problems in which transient effects are absent. This includes many classical problems which have been solved in references 95 and 68, and the reader is directed there for details. In this section some problems of soil mechanics are discussed in which the facility of the process for solving non-associated behaviour is demonstrated. 1~176 The whole subject of the behaviour of soils and similar porous media is one in which much yet needs to be done to formulate good constitutive models.
105
106
Inelastic and non-linear materials
For a fuller discussion the reader is referred to texts, conferences, and papers on the subject.101,102 One particular controversy centres on the 'associated' versus 'non-associated' nature of soil behaviour. In the example of Fig. 4.17, dealing with an axisymmetric sample, the effect of these different assumptions is investigated. 86 Here a Mohr-Coulomb law is used to describe the yield surface, and a similar form, but with a different friction angle, ~, is used in the plastic potential, thus reducing the plastic potential to the Tresca form of Fig. 4.8 when ~ -- 0 and suppressing volumetric strain changes. As can be seen from the results, only moderate changes in collapse load occur, although very appreciable differences in plastic flow patterns exist. Figure 4.18 shows a similar study carried out for an embankment. Here, despite quite different flow patterns, a prediction of collapse load was almost unaffected by the flow rate law assumed. Stiff platen
.
Prescribed load q
~ ~ ~ I
~. 90 ~80
.~1 ! L,J. ! iI' i ~L~ 1 I1 l
....
~
ill i I1 tl il,--
~
E40
!
19=15~ --
.11
~_.
--"
(x = C ' ~ $ N~ = 1 + sin 1 - sin
10-
J
~1
1.,.,
19= 45~ e~= 3.52 0=300, ~ = 3 - 3 5
Smooth platen associated and non-associat.ed)
~. 20 ~
.........
. . . . . . . .
o-60 ~ 50
I
=
~
~ ,
V;,,~;,~,','k/','~",,i,
, t B=I
~ ~------''"
00 1.0 2.0 3.0 4.0 5.0 6.0 7.0 Displacement v (m x 10"4)
Material properties c = 10 kN/m 2 = 45 ~ E = 2 x 105 kN/m 2 v = 0.25
(a)
(b)
!!iiii!i!iii!!ili! i!i!!!!i i ,
88
88 ~
~
-'~
i
!
88
88 X
"~
|
Associated (9 = r = 45 ~
I Non-associated (19=0 ~ r = 45 ~
(c) Fig. 4.17 Uniaxial, axisymmetric compression between rough plates: (a) mesh and problem; (b) pressure displacement result; (c)plastic flow velocity patterns.
Some special problems of brittle materials 107
~-
.,~--
.~t--
.~..--
~
~
~
~
~
~
~.
v
20
~
1
(a)
r
I+ +
(b)
1
2 3
4
Fig. 4.18 Embankment under action of gravity, relative plastic flow velocities at collapse, and effective shear strain rate contours at collapse: (a) associative behaviour; (b) non-associative (zero volume change) behaviour.
The non-associative plasticity, in essence caused by frictional behaviour, may lead to non-uniqueness of solution. The equivalent viscoplastic form is, however, always unique and hence viscoplasticity is on occasion used as a regularizing procedure.
+++++ii++++++++~++i i+++++++++~i +++i++++~~i+ii~+~i l ++++++++~+~+~+i ++~++++++~i ~+~+i ~+~+i++++~~~~+++i ++++++++I ++++++++++++++i +++++++i ++++++++++++I ++++++++++++++++++++++++++++ +ii ++++++++++++ i 4.10.1 The no-tension material
.....................................................................................................................................................................................~ .............................................................................. ........................................................................ ...............................................................
A hypothetical material capable of sustaining only compressive stresses and straining without resistance in tension is in many respects similar to an ideal plastic material. While in practice such an ideal material does not exist, it gives a reasonable
108 Inelasticand non-linearmaterials approximation of the behaviour of randomly jointed rock and other granular materials. While an explicit stress-strain relation cannot be generally written, it suffices to carry out the analysis elastically and wherever tensile stresses develop to reduce these to zero. The initial stress (modified Newton) process here is natural and indeed was developed in this context. ~~ The steps of calculation are obvious but it is important to remember that the principal tensile stresses have to be eliminated. The 'constitutive' law as stated above can at best approximate to the true situation, no account being taken of the closure of fissures on reapplication of compressive stresses. However, these results certainly give a clear insight into the behaviour of real rock structures.
An underground power station
Figure 4.19(a) and (b) shows an application of this model to a practical problem. 1~ In Fig. 4.19(a) an elastic solution is shown for stresses in the vicinity of an underground power station with 'rock bolt' prestressing applied in the vicinity of the opening. The zones in which tension exists are indicated. In Fig. 4.19(b) a no-tension solution is given for the same problem, indicating the rather small general redistribution and the zones where 'cracking' has occurred.
Reinforced concrete
A variant on this type of material may be one in which a finite tensile strength exists but when this is once exceeded the strength drops to zero (on fissuring). Such an analysis was used by Valliappan and Nath 1~ in the study of the behaviour of reinforced concrete beams. Good correlation with experimental results for underreinforced beams (in which development of compressive yield is not important) have been obtained. The beam is one for which test results were obtained by Krahl et al. 105 Figure 4.20 shows some relevant results. Much development work on the behaviour of reinforced concrete has taken place, with various plasticity forms being introduced to allow for compressive failure and procedures that take into account the crack-closing history. References 106 and 107 list some of the basic approaches on this subject. The subject of analysis of reinforced concrete has proved to be of great importance in recent years and publications in this field are proliferating. References 108-111 guide the reader to current practice in this field.
4.1 0.2 'Laminar' material and joint elements Another idealized material model is one that is assumed to be built up of a large number of elastic and inelastic laminae. When under compression, these can transmit shear stress parallel to their direction - providing this does not exceed the frictional resistance. No tensile stresses can, however, be transmitted in the normal direction to the laminae. This idealized material has obvious uses in the study of rock masses with parallel joints but has much wider applicability. Figure 4.21 shows a two-dimensional situation involving such a material. With a local coordinate axis x' oriented in the direction of
Some special problems of brittle materials
X
,#'~!
", "~1 ~, I ', / ,~Tension
1.1r
q~,lP
,,B.
/,,..
~
Arrow for tension J 1 ~75kg/cm I___...J. "
3
R.
4
./ .\t
.,. _ ..,, ,,: :' v. ,!,
')
(a)
1
k-
',
{ Cracked zones
t-
', ',"\
: I c'/,, ' / i
~lli~
,..~..,.-~L,L L .,jj < J t
(b) Fig. 4.19 Underground power station: gravity and prestressing loads. (a) Elastic stresses; (b)'no-tension' stresses.
109
110
Inelastic and non-linear materials 68 in
~ i " ' ~ ~
J__
c~x (Ib/in 2) -2400
"'--.J
-
~.."
(a) Mesh used
-1200
FAi --I-FI-
Cracking at maximum load
0
"
Compression
I
r
2 in 2 steel reinforcement
0
r ,
m
FB
-C
-
,.I
-150
I
0
I
Tension
Uncracked section
(b)
(c)
(d)
Fig. 4.20 Cracking of a reinforced concrete beam (maximum tensile strength 2001b/in2). Distribution of stresses at various sections?~ (a) Mesh used; (b) section AA; (c) section BB; (d) section CC.
the laminae we can write for a simple Coulomb friction joint Iq-x,y,I < #O-y, Cry, "-" 0
if Oy, __< 0 if ey, > 0
(4.178)
for stresses at which purely elastic behaviour occurs. In the above, # is the friction coefficient applicable between the laminae. If elastic stresses exceed the limits imposed the stresses have to be reduced to the limiting values given above. The application of the initial stress process in this context is again self-evident, and the problem is very similar to that implied in the no-tension material of the previous section. At each step of elastic calculation, first the existence of tensile stresses Cry,is checked and, if these develop, a corrective initial stress reducing these and the shearing stresses to zero is applied. If Cry, stresses are compressive, the absolute magnitude of the shearing stresses "rx,y, is checked again; if these stresses exceed the value given by Eq. (4.178) they are reduced to their proper limit. However, such a procedure poses the question of the manner in which the stresses are reduced, as two components have to be considered. It is, therefore, preferable to use the statements of relations (4.178) as definitions of plastic yield surfaces (F). The assumption of additional plastic potentials (Q) will now define the flow, and we
Some special problems of brittle materials 111
x (a)
3
(b) Fig. 4.21 'Laminar' material: (a) general laminarity; (b)laminar in narrow joint.
note that associated behaviour, with Eq. (4.178) used as the potential, will imply a simultaneous separation and sliding of the laminae (as the corresponding strain rates 4Ix,y, and ~y, are finite). Non-associated plasticity (or viscoplasticity) techniques have therefore to be used. Once again, if stress reversal is possible it is necessary to note the opening of the laminae, that is, the yield surface is made strain dependent. In some instances the laminar behaviour is confined to a narrow joint between relatively homogeneous elastic masses. This may well be of a nature of a geological fault or a major crushed rock zone. In such cases it is convenient to use narrow, generally rectangular elements whose geometry may be specified by mean coordinates of two ends A and B [Fig. 4.21 (b)] and the thickness. The element still has, however, separate
112 Inelasticand non-linear materials
points of continuity (1-4) with the adjacent rock m a s s . 112'113 Such joint elements can be simple rectangles, as shown here, but equally can take more complex shapes if represented by using isoparametric coordinates. Laminations may not be confined to one direction o n l y - and indeed the interlaminar material itself may possess a plastic limit. The use of such multilaminate models in the context of rock mechanics has proved very effective; ~14 with a random distribution of laminations we return, of course, to a typical soil-like material, and the possibilities of extending such models to obtain new and interesting constitutive relations have been highlighted by Pande and Sharma. ~15
::~:;~:: ~::i::::::::::::::::~::~:::J~i~~~::~:::::::::::.::::::i ~ ::::~::::::::i::i::::~:::;:::~:::: ~::~:~::~:':~::::::~i :~ ::;.:~::::~::~::~i':~::::~i::;::~:: :: ~::i:::::: i ~::::::::~::~::~::~::::::::::::~::~::~::~::~::::~::::::~::~:: ~~::i:i::~: ~~i:~i!~~::~~: ~: ~~::: ~~:::~::~~: : ::::~ : ~: : ::::~~s~::~::~~: ~::::::::: ~:::::::::i::::~~::~::~~:::~:~::~:~:~:~:~: :i~::~:::::::~::~::~::~~::i::~::::::::: ~::~::~::~:~:::::~:::::~::~:~::~::~:~~:: :~::~:: ~: ~:::::::::::::::::.~:,~:::~~:::::~::~~::~::~~:::~::~::::,:::::::::::::::::::::::: ::i~:::: :: ::::::~ ~::~:::~:~::~::~::~:::::~~:::~::~::::~::~:.~::~::::::~::::::~~::~i~:.~~::: ~? ~~::::j~::::::::::::i:::::::::: ::::::~::~::::~::~::i::~::i::~~::: : ~: i: : ~
In the preceding sections the general processes of dealing with complex, non-linear constitutive relations have been examined and some particular applications were discussed. Clearly, the subject is large and of great practical importance; however, presentation in a single chapter is not practical or possible. For different materials altemate forms of constitutive relations can be proposed and experimentally verified. Once such constitutive relations are available the processes of this chapter serve as a guide for constructing effective numerical solution strategies. Indeed, it is possible to build standard computing systems applicable to a wide variety of material properties in which new specifications of behaviour may be inserted. What must be restated is that, in non-linear problems: 1. non-uniqueness of solution may arise; 2. convergence can never be, a priori, guaranteed; 3. the cost of solution is invariably much greater than it is in linear solutions. Here, of course, the item of most serious concern is the first one, that is, that of nonuniqueness, which could lead to a physically irrelevant solution even if numerical convergence occurred and possibly large computational expense was incurred. Such non-uniqueness may be due to several reasons in elasto-plastic computations: 1. the existence of comers in the yield (or potential) surfaces at which the gradients are not uniquely defined; (to which we have already referred 2. the use of a non-associated formulation 2~ in Sec. 4.9.4); 3. the development of strain softening and localization. 1~8,119 The first problem is the least serious and can readily be avoided by modifying the yield (or potential) surface forms to avoid comers. A simple modification of the MohrCoulomb (or Tresca) surface expressions [Eq. (4.100)] is easily achieved by writing 55 F
=
where g(O) =
o m
sin q5 - c cos ~b+ g(O) 2K
(1 + K) - (1 - K) sin 30
(4.179)
Non-uniqueness and localization in elasto-plastic deformations G1
S BJ
/
S
t m
t
i i I i
I
1 1 1 1 1 1 1
: ' | |
@
~
.~176176
02
Fig. 4.22 H plane section of Mohr-Coulomb yield surface in principal stress space, with q~ = 25 ~ (solid line); smooth approximation of Eq. (4.179)(dotted line).
and K=
3 - sin~b 3 + sin~b
Figure 4.22 shows how the angular section of the Mohr-Coulomb surface in the FI plane (constant Cr,n) now becomes rounded. Similar procedures have been suggested by others. 12~An alternative to smoothing is to introduce a multisurface model and use a solution process which gives unique results for a comer. 16,30 Much more serious are the second and third possible causes of non-uniqueness mentioned above. Here, theoretical non-uniqueness can be avoided by considering the plastic deformation to be a limit state of viscoplastic behaviour in a manner we have already referred to in Sec. 4.9. Such a process, mathematically known as regularization, has allowed us to obtain many realistic solutions for both non-associative and strain softening behaviour in problems which are subjected to steady-state or quasi-static loading, as already shown. For fully transient cases, however, the process is quite delicate and much care is needed to obtain a valid regularization. However, on occasion (though not invariably), both forms of behaviour can lead to localization phenomena where strain (and displacement) discontinuities develop. 117-129 The non-uniqueness can be particularly evident in strain softening plasticity. We illustrate this in an example of Fig. 4.23 where a bar of length L, divided into elements of length h, is subject to a uniformly increasing extension u. The material is initially elastic with a modulus E and after exceeding a stress of ey, the yield stress softens (plastically) with a negative modulus H.
113
114
Inelastic and non-linear materials
I
I
!
~ U
h L
T
"
Oy
_1 r
(a) (y
s
h/L "--}" 0 h/L= I
h/L ,:~0.12 h/L = 0.25 (b)
L
r
./El-
J_
....................... _i ~=u/L
.y/E
Fig. 4.23 Non-uniqueness: mesh size dependence in extension of a homogeneous bar with a strain softening material. (Peak value of yield stress, Cry,perturbed in a single element.) (a) Stress cr versus strain e for material; (b) stress O versus average strain ~ ( - u/L) assuming yielding in a single element of length h.
The strain-stress relation is thus [Fig. 4.23(a)] tr = E e
if e < C r y / E - - e y
(4.180)
and for increasing e only,
(7-- O'y -- n (E-- Ey)
if e > ey
(4.181)
For unloading from any plastic point the material behaves elastically as shown. One possible solution is, of course, that in which all elements yield identically. Plotting the applied stress versus the elongation strain ~ = u/L the material behaviour curve is simply obtained identically as shown in Fig. 4.23(b) (h/L = 1). However, it is equally possible that after reaching the maximum stress O'y only one element (probably one with infinitesimally smaller yield stress owing to computer round-off) continues into the plastic range while all the others unload elastically. The total elongation strain is now given by u cr h ( c r - Or) = -- (4.182) L
E
LH
Non-uniqueness and localization in elasto-plastic deformations and as h tends to 0 then ~ tends to cr/E. Clearly, a multitude of solutions is possible for any arbitrary element subdivision and in this trivial example a unique finite element solution is impossible (with localization to a single element always occurring). Further, the above simple 'thought experiment' points to another unacceptable paradox implying the inadmissibility of the softening model specified with constant softening modulus. The difficulties are as follows. 1. The behaviour seems to depend on the size (h) of the subdivision chosen (also called a mesh sensitive result). Clearly this is unacceptable physically. 2. If the element size falls below a value given by h = H L / E only a catastrophic, brittle, behaviour is possible without involving an unacceptable energy gain. Similar difficulties can arise with non-associated plasticity which exhibits occasionally an effectively strain softening behaviour in some circumstances (see reference 130). The computational difficulties can be overcome to some extent by introducing viscoplasticity as a start to any computation. Such regularization was introduced as early as 197495 and was considered seriously by De Borst and co-authors. TM However, most of the difficulties remain as steady state is approached. The problem remains a serious line of research but two possible alternative treatments have emerged. The first of these is physically difficult to accept but is very effective in practice. This is the concept of properties which are labelled as non-local. In such an approach the softening modulus is made dependent on the element size. Many authors have contributed here, with the earliest being Bazant and co-workers. 125'126 Other relevant references are 131 and 132. The second approach, that of a concentrated discontinuity, is more elegant but, we believe, computationally more difficult. It was first suggested by Simo, Oliver and Armero in 1993133 and extended in later publications. 134-137 Both approaches allow strain and indeed displacement discontinuities to develop following the brittle failure behaviour on which we have already remarked. In the numerical application this limit is approached as element size decreases or alternatively when stress singularities, such as comers, trigger this type of behaviour. In the second approach, continuous plastic behaviour is not permitted and all action is concentrated on discontinuity lines which have to be suitably placed. A particular form of the non-local approach is illustrated in Fig. 4.24. Here we examine in detail a unit width of an element in which the displacement discontinuity is approximated. In the examples which we shall consider later this discontinuity is a slip one with the 'failure' being modelled as shown. However, an identical approach has been used to model strain softening behaviour of concrete in cracking. 125'126 The most basic form of non-local behaviour assumes that the work (or energy) expended in achieving the discontinuity must be the same whatever the dimension h of the element. This work is equal to O'ye y h ~
~ H = -~ryAU
(4.183)
If this work is to be identical in all highly strained elements we will require that H
= constant h Such a requirement is easy to apply in an adaptive refinement process.
(4.184)
115
116
Inelastic and non-linear materials .
<3===
h
._1
1
2
,==t> ,==C> a ==4:>
1
---f-
Ul_._.~~U2 (a)
ul/]u --~ ~ 12
h---~(~
Localized failure
(b)
/
a/Ik
t.,
Work dissipated in failure
per unit volume .....
1
y/H
=-I
EH
r
8
(c) Fig. 4.24 Illustration of a non-local approach (work dissipation in failure is assumed to be constant for all elements): (a) an element in which localization is considered; (b)localization; (c) stress-strain curve showing work dissipated in failure.
At this stage we can comment on the concentrated discontinuity approach of Bazant and co-workers. ~25'126In this we shall simply assume that the displacement increment of Eq. (4.183), that is, A U, is permitted to occur only on a discontinuity line and that its magnitude is strictly related to the energy density previously specified in Eq. (4.183). After considering the effects of large deformation we shall show in Sec. 6.7.2 how a very effective treatment and capture of discontinuity can be made adaptively.
Non-linearity may arise in many problems beyond those of solid mechanics, but the techniques described in this chapter are still universally applicable. Here we shall look again at one class of problems which is govemed by the quasi-harmonic field equations of Chapter 2.
Non-linear quasi-harmonic field problems 117 In some formulations it is assumed that q = -k(4~)V4~
(4.185)
which gives, then (with use of definitions from Sec. 2.6), Pq - H(~D)~
(4.186)
H - 3~ ( V N ) T k ( 0 ) V N dr2
(4.187)
where now H has the familiar form
In this form the general non-linear problem may be solved by direct iteration methods; however, as these often fail to converge it is frequently necessary to use a scheme for which a tangential matrix to 9 is required, as presented in Sec. 3.2.4 [see Eq. (3.25)]. The tangent for the form given by Eq. (4.185) is generally unsymmetric; however, special forms can be devised which lead to symmetry. 138 In many physical problems, however, the values of k in Eq. (4.185) depend on the absolute value of the gradient of Vq~, that is, V-
4(W0)TV0 dk
(4.188)
dV In such cases, we can write HT -where A-
OH(O)~b
= H + A
f J ( V N ) T [(V~b)Tk'Vq~] VN, d~2
(4.189)
(4.190)
and symmetry is preserved. Situations of this kind arise in seepage flow where the permeability is dependent on the absolute value of the flow velocity, 14~ in magnetic fields, 139'142-144 where magnetic response is a function of the absolute field strength, in slightly compressible fluid flow, and indeed in many other physical situations. 145 Figure 4.25 from reference 139 illustrates a typical non-linear magnetic field solution. While many more interesting problems could be quoted we conclude with one in which the only non-linearity is that due to the heat generation term Q [see Chapter 2, Eq. (2.69)]. This particular problem of spontaneous ignition, in which Q depends exponentially on the temperature, serves to illustrate the point about the possibility of multiple solutions and indeed the non-existence of any solution in certain non-linear cases. 146 Taking k = 1 and Q = ~ exp q~, we examine an elliptic domain in Fig. 4.26. For various values of 5, a Newton iteration is used to obtain a solution, and we find that no convergence (and indeed no solution) exists when ~ > ~crit exists; above the critical
118 Inelastic and non-linear materials
'
~Ii.
.
'
),..) ..\I
,
9
9 " -
9
-
"-,
-., ~.
9
. -
.
.
9
!i :.. '. ..;::::
..... '
,9
.
'. : : ,',,-_7...'.." ".~.. ~ ~ f :, t," ""-'~"' 9 ,., ~j~ 0.'~ It, l,
,
,.,~
','.'..~,.
.~..,,:.\.,:,.
9 z.-"..,
: : : "" ~t . . . . . .
-..,.. ,Irlq~.,.l,!.:;il-l-ll,. ,/
":,,l~:;,.
,',~'.,::..
9 "
,~:,,',..:.~: ..,
9
,.'
,,
~.
.
. . . . . . . .
...',
9
",,-.'..
9
.
.
..-
,: ..
9-
" ",
. . . .
,-.-
...
o
"..-.-"-;.:-
9
t,.:.:,:"
-
",,..
.
9
.','.,
,,:::.:. :':':,
9
li, :. ,'.: :-.~
.
": : " ' ' L . ' "
6 9
,
.
...:
I.. "' .'-: ':~'::' !.:': -
.
-'"
'
9 "
" i 9 . '1
Fig. 4.25 Magnetic field in a six-pole magnet with non-linearity owing to saturation, i39
value of c~the temperature rises indefinitely and spontaneous ignition of the material occurs. For values below this, two solutions are possible and the starting point of the iteration determines which one is in fact obtained. This last point illustrates that an insight into the problem is, in non-linear solutions, even more important than elsewhere.
~i~ii~!~iii~i~!~i~!~!~i~iii~i~!~!~ii~iii~i~i~i!~ii!~!~i!~!~!~i~!~i~iii~i~i~ii~i~i~i~i~ii~i!i~i~!ii~iii!~iiii~!i~iiiii~iii~iiiiii~!~ii~i~iiii~!~iii~i~i~i~i~i~i~i~ii~ill~ii!iilli!~i~i~!~i~ii~i~!~i!i~iiiii~iiiii!i~iiiiii~!
In this chapter we have considered a number of classical constitutive equations together with numerical algorithms which permit their inclusion in the formulations discussed in Chapter 2. These permit the solution to a wide range of practical problems in solid mechanics and geomechanics. The possibilities for models of real materials is endless
Concluding remarks
k 5.0 ~ B
4.0 I,...
~_ 3.0 E 0 2.0
Y~k ( ( (
lc
1.0
2' ( ( (
C <
2
Jx
,,...
0
0.2
0.6
1.0
(a)
~= 0.41
~= 3,41
Lower solution
x
Upper solution
x
(b) Fig. 4.26 A non-linear heat-generation problem illustrating the possibility of multiple or no solutions depending on the heat generation parameter 6; spontaneous combustion.146(a) Solution mesh and variation of temperature at point C; (b) two possible temperature distributions for ~ - 0.75.
119
120
Inelastic and non-linear materials and, thus, w e have not b e e n able to i n c l u d e m a n y o f the e x t e n s i o n s available in the literature. F o r e x a m p l e , the effect o f t e m p e r a t u r e c h a n g e s will n o r m a l l y affect the m a t e r i a l b e h a v i o u r t h r o u g h b o t h t h e r m a l e x p a n s i o n s as well as t h r o u g h the c h a n g e in the m a t e r i a l p a r a m e t e r s .
~iii~i~i~ !il ~i!i~! i! i i!ii!~i !i~!i!
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1. O.C. Zienkiewicz, M. Watson and I.E King. A numerical method of visco-elastic stress analysis. Int. J. Mech. Sci, 10:807-827, 1968. 2. J.L. White. Finite elements in linear viscoelastic analysis. In Proc. 2nd Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-68-150, pages 489-516. Wright Patterson Air Force Base, Ohio, October 1968. 3. R.L. Taylor, K.S. Pister and G.L. Goudreau. Thermomechanical analysis of viscoelastic solids. International Journal for Numerical Methods in Engineering, 2:45-79, 1970. 4. O.C. Zienkiewicz. The Finite Element Method in Engineering Science. McGraw-Hill, London, 2nd edition, 1971. 5. O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method, volume 2. McGraw-Hill, London, 4th edition, 1991. 6. B. Gross. Mathematical Structure of the Theories of Viscoelasticity. Herrmann & Cie, Paris, 1953. 7. R.M. Christensen. Theory of Viscoelasticity: An Introduction. Academic Press, New York, 1971 (reprinted 1991). 8. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann, Oxford, 6th edition, 2005. 9. T. Belytschko and T.J.R. Hughes, editors. Computational Methods for Transient Analysis. North-Holland, Amsterdam, 1983. 10. L.R. Herrmann and F.E. Peterson. A numerical procedure for viscoelastic stress analysis. In Proceedings 7th ICRPG Mechanical Behavior Working Group, Orlando, FL, 1968. 11. D. McHenry. A new aspect of creep in concrete and its application to design. Proc. ASTM, 43:1064, 1943. 12. T. Alfrey. Mechanical Behavior of High Polymers. Interscience, New York, 1948. 13. H.H. Hilton and H.G. Russell. An extension of Alfrey's analogy to thermal stress problems in temperature dependent linear visco-elastic media. J. Mech. Phys. Solids, 9:152-164, 1961. 14. O.C. Zienkiewicz, M. Watson and Y.K. Cheung. Stress analysis by the finite element method - thermal effects. In Proc. Conf. on Prestressed Concrete Pressure Vessels, London, 1967. Institute of Civil Engineers. 15. J. Lubliner. Plasticity Theory. Macmillan, New York, 1990. 16. J.C. Simo and T.J.R. Hughes. Computational Inelasticity, volume 7 of Interdisciplinary Applied Mathematics. Springer-Verlag, Berlin, 1998. 17. R. von Mises. Mechanik der Plastischen Form~inderung der Kristallen. Z. angew. Math. Mech., 8:161-185, 1928. 18. D.C. Drucker. A more fundamental approach to plastic stress-strain solutions. In Proceedings of the 1st U.S. National Congress on Applied Mechanics, pages 487-491, 1951. 19. W.T. Koiter. Stress-strain relations, uniqueness and variational theorems for elastic-plastic materials with a singular yield surface. Q. J. Appl. Math., 11:350-354, 1953. 20. R. Hill. The Mathematical Theory of Plasticity. Clarenden Press, Oxford, 1950. 21. W. Johnson and P.W. Mellor. Plasticity for Mechanical Engineers. Van Nostrand, New York, 1962. 22. W. Prager. An Introduction to Plasticity. Addison-Wesley, Reading, Mass., 1959.
References 23. W.E Chen. Plasticity in Reinforced Concrete. McGraw-Hill, New York, 1982. 24. D.C. Drucker. Conventional and unconventional plastic response and representations. Appl. Mech. Rev., 41:151-167, 1988. 25. G.C. Nayak and O.C. Zienkiewicz. Elasto-plastic stress analysis. Generalization for various constitutive relations including strain softening. International Journal for Numerical Methods in Engineering, 5:113-135, 1972. 26. W. Prager. A new method of analyzing stress and strains in work-hardening plastic solids. J. Appl. Mech., 23:493-496, 1956. 27. H. Ziegler. A modification of Prager's hardening rule. Q. Appl. Math., 17:55-65, 1959. 28. J. Mandel. Contribution throrique a l'rtude de l'rcrouissage et des lois de l'rcoulement plastique. In Proc. llth Int. Cong. of Appl. Mech., pages 502-509, 1964. 29. J. Lubliner. A maximum-dissipation principle in generalized plasticity. Acta Mechanica, 52:225-237, 1984. 30. J.C. Simo. Topics on the numerical analysis and simulation of plasticity. In EG. Ciarlet and J.L. Lions, editors, Handbook of Numerical Analysis, volume III, pages 183-499. Elsevier Science Publisher BV, 1999. 31. EJ. Armstrong and C.O. Frederick. A mathematical representation of the multi-axial Bauschinger effect. Technical Report RD/B/N731, C.E.G.B., Berkeley Nuclear Laboratories, R&D Department, 1965. 32. J.L. Chaboche. Constitutive equations for cyclic plasticity and cyclic visco-plasticity. Int. J. Plasticity, 5:247-302, 1989. 33. J.L. Chaboche. On some modifications of kinematic hardening to improve the description of ratcheting effects. Int. J. Plasticity, 7:661-678, 1991. 34. J.L. Lubliner. A simple model of generalized plasticity. International Journal of Solids and Structures, 28:769, 1991. 35. J.L. Lubliner. A new model of generalized plasticity. International Journal of Solids and Structures, 30:3171-3184, 1993. 36. E Auricchio and R.L. Taylor. Two material models for cyclic plasticity: nonlinear kinematic hardening and generalized plasticity. Int. J. Plasticity, 11:65-98, 1995. 37. J.E Besseling. A theory of elastic, plastic and creep deformations of an initially isotropic material. J. Appl. Mech., 25:529-536, 1958. 38. Z. Mr6z. An attempt to describe the behaviour of metals under cyclic loads using a more general work hardening model. Acta Mech., 7:199, 1969. 39. O.C. Zienkiewicz, C.T. Chang, N. Birani6 and E. Hinton. Earthquake response of earth and concrete in the partial damage range. In Proc. 13th Int. Cong. on Large Dams, pages 1033-1047, New Delhi, 1979. 40. O.C. Zienkiewicz, G.C. Nayak and D.R.J. Owen. Composite and 'Overlay' models in numerical analysis of elasto-plastic continua. In A. Sawczuk, editor, Foundations of Plasticity, pages 107122. Noordhoff, Dordrecht, 1972. 41. D.R.J. Owen, A. Prakash and O.C. Zienkiewicz. Finite element analysis of non-linear composite materials by use of overlay systems. Computers and Structures, 4:1251-1267, 1974. 42. Y. Yamada, N. Yishimura and T. Sakurai. Plastic stress-strain matrix and its application for the solution of elasto-plastic problems by the finite element method. Int. J. Mech. Sci., 10:343-354, 1968. 43. O.C. Zienkiewicz, S. Valliappan and I.P. King. Elasto-plastic solutions of engineering problems. Initial stress, finite element approach. International Journal for Numerical Methods in Engineering, 1:75-100, 1969. 44. EB. Hildebrand. Introduction to Numerical Analysis. Dover Publishers, 2nd edition, 1987. 45. G.C. Nayak and O.C. Zienkiewicz. Note on the 'alpha'-constant stiffness method for the analysis of nonlinear problems. International Journal for Numerical Methods in Engineering, 4:579-582, 1972.
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126 Inelasticand non-linear materials 135. J. Oliver. Modelling strong discontinuities in solid mechanics via strain softening constitutive equations. Part 1 Fundamentals. International Journal for Numerical Methods in Engineering, 39:3575-3600, 1996. 136. J. Oliver. Modelling strong discontinuities in solid mechanics via strain softening constitutive equations. Part 2 Numerical simulation. International Journal for Numerical Methods in Engineering, 39:3601-3623, 1996. 137. J. Oliver, M. Cervera and O. Manzoli. Strong discontinuities and continuum plasticity models: the strong discontinuity approach. Int. J. Plasticity, 15:319-351, 1999. 138. M. Muscat. The Flow of Homogeneous Fluids through Porous Media. Edwards, London, 1964. 139. A.M. Winslow. Numerical solution of the quasi-linear Poisson equation in a non-uniform triangle 'mesh'. J. Comp. Phys., 1:149-172, 1966. 140. R.E. Volker. Non-linear flow in porous media by finite elements. Proc. Am. Soc. Civ. Eng., 95(HY6), 1969. 141. H. Ahmed and D.K. Suneda. Non-linear flow in porous media by finite elements. Proc. Am. Soc. Civ. Eng., 95(HY6): 1847-1859, 1969. 142. M.V.K. Chari and P. Silvester. Finite element analysis of magnetically saturated D.C. motors. In IEEE Winter Meeting on Power, New York, 1971. 143. J.E Lyness, D.R.J. Owen and O.C. Zienkiewicz. The finite element analysis of engineering systems governed by a non-linear quasi-harmonic equation. Computers and Structures, 5:6579, 1975. 144. O.C. Zienkiewicz, J.F. Lyness and D.R.J. Owen. Three-dimensional magnetic field determination using a scalar potential. A finite element solution. IEEE Transactions on Magnetics, MAG-13(5): 1649-1656, 1977. 145. D. Gelder. Solution of the compressible flow equations. International Journal for Numerical Methods in Engineering, 3:35-43, 1971. 146. C.A. Anderson and O.C. Zienkiewicz. Spontaneous ignition: finite element solutions for steady and transient conditions. Trans ASME, J. Heat Transfer, pages 398-404, 1974.
Geometrically non-linear problems - finite deformation
In all our previous discussion we have assumed that deformations remained small so that linear relations could be used to represent the strain in a body. We now admit the possibility that deformations can become large during a loading process. In such cases it is necessary to distinguish between the reference configuration where initial shape of the body or bodies to be analysed is known and the current or deformed configuration after loading is applied. Figure 5.1 shows the two configurations and the coordinate frames which will be used to describe each one. We note that the deformed configuration of the body is unknown at the start of an analysis and, therefore, must be determined as part of the solution process - a process that is inherently non-linear. The relationships describing the finite deformation behaviour of solids involve equations related to both the reference and the deformed configurations. We shall generally find that such relations are most easily expressed using indicial notation (e.g. see Chapter 1, Sec. 1.2.1 or Appendix B of reference 1); however, after these indicial forms are developed we shall again return to a matrix form to construct the finite element approximations. The chapter starts by describing the basic kinematic relations used in finite deformation solid mechanics. This is followed by a summary of different stress and traction measures related to the reference and deformed configurations, a statement of boundary and initial conditions, and a brief overview of material constitution for finite elastic solids. A variational Galerkin statement for the finite elastic material is then given in the reference configuration. Using the variational form the problem is then cast into a matrix form and a standard finite element solution process is indicated. The procedure up to this point is based on equations related to the reference configuration. A transformation to a form related to the current configuration is performed and it is shown that a much simpler statement of the finite element formulation process results - one which again permits separation into a form for treating nearly incompressible situations. A mixed variational form is introduced and the solution process for problems which can have nearly incompressible behaviour is presented. This follows closely the developments for the small strain form given in Chapter 2. An alternative to the mixed form may also be given in the form of an enhanced strain model. 2-5 Here a fully
128 Geometrically non-linear problems- finite deformation
~(X, t)
.-X~,x~ Fig. 5.1 Referenceand deformed (current) configuration for finite deformation problems. mixed construction is shown and leads to a form which performs well in two- and three-dimensional problems. In finite deformation problems, loads can be given relative to the deformed configuration. An example is a pressure loading which always remains normal to a deformed surface. Here we discuss this case and show that by using finite element-type constructions a very simple result follows. Since the loading is no longer derivable from a potential function (i.e. conservative) the tangent matrix for the formulation is unsymmetric, leading in general to a requirement of an unsymmetric solver in a Newton solution scheme. The next three chapters concentrate on finite deformation forms for continuum problems where finite elements are used to discretize the problem in all directions modelled. In later chapters we shall consider forms for problems which have one (or more) small dimension(s) and thus can benefit from use of rod, plate and shell formulations of the type we shall discuss in Chapters 10 to 17 of this volume for small deformation situations.
5.2.1 Kinematics and deformation The basic equations for finite deformation solid mechanics may be found in standard references on the subject. 6-1~We begin by presenting a summary of the basic equations in three dimensions. Later we will specialize the equations to two dimensions to study problems of plane strain, plane stress and axisymmetry. A body has material points whose positions are given by the vector X in a fixed reference configuration, f2, in a three-dimensional space. In Cartesian coordinates the position vector is described in terms of its components as: X = XIEI;
I -- 1, 2, 3
(5.1)
Governing equations 129 where EI are unit orthogonal base vectors and summation convention is used for repeated indices of like kind (e.g. I). After the body is loaded each material point is described by its position vector, x, in the c u r r e n t deformed configuration, co. The position vector in the current configuration is given in terms of its Cartesian components as
X
i = 1, 2, 3
= xiei;
(5.2)
where ei are unit base vectors for the current time, t, and again summation convention is used.* In our discussion, common origins and directions of the reference and current coordinates are used for simplicity. Furthermore, in a Cartesian system base vectors do not change with position and all derivations may be made using components of tensors written in indicial form. Final equations are written in matrix form using standard transformations described in Chapter 2 (see also Appendix B of reference 1). The position vector at the current time is related to the position vector in the reference configuration through the mapping X i "-- ~ i ( X l ,
t)
(5.3)
Determination of ~)i is required as part of any solution and is analogous to the displacement vector, which we introduce next. When common origins and directions for the coordinate frames are used, a displacement vector may be introduced as the change between the two frames. Accordingly, Xi :
( ~ i l ( X l -Ai- U I )
(5.4)
where summation convention is implied over indices of the same kind and 3ii is a shifter between the two coordinate frames, and is defined by a Kronecker delta quantity such that 1 if/- I (~il -0 if/~ I (5.5) The shifter satisfies the relations (~il(~iJ -- (~IJ
(~il(~jl -- (~ij
and
(5.6)
where (~IJ and (~ij are Kronecker delta quantities in the reference and current configuration, respectively. Using the shifter, a displacement component may be written with respect to either the reference configuration or the current configuration and related through Ui - - (~itUI and UI -- 6 i t u i (5.7) and we observe that numerically u~ = U~, etc. Thus, either may be used equally to express finite element parameters. A fundamental measure of deformation is described by the deformation gradient relative to X t given by Fil - -
OXi OXl
=
O~ i OX I
(5.8a)
* As much as possible we adopt the notation that upper-case letters refer to quantities defined in the reference configuration and lower-case letters to quantities defined in the current deformed configuration. Exceptions occur when quantities are related to both the reference and the current configurations.
130 Geometricallynon-linear problems- finite deformation subject to the constraint J - det Fil > 0
(5.8b)
to ensure that material volume elements remain positive. The deformation gradient is a direct measure which maps a differential line element in the reference configuration into one in the current configuration as (Fig. 5.1) 0~) i dxi --- ~O X I d X l
-
-
Ftl d X l
(5.9)
Thus, it may be used to determine the change in length and direction of a differential line element. The determinant of the deformation gradient also maps a differential volume element in the reference configuration into one in the current configuration, that is dw - J dr2 (5.10) where dr2 is a differential volume element in the reference configuration and dw its corresponding form in the current configuration. The deformation gradient may be expressed in terms of the displacement as Fit -- (~i I -~-
Oui = (~i I -2i- u i , I OXI
(5.1 1)
and is a two-point tensor since it is referred to both the reference and the current configurations. Expanding the terms in Eq. (5.1 1), the deformation gradient components are given by f i i --
f21 f31
F22 F32
F23/ = f33J
/,/2,1 u3,1
(1 + u2,2) /13,2
u2,3 (1 + U3,3)
(5.12)
Using Fii directly complicates the development of constitutive equations and it is common to introduce deformation measures which are completely related to either the reference or the current configurations. For the reference configuration, the fight Cauchy-Green deformation tensor, C Ij, is introduced as
CIj = Fit Fig
(5.13)
Altematively the Green strain tensor, EIj, given as
EIj -- ~1 ( C i j - ~ig)
(5.14)
may be used. The Green strain may be expressed in terms of the displacements as ~Ou i -]-t~ij -__Ou E I j -- -21[(~i l "~'Tt ~t i ~ OX i Ox i 1
1 -- -- [(5ii Ui J -Jl" (~iJ Ui, l + Ui, I/1i, J]
2
_ 1_ JUt j + Uj,1 + UK,IUK j] 2
'
(5.15)
Governing equations 131 In the current configuration a common deformation measure is the left Cauchy-Green deformation tensor, b i j , expressed as
(5.16)
bij -- F i i F j i
eij,
The Almansi strain tensor,
is related to the inverse of
bij
as
1 eij -- ~((~ij -- b~ 1)
(5.17)
bij "- ((~ij - 2 e i j ) -1
(5.18)
or inverting by Generally, the Almansi strain tensor will not appear naturally in our constitutive equations and, thus, we will often use expressions in terms of bij directly.
5.2.2 Stress and traction for reference and deformed states , , , , ,
.
.
.
.
,
.
.
.
.
, .
.
.
.
.
.
.
.
.
Stress measures Stress measures the amount of force per unit of area. In finite deformation problems care must be taken to describe the configuration to which a stress is measured. The Cauchy (true) stress, aij, and the Kirchhoff stress, Tij, are symmetric measures of stress defined with respect to the current configuration. They are related through the determinant of the deformation gradient as Tij = J ffij
(5.19)
and often are the stresses used to define general constitutive equations for materials. Notationally, the first stress subscript defines the direction of a normal to the area on which the force acts and the second the direction of the force component. The second Piola-Kirchhoff stress, St j, is a symmetric stress measure with respect to the reference configuration and is related to the Kirchhoff stress through the deformation gradient as Tij - Fit S t j F j j
(5.20)
Finally, one can introduce the (unsymmetric) first Piola-Kirchhoff stress, Pit, which is related to St j through P i t --- F i j S j t (5.21) and to the Kirchhoff stress by Tij = Pit Fjt
(5.22)
Traction m e a s u r e s
For the current configuration traction is given by ti :
CTji n j
(5.23)
where nj are direction cosines of a unit outward pointing normal to a deformed surface. This form of the traction may be related to a reference surface quantity through force relations defined as ti d~/ = (~il TI dF (5.24)
132 Geometricallynon-linear problems- finite deformation where d7 and dF are surface area elements in the current and reference configurations, respectively, and Tt is traction on the reference configuration. Note that the direction of the traction component is preserved during the transformation and, thus, remains directly related to current configuration forces.
5.2.3 Equilibrium equations Using quantities related to the current (deformed) configuration, the equilibrium equations for a solid subjected to finite deformation are nearly identical to those for small deformation. The local equilibrium equation (balance of linear momentum) is obtained as a force balance on a small differential volume of deformed solid and is given by 8-1~ O0"ij
0Xi
(m)
+ p bj
(5.25)
-- p iJj (m)
where p is mass density in the current configuration, bj and vj is the material velocity vj =
0r 0t = k / -
tij
is body force per unit mass,
(5.26)
The mass density in the current configuration may be related to the reference configuration (initial) mass density, P0, using the balance of mass principle 8-1~ and yields Po = JP
(5.27)
Thus differences in the equilibrium equation from those of the small deformation case appear only in the body force and inertial force definitions. Similarly, moment equilibrium on a small differential volume element of the deformed solid gives the balance of angular momentum requirement for the Cauchy stress as O'ij -- O'ji
(5.28)
which is identical to the result from the small deformation problem given in Eq. (2.9a). The equilibrium requirements may also be written for the reference configuration using relations between stress measures and the chain rule of differentiation. 8 We will show the form for the balance of linear momentum when discussing the variational form for the problem. Here, however, we comment on the symmetry requirements for stress resulting from angular momentum balance. Using symmetry of the Cauchy stress tensor and Eqs (5.19) and (5.22) leads to the requirement on the first Piola-Kirchhoff stress eil Fjl = Fil Pjl
(5.29)
and subsequently, using Eq. (5.21), to the symmetry of the second Piola-Kirchhoff stress tensor SIj = SjI
(5.30)
Governing equations 133
5.2.4 Boundary conditions As described in Chapter 2 the basic boundary conditions for a continuum body consist of two types: displacement boundary conditions and traction boundary conditions. Boundary conditions generally are defined on each part of the boundary by specifying components with respect to a local coordinate system defined by the orthogonal basis, eit, i -- 1, 2, 3. Often one of the directions, say e 3' coincides with the normal to the surface and the other two are in tangential directions along the surface. At each point on the boundary one (and only one) boundary condition must be specified for all three directions of the basis. These conditions can be all for displacements (fixed surface), all for tractions (stress or free surface), or a combination of displacements and tractions (mixed surface). Displacement boundary conditions may be expressed for a component by requiting !
--I
(5.31a)
xi = xi
at each point on the displacement boundary, %. A quantity with a superposed bar, such as ~; again denotes a specified quantity. The boundary condition may also be expressed in terms of components of the displacement vector, ui. Accordingly, on 7u !
--I
--!
(5.31b)
U i - - U i -JI- gi --
l
where gl is the initial gap ~ i t ( X ' I - X i ) . The second type of boundary condition is a traction boundary condition. Using the orthogonal basis described above, the traction boundary conditions may be given for each component by requiting !
--!
(5.31c)
ti -- ti
at each point on the boundary, 7t. The boundary condition may be non-linear for loading such as pressure loads, as described later in Sec. 5.7. Many other types of boundary conditions can exist and in Chapter 7 we discuss one, namely that of c o n t a c t - t y p e conditions.
5.2.5 Initial conditions Initial conditions describe the state of a body at the start of an analysis. The conditions describe the initial kinematic and stress or strain states with respect to the reference configuration used to define the body. In addition, for constitutive equations with internal variables the initial values of terms which evolve in time must be given (e.g. initial plastic strain). The initial conditions for the kinematic state consist of specifying the position and velocity at some initial time, commonly taken as zero. Accordingly, xi(XI,
O) - - ~ i ( X I ,
0)
or
ui(Xl,
O) -
1.10(Xi)
(5.32a)
and vi(Xi,
O) = ~ i ( X l ,
are specified at each point in the body.
O) -
~3~
(5.32b)
134
Geometricallynon-linear problems- finite deformation The initial conditions for stresses are specified as (5.33)
o i j ( X l , O) = o ~
at each point in the body. Finally, as noted above the internal variables in the stressstrain relations that evolve in time must have their initial conditions set. For a finite elastic model, generally there are no internal variables to be set unless initial stress effects are included.
5.2.6 Constitutive equations - hyperelastic material We shall deal in more detail with constitutive equations for finite deformation materials in Chapter 6. Here we introduce the simplest form for an elastic material which can be used in a finite deformation formulation. A hyperelastic material is one where the stress is determined solely from the current state of deformation as described in Chapter 1, Eq. (1.11). We recall that the form deduces the constitutive behaviour from a stored energy function, W, from which the second Piola-Kirchhoff stress is computed using 8'1~
OW
SIJ = 2 ~
OW
(5.34)
= OEiJ
The simplest representation of the stored energy function is the Saint-Venant-Kirchhoff model given by
W ( E I j ) - - 1 ~CIJKL EIj EKL
(5.35)
where C IjKL are constant elastic moduli defined in a manner similar to the small deformation ones. Equation (5.34) then gives
(5.36)
SIJ = C1JKL EKL
for the stress-strain relation. While this relation is simple it is not adequate to define the behaviour of elastic finite deformation states. It is useful, however, for the case where strains are small but displacements (rotations) are large and we address this use later in Chapter 17. Other models for representing elastic behaviour at large strain are considered in Chapter 6. An alternative to the above exists in which we consider the stored energy expressed in terms of the deformation gradient Fit and deduce the first Piola-Kirchhoff stress from
P i l - OITr
(5.37)
OFii
Now there are nine independent values of stress and deformation that need to be used. The construction of a constitutive relation is also more difficult to construct although one could consider Eq. (5.35) expressed as 1 W ( E I j ) = W ( F i l ) -- ~ C I J K L ( F i l F i j -
1)(FkKFkt.- 1)
(5.38)
Variational description for finite deformation and perform the differentiation to obtain the stress. We will not follow this approach in this volume and leave to the reader the details of computing the derivatives expressed in Eq. (5.38) as well as the subsequent steps in a variational description.
In order to construct finite element approximations for the solution of finite deformation problems it is necessary to write the formulation in a Galerkin (weak) or variational form as illustrated many times previously. Here again we can write these integral forms in either the reference configuration or in the current configuration. The simplest approach is to start from a reference configuration since here integrals are all expressed over domains which do not change during the deformation process and thus are not affected by variation or linearization steps. Later the results can be transformed and written in terms of the deformed configuration. Using the reference configuration form variations and linearizations can be carried out in an identical manner as was done in the small deformation case. Thus, all the steps outlined in Chapter 2 can be extended immediately to the finite deformation problem. We shall discover that the final equations obtained by this approach are very different from those of the small deformation problem. However, after all derivation steps are completed a transformation to expressions integrated over the current configuration will yield a form which is nearly identical to the small deformation problem and thus greatly simplifies the development of the final force and stiffness terms as well as programming steps. To develop a finite element solution to the finite deformation problem we consider first the case of elasticity as a variational problem. Other material behaviour may be considered later by substitution of appropriate constitutive expressions for stress and tangent moduli - identical to the process used in Chapter 4 for the small deformation problem.
5.3.1 Reference configuration formulation A variational theorem for finite elasticity may be written in the reference configuration as10,11
P
FI(U) = / ~
W(CIj)dr2-
l"Iex t
(5.39a)
in which W(CIj) is a stored energy function for a hyperelastic material from which the second Piola-Kirchhoff stress is computed using (5.34).* When we consider material behaviour in this chapter we restrict attention to the Saint-Venant-Kirchhoff model given by Eq. (5.36); however, the results presented here are general and more complicated behaviour may be used as described, for example, in the next chapter. The potential for the external work is here assumed to be given by
I-Iext - f glPob~m) d~ + fr UI TI dF
(5.39b)
t
* The functional in Eq. (5.39a) may also be expressed in terms of the deformation gradient Fit and subsequent steps performed in terms of first Piola-Kirchhoff stress.
135
136
Geometricallynon-linear problems- finite deformation where 7) denotes specified tractions in the reference configuration and F t is the traction boundary surface in the reference configuration. Taking the variation of Eqs (5.39a) and (5.39b) we obtain ~(~CIJ S I j d~
- (~Flext = 0
(5.40a)
and
J~
(~FIex t - -
(~UI pOvt t.(m) dg2 +
fF ~UI Tt dF
(5.40b)
t
where ~UI is a variation of the reference configuration displacement (i.e. a virtual displacement) which is arbitrary except at the kinematic boundary condition locations, Fu, where, for convenience, it vanishes. Since a virtual displacement is an arbitrary function, satisfaction of the variational equation implies satisfaction of the balance of linear momentum at each point in the body as well as the traction boundary conditions. We note that by using Eq. (5.34) and constructing the variation of CIj, the first term in the integrand of Eq. (5.40a) can be expressed in alternate forms as l2 ( ~ c t j s I j
= (~EIjSIj
(5.41)
-- (~FiI F i j S I j
where symmetry of SIj has been used. The variation of the deformation gradient may be expressed directly in terms of the current configuration displacement as ~Fil --
O(~Ui
OX~
(5.42)
= (~ui, I
Using the above results, after integration by parts using Green's theorem, the variational equation may be written as (~l-I -- -- ~
(~Ui [ ( F i J a I j ) , I
-JI- ~iiPob~ m)] dS2 +
~
dl
(~Ui
[FijSij gi - (~iI~?I]dF - 0
t
(5.43) giving the Euler equations of (static) equilibrium in the reference configuration as (FijSIj),1%-
~ilPob~ m) = PiI, I ~- pob~ m)
= 0
(5.44)
and the reference configuration traction boundary condition
SIIFilNI
-- (SiITI --
PiINI
-- (~iITI -- 0
(5.45)
The variational equation (5.40a) is identical to a Galerkin method and, thus, can be used directly to formulate problems with constitutive models different from the hyperelastic behaviour above. In addition, direct use of the variational term (5.40b) permits non-conservative loading forms, such as follower forces or pressures, to be introduced. We shall address such extensions in Sec. 5.7.
Matrix form
At this point we can again introduce matrix notation to represent the stress, strain, and variation of strain. For three-dimensional problems we define the matrix for the second Piola-Kirchhoff stress as
S--[Sll,
$22, $33, S12, $23, $31]T
(5.46a)
Variational description for finite deformation
and the Green strain as E-JEll,
E22, E33, 2E12, 2E23, 2E31]r
(5.46b)
where, similar to the small strain problem, the shearing components are doubled to permit the reduction to six components. The variation of the Green strain is similarly given by 6 E = [~EI,,
~E22, ~E33, 25E,2,
2~E23, 2~E3,] r
(5.46c)
which permits Eq. (5.41) to be written as the matrix relation
(~Elj glj --- oeETS
(5.47)
The variation of the Green strain is deduced from Eqs (5.13), (5.14) and (5.42) and written as
1 0 ( ~OuX ~ E I j - - -~
Ii Fij --~ - ~ j Ell
= -~ ((~ui, l Fij --]-(~ui, J Ell )
(5.48)
Substitution of Eq. (5.48) into Eq. (5.46c) we obtain
F/l~Uz,1 Fiz(~Ui,2 0-E --
F/3(~ui'3
Fil(~Ui, 2 -~ Fi2t~ui,11 Fi2(~ui,3 -~ Fia(~ui,2 ~,Fi3(~ui,1 + Fil(~Ui,3
(5.49)
as the matrix form of the variation of the Green strain.
Finite element approximation
Using the isoparametric form developed in Chapter 2 and Appendix A (see also Chapters 4 and 5 of reference 1) we represent the reference configuration coordinates as
X i -- ~
ga ( ~ ) 2 ;
(5.50a)
a
where ~ are the three-dimensional natural coordinates ~1, ~2, ~3, Na are standard shape functions (see Appendix A or Chapters 4 and 5 of reference 1), and symbols a, b, c, etc. are introduced to identify uniquely the finite element nodal values from other indices. Similarly, we can approximate the displacement field in each element by
Ui -- ~
a
ga(~)bl a
(5.50b)
The reference system derivatives are constructed in an identical manner to that described in Chapter 2. Thus, Ui, I -- Na,ltt a (5.51) where explicit writing of the sum is omitted and summation convention for a is again invoked. The derivatives of the shape functions can be established by using standard routines to which the ~'~ coordinates of nodes attached to each element are supplied.
137
138 Geometricallynon-linear problems- finite deformation The deformation gradient and Green strain may now be computed with use of Eqs (5.11) and (5.15), respectively. Finally, the variation of the Green strain is given in matrix form as
" ~E
~"
--
FllNa,1
F21Na,1
F31Na,1
F12Na,2 F22Na,2 F32Na,2 F13Na,3 F23Na,3 F33Na,3 FllNa,2 -+- F12Na,1 F21Na,2 --I- F22Na,1 F31Na,2 -+- F32Na,1 FlzNa,3 -+- F13Na,2 F22Na,3 -at- F23Na,2 F32Na,3 -Jr-F33Na,2 F13Na,1 + FllNa,3 Fz3Na,1 -at- FzlNa,3 F33Na,1 -Jr"F31Na,3
Ba(~l
a
(5.52) (5.53)
where ]~a replaces the form previously defined Ba for the small deformation problem. Expressing the deformation gradient in terms of displacements it is also possible to split this matrix into two parts as ]~a - - B a "k-
BaNL
(5.54)
in which Ba is identical to the small deformation strain-displacement matrix and the remaining non-linear part is given by
" BauL =
Ul,lNa, 1 u2,1Na,l u3,1Na,1 Ul,2Na,2 u2,2Na,2 u3,2Na,2 Ul,3Na,3 Uz,3Na,3 u3,3Na,3 Ul,lNa,2 --I-Ul,2Na,1 u2,1Na,2 -+-u2,2Na,1 u3,1Na,2 -Jr"u3,2Na,1 Ul,2Na,3 "-JI-Ul,3Na,2 u2,2Na,3 .-I-u2,3Na,2 u3,2Na,3 -Jr u3,3Na,2 Ul,3Na,1 --~ Ul,lNa,3 u2,3Na,1 + u2,1Na,3 u3,3Na,1 +/~3,1Na,3
(5.55)
It is immediately evident that BauL is zero in the reference configuration and therefore that Ba ~ Ba. We note, however, that in general no advantage results from this split over the single term expression given in Eq. (5.52). The variational equation may now be written for the finite element problem by substituting Eqs (5.46a) and (5.52) into Eq. (5.40a) to obtain
,1-1-- (~fla)T ( f BTSd~ - fa) -- O where the external forces are determined from
(5.56)
~l-lex t a s
fa - f NaPob(m)d~ W fr N~TdF
(5.57)
t
with b (m) and ~i' the matrix form of the body and traction force vectors, respectively.
Transient problems
Using the d'Alembert principle we can introduce inertial forces through the body force as b (m) ~
b (m) - - ~, - - b (m) -- ~
(5.58)
Variational description for finite deformation where v is the material velocity vector defined in Eq. (5.26). Inserting Eq. (5.58) into Eq. (5.57) gives
fa ~ fa -- ./o Nap~ df2Vb
(5.59)
This adds an inertial term MabCebto the variational equation where the mass matrix is given in the reference configuration by
Mab -" f~ NaPoNbdf2l
(5.60)
For the transient problem we can introduce a Newton-type solution and replace Eq. (2.20a) by / ' 6
~I'1 = f - ./o l]~S dr2 - Mi, - 0
(5.61)
Applying the linearization process defined in Eq. (3.8) to Eq. (5.61) (without the inertia force)* we obtain Kr du (k) = ~Pl(k) (5.62a) with updates U (k+l) = U (k) +
du (k)
(5.62b)
Iteration continues until convergence is achieved with the process being identical to that introduced in Chapter 2. The tangent term is given by (omitting the iteration superscript) Kr =
f~ l~rI)rl~ dE2 + f --~-S Oflr d ~ o~ Of "- KM +
KG + KL
(5.63)
where the first term is the material tangent, KM, in which I)r is the matrix of tangent moduli. For the hyperelastic material we have OS i j 2 ~
OCKL
= 4
02 W
=
Oz W
OCIJOCKL OEIJOEKL
= CIJKL
(5.64)
which may be transformed to a matrix I)r as described in Chapter 2. The second term, Ko, defines a tangent term arising from the non-linear form of the strain-displacement equations and is often called the geometric stiffness. The derivation of this term is most easily constructed from the indicial form written as
( f~ O~EIJ Sij d~"~)dll~--(~l a ( f~ g a l(~ijgb j Slj d~'2)d~.l~--r a (g~b)G d~.g~ 0~
'
'
(5.65) Thus, the geometric part of the tangent matrix is given by
K~b - GabI * Extension to transient applications follows directly from the presentation given in Chapter 2.
(5.66a)
139
140 Geometricallynon-linear problems- finite deformation where
P
Gab -- ]o~ Na,I SIj Nb,j dr2
(5.66b)
The last term in Eq. (5.63) is the tangent relating to loading which changes with deformation (e.g. follower forces, etc.). We assume for the present that the derivative of the force term f is zero so that KL vanishes. In Sec. 5.7 we will consider a follower pressure loading which does give a non-zero KL tangent term.
5.3.2 Current configuration formulation ~
:
:
:
:
................................................................................................................................................................................................................
---
::: .............................................................................................................
The form of the equations related to the reference configuration presented in the previous section follows from straightforward application of the variational procedures and finite element approximation methods introduced previously in Chapter 2. However, the form of the resulting equations leads to much more complicated strain-displacement matrices, 1~, than previously encountered. To implement such a form it is thus necessary to reprogram completely all the element routines. We will now show that if the equations given above are transformed to the current configuration a much simpler process results. The transformations to the current configuration are made in two steps. In the first step we replace reference configuration terms by quantifies related to the current configuration (e.g. we use Cauchy or Kirchhoff stress). In the second step we convert integrals over the undeformed body to ones in the current configuration.* To transform from quantifies in the reference configuration to ones in the current configuration we use the chain rule for differentiation to write C~(" )
=
c~X I
C~(" ) ~X i c~xi C~Xl
=
C~(" )
~ F / l
c~xi
(5.67)
Using this relationship Eq. (5.48) may be transformed to ~Ez.l = ~1 ((SUi,j 7I- (~Uj,i) Fi I F j j = (5EijFilFjj
(5.68)
where we have noted that the variation term is identical to the variation of the small deformation strain-displacement relations by again using the notation t (~Cij --" ~1 ((~Ui,j .3t_(~Uj,i )
(5.69)
Equation (5.41) may now be written as (~Eil Slj i (~Eij Fil Fij Slj __ (~___CijTji__ (~Cijff ij J
and Eq. (5.40a) as
(5.70)
#
(~FI "-- ./o (Scijo'ij Jdff2 -
(~l"Iext "-- 0
(5.71)
* This latter step need not be done to obtain advantage of the current configuration form of the integrand. t We note that in finite deformation there is no meaning to ~ij itself; only its variation, increment, or rate can appear in expressions.
Variational description for finite deformation The second step is now performed easily by noting the transformation of the volume element given in Eq. (5.10) to obtain finally
(~II -- fw (~cij~
-
(~I'Iext -
0
(5.72a)
where w is the domain in the current configuration. The external potential I'Iex t given in Eq. (5.40b) may also be transformed to the current configuration using Eqs (5.24) and (5.27) to obtain
~l"lext -- /w ~uipb~m) dw + f~, ~uiTi d~
(5.72b)
The computation of the tangent matrix can similarly be transformed to the current configuration. The first term given in Eq. (5.63) is deduced from
~ (~EIJCIJKLdEKLdK2 -- f~ (~eijFil FjjCIJKL FkKFILdEkl dr2 (5.73)
-- fw ~'--CijCijkldCkldw where
JCijkl -- Fil Fjj FkK FILCIJKL
(5.74)
defines the moduli in the current configuration in terms of quantities in the reference state. Finally, the geometric stiffness term in Eq. (5.63) may be written in the current configuration by transforn~ng Eq. (5.66b) to obtain
aab = f Na,lSljNb, jd~-- fwNa,iO'ijNb,jdw
(5.75)
Thus, we obtain a form for the finite deformation problem which is identical to that of the small deformation problem except that a geometric stiffness term is added and integrals and derivatives are to be computed in the deformed configuration. Of course, another difference is the form of the constitutive equations which need to be given in an admissible finite deformation form.
Finite element formulation
The form of the variational problem in the current configuration is easily implemented as a finite element solution process. To obtain the shape functions and their derivatives it is necessary first to obtain the deformed Cartesian coordinates xi by using Eq. (5.4). After this step standard shape function routines can be used to compute the derivatives of shape functions, ONa/OXi.The terms in the variational equation can then be expressed in a form which is identical to that of the small deformation problem. Accordingly, the stress term is written as
fW (~EijO'ijdw -- 6fiT f~ B ~erdw
(5.76)
141
142 Geometricallynon-linear problems- finite deformation where B is identical to the form of the small deformation strain-displacement matrix, and Cauchy stress is transformed to matrix form as or--fill,
0"22,
0"33,
O"12,
0"23,
O"31
(5.77)
and involves only six independent components. The residual for the static problem of a Newton solution process is now given by ~I/1 =
f - f~ B r or dw = 0
(5.78)
The linearization step of the Newton solution process is performed by computing the tangent stiffness in matrix form. Transfomfing Eq. (5.73) to matrix form using the relations defined in Chapter 2, the material tangent is given by KM = fw BrDrB dw
(5.79)
where now the material moduli Dr are deduced by transforming the cijil moduli in the current configuration to matrix form. The form for Gabin Eq. (5.75) may be substituted into Eq. (5.66a) to obtain the geometric tangent stiffness matrix. Thus, the total tangent matrix for the steady-state problem in the current configuration is given by K~b = fw BarDrBb dw +
Gabl
(5.80)
and a Newton iterate consists in solving
KTdfl =
f - ~ B T or dw
(5.81)
where the external force is obtained from Eq. (5.72b) as
fa = fw Napb(m)dw + fT Nai d7
(5.82)
We can also transform the inertial force to a current configuration form by substituting Eqs (5.10) and (5.27) into Eq. (5.60) to obtain
Mab = f NaPoNbdQI = fw NapNb dwI
(5.83)
and thus, for the transient problem, the residual becomes III 1 - -
f - f~ B r or dw - Mv - 0
(5.84)
Linearization of this term is identical to the small deformation problem and is not given here. The development of displacement-basedfinite element models for three-dimensional problems may be performed easily merely by adding a few modifications to a standard linear form. These modifications include the following steps.
Two-dimensional forms 143
1. Use current configuration coordinates X i to compute shape functions and their derivatives. These are computed at nodes by adding current values of displacements fia to reference configuration nodal coordinates X~. 2. Add a geometric stiffness matrix to the usual stiffness matrix as indicated in Eq. (5.80). 3. Use the appropriate material constitution for a finite deformation model. 4. Solve the problem by means of an appropriate strategy for non-linear problems. It should be noted that the presence of the geometric stiffness and non-linear material behaviour may result in a tangent matrix which is no longer always positive definite (indeed, we shall discuss stability problems in Chapter 17 and this is a class of problems for which the tangent matrix can become singular as a result of the geometric stiffness term alone). Furthermore, use of displacement-based elements in finite deformation can lead to locking if the material has internal constraints, such as in nearly incompressible behaviour. It is then necessary again to resort to a mixed formulation to avoid such locking. The advantage of a properly constructed mixed form is that it may be used with equal accuracy for both the nearly incompressible problem as well as any compressible problem. In Sec. 5.5 we consider a mixed form which is a generalization to finite deformation of the one presented in Sec. 2.6 for small deformation problems.
i~i.M.si.i.~:~:~: ~: ~.;i.i~.is~z~.i~i.~.~.~.i.~.~: ~:~: ~: ~:.~i.T~i.i.iM~M..;!~ii.M~F@. i.MTiM.i~iiT.i!.iMi .i.i!i:i~i:~i.~:i~i.!.i:~i:i:i:i~:ii::~' i.i~i:i:~M.~::isTiiiiiii:iiiiiiiiiiiiiiiiii i !~i.!~i.!~iTi.~i!.i.i ........................... ii. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .i..`~i . . .`..~..~.i . . . . ..~i. ..i`..~i. . ~. .i.i.i..i i~i.i.i.i~i i i i i .i.i.i.i.i.i.i.i~i i i i i i i i i i.i i i i~i.i i i i i i i i i i i i i @i i i i i i i ~i.i i.i.i~/i.i.i~i.i.i ~~,i~,i/.~i.i~i.i@/i.~.i.i.i.i.i.i.i.i.i .i.i.i.~/i~i.~.~/
The three-dimensional form may be reduced to a two-dimensional form if loading, geometry, and material behaviour do not vary with a third coordinate. As in the linear theory presented in Chapter 1 we have three cases: plane strain, plane stress, and axisymmetric behaviour. A variational form is an invariant statement for a class of problems. Accordingly, it admits introduction of the basic quantities in the different coordinate frames and dimensions to define the above class of problems for finite deformation applications.
5.4.1 Plane strain The reduction to the two-dimensional form for plane strain is made by reducing the deformation gradient to the form
Fil
--
F21
F22
0
0
!]
=
u2,1
0
u12
(1 + U2,2) 0
!]
(5.85)
in which the displacements U 1 and u2 are functions of X1, X 2 only. A current configuration formulation for the finite element problem then follows from Sec. 5.3.2 by restricting the range of indices to two instead of three. Accordingly, the straindisplacement matrix is again identical to that of the small deformation problem given by Eq. (2.9b). Material constitution is specified by the three-dimensional model in which strains in the third direction (normal to the plane of deformation which is here taken as the 12-plane) are set to zero. This gives the same Dr merely restricted to
144
Geometrically non-linear problems - finite deformation
the terms in the two-dimensional problem (the upper 4 x 4 part for the ordering given above). Introducing a finite element approximation for u~ and uz in terms of shape functions Nb gives the strain-displacement matrix 0 Nb,x2
Bb =
0 0
Nb,x, ]
(5.86)
for use in the current configuration form. The differential volume for the plane strain problem is given by d~-
J dX1 dX2 with J - Fll F22- F12 F21
5.4.2 Plane stress To consider plane stress we need to account for the change in thickness and this may be included by taking the deformation gradient in the form (5.87)
0
F33
0
0
F33
Here the value for F33 is obtained from material constitution in which the stresses in the direction normal to the plane of deformation are zero. Namely, S31 = o3i ~-
0 for I, i -- 1, 2, 3
The resulting problem then gives the same Bb array as for plane strain but modified D~ tangent matrix obtained when satisfying the zero stress condition. We will discuss this aspect when we consider constitutive behaviour in Chapter 6, Sec. 6.2.4. The differential volume for the plane stress problem is given by dco =/-/3 J dX1 dX2 with J = (FI1
F22 --
F12/721) F33
Here/-/3 is the thickness of the slice in the reference configuration.
5.4.3 Axisymmetric with torsion For the full axisymmetric problem where coordinates are given by R = X~, Z = X2, and | in the reference configuration and r - x~, z = x2, and 0 in the current configuration, the deformation gradient is given as 12'13
Fi l "-"
F,R
F,Z
Z ,R
Z,Z
r O,R
r O,z
o]
=
r/R
=
Ur, z
0 ]
Uz,R r O,R
(1 + Uz,z) r 0,z
0 (1 + u r / R )
(1 + ul,1) U2,1
(1 + U2,2)
Ul,2
0
Xl (~,1
Xl r
0
(1 q- U l / X 1 )
(5.88) 1
A three-field, mixed finite deformation formulation
where the displacements are given by r
--
= X1 + Ul (X1, X2)
Xl
z - x2 - X2 + uz(X1, X2) 0- O+ r
(5.89)
X:)
Introduction of a finite element approximation for u 1, u2, and 0 and transformation to the current configuration gives the strain--displacement matrix
[Nb,x, Bb-- [ 0
0 Nb/Xl Nb,x2 0 0
0
Nb,x~ Nb,x,
0 0
0
(Xl Nb,x:,)
(Xl
0 1 0 Nb,x,)
J
(5.90)
Here the material constitution is identical to the three-dimensional problem. The differential volume for the axisymmetric problem is given by dw = 27rXl J dX1 dX2 with J -- (Ell F22 - F12 F21) F33
!!iiiiiii~iii~!!!@~i!~i~liBiiiiiiiii~!i~iiiiiiii~iiiiis~si!i8iiiiiiii!!i~i!~iii~iis!iiiiii~ii!iiii!ii~iiiii~!i~iiiiii!iii@~iiiii!!!iiii!8iii!i~!!~@ii!ii!i~iii~iiiii~iiiiiiiiiiiiiii~ii!iiiiiii!iiiiii~i!i~i@iii~i~i~i~iii~
A three-field, mixed variational form for the finite deformation hyperelastic problem is given by I-l(u, p, O) = / ~
[W(CIj) + p (J - 0)] d~2 - flext
(5.91)
where p is a pressure in the current (deformed) configuration, J is the determinant of the deformation gradient Fit, 0 is the volume in the current configuration for a unit volume in the reference state, W is the stored energy function expressed in terms of a (modified) fight Cauchy-Green deformation tensor CI], and l-lex t is the functional for the body loading and boundary terms given in Eq. (5.39b). The (modified) fight Green deformation tensor is expressed as CIj = FilPiJ
where Fi! -- f i j f J l
= (O1/3(~ij)(j-1/3fjl) -
(0)1,3 7
Nil
(5.92a)
(5.92b)
in which Fi~ is a volumetric and Ff~ a deviatoric part. We note that det F~ = 1 as required for a deviatoric (constant volume) state. This form of the variational problem has been used for problems formulated in principal stretches. 14 Here we use the form without referring to the specific structure of the stored energy function. In particular we wish to admit constitutive forms in which the volumetric and deviatoric parts are not split as in reference 14. The variation of Eq. (5.91) is given by (~l-I
--
-~(~CIJSIJ + 6p (J - 0) + (6J - 60) p dr2
- ~l"Iext
(5.93)
145
146 Geometricallynon-linear problems- finite deformation where a second Piola-Kirchhoff stress based on the modified deformation tensor is defined as OW OW __ (5.94) SlJ "--20~__,1J -- OF--,Ij with EIj "- 21 (~ IJ - (~IJ) Using Eq. (5.92a) the variation of the modified deformation tensor is given by (5.95)
(~CIj ~ ~Pil Fij + ~PiJ Fil in which
*Pil --"
(~) 1/3[
1 ('/9
*Fil "+" 5 Fil
t~J)]
0
J
Thus, upon noting that 9'1~
(~J = J ff-j I ~ f jj the first term of the integrand in Eq. (5.93) formally may be expanded as
6C-,IjSIj -- ~ P i l P i j S I j 1
(\~0
~J
=g-d
J
P i l P i j ~ l J -+-
(0) 1,3
(~FilPiS~l J
This expression again may be simplified by defining current configuration Kirchhoff and Cauchy stresses based on the modified deformation gradient as ~'ij = F i I S I j P j J
(5.97)
= O~ij
The definitions introduced for stress are consistent with using standard constitutive models in which the modified deformation tensor is used to compute stresses and material moduli. That is, we need not distinguish whether a standard displacement method or the three-field mixed model as given here is used. Also, we note from Eq. (5.67) that 6Fjj F~ 1
--
~SUj,kFkj F ~ 1
=
6Uj,k6kj
=
6u j, i
(5.98)
is the divergence of the variation in displacement. Thus, Eq. (5.96) simplifies to 3 C I j S- I j - 31 ( ~
1~0 O t ~iu i ( 7i - j - 3 3 i)j ~rkk - 3 u j ,j ) ~rii + 3 u ~,j ~r~~ - -~ --~ ~rkk + -~X
(5.99)
Substituting relations deduced above into Eq. (5.93) and noting symmetry of the Kirchhoff stress, a formulation in terms of quantities related to the deformed position may be written as ,
~0 (p - p) dg2
t~I-[ -'- ~ t~Eij [ o'ij -+- t~ij ( -~ p f + Jn ~p (J - 0) dQ
(5.100) -
t~l"lex t -"
0
q
where ~ei) is given by Eq. (5.69) and p = ~rii/3 defines a mean stress based on the Cauchy stress deduced according to Eq. (5.97). This variational equation may be transformed to integrals over the current configuration by replacing d~2 by doJ/J; however, this step is not an essential transformation.
A three-field, mixed finite deformation formulation
Finite element equations: matrix notation
The mixed method finite element approximation of the three-field variational form is expressed using deformation measures and stresses related to the current configuration. The development is very similar to that presented in Chapter 2 for the small deformation case. The reference coordinate and displacement fields are approximated by isoparametric interpolations as indicated in Eqs (5.50a) and (5.50b), respectively. These are used to compute the deformation gradient by means of Eqs (5.11) and (5.51). The pressure and volume are interpolated in a manner which is identical to the small deformation case as p - Np~ and 0 = N00 and for quadrilateral and brick elements are taken to be discontinuous between elements. Using the above approximation, Eq. (5.100) may be expressed in matrix form as ~H -- ~ r
B r ~r0 dr2 + ~ r
L
Np (J - 0) dr2 (5.101)
+ ~Or f a Nr (p - p) dr2 - ~Flext In this form of the finite deformation problem B again is identical to the small deformation strain--displacement matrix with a modified stress defined as 6, = s + (t5 - / 5 ) m
where
J /~ - ~ p
(5.102)
We note that inertia effects may again be included as described for the displacement model and the final result yields the discrete form of Eq. (5.101) given by
P+M~=f P. - Ko.~ = 0
(5.103a)
- K . o O + Ep = 0 where the arrays are given as
p -- f~ Br~ Kop --
f~ N~Np dr2 = Kpo
Pp=
1 ~nN~'OdQ (5.103b)
Ep =
Np J d~2
and force f and mass M are identical to the terms appearing in the displacement model presented previously. We can observe that the mixed model reduces to the displacement form if 0 = J and p - p at every point in the element. This would occur if our approximations for 0 and p contained all the terms appearing in results computed from the displacement-based deformations and, thus, again establishes the principle of limitation. 15 Moreover, if this occurred, any locking tendency in the displacement form would again occur in the mixed approach also.
147
148
Geometricallynon-linear problems- finite deformation To obtain a formulation free of locking it is again necessary to select approximations for pressure and volume which satisfy the mixed patch test count conditions as described in Chapters 10 and 11 of reference 1. Here, to approximate p and 0 in each element we assume that No = N p and for four-node quadrilateral and eight-node brick elements of linear order use constant (unit) interpolation. In nine-node quadrilateral and 27-node brick elements of quadratic order we assume linear interpolation. Linear interpolation in (1, ~2, ~3 or X1, X2, X3 can be used; however, Xl, x2, x3 should not be used since the interpolation becomes non-linear (since xi depend on ug) and the solution complexity is greatly increased from that indicated above. The second and third expressions in Eqs (5.103a) are linear in ~ and 0, respectively, and also are completely formed within a single element. Moreover, the coefficient matrix Kpo - Kop is symmetric positive definite when No - N p. Thus, a partial solution can be achieved in each element as = Ko-plpp
(5.104)
0 = KpTE p
An explicit method in time may be employed to solve the momentum equation: as was indeed used to solve examples shown at the end of Chapter 2. However, here we only consider further an implicit scheme which is applicable to either transient or static problems (see Chapters 2 and 3). A Newton scheme may be employed to solve Eq. (5.101). To construct the tangent matrix KT it is necessary to linearize Eq. (5.93). In indicial form, the Newton linearization may be assembled as d (31-I)
=
[(~CIjCIJKLdCKL+ ~d
j (5.105)
+ f 'p (dJ - dO) df2 + f dp ('J - 'O) d~ + d (6I'Iext) where dC"KL, dp, etc., denote incremental quantities and material tangent moduli are denoted by
OSij
(5.106)
2 0~_,KL -- CIjKL
The above integrals may also be expressed in quantities terms or current configuration terms in an identical manner as for the displacement model presented in Sec. 5.3.2. In this case the reference configuration moduli are transformed to the current configuration using 1C--'ijkl = ~ Fil F j j FkK FILCIJKL (5.107) Using standard transformations from indicial to matrix form the moduli for the current configuration may be written in matrix form as Dr. We can now write Eq. (5.105) in matrix form and obtain the set of equations which determine the parameters d~, dO and d~ as
Kou Koo Kpu -Kpo
-Kop 0
dO
d~
=
O
0
(5.108)
A three-field, mixed finite deformation formulation
where
Kuu -- [0 BTI)llB0dQ + Ko,
Kuo = ~ Brf)12 [~No] Ods = Ko~
Kup - fa BrmNpJ d~ - Kpr,
K o o - - / n [~N~'] /)22 [~No] Odf2
in which
IT}ll -- ldf)rld - 52 ( m & ~ + & a m r ) + 2 ( p - p ) I o -
(2) p-/5
mm r
2
I)12 -- ~ldDrm + ~tYd = I)T1 1
-
D22 -- ~m D r m -
1
T-
1
~p
with I0 defined by Eq. (2.61). Note also that the fight-hand side is zero in the second and third rows of Eq. (5.108) since the solution for pressure and volume parameters was determined exactly using Eq. (5.104). The geometric tangent term is given by
K~ b -- GabI where Gab -- f~ Na,iffijNb,j dr2
(5.109)
A solution to Eq. (5.108) may be formed by solving the third and second rows as
dO = K;/K ud d~
Ko-lpKoudfi + Ko-p1KoodO = (KoplK0u + KoplKooKp~Kpu)dfi
-
(5.110)
and substituting the result into the first row to obtain
Krdfi = [Kuu + KuoKp#Kp. + KupKoplKou + KupKolpKooKplKpu]dfi - f - P
(5.111) This result is obtained by inverting only the symmetric positive definite matrix Kpo, which we also note is independent of any specific constitutive model. Alternatively, if we define a mixed volumetric strain-displacement matrix as 1
B~ = ~NoW where W =
KpTKpu
the tangent matrix may be computed directly from Kr - [ [BTI)ll B + BTI)12Bv + BvTD21B + Bor [)22By] 0 d~2 + Ko J~2 --
E.
.:l
(5.112)
In this form the finite deformation formulation is similar to that developed in Sec. 2.5 for the small strain case.
149
150
Geometrically non-linear problems - finite deformation
An alternative finite element method to that just discussed is the fully mixed method in which strain approximations are enhanced. The key idea of the mixed-enhanced formulation is the parameterization of the deformation gradient in terms of a mixed and an enhanced deformation gradient from which a consistent formulation is derived. This methodology allows for a formulation which has standard-order quadrature and variational recoverable stresses, hence circumventing difficulties which arise in earlier enhanced strain methods. 2-4'~6-~8 Recently, an improved form for the enhanced strain hexahedral element has been proposed by Areias et al. 5 While the element has improved properties over that given by Simo e t al. 3 we prefer the approach presented here as a procedure to develop general element types. There is no need to separate any deformation gradient terms into deviatoric and mean parts as was necessary for the three-field approach discussed in the previous section. The mixed-enhanced formulation discussed here uses a three-field variational form for finite deformation hyperelasticity expressed as
H --- f[W(Fil ) + Pil(Fil-
~"iI)] d ~ 2 - Hex t
(5.113)
where Fit is the deformation gradient computed from displacements, F il is an independent deformation gradient, P iI is an independent Piola-Kirchhoff stress, W is an objective stored energy function in terms of P~I, and Hext is the loading term given by Eq. (5.39b). The stationary point of I-I is obtained by setting to zero the first variation of Eq. (5.113) with respect to the three independent fields. Accordingly,
(~II ~ f~ I(~FiIeilJl-(~PiI ( C~Wc~Pi I 19iI) ~-(~ei' (Fil - Fil)] d~'~-(~Hext- 0 (5.114) where P~t and Pit are mixed variables to be approximated directly. The reader will note that we now use the deformation gradient directly instead of the usual CI J, Et j, or bij symmetric forms. We will often use constitutive models which are expressed in these symmetric quantifies; however, we note that they are also implicitly functions of the deformation gradient through the definitions given in Sec. 5.2. Once again, at this point we may substitute a first Piola-Kirchhoff stress from any constitutive model in place of the derivative of the stored energy function OW/OPi1 in Eq. (5.114). Thus, the present form can be used in a general context. Finite element approximations to the mixed deformation gradient and the first PiolaKirchhoff stress are constructed directly in terms of local coordinates of the parent element using standard transformation concepts. Accordingly, we take (5.115a) and
[~)iI-- FTz/~-~1j-~lT)o~/~(~)
(5.115b)
A mixed-enhanced finite deformation formulation
where ~ denotes the natural coordinates ~1, so2, ~3, Greek subscripts are associated with the natural coordinates, and 79~ and ) r are the first Piola-Kirchhoff stress and deformation gradient approximations in the isoparametric (parent) coordinate space, respectively.* The arrays Jc~a and F'il used above are average quantities over the element volume, f2e. The average quantity J aa is defined as
Jo~A d~
J o~A = X
(5.116a)
e
where Jo~A is the standard Jacobian matrix as defined in Eq. (2.6b) (but now written for the reference coordinates), and Fil is defined as f i l = -~e
Fi l d f2
(5.116b)
e
The above form of approximation will ensure direct inclusion of constant states as well as minimize the order of quadrature needed to evaluate the finite element arrays and eliminate some sensitivity associated with initially distorted elements. The forms given in Eqs (5.115a) and (5.115b) are constructed so that the energy term of the physical and isoparametric pairs are equal. Accordingly, we observe that 79,~.T',~ -
Pil Fii
(5.117)
This greatly simplifies the integrations needed to construct the terms in Eq. (5.114). To construct the approximations we note that the tensor transformations for the mixed deformation gradient may be written in matrix form as -- A.9="
(5.118a)
and P = A-1T ~
(5.118b)
where A is a transformation to matrix form of the fourth-rank tensor given as
Aila3 -- ff'iA Jo~A J/31
(5.119)
The ordering for the matrix-tensor transformation for all the variables is described in Table 5.1. The approximation for the mixed deformation gradient may now be written as 1
~-, .._ ,r}/o+ - A [!~',1(~),r~/,--I-!~,2(~)t~] J
(5.120a)
and for the mixed stress as _
+ A-I
5.120b)
where j = det JaA and !~1, 1~2 define the functions to be selected in terms of natural coordinates. The functions suggested in reference 19 are given in Table 5.2. The terms /3~ and 7 0 ensure that constant stress and strain are available in the element. * Note the resulting transformed arrays are objective under a superposed rigid body motion, l0
151
152 Geometrically non-linear problems - finite deformation
Table 5.1 Matrix-tensor transformation for the nine-component form Row or column
1
2
3
4
5
6
7
8
9
iora I or/3
1 1
2 2
3 3
1 2
2 3
3 1
2 1
3 2
1 3
Table 5.2 Three-dimensional interpolations O~ /3
~l"?'
]~2Or
l 2 3
1 2 3
~3"Yl0
0
2 3 2 3
3 1 1 2
~1")/12 ~2"3/14 ~3711 ~1")/13
0 0 0 0
3
(2"Y15
0
1
1
2
~271 + ~3")/2 + ~2~3"Y3 ~10LI+ ~1~2 OL2q- ~1~3 OL3 ~''~4 + ~3'ff5 + ~1 ~3"Y6 ~2 OL4+ ~2~3 OL5+ ~1~2 OL6 ~1")/7+ ~2")/8 + ~1 ~2")/9 ~3OL7+ ~2~30s "Jr"~1~3OL9
The above construction is similar to that used to construct the Pian-Sumihara plane elastic element 2~ (see also Sec. 10.4.4 of reference 1). The enhanced parameters cz are added to the normal strains in Table 5.2 such that the resulting strain components are complete polynomials in natural coordinates. This is done to provide the necessary equations to enforce a near incompressibility constraint without loss of rank in the resulting finite element arrays. In addition, the enhanced parameters improve coarse mesh accuracy in bending dominated regimes.
Finite element equations: matrix notation
By isolating the equations associated with the variation of the first Piola-Kirchhoff stress tensor 6P in Eq. (5.114) some of the element parameters of the mixed-enhanced deformation gradient F may be obtained as
1 fn FdQ =~, and --
(/o)'L E r E 1 dF-1
l~fA -1 ( F - ~') dr2
(5.121a)
(5.121b)
e
where the box denotes integration over the element region defined by the isoparametric coordinates ~i. We note that the construction for I~1 and !~2 is such that integrals have the property j ; El d[-] - / D
E2 d[--] - / D
l ~ ' E 2 d[--] = 0
This greatly simplifies the construction of the partial solution given above. Use of the above definitions for ~" and P also makes the second term in the integrand of Eq. (5.113) zero, hence the modified functional 1~Iis expressed as
FI -- f~ W(Fil) dQ - I'Iext
(5.122)
A mixed-enhanced finite deformation formulation ^
The stationary condition of 1-I yields a reduced set of non-linear equations, in terms of the nodal displacements, ~, and the enhanced parameters, &, expressed as
O ~ i l a F i l d e - alIext - {(~a T
(~&T}
Pint(U, ) Penh(fi,~&)
f
= 0 (5.123)
where Pint is the internal force vector, Penh is the enhanced force vector, and f is the usual force vector computed from I'Iex t. Noting that the variations 3fi and 3& in Eq. (5.123) are arbitrary the finite element residual vectors are given by kI/u = f -- Pint(U, ~ ) = 0 klJc~ -- --Penh(U, &) = 0
(5.124)
A solution to these equations may now be constructed in the standard manner discussed in Chapter 2. Using a Newton scheme to linearize Eq. (5.123) we obtain
d ((~[l ) - fa (~lT i l o ~.,i~l O~.,j j dPjj + ~ow d((~'i I )dr2
R~j { d a
(5.125)
where I(u,, etc., are obtained by evaluating all the terms in the integrals in a standard manner, and the process is by now so familiar to the reader we leave it as an exercise. Using Eqs (5.124)-(5.125) we obtain the system of equations
Ru,]
on
(5.126)
where we note that the parameters & are associated with individual elements. We can use static condensation to perform a partial solution at the element level. E1 Here the situation is slightly different since the equations are non-linear. Thus, it is necessary to use the static condensation process in an iterative manner. Accordingly, given a solution a for some iterate in a Newton process we can isolate the part for each & and consider a local solution for the equation set d&(k) -- K,~,~ ^ -1 ~ (k)
(5.127a)
Iteration continues until ~I,~ is zero with updates (5.127b) performed on each element separately. Utilizing the final solution from Eq. (5.127a) an equivalent displacement model involving only the nodal displacement parameters is obtained as
where
153
154 Geometrically non-linear problems- finite deformation
The system of Eq. (5.128) is solved and the nodal displacements are updated in the usual manner for any displacement problem (see Chapter 2). Additional details and many example solutions using the above formulation, and its specialization to small deformations, may be found in references 19 and 22.
In the derivations presented in the previous sections it was assumed that the forces f were not themselves dependent on the deformation. In some instances this is not true. For instance, pressure loads on a deforming structure are in this category. Aerodynamic forces are an example of such pressure loads and can induce flutter. If forces vary with displacement then in relation (5.63) the variation of the forces with respect to the displacements has to be considered. This leads to the introduction of the load correction matrix K~. as originally suggested by Oden 23 and Hibbitt et al. 24 Here we consider the case where pressure acts on the current configuration and remains normal throughout the deformation history. If the pressure is given by p then the surface traction term in t~l"Iext is given by
f% t~ui fi d'7 = f% t~ui p ni d"),
(5.129)
where ni are the direction cosines of an outward pointing normal to the deformed surface. The computation of the nodal forces and tangent matrix terms is most conveniently computed by transforming the above expression to the surface approximated by finite elements. 25-27 In this case we have the approximation to Eq. (5.129) for a three-dimensional problem given in matrix notation by
t
/if,
t~UiPnidq/-- ~Jfla
l
1
ga]9(~l,~2 )
[(gc,~lXc)
X (g3,~2x,) ]
d~l d~2
(5.130)
where ~1, ~2 are natural coordinates of a two-dimensional finite element surface interpolation, P(~I, ~2) is a specified nodal pressure at each point on the surface, Xc are nodal coordinates of the deformed surface, and we have used the relation transforming surface area given in Eq. (2.18). A cross-product may be written in the alternate matrix forms Xc • x6 - f~cX6 - -'26Xc - f~Xc (5.131) where here R denotes a skew symmetric matrix given as
X--
i1/1
X03 --X30 -- Xxi1 ~X 2 X 1
(5.132)
Using the above relations the nodal forces for the 'follower' surface loading are given by
fa --
1
1
Nap(~I, ~2)Nc,~,N~,~2XcX~d~l d~2
(5.133)
Concluding remarks 155 Since the nodal forces involve the nodal coordinates in the current configuration explicitly, it is necessary to compute a tangent matrix KL for use in a Newton solution scheme. Linearizing Eq. (5.133) we obtain the tangent as --
0Ub
--
1 1
Nap(~I,
~2) [Nb,~lNc,~2
-
Nc,~tNb,~z]
f~c
d~l d~2 (5.134)
In general the tangent expression is unsymmetric; however, if the pressure loading is applied over a closed surface and is constant the final assembled terms a r e symmetric. 27 For cases where the pressure varies over the surface the pressure may be computed by using an interpolation
/~(~1, ~2) --- Na(~l, ~2)/~a
(5.135)
in which/~a are values of the known pressure at the nodes. Of course, these could also arise from solution of a problem which generates pressures on the contiguous surfaces and thus leads to the need to solve a coupled problem. ~ The form for two-dimensional plane problems simplifies considerably since in this case Eq. (5.130) becomes
i
(~uit i d~
-
t~[l a
/'
NaP(~l) (Nc,~tXc) x e3 d~l
(5.136)
1
where ~ is a one-dimensional natural coordinate for the surface side, e3 is the unit vector normal to the plane of deformation (which is constant), and P(~I) is now the force per unit length of surface side. For this case the nodal forces for the follower pressure load are given explicitly by
f. =
f 1 Nop(~l) { -x~'~l } d~l 1 Xl'~I
(5.137)
w h e r e Xi,~, are derivatives computed from the one-dimensional finite element interpolation used to approximate the element side. The case for axisymmetry involves additional terms and the reader is referred to reference 24 for details.
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This chapter presents a unified approach for all finite deformation problems. The various procedures for solution of the resulting non-linear algebraic system have followed those presented in Chapters 2 and 3. Although not discussed extensively in the chapter, the extension to consider transient (dynamic) situations is easily accomplished. The long-term integration of dynamic problems occasionally presents difficulties using the time integration procedures designed for linear problems (e.g. those discussed in Sec. 2.4 and in reference 1). Here schemes which conserve momentum and energy for hyperelastic materials can be considered as alternatives, and the reader is referred to literature on the subject for additional d e t a i l s . 28-34 We have also presented some mixed forms for developing elements which perform well at finite strains and with materials which can exhibit nearly incompressible behaviour. These elements are developed in a form which allow the introduction of
156
Geometricallynon-linear problems- finite deformation finite elastic and inelastic material models without difficulty. Indeed, we have shown that there is no need to decouple the constitutive behaviour between volumetric and deviatoric response as often assumed in many presentations. We usually find that transformation to a current configuration form in which either the Kirchhoff stress or the Cauchy stress is used directly will lead to a form which admits a simple extension of existing small deformation finite element procedures for developing the necessary residual (force) and stiffness matrices. An exception here is the presentation of the mixed-enhanced form in which all basic development is shown using the deformation gradient and first Piola-Kirchhoff stress. Here we could express final answers in a current configuration form also, but we leave these steps for the reader to perform.
!iiii•iiiiiiiiiiiiiiiiiii!iiiiii!ii•iiiiiiiii!iiii!ii•iiiiiiiii•iiii!iii!iiii!iiiiiiii•iii•iiiii!•iiiiiii•i•i•iiiiiiiiiiii••iiiiiiii!!i!i!!ii•i!ii•iiiiiiii•iii!iii•iii•ii!!iiiii•iiiiiiiii•iiiiii•iii•iiiiiiiiiiiiiiiiiiii
1. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann, Oxford, 6th edition, 2005. 2. J.C. Simo and E Armero. Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes. International Journal for Numerical Methods in Engineering, 33:1413-1449, 1992. 3. J.C. Simo, E Armero and R.L. Taylor. Improved versions of assumed enhanced strain tri-linear elements for 3d finite deformation problems. Computer Methods in Applied Mechanics and Engineering, 110:359-386, 1993. 4. S. Glaser and E Armero. On the formulation of enhanced strain finite elements in finite deformation. Engineering Computations, 14:759-791, 1996. 5. P.M.A. Areias, J.M.A. Crsar de S~i, C.A. Conceiq~o Anto6nio and A.A. Fernandes. Analysis of 3D problems using a new enhanced strain hexahedral element. International Journal for Numerical Methods in Engineering, 58:1637-1682, 2003. 6. L.E. Malvern. Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, Englewood Cliffs, NJ, 1969. 7. P. Chadwick. Continuum Mechanics. John Wiley & Sons, New York, 1976. 8. M.E. Gurtin. An Introduction to Continuum Mechanics. Academic Press, New York, 1981. 9. I.H. Shames and F.A. Cozzarelli. Elastic and Inelastic Stress Analysis. Taylor & Francis, Washington, DC, 1997. (Revised printing.) 10. J. Bonet and R.D. Wood. Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge University Press, Cambridge, 1997. ISBN 0-521-57272-X. 11. J.C. Simo and T.J.R. Hughes. Computational Inelasticity, volume 7 of Interdisciplinary Applied Mathematics. Springer-Verlag, Berlin, 1998. 12. C.C. Celigoj. An assumed enhanced displacement gradient ring-element for finite deformation axisymmetric and torsional problems. International Journal for Numerical Methods in Engineering, 43:1369-1382, 1998. 13. C.C. Celigoj. An improved 'assumed enhanced displacement gradient' ring-element for finite deformation axisymmetric and torsional problems. International Journal for Numerical Methods in Engineering, 50:899-918, 2001. 14. J.C. Simo and R.L. Taylor. Quasi-incompressible finite elasticity in principal stretches: continuum basis and numerical algorithms. Computer Methods in Applied Mechanics and Engineering, 85:273-310, 1991. 15. B. Fraeijs de Veubeke. Displacement and equilibrium models in finite element method. In O.C. Zienkiewicz and G.S. Holister, editors, Stress Analysis, Chapter 9, pages 145-197. John Wiley & Sons, Chichester, 1965. ,
References 157 16. U. Andelfinger, E. Ramm and D. Roehl. 2d- and 3d-enhanced assumed strain elements and their application in plasticity. In D. Owen, E. Ofiate and E. Hinton, editors, Proceedings of the 4th International Conference on Computational Plasticity, pages 1997-2007. Pineridge Press, Swansea, 1992. 17. P. Wriggers and G. Zavarise. Application of augmented Lagrangian techniques for non-linear constitutive laws in contact interfaces. Communications in Numerical Methods in Engineering, 9:813-824, 1993. 18. M. Bischoff, E. Ramm and D. Braess. A class of equivalent enhanced assumed strain and hybrid stress finite elements. Computational Mechanics, 22:443-449, 1999. 19. E.P. Kasper and R.L. Taylor. A mixed-enhanced strain method: Part 1 - Geometrically linear problems. Computers and Structures, 75(3):237-250, 2000. 20. T.H.H. Pian and K. Sumihara. Rational approach for assumed stress finite elements. International Journal for Numerical Methods in Engineering, 20:1685-1695, 1985. 21. E.L. Wilson. The static condensation algorithm. International Journal for Numerical Methods in Engineering, 8:199-203, 1974. 22. E.P. Kasper and R.L. Taylor. A mixed-enhanced strain method: Part 2 - Geometrically nonlinear problems. Computers and Structures, 75(3):251-260, 2000. 23. J.T. Oden. Discussion on 'Finite element analysis of non-linear structures' by R.H. Mallett and P.V. Mart;al. Proc. Am. Soc. Civ. Eng., 95(ST6):1376-1381, 1969. 24. H.D. Hibbitt, P.V. Marqal and J.R. Rice. A finite element formulation for problems of large strain and large displacement. International Journal of Solids and Structures, 6:1069-1086, 1970. 25. J.C. Simo, R.L. Taylor and P. Wriggers. A note on finite element implementation of pressure boundary loading. Communications in Applied Numerical Methods, 7:513-525, 1991. 26. K. Schweizerhof. Nitchlineare Berechnung von Tragwerken unter verformungsabhangiger belastung mitfiniten Elementen. Doctoral dissertation, U. Stuttgart, Stuttgart, Germany, 1982. 27. K. Schweizerhof and E. Ramm. Displacement dependent pressure loads in non-linear finite element analysis. Computers and Structures, 18(6): 1099-1114, 1984. 28. T.A. Laursen and V. Chawla. Design of energy conserving algorithms for frictionless dynamic contact problems. International Journal for Numerical Methods in Engineering, 40:863-886, 1997. 29. J.C. Simo and N. Tarnow. The discrete energy-momentum method. Conserving algorithm for nonlinear elastodynamics. Zeitschrifi ftlr Mathematik und Physik, 43:757-793, 1992. 30. J.C. Simo and N. Tarnow. Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics. Computer Methods in Applied Mechanics and Engineering, 100:63-116, 1992. 31. N. Tarnow. Energy and momentum conserving algorithms for Hamiltonian systems in the nonlinear dynamics of solids. PhD thesis, Department of Mechanical Engineering, Stanford University, Stanford, California, 1993. 32. O. Gonz~ilez. Design and analysis of conserving integrators for nonlinear Hamiltonian systems with symmetry. PhD thesis, Department of Mechanical Engineering, Stanford University, Stanford, California, 1996. 33. M.A. Crisfield and J. Shi. An energy conserving co-rotational procedure for non-linear dynamics with finite elements. Nonlinear Dynamics, 9:37-52, 1996. 34. U. Galvanetto and M.A. Crisfield. An energy-conserving co-rotational procedure for thedynamics of planar beam structures. International Journal for Numerical Methods in Engineering, 39:2265-2282, 1996.
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Material constitution for finite deformation
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In order to complete the finite element development for the finite deformation problem it is necessary to describe how the material behaves when subjected to deformation or deformation histories. In the previous chapter we considered elastic behaviour without introducing details on how to model specific material behaviour. Clearly, restriction to elastic behaviour is inadequate to model the behaviour of many engineering materials as we have already shown in previous applications. The modelling of engineering materials at finite strain is a subject of much research and any complete summary on the state of the art is clearly outside the scope of what can be presented here. In this chapter we present only some classical methods which may be used to model elastic, viscoelastic and elastic-plastic-type behaviours. The reader is directed to literature for details on other constitutive models (e.g. see references 1 and 2). We first consider some methods which may be used to describe the behaviour of isotropic elastic materials which undergo finite deformation. In this section we restrict attention to those materials in which a stored energy function is used; such behaviour is often called hyperelastic. Later we will extend this to permit the use of viscoelastic and elastic-plastic models and show that much of the material presented in Chapter 4 is here again useful. Finally, to permit the modelling of materials which are not isotropic or cannot be expressed as an extension to elastic behaviour (e.g. generalized plasticity models of Chapter 4) we introduce a rate f o r m - here again many options are possible. This latter form is heuristic and such an approach should be used with caution and only when experimental data are available to verify the behaviour obtained.
6.2.1 Isotropic elasticity-formulation in invariants We consider a finite deformation form for hyperelasticity in which a stored energy density function, W, is used to compute stresses. For a stored energy density expressed in terms of fight Cauchy-Green deformation tensor, CIj, the second Piola-Kirchhoff
Isotropic elasticity stress is computed by using Eq. (5.34). Through standard transformation we can also obtain the Kirchhoff stress a s 1 ' 3 ' 4
OW 70 = 2 b i k ~ Obkj
(6.1)
and thus, by using Eq. (5.19), also obtain directly the Cauchy stress. For an isotropic material the stored energy density depends only on three invariants of the deformation. Here we consider the three invariants (noting they also are equal to those for bij) expressed as 1,4
I
=
CKK
--
bkk
1 1 I I = 5(12 - CKLCLK) = 5(12
I I I = det CKL = det bkl
--
bklblk)
-
J2
(6.2) J = det FK I~
where
and write the strain energy density as (6.3)
W ( C K L ) -- W ( b k l ) =- W ( I , I I , J )
where we select J instead of I I I as the measure of the volume change. In this form the second Piola-Kirchhoff stress is computed as SIJ -- 2
OW OI OW 0 I I OI OCIJ i 0 I I OCIJ
t
OW OJ ] OJ OCIJ J
(6.4)
The derivatives of the invariants may be evaluated as (see Appendix B)
OI OCIj
--" (~lJ,
01I OCIj
OJ
--- I~IJ -- C I j ,
OCIj
=
1
JCl)
(6.5)
Thus, the stress is given by 0~
-O7 SIj = 2 [~lJ (I~ij - CIj)
0~
1
(6.6)
014
-g)
Using Eq. (5.20) the second Piola-Kirchhoff stress may be transformed to the Cauchy stress and gives
OW
-ffi2
O'ij ~ 7
[bij
(Ibij - bimbmj)
1
~J~ij]
OW
-ffi-i OW
(6.7)
159
160
Material constitution for finite deformation
A Newton-type solution process requires computation of the elastic moduli for the finite elasticity model. The elastic moduli with respect to the reference configuration are deduced froml'4 CIJKL
4
=
02 W
OSl J
OCIJOCKL
=
2
(6.8)
~
OCKL
thus from Eq. (6.6) the general form for the elastic moduli of an isotropic material is given by CIJKL -- 4 ISis, (I(~lJ -- C l J ) , - c~W
1
~JC#]
oaw
~w
-
012
0101I
OlOJ
c~W 0IIOI 02w _ OJOl
c~w 0112 02w OJOl I
oaw OIIOJ 02w OJ 2
(~KL
(I6KL --CKL) 1 ~JC;[
(6.9)
.
4 1 "Jr- [(~I J (~K L -- "~ ( (~I K (~J L -~- (~I L (~J K ) ,
J (Ci-j 1CK 1 -- 2C#KL) ]
OW OW
-g-f where
(6.10) C I)K L -_ 1 [CI-~Cj--1L _jr_C ; ; C j ~ ] The spatial elasticities related to the Cauchy stress are obtained by the push forward transformation (6.11)
JCijkl --- Ell F j j FkK FILCIJKL
which, applied to Eq. (6.9), gives
4
Cijkl "- ~[bij,
(I bij - bimbmj ),
1JSij]
"02W Ol 2
o2w 0IOII
02w 0II01
02w 0112
02w OIlOJ
o2w
oaw
o2w
- OJOI
OJOI
I
o2w OIOJ
bkl
(Ibkl -- bknbnl) 1
(6.12)
J (~kl
OJ 2
4 + -J1 [bijbkl - ~l (bikbjl + bilbjk)
J[(~ij(~kl -- 2 ~ijkl]]
OW
b77 ~W
~7 where Z'jkl -
1 [(~ikt~jI "4;- (~il(~jk]
(6.13)
Isotropic elasticity The above expressions describe completely the necessary equations to construct a finite element model for any isotropic hyperelastic material written in terms of invariants. All that remains is to select a specific form for the stored energy function W. Here, many options exist and we include below only a few very simple models. For others the reader is referred to literature on the subject.
Example 6.1
Volumetric behaviour
If we let the stored energy function be given by
W = ku U(J)
where kv is a scalar material parameter and J measures the volumetric deformation. Using Eq. (6.7) we obtain the stress in the current configuration given by crw) OU ij -- kv --~ ~ij
(6.14)
which is a pure hydrostatic stress (i.e. a pressure). Using Eq. (6.12) the elastic tangent moduli for this model are given by
: ijkl . = kv [( J ~:,
o,) (~ij(~kl -
-'[- - ' ~
,,]
2 ~'ijkl - ~
(6.15)
In the models given below we will assume that the volumetric behaviour is proportional to U (J) in which one of the following models is used: 1
1 (j2 _ 1) - ~lnJ
1 (J - 1) 2 -~ 1 (lnj)2
U(J) =
(6.16)
The derivatives of these give 89 OU __ OJ
J -
1
1) 1
and
ln,
1
~ ( l + J 5) 0 2 U __ .
OJ 2
1
5
(6.17)
We note that only the first of these models gives a pressure which approaches an infinite value when J ~ 0 and when J ~ c~. However, when J is near unity it may be approximated by J~l+~
0Ui
OXi
and all models give OU
OUi
OJ
OXi
and
02 U OJ 2
Thus, any of the models may be used when the deformations remain moderate.
161
162
Material
constitution
for finite
deformation I
I
i
Saint-Venant-Kirchhoff . ---
U=
....
,o
(j2_
- U = (J-
U=
1)/4-In(J)/2
i-
1)2/2
In 2
,_ ~.
i
1
-' ~
go ~
w,~
/
L/_,(-,,," - - -
9
'
. ~
~
-9
~i
, ~
'i ~
-'- - - - - ' - ' : - . . . . . . . . . . . . 11
11
i
i
,
,
,
,
,
,
:
'
:
l
i
i
,
,
,
'
I I
9
_,4
,~
i
~.-1
It
..
,
d)12
'~
'
I
'
I
0
Fig. 6.1
,
:
I
I
2
3 VIVo - V o l u m e
4
Volumetric deformation.
In Fig. 6.1 we show the behaviour of the above forms together with the results from the Saint-Venant-Kirchhoff model presented in the previous chapter. The material property for ku is chosen to match the small strain bulk modulus for small strain, with ku = K = E / ( 1 - 2u)/3. Results are normalized by E and u is equal to 0.25. It is clear that the model using U = l n 2 ( j ) / 2 is not useful in problems which have large dilation since the mean stress is getting smaller. For this stress state the other models are acceptable. However, we shall see next that when uniaxial stress states are considered the Saint-Venant-Kirchhoff model also becomes unacceptable for large strain cases.
Example 6.2 Compressible neo-Hookean material
As an example, we consider the case of a neo-Hookean material 5 that includes a compressibility effect. The stored energy density is expressed as W ( I , J) -- W(1)(I, J) + )~U(J)
(6.18)
1 W(1)(I, J) - ~#(I - 3 - 2 In J)
where the material constants ku - ,~ and # are selected to give the same response in small deformations as a linear elastic material using classical Lam6 parameters ~ and U ( J ) is one of the volumetric stored energy functions given in (6.16). The non-zero derivatives of W (~) are given by"
OW(1 )
oJ
-
~/z
1
-7
and
02W Oj 2
=
# j2
Isotropic elasticity Substitution of these into Eqs (6.7) and (6.12) gives 0.(1)_ #
ij
-j(bq - ~ij)
._,(1)
and
2#
t~ijkl = --f Zijkl
(6.19)
The final stress is obtained from the sum of Eqs (6.14) and (6.19) and, similarly, the tangent moduli from the sum of Eqs (6.15) and (6.19). We note that when J ~ 1 the small deformation result
(6.20)
Cijkl "~ /~(~ij (~kl "31- 2#~ijkl
is obtained and thus matches the usual linear elastic relations in terms of the Lam6 parameters. This permits the finite deformation formulation to be used directly for analyses in which the small strain assumptions hold as well as for situations in which deformations are finite.
Example 6.3 A modified compressible neo-Hookean material
As an alternative example, we consider the case of a modified neo-Hookean material which in the small strain limit is identical to the isotropic linear elastic model given in terms of bulk K (= ku) and shear G moduli. The stored energy density for this case is expressed as
W(I, J) = W(2)(I, J) + K U(J) W (2) (I, J) -
(6.21)
~1 G (J -2/3 I - 3)
For this model, the non-zero derivatives of W (2) a r e given by: 0 W (2)
OI
OW (2)
_ _1 G 2
_ 2 -5/3 gJ I
and
oJ
1 G j-2/3 2
02W (2)
012
OIOJ
02W (2)
02W (2)
OJOI
OJ 2
3
m
Substitution of these results into
02W (2)
- - ~2 J
1--~ 21
Eq. (6.7) gives
0-(2) ij -" -jG ([gij - g1 t~ij [)
(6.22)
where bij - J-2/3bij and [ = J-2/3I = bkk. We note that the term multiplying G / J is a deviatoric quantity, that is ~ -- 1 r bii
we thus define and simplify Eq. (6.22) to
~ = b~
k __ 0,
bij -- g1 (~ij[gkk
.(2) G ~ ij = 7 bd
(6.23)
163
164
Material constitution for finite deformation
Using the definitions for the derivatives of the stored energy function and introducing the above deformation measures the material moduli for the current spatial configuration are given as ..(=, 2G[ ] -- (~ijbkl (6.24) IJiJ kl __ - ~ Dmm(~ijk I __ -~(~ij~kl) 1 -d -- bd(~kl ~ Again, the results for the total stress and material moduli are obtained by combining the above with a model for volumetric behaviour. We note that when J ~ 1 the small deformation result becomes Cijkl "~ K (~ij(~kl Jr- 2 G (~ijkl -- "~1 (~ij (~kl)
(6.25)
and thus matches the usual linear elastic relations in terms of the bulk and shear moduli. In Fig. 6.2 we show the uniaxial response of the two forms of the neo-Hookean material together with those for the Saint-Venant-Kirchhoff model. The properties of the models are picked to match the small strain case for a modulus of elasticity E. For the neo-Hookean models we note that uniaxial behaviour involves both volumetric and distortional deformations, thus, it is necessary to use a model for U(J) in addition to each of the forms for W (i). The parameters for A, K and/z = G are selected to match the small strain values in terms of E and u with u -- 0.25. While the above models are classical and easily treated, they are not accurate for hyperelastic materials which exhibit increased stiffness with stretch. In order to treat such materials it is necessary to consider altemative models.
.... I
I
Saint-Venant-Kirchhoff . . . . Neo-Hookean ! Neo-Hookean (M)
!
_Z f |
r {D
o
L_
f
3
9t -
f
f
2
. . . . .
, | |
I
LU | I
ff
-1 -2
i | |
./I 0
1
I
2
I
~,- Stretch
3
I
4
I
5
Fig. 6.2 Uniaxial stretch. Saint-Venant-Kirchhoff and neo-Hookean material. (M) denotes a modified compressible material.
Isotropic elasticity
6.2.2 Isotropic elasticity-formulation in modified invariants In the previous example we introduced invariants based on the modified deformation gradient given by F i l -- j - 1 / 3 F i I (6.26) which yields the modified fight and left Cauchy deformation tensors
CIj = Fil Fij and
bij
--
Fil Fjl
(6.27)
respectively. The modified invariants given by and I " I - j-4/3 II
[ _ j-2/31
(6.28)
together with J may be used to construct stored energy functions as
W - wci, i l , J)
(6.29)
In addition to the modified neo-Hookean model given above, several forms in terms of W([) have been introduced to represent the behaviour of hyperelastic materials subjected to large stretch. A general form for such models depending only on I is given by
OW
-ff
OW
-rY
=
O[
-g J- [
(6.30)
which may be used in Eq. (6.7) to compute Cauchy stresses. Similarly, we recover
02W OlOJ ~W Oj 2
0
_ cOW_ [ - col 2 j-5/3 -5
2 j-5/3] __109j-2 [ J
+
[ 012
-4/9 -'32 j-5/3 [
] .~ -2 1-2 J
(6.31) which may be used in Eq. (6.12) to compute the tangent matrix. In the next two examples we consider two forms based on the first invariant [ and the volumetric effects from J.
Example 6.4
Yeoh model
The first extended form was proposed by Yeoh 6 and uses the stored energy function
W(3)_ 1 ~
[(I-- 3) +k 1([- 3)2-1--k2(1- 3)3]
(6.32)
for the deviatoric part and ku U (J) given in Example 6.1 for the volumetric part. The derivative with respect to the modified invariant for the Yeoh model is given by
oqW(3)
1 [l+2k, Cf 3)+3k2(i -
,.,,,
OI
_
3):]
(6.33)
165
166
Material constitution for finite deformation
Similarly the second derivative is given by
02W~3) 0i2 -- -2
(6.34)
[kl -k-6 k2([ - 3)]
These may be used in Eqs (6.30) and (6.7) to compute the Cauchy stress and in Eqs (6.31) and (6.12) to compute the tangent tensor. Indeed, the modifications to the expressions given for the modified neo-Hookean model in Example 6.3 are quite trivial.
Example 6.5 Arruda-Boyce model
The second extended form is due to Arruda and Boyce 7'8 and the stored energy expression is given by
1[
W (4) -- ~/~
([-
1
11,3
3) -t- 1-i~n (/2 - 9) + 525 n 2 (
where
- 27)
l
(6.35)
#
#=
99
1 + 3 + 175.:
for the deviatoric part and also uses ku U (J) given in Example 6.1 for the volumetric part. Typically # = G, ku = K, the linear elastic shear and bulk moduli, and n is the number of segments in the chain of the material molecular network structure. The derivatives of the stored energy function are 0W(4) 1 O[ = ~ / z and
o~ W (4)
O[2
1 =~#
[
1 l+~n[+
11 /21 175n 2
1 22 [~nn+175n2[]
(6.36)
(6.37)
which may be used in Eqs (6.30) and (6.7) to compute the Cauchy stress and in Eqs (6.31) and (6.12) to compute the tangent tensor. In Fig. 6.3 we show the results for a uniaxial stretch with the properties E, u = 0.49 and n = 2. The stress is normalized by E. Note the properties are now set to give a nearly incompressible state which is where the model is most applicable. The above models may be used in either the displacement model or with the threefield mixed form described in the previous chapter for situations where the ratio A/# or K~ G is large (i.e. nearly incompressible behaviour). Indeed this was an early use of the model. In addition, using Eqs (6.7) and (6.12) it is a simple task to develop any material model for which the stored energy function is expressed in terms of invariants.
6.2.3 Isotropic elasticity- formulation in principal stretches Other forms of elastic constitutive equations may be introduced by using appropriate expansions of the stored energy density function. As an alternative, an elastic formulation expressed in terms of principal stretches (which are the square root of the eigenvalues of C Ij o r bij) may be introduced. This approach has been presented by Ogden 9 and by Simo and Taylor. 10
Isotropic elasticity I
,
Arruda-Boyce
I I -I. . . . . . . .
,
,I
~-
I
I
~
L
II
I
!
I1) L_
J
0
"6-1
--
-t
I
j
J._
. . . . . .
-
I
I----
"__~L
r
__
' I--
--I
-t . . . . . . . . I I
co
I' I
-,'-
/
I I I
. . . . . . .
1
_L.
. . . . . . . I
-2
. . . . . . . .
-3
. . . . . . . .
L. . . . . . . . . I
-i-~,t---j
-4
-5
I I
I I I
0
L. . . . . . . .
0.5
i 1.5 ~,- Stretch
1
2
2.5
3
Fig. 5.3 Uniaxialstretch. Arruda-Boyce model. We first consider a change of coordinates given by (see Appendix B, Volume 1) (6.38)
Xi - - A m , i X m ,
where Am, i are direction cosines between two Cartesian systems. The transformation equations for a second-rank tensor, say bij, may then be written in the form bij -- Am,ibm,n,-/kn,j
(6.39)
To compute specific relations for the transformation array we consider the solution of the eigenproblem
bijqj-(n)
_
_
ttJ-(n) bn',
with
n -- 1 , 2 , 3
q(m)qk(n)
__ t~mn
(6.40)
where b, are the principal values of bij, and q:n) are direction cosines for the principal directions. The principal values of bij are equal to the square of the principal stretches, An, that is, bn -- )~n2
(6.41)
If we assign the direction cosines in the transformation equation (6.39) as An,j
-
q~n)
(6.42)
the spectral representation of the deformation tensor results and may be expressed as bij
"--
~
-2 (m) _(m) Amq i qj m
(6.43)
167
168
Material constitution for finite deformation
An advantage of a spectral form is that other forms of the tensor may easily be represented. For example, (m) (m)
qJ
bikbkj -- E / ~ 4 q i m
and
)~m2q[m' q~m)
b ~ l -" Z
(6.44)
m
Also, we note that an identity tensor may be represented as (~ij -- ~
qi(m) qj(m)
(6.45)
m
From Eq. (6.7) we can immediately observe that Cauchy and Kirchhoff stresses have the same principal directions as the left Cauchy-Green tensor. Thus, for example, the Kirchhoff stress has the representation Tmq[m) q~ m)
=
(6.46)
m
where ~-m denote principal values of the Kirchhoff stress. If we now represent the stored energy function in terms of principal stretch values as t~(A~, A2, A3) the principal values of the Kirchhoff stress may be deduced f r o m 2'9 0t~
(no sum)
Tm --- "~m O)km
(6.47)
The reader is referred to the literature for a more general discussion on formulations in principal stretches for use in general elasticity problems. 2'9'1~Here we wish to consider one form which is useful to develop solution algorithms for finite elastic-plastic behaviour of isotropic materials in which elastic strains are quite small. Such a form is useful, for example, in modelling metal plasticity.
Example 6.6 Logarithmic principal stretch form
A particularly simple result is obtained by writing the stored energy function in terms of logarithmic principal stretches. Accordingly, we take t~(A1, A2, A3) = //)(el, e2, e3) From Eq. (6.47) it follows that
where
em - - 1og()~m)
(6.48)
Ow
7"m --" O~ m
(6.49)
which is now identical to the form from linear elasticity, but expressed in principal directions. It also follows that the elastic moduli may be written as 2'9 (summation convention is not used to write this expression) 3
J Cijkl-
3
y~-'~[Cmn -- 2Tm(Smn] n ) ( nql) qi( m )qj( m ) (qk m=l n=l 3
1
-'1--2 E
m=l
3
Z n=l
n:~m
~ (m)
(n)_(m)
q}m)
(n)
(n)
(m)
gmntqi qj qk q[")+ _ qj ql, ql ]
(6.50)
Isotropic elasticity where
02//)
,,)k2 --T2-- ; Am r `,)in and
Cm,, - OemOe,, -
-
m - A.
gmn - -
0(7. m _ 7"n) ;
)km = )in
(6.51)
O~m
In practice the equal root form is used whenever differences are less than a small tolerance (say 10-8). Use of a quadratic form for w given by 1
2
w -- ~(K - ~G)[El "~- E2 "q" E3]2 "q- a [c 2 -~- C2 -~- E~]
(6.52)
yields principal Kirchhoff stresses given by
2
4 K + -gG
7"2 73
-
K-sG
4
-sG 2 5G
-
K+sG 2
K - 5G
(6.53)
K+4aj
in which the 3 • 3 elasticity matrix is given by a constant coefficient matrix which is identical to the usual linear elastic expression in terms of bulk and shear moduli. We also note that when roots are equal 0(% - "rn)
&m
4 2 = (K + =G) - (K - =G) = 2G ..5
(6.54)
.5
which defines the usual shear modulus form in isotropic linear elasticity.
6.2.4 Plane stress applications . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . .
-:::-
.
.
.
.
.
....
-
..=_:_.:_..:..
.................................
-
.
. . . . . . . . . . . . . . . . . . . . . . . . .
--: .........................
-=::-----:---
..........................................
-::::-
.............................................................................
The above constitutive models may be used directly for the two-dimensional plane strain and axisymmetric problems; however, for plane stress it is necessary to modify the constitutive terms to enforce the 0-33 = 0 condition and thus account for the/733 term of the deformation gradient described in Sec. 5.4.2. A local iteration form based on the second Piola-Kirchhoff stress (S) and the Green-Lagrange strain (E) was proposed by Klinkel and Govindjee. 11 The local iteration is based on a linearized form for the S33 stress and a Newton method. Accordingly, for each computation point in the element (i.e. the quadrature point used to compute the arrays) we obtain the deformation gradient given by Fil
=
I ~ l El2 ~ ] [~m 1 F22 -0 F33
0 1 F33
(6.55)
in which the components Fll, F12, Fel and Fee (Fm) are computed from the current displacements as described in Eq. (5.87) and/733 is computed from the material constitution as described next. A linearization of the second Piola-Kirchhoff stress Stj in terms of the Green-Lagrange strain EIj may be expressed as
s(k+l) m ~,(k) K,(k) IJ ~ I J nt- d ~ ' I J
169
170
Material constitution for finite deformation
where
d "c(k) IJ
c,(k)
-- "-'IJKL
d E~)L
where C/j/~L are the tangent moduli in the reference configuration for the given constitutive model. We can partition the above relations into two parts and write the result in matrix notation. The in-plane components are given by Sm--[S,l
322
812] T
and the stress normal to the plane of deformation $33.* Accordingly, the linearization may be written as ~,(k+l) ~'33
=
c(k) ~
"~ '1 d c(k)
(6.56a)
where
~dS~k3 )
=
/ t '(k) l '~'3m
t -'(k) ] ~"3333.1
, 4 ~ (k) u' L'33
(6.56b)
where Em contains the in-plane Green-Lagrange strains in the same order as for stresses. The tangent moduli are given by Cmm for the in-plane components, C3333 associated with $33 and E33 and Cm3,C3m are the moduli which couple the two. To obtain the value of E33 for the current solution step Klinkel and Govindjee propose a local Newton iteration given by
s(33k+l) --with the update
c(k) "1- C'(k) d '-z(k) = 0 ~-'3333 "-'33
~
]7(k+l) ]tT,(k) ]tT,(k) L.,33 - " L,33 + di_,33
(6.57a)
(6.57b)
where ,4 t~ ~,(k) ~-'33 "-- F33 d F33. Iteration continues until convergence to a specified tolerance is achieved. This form only involves the solution of the scalar equation as ~(k) ,
-
(k)
(6.58)
3333 ^
After convergence is achieved the tangent moduli Dr for the next equilibrium iteration [viz. Eqs (5.62a) and (5.63)] are given by
Dr = Cmm - Cm3C3~33C3m
(6.59)
The above form is efficient for problems formulated in terms of the reference configuration as described in Sec. 5.3.1; however, for spatial forms where the Cauchy stress is computed directly the above requires the constitutive model to be transformed to the reference configuration in order to perform the local iteration. * We restrict attention here to cases where the 3 direction defines a plane of symmetry such that given E13 = E23 -- 0 the stress components $13,823 are also zero.
Isotropic elasticity An alternative is to transform Eq. (6.57a) to the current configuration by noting -(k)
1 re(k)c(k) re(k)
and
33 "-- j ( k ~ 9 33 "33 "33
..(k)
1 re(k)re(k)r
t-;3333 -- - ~ ' 3 3
~(k) re(~)
"33 ~3333" 33 "33
(6.60)
This gives the local iteration Newton method written as -(k) ...(k) .aE33 (,) __ 0 33 + t-;3333
(6.61)
where the incremental normal strain for plane stress is given by re(k)
de~k3) = d. 33
re(k) 9 33
(6.62)
The iteration thus is achieved by solving the scalar equation to give _(k) 033 re(k) (k) 9 33 3333
(6.63)
re(*) F33 ( k~t33 + l"+")dF~3~ _)
(6.64)
d F(3k) with the update
This local iteration is continued until convergence is achieved, that is idF 33 (k)[-
(6.66)
where H3 is the thickness in the reference configuration. After convergence the tangent moduli of the current configuration are used in the next iteration. We can transform Eq. (6.56b) to obtain
do'm
d0-33)'-
[Dmm LO3m
Dm3]
D33J ( dc33 dem )
(6.67)
where D33 = c3333 and the remaining terms are the standard transformation of moduli to matrix form. The reduced plane stress moduli are now given by
Dr
= Dmm - Dm3D331D3m
(6.68)
The above form is superior to other approaches based on a linearization of the entire stress with the zero stress condition enforced at the element level (e.g. see references 12-14). The approach given above may be applied to any constitutive model. The only requirements are an available tangent array and a computation of the 0-33 stress component (see reference 11 for addition details and applications).
171
172 Material constitution for finite deformation
ii iii i!iiiiRii| i iH j i j i!iii!i iiiii!iliiiiiii!!ii!i iiiiiiiiiiiiiii ii ii i i i! iii!iiiii!!iii!
i'~,'~i~ili'.i~'i~i'!':,':i!~.#. i!i'.i'i !~#~:~!~i~i~:~:~!~i~!~ ii! !ii::~....iil ii!i !iiiiiii!iiii~.iiiiiii!iiiiii!i~.'i,ii'.ii,iii!!!ii' ilii!;~iii!iiii!i~iiii~;!i~!i~i~!iiii!i!iiii~!iii~:ii~ii~:!i!~!i!ii!ii~:z~!i!iiii~!ii~i ,',ii!i ', ".':~,i '"':','~:,',',"',"i:'",i'i',i',i!'~i',i'~'.'~!'i',i~i',i',!!iii'ii:i,i~i~i'.i~i~iiiii!iii~i ~i ii~,i~~.~.~,i'#~'.i;.'.i'.'.~i'.','.':~';',i';',~, i'"~',~' ".'~",',', ~, i~#,',',',i',i',ii'i~,~##.i~ii'',~';i~,i'.'.i
The theory of linear viscoelasticity presented in Sec. 4.2 can be easily extended to a finite deformation form. ]5'16 The isotropic form given by Eqs (4.13) to (4.16) is based on a split into deviatoric and mean stress-strain response where the stress is given by O'ij -- O'ij Jr- 0 "d
(6.69)
O'ij = p (Sij + sij
with p -- ~1 Crkk
1 Sij -- cr~j -- "~O'kk (~ij
and
To develop a finite deformation form we use a split of the second Piola-Kirchhoff stress into two parts as SIJ -- S~j -+- sdj (6.70) where Sij
-31(SKLCKL) C7 ) sdj -- SIj - 51 (SKLCKL) C7)
(6.71)
Applying the transformation given in Eqs (5.19) and (5.20) to (6.71) gives immediately Eq. (6.69) and thus represents a split of stress into mean and deviatoric parts in the current configuration. In the finite deformation generalization we shall assume that an elastic model is available in the form W = kv U (J) + W d (6.72) where U yields an elastic stress with the form S~j and W d an elastic deviatoric stress S~j given by S~j = 2 k v Sdlj = 2
OU
OCIJ
(6.73)
OW a
OCIJ
A viscoelastic model is then given by M
sdj - - . o S d j ' J i - Z , m
()(m)
(6.74)
~.~ij
m=l
where
M
with
1
0
(6.75)
m--0
and the partial stress Qxj satisfies the differential equation 1
I J 'F" -~m ~, I J Here
~m
iS
a relaxation time similar to that described in Sec. 4.2.
(6.76)
Plasticity models A numerical solution to the differential equation for /)(m) is given by Eqs (4.22) and (4.29) to (4.34). In addition the tangent moduli are given by those of the elastic model used to define W multiplied by appropriate constants, again as described for the linear model. iiiiiiiiiiiiiiiiiiiiiiiiiiii~ iiiiii!i~iiiiiii:~iii~'~'~'~!ii~i~iii~i!!ii~ii!~!~i iiii !i iiiiiiill ! iiiiii!i:~ iiil~'~'~ iiiii! ili iil iiilii ii iiiili ii i!i!i!iiii iii!iiiiiiiiiiii!i
i i
For isotropic materials, the modelling of elastic-plastic behaviour in which the total deformations are large may be performed by an extension of a hyperelastic formulation. In this case the deformation gradient is decomposed in a product form (instead of the additive form assumed in Chapter 4) written as 17-19 Fil--Fe^
isF31P
(6.77)
where Fe^ijis the elastic part and F j / t h e plastic part. The deformation picture is often shown as three parts, a reference state, a deformed state, and an intermediate state. The intermediate state is assumed to be the state of a point in a stress-free condition.* From this decomposition deformation tensors may be defined as
bi~ - Fe^ iK Fe^ jK
and
C~j-
FPlF~j
(6.78)
which when combined with Eq. (6.77) give the alternate representation bij -- F i l ( C ; j )
(6.79)
-1 Fjj
An incremental setting may now be established that obtains a solution for a time tn+l given the state at time tn. The steps to establish the algorithm are too lengthy to include here and the interested reader is referred to literature for details. 2'~6,2~ The components (bi~)n denote values of the converged elastic deformation tensor at time tn. We assume at the start of a new load step a trial value of the elastic tensor is determined from (bij)tn+l -
(6.80)
f i k ( b e k l ) n f jl
where an incremental deformation gradient is computed as
3~j - ( Fi r )n+ l ( FT1)n
(6.81)
A spectral representation of the trial tensor is then determined by using Eq. (6.43) giving e e 2 (m) tr (m) tr ( b i j ) n + l -- ~ - ~ ( ) ~ m ) n + l q i ' qj ' m
Owing to isotropy q~m),tr C a n be shown to equal the final Trial logarithmic strains are computed as (E~)n+l
--- 1 o g ( / ~ e ) n + l
directions
(6.82)
q}m).2
(6.83)
* The intermediate state is not a configuration, as it is generally discontinuous across interfaces between elastic and inelastic response.
173
174 Material constitution for finite deformation
and used with the stored energy function W (bei) to compute trial values of the principal Kirchhoff stress ('rtm)n+l. This may be used in conjunction with the return map algorithm (see Sec. 3.4.2) and a yield function written in principal stresses rm to compute a final stress state and any intemal hardening variables. This part of the algorithm is identical to the small strain form and needs no additional description except to emphasize that only the normal stress is included in the calculation of yield and flow directions. We note in particular that any of the yield functions for isotropic materials which we discussed in Chapter 4 may be used. The use of the return map algorithm also yields the consistent elastic-plastic tangent in principal space which can be transformed by means of Eq. (6.50) for subsequent use in the finite element matrix form. The last step in the algorithm is to compute the final elastic deformation tensor. This is accomplished from the spectral form and final elastic logarithmic strains resulting from the return map solution as 3 e
e
1
(bij)n+l -- y ~ e x p [ 2 ( - - C m ) n + l l q i
(m)
(m)
qj
(6.84)
m=l
The advantages of the above algorithm are numerous. The form again permits a consistent linearization of the algorithm resulting in optimal performance when used with a Newton solution scheme. Most important, all the steps previously developed for the small deformation case are used. For example, although not discussed here, extension to viscoplastic and generalized plastic forms for isotropic materials is again given by results contained in Secs 3.6.2 and 3.9. The primary difficulty is an inability to easily treat materials which are anisotropic. Here recourse to a rate form of the constitutive equation is possible, as discussed next.
In the previous sections we have assumed that the deformation gradient can be computed from Eq. (5.8a). In some formulations it is convenient to use an updated form (sometimes referred to as an updated Lagrangian method) as shown in Fig. 6.4. Here a time parameter t may be introduced to distinguish between the various configurations. Thus, when t is zero we describethe body in its initial configuration f2 by the coordinates XI. We assume then that the solution is carried out at a set of discrete times ti with the last known solution defined at tn, configuration ~o(n) shown in Fig. 6.4. It is then desired to compute the solution at tn+l, which will be the new current configuration, a/n+l). In this format the known 'reference configuration' may be defined as the body at tn, with coordinates x~~). The deformation gradient at t,+~ may now be defined as
Fi~n+')
=
049~n+,)~ (YXj(n) ~ (n) OXj
OX
I
f ( n ) l:;(n) -- Jij * j I
(6.85)
where f/~n) is an incremental deformation gradient associated with the reference configuration at xj-(") for the incremental time At -- t,+~ - t,
Incremental formulations (h(n)
Ar
X2, x2
co(n+1)
r
> x~, x~ Fig. 6.4 Incrementaldeformationmotionsand configurations.
To compute volumetric changes we use the determinant of Fq which is given by
det(Fff +',) = det(fi) ")) j(n+l)
._
j(n) j ( n )
det(F(7 ,)
(6.86)
where j(') is the determinant of the incremental deformation gradient. The forms given in Eqs (6.85) and (6.86) also permit an incremental update to advance the solution from t, to tn+l. For example, the fight Cauchy-Green deformation tensor given in Eq. (5.13) may be expressed in terms of the incremental quantities as C~71)
_ Ft(n+l)Fi~+l) -"
_ df (ijn ) f i (n) --
F~7) ~* (k"J)
(6.87)
--- C_( n ) F ( 1 ) t:;'( jn (n)
r
where cjk = f/~) Jik is the incremental fight Cauchy-Green deformation tensor.
Example 6.7 Incremental Saint- Venant-Kirchhoff model
For the Saint-Vcnant-Kirchhoff model the stress may now be dcternfined from (5.36) by substituting Eq. (6.87) into Eq. (5.14). The final result may be written in the form
5~71)
if(n)
-- "IJ
"Jv C I J K L
]t?(n)
(6.88)
A,..,KL
where '(n) __ ]tT'(n+ 1) __ ]7(n) AK L.,KL Z-,KL X-,KL __ 1/(-,(n+l) --
2~,"'KL
=
-2" k K
/-,(n)
-- ~'KL)
1 ]t?(n+l)Fl~+l)
1 /j~-,(n+l)]~-,(n+l)
" - "2~,* m K
((~kl _
* mL
fm;(n) f~l
b-,(n) p ( n )
(6.89)
- - Jt m K x m L )
(n)) __ p ( n + l ) F / ~ + l )
"kK
9 (n) Aekl
9 (n) is an incremental Almansi strain measure. Using this Eq. (6.88) may be where Aekl transformed to the current configuration giving the Kirchhoff stress as
7.~n+ 1) J
__ f i ( k n ) T ~ l ) f ( 1 ) +
j ( n + l ) t s i J kl''(n+l) .Aekl(n)
(6.90a)
175
176
Material constitution for finite deformation
and the Cauchy stress as .(n+l) tTij
__
--
1 r(n)cr~nt)fr ) + j(,,) Jik
_(,,+l) 9 (,) ISijkl Aekl
(6.90b)
Now all the steps to compute the finite element arrays follow identically those given in Chapter 5. The left Cauchy-Green deformation tensor may also be written in terms of the incremental deformation gradient as (n+l) 1) K,(n+l ) z.(n) lf,(n) bij -- f i(ln+ ~ jl = k "kI f~)Fl~l")
-- Jr ik f(1)h(n) Ukl
(6.91)
and, thus, may be updated directly and used directly in any of the elastic models included in Sec. 6.2 in their current configuration form. :i!!iiiii!"i"i~' ii!ii!~i~ z:~ili':i!li"iii"iiiii iV!i!i!i!iiii~~i~i~i~ii~i~i~i~i~i~i~i~ii?~ii~i:~ii~iii~i!!~i~i~i~i~/~i~ii~~~i~i~i~i~i~! U iiiiiiiiii',iii!!ii~d~l~iiiiiiiiiiii::::' ii!i:'::'::~iiiii!iiiiiiiiiiii ~':::,iiiiiiii!iiii !!iiiiiiiiiii"'i::'/~?,i:i::'!:i':i /i~,'~!ii~, :, i~,!~:~:i~i~/~i~!i~i~i~i~ii~,~',~/~:,',i',~:,:~,~!~i:,i:,i!':j,i~,~,i:,:'i~i'::',i:',:ii:, :~,:~,'~,iii:,:,i:,i:,~:i,:i,:~:,~!:i::!
The construction of a rate form for elastic constitutive equations deduced from a stored energy function is easily performed in the reference configuration by taking a time derivative of Eq. (5.34), which gives (6.92)
S I j = CIJKLE'KL
where, as before, CIJKL are moduli given by Eq. (6.8). The above result follows naturally from the notion of a derivative since SIJ = lim S i j ( t -[- T]) -- S i j ( t )
rico
~7
(6.93)
Such a definition is clearly not appropriate for the Cauchy or Kirchhoff stress since they are related to different configurations at time t + r/and t and thus would not satisfy the requirements of objectivity. 4'22 A definition of an objective time derivative may be computed for the Kirchhoff stress by using Eq. (5.20) and is sometimes referred to as the Truesdell rate 23 or equivalently a Lie derivative form. 24 Accordingly, to deduce an objective rate of the Kirchhoff stress we differentiate Eq. (5.20) with respect to time, obtaining ;l-ij -- F i l S I j F j j -~- F i l S l j F j j + F i l S l j F j j (6.94) Introducing the rate of deformation t e n s o r lij defined as Ell
:
O)~i ---- O2i OXj ____ OVi OXj ---- lij Fjl OX I
Ox j O X I
(6.95)
Ox j O X I
in which :tj -- vj is material velocity, the stress rate may be written as 3-ij -- Fil a l j F j j -Jr-likTkj
-1- Ti~ljk
(6.96)
Rate constitutive models
The objective stress, denoted as ~-ij, is then given by (6.97)
7"ij -- 7-ij -- likT"kj -- 7-ikljk -- Fil SIJ Fjj The rate of the second Piola-Kirchhoff stress may be transformed by noting
P~KL __ 51 (fkL FkK -Jr fkK FkL) _ 21 (ElK FkLlkt + Fkr FtLlkl) -- Fkr FlLkkt where
kkt = -21 (lkt + Ilk) --
(6.98)
21(Oql)k,
+ ~ ~OVl)
~
(6.99)
The form e'kl is identical to the rate of small strain. Furthermore we have upon grouping terms the rate of stress expression
~-ij -- JCijkl~kl
(6.100)
in which Cijkl is computed now by means of Eq. (5.74). Incremental forms may be deduced by integrating the rate equation. These involve objective approximations for the Lie derivative. 2'16 For example an approximation to the 'strain rate' may be computed from 16 1
(~ij)n+l/2 "~ ---~(Lkl)n+l/2 m E k l ( f fll)n+l/2
(6.101)
AEkl -- ~1 [(fkm)nWlflm)n+l -- (~kl] where
Om(ui)n+l ( f ij )n+a --" 6ij -~- OL O(Xj)n
(6.102)
with A(ui),+l = (ui),+l - (ui),. Similarly, an approximation to the Lie derivative of Kirchhoff stress may be taken as o
1
(7"ij)(n+l/2) ~-" --~( fik)n+l/2 [( f--1 km )n+l(7"mn)(n+l)(
f/nl)n+l
--
('r'kl)(n)] ( fjl)n+l/2
(6.103) Often simpler approximations are used to approximate the integral of the velocity gradient. Here
ft+At dt where
mu~ n-t-l) =
9 OIAU i(n+l/2) Ol)~n+l/2) ~__ (n+l/2)= ,~ (n+l/2) OXj OXj
lijdt ~ At
__
AI~; +1/2)
(6.104)
x~ n + l ) - X~n) a n d X~n'q-OL) = X~n) 21- OL(X~ n+l) _ (n+~) ,, (n+l) I__XUi -- OLI--XUi A
with 0 _< a _< 1 and also we n o t e
Au
i(n) - -
O.
--
X~n)) (6.105)
177
178 Material constitution for finite deformation
An explicit update for the Lie derivative may then be approximated as _(n+l) ( ,~o At,wij, (n+l) ,~, Tij
_
_(n) __ A / ~ + l / 2 )
7ij _ j(n+l/2)e(n+l/2) -" ijkl
...
,(n) _ _
Tkj
_(n)
l(n+l/2)
7ik A~jk
AE.~/+I/2)
(6.106)
in which Aekl is the symmetric part of Alkl. The Kirchhoff stress at tn+l may now be determined by solving Eq. (6.106). Other approximations may be used; however, the above are quite convenient. In the approximation a modulus array Cijkl must also be obtained. Here there is no simple form which is always consistent with the tangent needed for a full Newton solution scheme and, often, a constant array is used based on results from linear elasticity. Such models based on the rate form are referred to as hypoelastic and in cyclic loading can create or lose energy. Extension of the above to include general material constitution may be performed by replacing the strain rate by an additive form given, for example, as = ~e + ~p
(6.107)
for an elastic-plastic material. In this form we can again use all the constitutive equations discussed in Chapter 4 (including those which are not isotropic) to construct a finite element model for the large strain problem. Here, since approximations not consistent with a Newton scheme are generally used for the moduli, convergence generally does not achieve an asymptotic quadratic rate. Use of quasi-Newton schemes and line search, as described in Chapter 3, can improve the convergence properties and leads to excellent performance in many situations. Many other stress rates may be substituted for the Lie derivative. For example, the Jaumann-Zaremba stress rate form given as v
Tij "- ;l'ij --~)ikTkj -- T i k ~ j k -- ;i'ij --JdikTkj -~- Tik~)kj -- JCijkl~kl
(6.108)
may be used. This form is deduced by noting that the rate of deformation tensor may be split into a symmetric and skew-symmetric form as
lq
= ~ij -Ji- ~3ij
(6.109)
where ~bq is the rate of spin or vorticity. It is then assumed that the symmetric part of lij is small compared to the rate of spin. This form was often used in early developments of finite element solutions to large strain problems and enjoys considerable popularity even today.
6.7.1 Necking of circular bar In this example we consider the three-dimensional behaviour of a cylindrical bar subjected to tension. In the presence of plastic deformation an unstable plastic necking will occur at some location along a bar of mild steel, or similar elastic-plastic behaving
Numerical examples 179
(a)
(b)
Fig. 6.5 Necking of a cylindrical bar: eight-node elements. (a) Finite element model; (b) half-bar by symmetry. material. This is easily observed from the tension test of a cylindrical specimen which tapers by a small amount to a central location to ensure that the location of necking will occur in a specified location. A finite element model is constructed having the same taper, and here only one-eighth of the bar need be modelled as shown in Fig. 6.5(a). In Fig. 6.5(b) we show the half-bar model which is projected by symmetry and reflection and on which the behaviour will be illustrated. This problem has been studied by several authors and here the properties are taken as described by Simo and co-workers. 2'16'25 The one-eighth quadrant model consists of 960 eight-node hexahedra of the mixed type discussed in Sec. 10.5. The radius at the loading end is taken as R = 6.413 and a uniform taper to a central radius of Rc = 0.982 • R is used. The total length of the bar is L = 53.334 (giving a half length of 25.667). The mesh along the length is uniform between the centre (0) and a distance of 10, and again from 10 to the end. A blending function mesh generation is used (see Sec. 9.12, Volume 1) to ensure that exterior nodes lie exactly on the circular radius. This ensures that, as much as possible for the discretization employed, the response will be axisymmetric. The finite deformation plasticity model based on the logarithmic stretch elastic behaviour from Sec. 6.2.3 and the finite plasticity as described in Sec. 6.4 is used for the analysis. The material properties used are as follows: elastic properties are K = 164.21 and G = 80.1938; a J2 plasticity model in terms of principal Kirchhoff stresses ri with an initial yield in tension of ry = 0.45 is used. Only isotropic hardening is included and a saturation-type model defined by
- Hi e p + [r~y- ry] ( 1 - exp[-3eP]) with the parameters
Hi = 0.12924,
r~y = 0.715,
and
3 = 16.93
180
Material constitution for finite deformation
(a)
(b)
Fig. 6.6 Deformed configuration and contours for necking of bar: (a) first invariant (21); (b) second invariant (J2).
is employed. An alternative to this is a piecewise linear behaviour as suggested by some authors; however, the above model is very easy to implement and gives a smooth behaviour with increase in the accumulated plastic strain e p as the hardening parameter ~. In Fig. 6.6 we show the deformed configuration of the bar at an elongation of 22.5 per cent (elongation = 6 units). Figure 6.6(a) has the contours of the first invariant of Cauchy stress superposed and Fig. 6.6(b) those for the second invariant of the deviator stresses. It is apparent that considerable variation in pressure (first invariant divided by 3) occurs in the necked region, whereas the values of the second deviator invariant vary more smoothly in this region. A plot of the radius at the centre of the bar is shown in Fig. 6.7 for different elongation values. This example is quite sensitive to solve as the response involves an unstable behaviour of the necking process. Use of a full Newton scheme was generally ineffective in this regime and here a modified Newton scheme together with a BFGS (BroydenFletcher-Goldfarb-Shanno) secant update was employed (see Sec. 3.2.4). When near convergence was achieved the algorithm was then switched to a full Newton process and during the last iterations quadratic convergence was obtained when used with an algorithmic consistent tangent matrix as described in Secs 6.2.3 and 6.4.
6.7.2 Adaptive refinement and localization (slip-line) capture The simple discussion of localization phenomena given in Sec. 4.11 is sufficient we believe to convince the reader that with softening plastic behaviour localization and indeed rapid failure will occur inevitably. Similar behaviour will often be observed with ideal plasticity especially if large deformations are present. Here, however, 'brittle type' of failure will be replaced by collapse in which displacement can continue to increase without any increase of load. It is well known that during such continuing displacement
Numerical examples 181 .
-
0.5
.........................................
.
.
.
.
: ..........
: ..........
:
..........
-1
~
1.5 2.5
3.5 -40 '
1
2
3 4 5 Total elongation
6
7
Fig. 6.7 Neck radius versus elongation displacement for a half-bar.
1. the elastic strains will remain unchanged; 2. all displacement is confined to plastic mechanisms. Such mechanisms will (frequently) involve discontinuous displacements, such as sliding, and will therefore involve localization. To control and minimize the errors of the analysis it will be necessary to estimate errors and adaptively remesh in each step of an elasto-plastic computation. This, of course, implies a difficult and costly process. Nevertheless, many attempts to use adaptive refinement were made and references 26-34 provide a list of some successful attempts.
A d a p t i v e r e f i n e m e n t based on energy norm error estimates
A comprehensive survey of error estimation and h refinement in adaptivity is given in reference 35 (see Chapters 13 and 14). Most of the procedures there described could be applied with success to elasto-plastic analysis. One in particular, the recovery procedures for stress and strain, can be used very efficiently. Indeed, in references 36 and 37 the Superconvergent Patch Recovery (SPR) and Recovery by Equilibrium Patch (REP) methods are used successfully to estimate the errors. (The SPR method was introduced by Zienkiewicz and Zhu and described fully in references 38, 36 and 39. The REP method was presented by Boroomand and Zienkiewicz in references 40 and 41.) In Figs 6.8(a) and 6.8(b) we show an analysis of a tension strip using these procedures. It will be observed that as the load increases the refined mesh tends to capture a solution in which displacements are localized. In this solution triangular elements using quadratic displacements with three added bubble modes together with linear discontinuous pressures (T6/3B/3D) were used as these have an excellent performance in incompressibility and may be incorporated easily into an adaptive process. In this problem we have attempted here to keep the error to 5% in the relative energy norm.
182
Material constitution for finite deformation
_t_U
P
E= 100000 kg/cm 2 v=0.3 cm
P
(~y = 1000 kg/cm 2 (Von Mises criterion)
Fine mesh (a)
A
A
Mesh
Elems 65 179 134 178 242
Nodes ,,,
24 398 301 389 525
.....
(b) Fig. 6.8(a) Adaptive refinement applied to the problem of a perforated strip: (a) the geometry of the strip and a very fine mesh are used to obtain an 'exact' solution; (b) various stages of refinement aiming to achieve a 5% energy norm, relative, error at each load increment (quadratic elements T6/3B/3D were used); (c)local displacement results.
Numerical examples 183 1200 1000 e-o r cD
800 600 -
IT"
-~
Adaptive
400200
0
0
(c)
i 0.01
I l 0.02 0.03 Displacement
........J 0.04
Fig. 6.8(b) Continued
Alternate refinement using error indicators: discontinuity capture
In the illustrative example of the previous section we have shown how a refinement on the test of a specified energy norm error can indicate and capture discontinuity and slip lines. Nevertheless the process is not economical and may require the use of a very large number of elements. More direct processes have been developed for adaptive refinement in high-speed fluid dynamics where shocks presenting very similar discontinuity properties form (e.g. see reference 42). A brief summary of the procedure is given below. The processes developed are based on the recognition that in certain directions the unknown function that we are attempting to model exhibits higher gradient or curvatures. High degree of refinement can be achieved economically in high gradient areas with elongated elements. In such areas the smaller side of elements (hmin) is placed across the discontinuity, and the larger side (hmax) in the direction parallel to the discontinuity. We show such a directionality in Fig. 6.9. For determination of gradients and curvatures, we shall require a scalar function to be considered. The scalar variable which we frequently use in plasticity problems could be the absolute displacement value U = (UTU)1/2
(6.110)
The original refinement indicator of the type we are describing attempts to achieve an equal interpolation error ensuring that in the major and minor direction the equality
h2n Co2U
= h2ax Co2U
= constant
min
(6.111)
By fixing the value of the constant in the above equation and evaluating the approximate curvatures and the ratio of stretching hmax/hminof the function U immediately we have sufficient data to design a new mesh from the existing one. The procedures for such
184
Material constitution for finite deformation
Contour of 'principal' dependent variable Direction of principal variation
Elongated / elements
Fig. 6.9 Elongationof elements used to model the nearly one-dimensional behaviour and the discontinuity.
mesh generation are given in references 43 and 44, although other methods can be adopted. As an alternative to the above-mentioned procedure we can aim at limiting the first derivative of U by making OU h-z-- = constant (6.112) OXl For this procedure it is not easy to evaluate the stretching ratio. However, the firstderivative condition is useful for guiding the refinement. A plastic localization calculation based on Eq. (6.111) is shown in Fig. 6.10. This is an early example taken from reference 29. Here, purely plastic flow is shown, ignoring the elastic effects and the refinement is based on the second derivatives. Such a flow formulation (rigid plastic flow) is also frequently used in metal forming calculations. In the next example we shall use adaptivity based on the first derivatives. Figure 6.11 illustrates a load on a rigid footing over a vertical cut. Here a T6/3C element (triangular with quadratic displacement and continuous linear pressure) is used. Both coarse and adaptively refined meshes give nearly exact answers for the case of ideal plasticity. For strain softening with a plastic modulus H - - 5 0 0 0 answers appear to be mesh dependent. Here, we show how answers become almost mesh independent if H is varied with element size in the manner discussed in Sec. 4.11 [see Eq. (4.184)]. Figures 6.12 and 6.13 show, respectively, the behaviour of a rigid footing placed on an embankment and on a fiat foundation with eccentric loading. All cases illustrate the excellent discontinuity capturing properties of the adaptive refinement.
Concluding remarks 185
This chapter presents a summary of some models for use in finite deformation analyses. The scope of the presentation is limited but provides a description of requirements for use with many finite element formulations. We have not presented any discussion on coupled phenomena which can occur in problems, e.g. such effects of temperature, diffusion, creep, etc. For this, the reader is referred to the extensive literature on the subject for additional details.
(a)
(b)
(c) Fig. 6.10 Adaptive analysis of plastic flow deformation in a perforated plate: (a) initial mesh, 273 degrees of freedom; (b) final adapted mesh; (c) displacement of an initially uniform grid embedded in the material.
186
M a t e r i a l constitution for finite d e f o r m a t i o n
L r
i B=5m =I Rigid and
o It
o II
o II
o II
\\\\\ \\\\\\\\\\ \\\\\\\\\\ \\\\\\\\\\ \\\\\\\ \\\\\\\'~'~\ \\\\\\\\\\ \\\\\\\~,~\ \\\\\\~\ \\\\\\
lOre (a)
s
s
o
s
.
o
,,
t
,,
=,
,,
.
9
.
(c) /
2.25 I
.
. ,
.
.
.
.
.
.
.
.
.
.
. .
. .
.
.
iiii!i!i .
.
.
.
.
.
.
.
.
.
.
.
(d) 2.50
x m e s h 6 (adaptive mesh) 6C/3C, H = 0.0 A m e s h 3 (coarse bad mesh)
6c/3c, H = 0.0
2.25 -
coarse m e s h -. - - adaptive m e s h H = - 5 0 0 0 ...... adaptive m e s h H = - 5 0 0 0 *
0
H
Ioa~ ~
1.75 1. 50 L
2
i
4
= 2.0 [al]
I
8E
BCu (e)
\
(b)
,,
t
\\
\\\\\\\\\\
Uy=O tx=0
2.50
\
6
0
1.75
8
1.50
2
3
I
4 8E
I
5
6
BC. (f)
Fig. 6.11 Failure of a rigid footing on a vertical cut. Ideal von Mises plasticity and quadratic triangles with linear variation for pressure (T613C) elements are assumed. (a) Geometrical data; (b) coarse mesh; (c) final adapted mesh; (d) displacements after failure; (e) displacement-load diagrams for adaptive mesh and ideal plasticity (H = 0); (f) softening behaviour. Coarse mesh and adapted mesh results are with a constant H of -5000 and a variable H starting from -5000 at coarse mesh size.
Concluding remarks
4000 I IgIOUI ....................................
3500
.oil.
...............
llOIIOt
....
tt 9149149149 9
3000
9
2500[.
~
// ~
9 _ I / 1500 1 " /
'
~
"
.
~
.
.
.
.
.
.
.
.
.
.
.
.
.
.
adaptive and optimal
a_ 2000
...............coarse mesh - - - - . . - f i n e mesh
1000 I / 500
V
0 V
0
adaptive
I
0.1
I
0.2
I
0.3
I
A
B
D
E
C
I
I
0.4 0.5 0.6 Displacement
H
I
0.7
I
I
0.8
0.9
1.0
C
F Mesh
Elems
Nodes
A B C D E F G H
36 228 622 769 1577 1645 1521 1694
91 501 1309 1598 3212 3348 3100 3445
Fig. 6.12 A p - 6 diagram of elasto-plastic slope aiming at 2.5% error in ultimate load (15% incremental energy error) with use of quadratic triangular elements (T6/3B). Mesh A: u = 0.0 (coarse mesh). Mesh B: u = 0.025. Mesh C: u = 0.15. Mesh D: u = 0.3. Mesh E: u = 0.45. Mesh F: u = 0.6. Mesh G: u = 0.75. Mesh H: u = 0.9. The last mesh (mesh H, named as the 'optimal mesh')is used for the solution of the problem from the first load step, without further refinement.
187
188
Material constitution for finite deformation
.~=
4m
Rigid and rough footing
I o
II
i
o 0
!
I i I II
II
i
0
II
II
i
El
Ol
i
!
Uy=O tx=O
20 m
(a)
(b)
(c) Fig. 6.13 Foundation (eccentric loading); ideal von Mises plasticity. (a) Geometry and boundary conditions; (b) adaptive mesh; (c) deformed mesh using T6/1D elements (H = 0, ~, = 0.49).
References 189
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1. I.H. Shames and EA. Cozzarelli. Elastic and Inelastic Stress Analysis. Taylor & Francis, Washington, DC, 1997. (Revised printing.) 2. J.C. Simo. Topics on the numerical analysis and simulation of plasticity. In P.G. Ciarlet and J.L. Lions, editors, Handbook of Numerical Analysis, volume III, pages 183-499. Elsevier Science Publisher BV, 1999. 3. M.E. Gurtin. An Introduction to Continuum Mechanics. Academic Press, New York, 1981. 4. J. Bonet and R.D. Wood. Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge University Press, Cambridge, 1997. ISBN 0-521-57272-X. 5. M. Mooney. A theory of large elastic deformation. J. Appl. Physics, 1:582-592, 1940. 6. O.H. Yeoh. Characterization of elastic properties of carbon-black filled rubber vulcanizates. Rubber Chemistry and Technology, 63:792-805, 1990. 7. E.M. Arruda and M.C. Boyce. A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. Journal of the Mechanics and Physics of Solids, 41:389412, 1993. 8. G.A. Holzapfel. Nonlinear Solid Mechanics. A Continuum Approach for Engineering. John Wiley & Sons, Chichester, 2000. 9. R.W. Ogden. Non-linear Elastic Deformations. Ellis Horwood, Limited (reprinted by Dover, 1997), Chichester, England, 1984. 10. J.C. Simo and R.L. Taylor. Quasi-incompressible finite elasticity in principal stretches: continuum basis and numerical algorithms. Computer Methods in Applied Mechanics and Engineering, 85:273-310, 1991. 11. S. Klinkel and S. Govindjee. Using finite strain 3D-material models in beam and shell elements. Engineering Computations, 19:902-921, 2002. 12. T.J.R. Hughes and E. Carnoy. Nonlinear finite element formulation accounting for large membrane stress. Computer Methods in Applied Mechanics and Engineering, 39:69-82, 1983. 13. E. Dvorkin, D. Pantuso and E. Repetto. A formulation of the MITC4 shell element for finite strain elasto-plastic analysis. Engineering Computations, 1:17-40, 1995. 14. R. De Borst. The zero normal stress condition in plane stress and shell elastoplasticity. Communications in Applied Numerical Methods, 7:29-33, 1991. 15. J.C. Simo. On a fully three-dimensional finite-strain viscoelastic damage model: formulation and computational aspects. Computer Methods in Applied Mechanics and Engineering, 60:153-173, 1987. 16. J.C. Simo and T.J.R. Hughes. Computational Inelasticity, volume 7 of Interdisciplinary Applied Mathematics. Springer-Verlag, Berlin, 1998. 17. J. Mandel. Contribution throrique a l'rtude de l'rcrouissage et des lois de l'rcoulement plastique. In Proc. 11th Int. Cong. of Appl. Mech., pages 502-509, 1964. 18. E.H. Lee and D.T. Liu. Finite strain elastic-plastic theory particularly for plane wave analysis. J. Appl. Phys., 38, 1967. 19. E.H. Lee. Elastic-plastic deformations at finite strains. J. Appl. Mech., 36:1-6, 1969. 20. J.C. Simo. Algorithms for multiplicative plasticity that preserve the form of the return mappings of the infinitesimal theory. Computer Methods in Applied Mechanics and Engineering, 99:61112, 1992. 21. E Auricchio and R.L. Taylor. A return-map algorithm for general associative isotropic elastoplastic materials in large deformation regimes. Int. J. Plasticity, 15:1359-1378, 1999. 22. L.E. Malvern. Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, Englewood Cliffs, NJ, 1969. 23. C. Truesdell and W. Noll. The non-linear field theories of mechanics. In S. Fltigge, editor, Handbuch der Physik 111/3. Springer-Verlag, Berlin, 1965. 24. J.E. Marsden and T.J.R. Hughes. Mathematical Foundations of Elasticity. Dover, New York, 1994.
190
Material constitution for finite deformation 25. J.C. Simo and E Armero. Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes. International Journal for Numerical Methods in Engineering, 33:1413-1449, 1992. 26. M. Ortiz, Y. Leroy and A. Needleman. A finite element method for localized failure analysis. Computer Methods in Applied Mechanics and Engineering, 61:189-214, 1987. 27. R. De Borst, L.J. Sluys, H.B. Htihlhaus and J. Pamin. Fundamental issues in finite element analysis of localization of deformation. Engineering Computations, 10:99-121, 1993. 28. O.C. Zienkiewicz, H.C. Huang and M. Pastor. Localization problems in plasticity using the finite elements with adaptive remeshing. International Journal for Numerical Analysis Methods in Geomechanics, 19:127-148, 1995. 29. O.C. Zienkiewicz, M. Pastor and M. Huang. Softening, localization and adaptive remeshing. Capture of discontinuous solutions. Comp. Mech., 17:98-106, 1995. 30. J. Yu, D. Peric and D.R.J. Owen. Adaptive finite element analysis of strain localization problem for elasto plastic Cosserat continuum. In D.R.J. Owen et aL, editors, Computational Plasticity III: Models, Software and Applications, pages 551-566. Pineridge Press, Swansea, 1992. 31. P. Steinmann and K. Willam. Adaptive techniques for localization analysis. In Proc. A S M E Winter Annual Meeting 92, ASME, New York, 1992. 32. M. Pastor, J. Peraire and O.C. Zienkiewicz. Adaptive remeshing for shear band localization problems. Archive of Applied Mechanics, 61:30-91,1991. 33. M. Pastor, C. Rubio, P. Mira, J. Peraire and O.C. Zienkiewicz. Numerical analysis of localization. In G.N. Pande and S. Pietruszczak, editors, Numerical Models in Geomechanics, pages 339-348. A.A. Balkema, 1991. 34. R. Larsson, K. Runesson and N.S. Ottosen. Discontinuous displacement approximations for capturing localization. International Journal for Numerical Methods in Engineering, 36:20872105, 1993. 35. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann, Oxford, 6th edition, 2005. 36. O.C. Zienkiewicz and J.Z. Zhu. The superconvergent patch recovery (SPR) and a posteriori error estimates. Part 1: The recovery technique. International Journal for Numerical Methods in Engineering, 33:1331-1364, 1992. 37. O.C. Zienkiewicz, B. Boroomand and J.Z. Zhu. Recovery procedures in error estimation and adaptivity: adaptivity in linear problems. In P. Ladev~ze and J.T. Oden, editors, Advances in Adaptive Computational Mechanics in Mechanics, pages 3-23. Elsevier Science Ltd, 1998. 38. O.C. Zienkiewicz and J.Z. Zhu. The superconvergent patch recovery (SPR) and adaptive finite element refinement. Computer Methods in Applied Mechanics and Engineering, 101:207-224, 1992. 39. O.C. Zienkiewicz and J.Z. Zhu. The superconvergent patch recovery (SPR) and a posteriori error estimates. Part 2: Error estimates and adaptivity. International Journal for Numerical Methods in Engineering, 33:1365-1382, 1992. 40. B. Boroomand and O.C. Zienkiewicz. Recovery by equilibrium patches (REP). International Journal for Numerical Methods in Engineering, 40:137-154, 1997. 41. B. Boroomand and O.C. Zienkiewicz. An improved REP recovery and the effectivity robustness test. International Journal for Numerical Methods in Engineering, 40:3247-3277, 1997. 42. O.C. Zienkiewicz, R.L. Taylor and P. Nithiarasu. The Finite Element Method for Fluid Dynamics. Butterworth-Heinemann, Oxford, 6th edition, 2005. 43. J. Peraire, M. Vahdati, K. Morgan and O.C. Zienkiewicz. Adaptive remeshing for compressible flow computations. Journal of Computational Physics, 72:449-466, 1987. 44. O.C. Zienkiewicz and J. Wu. Automatic directional refinement in adaptive analysis of compressible flows. International Journal for Numerical Methods in Engineering, 37:2189-2219, 1994.
Treatment of constraints - contact and tied interfaces
In many problems situations arise where the position of parts of the boundary of one body coincide with those of another part of the boundary of the same or another body. Such problems are commonly called contact problems. Finite element methods have been used for many years to solve contact problems. 1-32 The patch test has also been extended to test consistency of contact developments. 33 Contact problems are inherently non-linear since, prior to contact, boundary conditions are given by traction conditions (often the traction being simply zero) whereas during 'contact' kinematic constraints must be imposed to prevent penetration of one boundary through the other, called the impenetrability condition. In addition, the constraints must enforce traction continuity between the bodies during persistent contacts. In this chapter we consider modelling of the interaction between one or more bodies that come into contact with each other. Such contact problems are among the most difficult to model by finite elements and we summarize here only some of the approaches which have proved successful in practice. In general, the finite element discretization process itself leads to surfaces which are not smooth and, thus, when large sliding occurs, the transition from one element to the next leads to discontinuities in the response- and in transient applications can induce non-physical inertial discontinuities also. For quasi-static response such discontinuity leads to difficulties in defining a unique solution and here methods of multi-surface plasticity prove useful. 34 We include in the chapter some illustrations of performance for many of the formulations and problem classes discussed; however, the range is so broad that it is not possible to cover a comprehensive set. Here again the reader is referred to cited literature for additional insight and results. The solution of a contact problem involves: (a) using a search algorithm to identify points on a boundary segment that interact with those on another boundary segment and (b) the insertion of appropriate conditions to prevent the penetration and correctly transmit the traction between the bodies. Figure 7.1 shows a typical situation in which one body is being pressed into a second body. In Fig. 7.1(a) the two objects are not in contact and the boundary conditions are specified by zero traction conditions for both bodies. In Fig. 7.1 (b) the two objects are in contact along a part of the boundary
192 Treatment of constraints-contact and tied interfaces
i (a)
I.,......_
(b) Fig. 7.1 Contactbetweentwo bodies:(a) no contactcondition;(b) contactstate. segment and here conditions must be inserted to ensure that penetration does not occur and traction is consistent. Along this boundary different types of contact interaction can be modelled, a simple case being africtionless condition in which the only non-zero traction component is normal to the contact surface. A more complex condition occurs in which traction tangential to the surface can be generated by frictional conditions. The simplest frictional condition is a Coulomb model where, in a slip condition, the tangential traction is directly proportional to the normal traction. If the magnitude of the tangential traction is less than the limit condition the points on the surface are assumed to stick. Overall the frictional problem leads to a stick-slip-type response. We shall consider this condition in more detail later; however, first we consider the process of imposing a contact condition for the frictionless problem. Even in this form there are several aspects to consider for the finite element problem. In modelling contact problems by finite element methods immediate difficulties arise. First, it is not possible to model contact at every point along a boundary. This is primarily because of the fact that the finite element representation of the boundary is not smooth. For example, in the two-dimensional case in which boundaries of individual elements are straight line segments as shown in Fig. 7.2 nodes A and B are in contact with the lower body but the segment between the nodes is not in contact. Also finite element modelling results in non-unique representation of a normal between the two bodies and, again because of finite element discretization, the normals are not continuous between elements. This is illustrated also in Fig. 7.2 where it is evident that the normal to the segment between nodes A and B is not the same as the negative normal of the facets around node C (which indeed are not unique at node C). Before proceeding to methods appropriate for large deformation problems we consider the case where the nodes on one surface interact directly with those on the other surface and describe conditions that may be introduced to prevent penetration of the bodies.
Node-node contact: Hertzian contact
Fig. 7.2 Contact by finite elements. ii!i~i~i!i~i~~i~ !i~i~!~iii!i!~i~:~i~iil~ii7~ ~!~~ii~iiii~~i~iiiii!i lili~!ilii iili'~~~'~iii i~ili ii!i!iii ii i ililii i i iiilili ili!iii~~'~'~iiiliilili i!ii i~ililiii~iiiii!i i~iiiiili iliii!~i~ii ii ii ii iiiii!i iilil!i~~i!iiii lili liii!ii~~iiii~ il~iiililii lilii ili~!i!i~~~i!i!ii lii i~iiiilii!ili li!ilili li!i iiiii! ~i!iii!ilili~!i!i!i!iiiilil~i ilii i!i!~i~ ii!i ii i ilii!ii!iii~ilili ilii iili i ii ii!i!ii ii ii!iiili!i iliili!!ii !i!iiiii ilili!i ili ii iMi iiii iiili !ilii ililili!li!ili!i iii !iiii!i!!ii'i i i li!i i!i!i!iiii i lili iii!i i i i i i i ililili!i
7.2.1 Geometric modelling For applications in which displacements on the contact boundary are small it is sometimes possible to model the contact by means of nodes (this form is applicable to Hertzian contact problems35). For this to be possible, the finite element mesh must be constructed such that boundary nodes on one body, here referred to as slave nodes, align with the location of the boundary nodes on the other body, referred to as master nodes, to within conditions acceptable for small deformation analysis. Such conditions may also be extended for cases where the boundary of one body is treated as fiat and rigid (unilateral contact). A problem in which such conditions may be used is the interaction between two half discs (or hemispheres) which are pressed together along the line of action between their centres. A simple finite element model for such a problem is shown in Fig. 7.3(a) where it is observed that the horizontal alignment of potential contact nodes on the boundary of each disc are identical. The solution after pressing the bodies together is indicated in Fig. 7.3(b) and contours for the vertical normal stress are shown in Fig. 7.3(c). It is evident that the contours do not match perfectly along the vertical axis owing to lack of alignment of the nodes in the deformed position. However, the mismatch is not severe, and useful engineering results are possible. Later we will consider methods which give a more accurate representation; however, before doing so we consider the methods available to prevent penetration. The determination of which nodes are in contact for the problem shown in Fig. 7.3 can be monitored simply by comparing the vertical position of each node pair and, thus, a finite element model may be treated as a simple two-node element. Denoting the upper disc as slave b o d y ' s ' and the lower one as master body 'm' we can monitor the vertical gap given by g
=
x 2 -
,-m
x 2
=
-
X 2 ) -~- (bl~ - - b/2 )
(7.1)
Thus, the solution of each contact constraint is treated as g-
> 0; 0; < 0;
No contact Contact Penetration
(7.2)
193
194
Treatment of constraints - contact and tied interfaces
(a)
(b)
(c) Fig. 7.3 Contact between semicircular discs: node-node solution. (a) Undeformed mesh; (b) deformed mesh; (c) vertical stress contours.
We note that penetration can exist for any solution iteration in which the constraint condition is not imposed. Thus, the next step is to insert a constraint condition for any nodal pair (element) in which the gap g is negative or zero (here some tolerance may be necessary to define 'zero'). There are many approaches which can be used to insert a constraint. Here we discuss use of a Lagrange multiplier form, penalty approaches, and an augmented Lagrangian approach. 7'36
7.2.2 Contact models
Lagrange multiplier form
A Lagrange multiplier approach is given simply by multiplying the gap condition given in Eq. (7.1) by the multiplier. Accordingly, we can write for each nodal pair for which a contact constraint is assigned a variational term I-Ic = f r tr~(xS- xm) dr' ~,~ )~ng
(7.3)
c
where tr is the surface traction, x s is the position on the surface of the slave body, x m is the position on the surface of the master body,/~n is a Lagrange multiplier force and g is the gap given by Eq. (7.1). We then add the first variation of Plc to the variational
Node-node contact: Hertzian contact 195
equations being used to solve the problem. The first variation to Eq. (7.3) is given as
(~/~7 (~)kn]
(~II c -- (~)kn g "11-( (~as2 - (~a ~ ) ,,~n -- [(~a ~
--'~n g
(7.4)
and thus we identify "~n as a 'force' applied to each node to prevent penetration. Linearization of Eq. (7.4) produces a tangent matrix term for use in a Newton solution process. The final tangent and residual for the nodal contact element may be written as
0 l
0 --I
--
~d/~7 [,d,~n
=
,~n -g
(7.5)
and is accumulated into the global equations in a manner identical to any finite element assembly process. It is evident that the equations in this form introduce a new unknown for each contact pair. Also, as for any Lagrange multiplier approach, the equations have a zero diagonal for each multiplier term, thus, special care is needed in the solution process to avoid division by the zero diagonal. Of course in a contact state, one could select one of the parameters, say ~ , as a primary variable and directly satisfy the gap constraint by making ~ ' - :~. This approach is called c o n s t r a i n t e l i m i n a t i o n and may be used to reduce the number of overall unknowns. In the simple frictionless node-to-node contact case it is simple to implement as no transformations are needed to write the constraint equation. In a general case, however, the approach can become quite cumbersome and it is often simpler to use the Lagrange multiplier form directly or to consider other related approaches. If the global tangent matrix has its non-zero sparse structure defined for the case when all the specified contact elements are active (e.g. the tangent matrix defined by Eq. (7.5) can be inserted without adding new non-zero terms) then a full contact analysis may be performed using Eq. (7.5) when g < 0 and using the alternate tangent matrix and residual
[i0i] 0 0
~dt/~n td,~n
{0}
=
0 0
(7.6)
for nodal pairs when g > 0. However, if a large number of possible contact pairs are inactive (i.e. g > 0) it is more efficient to recompute the sparse structure of the global tangent matrix to just accommodate the active contact pairs (i.e. those for which g < 0). This step can be performed by determining all the active pairs prior to computing the tangent arrays.
Perturbed Lagrangian
The problem related to the zero diagonal may be resolved by considering a p e r t u r b e d L a g r a n g i a n form where 1
I-l c -- ,~n g - - ~
2
)k n
(7.7)
in which t~ is a parameter to be selected. As t~ --+ oo the perturbed Lagrangian method 9converges to the same functional as the standard Lagrange multiplier method. The first
196 Treatment of constraints- contact and tied interfaces
variation of 1-Ic becomes (~l"I c - -
(~)k n
(1) g -
- )k n /s
+
[(~/~
-
(~/42n1,~ n
(7.8)
and again we identify A, as a 'force' applied to each node to prevent penetration. Linearization of Eq. (7.4) produces a tangent matrix term for use in a Newton solution process. The final tangent and residual for the nodal contact element may be written as
i0 0 ll/d / { 0
0
dt/~'
1 -1
-1/t~
"~n - g Jr- )kn/~
--
d)~n
}
(7.9)
which is added into the equations in a manner identical to the Lagrange multiplier form. It is also possible to eliminate/~n directly from Eq. (7.8) giving /~n - -
/'~
Substitution into Eq. (7.9) and eliminating
I ~-~
(7.10)
g -- t~ (Y~ - Y~)
dan gives the reduced form
-~]~duS21=l
/~n
(7.11)
In a perturbed Lagrangian approach the final gap will not be zero but becomes a small number depending on the value of the parameter ~ selected. Thus, the advantage of the perturbed Lagrangian method is somewhat offset by a need to identify a value of the parameter that gives an acceptable answer. Indeed, in a complex problem this is not a trivial task, especially for problems involving contact between structural elements (e.g. rods, plates, or shells) and solid elements. This can be avoided in part by modifying Eq. (7.9) to read 0
0
1 -1
dti~'
-1/~
perturbed tangent
=
d)~n
A,
(7.12)
-g
Here this form is called a method and is a combination of the perturbed Lagrangian tangent matrix with the Lagrange multiplier residual. As such it is not a consistent linearization of any functional and there is some loss in convergence rate in solving the overall non-linear problem. Moreover, it is not possible to directly solve for )k n in each element and an iterative update must be used with
d/~n -- ~ (g + da~2- daT) /~ n <"- )k n "Jl" d ~n
(7.13)
In the above the incremental displacements are those from the last global solution; however, the same form may be used in Eq. (7.12) to give the reduced problem
[ ~-~ -~3 {duS2}-{ A~+~g
(7.14)
The method does, however, converge to a solution in which g approaches zero when is large enough. Thus the difficulty of selecting an appropriate value for e; is still not fully resolved.
Tied interfaces
Penalty function form
An alternative approach which avoids the difficulties of dealing with a zero diagonal from a Lagrange multiplier method is the classical penalty method. In this method the contact term is given by I-I - -
(7.15)
1 ~ ng 2
where ~ is a penalty parameter. The matrix equation for a nodal pair is now given by I t~-~ -~1 { d- t ~ g} } d=~ '{
~g
(7.16)
For the scalar problem considered here the penalty and the perturbed Lagrangian methods lead to identical reduced problems. However, when multi-point constraints are considered and independent approximations are taken for )~n and u the two methods are different unless the limitation principle is satisfied (i.e. the An includes all the terms in the expression for g). Thus, in practice, the use of the perturbed Lagrangian method is preferred. This is especially crucial for more complex methods in treating contact problems, such as mortar methods or other surface-to-surface treatments. 3~,32,37
Augmented Lagrangian form
A compromise between the perturbed Lagrangian or penalty methods and the Lagrange multiplier method may be achieved by using an iterative update for the multiplier combined with a penalty-like form. We write the augmented form a s 38 -~
~dti~'
An~ + t~g
(7.17)
where an update to the Lagrange multiplier is computed by using 7 )~kn+l = ,~ + t~g
(7.18)
Such an update may be computed after each Newton iteration or in an added iteration loop after convergence of the Newton iteration. In either case a loss of quadratic convergence in solving the global non-linear problem results for the simple augmented strategy shown. Improvements to super-linear convergence are possible as shown by Zavarise and Wriggers, 39 and a more complex approach which restores the quadratic convergence rate may be introduced at the expense of retaining an added variable. 2~ In general, however, use of a fairly large value of the penalty parameter in the simple scheme shown above is sufficient to achieve good solutions with few added iterations. In summary, we find the Lagrange multiplier form to be the only one that does not require the identification of an appropriate value for the n parameter. Furthermore, in a Newton solution algorithm the Lagrange multiplier form leads to optimal satisfaction of the impenetrability condition in a minimum number of iterations.
~iiii~iiiiiii~i~i~i~i~i~i~i~i~iiii~i~i~i~i}i~i~i~i~i~i~i!!iiiiiiii~i~i~i~i~i~i~i~i~i~i~iiiiii~iiii~iii~i~i~i~i!~!i~i!!~!ii~!i!ii~!iiiiiiiii~iii!i~ii~i~iiiii~!iiiii~i~i~i~i~!~!!i~i!~iiiiii!i~iiii~i~i~i!~!~i~!~!~!ii!i~i!i
Before describing generalizations to the above treatment of nodal contact problems we consider a technique to connect regions in which the finite element mesh is different
197
198 Treatment of constraints- contact and tied interfaces
B
Region 2
Region 1
Fig. 7.4 Tied interface for a two region problem.
in each region. A simple case of this type of situation is shown in Fig. 7.4 for a beam loaded by an end traction. The region near the load is described by a finer mesh than that used for the more remote portions. The model now requires the introduction of an interface to 'tie' the two parts together. Thus along the boundary 'AB' it is necessary to have xl I --" X21 and tl I +t21 =0 (7.19) AB
AB
AB
AB
where x ~ is the deformed position and t i the traction for the interface between the regions. To impose these conditions we may introduce the Lagrange multiplier functional I-Ii -- fF AT (xl -- X2) dl-'1 I in which we identify the multiplier as --
t 1
_
(7.20)
_t 2
We use a standard finite element approximation to define the positions Xi . For the approximation of )~ we can consider several alternative approximations. Here we consider the approximation ,~ ~,~ ~ N a (~))t a (7.21 ) a where Na are also standard shape functions. An approximation to the integral (7.20) may be given using quadrature points located at the nodes of region 1. Accordingly, we have I-ll ,~ ~ T (;~1 _ ~ Nb(~a)X 2) aa (7.22)
b
where ~a is the location on the surface of region 2 for the node a of region 1, Aa is the surface Jacobian for node a, and we assume unit quadrature weights. The location for each ~a can be obtained using a closestpoint projection where c-
1(~1_ ~
Nb(~)s b
Nc(,)s
r (~a1 _ ~ c
__ min
(7.23)
Tied interfaces
For an interface of a three-dimensional problem this gives the two equations Oc
T
b T =
-
gr
ONce2
gr
c
Oc
0~
ONc ~2 _
b
(7.24)
c -
c
where ~ and r/are the parent coordinates of the surface facet and g = x~1a - E b NbX2. For a two-dimensional problem we omit the second equation and consider only the parent coordinate ~. We note that T~ and T o2 are tangent vectors to the facet describing the surface of region 2. The equations (7.24) are in general non-linear and may be solved using a Newton method given as
Ad~ = - R
where d6, = (d~, d~7)r, R = (c,~, c,,7)r and A
~..
(T~Tr + k~)
(TffT,7 + k~,7)]
.(TrTr + kor
(T r T o +
k~)J
with
k~ - gr y ~ Nc,~f~2, k~,7_ gr ~ C
Nc,r
= k,7~ and k,m = gr ~
-2 Nc,~Xc
C
Once ~ is known the functional given in Eq. (7.20) may be satisfied using the Lagrange multiplier form or any of the other methods described above for the nodenode contact problem.
Example 7.1 Two-dimensional tied interface using
linear elements
Consider the example of a two-dimensional problem in which the edges of elements are linear segments. The interpolation for the positions is given by
1 -~)~ x 2 = N1 (~)~2 + Nz(~)ff2 = 5(1
1 + ~)~2 + 5(1
The tangent vector for the edge is given by
and is constant over the whole element edge. The closest point projection gives OC
0~
(~1
Xa
1
~(1
1
~2) T
~),Z~ ~{l+~)x~
is a linear relation in ~ and gives the solution
(~lXa __ 21(~2"[-~2))
T
T~
W~ 0
199
200 Treatment of constraints- contact and tied interfaces The interface functional for node a is given by I'I, "~ X T x~1a
_ NI(~a)]K~
N2(~a)X~)
Aa
where Aa is half the area of the one or two elements adjacent to node a. The variation of the functional gives -Nl(~a)Xa
(SX~a]
-N2(~a)*a_
Aa
g
and is used to defined the element residual vector. Similarly,
~]
/
0
0
0
- N 1 (~a)I
0
0 0
0 0
-N2 (~ )I I
-Nl(~a)I
-N2(~a)I
I
0
Aa
LdXaJ defines the tangent array. We note the structure of the arrays is identical to that obtained for the node-node contact treatment. Accordingly, any of the other solution methods may also be employed to formulate the interface arrays.
7.4.1 Geometric modelling A simple form for contact between bodies in which nodes on the surface of one body do not interact directly with nodes on a second body may be defined by a node-surface treatment similarly to that used in the previous section for a tied interface. A twodimensional treatment for this case is shown in Fig. 7.5 where a node, called the slave node, with deformed position s can contact a segment, called the master surface, defined by an interpolation Xm "-- ~ Na(~):K m (7.25) a
where ~ is equal to ~, r/in three dimensions and to ~ in two dimensions. This interpolation may be treated either as the usual interpolation along the boundary facets of elements describing the target body as shown in Fig. 7.5(a) or by an interpolation which
Node-surface contact
w
S Fig. 7.5 Node-to-surface contact: (a) contact using element interpolations; (b) contact using 'smoothed' interpolations.
smooths the slope discontinuity between adjacent element surface facets as shown in Fig. 7.5(b). A contact between the two bodies occurs when gn, the gap shown in Fig. 7.6, becomes zero. The determination of a contact state requires a search to find which target facet is a potential contact point on the master surface and computation of the associated gap gn and contact position ~ for each one. 6'40'41 If the gap is positive no contact condition exists and, thus, no modification to the governing equations is required. If the gap is negative a 'penetration' of the two bodies has occurred and it is necessary to modify the equilibrium equations to make the gap zero and to define the contact tractions (or nodal forces) that occur. The determination of the gap gn and the point ~ on the target (master) facet may be obtained by solving the constraint equation c ( g , , , ~ ) = ~,s
-
Xm (~)
-
g,, n = ~,s -
Na (,~)s - gn n = 0
~
(7.26)
G
where n is a unit 'normal' vector (i.e. n r n -- 1) that is defined relative to the master contact surface. This is an alternative to the closest point projection presented in the previous section and permits a more general treatment of contact. Thus, in a finite
Master
n
Fig. 7.6 Node-to-surface contact: gap and normal definition.
201
202
Treatment of constraints- contact and tied interfaces element approximation the vector n may be computed based on either the master or the slave surface but, in general, will not be normal to both. In Eq. (7.26), however, we define the contact state such that a normal vector always points outward from the master surface. In this way the normal gap is always the distance from the master to the slave surface. A Newton solution method may be used to find a solution to Eq. (7.26). Linearizing, we solve for iterates from
k (X,~ + gn n,,) d~ k + n k dgkn : c(g k, ~k) with updates
,,
=g,,+dg,,
and
-
(
(7.27)
+d~ff
until the constraint equation c is satisfied to within a specified tolerance. Alternatively, the problem may be split into two parts by premultiplying Eq. (7.26) by the transpose of lim and ti where nm is a unit normal to the master surface and ti are unit tangent vectors orthogonal to n (when defined on the slave surface n ~ nm). It is not necessary that these vectors be unit vectors but they must satisfy the orthogonality relations t/rn __ r m I T m[ ~0 n mX,~ ~J=t~ _ nmX,rI t~=t~ where X,~ m = Z
~m Na'~ xa
and X,r/ m : Z
a
Also note that
t~ does not need to be orthogonal to t2. gn
"= limT ( X s ~ y ~ a
Na,n Xa ~m a
g a ( ~ c ) X ~m a
(7.28)
In this approach we obtain
)/(liT")
(7.29a)
and ~Jc is found by solving
tf ~ = tf (~. - y ~ Na(~)f~ m)
(7.29b)
a
for i -- 1, 2 in three dimensions and i = 1 in two dimensions. In a general setting the equation for ~ is non-linear and a Newton method may be used to find a solution; however, once the contact point ~c is determined the expression (7.29a) for the gap gn is linear.
Normal and tangent vector definitions
As noted above n is defined here to be a unit normal vector that always points outward from the master body contact surface. The normal to a master surface for a threedimensional problem may be computed from the cross product of two vectors that are tangent to the contact surface. Accordingly, for a three-dimensional finite element facet we can use m Nm "- x,~ • x,rI (7.30a)
Node-surface contact 203
and in two dimensions m
Nm = x,~ •
ez
= Tm •
ez =
-ez
x Tm -
Ez Tm,~
z:I 0 0] -1
(7.30b)
where ez is a unit vector normal to the plane of deformation and ' x ' denotes the vector cross product. The unit normal is then defined by Nm
nm=
IINmll- (NTNm)1/2
where
IINmll
(7.31)
A 'normal' to the slave surface (with outward direction relative to the master body) may be determined by summing the normals to the elements surrounding a slave node s as
es
Ns -
~
-
,
XS~ e X
s'et I X,r
e=l
(7.32a)
~=~s
for three dimensions or 2 Ns'--
•215
~--~ x~e I e=l
(7.32b)
~=~s
for two dimensions. The unit normal is then obtained using Eq. (7.31) with Ns now
replacing Nm.
The variation of a unit normal vector is computed from Eq. (7.31) and given by ~n =
1
IINII
( I - n n ~) 0"N
(7.33)
where I is an identity matrix and N is computed from Eq. (7.30a) or (7.32a) for threedimensional problems and Eq. (7.30b) or (7.32b) for two-dimensional ones. The variation, 6N, is computed directly from the definition of N. We note from Eq. (7.33) that the unit normal is orthogonal to its variation so that n r 6n -- 0
(7.34)
For two-dimensional problems the variation of n may be simplified by noting the spectral decomposition of the identity matrix may be written as I = n nr + t tr
(7.35)
where t is the unit tangent orthogonal to n and ez. Thus, the variation of the unit normal vector in two dimensions may be written as 1 ~n- ~ t tr 6NIINII
1
IINII
t n r 0-I'
(7.36)
where from either Eq. (7.30b) or (7.32b) we note that 6N = Ez o"I'
(7.37)
204
Treatment of constraints- contact and tied interfaces
In computing ~ and tangent arrays for a finite element representation of contact it is necessary to obtain variations and increments of tangent vectors. These are computed in an identical manner to the normal vector. Thus, for a unit tangent we can write Ti) 6ti = 6
1
liT/II
liT/II
-
(I - tit T) t~Ti
(7.38a)
In two dimensions the use of the spectral representation of the identity simplifies the representation to ~t = ~
1
n n r 6I'
(7.38b)
Example 7.2 Normal vector to 2D linear master facets
Consider a two-dimensional problem modelled by four-node quadrilateral or threenode triangular elements in which the edges are linear segments and the interpolations for Na are given by N1 = i1( 1 -
~)
N2 = 1(1 + ~ )
and
The tangent vector to a master facet shown in Fig. 7.7(a) may be defined as m
Tm -- x,~ =
~(~, _
~,)
and, thus, the interpolation for x m may be written as x m (r = ~1 ( ~ 7 + ~ ' ) + Tm
The normal vector Nm : Tm • ez = EzTm
gives components N~n = T2m and N~' = - Tim. We note that in this case the vectors are constant over the entire facet. Unit normal and tangent vectors are given by Nm
Tm
tm
and
nm -- IINm II
-
IITm II
where we note that IITmII - IINmII. Thus, the relation (7.26) for c is linear in gn and and the solution is given by T gn -- nmg0
and ~c =
1 (/~,~n .q_ ~m
where
tmrgo tmrTm
=
~
go = s - ~ 1
T
1
t m g o -- ~ t m
IlTmll
IlNmll
x2 )
r
go
Example 7.3 Normal vector to 2D linear slave facets
If we consider the two-dimensional problem with edges of slave elements as shown in Fig. 7.7(b) a normal vector may be written using Eq. (7.32b) as Ns = [~(:~ - s
+ i
~(:~ - :i 2) x ez = Ts x ez - EaT~
Node-surface contact
1
Tm
n
2
(a) Normal to master facet
(b) Normal to slave node
Fig. 7.7 Node-to-surfacecontact: normal vector description. which gives components N~ = T~ and N~ = - T1s. Again, we obtain the vectors
Ns
Ts
n~ = IINsII
and
ts -
IITs II
which are independent of ~. The solution for the gap gn and contact point ~c are now given by n mT (/~s _
gn--
~m N1 (~c)Xl
_
_
~m N2 (~c)X2))
T llm lls
and ~c -
n mTg 0
--
T llm lls
ts~g0 tTTm
where rim, Tm and go are identical to the vectors defined in Example 7.2.
7.4.2 Contact modelling - frictionless case For a frictionless contact only normal tractions are involved on the surfaces between the two bodies; thus sliding can occur without generation of tangential forces and the contact traction is given by tr = ~ , n (7.39) where ~, is the magnitude of a normal traction applied to the contact target and n is a unit normal directed outward relative to the master surface. This case can be included by appending the variation of a Lagrange multiplier term to the Galerkin (weak) form describing equilibrium of the problem for each contact slave node. Accordingly, again using nodal quadrature on the slave surface we obtain
FIc -- f r grt dF ~ ()~nnr) (gnn) ac = )~n gn ac
(7.40)
where Ac is a surface area associated with the slave node and for the solution point of Eq. (7.26), denoted by ~c, the gap is given by
gn n -- Xs
-- xm (~c)
(7.41)
205
206 Treatment of constraints- contact and tied interfaces
In the development summarized here, for simplicity the surface area term is based on the reference configuration and kept constant during the analysis. Thus, the traction measure )k n is a reference surface measure which must be scaled by a ratio of the current surface area to obtain the magnitude of the Cauchy traction in the deformed state. Use of the Lagrange multiplier form introduces an additional unknown A, for each master-slave contact pair. Since a contact traction interacts with both bodies it must be determined as part of the solution of the global equilibrium equations. Of course, we again could eliminate the contact tractions by using aperturbed Lagrangian or penalty form for the constraint in a manner similar to that used for treating node-node contact. However, even then the problem is more complex as we do not know a priori which master facet interacts with a specified contact slave node. Thus, for each contact state a search to establish the active set of pairs is necessary in order to compute the non-zero structure of the problem tangent matrix. This implies that the non-zero structure of the tangent matrix will change during the solution of any contact problem and continual updates are required to describe the sparse structure (or profile) of the global matrix.
Contact residual
The variation of the potential given in Eq. (7.40) may be expressed as (7.42)
(~I'Ic -- (rgn An --F 6An gn) Ac
and used to compute the contact residual array. The result appears to be nearly identical to that obtained for the node-to-node contact. However, the relationship between the ~gn and 3~ terms with the 3g (or &i) is more complex and must be determined from a variation of the gap relation (7.26). Formally, this is given by n ~gn + T~ ~ + T~ ~ = ,Sf~s- ~
Na ( ~ ) ( ~ m
__
gn ~n
(7.43)
a
where T~m __ X,~ m --- y ~
ga,~ X~m a
and
T~
m! ~ ~ --" X,r
Na,r I X~m a a
with T~ and T~ being tangent vectors to the master surface at point ~. The expression for ~gn may now be obtained in the same manner used to compute g, thus, T (nmn) ~gn
= n mT ( ~ X s - - ~ N a ( ~ ) ~ Y , m a
gn~II)
(7.44)
This may be written in matrix notation as
3g,, =
(7.45)
~xTu n
where Un is a vector that defines the distribution of contact forces and s is a vector of coordinates for the set of nodes involved in the contact constraint equation (7.26). Thus the variation of Flc may be written in matrix form as
,nc-t
'
, n l n nac} gn A~
(7.46)
Node-surface contact
An alternate form to compute the variation of the gap may be given by multiplying Eq. (7.43) by the transpose of n to give
~gn : nT (~Xs - ~
Na(~) t~m
a
-
T~ ~ -
T~ ~7)
(7.47)
where we have used Eq. (7.34). In constructing models in which the normal is based on the slave surface geometry we will make use of this alternate form. The computation of d;~ proceeds by premultiplying Eq. (7.43) by the unit tangent vectors ti to obtain t/~T~ ~ + f i T : &l : f f ( ~ s
E
ga(~)(~xm) -- gn tf~n;
i : 1,2
(7,48)
a
We note that when n = n(~), its variation will contribute to the coefficients of 6~ and &7To make the steps clearer we consider next the formulation in two dimensions, first for the case where the normal n is computed from the master surface geometry and, second, the case where it is computed from the slave surface geometry.
Example 7.4
Contact forces for normal to 2D master surface
As an example we consider the case where the edge of a two-dimensional element is a linear segment as shown in Fig. 7.7(a). In this case the tangent vector may be taken as the Tm defined in Example 7.2 and gives the normal vector Nm --
Ezx,~ = ~1 Ez(s
- x ~m) 1 _ EzTm
With this definition the unit normal vector nm given by Eq. (7.31) is orthogonal to Tm and, thus, 6gn may be determined from 2
~gn -- nmT ((~ s -- ~-~ g a ( ~ 7 ) -- U nT ~X where Un
"~
( Nl,nm/ -N2(~)nm nm
a--1
and
6x --
6~' ~Xs
Indeed ~ need not be computed at this point. We will find, however, it is required in order to compute the tangent. The variation of FIc is now given by
(~FIc~[(~xT
Example 7.5
')~n]I)~nUnac ac
Contact forces for normal to 2D slave surface
As a second example we consider the case where the edges of a two-dimensional element are linear segments as shown in Fig. 7.7(b). In this case the tangent vector may be taken as Ts as defined in Example 7.3 and gives the normal vector
N s = ( ~ - ~ ) x ez = Ez ( ~ - ~ ) = EzTs
207
208 Treatment of constraints- contact and tied interfaces
Since Ts is independent of ~, its variation is given by yrs
=
-
Using this and previously defined quantities, we can define the variation of the unit normal as 1 T ~ ts W n (~X ~n, = - ~ ts n sr6Ts -IIN~]I
where 9 o o
1 w.
=
0 ns -ns
IINsll
and
~, -
~s ~ ~,~
From Eq. (7.44) we obtain the variation of the gap as ~g. = 6s (Vn -q--g. k. W.) = 6s U. where
- N 1 (~)nm -N2(~)nm
1
am
Vn -" nsTn m
and
kn --
tTnm nTnm
0
0 The variation of I-Ic may be expressed as
6I-Ic = [&2T ~An] {/~nUnAc gnAc } Example 7.6 Contact forces for normal to 2D slave surface - alternative form
In the previous two examples we obtained a form in which the distribution of the contact forces depends on the normal to the master surface Nm. For the two-dimensional problem in which element edges are straight line segments the normal to the master surface can have a sudden discontinuity between adjacent elements. This will lead to a discontinuity in the normal traction An when sliding from one facet to the next occurs. The effects of the discontinuity may be mitigated by modifying the definition of the variation of the gap in Example 7.5. Multiplying Eq. (7.43) by n sT we obtain ~ g . _ nsr (~s
-
N1 (~)~s
-- N 2 ( ~ ) ~ s
TTm~
- n~
Computing 3~ from the expression given in Example 7.3 yields 1
~ = tTTm [3ts~(is -- Nl(~)i~ - N2(~)i~) + (~if - N1 (~)3~]~T -1
--" ~ ( ~ i T
tsTTm
(Vt -~-
gn W n )
Nz(~)~i~T)ts]
Node-surface contact 209
where W, is defined in Example 7.5 and
Vt--
--N1 (~)ts -Nz(~)ts ts 0 0
The expression for the variation of g, may now be written in the matrix form ~gn = ~ T [Vs + kt(Vt -+-gnWn)] where
-- (~XTUn
-Nl(~)ns -N2(~)ns ns 0 0
Vs -
nsrTm
kt = t~s Tm
and
This form may be used to give the contact residual as above and when linearized leads to a symmetric tangent matrix for use in a Newton solution algorithm. If we restrict the variation to one where we set 3~ to zero, which is also an admissible variation, the distribution of contact forces is given by the form
~gn -" (~]KT~Jn
where
IJn = Vs
which yields the variation of I-Ic as
(~I-Ic = [(~T
(~'~n]
{/~nOnAc} gn ac
This form has a distribution dependent on the normal to the slave surface and with no forces acting on the slave nodes adjacent to s that are used to compute the normal. This is physically more realistic; however, we shall find that the tangent matrix will not be symmetric. This is a distinct disadvantage unless friction is also included where, normally, the tangent matrix also will be unsymmetric during slip.
Contact
tangent
To compute the contact tangent array we linearize the variation of the potential ~l-Ic and obtain (7.49) d(3I-lc) = (~gn dan + (~/~n dgn + )in d(~gn)) Ac Except for the last term the structure is identical to that obtained for the node-to-node contact problem. The increment to the normal gap, dgn, is obtained from Eq. (7.44) by replacing the variation 3 by the increment d. To obtain a symmetric tangent the computation of d(~gn) proceeds from Eq. (7.43) as T (nmn) d(3gn) = - (3nrnm) dgn - 3gn (nmdn) r
-
gn (n rd(~n))
-
__ ~-~((~T Na,~,)(nm~ d~,m) - Z((~XamT nm)(Nf~d~) a
--
~T
a T
m
(nmX,~) d ~
(7.50)
210 Treatment of constraints- contact and tied interfaces
where
a(Na.-
[ae.
and
Na.,}Na'
{ "
T m (nmX,m~) (nmrx,~)] { d { }
~r(nmr x,~) d~ - [~, &7] .(nmrX,~) (nmrx,%o)] In matrix form
d(~g.)
--
(~]~T
dr/
Ko d~,
(7.51)
The final form for the tangent may be written as
t~i'T ~/~n]
d(~l-lc)--[
where
'
~gn = ~ r On
n]
[t~nKG L
and
,7.,2,
dgn = Urn d~
For cases where U, - U, the tangent will be symmetric. To illustrate the process we again consider the two dimensional problems.
Example 7.7 Tangent for 2D linear master surface
For this case we note that
x,~ m
0
__
and
~nrnm --
nmdn - 0
so that Eq. (7.50) simplifies to T ~ d((Sgn) -- -- ~ n m
(ga,~
d~,m) -- ~(r
a :
gn nmr
T d T m -- o~rTnm d ~ m
--~n
ga,~) nm
d~
-- gn nmT d(~nm)
a
d(6nm)
where since the Na,~ are constants OCTm -
~
Na, ~ (Sx7
and
dTm -- ~
a
Na,~ d i m a
Using Eqs (7.33) and (7.34) it is easy to show that nrmd(6nm) =
1
]]Nmll20~mr (tm tm r)
1
d N m --
r (nm n m) d T m liNml[2 ~Tm T
In matrix form 6~ is given by 1
~ = IINmII where
Vt =
(Vt+gnDn)(~'x--
--N1 (~)tm} -N2 (~)tm tm
D~
---
1
IINmII
Vh~s
1 {am} 2 IINmII
am
0
Node-surface contact
and s is as defined in Example 7.4. Thus, the geometric stiffness term may be written as
KG --
-- Vh
D~n -
Dn V~ + gn Dn D~n
which is symmetric. The final tangent is given by Eq. (7.52) with IJn equal to the Un given in Example 7.4.
Example 7.8 Tangent for normal to 2D slave surface
For this case Eq. (7.50) simplifies to T
T
(nmns) d(~gn) = - ~ n m d T _ OeTmnmT d~ - gn nmd(~ns) r - t~gnnmdns - ~n~ nm dgn Using the definitions for
Vn
and
W n
from Example 7.5 we obtain
1 T T 7" nmdn~ = k,, W n df~ nmns 1 r dTm = D m r ds nm T nmns
and
d~ --. ( V T .-Ji-gnkn W n ) dv~ : V T dx with similar expressions for the terms with variations. In the above -Nl(~)ts -N2(~)ts ts 0
1 Vt = r nmns
--rim
-m
1 and Dm -T 2 nmns
0 0
0
and ~ nmrd(Jn~) = T nmns
1 6Tr(ns r k~(ts r ) IINs 112 ns ns + ns tsr) dTs
In matrix notation we may write the geometric term as Kc = - Dm V T - Vh Dm r + kn ( W n
U T -- Un
W T)
']- gn ( W n W T -- kn ( W t Wn + Wn W t ) ) where Wt m
IINsll
0 0 0 --is
ts Again the geometric term is symmetric and with U~ Eq. (7.52) is also symmetric.
--
Un
the tangent given by
211
212 Treatment of constraints - contact and tied interfaces
Example 7.9 Tangent for normal to 2D slave surface - alternative form
The tangent follows from Example 7.6 where
<~gn- nrs (<5s
NI(~)~x~n - N2(()<~x~)
-
(~xT~Jn
-
and gives
d(,gn) : dnrs ( ' f ~ s - NI(~)'~,~'- N2(~)'s
- 89 r ('s
'~')d,
For this case we have
dn~ = - ~
1
IIN~II
ts nrs dTs - -- ts Wrn d i
with W, given in Example 7.5 and d~ - V ~ d i as described in Example 7.8. Thus the geometric term is given by =
where Vt-
-v,
-Nl(~)ts -N2(~)ts ts 0 0
Wn
and
--
1 Ds - ~
-ns ns 0 0 0
with Vt defined in Example 7.8. The tangent matrix is given by Eq. (7.52) and is now clearly unsymmetric due to the form of Un and Un as well as the form of K~. All of the above forms suffer some solution irregularity during sliding from one facet to another. In the form where normals are defined relative to the master surface an expedient solution is to use concepts from multi-surface plasticity to define a 'continuous' approximation for the normal. This leads to additional considerations which are not given here and are left for the reader to develop (see reference 21, 40 or 41). It is also possible to 'smooth' the master surface using continuous interpolation across facets. 41-43 Extension to three-dimensional problems is straightforward and involves addition terms related to second derivatives of shape functions unless three-node triangular facets are used. Extension to include frictional effects is described next for a simple Coulomb model. For other models the reader is referred to the literature for additional details.34,40,41,44-57
7.4.3 Contact modelling - frictional case For frictional contact both normal and tangential tractions are involved on the surfaces between the two bodies. Thus, the contact traction for a three-dimensional problem is expressed as 2 A : ,~n n + Z / ~ t i ti -- An n + "r (7.53a) i=l
Node-surface contact
where ti form a pair of tangent vectors on the contact surface. For a two-dimensional case this simplifies to A, --- ,~nn -k- )~tt (7.53b) In a finite element setting the tangent vectors are obtained during the computation of the normal as described above. For example, in a three-dimensional problem in which the normal is defined for a master facet the tangent vectors may be taken as Tm
-
x ,m ~
and
T m 2 -
(7.54)
x , m7
with unit vectors defined in the usual way. In general the tangential traction components are dependent upon the magnitude of the normal traction as well as upon the amount of sliding, lubrication, temperature and other effects that occur. For a detailed description of various models the reader is referred to references 58, 40 and 41. Here we restrict our attention to a simple Coulomb model in which the tangential forces satisfy a stick-slip behaviour. For a stick condition the solution is obtained using the contact functional 2
FIs t - AT g st a c -
)kti t f g S t ) a c
(An nTgst-+ - Z
(7.55)
i=1
In the stick case the total gap, gSt, is determined from the constraint equation
C--X s _ E ga(~o) x m - gSt _ 0 a
(7.56)
in which ~0 defines a fixed point on the surface at which either initial contact is made or a previously sliding state stops. In a discrete setting in which a solution is sought at time tn+l the value of ~0 = ~n, the value obtained from the solution at time tn. For a stick state a variation of the gap is given by
~gSt : ~'Xs -- Y~ Na(~o) ~,ma a
(7.57)
The residual equations for a three-dimensional problem are obtained from 2
(~I"ls t - ((~,)kn(nTg st) at- An ~(nTg st) d" E (~)~ti(t/re st) + i=1
2
~
Ati 6(t/reSt)) ac
i=1
~n ~js, + )kt 1J~J~'~ + ~t2~J~
_ [(~T (~n (~,~t1 (~,~t2]
in which ~n and plifies to
"~ti are
gn gtl gt2
Ac (7.58a)
Lagrange multipliers. For the two-dimensional case this sim-
t~[-lSt._ [~T
t~An t~At]
(~n~JSnt-JI-)~t~JttI Ac gn gt
(7.58b)
213
214 Treatment of constraints-contact and tied interfaces
Alternatively, a perturbed Lagrangian form may be written and the contact forces may then be computed directly in terms of the gap relations. Since the multiplier forces are scalars this often leads to a form which is identical to a penalty method. In a simple Coulomb friction model a stick state persists whenever
II,t~t II = (A~ 'r
1/2 <
#
IAn I
(7.59)
The 'friction' parameter # is a positive material constant that depends on the properties of the contacting surfaces. When IIXt II reaches its limit value 'sliding' occurs and the tangential tractions are given by Xt = - # A n
g;l II1~/11
Agll
~ -#A,
(7.60)
IIAg~/ll
in which for a discrete time increment At -- t.+l - t., the gap increment is defined by Ag~l -- gt I (tn+l) - gt I (tn). AS shown in Fig. 7.8 an increment of tangential slip may be defined by 41 m m m dg~/ = x,~ d~ + x,, d~7 = T 1 d~ + T 2 d~7 (7.61) The incremental slip is then given by Ag~/ --
l
tn+l
dg~/
(7.62)
d tn
Thus, the movement of the contact point on the surface is described by the parameters and r/. The location for every contact point is described by Eq. (7.26). Accordingly, during persistent contact we will always have t/~ Ixs
-
:
x m(~)]
(7.63)
0
from which differentials, variations and increments may be computed. For a finite element surface approximation we can compute 3~ from Eq. (7.43) and the other quantifies using the same expression with variations replaced by the appropriate terms. I
/ / / /
.
T';
./-" ~ ~ y 9
~'c+ d~,
~_...._~.-. Z......... s
~'
/ /
/ Fig. 7.8
Increment of tangential slip.
/
9
P,c
T';'
'
=,,-
~
~
.
Node-surface contact 215
In a discrete setting the first iteration in each time (load) increment is assumed to be a stick state. If at the end of the iteration liXt II > mlA, I the state is changed to slip for the next iteration. In many instances some of the conditions assumed as slip will give solutions in which it is necessary to change the state back to stick. Indeed, due to the discrete nature of the solution process and that of the finite element approximation for the contacting surfaces it may occur that there is no fully consistent solution. In such cases it may be necessary to 'accept' a solution after a set number of iterations (provided the 'error' is sufficiently small). A contact functional II st does not exist for a sliding state; however, a Galerkin equation for the contact equations may be written directly as
(~l"Isl -- ((~/~ngsl .~_(~gS,/~n "lt- ~
2
~
2
(~gtS!Ai j/~tj) Ac
i=1 j=l
--" [SxT 5/~n] { /~n~'Jsl "ll-/~tl~JTlgn + where
Aij
"--
(7.64)
At2~J~l} Ac
fftj and
5g;IA,i + 6gt~Azi - 5s
i = 1, 2
The normal component terms depend on the current value of ~ and are computed in an identical manner to that described above for the frictionless case. When sliding occurs the tangential tractions )~ti are not independent Lagrange multipliers but are computed from the Coulomb model in terms of the normal component and the geometric properties defining Agff. The tangent matrix for a stick-slip behaviour is computed in two parts. For the stick state we compute the tangent terms from Eq. (7.58a) and for the slip state from Eq. (7.64). The tangent for a stick state has the form
ws,
, ,211 Un 's' U T'st
0
uL 's'
o
L
o 0o
rJsAl
d~,
d)~t d /~t:
o
AC (7.65)
where
.r K~ -- /~n K~n --]--/~tlK~I +/~,2 K~2 with
d(5(nrg)) "-- e~T-rst OX l~Gn dx T
and
d (5(t[ g)) - 5s r K~i ds v
For the slip case the final tangent has the form
d(,m) = [a,z~
[(~.
L
Kg)
uT'sl
~ 13't)~{ j ds }
(Os, + a#O~ I + p# ,2 | 0
d/~n
Ac
(7.66)
216
Treatment of constraints- contact and tied interfaces
where c~ and/3 are proportions of the Atl and A,2 components satisfying Coulomb relation for slip. In a two-dimensional problem/3 = 0 and c~ = 4-1 with the sign depending on the direction of sliding. To illustrate the process of computing the tangent for each state we again consider some two-dimensional examples.
Example 7.10 Residual and tangent for stick - normal to master surface
The gap for the stick case is given by
gSt _ Xs -- N1 (~o)~,~n -- N2(~o):~n
and its variation by
6gst - 6~s - N1 (~0)6~,~' -- N2(~0)6~n with a similar expression for dg st. From Eqs (7.58a) and (7.58b) we obtain for the normal residual ~ ( n mTg ) -- ~gTnm -- ~ o r T T n m
tmrg = 6 i T (Vno - gt Dn) = 6XTUno
IINmll
where D~ is defined in Example 7.7. Similarly, for the tangential residual we obtain 1
T
~(tmg) = ~grt m + where in the above
(
Vn0 --
IINmll
T = ~iT (Vto -l- gn Do) = ~s ~TTnm nmg
- N 1 (~o)nm ) N2 (~0) nm nm
N1 (~0) tm
and
V t o - { - N2(~0) tm tm
The geometric term for the normal tangent is given by d (~n m T g)
-
-
T + d/gTdnm + gT d (~nm) ~Snmdg
where T
~nmdg -
T
IINmll
T
T
~Tmnm tmdg = - ~,~rDnVto ds
and
1( r T T T dTm) d(~nm) = IINm II2 tm 6Tm ~ ( n / t m +nm t m) dTm - nm ~Tmnm n m = tm ~s (DoDf + DtDnr) d~ - nm &~rDnD~ d~,
Node-surface contact
where I)t = 2 IINm II
A similar computation for the tangential geometric stiffness gives d (~tm~g) - ~tm r d g + ~gr d tm + gr d (~tm) with T T CStm d g -- 6xT Dn Vno d,2
and d(~tm) - - nm aX,r ( D n D f + DtDnr ) dff, -- tm aXrDnDnr dff,
Thus the final form of the geometric stiffness is given by )k" K ~ -- -- )kn (Dn Vt~ -t-- Vt0Dn) -!- /~t (Dn VnT0 + VnoDn) + (/~ngt -- /~tgn)(D~Dt r + DtDnr) - ()~ngn q" Atgt) Dn DT
The residual for stick may be written in matrix form as
aFlc- [&27" aA,, t~)kt]
)kn Un0 + )kt Uto }
gn gt
and the tangent as
d ( r n c ) - [6~r
/
6~n
[
Uno Uto o o
o/
Example Z 11 Residual and tangent for slip - normal to master surface
The residual and tangent for the normal gap and normal traction component is identical to that for a frictionless behaviour. Thus, the residual is the same as that presented in Example 7.4 and the tangent as given in Example 7.7. For the tangential behaviour, the variation of the tangential slip is given by ~ g f l - ~ IITmll- 6 t ~ ( ~ s - ~
a
N . ~ ) - (~Y~s- ~
= ~ r V h _ &~r (V, + gn D~)
a
N.~s
217
218 Treatment of constraints-contact and tied interfaces
where, noting that IlNmII -- IlTmII, the vectors for Vh, Vt and Dn are defined in Example 7.7. Expanding this relation gives
d(ag/) - atom(ds - ~
Nads
- Tm d{) + (as - E
a q- (Xs -- E Naxm)Td(atm) a
a
Naas
Inserting the definitions for terms and writing in matrix notation gives
d (~gt t) = t$~rK~td,~ with in which
K~t -- Dn VT + Vn DT -- DtVff - gn(Dn DT + DtDn~) ~ Dt
1 { -tm }tm
IINm II
Thus, the final form of the residual for slip is given by
and that for the tangent is
d ( tSl-Ic) -- [(5~ T (~)kn] ['~ "UTnKSGI (Un "~"0~ #Ut)] /dd~~l where c~ = + 1 depending on the direction of sliding and A. K~ =/~n (K~n -if-o~# K~t)
~i!iiiii~ii!i!~i~!iiii!~iii~iiii~iiii~i!~iiii~i!i~iiiii~i~iiiii~ii~iiiiiii~iii~ii~iiii~iii!~iiiiiiiiiii!iiiii!ii~i!ii!iiiii~i~!~iii~ii~!i!iiii~iiii~i!~iii~iii~iiiii~iii~iiiiiiiiiiiii!iiii~ii~iii!i~iiiii!iii!iiii~iiiii!i The above treatment for contact may be generalized to a surface-to-surface treatment in which behaviour over a facet on the slave body interacts with one or more facets on the master body. An early attempt at defining appropriate segments for twodimensional applications was presented by Simo et al. 9 and for three-dimensional ones by Papadopoulos and Taylor. 59 Both of these approaches had practical difficulties in large deformation and sliding situations. More recently developments have utilized so-called mortar methods. Mortar methods have their roots in domain decomposition in which subdomains are joined using appropriate tied interfaces as introduced in Sec. 7.3. Mortaring relates to how the Lagrange multiplier is approximated such that accuracy and stability are maintained. A brief description for tied interfaces is given in Chapter 12 of reference 60 and a detailed mathematical presentation is given in reference 61. The basic treatment for contact problems relies on a proper definition of the
Numerical examples 219 gap relation and appropriate quadrature over the contact surface facets. At the present time the work of Puso and Laursen 31'32'62 presents the most comprehensive treatment for this approach. The details of the construction are quite involved and the reader is referred to the cited papers for the additional details and results. iiiiiiiiiiii,,iiii~~~i~~ii!6!, iiii~~ii~~ii i~i i,~,,~!~!UW~,,,~ iiiiiii ~i~ililili i i iiliiii i iliilii i i iilili i i iii i !iiil|ili~i i~li i li!iliMmili i i iiiipii i i i il!ili i i ili~ili~l~iili ii i!ii!ili li i~i !ililiiiiiiiiiii!iiiiiiii li~ ~iiiliiiiiiiiiiiii ~ iiiiiiiliiiiiiiii!iil ~ i•iiii•iii••i•••••iiii•i•••iiiiiiiiiiiiiiiiiiiiiii•i•,i~ii,,i,i•,i,ii!•ii•ii,,,,,,,~, ii•••••i~,,,, ii•i•~i•••••i•~iii ~ ! iiiii~ii iiii i i i i i i iiil iiliiiii ililiii i
7.6.1 Contact between two discs As a first example here we consider the contact problem previously solved using a nodenode approach. In that case we observed a small but significant discontinuity between the contours of vertical stress between the bodies, indicating that traction is not correctly transmitted across the section. Here we use the node-surface method given above in which the contact area of each body is taken as the boundary of elements. The solution is achieved by using a penalty method and a two-pass solution procedure where on the first pass one body is the slave and the other the master and on the second pass the designation is reversed. This approach has been shown to be necessary in order to satisfy the mixed patch test for contact. 33 The results using this approach are shown in Fig. 7.9. For the solution, the two-dimensional plane strain finite deformation displacement element described in Sec. 5.3.2 is used with material behaviour given by the neo-Hookean hyperelastic model described in Sec. 6.2.1. Properties are: E = 100 000 for the upper body and E = 1000 for the lower body. A Poisson ratio of u = 0.25 is used to compute Lam6 parameters /~n and #. As can readily be seen in the figure the results obtained are significantly better than those from the node-to-node analysis.
7.6.2 Contact between a disk and a block .
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As a second example we consider the interaction between a semicircular disk and a rectangular block. The disk has a radius of 10 units with the material modelled by a neo-Hookean material with initial modulus E = 100 and Poisson ratio u = 0.25.
Fig. 7.9
Contact between semicircular disks: vertical contours for node-to-surface solution.
220 Treatment of constraints-contact and tied interfaces
..=--.,.=-=.--.-.nl====...m==.,.....m=====u=.=lmm.....=l==.u=.=mmmm.,m..===. I I I I I I I I I I I I I I I I I I I I I I l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l ,......................,..,.................,.,...,.....,..................,., , . u n H . H n n l n n m n m . . . . . . . N n . n u m u H . . . . , . . N l u l l i n , u l . n i u N . . . = , , = = l = l n u n n n l m m n n l n u m = l l , n l , l H n m n m , l n H i n l l n n n n l n u l l D n H m l IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIlllllllllllllllllllll l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l IIIIIIIIIIIIIIIIIIIIIIIIIIIlllllllllllllllllllllllllllllllllllllllllllllllllll I I I I I l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l I l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l I I I l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l i l l l l l l l l l l l l l l ,........,.......,..,.,...............,...........,...............,...,....,., I I I I I I = l l , l l l l l l l l l l l l l l l l l l l l l l l l l l , l l l l l l l l l l l l l l l l l l l l l l l l l l = l l l l l l l l , l l l l l I I I I I I I I I I I I I I I I I l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l = l l , l l l l l l l l l l l l l l l l l l l I I I I I I I I l l l l l l l l l l l l l , l l l l l l l l l l l l l l l l l l l l l l l l = l l l l l l l l l l l l l l l l l l l l l l l l , l l i l l l I l l l l l l l l l l l l l l l l , l l l l l l l l l l l l l l l l l = l l l l l l l l l l l l l l l l l l l l l l , l l l l l l l = l l l l l l l l l l l ,.....,.........,..,..,...,....................,.,.........................,., I I I I l l l l l l l l l l l l l , l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l , l l l l l l l l l l l l l l l l l I I = l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l , l l l l l l l l l l ,i,,==,,,,.,.,===,....=,,,,..,,,,,,=,=.,,=.,.......,,,,.,=,,,,,,,...,,.=...,., :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ,N...,.=.Hu=..in.===nnnu,..====.......,n..=.H=mnm...=l.=......=n.==, I=..=HN=...=nl.......=.U==..a.=n=l..=...=..........n=H,=.Un.===,=...,., ,...............=........•..=.......=.........,=........=........=......==..., IIIIIIIIIIIIIIIIIIIIIl,llllllllllllllll=lll,llllllllllllllllllllllllllllllllll .......................=.•...=...............=.................=......=....... ,.=.=m,==N===.=.................=...===,,==...=..,......===.=...=..==,,...,=, ,=---..-=.-.-.=-=,.=..-=..=.=.....=.=........=.......=n.=......,......u==,., ,,,,,,=,,,=,,,•,,,,=,..,,,,,,,,=,,,,,.,,,=,,==..=,.,,,,,=,=,,,,,,=,=,,,==,•,=, ,.,......,,,,,,,.....=,...........,..,,=,=,................,,...,.===,,=....., Ilillilllimlillllllmllllmllllllllmlllilmlllllmlllllmmllllll=llllillmimilllllll ,..==,.H..=.===......====.....,,.=m=..............=.=...==.,===...=..==....., ,...==.=.=.....................=..=.......=..............•.....=.............,
l'==llllllllllllB,llllllllllll=llllllll==ll,llll=lllll,lllml=lllllllll=,llllll
(b) Deformed mesh
(c) 0"1 stress
(d) 0"2 stress
Fig. 7.10 Contact between a disk and a block- frictionless solution.
The block has a width of 30 units and a height of 15 units. The block is also modelled by a neo-Hookean material with initial modulus E =- 10 and Poisson ratio L, -- 0.45. An initial gap of 0.2 units exists between the disk and the block. The disk is punched into the block by an overall vertical motion of 4 units, thus leading to a large deformation of the block. A solution assuming frictionless contact was achieved using the Lagrange multiplier approach in 40 steps by imposed displacements along the top of the disk. In the analysis the disk is treated as the slave body. The initial and final configuration of the mesh used in the analysis is shown in Fig. 7.10 along with contours for the major and minor principal stresses (a coarser mesh is used to allow display of the mesh and the contours on the same figure). In Fig. 7.11 we show the pressure distribution obtained from the Lagrange multipliers acting on the slave nodes in contact. The result has a small oscillatory behaviour despite the rather fine discretization on the contact surface. This is characteristic of the action between slave nodes and the projection points on the master surface and emphasizes the need for improved treatment by surface-to-surface methods. Nevertheless, it is possible to obtain necessary engineering results for design from such treatment and today the node-to-surface form is the one available in most research and commercial computer programs.
Numerical examples lO
(9 L
=
7
6
(U i_
m 5 I
~- 4
0
-40 Fig. 7.11
-30
-20
-10 0 10 20 0 - Angle (degrees)
30
40
Contact between a disk and a block - contact pressure.
7.6.3 Frictional sliding of a flexible disk on a sloping block As an example which includes friction we consider the forced motion of a flexible half disk along a sloping disk. The disk is fixed along its top edge which is subjected to the imposed displacement L/1
-
-
t; 0 < t
and
u2 -
t; 4;
O_
The disk has a radius of 3 units and is composed of a neo-Hookean material with material parameters E = 8 and u = 0.35. The sloping block has a width of 10 units with a height of 10 units at the left side and 5 units at the fight side and is also modelled by a neo-Hookean material with parameters E -- 20 and u -- 0.25. The centre of the disk is initially located at x] = 4 and x2 = 11.6. The solution is obtained using Lagrange multiplier form with a Coulomb friction model of # = 0.0 (frictionless) and # = 0.2. A set of views of the deformed positions is shown in Fig. 7.12 and the resultant force history in Fig. 7.13. The peak normal force occurs at the time where the vertical motion becomes constant. Note that friction slightly increases the value of the peak load.
7.6.4 Upsetting of a cylindrical billet To illustrate performance in highly strained regimes, we consider large compression of a three-dimensional cylindrical billet. The initial configuration is a cylinder with radius
221
222 Treatment of constraints- contact and tied interfaces
-..__
L (a) Initial configuration
(b) Position at t = 2
(c) Position at t = 4
(d) Position at t = 6
? Fig. 7.12 Configurationsfor a frictional sliding.
r = 10 and height h = 15. The mesh consists of 459 eight-node hexahedral elements based on the mixed-enhanced formulation presented in Sec. 5.6. The billet is loaded via displacement control on the upper surface, while the lower edge is fully restrained. A Newton solution process is used in which the upper displacement is increased by increments of displacement equal to 0.25 units. To prevent penetration with the rigid base during large deformations a simple nodeon-node penalty formulation with a penalty parameter k = 106 is defined for nodes on the lower part of the cylindrical boundary. A neo-Hookean material model with )~n "-- 104 and # = 10 is used for the simulation. Figure 7.14 depicts the initial mesh and progression of deformation.
Numerical examples 0 -2
-2
-4
-4
-8
2_ ~-a
u-~ -10
-10
-12
-12
L
o -~ -6
E
0
Z
I
-14
Fig. 7.13
Fig. 7.14
o -6
1
2
3
4 5 t - Time (a) Frictionless
6
7
-14
0
1
2
4 5 6 t - Time (b) Friction/z = 0.2
Resultant force history for sliding disk.
V= 0
V = --I
V ~ ~2
V ~ --:3
v = -4
V .-~ "*~5
V~ ~:6
V ~ --7
Initial and final configurations for a billet.
3
7
J
8
223
224 Treatment of constraints- contact and tied interfaces
In the preceding three chapters we have presented methods to i m p l e m e n t basic strategies for solving general finite deformation problems in solid mechanics. A variational structure easily implemented for both two- and three-dimensional problems which can include nearly incompressible behaviour has been given. In addition we have shown how various constitutive models can be included to represent elastic, viscoelastic and elastic-plastic behaviour. Finally in this chapter we have included an introduction to constraining interactions resulting from intermittent contact between contiguous bodies. The formulation of methods to model contact is an active area of research and m a n y new procedures are being introduced. The reader is encouraged to consult recent research on the topic.
1. S.K. Chan and I.S. Tuba. A finite element method for contact problems of solid bodies - Part I. Theory and validation. Int. J. Mech. Sci., 13:615-625, 1971. 2. S.K. Chan and I.S. Tuba. A finite element method for contact problems of solid bodies- Part II. Application to turbine blade fastenings. Int. J. Mech. Sci., 13:627-639, 1971. 3. J.J. Kalker and Y. van Randen. A minimum principle for frictionless elastic contact with application to non-Hertzian half-space contact problems. J. Engr. Math., 6:193-206, 1972. 4. A. Francavilla and O.C. Zienkiewicz. A note on numerical computation of elastic contact problems. International Journal for Numerical Methods in Engineering, 9:913-924, 1975. 5. T.J.R. Hughes, R.L. Taylor, J.L. Sackman, A. Curnier and W. Kanoknukulchai. A finite element method for a class of contact-impact problems. Computer Methods in Applied Mechanics and Engineering, 8:249-276, 1976. 6. J.O. Hallquist, G.L. Goudreau and D.J. Benson. Sliding interfaces with contact-impact in large scale Lagrangian computations. Computer Methods in Applied Mechanics and Engineering, 51:107-137, 1985. 7. J.A. Landers and R.L. Taylor. An augmented Lagrangian formulation for the finite element solution of contact problems. Technical Report SESM 85/09, University of California, Berkeley, 1985. 8. P. Wriggers and J.C. Simo. A note on tangent stiffness for fully nonlinear contact problems. Comm. Appl. Num. Meth., 1:199-203, 1985. 9. J.C. Simo, P. Wriggers and R.L. Taylor. A perturbed Lagrangian formulation for the finite element solution of contact problems. Computer Methods in Applied Mechanics and Engineering, 50: 163-180, 1985. 10. J.C. Simo, P. Wriggers, K.H. Schweizerhof and R.L Taylor. Finite deformation post-buckling analysis involving inelasticity and contact constraints. International Journal for Numerical Methods in Engineering, 23:779-800, 1986. 11. K.-J. Bathe and A.B. Chaudhary. A solution method for planar and axisymmetric contact problems. International Journal for Numerical Methods in Engineering, 21:65-88, 1985. 12. K.-J. Bathe and P.A. Bouzinov. On the constraint function method for contact problems. Computers and Structures, 64(5/6): 1069-1085, 1997. 13. J.J. Kalker. Contact mechanical algorithms. Comm. Appl. Num. Meth., 4:25-32, 1988. 14. N. Kikuchi and J.T. Oden. Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, volume 8. SIAM, Philadelphia, 1988.
References 225 15. H. Parisch. A consistent tangent stiffness matrix for three dimensional non-linear contact analysis. International Journal for Numerical Methods in Engineering, 28:1803-1812, 1989. 16. D.J. Benson and J.O. Hallquist. A single surface contact algorithm for the post-buckling analysis of shell structures. Comp. Meth. Appl. Mech. Engr., 78:141-163, 1990. 17. T. Belytschko and M.O. Neal. Contact-impact by the pinball algorithm with penalty and Lagrangian methods. International Journal for Numerical Methods in Engineering, 31:547572, 1991. 18. R.L. Taylor and P. Papadopoulos. On a finite element method for dynamic contact-impact problems. International Journal for Numerical Methods in Engineering, 36:2123-2139, 1992. 19. Z. Zhong and J. Mackerle. Static contact problems - a review. Engineering Computations, 9:3-37, 1992. 20. J.-H. Heegaard and A. Curnier. An augmented Lagrangian method for discrete large-slip contact problems. International Journal for Numerical Methods in Engineering, 36:569-593, 1993. 21. T.A. Laursen and S. Govindjee. A note on the treatment of frictionless contact between nonsmooth surfaces in fully non-linear problems. Communications in Numerical Methods in Engineering, 10:869-878, 1994. 22. T.A. Laursen and V. Chawla. Design of energy conserving algorithms for frictionless dynamic contact problems. International Journal for Numerical Methods in Engineering, 40:863-886, 1997. 23. P. Papadopoulos and R.L. Taylor. A mixed formulation for the finite element solution of contact problems. Computer Methods in Applied Mechanics and Engineering, 94:373-389, 1992. 24. P. Papadopoulos, R.E. Jones and J.M. Solberg. A novel finite element formulation for frictionless contact problems. International Journal for Numerical Methods in Engineering, 38:2603-2617, 1995. 25. P. Papadopoulos and J.M. Solberg. A Lagrange multiplier method for the finite element solution of frictionless contact problems. Mathematical and Computer Modelling, 28:373-384, 1998. 26. J.M. Solberg and P. Papadopoulos. A finite element method for contact/impact. Finite Elements in Analysis and Design, 30:297-311, 1998. 27. E. Bittencourt and G.J. Creus. Finite element analysis of three-dimensional contact and impact in large deformation problems. Computers and Structures, 69:219-234, 1998. 28. M. Cuomo and G. Ventura. Complementary energy approach to contact problems based on consistent augmented Lagrangian formulation. Mathematical and Computer Modelling, 28:185-204, 1998. 29. C. Kane, E.A. Repetto, M. Ortiz and J.E. Marsden. Finite element analysis of non smooth contact. Computer Methods in Applied Mechanics and Engineering, 180:1-26, 1999. 30. I. Paczelt, B.A. Szabo and T. Szabo. Solution of contact problem using the hp-version of the finite element method. Computers and Mathematics with Applications, 38:49-69, 1999. 31. M.A. Puso and T.A. Laursen. A mortar segment-to-segment contact method for large deformation solid mechanics. Computer Methods in Applied Mechanics and Engineering, 193:601-629, 2004. 32. M.A. Puso and T.A. Laursen. A mortar segment-to-segment frictional contact method for large displacements. Computer Methods in Applied Mechanics and Engineering, 193:4891-4913, 2004. 33. R.L. Taylor and P. Papadopoulos. A patch test for contact problems in two dimensions. In P. Wriggers and W. Wagner, editors, Nonlinear Computational Mechanics, pages 690-702. Springer, Berlin, 1991. 34. T.A. Laursen and V.G. Oancea. Automation and assessment of augmented lagrangian algorithms for frictional contact problems. J. Appl. Mech, 61:956-963, 1994. 35. S.P. Timoshenko and J.N. Goodier. Theory of Elasticity. McGraw-Hill, New York, 3rd edition, 1969. 36. D.G. Luenberger. Linear and Nonlinear Programming. Addison-Wesley, Reading, Mass., 1984.
226
Treatment of constraints-contact and tied interfaces 37. B.I. Wohlmuth. A mortar finite element method using dual spaces for the Lagrange multiplier. Society for Industrial and Applied Mathematics, 38:989-1012, 2000. 38. K.J. Arrow, L. Hurwicz and H. Uzawa. Studies in Non-Linear Programming. Stanford University Press, Stanford, CA, 1958. 39. G. Zavarise and P. Wriggers. A superlinear convergent augmented Lagrangian procedure for contact problems. Engineering Computations, 16:88-119, 1999. 40. T.A. Laursen. Computational Contact and Impact Mechanics. Springer, Berlin, 2002. 41. P. Wriggers. Computational Contact Mechanics. John Wiley & Sons, Chichester, 2002. 42. R.L. Taylor and P. Wriggers. Smooth surface discretization for large deformation frictionless contact. Technical Report UCB/SEMM-99/04, University of California, Berkeley, February 1999. 43. M.A. Puso and T.A. Laursen. A 3d contact smoothing algorithm using Gregory patches. International Journal for Numerical Methods in Engineering, 54:1161-1194, 2002. 44. A.B. Chaudhary and K.-J. Bathe. A solution method for static and dynamic analysis of threedimensional contact problems with friction. Computers and Structures, 24:855-873, 1986. 45. J.-W. Ju and R.L. Taylor. A perturbed Lagrangian formulation for the finite element solution of nonlinear frictional contact problems. Journal de Mgcanique Th~orique et Appliquge, 7(Supplement, 1):1-14, 1988. 46. A. Curnier and P. Alart. A generalized Newton method for contact problems with friction. Journal de M~canique Thgorique et Appliqu~e, 7:67-82, 1988. 47. P. Wriggers, T. Vu Van and E. Stein. Finite element formulation of large deformation impactcontact problems with friction. Computers and Structures, 37:319-331, 1990. 48. P. Alart and A. Curnier. A mixed formulation for frictional contact problems prone to Newton like solution methods. Computer Methods in Applied Mechanics and Engineering, 92:353-375, 1991. 49. J.C. Simo and T.A. Laursen. An augmented Lagrangian treatment of contact problems involving friction. Computers and Structures, 42:97-116, 1992. 50. T.A. Laursen and J.C. Simo. A continuum-based finite element formulation for the implicit solution of multibody, large-deformation, frictional, contact problems. International Journal for Numerical Methods in Engineering, 36:3451-3486, 1993. 51. T.A. Laursen and J.C. Simo. Algorithmic symmetrization of Coulomb frictional problems using augmented Lagrangians. Computer Methods in Applied Mechanics and Engineering, 108:133-146, 1993. 52. A. Heege and P. Alart. A frictional contact element for strongly curved contact problems. International Journal for Numerical Methods in Engineering, 39:165-184, 1996. 53. C. Agelet de Saracibar. A new frictional time integration algorithm for large slip multibody frictional contact problems. Computer Methods in Applied Mechanics and Engineering, 142:303-334, 1997. 54. W. Ling and H.K. Stolarski. On elasto-plastic finite element analysis of some frictional contact problems with large sliding. Engineering Computations, 14:558-580, 1997. 55. E Jourdan, P. Alart and M. Jean. A gauss-seidel like algorithm to solve frictional contact problems. Computer Methods in Applied Mechanics and Engineering, 155:31-47, 1998. 56. C. Agelet de Saracibar. Numerical analysis of coupled thermomechanical frictional contact. Computational model and applications. Archives of Computational Methods in Engineering, 5(3):243-301, 1998. 57. G. Pietrzak and A. Curnier. Large deformation frictional contact mechanics: continuum formulation and augmented Lagrangian treatment. Computer Methods in Applied Mechanics and Engineering, 177:351-381, 1999. 58. J.T. Oden and J.A.C. Martins. Models and computational methods for dynamic friction phenomena. Computer Methods in Applied Mechanics and Engineering, 52:527-634, 1985. 59. P. Papadopoulos and R.L. Taylor. A mixed formulation fo the finite element solution of contact problems. Computer Methods in Applied Mechanics and Engineering, 94:373-389, 1992.
References 227 60. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann, Oxford, 6th edition, 2005. 61. B.I. Wohlmuth. Discretization Methods and Iterative Solvers Based on Domain Decomposition. Springer-Verlag, Heidelberg, 2001. 62. M.A. Puso and T.A. Laursen. Mesh tying on curved interfaces in 3d. Engineering Computations, 20:305-319, 2003.
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In
it is
we
Many situations are encountered where treatment of the entire system as deformable bodies is neither necessary nor practical. For example, the frontal impact of a vehicle against a barrier requires a detailed modelling of the front part of the vehicle but the primary function of the engine and the rear part is to provide inertia, deformation being negligible for purposes of modelling the frontal impact. A second example, from geotechnical engineering, is the modelling of rock mass landslides or interaction between rocks on a conveyor belt where deformation of individual blocks is secondary. this chapter we consider briefly the study of the first class of problems and in the next chapter the second type in much more detail. The above problem classes divide themselves into two further subclasses: one where necessary to include some simple mechanisms of deformation in each body (e.g. an individual rock piece) and the second in which the individual bodies have no deformation at all. The first class is calledpseudo-rigid body deformationl and the second rigid body behaviour. ~-Here we wish to illustrate how such behaviour can be described and combined in a finite element system. For the modelling of pseudo-rigid body analyses follow closely the work of Cohen and Muncaster ~ and the numerical implementation proposed by Solberg and Papadopoulos. 3 The literature on rigid body analysis is extensive, and here we refer the reader to papers for additional details on methods and formulations beyond those covered here. 4-21
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~',~ ~', ~ composed ~'~',~ ~ ~',~,~',~,~,~,~ ~ ~'~', ,~', ','~',~,~'systems ,~',~,~ ~~ this section we consider the analysis of which;~ are of many small bodies, each of which is assumed to undergo large displacements and a uniform In deformation.* The individual bodies which we consider are of the types shown in 8.1. In particular, a faceted shape can be constructed directly from a finite element discretization in which the elements are designated as all belonging to a single solid Fig. object or the individual bodies can be described by simple geometric forms such as or ellipsoids. discs * body.
Higher-order
approximations
can
be
included
using
polynomial
approximation
for
the
deformation
of each
Pseudo-rigid motions 229
(a)
(b)
Fig. 8.1 Shapesfor pseudo-rigid and rigid body analysis: (a) ellipsoid; (b) faceted body. A homogeneous motion of a body may be written as r
t) = ri(t) -+- Fii(t) [XI - RI]
(8.1)
in which XI is position, t is time, RI is some reference point in the undeformed body, ri is the position of the same point in the deformed body, and Fil is a constant deformation gradient. We note immediately that at time zero the deformation gradient is the identity tensor (matrix) and Eq. (8.1) becomes r
O) - -
ri(O) -+- tSiI [ X I -- RI] = ri(O) + t ~ i l X l -- tSilRl ~ t S i l X l
(8.2)
where ri(O) "-- t~iiRi by definition. The behaviour of solids which obey the above description is sometimes referred to as analysis ofpseudo-rigid bodies. 1 A treatment by finite elements has been considered by Solberg and Papadopoulos, 3 and an alternative expression for motions restricted to incrementally linear behaviour has been developed by Shi, and the method is commonly called discontinuous deformation analysis (DDA). 22 The DDA form, while widely used in the geotechnical community, is usually combined with a simple linear elastic constitutive model and linear strain-displacement forms which can lead to large errors when finite rotations are encountered. Once the deformation gradient is computed, the procedures for analysis follow the methods described in Chapter 5. It is, of course, necessary to include the inertial term for each body in the analysis. No difficulties are encountered once a shape of each body is described and a constitutive model is introduced. For elastic behaviour it is not necessary to use a complicated model, and here use of the St Venant-Kirchhoff relation is adequate - indeed, if large deformations occur within an individual body the approximation of homogeneous deformation generally is not adequate to describe the solution. The primary difficulty for this class of problems is modelling the large number of interactions between bodies by contact phenomena and here the reader is referred to Chapters 7 and 9 and references on the subject for additional information on contact and other details. 22'23
230
Pseudo-rigidand rigid-flexible bodies
The pseudo-rigid body form can be directly extended to rigid bodies by using the polar decomposition on the deformation tensor. The polar decomposition of the deformation gradient may be given as 24-26 Fil -- A i j U j I
where
AilAij
"- (~lJ
and
AilAjl
= ~Sij
(8.3)
Here AiI is a rigid rotation* and UIj is a stretch tensor (which has eigenvalues Am as defined in Chapter 5). In the case of rigid motions the stretches are all unity and UIj simply becomes an identity. Thus, a rigid body motion may be specified as (gi(Xl, t) = ri(t) + A i l ( t ) [XI - RI]
(8.4)
q~(X, t) = r(t) + A ( t ) [ X - R]
(8.5)
or, in matrix form, as Alternatively, we can express the rigid motion using Eq. (8.1) and impose constraints to make the stretches unity. For example, in two dimensions we can represent the motion in terms of the displacements of the vertices of a triangle and apply constraints that the lengths of the triangle sides are unchanged during deformation. The constraints may be added as Lagrange multipliers or other constraint methods and the analysis may proceed directly from a standard finite element representation of the triangle. Such an approach has been used in reference 27 with a penalty method used to impose the constraints. Here we do not pursue this approach further and instead consider direct use of rigid body motions to construct the formulation. For subsequent use we note the form of the variation of a rigid motion and its incremental part. These may be expressed as A
3~b = 3r + 30A [X - R] dO - dr + dOA [X - R]
Using Eq. (8.5) these may be simplified to 3q~ - 3 r - ~30 dc~ - d r - ~dO
where
y = x- r (8.6)
where dO and ~0 are incremental and variational rotation vectors, respectively. In a similar manner we obtain the velocity for the rigid motion as -/"-
~w
(8.7)
in which f is translational velocity and w angular velocity, both at the centre of mass. The angular velocity is obtained by solving h = a,A * Often literature denotes this rotation as Rii; however, here we use to avoid confusion use Ail to denote rotation.
(8.8) RI
as a position of a point in the body and
Rigid motions 231 or
A-
A~
(8.9)
where f~ is the reference configuration angular velocity. 8 This is clearer by writing the equations in indicial form given by ~kil -- w i j A j l
(8.10)
-- A i j ~ ' 2 j l
where the velocity matrices are defined in terms of vector components and give the skew symmetric form Wij =
W3 --W 2
0
--W1
W1
0
(8.11)
and similarly for f2lj. The above form allows for the use of either the material angular velocity or the spatial one. Transformation between the two is easily performed since the rigid rotation must satisfy the orthogonality conditions ATA = A A T = I
(8.12)
at all times. Using Eqs (8.8) and (8.9) we obtain -- A ~ A T
(8.13)
= ATcDA
(8.14)
or by transforming in the opposite way
8.3.1 Equations of motion for a rigid body If we consider a single rigid body subjected to concentrated loads fa applied at points whose current position is Xa and locate the reference position for R at the centre of mass, the equations of equilibrium are given by conservation of linear momentum = ~
fa = f;
p = mr
(8.15)
a
where p defines a linear momentum, f is a resultant force and total mass of the body is computed from P
m - ]
J.
P0 dV
(8.16)
and conservation of angular momentum ~" - - Z
(Xa - -
r) • fa -- m;
Jr = I[w
(8.17)
a
where rr is the angular momentum of the rigid body, m is a resultant couple and ]I is the spatial inertia tensor.
232
Pseudo-rigidand rigid-flexible bodies The spatial inertia tensor (matrix) ITis computed from ]I = A J A T
(8.18)
where J is the inertia tensor (matrix) computed from an integral on the reference configuration and is given by ,~ - J~ P0 [(YTY) I -- yyT] dV
where
Y - X - R
(8.19)
Thus, description of an individual rigid body requires locating the centre of mass R and computing the total mass m and inertia matrix Jl. It is then necessary to integrate the equilibrium equations to define the position r and the orientation of the body A.
8.3.2 Construction from a finite element model If we model a body by finite elements, as described throughout this volume, we can define individual bodies or parts of bodies as being rigid. For each such body (or part of a body) it is then necessary to define the total mass, inertia matrix, and location of the centre of mass. This may be accomplished by computing the integrals given by Eqs (8.16) and (8.19) together with the relation to determine the centre of mass given by mR -- f n poX d V
(8.20)
In these expressions it is necessary only to define each point in the volume of an element by its reference position interpolation X. For solid (e.g. brick or tetrahedral) elements such interpolation is given by Eq. (5.50a) which in matrix form becomes (omitting the summation symbol) X = N.X~
(8.21)
This interpolation may be used to determine the volume element necessary to carry out all the integrals numerically by quadrature. 28 The total mass may now be computed as
m=~-'~(f~ podV) e
(8.22)
e
where ~"2 e is the reference volume of each element e. Use of Eq. (8.21) in Eq. (8.20) to determine the centre of mass now gives
R - --I _ m
e
PoN,~ dV) X~
(8.23)
e
and finally the reference inertia tensor (matrix) as J = E
M~;~ [(YSY~)I- Y~Y~] ; e
Y~ = X~ - R
(8.24)
Rigid motions 233 where
Mea3 = f~ PoN"N~dV
(8.25)
e
The above definition of Y, tacitly assumes that y ~ N, - 1. If other interpolations are used to define the shape functions (e.g. hierarchical shape functions) it is necessary to modify the above procedure to determine the mass and inertia matrix.
8.3.3 Transient solutions The integration of the translational rigid term r may be performed using any of the methods described in reference 28 or indeed by other methods described in the literature. The integration of the rotational part can also be performed by many schemes; however, it is important that updates of the rotation produce discrete time values for rigid rotations which retain an orthonormal character, that is, the An must satisfy the orthogonality condition given by Eq. (8.12). One procedure to obtain this is to assume that the angular velocity within a time increment is constant, being measured as
w(t) ~
(,~t3n+c~ =
•At
(8.26)
in which At is the time increment between tn and tn+l, 0 is the increment of rotation during the time step, and 0 < c~ < 1. The approximation (at,~nq_c~ --
(8.27)
(1 -- OL)(,~3 n -'1- OL(aL3n+ 1
is used to define intermediate values in terms of those at tn and tn+l. Equation (8.8) now becomes a constant coefficient ordinary differential equation which may be integrated exactly, yielding the solution A(t) -- exp[0(t - tn)/At]An
tn < t < tn-t-1
(8.28)
In particular at tn+~ we obtain An+c~ - exp[ce0]An This may also be performed using the material angular velocity f~.8 Many algorithms exist to construct the exponential of a matrix, and the closed-form expression given by the classical formula of Euler and Rodrigues (e.g. see Wittaker 29) is quite popular. This is given by exp[0] - I +
1010 + _1 sin 2 101/202 101 2 [1012/2]
sin
where
101 =
[0T0 ] 1/2
(8.29)
This update may also be given in terms of quaternions and has been used for integration of both rigid body motions as well as for the integration of the rotations appearing in three-dimensional beam formulations (see Chapter 17). 8'30'31Another alternative to the
234
Pseudo-rigidand rigid-flexible bodies direct use of the exponential update is to use the Cayley transform to perform updates for A which remain orthonormal. Once the form for the update of the rigid rotation is defined any of the integration procedures defined in reference 28 may be used to advance the incremental rotation by noting that 0 or 19 (the material counterpart) are in fact the change from time t~ to tn+~. The reader also is referred to reference 8 for additional algorithms directly based on the GN 11 and GN22 methods. 28 Here forms for conservation of linear and angular momentum are of particular importance.
In some analyses the rigid body is directly attached to flexible body parts of the problem [Fig. 8.2(a)]. Consider a rigid body that occupies the part of the domain denoted a s ~2 r and is 'bonded' to a flexible body with domain f2f. In such a case the formulation to 'bond' the surface may be performed in a concise manner using Lagrange multiplier constraints. We shall find that these multiplier constraints can be easily eliminated from the analysis by a local solution process, as opposed to the need to carry them to the global solution arrays as was the case in their use in contact problems (see Sec. 5.3).
8.4.1 Lagrange multiplier constraints A simple two-dimensional rigid-flexible body problem is shown in Fig. 8.2(a) in which the interface will involve only three-nodal points. In Fig. 8.2(b) we show an exploded view between the rigid body and one of the elements which lies along the rigid-flexible interface. Here we need to enforce that the position of the two interface nodes for the element will have the same deformed position as the corresponding point on the rigid body. Such a constraint can easily be written using Eq. (8.4) as Ca = r(t) + A(t) [X~ - R] - x~(t) = 0
(8.30)
in which the subscript c~ denotes a node number. We can now modify a functional to include the constraint using a classical Lagrange multiplier approach in which we add the term 1-Irf = )~C~ = ,,~ [x~(t) - r(t) - A(t) [X~ - R]]
(8.31)
0
(a)
(b)
Fig. 8.2 Lagrange multiplier constraint between flexible and rigid bodies: (a)rigid-flexible body; (b) Lagrange multipliers.
Connecting a rigid body to a flexible body 235 Taking the variation we obtain
aFlrf = aA~ [x~ - r - A [X~ - R]] + A~ [ax~ - 6r - a0A [X~ - R]]
(8.32)
From this we immediately obtain the constraint equation and a modification to the equilibrium equations for each flexible node and the rigid body. Accordingly, the modified variational principle may now be written for a typical node a on the interface of the rigid body as ~I'I "4- ~I'Irf :
[~Xtz
6x~
~r
dO ~X,]
M.~v~ + M ~ v ~ + P. - f~, M ~ v ~ + M~av/~ + P~ - f~ + X~ #-f-X~
=0
(8.33)
e-m-yTA x~ - r -
A [ X ~ - R]
where y~ = x~ - r are the nodal values of y, fi are any other rigid body nodes connected to node a and #, v are flexible nodes connected to node a. Since the parameters x~ enter the equations in a linear manner we can use the constraint equation to eliminate their appearance in the equations. Accordingly, from the variation of the constraint equation we may write
6x~ = [I, y~]
(8.34)
60
which permits the remaining equations in Eq. (8.33) to be rewritten as
Mv~v~ + Mv~'b~ + P~ - f~ ~1"1 + ~I-Irf -- [~Xtt
&
~0]
p -- f + M~jr
+ M~av~ + P~ - f~
= 0
"/r - m - ~,T(M,~,% + M,~zi'a + P~ - f~)
(8.35)
For use in a Newton solution scheme it is necessary to linearize Eq. (8.35). This is easily achieved
d(~I'I)+d(~I-Irf)= [ax~
ar I
30]
(K~)T
(Ktzfl)T
~X~
0
(Kav)T (Ka/~)T K.~ --yT(Kav)T --~T(Ka/~)T 0
qx~ KTo
dr
dO
(8.36)
236 Pseudo-rigidand rigid-flexible bodies Once again this form may be reduced using the equivalent of Eq. (8.34) for an incremental dx~ to obtain
d(~Fl) + d(~l-Irf )
[~x,
8r ~0]
r
(K~)T --(Kvr162 (KIzv)T - (K,~)T~'~ / (K~v)T K.~ + (K~3)T ,,T K [K~ + L--YT(K~.)T -Y~( ~;~)T
x x
l
| (8.37)
dr
dO
]fdxv 1 ~d;
Combining all the steps we obtain the set of equations for each rigid body as I
(Kvv)T
(Kv~)T
(K~)T
[K~ + (K~)T]
_~,T (Kav)T
_~,T (Ka~)T
-- (Ktzfl)T~r/3
--(Ka/~)T~r/3 / [K~ + :9~(K~)T:9~IJ
(8.38)
=
m - ~ + ~'T'I'~ OL
in which ~ and ~ . are the residuals from the finite element calculation at node a and #, respectively. We recall from Chapter 5 that each is given by a form (8.39) which is now not zero since total balance of momentum includes the addition of the
~a.
The above steps to compute the residual and the tangent can be performed in each element separately by noting that
,,ks = ~
Xe
(8.40)
e
where Xe denotes the contribution from element e. Thus, the steps to constrain a flexible body to a rigid body are once again a standard finite element assembly process and may easily be incorporated into a solution system. The above discussion has considered the connection between a rigid body and a body which is modelled using solid finite elements (e.g. quadrilateral and hexahedral elements in two and three dimensions, respectively). It is also possible directly to connect beam elements which have nodal parameters of translation and rotation. This is easily performed if the rotation parameters of the beam are also defined in terms of the rigid rotation A. In this case one merely transforms the rotation to be defined relative to the reference description of the rigid body rotation and assembles the result directly into the rotation terms of the rigid body. If one uses a rotation for both the beam and the rigid body which is defined in terms of the global Cartesian reference configuration no transformation is required. Shells can be similarly treated; however, it is best then to define the shell directly in terms of three rotation parameters instead of only two at points where connection is to be performed. 32'33
Multibody coupling by joints
Often it is desirable to have two (or more) rigid bodies connected in some specified manner. For example, in Fig. 8.3 we show a disc connected to an arm. Both are treated as rigid bodies but it is desired to have the disc connected to the arm in such a way that it can rotate freely about the axis normal to the page. This type of motion is characteristic of many rotating machine connections and it as well as many other types of connections are encountered in the study of rigid body motions. 4'34 This type of interconnection is commonly referred to as a joint. In quite general terms joints may be constructed by a combination of two types of simple constraints: translational constraints and rotational constraints.
8.5.1 Translation constraints The simplest type of joint is a spherical connection in which one body may freely rotate around the other but relative translation is prevented. Such a situation is shown in Fig. 8.3 where it is evident the spinning disc must stay attached to the rigid arm at its axle. Thus it may not translate relative to the arm in any direction (additional constraints are necessary to ensure it rotates only about the one axis - these are discussed in Sec. 8.5.2). If a full translation constraint is imposed a simple relation may be introduced as C j - - x (a) - x (b) - - 0 (8.41) where a and b denote two rigid bodies. Thus, addition of the Lagrange multiplier constraint (8.42) l - [ j - - ,,~jT[x(a) -- X(b)] imposes the spherical joint condition. It is necessary only to define the location for the spherical joint in the reference configuration. Denoting this as Xj (which is common
Fig. 8.3 Spinning disc constrained by a joint to a rigid arm.
237
238
Pseudo-rigidand rigid-flexible bodies to the two bodies) and introducing the rigid motion yields a constraint in terms of the rigid body positions as l-lj -- )kT[r (a) + A ( a ) ( x j
J
- R (a)) - r (b) - A(b)(xj -- R(b))]
(8.43)
The variation and subsequent linearization of this relation yields the contribution to the residual and tangent matrix for each body, respectively. This is easily performed using relations given above and is left as an exercise for the reader. If the translation constraint is restricted to be in one direction with respect to, say, body a it is necessary to track this direction and write the constraint accordingly. To accomplish this the specific direction of the body a in the reference configuration is required. This may be computed by defining two points in space X l and X2 from which a unit vector V is defined by V=
X2-Xl
IXz - Xll
(8.44)
The direction of this vector in the current configuration, v, may be obtained using the rigid rotation for body a v = A(a)v (8.45) A constraint can now be introduced into the variational problem as
I-lj =/kj{VT(A(a))T[r (a) + A(a)(xj - R (a)) - r O) - A~
- R(b))]}
(8.46)
where, owing to the fact there is only a single constraint direction, the Lagrange multiplier is a scalar/~j and, again, Xj denotes the reference position where the constraint is imposed. The above constraints may also be imposed by using a penalty function. The most direct form is to perturb each Lagrange multiplier form by a penalty tenn. Accordingly, for each constraint we write the variational problem as 1
2kj AJ
~
1-Ij -- )~jCj
2
(8.47)
where it is immediately obvious that the limit kj ---> oo yields exact satisfaction of the constraint. Use of a large kj and variation with respect to )~j give
E 1 ]
~%j C j - ~ ) ~ j
=0
(8.48)
and may easily be solved for the Lagrange multiplier as
)~j = kjCj
(8.49)
which when substituted back into Eq. (8.47) gives the classical form
nj -
kj
[cj
]2
(8.50)
The reader will recognize that Eq. (8.47) is a mixed problem, whereas Eq. (8.50) is irreducible. An augmented Lagrangian form is also possible following the procedures described in reference 28 and used in Chapter 7 for contact problems.
Multibody coupling by joints 239
8.5.2 Rotation constraints A second kind of constraint that needs to be considered relates to rotations. We have already observed in Fig. 8.3 that the disc is free to rotate around only one axis. Accordingly, constraints must be imposed which limit this type of motion. This may be accomplished by constructing an orthogonal set of unit vectors V I in the reference configuration and tracking the orientation of the deformed set of axes for each body as V~ c) --
~ilA(C)Vl
for c = a, b v~Vj - 5ij
(8.51)
A rotational constraint which imposes that axis i of body a remains perpendicular to axis j of body b may then be written as (v!a))Tv t
(b) - j
vT(A(a))TA(b)vj
-- 0
(8.52)
Example 8.1: Revolute joint
As an example, consider the situation shown for the disc in Fig. 8.3 and define the axis of rotation in the reference configuration by the Cartesian unit vectors EI (i.e. V I = EI). If we let the disc be body a and the arm body b the set of constraints can be written as (where v3 is axis of rotation) X (a) __ X (c)
Cj =
~,*l[w(a)~Tw(b)J "3 = 0
(8.53)
l'w(a)~Tw(b) k*2 I *3
and included in a formulation using a Lagrange multiplier form
rlj = )~jT Cj
(8.54)
The modifications to the finite element equations are obtained by appending the variation and linearization of Eq. (8.54) to the usual equilibrium equations. Here five Lagrange multipliers are involved to impose the three translational constraints (spherical joint) and the angle constraints for the rotating disc. The set of constraints is known as a revolute joint, e
8.5.3 Library of joints Translational and rotational constraints may be combined in many forms to develop different types of constraints between rigid bodies. For the development it is necessary to have only the three types of constraints described above. Namely, the spherical joint, a single translational constraint, and a single rotational constraint. Once these are available it is possible to combine them to form classical constraint joints and here the reader is referred to the literature for the many kinds commonly encountered. 2,4,7,35 The only situation that requires special mention is the case when a series of rigid bodies is connected together to form a closed loop. In this case the method given above can lead to situations in which some of the joints are redundant. Using Lagrange multipliers this implies the resulting tangent matrix will be singular and, thus, one
240
Pseudo-rigid and rigid-flexible bodies cannot obtain solutions. Here a penalty method provides a viable method to circumvent this problem. The penalty method introduces elastic deformation in the joints and in this way removes the singular problem. If necessary an augmented Lagrangian method can be used to keep the deformation in the joint within required small tolerances. An alternative to this is to extract the closed loop rigid equations from the problem and use singular valued decomposition 36 to identify the redundant equations. These may then be removed by constructing a pseudo-inverse for the tangent matrix of the closed loop. This method has been used successfully by Chen to solve single loop problems. 35
'iii!iiiiiii iiiiiiiiiiii'ii,i'~i~ ,i~:~:',~::~::~iiiiiii'i,~:~i : :::~~i~,i,i:~,':ii':~iiii'i',i{iiii::~::::~'~'i~ii'~,iiiii::::~i~i'i~il~,ii,i~','i~,'i',i~',il!ili'~,i',iliiiiiii! i i:,i:,iiiiiiii~,~,',i~i{i'i,{!'!,!~,i ~!',iiiiiiiiii 'i,i ~""i~',~{'~,'!i~~:~ !i'i:~i',~'i~i'i i'~,i'~i,iiii'i!i'i,'i ':,,i'i,~i~,'il~,i!iiiiliiiiiiii' i i~i~,i',i!'i,:'iii'',',:iii!iii~'~'':iiili',iii',ii!'iii'~i',i',!i~,:,il,i',:{:i''i,~,i,i:ii!:i'',:,iii",','i, iii'!~,:',:ii',ii'"i~,,'~ili:,l'i,'ii'i:{~iiiiii'','i:,,',i!!,i~,i:i~:,'i:':,i:ii:i:,:i:,i::iif'~,i,'!',!i!'ii~'i,i'::,'i:~',,i'::',!,!iii!ii{i:i:,i!iii:,!:,i!':!:i{::i','i:i"~,ii'iiii!':i'~,'~,:!,i~:,i':i',i:i',,i','!'~,,i'i:!i~,,',',ii:: i"i'':,~,i i'!i,'','!~ii',i:,i' : ii',i'::,{:iiii'i':,,',i!:,:,i:,ii:!ii:i:' '~!~,i::ii':i':~ !i~i:'i :i,i','':iii',i,,'~~!: ':ii': i'
8.6.1 Rotating disc As a first example we consider a problem for the rotating disc on a rigid arm which is attached to a deformable base as shown in Fig. 8.4. The finite element model is constructed from four-node displacement elements in which a St Venant-Kirchhoff material model is used for the elastic part. The elastic properties in the model are E = 10 000 and z~ = 0.25, with a uniform mass density P0 = 5 throughout. The disc and arm are made rigid by using the procedures described in this chapter. The disc is attached to the arm by means of a revolute joint with the constraints imposed using the Lagrange multiplier method. The rigid arm is constrained to the elastic support by using the local Lagrange multiplier method described in Sec. 8.4. The problem is excited by a constant vertical load applied at the revolute joint and a torque applied to spin the disc. Each load is applied for the first 10 units of time. The mesh and configuration are shown in Fig. 8.4(a). Deformed positions of the model are shown at 2.5 unit intervals of time in Fig. 8.4(b)-(h). A marker element shows the position of the rotating disc. The displacements at the revolute joint and the radial exterior point at the marker element location are shown in Fig. 8.5.
8.6.2 Beam with attached mass ....................................................................................
--: .....................................................................................................................................................................................
As a second example we consider an elastic cantilever beam with an attached end mass of rectangular shape. The beam is excited by a horizontal load applied at the top as a triangular pulse for two units of time. The rigid mass is attached to the top of the beam by using the Lagrange multiplier method described in Sec. 8.4 and here it is necessary to constrain both the translation and the rotation parameters of the beam. The beam is three dimensional and has an elastic modulus of E = 100 000 and a moment of inertia in both directions of I1~ = 122 = 12. The beam mass density is low, with a value of /90 = 0.02. The tip mass is a cube with side lengths 4 units and mass density/90 = 1. The shape of the beam at several instants of time is shown in Fig. 8.6 and it is clear that large translation and rotation is occurring and also that the rigid block is correctly following a constrained rigid body motion.
Numerical examples 241
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Fig. 8.4 Rigid-flexible model for spinning disc: (a) problem definition, solutions at time; (b) t = 2.5 units; (c) t - 5.0 units; (d) t - 7.5 units; (e) t -- 10.0 units; (f) t - 12.5 units; (g) t - 15.0 units; (h) t - 17.5 units.
242
Pseudo-rigid and rigid-flexible bodies
15, -,r
0.5
E o
0
r 03
-0.5
A
A
l!
h
1 I I---Horizontall 10~"'Vertical I
A
".......",..-'I .- ::, ,,:I,- 1 0
4_~,,-
L
---Vertical J _ . -1 10 20 0 Time (a)
30
40
-15 -20
/""%
-.
'
0
10
'
20 Time (b)
30
40
Fig. 8.5 Displacements for rigid-flexible model for spinning disc. Displacement at: (a) revolute; (b) disc rim.
(a)
(c)
(b)
(e)
(f)
(g)
(d)
(h)
(i)
Fig. 8.6 Cantilever with tip mass: (a) t -- 2 units; (b) t -- 4 units; (c) t -- 6 units; (d) t - 10 units; (e) t -- 12 units; (f) t - 14 units; (g) t - 16 units; (h) t - 18 units; (i) t - 20 units.
1. H. Cohen and R.G. Muncaster. The Theory of Pseudo-rigid Bodies. Springer, New York, 1988. 2. A.A. Shabana. Dynamics of Multibody Systems. John Wiley & Sons, New York, 1989. 3. J.M. Solberg and P. Papadopoulos. A simple finite element-based framework for the analysis of elastic pseudo-rigid bodies. International Journal for Numerical Methods in Engineering, 45:1297-1314, 1999. 4. D.J. Benson and J.O. Hallquist. A simple rigid body algorithm for structural dynamics programs. International Journal for Numerical Methods in Engineering, 22:723-749, 1986.
References 243 5. R.A. Wehage and E.J. Haug. Generalized coordinate partitioning for dimension reduction in analysis of constrained dynamic systems. Journal of Mechanical Design, 104:247-255, 1982. 6. A. Cardona and M. Geradin. Beam finite element nonlinear theory with finite rotations. International Journal for Numerical Methods in Engineering, 26:2403-2438, 1988. 7. A. Cardona, M. Geradin and D.B. Doan. Rigid and flexible joint modelling in multibody dynamics using finite elements. Computer Methods in Applied Mechanics and Engineering, 89:395418,91. 8. J.C. Simo and K. Wong. Unconditionally stable algorithms for rigid body dynamics that exactly conserve energy and momentum. International Journal for Numerical Methods in Engineering, 31:19-52, 1991. [Addendum: 33:1321-1323, (1992).] 9. H.T. Clark and D.S. Kang. Application of penalty constraints for multibody dynamics of large space structures. Advances in the Astronautical Sciences, 79:511-530, 1992. 10. G.M. Hulbert. Explicit momentum conserving algorithms for rigid body dynamics. Computers and Structures, 44:1291-1303, 1992. 11. M. Geradin, D.B. Doan and I. Klapka. MECANO: a finite element software for flexible multibody analysis. Vehicle System Dynamics, 22:87-90, 1993. Supplement issue. 12. S.N. Atluri and A. Cazzani. Rotations in computational solid mechanics. Archives of Computational Methods in Engineering, 2:49-138, 1995. 13. O.A. Bauchau, G. Damilano and N.J. Theron. Numerical integration of nonlinear elastic multibody systems. International Journal for Numerical Methods in Engineering, 38:2727-2751, 1995. 14. J.A.C. Ambr6sio. Dynamics of structures undergoing gross motion and nonlinear deformations: a multibody approach. Computers and Structures, 59:1001-1012, 1996. 15. R.L. Huston. Multibody dynamics since 1990. Applied Mechanics Reviews, 49:$35-$40, 1996. 16. O.A. Bauchau and N.J. Theron. Energy decaying scheme for non-linear beam models. Computer Methods in Applied Mechanics and Engineering, 134:37-56, 1996. 17. O.A. Bauchau and N.J. Theron. Energy decaying scheme for non-linear elastic multi-body systems. Computers and Structures, 59:317-331, 1996. 18. C. Bottasso and M. Borri. Energy preserving/decaying schemes for nonlinear beam dynamics using the helicoidal approximation. Computer Methods in Applied Mechanics and Engineering, 143:393-415, 1997. 19. O.A. Bauchau. Computational schemes for flexible, nonlinear multi-body systems. Multibody System Dynamics, 2:169-222, 1998. 20. O.A. Bauchau and C.L. Bottasso. On the design of energy preserving and decaying schemes for flexible nonlinear multi-body systems. Computer Methods in Applied Mechanics and Engineering, 169:61-79, 1999. 21. O.A. Bauchau and T. Joo. Computational schemes for non-linear elasto-dynamics. International Journal for Numerical Methods in Engineering, 45:693-719, 1999. 22. G.-H. Shi. Block System Modelling by Discontinuous Deformation Analysis. Computational Mechanics Publications, Southampton, 1993. 23. E.G. Petocz. Formulation and analysis of stable time-stepping algorithms for contact problems. PhD thesis, Department of Mechanical Engineering, Stanford University, Stanford, California, 1998. 24. M.E. Gurtin. An Introduction to Continuum Mechanics. Academic Press, New York, 1981. 25. L.E. Malvern. Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, Englewood Cliffs, NJ, 1969. 26. J. Bonet and R.D. Wood. Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge University Press, Cambridge, 1997. ISBN 0-521-57272-X. 27. J.C. Garcia Orden and J.M. Goicolea. Dynamic analysis of rigid and deformable multibody systems with penalty methods and energy-momentum schemes. Computer Methods in Applied Mechanics and Engineering, 188:789-804, 2000.
244 Pseudo-rigidand rigid-flexible bodies 28. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann, Oxford, 6th edition, 2005. 29. E.T. Wittaker. A Treatise on Analytical Dynamics. Dover Publications, New York, 1944. 30. J.H. Argyris and D.W. Scharpf. Finite elements in time and space. Nuclear Engineering and Design, 10:456--469, 1969. 31. A. Ibrahimbegovic and M. A1 Mikdad. Finite rotations in dynamics of beams and implicit timestepping schemes. International Journal for Numerical Methods in Engineering, 41:781-814, 1998. 32. J.C. Simo. On a stress resultant geometrically exact shell model. Part VII: Shell intersections with 5/6 DOF finite element formulations. Computer Methods in Applied Mechanics and Engineering, 108:319-339, 1993. 33. P. Betsch, F. Gruttmann and E. Stein. A 4-node finite shell element for the implementation of general hyperelastic 3d-elasticity at finite strains. Computer Methods in Applied Mechanics and Engineering, 130:57-79, 1996. 34. H. Goldstein. Classical Mechanics. Addison-Wesley, Reading, 2nd edition, 1980. 35. A.J. Chen. Energy-momentum conserving methods for three dimensional dynamic nonlinear multibody systems. PhD thesis, Department of Mechanical Engineering, Stanford University, Stanford, California, 1998. (Also SUDMC Report 98--01.) 36. G.H. Golub and C.F. Van Loan. Matrix Computations. The Johns Hopkins University Press, Baltimore MD, 3rd edition, 1996.
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In the previous chapters we have considered the solution of solid mechanics problems from the view of a continuum. A computational framework which allows for some level of displacement field discontinuity to be represented a priori might be better suited to model particular phenomena, for example the behaviour of jointed rock or the granular material flow in silos. The treatment of these classes of problems is more naturally related to discrete element methods, in which distinctly separate material regions interacting with other discrete elements in some way are considered. A number of more complex models for both solid materials as well as for contact interactions have been formulated in the context of a discrete element methodology, with successful applications in many fields of science and engineering. Moreover, most media are discontinuous at some level of observation (nano, micro, meso, macro), where the continuum assumptions cease to apply. This happens when the scale of the problem becomes similar to the characteristic length scale of the associated material structure and surface interaction laws between bodies or particles are invoked instead of a homogenized continuum constitutive law. The computational modelling of inherently discontinuous media requires the discrete nature of discontinuities to be taken into account. Discontinuities can be either pre-existing (e.g. joints, bedding planes, interfaces, planes of weakness, construction joints) or they evolve (e.g. in the case of cohesive frictional materials, where the growth and coalescence of micro cracks eventually appear in a form of a macro-crack). Many structures, structural systems or structural components comprise discrete discontinuities, appearing either in a highly regular or structured manner or they are of a heterogeneous nature. An obvious example of structured discontinua is brick masonry or jointed rock structures where the displacement discontinuities commonly occur at block interfaces without necessarily rendering structures unsafe. Such problems are best considered with discrete element methods. The term discrete element methods will here be understood to comprise different techniques suitable for a simulation of dynamic behaviour of systems of multiple rigid, simply deformable (pseudo-rigid) or fully deformable separated bodies of simplified or arbitrary shapes, subject to continuous changes in the contact stares and varying contact forces, which in turn influence the subsequent movement of the bodies. Such problems * This chapter was contributed by Professor Nenad Bidanid, University of Glasgow, UK.
246 Discreteelement methods
are non-smooth in space (separate bodies) and in time (jumps in velocities upon collisions) and the unilateral constraints (non-penetrability) need to be considered. A system of bodies changes its position continuously under the action of external forces and interaction forces between bodies, which may eventually lead to a steady-state configuration, once static equilibrium is achieved. For rigid bodies, the contact interaction law is the only constitutive law considered, while the continuum constitutive law (e.g. elasticity, plasticity, damage, fracturing) needs to be included for deformable bodies. Computational modelling of multi-body contacts (both the contact detection and contact resolution) represents the dominant feature in discrete element methods, as the number of bodies considered may be very large. If the number of potential contact surfaces is relatively small (e.g. non-linear finite element analysis of contact problems) it is convenient to define groups of nodes, segments or surfaces which belong to a possible contact set a priori. These geometric attributes can then be continuously checked against one another and the kinematic resolution can be treated in a very rigorous manner. Bodies which are possibly in contact may be internally discretized by finite elements (Fig. 9.1), and their material behaviour can essentially be of any complexity (viz. Chapter 5). The category of discrete element methods specifically refers to simulations involving a large number of bodies where the contact locations and conditions cannot be defined in advance and need to be continuously updated as the solution progresses. Discrete element methods are most frequently applied to macroscopically discrete system of bodies (jointed rock, granular flow) but have also been successfully utilized in a microscopic setting, where very simple interaction laws between individual particles provide the material behaviour observed at a homogenized, macroscopic level. The discrete element method is most commonly defined 1 as a computational modelling framework which 1. allows finite displacements and rotations of discrete bodies, including complete detachment; 2. recognizes new contacts automatically, as the calculation progresses.
Fig. 9.1 System of rigid or deformable bodies, discretization of bodies into finite elements, continuously changing configurations and possible fragmentation.
Early DEM formulations There exist many methods (e.g. DEM discrete element method, RBSM rigid block spring method, DDA discontinuous deformation analysis, DEM/FEM combined discrete/finite elements, NSCD non-smooth contact dynamics), which belong to a broad family of discrete element methods. 2-4 Although these methods appear under different names and each of them is developing in its own fight, there are many unifying aspects and a more general framework is emerging, which allows for an equivalence between these apparently different methodologies to be recognized. Possible classification may be based on the manner these methods address: (a) detection of contacts, (b) treatment of contacts (rigid, deformable), (c) deformability (constitutive law) of bodies in contact (rigid, deformable, elastic, elasto-plastic, etc.), (d) large displacements and large rotations, (e) number (small or large) and/or distribution (loose or dense packing) of interacting bodies considered, (f) consideration of the model boundaries, (g) possible subsequent fracturing or fragmentation and (h) time-stepping integration schemes (explicit, implicit). Discrete element methods are also used for problems where the discrete nature of the emerging discontinuities needs to be taken into account. Application ranges from modelling problems of a discontinuous behaviour a priori (granular and particulate materials, silo flow, sediment transport, jointed rocks, stone or brick masonry) to problems where the modelling of transition from a continuum to a discontinuum is more important. Increased complexity of different discontinuous models is achieved by incorporating the deformability of solid material and/or by more complex contact interaction laws, as well as by the introduction of some failure or fracturing criteria controlling the solid material behaviour and the emergence of new discontinuities.
The initial formulation of the discrete element method, originally termed distinct element method or DEM, 5 was based on the assumption of rigid circular bodies in two dimensions with deformable contacts. The overall solution scheme for the DEM is straightforward, typically formulated in an explicit time-stepping format. Movement of bodies is driven by external forces (Fig. 9.2) and varying contact forces (normal and tangential contact forces, proportional to current overlap and viscous contact forces proportional to the relative velocities in the normal and tangential direction). The method considers each body in turn and at any given time determines all forces (external or contact) acting at it. Out of balance forces (or moments) induce accelerations (translational or rotational), which then determine the movement of that body during the next time step. The simplest computational sequence for the DEM (often formulated in an explicit 'leap-frog' format, see Table 9.1) typically proceeds by solving the equations of motion of a given discrete element and updating contact force histories as a consequence of contacts between different discrete elements and/or resulting from contacts with model boundaries. The Rigid Bodies Spring Model, RBSM, 7 was proposed early as a generalized limit plastic analysis framework. Solid structures are assumed to be assemblies of rigid blocks, interconnected by discrete deformable interfaces with distributed (elastic) normal and tangential springs.
247
248
Discrete element methods
F3
ma
~. =/~kn
F2 (a)
F4 (b)
Fig. 9.2 (a) Discrete element bodies (particles)in contact, giving rise to axial and tangential contact forces. Force magnitudes related to the relative normal and tangential velocity and to relative normal and tangential velocity at the contact point; (b) arbitrary particle shapes as assemblies of clustered particles of simple shapes.
The stiffness matrix is obtained by considering rigid bodies to be connected by distributed normal and tangential springs with stiffness values k, and ks per unit length, respectively. The rigid displacement field within an arbitrary two-dimensional block is expressed in terms of the centroid displacements and rotation (u, v, O)r. Centroid degree offreedoms (Ui, 1)/, Oi) T and (uj, vj, Oj) r ofthe two neighbouring blocks (Fig. 9.3) with centroids located at (x ~ yO) and (x ~ yO), respectively, define independently the displacements at a common interface point P
~C I I
Di
C
Fig. 9.3 Two rigid blocks with an elastic interface contact in RBSM.
Early DEM formulations Table 9.1
Simple discrete element method algorithm (after Cundall and Strack 6)
ny (y~,y2) \ ei - - y i - x i - - ( c o s a D -ti -- (sin a, -- cos a)
sin
a)
Body centre (Xl, X2), (Yl, Y2)
1 ,.
~,, ( ,
Translational velocity xi, j;i
xi, 0 = const
Rate of rotation
"~i, O" = const
I
n-1
n
i
At
Ox,Oy
Time
n+l
n+2
A. F O R C E - D I S P L A C E M E N T LAW (1) Relative velocities (2) Relative displacements
J~i -- (Jci - Yi) - (Ox gx -[-Oyey)ti h : X i e i , k -- Xiti An = nat, As = sat
(3) Contact force increments
A F . = kn(An +/~h), A F s = k s ( A s +/3k)
(4) Total forces (5) Check for slip
F . = Fn-~ +
AFn, Fs =
F~"-~ +
AFs
Fs = min ( Fs , C + Fn tan q~)
If not converged go to (1), else Mx -- ~-~FxRx, My -- ~ - ] F y R y
(6) Compute moments B. EQUATIONS OF MOTION (1) Assume force and moment constant over At (2) Acceleration (3) Velocity
A t = (t n+l/2 -- t n-l~2) m Jci -- Y]~ Fi, I 0 : ~]~ Mi jcn+l/2 __ ion-l~2 0/+1/2 i "~- J(i At, 0 n-1/2 Jr- OAt At : (t n+l - t n)
(4) Assume velocities constant over At (5) Displacements (6) Rotation
~7+' = x~ + ~7+'/:at on+ 1 _. On _[_ on+ 1/2 A t
If not converged go to A (6), else Next time step: Go to A
C. TIME I N C R E M E N T
Up
--
Qpu
v;] U-
[Ui
1 QP-
Ui
0
i 1 0
0
0i
Hj
_ ( y ~ _ yO) (x~ - x ~
Vj
Oj] T
oo 0 1 o
0 o 1
o ]
0 -(y~-y~ (x~-x~
(9.1)
249
250 Discreteelement methods and the relative displacements at the location P are expressed as
10]
-- MUp = M R Q p u -- B u with M = [ ; 1
0 -1
0
and Up : l~Up - R Q p u
(9.2)
after Up is projected fJp aligned to the local coordinate system along the interface. The constitutive relation in plane stress can be written as of=Dr with D -
~r=
[
O'n,
ks
7"s
IT
h (1 - 2u) (1 + u)
E ks= h(1 + u )
(9.3)
where h is the sum of shortest distances between the two block centroids to the contact line. This projected distance h is also used to evaluate approximate normal and shear strain components through ep-
I l[ n] 1
% e
-h
6s = h r P
(9.4)
Applying the virtual work principle along the interface leads to f=
[~sBrDBds]u-Ku
(9.5)
Generalization to the three-dimensional situation is straightforward and the method can clearly be interpreted as a similar method to the finite element method with joint (interface) elements of zero thickness, the only difference coming from the assumption that the overall elastic behaviour is represented only by the distributed stiffness springs along interfaces. Given some criterion, the contact springs may be deactivated, so that an interface becomes a discontinuity and the progressive failure is modelled by following evolving discontinuities, through cracks and/or slipping at the interfaces between rigid blocks.
A principal algorithmic issue of the discrete element method represents a detection of bodies in contact followed by the evaluation of the contact forces (both a magnitude and a direction) emanating from the contact. The contact detection problem generally can be stated as one of finding a contact or overlap of a given contactor body with a number of bodies from a target set of N bodies in R" space but the strategies for contact detection are intimately related to the geometric characterization and topological attributes of interacting bodies. If the interacting bodies are of very simple geometry (e.g. circular in two dimensions or spherical in three dimensions) an algorithmic check for a possible overlap is simple and the definition of the tangential contact plane is unambiguous. Rigid bodies of more complex shapes can be approximated by forming convenient clusters of rigidly connected circular or spherical shapes, while the contact detection
Contact detection =
"
NG
J=i
0
h
I=i
I
..........
1..._
cell location for each body i= (int)[(xm/h) + 1]j= (int)[(Ym/k) + 1] Xm, Ym coordinates body centroid
Fig. 9.4 Hashing or binning algorithm for simple particle shapes and clustered particles. and resolution of the tangential contact plane remain the same as for individual bodies. However, this may be a crude approximation, and when interacting bodies of arbitrary geometry are considered, the algorithmic complexity of the contact detection and the associated definition of the contact plane between the two bodies increase significantly. The efficiency of these algorithms is crucial, as the conceptually simple procedure to test the possibility of contact of a body with all other bodies at every time step becomes highly uneconomical, once the number of bodies becomes large. Contact search algorithms are typically based on so-called body-based search or a space-based search. In the former, only the space in the vicinity of the specified discrete element is searched (and the search repeated only after a number of time steps), whereas the latter implies a subdivision of the total searching space into a number of overlapping
windows. For arbitrary geometric shapes, most algorithms typically employ a two phase strategy, where the bodies are first approximated by simpler geometric constructs (bounding boxes or bounding spheres) which encircle the actual body and a list of possible contact pairs is established via an efficient global neighbour or region search algorithm. This is then followed by a detailed local contact resolution phase, where the potential contact pairs are examined by considering the actual body geometries. This phase is strongly linked with the manner that the geometry of actual bodies is characterized.
9.3.1 Global neighbour or region search An example of the region search represents the boxing algorithm, g also referred to as hashing or binning algorithm.
251
252
Discrete element methods
The entire computational domain is typically subdivided into regular cells, and a list of bodies overlapping a given cell is established via the contact detection of square (2D) or cube (3D) regions. The contact resolution phase for a given body then comprises a detailed check for possible contact with all bodies which share the same cell and the check is usually extended to a list of bodies associated with neighbouring cells (i.e. 8 cells in two dimensions, 26 cells in three dimensions). The relationship between the cell size and the maximum size of a body is important for the overall efficiency. If the cell size is large compared to the body size, the initial search is fast, but many bodies are listed as potential contact pairs and the contact resolution phase is thereby extensive. However, if the cell size is small, the initial search is computationally more demanding, but this results in a smaller number of potential contact pairs and consequently a faster contact resolution phase. A balance is reached with cell sizes that are approximately of the size of the largest body in the system. Efficient contact detection algorithms and powerful data representation concepts are often borrowed from other disciplines, notably computer graphics, 9 with compact data representation techniques to describe the current geometric position of a discrete element - e.g. nodes, sides or faces. The decomposition of the computational space and various cell data representation for a large number of contactor objects (binary tree, quad tree, direct evidence, combination of direct evidence, rooted trees, alternating data trees) are usually adopted. 8'1~ Algorithmic issues and details of the associated data structures are quite involved and there is a non-linear relationship between the number of cells and the total number of bodies. Linear complexity contact detection algorithms are desirable and have been shown to be essential for simulations involving a very large number of bodies - e.g. the NBS (no binary search) algorithm for bodies of similar sizes has a total contact detection time proportional to the total number of bodies, irrespective of the particle packing density. 11 For simple shapes very efficient data structures map a minimum set of parameters which uniquely define a domain in Rn into a representative point in an associated R2n space (Fig. 9.5)- for example, a one-dimensional segment (a-b) is mapped into a representative point in two-dimensional space, with coordinates (a, b), or a two-dimensional rectangle of a size ( X m i n - - Xmax) and (Ymin -- Ymax) is mapped to a representative point in a four-dimensional space (Xmin, Ymin, Xmax, Xmax). Alternative representation schemes are also possible, e.g. by characterizing a rectangular domain in R 2 by the starting point coordinates (Xmin, Ymin) and the two rectangle sizes (hx, hy) followed by a mapping into an associated R 4 space (X~n, Ymin,hx, hy). As the representation of the physical domain is reduced to a point, region search algorithms are more efficient in the mapped R 2n spaces than in the physical R n space.
9.3.2 Contact resolution After the list of potential contact pairs is established through the global neighbour search, a detailed contact resolution algorithm is required, which will in turn depend on the way the detailed body geometries are defined. The contact resolution phase searches through potential pairs and if the actual contact is established, the algorithm needs to define the orientation of the contact plane, so that a local (n, t, s) coordinate
Contact detection Body 2 :
i
::Body 1
!
9
,
1 , , , ,
, , , , ,
I
~s
Body 3
i
X2S XlE
X3S X2E
X3E
in 2D space Body 1 ~ [x 1s, Xl E] Body 2 ~ [X2s,X2E] Body 3 ~ [X3s,X3E] XEnd
,.~
~
X3E X2E
tnd
X~E
XlS
X2S
X3S
Xstart
~ ............... - - ~ . - - o P4 P2 ~
P6 '" ..................
iI
o
ms
x"
in 4D space Body 1 ~ [P1,P2] Body 1 ~ [P1,P2] Body 1 -~ [P1,P2] Fig. 9.5 Mapping of a segment from one-dimensional space into a point in an associated two-dimensional space and mapping of a box in two dimensions into an associated four-dimensional space 9
system can be determined and the conditions for impenetrability or sliding can be properly applied. Body geometry characterization can be categorized 16 into three main groups: (a) polygon or polyhedron representation, (b) implicit continuous function representation (elliptical or general superquadrics) and (c) discrete function representation (DFR). Polygonal representation in two dimensions defines a body in terms of comers and edges, and there are a number of algorithms to determine an intersection of two
253
254
Discreteelement methods
Fig. 9.6 Definition of the contact plane: a unique definition for the corner to edge case (a), the edge to edge case (b) and an ambiguous situation for the corner-to-corner (c) contact problem (after Hogue16).
coplanar polygons. Convex polygons simplify the algorithm, as concave comers imply a possibility of multiple contact points. There are no difficulties to define the orientation of the contact plane when considering the comer-to-edge or the edge-to-edge contact, as the contact plane normal is uniquely defined by the edge normal. Difficulties arise when a comer-to-comer contact needs to be resolved (Fig. 9.6), as the orientation of the contact plane is not uniquely defined. This problem can be regularized by the rounding of comers 17 to ensure continuous changes of the contact plane outer normals. The generalization in three dimensions can be realized by the common plane concept, 17 which 'hovers' between the two bodies coming into a comer-to-comer contact, and the actual orientation of this common plane is found by a local optimization problem of maximizing the gap between the plane and a set of closest comers. Another possible procedure (restricted to a two-dimensional situation) utilizes an optimum triangularization of the space between the polygons, 18 whereby a collapse of a triangle indicates an occurrence of contact. A continuous implicit function representation of bodies, e.g. elliptical particles in two dimensions, 19'2~ellipsoids in three dimensions, 21'22 or superquadrics (Fig. 9.7) in two and three dimensions, 23-25 provides an opportunity to employ a simple analytical check (i.e. inside-outside) to identify whether a given point lies inside or on the boundary (~b(x, y) < 0) or outside (~b(x, y) > 0) of the body where ~(x , y)= (x) ~' + (b) & a
1
(9.6)
Unlike a polygonal representation, it is now significantly more difficult to solve for a complete intersection of overlapping superquadrics, and the solution is normally found by discretizing one of the surfaces into facets and nodes, and the contact for a specific node on one body can be verified through the inside-outside analytical check with respect to the functional representation of the other body. A discrete functional representation (DFR) describes the body boundary with a parametric function in one parameter. The concept of the DFR 26 replaces the continuous implicit function representation of bodies by the set of pre-evaluated function values on a background grid for the inside-outside check, which is used as an algorithmic look-up table. As such discrete function values at the grid nodes can be arbitrary, a grid (or cage) of cells can also be used to model an arbitrarily shaped body - including bodies with holes. The DFR concept in contact detection is illustrated through the polar DFR descriptor in two dimensions 27 (Fig. 9.8) where, following the global region search for possible neighbours, the local contact is established by transforming the local coordinates of the approaching comer Pi of a body i into the polar coordinates
Contact detection
r
84
Fig. 9.7 Superquadrics in three dimensions (reproduced from Hogue~6).
~
Ri
Nj
Fig. 9.8 Contact detection and the polar discrete functional representation (DFR) of bodies' geometry.
255
256
Discreteelement methods
of the other body Pij and checking if an intersection between the segments (Oj P/J) and (Mj Nj) can be found.
In the case of rigid bodies of simple shapes, an event-by-event simulation strategy can be applied in order to strictly satisfy the condition of impenetrability. In such cases, if the contact time is considered to be infinitely short, time instants of collisions can be calculated exactly and momentum exchange methodologies used to determine post-impact velocities, with possible energy loss accounted for by a coefficient of restitution or by friction losses. In either case there are non-smooth step changes and reversals in velocities. Such methodologies are used for molecular dynamics (MD) simulations with a very large number of particles. Although the event-driven algorithms work well for loose (gas-like) assemblies of particles, for dense configurations these lead to an effective solution locking or inelastic collapse, 28 manifested in critically slow simulations.
9.4.1 Regularization of non-smooth contact In the case of deformable bodies the contact time is finite and contact forces vary for the duration of the contact. In computational simulations it is necessary to regularize the non-smooth nature for the impenetrability and friction conditions. Constraints on impenetrability during the contact between the two bodies f2c and f2t require that the gap between them must be non-negative. In frictionless and cohesionless contacts, only a compressive interaction force Fn exists and this interaction force vanishes for an inactive contact described by the Signorini condition g > 0. The infinitely steep 'non-smooth' graph (viz. Fig. 9.9) is regularized by assuming that the interaction force Fn is a function of the gap violation and is replaced by a penalty formulation, with a linear or non-linear penalty coefficient. Non-smooth relations also exist if the interaction law considers a tangential friction force Fs, related and opposed to the relative sliding velocity k. For a Coulomb friction law, there exists a threshold tangential force, proportional to the normal interaction force Fs = # Fn, before any sliding can occur, corresponding again to an infinitely steep graph.
9.4.2 Contact constraints between bodies Once the contact between discrete elements is detected as a geometric overlap, the actual contact forces have to be evaluated, which are then used for the subsequent motion of the discrete elements controlled by the governing dynamic equilibrium equations. Contact forces follow from an imposition of contact constraints at contacting points. In variational formulations, a constraint functional FIc can therefore be added to the functional of the unconstrained system using a penalty function, Lagrange multiplier, augmented Lagrangian or perturbed Lagrangian form, as discussed in Sec. 7.2.
Contact constraints and boundary conditions 257
Fn
S
g i
g>-O
Fn>-O
Fng>- O
~Fn
f (gm k
g v
F =-kg >_0
Fn = f (gm) >_ 0
~=llFn
~-s=~F,
~ =/IFn
P,=~F,
IIF~II~~F~
F~=~
i IIF~II---~Fn
Fig. 9.9 Non-smooth treatment of normal contact, regularized treatment of normal contact (linear and nonlinear penalty term) and non-smooth and regularized treatment of frictional contact.
258 Discreteelement methods
'',,
%
%%%
n
t
g = (x i - x j ) . n < 0
g= (xi-xj).n>
O
Fi 9. 9.10 Determination of the contact surface and its local n - t coordinate system.
Most discrete element formulations utilize the penalty function concept. The information about the position and the orientation normal n of the contact surface (Fig. 9.10) as well as the current geometric overlap or penetration of contactor objects is used to establish the direction and the intensity of the contact forces between contactor objects at any given time. The impenetrability condition is formulated through the gap function g = [x i - xJ] 9n _< 0, which defines the relative displacement in the normal direction Un = [ u i - u j ] 9 n and the tangential direction ut = [ u i - u j ] 9 t. The resolution of the total contact traction tr into normal and tangential components t~ = (tc. n)n + (t~. t)t = t~' n + t [ t is then integrated over the contact surface to obtain the normal F, and tangential component Ft of the contact force. In the case of a comer-to-comer contact, no rigorous analytical solution exists, and rounding of comers for arbitrary shaped bodies leads to an approximate Hertzian solution. In the case of a non-frictional contact (i.e. normal contact force only) an elegant resolution to the comer-to-corner problem in two dimensions is provided by the contact energy potential algorithm. 29 Contact energy W is assumed to be a function of the overlap area between the two bodies W (A) and the contact force is oriented in the direction which corresponds to the highest rate of reduction of the overlap area A. As the overlap area is relative to bodies f2i and f2j, it can be expressed as a function of position of the comer point Xp and the rotational angle 0 with respect to the starting reference frame. The procedure (Fig. 9.11) fumishes a robust, unambiguous orientation of the contact plane in the two-dimensional comer-to-comer contact case, running through the intersection points g and h, and the contact force over the contact surface bw is applied through the reference contact point shifted by a distance d = Mo/11Fn11from the comer P, where the force Fn and the moment Mo are defined from the contact energy potential as
Fn=
OW(A) OA(xp, O) OW(A) OA(xp, O) and Mo= OA Oxp OA O0
Contact constraints and boundary conditions
A(Xp, O)
bw=
" ..... g
;
h
Fig. 9.11 Corner-to-cornercontact, based on energy potential (after Feng and Owen29). Different choices for the potential function (Table 9.2) are capable of reproducing various traditional models for contact forces.
9.4.3 Contact constraints on model boundaries Treatment of model boundaries represents an important aspect in the discrete element methods. Boundaries can be formulated either as real physical boundaries, or through virtual constraints. In the so-called periodic boundary, often adopted in the DEM analysis of granular media, a virtual constraint implies that the particles (or bodies) 'exiting' on one side of the computational domain with a certain velocity are reintroduced on the other side, with the same, now 'incoming', velocity. The use of periodic boundaries excludes capturing of any localization phenomena. In cases of particle assemblies, flexible and hydrostatic boundaries have also been employed. A flexible boundary 3~ framework is equivalent to physically 'stringing' together particles on the perimeter of the discrete element assembly, forming a boundary network. Flexible boundaries are mostly used to simulate the controlled stress boundary condition t Y i j = t7 C , where the traction vector tim _ _ Am crijnjC m is distributed over the centroids of a triangular facet Am, that is formed by three boundary particles. The hydrostatic boundary 31 can be interpreted as a virtual wall of pressurized fluid imagined to surround granular material particles and the desired hydrostatic pressure at the centroid of the intersection area between the particle and virtual wall. Table 9.2 Contact energy potential functions (after Feng and Owen 29)
W (A )
IIF.II
Hertz-type form
kn A 2 kn A 3/2
kn bw kn A 1/2bw
General power form
1 A mkn
Linear form
m
kn Am-1 bw
259
260 Discreteelement methods
The simplest form of a physical boundary representation in two dimensions is given by the definition of line segments (often referred to as 'walls' in the DEM context), and the kinematics of the contact between the particle and the wall is again resolved in the penalty format. Frequently, individual particles are declared as immovable, thereby creating an efficient way of representing and characterizing a rigid boundary, without any changes in the contact detection algorithm. An interesting idea is the finite wall (FW) method, 32'33 with the boundary surface triangulated into a number of rigid planar elements, which are then represented by inscribed circles and subsequently used in the contact detection analysis between the particles and the boundary.
Consideration of deformability increases the complexity for the analysis of multibody systems, where the bodies represent fully deformable media or belong to a class of constrained media, corresponding to restricted deformability. The deformability of individual discrete elements was initially dealt with by subdividing the body into triangular constant strain elements, 34 which can be identified as an early precursor of today's combined finite/discrete element modelling. Further developments of the discrete element methods include simple body deformability, so that a displacement at any point within a simply deformable element can be expressed by U i = U 0 "-["~.)ijXj + EijX J, where u ~ are the rigid body displacements of the element centroid, ~gj, Eij are the rotation and strain tensor respectively and x~ represent local coordinates of the point, relative to element centroid. It should be noted that this deformability statement implies a spatially constant deformation gradient that is equivalent to the class of pseudo-rigid bodies 35 discussed in Chapter 8. Displacements of body centroids follow from balance equations for the translation of the centre of mass in direction ~-~i Fi -" nit, and the rotation about the centre of mass ~-~i M/c - I ~Ji. The simply deformable discrete elements 17 comprise generalized strain modes ek, independent of the rigid body modes mk)~ = ~ - ~r~, where m ~ is the generalized mass, ~rk is a generalized applied stresses and crk/ generalized internal stresses, corresponding to strain modes. The discrete element deformation field (displacements relative to the centroid) can also be expanded in terms of the eigenmodes of the generalized eigenvalue problem, associated with the discrete element stiffness and mass matrix, giving rise to the Modal Expansion Discrete Element Method, 36'37 where the 'deformability' equations are uncoupled (due to the orthogonality of eigenvectors) and appear as modal equations written with respect to a non-inertial frame of reference. In the practical realization and implementation of the discrete element methodology the deformability of a discrete element of an arbitrary shape is either described by an intemal division into finite elements (discrete finite elements and/or combined finite/discrete elements) or by the polynomial expansion of a given order for the displacement field (discontinuous deformations analysis).
9.5.1 Combined finite/discrete element method A combination of discrete and finite elements was employed in the discrete finite element approach by Ghaboussi. 38 The combined finite/discrete element approach of
Block deformability 261 Munjiza e t al. 39'40 introduces bodies deformability through finite element expansion of the displacement field within a discrete element, while the contact between the discrete elements is solved again in an explicit transient dynamic setting. The overall algorithmic framework for a combined finite element/discrete element framework (Table 9.3) proceeds by solving the balance equations, while updating force histories as a consequence of contacts between different discrete element domains, internally subdivided into finite elements. A combined finite element/discrete element method is easily extended to problems comprising progressive fracturing and fragmentation, as complex constitutive relations can be utilized within discrete elements. Large displacements and rotations of discrete domains, internally discretized by finite elements, have been formulated in a generalized updated Lagrangian (UL) format by Barbosa and G h a b o u s s i . 41 The contact forces were obtained using a penalty form (concentrated and distributed contacts) and transformed into equivalent nodal forces on the finite element mesh. The equations of motion for each of the deformable discrete elements (assuming also a presence of mass proportional damping (C = ctMt)) are then expressed as MtU
+ o~MtU .
fetxt .+ fctont.
fitnt.
fetxt _1_ ftont
~-~/
Ja
t T t0"k df2k (Bk)
(9.7)
In the computational solution Barbosa and Ghaboussi 41 used a central difference scheme for integrating the incremental updated Lagrangian formulation, which neglects the non-linear part of the stress-strain relationship. Evaluation of the internal force vector fi
nt
At-
~-~~ k
~+At
(Btk+At)T--t+Atdak ~
at the new time station t + At recognizes the continuous changing of the configuration, as the Cauchy stress at t + At cannot be evaluated by simply adding a stress increment Table 9.3 Pseudo code for the combined discrete/finite element method, small displacement analysis, including material non-linearity (after Petrinid 12) (1) Increment from the time station t -- tn current displacement state Un external load vector, contact forces -nFeXt, FnC ~ ~-ext internal force, e.g. Finnt = ff2 BT O'n df2 (2) Solve for the displacement increment from M / i n + F int - - 1 ~ ext --n --n tin+l/2 = M -l(~'extn - Fint)n At + iln-1/2 Un+l = Un + iln+l/2At for an explicit time-stepping scheme (3) Compute the strain increment A~n+l = f(Aun+l) (4) Check the total stress predictor O'*+1 = 0"n + D A~Tn+l against a failure criterion, e.g. hardening plasticity qS(0"n+l, N) = 0 (4) Compute inelastic strain increment 'A~-inel e.g. using an associated plastic flow rule - " " n + 1' A ~inel (5) Update stress state 0"n+l = D(A~n+I - "'~'n+l ) (6) Establish contact states between discrete element domains at tn+l and the associated contact forces Fn+ 1 (7) n ~ n + l ,
go to step (1)
262 Discreteelement methods due to straining of the material to the Cauchy stress at t and the effects of the rigid body rotation on the components of the Cauchy stress tensor need to be accounted for [viz. incremental form (6.90b) or Jaumann-Zaremba rate form (6.109)]. For inelastic analyses, care needs be given to the objectivity of the adopted constitutive law. Advanced combined discrete/finite element frameworks 42 include a rigorous treatment of changes in configuration, evaluation of the deformation gradient and the objective stress measures.
9.5.2 Discontinuous deformation analysis The discontinuous deformation analysis (DDA) employs a general polynomial approximation of the displacement field superimposed over the centroid movement for each discrete body. 43'44 The DDA development followed the formulation of the Keyblock Theory 45 and was used for simulating the behaviour of a jointed deformable rock. Blocks of arbitrary shapes with convex or concave boundaries, including holes, are considered. The original framework under the name the 'DDA method' comprises a number of distinct features: (a) the assumption of the order of the displacement approximation field over the whole block domain, (b) the derivation of the incremental equilibrium equations on the basis of the minimization of potential energy, (c) the block interface constitutive law (Mohr-Coulomb) with tension cutoff and (d) use of a special implicit time-stepping algorithm. The original implementation has been expanded and modified by many other researchers. The method is realized in an incremental form and it deals with the large displacements and deformations as an accumulation of small displacements and deformations. The issue of inaccuracies when large rotations are occurring has been recognized and several partial remedies have been proposed. 46'47 The DDA method represents an alternative way of introducing solid deformability into the discrete element framework, and block sliding and separation is considered along predetermined discontinuity planes at block boundaries. The initial formulation was restricted to simply deformable blocks (constant strain state over the entire block of arbitrary shapes in two dimensions, similar to a constant deformation gradient as in pseudo-rigid bodies, Fig. 9.12). The first-order polynomial displacement field for the block [u ' v] ir 1st is equivalent to the three displacement components of the block centroid, augmented by the displacement field with a clear physical meaning of three constant strain states, i.e. it represents a constrained medium capable only of sustaining a spatially constant displacement gradient. The six deformation variables are denoted by the block deformation vector D~st.
[;]
_ [0 st 9
nZlst =
-
L.J 1st
[u0
0 1
--(Y--Yo) (x - Xo)
1)o d/)0 ~x
(X--Xo) 0
Cy 7xy]ir
0 (Y - Yo)
(y--yo)/2] Dilst (x - Xo)/2]i (9.8)
Tilst Dilst
An increase of block deformability characterization can be achieved by increasing the order of the displacement polynomial used, leading to a correspondingly larger number
Block deformability 263
Di = [ u, v, q~, ex, % exy]
ii
~ao ~
bo
81 bl
",
[;1:[,o ~ ; o o~; ~~ ~ x
,.
I
82
:Ox~~o ~I ~
y
~o,
53
a4 b4 85 .55_
Fig. 9.12 Deformation variables for the first- and second-order polynomial approximation in discontinuous deformation analysis.
of block deformation variables (higher-order DDA, where higher-order strain fields are added for blocks of arbitrary shape). By increasing the order of the displacement field, the block medium becomes less constrained and gradually moves towards full deformability. A second-order approximation for the block displacement field requires 12 deformation variables, which can again be given a recognizable physical meaning. The formulation implies a linearly varying displacement gradient across the block domain and the deformation parameters comprise the centroid displacements and rotation, strain tensor components at the centroid, as well as the spatial gradients of the strain tensor components, i.e.
D~nd--[U0
Vo
0 ----C x
dp0
0
~y
0
")/xy ~x,x
~y,x
~xy,x
~x,y
~y,y
'ffxy,y ] iT
(9.9) For a higher-order approximation of order n for the block displacement field, 48 the spatial distribution of the deformation gradient is of the order (n - 1) and a clear physical interpretation of the deformation variables is no longer plausible. The generalized block deformation variables are defined as U
=
dl at- d3x + dsy + d7x2 ~- d9xy + dllY 2 -+-... + dr_l xn + . . . + dm_lY n d2 4- d4x + d6y + d8x2 + dloxy + dl2y 2 + . . . + drx n + . . . + dmy n
Iul i
i
-
kJ
i
T.thD.t h
i
Dnth
d3
.
(9.10)
nth
Once the block displacement field is approximated with a finite number of generalized deformation variables, the associated block strain and block stress field can be expressed
264
Discrete
element
methods
in a manner similar to finite elements as
ei -- BiDi Eiei -- EiBiDi
ori =
(9.11)
For a system of N blocks, all block deformation variables (n variables per block, depending on the order of the approximation) are assembled into a set of system deformation variables (N x n) and the incremental equilibrium equations are derived from the minimization of the total potential energy FI comprising contributions from the block strain energy Fie, energy from the external concentrated and distributed loads FI p, Flq, interblock contact energy FIe, block initial stress energy H~, as well as the energy associated with an imposition of displacement boundary conditions using a penalty approach Fib I-I = He ~- lip if- I-Iq d- 1-Ic d-- I'I~r q- lib
(9.12)
Components of the stiffness matrix and the load vector are obtained by the usual process of the minimization of the potential energy
Kii -- f~i BTEiBi dr2
(9.13)
The global system stiffness matrix (Fig. 9.13) contains (n x n) submatrices Kii and non-zero submatrices Kij are included when the blocks i and j are in active contact and D comprises displacement variables of all blocks considered in the system. The interblock contact conditions of impenetrability and Mohr-Coulomb friction can be interpreted as block displacement constraints, which are algorithmically reduced to an interaction problem between a vertex of one block with the edge of another block. Denoting the increments of deformation variables of the two blocks in contact by I)i and Dj respectively, the penetration of the vertex in the direction normal to the block edge can be expressed as a function of these deformation increments and various algorithmic approaches can be adopted for an implicit imposition of the impenetrability condition (Fig. 9.14). Ifthepenaltyformat is adopted to impose contact constraints, additional terms appear both in the global DDA stiffness matrix as well as in the RHS load vector (Table 9.4),
Kll
K21 9
Kr,,~
Mlm
D1
fl
K12
"'"
i
"'"
:
"
"
Kn~2
"'"
r~,,,
Dr,
fr,,
K22
.."
K
K2r,,
D2
D
f2
=
Fig. 9.13 Assembly process in the DDA analysis.
f
Block deformability
C
~i
V
Fig. 9.14 Non-Penetrationand frictional contact constraint in DDA, point-to-edge contact.
Table 9.4 Additional terms in the DDA stiffness matrix and the load vectors as a result of contact between bodies i and j Normal non-penetration constraint Kii : Kii + pHi HT T Kij - H i G j
fi = fi - p C Hi
Kji ~-- G j n f
fj -- fj - p C G j
Kjj = Kjj + pGjGj T
Frictional constraint Kii -- Kii Jr p n f r H f r'T Kij "- U f r G f r'T
fi -- fi - p c f r n f r
Kji = G f r a f r,T
fj ~- fj - p C fr G f r
_l-~fr f~ fr, T
K j j -- K j j at- lJ~J j IJ j
and the terms differ depending on the nature of the constraint (normal gap or frictional constraint). The DDA method typically adopts an implicit algorithm and uses so-called openclose iterations which proceed until the global equilibrium is satisfied (norm of the outof-balance forces within some tolerance) and a near zero penetration condition satisfied at all active contact positions. For the normal gap condition convergence implies that the identified set of contacts does not change between iterations, whereas for the frictional constraint it implies that the changes in the location of the projected contact point remains within a given tolerance. For complex block shapes, the convergence may be very slow, as both activation and deactivation of contacts during the iteration process are possible.
9.5.3 Block fracturing and fragmentation Consideration of block deformability allows for a more precise determination of stress states throughout the discrete element. Block interface and through-block fracturing
265
266
Discrete element methods
and fragmentation were introduced to the discrete element method in the form of brittle fracturing. 49-52 Later models recognized the need to regularize the strain softening response for quasi-brittle materials, prior to eventual separation through cracking and/or shear slip). Constitutive models adopted for a pre-fragmentation stage are often based on concepts of continuum damage mechanics, regularized strain softening plasticity formulations or are formulated using some higher-order continuum theory. 53 Computational issues for discrete element methods, when more complex formulations for inelastic constitutive models are adopted, are the same as in the continuum FEM context computational inelasticity, where the consistent linearization procedure has affected the application of plasticity of fracture criteria in the DEM context. Moreover, every time a partial fracture takes place, the discrete element changes its overall geometry while a complete fracture leads to a creation of two or more discrete elements, there is a need for automatic remeshing of the newly obtained domains (Fig. 9.15). An unstructured remeshing technique can be applied, where the new mesh orientation and density is decided upon based on the distribution of the residual material strength, or some other state variable. 39 An additional algorithmic problem arises upon separation, as the 'book-keeping' of neighbours and updating of the discrete element list are needed whenever a new (partial or complete) failure occurs. In addition, there is also a need to transfer state variables (plastic strains, equivalent plastic strain, dissipated energy, damage variables) from the original deformable discrete element to the newly created deformable discrete elements. Fragmentation frameworks are also used with DEM implementations which consider clusters of particles bonded together to represent a solid body of a complex shape. The bond stiffness terms are derived on the basis of equivalent continuum strain energy. 54'55 In these lattice-like models of solids, fracturing (Fig. 9.16) is introduced through breakage of interparticle lattice bonds, which may be treated as a simple normal bond (truss elements) and/or parallel bond (beam elements), which can be seen as a microscopic representation of the Cosserat continuum. Bond failure is typically considered on the basis of limited strength, but some softening lattice models for quasi-brittle material have also considered a gradual reduction of strength, i.e. softening, before a complete
\
Fracture orientation J
/
/
N2
'
Remeshing with through-element fracture
Remeshing with interelement fracture
Fig. 9.15 Elementand node-based local remeshing algorithm in a two-dimensional context (Munjiza et al.;39 Feng and 0wen29), following the weighted local residual strength concept.
Time integration for discrete element methods 267
!
,!.!,!-..
l
i~,.
.
..:!!! 8 4
..~.. ~
Fig. 9.16 Fracturing of the notched concrete beam modelled by jointed particulate assembly, with normal contact bond of limited strength (Itasca PFC 2DS6).
breakage of the bond takes place. Despite use of very simple bond failure rules, bonded particulate assemblies have been shown to reproduce macroscopic manifestations of softening, dilation, and progressive fracturing. Simulations using bonded particle assemblies typically use an explicit time-stepping scheme, i.e. the overall stiffness matrix is never assembled and steady-state solutions are obtained through dynamic relaxation (local or global damping by viscous or kinetic means). A jointed particulate medium is very similar to the more recent developments in lattice models for heterogeneous fracturing media. 57-59 Adaptive continuum/discontinuum strategies are also envisaged, where the discrete elements are adaptively introduced into a continuum model, if and when conditions for fracturing are met. Discrete element methods are also coupled with fluid flow methods and a number of combined finite/discrete element simulations of various multi-field problems have been reported.
Governing balance equations are integrated in time using a time-stepping scheme, such as the GN22 method discussed in Chapter 2 and in reference 60. Both the usual DEM and the DDA time-stepping schemes can be interpreted as members of the GN22 family of algorithms and only the issues pertaining to their use in a discrete element context will be discussed here. The traditional DEM framework implies a conditionally stable explicit time-stepping scheme. Local numerical dissipation is sometimes introduced to avoid any artificial increase in the contact energy. 61 Such modified temporal operators do not affect the size of the critical time step, and the choice of the actual computational time step is controlled by accuracy requirements resulting from the numerical energy dissipation added to the system. An energy balance check is desired, for the purpose of monitoring possible creation of spurious energy, as well as monitoring energy dissipation during fracturing, where the inclusion of softening imposes severe limits on the admissible time step. When inertial effects are omitted in the DDA context, the resulting system of equilibrium equations may be singular when blocks are separated or have insufficient
268
Discreteelement methods
constraints. The regularity of the system stiffness matrix is restored by adding very soft spring stiffness to the block centroid deformation variables. Such modification is not necessary when inertial effects are included. The DDA framework typically utilizes a particular type of the generalized collocation time integration scheme 62 M
Stn+ol "t- C Xn+c~ + K Xn+o~ - -
f.+~
(1 - a) Stn 4- O/Stn+l fn+~ -- (1 -- a) fn + afn+l 1 (aAt)2[(1 _/32) St. +/32 Stn-t-1]
Stn+a
--
(9.14)
Xn+~ -- Xn + a At[(1 - ill) Stn +/~1Stn+l] which, for the case a - 1, leads to a recursive algorithm 2 M + 2/~1 C] Xn 2 M + 2/31 C + K] fl2At [/~2At i /~2At Xn+l =fn+l+ [fl2At2
1
M+ ( 2/31-1) C]xn+ [(~2-1)
/31
M+At(~22-1)
CI
_st"
(9.15)
A specific choice of the time integration parameters /~1 - - /~2 - - 1 represents an implicit, unconditionally stable scheme. The DDA time-stepping algorithm is also referred to as Right Riemann in accelerations, 63 as both the new displacement and velocity depend only on the acceleration at the end of the time increment as Xn+1 - -
Xn
Xn+l - - Xn
-~- At xn + g1 A t 2 St.+1 "-t- At Stn+ 1
(9.16)
which may also be expressed as 1 Xn+ 1 - - X n -~- ~
1
At [~:. +
Xn+l]
(9.17)
~t.+l - A-t [/~.+l -/~n]
For this choice of time integration parameters the coefficient matrix associated with the acceleration vector St. vanishes, hence
i+xTC+I
xT M+ TC Xn+
i+C
Xn+l= L+I
(9.18)
The effective stiffness matrix (( includes the inertia and damping terms and the effective load vector fn+l accounts for the velocity at the start of the increment. Thus, the next solution Xn+l is obtained from (9.19) Xn+ 1 ~ l ~ - l ? n + 1 Using Eq. (9.17) the next rate of deformation vector (velocity) then equals 2 X~+l = ~ [X.+l - x.] - ~:.
(9.20)
Time 1,4
0.8
'
t
I
I
I
I
methods I
I
atio
_
~u~ 0.4
t_
.~ 0 . 6 -
0.3 0.2
0.4t
0.20.01
J
element
0.5
~ 9 0.8-
0
I
discrete
0.6
1
"0
Q..
I
for
0.7
1.2=
integration
'
,
,I
O. 1
,
,
,I
,
1
\
0.1 ,
~I--
10
,
,
,I
1 O0
,
,
,
1000
0 ~,
0
i
1
i
2
i
3
i
4
i
i
5 6 f~
i
7
i
8
i
9 10
Fig. 9.17 Spectralradius and algorithmic damping for the DDA time integration scheme (generalized Newmark scheme GN22/~i = ~2 = I) (reproduced from Doolin and Sitar63).
which is then used for the start of the next time increment. We note that in this form it is never necessary to compute the acceleration during the solution process. If necessary, the acceleration may be computed to interpret the inertial effects in the solution. The effective stiffness matrix is regular due to the presence of inertia terms and the block separation can be accounted for without a need to add artificial springs to block centroid variables, even when blocks are separated. The time-stepping procedure can also be used to obtain a steady-state solution through a dynamic relaxation process. The steady-state solution can be obtained by the use of the so-called kinetic damping, i.e. by simply setting all block velocities to zero at the beginning of every increment. From the analysis of spectral radii for the above time integrator (Fig. 9.17), it is clear that the scheme is associated with a very substantial numerical damping, 63'64 which is otherwise absent in the central difference scheme. Moreover, predictor-corrector or even predictor-multiple corrector schemes are adopted in simulations of granular materials, 65 in order to capture the high frequency events, like the collapse of arching or avalanching. The discrete element method simulations are computationally intensive and there is a need for the development of parallel processing procedures. The explicit time integration procedure is a naturally concurrent process and can be implemented on parallel hardware in a highly efficient form. 66'67 Much of the computational effort devoted to contact detection and the search algorithms, which is formulated principally for sequential implementation, has to be restructured to suit parallel computation. Efficient solution procedures are employed to minimize communication requirements and maintain a load balanced processor configuration. Parallel implementations of DEM are typically made on both workstation clusters as well as multi-processor workstations. In this way the effort of porting software from sequential to parallel hardware is usually minimized. The program development often utilizes the sequential programming language with the organization of data structure suited to multi-processor workstation environment, where every processor independently performs the same basic DEM algorithm on a subdomain and only communicates with other subdomains via interface data. In view of the large mass of time dependent data produced by a typical FEM/DEM simulation, it is essential that some means be available of continually visualizing the solution process. Visualization is
269
270
Discreteelement methods
particularly important in displaying the transition from a continuous to a discontinuous state. Visual representation is also used to monitor energy balance, as many discretization concepts both in space (stiffness, mass, fracturing, fragmentation, contact) and time can contribute to spurious energy imbalances.
ii~liJ~ili~i i~i~ i ililiU':~"~i. .i.i.!.i.ii;i.iiii.iiiiii;iilJiJii~, ~ ~'iJ~i~'i:~' i~ii~ililiiiii~il iii~iiliiiii~i~~i~i~i~i;i~i~i i !i~i i i ililJiiii~iiiiiiiiiiiiii!i! i i iJi~i~'i:::::!~!~i :ii!iiii~i~iil~ii?~ :ii~iiiiii!i i ~iiiiiilJi~i i~i~i~iiii!ii~iiiiiiiiiiii ~i~iiiiiiiii i ii~iiiii i ~'~'~iiii~ii~~iliiililil;iiiii!iiiiiiiiii!iiii~ i~iiiiilii!~iiiiiiiiii!i!iiii ii!i E~i~iiii iJi!i~i~i~!~d~iii~ iiiiliiiiiliii~iii~iiii iiiiiiiii~i i~ii~iii!ii!i i i':':i::i i i i ~!il:i~::i:'~i~'!i :':i'ii~iiii!i iiii~iiiiii!iiiii;iiii i~i ~i~iiiiiiiiii~ii!i ! iliiiiiiili i i~ilil ii~!iiliii~ i~~i~iiii'i:':i'~i!;iii i iiiii~ii!i ~ iiiiiidii!iilil~i~iiiii:' l:':iiiiiiiiiii~ ~ i~ii~i!if:~:':'~iiiliiiliiiiiiliiii~i~i;iiii~iilii~ l iiiii!iiiiiliiiiiii~i!iliiiii~~ii~i~i~!ii~!iilii~!~~i iii~i~!~i~!!~ ',!',i',!':i~i'~i',:~ii '~i~',!i~','i~i:~,"!~i~i~!i~:~i~!~i~i~iii~i~i'!,~i'~~i',i~i'i,~~ii~i!~i'~i~',i',i~',i:'i~i~i~i~,i~i,~ii~,i'i,~!i~'i~,i~',i',~i~i'~,!',i!'i~!'~!~, ,i'~,i,~'i~i',i~',ii'i!i~',~i~',i,'i!~,!ii',i',i~,i',ii~i~, ~,ii'~'~ii'~,,i',i'~,iili'~i~, '~'~il:i',i~i',i'i',i~,,!iii',iil!',i!ii!i'~i':iii~,!i~,~i~:i' i','~i~'~/i~,,iii!i~ ii!~,',',i}i'~,,i'!},~i~',i'i',~,i,',ii'i',i!',i'!,iili',i~,~,!i:,',i'~iii'i',i',~i'i:',iiii~,~:~,i~,i',i'i,ii!'',i,i'~i:,ili'iii,',:,i'i,i~,i!'',i!'~'~'/~i',i'~'~:,'~,i ~,~,~,~,~i~::~',i',i:!i':,~',~i~,~i~:,!,~~,',~,','~i':',~,i~,~,~~:~:~/~:~i~iii~ii~iii~i!~:~iii~::~ii~ ,',',i~i',':i',ii~,~/,',i ,,~i',':',',',i',~,',~i',ii~'~,',',~'~i':',',',,~'/:'i,'::', ~''~:
There are several other modelling frameworks which have been proposed as variations of the existing discrete element methods, e.g. the modified distinct element method (MDEM). 68 It is also interesting to observe a convergence of methodologies designed to deal with a transition from a continuum to a discontinuum. For example, in a continuum setting, the pre-existing macro discontinuities are traditionally accounted for through the use of interface (or joint) elements, 34 which may be used to model crack formation as well as shearing at joints or predetermined planes of weakness. 69 Joint planes are assumed to be of a discrete nature, their location and orientation are either given, or are fixed once they are formed. The term discrete cracking is adopted, as opposed to the term smeared cracking, where the localized failure at an integration point level is considered in a locally averaged sense. Natural extensions are interface constitutive models which account for a combination of cracking and Coulomb friction, which have been frequently formulated as two parameter failure surfaces in the context of computational plasticity. The non-smooth contact dynamics method (NSCDM) 7~ is closely related to both the combined FEM/DEM and the DDA, but it comprises significant differences, as the unilateral Signorini condition and the dry friction Mohr-Coulomb model are employed without resorting to contact regularization. For multi-body contact, with jumps in the velocity field, it is not possible to define the acceleration as a usual derivative of a smooth function and the non-smooth time discretized form of dynamic equations is obtained by integrating the balance equations, so that the velocities become the primary unknowns
ftn+l M (/r
-- Zk.) = J
tn
Xn+~ = x~ +
ftn+l f (t, X,/~) dt +
t,~+l
fd tn
r dt dtn
(9.21)
~: dt
The force impulse on the RHS is split into two parts, where the first part t,t"+' f(t, x, x) dt (which excludes the contact forces) is considered continuous, whereas the second part
f
t=+, r dt
(representing contact force contributions to the total impulse) is replaced by the mean value impulse 1 ft.+l = r dt rn+l "~ Jt,~
Unifying aspects of discrete element methods 271 over the finite time interval. In physical terms this implies that the actual contact interaction history is only accounted for in an averaged sense over the time interval. This has a consequence that the fine details of the contact history are discarded, which are either impossible to characterize due to insufficient data available, or inconsequential in terms of their effect on the overall behaviour of the multi-body system. Different time-stepping algorithms may be adopted and typically a low-order implicit time-stepping scheme is used. Resolution for contact kinematics then considers the relationship between the relative velocities of contacting bodies and the mean value impulse at discrete contact locations ]Krel -- Xrel-free "q- A t
K~rn+l
(9.22)
where the first part represents the 'free' relative velocity (without the influence of contact forces) and the second part comprises 'corrective' velocities emanating from contacts. The actual algorithm is realized in a similar manner to the closest point projection stress return schemes in computational plasticity- a 'free predictor' for relative velocities is followed by iterations to obtain 'iterative corrector' values for the mean value impulses, such that the inequality constraints (both Signorini and MohrCoulomb) are satisfied in a similar manner as the plastic consistency condition is iteratively satisfied in computational plasticity algorithms. The admissible domains in contact problems are generally non-convex and it is argued that it is necessary to treat the contact forces in a fully implicit manner, whereas other forces can be considered either explicitly or implicitly, 73 leading to either implicit/implicit or explicit/implicit algorithms.
There are many similarities in the apparently different discrete element methodologies, which partly stem from the fact that a given methodology is often presented as a package of specific choices, e.g. comprising a choice for the manner the bodies' deformability is treated, a way the governing equations are arrived at, a specific time integration scheme or a special way of dealing with unilateral constraints. For example, the DDA framework is often perceived as a methodology which specifically deals with simply deformable (constant strain) domains of arbitrary shapes, uses a very special case of implicit time-stepping scheme and treats contact constraints through a series of openclose iterations. Conversely, the DEM framework is often presented as a methodology for rigid particles of simple shapes, where the balance equations are integrated by an explicit scheme, with a penalty format for contacts. These perceived restrictions seem arbitrary, as there is nothing wrong, for example, in integrating the DEM equations with an implicit scheme or solving the DDA equations by an explicit method. It is also interesting to observe that the choice of a contact detection scheme was never perceived fixed and many contact detection algorithms have been applied. It can be observed that the major developments for all discrete element methods have primarily been associated with a characterization of bodies' enhanced deformability. While the higher-order DDA increases the order of polynomial approximation for the displacement field, other attributes of the method remain the same. In the same spirit
272
Discrete element methods
the combined finite/discrete element again enhances the description of deformability, while the other algorithmic features remain the same. It is therefore perhaps most appropriate to leave all other attributes aside and concentrate on the characterization of bodies' deformability and the treatment of geometrically non-linear motions as the basis for any comparison or equivalence and a potential identification of common, unifying concepts of different discrete element methodologies. In that context it is interesting to return to the background of the theory of pseudo-rigid bodies, 35 as discussed in Chapter 8. As in all discrete element methodologies, the theory is concerned with large-scale motions of deformable bodies. It was presented simultaneously as a generalization of the classical rigid body mechanics for bodies with some added deformability, as well as a 'restriction' (or coarse version in the terminology of reference 35) of a fully deformable continuum. Clearly, a hierarchy of theories emerges, depending on the degree of deformability added to the rigid body or on the class of restrictions introduced to the fully deformable continuum body. This is where the relationship between the theory of pseudo-rigid bodies and the discrete element methods provides a powerful link that allows for a rational comparison of different assumptions and an assessment of consequences of approximations adopted. In that spirit, the equivalence of the equations of motion between the elastic pseudorigid bodies and constant strain finite element approximations was confirmed and it was argued TM that the pseudo-rigid bodies can be viewed as a generalization of the low-order DDA in the regime of finite (as opposed to stepwise linearized as in DDA) kinematics. The crucial equivalence stems from the spatially homogeneous deformation gradient present in both cases, which can again be viewed as an enrichment of deformability given to the rigid body, or as a restriction imposed to the fully deformable continuum. The availability of a rational theory has an additional benefit by fully explaining manifestations of excessive volume changes for large rotations, discussed earlier in Sec. 9.7. Similarly, the higher-order DDA and higher-order pseudo-rigid bodies 75 can be unified in the sense that the deformation gradient F varies linearly over the body, i.e. F = F ~1) (X - X0) + F ~~
(9.23)
where F ~~ is the deformation gradient at X - X0 and F ~1) is the derivative of the deformation gradient with respect to X, assumed constant. Clearly, if F ~1) - 0, the simple pseudo-rigid body (or the lower-order DDA) is recovered. The extension to general higher-order approximations is plausible and again can be interpreted as enrichments of deformability or restrictions of the fully deformable continuum. Another rational interpretation of discrete element methods was developed based on Cosserat points theory 76 which moves away from algorithmic differences and concentrates on similarities between the DEM and DDA methods when they are recast using Hamilton's principle of least action.
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There is a degree of commonality of novel ideas in terms of describing the block deformability in discrete element methods and novel developments in the continuumbased techniques. The numerical manifold method of Shi 77 and Chen et al. 78 advocates
References 273 similar ideas to the ones advocated in the meshless 79 or the partition of unity methods 8~ in dealing with emerging discontinuities. Similar to the meshless methods, the manifold method identifies a cover displacement function Ci and the cover weighting function wi, where the geometry of the actual blocks ~"2 i is utilized for numerical integration purposes over a background grid. The treatment of any emerging discontinuities is envisaged by introducing the concept of effective cover regions, where there is a need to introduce n independent covers, if a cover intersects n disconnected domains. These concepts point to a range of possibilities in the simulation of progressive discontinuities in quasi-brittle materials. Discrete element methods increasingly appear in formulations and applications in multi-field or multi-physics problems, in particular in the area of the coupled fluid flow in discontinuous, jointed media. These discontinuous modelling frameworks are promising especially in the context of fragmentation and in the microscopic simulation of the behaviour of heterogeneous materials, where simple constitutive laws at the macro, meso or nano level directly generate manifestations of complex macroscopic behaviour, such as plasticity or fracture. 81 Increased computing power and efficient contact detection algorithms will not only allow modelling of progressive fracturing of continua and a transition to discontinuum including a fragmented state but will also encourage the development of discrete micro-structural models where an internal length scale may be intrinsically incorporated into the model. Moreover the large-scale simulations with adaptive multi-scale material models are increasingly feasible, where different regions are accounted for at a different scale of observation. Some of these aspects are considered in the next chapter.
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In the previous chapters we considered the domain to be a continuum, a rigid multibody system or a set of discrete elements. In the study of continuum problems we developed finite element approximations based on approximation of the displacement, stress and strain fields at each point in the domain. While such approximation is general there are instances when it is difficult to obtain viable solutions economically. Many such situations arise when one or two dimensions of the domain are small compared to the others. For example, when two dimensions are small we have a very slender cross-section which is translated along a one-dimensional axis as shown in Fig. 10.1. Such a form is herein called a rod and consists of a member which carries axial, shear, moment and torsion force resultants. When one dimension is small compared to the other two we have either a plate theory for initially flat surfaces or a shell theory for general curved surfaces. In this chapter we consider the behaviour of rods. Plate and shell problems will be considered in subsequent chapters. Bending of rods is generally associated with a beam theory such as the classical Euler-Bernoulli theory studied in introductory strength of materials. ~-3 If one attempts to model a rod with a standard three-dimensional finite element model there are two aspects which give difficulty. One is purely numerical and associated with large roundoff errors when attempting to solve the simultaneous equations. 4 The other is a new form of locking in interactions between bending, shear and axial behaviour when loworder elements are used. Often a much more economical solution is to use a structural mechanics approach in which the problem is formulated as a one-dimensional problem along the axis of the rod. Using this approach and appropriate interpolation forms one can avoid numerical difficulties associated with round-off and locking. In this chapter we consider approximation for two classical rod theories. The first combines the Euler-Bernoulli theory of bending with axial and torsion theories. In this theory the deformation field is restricted to axial, torsion and bending strains. The application of the Euler-Bernoulli theory is usually restricted to situations where dimensions along the axis of the rod are at least ten times those of the transverse dimensions. In the second form we consider the Timoshenko theory of bending together with the axial and torsion theories. The Timoshenko theory adds a transverse shear deformation to the other strains and is applicable when the length to cross-section dimensions are above five (when smaller than this the continuum theory becomes viable).
Governing equations 279
Fig. 10.1 Slenderrod. Care should always be used when measuring the length parameter. Distances between sudden changes in cross-section or loads should be considered. Also, when transient effects are included the frequency content of the solution will establish which natural 'modes' are active and the length between modal zeros also needs to be considered. Use of the Timoshenko theory can lead to 'locking' effects when the theory is applied to cases where the Euler-Bernoulli theory also could be used. However, it is desirable to have a single formulation which remains valid throughout the range of length to cross-section considerations and for this the Timoshenko theory should be used. In this chapter we show how finite element approximations, which are free from locking effects, can be developed for the Timoshenko theory. These are also useful when we consider plate and shell problems in the later chapters. Indeed one of our goals for presenting the rod theory is to develop an understanding of the delicate nature of 'locking' when structural mechanics formulations, including those of plate and shell problems, are solved by finite element methods.
We consider a straight rod with the axis of definition in the x direction and the crosssection A in the y - z plane (Fig. 10.1).* In the study of rods we will assume that the primary stress components are the normal stress Ox and shear stresses 7"xy , Txz acting on each cross-section. The remaining stresses, while they can exist, are of less importance and their effects are either ignored or are included as applied boundary loads to the rod as indicated next.
10.2.1 Equilibrium equations We first consider the static behaviour of a rod where the local balance of momentum equations at each point of a body are expressed as * In the study of structural mechanics formulations we shall revert to the notation where coordinates are denoted by x, y, z. A similar type of notation will also be introduced for displacements, strains and stresses.
280
Structural mechanics problems in one dimension- rods
O~x
Or,~
O-r=,
-~x + - ~ y + - ~ z + b x = 0
+N+W+b,,-o 0r
Orv~.
(lO.la)
Oo-~.
and
T~y = <~,~; Zxz = ~
and Zyz = %,
(10. lb)
The balance of momentum equations for a rod are obtained by integrations of Eq. (10.1 a) over the cross-section A. Accordingly, for the equation in the x direction we write (10.2)
--if-fix + -~y + --~-z + bx d A = 0
We consider the case for which the cross-section is either constant or varies slowly enough along the x direction such that we can use
/aOI /a/a I -~x dA ~ Ox
f dA;
~ dA =
f ny dS and
~ dA ---
f nz dS (10.3)
where S is the perimeter of the cross-section. Thus, after integration over the crosssection Eq. (10.2) is given by oP + qx = 0
(10.4a)
Ox where P is an axial force resultant and qx an applied load in the x direction defined by P--facrxdA
and
qx--fabxdA+~s(ryxny+yzxnz)dS,
(10.4b)
respectively. Similarly, integrating the equations for the y and z directions gives OSy -~x -~- qy
"-
0
and
OS z
~
(10.5a)
+ qz - 0
where Sy - f A 7Yx
dA; qY -- fA by dA -1- fs (ffyny + yzynz) dS
S z -- f A T z x d a ;
(lO.5b)
qz = fA bz dA + fs (yyzny -1- crznz) dS
In the above Sy, S z are transverse shear force resultants and qy, qz are applied loads per unit length. Moment resultants acting on the rod are recovered from the pair of equations
Z
[Ox +--~-y +--~z +bx
dA=
0
(10.6)
Governing equations Evaluating the integrals using Eq. (10.3) yields the relations
OM z
-t- Sy + m z -- 0 and
OMy Ox
Sz + my = 0
(10.7a)
where
Mz = My --
fA y a ~ d A ;
mz -- -- fa y b x d A -
f A z Ox dA;
my--
f s Y (7yxny +Yzxnz) dS
L zbxdA-Ffs
(lO.7b)
z(7yxny +yzxnz) dS
define the bending moment resultants My, Mz and loading couple per unit length
my, m z.
One final couple, a torque, can also act on a cross-section. The torsion resultant Mz = T is computed from shear stresses acting on the cross-section as
T = fa (yr~z -- Z~-xy) dA
(10.8)
Guided by the solution of the Saint-Venant torsion problem (see, for example, reference 5 and Sec. 7.5.1 of reference 6) we neglect ay, az and ryz in the equilibrium equations, thus the integrated equilibrium equation can be obtained from the local form
fA[Y(OT"xz -+-bz) - z (07"xy ~k Ox + b y ) ] d A - 0 This yields the result
OT 0x
+ mx = 0
(10.9)
(10.10a)
where mx is a distributed loading torque per unit length given by
mz -- fA [y bz - z by] dA
(lO.lOb)
10.2.2 Kinematics The development of a structural mechanics theory of rods is based on the assumed kinematic field for each cross-section. Here we will consider only the simplest theories where it is assumed that a plane cross-section remains plane throughout the deformation process. Accordingly, we assume a displacement field expressed as
Ux(X, y, z) = u(x) -4- z Or(x) - y Oz(x) Uy(X, y, z) = v(x) - z Ox(x) Uz(X, y, z) = w(x) + y Ox(x)
(10.11)
where u, v, to are displacements of the axis defining the rod and Ox, Oy, 0z are small rotation angles about the coordinate axes. A warping function can be added to improve
281
282
Structural mechanics problems in one dimension- rods
the kinematic approximation as shown by Klinkel and Govindjee; 7 however, this complicates the formulation and we choose instead to use the simple form shown above. Based on the displacements given in Eq. (1 0.1 1) we obtain the non-zero strains On OOy O0 z ex = -~x + z -~x - y ~ -- e ( x ) + z Xy(X) - y Xz(X) 2 e xy --
-~x -- Oz
-- Z -~x = ")'y( x ) - z X x ( x )
2exz =
~
+ y -~x -- ~/z(X) + y X~(x)
+ Oy
(10.12)
The strain m e a s u r e s )(y and Xz are related to the reciprocal radii of curvature of the initially straight bar in y and z directions, respectively. Since the radii of an initially straight bar are infinite, the Xy, Xz terms are in fact changes in curvature of the rod. In the displacements assumed above we have ignored the actual distortion of the cross-section due to transverse shear and torsion resultants. It will be necessary to account for this omission later through the use of correction factors.
10.2.3 Transient behaviour Using the assumed form for the displacement field, inertia effects may be included in the theory by replacing the body forces using the d' Alembert principle. Accordingly, bx ~
bx - p ( i i + zOy - yOz)
by ~
by - p (ii - z Ox)
bz ~
bz - p(ff) + y Ox)
(10.13)
may be inserted into the relations defined in Sec. 10.2.1 to yield the momentum balance equations for the transient case as OP
= p A (u" + ZOy - yOz)
--+qx
Ox OSy
-t- qy = p A (i) - zOx)
Ox OSz ~+qz--pA Ox
(f~+YOx)
for the force resultants. Placing the x axis at the centroid of the cross-section gives 37 = g - 0 and simplifies the above to OP ~+qx = pAil Ox OSy -1--qy -- ,o A iJ Ox OSz +qz= pAf~ Ox
(10.14)
Governing equations The balance of momentum for the couples is given by
OMz -~- Sy "41-mz -- p (Iz Oz - Iyz Oy) Ox
OMy Ox
S z--I-my - p (lyOy -- Iy zOz)
(10.15)
OT ~+mx=PJOx Ox
where the area inertia array is given by
Iy "- fa Z2 dA;
and
Iz -" fA y2 dA
and the polar inertia by
Iyz = fa y z d A
g=Iy+Iz
The orientation of the axes may always be placed such that Iyz = 0; in the sequel we assume this form to simplify the presentation of numerical approximations.
10.2.4 Elastic constitutive relations Based on the assumption that O-y,O'z, and ~-yzare zero and x defines a material symmetry direction, the constitutive equations for an orthotropic linear elastic material are given in matrix form by
7"xy
7-xz
=
[i ~ Gy 0
Gz
2exy 2exz
}
(10.16)
where E is the elastic modulus in the x direction and Gy, G z are the shear moduli for the y and z directions. Here we are interested in procedures for developing a finite element approximation to the theory and, thus, we assume the properties are constant in the rod. Inserting the constitutive relations in the expressions for the axial force resultant and assuming the x axis at the cross-section centroid we obtain
P = fa E (e(x) + z ~u
- y Xz(X)) dA -- EA e(x)
(10.17)
where EA is the elastic axial stiffness of the rod cross-section. Repeating the above for the bending moments we obtain (assuming lyz = O)
My - Ely )(,y(X)
and
Mz = EIz Xz(X)
(10.18)
In the above Ely and E Iz are the elastic bending stiffness of the rod cross-section computed with respect to the centroid of the cross-section. The relation for the torsion is expressed as
T = ~x GJ Xx(X)
(10.19)
283
284
Structural mechanics problems in one dimension- rods
where ~x is a correction factor accounting for the difference between the assumed displacement field and the true warping of the cross-section and
G J -- ~ (Gz y2 + Gy Z2) d A The term ~x G J is the elastic torsional stiffness of the cross-section. Finally, the constitutive behaviour for the shear resultants is given by Sy
-- Ny Gy A 'fly
(10.20)
Sz = roz GzA 3'z where/,i;y and ~z are correction factors accounting for the cross-section distortion.
For the assumed form of the constitution and location of the x axis the behaviour of the rod separates into three distinct types of problems as shown in Table 10.1. The equations given above for elastic rods constitute those for the Timoshenko theory of beams 8 combined with the axial and torsion problems. The Timoshenko theory includes transverse shearing strains as well as changes in curvature in the behaviour of the beam. An alternative theory exists for which the transverse shear deformation becomes negligible and from ",/y = "Yz ~ 0 we may use
Oy =
Ow Ox
Ov Oz = Ox
and
(10.21)
In addition for the slender case the couple loading my, m z and the rotatory inertia terms plyOy, fllzOz are also usually ignored in the equilibrium equations. This is generally called the Euler-Bernoulli theory of beams. Noting that the shear forces are now computed from the simple equilibrium relations Sy--
OMz OX
and
Sz =
oqMy Ox
the relations for the beam may be written as shown in Table 10.2. In the next section we consider the weak form for each type. Table 10.1 Equations for a rod with Timoshenko beam theory for bending Type
Momentum balance
Strain-displacement
Constitution
1. Axial:
OP -fffx + qx = P n it';
Ou e ( x ) --- -z--" tyx
P -- E A e ( x )
2. Torsion:
OT Ox + mx = p J Ox;
OOx. Xx = Ox
T = t~x G J Xx(X)
3. Bending:
OSy Ox ~ q y = p A iJ;
Ou "flY -- -~x
OMz Ox
OSz Ogy
+ Sy + mz = p lz Oz"
+ qz = P A fb; - Sz -Jr-my = p l y Oy"
Oz"
OOz
Xz = Ox "
Ow
"flz -- ~ x "1- Oy" ~Y -
OOy . Ox '
S y --- ~ y G y A "fly Mz = E l z Xz(X) Sz = ~z G z A "flz My = E l y Xy(X)
Weak (Galerkin) forms for rods 285 Table 10.2 Equations for Euler-Bernoulli beam theory Momentum balance
Strain--displacement
OZMz -
Constitution
OZv
Ox----5---t- qy = p a i~;
02My Ox "b qz = p A fi)"
Xz =
Ox 2"
Mz = E I z Xz(X)
X.y --
02 w Ox 2 "
M y = E Iy Xy (X)
For the form given above we can construct weak (Galerkin) forms individually for each of the problem types. Each of the forms given below is initially given in terms of a three-field variational form of Hu-Washizu type. Although each of the forms given could include displacement boundary conditions as natural conditions [i.e. see Eq. (1.38)], we choose to enforce these as essential conditions as described in Chapter 2. The three-field form will be used to construct mixed finite element approximations for the rod. From the three-field form a two-field mixed form of Hellenger-Reissner type and an irreducible form of displacement type may also be obtained for use in finite element approximations.
10.3.1 Axial weak form For the axial behaviour we write a three-field variational form as
51-l~(u, P , r
fL [5r ( P ( r
+I
P ) + S u ( p A i i - q x ) 1 dx (10.22)
(~
=0
where 5u, 3e and 5P are arbitrary functions over the length of the rod L; OLp denotes the ends of the bar at which the axial force is specified and P = / 3 (e) is the constitutive equation. After integration by parts we recover all the equations for the axial behaviour except the displacement boundary condition u -- ti on OLu and the initial conditions u(x, O) = ao and ti(x, 0) = ~70. As noted in Chapter 1 we can recover a Hellinger-Reissner type form provided we can find the inverse constitutive relation such that
P = P(e)
--+
~ - g(P)
(10.23)
With this substitution Eq. (10.22) becomes
51"Ip(u, P) --
5P-~x + ~
P + 5u (pA ii - qx) dx (10.24) -o
286
Structural mechanics problems in one dimension - rods
An irreducible form for the axial behaviour is obtained from Eq. (10.22) by enforcing the strain-displacement equation at each point of the rod. Accordingly, we substitute e
Ou Ox
--
~
=
~
and
d;e =
&Su Ox
=
~
into Eq. (10.22) and obtain ~Flu(u)-
fr ~
L-~-xP ( ~ ) + ~ S u ( p a u ' - q x
,]~x- (~u~
)]ot, p = 0
(10.25)
Boundary conditions
The boundary conditions for the axial response involve specification of the force condition P - 15 or the displacement condition u = ~ at each end of the bar. In addition it is often required to impose a symmetry condition as u = 0 or an asymmetry condition P = 0 when solving problems.
10.3.2 Torsion weak form For the torsion behaviour we write a three-field variational form as
~I'IT(Ox, T, Xx) = /
[~SXx (7'(Xx) - T ) + ~50x (pJ Ox - mx)] dx
+ f~ ['~ ~ox~~176 ~x)+ O~OXox ~1~x_ (~Ox~)loL,
=0 (10.26)
where ~50x, ~SXx and ~T are arbitrary functions over the length of the rod L; and OL T denotes the ends of the bar at which the torque is specified. After integration by parts we recover all the equations for the torsion behaviour except the displacement boundary condition Ox -- Ox on OLx and the initial conditions Ox(X, O) = OxO and
Ox(x, o) = Oxo. Similar to the axial problem, we may recover a Hellinger-Reissner form by finding an inverse constitutive relation of the form
Xx - ~x(T)
(10.27)
Alternatively, an irreducible form for the torsion behaviour may be obtained by enforcing the strain-displacement equation at each point of the rod. Accordingly, we substitute
OOx Xx = Ox - X.x
and
&50x ~SXx = Ox = ~5~x
into Eq. (10.26) and obtain
~no~(Ox) =
/~ r~176 ~<:,~x~>+ ~o~,~,o.,,Ox-qx)] L cox
dx-
T)I,~,_. .,
(6'0.~
-- 0
(10.28)
Weak (Galerkin) forms for rods 287
Boundary conditions
The boundary conditions for the torsion response are identical to those for the axial problem, namely a force condition T = 7~, a displacement condition Ox = Ox, a symmetry condition as 0~ = 0 or an asymmetry condition T = 0.
10.3.3 Bending weak forms For the bending problem we have two cases to consider: the Euler-Bernoulli beam theory and the Timoshenko beam theory. As given above the bending of the rod occurs in two planes but, with the exception of the signs, leads to identical sets of equations. Accordingly, in the following we consider only the bending problem for the x - z plane in terms of the displacements w, 0y, the forces My, Sz and strains Xy, "h (with "Yz = 0 for the Euler-Bernoulli problem). The behaviour in the x - y plane is obtained by making appropriate changes in the notation and signs, but otherwise is identical.
Euler-Bernoulli beam
We begin by considering the Euler-Bernoulli theory since, like axial force and torsion problems above, the weak form depends on one resultant My, one strain measure Xy, and one displacement component w. The three-field weak form for the Euler-Bernoulli theory is given by (~l-Ixy(tO ,
My, ~y) -- J~L [(~y (I~/Iy(~Y) -- My) + (Sto (pAl b-qz)] dx -/]
[(SMy (02w-~X2--~-~y)-]- 026tOOx 2
My]
dx
(10.29)
+ ( --~-x O6wl~ Y) loM,, - (6w ~z) los ~.. o This weak form differs from the previous forms in two ways. First, the strain measure involves second derivatives and hence for the form given in Eq. (10.29) will require C 1 continuous interpolation for the finite element solution. Second, there are two boundary conditions at each end of a beam of length L. A Hellinger-Reissner form may be obtained by expressing the curvature in terms of the bending moment as Xy -- 2y(My) (10.30) and substituting the result into Eq. (10.29) to give
(~I--IMy( W , M y) __ f LL[ _ (~M y O2 OXW 2
02OX(~tO 2 My nu t~W (pA
-- fL (~My 2y(My)dx + (O(Sw -
---~x M Y) IoM,
fb - qz) ] dx - ((Sw Sz) I
OSz =
0
(10.31)
288
Structural mechanics problems in one dimension - rods
An irreducible form for the Euler-Bemoulli beam is obtained by substituting OZw ~y --
02~w
Ox 2 = ~ y
and
~)~y "-
Ox 2
-- ~2y
into Eq. (10.29) to obtain (~I-Iw(t~
~ -
fL
[ 0 2OX2 (~w l~y(~y)
--~- (~w ( p A ff) - q z ) ]
dx (10.32)
+ (O~ll)l~"Iy) I -- ((~wSz) I --0 ' OMy
OSz
Note for the linear elastic form described in Eq. (10.18) and Table 10.1
My - -
l~y(~(~y) = E l y
~y -- - Ely
021/3 Ox 2
and, thus, when ~w = w the first term in Eq. (10.32) is positive as required for a form obtained from the minimum potential energy principle. The options for boundary conditions for bending are more numerous than for the axial or torsion problem since there are two conditions to be imposed at each end of the bar. The options are Natural condition Sz =
My - -
Sz;
Essential condition w-go Ow Ox =-Oy
]~"ly ;
These conditions may be combined to form typical boundary types described in Table 10.3. Table 10.3 Typical boundary conditions for Euler-Bernoulli and Timoshenko beam bending
BC
type
Symbol
used
Fixed: Pinned: Roller: Free:
/~//// %-~ 1
Euler-Bemoulli condition
Timoshenko condition
w = w;
c~w -g-xx = -0,,
w = w;
Or = O~
w -- Co;
My = if/Iv
w -- if_);
My : iVly
w -- if);
My = l~/ly
w -- (o;
My = l~'ly
Sz : Sz;
My -- iVly
Sz -- Sz"
My -- ff/ly
S z "- O;
Oy -- Oy
w -" O;
My -- O
Symmetry:
&=o;
Ow-- --Oy
Asymmetry:
w=O;
My - - 0
Weak (Galerkin) forms for rods 289
Timoshenko beam
The three-field weak form for the Timoshenko beam theory involves two displacement components, two forces and two strains as given by
(~l-l,yz(ll.),Oy, Sz, My, ')'z,Xy)'~ L" [(~X.y(l~y(~..y, ")/z)-- My)-Jr-(~")/z(Sz(X.y, ")/z)-- Sz)] dx -I- f [(SW(pAfO-qz)-t-(50y (ply Oy--my)] dx
+
(0,y
dx
+Oy
[ O~Oy -]"fL [--~X My--I- ( ---~x O(Sw -Jr(~Oy) Sz] dx
(,0yMy)10~y ('~ ~z)Iosz o
(10.33)
For this formulation the reduction to a Hellinger-Reissner form requires determination of a pair of inverse relations such that
X.y -- 2y(My, Sz)
and
7z - "r
Sz)
(10.34)
This greatly increases the complexity of the problem, and except for the linear elastic problem, has not been used extensively. For this form
(~l"l,yz(W, Oy, Sz, My, 7z, ~.y) "- /r" [(~W(pA fO -- qz) + (~Oy(ply Oy -- my)] dx
+f~[,~yO0y o~ 0y)I ax ~x+~Sz(~x+ -~xMy+(-~x
+~50y) Szldx
(10.35)
--(t~Oy l~/lY)lOMy- ((~WSz) IoSz-- 0 An irreducible form is obtained by an exact satisfaction of the strain-displacement equations, as described previously for other problem forms. Accordingly, we obtain for the Timoshenko beam the relation
(51-1Or(W,Oy) = fLL[ ~O(~Oy IVIy(Xy, ~z) nt- ( ~O 'nt-W(50y) ISz(Xy, ~z) dx + / i [(Sw(pA fO - qz) -Jr-(50y(ply Oy -- my)] dx
(,Oy~,y)lo~y
where
2y -- OOy/OXand X/z - Ow/Ox + Oy.
0sz
=0
(10.36)
290
Structural mechanics problems in one dimension - rods
Boundary conditions
The options for boundary conditions are similar to those for the Euler-Bernoulli theory except Or replaces Ow/Ox. The options are Natural condition
Essential condition
Sz = Sz;
w = ff)
My = l~y;
Or -- Oy
These conditions may be combined to form the same boundary condition types as in the Euler-Bernoulli theory as described in Table 10.3.
The finite element solution of rod problems involves finding one-dimensional interpolations to approximate each of the functions appearing in the weak forms. We begin by considering the Euler-Bernoulli bending problem since, as we will show later, it has implications on the approximations for the axial behaviour of the rod when either non-linear material behaviour or large displacements are considered.
10.4.1 Beam bending - irreducible form The Euler-Bernoulli beam is the first case we have considered which requires C 1 continuity in the approximation for the transverse displacement w. In all previous problems considered we were able to use C O approximation. If we first consider the static behaviour where 03 is negligible we can find interpolations which give exact solutions at the interelement nodes. Subsequently these may be used in transient analysis; however then the solution will no longer be exact at the interelement nodes. In order to obtain exact interelement nodal solutions for ordinary differential equations the interpolation functions for the weight function ~Sw must satisfy the exact solution of the adjoint differential equation of the weak form (see Tong 9 or Appendix H in reference 6). The adjoint equation is obtained by integrations on the weak form until all derivatives of the solution variable w are removed. For the case where the linear elastic constitutive equation (1 0.1 8) is used and Ely is constant the adjoint equation is given by
da~w
EIy dx 4 = 0
(10.37)
and the exact solution is a cubic polynomial in x. In a Galerkin solution we will use the same form of interpolation for both ~w and w.
Hermite interpolation for beam displacement
To construct cubic functions for each element we express the interpolation in the natural coordinate ~ and use the coordinate interpolation 1
x = 5 (1 - ~)s
1
+ 5 (1 + ~)s
= NI(~)s
with - 1 < ~ ~ 1 and s the coordinate for node a.
+ N2(~)s
(10.38)
Finite element solution: Euler-Bernoulli rods 291
Since second derivatives of w appear in the weak form (10.32) the interpolation for w must have C 1 continuity, indicating that both the function and its first derivative must be continuous in the entire domain L. The most logical manner to satisfy this requirement is to use Hermite interpolation forms. 1~A Hermitian polynomial of order 2n + 1 is written for a segment in terms of end values of the function and its first n-order derivatives, thus making it trivial to satisfy C n continuity. The lowest-order Hermite interpolation involves a complete polynomial of cubic order with the desired C 1 continuity needed for the Euler-Bemoulli beam shape functions. The interpolation in terms of ~ is given by
2[
O al
tO(~, t) -- E Ha(~ a=l
tOa(t) -}- Ha(1)(~) --~--x(t)
fib(t)
= N~
(10.39)
where the superscript b denotes bending,
/4a~~
1 { 2-3~c+~3;
= ~
L e
H(1) (~) -- T
a=l
2 + 3 ~ - ~;
{
a-2'
(10.40a)
1 -- ~ - ~2 + ~3. a = l --1 - ~ + ~2 + ~31, a - - 2
with
Nb(~) __--[n~ 0), H(1), H2(0), n(1)] ; ub =
01~ 1
1~1'
OX '
01~2 ]
ff132' --'~X J
and Le - - 3~2 -- 3~1 the element length. The chain rule OW
Ow Ox 1 Ow Ox o~ = g Le-b-fx
may be used to compute derivatives of w as
Ow
20w
OX -- L e 0 ~ '
02 w
40Zw
Ox 2
Z 20~ 2
and
03w
8 03w
Ox3
L3 0~3
and yields
OH~a~ Ox
3 2 Le
{
- - 1 + ~ 2" a = l 1 - ~ 2", a = 2
and
OH (1)
0x
1
--4
-1-2~+3~
2" a = l
-1+2~+3~2;
a=2 (10.40b)
for first derivatives, 02Ha<~ Ox 2
6 I~2e
{-~;
a=l a-2
and
02Ha(1) 0x 2
1 L--~{
-1 +3~; 1+3~;
a-1 a=2 (10.40c)
292
Structural mechanics problems in one dimension- rods
for second derivatives and 03Ha(~
12 {
Ox 3 = ~ e 3
1; -1;
a--1 a-2
03Ha(1) and
6 ( 1; 1;
Ox 3 = ~ e 2
a--1 a-2
(10.40d)
for third derivatives. The third derivatives are only needed to compute shear force in the element. While the C 1 continuity is easily observed from Figures 10.2 and 10.3, we note that the second and third derivatives are discontinuous and can adversely affect the accuracy of moments and shears in the elements.
Strain-displacement relations
Using the Hermite interpolations in Eq. (10.40c) the finite element interpolation for the bending strain is given by
2
02w Xy ---
OX 2 .__ __ y ~
OX 2
OX 2
a=l
/ / 01~)a
__ B b fib
(10.41)
-~X
in which B b is the strain-displacement matrix for bending.
Stiffness and load arrays
If we consider the weak form (10.28) for an element of length Le we obtain
61"Ie = 6f]ar f
[Bar My(Baf~b) dx + Nar (p A Na(ja - qz) dx] (10.42)
e
-C~
~O-
(a) H (~ functions
1
(b) First derivative
..-"3
-o I I I I I
i
(c) Second function Fig. 10.2 Hermite interpolation on two elements - H (~ functions.
I
6 I I I I I
(d) Third derivative
I
(3-
Finite element solution: Euler-Bernoulli rods 293
(a) H O) functions
(b) First derivative
I
I
?/ ,v
?
/"J"0-
, /
i I I I I I
i I I I I I
-<3
(b
I I
O-
t..,." (c) Second function
(d) Third derivative
Fig. 10.3 Hermite interpolation on two elements - H (1) functions.
where pb is the stress divergence vector, M b is the mass matrix and fb is the force vector for the element. For a given constitutive behaviour a Newton method may be used to linearize the expression for moment. For a local constitutive equation that has the linearized form
dox = ET dex where Er is the tangent modulus, the linearized moment is given by
~dw dMy = - Ely Ox2 = EIyBbdfl b Thus applying a Newton method to Eq. (10.42) yields the form
d(t~l-I)e -- t~flbT [ fL (BbT EIyBb + cM NbT p A Nb) dxI dfllb e
=
+
where
d~
=
CM
dr1
is obtained from the time integration scheme used (e.g. GN22 in Chapter 2).
294
Structural mechanics problems in one dimension - rods
Example 10.1
Bending stiffness, mass and load matrices
Using the approximation (10.42) for a linear elastic constitutive equation given by Eq. (10.18) with Ely constant, the element stiffness matrix is given by 6
Ke
2EIy L3e
=
-
3
Le
-6
3 Le]
-
I
~~Le 6
2L 2 3 Le
BEe 6
3 tel
Le
L2e
3Le
2L2I
(10.43a)
the mass matrix by
Me=
1
pALe
420
156
22L e
54
- 13 t e l
22Le
4L 2
13Le
- 3 L 21
54
13 Le
156
-3L 2
22Le
-13Le
22
(10.43b)
tel
4L 2 J
and for constant qz over the element the load vector by re__ 1--2 1 qz L e216L e ,
-- 1 6, L e
1] r
,
10.4.2 Axial deformation -irreducible form ....
----:-:-:
...........
- :
--::-::---:
.............
--
....................
- : : : :
-
.......
::::
--
. . . . . . .
::
::
------::
: ~ .
........
:
- - : : - :
r:::
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
-. . . . . . . . . . .
~:--:
. . . . .
: ........
(10.43c)
. . . . . . . . . . . . . . . . . . . . . . .
--:-:
..................................
-. . . . . . . . . . . . . . . . . . . . .
For a linearly elastic rod the axial constitutive behaviour is given by
P=EA~
0u
Ox
For the case where EA is constant the adjoint differential equation obtained for the static problem using Eq. (10.25) is given by
EA
cO2& Ox2
=
(10.44)
0
which yields a linear solution in the rod. This is adequate provided the rod is linearly elastic. However, we note that both bending and axial behaviour of a rod depend on the axial stress Crx, consequently it is important that the approximation for axial behaviour has strains which are of the same order as those for bending. In the development of the irreducible solution for the Euler-Bernoulli theory the bending strains Xy varied linearly along the length of each element. To have compatible behaviour we need to have the axial strain ex vary linearly also. Accordingly, we use the hierarchic interpolation 1
1
~u = 5(1 - ~)t~/~ 1 + 5(1 + ~)t~/~ 2 -~- (1 -- ~2) m(t~/~3) __
Na
(~a
(10.45)
where ~a = (/~1, /~2, /~3) T. Equation (10.38) is again employed for the x coordinate interpolation. The same form of interpolation is used for u. Using this interpolation
Finite element solution: Euler-Bernoulli rods 295
the axial strain in an element of length
Le is given by
1
e = L-7(ti2 -/~1
-
-
4 ~ Ati3)
=
Bafil a
(10.46)
Substituting the above results into the weak form (10.25) yields
t~I'Ie-- t~flar[ ( fLe BaTI)aBadx) fla -- fLe NaTqx dX] in which I) a
-- [EA]
(10.47)
is the axial stiffness matrix.
Example 10.2 Axial stiffness and load arrays
If E A is constant in
Le the element stiffness is given by Ka
--
ai!0 0]
3Le
0
16
and for constant qx the element load by
fa l {3} = gqxLe
3 6
We note for the static linearly elastic problem that Ati3 depends only on the distribution of the load on the element and does not affect the values/il and 52, thus for uniform cross-sections the interelement nodal values will be exact. If E A varies along the element or the material is non-linear this will not be true and A53 will interact with the end values, which will then no longer be exact.
10.4.3 Torsion deformation- irreducible form The constitutive equation for the torsional behaviour of a linear elastic rod is given by
T = ~xGJ
COOx Ox
For a constant ~x G J along the length of an element the adjoint differential equation obtained for the static behaviour is determined from Eq. (10.28) to be
~xGJ
c~Ox = 0 cox2
which has the linear solution given by
5Ox = ~1(1 - ~) ~0~1 + ~1(1 + ~) 50x2 An identical expression is used to represent Ox in terms of Ox1 and Ox2. Note that a linear form gives a constant strain which is identical to the form obtained for shear strains from the Hermite interpolation in the Euler-Bernoulli theory.
296
Structural mechanics problems in one dimension - rods
Inserting the above interpolations into the weak form (10.28) for a single element gives
'1-I~9~--['0xl '/~x2]{~'el [11] (Lll ~;xGJ(~)d~)Ill] } { ~xl0x2} (10.48) --[(~Ox1 (~0x2]I1te fl 1 {(1--~)} (1+r m x ( ~ ) d ~] where integrals are carried out over the coordinate (.
Example 10.3
Torsionstiffness and load arrays
For an element of length L e and constant torsional stiffness the stiffness matrix from Eq. (10.48) is given by GJ
Kt __~x
[o ~
and for constant mx the element load by - - ~ mx L e
1
10.4.4 Beam bending- mixed form The three-field weak form given by Eq. (10.29) may be used to obtain exact interelement nodal solutions of the finite element formulation of elastic problems in which the EIy varies along the length of the element. Accordingly, from the weak form we obtain the adjoint differential equation for 6My given by
o~Uy 0x 2
= 0
(10.49)
for which the exact solution is a linear function over the element length that may be given by the interpolation
~My-- N1(~) ~/~yl+ N2(~)~/~y2
(10.50)
where Na(O are defined in Eq. (10.38). Considering an individual element of length Le and performing an integration by parts on the derivatives of w and ~w gives the weak form for the static problem
e
Finite element solution: Euler-Bernoulli rods 297
If we use a curvature )~y(My) which satisfies the constitutive equation
l~y()~y) -- My and a real moment field which satisfies
02My t_ qz = O Ox2 the weak form for the static problem simplifies to nk,(m M y ) -
-
-
, ~ l + ( OxO Mym) I :o m )l OMy +
-- fLL (~My~)~y(My)dx
)l
e
(10.52)
Using the shape functions given in Eq. (10.38) the real moment field may be expressed as
My = NI(~)/~'/yl + N2(~)/~'/y2 + l~y(qz) (10.53) where My(qz) is a particular solution due to the loading qz acting on the element. In this form only end values for Wa and Owa/Ox and their variation appear and no internal interpolation is needed to obtain exact interelement solutions at nodes. Note that in this form the moments are not continuous across element ends and, thus, Myl and My2 may be eliminated at the element level. For the linear elastic case the determination of 2y(My) is trivial and is given by the simple relation
~y(My) --
1
My
Ely
If the material behaviour is non-linear then the determination is more complex and,
indeed, may not be possible to obtain. We consider this further in Sec. 10.4.5. Also, when inertia effects are included an interpolation for w must be used; however, this becomes a case for which the interelement solution is no longer exact. The form given above has been used by several authors, see, for example, references 11-25.
Example 10.4 Element array for variable Ely
Let us consider the example of a uniformly loaded element with length L e that has a variable moment of inertia given by 1
ely
= N1 (~)
1
Elyl
+ N2(~)
1
EIy
A particular solution for a uniformly loaded element may be expressed as
l~y -- g1 qz L e2 (1
-
r
With the above forms
~ ]i Na -~y 1 (NbMb-+l~y)dx fLe (~My~y(My) dx - 5Ma e
= d'My
(vM,
Lv, v 4{
}+{:"'
298
Structural mechanics problems in one dimension - rods
where
V __.
12EIyl
[3 :]
+1---'-2
[',
fM--"
and
qz" e 120
3
2
+ E~y2
E I y l Jr" EIy2
The element boundary terms give
_(SMyOW
[ ~x)lo~+(&SMyw) ox o~-Ce
1 [i~/~y 1
1 ~/~y2] --1
-Le 0
-1 1
0]
0y1
Le
1132 Or2
= 61~yrG~ b The other element boundary terms give
_(o~w ~x MY)IoL+ (Sw OMy
o1)i
where t', is a force vector resulting becomes
OL
-- ~fl bT [GT1VIy -Jr fu]
from/~y and for the particular solution given above
f ~ = ' ~qzLe [1
0
1 0] T
When substituted back into Eq. (10.52) the above results give an element weak form ab fu 5HeMy__.[~fibT ~iIT] ([0G ~3 (l~yl__ (fMI)
After ~ly
(10.54)
is eliminated for each element we obtain the 'stiffness' matrix K e = GrV-1G
and load vector
fe __ fu + GTv-lfM
which may be assembled in an identical manner as in the irreducible form.
Example 10.5 Cantilever tapered bar- Euler-Bernoulli theory
To illustrate the performance of the irreducible and mixed forms presented we consider a cantilever beam that is loaded by a uniform load qz as shown in Fig. 10.4. The bar has a variable bending stiffness given in reciprocal form by
1
1
1
Ely -- EIo + (Ell where ~7 = respectively.
x/L
1
Ell)(4r/-7]2)
and the stiffness at the support and free end are
E Io
and
E I1,
Finite element solution: Euler-Bernoulli rods 299
lllllIllll qz
L
k
~'r
L
X
d
Fig. 10.4 Taperedcantilever beam. The exact solution for the displacement and slope are given by
W(X)
=
qzL4
~--~ qzL4 ( 1 Ell
24EIo [8/-]3. /]4 __ 24r/2] +
1 ) Ell
x [r/6- lZr/5 + 50r/4 - 8@/3]
Ox
=
6EIo
[6rl2- r/3- 12r/] +-i~qzL 4
EI,
Ell
x [3r/5- 30r/4 + lOOt/3- 12@/2] and those for moment and shear by
M -- ~1 qz L 2 [4 r l - r/2 - 4]
~
S = qzL [2 - r/] In Fig. 10.5 we present the solution using two equal length elements and an irreducible (displacement) and mixed solution as described above. The results for the mixed solution are exact whereas those for the irreducible solution have error in all quantities, although those for the displacement and slope are very small. Using more elements will lead to convergent results for the irreducible form.
10.4.5 Inelastic behaviour of rods The inelastic behaviour of a three-dimensional rod requires simultaneous consideration of the axial, bending and torsion behaviour. For the two-dimensional case the torsion behaviour can be omitted and we can consider only axial and bending interaction. Let us consider the case where the constitutive behaviour is elastic-plastic as described in Chapter 4. The constitutive equation in one dimension may be given as Crx = E (ex
exp)
(10.55a)
where E is the elastic modulus and exP is the plastic strain. We assume an associative form and a yield function given by F(cr, ~) =
] 1/2
2(~) crx2 - Cry
_< 0
(10.55b)
300
Structural
mechanics
problems
in o n e d i m e n s i o n
- rods
3
2.5 I I - - Exac,
i
21- ......
t
,
I
.,w
zs
/
2
I ...... ,
~~
o.s
' I-/ ....
/ / o.sp-/--,,I t
0
0 5 (a) D , s l eI/aLe ment~ 1 5
0.5 It-- Ex.~ I
o~ - 0 . 5
I
II . o
/
I-
/
. . . . .
I
. . . . .
I
ii
-1.5 [ - - ~ - - [ - ~ ' ~
i
Disp. FE I Mixed FEJ
. . . .
I .1i
9 . . . . . .
, /<~ L~____• . . . . .
-1 / . . . . . :~"
_ 7
i '
....
.....
. . . . .
r ..... i
; ..... ' . . . . .
2
i
. . . .
I 2 ...... i
~ ......
x/L
-~ ...... i
i I 9 ...... i
t
9
L I
.L I
_L I
3 - -% . . . .
L
J_
.1
-~
. . . . ~.
~
2
i
i ......
1
1.5
' . . . . . .
1.5
I
i I + ..... i
......
1
. . . . . .
. . . . . .
0 1
.....
- ', I' ~ ~.~ - - - ~ - - - r ~- z - . ~ -"k - - r-'
-2.5 0.5
i
~=
-r, ......
~
x/L1 (b) Slope
3.5
I . . . . . .
I
-
0.5
4
'
i
iT.... -i' ' ..... .... +i ..... I
4"5
I -I-
I I
-/-
I
0
o.~ 0
/i
0
0-
0
i
II = Oisp. FE | I I o Mixed FEJ
0
~
'
' r T' I
t'
~--<-~:-i'~---~ ,,- . . . . .
1.5 inExact
I' -r -r'
-t'
t------'~--
I
I
I
0.5
1
1.5
(c) Moment
2
x/L
;
(d) Shear
Fi 9. 10.5 Solution for tapered beam using irreducible (displacement) and mixed solutions - Euler-Bernoulli theory.
where O y is the uniaxial yield stress and n an isotropic hardening parameter. Thus, the plastic strain is given by
OF Ox gP = 5' b-~-~.~= ~ Icrxl
(10.55c)
where Icrxl = [Crx2]~/2. For simplicity we consider the linear hardening form given by O'y "-- O"0 JR H i N,
(10.55d)
where Hi is the hardening modulus and n is computed from (10.55e) A numerical solution to the plasticity equations may be constructed using the retum map algorithm presented in Sec. 4.4.2. This will give a sequence of stresses at each 'time' step tn given by
a(~n)--E(r
xp(n) )
which linearized gives the incremental form dcr(x") = E(~") de(x")
(10.56)
Finite element solution: Euler-Bernoulli rods 301
Irreducible form for inelastic problems
For a two-dimensional problem based on an irreducible form the above result for inelastic stress gives the force resultants p(n)=
JA E(e(xn) __ ex _p(n ) )dA (10.57)
M~ n) -- -where
Z E (e~n) - e x ) d a p(n)
fA
Ex(n)__E(0n)+ Z/l~(n) y'~
(10.58)
For a Newton solution a linearization of Eq. (10.57) will give the tangent matrix for the cross-section as z Er
(10.59)
z 2 Er]
where Er is the material tangent modulus obtained from the local constitutive equation [e.g. from Eq. (10.55a)] at each computation point on the cross-section. Note that the tangent matrix will no longer be diagonal in all situations and, thus, itis necessary to consider the axial and bending problems simultaneously. The arrays for each element of the coupled problem are deduced from 51-Ie - 5u T f
JLe
(Bra-
Nrq) dx = - - S U r e
(10.60)
where B--
I~a
0
1
B b ' P-
/ MyP / ' N - INa
0
]
Nb ' q - -
(/qxqz
and
fia-- ( l~a /~a / Off~a/OX
A linearization for a Newton method gives the element tangent matrix
K~ = / "
BrDrBdx
,/Le
The solution process now follows a standard procedure for solving non-linear problems. For each time tn 1. Initialize an equilibrium iteration parameter k = 0 and assume an elastic behaviour at all points to compute a trial displacement -- (0) where the superscript indicates an iteration number k. 2. For each quadrature point used to compute element arrays perform a numerical integration over the cross-section to compute P, My and Dr. This requires storage of the plastic solution parameters for each solution point on each cross-section. 3. Use P, My and Dr to compute the element arrays K
U(n)
u(~+l) ._ U(k) q_ du (k) 4. Check convergence. If converged set n = n + 1 and go to step 1; else set k = k + 1 and go to step 2.
302
Structural mechanics problems in one dimension - rods
Fig. 10.6 Typical discontinuous strain distribution in rod element.
Mixed form for inelastic problems
The development of the inelastic finite element approximation given above leads to a very simple implementation; however, when used with just a few elements the inaccuracy in force resultants can lead to significant solution error. As shown above for the elastic bending case very accurate force prediction is obtained when we are able to exactly determine the curvatures from the constitutive equation, thus leading to a Hellinger-Reissner-type weak form. In the inelastic case it is necessary to determine the exact axial strain and curvature over the entire element in order to reduce the problem to an equivalent two-field form and this is not possible in general. Thus it is necessary to return to the three-field weak form and introduce an approximation for the axial strain and curvature in addition to force, moment and displacement variables. The approximation of strains may be performed in several ways; however, dividing the element into segments in which the axial strain and curvature are assumed constant in each segment as shown in Fig. 10.6 is simple and gives good results. The shape functions for the strains may be expressed as
6= ).,Nj(~)gj; J X.y =
ENj(~)~yj;
j = 1, N (10.61)
J -- 1, N
J
where
Nj(~) -- 1 for ~j < ~ _< ~j+l
(10.62)
The values of ~j are selected with ~l = - 1 and ~N+l -- 1. The three-field weak form for the axial and bending behaviour may be written in matrix form as (~i--i e _
[(~T
(~gT
f (10.63)
= - oq3~ ,I,
Finite element solution: Euler-Bernoulli rods 303
where G = IO a
O] ; Gb
1 fr bj -- -~Le
I10 1 0 5(1_,~)
j~j
Ol~)a / OX
1 0 ] d~; 5(l_~c)
]~ y2
~ YJ
sp denotes the particular solution for the resultants and g(e) denotes the constitutive equation expressions in terms of the strain approximations. The constitutive equations are computed from
s-f~ I! ~ I(/A{) 1
e
)
1
i( 1 - ~) 1 ~(1 +~)
(10.64)
Crx(ex)dA dx
Z
which may be evaluated using trapezoidal quadrature* over each segment of the element to give S - - "~Le
1
~(1
N
o ] (/A{ / 1
- ~l) 1 ~(1 -q- ~l)J
l=l
O x ( e x ( ~ l ) ) dA
Z
)
W1
(10.65)
where (l = (xj + Xj+l)/2, WI = 2 / N and Cx (~l) --" el "~- Z ~yl
Linearizing Eq. (10.64) using the incremental stress given by Eq. (10.56) gives the element tangent matrix K~w =
Ii 0] 0
BeT
Be
k
(10.66)
where k is the constitutive tangent given by k=
Ii, 01 .
".
..i
k
n
and kj =
1
ET [1 Z]
dA
Z
The integral over A is performed using numerical integration. 14The constitutive parameters may be eliminated to give
I~ e =
[~0_~v]
where V - Ber Iil
... ...
01 Be
kn
* Other forms based on ~j placed by Gauss or Gauss-Lobatto quadrature1~ may also be used.
(10.67)
304 Structural mechanics problems in one dimension- rods 250 ~,
/k L/2
-
f l
Fig. 10.7 Simplysupported beam with central point load.
Finally, the tangent matrix for the element is obtained by eliminating the force parameters to give K e = GTv-I
G
Similar reductions may be made for any non-zero force components in ~ . In the above formulation the only displacement parameters obtained are at the ends of each element. For plotting displacements along the length of the element may be obtained by integrating the curvature values. ~1'~2,~5
Example 10.6 Simplysupported beam with point load
As an example problem we consider a simply supported beam with a central point load as shown in Fig. 10.7. To show the advantages of the mixed formulation presented above we allow the beam to have elasto-plastic behaviour. The entire beam is modelled with two elements (one element for each symmetric half); five curvature stations per element are used along the beam axis. A rectangular cross-section is considered with 10 Gauss-Lobatto points through the depth to permit modelling of the spread of the plastic zone. The properties for the analysis are shown in Table 10.4. The central load is allowed to vary using a load control strategy (e.g. see Chapter 2 or reference 26). For the comparison we also consider the solution using an irreducible model with cubic Hermite polynomial shape functions. Solutions for two, four, and eight elements for the length are used (one, two and four on each half length). In Fig. 10.8 we show the force--displacement relation, deformed shape and distribution of moment and curvature along the length of the beam at the last computed load state for each analysis. The displacement model permits only linear change, whereas the mixed model presented here allows for arbitrary change at each axial station used (five in the present case). The superiority of the mixed form is evident in the force-displacement, the computed deformed shape and the moment and the curvature distribution. Table 10.4
Properties for inelastic beam
Length Depth Width Elastic modulus Yield stress Hardening modulus
L = 180 h = 10 b = 10 E = 29 000 O'y = 50 H i s o = 290 (1%)
Finite element solution: Timoshenko rods 1.6
1.4
i
~-" ' "
/ ... . . . . EL--. ........ /~. -=.~ ...........
1.2
/.oo"~
0.8 j
0.6
0.4
0.2
--,,- Undeformed M F 2 elements -- . DF 2 elements -- - DF 4 elements ..... DF 8 elements
elements DF 2 elements - - DF 4 elements .... DF 8 elements 4 5 6
/
-
00
1
2
3
Midspan deflection
(a) Force-Displacement
x 104 1/
.
.
.
.
.
---4 --~.-o-
I" 0.5 .!'\ \
o
i 41
i
\
-1
M'F 2 elements I DF 9 2 elements I 'i DF 4 elements I! DF 8 elements )'. .
\'~
=-0.5
\ \ k~ ' , ~ ~
1
• 10-3
I S / /
/
1
0
.......
(b) Deformed shape ~ .......................
p .......
-1
~-a
. .
~-4 -5
-6 I
II -1.5
0
. . . . ' . . . . 20 40 60 80 100 120 140 160 180 Distance from left support
8
o
--~
9DF 2 elements
---~- DF 4 elements DF 8 elements
2'0 4'0 6'0 8'0 ~oo ~0 ~40 16~ ~8o Distance from left support
(d) Curvature
(c) Moment
Fig. 10.8 Simply supported, point loaded beam - inelastic solution. DF = displacement formulation; MF = mixed formulation.
The development of finite element models for the Timoshenko theory, which includes the primary effects from both axial and sheafing strains, is developed from the functional (10.36) for an irreducible theory or from Eq. (10.33) for a mixed theory. The highest derivative in either form is first order, consequently C Ofunctions may be used in the approximation. However, the approximation can be quite sensitive if it is desired that the Timoshenko form accurately solve problems in which the transverse shear deformations are small compared to the bending strains. Below we present some successful mixed approximations which are free from locking. We begin by considering the irreducible form for bending.
10.5.1
Beam
bending
- irreducible
form
Since the highest derivatives of w and Oy in the functional (10.36) are only first order, both may be interpolated by C O functions. The use of the Timoshenko theory permite the solution of problems for which the length to cross-section dimensions are
305
306
Structural mechanics problems in one dimension - rods
somewhat larger than for the Euler-Bernoulli theory. It is desirable, however, to have a finite element implementation of the Timoshenko theory which permits the solution of problems at the 'thin' limit where the Euler-Bernoulli theory is applicable. The choice of appropriate functions is quite delicate if we wish to avoid 'locking' between shear and bending response as the beam approaches the thin limit. For example, as we show next the use of low-order/equal-order C Ointerpolation for the transverse displacement will lead to an element which as the beam becomes slender (i.e. w and rotation > 5, where L is a characteristic length and d a cross-section dimension). Later we consider an approach which avoids such locking. Moreover these developments are also useful when we consider the study of plate problems in the next chapters.
L/d
Or
locks
Equal-order interpolation
Use of an equal-order interpolation for the transverse displacement and the rotation is expressed as
/W/ Oy __Z a
where Wa and
Na(~) Ilia} Oya --- N~b
(10.68)
Oyaare the values at node a with Ua -- Oya
The curvature and shear strain in each element are given by
dNa Oya
{~(Y}~z:~a ~ idNadx
]
--'~X l~a "]-NaOya
=Bbfib
(10.69)
where the strain displacement matrix B b is given by o
B ba= [ NadX dNa_~x
(10.70)
The material elastic property array may be written as
o -[o y z zA]
(10.71)
Using the above arrays in Eq. (10.36) the element stiffness and load array of the static problem are computed from
'l"I w --'llbT[(fLeBbTDbBbdx)tlb--fLeNTqzdX] Example 10.7 Linear interpolation
As an example we consider the case where w and linear interpolation w
tbl
0y in each element are given by the tb2
IOy} "-'Nl(~) IOyll'#c'N2(') {Oy21 where N1 = (1 - ~)/2 and N2 = (1 + ~c)/2.
(10.72)
element
Finite
solution:
Timoshenko
rods
Evaluating the integrals in Eq. (10.72) analytically (or by a 2-point Gauss quadrature) gives the element stiffness o
Ke
Ely --te
o
i
0 -I
o
0
1 0
6
~yaya -t- 6Le
O11
0 0
2LZeNge 3Le 6 L2e 3Le
L3Le
Le 3te
2Le2/
and for a uniform load qz the element force vector
fe -- ~qzLe 1 [1
0
1 O] T
Evaluating the element stiffness by a reduced 1-point quadrature gives
K
e
_..
0 0
Ely Le
0
0 1 0 0 0 --1 0
o
JF 4-2Le--4 4 e t2e 2te
0 -1 Nyaya 0 -Jr- 4Le 1
-2Le] t2e J 4 2Lel 2L e Z 2 J
2Le Z2
I-2Z e
The only difference between exact and reduced integration appears in the K22, K24, K42 and K44 terms in the shear modulus matrix. These seemingly small differences have a significant effect on the element performance as shown in Fig. 10.9 in which we solve a cantilever beam using 20 elements for different span L to depth h ratios for a rectangular cross-section. The use of exact integration leads to a solution which 'locks' as the beam becomes slender, whereas the reduced integration form shows no locking for the range plotted. . 4
.
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,,
,
1.2
1
.....
. i . . ~ o , ,
9v -
,,,
-
,, o . . . ~
1,w
. 9
9
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L
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0.8
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.~ . . . . . . ~1 i i i i i
Exact
: : : :: .............
: .x: : : :
: : :::
" " :X;
9. . . . .....
. ,,, . . . .
v
. .
FE-2pt.
FE-lpt. . . . .
,,,
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i i 1111
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...... i i iiii .
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~
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0 100
.
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i i ::::
. .
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:-~ ~'1 .:~- \
...............
. .
i i iii i i iii .............
: : : : ~ : ::::; 102
101
: : :::"
: : : :: "m--
_
: : ::: . . . . _, 103
log ( L / h ) Fig. 1 0 . 9 Cantilever beam solved using e q u a l - o r d e r interpolation on linear elements. d i s p l a c e m e n t for Timoshenko beam theory to that of Euler-Bernoulli theory.
W/WEBis
ratio of tip
307
308
Structural mechanics problems in one dimension - rods
Exact nodal solution form
As in the Euler-Bernoulli theory we first consider the static behaviour for the linear elastic Timoshenko beam. Our goal is to determine the interpolation functions which give exact interelement nodal solutions when the material properties are constant along the element length. The requirement is for ~w and (~Oy to satisfy the homogeneous adjoint equations of the functional given in Eq. (10.36). For EIy and ~;zGz A constant the equations are given by
d260y (d6w ) Ely dx 2 ' + t~zGzA ~ + 3Oy - 0
d ( d~w (50y\) r~zGzA dx i\ --d2-x + - 0
(10.73)
Differentiating the first equation and combining with the second gives the simpler requirements
d 3(~Oy _ 0
and
d 2(5w
-
d(50y +
-0
dx 3 dx 2 dx which imply ~0y is a quadratic polynomial and ~w a cubic one. The expression for ~50y may be given as a Lagrange interpolation in the natural coordinates ~ as 60y
_
21 (~2 __ ~)60yl + g1 (~2 "k- ~)(~0y2 "]- (1 - ~2)t~y 3
(10.74)
where ~0y I and t~0y2 are the interelement nodal parameters and t~0y3 is a mid-length parameter. Using the above interpolation for 3Oy in the differential equation for 3w gives an interpolation 6w = g1 (1 - ~) 6t~ + g1 (1 + ~) (~1~2 (10.75) __ n e (1 __ ~2) [(gI _ _ 1 ~) (~0y1 __ (8I + ~2 ~) ~0y2 -'1- gI 6 / v~ y 3 lj
The parametric relation for x given in Eq. (10.38) is then used to construct the derivatives. We note that the interpolation for 3w is linked to the parameters of ~50y. The earliest appearance of linked interpolation appears in a paper by Fraeijs de Veubeke. 27 Alternative expressions have been deduced by Tessler and Dong, 28 Crisfield 29 and Stolarski et al. 3~ We shall find such linked interpolation forms are also very useful in the construction of plate bending elements. The interpolation for the variables may also be given in hierarchical form by substituting 1 ~Oy3 -- (~(A0y3) -~- ~((~0yl -~- (50y2) into Eqs (10.74) and (10.75) to obtain d~0y _ ~1 (1 - ~)(~0yl -i" ~1 (1 + ()t~0y2 "q- (1 - ~2)(~(m0y3) = NI(~)t~0yl + N2(~)60y2 + N3(()6(m0y2) = N060
(10.76a)
and d~w = ~1 (1 - ~) &~l + ~1 (1 + ~) ~l~2 1
-- L e (1 - ~2) [8 ((50Y 1 -- (50y2) -'[-" -~ ~(~(A0y3) ]
= Nl~t01 -+- Nwl(~Oyl -I- N2~1~2 -[- Nwz(5Oy2 Jr- Nw3(5(AOy3) = Nwfl b
(10.76b)
Finite element solution" Timoshenko rods
where
Nwl N2 Nw2 Nw3]
Nw-[N1
ub-
[/~1
0yl
mOy3] T
1~2 0y2
(10.77)
This form has some useful advantages which we will exploit later. The same form of interpolation is used for w and Or in a Galerkin solution. Using the hierarchical interpolations in the strain-displacement relations gives 1
X.y = te joy2 -- 0yl -- 4 ~cA0y3] 1
1
-
(10.78)
2
")/z -- ~e (1~2 -- l~l) -~- 2 (Oyl -4- 0y2) -q- ~ A0y 3 The finite element strain-displacement relations may be written in the standard matrix form
{/ ~)(y
")'z
--
[~
1
,
1
1,
0,
1
1,
-~e -- , ~L e , 1 , gLe ,
2
=
gLe
(10.79)
Using the above arrays in Eq. (10.36) the functional of the static problem for an individual element is given by
BbrDbBbdx) flb-- fL Nrqzdx]
'FIw = ' f i b r [ ( f L e
(10.80)
e
Example 10.8 Stiffness and load arrays for uniform Timoshenko beam
If we evaluate Eq. (10.80) for an element of length Le with Ely and ~zGzA constant the element stiffness matrix is given by 0
R
e
EIy 3Le
o o
0 3 0 -3 0
0 0 0 0 0
0 0 0 O 16
-18Le
-18Le 9L 2 18Le 9L 2
[.-24Le
12L 2
36
+ ~zGzA 36Le
0 -3 0 3 0
-18Le -36
-18Le - 2 4 L e ] 18Le 9LZe 12Le / 36 18Le 24Le| laLe 9L2e 12L2 / 24Le 12Le2 16L 2 J --36
(10.81)
and for constant qz the element load vector
e -- -q z L122
[6L
e.
-1 6Le . . .
1
0] r
(10.82)
which, except for the last 0, is identical to the load vector of the Euler-Bernoulli theory.
309
310
Structural mechanics problems in one dimension - rods
Since the AOy 3 is associated with each element individually, it may be eliminated and the reduced stiffness may be written as 2I OLLe 1 aLZ(a +/3)
OL 1
Ke = 12EIy L3
~ "~ceLe -a
g
1 aLe
where
1
a=~;
--~ 1
~ aLe
~1 a L e
a
1 o~L2(a_ 2/3) -~
~1 aLe
"7
/3=
and
~OLLe 2
1
]
-~aLe(a - 2~)[ 1 o~Le
(10.83)
1 a L 2e (a -'t-/3) .] -g
12EIy ~zGz AL2
"7=
1 + "7 4(1 + 7) As the beam becomes 'thin' 7 -+ 0 and this gives a ~ 1 and/3 --+ 0 which when substituted into Eq. (10.83) we immediately observe that the stiffness converges to that of the Euler-Bernoulli theory and, thus, the element will not lock.
Timoshenko theory- constant strain form
It is possible to use the displacements given by Eqs (10.76a) and (10.76b) in which the A/93 terms are omitted. Accordingly, with the interpolation given by 1 1 1 W = 7 ( | -- ~) 1B 1 -'[- 7 ( 1 + ~) 1~ 2 -- ~ t e Or _-- ~1 (1 -- ~)[~yl "q- ~1 (1 + ~)Or2
(1
-- ~2) (0y 1 _ 0y2) (10.84)
the strain--displacement relations are given by the constant relations 1
-
Xy = ~ (Oy2 - Oyl) 1
1
(10.85)
")/z "-- ~ e (1~2 -- /~1) + ~ (0yl "~- 0y2)
The finite element strain--displacement relations again may be written in the standard matrix form ()(y} __ 1 [ O, 1-1' O, 11,] filb- - B b f l b (10.86)
7z
--~e--1,
.~te,
1,
.~teJ
where now the fib contains only the end values of w and Or. Inserting the above arrays in Eq. (10.80) yields the element stiffness and load relations
K e = BbT(fLDbdx) Bb and f e f=L T Nwqz dx e
(10.87)
e
where Nw is given by Eq. (10.77) with the last entry removed.
Example 10.9 Stiffness~loadfor uniform constant strain Timoshenko beam If we evaluate Eq. (10.87) for an element of length Le with Ely and xozGzA constant the element stiffness matrix is given by ~(e__ Ely
3Le
0
0 0 3
0
0 0 --3
0
0
-3
3
I~zaza
+ 4L-----~
I[ ?_4 4
e
-2Le L2
2Ze
L-2Le
t 2
-4
2te
4
2L e
-2Le']
L2 I
2Le| (10.88)
t 2 J
Finite element solution: Timoshenko rods 311
and for constant qz the element load vector
fe :
qz L2 12 [6Le,
-1,
6Le,
1It
(10.89)
We note that the above stiffness is identical to that obtained in Example 10.7 by reduced integration; however, in the present case we were able to use exact integration and hence the linked form will be more useful when we consider plate bending problems. In addition, the linked form gives forces on the nodal rotations which improves the solution obtained.
Timoshenko theory - enhanced assumed strain form
The constant strain form of the displacement interpolation has the advantage that the transient behaviour involves the interelement nodes only, whereas use of the added internal form complicates the dynamics. The advantages of the form which gives exact nodal answers has significant accuracy advantages. Using an enhanced assumed strain f o r m 31-33 c a n restore the advantage of the full interpolation without complicating the dynamics. In the enhanced assumed strain form we use the constant strain form of the displacements w and 0r_given by Eq. (10.84) and enhance the strains with the effects which arise from the A 0r3. Thus, we express the strains as
~bIX.yl = "Yz
=
ll
~e
(w2
-
-
(Oy2--Oyl) ~} 6z 1 + (Oyl-+"Or2)114~
Wl) -4- ~ Le
(10.90)
= Bbfi b + Bb,, 6L
where B b is the strain-displacement matrix given in Eq. (10.86) and c~ is the enhanced strain parameter. The elimination of the enhanced parameter at the element level now proceeds as given for the static case and is not affected by inertial effects.
10.5.2 "rimoshenko - mixed form A mixed form approximation for the Timoshenko beam which gives exact interelement nodal behaviour may be constructed in a similar way as we performed for the Euler-Bernoulli theory in Sec. 10.4.1. Accordingly, considering the static problem the approximations are deduced from d<SSz _ dx
0
and
d<SMy ~ + ~SSz = 0 dx
(10.91)
which are satisfied by 6My - N1 (~) r
+ N2(~) <SMy2
where Na (~) are defined in Eq. (10.38).
and
1 ~Sz = - 7 (6My2 - 6Myl) Le
(10.92)
312
Structural mechanics problems in one dimension - rods
In order to obtain exact interelement nodal displacements for the mixed formulation we let My = NI(~)/~vl + N2(~)/17f,,2 q'-/~v(~) 1 S: = L--~e(My2 - M,.l) + S:(~)
(10.93)
where/f/,, and S: are particular solutions which satisfy dSz dx
t- qz - 0
d]lT'/y dx
and
Sz + my = 0
In addition we employ curvature approximations which satisfy the constitutive equations. Accordingly, for the linear elastic problem we use 1 E---~yM~, and 7 : = "
Xv= "
1
Sz
n~GzA
With these approximations, after integrating Eq. (10.33) by parts we obtain the weak form for the static problem
(~l"IeMy(IlO, Oy,My)-- (,My Oy)[OLe"4-((~OyMy)]OLe"J[-"((~Sz 11.))IO'e+ (6W am)OLe e
I. My
az A
(10.94)
which is a similar form to that deduced for the Euler-Bernoulli beam in Eq. (10.52).
Example 10.10
Timoshenko theory mixed stiffness matrix
Let us consider the example of a uniformly loaded element with length L e that has a variable moment of inertia given in Example 10.4 and a shear stiffness given by
1 t~zGzA
-- N1 (~)
1 xozGzA1
+ N2(~)
1 ~zGzA2
The particular solutions for the uniformly loaded element again may be expressed as
l f 4 y - - g1q z L e2( 1 - ~ : )
and
Sz--
1
iqzLe
The arrays for V, G, fM and fu for the bending response are identical to those of the weak form (10.54) given in Example 10.4. The remaining term from the shear response is given by
e6SzTz(Sz) dx = g(6My2 -
e xozGzAd x ( M y 2
-
Myl)
e azGzA Szdx
= [~/~yl =
- 6My~)
(w
l~y 5~y~](rWllL Wzl W12]W{/~Yl 22j 1}-Jr { ,S1 fs2}) +
Finite element solution: Timoshenko rods
where
1(,
W
=
2Le
~zGzA1
fs
=
1-5q: Le
1
2(
+
1
)E
~zGzJ~2
1
-1
1
1
-1
~zGzA2 - ~ z G z a l ) ( - l l
1
and
1
Thus, the weak form for an element is given by
(ll~Iy}--(fM-~-fs))
Gr ~b -(V q--W)I After ~VIyis eliminated for each element we obtain the Ke
-
-
L
'stiffness' matrix
G r (V + W ) - I G
and load vector
f~ - f. + G ~ (V + W) -~ (fM + Is) which may be assembled in an identical manner as in the irreducible form. We note that the construction of the element array for the Timoshenko mixed form merely adds the effects of the shear to the terms of the Euler-Bemoulli theory. Another difference, however, is the nodal rotation Or is no longer directly related to the derivative of the transverse displacement w.
Example 10.11
Cantilever tapered bar - Timoshenko theory
To illustrate the performance of the irreducible and mixed forms presented for the Timoshenko beam theory we consider a cantilever beam that is loaded by a uniform load qz as shown in Fig. 10.10. The bar has a variable bending and shear stiffness given by
Ely =
2 EIo 2 kGAo and t~zGzArl rl
where r / - x / L and the stiffness values at the support are twice the free end value of
EIo.
IW J
qz
L
Fig. 10.10 Taperedcantilever beam.
313
314
Structural mechanics problems in one dimension - rods
The exact solution for the displacement and slope are given by qz L4
1/)(x) -
[3/~5 _ 2 0 T~4 --t- 4 0 ~73 - 55 ~7 + 32]
240EI0
qz L2 [3 r/2 _ r/3 _ 2] + 6kGAo Oy = qzL3 [at/4 - 16@ -t- 2 4 r / 2 - 11] 48Eio
and those for moment and shear by ~1 qzL 2 [ 4 q -
M=
r/2-4]
S = qz L [2 - q]
In Fig. 10.11 we present the solution using two equal length elements and an irreducible (displacement) solution with cubic w and quadratic Or [FE(3)], an irreducible solution with quadratic w and linear Or [FE(2)] and the mixed solution as described above. The results for the mixed solution are exact whereas those for the other solu-
O.l,o..F I I
" *
Disp. FE(3) Disp. FE(2)
S*t
0.08 0.06
.................
004
.....
,
.
.
.
' ,...
. . . . . . . . . . . . . .
,
.
, '
.
;p,,
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II
II }! - r
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;~'. . . . . . .
.
.
.
~)isp. FE(3) I 9 Ois'p. FEi2il
.
- ....
" .....
' .....
.
1.2
1.4
1.6 xlL (a) Displacement
I
Disp. FE(3) I Disp,'li:E(2)l-
:
1.8
:
-0.12
2
1.2
1.4
1.6 x/L (b) Slope
'
~ '
1.2
:
, , . . . . . . . . . . . . . . .
1
,
I
~: . . . . . . . . . . . . . . . . . . . . ',
.
.
.
; '
1.8
2
I re:Exact | I = .Disp. FE(3)~
I -r Disp. FE(2)~ o
.
,Mixed
FE
|
0.8
--0.1
..j
--0.2
0.6
--0.3
0.4
-"0.4
0.2
-"0.5
--0.6
.
e .
-0.1 [
u, I I - - Exact:
~>"
. .
o
.
,
0.02 0
. .
~-0.061" .
,
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.
.
-0.04 I. . . . .
,
.
;,, r,5,'" .
'~ . \\ .
j
;f-t . . . . . . . . . .
.
,~ .
f.,
-
1
Fig. 10.11 theory.
1.2
-
-, ,
.
.
.
.
.
.
.
. ,
.
.
.
.
.
.
.
,
,
,
1.6
1.8
Solution for tapered
0
.
. . . . .
,
1.4
x/L (c) Moment
.
.....
beam using
2
-0.2
1
". . . . . ,. . . . . . . . .
1.2
i - - -e'- --" i,,-"- -
"
~
-
,
1.6
1.8
-, . . . . . . . . . . . . . . . . . . . . .
1.4
x/L
~
(d) Shear
irreducible
(displacement)and
mixed solutions -Timoshenko
2
Finite element solution: Timoshenko rods 315
tions have error in all quantifies, although those for the displacement and slope are quite small.
10.5.3 Discrete Kirchhoff constraints .................................................................
- ............
- .....................
- .....................
-.........
=
................
=
........
. ..........
~ -:
.....................................................
. ......................
. ..............
-~- .....................................
: ......................................
- ........................................................................
In the previous discussion we have considered formulations for the Euler-Bernoulli and Timoshenko theories of beams. Using appropriate interpolations in the Timoshenko theory we have shown that it is possible to obtain a viable solution for situations in which the Euler-Bemoulli theory is accurate. We consider here an alternative to the previous approach by considering a reduced form of the functional for the Timoshenko theory to directly solve the thin beam case. In this form we use an irreducible form for the bending strains, set the constitutive behaviour for shear to zero, and introduce constraints to enforce zero shear strains. Initially this approach was applied to the study of thin plates based on the Kirchhoff theory and, thus, the approach is termed a discrete Kirchhoff method. In this section we illustrate the approach by considering a beam example. The reduced functional is given by
(~I'I ( u3, Oy ) -- fL ~O~Oy l~/Iy ( X.y ( Oy ) ) d x (10.95)
-- (~OyI~Y) IOMy --
OSz =
and is valid if we introduce appropriate constraints to satisfy (discretely)
Ow -~- Oy Ox
= 0
(10.96)
To solve the problem posed by Eqs (10.95) and (10.96) we can 1. approximate w and Oy by independent interpolations of Co continuity as w - Nw'~'
and
Oy = N00
(10.97)
2. impose a discrete approximation to the constraint of Eq. (10.96) and solve the problem resulting from substitution of Eq. (10.96) into Eq. (10.95) by either discrete elimination, use of suitable Lagrangian multipliers, or penalty procedures. In the application of the so-called discrete Kirchhoff constraints, Eq. (10.96) is approximated by point (or subdomain) collocation and direct elimination is used to reduce the number of nodal parameters. Of course, other means of imposing the constraints could be used with identical effect and we shall return to these in the next chapter. However, direct elimination is advantageous in reducing the final total number of variables and can be used effectively.
Example 10.12 One-dimensionalbeam example
We illustrate the process to impose discrete constraints on a simple, one-dimensional, example of a beam shown in Fig. 10.12. In this, initially the displacements and rotations
316
Structural mechanics problems in one dimension- rods
./-
0
Constraint
/x,
1
0
~a
(3
A
3
2
~13
0
(3
1
2
Fig. 10.12 A beam element with independent, Lagrangian, interpolation of w and Oy with constraint OwlOx + Oy - 0 applied at points x.
are taken as determined by a quadratic interpolation of an identical kind and we write in place of Eq. (10.97),
IWlOy
3
- ~ . ~ Na
a-1
Ill)aloa
(10.98)
where a are the three element nodes. The constraint is now applied by point collocation at coordinates x~ and x~ of the beam; that is, we require that at these points
Ow -~x + Oy - 0
(10.99)
This can be written by using the interpolation of Eq. (10.98) as two simultaneous equations
3
3
a=l
a=l
ENa(Xa)Wa-~L~Na(xa)Oa-O 3
E
a=l
where
3
N~a(Xfl)Ul)a-'1"Y~
[
N~ (x~) -- N~ (x) x=x~
a=l
and
Equations (10.100) can be used to eliminate we have
3 a=l
Aa
/tOa/ 0a
--
0 where Aa
(10.100)
Ua(xfl)Oa= 0
aN/[
N" (x~) - ~
x=x~
tO3 and 03. Writing Eqs (10.100) explicitly
- [Na'(Xa)'
[Na,(X•) '
Na(xa) 1 Na(xe) j
(10.101)
Substitution of the above into Eq. (10.98) results directly in shape functions from which the centre node has been eliminated, that is,
2 113 1~a ( 0y / - - E ( l~)a 0a } a=l
(10.102)
Forms without rotation parameters
with Na -- Na I - N3 A31Aa; a - 1, 2
where I is a 2 x 2 identity matrix. If these functions are used for the beam, we arrive at an element that is convergent. Indeed, in the particular case where x~ and x 9 are chosen to coincide with the two Gauss quadrature points the element stiffness coincides with that given by a displacement formulation involving a cubic w interpolation as described above for the irreducible form. In fact, the agreement is exact for a uniform beam.
It is possible to formulate the Timoshenko beam theory without direct use of rotation parameters. Such an approach has advantages for problems with large rotations where use of rotation parameters leads to introduction of trigonometric functions (e.g. see Chapter 17). Here we consider the case of a straight beam in two dimensions where each element is defined by coordinates at the two ends. Starting from a four-node rectangular element in which the origin of a local Cartesian coordinate system passes through the centroid of the element we may write interpolations as (Fig. 10.13)
x -- Ni(~, T]).~i 7I- Nj(~, ~?)2j + Nk(~, ~7)2k + NI(~, ?]).~l y = Ni(~, ?])Yi -']- Nj(~, ?])yj -Jl" N~(~, rl)Yk + NI(~, ~])Yl
(10.103)
in which Ni, etc. are the usual four-node bilinear shape functions. Noting the rectangular form of the element, these interpolations may be rewritten in terms of alternative parameters [Fig. 10.13(b)] as x = N1 (~)21 + N2 (~)3~2
y-~~7[Nl(~)t'~ +
N2(~)t'2]
(10.104)
where shape functions are NI(~) = ~1 (1
-
~)
,
N2(~)
--
~1 (1 + ~)
(10.105)
and new nodal parameters are related to the original ones through 1 -~1 = ~(3~i + X2--~~l(3~j +
-~l) and t'l -" Yl - Yk 2k) and t"2 -- Yk -- Yj
(10.106)
Since the element is rectangular t'l = t'2 = t" [Fig. 10.13(b)]; however, the above interpolations can be generalized easily to elements which are tapered. We can now use isoparametric concepts to write the displacement field for the element as
E
r~t u -- N1 (~) /~1 -~- ~ A/~I v -- NI(~)
1~1 + ~ A I ~ I
]
[
rF "~ N2 (~) /~2 "~ ~ A/~2 + N2(~)
1~2 -~- ~ A I ~ 2
1
(10.107)
317
318
Structural mechanics problems in one dimension - rods
(u., w.)
'I
I
i
(a)
i-o,
t
~.--x J
o
(,~u2, ~xv2)
(b)
(,aUn, ,aVn)
on
,,
l
i
(Un, Vn)
(c) Fig. 10.13 One-dimensional bending of planar beams: (a) geometry, Q4 element; (b) geometry, no rotation parameters; (c) joining elements with different thickness. in which/-is a 'thickness' parameter chosen to permit elements of different cross-section to be joined at a common node n [Fig. 10.13(c)]. It is evident that the above interpolations are identical to those originally written for the quadrilateral element. Only the parameters are different. Based on results for the incompressible problem, we also know the element will not perform well in bending situations because of 'shear locking', especially when the aspect ratio of the element length to depth becomes very large. In order to improve the behaviour we introduce a three-field approximation by using the enhanced strain concept described in reference 6. Accordingly, the mixed strain approximation will be taken as
~7
Cy =
Ni'x 0
%y
0
0
-~Ni'xO 1 -Ni t
0 Nix,
10
/~i
t Ni
ma i Ui
~]t I ~-~Ni,xj
mVi
Jr-
I0~
~] ~/~12
0 0 0 7] 0
~3
0
~4
(lO.lO8)
where fli are parameters of the enhanced strains. 6 The remainder of the development is straightforward and is left as an exercise for the reader. We do note that here it is not necessary to use a constitutive equation which has been reduced to give zero stress in the through-thickness (y) direction. By including additional enhanced terms in the
Moment resisting frames thickness direction one may use the three-dimensional constitutive equations directly. Such developments have been pursued for plate and shell applications. 34-37 We note that while the above form can be used for fiat surfaces and easily extended for smoothly curved surfaces it has difficulties when 'kinks' or multiple branches are encountered as then there is no unique 'thickness' direction. Thus, considerable additional work remains to be done to make this a generally viable approach.
The formulation given above may be used to solve moment resisting flame problems such as the plane frame shown in Fig. 10.14(a). Using a global coordinate system x' = (x', y', z') a transformation between the local frame x used in the above description of axial, bending and torsion is given by [Fig. 10.14(b)] x = Lx'
(10.109)
where L is an array of direction cosines. The same form may be used to transform the degrees of freedom for the problem. In a general three-dimensional setting we have
where Ua - -
Ua
,
/0"" / 0~ Oyo
with similar relations for 8' and 0'. ZI
>>y'
///
//
///
//
(a) Two-dimensional frame
(b) Coordinate system
Fig. 10.14 Momentresisting frame and coordinate transformation.
319
320
Structural mechanics problems in one dimension - rods
Using the above transformation, the residuals for each element are transformed by
and the tangent matrix by
K;,,
Koo j
{si}{s:} o] kKo. KooJ Lr
(10.112)
In this chapter we have summarized the basic steps needed to formulate and solve rod problems by the finite element method. We have focused attention on the development of accurate interpolation of the variables which, in certain circumstances, give exact solutions at interelement nodes. In addition, for the Timoshenko beam theory, we have shown how to obtain interpolation functions which do not 'lock' in applications to thin beam problems. This latter aspect will be exploited in the solution of plate and shell problems in the next chapters.
1. G. Wempner. Mechanics of Solids, with Applications to Thin Bodies. McGraw-Hill, New York, 1973. 2. E.P. Popov and T.A. Balan. Engineering Mechanics of Solids. Prentice-Hall, Upper Saddle River, NJ, 2nd edition, 1999. 3. F.P. Beer, Jr., E.R. Johnson and W.E. Clausen. Mechanics of Materials. McGraw-Hill, Boston, 3rd edition, 2002. 4. J. Demmel. Applied Numerical Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1997. 5. S.P. Timoshenko and J.N. Goodier. Theory of Elasticity. McGraw-Hill, New York, 3rd edition, 1969. 6. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann, Oxford, 6th edition, 2005. 7. S. Klinkel and S. Govindjee. Anisotropic bending-torsion coupling for warping in a non-linear beam. Computational Mechanics, 31:78-87, 2003. 8. S.P. Timoshenko and S. Woinowski-Krieger. Theory of Plates and Shells. McGraw-Hill, New York, 2nd edition, 1959. 9. P. Tong. Exact solution of certain problems by the finite element method. Journal of AIAA, 7:179-180, 1969. 10. M. Abramowitz and I.A. Stegun, editors. Handbook of Mathematical Functions. Dover Publications, New York, 1965. 11. V. Ciampi and U Carlesimo. A nonlinear beam element for seismic analysis of structures. In Proc. European Conference on Earthquake Engineering, pages 73-80, Lisbon, June 1986.
References 321 12. E. Spacone, V. Ciampi and EC. Filippou. Mixed formulation of nonlinear beam finite element. Computers and Structures, 58:71-83, 1995. 13. E. Spacone, F.C. Filippou and EE Taucer. Fiber beam-column model for non-linear analysis of R/C frames. 1. Formulation. Earthquake Engineering and Structural Dynamics, 25:711-725, 1996. 14. E. Spacone, F.C. Filippou and EE Taucer. Fiber beam-column model for non-linear analysis of R/C frames. 2. Applications. Earthquake Engineering and Structural Dynamics, 25:727-742, 1996. 15. A. Neuenhofer and F.C. Filippou. Evaluation of nonlinear frame finite element models. J. Structural Engineering, ASCE, 123:958-966, 1997. 16. A. Neuenhofer and EC. Filippou. Geometrically nonlinear flexibility-based frame finite element. J. Structural Engineering, ASCE, 124:704-711, 1998. 17. M. Petrangeli and V. Ciampi. Equilibrium based iterative solution for the non-linear beam problem. International Journal for Numerical Methods in Engineering, 40:423-437, 1997. 18. M. Petrangeli, P.E. Pinto and V. Ciampi. Fiber element for cyclic bending and shear of RC structures. I. Theory. J. Engineering Mechanics, ASCE, 125:994-1001, 1999. 19. M. Schultz and EC. Filippou. Non-linear spatial Timoshenko beam element with curvature interpolation. International Journal for Numerical Methods in Engineering, 50:761-785,2001. 20. M. Schultz and EC. Filippou. Generalized warping torsion formulation. J. Structural Engineering, ASCE, 124:339-347, 1998. 21. A. Ayoub and EC. Filippou. Nonlinear finite-element analysis of RC shear panels and walls. J. Structural Engineering, ASCE, 124:298-308, 1998. 22. A. Ayoub and F.C. Filippou. Mixed formulation of bond-slip problems under cyclic loads. J. Structural Engineering, ASCE, 125(ST6):661-671, 1999. 23. A. Ayoub and F.C. Filippou. Mixed formulation of nonlinear steel-concrete composite beam element. J. Structural Engineering, ASCE, 126:371-381, 2000. 24. R.L. Taylor, EC. Filippou and A. Saritas. Finite element solution of beam problems. Computational Mechanics, 31, 2003. 25. K.D. Hjelmstad and E. Taciroglu. Mixed variational methods for finite element analysis of geometrically non-linear, inelastic Bernoulli-Euler beams. Communications in Numerical Methods of Engineering, 19:809-832, 2003. 26. M.A. Crisfield. Non-linear Finite Element Analysis of Solids and Structures, volume 2. John Wiley & Sons, Chichester, 1997. 27. B. Fraeijs de Veubeke. Displacement and equilibrium models in finite element method. In O.C. Zienkiewicz and G.S. Holister, editors, Stress Analysis, Chapter 9, pages 145-197. John Wiley & Sons, Chichester, 1965. 28. A. Tessler and S.B. Dong. On a hierarchy of conforming Timoshenko beam elements. Computers and Structures, 14:335-344, 1981. 29. M.A. Crisfield. Finite Elements and Solution Procedures for Structural Analysis, Vol. 1, Linear Analysis. Pineridge Press, Swansea, 1986. 30. H.K. Stolarski, N. Carpenter and T. Belytschko. Bending and shear mode decomposition in C O structural elements. Journal of Structural Mechanics, ASCE, 11(2): 153-176, 1983. 31. J.C. Simo and M.S. Rifai. A class of mixed assumed strain methods and the method of incompatible modes. International Journal for Numerical Methods in Engineering, 29:1595-1638, 1990. 32. U. Andelfinger, E. Ramm and D. Roehl. 2d- and 3d-enhanced assumed strain elements and their application in plasticity. In D. Owen, E. Ofiate and E. Hinton, editors, Proceedings of the 4th International Conference on Computational Plasticity, pages 1997-2007. Pineridge Press, Swansea, 1992. 33. M. Bischoff, E. Ramm and D. Braess. A class of equivalent enhanced assumed strain and hybrid stress finite elements. Computational Mechanics, 22:443-449, 1999.
322
Structural mechanics problems in one dimension- rods 34. N. Btichter, E. Ramm and D. Roehl. Three-dimensional extension of non-linear shell formulations based on the enhanced assumed strain concept. International Journal for Numerical Methods in Engineering, 37:2551-2568, 1994. 35. M. Braun, M. Bischoff and E. Ramm. Nonlinear shell formulations for complete threedimensional constitutive laws include composites and laminates. Computational Mechanics, 15:1-18, 1994. 36. P. Betsch, F. Gruttmann and E. Stein. A 4-node finite shell element for the implementation of general hyperelastic 3d-elasticity at finite strains. Computer Methods in Applied Mechanics and Engineering, 130:57-79, 1996. 37. M. Bischoff and E. Ramm. Shear deformable shell elements for large strains and rotations. International Journal for Numerical Methods in Engineering, 40:4427--4449, 1997.
Plate bending approximation: thin (Kirchhoff) plates and
continuity requirements
iiiiii~,iil!ii{i',~ii~~ili~i::~,'::'~iii{~{i!~:~::'~'~ill!!',!'i'~i{~i'ii'~;i~~:,~,i:~~''i!~,!iiliii{i!ii~,!iilii"~::'~i:!i:i~il,~i~ili!il~iiiii:::::: ~'i~'~i!i'~',iiii~,',iii!i~i!i~,i:,~i',i:,i~'i:::':i~!'~i'ii{~iiii!ii'~i~iiiiiiiiiii',!'i::::::,i::::iiii~i ~,i~,ii''~,i'~'~,'i'~,'~,i'{,!i'~!~,!iiiiii'i~~iii',i i~,}ili i',~,i~i,~'i}~,il'~iiiiiiiii!iii'i',,il'~'~i!iilii',iiii}iilii'~i'i,'i,iii',iiii}~ii'~i'~~,i',i~'il,i'~'i~!'~ii'iil,i!i','~iiiiiiiiiiiiiiiiiiiiiil~,'~iii~iililii',~,i ~'i~i'~'~i,:i{iii iili~{i'iiii,i~,i'i',,ili'i~i~i'~i'!!i,~i;ili'~,i~i;ii',i~i~'~,i'~,i~,i'',i~iii''~,i'i',i~i'~i~'~iliiii{i!i{iili'~il'i,i',iiiii'!ii',i,'li:,i', i!i',ii!ii' ', i,'i'~',i ',~' !i,!'i',i,'~'~i{,':i'ii'~:i?!'i{,~''~,'{i,'~'~!'i!ii,!',ii'lii,i!ii!iii!il}i'{~','~i,{i{i',!ii!i~iil','~i!ii',iii~,',i'i,l'~i 'i'~i,lilii!i!li! !'}i,i'i,'~',~i'~i~i{~,i~~{~, '~~,i~i{~~i~i,'~'',~i'~,'~i'~i{i~,','~'~i'~i!',i',i~{'i~i'i!'~'i!i'~,'i,i!i'~~i'i~i''i~,'~i'ii,i',!ii'~~ii'ii,i''~,i!'~,',!
The subject of bending of plates and indeed its extension to shells was one of the first to which the finite element method was applied in the early 1960s. At that time the various difficulties that were to be encountered were not fully appreciated and for this reason the topic remains one in which research is active to the present day. Although the subject is of direct interest only to applied mechanicians and structural engineers there is much that has more general applicability, and many of the procedures which we shall introduce can be directly translated to other fields of application. Plates and shells are but a particular form of a three-dimensional solid, the treatment of which presents no theoretical difficulties, at least in the case of elasticity. However, the thickness of such structures (denoted throughout this and later chapters as t) is very small when compared with other dimensions, and complete three-dimensional numerical treatment is not only costly but in addition often leads to serious numerical ill-conditioning problems. To ease the solution, even long before numerical approaches became possible, several classical assumptions regarding the behaviour of such structures were introduced. Clearly, such assumptions result in a series of approximations. Thus numerical treatment will, in general, concern itself with the approximation to an already approximate theory (or mathematical model), the validity of which is restricted. On occasion we shall point out the shortcomings of the original assumptions, and indeed modify these as necessary or convenient. This can be done simply because now we are granted more freedom than that which existed in the 'pre-computer' era. The thin plate theory is based on the assumptions formalized by Kirchhoff in 1850,1 and indeed his name is often associated with this theory, though an early version was presented by Sophie Germain in 1811. 2-4 A relaxation of the assumptions was made by Reissner in 19455 and in a slightly different manner by Mindlin 6 in 1951. These modified theories extend the field of application of the theory to thick plates and we shall associate this name with the Reissner-Mindlin postulates. It turns out that the thick plate theory is simpler to implement in the finite element method, though in the early days of analytical treatment it presented more difficulties. As it is more convenient to introduce first the thick plate theory and by imposition of
324
Plate bending approximation: thin (Kirchhoff) plates and C1 continuity requirements additional assumptions to limit it to thin plate theory we shall follow this path in the present chapter. However, when discussing numerical solutions we shall reverse the process and follow the historical procedure of dealing with the thin plate situations first in this chapter. The extension to thick plates and to what turns out always to be a mixed formulation will be the subject of Chapter 12. In the thin plate theory it is possible to represent the state of deformation by one quantity w, the lateral displacement of the middle plane of the plate. Thus we find that thin plates share some of the same characteristics as the Euler-Bemoulli beam theory considered in the previous chapter. Clearly, such a formulation is irreducible. The achievement of this irreducible form introduces second derivatives of w in the strain definition and continuity conditions between elements have now to be imposed not only on this quantity but also on its derivatives (C1 continuity). This is to ensure that the plate remains continuous and does not 'kink'.* Thus at nodes on element interfaces it will always be necessary to use both the value of w and its slopes (first derivatives of w) to impose continuity, again this is similar to the treatment of Euler-Bemoulli beam theory. Determination of suitable shape functions is now much more complex than those needed for Co continuity or for beams. Indeed, as complete slope continuity is required on the interfaces between various elements, the mathematical and computational difficulties often rise disproportionately fast. It is, however, relatively simple to obtain shape functions which, while preserving continuity of w, may violate its slope continuity between elements, though normally not at a node where such continuity is imposed, t If such chosen functions satisfy the 'patch test' then convergence will still be found. 7 The first part of this chapter will be concemed with such 'non-conforming' or 'incompatible' shape functions. In later parts new functions will be introduced by which continuity can be restored. The solution with such 'conforming' shape functions will now give bounds to the energy of the correct solution, but, on many occasions, will yield inferior accuracy to that achieved with non-conforming elements. Thus, for practical usage the methods of the first part of the chapter are often recommended. The shape functions for rectangular elements are the simplest to form for thin plates and will be introduced first. Shape functions for triangular and quadrilateral elements are more complex and will be introduced later for solutions of plates of arbitrary shape or, for that matter, for dealing with shell problems where such elements are essential. The problem of thin plates is associated with fourth-order differential equations leading to a potential energy function which contains second derivatives of the unknown function. It is characteristic of a large class of physical problems and, although the chapter concentrates on the structural problem, the reader will find that the procedures developed will also be equally applicable to any problem which is of fourth order. The difficulty of imposing C1 continuity on the shape functions has resulted in many alternative approaches to the problems in which this difficulty is side-stepped. Several possibilities exist. Two of the most important are:
* If 'kinking' occurs the second derivative or curvature becomes infinite and squares of infinite terms occur in the energy expression. t Later we show that even slope discontinuity at the node may be used.
The plate problem: thick and thin formulations 325 1. independent interpolation of rotations q~ and displacement w, imposing continuity as a special constraint, often applied at discrete points only; 2. the introduction of Lagrange multiplier variables or indeed other variables to avoid the necessity of C1 continuity. Both approaches fall into the class of mixed formulations and we shall discuss these briefly at the end of the chapter. However, a fuller statement of mixed approaches will be made in the next chapter where both thick and thin approximations will be dealt with simultaneously. i~!iiiiif~!ii!!i i!:i!i i ~ii~l~;~i:i ii~~'~'~!il:i!i~ ii i~i~l~'~~i!ii i iili ii'~~'~''~'~~"~~''i i i ii i i~i i iili i!ii~.!.i i~li iili i':~'~iil'i i!i i iiili!i~~ii!iiiii i ii!i!ii i!iil!i:~:~ ii~i i i!ii~ii~i~i i!iil~ii i~ii ie~i ili~!~,i!~i ,,~~,iii~i i!i!i,~i~,i~,iii~i~i i~ii!~iiii ~i ili~ii~i ii~iifi!ii!l!i i!ii!iiii~!~ ~i~~!ii,~,,, i ii~i !i!l~i~i i~iili ii,~~i,!!!i, !ii!~i i ~i i i!i i!~ii i i ii!ii i i ii i,l~,~i ~i~i!~i !~ii ii,~,i~li!,~iii ~i ii!ii~lili~i ii~i"l~i l'~ ,iii~i i i iil!!ii!i!i!ii i~iiliii~ii i iiil~i iili i ,!i~;~i,i~i i ili~i ii~ iiii~~i ii!i!i i~lii i iii iili ii!iliiii ii i!ii !i!ii i ii !i!i!!!ii!ii iii i ilii~!i ~i i iiii!i!li i~i
11.2.1 Governing equations The mechanics of plate action can be illustrated in one dimension by considering a plate of infinite extent in one dimension (here assumed the y) and considering equations similar to those developed for a beam in Sec. 10.2. Here we consider the problem of cylindrical bending of plates. 2 In this problem the plate is to be loaded and supported by conditions independent of y. In this case we may analyse a strip of unit width subjected to some stress resultants Mx, Px, and Sx, which denote x-direction bending moment, axial force and transverse shear force, respectively,* as shown in Fig. 11.1. For cross-sections that are originally normal to the middle plane of the plate we can use the approximation that at some distance from points of support or concentrated loads plane sections will remain plane during the deformation process. The postulate that sections normal to the middle plane remain plane during deformation is thus the first and most important assumption of the theory of plates (and indeed shells). To this is added the second assumption. This simply observes that the direct stresses in the normal direction, z, are small, that is, of the order of applied lateral load intensifies, q, and hence direct strains in that direction can be neglected. This 'inconsistency' in approximation is compensated for by assuming a plane stress condition in each lamina. With these two assumptions it is easy to see that the total state of deformation can be described by displacements u and w of the middle surface (z = 0) and a rotation ~bx of the normal (Fig. 11.1). Thus the local displacements in the directions of the x and z axes are taken as
Ul(X, z) = u(x) + Z~x(X)
u3(x, z) = w(x)
and
(11.1)
Immediately the strains in the x and z directions are available as
Ou
OUl Cx ~
tc)qX
"-
Ox
-t-Z~
OCx Ox (11.2)
Cz=0
Oul %Z=Oz+
C~ 3
Ox
=
0//3
Ox
+4~x
* Here we change our notation slightly from that used for beams in order to conform to the notation commonly used for plates.
326
Plate bending approximation: thin (Kirchhoff) plates and C1 continuity requirements I)X
qz
llllll ~
x
7 I A'
.A" I I
P~
I
~A' Fig. 11.1 Displacements and force resultants for cylindrical bending of a plate. The non-zero strains are identical to those obtained for bending of beams. For the cylindrical bending problem a state of linear elastic, plane stress for each lamina yields the stress-strain relations E
~rx - - 1 -
and
u a ex
~'xz = G % z
The stress resultants are obtained as Px --
f
t/2
On Crx d z = B
J -t/2 Sx =
ax Txz d z = n G t
d -t/2 Mx =
f
t/2
J -t/2
(O;x ) + C~x
(11.3a)
Od;x Crx z d z = D
OX
where B is the in-plane plate stiffness and D the bending stiffness for an isotropic elastic material and are computed from B =
Et 1 -
ua
and
D =
Et 3
12(1
-
ua)
(11.3b)
with u Poisson's ratio, E and G direct and shear elastic moduli, respectively.* Three equations of equilibrium complete the basic formulation. These equilibrium equations may be computed directly from a differential element of the plate or by integration over the thickness of the local equilibrium equations as was performed for * A constant ~ has been added here to account for the fact that the shear stresses are not constant across the section. A value of ~ = 5 / 6 is exact for a rectangular, homogeneous section and corresponds to a parabolic shear stress distribution.
The plate problem: thick and thin formulations 327 the beam in Sec. 10.2.1. Using the latter approach and assuming zero inertial forces we have for the axial resultant
it~2 i~O.x ~xz ] 0 ill2 d --t/2 "~X -Ji--~Z -Jr-bx dz = ~ .J-t/2 a~ dz Oex Ox
+
it~2 bx dz J -t/2
+ Txz
I ,/2
Txz
I = -t/2
0
+qx = 0 (11.4a)
where qx is an axial load similar to that obtained for the beam. Similarly, the shear resultant follows from
f,, r xzOo z ] .J--t~2 L OX -JI--~Z "JI-bz OSx Ox
oft/d-t~2
dz = -~x
~-~z dz +
d-t~2 bx dz
+ az
It/2 I--t/2 - crz
= 0
-t-qz = 0 (11.4b)
where the transverse loading qz arises from the body force and the resultant of the normal traction on the top and/or bottom surfaces. Finally, the moment equilibrium is deduced from
/t/2[OO.xOT.xz z + ,I-t~2 ~X ~ OMx Ox
+ bx
]
dz =
o/t/2 z Crx dz ~ J-t/2
-
it~2 rxz dz ,I-t~2
+
ill2 z bx dz ,I-t~2
- 0
Sx + m ~ = 0
(11.4c)
Generally, mx loads are not included in plate theory except as an artifice to introduce the d' Alembert inertial forces. As we found was true for beams, in the elastic case of a plate it is easy to see that the in-plane displacements and forces, u and Px, decouple from the other terms and the problem of lateral deformations can be dealt with separately. We shall thus only consider bending in the present chapter, returning to the combined problem, characteristic of shell behaviour, in later chapters. Equations (11.1)-(11.4c) are typical for thick plates, and the thin plate theory adds an additional assumption. This simply neglects the shear deformation and puts G = c~ (or %z = 0). Equation (11.3a) thus becomes Ow Ox
+ ~x = 0
(11.5)
This thin plate assumption is equivalent to stating that the normals to the middle plane remain normal to it during deformation and is the same as the Bernoulli-Euler assumption for thin beams considered in Chapter 10. The thin, constrained theory is very widely used in practice and proves adequate for a large number of structural problems, though, of course, should not be taken literally as the true behaviour near supports or where local load action is important and is three dimensional. In Fig. 11.2 we illustrate some of the boundary conditions imposed on plates and immediately note that the diagrammatic representations of simple support as a knife
328
Plate bending approximation: thin (Kirchhoff) plates and C1 continuity requirements Singularity j disregarded
(a) Built-in support (clamped) with u= v= w= 0, 0=0
(b) Free edge with M = 0, S=0(P=0)
Singularity
Rigid
disregarded
~/M=0 w=0
Real approximation
Conventional illustration
Plate " - - . ~ s ) _ (n)
or 0x = 0s = 0
w- 0
SS1 (soft support) SS2 (hard support)
(c) Simply supported condition Fig. 11.2 Support (end) conditions for a plate. Note: the conventionally illustrated simple support leads to infinite displacement - reality is different.
edge would lead to infinite displacements and stresses. Of course, if a rigid bracket is added in the manner shown this will alter the behaviour to that which we shall generally assume. The one-dimensional problem of plates and the introduction of thick and thin assumptions translate directly to the general theory of plates. In Fig. 11.3 we illustrate the extensions necessary and write, in place of Eq. (11.1) (assuming u0 and v0 to be zero), Ux -- z #Px(X, y)
Uy = Z ~y(X, y)
uz = w ( x , y)
(11.6)
where we note that displacement parameters are now functions of x and y. It is sometimes advantageous to replace q5x and qSyby rotations about the x and y coordinates in a manner used for the beam developments. Thus,
-~ {~
4,=TO
or
IOy}-[o oJ Ox
(11.7)
The plate problem: thick and thin formulations
~
z(~
x(u)
y(v) <
(a) Displacement, rotations and load x
7
j
<
(b) Force resultants (on-faces) Fig. 11.3 Definitions of variables for plate approximations.
may be substituted at any time in the development. We do not make this change here as it complicates the expressions for the governing equations. We normally would make the substitution on nodal parameters after arrays are formulated. The strains may now be separated into bending (in-plane components) and transverse shear groups and we have, in place of Eq. (11.2), "0
--Z
~xy
"
0
o
g;y
0
0
.Oy and
o
{~;}----ZL~
(11.8a)
V w + r
(11.8b)
Ox_
Ow 7-
L,.)/y z
--
OW
+
--
We note that now in addition to normal bending moments Mx and My, now defined by expression (11.3a) for the x and y directions, respectively, a twisting moment arises defined by Mxy --
i
t/2
Txy dz
(11.9)
J -t/2
Introducing appropriate constitutive relations, all moment components can be related to displacement derivatives. For isotropic elasticity we can thus write, in place of Eq. (11.3a), M =
My Mxy
-- D/205
(11.10a)
329
330
Plate bending approximation: thin (Kirchhoff) plates and C~ continuity requirements where, assuming plane stress behaviour in each layer, D=D
v 0
1 0
(11.lOb) (l-v)/2
in which v is Poisson's ratio and D is defined by the second of Eqs (11.3b). Further, the shear force resultants are
s{Sx} Sy
--- Of. ( V W "JI- ~ )
(11.10c)
For isotropic elasticity c~ = t~ G t I
(11.10d)
where I is a 2 x 2 identity matrix (though here we deliberately have not related G to E and u to allow for possibly different shear rigidities). Of course, the constitutive relations can be simply generalized to anisotropic or inhomogeneous behaviour such as can be manifested if several layers of materials are assembled to form a composite. T h e only apparent difference is the structure of the D and c~ matrices, which can always be found by simple integration. The governing equations of thick and thin plate behaviour are completed by writing the equilibrium relations. Again omitting the 'in-plane' behaviour we have, in place of Eq. (11.4b), '
+ q = ~'Ts + q = 0
Sy
(ll.lla)
and, in place of Eq. (11.4c),
0 --O00yl{}{ SX 0
My
--
Mxy
Oy
Ox.l
+
=/2TM + S = 0
(11 11 b)
Sy
Equations (11.10a)-(11.11 b) are the basis from which the solution of both thick and thin plates can start. For thick plates any (or all) of the independent variables can be approximated independently, leading to a mixed formulation which we shall discuss in Chapter 12 and also briefly in Sec. 11.16 of this chapter. For thin plates in which the shear deformations are suppressed Eq. (11.10c) is rewritten as V w + dp = O (11.12) and the strain-displacement relations (11.8a) become 02w
e = -zEVw
= -z
02w Oy2 02w 20xOy
~ = -zx
(11.13)
The plate problem: thick and thin formulations 331 where X is the matrix of changes in curvature of the plate. Using the above form for the thin plate, both irreducible and mixed forms can now be written. In particular, it is an easy matter to eliminate M, S and q~ and leave only w as the variable. Applying the operator ~rT to expression (11.11 a), inserting Eqs (11.10a) and (11.11 a) and finally replacing q~ by the use of Eq. (11.12) gives a scalar equation (f--.V)TDf-.,Vw
(11.14a)
+ q -- 0
where, using Eq. (11.13), [0 2 (~V)--
OqX2,
02
2 02 ]
Oy2,
OxOy
In the case ofisotropy with constant bending stiffness D this becomes the well-known biharmonic equation of plate flexure
D
(~w a4w ~w) ~ + 20x20y-~-7 + ff~y4 + q --0
(11.14b)
11.2.2 The boundary conditions The boundary conditions which have to be imposed on the problem (see Figs 11.2 and 11.4) include the following classical conditions.
1. Fixed boundary, where displacements on restrained parts of the boundary are given specified values.* These conditions are expressed as W -- ~l);
Cn = ~n
Cs -- r
and
Here n and s are directions normal and tangential to the boundary curve of the middle surface. A clamped edge is a special case with zero values assigned. 2. Traction boundary, where stress resultants Mn, Mns and Sn (conjugate to the displacements On, 0s and w) are given prescribed values. Mn=l~n;
Mns---l~ns
and
S~ -- Sn
A free edge is a special case with zero values assigned. 3. 'Mixed' boundary conditions, where both traction and displacement components can be specified. Typical here is the simply supported edge (see Fig. 11.2). For this, clearly, M, = 0 and w = 0, but it is less clear whether Mns or Cs needs to be given. Specification of Mns = 0 is physically a more acceptable condition. This should be adopted for thick plates. Thus options for simply supported edges are Type
Conditions
SS1
w=0,
Mn=0,
Mns=O
SS2
w
Mn --- O,
Cs -- 0
- - 0,
* Note that in thin plates the specification of w along s automatically specifies ~bs by Eq. (11.12), but this is not the case in thick plates where the quantities are independently prescribed.
332
Plate bending approximation: thin (Kirchhoff) plates and C1 continuity requirements
(a)
Boundarytractions Mn, Mns,Sn
and corresponding displacements
F~...L~
i i,
....- -~
(b)
On, 0s, W
Fig. 11.4 Boundary traction and conjugate displacement. Note: the simply supported condition requiring Mn = O, ~s -- 0 and w = 0 is identical at a corner node to specifying ~bn = ~bs -- O, that is, a clamped support. This leads to a paradox if a curved boundary (a)is modelled as a polygon (b).
In thin plates ~b~is automatically specified from w and we shall find certain difficulties, and indeed anomalies, associated with this assumption. 8'9 For instance, in Fig. 11.4 we see how a specification of q5s = 0 at comer nodes implicit in thin plates formally leads to the prescription of all boundary parameters, which is identical to boundary conditions of a clamped plate for this point. Thus, for a curved simply supported edge a unique normal to each node must be specified and used to specify one of the mixed conditions given above.
11.2.3 The irreducible, thin plate approximation The thin plate approximation when cast in terms of a single variable w is clearly irreducible and is in fact typical of a displacement formulation. The equations (11.11 a) and (11.1 l b) can be written together as
-- (.~V)TM -t- q -- 0
(11.15)
and the constitutive relation (11.10a) can be recast by using Eq. (11.12) as M = -Ds
(11.16)
The derivation of the finite element equations can be obtained either from a weak form of Eq. (11.15) obtained by weighting with an arbitrary function (say v = N~) and integration by parts (done twice) or, more directly, by application of the virtual work equivalence. Using the latter approach we may write the internal virtual work for the plate as (~l"Iin t "-
f (t~r162 d ~ - - f~ ~w(CV)TD(ff_,V)w d~
(11.17)
The plate problem: thick and thin formulations 333
where Q denotes the area of the plate reference (middle) surface and D is the plate stiffness, which for isotropy is given by Eq. (11.10b). Similarly the external work is given by 2 ~l-lex t - -
(~W q dr2 +
(~n
Mn dF +
n
(Sd)s Mns dF + t
~w
Sn
dI"
(11.18)
s
where /~7'/n, t~/lns,an are specified values and Fn, Ft and Fs are parts of the boundary where each component is specified. For thin plates with straight edges Eq. (11.12) gives immediately ~)s - - Ow/Os and thus the last two terms above may be combined as
fFt~/)sl~nsdI~2I-fFs(~ll)SndI~--fFs(~ll)(Sn-JI-
)
OMns
0s
dF
+ ~ (~U)iRi
(11.19)
i
where Ri are concentrated forces arising at locations where comers exist (see Fig. 11.2). 2 Substituting into Eqs (11.17) and (11.18) the discretization w = N~t
(11.20)
where ~ are appropriate parameters, we can obtain for a linear case standard displacement approximation equations Kfi-f
(11.21)
with Kfi= (LBTDBdf2)
fi---LBTMdf2
(11.22)
and f = L NTq dr2 + fb
(11.23)
where fb is the boundary contribution to be discussed lal~er and M = -DBfi
(11.24)
B = (s
(11.25)
with N
334
Plate bending approximation: thin (Kirchhoff) plates and C1 continuity requirements
I
I !
Y.~
I !
T2
A
v
X
i
Fig. 11.5 Continuity requirement for normal slopes.
It is of interest, and indeed important to note, that when tractions are prescribed to non-zero values the force term fb includes all prescribed values of M,,, Mn~ and S, irrespective of whether the thick or thin formulation is used. The reader can verify that this term is P fb = /,. (N, Mn + NsTM,s - + N TS,,) dF (11.26a) T
-
where Mn, M,,s and S, are prescribed values and for thin plates [though, of course, relation (11.26a) is valid for thick plates also]" N,=
ON 0N On and N ~ = 0s
(11.26b)
The reader will recognize in the above the well-known ingredients of a displacement formulation (see Chapter 2 and reference 7) and the procedures are almost automatic once N is chosen.
11.2.4 Continuity requirement for shape functions (C1 continuity) In Sec. 11.3-11.13 we will be concerned with the above formulation [starting from Eqs (11.17) and (11.18)], and the presence of the second derivatives indicates quite clearly that we shall need C1 continuity of the shape functions for the irreducible, thin plate, formulation. This continuity is difficult to achieve and reasons for this are given below. To ensure the continuity of both w and its normal slope across an interface we must have both w and aw/On uniquely defined by values of nodal parameters along such an interface. Consider Fig. 11.5 depicting the side 1-2 of a rectangular element. The normal direction n is in fact that of y and we desire w and cgw/Oy to be uniquely determined by values of w, O~v/Ox, Ow/Oy at the nodes lying along this line. To show the C1 continuity along side 1-2, we write
w -- A1 Jr- A2x + A3y + ' " and
(11.27)
Ow
Oy
-" B1 -t-- B2x -4- B3y + . . .
(11.28)
with a number of constants in each expression just sufficient to determine a unique solution for the nodal parameters associated with the line.
The plate problem: thick and thin formulations Thus, for instance, if only two nodes are present a cubic variation of w should be permissible noting that Ow/Ox and to are specified at each node. Similarly, only a linear, or two-term, variation of Ow/Oy would be permissible. Note, however, that a similar exercise could be performed along the side placed in the y direction preserving continuity of Ow/Ox along this. Along side 1-2 we thus have Ow/Oy, depending on nodal parameters of line 1-2 only, and along side 1-3 we have Ow/Ox, depending on nodal parameters of line 1-3 only. Differentiating the first with respect to x, on line 1-2 we have ~ w/Ox Oy, depending on nodal parameters of line 1-2 only, and similarly, on line 1-3 we have ~ w/Oy Ox, depending on nodal parameters of line 1-3 only. At the common point, 1, an inconsistency arises immediately as we cannot automatically have there the necessary identity for continuous functions
0=w
=
O:w
OxOy OyOx
(11.29)
for arbitrary values of the parameters at nodes 2 and 3. It is thus impossible to specify simple polynomial expressions for shape functions ensuring full compatibility when only w and its slopes are prescribed at c o m e r n o d e s . 10 Thus if any functions satisfying the compatibility are found with the three nodal variables, they must be such that at comer nodes these functions are not continuously differentiable and the cross-derivative is not unique. Some such functions are discussed in the second part of this chapter. 11-16 The above proof has been given for a rectangular element. Clearly, the arguments can be extended for any two arbitrary directions of interface at the comer node 1. A way out of this difficulty appears to be obvious. We could specify the crossderivative as one of the nodal parameters. For an assembly of rectangular elements, this is convenient and indeed permissible. Simple functions of that type have been suggested by Bogner et al. 17 and used with some success. Unfortunately, the extension to nodes at which a number of element interfaces meet with different angles (Fig. 11.6) is not, in general, permissible. Here, the continuity of cross-derivatives in several sets of orthogonal directions implies, in fact, a specification of all second derivatives
at a node. This, however, violates physical requirements if the plate stiffness varies abruptly from element to element, for then equality of moments normal to the interfaces cannot
Fig. 11.6 Nodeswhere elements meet in arbitrary directions.
335
336
Plate bending approximation: thin (Kirchhoff) plates and C1 continuity requirements be maintained. However, this process has been used with some success in homogeneous plate situations 18-25 although Smith and Duncan TMcomment adversely on the effect of imposing such excessive continuities on several orders of higher derivatives. The difficulties of finding compatible displacement functions have led to many attempts at ignoring the complete slope continuity while still continuing with the other necessary criteria. Proceeding perhaps from a naive but intuitive idea that the imposition of slope continuity at nodes only must, in the limit, lead to a complete slope continuity, several successful, 'non-conforming', elements have been developed. ~2'26-4~ The convergence of such elements is not obvious but can be proved either by application of the patch test or by comparison with finite difference algorithms. We have discussed the importance of the patch test extensively in reference 7 and additional details are available in Sec. 11.7 and in references 41-43. In plate problems the importance of the patch test in both design and testing of elements is paramount and this test should never be omitted. In the first part of this chapter, dealing with non-conforming elements, we shall repeatedly make use of it. Indeed, we shall show how some successful elements have developed via this analytical interpretation. 44-49
Non-conforming shape functions
iiliiii!ii!ilii!i!iii!li!ii!i;ii!ilii~iiii~~~!iil ii!ii!!i!!iii!iii!iiiiiii!iii;i!i!iiiii!!iii!iiiii!!iiiil!!iiilii!i!!iiiil!iii!i!ii!i!iiiiii!iiiiiliiiiiiiiiii ii!iiii!i i i i !i!i!iill!illiiiiii iii!!ii Shape functions
Consider a rectangular element of a plate 1234 coinciding with the xy plane as shown in Fig. 11.7. At each node, a, displacements Ua are introduced. These have three components: the first a displacement in the z direction, wa, the second a rotation about the x axis, (Ox)a, and the third a rotation about the y axis, (Oy)a. The nodal displacement vectors are defined below as Ua. The element displacement will, as usual, be given by a listing of the nodal displacements, now totalling twelve:
ol ~e __
U2 U3
~14
with
!1a --
Oxa
(11.30)
~ya
A polynomial expression is conveniently used to define the shape functions in terms of the 12 parameters. Certain terms must be omitted from a complete fourth-order * Note that we have changed here the convention from that of Fig. 11.3 in this chapter by using Eq. (11.7). This allows transformations needed for shells to be carried out in an easier manner. However, when manipulating the equations of Chapter 12 we shall return to the original definitions using the ~b of Fig. 11.3. Similar difficulties are discussed by Hughes. 5~
Rectangular element with corner nodes (12 degrees of freedom) 337 2a J
S
y
3 7
4 J-"
Y
P
X
L/; ;__ Forces and corresponding displacements
Fig. 11.7 A rectangular plate element. polynomial. Writing W - - OL1 --[- O~2X --[- ol3y + OL4x2 -'[-"C e s x y "1- o~6y 2 q- OL7X3 +
-1- o~9xy2 -F o~loY3 + OLllX3y --[- OL12xy3
o~8x2y (11.31)
= pot has certain advantages. In particular, along any x constant or y constant line, the displacement w will vary as a cubic. The element boundaries or interfaces are composed of such lines. As a cubic is uniquely defined by four constants, the two end values of slopes and the two displacements at the ends will therefore define the displacements along the boundaries uniquely. As such end values are common to adjacent elements continuity of w will be imposed along any interface. It will be observed that the gradient of w normal to any of the boundaries also varies along it in a cubic way. (Consider, for instance, values of the normal Ow/Ox along a line on which x is constant.) As on such lines only two values of the normal slope are defined, the cubic is not specified uniquely and, in general, a discontinuity of normal slope will occur. The function is thus 'non-conforming'. The constants C~l to c~12 can be evaluated by writing down the 12 simultaneous equations linking the values of w and its slopes at the nodes when the coordinates take their appropriate values. For instance, ff13a :
~OyaI ~ I OXo
Ot,1 -Jr Ot,2 X a " [ - o ~ 3 Y a ' [ - ' ' "
"-- Oxa " - OL3 "[" OL5Xa ~
- " Oya --" ~ OL2 ~
"'"
01.5 Y a . . . .
Listing all 12 equations, we can write, in matrix form, ~e __ C o t
(11.32)
338 Plate bending approximation: thin (Kirchhoff) plates and C1 continuity requirements where C is a 12 • 12 matrix depending on nodal coordinates, and a is a vector of the 12 unknown constants. Inverting we have Ot - - C - I n e
(11.33)
This inversion can be carried out by computer or, if an explicit expression for the stiffness, etc., is desired, it can be performed algebraically. This was in fact done by Zienkiewicz and Cheung. 26 It is now possible to write the expression for the displacement within the element in a standard form as U --= tO :
where
P = (1,x,
y,x
2,
xy,
NU e -- pC-in
e
(11.34)
y2 , x 3 , x2y , xy2 , y3 , x3y , xy3)
/o4 +6xo7
+6xyo /
The form of the B is obtained directly from Eqs (11.20) and (11.25). We thus have
~7'W
--
Ol6
+2x
a5
+ 4 x a8
+ 6 y alo +
OL9
+4y
6xy alz
Ol9 -+- 6X2Olll
+ 6y2a12
We can write the above as / 2 V w = Q a = QC-1 fie = Bfi~ and thus B = QC-1
(11.35a)
in which
Q-
[ 0 0 0 2 0 0 6x 2y 0 0 6xy O] 0 0 0 0 0 2 0 0 2x 6y 0 6xy 0
0
0
0
2
0
0
4x
4y
0
6x 2
(11.35b)
6y 2
It is of interest to remark now that the displacement function chosen does in fact permit a state of constant strain (curvature) to exist and therefore satisfies one of the criteria of convergence stated in reference 7.* An explicit form of the shape function N was derived by Melosh 36 and can be written simply in terms of normalized coordinates. Thus, we can write for any node
brla(1 - ~72) --a~a(1-- )
~(1 + ~o)(1 + %)
(11.36)
with normalized coordinates defined as: = r/-
X -- X c
a
where
~0 " - ~ a
Y - Yc where 70 b
--
T~a
This form avoids the explicit inversion of C; however, for simplicity we pursue the direct use of polynomials to deduce the stiffness and load matrices. * If a7 to al2 are zero, then the 'strain' defined by second derivatives is constant. By Eq. (11.32), the corresponding fie can be found. As there is a unique correspondence between ~e and a such a state is therefore unique. All this presumes that C-1 does in fact exist. The algebraic inversion shows that the matrix C is never singular.
Rectangular element with corner nodes (12 degrees of freedom) 339
Stiffness and load matrices
Standard procedures can now be followed, and it is almost superfluous to recount the details. The stiffness matrix relating the nodal forces (given by lateral force and two moments at each node) to the corresponding nodal displacement is Ke -- I BTDB dx dy dr2
(11.37a)
e
or, substituting Eq. (11.35a) into this expression, K e -- C -T
(f_if_ a
QTDQ dx dy
a
)
(11.37b)
C -1
The terms not containing x and y have now been moved outside the integration operation. If D is constant the terms within the integration can be multiplied together and integrated explicitly without difficulty. The external forces at nodes arising from distributed loading can be computed from expression (11.23). The contribution of these forces to each of the nodes is
fa "--
{'-} fOxa
or, by Eq. (11.34), f _ _c-T The integral is again evaluated simply.
(11.38a)
NTaq dx dy
"--
f Oya
b
a
f_ fa b
a
pTq dx dy
(11.38b)
Example 11.1 Uniform load on 12 DOF rectangle
If we consider a rectangular element with the dimensions a and b shown in Fig. l 1.7 and subjected to uniform loading q, evaluation of Eq. (11.38a) (or Eq. [ 11.38b)] yields the element nodal load vector given by fl-
lqab
b -a
, f2--~qab
-b -a
, f3-
qab
b a
, f4---~qab
-b a
We note that the force associated with l/)a coincides with physical intuition as one-fourth the total load on the element. The moment type forces associated with the rotations is similar to that found for the Euler-Bernoulli beam in Example 10.1 of Sec. 10.4.1. If a physically 'lumped'-type loading is considered these moment forces are simply ignored. The vector of nodal plate forces due to initial strains and initial stresses can also be found in a similar way. It is necessary to remark in this connection that initial strains, such as may be due to a temperature rise, is seldom confined in its effects on curvatures. Usually, direct (in-plane) strains in the plate are introduced additionally, and the complete problem can be solved only by consideration of a plane stress (membrane) problem as well as that of bending.
340 Plate bending approximation: thin (Kirchhoff) plates and Q continuity requirements
The rectangular element developed in the preceding section passes the patch test 4~ and is always convergent. However, it cannot be easily generalized into a quadrilateral shape. Transformation of coordinates of the isoparametric type described in Chapter 2 can be performed but unfortunately now it will be found that the constant curvature criterion is violated. As expected, such elements behave badly but convergence may still occur providing the patch test is passed in the curvilinear coordinates. 7 Henshell e t al. 4~ studied the performance of such an element (and also some of a higher order) and concluded that reasonable accuracy is attainable. Their paper gives all the details of transformations required for an isoparametric mapping and the resulting need for numerical integration. Only for the case of a parallelogram is it possible to achieve states of constant curvature exclusively using functions of ~ and rl and for this case the patch test is satisfied. For a parallelogram the local coordinates can be related to the global ones by the explicit expression (Fig. 11.8) x - y cot c~ y csc ac~
(11.39)
and all expressions for the stiffness and loads can therefore also be derived directly. Such an element is suggested in the discussion in reference 26, and the stiffness matrices have been worked out by Dawe. 28 A somewhat different set of shape functions was suggested by Argyris. 29
~,, , , , ,~i,i!~!i, ~!i:~,ii,i~,~ii~,~~,~~~~~,,,~!i,:!,,,i~i,,,!:,ii,,,!i,,!,,!,,,!, ~i! i!!!ii iii!iiiii!ii!iiii l!i !iiili!iil
. . .!i.. ..!,. .,.,i!ii!, i! ii.... !ii?,ii ii.i...!!i.ii.i!! ....
At first sight, it would seem that once again a simple polynomial expansion could be used in a manner identical to that of the previous section. As only nine independent movements are imposed, only nine terms of the expansion are permissible. Here an immediate difficulty arises as the full cubic expansion contains 10 terms [Eq. (11.31) ~I~,y
///i "
L
2~
Fig. 11.8 Parallelogramelement and skew coordinates.
x,q
_1
Triangular element with corner nodes (9 degrees of freedom) 3
P (L1, L2, L3) L1 =
area P23 etc. area 123 "
fArea L~ L b L~ dA = (a + al b +b! cc!+ 2)!2A A = area 1 2 3
1
2
Fig. 11.9 Area coordinates.
with OLll - - OL12 --" 0 ] and any omission has to be made arbitrarily. To retain a certain symmetry of appearance all 10 terms could be retained and two coefficients made equal (for example, c~8 = c~9) to limit the number of unknowns to nine. Several such possibilities have been investigated but a further, much more serious, problem arises. The matrix corresponding to C of Eq. (11.32) becomes singular for certain orientations of the triangle sides. This happens, for instance, when two sides of the triangle are parallel to the x and y axes respectively. An 'obvious' alternative is to add a central node to the formulation and eliminate this by static condensation. This would allow a complete cubic to be used, but again it was found that an element derived on this basis does not converge to correct answers. Difficulties of asymmetry can be avoided by the use of area coordinates described in reference 7. These are indeed nearly always a natural choice for triangles, see (Fig. 11.9).
Shape functions
As before we shall use polynomial expansion terms, and it is worth remarking that these are given in area coordinates in an unusual form. For instance, OzlL1 --I-"0~2L2 --1-r gives the three terms of a
(11.40)
complete linear polynomial and
o~1L21+ 0~2L2 -k- OL3L2 + o~4L1L2 + o~sL2L3 "-F oL6L3L1
(11.41)
gives all six terms of a quadratic (containing within it the linear terms).* The 10 terms of a cubic expression are similarly formed by the products of all possible cubic combinations, that is,
L~, L32, L~, L21L2,L21L3,L2L3, L22L1,L2L1, L~L2, nln2n3
(11.42)
For a 9 degree-of-freedom element any of the above terms can be used in a suitable combination, remembering, however, that only nine independent functions are needed * However, it is also possible to write a completequadratic as c~lLl + c~2L2+ ~3L3 + oL4LIL2 + o~5L2L3+ c~6L3L1 and so on, for higher orders. This has the advantage of explicitly stating all retained terms of polynomialsof lower order.
341
342
Plate bending approximation: thin (Kirchhoff) plates and C~ continuity requirements and that constant curvature states have to be obtained. Figure 11.10 shows some functions that are of importance. The first [Fig. 11.10(a)] gives one of three functions representing a simple, unstrained rotation of the plate. Obviously, these must be available to produce the rigid body modes. Further, functions of the type L 2L2, of which there are six in the cubic expression, will be found to take up a form similar (though not identical) to Fig. 11.10(b). The cubic function L ~L2L3 is shown in Fig. 11.10(c), illustrating that this is a purely internal (bubble) mode with zero values and slopes at all three comer nodes (though slopes are not zero along edges). This function could thus be useful for a nodeless or internal variable but will not, in isolation, be used as it cannot be prescribed in terms of comer variables. It can, however, be added to any other basic shape in any proportion, as indicated in Fig. 11.10(b). The functions of the second kind are of special interest. They have zero values of w at all comers and indeed always have zero slope in the direction of one side. A linear combination of two of these (for example, L22Ll and L~L3) are capable of providing any desired slopes in the x and y directions at one node while maintaining all other nodal slopes at zero. For an element envisaged with 9 degrees of freedom we must ensure that all six quadratic terms are present. In addition we select three of the cubic terms. The
2 (a) I
(b)
1
(c)
(d)
3
1/2L1 L2 L3
"
3
1
Fig. 11.10 Some basic functions in area coordinate polynomials.
v
v
v
l2
Triangular element with corner nodes (9 degrees of freedom) quadratic terms ensure that a constant curvature, necessary for patch test satisfaction, is possible. Thus, the polynomials we consider are P = ILl, L2, L3, L1L2, L2L3, L3L,, L~L2, L~L3, L~L,] and we write the interpolation as w = Pc~
(11.43)
where c~ are parameters to be expressed in terms of nodal values. The nine nodal values are denoted as
[fVa, Ox~, 0y~]= fVa, --~Y a'
OX a ; a = 1 , 2 , 3
Upon noting that
o 0
[OL 1 "~
-~y
m[bl
01_,2 OL3
I-;/1 01.,2 ox OL3 ox
k-~y
where
Oy
Cl
o b2 c2
(11.44)
Oy
2A = blC2 - b2Cl; ba -- Yb -- Yc; Ca "-- Xc -- Xb
with a, b, c a cyclic permutation of indices, we now determine the shape function by a suitable inversion [see Sec. 11.3, Eq. (11.34)], and write for node a
N T --
I
3L2--2L3 I 1 L2(bjLk - bkLj) + ~(bj - bk)L1L2L3 1 -- Ck)L1L2L3 L2(cjLk - CkLj) + ~(cj
(11.45)
Here the term L1L2L3 is added to permit constant curvature states. The computation of stiffness and load matrices can again follow the standard patterns, and integration of expressions (11.22) and (11.23)can be done exactly using the general integrals given in Fig. 11.9. However, numerical quadrature is generally used and proves equally efficient (see Sec. 2.3). The stiffness matrix requires computation of second derivatives of shape functions and these may be conveniently obtained from - eYN.
I 02N.
02Na
-0-~-
OxOy
02Na c92Na 022
=
1
[bl
4A 2 Cl
b2 b3] c2
c3
~Na
c~Na -
OL 2
0L10L2
0L10L3
oaG
O:Na
0aNa
OL2OL1
OL 2
0/_,20/_,3
02No
02Na
~Na
OL3OL1
OL30L2
in which Na denotes any of the shape functions given in Eq. (11.45).
Cl
b2 c2 b3 c3 .
(11.46)
343
344
Plate bending approximation: thin (Kirchhoff) plates and C1 continuity requirements The element just derived is one first developed in reference 12. Although it satisfies the constant strain criterion (being able to produce constant curvature states) it unfortunately does not pass the patch test for arbitrary mesh configurations. Indeed, this was pointed out in the original reference (which also was the one in which the patch test was mentioned for the first time). However, the patch test is fully satisfied with this element for meshes of triangles created by three sets of equally spaced straight lines. In general, the performance of the element, despite this shortcoming, made the element quite popular in some practical applications. 38 It is possible to amend the element shape functions so that the resulting element passes the patch test in all configurations. An early approach was presented by Kikuchi and Ando 51 by replacing boundary integral terms in the virtual work statement of Eq. (11.18) by
~l"lext'- f ~ t o q d Q + ~ - ~
[Ji
On
O~)On Mn(w) dI"] (11.47)
e
+
+
dr +
s
dr n
in which Fe is the boundary of each element e, Mn (w) is the normal moment computed from second derivatives of the w interpolation, and s is the tangent direction along the element boundaries. The interpolations given by Eq. (11.45) are Co conforming and have slopes which match those of adjacent elements at nodes. To correct the slope incompatibility between nodes, a simple interpolation is introduced along each element boundary segment as On = ( 1 - s
-~x ny + -~y anX +
-~x bnY + -~y bn~
where s' is 0 at node a and 1 at node b, and n~ and ny are direction cosines of the ab side with respect to the x and y axes, respectively. The above modification requires boundary integrals in addition to the usual area integrals; however, the final result is one which passes the patch test. Bergen 44'46'47 and Samuelsson 48 also show a way of producing elements which pass the patch test, but a successful modification useful for general application with elastic and inelastic material behaviour is one derived by Specht. 49 This modification uses three fourth-order terms in place of the three cubic terms of the equation preceding Eq. (11.43). The particular form of these is so designed that the patch test criterion which we shall discuss in detail later in Sec. 11.7 is identically satisfied. We consider now the nine polynomial functions given by
P - ILl, L2, L3, L1L2, L2L3, L3L1, 1 {3(1 - #3)L1 L2L2 + 5L1L2L3 1 {3(1 -#1 )L 2 L2L3 + sL1L2L3 L~L1 + 5L1L2L31 {3(1 - #2)L3 -
( 1 + 3 # 3 ) L 2 + ( 1 + 3 # 3 ) L 3} (1 + 3# 1)L 3 + (1 + 3#l)L 1}
,
,
(11.49)
(1 + 3#2)L1 + (1 + 3#2)L2}]
where 12 - l~ ~a
--
1a2
(11.50)
Triangular element of the simplest form (6 degrees of freedom) 345 and la is the length of the triangle side opposite node a and a, b, c is a cyclic permutation.* The modified interpolation for w is taken as w = Pa
(11.51)
and, on identification of nodal values and inversion, the shape functions can be written explicitly in terms of the components of the vector P defined by Eq. (11.49) as N~ =
( ea - ea+3 + ec+3 -Jc"2 ( ea+6 - ec+6) } --bb (Pc+6 - Pc+3) - bcPa+6 --Cb (Pc+6 -- Pc+3) - ccPa+6
(11.52)
where a, b, c are the cyclic permutations of 1, 2, 3. Once again, stiffness and load matrices can be determined either explicitly or using numerical quadrature. The element derived above passes all the patch tests and performs excellently. 41 Indeed, if the quadrature is carried out in a 'reduced' manner using the three quadrature points and weights 7 Ll 1/2 1/2 0
L2
L3
W
1/2 0 1/3 0 1/2 ~/3 1/2 1/2 1/3
then the element is one of the best thin plate triangles with 9 degrees of freedom that is currently available, as we shall show in the section dealing with numerical comparisons.
If conformity at nodes (C1 continuity) is to be abandoned, it is possible to introduce even simpler elements than those already described by reducing the element interconnections. A very simple element of this type was first proposed by Morley. 3~ In this element, illustrated in Fig. 11.11, the interconnections require continuity of the displacement w at the triangle vertices and of normal slopes at the element mid-sides. With 6 degrees of freedom the expansion can be limited to quadratic terms alone, which one can write as w--[L1,
L2,
L3,
L,Le,
L2L3,
L3L1]oz
(11.53a)
Identification of nodal variables and inversion leads to the following shape functions: for comer nodes Na -- Z a - Z a ( l - Z a ) -
babc - CaCc babb - CaCb b 2 ..jr_c 2 t b (1 -- t b ) -b c2 -Ji- c c2 t c (1 - z c )
(11.53b) * The constants/z a are geometric parameters occurring in the expression for normal derivatives. Thus on side the normal derivative is given by
0 la [ ~ 0 '~-70 '~70 (0 On -- 4A ~ o +,.,~c --2~,,~a +tza OLc
0 )] OLb
la
346
Plate bending approximation: thin (Kirchhoff) plates and Q continuity requirements 1
w
r
2
/
4
+
Fig. 11.11 The simplest non-conforming triangle, from Morley,3~with 6 degrees of freedom. and for 'normal gradient' nodes -
N.+3 -
2A v/b a +
L,, (1 -
L,,)
(11.5
3c)
where the symbols are identical to those used in Eq. (11.44) and subscripts a, b, c are a cyclic permutation of 1, 2, 3. Establishment of stiffness and load matrices follows the standard pattem and we find that once again the element passes fully all the patch tests required. This simple element performs reasonably, as we shall show later, though its accuracy is, of course, less than that of the preceding ones. It is of interest to remark that the moment field described by the element satisfies exactly interelement equilibrium conditions on the normal moment Mn, as the reader can verify. Indeed, originally this element was derived as an equilibrating one using the complementary energy principle, 52 and for this reason it always gives an upper bound on the strain energy of flexure. This is the simplest possible element as it simply represents the minimum requirements of a constant moment field. An explicit form of stiffness routines for this element is given by Wood. 31
The patch test in its different forms is generally applied numerically to test the final form of an element. 7 However, the basic requirements for its satisfaction by shape functions that violate compatibility can be forecast accurately if certain conditions are satisfied in the choice of such functions. These conditions follow from the requirement that for constant strain states the virtual work done by internal forces acting at the discontinuity must be zero. Thus if the tractions acting on an element interface of a plate are (see Fig. 11.4)
Mn,
Mns
and
Sn
(11.54)
and if the corresponding mismatch of virtual displacements are m(/) n ~
m
,
A0~ ~ A
~
and
Aw
(11.55)
The patch test-an analytical requirement then ideally we would like the integral given below to be zero, as indicated, at least for the constant stress states:
fFFMnA~ndI"JrfF MnsA~)sdI"-'~SnA1/)dI" -J; : 0 e
e
(11.56)
e
The last term will always be zero identically for constant Mx, My, Mxy fields as then Sx = Sy = 0 [in the absence of applied couples, see Eq. (11.1 lb)] and we can ensure the satisfaction of the remaining conditions if fr A~bndF=0
and
fr A~bsdF=0
e
(11.57)
e
is satisfied for each straight side Fe of the element. For elements joining at vertices where Ow/On is prescribed, these integrals will be identically zero only if anti-symmetric cubic terms arise in the departure from linearity and a quadratic variation of normal gradients is absent, as shown in Fig. 11.12(a). This is the motivation for the rather special form of shape function basis chosen to describe the incompatible triangle in Eq. (11.49), and here the first condition of Eq. (11.57) is automatically satisfied. The satisfaction of the second condition of Eq. (11.57) is always ensured if the function w and its first derivatives are prescribed at the comer nodes. For the purely quadratic triangle of Sec. 11.6 the situation is even simpler. Here the gradients can only be linear, and if their value is prescribed at the element mid-side as shown in Fig. 11.11 (b) the integral is identically zero. The same arguments apparently fail when the rectangular element with the function basis given in Eq. (11.34) is examined. However, the reader can verify by direct algebra that the integrals of Eqs (11.57) are identically satisfied. Thus, for instance, a-~Y d x = O
when y = + b
and Ow/Oy is taken as zero at the two nodes (i.e. departure from prescribed linear variations only is considered).
Compatible base
Not permissible
I Quadratic I iv'~'~~...~
~
Linear
Cubic
(a)
(b)
Fig. 11.12 Continuity condition for satisfaction of patch test [f (/gw / an) d5 = 0]; variation of c3vv / an along side. (a) Definition by corner nodes (linear component compatible); (b) definition by one central node (constant component compatible).
347
348
Plate bending approximation: thin (Kirchhoff) plates and C1 continuity requirements
The remarks of this section are verified in numerical tests and lead to an intelligent, a priori, determination of conditions which make shape functions convergent for incompatible elements.
The various plate bending elements already derived- and those to be derived in subsequent sections - have been used to solve some classical plate bending problems. We first give two specific illustrations and then follow these with a general convergence study of elements discussed.
Example 11.2 Deflections and moments for clamped square plate
The solution of a clamped plate subjected to uniform loading q was a topic of considerable study during the early 1900s. 53,54An accurate numerical solution in series form was given by Hencky 55'56 and recently evaluated using the form given by Wojtaszak 57 by Taylor and Govindjee 58 to obtain correct solution values to several significant figures. Figure 11.13 shows the deflections and moments in a square plate clamped along its edges and solved by the use of the rectangular element derived in Sec. 11.3 and a uniform mesh. 26 Table 11.1 gives numerical results for a set of similar examples solved I
i
f ~1, 10
(wDIqL 4) x 103
" 1:],,.. "~"
'M:Y~ o ..if'~
i 1~" ~
,(3" ~ Deflections w
--5 D "-4
-_~
--2
~Ib
",
## #(
:
9
../,
I
-0
#~
- 2
(MxlqL 2) x 102 ""
-3
Bending moments
-,~~-~t'
o/
9
16 x 16 mesh finite difference solution (Southwell, 1946) 45 . . . . . . . o 6 x 6 division into finite elements ....---..-,a, 4 x 4 division into finite elements - - - O 2 x 2 division into finite elements Fig. 11.13 A square plate with clamped edges; uniform load q; square elements.
Numerical examples 349 Table 11.1
Computed central deflection of a square plate for several meshes (rectangular elements) 39 Simply supported plate Total number of nodes
Mesh 2 x 2 4 x 4 8 x 8 16 x 16 Series (Timoshenko)
9 25 81 169
Clamped plate
a*
/3t
a*
0.003446 0.003939 0.004033 0.004050 0.004062
0.013784 0.012327 0.011829 0.011715 0.01160
0.001480 0.001403 0.001304 0.001283 0.001265
0.005919 0.006134 0.005803 0.005710 0.00560
* Wmax --- aqL4/D for uniformly distributed load q. t Wmax-- flpL2/D for central concentrated load P.
Note: Subdivision of whole plate given for mesh. Table 11.2
Comer supported square plate Point 2
Point 1 Method
Mesh
w
Mx
w
Mx
Finite element
2• 4x4 6•
0.0126 0.0165 0.0173 0.0180 0.0170
0.139 0.149 0.150 0.154 0.140
0.0176 0.0232 0.0244 0.0281 0.0265
0.095 0.108 0.109 0.110 0.109
Marcus 59 Ballesteros and Lee Multiplier
qL4/D
qL 2
qL4/D
qL 2
Note: Point 1, centre of side; point 2, centre of plate.
with the same element, and Table 11.2 presents another square plate with more complex boundary conditions. 39 Exact results are available here and comparisons are made. 59'6~
Example 11.3 Skewed slab bridge
Figures 11.14 and 11.15 show practical engineering applications to more complex shapes of slab bridges. In both examples the requirements of geometry necessitate the use of a triangular element - with that of reference 12 being used here. Further, in both examples, beams reinforce the slab edges and these are simply incorporated in the analysis on the assumption of concentric behaviour.
Example 11.4 Comparison of convergence behaviour
In Fig. 11.16(a)-(d) we show the results of a convergence study for a square plate with simply supported and clamped edge conditions using various triangular and rectangular elements and two load types. This type of diagram is commonly used for assessing the behaviour of various elements, and we show on it the performance of the elements already described as well as others to which we shall refer to later. Table 11.3 gives the key to the various element 'codes' which include elements yet to be described. 63-66
Example 11.5 Energy convergence in a skew plate
The comparisons in Example 11.4 single out only one displacement and each plot uses the number of mesh divisions in a quarter of the plate as abscissa. It is therefore difficult to deduce the convergence rate and the performance of elements with multiple nodes. A more convenient plot gives the energy norm Ilul I, versus the number of degrees of freedom N on a logarithmic scale (see also reference 7). We show such a comparison Text continued
on p a g e
356
350
I,I
.;.
~,.,
I
J
/i,,v
,~.~ ~" ' / "
'
!x
.~
!
I >, ,+
.I
t I
I
-
t
~,
'1 E
I'-'%
J \
I
J
/,~ ~''!
~,"
~.
0
~ rr
~
,
"
;
I I
I
I
,
I
"~ c~
!,
I
'~
§ ! .f, j I ~ ! I',,'1.,( . .,~I - / ,
~,
.," L .
/, ~r .~,
<
'-IN1 ~ , , -
\.I,•
I
i
Plate bending approximation: thin (Kirchhoff) plates and C1 continuity requirements
o
o 9
I~
"13 r
lib
~<
r
IE
0
"-"
_=-"
o
c-"~"
-o
e o E
c~a
~
__9.0
Fig. 11.14 A skew, curved, bridge with beams and non-uniform thickness; plot of principal moments under dead load.
Numerical examples
3ft
3ft
~'1
I'"1
(a)
72o-20
. (,, \
3ft 9
3ft 8ft
- -.-.~ . . . . .
(b)
3ff
tOM,o,
~
~ ,l~ .O . Mxy-10
-40
-30 -40
(c)
Fig. 11.15 Castleton railway bridge: general geometry and details of finite element subdivision. (a) Typical actual section; (b) idealization and meshing; (c) moment components (ton ft -~) under uniform load of 150 Ib ft -2 with computer plot of contours.
351
352
/
/
/
/
d
I
~11
'
'.~
---'-
(%) ~M u! JoJJ3
I
0
(%) ~M u! JoJJ3
I
i
I,
I
R
tO
~
0
r
"4"
u~
T
O
T
0
r
c
E
("
.~-
-~
(D
c/)
m ~6 ~ rr m
1.-
0~
Plate bending approximation: thin (Kirchhoff) plates and C~ continuity requirements
O
N / 0rn f aO
u')
.m, EL
ch
UD
E
m
o Q..
"o cu
IE
o
-o
O"
Q-
---~
Fig. 11.16 (a) Simply supported uniformly loaded square plate.
0
p
,"
0 t
/
//" /
N_ 0
me ,, J
m
I
I ~I I
i C~
"
I
T
~
I
if)
Im
I-
I-
E
r" ~
rr
c~ 0
--~
-c; ,..
oo
(:u 4_,
L.) cO u
Q
..Q
t~
Q_ E
cz.
(3.) .i.-, t...
(3J
(1)
.i.-, (-~ I._. .l-, ,-
u
-,-, c-(3)
14)
r
CO
O)
0""
0
,,,
",.~,
i~ I i
'
U1
I!
~~_ 0 m
_L.:~
:._~:!.~,.~~ ....
~ L~ m
(%) o~ u! JoJJ3
i
=
II
'
I I
0
(%) o/. u! JoJJ3
u_
Numerical examples
Fig. 11.17 Cont. (b) simply supported square plate with concentrated central load.
353
354
h
9
El
I.q
f I
!
I i I
i
""
1~_ ~__~ , . . . .
I
u! JoJJ3
0
I
f 141
0 I
I
IZ)
~ ~.,~-i l'"~~, "
tl
(%) o•
,,//
O
/'---
141
0 I
T
0
r
"d"
E
.~
m
(/) r-
L
~)
Plate bending approximation: thin (Kirchhoff) plates and Q continuity requirements
"
O
(%) ~ u! Jo~3
E
u
u__
v
"0 Q_
0
>,
o
QJ
u'l
CL
u ro
Fig. 11.16 Cont. (c) clamped uniformly loaded square plate.
0
~
iI
if)
!
II
l
1
'
%
i
~9 i~ ..
_1~,~__. I
-
(%) ~,~ u! JoJ.~3
,,
gr %%
~"...! 0
(%) o,~ u! JoJJ3
0
T
(,O
O ,e'-
O'J
CO I~ (s
Ul
~
E:
rr v
u) ,.i..., e-
E
I-
u_
~D
Q
u
E
Q.
"o
c'-
"(:3 o) ,ca .i-, (.-
ca 4-,, ca~ L)
o
ca
-d
(3) (1,1 En ~D c-(1,) U '.-.q) CL
oL_
.c_
C: <]) L)
t13
"t:3
(:]J L)
E
.i--,
(]J q)
Fig. 11.16 Cont. (d) clamped square plate with concentrated central load. Percentage error in central displacement (seeTable 11.3 for key).
Numerical examples
355
356
Plate bending approximation: thin (Kirchhoff) plates and C1 continuity requirements Table 11.3 List of elements for comparison of performance in Fig. 11.16: (a) 9 degree-offreedom triangles; (b) 12 degree-of-freedom rectangles; (c) 16 degree-of-freedom rectangle Code
Reference
Symbol
(a) BCIZ 1
Bazeleyet
PAT BCIZ 2 (HCT) DKT
Specht49 Bazeleyet al. 12 Cloughand Tocher1~ Stricklin et al. 61 and Dhatt62
(b) ACM Q 19 DKQ (c) BF
Descriptionand comment
[3
Displacement, non-conforming (fails patch test) Displacement, non-conforming Displacement, conforming
o
Discrete Kirchhoff
Zienkiewicz and Cheung26 Clough and Felippa15 Batoz and Ben Tohar61
Z~ o F-q
Displacement, non-conforming Displacement, conforming Displacement, conforming
Bogner et al. 17
o
Displacement, conforming
al. 12
A
for s o m e e l e m e n t s in Fig. 11.17 for a p r o b l e m of a slightly skewed, simply supported plate. 9 It is of interest to observe that, owing to a singularity in the obtuse angle c o m e r s , both high- and low-order e l e m e n t s converge at almost identical rates (though, of course, the f o r m e r give better overall accuracy). 7 Different rates of c o n v e r g e n c e would, of course, be obtained if no singularity existed.
100 80
O,
40 LE o = 20
-.~C ,.r
".oB
e-.
.~_ 10 ,_ 8
"E~ O
O.
--~ 4
D~'eb ~
20
40 80100 200 400 Number of degrees of freedom
800 1000
A Fifth-order conforming triangle1923 B Low-order conforming element (p=2)1o, 11 C Hybrid82
Fig. 11.17 Rateof convergence in energy norm versus degree of freedom for three elements: the problem of a slightly skewed, simply supported plate (80~ with uniform mesh subdivision.9
Singular shape functions for the simple triangular element 357
Conforming shape functions with nodal singularities
It has already been demonstrated in Sec. 11.3 that it is impossible to devise a simple polynomial function with only three nodal degrees of freedom that will be able to satisfy slope continuity requirements at all locations along element boundaries. The alternative of imposing curvature parameters at nodes has the disadvantage, however, of imposing excessive conditions of continuity (although we will investigate some of the elements that have been proposed from this class). Furthermore, it is desirable from many points of view to limit the nodal variables to three quantities only. These, with simple physical interpretation, allow the generalization of plate elements to shells to be easily interpreted also. It is, however, possible to achieve C1 continuity by provision of additional shape functions for which, in general, second-order derivatives have non-unique values at nodes. Providing the patch test conditions are satisfied, convergence is again assured. Such shape functions will be discussed now in the context of triangular and quadrilateral elements. The simple rectangular shape will be omitted as it is a special case of the quadrilateral.
Consider for instance either of the following sets of functions: LaL~L2c(Lc - Lb)
(11.58)
8bc -- (La q- Lb)(Lb -Jr"Lc) or
LaL2L2(1 at- La) 8bc -- (La + Lb)(Lb + Lc)
(11.59)
in which once again a, b, c are a cyclic permutation of l, 2, 3. Both have the property for a = 1 that along two sides (1-2 and 1-3) of a triangle (Fig. 11.18) their values and the values of their normal slope are zero. On the third side (2-3) the function is zero but a normal slope exists. In both, its variation is parabolic. Now, all the functions used to define the non-conforming triangle [see Eq. (11.43)] were cubic and hence permit also a parabolic variation of the normal slope which is not uniquely defined by the two end nodal values (and hence resulted in non-conformity). However, if we specify as an additional variable the normal slope of w at a mid-point of each side then, by combining the new functions 8bc with the other functions previously given, a unique parabolic variation of normal slope along interelement faces is achieved and a compatible element results.
358
Plate bending approximation: thin (Kirchhoff) plates and C1 continuity requirements 3
1
t
LIL2L2(I+L1)
2
~.~
3
1
L,L2L~(L3-L2)
2
Fig. 11.18 Somesingulararea coordinatefunctions. Apparently, this can be achieved by adding three such additional degrees of freedom to expression (11.43) and proceeding as described above. This will result in an element shown in Fig. 11.19(a), which has six nodes, three comer ones as before and three additional ones at which only normal slope is specified. Such an element requires the definition of a node (or an alternative) to define the normal slope and also involves assembly of nodes with differing numbers of degrees of freedom. It is necessary to define a unique normal slope for the parameter associated with the mid-point of adjacent elements. One simple solution is to use the direction of increasing node number of the adjacent vertices to define a unique normal. Another alternative, which avoids the above difficulties, is to constrain the mid-side node degree of freedom. For instance, we can assume that the normal slope at the centre-point of a line is given as the average of the two slopes at the ends. This, after suitable transformation, results in a compatible element with exactly the same degrees of freedom as that described in previous sections [see Fig. 11.19(b)]. The algebra involved in the generation of suitable shape functions along the lines described here is quite extensive and will not be given fully. First, the normal slopes at the mid-sides are calculated from the basic element shape functions [Eq. (11.45)] as
Ewl I wljT
-'- Z u e
(11.60)
Similarly, the average values of the nodal slopes in directions normal to the sides are calculated from these functions:
a O la O pl]
4
"~-5
-~-
--zfie
The contribution of the e functions to these slopes is added in proportions of and is simply (as these give unit normal slope) "7-- [7,
72
%IT
(11.61) ebc
- - ")'a
(11.62)
Singular shape functions for the simple triangular element 359 3 6
5
1
2
(a)
(b)
Degrees of freedom
0
[]
(c)
(
W,
ay' ~)--x i
w,
O n ' ~n ~s i
( aw aw a2w a2w a2w) 0 w,-a--~, ay, ax---~,ay---~,c3xc~yi
n
(e)
(d) Fig. 11.19 Various conforming triangular elements.
On combining Eq. (11.45) and the last three relations we have ZU e "-- ZU e + "/
(1 1.63)
from which it immediately follows on finding "7 that tO -- N~ e -+- [E23, E31, ~12] ( Z -
Z)~e
(11.64)
in which N Oare the non-conforming shape functions defined in Eq. (11.45). Thus, new shape functions are now available from Eq. (11.64).
360 Plate bending approximation: thin (Kirchhoff) plates and C~ continuity requirements An alternative way of generating compatible triangles was developed by Clough and Tocher. 1~ As shown in Fig. 11.19(a) each element triangle is first divided into three parts based on an internal point p. For each abp triangle a complete cubic expansion is written involving 10 terms which may be expressed in terms of the displacement and slopes at each vertex and the mid-side slope along the ab edge. Matching the values at the vertices for the three subtriangles produces an element with 15 degrees of freedom: 12 conventional degrees of freedom at nodes 1, 2, 3 and p; and three normal slopes at nodes 4, 5, 6. Full C~ continuity in the interior of the element is achieved by constraining the three parameters at the p node to satisfy continuous normal slope at each internal mid-side. Thus, we achieve an element with 12 degrees of freedom similar to the one previously outlined using the singular shape functions. Constraining the normal slopes on the exterior mid-sides leads to an element with 9 degrees of freedom [see Fig. 11.19(b)]. These elements are achieved at the expense of providing non-unique values of second derivatives at the comers. We note, however, that strains are in general also non-unique in elements surrounding a node (e.g. constant strain triangles in elasticity have different strains in each element surrounding each node). In the previously developed shape functions ebc an infinite number of values to the second derivatives are obtained at each node depending on the direction the comer is approached. Indeed, the derivation of the Clough and Tocher triangle can be obtained by defining an altemative set of e functions, as has been shown in reference 12. As both types of elements lead to almost identical numerical results the preferable one is that leading to simplified computation. If numerical integration is used (as indeed is always strongly recommended for such elements) the form of functions continuously defined over the whole triangle as given by Eqs (11.45) and (11.64) is advantageous, although a fairly high order of numerical integration is necessary because of the singular nature of the functions.
An element that presents a considerable improvement over the type illustrated in Fig. 11.19(a) is shown in Fig. 11.19(c). Here, the 12 degrees of freedom are increased to 18 by considering both the values of w and its cross derivative cO2w/OsOn,in addition to the normal slope Ow/On, at element mid-sides.* Thus an equal number of degrees of freedom is presented at each node. Imposition of the continuity of cross derivatives at mid-sides does not involve any additional constraint as this indeed must be continuous in physical situations. The derivation of this element is given by Irons 14 and it will suffice here to say that in addition to the modes already discussed, fourth-order terms of the type illustrated in Fig. 11.10(d) and 'twist' functions of Fig. 11.18(b) are used. Indeed, it can be simply verified that the element contains all 15 terms of the quartic expansion in addition to the 'singularity' functions. * This is, in fact, identical to specifying both
Ow/On and Ow/Os at the
mid-side.
Compatible quadrilateral elements 361
Any of the previous triangles can be combined to produce 'composite' compatible quadrilateral elements with or without internal degrees of freedom. Three such quadrilaterals are illustrated in Fig. 11.20 and, in all, no mid-side nodes exist on the external boundaries. This avoids the difficulties of defining a unique parameter and of assembly already mentioned. In the first, no internal degrees of freedom are present and indeed no improvement on the comparable triangles is expected. In the following two, 3 and 7 internal degrees of freedom exist, respectively. Here, normal slope continuity imposed in the last one does not interfere with the assembly, as intemal degrees of freedom are in all cases eliminated by static condensation. 67 Much improved accuracy with these elements has been demonstrated by Clough and Felippa. 15 An alternative direct derivation of a quadrilateral element was proposed by Sander 68 and Fraeijs de Veubeke. 13,16This is along the following lines. Within a quadrilateral of Fig. 11.21(a) a complete cubic with 10 constants is taken, giving the first component of the displacement which is defined by three functions. Thus,
(a)
(b) Three internal degrees of freedom
(c) Seven internal degrees of freedom
Fig. 11.20 Some composite quadrilateral elements.
j
m
Y'
J
I wc
y lk
~'
Y
"
/\/
"- x"
)m
>J
k i
(a)
r
x
i
io-"--
(b)
Fig. 11.21 The compatible functions of Fraeijs de Veubeke.13,16
(c)
M
362
Plate bending approximation: thin (Kirchhoff) plates and C1 continuity requirements W -- W a-~- W b-~- tOc
(11.65)
W a -- O~1 -~- O~2X -~---.--~ CelOY3
The second function w b is defined in a piecewise manner. In the lower triangle of Fig. 11.21(b) it is taken as zero; in the upper triangle a cubic expression with three constants merges with slope discontinuity into the field of the lower triangle. Thus, in
jkm, tOb -- O~llY '2 + OZl2Y'3 + OZl3X'y '2
(11.66)
in terms of the locally specified coordinates x' and y'. Similarly, for the third function, Fig. 11.21 (c), w e = 0 in the lower triangle, and in i mj we define
to c -'-Ctl4Y 'r2 -k-Cel5Y "3 -if-Ctl6x"y
'r2
(11.67)
The 16 external degrees of freedom are provided by 12 usual comer variables and four normal mid-side slopes and allow the 16 constants c~ to O~16to be found by inversion. Compatibility is assured and once again non-unique second derivatives arise at comers. Again it is possible to constrain the mid-side nodes if desired and thus obtain a 12 degree-of-freedom element. The expansion can be found explicitly, as shown by Fraeijs de Veubeke, and a useful element generated. 16 The element described above cannot be formulated if a comer of the quadrilateral is re-entrant. This is not a serious limitation but needs to be considered on occasion if such an element degenerates to a near triangular shape.
ii!iiii~HH~ iii~i~i~iiii~' iii~~i;~i!iiii~~ii~i~'i~~iii'~~i!iii ' ~~i~~ii ~i~iiiii!iii~iiiiiiiiiiiiliiiiiiiiiiiiiiiiiiiiiiii~ iiiiii~ ~'3iiiiiiiiii!iiii~i~iiiiiiiiiiiiii~i~ ~~i'li ~iii~iiii~!iiii~~i~ii~ii!i?i~!i~i!ii~i~i~i~iiiii~iiiiii~iiii ~i ~ i ~ ~i ii ~ ii ~ i ili i ii iHii i i i ~iiii~iiiii~ii~iiiii~iii~!i!i!i!i!ii~!!~!ii!i~ii~i~iiiii!i~i!!~!!!iiiiii~iiii!i!i!!~i!~iii~i~i!i~i~iii~iii~i~ii!i!ii~iiii!i!iiii~ii~i~iii~i!ii!~i~i!!i~i~i~i~i~i!i!i~i~ii~iii~!iiii!iii~i!iiii~ii~i~i!i!!~!~iiiii~i!iii~i!i The performance of some of the conforming elements discussed in Secs 11.10-11.12 is shown in the comparison graphs of Fig. 11.16. It should be noted that although monotonic convergence in energy norm is now guaranteed, by subdividing each mesh to obtain the next one, the conforming triangular elements of references 11 and 12 perform almost identically but are considerably stiffer and hence less accurate than many of the non-conforming elements previously cited. To overcome this inaccuracy a quasi-conforming or smoothed element was derived by Razzaque and Irons. 33'34 For the derivation of this element substitute shape functions
are used. The substitute functions are cubic functions (in area coordinates) so designed as to approximate in a least-square sense the singular functions e and their derivatives used to enforce continuity [see Eqs (11.58)-(11.64)], as shown in Fig. 11.22. The algebra involved is complex but a full subprogram for stiffness computations is available in reference 33. It is noted that this element performs very similarly to the simpler, non-conforming element previously derived for the triangle. It is interesting to observe that here the non-conforming element is developed by choice and not to avoid difficulties. Its validity, however, is established by patch tests.
Hermitian rectangle shape function 363 3
Discontinuity
...e,~o~oO~ ~
b_2 ~ ei
~'~176176 f~~~~"~O'q.
Zero boundary slope
Li L2L2 (1 + L,) ei = (Li+ Lj) (Li+ Lk)
E/* = 1/6 Li (2 Li- 1) (L i- 1)
Fig. 11.22 Least-square substitute cubic shape function e* in place of rational function e for plate bending triangles.
Conforming shape functions with additional degrees of freedom
With the rectangular element of Fig. 11.7 the specification of OZw/OxOy as a nodal parameter is always permissible as it does not involve 'excessive continuity'. It is easy to show that for such an element polynomial shape functions giving compatibility can be easily determined. A polynomial expansion involving 16 constants [equal to the number of nodal parameters Wa, (Ow/OX)a, (Ow/Oy)a and (omw/OxOy)a] could, for instance, be written
364
Plate bending approximation: thin (Kirchhoff) plates and C1 continuity requirements retaining terms that do not produce a higher-order variation of w or its normal slope along the sides. Many alternatives will be present here and some may not produce invertible C matrices [see Eq. (11.33)]. An alternative derivation uses Hermitian polynomials used for shape functions of the Euler-Bemoulli beam (see Sec. 10.4.1) which permit the writing down of suitable functions directly. It is easy to verify that the following shape functions
Na-
[H~a~176
H~~
H~a~)(x)H(a~
H~al)(x)H~al)(y)] (11.68)
correspond to the values of //3,
Ow
Ow
Ox'
Oy' OxOy'
OZw
specified at the comer nodes, taking successively unit values at node a and zero at other nodes. An element based on these shape functions has been developed by Bogner et al. 17 and used with success. Indeed it is the most accurate rectangular element available as indicated by results in Fig. 11.16. A development of this type of element to include continuity of higher derivatives is simple and outlined in reference 18. In their undistorted form the above elements are, as for all rectangles, of very limited applicability.
If continuity of higher derivatives than first is accepted at nodes (thus imposing a certain constraint on non-homogeneous material and discontinuous thickness situations as explained in Sec. 11.2.4), the generation of slope and deflection compatible elements presents less difficulty. Considering as nodal degrees of freedom tO,
Ow Ox '
Ow Oy '
OZw Ox2'
OZw OxOy '
OZw Oy2'
a triangular element will involve at least 18 degrees of freedom. However, a complete fifth-order polynomial contains 21 terms. If, therefore, we add three normal slopes at the mid-side as additional degrees of freedom a sufficient number of equations appear to exist for which the shape functions can be found with a complete quintic polynomial. Along any edge we have six quantities determining the variation of w (displacement, slopes, and curvature at comer nodes), that is, specifying a fifth-order variation. Thus, this is uniquely defined and therefore w is continuous between elements. Similarly, Ow/On is prescribed by five quantities and varies as a fourth-order polynomial. Again this is as required by the slope continuity between elements. If we write the complete quintic polynomial as* l/3 - - OZ1 + OL2X " 1 - " " " "1- Ct21Y 5
(11.69)
* For this derivation use of simple Cartesian coordinates is recommended in preference to area coordinates. Symmetry is assured as the polynomial is complete.
The 21 and 18 degree-of-freedom triangle 365 proceed along the lines of the argument used to develop the rectangle in Sec. 11.3 and write llO a - -
Ol21Ya5
- - OL2 - - ~ - . . . . q - O~20Ya 4
0Xo
ow I 0ya
OX 2
OL 1 -~- OL 2 X a -~- . . . -~-
i
- - Ce3 - + - ' ' ' - + -
5Ce21Y 4
= 2ot4 -+--..-+- 2ot19y3
and so on, and finally obtain an expression ~e __ CCI~
(11.70)
in which C is a 21 x 21 matrix. The only apparent difficulty in the process that the reader may experience in forming this is that of the definition of the normal slopes at the mid-side nodes. However, if one notes that ~ (11.71) Ow Ow + sin ~ bOw Oy On = cos q50x in which 4) is the angle of a particular side to the x axis, the manner of formulation becomes simple. It is not easy to determine an explicit inverse of C, and the stiffness expressions, etc., are evaluated as in Eqs (11.22)-(11.25) by a numerical inversion. The existence of the mid-side nodes with their single degree of freedom is an inconvenience. It is possible, however, to constrain these by allowing only a cubic variation of the normal slope along each triangle side. Now, explicitly, the matrix C and the degrees of freedom can be reduced to 18, giving an element illustrated in Fig. 11.19(e) with three comer nodes and 6 degrees of freedom at each node. Both of these elements were described in several independently derived publications appearing during 1968 and 1969. The 21 degree-of-freedom element was described independently by Argyris et al., 23 Bell, 19 B o s s h a r d , 22 and Visser, 24 listing the authors alphabetically. The reduced 18 degree-of-freedom version was developed by Argyris et al., 23 Bell, 19 Cowper et al., 21 and Irons. 14 An essentially similar, but more complicated, formulation has been developed by Butlin and Ford, 2~and mention of the element shape functions was made earlier by Withum 69 and Felippa. 7~ It is clear that many more elements of this type could be developed and indeed some are suggested in the above references. A very inclusive study is found in the work of Zenisek, 71 Peano, 72 and others. 73-75 However, it should always be borne in mind that all the elements discussed in this section involve an inconsistency when discontinuous variation of material properties occurs. Further, the existence of higherorder derivatives makes it more difficult to impose boundary conditions and indeed the simple interpretation of energy conjugates as 'nodal forces' is more complex. Thus, the engineer may still feel a justified preference for the more intuitive formulation
366
Plate bending approximation: thin (Kirchhoff) plates and C1 continuity requirements involving displacements and slopes only, despite the fact that very good accuracy is demonstrated in the references cited for the quartic and quintic elements.
Avoidance of continuity difficulties in mixed and constrained elements
Equations (11.10a)-(11.11 b) of this chapter provide for many possibilities to approximate both thick and thin plates by using mixed (i.e. reducible) forms. In these, more than one set of variables is approximated directly, and generally continuity requirements for such approximations can be of either C I or Co type. The options open are large and indeed so is the number of publications proposing various alternatives. We shall therefore limit the discussion to those that appear most useful. To avoid constant reference to the beginning of this chapter, the four governing equations (11.10a)-(11.1 l b) are rewritten below in their abbreviated form with dependent variable sets M, q~, S, and w" M - Ds
= 0
1
-S-4,-Vw =0 a Z~TM + S = 0
(11.72)
v T s q- q - - 0 in which a = t~ Gt. To these, of course, the appropriate boundary conditions can be added. For details of the operators, etc., the fuller forms previously quoted need to be consulted. Mixed forms that utilize direct approximations to all the four variables are not common. The most obvious set arises from elimination of the moments M, that is ~TI)~D
"-[- S -- 0
1 -S - ~b- Vw O~
= 0
(11.73)
v T s -[- q = 0 and is the basis of a formulation directly related to the three-dimensional elasticity consideration. This is so important that we shall devote Chapter 12 entirely to it, and, of course, there it can be used for both thick and thin plates. We shall, however, return to one of its derivations in Sec. 11.18. One of the earliest mixed approaches leaves the variables M and w to be approximated and eliminates S and ~b. The form given is restricted to thin plates and thus a = o~ is taken. We now can write for the first two of Eqs (11.72), D-1M + Z~Vw = 0
(11.74)
Mixed formulations- general remarks and for the last two of Eqs
(11.72), V TETM - q = 0
(11.75)
The approximation can now be made directly putting M
-
NM]~/I
and
w = Nw#
(11.76)
where M and ~ list the nodal (or other) parameters of the expansions, and NM and Nw are appropriate shape functions. The approximation equations can, as is well known, be made either via a suitable variational principle or directly in a weighted residual, Galerkin form, both leading to identical results. We choose here the latter, although the first presentations of this approximation by H e r r m a n n 76 and o t h e r s 52'77-84 all use the Hellinger-Reissner principle. A weak form from which the plate approximation may be deduced is given by
~51"I= f rM (D-1M + ff.,Vw) d~+ f ~Sw (VT ff.,TM - q) d~+~5I-lbt = O ( 1 1 . 7 7 ) where ~nbt describes appropriate boundary condition terms. weighting approximations 6M = NMrl~I
and
Using the Galerkin
5w = NwSff
(11.78)
gives on integration by parts the following equation set (11.79) where
A= ~
NTD-1NM dr2
fl =
(VNw) T
l~ns
dF
(11.80)
t
C = ~ (ENM)TX7Nw dr2
=
Nwq dr2 +
NwS~dF t
where /~n and t~ns are the prescribed boundary moments, and Sn is the prescribed boundary shear force. Immediately, it is evident that only Co continuity is required for both M and w interpolation,* and many forms of elements are therefore applicable. Of course, appropriate patch tests for the mixed formulation must be enforced 43 and this requires a necessary condition that
nm >__nw
(11.81)
where nm stands for the number of parameters describing the moment field and nw the number in the displacement field. * It should be observed that, if Co continuity to the whole M field is taken, excessive continuity will arise and it is usual to ensure the continuity of Mn and Mnsat interfaces only.
367
368
Plate bending approximation: thin (Kirchhoff) plates and C1 continuity requirements Many excellent elements have been developed by using this type of approximation, though their application is limited because of the difficulty of interconnection with other structures as well as the fact that the coefficient matrix in Eq. (11.79) is indefinite with many zero diagonal terms. Indeed, a similar fate is encountered in numerous 'equilibrium element' forms in which the moment (stress) field is chosen a priori in a manner satisfying Eq. (11.75). Here the research of Fraeijs de Veubeke 83 and others 3~ has to be noted. It must, however, be observed that the second of these elements 3~ is in fact identical to the mixed element developed by Herrmann 77 and Hellan 85 (see also reference 52).
ii......... il!iiii!iiiiiiiiiiiiii !::,!,~ i!ii!ilili~i~i~iiiiiiiii!i ~,,~!~ ii~i~iliiiii!iii!iiiil ~!i~i~iiiiiiii~~i~i~i~i~i~iiliiiii !!!,,,:H~,~!:~,,:!:p~,~,:!:, !iilili!iiii~i~iiiilii~ii~~il!iiliiiiiii!iii!i~~,~,il~iiiiiiiiiiiiiiiiiiT iiil!i!iil~~iiiiiliiii!iiii!iiiiif!i!iii~'~'~iii!ii!!iiliiiiiiiii!iiiiiiiiii!i!i!ii!i iiiiiii!i!iiii e iiililiill ! ~'~'~'iiililiiiiiiiiii , iiiiiii ~ iiiiiiiil t iiifill~i!iiiil!iiilliiii!iiiiiiiliiiii!ii~ii!ii!iiiii!i!ii!iiii........................ iiiiiiiili!ii!iii!iiiiiiiiiiiiiliii!iiliiii............................................................................................. i iiiiii!iiiililiiliiii!iiliiii!iii!iiliiiiiiiiiiiiiiili!ii!iliiiiiiiiliiiliiiiiiiliiiiiiiiiiiili!iliiiiiiliil!iiliiii!iiiiiiiiiiii !i!.!il. .ii.iiiii . .iiiliil . .ii.ii!iiii . .i!i!iil . .!i.ii!ii . .iiii.iilii . .iiiiiil . .il.iiiiiii . .il.iilil . .i!iii!iii . . .iiiilii . .il.ii!il.iil.!i............. .i.ili.liliiiiiiii....................................... iil~!iiii!iiii!iiliiiiiiiiiiiiiii!i~!i!i!~iiiiliiiiiiiliiiiilii!i!i!i ,..... ii~iiiliiiiiiiii!iil!!i,!ii~~iil~ii~ii~ii~
Hybrid elements are essentially mixed elements in which the field inside the element is defined by one set of parameters and the one on the element frame by another, as shown in Fig. 11.23. The latter are generally chosen to be of a type identical to other displacement models and thus can be readily incorporated in a general program and indeed used in conjunction with the standard displacement types we have already discussed. The internal parameters can be readily eliminated (being confined to a single element) and thus the difference from displacement forms are confined to the element subprogram. The original concept is attributable to Pian 86'87 who pioneered this approach, and today many variants of the procedures exist in the context of thin plate theory. 65,88-97 In the majority of approximations, an equilibrating stress field is assumed to be given by a number of suitable shape functions and unknown parameters. In others, a mixed stress field is taken in the interior. A more refined procedure, introduced by Jirousek, 65'97 assumes in the interior a series solution exactly satisfying all the differential equations involved for a homogeneous field. w and ~)wt;~nd e f i ~ on frame by usual connection
/
Interior field defined by independent parameters
Fig. 11.23 Hybridelements.
Singularity (crack)
Discrete Kirchhoff constraints 369
All procedures use a suitable linking of the interior parameters with those defined on the boundary by the 'frame parameters'. The procedures for doing this are described in Chapter 12 of reference 7 in the context of elasticity equations, and only a small change of variables is needed to adapt these to the present case. We leave this extension to the reader who can also consult appropriate references for details. Some remarks need to be made in the context of hybrid elements. Remark: The first is that the number of internal parameters, n~, must be at least as large as the number of frame parameters, nF, which describe the displacements, less the number of rigid body modes if singularity of the final (stiffness) matrix is to be avoided. Thus, we require that
n~ > n F - 3
(11.82)
for plates. Remark: The second remark is a simple statement that it is possible, but counterproductive, to introduce an excessive number of internal parameters that simply give a more exact solution to a 'wrong' problem in which the frame is constraining the interior of an element. Thus additional accuracy is not achieved overall. Remark: Most of the formulations are available for non-homogeneous plates (and hence non-linear problems). However, this is not true for the Trefftz-hybrid elements 65'97where an exact solution to the differential equation needs to be available for the element interior. Such solutions are not known for arbitrary non-homogeneous interiors and hence the procedure fails. However, for homogeneous problems the elements can be made much more accurate than any of the others and indeed allow a general polygonal element with singularities and/or internal boundaries to be developed by the use of special functions (see Fig. 11.23). Obviously, this advantage needs to be bome in mind.
A number of elements matching (or duplicating) the displacement method have been developed and the performance of some of the simpler ones is shown in Fig. 11.16. Indeed, it can be shown that many hybrid-type elements duplicate precisely the various incompatible elements that pass the convergence requirement. Thus, it is interesting to note that the triangle of Allman 96 gives precisely the same results as the 'smoothed' Razzaque element of references 33 and 34 or, indeed, the element of Sec. 11.5.
Another procedure for achieving excellent element performance is achieved as a constrained (mixed) element. Here it is convenient (though by no means essential) to use a variational principle to describe the first and third of Eqs (11.73). This can be written simply as the minimization of the functional n =
1/sTls o /wq +Hbt nimum
( z : o ) q ) ( z : o ) dr2 + ~
a
(11.83)
370
Plate bending approximation: thin (Kirchhoff) plates and C~ continuity requirements subject to the constraint that the second of Eqs (11.73) be satisfied, that is, 1
-S - O- Vw = 0
(11.84)
o~
We shall use this form for general thick plates in Chapter 12, but in the case of thin plates with which this chapter is concerned, we can specialize by putting c~ = cx~ and rewrite the above as
'L
I1 -- ~
(ff,,0)TD(s
dr2-
L
wq dr2 +
1-Ibt =
minimum
(11.85)
subject to 0 + ~Tw = 0
(11.86)
and we note that the explicit mention of shear forces S is no longer necessary. To solve the problem posed by Eqs (11.85) and (11.86) we can 1. approximate w and 0 by independent interpolations of Co continuity as w = Nw~r
and
0 = N00
(11.87)
2. impose a discrete approximation to the constraint of Eq. (11.86) and solve the minimization problem resulting from substitution of Eq. (11.87) into Eq. (11.85) by either discrete elimination, use of suitable Lagrangian multipliers, or penalty procedures. In the application of the so-called discrete Kirchhoff constraints, Eq. (11.86) is approximated by point (or subdomain) collocation and direct elimination is used to reduce the number of nodal parameters. Of course, the other means of imposing the constraints could be used with identical effect and we shall return to these in the next chapter. However, direct elimination is advantageous in reducing the final total number of variables and can be used effectively. The procedure for constructing the discrete Kirchhoff relations was presented in Sec. 10.5.3 and applied to beams. For two-dimensional plate elements the situation is a little more complex, but if we imagine x to coincide with the direction tangent to an element side, precisely identical elimination to that presented in Sec. 10.5.3 enforces complete compatibility along an element side when both gradients of w are specified at the ends. However, with discrete imposition of the constraints it is not clear a priori that convergence will always o c c u r - though, of course, one can argue heuristically that collocation applied in numerous directions should result in an acceptable element. Indeed, patch tests turn out to be satisfied by most elements in which the w interpolation (and hence the Ow/Os interpolation) have Co continuity. The constraints frequently applied in practice involve the use of line or subdomain collocation to increase their number (which must, of course, always be less than the
Rotation-free elements
number of remaining variables) and such additional constraint equations as Ir ~
+ ~bs e
lax--
L(~ L(~
+ ~x
e
I~2y =--
e
-1- ~y
ds = 0
) )
d~ = O
(11.88)
dr2 = 0
are frequently used. The algebra involved in the elimination is not always easy and the reader is referred to original references for details pertaining to each particular element. The concept of discrete Kirchhoff constraints was first introduced by Wempner et al., 98 Stricklin et al., 61 and Dhatt 62 in 1968-69, but it has been applied extensively since.
99-110
In particular, the 9 degree-of-freedom triangle 99'100 and the complex semi-loof element of Irons 102 are elements which have been successfully used. Figure 11.24 illustrates some of the possible types of quadrilateral elements achieved in these references. iliii!ii!iii! iiiiiiiiiiiii~,7,~~i::~'~::::i~i~i~!iii~ ~~:'~~':':~'~'!i,7ii~ii'~,!~~'~'~'::i!i!i7 ~~:~::'::':::4i!',iil','!,:,!iiiii!~,~~~':~:::::~iiii i iiiiii~il~7iiiii~,~:~:iiiiiiliiii!!i~,i!i~,i',i~:~i~ii~i :i iiiii~~i777 ~iiiiiilii~iii!i!i',iiF'~:~:i!i!!iiiiiii',!iiil ~,!ii',iii!!i!i !iliTi',iiililiii!iiii~iiii~,:i~!!iiii":"i !i ~iiiii'~i,~,i~iiili!'~'~77, ii iili~iii~:~i!ii'!,iiiiiiii',iii'~i',ii7ii'~,~:ill~i~iiii::~i,iiil:i~,ilii!i!iiiiiiiiiiiiiiiiiiiiiiii!i!iiiiiiiiiiiiiiiiiTii!iiii!',!iiiiiiiiiiiiiii!iilii7iiiiii!i!i!i! ii':i!iiiiii iiiii7i!ii'i,7iiiiiiiiiiiiiiiiii!iiiii!i!iiiiiiiiiiiiiiili iii iliiili!iliiii~,ii'i, ~!i!:, iiliiil7iiiiiiiiiiTiiT, iTii!i!!ii ii!iiiiiiii!ii 7iiiTiiiiiT!i!iiiii iiiiiiiiiiiii!7~i!ii!iliiii7iiii~,iii'~,,i'i,Ti7i~i ',i'~,i,'i!i'i!,7i:,iliiiii' ~ i,!iiii~ii,7~i~,1!ii!i7,',ii~i'1 , iiiiii~iiiiiiiiii7~iii~iiiiiiiiiiiiiiiiiii!7iiiiiiii~7iiii!ii~iiiiiii~7i~i!ii~iiiii~i~i~ii~i~ii7i7i~i~i:~iii~i7i~ii!i:~i~7~!i~i~!~ii!iiii!iiii~i~i~i~iiiiiiiii77iii~7iiiiiiiiiii~7i~ii~i~i~i~iiiiiiiiiii~i~i77i~iiiiiiiiiiii
It is possible to construct elements for thin plates in terms of transverse displacement parameters alone. Nay and Utku used quadratic displacement approximation and minimum potential energy to construct a least-square fit for an element configuration shown in Fig. 11.25(a). lll The element is non-conforming but passes the patch test and therefore is an admissible form. An alternative, mixed field, construction is given by Ofiate and Zfirate for a composite element constructed from linear interpolation on each triangle, ll2'll3 In this work a mixed variational principle is used together with a special approximation for the curvature. We summarize here the steps in the better approach. A three-field mixed variational form for a thin plate problem based on the Hu-Washizu functional may be written as
'/a
1-I -- -~
x T D x dA -
/a
[ ( E V ) w - X] dA - j a wq dA + l"Ibt
(11.89)
where now X and M are mixed variables to be approximated, (E~7)w are again second derivatives of displacement w given in Eq. (11.13) and integration is over the area of the plate middle surface. Variation of Eq. (11.89) with respect to X gives the discrete constitutive equation 31-Ix
/ dA
~XT [Dx - M] dA -- 0
(11.90)
e
where Ae is the domain of the patch for the element. Two alternatives for Ae are considered in reference 112 and named BPT and BPN as shown in Figs 11.25(a) and 11.25(b), respectively. For the BPT form the integration is taken over the area of the element 'abc' with area Ap and boundary Fp. For the type BPN integration is over
371
372
Plate bending approximation: thin (Kirchhoff) plates and C1 continuity requirements Virgin 0
Constrained "I
'
x2
x2
x2
x2
(~ .
,, o 24 DOF
(a) "
o
1 1
""
I
1 1
(b)
[~~
1 0 . 25 D O F
"
;" 1
c
1 .
.
J ' 16 D O F
~ 11
1
E:~
1 o.,
(c)
fl .
.
--.
;." 1
11 "
irons 102
t 16 DOF
[] 11
..
j'3
""
27 DOF
'i
(
' Lyons 103
. . . . . . '
c
'1
irons 102
"
c
16 DOF
Lyons 1~
~3 (d)
23 DOF xr 11
~
11
~
(e)
[] t
I
I
~,, 11•
11
27 DOF DOF o
12 DOF
~I
~
Degrees of freedom Nodal DOF [w, 0x, 0y] Nodal DOF [w] Nodal DOF [w, On] Nodal [On] Nodal DOF [On, 0s]
[]
I Irons 102 (semi-Ioof)
I
o
16 DOF
I
1 point constraint x 1, etc. 3 integral constraints .1"3,etc.
Fig. 11.24 A series of discrete Kirchhoff theory (DKT)-type elements of quadrilateral type.
the more complex area Aa with boundary Fa. Each, however, is simple to construct. Similarly, variation of Eq. (11.89) with respect to moment gives the discrete curvature relation ~l-IM -- / dA
~M T [(~7)w e
- ~i~] dA - 0
(11.91)
Rotation-free elements e
e
b
d (a)
(b)
Fig. 11.25 Elementsfor rotation-free thin plates: (a) patch for Nay and Utku procedureTM BPT triangle; and (b) patch for BPN triangle. 112
Finally, the equilibrium equations are obtained from the variation with respect to the displacement, and are expressed as
6nw - f
[(/2V)6w]TMdA-fa
6 w q d A + 6rlbt - O
e
(11.92)
e
A finite element approximation may be constructed in the standard manner by writing M--
NmI~/Ia,
x-
NXa~(.a and
ll) -- NW'wa
(11.93)
The simplest approximations are for N a = NaX = 1 and linear interpolation over each triangle for N~~ Equation (11.90) is easily evaluated; however, the other two integrals have apparent difficulty since a linear interpolation yields zero derivatives within each triangle. Indeed the curvature is now concentrated in the 'kinks' which occur between contiguous triangles. To obtain discrete approximations to the curvature changes an integration by parts (i.e. application of Green's theorem) is used to rewrite Eq. (11.91) as
fA ~ T ~ u dA + f r 6MTgVw dF -- 0 e
e
where g
0]
ny
-
Lny
(11.94)
(11 95)
nx
is a matrix of the direction cosines for an outward pointing normal vector n to the boundary F e and V w --
'
W,y
(11.96)
In these expressions l-'e is the part of the boundary within the area of integration Ae. Thus, for the element type BPT it is just the contour 1-'p as shown in Fig. 11.25(a).
373
374
Plate bending approximation: thin (Kirchhoff) plates and C~ continuity requirements For the element type BPN no slope discontinuity occurs on the boundary F a shown in Fig. 11.25(b); however, it is necessary to integrate along the half sides of each triangle within the patch bounded by Fa. The remainder of the derivation is now straightforward and the reader is referred to reference 112 for additional details and results. In this paper results are also presented for thin shells. We note that the type of element discussed in this section is quite different from those presented previously in that nodes exist outside the boundary of the element. Thus, the definition of an element and the assembly process are somewhat different. In addition, boundary conditions need some special treatments to include in a general manner. 112 Because of these differences we do not consider additional members in this family. We do note, however, that for explicit dynamic programs some advantages occur since no rotation parameters need be integrated. Results for thin shells subjected to impulsive loading are particularly noteworthy. 112
iiiiiiiii!!i~i!ii~ii~ii!ii~iiiiii~iiiiiiii~iiiii~i~iiiiiiiiiiii~iii~iiiii~iiiiiiii!~iiii~i~iii~iii~iiiiiiiii!i~iii~iiiiiiiiiiiii!i!ii~i~iiii~i~i~iiiii~ii~ii~i~iiiiii~iiiiii~iiiiii~%iiii~iiii~iiiiiii~iiiiiii~iiiiiiii~iiiiii~ii~i~ ii~ii!ii!~ i!~i~iiii!iii!i ili !~iii!i2Q ~iiiii!!i~ i ~li~ ~|~ C~!~~ i~~i~it~!ii~iih~h~| O UiF i~!!!~!~~iiiiiiii~~i~!iiiiiiii ii~iiii~iii~ii~iiiiii iiiiii~~i~i!~liiiii!ii~i~!i~iiii!iiiii~iii!iiii!i~i~i~i~i~i~iiiiiiii The preceding discussion has assumed the plate to be a linear elastic material. In many situations it is necessary to consider a more general constitutive behaviour in order to represent the physical problem correctly. For thin plates, only the bending and twisting moment are associated with deformations and are related to the local stresses through
M =
My Mxy
=
O-y z dz a -t/2
(11.97)
Txy
Any of the material models discussed in Chapter 4 which have symmetric stress behaviour with respect to strains may be used in plate analysis provided an appropriate plane stress form is available (either analytically or through iterative reduction of the three-dimensional equations). The symmetry is necessary to avoid the generation of in-plane force resultants - which are assumed to decouple from the bending behaviour. If such conditions do not exist it is necessary to use a shell formulation as described in Chapters 13-16. In practice two approaches are considered- one dealing with the individual lamina using local stress components ~rx, O'y a n d Txy and the other using plate resultant forces Mx, My and Mxy directly.
Numerical integration through thickness
The most direct approach is to use a plane stress form of the stress-strain relation and perform the through-thickness integration numerically. In order to capture the maximum stresses at the top and bottom of the plate it is best to use Gauss-Lobattotype quadrature formulae TM where integrals are approximated by 1
J[ f(()d~
1
N-1 f ( - 1 ) W 1 + Z f ( ~ n ) W ~ + f(1)WN
n=2
(11.98)
These formula differ from the typical Gaussian quadrature considered previously and use the end points on the interval directly. This allows computation of first yield to be
Inelastic material behaviour Table 11.4 Gauss-Lobatto quadrature points and weights
N
~,,
3
4-1.0
1/3
0.0
4/3
4 5
W.
4-1.0
1/6
4- 0q/0-~.2
5/6
4-1.0
0.1
4-~
49/90 0.0
6
64/90
4-1.0
1/15
q-~//~
0.6/[tl (1 - t o ) 2]
•
0.6/[t2 ( 1 - t o ) z]
to = V/ff
tl = (7 + 2to)/21 t2 = ( 7 - 2to)/21
more accurate. The values of the parameters ~n and Wn are given in Table 11.4 up to the six-point formula. Parameters for higher-order formulae may be found in reference 114. Noting that the strain components in plates [see Eq. (11.8a)] are asymmetric with respect to the middle surface of the plate and that the z coordinate is also asymmetric we can compute the plate resultants by evaluating only half the integral. Accordingly, we may use M =
My xy
= 2
Oy ,!o
z dz
(11.99)
Txy
and here a six-point formula or less will generally be sufficient to compute integrals. Equation (11.21) is replaced by the non-linear equation given as ~ ( w ) ----f -- L BTM dff2 = 0
(11.100)
The solution process (for a static case) may now proceed by using, for instance, a Newton scheme in which the t a n g e n t m o d u l i for the plate are obtained by using the tangent moduli for the stress components as dM =
dMy dMxy
=
2
dz
12Vdw
-
DT(LV)dw
(11.101)
ao
where D~ps)(z) is the tangent modulus matrix of a plane stress material model at each lamina level z, and DT is the resulting bending tangent stiffness matrix of the plate. The Newton iteration for the displacement increment is computed as K(Tk)d~(k) = ffj(k) with iterative updates
~Tv(k+1) _- ~TV(k) ~-
d~r (k)
(11.102) (11.103)
375
376
Plate bending approximation: thin (Kirchhoff) plates and C~ continuity requirements until a suitable convergence criterion is satisfied. previously defined for solids.
This follows precisely methods
Resultant constitutive models
A resultant yield function for plates with Huber-von Mises-type material is given by ~15 F(M)-
( M 2 + M 2 - M~M,, + 3M~,) - M2(tQ _< 0
(11.104)
where ~ is an 'isotropic' hardening parameter and Mu denotes a uniaxial yield moment and which for homogeneous plates is generally given by 1 Mu -- ~t2Cry(~)
(11.105)
in which or,, is the material uniaxial yield stress in tension (and compression). We observe that, in the absence of hardening, Mu is the moment that exists when the entire cross-section is at a yield stress. . . . . . . . . .].::,. .~.,~. .~:~U ~::~:~!:' 9 ]:"~::~...................... ~ i:~'i~:~N !iiiiiii!iii!i~ n iii!i!i ~i ii'::'~li'l~ '::iiii!ii !ili!ii::':'~ION :i~iiiiTi!!iiiiiiiiiiiiiiii!iiiiiiiiii!iiiii!iiiiiiiiiiiiiiii!iiii":' V ~ m ~i rk~ iii',:ii:i i ii::!iiiiii!',i',ii!!'wi!iiii!!iii ,!i,'!~ii!i!:'i~~ii',,i!i:,iilf~ ~i ! ,,i!ii,:iii!ii'~'~'iiii!ii~iii !iil!i!iiii ~i|~ iiiiiiiiilli~,:i~':'~'iiiiiiiiii:iiiii:i:i:i!i:!ii!iiiii!: !:i:!i!'i~,i:iiiii:iiii' : ~!iiii::~:!: ,~': i:'~'~i',:ii!:i:i~i!:!ii:ii :i:i!i:,!:ii:i:ii!i :iiii!ii!iiiiii:i: i:i'~i:ilii ii~i~ii:i:i:ii:i~ii~ii~i~,!:i'i~i~!:ii'!i~iiii~iii',iiiii!ii'~'ii'!:~i!ii!:~, :,i i!iiiiiiil ...... ::~,~i,, : i!:iiiiiii:::::::u,::i 7:!:!:!:ii::i iii i!'.......iili'.........~.........iiiil/-::~i..............iii~i 'i :i ::/~!:iiii~::il/:i,-~::: .....'i~ ?:!:',:!i'..... : 'iiiii'~i',u:,i ii:~: '::m:e i:R~~ ,~i ~:ii~i~iiii~ii~i~i~iii!iiiiiiii~iiiiiiiiiiiiii!iiii~iiii!:~!!~iiiii~:~i~:~::i::~:ii~:~i:~i!ii~:i~:i:~:::~i~:!~:~i~iiiii::ii!ii:~i~i~:i::i!
The extensive bibliography of this chapter outlining the numerous approaches capable of solving the problems of thin, Kirchhoff, plate flexure shows both the importance of the subject in structural engineering - particularly as a preliminary to shell analysis and the wide variety of possible approaches. Indeed, only part of the story is outlined here, as the next chapter, dealing with thick plate formulation, presents many practical alternatives of dealing with the same problem. We hope that the presentation, in addition to providing a guide to a particular problem area, is useful in its direct extension to other fields where governing equations lead to C1 continuity requirements. Users of practical computer programs will be faced with a problem of 'which element' is to be used to satisfy their needs. We have listed in Table 11.3 some of the more widely known simple elements and compared their performance in Fig. 11.16. The choice is not always unique, and much more will depend on preferences and indeed extensions desired. As will be seen in Chapter 13 for general shell problems, triangular elements are an optimal choice for many applications and configurations. Further, such elements are most easily incorporated if adaptive mesh generation is to be used for achieving errors of predetermined magnitude.
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1. G. Kirchhoff. Ober das Gleichqewicht und die Bewegung einer elastichen Scheibe. J. Reine und Angewandte Mathematik, 40:51-88, 1850. 2. S.P. Timoshenko and S. Woinowski-Krieger. Theory of Plates and Shells. McGraw-Hill, New York, 2nd edition, 1959. 3. L. Bucciarelly and N. Dworsky. Sophie Germain, an Essay on the History of Elasticity. Reidel, New York, 1980.
References 377 4. E. Reissner. Reflections on the theory of elastic plates. Appl. Mech. Rev., 38:1453-1464, 1985. 5. E. Reissner. The effect of transverse shear deformation on the bending of elastic plates. J. Appl. Mech., 12:69-76, 1945. 6. R.D. Mindlin. Influence of rotatory inertia and shear in flexural motions of isotropic elastic plates. J. Appl. Mech., 18:31-38, 1951. 7. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann, Oxford, 6th edition, 2005. 8. I. Babugka and T. Scapolla. Benchmark computation and performance evaluation for a rhombic plate bending problem. International Journal for Numerical Methods in Engineering, 28:155180, 1989. 9. I. Babugka. The stability of domains and the question of formulation of plate problems. Appl. Math., pages 463-467, 1962. 10. B.M. Irons and J.K. Draper. Inadequacy of nodal connections in a stiffness solution for plate bending. Journal of AIAA, 3:5, 1965. 11. R.W. Clough and J.L. Tocher. Finite element stiffness matrices for analysis of plate bending. In Proc. 1st Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-66-80, pages 515-545, Wright Patterson Air Force Base, Ohio, October 1966. 12. G.E Bazeley, Y.K. Cheung, B.M. Irons and O.C. Zienkiewicz. Triangular elements in bending -conforming and non-conforming solutions. In Proc. 1st Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-66-80, pages 547-576, Wright Patterson Air Force Base, Ohio, October 1966. 13. B. Fraeijs de Veubeke. Bending and stretching of plates. Special models for upper and lower bounds. In Proc. 1st Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-66-80, pages 863-886, Wright Patterson Air Force Base, Ohio, October 1966. 14. B.M. Irons. A conforming quartic triangular element for plate bending. International Journal for Numerical Methods in Engineering, 1:29-46, 1969. 15. R.W. Clough and C.A. Felippa. A refined quadrilateral element for analysis of plate bending. In Proc. 2nd Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-68-150, pages 399-440, Wright Patterson Air Force Base, Ohio, October 1968. 16. B. Fraeijs de Veubeke. A conforming finite element for plate bending. International Journal of Solids and Structures, 4:95-108, 1968. 17. EK. Bogner, R.L. Fox and L.A. Schmit. The generation of interelement-compatible stiffness and mass matrices by the use of interpolation formulae. In Proc. 1st Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-66-80, pages 397-443, Wright Patterson Air Force Base, Ohio, October 1966. 18. I.M. Smith and W. Duncan. The effectiveness of nodal continuities in finite element analysis of thin rectangular and skew plates in bending. International Journal for Numerical Methods in Engineering, 2:253-258, 1970. 19. K. Bell. A refined triangular plate bending element. International Journal for Numerical Methods in Engineering, 1:101-122, 1969. 20. G.A. Butlin and R. Ford. A compatible plate bending element. Technical Report 68-15, University of Leicester Engineering Department, 1968. 21. G.R. Cowper, E. Kosko, G.M. Lindberg and M.D. Olson. Formulation of a new triangular plate bending element. Trans. Canad. Aero-Space Inst., 1:86-90, 1968 (see also NRC Aero Report LR514, 1968). 22. W. Bosshard. Ein neues vollvertr~igliches endliches Element for Plattenbiegung. Mt. Ass. Bridge Struct. Eng. Bull., 28:27-40, 1968. 23. J.H. Argyris, I. Fried and D.W. Scharpf. The TUBA family of plate elements for the matrix displacement method. The Aeronaut. J., Roy. Aeronaut. Soc., 72:701-709, 1968. 24. W. Visser. The finite element method in deformation and heat conduction problems. Dr. Wiss. dissertation, Technische Hochschule, Delft, 1968.
378
Plate bending approximation: thin (Kirchhoff) plates and C1 continuity requirements 25. B.M. Irons, J.G. Ergatoudis and O.C. Zienkiewicz. Comments on 'complete polynomial displacement fields for finite element method' (by P.C. Dunne). Trans. Roy. Aeronaut. Soc., 72:709, 1968. 26. O.C. Zienkiewicz and Y.K. Cheung. The finite element method for analysis of elastic isotropic and orthotropic slabs. Proc. Inst. Civ. Eng., 28:471-488, 1964. 27. R.W. Clough. The finite element method in structural mechanics. In O.C. Zienkiewicz and G.S. Holister, editors, Stress Analysis, Chapter 7. John Wiley & Sons, Chichester, 1965. 28. D.J. Dawe. Parallelogram element in the solution of rhombic cantilever plate problems. J. Strain Anal., 1:223-230, 1966. 29. J.H. Argyris. Continua and discontinua. In Proc. 1st Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-66-80, pages 11-189, Wright Patterson Air Force Base, Ohio, October 1966. 30. L.S.D. Morley. On the constant moment plate bending element. J. Strain Anal., 6:20-24, 1971. 31. R.D. Wood. A shape function routine for the constant moment triangular plate bending element. Engineering Computations, 1:189-198, 1984. 32. R. Narayanaswami. New triangular plate bending element with transverse shear flexibility. Journal of AIAA, 12:1761-1763, 1974. 33. A. Razzaque. Program for triangular bending element with derivative smoothing. International Journal for Numerical Methods in Engineering, 5:588-589, 1973. 34. B.M. Irons and A. Razzaque. Shape function formulation for elements other than displacement models. In C.A. Brebbia and H. Tottenham, editors, Proc. of the International Conference on Variational Methods in Engineering, volume II, pages 4/59-4/72. Southampton University Press, 1973. 35. J.E. Walz, R.E. Fulton and N.J. Cyrus. Accuracy and convergence of finite element approximations. In Proc. 2nd Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-68-150, pages 995-1027, Wright Patterson Air Force Base, Ohio, October 1968. 36. R.J. Melosh. Structural analysis of solids. J. Structural Engineering, ASCE, 4:205-223, August 1963. 37. A. Adini and R.W. Clough. Analysis of plate bending by the finite element method. Technical Report G-7337, Report to National Science Foundation USA, 1961. 38. Y.K. Cheung, I.P. King and O.C. Zienkiewicz. Slab bridges with arbitrary shape and support conditions. Proc. Inst. Civ. Eng., 40:9-36, 1968. 39. J.L. Tocher and K.K. Kapur. Comment on basis of derivation of matrices for direct stiffness method (by R. Melosh). Journal of AIAA, 3:1215-1216, 1965. 40. R.D. Henshell, D. Waiters and G.B. Warburton. A new family of curvilinear plate bending elements for vibration and stability. J. Sound Vibr., 20:327-343, 1972. 41. R.L. Taylor, O.C. Zienkiewicz, J.C. Simo and A.H.C. Chan. The patch test - a condition for assessing FEM convergence. International Journal for Numerical Methods in Engineering, 22:39-62, 1986. 42. O.C. Zienkiewicz, S. Qu, R.L. Taylor and S. Nakazawa. The patch test for mixed formulations. International Journal for Numerical Methods in Engineering, 23:1873-1883, 1986. 43. O.C. Zienkiewicz and D. Lefebvre. Three field mixed approximation and the plate bending problem. Comm. Appl. Num. Meth., 3:301-309, 1987. 44. P.G. Bergen and L. Hanssen. A new approach for deriving 'good' element stiffness matrices. In J.R. Whiteman, editor, The Mathematics of Finite Elements and Applications, pages 483-497. Academic Press, London, 1977. 45. R.V. Southwell. Relaxation Methods in Theoretical Physics. Clarendon Press, Oxford, 1st edition, 1946. 46. P.G. Bergan and M.K. Nygard. Finite elements with increased freedom in choosing shape functions. International Journal for Numerical Methods in Engineering, 20:643-663, 1984.
References 379 47. C.A. Felippa and EG. Bergan. A triangular plate bending element based on energy orthogonal free formulation. Computer Methods in Applied Mechanics and Engineering, 61:129-160, 1987. 48. A. Samuelsson. The global constant strain condition and the patch test. In R. Glowinski, E.Y. Rodin and O.C. Zienkiewicz, editors, Energy Methods in Finite Element Analysis, Chapter 3, pages 49-68. John Wiley & Sons, Chichester, 1979. 49. B. Specht. Modified shape functions for the three node plate bending element passing the patch test. International Journal for Numerical Methods in Engineering, 26:705-715, 1988. 50. T.J.R. Hughes. The Finite Element Method: Linear Static and Dynamic Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1987. 51. F. Kikuchi and Y. Ando. A new variational functional for the finite element method and its application to plate and shell problems. Nuclear Engineering and Design, 21(1):95-113, 1972. 52. L.S.D. Morley. The triangular equilibrium element in the solution of plate bending problems. Aero. Q., 19:149-169, 1968. 53. W. Ritz. Ober eine neue Methode zur Ltisung gewisser variationsproblem der mathematischen physik. Journal far die reine und angewandte Mathematik, 135:1-61, 1908. 54. B.G. Galerkin. Series solution of some problems in elastic equilibrium of rods and plates. Vestn. Inzh. Tech., 19:897-908, 1915. 55. H. Hencky. Der Spannungszustand in rechteckigen Platten. Technical ReportVI, 94 S, Miinchen, 1913. 56. H. Hencky. Der Spannungszustand in rechteckigen Platten. PhD thesis, Darmstadt, Published by R. Oldenbourg, Munich and Berlin, Germany, 1913. 57. I.A. Wojtaszak. The calculation of maximum deflection, moment, and shear for uniformly loaded rectangular plate with clamped edges. J. Applied Mechanics, ASME, 59:A173-A176, 1937. 58. R.L. Taylor and S. Govindjee. Solution of clamped rectangular plate problems. Communications in Numerical Methods of Engineering, 20:757-765, 2004. 59. H. Marcus. Die Theorie elastisher Geweve und ihre Anwendung auf die Berechnung biegsamer Platten. Springer, Berlin, 1932. 60. P. Ballesteros and S.L. Lee. Uniformly loaded rectangular plate supported at the comers. Int. J. Mech. Sci., 2:206-211, 1960. 61. J.A. Stricklin, W. Haisler, P. Tisdale and K. Gunderson. A rapidly converging triangle plate element. Journal of AIAA, 7:180-181, 1969. 62. G.S. Dhatt. Numerical analysis of thin shells by curved triangular elements based on discrete Kirchhoff hypotheses. In W.R. Rowan and R.M. Hackett, editors, Proc. Symp. on Applications of FEM in Civil Engineering, Vanderbilt University, Nashville, Tennessee, 1969. ASCE. 63. J.L. Batoz, K.-J. Bathe and L.W. Ho. A study of three-node triangular plate bending elements. International Journal for Numerical Methods in Engineering, 15:1771-1812, 1980. 64. M.M. Hrabok and T.M. Hrudey. A review and catalogue of plate bending finite elements. Computers and Structures, 19:479-495, 1984. 65. J. Jirousek and L. Guex. The hybrid-Trefftz finite element model and its application to plate bending. Int. J. Num. Meth. Eng., 23:651-693, 1986. 66. A. Razzaque. Finite element analysis of plates and shells. PhD thesis, Civil Engineering Department, University of Wales, Swansea, 1972. 67. E.L. Wilson. The static condensation algorithm. International Journal for Numerical Methods in Engineering, 8:1974, 199-203. 68. G. Sander. Bournes suprrieures et infrrieures dans l'analyse matricielle des plates en flexiontorsion. Bull. Soc. Royale des Sci. de Liege, 33:456--494, 1974. 69. D. Withum. Berechnung von Platten nach dem Ritzchen Verfahren mit Hilfe dreieckftirmiger Meshnetze. Technical report, Mittl. Inst. Statik, Technische Hochschule, Hanover, 1966.
380 Plate bending approximation: thin (Kirchhoff) plates and C1 continuity requirements 70. C.A. Felippa. Refined finite element analysis of linear and non-linear two-dimensional structures. PhD dissertation, Department of Civil Engineering, SEMM, University of California, Berkeley, 1966. Also: SEL Report 66-22, Structures Materials Research Laboratory. 71. A. Zanisek. Interpolation polynomials on the triangle. International Journal for Numerical Methods in Engineering, 10:283-296, 1976. 72. A.G. Peano. Conforming approximation for Kirchhoff plates and shells. International Journal for Numerical Methods in Engineering, 14:1273-1291, 1979. 73. J.J. Grel. Construction of basic functions for numerical utilization of Ritz's method. Numerische Math., 12:435-447, 1968. 74. G. Birkhoff and L. Mansfield. Compatible triangular finite elements. J. Math. Anal. Appl., 47:531-553, 1974. 75. C.L. Lawson. Cl-compatible interpolation over a triangle. Technical Report RM 33-770, NASA Jet Propulsion Laboratory, Pasadena, California, 1976. 76. L.R. Herrmann. Finite element bending analysis of plates. In Proc. 1st Conf. Matrix Methods in Structural Mechanics, AFFDL-TR-66-80, pages 577-602, Wright Patterson Air Force Base, Ohio, 1965. 77. L.R. Herrmann. Finite element bending analysis of plates. J. Engineering Mechanics, ASCE, 94(EM5): 13-25, 1968. 78. W. Visser. A refined mixed type plate bending element. Journal of AIAA, 7:1801-1803, 1969. 79. J.C. Boot. On a problem arising from the derivation of finite element matrices using Reissner's principle. International Journal for Numerical Methods in Engineering, 12:1879-1882, 1978. 80. A. Chaterjee and A.V. Setlur. A mixed finite element formulation for plate problems. International Journal for Numerical Methods in Engineering, 4:67-84, 1972. 81. J.W. Harvey and S. Kelsey. Triangular plate bending elements with enforced compatibility. Journal of AIAA, 9:1023-1026, 1971. 82. B. Fraeijs de Veubeke and O.C. Zienkiewicz. Strain energy bounds in finite element analysis by slab analogy. J. Strain Anal., 2:265-271, 1967. 83. B. Fraeijs de Veubeke. An equilibrium model for plate bending. International Journal of Solids and Structures, 4:447-468, 1968. 84. J. Bron and G. Dhatt. Mixed quadrilateral elements for plate bending. Journal of AIAA, 10: 1359-1361, 1972. 85. K. Hellan. Analysis of elastic plates in flexure by a simplified finite element method. Technical Report Civ. Eng. Series 46, Acta Polytechnica Scandinavia, Trondheim, 1967. 86. T.H.H. Pian. Derivation of element stiffness matrices by assumed stress distribution. Journal of AIAA, 2:1332-1336, 1964. 87. T.H.H. Pian and P. Tong. Basis of finite element methods for solid continua. International Journal for Numerical Methods in Engineering, 1:3-28, 1969. 88. R.J. Allwood and G.M.M. Comes. A polygonal finite element for plate bending problems using the assumed stress approach. International Journal for Numerical Methods in Engineering, 1:135-160, 1969. 89. B.E. Greene, R.E. Jones, R.M. McLay and D.R. Strome. Generalized variational principles in the finite element method. Journal of AIAA, 7:1254-1260, 1969. 90. P. Tong. New displacement hybrid models for solid continua. International Journal for Numerical Methods in Engineering, 2:73-83, 1970. 91. B.K. Neale, R.D. Henshell and G. Edwards. Hybrid plate bending elements. J. Sound Vibr., 22:101-112, 1972. 92. R.D. Cook. Two hybrid elements for analysis of thick, thin and sandwich plates. International Journal for Numerical Methods in Engineering, 5:277-299, 1972. 93. C. Johnson. On the convergence of a mixed finite element method for plate bending problems. Num. Math., 21:43-62, 1973. 94. R.D. Cook and S.G. Ladkany. Observations regarding assumed-stress hybrid plate elements. International Journal for Numerical Methods in Engineering, 8:513-520, 1974.
References 381 95. I. Torbe and K. Church. A general quadrilateral plate element. International Journal for Numerical Methods in Engineering, 9:855-868, 1975. 96. D.J. Allman. A simple cubic displacement model for plate bending. International Journal for Numerical Methods in Engineering, 10:263-281, 1976. 97. J. Jirousek. Improvement of computational efficiency of the 9 dof triangular hybrid-Trefftz plate bending element. International Journal for Numerical Methods in Engineering, 23:2167-2168, 1986. (Letter to editor.) 98. G.A. Wempner, J.T. Oden and D.K. Cross. Finite element analysis of thin shells. Proc. Am. Soc. Civ. Eng., 94(EM6):1273-1294, 1968. 99. G. Dhatt. An efficient triangular shell element. Journal of AIAA, 8:2100-2102, 1970. 100. J.L. Batoz and G. Dhatt. Development of two simple shell elements. Journal of AIAA, 10: 237-238, 1972. 101. J.T. Baldwin, A. Razzaque and B.M. Irons. Shape function subroutine for an isoparametric thin plate element. International Journal for Numerical Methods in Engineering, 7:431-440, 1973. 102. B.M. Irons. The semi-Loof shell element. In D.G. Ashwell and R.H. Gallagher, editors, Finite Elements for Thin Shells and Curved Members, Chapter 11, pages 197-222. John Wiley & Sons, Chichester, 1976. 103. L.ER. Lyons. A general finite element system with special analysis of cellular structures. PhD thesis, Imperial College of Science and Technology, London, 1977. 104. R.A.E Martins and D.R.J. Owen. Thin plate semi-Loof element for structural analysis including stability and structural vibration. International Journal for Numerical Methods in Engineering, 12:1667-1676, 1978. 105. J.L. Batoz and M.B. Tahar. Evaluation of a new quadrilateral thin plate bending element. International Journal for Numerical Methods in Engineering, 18:1655-1677, 1982. 106. J.L. Batoz. An explicit formulation for an efficient triangular plate bending element. International Journal for Numerical Methods in Engineering, 18:1077-1089, 1982. 107. M.A. Crisfield. A new model thin plate bending element using shear constraints: A modified version of Lyons' element. Computer Methods in Applied Mechanics and Engineering, 38: 93-120, 1983. 108. M.A. Crisfield. A qualitative Mindlin element using shear constraints. Computers and Structures, 18:833-852, 1984. 109. M.A. Crisfield. Finite Elements and Solution Procedures for Structural Analysis, Vol. 1, Linear Analysis. Pineridge Press, Swansea, 1986. 110. G. Dhatt, L. Marcotte and Y. Matte. A new triangular discrete Kirchhoff plate-shell element. International Journal for Numerical Methods in Engineering, 23:453-470, 1986. 111. R.A. Nay and S. Utku. An alternative for the finite element method. Variational Methods in Engineering, 1, 1972. 112. E. Ofiate and E Z~rate. Rotation-free triangular plate and shell elements. International Journal for Numerical Methods in Engineering, 47:557-603, 2000. 113. E. Ofiate and G. Bugeda. A study of mesh optimality criteria in adaptive finite element analysis. Engineering Computations, 10:307-321, 1993. 114. M. Abramowitz and I.A. Stegun, editors. Handbook of Mathematical Functions. Dover Publications, New York, 1965. 115. G.S. Shapiro. On yield surfaces for ideal plastic shells. In Problems of Continuum Mechanics, pages 414-418. SIAM, Philadelphia, 1961.
ii~i~i~i~ii!~i~:~i~i!~i!i~i~i~i~ii~i:i!i!ii~:~i~!i~i~i~i~.i~i~i~i~i~!~!i~
'Thick' Reissner-Mindlin platesirreducible and mixed formulations ':',
We the the
i ' ~?~i , i ~7?~z~=~i 2 ~ i~i~:~=~=~: i'=='~'==|~~dUC~|O=~i ~!~i~i~i~=~i=~:~i!~i~=""~' i~i~i~i~!~i~i~i!~i~,~' i~~ i~;i~ .......~ii ~ 2 ~ : ~ : ~ ~ = ~ = :~~~=~:~~:=~=:~=~=~ ~ i{ ~ {i ~ { iili ~ { ~ ~i ~ { ~ ~ i
i
have already introduced in Chapter 11 the full theory of thick plates from which thin plate, Kirchhoff, theory arises as the limiting case. In this chapter we shall show how the numerical solution of thick plates can easily be achieved and how, in limit, an alternative procedure for solving all problems of Chapter 11 appears. The development of approximations to the Timoshenko beam form discussed in Chapter 10 plays an important role in the development of viable solutions of the thick, ReissnerMindlin, theory that work in 'thin' plate applications. To ensure continuity we repeat below the governing equations [Eqs (11.10a)(11.1 l b), or Eqs (11.72)]. Referring to Fig. 11.3 of Chapter 11 and the text for definitions, we remark that all the equations could equally well be derived from full three-dimensional analysis of a flat and relatively thin portion of an elastic continuum illustrated in Fig. 12.1. All that it is now necessary to do is to assume that whatever form of the approximating shape functions in the xy plane those in the z direction are only linear. Further, it is assumed that Crz stress is zero, thus eliminating the effect of vertical strain.* The first approximations of this type were introduced quite early 1'2 and the elements then derived are exactly of the Reissner-Mindlin type discussed in Chapter 11. The equations from which we shall start and on which we shall base all subsequent discussion are the moment constitutive equation [see Eqs (11.10a) and (11.72)], M-
D/~t~ = 0
(12.1a)
where D is the matrix of bending rigidities, the shear constitutive equation [see Eqs (11.10c) and (11.72)] 1 -S OL
- q5-
Vw
= 0
(12.1b)
where c~ - nGt is the shear rigidity, the moment equilibrium (angular momentum) equation [see Eqs ( 11.1 lb) and ( 11.72)] ETM + S = 0 *
Reissner includes the effect of O'z in bending but, for simplicity, this is disregarded here.
(12.1c)
Introduction
i 55/
(w, ,~, %)
Fig. 12.1 An isoparametric three-dimensional element with linear interpolation in the transverse (thickness) direction and the 'thick' plate element.
and the shear equilibrium equation [see Eqs (11.1 la) and (11.72)]
~rTs + q = 0
(12.1d)
In the above the moments M, the transverse shear forces S, and the elastic matrices D are as defined in Chapter 11, and -0 s
0
0 0 ~yy
0
0
_Oy
Ox_
(12.2)
defines the strain-displacement operator on rotations qS, and its transpose the equilibrium operator on moments, M. Boundary conditions are of course imposed on w and ~b or the corresponding plate forces S,, M,, M,s in the manner discussed in Sec. 11.2.2. It is convenient to eliminate M from Eqs (12.1 a)-(12.1d) and write the system of three equations [Eqs (11.73)] as
LTDL~b + S = 0 1 --S - q~- ~'w = 0
OL
(12.3)
vTS+q =0
This equation system can serve as the basis on which a mixed discretization is builtor alternatively can be reduced further to yield an irreducible form. In Chapter 11 we
383
384 'Thick' Reissner-Mindlin plates-irreducible and mixed formulations have dealt with the irreducible form which is given by a fourth-order equation in terms of w alone and which could only serve for solution of thin plate problems, that is, when c~ = c~ [Eq. (11.14a)]. On the other hand, it is easy to derive an alternative irreducible form which is valid only if c~ ~ cx~. Thus, the shear forces can be eliminated yielding two equations: / ~ T D / ~ -4- ct (VW -4- ~b) -- 0 V v [c~ ( V w + qS)] + q -- 0
(12.4)
This is an irreducible system corresponding to minimization of the total potential energy
I'I : 7
(s
12
D/2th dr2 + ~
( V w -+- ~t~)T OL(Vl/) + ~ ) d r 2 (12.5)
/.
- / ~ w q dr2 + Flbt = minimum as can easily be verified. In the above the first term is simply the bending energy and the second the shear distortion energy [see Eq. (11.83)]. Clearly, this irreducible system is only possible when a ~ oo, but it can, obviously, be interpreted as a solution of the potential energy given by Eq. (11.83) for 'thin' plates with the constraint of Eq. (11.84) being imposed in a penalty manner with a being now a penalty parameter. Thus, as indeed is physically evident, the thin plate formulation is simply a limiting case of such analysis. We shall see that the penalty form can yield a satisfactory solution only when discretization of the corresponding mixed formulation satisfies the necessary convergence criteria. The thick plate form now permits independent specification of three conditions at each point of the boundary. The options which exist are: w (/)n q5s
or or or
Sn M,, M.~,
in which the subscript n refers to a normal direction to the boundary and s a tangential direction. Clearly, now there are many combinations of possible boundary conditions. A 'fixed' or 'clamped' situation exists when all three conditions are given by displacement components, which are generally zero, as
W= n
=0
and a free boundary when all conditions are the 'resultant' components
S.=Mn =Mns=O When we discuss the so-called simply supported conditions (see Sec. 11.2.2), we shall usually refer to the specification w = 0
and
M~ = M~s = 0
The irreducible formulation-reduced integration
385
as a 'soft' support, SS 1 (and indeed the most realistic support), and to w = 0
Mn = 0
and
q~ = 0
as a 'hard' support, SS2. The latter in fact replicates the thin plate assumptions and, incidentally, leads to some of the difficulties associated with it. Finally, there is an important difference between thin and thick plates when 'point' loads are involved. In the thin plate case the displacement w remains finite at locations where a point load is applied; however, for thick plates the presence of shearing deformation leads to an infinite displacement (as indeed three-dimensional elasticity theory also predicts). In finite element approximations one always predicts a finite displacement at point locations with the magnitude increasing without limit as a mesh is refined near the loads. Thus, it is meaningless to compare the deflections at point load locations for different element formulations and we will not do so in this chapter. It is, however, possible to compare the total strain energy for such situations and here we immediately observe that for cases in which a single point load is involved the displacement provides a direct measure for this quantity.
!:7i7i:~i~i:i~:~i~i7~i7i~!~!~{!i~ii~!~!~!i ...7.~i.~i.~.!~i i:.~:{i~i!{!i!~i~!i~i:~:i~ii~i~:~ii~i~i~i!i~:~i~i7~ii~i~::~:~;~i~i~;~i~i7~:~i!~:~:~:~:~:~.............. :!~:~::~:~:~7~:~::~!~::~::~+::~::~::~:~7:ii:~i~iii~::~::~i~::~:!i~i~i~ii!i!i!~!~!i!~ii7iii!i~7~:~ii!i!~:~i~i~i~i!i::~:~:~:~:~::~i~i~:~:~i~::~::~:~ii~i~::::~::~i~i~iiiii::::~::~i77:~i~i~i~7~!::~::!i~:::~i~!i!:::~::~i~ :~7i~:~ii~7!iii!i!i~!iii:~:~:i7~:~7i~7~
The procedures for discretizing Eq. (12.4) appear to be straightforward. However, we will find that the process is very sensitive. First, we consider standard isoparametric interpolation in which the two displacement variables are approximated by shape functions and parameters as 0 5 - No~b
and
w = Nw~
(12.6)
We recall that the rotation parameters 05 may be transformed into physical rotations about the coordinate axes, O, using Eq. (1 1.7). The parameters 0 are often more convenient for calculations and are essential in shell developments. T h e approximation equations are now obtained directly by the use of the total potential energy principle [Eq. (1 2.5)], the Galerkin process on the weak form, or by the use of virtual work expressions. Here we note that the appropriate generalized strain components, corresponding to the moments M and shear forces S, are r
--
~ b -
(s162
(12.7a)
and (12.7b)
e~ - V w + 05 - VNw~r + N4,~b We thus obtain the discretized problem
(L and
dO+ L N:~
=
386 'Thick' Reissner-Mindlin plates -irreducible and mixed formulations
or simply [Kww Kw,] { ~ } Kow K,,.]
= Kfi = (Kb + Ks) fi =
{f~}--f fo
(12.8a)
with
[: oo]
,,:
q)y, Ks
(12.8b)
[K;~ = [K~
K;o ] K~J
where the arrays are defined by b = fn (s Ko,
= L s1
dg2 K~,w = fn (VNw)T oLVN,,, dr2 K~,w = fn NTaVN"-' dr2 = (K~0)T
so aa
(12.8c)
and forces are given by --
Nw ~SndF
Nwq d ~ +
(12.8d)
s
fo-/
N~I~ dr m
where Sn is the prescribed shear on boundary Fs, and 1~ is the prescribed moment on boundary Fm. The formulation is straightforward and there is little to be said about it a priori. Since the form contains only first derivatives apparently any Co shape functions of a two-dimensional kind can be used to interpolate the two rotations and the lateral displacement. Figure 12.2 shows some rectangular (or with isoparametric distortion,
o Node with two rotation parameters ~) r-1 Node with one lateral displacement parameter w
QS Fig. 12.2 Someearlythick plate elements.
QL
QH
The irreducibleformulation-reduced integration quadrilateral) elements used in the early work. ~-3 All should, in principle, be convergent as Co continuity exists and constant strain states are available. In Fig. 12.3 we show what in fact happens with a fairly fine subdivision of quadratic serendipity and Lagrangian rectangles as the ratio of span to thickness, L / t , varies. Here L is a characteristic length of the plate and may be a side length, a loading length or a normal mode characteristic. We note that the magnitude of the coefficient a is best measured by the ratio of the bending to shear rigidities and we could assess its value in a non-dimensional form. Thus, for an isotropic material with a = G t this ratio becomes Et 3
(12.9)
~
Obviously, 'thick' and 'thin' behaviour therefore depends on the L / t ratio. Clamped edge 0.0016 0.0015 Exact thin plate 0.0014 Exact thin plate sol .00406 0.0013 ",'~,, solution 0.00127 0.0012 0.0011 0.0010 llfl I-''~--- ~lijj l I:~ ~ ~ J I III I i l,,~J i J 0.0009 102 103 104 lO 4 101 102 103 L/t L/t
Simply supported
0.0044 0.0043 k 0.0042 0.0041 0.0040 0.0039 0.0038 0.00371111
101
(a)
QS-R , QS-N . . . . . . .
0.0044
- •
2 x 2 Gaussian integration of all terms 3 x 3 Gaussian integration of all terms
Simply supported ..........
0.0043~~._ 0.0042 0.0041 0.0040 0.0039 0.0038 0.0037 I 101
(b)
Clamped edge
0.001!~6
Exact thin plate solution 0.00406
" "= " " '= =' " ' " "= " ' "= ="
'
0.001 0.001 0.001
'
Exact thin plate solution 0.00127 ~
0.0012 0.0011 I-0.00101-
9
""
/
I I I J~ ~ ~ I~m ~ ~
QS-R QS-N . . . . . . .
102
L/t
103
104
0.0009111 I 101
III
102
I
L/t
i Ill l l 103
104
2 x 2 Gaussian integration of all terms 3 x 3 Gaussian integration of all terms
Fig. 12.3 Performance of (a) quadratic serendipity (QS) and (b) Lagrangian (QL) elements with varying span-to-thickness L/t, ratios, uniform load on a square plate with 4 x 4 normal subdivisions in a quarter. R is reduced 2 x 2 quadrature and N is normal 3 x 3 quadrature.
387
388
'Thick' Reissner-Mindlin plates -irreducible and mixed formulations
It is immediately evident from Fig. 12.3 that, while the answers are quite good for smaller L / t ratios, the serendipity quadratic fully integrated elements (QS) rapidly depart from the thin plate solution, and in fact tend to zero results (locking) when this ratio becomes large. For Lagrangian quadratics (QL) the answers are better, but again as the plate tends to be thin they are on the small side. The reason for this 'locking' performance is similar to those considered for the nearly incompressible problem 4 and the Timoshenko beam problem (viz. Chapter 10). In the case of plates the shear constraint implied by the third of Eq. (12.3), and used to eliminate the shear resultant, is too strong if the terms in which this is involved are fully integrated. Indeed, we see that the effect is more pronounced in the serendipity element than in the Lagrangian one. In early work the problem was thus mitigated by using a reduced quadrature, either on all terms, which we label R in the figure, 5'6 or only on the offending shear terms selectively 7,8 (labelled S). The dramatic improvement in results is immediately noted. The use of reduced quadrature to develop a four-node plate element has been revisited recently by Gruttmann and Wagner using a stabilized form. 9 The same improvement in results is observed for linear quadrilaterals in which the full (exact) integration gives results that are totally unacceptable (as shown in Fig. 12.4), but where a reduced integration on the shear terms (single point) gives excellent performance, 1~ although a careful assessment of the element stiffness shows it to be rank deficient in an 'hourglass' mode in transverse displacements. (Reduced integration on all terms gives additional matrix singularity.) A remedy thus has been suggested; however, it is not universal. We note in Fig. 12.3 that even without reduction of integration order, Lagrangian elements perform better in the quadratic expansion. In cubic elements (Fig. 12.5), however, we note that (a) almost no change occurs when integration is 'reduced' and (b), again, Lagrangian-type elements perform very much better.
~"
0.0044 0.0043 0.0042 0.0041
#
o.oo4o-
Simply supported
0.0016 0.0015 0.0014 _ ~ 0.0013 0.0012 0.0011 0.0010
Clamped edge
m
Exact thin plate solution 0.00406 ~
0.0039 0.0038 0.0037 ~ t i I~ i i IIII 101 102 103 L/t L-R L-N . . . . . . .
l
104
0.0009
I I
101
Exact thin plate solution 0.00127
I l=~l
102
I
L/t
Iiii 103
I
2 x 2 flexure integration - 1 x 1 shear integration 2 x 2 integration of all terms- this gives poor results, and diverges rapidly as L/t increases
Fig. 12.4 Performanceof bilinear elementswith varyingspan-to-thickness,L/t, values.
10 4
The irreducible
0.0044~
~
~176176176 f
(a)
~
102
integration
Clamped edge
Simply supported
0.0043 0.0042 Exact thin plate 0.0041 I" ~,,...~solution 0.00406 0.0039 0.0038 0.0037 I I 101
formulation-reduced
_•
0"0016 I 0.0015 0.0014 0.0013
Exact thin plate solution 0.00127
0"0012f
Ill 103
0.0011 0.0010 I 0.0009 t t i Ill i~ Itl i I 101 104 102 103 104
L/t L/t QS-R 3 x 3 Gaussian integration of all terms QS-N . . . . . . . 4 x 4 Gaussian integration of all terms
0.0044 0.0043 ~-.a 0.0042
Simply supported
Clamped edge
Exact thin plate
0.0015, 0. 0010"0016 f,3 0.0014
Exact thin plate
"-,,._ solution 0.00127 0.0012 ~ 0.0041 0.0038 " ...... s.oI.ution0.00406 ~t, 0.0040 0"0012f 0.0011 0.0039
0.0010 0.0009 I i I ill i I ill i i 102 103 104 101 102 103 104 L/t L/t QL-R 3 x 3 Gaussian integration of all terms QL-N . . . . . . . 4 x 4 Gaussian integration of all terms
0.0037 101 (b)
lal
i ilJ i
t
Fig. 12.5 Performanceof cubic quadrilaterals: (a) serendipity (QS) and (b) Lagrangian (QL) with varying span-to-thickness, Lit, values.
In the late 1970s many heuristic arguments were advanced for devising better elements,l 1-14all making use of reduced integration concepts. Some of these perform quite well, for example the so-called 'heterosis' element of Hughes and Cohen ]1 illustrated in Fig. 12.3 (in which the serendipity-type interpolation is used on w and a Lagrangian one on qS), but all of the elements suggested in that era fail on some occasions, either locking or exhibiting singular behaviour. Thus such elements are not 'robust' and should not be used universally. A better explanation of their failure is needed and hence an understanding of how such elements could be designed. In the next section we shall address this problem by considering a mixed formulation. The reader will recognize here arguments used in the nearly incompressible problem in Chapter 2 and in reference 4 which led to a better understanding of the failure of some straightforward elasticity elements as incompressible behaviour was approached. The situation is completely parallel here.
389
390 'Thick' Reissner-Mindlin plates-irreducible and mixed formulations
i"i!~ii'iDi'~"i"'~ii~ ii"~'~i"i~'~'i~!i'~i"'~i'"i~i'!~i~'!'i~~ ii'~i'i~~! i'!'ii~i'il!i'~i~~ii~i~i~i i~iJ~: ~!~~i~i~i!~i !i~i!i~ii~!i!~i i~i!ii~i!i!i!i!i~!i~i'!i~i!i'ii~'th ii'~i9!i~ii~ki~ !i!i~!i~~i i~~ii~~i!ii!!iii!iP!~~i i~i,i~!ili~' ii~i!i~ii!~iii~ii~i~!i~ii!i~i!i~i~i~i~!........... ii~i~i i~i~i~!i................ i~i:~ii!ii~i 'i~ii!~!ii!iii~ii!iii!ii!i!ii!iiiiiii~iii!i~iii~ii!i!i!iiiii ........................................................
12.3.1 The approximation The problem of thick plates can, of course, be solved as a mixed one starting from Eq. (12.3) and approximating directly each of the variables w, q~ and S independently. Using Eq. (12.3), we construct a weak form as _~oSwlVTS+q] d ~ = 0 f~q~T [/~TD/~q~ + S] dff2
0
(12.10)
fa~sT[----c~Is+q~+VW] d f 2 = 0 We now write the independent approximations, using the standard Galerkin procedure, as
w = N,L,~r d;w = N w ~
q~ = N~,tp
and
d;q~ = N~q~
S = NsS
and
~S = N ~ S
(12.11)
though, of course, other interpolation forms can be used, as we shall note later. After appropriate integrations by parts of Eq. (12.10), we obtain the discrete symmetric equation system
E
KCb H r
~
=
f'~
(12.12)
where Kb : f~ (s162
D (Z2Nr
E = f~ N~VNw d~ (12.13) C - f~ N~N~ dr2 H -- - f N r l N s dr2 and where fw and fr are as defined in Eq. (12.8d). The above represents a typical three-field mixed problem of the type discussed in Sec. 10.5.1 of reference 4, which has to satisfy certain criteria for stability of approximation as the thin plate limit (which can now be solved exactly) is approached. For this limit we have a = c~ and H = 0 (12.14)
Mixed formulation for thick plates 391 In this limiting case it can readily be shown that necessary criteria of solution stability for any element assembly and boundary conditions are that nr + nw
n o + n~ > ns
or
C~p =
ns > nw
or
/3p ---- ~
and
ns
ns
nw
> 1
> 1
(12.15a)
(12.15b)
where n 0, n~ and nw are the number of q~, S and ~ parameters in Eqs (12.11). When the count condition is not satisfied then the equation system either will be singular or will lock. Equations (12.15a) and (12.15b) must be satisfied for the whole system but, in addition, they need to be satisfied for element patches if local instabilities and oscillations are to be avoided. 15-17 We remind the reader that Eqs (12.15a) are (12.15b) are necessary conditions; however, they are not sufficient conditions. It is always necessary to conduct consistency and stability tests to ensure the proposed element passes the complete mixed patch test. 4 The above criteria will, as we shall see later, help us to design suitable thick plate elements which show convergence to correct thin plate solutions.
12.3.2 Continuity requirements The approximation of the form given in Eqs (12.12) and (12.13) implies certain continuities. It is immediately evident that Co continuity is needed for rotation shape functions N o (as products of first derivatives are present in the approximation), but that either Nw or Ns can be discontinuous. In the form given in Eq. (12.13) a Co approximation for w is implied; however, after integration by parts a form for Co approximation of S results. Of course, physically only the component of S normal to boundaries should be continuous, as we noted also previously for moments in the mixed form discussed in Sec. 11.16. In all the early approximations discussed in the previous section, Co continuity was assumed for both tp and w variables, this being very easy to impose. We note that such continuity cannot be described as excessive (as no physical conditions are violated), but we shall show later that successful elements also can be generated with discontinuous w interpolation (which is indeed not motivated by physical considerations). For S it is obviously more convenient to use a completely discontinuous interpolation as then the shear can be eliminated at the element level and the final stiffness matrices written simply in standard q~, ,~ terms for element boundary nodes. We shall show later that some formulations permit a limited case where c~-1 is identically zero while others require it to be non-zero. The continuous interpolation of the normal component of S is, as stated above, physically correct in the absence of line or point loads. However, with such interpolation, elimination of S is not possible and the retention of such additional system variables is usually too costly to be used in practice and has so far not been adopted. However, we should note that an iterative solution process applicable to mixed forms can reduce substantially the cost of such additional variables. TM
392
'Thick' Reissner-Mindlin p l a t e s - i r r e d u c i b l e and mixed formulations (zx = two S variables) Irreducible - with shear integration at 2 x 2 Gauss points
=
Mixed - discontinuous shear interpolation with shear nodes at 2 x 2 Gauss points
Fig. 12.6 Equivalence of mixed form and reduced shear integration in quadratic serendipity rectangle.
12.3.3 Equivalence of mixed forms with discontinuous S
interpolation and reduced (selective) integration
An equivalence of penalized mixed forms with discontinuous interpolation of the constraint variable and of the corresponding irreducible forms with the same penalty variable may be demonstrated following work of Malkus and Hughes for incompressible problems. 19Indeed, an exactly analogous proof can be used for the present case, and we leave the details of this to the reader; however, below we summarize some equivalencies that result. Thus, for instance, we consider a serendipity quadrilateral, shown in Fig. 12.6 (a), in which integration of shear terms (involving c~) is made at four Gauss points (i.e. 2 • 2 reduced quadrature) in an irreducible formulation [see Eqs (12.8a)-(12.8d)], we find that the answers are identical to a mixed form in which the S variables are given by a bilinear interpolation from nodes placed at the same Gauss points. This result can also be argued from the limitation principle first given by Fraeijs de Veubeke. 2~ This states that if the mixed form in which the stress is independently interpolated is precisely capable of reproducing the stress variation which is given in a corresponding irreducible form then the analysis results will be identical. It is clear that the four Gauss points at which the shear stress is sampled can only define a bilinear variation and thus the identity applies here. The equivalence of reduced integration with the mixed discontinuous interpolation of S will be useful in our discussion to point out reasons why many elements mentioned in the previous section failed. However, in practice, it will be found equally convenient (and often more effective) to use the mixed interpolation explicitly and eliminate the S variables by element-level condensation rather than to use special integration rules. Moreover, in problems where the material properties lead to coupling between bending and shear response (e.g. elastic-plastic behaviour) use of selective reduced integration is not convenient. It must also be pointed out that the equivalence fails if c~varies within an element or indeed if the isoparametric mapping implies different interpolations. In such cases the mixed procedures are generally more accurate. ii!ii i iiiiiiiiiiliziil~i~i i'~'ii'iii~ii! ~i~'~'~'~'~,iiliiiiiiii~i !i i~'~':~i, iiiiiii iiiiii!i'~,',~,',~"i~'i'ii:::'i!iiiil ii!iiiiilii~ilii~i~iii~i iiii!ii~i i i i ihiiiiiiiii~!J~i~i i!ii i i i i~::'::ii iiiiiiiiii i iiilii :i~~,iiii i i i i iiiiiiiiiiiiii!~iiiiiiiiii~i ~:::'ii i i i i i i i i i i i iiii~iiiiil~ii~i~ iiiiiiii i i! i i !iii~::'ili !iiiiiiiiiiii i ~J~iiiii i i i i i!~i i ~:"::~!iiiiili ~iliii~diili!i~i~i !Eiiiili~iiiiiii ~!i!i~~:~:: i:iii~iiiiiiiiiiiiiiiiili ~i i i iilili i iiiiiiiiii~iiii~iii i~:'i:::iiii~li i !iiiii~iiiiiiiili i~i i i iiiili~iiiiiii il~iiiiiiiiiiiiii~i~i~il!iii!!iiiii!i~iiiiiili~iililiiiliiiiiiiiiliiiiiiiiiilililiililililiiii~iiii~iiiiiliiiliiiiiili!iiiiiiiiii!iilii i i i i i i~i~iiiiili i ~il>ii!i:!i!i!iilii ::ii i i i i:i i i i !i ~i i~i!ii iilili i ::ii!i i i ii!iiiiiiiiiiii i i i! i !i!iiii!iiiiiiiiiiiiiiiiiiiiiiiiii! ~i iiiiili::::!iiiiiiiiiiii::iiiiiiiiiiii::iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii i i :i!i::i!iiii iiiii!iiiiiiiii:i;iiiiiiiii i iiii ii !i il i ii i~i!ii~i~i~i!i~i::::: :i!i:ii:i i i iii!!:i:~ii i!i:il ii ii i!i i i!i i!:i~:!i!i!~ii!i~i!i:ii !ii l:ilili: i i:iiii!iiii :ili :iii i!~i!ii:ii:l!i i li!iii i i i i !i i ii~ii!i i iiiil::iiiiiii i i :!::!:!::::~iiiii i!i iiiiiiiiiiliiiiliiil ii!iliii ~iii:ii:i i i i i i!!iiiiiiiiiiliiiliili!iiiiii i i i::ii!i !ii ii!il i!i ii ii i i i!:i:i i i:::i:iiiii i i i i i::ii i !i i i i i!::ii !iiii i i::::iililiiiiil!ii !i::i!ili!i !ii::!ii!iiiiiiiiiiili !i i i!::ii i!i i i i ?:iiiiiiilli ii
12.4.1 Why elements fail The nature and application of the patch test have changed considerably since its early introduction. As shown in references 15-17 and 21-25 this test can prove, in addition
The patch test for plate bending elements to consistency requirements (which were initially the only item tested), the stability of the approximation by requiting that for a patch consisting of an assembly of one or more elements the stiffness matrices are non-singular whatever the boundary conditions imposed. To be absolutely sure of such non-singularity the test must, at the final stage, be performed numerically. However, we find that the 'count' conditions given in Eqs (12.15a) and (12.15b) are necessary for avoiding such non-singularity. Frequently, they also prove sufficient and make the numerical test only a final confirmation. ]6'17 We shall demonstrate how the simple application of such counts immediately indicates which elements fail and which have a chance of success. Indeed, it is easy to show why the original quadratic serendipity element with reduced integration (QS-R) is not robust. In Fig. 12.7 we consider this element in a single-element and four-element patch subjected to so-called constrained boundary conditions, in which all displacements on the external boundary of the patch are prescribed and a relaxed boundary condition in which only three displacements (conveniently two 0s and one w) eliminate the rigid body modes. To ease the presentation of this figure, as well as in subsequent tests, we shall simply quote the values of C~p and/3p parameters as defined in Eqs (12.15a) and (12.15b) with subscript replaced by C or R to denote the constrained or relaxed tests, respectively. The symbol F will be given to any failure to satisfy the necessary condition. In the tests of Fig. 12.7 we note that both patch tests fail with the parameter c~c being less than 1, and hence the elements will lock under certain circumstances (or show a singularity in the evaluation of S). A failure in the relaxed tests generally
Relaxed
Constrained
(a)
(xc = 0/8 (F)
0f,R = (24-3)/8 .- 21/8
I~c = 810
JIB = 8/(8-1) = 8/7
Boundaries constrained I F Failure I Only 3 DOF
Boundaries constrained
~ constrained 0CC = 15/32
(F)
E E
A
A
A
A
A
A
Z&.
A
A
A
A
Z~
Z~
A
A A
B
on this boundary
~ R = (63-3)/32 = 60/32
13R=
E (b)
Fig. 12.7 'Constrained' and 'relaxed' patch test/count for serendipity (quadrilateral). (In the C test all boundary displacements are fixed. In the R test only three boundary displacements are fixed, eliminating rigid body modes.) (a) Single-element test; (b) four-element test.
393
394 'Thick' Reissner-Mindlin plates -irreducible and mixed formulations
predicts a singularity in the final stiffness matrix of the assembly, and this is also where frequently computational failures have been observed. As the mixed and reduced integration elements are identical in this case we see immediately why the element fails in the problem of Fig. 12.3 (more severely under clamped conditions). Indeed, it is clear why in general the performance of Lagrangiantype elements is better as it adds further degrees of freedom to increase nr (and also nw unless heterosis-type interpolation is used), ll In Table 12.1 we show a list of the C~p and/3p values for single- and four-element patches of various rectangles, and again we note that none of these satisfies completely the necessary requirements, and therefore none can be considered robust. However, it is interesting to note that the elements closest to satisfaction of the count perform best, and this explains why the heterosis elements 26 are quite successful and indeed why the Lagrangian cubic is nearly robust and often is used with s u c c e s s . 27 Of course, similar approximation and counts can be made for various triangular elements. We list some typical and obvious ones, together with patch test counts, in the first part of Table 12.2. Again, none perform adequately and all will result in locking and spurious modes in finite element applications. We should note again that the failure of the patch test (with regard to stability) means that under some circumstances the element will fail. However, in many problems a reasonable performance can still be obtained and non-singularity observed in its performance, providing consistency is, of course, also satisfied.
Numerical patch test
While the 'count' condition of Eqs (12.15a) and (12.15b) is a necessary one for stability of patches, on occasion singularity (and hence instability) can still arise even with its satisfaction. For this reason numerical tests should always be conducted ascertaining the rank sufficiency of the stiffness matrices and also testing the consistency condition. In Chapter 9 of reference 4 we discussed in detail the consistency test for irreducible forms in which a single variable set u occurred. It was found that with a second-order operator the discrete equations should satisfy at least the solution corresponding to a linear field u exactly, thus giving constant strains (or first derivatives) and stresses. For the mixed equation set [Eq. (12.3)] again the lowest-order exact solution that has to be satisfied corresponds to: 1. constant values of moments or Z~q5and hence a linear th field; 2. linear w field; 3. constant S field. The exact solutions for which plate elements commonly are tested and where full satisfaction of nodal equations is required consist of: 1. arbitrary constant M fields and arbitrary linear q~ fields with zero shear forces (S = 0); here a quadratic w form is assumed still yielding an exact finite element solution; 2. constant S and linear w fields yielding a constant q~ field. The solution requires a distributed couple on the fight-hand side of the first of Eq. (12.3) and this was not included in the original formulation. A simple procedure is to disregard the satisfaction of the moment equilibrium in this test. This may be done simply by inserting a very large value of the bending rigidity D.
The patch test for plate bending elements 395 Table 12.1
Quadrilateral mixed elements: patch count Single-element patch
Element
Four-element patch
Reference
ac
/~c
~R
~R
~C
~C
~R
Q4S15,8,1o
0
2
9
2
3
8
24
8
(F)
(F)
15 3"2 (F)
32 -5-
60 32
32 20
32 "9-
72 32
32 20
23 3-'2
32 "5-
68 3-'2
32 2"-0
27 7-2
72 -if"
9672
7232
I
] I
(F)
Q8S45
0 i (F)
8 0
21 "-if"
8 7
Q9S47'8
3 g
8 T
16 "8"
8 8
(F)
Q9HS411
2
(F) Q 12S95
0 Yg
(F)
8 0
15 T
8 7
18 "0"
23 1-8
18 1--i
(F)
Q 16S927
27 32
(F)
(F)
12 1-8
18 4
45 1--8
18 ~
75 72
72 2-5
150 7-2-
72 5-0
Q 4 S 1B 128 Q 4 S 1B 1L29.3~
2 2
2 0
11 2
2 3
!1 T
8 T
32 'if
8
r~A ~ O ze'~Dor31 , Ol.L,
4
4 i
13 T
4 ~
19 1-6
16 T
40 1-6
16 -g
(F) II
[ ] I ~
* I
(F)
F Failure to satisfy necessary conditions.
12.4.2 Design of some useful elements The simple patch count test indicates how elements could be designed to pass it, and thus avoid the singularity (instability). Equation (12.15b) is always trivial to satisfy for elements in which S is interpolated independently in each element. In a single-element test it will be necessary to restrain at least one ,~' degree of freedom to prevent rigid body translations. Thus, the m i n i m u m number of terms which can be included in S for each element is always one less than the number of,~, parameters in each element. As patches with more than one element are constructed the number of w parameters will increase proportionally with the number of nodes and the number of shear constraints increase by the number of elements. For both quadrilateral and triangular elements the requirement that ns > nw - 1 f o r no b o u n d a r y restraints ensures that Eq. (12.15b) is satisfied on all patches for both constrained and relaxed boundary conditions. Failure to satisfy
396
'Thick' Reissner-Mindlin plates-irreducible and mixed formulations Table 12.2
Triangular mixed elements: patch count Single-element patch
Element
Six-element patch
Reference
aC
tiC
O~R
fiR
OLC
tiC
O~R
fiR
T3 S 1
0 ~
2 ~
6 ~
2 ~
3 1~
12 -F
18 1-~
12 -K
36 if
54 3-3
36 1-8
57 3-3
36 l~
108 3--K
36 3-3
(F)
T 6S3
/z
0 ~
(F)
6 0
15 ~
6 5
(F)
T 10 S 3
3 g
21 3-3 (F)
6 ~
27 -(
(F)
6 ~ (F)
II
T 3 S 1B 1L 32'33
~
~
~
~
2
20 1-2
12 "6-
35 1"2
12 I-T
T3S1B1A34
2
2
8
2
15
12
30
12
2
2
17
2
33
12
66
12
T 6 S 1B 1
2
2
8
(F)
T6S3B317
6
6 ~
21 -'C
6 g
(F)
75 3"g
36 "7-
108 3--6-
36
F Failure to satisfy necessary conditions.
this simple requirement explains clearly why certain of the elements in Tables 12.1 and 12.2 failed the single-element patch test for the relaxed boundary condition case. Thus, a successful satisfaction of the count condition requires now only the consideration of Eq. (12.15a). In the remainder of this chapter we will discuss two approaches which can successfully satisfy Eq. (12.15a). The first is the use of discrete collocation constraints in which the third of Eq. (12.3) is enforced at preselected points on the boundary and occasionally in the interior of elements. Boundary constraints are often 'shared' between two elements and thus reduce the rate at which ns increases. The other approach is to introduce bubble or enhanced modes for the rotation parameters in the interior of elements. Here, for convenience, we refer to both as a 'bubble mode' approach. The inclusion of at least as many bubble modes as shear modes will automatically satisfy Eq. (12.15a). This latter approach is similar to that used in Sec. 12.7 of Volume 1 to stabilize elements for solving the (nearly) incompressible problem and is a clear violation of 'intuition' since for the thin plate problem the rotations appear as derivatives of w. Its use in this case is justified by patch counts and performance.
Elements with discrete collocation constraints
~iiii~z~ iiiiii!i~i iiii~ili~2 ii~i~i~ii~5 iiii!~i~iiiliiie~ ii!~i~i~!!~ i~iiiiiiim~ i!iiiiliiii~ii!iiii!iiiiii~iiiiiliili!i!i!iii!i~W|~h i!i!iil!ii!iliiiii~iiii~~iiilii?ii!~iiii~i~iti~iSiiii~ii!i!liliiiiliriii~ii!itiii~ii!i!iiiiiiM i~iiiiO~!~ iiiiiiii~!ij~iiO iiii!iiiC~ iiiiiiiiitiiiiiIiiO~R ~ilili!iiiilii!i!iiiiii iiiiiiiiii~iiCO iiii!iiiiii~iiiliiii~iiii!iiii i~iiiiiii!iiiii!iii!ii!i!i~i~iiiiiii~ii~iii!iiii!ii!i!iiiiiiiiii i~iliiii~iiiiiiiiiiiiiiiiii!iiiiiil
12.5.1 General possibilities of discrete collocation constraintsquadrilaterals The possibility of using conventional interpolation to achieve satisfactory performance of mixed-type elements is limited, as is apparent from the preceding discussion. One feasible alternative is that of increasing the element order, and we have already observed that the cubic Lagrangian interpolation nearly satisfies the stability requirement and often performs well. 3'827 However, the complexity of the formulation is formidable and this direction is not often used. A different approach uses collocation constraints for the shear approximation on the element boundaries, thus limiting the number of S parameters and making the patch count more easily satisfied. This direction is indicated in the work of Hughes and Tezduyar, 35 Bathe and co-workers, 36'37 and Hinton and Huang, 38'39 as well as in generalizations by Zienkiewicz et al., 4~ and others. 41-48 The procedure bears a close relationship to the so-called DKT (discrete Kirchhoff theory) developed in Chapter 11 (see Sec. 11.18) and indeed explains why these, essentially thin plate, approximations are successful. The key to the discrete formulation is evident if we consider Fig. 12.8, where a simple bilinear element is illustrated. We observe that with a Co interpolation of q~ and w, the shear strain 0w
Ox +q~x
7X-
(12.16)
is uniquely determined at any point of the side 1-2 (such as point I, for instance) and
Sx interpolation 1
2
IV
Yt
4
3
X
z~ Sy node I> Sx node
SF interpolation
ii
Fig. 12.8 Collocation constraints on a bilinear element: independent interpolation of 5x and 5y.
397
398
'Thick' Reissner-Mindlin plates-irreducible and mixed formulations that hence [by Eq. (12.1b)]
Sx = OCyx
(12.17)
is also uniquely determined there. Thus, if a node specifying the shear resultant distribution were placed at that point and if the constraints [or satisfaction of Eq. (12.1b)] were only imposed there, then 1. the nodal value of Sx would be shared by adjacent elements (assuming continuity of a); 2. the nodal values of S~ would be prescribed if the t~ and # values were constrained as they are in the constrained patch test. Indeed if a, the shear rigidity, were to vary between adjacent elements the values of S~ would only differ by a multiplying constant and arguments remain essentially the same. The prescription of the shear field in terms of such boundary values is simple. In the case illustrated in Fig. 12.8 we interpolate independently
Sx = NsxSx
and
Sy -- NsySy
(12.18)
using the shape functions 1
Nsx
Yl - YlII
[Y - ylII,
1
x,,-x]
Nsy -Xll
~
Yl - Y]
XIV
(12.19)
as illustrated. Such an interpolation, of course, defines Ns of Eq. (12.11). The introduction of the discrete constraint into the analysis is a little more involved. We can proceed by using different (Petrov-Galerkin) weighting functions, and in particular applying a Dirac delta weighting or point collocation to Eq. (12.1b) in the approximate form. However, it is advantageous here to return to the constrained variational principle [see Eq. (11.83)] and seek stationarity of n = l f a (s
oLq da + ~l fa s T 1c~S d~2 - fa w q df2 + Hbt = stationary (12.20)
where the first term on the fight-hand side denotes the bending and the second the transverse shear energy. In the above we again use the approximations
S
NsS
Ns=[Nsx, Nsy]
(12.21)
subject to the constraint Eq. (12.1 b)" S = c~ (Vw + qS)
(12.22)
being applied directly in a discrete manner, that is, by collocation at such points as I to I V in Fig. 12.8 and appropriate direction selection. We shall eliminate S from the computation but before proceeding with any details of the algebra it is interesting to
Elements with discrete collocation constraints
~CC= 0/0 0/0
Or.R= 9/4 139 4/3
(a)
ok; = 3/4(F) 13c = 4
o.R = 24/12 13R = 12/8
(b)
Fig. 12.9 Patch test on (a) one and (b) four elements of the type given in Fig. 12.8. (Observe that in a constrained test boundary values of S are prescribed.) I
i
zk
O and w interpolation
Sxinterpolation
and collocation nodes
Syinterpolation
and collocation nodes
I
U
Typical shape function
Fig. 12.10 The quadratic Lagrangian element with collocation constraints on boundaries and in the internal domain .38,39
observe the relation of the element of Fig. 12.8 to the patch test, noting that we still have a mixed problem requiting the count conditions to be satisfied. (This indeed is the element of references 36 and 37.) We show the counts on Fig. 12.9 and observe that although they fail in the four-element assembly the margin is small here (and for larger patches, counts are satisfactory).* The results given by this element are quite good, as will be shown in Sec. 12.9. The discrete constraints and the boundary-type interpolation can of course be used in other forms. In Fig. 12.10 we illustrate the quadratic element of Huang and Hinton. 38'39 Here two points on each side of the quadrilateral define the shears Sx and Sy but in * Reference 37 reports a mathematical study of stability for this element.
399
400
'Thick' Reissner-Mindlin plates-irreducible and mixed formulations Table 12.3 Elements with collocation constraints: patch count. Degrees of freedom: [--1, w - 1" 0 , ~ b - 2; l~, S - 1" n, 4,,, - 1 Single-element patch Element
Four-element patch
Reference
o~c
/~c
O~R
/OR
O~C
/~C
O~R
~R
Q9D1238,39
g3
T4
24 12
y12
27 24
"9-24
72 40
40 24
Q9 D 10
3 2
T
2
24
10
27
16
72
32
(F) 0
0
]-6
21
-8-
8
1-6
15
-if
3-2
2~
Q8D8
i
o
-8-
7
T
5
8
60
24
Q5D6
2
3
2 T
12 T
6 4
15 1-2
12 -5-
36 2--0
20 1-2
Q4D436, 37
0
0
8
4
3
4
24
12
2-4
2--]
(F) Single-element patch
Six-element patch
T6D640
0
0 0
15 T
6 5
21 12
12 'if
43 24
24 23
T3D340,44
0 i
0 ~
9 ~
3 2
9 ~
6 T
45 T2
12 -C
F Failure to satisfy necessary conditions.
addition four internal parameters are introduced as shown. Now both the boundary and internal 'nodes' are again used as collocation points for imposing the constraints. The count for single-element and four-element patches is given in Table 12.3. This element only fails in a single-element patch under constrained conditions, and again numerical verification shows generally good performance. Details of numerical examples will be given later. It is clear that with discrete constraints many more altematives for design of satisfactory elements that pass the patch test are present. In Table 12.3 several quadrilaterals and triangles that satisfy the count conditions are illustrated. In the first a modification of the Hinton-Huang element with reduced intemal shear constraints is shown (second element). Here biquadratic 'bubble functions' are used in the interior shear component interpolation, as shown in Fig. 12.11. Similar improvements in the count can be achieved by using a serendipity-type interpolation, but now, of course, the distorted performance of the element may be impaired (for reasons we discussed in Volume 1, Sec. 9.7). Addition of bubble functions on all the w and q5 parameters can, as shown,
Elements with discrete collocation constraints 401
make the Bathe-Dvorkin fully satisfy the count condition. We shall pursue this further in Sec. 12.6. All quadrilateral elements can, of course, be mapped isoparametrically, remembering of course that components of shear S~ and S~ parallel to the ~, r/coordinates have to be used to ensure the preservation of the desirable constrained properties previously discussed. Such 'directional' shear interpolation is also essential when considering triangular elements, to which the next section is devoted. Before doing this, however, we shall complete the algebraic derivation of element properties.
12.5.2 Element matrices for discrete collocation constraints The starting point here will be to use the variational principle given by Eq. (12.20) with the shear variables eliminated directly. The application of the discrete constraints of Eq. (12.22) allows the 'nodal' parameters S defining the shear force distribution to be determined explicitly in terms of the and q5 parameters. This gives in general terms (12.23) in each element. For instance, for the rectangular element of Fig. 12.8 we can write S/=oL Sy
- c~
S Ill
- - oL
[/1)2-- /1)1+ ~xl + @x2] a
2
b
+
IV
-- c~
(12.24)
[l~3 -- tO4+ ~x3-~-~x4] a
Sy
2
2
[l~l -- l~4 ~Yl -t-~Y4] b
+
2
which can readily be rearranged into the form of Eq. (12.23) as
1 !a0 ab ! Qw-~-~ -b
where
b
0
0
0
0
gOl
SI l
.
Six I I
'
IV
Sy
1 0
1
0
!
1 0
Six ~ __
1!000 0 and Q ~ - ~
~g _
JPl
Ul)2
.
~1)3
'
flU4
~ _
~ 2
.
~ 3
'
~) 4
era)
402
'Thick' Reissner-Mindlin plates-irreducible and mixed formulations
/ N&'bubble'
Fig. 12.11 A biquadratic hierarchical bubble for 5x.
Including the above discrete constraint conditions in the variational principle of Eq. (12.20) we obtain
'L
H - ~
(EN~q~) T Ds
5d ~
+ ~l f ~ [N~ (Qe4~ ~ + Q~r)]Tc~[Ns (Qe q5 ~ + Qm~)] ds2 -
(12.25)
fa wq d~ + Hbt -- minimum
This is a constrained potential energy principle from which on minimization we obtain the system of equations
Kr
Kr162
re)
(12.26)
The element contributions are Kww -- QwKs~Qw T
K~
T
=
=
Kr162 -- ./o (EN4,)TD(E,N,~) dr2 +
Q~KssQ~
(12.27)
Kss - [
Jn
with the force terms identical to those defined in Eq. (12.8d). These general expressions derived above can be used for any form of discrete constraint elements described and present no computational difficulties. In the preceding we have imposed the constraints by point collocation of nodes placed on external boundaries or indeed the interior of the element. Other integrals could be used without introducing any difficulties in the final construction of the stiffness matrix. One could, for instance, require integrals such as
fr W [Ss -
Elements with discrete collocation constraints 403
on segments of the boundary, or +
d.
_
o
in the interior of an element. All would achieve the same objective, providing elimination of the Ss parameters is still possible. The use of discrete constraints can easily be shown to be equivalent to use of substitute shear strain matrices in the irreducible formulation of Eq. (12.8a). This makes introduction of such forms easy in standard computer programs. Details of such an approach are given by Ofiate et al. 47'48
12.5.3 Relation to the discrete Kirchhoff formulation In Chapter 11, Sec. 11.18, we have discussed the so-called discrete Kirchhoff theory (DKT) formulation for beams in which the Kirchhoff constraints [i.e. Eq. (12.22) with c~ = o0] were applied in a discrete manner. The reason for the success of such discrete constraints was not obvious previously, but we believe that the formulation presented here in terms of the mixed form fully explains its basis. It is well known that the study of mixed forms frequently reveals the robustness or otherwise of irreducible approaches. In Chapter 11 of reference 4 we explained why certain elements of irreducible form perform well as the limit of incompressibility is approached and why others fail. Here an analogous situation is illustrated. It is clear that every one of the elements so far discussed has its analogue in the DKT form. Indeed, the thick plate approach we have adopted here with c~ ~ c~ is simply a penalty approach to the DKT constraints in which direct elimination of variables was used. Many opportunities for development of interesting and perhaps effective plate elements are thus available for both the thick and thin range. We shall show in the next section some triangular elements and their DKT counterparts. Perhaps the whole range of the present elements should be termed 'QnDc' and 'TnDc' (discrete Reissner-Mindlin) elements in order to ease the classification. Here 'n' is the number of displacement nodes and 'c' the number of shear constraints.
12.5.4 Collocation constraints for triangular elements Figure 12.12 illustrates a triangle in which a straightforward quadratic interpolation of q5 and w is used. In this we shall take the shear forces to be given as a complete linear field defined by six shear force values on the element boundaries in directions parallel to these. The shear 'nodes' are located at Gauss points along each edge and the constraint collocation is made at the same position. Writing the interpolation in standard area coordinates 4 we have 3
S--ZtaSa a=l
(12.28)
404
'Thick' Reissner-Mindlin plates-irreducible and mixed formulations
/~
",~ "
~,d
/~
_ ~Snodes
\s,2*~
3 (a) The parameters (0 = 12 DOF, w= 6 DOF and S = 6 DOF)
2 e2 =
n0en,
v~ors
(b) .Area coordinates and not~ion
Fig. 12.12 TheT6D6triangularplateelement. where Sa are six parameters which can be determined by writing the tangential shear at the six constraint nodes. Introducing the tangent vector to each edge of the element as
eb --- 1,eby
(12.29)
a tangential component of shear on the b-edge (for which Lb = O) is obtained from Sb = S. eb
(12.30)
Evaluation of Eq. (12.30) at the two Gauss points (defined on interval 0 to 1) 1
1
gl = 2 ~ / ~ ( ~ / ~ - 1) and g2 = 243-~-(4r3t- 1)
(12.31)
yields a set of six equations which can be used to determine the six parameters a in terms of the tangential edge shears Sbl and Sba. The final solution for the shear interpolation then becomes
3 La [ ecy,--eby] {glSbl-]-g2Sb2} S --- Za=l ~ a --ecx, ebx.] glScl + g2Sc2
(12.32)
in which a, b, c is a cyclic permutation of 1, 2, 3. This defines uniquely the shape functions of Eq. (12.11) and, on application of constraints, expresses finally the shear
Elements with rotational bubble or enhanced modes
<1
J
Fig. 12.13 The T3D3 (discrete Reissner-Mindlin) triangle of reference 44 with w linear, q~ constrained quadratic, and 5b constant and parallel to sides.
field of nodal displacements ~ and rotations q5 in the manner of Eq. (12.23). The full derivation of the above expression is given in reference 40, and the final derivation of element matrices follows the procedures of Eqs (12.25)-(12.27). The element derived satisfies fully the patch test count conditions as shown in Table 12.3 as the T6D6 element. This element yields results which are generally 'too flexible'. An alternative triangular element which shows considerable improvement in performance is indicated in Fig. 12.13. Here the w displacement is interpolated linearly and 4) is initially a quadratic but is constrained to have linear behaviour along the tangent to each element edge. 44 Only a single shear constraint is introduced on each element side with the shear interpolation obtained from Eq. (12.32) by setting Sbl -- Sb2 = Sb
(12.33)
The 'count' conditions are again fully satisfied for single and multiple element patches as shown in Table 12.3. This element is of particular interest as it turns out to be the exact equivalent of the DKT triangle with 9 degrees of freedom which gave a satisfactory solution for thin plates. 49-51 Indeed, in the limit the two elements have an identical performance, though of course the T3D3 element is applicable also to plates with shear deformation. We note that the original DKT element can be modified in a different manner to achieve shear deformability 5= and obtain similar results. However, this element as introduced in reference 52 is not fully convergent.
As a starting point for this class of elements we may consider a standard functional of Reissner type given by FI - ~
(Eo)T DEO dr2
2
c~-lS dr2 + f~ S T (Vw + qS) dr2 (12.34)
-
J~ w q dr2 + I-Ibt -- stationary
in which approximations for w, q5 and S are required.
405
406
'Thick' Reissner-Mindlin plates-irreducible and mixed formulations
Three triangular elements designed by introducing 'bubble modes' for rotation parameters are found to be robust, and at the same time excellent performers. None of these elements is 'obvious', and they all use an interpolation of rotations that is of higher or equal order than that of w. Figure 12.14 shows the degree-of-freedom assignments for these triangular elements and the second part of Table 12.2 shows again their performance in patches. The quadratic element (T6S3B3) was devised by Zienkiewicz and Lefebvre 17 starting from a quadratic interpolation for w and q~. The shear S is interpolated by a complete
(a) Zienkiewicz and Lefebvre 17
(b) XU 32
(c) Arnold and Falk 34
O 2 rotation DOF (0) r-] 1 displacement DOF (~,) z~ 2 shear force DOF (S) Fig. 12.14 Three robust triangular elements: (a) the T6S3B3 element of Zienkiewicz and Lefebvre;17 (b) the T6S1B1 element of Xu;32 (c) the T3S1B1A element of Arnold and Falk.34
Elements with rotational bubble or enhanced modes 407
linear polynomial in each element, giving here six parameters, S. Three hierarchical quartic bubbles are added to the rotations giving the interpolation 6
r --- ~
3
ga(tk)~Da + ~
a=l
ANb(Lk)A~)b
(12.35)
b=l
where Na (Lk) are conventional quadratic isoparametric interpolations on the six-node triangle 4 and shape functions for the quartic bubble modes are given as
ANb(Lk) : Lb(L1L2L3) Thus, we have introduced six additional rotation parameters but have left the number of w parameters unaltered from those given by a quadratic interpolation. This element has very desirable properties and good performance when the integral for Kb in Eq. (12.13) is computed by using seven point quadrature 4 and the other integrals are computed by using four points. Some improvement is gained using a mixed form in which the bending moments are approximated as discontinuous quadratic interpolations for each component and the first integral in Eq. (12.34) is replaced as (F_.,q~)TDF_.,q~dr2 --~
(F_.,~b)TM dr2 -
(12.36)
All other terms in Eq. (12.34) remain the same. The mixed element computed in this way is denoted as T6S3B3M in subsequent results. We shall show later that optimal performance can be attained using 'linked' interpolation on this element. Since the T6S3B3 type elements use a complete quadratic to describe the displacement and rotation field, an isoparametric mapping may be used to produce curved-sided elements and, indeed, curved-shell elements. Furthermore, by design this type passes the count test and by numerical testing is proved to be quite robust when used to analyse both thick and thin plate problems. ~7'53 Since the w displacement interpolation is a standard quadratic interpolation the element may be joined compatibly to tetrahedral or prism solid elements which have six-node faces. We shall show later that optimal performance can be attained for this element by using 'linked' interpolation for w with additional enhanced strain modes - however, using this form the compatability between any attached tetrahedral elements is lost. A linear triangular element [T3S 1B 1 - Fig. 12.14(b)] with a total of 9 nodal degrees of freedom adds a single cubic bubble to the linear rotational interpolation and uses linear interpolation for w with constant discontinuous shear. This element satisfies all count conditions for solution (see Table 12.2); however, without further enhancements it locks as the thin plate limit is approached. 32 As we have stated previously the count condition is necessary but not sufficient to define successful elements and numerical testing is always needed. In a later section we discuss a 'linked interpolation' modification which also makes this element robust. A third element employing bubble modes [T3S 1B 1 A - Fig. 12.14(c)] was devised by Arnold and Falk. 34 It is of interest to note that this element uses a discontinuous (nonconforming) w interpolation with parameters located at the mid-side of each triangle. The rotation interpolation is a standard linear interpolation with an added cubic bubble. The shear interpolation is constant on each element. This element is a direct opposite of
408 'Thick' Reissner-Mindlin plates-irreducible and mixed formulations the triangular element of Morley discussed in Chapter 11 in that location of the displacement and rotation parameters is reversed. The location of the displacement parameters, however, precludes its use in combination with standard solid elements. Thus, this element is of little general interest. The introduction of successful bubble modes in quadrilaterals is more difficult. The first condition examined was the linear quadrilateral with a single bubble mode (Q4S1B1). For this element the patch count test fails when only a single element is considered but for assemblies above four elements it is passed and much hope was placed on this condition. 29'3~Unfortunately, one singular mode with a zero eigenvalue persists in all assemblies when the completely relaxed support conditions are considered. Despite this singularity the element does not lock and usually gives an excellent performance. 3~ To avoid, however, any singularity it is necessary to have at least three shear stress components and a similar number of rotation components of bubble form. No simple way of achieving a three-term interpolation exists but a successful four-component form was obtained by Auricchio and Taylor. 31 This four-term interpolation for shear is given by
s=
jo jo
jo jo
jo,~ jo~l jo,~ jo~j
s~ s3 S4
(12.37)
The Jacobian transformation jo is identical to that introduced when describing the Pian-Sumihara element 4,54 and is computed as
jo _ ~,~1~:~-o,
J~ - y,~l~:~=o,
jo _ x,~l~:~:o J~ y,n [~=,7=0
(12.38)
To satisfy Eq. (12.15a) it is necessary to construct a set of four bubble modes. An appropriate form is found to be ANbub
--
1Nb (~, ~l) I J~ -J~
--
j
_jo,
J~
jo 1 _ j o ~
-J~ jOlCj
(12.39)
in which j is the determinant of the Jacobian transformation J (i.e. not the determinant of j0) and Nbub -- (1 -- ~2)(1 -- ~ ) is a bubble mode. Thus, the rotation parameters are interpolated by using
4 0 -- ~Na~i)a + mNbub m~Dbub
Na are
a=l
(12.40)
the standard bilinear interpolations for the four-node quadrilateral. where The element so achieved (Q4S2B2) is stable.
iiiiijiiiiiiiiiiiiiiiiiiilili~i~ ~''ii!i'~'~~,ii'~ii'!,~iii'~i'!iii!i!i~:!il'~i~'~'~ii!~~' iii~'~'!i:i'i~!ii'i~iiiii~iiii~iliiiiiiiili~~i~j~ i~i!iiiiiiii~ li!iJi~iiiili!iiiii!i!iiiii~iii~iiii!i~!ii~ii!i~!i!ii!ii!!ii~!iii~!i~!ii~iiiii~Jiii!~!i~i!i~ii~iiii~i!ii!ii~i~i~i~i!i!ii~i~i~i~i~i!ii~iiii~i~i~i~i~ii~i!ij~~~~i~iii!i~ii~i~i!i~i~ii~ii!i In the previous section we outlined various procedures which are effective in ensuring the necessary count conditions and which are, therefore, essential to make the elements
Linked interpolation - an improvement of accuracy
'work' without locking or singularity. In this section we shall try to improve the interpolation used to increase the accuracy without involving additional parameters. The reader will here observe that in the primary interpolation we have used equalorder polynomials to interpolate both the displacement (w) and the rotations (05). Clearly, if we consider the limit of thin plate theory
0 {::}
(12.41)
and hence one order lower interpolation for q5 appears necessary. To achieve this we introduce here the concept of linked interpolation in which the primary expression is written as q 5 - Noq5 (12.42a) and w - Nw~ + N cq~
(12.42b)
This is precisely the form we obtained when seeking interpolations for the Timoshenko beam that gave exact interelement nodal results and forms the basis from which we now develop an interpolation for the Reissner-Mindlin plate theory. Such an expression ensures that a higher-order polynomial can be introduced for the representation of w without adding additional element parameters. This procedure can, of course, be applied to any of the elements we have listed in Tables 12.1 and 12.2 to improve the accuracy attainable. We shall here develop such linking for two types of elements in which the essential interpolations are linear on each edge and one element where they are quadratic. We thus improve the triangular T3S1B1 and by linking L to its formulation we arrive at T3S 1B 1L. The same procedure can, of course, be applied to the quadrilaterals Q4S 1B 1 and Q4S2B2 of which only the second is unconditionally stable and add the letter L. In the context of thick plates linked interpolation on a three-node triangle was introduced by Lynn and co-workers 55'56 and first extended to permit also thin plate analysis by X u . 32'57 Similar interpolations have also been used by Tessler and Hughes 58'59 and termed 'anisoparametric'. Additional presentations dealing with the simple triangular element with 9 degrees of freedom in its reduced form [see Eq. (12.66)] have been given by Taylor and co-workers. 33'44'6~ Quadrilateral elements employing linked interpolation have been developed by Crisfield, 61 Zienkiewicz and co-workers, 28'3~and Auricchio and Taylor. 31
12.7.1 Linking function for linear triangles and quadrilaterals Based on the linked form for beams given in Sec. 10.5.1 the function Nwr for the linear triangle T3 and the linear quadrilaterals may be expressed as nel
11) -- NaffJa -
-~ Z(Uu,r162 a=l
-- Csc)
(12.43)
409
410 'Thick' Reissner-Mindlin plates-irreducible and mixed formulations
where nel is the number of vertex nodes on the element (i.e. 3 or 4), lbc is the length of the b-c side, ~sb is the rotation at node b in the tangent direction of the ath side, and (Nwq~) a are shape functions defining the quadratic w along the side but still maintaining zero w at comer nodes. For the triangular element these are the shape functions identical to those arising in the plane six-node element at mid-side nodes and are given by 4
LzL3,
N ~ q s = 4 [L1L2,
L3L1]
(12.44)
and for the quadrilateral element these are the shape functions for the eight-node serendipity functions given by N 0_~1 [(1-~2)(1-~),
(l+O(1-~),
(1-~2)(1+~),
(1-0(1-~)] (12.45)
Example 12.1 Derivation of linked function for three-node
triangle
The development of a shape function for the three-node triangular element with a total of 9 degrees of freedom is developed using the fact that both shear and moment in the equivalent Timoshenko beam form were constant. A process to develop a linked interpolation for the plate transverse displacement, w, can start from a full quadratic expansion written in hierarchical form. Thus, for a triangle we have the interpolation in area coordinates w = Lltbl + L2t02 q" L3t03 -if-4L1L2c~I + 4L2L3c~2 -k-4L3Llt~3
(12.46)
where l~ a a r e the nodal displacements and (~a a r e hierarchical parameters. The hierarchical parameters are then expressed in terms of rotational parameters. Along any edge, say the 1-2 edge (where L3 = 0), the displacement is given by w = Llff~l + L2to2 -q-4L1L2C~l
(12.47)
The expression used to eliminate c~ is deduced by constraining the transverse edge shear to be a constant. Along the edge the transverse shear is given by
Ow
712 -- ~
+ (fis
(12.48)
where s is the coordinate tangential to the edge and ~bs is the component of the rotation along the edge. The derivative of Eq. (12.47) along the edge is given by 01/)
OS
=
/ ~ 2 - tO1 112
+4
L 2 - L1 ~ c~1 112
(12.49)
where 1~2 is the length of the 1-2 side. Assuming a linear interpolation for 4~s along the edge we have C~s -- L l ~ l + L2~s2 (12.50) which, after noting that L1 + L2 -- 1 along the edge, may also be expressed as qSs = i~(~s, + ~s2) + il(~s, - ~s2)(L2 - L,)
(12.51)
Linked interpolation - an improvement of accuracy
The transverse shear may now be given as ")/12 =
W2 -- 1/31 +
/12
1(~sl
1
+ ~s2) +
~1 + ~ ( ~ s l -- ~s2)
] (L2
-- L1)
(12.52)
Constraining the strain to be constant gives
l,~ (~s, - ~s=)
(~1 =
(12.53)
8
yielding the 'linked' edge interpolation
1 w -- Lltbl + L2t~2 + 5ll2L1L2(~sl
- ~s:)
(12.54)
The normal rotations may now be expressed in terms of the nodal Cartesian components by using t~s -- COS ?/)12~)x + sin ~12~)y (12.55) where ~12 is the angle that the normal to the edge makes with the x axis. Repeating this process for the other two edges gives the final interpolation for the transverse displacement. A similar process can be followed to develop the linked interpolations for the quadrilateral element. The reader can verify that the use of the constant 1/8 ensures that constant shear strain on the element side occurs. Further, a rigid body rotation with ~a -- ~b in the element causes no straining. Finally, with rotation (~a being the same for adjacent elements Co continuity is ensured.
12.7.2 Linked interpolation for quadratic elements The interpolation for the quadratic element shown in Fig. 12.15 proceeds in a similar way to the linear form just presented. The development of the quadratic element, which
~3
I--1 w
~x
~y
C) wb I-1
Fig. 12.15 Quadratic linked element.
411
412
'Thick' Reissner-Mindlin plates-irreducible and mixed formulations is similar to the Zienkiewicz-Lefebvre element described above in Sec. 12.6, was presented by Auricchio and Lovadina. 62 The element was extended to include anisotropic materials and coupling with in-plane displacements by Taylor and Govindjee. 63 The development starts with quadratic interpolation for the displacement w and rotation ~b. The transverse displacement is then increased to a complete cubic using the linked concept in which w is written as 6
tO -- ~
Na ( L k ) ffOa -t- Nbub/~bub
a=l
1
(12.56)
3
+ ~ ~
labLaLb(Lb -- La)tT(~a
-k- @b -- 2 @a+3); a -- 1, 2, 3; b -- 2, 3, 1
a=l
with
(12.57)
Nbub ~-~ L1L2L3
Here//)bub is an hierarchical parameter. We note that the bubble mode can be retained as a displacement degree of freedom or its gradient added as an enhanced mode to shearing strains. As noted above the rotation interpolation is given by the quadratic form 6
~p -- ~
(12.58)
Na (Lk)~t~a
a=l
To satisfy the mixed patch test the strains for the linked element need additional enhanced terms which are computed from the derivatives and values of additional displacement functions. Accordingly, we consider the enhanced curvature given by
(t)en
"0Na
Ox
Ox X =
Xv
XxY
0
-Z6 a=l
ONa
.Oy where
,~
X
~X
ONa
en
ONa
(12.59)
Ox
Oy
n
Ox
3
(/)en __ Z ( L b
~)Nbub/~ -1- (VNbub)Nbub ~bub
(12.60)
b=l
in which/3 and (~bubadd seven parameters. The enhanced transverse shear strain is given by
0113 ")/ --
")/y
0//3
-t- dpen
N++y
where we assume that Wbub is treated as a displacement parameter.
(12.61)
Discrete 'exact' thin plate limit Finally, the shear interpolation is given by
S .+-y~ 3LaSa + ~rNbubSbub
(12.62)
a=l
For the form given above each node has 3 degrees of freedom (~a, Cxa and Cya), one internal nw, seven internal nr and seven internal nx. The element satisfies the count condition of the mixed patch test for all element configurations including a single element with all external degrees of freedom restrained.
Example 12.2 Clampedplate by linked elements
For a clamped plate a solution based on the Reissner-Mindlin plate theory has been given by Chinosi and Lovadina. 64 Using an inverse method we may write the solution as with
LO --- //)b -~- tOs //)b-
~1
x3 (x
__
r
= --V//) b
1)3 y3 ( y _ 1)3
t2
Ws = - 6a(1 - u)
V2//)b
which gives the load q -- D [12y(y - 1)(5x(x - 1) + 1)(2y2(y - 1) 2 + x(x - 1)(5y(y - 1) + 1) + 12x(x - 1)(5y(y - 1) + 1)(2x2(x - 1) 2 + y(y - 1)(5x(x - 1) + 1)] This solution is used to illustrate the convergence of the energy error for a clamped plate as shown in Fig. 12.16.
Discretization of Eq. (12.34) using interpolations of the form* S - NsS
w - N~,~ + N~+q~,
(12.63)
where here we use the abbreviated notation b (instead of bub) for bubble modes, leads to the algebraic system of equations
o
o
o
KL]
Kbr Kbb KTb] Ksw
K~
Ksb Ks~
b S
0
(12.64)
0
where, for simplicity, only the forces fw and fr due to transverse load q and boundary conditions are included [see Eq. (12.8d)]. The arrays appearing in Eq. (12.64) are given by * The term Nwr will be exploited in the next part of this section and thus is included for completeness.
413
414
'Thick' Reissner-Mindlin
plates-irreducible
and mixed formulations
Kr162= ~ (s162162
dr2,
Kss = - fa Ns~
Kbr -- jf~ (s162
dr2,
Ksr f~ NT [VNwr N~] dr2,
Kbb = fa (s
dS2,
Ksb -- --
dr2,
(12.65)
N s Nb dr2,
Ksw = ~ Ns~VNwdr2 Adopting a static condensation process at the element level 65 in which the internal rotational parameters are eliminated first, followed by the shear parameters, yields the element stiffness matrix in terms of the element @ and q~ parameters given by T
-1
T -1 -KswAss Asr
KswAss Ks~
] { @/
T -1 T -1 [.-As~Ass Ksw -ar162 + AscAss Asc]
q~
( -
fw /
(12.66)
fr
in which KsbK~-~KTb- Kss,
Ass Ar162-
T
-1
K b c K b b K b r --
Asr = KsbKbb~Kbr -- Ksr
(12.67)
Kr162
This solution strategy requires the inverse of Kbb and Ass only. In particular, we note that the inverse of Ass can be performed even if Kss is zero (provided the other term is non-singular). The vanishing of Kss defines the thin plate limit. Thus, the above
10- 5
:...:..:.-.: ! ! ! ! ! ! ! !~!! ~ !~! ~! !! ! !! ! .:.!~!-.:!!~!. . !. . !. .!. .! ! ! ! !: .!. !. . ! ! :! ~ .......
........
.~. ....
".....
".....
N" " "" : "" "'" ........
::::::::::::::::::::::::::::::::::::
t_ o ID >, I1) tLU I
tu
::::::::.:::.'.::: ........ : ..... ! ! ~: ! ! ! ! ~ ~
10-7
" ..............
:::::::::::::::::::::::::::::::::::::::::::::::
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Elements/side
Energy error for quadratic linked element. Clamped plate.
. . . . . . . .
!i!i! 10 2
Performance of various 'thick' plate elements - limitations of thin plate theory
strategy leads to a solution process in which the thin plate limit is defined without recourse to a penalty method. Indeed, all terms in the process generally are not subject to ill-conditioning due to differences in large and small numbers. In the context of thick and thin plate analysis this solution strategy has been exploited with success in references 31 and 33.
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The performance of both 'thick' and 'thin' elements is frequently compared for the examples of clamped and simply supported square plates, though, of course, more stringent tests can and should be devised. Figure 12.17(a)-12.17(d) illustrates the behaviour of various elements we have discussed in the case of a span-to-thickness ratio (L/t) of 100, which generally is considered within the scope of thin plate theory. The results are indeed directly comparable to those of Fig. 11.16 of Chapter 11, and it is evident that here the thick plate elements perform as well as the best of the thin plate forms. It is of interest to note that in Fig. 12.17 we have included some elements that do not fully pass the patch test and hence are not robust. Many such elements are still used as their failure occurs only occasionally - although new developments should always strive to use an element which is robust. All 'robust' elements of the thick plate kind can be easily mapped isoparametrically and their performance remains excellent and convergent. Figure 12.18 shows isoparametric mapping used on a curved-sided mesh in the solution of a circular plate for two elements previously discussed. Obviously, such a lack of sensitivity to distortion will be of considerable advantage when shells are considered, as we shall show in Chapter 15. Of course, when the span-to-thickness ratio decreases and thus shear deformation importance increases, the thick plate elements are capable of yielding results not obtainable with thin plate theory. In Table 12.4 we show some results for a simply supported, uniformly loaded plate for two L/t ratios and in Table 12.5 we show results for the clamped uniformly loaded plate for the same ratios. In this example we show also the effect of the hard and soft simple support conditions. In the hard support we assume just as in thin plates that the rotation along the support (~bs) is zero. In the soft support case we take, more rationally, a zero twisting moment along the support (see Chapter 11, Sec. 11.2.2). It is immediately evident that: 1. the thick plate (L/t = 10) shows deflections converging to very different values depending on the support conditions, both being considerably larger than those given by thin plate theory; 2. for the thin plate (L/t = 1000) the deflections converge uniformly to the thin (Kirchhoff) plate results for hard support conditions, but for soft support conditions give answers about 0.2% higher in centre deflection.
415
416
'Thick' Reissner-Mindlin plates-irreducible and mixed formulations 1.10 DRM
1.05
~
a-T'_~_ ~ i
M= 2
1.00
0.95
0.90
i/
I
1
2
I
I
3 4 Mesh density M
I
I
5
I
6
I
I
78910
(a) Centre displacement normalized with respect to thin plate theory for simply supported, uniformly loaded square plate
1.10 L. r~
i
!
J,,q
1.05
1 .OO
0.95
0.90 t/ 1
t
#/t . ' / 2
" / If I 3 4 Mesh density M
I 5
I 6
I I I 78910
I
(b) Moment at Gauss point nearest centre (or central point) normalized by centre moment of thin plate theory for simply supported, uniformly loaded square plate Fig. 12.17 Convergencestudy for relatively thin plate (Lit = 100)' (a) centre displacement (simply supported, uniform load, square plate); (b) moment at Gauss point nearest centre (simply supported, uniform load, square plate). Tables 12.1 and 12.2 give keys to elements used.
It is perhaps an insignificant difference that occurs in this example between the support conditions but this can be more pronounced in different plate configurations. In Fig. 12.19 we show the results of a study of a simply supported rhombic plate with L/t = 100 and 1000. For this problem an accurate Kirchhoff plate theory solution
Performance of various 'thick' plate elements - limitations of thin plate theory
1.10
i!
44 44 44
\ '4
1.05
,J "7
a
--
1.00 Q4S 0.95
0.90
1
2
3 4 Mesh density M
5
6
78910
(C) Centre displacement normalized with respect to thin plate theory for clamped, uniformly loaded square plate
1.10
i
!
J "7
1.05
=2
1.00
Q9R 0.95 t ## #
0.90
1
2
3 4 Mesh density M
5
6
78910
(d) Moment at Gauss point nearest centre (or central point) normalized by centre moments of thin plate theory for clamped, uniformly loaded square plate Fig. 12.17 Cont. (c) Centre displacement (clamped, uniform load, square plate); (d) moment at Gauss point nearest centre (clamped, uniform load, square plate).
is available, 66 but as will be noticed the thick plate results converge uniformly to a displacement nearly 4% in excess of the thin plate solution for all the L / t = 1O0 cases. This problem is illustrative of the substantial difference that can on occasion arise in situations that fall well within the limits assumed by conventional thin plate theory
417
418
'Thick' Reissner-Mindlin plates- irreducible and mixed formulations
NEL = 2
NEL =
NEL = 24
NEL = 6
(a)
~24
c_,. 60 i--(~
"E0
"E
~.
(b)
e~m 092040 p~,,,,..,., a 0 Q9D:1234. :] 35
.c,. 6
~oo2oo~~
Degrees of freedom
~.
o T6S3B3 ~s
~i: 5oloo2oo~4oo
Degrees of freedom
Fig. 12.18 Mapped curvilinear elements in solution of a circular clamped plate under uniform load' (a) meshes used; (b) percentage error in centre displacement and moment.
Table 12.4 Centre displacement of a simply supported plate under uniform load for two L/t ratios; E = 1 0 . 9 2 , / / = 0.3, L = 10, q = 1
L/t Mesh, M
-- 10; w x 10 -1
L/t
= 1000; w x 10 -7
hard support
soft support
hard support
soft support
4.2626 4.2720 4.2727 4.2728 4.2728 4.2728
4.6085 4.5629 4.5883 4.6077 4.6144
4.0389 4.0607 4.0637 4.0643 4.0644 4.0624
4.2397 4.1297 4.0928 4.0773 4.0700
2 4 8 16 32 Series
Table 12.5 Centre displacement of a clamped plate under uniform load for two L / t ratios; E = 10.92,/) = 0.3, L = 10, q = 1 ,,
Mesh, 2 4 8 16 32 Series
M
L/t
= 10; w x 10 - ] 1.4211 1.4858 1.4997 1.5034 1.5043 1.49929
L/t
= 1000; w • 10 -7 1.1469 1.2362 1.2583 1.2637 1.2646 1.2653
Inelastic material
100
300
Number of degrees of freedom -N 700 1000 3000
I
0.43
-
0.42
I
I
behaviour
7000 10000
I
I
I
L/t = 100, t h r e e - d i m e n s i o n a l solution s2
. . . . / / Kirchhoff plate theorysl S Benchmark
L/t = 100
L/t = 1000
0.41
0.40
-
L/t= 100
zx On uniformly
x
0.39
-
0.38
-
0.37
-
[] O n adaptively refined m e s h of Fig. 5.19
E = 103 v=0.3
L.
r 0.36
L , I
2.0
subdivided
mesh
2.5
,._I
"I I
3.0
,
Log
I
N
3.5
I
4.0
Fig. 12.19 Skew30~ simply supported plate (soft support); maximum deflection at A, the centre for various degrees of freedom N. The triangular element of reference 17 is used.
(L/t = 100), and for this reason the problem has been thoroughly investigated by Babu~ka and Scapolla, 67 who solve it as a fully three-dimensional elasticity problem using support conditions of the 'soft' type which appear to be the closest to physical reality. Their three-dimensional result is very close to the thick plate solution, and confirms its validity and, indeed, superiority over the thin plate forms. However, we note that for very thin plates, even with soft support, convergence to the thin plate results occurs.
We have discussed in some detail the problem of inelastic behaviour in Sec. 11.19 of Chapter 11. The procedures of dealing with the same situation when using the
419
420
'Thick' Reissner-Mindlin plates -irreducible and mixed formulations
Reissner-Mindlin theory are nearly identical and here we will simply refer the reader to the literature on the subject 68'69 and to the previous chapter.
The simplicity of deriving and using elements in which independent interpolation of rotations and displacements is postulated and shear deformations are included assures popularity of the approach. The final degrees of freedom used are exactly the same type as those used in the direct approach to thin plate forms in Chapter 14, and at no additional complexity shear deformability is included in the analysis. If care is used to ensure robustness, elements of the type discussed in this chapter are generally applicable and indeed could be used with similar restrictions to other finite element approximations requiting C1 continuity in the limit. The ease of element distortion will make elements of the type discussed here the first choice for curved element solutions and they can easily be adapted to non-linear material behaviour. Extension to geometric non-linearity is also possible; however, in Y i
i
32 elements q = 32.90% e= 1.09
1.o
-
A ~
,
..]
,
,-
v
I
Mesh 1 ........
SSl 98 elements SSl
SSI
;
6.39%
= 1.26
SSI Soft support Mesh 2
902 elements
4.24% 1.18
Mesh 3 Fig. 12.20 Simply supported 30 ~ skew plate with uniform load (problem of Fig. 12.19); adaptive analysis to achieve 5% accuracy; Lit = 100, ~, = 0.3, six-node element T6S3B3;17 ~ = effectivity index, r/= percentage error in energy norm of estimator.
References
-0.5
100
200
I
I
Number of degrees of freedom - N 500 1000 3000 5000 10000 I
I
I
I
30000
I
I
-
50.0
-
30.0
- 40.0
Uniform refinement -1.0
- 20.0
0.35
t-
, m
-
= -1.5
v
Adaptive mesh refinement
o
$
-5.o
o
1.0 "
-
1.0
-2.0
I 2.0
I 2.5
I 3.0 log N
I 3.5
3.0
-2.0 -
-2.5
10.0 o~"
I 4.0
-.~ m
rr
1.0
! 4.5
Fig. 12.21 Energy norm rate of convergence for the 30 ~ skew plate of Fig. 12.19 for uniform and adaptive refinement; adaptive analysis to achieve 5% accuracy.
this case the effects of in-plane forces must be included and this renders the problem identical to shell theory. We shall discuss this more in Chapter 11. In Chapters 14 and 15 of Volume 1 we discussed the need for an adaptive approach in which error estimation is used in conjunction with mesh generation to obtain answers of specified accuracy. Such adaptive procedures are easily used in plate bending problems with an almost identical form of error estimation. 7~ In Figs 12.20 and 12.21 we show a sequence of automatically generated meshes for the problem of the skew plate. It is of particular interest to note: 1. the initial refinement in the vicinity of comer singularity; 2. the final refinement near the simply support boundary conditions where the effects of transverse shear will lead to a 'boundary layer'. Indeed, such boundary layers can occur near all boundaries of shear deformable plates and it is usually found that the shear error represents a very large fraction of the total error when approximations are made.
1. S. Ahmad, B.M. Irons and O.C. Zienkiewicz. Curved thick shell and membrane elements with particular reference to axi-symmetric problems. In Proc. 2nd Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-68-150, pages 539-572, Wright Patterson Air Force Base, Ohio, October 1968. 2. S. Ahmad, B.M. Irons and O.C. Zienkiewicz. Analysis of thick and thin shell structures by curved finite elements. International Journal for Numerical Methods in Engineering, 2:419--451, 1970.
421
422
'Thick' Reissner-Mindlin plates- irreducible and mixed formulations 3. S. Ahmad. Curved finite elements in the analysis of solids, shells and plate structures. PhD thesis, Department of Civil Engineering, University of Wales, Swansea, 1969. 4. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann, Oxford, 6th edition, 2005. 5. O.C. Zienkiewicz, J. Too and R.L. Taylor. Reduced integration technique in general analysis of plates and shells. International Journal for Numerical Methods in Engineering, 3:275-290, 1971. 6. S.E Pawsey and R.W. Clough. Improved numerical integration of thick slab finite elements. International Journal for Numerical Methods in Engineering, 3:575-586, 1971. 7. O.C. Zienkiewicz and E. Hinton. Reduced integration, function smoothing and non-conformity in finite element analysis. J. Franklin Inst., 302:443-461, 1976. 8. E.D.L. Pugh, E. Hinton and O.C. Zienkiewicz. A study of quadrilateral plate bending elements with reduced integration. International Journal for Numerical Methods in Engineering, 12: 1059-1079, 1978. 9. E Gruttmann and W. Wagner. A stabilized one-point integrated quadrilateral Reissner-Mindlin plate element. International Journal for Numerical Methods in Engineering, 61:2273-2295, 2004. 10. T.J.R. Hughes, R.L. Taylor and W. Kanoknukulchai. A simple and efficient finite element for plate bending. International Journal for Numerical Methods in Engineering, 11:1529-1543, 1977. 11. T.J.R. Hughes and M. Cohen. The 'heterosis' finite element for plate bending. Computers and Structures, 9:445-450, 1978. 12. T.J.R. Hughes, M. Cohen and M. Harou. Reduced and selective integration techniques in the finite element analysis of plates. Nuclear Engineering and Design, 46:203-222, 1978. 13. E. Hinton and N. Biranir. A comparison of Lagrangian and serendipity Mindlin plate elements for free vibration analysis. Computers and Structures, 10:483-493, 1979. 14. R.D. Cook. Concepts and Applications of Finite Element Analysis. John Wiley & Sons, Chichester, 1982. 15. R.L. Taylor, O.C. Zienkiewicz, J.C. Simo and A.H.C. Chan. The patch t e s t - a condition for assessing FEM convergence. International Journal for Numerical Methods in Engineering, 22:39-62, 1986. 16. O.C. Zienkiewicz, S. Qu, R.L. Taylor and S. Nakazawa. The patch test for mixed formulations. International Journal for Numerical Methods in Engineering, 23:1873-1883, 1986. 17. O.C. Zienkiewicz and D. Lefebvre. A robust triangular plate bending element of the ReissnerMindlin type. International Journal for Numerical Methods in Engineering, 26:1169-1184, 1988. 18. O.C. Zienkiewicz, J.P. Vilotte, S. Toyoshima and S. Nakazawa. Iterative method for constrained and mixed approximation. An inexpensive improvement of FEM performance. Computer Methods in Applied Mechanics and Engineering, 51:3-29, 1985. 19. D.S. Malkus and T.J.R. Hughes. Mixed finite element methods in reduced and selective integration techniques: a unification of concepts. Computer Methods in Applied Mechanics and Engineering, 15:63-81, 1978. 20. B. Fraeijs de Veubeke. Displacement and equilibrium models in finite element method. In O.C. Zienkiewicz and G.S. Holister, editors, Stress Analysis, Chapter 9, pages 145-197. John Wiley & Sons, Chichester, 1965. 21. K.-J. Bathe. Finite Element Procedures. Prentice-Hall, Englewood Cliffs, NJ, 1996. 22. W.X. Zhong. FEM patch test and its convergence. Technical Report 97-3001, Research Institute Engineering Mechanics, Dalian University of Technology, 1997 (in Chinese). 23. W.X. Zhong. Convergence of FEM and the conditions of patch test. Technical Report 97-3002, Research Institute Engineering Mechanics, Dalian University of Technology, 1997 (in Chinese).
References 423 24. I. Babu~ka and R. Narasimhan. The Babu~ka-Brezzi condition and the patch test: an example. Computer Methods in Applied Mechanics and Engineering, 140:183-199, 1997. 25. O.C. Zienkiewicz and R.L. Taylor. The finite element patch test revisited: a computer test for convergence, validation and error estimates. Computer Methods in Applied Mechanics and Engineering, 149:523-544, 1997. 26. T.J.R. Hughes and W.K. Liu. Implicit-explicit finite elements in transient analyses. Part I and Part II. J. Appl. Mech., 45:371-378, 1978. 27. K.-J. Bathe and L.W. Ho. Some results in the analysis of thin shell structures. In W. Wunderlich et al., editors, Nonlinear Finite Element Analysis in Structural Mechanics, pages 122-156. Springer-Verlag, Berlin, 1981. 28. O.C. Zienkiewicz, M. Huang, J. Wu and S. Wu. A new algorithm for the coupled soil-pore fluid problem. Shock and Vibration, 1:3-14, 1993. 29. O.C. Zienkiewicz, Z. Xu, L.F. Zeng, A. Samuelsson and N.-E. Wiberg. Linked interpolation for Reissner-Mindlin plate elements. Part I - a simple quadrilateral element. International Journal for Numerical Methods in Engineering, 36:3043-3056, 1993. 30. Z. Xu, O.C. Zienkiewicz and L.F. Zeng. Linked interpolation for Reissner-Mindlin plate elements. Part I I I - an alternative quadrilateral. International Journal for Numerical Methods in Engineering, 36:3043-3056, 1993. 31. F. Auricchio and R.L. Taylor. A shear deformable plate element with an exact thin limit. Computer Methods in Applied Mechanics and Engineering, 118:393-412, 1994. 32. Z. Xu. A simple and efficient triangular finite element for plate bending. Acta Mechanica Sinica, 2:185-192, 1986. 33. F. Auricchio and R.L. Taylor. A triangular thick plate finite element with an exact thin limit. Finite Elements in Analysis and Design, 19:57-68, 1995. 34. D.N. Arnold and R.S. Falk. A uniformly accurate finite element method for Mindlin-Reissner plate. Technical Report IMA Preprint Series No. 307, Institute for Mathematics and its Application, University of Maryland, 1987. 35. T.J.R. Hughes and T. Tezduyar. Finite elements based upon Mindlin plate theory with particular reference to the four node bilinear isoparametric element. J. Appl. Mech., 46:587-596, 1981. 36. E.N. Dvorkin and K.-J. Bathe. A continuum mechanics based four node shell element for general non-linear analysis. Engineering Computations, 1:77-88, 1984. 37. K.-J. Bathe and A.B. Chaudhary. A solution method for planar and axisymmetric contact problems. International Journal for Numerical Methods in Engineering, 21:65-88, 1985. 38. H.C. Huang and E. Hinton. A nine node Lagrangian Mindlin element with enhanced shear interpolation. Engineering Computations, 1:369-380, 1984. 39. E. Hinton and H.C. Huang. A family of quadrilateral Mindlin plate elements with substitute shear strain fields. Computers and Structures, 23:409-431, 1986. 40. O.C. Zienkiewicz, R.L. Taylor, P. Papadopoulos and E. Ofiate. Plate bending elements with discrete constraints; new triangular elements. Computers and Structures, 35:505-522, 1990. 41. K.-J. Bathe and F. Brezzi. On the convergence of a four node plate bending element based on Mindlin-Reissner plate theory and a mixed interpolation. In J.R. Whiteman, editor, The Mathematics of Finite Elements and Applications, volume V, pages 491-503. Academic Press, London, 1985. 42. H.K. Stolarski and M.Y.M. Chiang. On a definition of the assumed shear strains in formulation of the C Oplate elements. European Journal of Mechanics, A/Solids, 8:53-72, 1989. 43. H.K. Stolarski and M.Y.M. Chiang. Thin plate elements with relaxed Kirchhoff constraints. International Journal for Numerical Methods in Engineering, 26:913-933, 1988. 44. P. Papadopoulos and R.L. Taylor. A triangular element based on Reissner-Mindlin plate theory. International Journal for Numerical Methods in Engineering, 30:1029-1049, 1990. 45. R.S. Rao and H.K. Stolarski. Finite element analysis of composite plates using a weak form of the Kirchhoff constraints. Finite Elements in Analysis and Design, 13:191-208, 1993.
424
'Thick' Reissner-Mindlin plates-irreducible and mixed formulations 46. K.-J. Bathe, M.L. Bucalem and E Brezzi. Displacement and stress convergence of four MITC plate bending elements. Engineering Computations, 7:291-302, 1990. 47. E. Ofiate, R.L. Taylor and O.C. Zienkiewicz. Consistent formulation of shear constrained Reissner-Mindlin plate elements. In C. Kuhn and H. Mang, editors, Discretization Methods in Structural Mechanics, pages 169-180. Springer-Verlag, Berlin, 1990. 48. E. Ofiate, O.C. Zienkiewicz, B. Su~irez and R.L. Taylor. A general methodology for deriving shear constrained Reissner-Mindlin plate elements. International Journal for Numerical Methods in Engineering, 33:345-367, 1992. 49. G.S. Dhatt. Numerical analysis of thin shells by curved triangular elements based on discrete Kirchhoff hypotheses. In W.R. Rowan and R.M. Hackett, editors, Proc. Symp. on Applications of FEM in Civil Engineering, Vanderbilt University, Nashville, Tennessee, 1969. ASCE. 50. J.L. Batoz, K.-J. Bathe and L.W. Ho. A study of three-node triangular plate bending elements. International Journal for Numerical Methods in Engineering, 15:1771-1812, 1980. 51. J.L. Batoz. An explicit formulation for an efficient triangular plate bending element. International Journal for Numerical Methods in Engineering, 18:1077-1089, 1982. 52. J.L. Batoz and E Lardeur. A discrete shear triangular nine d.o.f, element for the analysis of thick to very thin plates. International Journal for Numerical Methods in Engineering, 28:533-560, 1989. 53. T. Tu. Performance of Reissner-Mindlin elements. PhD thesis, Rutgers University, Department of Mathematics, 1998. 54. T.H.H. Pian and K. Sumihara. Rational approach for assumed stress fnite elements. International Journal for Numerical Methods in Engineering, 20:1685-1695, 1985. 55. L.E Greimann and EE Lynn. Finite element analysis of plate bending with transverse shear deformation. Nuclear Engineering and Design, 14:223-230, 1970. 56. EE Lynn and B.S. Dhillon. Triangular thick plate bending elements. In Proceedings 1st International Conference on Structural Mechanics in Reactor Technology, page M 6/5, Berlin, 1971. 57. Z. Xu. A thick-thin triangular plate element. International Journal for Numerical Methods in Engineering, 33:963-973, 1992. 58. A. Tessler and T.J.R. Hughes. A three node Mindlin plate element with improved transverse shear. Computer Methods in Applied Mechanics and Engineering, 50:71-101, 1985. 59. A. Tessler. A C o anisoparametric three node shallow shell element. Computer Methods in Applied Mechanics and Engineering, 78:89-103. 60. R.L. Taylor and E Auricchio. Linked interpolation for Reissner-Mindlin plate elements: Part I I a simple triangle. International Journal for Numerical Methods in Engineering, 36:3057-3066, 1993. 61. M.A. Crisfield. Non-linear Finite Element Analysis of Solids and Structures, volume 1. John Wiley & Sons, Chichester, 1991. 62. E Auricchio and C. Lovadina. Analysis of kinematic linked interpolation methods for ReissnerMindlin plate problems. Computer Methods in Applied Mechanics and Engineering, 190: 2465-2482, 2001. 63. R.L. Taylor and S. Govindjee. A quadratic linked plate element with an exact thin plate limit. Technical Report UCB/SEMM-02/10, University of California, Berkeley, November 2002. 64. C. Chinosi and C. Lovadina. Numerical analysis of some mixed finite element methods for Reissner-Mindlin plates. Computational Mechanics, 16:36-44, 1995. 65. E.L. Wilson. The static condensation algorithm. International Journal for Numerical Methods in Engineering, 8:1974, 199-203. 66. L.S.D. Morley. Skew Plates and Structures. Macmillan, New York, 1963. International Series of Monographs in Aeronautics and Astronautics. 67. I. Babu~ka and T. Scapolla. Benchmark computation and performance evaluation for a rhombic plate bending problem. International Journal for Numerical Methods in Engineering, 28: 155-180, 1989.
References 425 68. E Papadopoulos and R.L. Taylor. Elasto-plastic analysis of Reissner-Mindlin plates. Appl. Mech. Rev., 43(5):$40-$50, 1990. 69. E Papadopoulos. On the Finite Element Solution of General Contact Problems. PhD dissertation, Department of Civil Engineering, University of California at Berkeley, Berkeley, USA, 1991. 70. O.C. Zienkiewicz and J.Z. Zhu. Error estimation and adaptive refinement for plate bending problems. International Journal for Numerical Methods in Engineering, 28:2839-2853, 1989.
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is
a if
shell is, in essence, a structure that can be derived from a plate by initially forming the middle surface as a singly (or doubly) curved surface. The same assumptions as used in thin plates regarding the transverse distribution of strains and stresses are again valid. However, the way in which the shell supports external loads is quite different from that of a fiat plate. The stress resultants acting on the middle surface of the shell now have both tangential and normal components which carry a major part of the load, a fact that explains the economy of shells as load-carrying structures and their well-deserved popularity. The derivation of detailed governing equations for a curved shell problem presents many difficulties and, in fact, leads to many alternative formulations, each depending on the approximations introduced. For details of classical shell treatment the reader referred to standard texts on the subject, for example the well-known treatise by Fltigge 1 or the classical book by Timoshenko and Woinowski-Krieger. 2 In the finite element treatment of shell problems to be described in this chapter the difficulties referred to above are eliminated, at the expense of introducing a further approximation. This approximation is of a physical, rather than mathematical, nature. In this it is assumed that the behaviour of a continuously curved surface can be adequately represented by the behaviour of a surface built up of small fiat elements. Intuitively, as the size of the subdivision decreases it would seem that convergence must occur and indeed experience indicates such a convergence. It will be stated by many shell experts that when we compare the e x a c t solution of shell approximated by fiat facets to the exact solution of a truly curved shell, considerable differences in the distribution of bending moments, etc., occur. It is arguable this is true, but for simple elements the discretization error is approximately of the same order and excellent results can be obtained with fiat shell element approximation. The mathematics of this problem is discussed in detail by Ciarlet. 3 In a shell, the element generally will be subject both to bending and to 'in-plane' force resultants. For a fiat element these cause independent deformations, provided the local deformations are small, and therefore the ingredients for obtaining the necessary stiffness matrices are available in the material already covered in the preceding chapters.
~
Introduction
In the division of an arbitrary shell into flat elements only triangular elements can be used for doubly curved surfaces. Although the concept of the use of such elements in the analysis was suggested as early as 1961 by Greene et al., 4 the success of such analysis was hampered by the lack of a good stiffness matrix for triangular plate elements in bending. 5-8 The developments described in Chapters 11 and 12 open the way to adequate models for representing the behaviour of shells with such a division. Some shells, for example those with general cylindrical shapes, can be well represented by flat elements of rectangular or quadrilateral shape provided the mesh subdivision does not lead to 'warped' elements. With good stiffness matrices available for such elements the progress here has been more satisfactory. Practical problems of arch dam design and others for cylindrical shape roofs have been solved quite early with such subdivisions. 9'1~ Clearly, the possibilities of analysis of shell structures by the finite element method are enormous. Problems presented by openings, variation of thickness, or anisotropy are no longer of consequence. A special case is presented by axisymmetrical shells. Although it is obviously possible to deal with these in the way described in this chapter, a simpler approach can be used. This will be presented in Chapters 14-16. As an alternative to the type of analysis described here, curved shell elements could be used. Here curvilinear coordinates are essential and general procedures in Chapter 5 of reference 11 can be extended to define these. The physical approximation involved in fiat elements is now avoided at the expense of reintroducing an arbitrariness of various shell theories. Several approaches using a direct displacement approximation are given in references 12-32 and the use of 'mixed variational principles' is given in references 33-36. A very simple and effective way of deriving curved shell elements is to use the socalled 'shallow' shell theory approach. 14'15'37'38 Here the variables u, v, w define the tangential and normal components of displacement to the curved surface. If all the elements are assumed to be tangential to each other, no need arises to transfer these from local to global values. The element is assumed to be 'shallow' with respect to a local coordinate system representing its projection on a plane defined by nodal points, and its strain energy is defined by appropriate equations that include derivatives with respect to coordinates in the plane of projection. Thus, precisely the same shape functions can be used as in flat elements discussed in this chapter and all integrations are in fact carried out in the 'plane' as before. Such shallow shell elements, by coupling the effects of membrane and bending strain in the energy expression, are slightly more efficient than fiat ones where such coupling occurs on the interelement boundary only. For simple, small elements the gains are marginal, but with few higher-order large elements advantages appear. A good discussion of such a formulation is given in reference 23. For many practical purposes the fiat element approximation gives very adequate answers and also permits an easy coupling with edge beam and rib members, a facility sometimes not present in a curved element formulation. Indeed, in many practical problems the structure is in fact composed of fiat surfaces, at least in part, and these can be simply reproduced. For these reasons curved, general, thin shell forms will not be discussed here and instead a general formulation of thick curved shells (based directly on three-dimensional behaviour and avoiding the shell equation ambiguities)
427
428
Shells
as an
assembly
of flat
elements
will be presented in Chapter 15. The development of curved elements for general shell theories can also be effected in a direct manner; however, additional transformations over those discussed in this chapter are involved. The interested reader is referred to references 39 and 40 for additional discussion on this approach. In many respects the differences in the two approaches are quite similar, as shown by Bischoff and R a m m . 41 In most arbitrary-shaped, curved shell elements the coordinates used are such that complete smoothness of the surface between elements is not guaranteed. The shape discontinuity occurring there, and indeed on any shell where 'branching' occurs, is precisely of the same type as that encountered in this chapter and therefore the methodology of assembly discussed here is perfectly general.
illi~,ili!i~,i iiiii!i!! ! iiii!iiiii',iiiiiii!iitii' '~'~ii~~i,i~!iiiii'ii~ii i i i'~i~,'~i~i'i~i!iiil !~,i'~iiiiiil ',!i iliii~!ii',iii l '~i!~!i! liii~',iiii!'i~i~,i',i!~ii ',~'~iiiiii'i,i~i~!!i',~'i's~, ,il~ilii!!iiiiiiii!'~~'i'~ii,ii!~i~,iiii i !~'~!i~,i~'i~i~il,ili~ii!ili i!i!i~,,,,,,,~'~,i:,i::,iliiiiiiiiii !'~i~!'~i~i~ii',iii ~ii~,' ,!!~~ii': i!.ii!!iiii',~',i~,i~:ii!i~i!~:!i '~i i!'::i'~'.~ii~:!~, !~!'~i'.~,:,~ii~,':::i!!!i~,~i~!~!i'::~i!~i~,i::iiii!!i'~,~,~i~''!:i~,iii! ~i !ii!!~:,~,~ii!ii~i i i'~i',~,i'~,i'~!i'~!ii!iiiiii~iiiiiii! ~i~i~il i~~,i',~i!i~,i'i,i~i'i'!!ii, ~'~i~iii' , ,'~,:i!!i~~i~,~.~i'~i'i,~'!i,~i~,:,i' i ~iiiiii~i',~',::iiiii' i~,~',~,~,i!ili ':::,~ii!ii~' i'~i~,~:~,~,i:i',~i',i:.ii~i~iili:,i
Consider a typical polygonal flat element in a local coordinate system xyz subject simultaneously to 'in-plane' and 'bending' actions (Fig. 13.1). Taking first the in-plane (plane stress) action, we know that the state of strain is uniquely described in terms of the fi and ~ displacement of each typical node a. The minimization of the total potential energy led to the stiffness matrices described there and gives 'nodal' forces due to displacement parameters tiP as
(~e)p:(Ke)pflPwithflP__{~a}
lJa
f P = {fy~a} F~a
(13.1)
Similarly, when bending was considered in Chapters 11 and 12, the state of strain was given uniquely by the nodal displacement in the ~ direction (~) and the two rotations 0~ and 09. This resulted in stiffness matrices of the type (~e)b
(i~e)b~lb
with
-b Ua
Oyca O~a
~
~b __
Myca M~a
(13.2)
Before combining these stiffnesses it is important to note two facts. The first is that the displacements prescribed for 'in-plane' forces do not affect the bending deformations and vice versa. The second is that the rotation 0~ does not enter as a parameter into the definition of deformations in either mode. While one could neglect this entirely at the present stage it is convenient, for reasons which will be apparent later when assembly is considered, to take this rotation into account and associate with it a fictitious couple M~. The fact that it does not enter into the minimization procedure can be accounted for simply by inserting an appropriate number of zeros into the stiffness matrix. Redefining the combined nodal displacement as
Ila -- [bla
13a
(13a
Oyca
F~a
M2a
0 S~a
0 ~a ] T
(13.3)
and the appropriate nodal 'forces' as
f.e--[fyr a
Ego
Ms; a
M~a]T
(13.4)
* In dealing with shells it is now convenient to use the rotation parameters 0 instead of the director angle q$. This is accomplished using Eq. (11.7).
Transformation to global coordinates and assembly of elements 429
M~.a
,J Z
rX
a
v
In plane forces
In plane deformations
J
~
(~0~n ~
P"P'-
X
Bending forces
Bending deformations Fig. 13.1 A flat element subject to 'in-plane' and 'bending' actions.
we can write
i ~ e ~ __ ~e
(13.5)
The stiffness matrix is now made up from the following submatrices _
~'ab =
-p Kab
" 0 : 0
0 0
0 0
9
,
0
0
"
" 0
0
O:
0
0
0
0"
I4~,bb
9
" 0 : 0 :0
9
9
.
0
0
(13.6)
9
0
9
0
9 0
if we note that LIa - - I L l p
Ua
Oa]
(13.7)
The above formulation is valid for any shape of polygonal element and, in particular, for the two important types illustrated in Fig. 13.1.
The stiffness matrix derived in the previous section used a system of local coordinates as the 'reference plane', and forces and bending components also are originally derived for this system.
430
Shells as an assembly of flat elements
z
x
z~
y
r
x
Fig. 13.2 Local and global coordinates.
Transformation of coordinates to a common global system (which will be denoted by xyz with the local system still ~ ) will be necessary to assemble the elements and to write the appropriate equilibrium equations. In addition it will be initially more convenient to specify the element nodes by their global coordinates and to establish from these the local coordinates, thus requiting an inverse transformation. All the transformations are accomplished by a simple process. The two systems of coordinates are shown in Fig. 13.2. The forces and displacements of a node transform from the global to the local system by a matrix T giving Ua = T u a
fa = Tfa
(13.8)
in which
with A being a 3 • 3 matrix of direction cosines between the two sets of axes, 42'43 that is, [cos(~, x) A-cos(~,x) cos(~, x)
cos(:~, y) cos(~,y) cos(~, y)
cos(~, z)] cos(~,z) cos(~, z)
-A~x Aycy A2z1
A~x A)y A)z _A~x A~y A~z.]
(13.10)
where cos(:~, x) is the cosine of the angle between the ~ axis and the x axis, and so on. By the rules of orthogonal transformation the inverse of T is given by its transpose; thus we have Ua -- TTUa
fa -- TTfa
(13.11)
Local direction cosines 431
which permits the stiffness matrix of an element in the global coordinates to be computed as Keb -- TTI(aeb T (13.12) -e in which Kab is determined by Eq. (13.6) in the local coordinates. The determination of the local coordinates follows a similar pattern. The relationship between global and local systems is given by
= A
{x x0} y z
y0 z0
(13.13)
where x0, Y0, z0 is the distance from the origin of the global coordinates to the origin of the local coordinates. As in the computation of stiffness matrices for flat plane and bending elements the position of the origin is immaterial, this transformation will always suffice for determination of the local coordinates in the plane (or a plane parallel to the element). Once the stiffness matrices of all the elements have been determined in a common global coordinate system, the assembly of the elements and forces follow the standard solution pattern. The resulting displacements calculated are referred to the global system, and before the stresses can be computed it is necessary to change these to the local system for each element. The usual stress calculations for 'in-plane' and 'bending' components can then be used.
The determination of the direction cosine matrix A gives rise to some algebraic difficulties and, indeed, is not unique since the direction of one of the local axes is arbitrary, provided it lies in the plane of the element. We shall first deal with the assembly of rectangular elements in which this problem is particularly simple; later we shall consider the case for triangular elements arbitrarily orientated in space.
13.4.1 Rectangular elements Such elements are limited in use to representing a cylindrical or box type of surface. It is convenient to take one side of each element and the corresponding 2 axis parallel to the global x axis. For a typical element 1234, illustrated in Fig. 13.3, it is now easy to calculate all the relevant direction cosines. Direction cosines of 2 are, obviously, A~x - 1
A~y =
A~z- 0
(13.14)
The direction cosines of the ~ axis have to be obtained by consideration of the coordinates of the various nodal points. Thus, Ayx - - 0 A~y - -
Y4 - Y~ V/(Y4 - y l ) 2 -k- (Z4 -- Zl) 2
A~z =
Z4 -- Zl V/(Y4 -- y l ) 2 -k- (Z4 -- Z 1) 2
(13.15)
432
Shells as an assembly of flat elements
z
l j j
j
7
Y 84
(a) z
'
4
Y
-
....................~
(b)
Vertical ~ o n
Y
ij
Fig. 13.3 A cylindrical shell as an assembly of rectangular elements: local and global coordinates.
are simple geometrical relations which can be obtained by consideration of the sectional plane passing vertically through 1 - 4 in the z direction. Similarly, from the same section, we have for the ~: axis A~x - 0 Zl - z4 A~y : A~z
=
V/(y 4 _ Yl) 2 "-I-( Z 4
= -Ayz -- Zl) 2
(13.16)
Y4 -- Yl -- A~y v / ( Y 4 - yl) 2 -+-(Z4- Zl) 2 - -
Clearly, the numbering of points in a consistent fashion is important to preserve the correct signs of the expression.
13.4.2 Triangular elements arbitrarily orientated in space An arbitrary shell divided into triangular elements is shown in Fig. 13.4(a). Each element has an orientation in which the angles with the coordinate planes are arbitrary. The problem of defining local axes and their direction cosines is therefore more complex
Local direction cosines 433
! ~
~
Y
(a) X
z
(b)
.X
84
Fig. 13.4 (a) An assemblage of triangular elements representing an arbitrary shell; (b)local and global coordinates for a triangular element.
than in the previous simple example. The most convenient way of dealing with the problem is to use some properties of geometrical vector algebra. One arbitrary but convenient choice of local axis direction is given here. We shall specify that the 2 axis is to be directed along the side 1-2 of the triangle, as shown in Fig. 13.4(b). The vector V2~ defines this side and in terms of global coordinates we have
V21 - -
{x2 x,} {x21} Y2
Yl
Z2
Zl
=
Y21 Z21
(13.17)
434
Shells as an assembly of flat elements
The direction cosines are given by dividing the components of this vector by its length, that is, defining a vector of unit length
V~ --
A~y = A~z ~
Y21 z21
with
121 --- 4X21 + y21 -q"
221
(1 3.1 8)
Now, the ~ direction, which must be normal to the plane of the triangle, needs to be established. We can obtain this direction from a 'vector' cross-product of two sides of the triangle. Thus,
fY21Z31--z21Y31} fYZ123} "- ]zx123
V~--V21 x V31-- ]z21x31- x21z31 kx21Y31 -- y21x31
kxyl23
(13.19)
represents a vector normal to the plane of the triangle whose length, by definition, is equal to twice the area of the triangle. Thus,
l~ = 2A = 4(YZ123)2 d- (zx123)2 q-" (xy123)2 The direction cosines of the ~ axis are available simply as the direction cosines of V~, and we have a unit vector V~ -"
{
fY21Z31--Z21Y31}
}
A~x A r:y
A~z
-"
1 ] z21x31 ~'~
xalz31
(13.20)
kx21Y31 Y21X31
Finally, the direction cosines of the y axis are established in a similar manner as the direction cosines of a vector normal both to the ~ direction and to the ~ direction. If vectors of unit length are taken in each of these directions [as in fact defined by Eqs (13.18)-(13.20)] we have simply
v~ =
{A~x} {A~yAxz-A~zA2y} A~y = v~ • v~ = A~zA~x - A~xA~z
Ayz
A~xA~y- A~yAycx
(13.21)
without having to divide by the length of the vector, which is now simply unity. The vector operations involved can be written as a special computer routine in which vector products, normalizing (i.e. division by length), etc., are automatically carried out 44 and there is no need to specify in detail the various operations given above. In the preceding outline the direction of the 2 axis was taken as lying along one side of the element. A useful alternative is to specify this by the section of the triangle plane with a plane parallel to one of the coordinate planes. Thus, for instance, if we desire to erect the 2 axis along a horizontal contour of the triangle (i.e. a section parallel to the xy plane) we can proceed as follows. First, the normal direction cosines v~ are defined as in Eq. (13.20). Now, the matrix of direction cosines of 2 has to have a zero component in the z direction and thus we have (13.22)
'Drilling' rotational stiffness-6 degree-of-freedom assembly 435 As the length of the vector is unity (13.23) and as further the scalar product of the v~ and v~ must be zero, we can write
A~xA~x + AycyA~y --0
(13.24)
and from these two equations v~ can be uniquely determined. Finally, as before vy - v~ x v~
(13.25)
It should be noted that this transformation will be singular if there is no line in the plane of the element which is parallel to the xy plane, and some other orientation must then be selected. Yet another alternative of a specification of the ~ axis is given in Chapter 15 where we discuss the development of 'shell' elements directly from the three-dimensional equations of solids.
In the formulation described above a difficulty arises if all the elements meeting at a node are co-planar. This situation will happen for flat (folded) shell segments and at straight boundaries of developable surfaces (e.g. cylinders or cones). The difficulty is due to the assignment of a zero stiffness in the O~i direction of Fig. 13.1 and the fact that classical shell equations do not produce equations associated with this rotational parameter. Inclusion of the third rotation and the associated 'force' F~i has obvious benefits for a finite element model in that both rotations and displacements at nodes may be treated in a very simple manner using the transformations just presented. If the set of assembled equilibrium equations in local coordinates is considered at such a point we have six equations of which the last (corresponding to the 0~ direction) is simply 0 0~ = 0
(13.26)
As such, an equation of this type presents no special difficulties (solution programs usually detect the problem and issue a warning). However, if the global coordinate directions differ from the local ones and a transformation is accomplished, the six equations mask the fact that the equations are singular. Detection of this singularity is somewhat more difficult and depends on round-off in each computer system. A number of alternatives have been presented that avoid the presence of this singular behaviour. Two simple ones are: 1. assemble the equations (or just the rotational parts) at points where elements are co-planar in local coordinates (and delete the 00~ = 0 equation); and/or 2. insert an arbitrary stiffness coefficient ko~ at such points only.
436
Shells as an assembly of flat elements
This leads in the local coordinates to replacing Eq. (13.26) by
ko~0~b - 0 w
(13.27)
which, on transformation, leads to a perfectly well-behaved set of equations from which, by usual processes, all displacements now including 0~ are obtained. As O~i does not affect the stresses and indeed is uncoupled from all equilibrium equations any value of k0~ similar to values already in Eq. (13.6) can be inserted as an external stiffness without affecting the result. These two approaches lead to programming complexity (as a decision on the coplanar nature is necessary) and an alternative is to modify the formulation so that the rotational parameters arise more naturally and have a real physical significance. This has been a topic of much study 45-57 and the 0~ parameter introduced in this way is commonly called a drilling degree of freedom, on account of its action to the surface of the shell. An early application considering the rotation as an additional degree of freedom in plane analysis is contained in reference 15. In reference 8 a set of rotational stiffness coefficients was used in a general shell program for all elements whether co-planar or not. These were defined such that in local coordinates overall equilibrium is not disturbed. This may be accomplished by adding to the formulation for each element the term
I'I* = I-I -3t- J~2 olnEtn (0~ -- 0~) 2 d~
(13.28)
in which the parameter c~, is a fictitious elastic parameter and 0~ is a mean rotation of each element which permits the element to satisfy local equilibrium in a weak sense. The above is a generalization of that proposed in reference 8 where the value of n is unity in the scaling value t n. Since the term will lead to a stiffness that will be in terms of rotation parameters the scaling indicated above permits values proportional to those generated by the bending rotations - namely, proportional to t cubed. In numerical experiments this scaling leads to less sensitivity in the choice of c~,. For a triangular element in which a linear interpolation is used for 0~ minimization with respect to O~a leads to the form
{
1 tn [ 1 - - 0 . 5 - ! 1 5 ] {0~1 / m~2 = ~C~n E A --0.5 1 -- 5 0~2 M~3 -0.5 -0.5 0~3
(13.29)
where OLn is yet to be specified. This additional stiffness does in fact affect the results where nodes are not co-planar and indeed represents an approximation; however, effects of varying c~, over fairly wide limits are quite small in many applications. For instance in Table 13.1 a set of displacements of an arch dam analysed in reference 4 is given for various values of c~1. For practical purposes extremely small values of Ogn are possible, providing a large computer word length is used. 58
Table 13.1 Nodalrotationcoefficientin dam analysis4 c~1 1.00 0.50 0.10
0.03
0.00
Radialdisplacement(mm) 61.13 63.551 64.52 64.78 65.28
'Drilling' rotational stiffness-6 degree-of-freedom assembly 437 zt
18~ Free edge
Symmetry plane
Symmetry plane
load Unit load
--
\
Symmetry plane R= 10, t= 0.04 E = 6.825 x 107. v = 0.3 Fig. 13.5 Sphericalshell test problem.59
The analysis of the spherical test problem proposed by MacNeal and Harter as a standard t e s t 59 is indicated in Fig. 13.5. For this test problem a constant strain triangular membrane together with the discrete Kirchhoff triangular plate bending element is combined with the rotational treatment. The results for regular meshes are shown in Table 13.2 for several values of c~3 and mesh subdivisions. The above development, while quite easy to implement, retains the original form of the membrane interpolations. For triangular elements with comer nodes only, the membrane form utilizes linear displacement fields that yield only constant strain terms. Most bending elements discussed in Chapters 11 and 12 have bending strains with higher than constant terms. Consequently, the membrane error terms will dominate the behaviour of many shell problem solutions. In order to improve the situation it is desirable to increase the order of interpolation. Using conventional interpolations this implies the introduction of additional nodes on each element; however, by utilizing a Table 13.2
Sphere problem: radial displacement at load OL3
Mesh
value
10.0
1.00
0.100
0.010
0.001
0.000
4 • 4
0.0639
0.0919
0.0972
0.0979
0.0980
0.0980
8 • 8
0.0897
0.0940
0.0945
0.0946
0.0946
0.0946
16 • 16
0.0926
0.0929
0.0929
0.0929
0.0930
0.0930
438
Shells as an assembly of flat elements
drill parameter these interpolations can be transformed to a form that permits a 6 degreeof-freedom assembly at each vertex node. Quadratic interpolations along the edge of an element can be expressed as (13.30)
U(~) -- NI(~)UI1 "l- N2(~)UI2 -I" N3(~)Au3
where Ul are nodal displacements (ill, ~l) at an end of the edge (vertex), similarly U2 is the other end, and Aul3 are hierarchical displacements at the centre of the edge (Fig. 13.6).
T (a) n
" Av / -1 (b)
~
= Ul ~2
Fig. 13.6 Construction of in-plane interpolations with drilling parameters.
'Drilling' rotational stiffness-6 degree-of-freedom assembly 439 The centre displacement parameters may be expressed in terms of normal (Au n) and tangential ( m~lt) components as AUI 3 --
Afinn q-
Afitt
(13.31)
where n is a unit outward normal and t is a unit tangential vector to the edge: n=
{ cos u } sinu
and t = { - sinu~ cosu ]
(13.32)
where u is the angle that the normal makes with the 2 axis. The normal displacement component may be expressed in terms of drilling parameters at each end of the edge (assuming a quadratic expansion). 45'53 Accordingly,
AUn
1 -- g112(0~2 -- 0~1)
(13.33)
in which 112 is the length of the 1-2 side. This construction produces an interpolation on each edge given by fi(~) -- N l ( ~ ) f i l -}- N2(~)fi2 -+" N 3 ( ~ ) [8112(022 - 021) +
Afitt]
(13.34)
The reader will undoubtedly observe the similarity here with the process used for linked interpolation for the bending element (see Sec. 12.7). The above interpolation may be further simplified by constraining the Afit parameters to zero. We note, however, that these terms are beneficial in a three-node triangular element. If a common sign convention is used for the hierarchical tangential displacement at each edge, this tangential component maintains compatibility of displacement even in the presence of a kink between adjacent elements. For example, an appropriate sign convention can be accomplished by directing a positive component in the direction in which the end (vertex) node numbers increase. The above structure for the in-plane displacement interpolations may be used for either an irreducible or a mixed element model and generates stiffness coefficients that include terms for the 0~ parameters as well as those for fi and ~. It is apparent, however, that the element generated in this manner must be singular (i.e. has spurious zero-energy modes) since for equal values of the end rotation the interpolation is independent of the 02 parameters. Moreover, when used in non-flat shell applications the element is not free of local equilibrium errors. This later defect may be removed by using the procedure identified above in Eq. (13.28), and results for a quadrilateral element generated according to this scheme are given by Jetteur 54 and Taylor. 55 A structure of the plane stress problem which includes the effects of a drill rotation field is given by Reissner 6~ and is extended to finite element applications by Hughes and Brezzi. 61 A variational formulation for the in-plane problem may be stated as l-Ie(fi, 0~, 7-) -- -~
sTDs dr2 +
r(a~iy- 02) d ~
(13.35)
where r is a skew symmetric stress component and w~y is the rotational part of the displacement gradient, which for the 2~ plane is given by O~ a~y = 02
0~ O~
(13.36)
440
Shells as an assembly of flat elements
In addition to the terms shown in Eq. (13.35), terms associated with initial stress and strain as well as boundary and body load must be appended for the general shell problem as discussed in Chapter 2 and also in reference 11. A variation of Eq. (13.35) with respect to 7-gives the constraint that the skew symmetric part of the displacement gradients is the rotation 0~. Conversely, variation with respect to 0~ gives the result that 7- must vanish. Thus, the equations generated from Eq. (13.35) are those of the conventional membrane but include the rotation field. A penalty form of the above equations suitable for finite element applications may be constructed by modifying Eq. (13.35) to l i d -- l i d -
L'
a ~ E t 72 dft
(13.37)
where a~ is a penalty number. It is important to use this mixed representation of the problem with the mixed patch test to construct viable finite element models. Use of constant 7- and isoparametric interpolation of 0~ in each element together with the interpolations for the displacement approximation given by Eq. (13.34) lead to good triangular and quadrilateral membrane elements. Applications to shell solutions using this form are given by Ibrahimbegovic et al. 57 Also the solution for a standard barrel vault problem is contained in Sec. 13.8. !iiiiiiiiiiiiiiiiiiiiii iiiiiiiiil i~iii~,~,,i~iliii~ ~i~ ~~'~"~'~~iiiiiiilili!iii~,~,~,',"iiiiiiiiii~iiiiiiiii'~~"~'~'~i'~'ii:~:~:ii~ii~i~~i~i iiiiiiiii!iiiii~i iili i i~i~i iiiiiiiiiiiii i~'~i~iiii~ii~iiii~iii~liiiiii~'~'~:~:iiii~iii~i~iiiii~iiiiii! i iiiiiii i ~i~i~ii ~i iiiii~ii:i~:ii~iiilili : i~iii~iiiiiiiilil~i~i~iiiiilii~'~':~i~iii~i~iiiii i iiiiliiiiiiii~iiiiiii~::iiiiii~ ~:iiii!ii~iii~ii!~~i~i!i~~iiiiiiil i~iiiiiiiiiiiii!iilii~iiiiiiiii~ii~iii~il ~!iiiiiii~i~i~iiiiilill ~iiiiiiiii i iiiiiiiiiiiiii ~iiiiiiiiii~ii~iiiii ~i~ii~iii!i~~~iiii~iiiiiiiliiiiiiiiiiiiiiiiiiiiiiii~iiiiliii!!i~ii i~iii~iiiii~ilil ~iiiiiliiiiiiiii~i~iiil~iii:ii~i~:ii~ii~iiiii ~i i~i!i~iiiiiii il!iiiiiiiiiiiiiiii!iiiiliiiii~iil~iiii~i~!~~ii~iiiiiiiii~i~~iiiii~ iiiiiiiiliiiiii
Many of the difficulties encountered with the nodal assembly in global coordinates disappear if the element is so constructed as to require only the continuity of displacements u, o, and w at the comer nodes, with continuity of the normal slope being imposed along the element sides. Clearly, the corner assembly is now simple and the introduction of the sixth nodal variable is unnecessary. As the normal slope rotation along the sides is the same both in local and in global coordinates its transformation there is unnecessary - although again it is necessary to have a unique definition of parameters for the adjacent elements. Elements of this type arise naturally in hybrid forms ~1 and we have already referred to a plate bending element of a suitable type in Sec. 11.6. This element of the simplest possible kind has been used in shell problems by Dawe 26 with some success. A considerably more sophisticated and complex element of such type is derived by Irons and named 'semi-loof'. 3~ This element is briefly mentioned in Chapter 9 and although its derivation is far from simple it performs well in many situations. ~i!il~i~i~i1!i!ii3!iiii~!~~Zi ~iii7i~i~i~i~i~ii~Ci~iih~i!~i!i~i~i~i~|C~i~i~i~i!~!iii~iii~iii!ii~!~iiiiiii!iiii~'~iii~iii~iieiii~ii!i!i~i!i~ii~i!iii|iiiiiiiiiiiiiiiiiiiiiiiiiiii!iiiiiiiiiiiiiiiiii~iiiiiii i!im i~ii e ~ iiiiiiiiiiiiiiiiiliiiiiiiiiiiiiiiiiiiii iiiiiiiiiiiiiiiii~iiiiiiiiiiii ili!i!ili!iiiiiiiiiiiiiiiiiiii!iiiiiiiiiiil i~i~ii~iiiiiiii!iiiiiiiiiiiiiiiiiiiiiiiiiiii i~ii ii!iiiiiiii!iil !i!~iii!iiiiii!iiii!iiiiiiiiiliiiiiil ~!ii!iiiii!iiiillii!iiiiiiiiiiiiiiiiiiiiii iiiiiliiiiiii i!iil!iiiii!i!i!iiiiiiiililili!ill iiiiiiiiiii!iii illill!!iii!i!iiiii!iiiiii!i!iiiii~i!ili!ili!iii!!iiii ililililili iiii!iii!i!iiiiiiiiiiii!iii!iiiii iiiiii!ii!ii!iiiii!il!ili!iiiii!ii!iiiiiiiiiiiii! Numerous membrane and bending element formulations are now available, and, in both, conformity is achievable in flat assemblies. Clearly, if the elements are not coplanar conformity will, in general, be violated and only approached in the limit as smooth shell conditions are reached. It would appear consistent to use expansions of similar accuracy in both the membrane and bending approximations but much depends on which action is dominant. For thin shells, the simplest triangular element would thus appear to be one with a linear
Practical examples
441
in-plane displacement field and a quadratic bending displacement- thus approximating the stresses as constants in membrane and in bending actions. Such an element is used by Dawe 26 but gives rather marginal (though convergent) results. In the examples shown we use the following elements which give quite adequate performance. Element A: This is a mixed rectangular membrane with four comer nodes (Sec. 10.4.4 of reference 11) combined with the non-conforming bending rectangle with four comer nodes (Sec. 11.3). This was first used in references 9 and 10. Element B: This is a constant strain triangle with three nodes combined with the incompatible bending triangle with 9 degrees of freedom (Sec. 11.5). Use of this in the shell context is given in references 8 and 62. Element C: In this a more consistent linear strain triangle with six nodes is combined with a 12 degree-of-freedom bending triangle using shape function smoothing. This element has been introduced by Razzaque. 63 Element D: This is a four-node quadrilateral with drilling degrees of freedom [Eq. (13.34) with A n t constrained to zero] combined with a discrete Kirchhoff quadrilateral. 55,64
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The first example given here is that for the solution of an arch dam shell. The simple geometrical configuration, shown in Fig. 13.7, was taken for this particular problem as results of model experiments and alternative numerical approaches were available. A division based on rectangular elements (type A) was used as the simple cylindrical shape permitted this, although a rather crude approximation for the fixed foundation had to be used. Water load as discrete point loads (E
106~
_
3o m
I
~'" Nodal loads
elements \
,.~c~-Y-~,'~"
~L_L.-.~
Actual foundation line
Actual foundation line Coarse mesh
Fig. 13.7 An arch dam as an assemblyof rectangular elements.
Fine mesh
442
Shells as an assembly of flat elements 30
\\
25
20
E
" 15 -r"
/ 10
0
2
6 4 Deflections (mm)
8
--.-o-- Finite element solution (coarse) -.-'P-- Finite element solution (fine) . . . . . . Trial load solution (USBR) (Poisson's ratio v -- O.15)
Fig. 13.8 Arch dam of Fig. 13.7: horizontal deflections on centre-line.
Two sizes of division into elements are used, and the results given in Figs 13.8 and 13.9 for deflections and stresses on the centre-line section show that little change occurred by the use of the finer mesh. This indicates that the convergence of both the physical approximation to the true shape by fiat elements and of the mathematical approximation involved in the finite element formulation is more than adequate. For comparison, stresses and deflection obtained using the USBR trial load solution (another approximate method) are also shown. A large number of examples have been computed by Parekh 62 using the triangular, non-conforming element (type B), and indeed show for equal division a general improvement over the conforming triangular version presented by Clough and Johnson. 7 Some examples of such analyses are now shown. A doubly curved arch dam was similarly analysed using the triangular flat element (type B) representation. The results show an even better approximation. 8
Practical examples 443 Downstream face
Crest
\
.......................
.
Upstream face ..~.
.
.
30
......
.
25
.
/"
- - o - - Finite element solution (coarse) Finite element solution (fine)
"~l~ ~
........ Trial load solution (USBR) (Poisson's ratio v = 0.15)
...............
15
i
. . . . . . . . .
,.
-50 ~ 0 - 3 0 - 2 0 - ! , 0 0 10 ~ 30 40 50 60 Vertical stresses on crown section (+ tension) (kg/cm 2)
:::::::::::::::
Fig. 13.9 Arch dam of Fig. 13.7: vertical stresses on centre-line.
,1=, o 9
l~ T M r "-I ~4 ft
>x
r
i
I r"
(a)
137 ft
1.6
U) r
Actual . . . . . Assumed
._r 1.2 ._o r 0.8 o 0.4 e = 0 0
'
. . . . .
r -0.4
a. -0.8 (b)
~l
'|
,,1.
.
.
.
i
,
0
30
60
90 120 150 180 e (deg)
Fig. 13.10 Cooling tower: geometry and pressure load variation about circumference.
13.8.1 Cooling tower This problem of a general axisymmetric shape could be more efficiently dealt with by the axisymmetric formulations to be presented in Chapters 11 and 12. However, here this example is used as a general illustration of the accuracy attainable. The answers
444
Shells as an assembly of flat elements
lOft 15ft
CXl 4.., 0')
i~-~
r
12 at 15 ~
I
Fig. 13.11 Cooling tower of Fig. 13.10: mesh subdivisions.
against which the numerical solution is compared have been derived by Albasiny and Martin. 65 Figures 13.10 to 13.1 2 show the geometry of the mesh used and some results for a 5 inch and a 7 inch thick shell. Unsymmetric wind loading is used here.
13.8.2 Barrel vault This typical shell used in many civil engineering applications is solved using analytical methods by Scordelis and L o 66 and Scordelis. 67 The barrel is supported on rigid diaphragms and is loaded by its own weight. Figures 13.13 and 13.14 show some comparative answers, obtained by elements of type B, C and D. Elements of type C are obviously more accurate, involving more degrees of freedom, and with a mesh
Practical examples 100
5 in shell
o
9
"\
.... Finite e l e m e n t
. . . . Albasiny and Martin65
.~
N -100
-200
/L
N1 =__=._=- .
-300 -250
0
I
250
500 Ib/ft
750
1000
1250
(a)
100
7 in s h e ~ r
-100
i ~ 5 in shell
-200 ~ -300
0
~--
,, , Finite e l e m e n t . . . . Albasiny and Martin 65
0.001 0.002 0.003 0.004 0.005 0.006 0.007 ft
(b) lOO
N -100 7 in shell -200 -300
" ~ ~ ' ~ ' ~ ~"
Finite e l e m e n t
. . . . Albasiny and Martin 65 30
20
Ib ft/ft
10
5 in s h e ~
~ 0
i
10
|
0
(c)
Fig. 13.12 Cooling tower of Fig. 13.10: (a) membrane forces at ~ -- 0~ N1, tangential; N2, meridian; (b) radial displacements at ~ -- 0~ (c) moments at 8 -- 0~ MI, tangential; M2, meridian.
445
446
Shells as an assembly of flat elements z,w 9
y,v
x,u
E = 3 x 103 k/in 2 v=O
9' = 0.09 k/ft2 Support rigid diaphram U=0
e edge
\~' ~,40~ ~ ~
I
W=0
(a) _._ Analytical
I
2O
A o
4O
8 x 12 Mesh EI.B 12x18Mesh
E] 3 x 3 Mesh EI.Dc 9 8x 12 Mesh EI.D
0.1
w(ft) 0.2
0.3 (b)
40 -n ---->- r
i !
i
-0.005
v(~) -0.010 -0.015
(c) Fig. 13.13 Barrel (cylindrical) vault: flat element model results. (a) Barrel vault geometry and properties; (b) vertical displacement of centre section; (c)longitudinal displacement of support.
Practical examples 447
M~ r
M2 -1
0
10
(a)
20
$
30
40
m
~
m
l ~ ~ 10j
0
,i20
I
30
40
(b) Fig. 13.14 Barrel vault of Fig. 13.13. (a) M1, transverse; M2, longitudinal; centre-line moments; (b) M12, twisting moment at support.
of 6 • 6 elements the results are almost indistinguishable from analytical ones. This problem has become a classic on which various shell elements are compared and we shall return to it in Chapter 15. It is worthwhile remarking that only a few, secondorder, curved elements give superior results to those presented here with a fiat element approximation.
13.8.3 Folded plate structure As no analytical solution of this problem is known, comparison is made with a set of experimental results obtained by Mark and Riesa. 68
448 Shells as an assembly of flat elements
,~~
I
4.082 in
,..(bq,Y//'"'--7 ~ ~,~,/~....Plate 3
in rib
'<'J/~45~ F"ate 2i
.c: ;~'--_~~_ 0.2 in ~ LI~"
Plate 1
I
L
Plate 1 2 3
Verticalload (Ib/in2) 0.009 0.009 0.129 .
f//////
Rigid diaphragm
Beam (rib) CE
Fig. 13.15 A folded plate structure;68 modelgeometry,loading and mesh,E - 3560 Ib/in2, z) - 0.43.
Practical
0
Scale (Ib) 0.1 0.2
1
I
Scale (10 -3 in) 0 100 200
CE
I
examples
I
I
I
I
~
58
-oj Crown Edge
7.62
3.81
I
(a)
0.1 in
I
i !
I 3.81
E
y
i 3.81
..........
...--,,,-,-~..- ~
0.lib ," r
c~
om
7.62
~
(b)
O)
(c)
0.1 in
Fig. 13.16 Foldedplate of Fig. 13.15; moments and displacements on centre section. (a)Vertical displacements along the crown; (b)longitudinal moments along the crown; (c) horizontal displacements along edge.
This example presents a problem in which actual flat finite element representation is physically exact. Also a frame stiffness is included by suitable superposition of beam elements - thus illustrating also the versatility and ease by which different types of elements may be used in a single analysis.
449
450
Shells as an assembly of flat elements Figures 13.15 and 13.16 show the results using elements of type B. Similar applications are of considerable importance in the analysis of box type bridge structure s, etc.
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1. W. Fltigge. Stresses in Shells. Springer-Verlag, Bedim 1960. 2. S.P. Timoshenko and S. Woinowski-Krieger. Theory of Plates and Shells. McGraw-Hill, New York, 2nd edition, 1959. 3. P.G. Ciarlet. Conforming finite element method for shell problem. In J.R. Whiteman, editor, The Mathematics of Finite Elements and Application, volume II. Academic Press, London, 1977. 4. B.E. Greene, D.R. Strome and R.C. Weikel. Application of the stiffness method to the analysis of shell structures. In Proc. Aviation Conf. of American Society of Mechanical Engineers, Los Angeles, March 1961. ASME. 5. R.W. Clough and J.L. Tocher. Analysis of thin arch dams by the finite element method. In Proc. Int. Symp. on Theory of Arch Dams. Pergamon Press, Oxford, 1965. 6. J.H. Argyris. Matrix displacement analysis of anisotropic shells by triangular elements. J. Roy. Aero. Soc., 69:801-805, 1965. 7. R.W. Clough and C.P. Johnson. A finite element approximation for the analysis of thin shells. J. Solids Struct., 4:43-60, 1968. 8. O.C. Zienkiewicz, C.J. Parekh and I.P. King. Arch dams analysed by a linear finite element shell solution program. In Proc. Int. Symp. on Theory of Arch Dams. Pergamon Press, Oxford, 1965. 9. O.C. Zienkiewicz and Y.K. Cheung. Finite element procedures in the solution of plate and shell problems. In O.C. Zienkiewicz and G.S. Holister, editors, Stress Analysis, Chapter 8. John Wiley & Sons, Chichester, 1965. 10. O.C. Zienkiewicz and Y.K. Cheung. Finite element methods of analysis for arch dam shells and comparison with finite difference procedures. In Proc. Int. Symp. on Theory of Arch Dams, pages 123-140. Pergamon Press, Oxford, 1965. 11. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann, Oxford, 6th edition, 2005. 12. R.H. Gallagher. Shell elements. In World Conf. on Finite Element Methods in Structural Mechanics, Bournemouth, England, October 1975. 13. F.K. Bogner, R.L. Fox and L.A. Schmit. A cylindrical shell element. Journal of AIAA, 5:745-750, 1966. 14. J. Connor and C. Brebbia. Stiffness matrix for shallow rectangular shell element. Proc. Am. Soc. Civ. Eng., 93(EM1):43-65, 1967. 15. A.J. Carr. A refined element analysis of thin shell structures including dynamic loading. Technical Report SEL Report 67-9, University of California, Berkeley, 1967. 16. S. Utku. Stiffness matrices for thin triangular elements of non-zero Gaussian curvature. Journal of AIAA, 5:1659-1667, 1967. 17. G. Cantin. Strain-displacement relationships for cylindrical shells. Journal of AIAA, 6: 1787-1788, 1968. 18. G. Cantin and R.W. Clough. A curved, cylindrical shell finite element. AIAA Journal, 6(6): 1057-1062, 1968. 19. G. Bonnew, G. Dhatt, Y.M. Giroux and L.P.A. Robichaud. Curved triangular elements for analysis of shells. In Proc. 2nd Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-68-150, Wright Patterson Air Force Base, Ohio, October 1968. 20. G.E. Strickland and W.A. Loden. A doubly curved triangular shell element. In Proc. 2nd Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-68-150, Wright Patterson Air Force Base, Ohio, October 1968.
References 451 21. B.E. Greene, R.E. Jones and D.R. Strome. Dynamic analysis of shells using doubly curved finite elements. In Proc. 2nd Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-68150, Wright Patterson Air Force Base, Ohio, October 1968. 22. S. Ahmad. Curved finite elements in the analysis of solids, shells and plate structures. PhD thesis, Department of Civil Engineering, University of Wales, Swansea, 1969. 23. G.R. Cowper, G.M. Lindberg and M.D. Olson. A shallow shell finite element of triangular shape. International Journal of Solids and Structures, 6:1133-1156, 1970. 24. G. Dupuis and J.J. Goal. A curved finite element for thin elastic shells. International Journal of Solids and Structures, 6:987-996, 1970. 25. S.W. Key and Z.E. Beisinger. The analysis of thin shells by the finite element method. In High Speed Computing of Elastic Structures, volume 1, pages 209-252. University of Liege Press, 1971. 26. D.J. Dawe. The analysis of thin shells using a facet element. Technical Report CEGB RD/B/ N2038, Berkeley Nuclear Laboratory, England, 1971. 27. D.J. Dawe. Rigid-body motions and strain-displacement equations of curved shell finite elements. Int. J. Mech. Sci., 14:569-578, 1972. 28. D.G. Ashwell and A. Sabir. A new cylindrical shell finite element based on simple independent strain functions. Int. J. Mech. Sci., 4:37-47, 1973. 29. G.R. Thomas and R.H. Gallagher. A triangular thin shell finite element: linear analysis. Technical Report CR-2582, NASA, 1975. 30. B.M. Irons. The semi-Loof shell element. In D.G. Ashwell and R.H. Gallagher, editors, Finite Elements for Thin Shells and Curved Members, Chapter 11, pages 197-222. John Wiley & Sons, Chichester, 1976. 31. D.G. Ashwell. Strain elements with application to arches, tings, and cylindrical shells. In D.G. Ashwell and R.H. Gallagher, editors, Finite Elements for Thin Shells and Curved Members, Chapter 6, pages 91-111. John Wiley & Sons, Chichester, 1976. 32. N. Carpenter, H. Stolarski and T. Belytschko. A fiat triangular shell element with improved membrane interpolation. Communications in Applied Numerical Methods, 1:161-168, 1985. 33. C. Pratt. Shell finite element via Reissner's principle. International Journal of Solids and Structures, 5:1119-1133, 1969. 34. J. Connor and G. Will. A mixed finite element shallow shell formulation. In R.H. Gallagher et al., editors, Advances in Matrix Methods of Structural Analysis and Design, pages 105-137. University of Alabama Press, 1969. 35. L.R. Herrmann and W.E. Mason. Mixed formulations for finite element shell analysis. In Conf. on Computer-Oriented Analysis of Shell Structures, Paper AFFDL-TR-71-79, June 1971. 36. G. Edwards and J.J. Webster. Hybrid cylindrical shell elements. In D.G. Ashwell and R.H. Gallagher, editors, Finite Elements for Thin Shells and Curved Members, Chapter 10, pages 171-195. John Wiley & Sons, Chichester, 1976. 37. H. Stolarski and T. Belytschko. Membrane locking and reduced integration for curved elements. J. Appl. Mech., 49:172-176, 1982. 38. Ph. Jetteur and F. Frey. A four node Marguerre element for non-linear shell analysis. Engineering Computations, 3:276-282, 1986. 39. J.C. Simo and D.D. Fox. On a stress resultant geometrically exact shell model. Part I: Formulation and optimal parametrization. Computer Methods in Applied Mechanics and Engineering, 72:267-304, 1989. 40. J.C. Simo, D.D. Fox and M.S. Rifai. On a stress resultant geometrically exact shell model. Part II: The linear theory; computational aspects. Computer Methods in Applied Mechanics and Engineering, 73:53-92, 1989. 41. M. Bischoff and E. Ramm. Solid-like shell or shell-like solid formulation? A personal view. In W. Wunderlich, editor, Proc. Eur. Conf. on Comp. Mech. (ECCM'99 on CD-ROM), Munich, September 1999.
452
Shells as an assembly of flat elements 42. L.E. Malvern. Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, Englewood Cliffs, NJ, 1969. 43. I.H. Shames and EA. Cozzarelli. Elastic and Inelastic Stress Analysis. Taylor & Francis, Washington, DC, 1997. (Revised printing.) 44. S. Ahmad, B.M. Irons and O.C. Zienkiewicz. A simple matrix-vector handling scheme for threedimensional and shell analysis. International Journal for Numerical Methods in Engineering, 2:509-522, 1970. 45. D.J. Allman. A compatible triangular element including vertex rotations for plane elasticity analysis. Computers and Structures, 19:1-8, 1984. 46. D.J. Allman. A quadrilateral finite element including vertex rotations for plane elasticity analysis. International Journal for Numerical Methods in Engineering, 26:717-730, 1988. 47. D.J. Allman. Evaluation of the constant strain triangle with drilling rotations. International Journal for Numerical Methods in Engineering, 26:2645-2655, 1988. 48. P.G. Bergan and C.A. Felippa. A triangular membrane element with rotational degrees of freedom. Computer Methods in Applied Mechanics and Engineering, 50:25-69, 1985. 49. P.G. Bergan and C.A. Felippa. Efficient implementation of a triangular membrane element with drilling freedoms. In T.J.R. Hughes and E. Hinton, editors, Finite Element Methods for Plate and Shell Structures, volume 1, pages 128-152. Pineridge Press, Swansea, 1986. 50. R.D. Cook. On the Allman triangle and a related quadrilateral element. Computers and Structures, 2:1065-1067, 1986. 51. R.D. Cook. A plane hybrid element with rotational d.o.f, and adjustable stiffness. International Journal for Numerical Methods in Engineering, 24:1499-1508, 1987. 52. T.J.R. Hughes, L.P. Franca and G.M. Hulbert. A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective-diffusive equations. Computer Methods in Applied Mechanics and Engineering, 73:173-189, 1989. 53. R.L. Taylor and J.C. Simo. Bending and membrane elements for analysis of thick and thin shells. In G.N. Pande and J. Middleton, editors, Proc. NUMETA 85 Conf., volume 1, pages 587-591. A.A. Balkema, Rotterdam, 1985. 54. Ph. Jetteur. Improvement of the quadrilateral JET shell element for a particular class of shell problems. Technical Report IREM 87/1, Ecole Polytechnique Federale de Lausanne, February 1987. 55. R.L. Taylor. Finite element analysis of linear shell problems. In J.R. Whiteman, editor, The Mathematics of Finite Elements and Applications VI, pages 191-203. Academic Press, London, 1988. 56. R.H. MacNeal and R.L Harter. A refined four-noded membrane element with rotational degrees of freedom. Computers and Structures, 28:75-88, 1988. 57. A. Ibrahimbegovic, R.L. Taylor and E.L. Wilson. A robust quadrilateral membrane finite element with drilling degrees of freedom. International Journal for Numerical Methods in Engineering, 30:445--457, 1990. 58. R.W. Clough and E.L. Wilson. Dynamic finite element analysis of arbitrary thin shells. Computers and Structures, 1:1971, 33-56. 59. R.H. MacNeal and R.L. Harter. A proposed standard set of problems to test finite element accuracy. Journal of Finite Elements in Analysis and Design, 1:3-20, 1985. 60. E. Reissner. A note on variational theorems in elasticity. International Journal of Solids and Structures, 1:93-95, 1965. 61. T.J.R. Hughes and F. Brezzi. On drilling degrees-of-freedom. Computer Methods in Applied Mechanics and Engineering, 72:105-121, 1989. 62. C.J. Parekh. Finite element solution system. PhD thesis, Department of Civil Engineering, University of Wales, Swansea, 1969. 63. A. Razzaque. Finite element analysis of plates and shells. PhD thesis, Civil Engineering Department, University of Wales, Swansea, 1972. 64. J.L. Batoz and M.B. Tahar. Evaluation of a new quadrilateral thin plate bending element. International Journal for Numerical Methods in Engineering, 18:1655-1677, 1982.
References 453 65. E.L. Albasiny and D.W. Martin. Bending and membrane equilibrium in cooling towers. Proc. Am. Soc. Civ. Eng., 93(EM3):1-17, 1967. 66. A.C. Scordelis and K.S. Lo. Computer analysis of cylindrical shells. J. Am. Concr. Inst., 61:539-561, 1964. 67. A.C. Scordelis. Analysis of cylindrical shells and folded plates. In Concrete Thin Shells, Report SP 28-N. American Concrete Institute, Farmington, Michigan, 1971. 68. R. Mark and J.D. Riesa. Photoelastic analysis of folded plate structures. Proc. Am. Soc. Civ. Eng., 93(EM4):79-83, 1967.
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The problem of axisymmetric shells is of sufficient practical importance to include in this chapter special methods dealing with their solution. While the general method described in the previous chapter is obviously applicable here, it will be found that considerable simplification can be achieved if account is taken of axial symmetry of the structure. In particular, if both the shell and the loading are axisymmetric it will be found that the elements become 'one dimensional'. This is the simplest type of element, to which little attention was given in earlier chapters. The first approach to the finite element solution of axisymmetric shells was presented by Grafton and Strome. ~ In this, the elements are simple conical frustra and a direct approach via displacement functions is used. Refinements in the derivation of the element stiffness are presented in Popov et al. 2 and in Jones and Strome. 3 An extension to the case of unsymmetrical loads, which was suggested in Grafton and Strome, is elaborated in Percy et al. 4 and others. 5'6 Later, much work was accomplished to extend the process to curved elements and indeed to refine the approximations involved. The literature on the subject is considerable, no doubt promoted by the interest in aerospace structures, and a complete bibliography is here impractical. References 7-15 show how curvilinear coordinates of various kinds can be introduced to the analysis, and references 9 and 14 discuss the use of additional nodeless degrees of freedom in improving accuracy. 'Mixed' formulations have found here s o m e u s e . 16 Early work on the subject is reviewed comprehensively by Gallagher 17'18 and Stricklin. 19 In axisymmetric shells, in common with all other shells, both bending and 'in-plane' or 'membrane' forces will occur. These will be specified uniquely in terms of the generalized 'strains', which now involve extensions and changes in curvatures of the middle surface. If the displacement of each point of the middle surface is specified, such 'strains' and the internal stress resultants, or simply 'stresses', can be determined by formulae available in standard references dealing with s h e l l theory. 2~
As a simple example of an axisymmetric shell subjected to axisymmetric loading we consider the case shown in Figs 14.1 and 14.2 in which the displacement of a point
Straight element 455 Z
Fig. 14.1 Axisymmetric shell, loading, displacements, and stress resultants; shell represented as a stack of conical frustra.
~r
Fig. 14.2 An element of an axisymmetric shell.
on the middle surface of the meridian plane at an angle ~b measured positive from the x axis is uniquely determined by two components fi and 0J in the tangential (s) and normal directions, respectively. Using the Kirchhoff-Love assumption (which excludes transverse shear deformations) and assuming that the angle ~b does not vary (i.e. elements are straight), the four strain components are given by 2~ ~s G0
Xs
Xo
d~/ds [fi cos ~b - t~ sin ~b] / r -d2~/ds 2 - ( d t ~ / d s ) cos ~b/r
(14.1)
456 Curved rods and axisymmetric shells
This results in the four internal stress resultants shown in Fig. 14.1 that are related to the strains by an elasticity matrix D:
Ns No
or=
Ms
= De
(14.2)
Mo For an isotropic shell the elasticity matrix becomes
D-
Et 1-~
1
u
0
0
-]
u 0
1 0
0 t2/12
0 ut2/12
0
ut2/12
t2/12J
I
(14.3)
the upper part being a plane stress and the lower a bending stiffness matrix with shear terms omitted as 'thin' conditions are assumed.
14.2.1 Element characteristics - axisymmetrical loads Let the shell be divided by nodal circles into a series of conical frustra, as shown in Fig. 14.2. The nodal displacements at points 1 and 2 for a typical 1-2 element will have to define uniquely the deformations of the element via prescribed shape functions. At each node the radial and axial displacements, u and w, and a rotation, ~b, will be used as parameters. From virtual work by edge forces we find that all three components are necessary as the shell can carry in-plane forces and bending moments. The displacements of a node i can thus be defined by three components, the first two being in global directions r and z, Ua --
ll)a ~a
(14.4)
The simplest elements with two nodes, 1 and 2, thus possess 6 degrees of freedom, determined by the element displacements ~2
(14.5)
The displacements within the element have to be uniquely determined by the nodal displacements u e and the position s (as shown in Fig. 14.2) and maintain slope and displacement continuity. Thus in local (s) coordinates we have ( u } - N(s)file
(14.6)
Based on the strain-displacement relations (14.1) we observe that ~ can be of Co type while t~ must be of type C1. The simplest approximation takes ~ varying linearly with
Straight element s and ~ as cubic in s. We shall then have six undetermined constants which can be determined from nodal values of u, w and ~b. At the node a,
/
~a //)a
/
F[COSr O
--
sin,
--sine
COS~)
(dfo/ds)a
0
!1/~
-- Tfia
ffOa @a
(14.7)
Introducing the interpolations
-- NI(~)Ul + N2(~)u2 t/) = H(0) (~)li) 1 + H2(0) (~)1~ 2 -t- -~- H(1) (~)-~s 1
-(1)
+n~
dt~]]
( 0 ~ -s 2
(14.8) (14.9)
where Na are the usual linear interpolations in ~ ( - 1 <_ ~ < 1) N1
1
- - ~ (1 -
~)
and
N2
1
- - ~ (1 +
~0
and Ha~~ and Ha(l) are the Hermitian interpolations defined in Sec. 10.4 and repeated here for completeness 1
n2(O~)--41 (2 + 3,~ _ ),~3
and
and
H(1) = 1 (1 - ~ __ ~2 + ~3)
n (1, __ ~1 ( - 1 - ~ ' +
and
~2 + ~3 )
in which, placing the origin of the meridian coordinate s at the 1 node, 1
s = N2(~)L = ~(1 + ~)L
Z
The global coordinates for the conical frustrum may also be expressed by using the Na interpolations as r = N1 (~)?1 + N 2 ( ~ ) 7 2
(14.10)
z = Nl(~)~l + N2(~)z2 and used to compute the length L as L - V/(?2 - 71)2 + (z2 - zl) 2 Writing the interpolations as
-'-
0 01/ 0a /
n(0)
Ha(1)
113a
(dfo/ds)a
-- NaLla
(14.11)
457
458 Curved rods and axisymmetric shells
we can now write the global interpolation as u -
(it} to
= [SIT
l~2T]fi e =
N0 e
(14.12)
From Eq. (14.12) it is a simple matter to obtain the strain matrix B by use of the definition (14.1). This gives e = Bfi e -- [I~IT
]~2T] ~e
(14.13)
in which, noting from Eq. (14.7) that u = cos ~b~ - sin ~b~, we have
Ba - -
dNa/ds Na cos r 0
0
0 - H a ~~ sin r _ d 2 HalO)/ds2
0 - H a ~1)sin ~b/r _ d2 Ha(1)/ds2
cos~b/r
-(dH~a~
(14.14)
-(dHCal)/ds) cos~b/r
Derivatives are evaluated by using dNa 2 dNa = ds L d~
d 2Na 4 d 2Na = ds 2 L 2 d~2
and
with similar expressions for Ha~ and Ha~l). Now all the 'ingredients' required for computing the stiffness matrix (or load, stress, and initial stress matrices) by standard formulae are known. The integrations required are carried out over the area, A, of each element, that is, with dA = 27rr ds = 7rrL d~ (14.15) with ~ varying from - 1 to 1. Thus, the stiffness matrix K becomes, in local coordinates, Kab --
7rL
f
l 1
Bra DBbr d~
(14.16)
On transformation, the stiffness Kab of the global matrix is given by Kab -- T T l~f~abT
(14.17)
Once again it is convenient to evaluate the integrals numerically and the form above is written for Gaussian quadrature. 24 Grafton and Strome 1 give an explicit formula for the stiffness matrix based on a single average value of the integrand (one-point Gaussian quadrature) and using a D matrix corresponding to an orthotropic material. Percy et al. 4 and Klein 5 used a seven-point numerical integration; however, it is generally recommended to use only two points to obtain all arrays (especially if inertia forces are added, since one point then would yield a rank deficient mass matrix). It should be remembered that if any external line loads or moments are present, their full circumferential value must be used in the analysis, just as was the case with axisymmetric solids discussed in Chapter 2.
Straight element 459
14.2.2 Additional enhanced mode A slight improvement to the above element may be achieved by adding an e n h a n c e d strain mode to the r component. Here this is achieved by following the procedures outlined in Chapter 10 and in reference 24, and we can observe that the necessary condition not to affect a constant value of N~ is given by 27r
Csen r ds = TrL
1
e en s r d~=O
(14.18)
where een denotes the enhanced strain component. A simple mode may thus be defined
as
Ee n -
--OLen -- nen~en r
(14.19)
in which men is a parameter to be determined. For the linear elastic case considered above the mode may be determined from
G
[Ken
ule/ - {~/
GKT]{~en
where Ken -- 27rL
f
Ga = 27rL
(14.20)
l 1 1
BenOll Ben r d~
(14.21) BenDBa r d~
Now a partial solution may be performed by means of static condensation 25 to obtain the stiffness for assembly I~ = K - G r K ~ G
(14.22)
The effect of the added mode is most apparent in the force resultant Ns where solution oscillations are greatly reduced. This improvement is not needed for the purely elastic case but is more effective when the material properties are inelastic and the oscillations can cause errors in behaviour, such as erratic yielding in elasto-plastic solutions.
14.2.3 Examples and accuracy In the treatment of axisymmetric shells described here, continuity between the shell elements is satisfied at all times. For an axisymmetric shell of polygonal meridian shape, therefore, convergence will always occur. The problem of the physical approximation to a curved shell by a polygonal shape is similar to the one discussed in Chapter 13. Intuitively, convergence can be expected, and indeed numerous examples indicate this.
460 Curved rods and axisymmetric shells When the loading is such as to cause predominantly membrane stresses, discrepancies in bending moment values exist (even with reasonably fine subdivision). Again, however, these disappear as the size of the subdivisions decreases, particularly if correct sampling is used (see Chapter 10). This is necessary to eliminate the physical approximation involved in representing the shell as a series of conical frustra. Figures 14.3 and 14.4 illustrate some typical examples taken from the Grafton and Strome paper which show quite remarkable accuracy. In each problem it should be noted that small elements are needed near free edges to capture the 'boundary layer' nature of shell solutions.
3.0, r
z: r
2.5~
. u
o~~ I
O X r to ..,.
Theoretical % error max. deflection
2.0t
1.5
~\
&- - -
,~
o-..-.-- Case 2 11.1 9
Theoretical % error max. moment
Case I 31.7 Case 3 3.1
&---
Case I 28.8
o-----
Case 2 11.0
.
Case 3 2.5
60 r-
1.0
ID
~-0.5 ID
50
._~
40
9
"0
"o
ft.
-0.5 _ 0.1 0.2 0.3 0.4
0.5 0.6 0.7 x ~ inches Deflections : ~ ~0
E = 107 ib/in 2 v = 0.30 1 Ib/in
t = 0.01 in
0 - 10
0.1 0.2 0.3 0.4 0.5 0.6 0.7 x ~ inches Meridional moment
1.0 in,,.~ . . . . _1_~,__ _.___ 2.0 in __ 2.0 "in _[
Case I = 5 at 0i0.1"2inin= ~Case 2 ~ 10 at
Nodal divisions
Case 3 20 at 0.05 in Fig. 14.3 A cylindrical shell solution by finite elements, from Grafton and Strome. I
Curved elements spacing at 0.5 ~ interval
,
Nodal
M = 1 in Ib/in
t 1.0 ~ interval 2.0 ~ interval
30 ~
t=l
in
~~176
4 a t 10 ~ nterval
E = 107 Ib/in 2 v = 0.33
2.0
-9
o
1.5-
~
1.0 Theoretical o
FEM
1
_
X to e-
Moment
... 1.0e-
0.5
E e~
.-= "o
0.5 -
t'tl tO N
Displace~int
45
..--,, L_
o
-r-
0
-0.5 Angle between normal to Fig. 14.4
0
- -0.25
surface and axis of rotation ~ degrees
A hemispherical shell solution by finite elements, from Grafton and Strome. 1
Use of curved elements has already been described in the context of analyses that involved only first derivatives in the definition of strain. Here second derivatives exist [see Eq. (14.1)] and require additional effort to compute the strains. It was previously mentioned that many possible definitions of curved elements have been proposed and used in the context of axisymmetric shells. The derivation used here is one due to Delpak 14 and is of the subparametric type. 24
461
462
Curved rods and axisymmetric shells I-~
r2- q
Az ]7~ ~
-
)2
)1 r
(a)
(b)
Fig. 14.5 Curved, isoparametric, shell element for axisymmetric problems: (a) parent element; (b) curvilinear coordinates.
The basis of curved element definition is one that gives a common tangent between adjacent elements (or alternatively, a specified tangent direction). This is physically necessary to avoid 'kinks' in the description of a smooth shell. If a general curved form of a shell of revolution is considered, as shown in Fig. 14.5, the expressions for strain quoted in Eq. (14.1) have to be modified to take into account the curvature of the shell in the meridian plane. 2~ These now become
~---
es
d~/ds + #o/Rs
e0
[fi cos ~b - t~ sin ~] / r -dZlig/ds 2 - d(fi/Rs)/ds
Xs Xo
__
(14.23)
- [(dt~/ds + ~/R~)] cos ~b/r
In the above the angle ~ is a function of s, that is, dr ds
= cos ~
and
dz - - = sin ds
Rs is the principal radius in the meridian plane, and the second principal curvature radius Ro is given by r = Ro sin The reader can verify that for Rs = cxz Eq. (14.23) coincides with Eq. (14.1).
14.3.1 Shape functions for a curved element We shall now consider the 1-2 element to be curved as shown in Fig. 14.5(b), where the coordinate is in 'parent' form ( - 1 _< ~ _< 1) as shown in Fig. 14.5(a). The coordinates
Curved elements 463
and the unknowns are 'mapped' in the manner of Chapter 2. As we wish to interpolate a quantity with slope continuity we can write for a typical function g(~) 2
+
L --
(14.24)
where again the Hermitian interpolations have been used. We can now simultaneously use these functions to describe variations of the global displacements u and w as
2
]
U-- Z In(O)(~)~'la+ o(1) du
I d-~ a
a-1
W= ~ a=l
H(aO)(~)Wa+ H(a1)dfo i] ~
(14.25)
a
and of the coordinates r and z which define the shell (mid-surface). Indeed, if the thickness of the element is also variable the same interpolation could be applied to it. Such an element would then be isoparametric. Accordingly, we can define the geometry as r--
2
~ [na(~ a=l
a + Ha(1) d r I d-~ a
'0)(~)~ a +
Z "-- ~
]
(14.26)
/_./(1) _.~
a=l
a
and, provided the nodal values in the above can be specified, a one-to-one relation between ~ and the position on the curved element surface is defined [Fig. 14.5(b)]. While specification of ra and Za is obvious, at the ends only the slope c o t 2/3a - -
I
dz a
(14.27)
is defined. The specification to be adopted with regard to the derivatives occurring in Eq. (14.26) depends on the scaling of ~ along the tangent length s. Only the ratio d r II _- (dr/d~)a
dz I
a
(dz/d~)a
(14.28)
is unambiguously specified. Thus (dr/d~)a or (dz/d~)a can be given an arbitrary value. Here, however, practical considerations intervene as with the wrong choice a very uneven relationship between s and ~ will occur. Indeed, with an unsuitable choice the shape of the curve can depart from the smooth one illustrated and loop between the end values. * One immediate difference will be observed from that of the previous formulation. Now both displacement components vary in a cubic manner along an element while previously a linear variation of the tangential displacement was permitted. This additional degree of freedom does not, however, introduce excessive constraints provided the shell thickness is itself continuous.
464 Curved rods and axisymmetric shells
To achieve a reasonably uniform spacing it suffices for well-behaved surfaces to approximate dr Ar r2 - ?1 dz Az ~:2 - ~:1 ~ = or ~ ~ = (14.29) 2 a( 2 using whichever is largest and noting that the whole range of ~ is 2 between the nodal points.
14.3.2 Strain expressions and properties of curved elements The variation of global displacements are specified by Eq. (14.25) while the strains are described in locally directed displacements in Eq. (14.23). Some transformations are therefore necessary before the strains can be determined. We can express the locally directed displacements fi and ~ in terms of the global displacements by using Eq. (14.7), that is, {fi}_
[cos~ [-sin~
sin~l {u}=~, cos~J w
u
(14.30)
where ~ is the angle of the tangent to the curve and the r axis (Fig. 14.5). We note that this transformation may be expressed in terms of the ~ coordinate using Eqs (14.27) and (14.28) and the interpolations for r and z. With this transformation the continuity of displacement between adjacent elements is achieved by matching the global nodal displacements Ua and fVa. However, in the development for the conical element we have specified continuity of rotation of the cross-section only. Here we shall allow usually the continuity of both s derivatives in displacements. Thus, the parameters du/ds and d w/ds will be given common values at nodes. As duds du = ~d---7 ds d~
and
dw ds dw ds d~ = d~
(14.31)
/dr/2 /dz/2
where
No difficulty exists in substituting these new variables in Eqs (14.25) and (14.30) which now take the form fi=N(~)fi e
with
~1 a - -
[bl a
113a
(d~/ds)a
(dw/ds)a] r
(14.32)
The form of the 2 x 4 shape function submatrices Na can now be explicitly determined by using the above transformations in Eq. (14.25). 14 We note that the meridian radius of curvature Rs can be calculated explicitly from the mapped, parametric, form of the element by using
Rs --
[(dr/d~)2 + (dz/d~)213/2
(dr/d~)(d2z/d~2) _ (dz/d~)(d2r/d~2) in which all the derivatives are directly determined from expression (14.26).
(14.33)
Curved elements 465
If shells that branch or in which abrupt thickness changes occur are to be treated, the nodal parameters specified in Eq. (14.32) are not satisfactory. It is better to rewrite these as lla--[bla Ul)a @a (du/ds)a] T (14.34) where ~ba, equal to (dfo/ds)a, is the nodal rotation, and to connect only the first three parameters. The fourth is now an unconnected element parameter with respect to which, however, the usual treatment is still carried out. Transformations needed in the above are implied in Eq. (14.7). In the derivation of the B matrix expressions which define the strains, both first and second derivatives with respect to s occur, as seen in the definition of Eq. (14.23). If we observe that the derivatives can be obtained by the simple (chain) rules already implied in Eq. (14.31), for any function F we can write
d~ = ds d~
d~2 = ds 2
( s)2 d-~
+--~s
d~2/
(14.35)
and all the expressions of B can be found. Finally, the stiffness matrix is obtained in a similar way as in Eq. (14.16), changing the variable ds ds - ~-~ d~ (14.36) and integrating ~ within the limits - 1 and + 1. Once again the quantities contained in the integral expressions prohibit exact integration, and numerical quadrature must be used. As this is carried out in one coordinate only it is not very time consuming and an adequate number of Gauss points can be used to determine the stiffness (generally three points suffice). Initial stress and other load matrices are similarly obtained. The particular isoparametric formulation presented in summary form here differs somewhat from the altematives of references 15, 7, 8 and 13 and has the advantage that, because of its isoparametric form, rigid body displacement modes and indeed the states of constant first derivatives are available. Proof of this is similar to that contained in Sec. 5.5 of reference 24. The fact that the forms given in the altemative formulations have strain under rigid body nodal displacements may not be serious in some applications, as discussed by Haisler and S t r i c k l i n . 26 However, in some modes of non-axisymmetric loads (see Chapter 16) this incompleteness may be a serious drawback and may indeed lead to very wrong results. Constant states of curvature cannot be obtained for a finite element of any kind described here and indeed are not physically possible. When the size of the element decreases it will be found that such arbitrary constant curvature states are available in the limit.
14.3.3 Additional nodeless variables As in the straight frustrum element, addition of nodeless (enhanced) variables in the analysis of axisymmetric shells is particularly valuable when large curved elements are capable of reproducing with good accuracy the geometric shapes. Thus an addition of
466
Curved rods and axisymmetric shells
N4 -1
= ~
1
Fig. 14.6 Internal shape functions for a linear element. a set of internal, hierarchical, element variables n
(14.37) b=l
to the definition of the normal displacement defined in Eq. (14.6) or Eq. (14.25), in which Afib is a set of internal parameters and Nb is a set of functions having zero values and zero first derivatives at the nodal points, allows considerable improvement in representation of the displacements to be achieved without violating any of the convergence requirements. For tangential displacements the requirement of zero first derivatives at nodes could be omitted. Webster also uses such additional functions in the context of straight elements. 9 In transient situations where these modes affect the mass matrix one can also use these functions as a basis for developing enhanced strain modes (see Sec. 14.2) since these by definition do not influence the assumed displacement field and, hence, the mass and surface loading terms. Whether the element is in fact straight or curved does not matter and indeed we can supplement the definitions of displacements contained in Eq. (14.25) by Eq. (14.37) for each of the components. If this is done only in the displacement definition and not in the coordinate definition [Eq. (14.26)] the element now becomes of the category of subparametric.* As proved in Chapter 5 of reference 24, the same advantages are retained as in isoparametric forms. The question as to the expression to be used for additional, internal shape functions is of some importance though the choice is wide. While it is no longer necessary to use polynomial representation, Delpak does so and uses a special form of Legendre polynomial (hierarchical functions). The general shapes are shown in Fig. 14.6. A
* While it would obviously be possible to include the new shape function in the element coordinate definition, little practical advantage would be gained as a cubic represents realistic shapes adequately. Further, the development would then require 'fitting' the A~b and A~b for coordinates to the shape, complicating even further the development of derivatives.
Curved elements 467
ct 30
~"
\
p = 1 Ib/in 2
20
t,.m
00
-10
30 ~
35 ~
-10
25 ~
20 ~
15 ~
10 ~
5~
0~
7
-20 t-
-a -30
-40 'Q,,,,W...~ :.._..o_~ r...l::x...W~ :) -50
35 ~
30 ~ O
v
25 ~
20 ~
15 ~
10 ~
5~
0o
Theoretical" T i m o s h e n k o and W e i n o w s k i - K r i e g e r 21 Delpak TM Zienkiewicz et aL 24 (10 elements)
Fig. 14.7
Spherical dome under uniform pressure.
A series of examples shown in Figs 14.7-14.9 illustrate the applications of the isoparametric curvilinear element of the previous section with additional internal parameters. In Fig. 14.7 a spherical dome with clamped edges is analysed and compared with analytical results of reference 21. Figures 14.8 and 14.9 show, respectively, more complex examples. In the first a torus analysis is made and compared with alternative finite element and analytical results. 12'13'27-29 The second case is one where branching occurs, and here alternative analytical results are given by Kraus. 3~
468
Curved rods and axisymmetric shells
subdivision (18 elements)
0.5 in
(a)
0.4 --9 0.3
0
X v
0.2 D
o.1
I
0 -90
-70
I
-50
I
-30
I
-10
I
~o
10
I
30
I
50
I
70
90
Chan and Firmin TM Giannini and Miles TM
--o--
r 9
Delpak 14 Zienkiewicz e t al. 24
(b)
Fig. 14.8 Toroidal shell under internal pressure: (a) element subdivision; (b) radial displacements.
~i~i!i~1~!~i!i4i4i~!i!i~!~ii~i~!~!!i~i~!~i~dep~d~~!~!~ii~i~ii~i!iii~!!i~i~i~i~i~i~i~i~i~i~iii!i~ii~i~i~i~i~ii~i~i!i~i~i~i~ii!~i~i~i~i~i~i~i~i~iii! In Chapters 10 to 12 we discussed the use of independent slope and displacement interpolation in the context of beams and plates, respectively. Continuity was assured by the introduction of the shear force as an independent mixed variable which was defined within each element. The elimination of the shear variable led to a penalty type formulation in which the shear rigidity played the role of the penalty parameter. The
Independent
71
~
~
interpolation
~ ~,O...."Oo.~1761.76 m
m v
slope-displacement
9
.:
I
I - 10
3 210 -90
I -70
I -50
-30
! 10
~o
I
30
I
50
I
70
90
(c) 20 irQ. 18
_ ~176176176176 -....
16 14 --~ 12 10
-
~""'~'"N,
_
-90
~ 1 7 6 1 7 6.....,o. ..... v.o ..... ~ ........ q/,..
i
-70
I
-50
I
I
I
-30
-10
(d)
IF~
10
I
30
I
50
I
70
90
Chan and Firmin TM ---e--
Giannini and Miles 13
......o ..... Delpak TM 9 Zienkiewicz et al. 24 v
Jordan 29 9 Sanders and Liepins 28
Fig. 14.8 Cont. (c)in-plane stress resultants; (d)in-plane stress resultants.
equivalence of the number of parameters used in defining the shear variation and the number of integration points used in evaluating the penalty terms was demonstrated there in special cases, and this justified the success of reduced integration methods. This equivalence is not exact in the case of the axisymmetric problem in which the radius, r, enters the integrals, and hence slightly different results can be expected from the use of the mixed form and simple use of reduced integration. The differences become greatest near the axis of rotation and disappear completely when r --+ c~
469
470 Curvedrods and axisymmetric shells
N Linear = elements29~p
l
- 20.0 I
~10.0 I-
0.0
~---F 20.0
+1
....
o I +3
VII
i
O-----
I-- VII: elements Delpak14
Ws~~adiea/ent n 4 ~ i of cylinder)(in) " " [
OlO[
/
0.06 :~
~ -00 .4 ,__,~i
-6
-4
o Fig. 14.9 Branchingshell.
i I _0.121 I I I I
-2
0 Z(in)
2
Analytical: Kraus3~ Delpak14 9 Zienkiewiczet aL24
4
6
N
Independent slope-displacement interpolation 471 where the axisymmetric form results in an equivalent beam (or cylindrical bending plate) element. Although in general the use of the mixed form yields a superior result, for simplicity we shall here derive only the reduced integration form, leaving the former to the reader as an exercise accomplished following the rules of Chapter 10. In what follows we shall develop in detail the simplest possible element of this class. This is a direct descendant of the linear beam and plate elements. 27'31 (We note, however, that the plate element formulated in this way has singular modes and can on occasion give completely erroneous results; no such deficiency is present in the beam or the axisymmetric shell.) Consider the strain expressions of Eq. (14.1) for a straight element. When using these the need for C1 continuity was implied by the second derivative of w existing there. If now we use d~ ds = -q5
(14.38)
the strain expression becomes dFt / ds
Cs C0
[~ cos ~ - t~ sin ~] / r
d /as
Xs
(14.39)
cos ~ / r
Xo
As ~bcan vary independently, a constraint has to be imposed: dli) C(t~, ~b) = ds + ~b = 0
(14.40)
This can be done by using the energy functional with a penalty multiplier a. We can thus write 2 l"I -- 71"
~TD~ r ds + 7r
a
~
+ ~
r ds + I'Iex t
(14.41)
where I"Iext is a potential for boundary and loading terms and e and D are defined as in Eq. (14.3). Immediately, a can be identified as the shear rigidity: a = ~Gt
where for a homogeneous shell ~ = 5/6
(14.42)
The penalty functional (14.41) can be identified on purely physical grounds. Washizu 22 quotes this on pages 199-201, and the general theory indeed follows that earlier suggested by Naghdi 23 for shells with shear deformation. With first derivatives occurring in the energy expression only Co continuity is now required for the interpolation of u, w, and ~b, and in place of Eqs (14.6)-(14.12) we can write directly fl--
fro
=
~) T
Na(~)Tfla a=l
(14.43)
~, Wa
where for Na (~) we can use any one-dimensional Co interpolation. 24 Once again, an isoparametric transformation could be used for curvilinear elements with strains defined
472
Curved rods and axisymmetric shells
by Eq. (14.23), and a formulation that we shall discuss in Chapter 15 is but an alternative to this process. If linear elements are used, we can write the expression without consequent use of isoparametric transformation. Indeed, we can replace the interpolations in Eq. (14.9) and now simply use u = Nl(~)fil + Nz(~)fi2 w = N~(~c)t~ 4- Nz(~)w2
(14.44)
and evaluate the integrals arising from expression (l 4.4 I) at one Gauss point, which is sufficient to maintain convergence and yet here does not give a singularity. This extremely simple form will, of course, give very poor results with exact integration, even for thick shells, but now with reduced integration shows excellent performance. In Figs 14.7-14.9 we superpose results obtained with this simple, straight clement, and the results speak for themselves.
I J I P = 1.0 Iblin2 !
t=o.1
I
r
r
" 10.0
=
t 0 0.05 -
1.0
2.0
I
I
3.0
4.0
r(in) 5.0
6.0
7.0
8.0
9.0. 10.0
.
-
r
.~. 0.10 ~" 0.15
Theoretical
-
0.20-10
4 elements 9 8 | 9 16 .)
9
E = 1.0 x 107 Ibf/in 2 v=03
.&
-8 -6 -2-4 "7,
0
+2 +4
1.0 i
2.0 ,
3.0 ,
4.0 i
r (in) 5.0 6 . ~ , _Jir
~1~ , 70
8.0 ,
9.0 ,
+6 +8
Fig. 14.10
Bending of a circular plate under uniform load; convergence study.
10.0 ,
References 473
For other examples the reader can consult reference 27, but in Fig. 14.10 we show a very simple example of a bending of a circular plate with use of different numbers of equal elements. This purely bending problem shows the type of results and convergence attainable. Interpreting the single integrating point as a single shear variable and applying the patch test count of Chapter 12, the reader can verify that this simple formulation passes the test in assemblies of two or more elements. In a similar way it can be verified that a quadratic interpolation of displacements and the use of two quadrature points (or a linear shear force) will also result in a robust element of excellent performance. One final word of caution when the element is used in transient analyses is in order. Here it is necessary to compute a mass matrix which can be deduced from the term (~l"lin = 27r
~u p t i~ + ~w p t ib + ~c~ p -i2
r ds
(14.45)
Evaluation of this integral with a single quadrature point will lead to a rank deficient mass matrix, which when used with any time stepping scheme can lead to large numerical errors (generally after many time steps have been computed). Accordingly, it is necessary to compute the mass matrix with at least two quadrature points (nodal quadrature giving immediately a diagonal 'lumped' mass).
i!i!iii•iiiiiiiiii•i!iii••iiiiii•i•i!i!i!iii!ii•i•i!iii•i•iiti•iitiiiiii•iiiiii!iii!i!iii•iiiiiiiiiiiiii!iiiii!iii•iiii!i•ii•i•i•iii@!i!iiiiii•iiiii•!iiiii•i•iii!i!ii•i•iiiii!iii•iiiiitiiiii•iii•iiiiit!iiiiiii!iiiiiiiii
1. EE. Grafton and D.R. Strome. Analysis of axi-symmetric shells by the direct stiffness method. Journal of AIAA, 1:2342-2347, 1963. 2. E.P. Popov, J. Penzien and Z.A. Liu. Finite element solution for axisymmetric shells. Proc. Am. Soc. Civ. Eng., EM5:119-145, 1964. 3. R.E. Jones and D.R. Strome. Direct stiffness method of analysis of shells of revolution utilizing curved elements. Journal of AIAA, 4:1519-1525, 1966. 4. J.H. Percy, T.H.H. Pian, S. Klein and D.R. Navaratna. Application of matrix displacement method to linear elastic analysis of shells of revolution. Journal of AIAA, 3:2138-2145, 1965. 5. S. Klein. A study of the matrix displacement method as applied to shells of revolution. In Proc. 1st Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-66-80, Wright Patterson Air Force Base, Ohio, October 1966. 6. R.E. Jones and D.R. Strome. A survey of analysis of shells by the displacement method. In Proc. 1st Conf Matrix Methods in Structural Mechanics, volume AFFDL-TR-66-80, Wright Patterson Air Force Base, Ohio, October 1966. 7. J.A. Stricklin, D.R. Navaratna and T.H.H. Pian. Improvements in the analysis of shells of revolution by matrix displacement method (curved elements). Journal of AIAA, 4:2069-2072, 1966. 8. M. Khojasteh-Bakht. Analysis of elastic-plastic shells of revolution under axi-symmetric loading by the finite element method. Technical Report SESM 67-8, University of California, Berkeley, 1967. 9. J.J. Webster. Free vibration of shells of revolution using ring elements. J. Mech. Sci., 9:559-570, 1967. 10. S. Ahmad, B.M. Irons and O.C. Zienkiewicz. Curved thick shell and membrane elements with particular reference to axi-symmetric problems. In Proc. 2nd Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-68-150, pages 539-572, Wright Patterson Air Force Base, Ohio, October 1968.
474 Curved rods and axisymmetric shells 11. E.A. Witmer and J.J. Kotanchik. Progress report on discrete element elastic and elastic-plastic analysis of shells of revolution subjected to axisymmetric and asymmetric loading. In Proc. 2nd Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-68-150, pages 1341-1453, Wright Patterson Air Force Base, Ohio, October 1968. 12. A.S.L. Chan and A. Firmin. The analysis of cooling towers by the matrix finite element method. Aeronaut. J., 74:826-835, 1970. 13. M. Giannini and G.A. Miles. A curved element approximation in the analysis of axi-symmetric thin shells. International Journal for Numerical Methods in Engineering, 2:459-476, 1970. 14. R. Delpak. Role of the curved parametric element in linear analysis of thin rotational shells. PhD thesis, Department of Civil Engineering and Building, the Polytechnic of Wales, 1975. 15. P.L. Gould and S.K. Sen. Refined mixed method finite elements for shells of revolution. In Proc. 3rd Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-71-160, WrightPatterson Air Force Base, Ohio, 1972. 16. Z.M. Elias. Mixed finite element method for axisymmetric shells. International Journal for Numerical Methods in Engineering, 4:261-272, 1972. 17. R.H. Gallagher. Analysis of plate and shell structures. In W.R. Rowan and R.M. Hackett, editors, Applications of Finite Element Method in Engineering, pages 155-205. ASCE, Vanderbilt University, Nashville, Tennessee, 1969. 18. R.H. Gallagher. Finite Element Analysis: Fundamentals. Prentice-Hall, Englewood Cliffs, NJ, 1975. 19. J.A. Stricklin. Geometrically nonlinear static and dynamic analysis of shells of revolution. In High Speed Computing of Elastic Structures, pages 383-411. University of Liege, 1976. 20. V.V. Novozhilov. Theory of Thin Shells. Noordhoff, Dordrecht, 1959. (English translation.) 21. S.P. Timoshenko and S. Woinowski-Krieger. Theory of Plates and Shells. McGraw-Hill, New York, 2nd edition, 1959. 22. K. Washizu. Variational Methods in Elasticity and Plasticity. Pergamon Press, New York, 3rd edition, 1982. 23. P.M. Naghdi. Foundations of elastic shell theory. In I.N. Sneddon and R. Hill, editors, Progress in Solid Mechanics, volume IV, Chapter 1. North Holland, Amsterdam, 1963. 24. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann, Oxford, 6th edition, 2005. 25. E.L. Wilson. The static condensation algorithm. International Journal for Numerical Methods in Engineering, 8:1974, 199-203. 26. W.E. Haisler and J.A. Stricklin. Rigid body displacements of curved elements in the analysis of shells by the matrix displacement method. Journal of AIAA, 5:1525-1527, 1967. 27. O.C. Zienkiewicz, J. Bauer, K. Morgan and E. Ofiate. A simple element for axi-symmetric shells with shear deformation. International Journal for Numerical Methods in Engineering, 11:1545-1558, 1977. 28. J.L. Sanders Jr. and A. Liepins. Toroidal membrane under internal pressure. Journal of AIAA, 1:2105-2110, 1963. 29. F.E Jordan. Stresses and deformations of the thin-walled pressurized toms. J. Aero. Sci., 29: 213-225, 1962. 30. H. Kraus. Thin Elastic Shells. John Wiley & Sons, New York, 1967. 31. T.J.R. Hughes, R.L. Taylor and W. Kanoknukulchai. A simple and efficient finite element for plate bending. International Journal for Numerical Methods in Engineering, 11:1529-1543, 1977.
Shells as a special case of three-dimensional analysis-
Reissner-Mindlin assumptions ',i',i',i' iiii~li!,i~i~i~ ~i~l~,i~:~ii~::~iii ~iiiiiii lii'~:i~~:il:i!i:~,iiiiiiil !i!,~i~i::,::i~i:'i~ii'i,ii!!i~:~!i'~ilii'!,i~iili~iilii'iiii' ~i i~ii,~i~i"~"i '~i'~~ili~i'~ilii!iil l~!'i,li'~~:i',i:'i,~'~, ii~iiiiiii ili!iiiii' '~~,iiiii',ii!iiiiii ,ii'i~iiiil~~ii,iiliiiiilil iiiiliiiiiiiiiiil iiiiiiiiiiiiiiiiiilliili!i iiilil !iiiiiiii!iil !i!i!iiiiiiil i!iiiil i !iiiiiili!iiiiiiiiiii !i!iiiiiiiiiiiii!i!i!iiiii iiii! iliiiiiiliii!iii!iii !iiiiiiiiiiiiiilil iiiiiiiii iliil ilili!i!i!iiii i~iiiiiiil i iilliii!iiiiiii!iii!ili!iiiiiiiii!iiii!iiiiiiiii~i~,i!iiiii i!iliiiiiilii~i'~iiiiill i,iiiiiiii!iiiiiiliiiiii!ii iiii'i',i~',~ii!i~!i~ii,i!i!~i!iiiiiiiil li'~iiiiiiiiiiiiiiiliiiiii!i',!'~,i~'i,i!~i:i~ili!~i,!i~ilil~iiiii!i! i'~iliiiiiiiiiiiii!'ii!ii',i!,if!',~iiliii i~i',~i',iili~,iiiii'i!iii ~,!ii'~,iiiiiiii! iliilii iiiiiiiiiiiiiil ii!iiiiiiii' ~ii"i' ,~ii
In the analysis of solids the use of isoparametric, curved, two- and three-dimensional elements is particularly effective, as illustrated in Chapters 2 and 4 and presented in reference 1. It seems obvious that use of such elements in the analysis of curved shells could be made directly simply by reducing their dimension in the thickness direction as shown in Fig. 15.1. Indeed, in an axisymmetric situation such an application is illustrated in the example of Fig. 9.25 of reference 1. With a straightforward use of the three-dimensional concept, however, certain difficulties will be encountered. In the first place the retention of 3 displacement degrees of freedom at each node leads to large stiffness coefficients from strains in the shell thickness direction. This presents numerical problems and may lead to ill-conditioned equations when the shell thickness becomes small compared with other dimensions of the element. The second factor is that of economy. The use of several nodes across the shell thickness ignores the well-known fact that even for thick shells the 'normals' to the mid-surface remain practically straight after deformation. Thus an unnecessarily high number of degrees of freedom has to be carried, involving penalties of computer time. In this chapter we present specialized formulations which overcome both of these difficulties. The constraint of straight 'normals' is introduced to improve economy and the strain energy corresponding to the stress perpendicular to the mid-surface is ignored to improve numerical conditioning. 2-4 With these modifications an efficient tool for analysing curved thick shells becomes available. The accuracy and wide range of applicability of the approach is demonstrated in several examples.
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i
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The reader will note that the two constraints introduced correspond precisely to the so-called Reissner-Mindlin assumptions already discussed in Chapter 12 to describe
476
Shells as a special 3-D case
Fig. 15.1 Curved, isoparametric hexahedra in a direct approximation to a curved shell.
the behaviour of thick plates. The omission of the third constraint associated with the thin plate theory (normals remaining normal to the mid-surface after deformation) permits the shell to experience transverse shear deformations - an important feature of thick shell situations. The formulation presented here leads to additional complications compared with the straightforward use of a three-dimensional element. The elements developed here are in essence an alternative to the processes discussed in Chapter 12, for which an independent interpolation of slopes and displacement are used with a penalty function imposition of the continuity requirements. The use of reduced integration is useful if thin shells are to be dealt with- and, indeed, it was in this context that this procedure was first discovered. 5-8 Again the same restrictions for robust behaviour as those discussed in Chapter 12 become applicable and generally elements that perform well in plate situations will do well in shells.
Shell element with displacement and rotation parameters
15.2.1 Geometric definition of an element Consider a typical shell element illustrated in Fig. 15.2. The external faces of the element are curved, while the sections across the thickness are generated by straight lines. Pairs of points, atop and abottom, each with given Cartesian coordinates, prescribe the shape of the element. Let ~, r/be the two curvilinear coordinates in the mid-surface of the shell and let be a linear coordinate in the thickness direction. If, further, we assume that ~, 7?, ~ vary between - 1 and 1 on the respective faces of the element we can write a relationship between the Cartesian coordinates of any point of the shell and the curvilinear coordinates in the form
{i}
--
Ya
2
top
+
2
Ya
/
(15.1)
bottom
Here Na (~, ~?) is a standard two-dimensional shape function taking a value of unity at the top and bottom nodes a and zero at all other nodes (see Appendix A). If the basic Z;=l 1
z
l/
y
~'=-1
~",,,,,j,~'~ r/=-I
~-1
=-'-"---~ X
1(
0
(
0
(
O(
-1(
0
1
Fi9. 15.2 Curvedthick shell elements of various types.
L1 L2 L3 coordinates
477
478
Shells as a special 3-D case
Vla Fig. 15.3 Local and global coordinates.
functions Na are derived as 'shape functions' of a 'parent', two-dimensional element, square or triangular* in plan, and are so 'designed' that compatibility is achieved at interfaces, then the curved space elements will fit into each other. Arbitrary curved shapes of the element can be achieved by using shape functions of higher order than linear. Indeed, any of the two-dimensional shape functions of Chapter 2 or Appendix A can be used here. The relation between the Cartesian and curvilinear coordinates is now established and it will be found desirable to operate with the curvilinear coordinates as the basis. It should be noted that often the coordinate direction ~ is only approximately normal to the mid-surface. It is convenient to rewrite the relationship, Eq. (15.1), in a form specified by the 'vector' connecting the upper and lower points (i.e. a vector of length equal to the shell thickness t) and the mid-surface coordinates. Thus we can rewrite Eq. (15.1) as (Fig. 15.3)
{?}
= ~
Na(~, 7)
Ya Za
where
.~a Za
--~
l/{a} Ya Za
top
+
.~a Za
/
and
+
1 /
~(Vaa
(15.2)
mid
V3a--
bottom
.~a Za
(15.3)
--Ya top
Za
bouom
with V3a defining a vector whose length represents the shell director and has length of shell thickness. * Area coordinates Lk would be used in this case in place of (, r/. 1
Shell element with displacement and rotation parameters 479 For relatively thin shells, it is convenient to replace the vector V3a by a unit vector V3a in the direction normal to the mid-surface. Now Eq. (15.2) is written simply as
{X} y Z
= y~Na(~,zl)
Ya Za
mid
+r I
where ta is the shell thickness at the node a. Construction of a vector normal to the mid-surface is a simple process (see Sec. 13.4.2).
15.2.2 Displacement field The displacement field is now specified for the element. As the strains in the direction normal to the mid-surface will be assumed to be negligible, the displacement throughout the element will be taken to be uniquely defined by the three Cartesian components of the mid-surface node displacement and two rotations about two orthogonal directions normal to the nodal vector V3a. If these two orthogonal directions are denoted by unit vectors Via and V2a with corresponding rotations ~a and fla (see Fig. 15.3), we can write, similar to Eq. (15.2) but dropping the subscript 'mid' for simplicity,
{U} P 113
--y~Na(~,?])
(/Oa} 1 ,la ~)a 113a
-~"-~ta
(15.4)
,
from which the usual form is readily obtained as
In/ v
//3
_ N~e;
~ e __
/elu 9
~a Va with
~e Un
~e __
/1)a (~a
(15.5)
where u, v and w are displacements in the directions of the global x, y and z axes. As an infinity of vector directions normal to a given direction can be generated, a particular scheme has to be devised to ensure a unique definition. Some such schemes were discussed in Chapter 13. Here another unique alternative will be given, 3'5 but other possibilities are open. 8 Here V 3 a i s the vector to which a normal direction is to be constructed. A coordinate vector in a Cartesian system may be defined by x = Xex + yey + Zez
(15.6)
in which ex, ey and ez are three (orthogonal) base vectors. To find the first normal vector we find the minimum component of V3a and construct a vector cross-product with the unit vector in this direction to define Via. For example, if the x component of V3a is the smallest one we construct Via = ex • V3a
(15.7)
480
Shells as a special 3-D case
where ex=[1
0
0] T
is the form of the unit vector in the x direction. Now Via-"
Via
IVlal
where ]Vial- ~/VTaVIa
(15.8)
defines the first unit vector. The second normal vector may now be computed from V2a - V3a X Via
(15.9)
and normalized using the form in Eq. (15.8). We have thus three local, orthogonal axes defined by unit vectors u la, u and u (15.10) Once again if Na are Co functions then displacement compatibility is maintained between adjacent elements. The element coordinate definition is now given by the relation Eq. (15.2) and has more degrees of freedom than the definition of the displacements. The element is therefore of the 'superparametric' kind ~ and the constant strain criteria are not automatically satisfied. Nevertheless, it will be seen from the definition of strain components involved that both rigid body motions and constant strain conditions are available. Physically it has been assumed in the definition of Eq. (15.4) that no strains occur in the 'thickness' direction ~. While this direction is not always exactly normal to the midsurface it still represents a good approximation of one of the usual shell assumptions. At each mid-surface node a of Fig. 15.3 we now have the five basic degrees of freedom, and the connection of elements will follow precisely the patterns described in Chapter 13 (Secs 13.3 and 13.4).
15.2.3 Definition of strains and stresses To derive the properties of a finite element the essential strains and stresses need first to be defined. The components in directions of orthogonal axes related to the surface (constant) are essential if account is to be taken of the basic shell assumptions. Thus, if at any point in this surface we erect a normal 5~with two other orthogonal axes ~ and tangential to it (Fig. 15.3), the strain components of interest are given simply by the three-dimensional relationships in Chapter 2: c~
~,~ (15.11)
with the strain in direction ~ neglected so as to be consistent with the usual shell assumptions. It must be noted that in general none of these directions coincide with
Shell element with displacement and rotation parameters 481 those of the curvilinear coordinates ~, ~7, ~, although ~, ~ are in the ~r/plane (ff = constant).* The stresses corresponding to these strains are defined by a matrix (r and for elastic behaviour are related to the usual elasticity matrix I). Thus o'~ o'~
0-9 7"~
-- I ) ( ~ - ~o) + &o
(15.12)
where ~0 and &0 represent any 'initial' strains and stresses, respectively. The 5 x 5 matrix D can now include any anisotropic properties and indeed may be prescribed as a function of ~ if sandwich or laminated construction is used. For the present we shall define it only for an isotropic material. Here E I) = 1 - / / 2
1
u
0
0
0
u 0
1 0 0 0
0 (1 - u)/2 0 0
0 0 to(1 - u)/2 0
0 0 0 to(1 - u)/2
(15.13)
in which E and u are Young's modulus and Poisson's ratio, respectively. The factor t~ included in the last two shear terms is taken as 5/6 and its purpose is to improve the shear displacement approximations (see Chapter 11). From the displacement definition it will be seen that the shear distribution is approximately constant through the thickness, whereas in reality the shear distribution for elastic behaviour is approximately parabolic. The value t~ = 5/6 is the ratio of relevant strain energies. It is important to note that this matrix is not defined by deleting appropriate terms from the equivalent three-dimensional stress matrix. It must be derived by substituting cr~ - 0 into the three-dimensional constitutive equations and performing suitable elimination so that this important shell assumption is satisfied. This is similar to the procedure for deriving plane stress behaviour in two-dimensional analyses.
15.2.4 Element properties and necessary transformations The stiffness m a t r i x - and indeed all other 'element' property matrices - involves integrals over the volume of the element, which are quite generally of the form f
I-I dx dy dz
(15.14)
e
where the matrix H is a function of the coordinates. For instance, in the stiffness matrix H-
l]rl)l]
(15.15)
* Indeed, these directions will only approximately agree with the nodal directions Via, V2a previously derived, as in general the vector V3a is only approximately normal to the mid-surface.
482
Shells as a special 3-D case
and with the usual definition _ B~e
(15.16)
we have fl defined in terms of the displacement derivatives with respect to the local Cartesian coordinates 2, y, z by Eq. (15.11). Now, therefore, two sets o f transformations are necessary before the element can be integrated with respect to the curvilinear coordinates ~, r/, ~. First, by identically the same process as that used for isoparametric elements, the derivatives with respect to the x, y, z directions are obtained. As Eq. (15.4) relates the global displacements u, v, w to the curvilinear coordinates, the derivatives of these displacements with respect to the global x, y, z coordinates are given by a matrix relation" - Ou
Ov
Ow "
- Ou
Ox Ou
Ox Ov
Ox Ow
o~
o~
o~
Ou
Ov
Ow
cgy Ou .&
Oy Ov &
cgy Ow &
=j-1
Ov
Ow"
o0o0o0 Ou
Ov
Ow
.o~
o~
or
(15.17)
In this, the Jacobian matrix is defined as
J
_~
Ox Oy &o~ o~ o~ Ox Oy Oz o,7o,7o,1 Ox Oy Oz _or o~ o~.
(15.18)
and calculated from the coordinate definitions of Eq. (15.2). Now, for every set of curvilinear coordinates the global displacement derivatives can be obtained numerically. A second transformation to the local displacements 2, y, z will allow the strains, and hence the I~ matrix, to be evaluated. The directions of the local axes can be established from a vector normal to the ~rl mid-surface (~ = 0). This vector can be found from two vectors 0x/0~ and 0x/Oq that are tangential to the mid-surface. Thus "Ox "
V3=
_
~
Oz
x
"Ox -
O y cgz
Oy
&Ox
&Ox
Oz
&Oy
&Oy
~
Oy cgz
(15.19)
We can now construct two perpendicular vectors V 1 and V2 following the process given previously to describe the 2 and y directions, respectively. The three orthogonal
Shell element with displacement and rotation parameters 483 vectors can be reduced to unit magnitudes to obtain a matrix of vectors in the 2, y, z directions (which is in fact the direction cosine matrix) given as 0 = [v l,
(15.20
The global derivatives of displacement u, v and w are now transformed to the local derivatives of the local orthogonal displacements by a standard operation "0~
O~
Ofv"
O2 O~
O2 O~
O2 OW
O~
O~
OW
I ~176
_0~
O~
0~ .
L&
Ox
= 0r
Ox
O__y_v Ow
0
(15.21)
O_2v Ow
&
&.
From this the components of the fl matrix can now be found explicitly, noting that five degrees of freedom exist at each node: -- Bfi e m
(15.22)
where the form of ~e is given in Eq. (15.5). The infinitesimal volume is given in terms of the curvilinear coordinates as dx dy dz - det J d~ d~7dff - j d~ d~7dff
(15.23)
where j = det J. This standard expression completes the basic formulation. Numerical integration within the appropriate limits is carried out in exactly the same way as for three-dimensional elements using Gaussian quadrature formulae. 1'9 An identical process serves to define all the other relevant element matrices arising from body and surface loading, inertia matrices, etc. As the variation of the strain quantities in the thickness, or ~ direction, is linear, two Gauss points in that direction are sufficient for homogeneous elastic sections, while two to four in the ~, r/directions are needed for parabolic and cubic shape functions
Na. It remarked here that, in fact, the integration with respect to ( can be performed explicitly if desired, thus saving computation time. 2'5
15.2.5 Some remarks on stress representation The element properties are now defined, and the assembly and solution are in standard form. It remains to discuss the presentation of the stresses, and this problem is of some consequence. The strains being defined in local direction, &, are readily available. Such components are indeed directly of interest but as the directions of local axes are not easily visualized (and indeed may not be continuously defined between adjacent elements) it is sometimes convenient to transfer the components to the global system
484
Shells as a special 3-D case
using the standard transformation
[fix 7"xy Txz] 7-yx O'y "ryz| LT"zx 7"zy O'zJ
[0"2 T.rcS, 7"2~1
9
9
=0
"ry~ oy L7~2
T~
"rye| Or
(15.24)
cr~j
This transformation should be performed only for elements which belong to the approximation for the same smooth surface and/or same material. In a general shell structure, the stresses in a global system do not, however, give a clear picture of shell surface stresses. It is thus convenient always to compute the principal stresses (or invariants of stress) by a suitable transformation. Regarding the shell stresses more rationally, one may note that the shear components ~-~ and ~-~ are often zero on the top and bottom surfaces and this may be noted when making the transformation of Eq. (15.24) before converting to global components to ensure in this case that the principal stresses lie on the surface of the shell. The values obtained directly for these shear components are the average values across the section. The maximum transverse shear on a solid cross-section occurs on the mid-surface and is equal to about 1.5 times the average value.
For axisymmetric shells the formulation is simplified. Now the element mid-surface is defined by only two coordinates ~, r/and a considerable saving in computer effort is obtained. 2 The element now is derived in a similar manner by starting from a two-dimensional definition of Fig. 15.4.
m Za
i 1
Zab
i 1 i 1 1 i i
(a)
r
(b)
Fig. 15.4 Coordinates for an axisymmetric shell' (a) coordinate representation; (b) shell representation.
-r
Special case of axisymmetric, curved, thick shells 485
Equations (15.1) and (15.2) are now replaced by their two-dimensional equivalents defining the relation between coordinates as
z
Za top
2 ?a
+ 1 - ~7 2 / zr~/ bottom)
(15.25)
1
"~ Y~ga(~) ({~a)mid~t--2~taV3a) with
COS2~3a} sin 2/2a
V3a --
in which 2/3a is the angle defined in Fig. 15.4(b) and ta is the shell thickness. Similarly, the displacement definition is specified by following the lines of Eq. (15.4). Here we consider the case of axisymmetric loading only. Non-axisymmetric loading is addressed in Chapter 16 along with other schemes which permit treatment of problems in a reduced manner. Thus, we specify the two displacement components as
{ul=~-~Nat1) ((bla 1113 + -2~Tta ( a
-)C sinO~3~a Sa
~)a)
(15.26)
In this (~a is the rotation illustrated in Fig. 15.5, and fia, /13aare the displacements of the middle surface node. Global strains are conveniently defined by the relationship 1~
Ou ~r
e=
Ez e0
~/rz
Or Ow
=
Oz
(15.27)
u_
OU
gz
r
011)
These strains are transformed to the local coordinates and the component normal to r/ (r/--- constant) is neglected. All the transformations follow the pattern described in previous sections and need not be further commented on except perhaps to remark that they are now carried out only between sets of directions ~, ~, r, z, and ?, ~, thus involving only two variables. Similarly the integration of element properties is carried out numerically with respect to ~ and r/only, noting, however, that the volume element is dx dy dz = 2rcr det J d~ d~7dO = 2rrr j d~ d~7dO
(15.28)
By suitable choice of shape functions Na (~), straight, parabolic, or cubic shapes of variable thickness elements can be used as shown in Fig. 15.6.
486
Shells as a special 3-D case
ZA
Fig. 15.5 Global displacements in an axisymmetric shell.
~ = ' ~ . . .--.. .j,
,,=,
~=-1
1
(a)
3,,
""-,
r/
"'~'----.
=-1
,
(b)
1
-.
1
r/
'
,~--- -.. ~=-~ 3 (c)
1
Fig. 15.6 Axisymmetric shell elements: (a)linear; (b) parabolic; (c) cubic.
= -1
Convergence 487
The transformations necessary in this chapter are somewhat involved and the programming steps are quite sophisticated. However, the application of the principle involved is available for thick plates and readers are advised to first test their comprehension on such a simple problem. Here the following obvious simplifications arise. 1. ~ = 2 z / t and unit vectors v l, v2 and v3 can be taken as ex, ey, and ez, respectively. 2. t~ a and/~a are simply the r o t a t i o n s ~y and 0x, respectively (see Chapter 12). 3. It is no longer necessary to transform stress and strain components to a local system of axes 2, y, z and global definitions x, y, z can be used throughout. For elements of this type, numerical thickness integration can be avoided and, as an exercise, readers are encouraged to derive the stiffness matrices, etc., for, say, linear, rectangular elements. Forms will be found which are identical to those derived in Chapter 12 with an independent displacement and rotation interpolation and using shear constraints. This demonstrates the essential identity of the alternative procedures.
Whereas in three-dimensional analysis it is possible to talk about absolute convergence to the true exact solution of the elasticity problem, in equivalent plate and shell problems such a convergence cannot happen. As the element size decreases the so-called convergent solution of a plate bending problem approaches only to the exact solution of the approximate model implied in the formulation. Thus, here again convergence of the above formulation will only occur to the exact solution constrained by the requirement that straight 'normals' remain straight during deformation. In elements of finite size it will be found that pure bending deformation modes are nearly always accompanied by some shear strains which in fact do not exist in the conventional thin plate or shell bending theory (although quite generally shear stresses may be deduced by equilibrium considerations on an element of the model, similar to the manner by which shear stresses in beams are deduced). Thus large elements deforming mainly under bending action (as would be the case of the shell element degenerated to a fiat plate) tend to be appreciably too stiff. In such cases certain limits of the ratio of size of element to its thickness need to be imposed. However, it will be found that such restrictions often are relaxed by the simple expedient of reducing the integration order. 5 Figure 15.7 shows, for instance, the application of the quadratic eight-node element to a square plate situation. Here results for integration with 3 x 3 and 2 • 2 Gauss points are given and results plotted for different thickness-to-span ratios. For reasonably thick situations, the results are similar and both give the additional shear deformation not available by thin plate theory. However, for thin plates the results with the more exact integration tend to diverge rapidly from the now correct thin plate results whereas the reduced integration still gives excellent results. The reasons for this improved performance are fully discussed in Chapter 12 and the reader is referred there for further plate examples using different types of shape functions.
488
Shells as a special 3-D case
t=0-1
1.13
a
o.os
Thin plate
~a.
0.01 ~,,,
exact
, ,o"""
t=0.1
~.,~
-
...-- -
~~
.0
,,., \
~ ....... 9s
O ~0~
0.01
"''"~N.
" n-'" ( ~ 0.005
0.5 ~'~
0.005
~'~
(b)
~'/"
I
,~
I
I
I
0
I
2
I
4
I
6
I
8
'
10
Fig. 15.7 A simply supported square plate under uniform load qo: plot of central deflection Wc for eight-node elements with (a) 3 x 3 Gauss point integration and (b) with 2 x 2 (reduced) Gauss point integration. Central deflection is r for thin plate theory.
All the formulations presented in this chapter can of course be used for all non-linear materials. The procedures are similar to those mentioned in Chapters 11 and 12 dealing with plates. Now it is only necessary to replace Eqs (15.12) and (15.13) by the appropriate constitutive equation and tangent operator, respectively. In this case it is necessary always to perform the through thickness integration numerically since a priori knowledge of the behaviour will not be available. Any of the constitutive models described in Chapter 4 may be used for this purpose provided appropriate transformations are made to make cr~ zero (viz. Sec. 6.2.4).
A limited number of examples which show the accuracy and range of application of the axisymmetric shell formulation presented in this chapter will be given. For a fuller selection the reader is referred to references 2-8.
15.7.1 Spherical dome under uniform pressure The 'exact' solution of shell theory is known for this axisymmetric problem, illustrated in Fig. 15.8. Twenty-four cubic-type elements are used with graded size more closely spaced towards supports. Contrary to the 'exact' shell theory solution, the present
Some shell e x a m p l e s 1 Ib/in
40
ci
~p = 1 Ib/in 2
z
,
30 i
I
t=3in
20 10
- 1035
Case II
Case l
30
25
I,
!
20
15
I^ 10
.
5
0
0 Analytical Finite element o Case l z~ Case II
10 ~N 20
~~
30 40 o
5U35"
30
25
20
15
10
5
0
Fig. 15.8 Spherical dome under uniform pressure analysed with 24 cubic elements (first element subtends an angle of O.1~ from fixed end, others in arithmetic progression).
formulation can distinguish between the application of pressure on the inner and outer surfaces as shown in the figure.
15.7.2 Edge loaded cylinder A further axisymmetric example is shown in Fig. 15.9 to study the effect of subdivision. Two, six, or 14 cubic elements of unequal length are used and the results for both of the finer subdivisions are almost coincident with the exact solution. Even the two-element solution gives reasonable results and departs only in the vicinity of the loaded edge. Once again the solutions are basically identical to those derived with independent slope and displacement interpolation in the manner presented in Chapter 12.
15.7.3 Cylindrical vault This is a test example of application of the full process to a shell in which bending action is dominant as a result of supports restraining deflection at the ends (see also Sec. 13.8.2).
489
490
Shells as a special 3-D case
e-
e-
|
E
E
e
o4
r
v-
9
ID
E
i,i i,i qr
x<~
31
Theoretical
I_ r
1
-1 0
0.2
0.4
0.6 Z
10 El
_1
lO in dia.
0.8
o
-J
1.0
1.2
0.6 0.8 1.0 Z u = radial displacement (10 -3) in My, = meridional moment (Ib/in) E = 107 Ib/in 2 v=0.3
1.2
60
40
:~
~
--
2o 1 O'
-20
'
0
0.2
0.4
Fig. 15.9 Thin cylinder under a unit radial edge load.
In Fig. 15.10 the geometry, physical details of the problem, and subdivision are given, and in Fig. 15.11 the comparison of the effects of 3 x 3 and 2 x 2 integration using eight-node quadratic elements is shown on the displacements calculated. Both integrations result, as expected, in convergence. For the more exact integration, this is rather slow, but, with reduced integration order, very accurate results are obtained, even with one element. The improved convergence of displacements is matched by rapid convergence of stress components.
492
Shells as a special 3-D case
40 ~
30 ~
~(a) ~
20 ~
~
",~
20 ~
!
30 ~
40 ~
/
"~.
40*
40* 20*
10 ~ 00 10 ~ Axial at support
20*
0.00
3
0
0.005 3=
0.0101
0.010
Original element Integration for reference 3
shear reduced
9
El
Fig. 15.11
Mesh
Simple 2 x 2
(a)
o
,,
integration
x
(b)
-
+
(c)
~,
(d)
o
Displacement (parabolic element), cylindrical shell roof.
has been applied to the doubly curved dams illustrated in Chapter 9 of Volume 1 (Fig. 9.28). Indeed, exactly the same subdivision is again used and results reproduce almost exactly those of the three-dimensional solution. 4 This remarkable result is achieved at a very considerable saving in both degrees of freedom and computer solution time. Clearly, the range of application of this type of element is very wide.
15.7.5 Pipe penetration and spherical cap ..................
- = .
..............
. = =
............
-~:~----~
.................
--4-:
..............
-~- ..........................
-.
...................
- ..................................
The last two examples, a pipe penetration 12 shown in Figs 15.12 and 15.13 and a spherical cap 8 shown in Fig. 15.14, illustrate applications in which the irregular shape of elements is used. Both illustrate practical problems of some interest and show that with reduced integration a useful and very general shell element is available, even when the elements are quite distorted.
Concluding remarks
I S'r
493
i E = 30 x 10 s Ib/in 2 v=0.3
I I I
,, !
t = 0.05 in
! !
,, ', ,,
~_
19.5
Two layers of true three-dimensional elements
1 !
',
~_
O* line
! ,,, +,, . . . . . .
.
I
S
s"*
"1 I I I
,
, !
,
r
I ',
=Z b
~.
10 in
--
~
oJr"
\,,
',,
%
o
/
x
/-/
~[_
. . . . .
\
it=0"1 in
~
Y
=501bin 2
19.5 in
;"
Fig. 15.12 An analysis of cylinder intersection by means of reduced integration shell-type elements.12
!iijiljii iiii iiiiiiii+ ili i
~~+~+~+~+~+~+~~+~~G~+~
i
i i~iii~+~i~i~ii~!!~!~~ii~i~!~!~+~i~i~!~i~ ,,ii,,ii,,i,,,~,,~,i~
i!ii illii!ii!iiiiiii+iiiii!iiiiiiiiiiiiiiii!ii!!iii i
The elements described in this chapter using degeneration of solid elements are shown in plate and axisymmetric problems to be nearly identical to those described in Chapters 12 and 14 where an independent slope and displacement interpolation is directly used in the middle plane. For the general curved shell the analogy is less obvious but clearly still exists. We should therefore expect that the conditions established in Chapter 12 for robustness of plate elements to be still valid. Further, it appears possible that other additional conditions on the various interpolations may have to be imposed in curved element forms. Both statements are true. The eight- and nine-node elements which we have shown in the previous section to perform well will fail under certain circumstances and for this reason many of the more successful plate elements also have been adapted to the shell problem. The introduction of additional degrees of freedom in the interior of the eight-node serendipity element was first suggested by Cook 13'14 and later by Hughes 15-~7 without, however, achieving complete robustness. The full Lagrangian cubic interpolation as shown in Chapter 12 is quite effective and has been shown to perform well. However, the
494
Shells as a special 3-D case 25
I
~
~. 20 % v
u~
~
15 -~o
I
o
Experimental
}
x
Experimental
}
---
Finite element
Finite element
I
Outside surface
_
Inside surface
_
10
_
0 0 "i"
5 . . . . . . . . .
I
1
,
I,,
I
2
,
3
Distance from intersection (in) (a)
20 ~
!o
15
% ~
x
10
~
---
i Experimental Finite element
Experimental
Finite element
t
i Outside surface Inside surface
s IA x- ~ - . x .
-5
~
i t i 1 2 3 Distance from intersection (in)
(b) Fig. 15.13 Cylinder-to-cylinderintersections of Fig. 15.12" (a) hoop stresses near 0~ line; (b) axial stresses near 0~ line.
best results achieved to date appear to be those in which 'local constraints' are applied (see Sec. 12.5) and such elements as those due to Dvorkin and Bathe, 18 Huang and Hinton, ~9 and Simo eta/. 2~ fall into this category. While the importance of transverse shear strain constraints is now fully understood, the constraints introduced by the 'in-plane' (membrane) stress resultants are less amenable to analysis (although the elastic parameters Et associated with these are of the same order as those of shear G t). It is well known that membrane locking can occur in situations that do not permit inextensional bending. Such locking has been thoroughly discussed 22-24 but to date the problem has not been rigorously solved and further developments are required. Much effort is continuing to improve the formulation of the processes described in this chapter as they offer an excellent solution to the curved shell problem. 24-27
References
I
t= 2.36
r = 56.30'Y /
I
(a)
/
,,,jOo~/,
39,
E = 10 0.2
/
/
I
z
v =
~
i
(b)
10 u
~'=0
0.05
_o--- o . - - - 0 - - -
30 J
-'o" -~L" "0~ ~
/
~
.m_
fie
39 ~
....o.- " " ..o-- " "
3 x 3 integration
t.-
~i
20 I
Exact
2 x 2 integration - - ~
0.10
Z
~
=.
9
-
9
9m zx Mesh (a) 9
o Mesh (b)
Fig. 15.14 A spherical cap analysis with irregular is0parametric shell elements using full 3 x 3 and reduced 2 x 2 integration.
1. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann, Oxford, 6th edition, 2005. 2. S. Ahmad, B.M. Irons and O.C. Zienkiewicz. Curved thick shell and membrane elements with particular reference to axi-symmetric problems. In Proc. 2nd Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-68-150, pages 539-572, Wright Patterson Air Force Base, Ohio, October 1968.
495
496
Shells as a special 3-D case
3. S. Ahmad. Curved finite elements in the analysis of solids, shells and plate structures. PhD thesis, Department of Civil Engineering, University of Wales, Swansea, 1969. 4. S. Ahmad, B.M. Irons and O.C. Zienkiewicz. Analysis of thick and thin shell structures by curved finite elements. International Journalfor Numerical Methods in Engineering, 2:419-451, 1970. 5. O.C. Zienkiewicz, J. Too and R.L. Taylor. Reduced integration technique in general analysis of plates and shells. International Journal for Numerical Methods in Engineering, 3:275-290, 1971. 6. S.E Pawsey and R.W. Clough. Improved numerical integration of thick slab finite elements. International Journal for Numerical Methods in Engineering, 3:575-586, 1971. 7. S.E Pawsey. The analysis of moderately thick to thin shells by the finite element method. PhD dissertation, Department of Civil Engineering, University of California, Berkeley, 1970 (also SESM Report 70-12). 8. J.J.M. Too. Two dimensional plate, shell and finite prism isoparametric elements and their application. PhD thesis, Department of Civil Engineering, University of Wales, Swansea, 1970. 9. M. Abramowitz and I.A. Stegun, editors. Handbook of Mathematical Functions. Dover Publications, New York, 1965. 10. I.S. Sokolnikoff. The Mathematical Theory of Elasticity. McGraw-Hill, New York, 2nd edition, 1956. 11. A.C. Scordelis and K.S. Lo. Computer analysis of cylindrical shells. J. Am. Concr. Inst., 61: 539-561, 1964. 12. S.A. Bakhrebah and W.C. Schnobrich. Finite element analysis of intersecting cylinders. Technical Report UILU-ENG-73-2018, University of Illinois, Civil Engineering Studies, 1973. 13. R.D. Cook. More on reduced integration and isoparametric elements. International Journal for Numerical Methods in Engineering, 5:141-142, 1972. 14. R.D. Cook. Concepts and Applications of Finite Element Analysis. John Wiley & Sons, Chichester, 1982. 15. T.J.R. Hughes and M. Cohen. The 'heterosis' finite element for plate bending. Computers and Structures, 9:445-450, 1978. 16. T.J.R. Hughes and W.K. Liu. Non linear finite element analysis of shells. Part I. Computer Methods in Applied Mechanics and Engineering, 26:331-362, 1981. 17. T.J.R. Hughes and W.K. Liu. Non linear finite element analysis of shells. Part II. Computer Methods in Applied Mechanics and Engineering, 27:331-362, 1981. 18. E.N. Dvorkin and K.-J. Bathe. A continuum mechanics based four node shell element for general non-linear analysis. Engineering Computations, 1:77-88, 1984. 19. E.C. Huang and E. Hinton. Elastic, plastic and geometrically non-linear analysis of plates and shells using a new, nine-noded element. In P. Bergan et al., editors, Finite elements for Non Linear Problems, pages 283-297. Springer-Verlag, Berlin, 1986. 20. J.C. Simo and D.D. Fox. On a stress resultant geometrically exact shell model. Part I: Formulation and optimal parametrization. Computer Methods in Applied Mechanics and Engineering, 72: 267-304, 1989. 21. J.C. Simo, D.D. Fox and M.S. Rifai. On a stress resultant geometrically exact shell model. Part II: The linear theory; computational aspects. Computer Methods in Applied Mechanics and Engineering, 73:53-92, 1989. 22. H. Stolarski and T. Belytschko. Membrane locking and reduced integration for curved elements. J. Appl. Mech., 49:172-176, 1982. 23. H. Stolarski and T. Belytschko. Shear and membrane locking in curved C Oelements. Computer Methods in Applied Mechanics and Engineering, 41:279-296, 1983. 24. R.V. Milford and W.C. Schnobrich. Degenerated isoparametric finite elements using explicit integration. International Journal for Numerical Methods in Engineering, 23:133-154, 1986.
References 497 25. D. Bushnell. Computerized analysis of shells -goveming equations. Computers and Structures, 18:471-536, 1984. 26. M.A. Cilia and N.G. Gray. Improved coordinate transformations for finite elements: the Lagrange cubic case. International Journal for Numerical Methods in Engineering, 23:1529-1545, 1986. 27. S. Vlachoutsis. Explicit integration for three dimensional degenerated shell finite elements. International Journal for Numerical Methods in Engineering, 29:861-880, 1990.
ii iii!lK:i,i~i!i Ci~i,~iiiliCi~ii~i!ilii~ii iii~iiliii&lii iii!ii!iii~i!~iilii~liS? iiiiii!i~iii~i{~
Semiianalytical finite element processes- use of orthogonal functions and 'finite strip' methods ii of for
the to
a of use not is
|~i~iiiiii~|~d~i~i!Rii
i
!i
i l i i i iiii
Standard finite element methods have been shown to be capable, in principle, of dealing with any two- or three- (or even four-)* dimensional situations. Nevertheless, the cost solutions increases greatly with each dimension added and indeed, on occasion, overtaxes the available computer capability. It is therefore always desirable to search alternatives that may reduce computational effort. One such class of processes of quite wide applicability will be illustrated in this chapter. In many physical problems the situation is such that the geometry and material properties do not vary along one coordinate direction. However, the 'load' terms may still exhibit a variation in that direction, preventing the use of such simplifying assumptions as those that, for instance, permitted a two-dimensional plane strain or axisymmetric analysis to be substituted for a full three-dimensional treatment. In such cases it is possible still to consider a 'substitute' problem, not involving the particular coordinate (along which the geometry and properties do not vary), and to synthesize true answer from a series of such simplified solutions. The method to be described is of quite general use and, obviously, is not limited structural situations. It will be convenient, however, to use the nomenclature of structural mechanics and to use potential energy minimization as an example. We shall confine our attention to problems of minimizing a quadratic functional for linear elastic material. The interpretation of the process involved as the application partial discretization leading to semi-discrete forms ~ followed (or preceded) by the of a Fourier series expansion should be noted. Let (x, y, z) be the coordinates describing the domain (in this context these do necessarily have to be the Cartesian coordinates). The last one of these, z, is the coordinate along which the geometry and material properties do not change and which limited to lie between two values
O
iiiiiiiiiii iiii i iLi i
boundary values are thus specified at z = 0 and z - a. See finite elements in the time domain in reference 1.
Introduction 499
We shall assume that the shape functions defining the variation of displacement u can be written in a separation of variables form as u = N(x,
y, z)fie = ~ ( N(x, y) cos ~17rz + a
l~l(x, y) sin
l=l
17rz) (fil )e a
L
(16.1)
"- E Nl (X, y, l=l
Z) (Lll) e
In this type of representation completeness is preserved in view of the capability of Fourier series to represent any continuous function within a given region (naturally assuming that the shape functions N and 1~Iin the domain x, y satisfy the same requirements). The loading terms will similarly be given a form b =
~-~ (17rZ cos
l=l
[)l a
+ sin
17rZ~l )
~~
=
a
b l (x, y, z)
(16.2)
/=1
for body force, with similar form for any concentrated loads and boundary tractions. Indeed, initial strains and stresses, if present, would be expanded again in the above form. Finally, we restrict our attention to linear elastic materials in which the constitutive equation is given by tr = D(e - e0) + tr0 (1 6.3) with stress and strain ordered as
zx] ~--" [EX Cy Cz ")/xy 'ffyz ~ZX]T and the elastic material matrix has a z-symmetry direction such that
O --
-Dll DE1 D31 D41
0 0
D12 D22
D13 D23
D32
D42 0 0
D33
D14 D24 D34
0 0 0
0 0 0
D43 0
D44 0
0 D55
0
0
D65
0 D56 D66
(16.4)
Applying the standard process for the determination of the element contribution to the equation minimizing the potential energy H, and limiting our attention to the contribution of body forces b only, we can write tgI'I _ Ke
O~le --
/~ /fie/ .
fiLe
-]-
.
-- 0
(16.5)
fLe
In the above, to avoid summation signs, the vectors fie, etc., are expanded, listing the contribution of each value of I separately.
500
Semi-analyticalfinite element processes Now a typical submatrix of K e is
J~(B/)TD B m dr2 dz
( K l m ) e --
(16.6)
and a typical term of the 'force' vector becomes
(f/)e __ f ~ ( N / ) T b / d ~
(16.7)
Without going into details, it is obvious that the matrix given by Eq. (16.6) will contain the following integrals as products of various submatrices: 11 - -
fO a sin 17rz cos mTrz dz a a
12 -"
fO a sin ~17rz sin ~m Trz dz a a
13 --
fo a cos ~17rz cos ~m Trz dz a a
(16.8)
These integrals arise from products of the derivatives contained in the definition of B l and, owing to the well-known orthogonality property, give
1,a
/2 = / 3 -
0 '
forl = m
for / ~- m .9 . 1,.m .= 1. 2,
(16.9)
The first integral 11 is only zero when l and m are both even or odd numbers. The term involving 11, however, vanishes in many applications because of the structure of B I. This means that K e becomes a block diagonal matrix and that the assembled final equations of the system have the form
Kll
~1
K22
fl
~2 9
KLL
fiL
f2 ~
.
(16.10)
f?
and the large system of equations splits into L separate problems: K l l ( l I _jr_fl __ 0
(16.11)
-- f (B/a)TDB / dr2
(16.12)
in which Klal
Further, from Eqs (16.7) and (16.2) we observe that owing to the orthogonality property of the integrals given by Eqs (16.8), the typical load term becomes simply
fa/ "~
f (NI)Tb/ dr2
(16.13)
Prismatic bar
This means that the force term of the/th harmonic only affects t h e / t h system of Eq. (16.11) and contributes nothing to the other equations. This extremely important property is of considerable practical significance for, if the expansion of the loading factors involves only one term, only one set of equations need be solved. The solution of this will tend to the exact one with increasing subdivision in the xy domain only. Thus, what was originally a three-dimensional problem now has been reduced to a two-dimensional one with consequent reduction of computational effort. The preceding derivation was illustrated on a three-dimensional, elastic situation. Clearly, the arguments could equally well be applied for reduction of two-dimensional problems to one-dimensional ones, etc., and the arguments are not restricted to problems of elasticity. Any physical problem governed by a minimization of a quadratic functional or by linear differential equations is amenable to the same treatment. A word of warning should be added regarding the boundary conditions imposed on u. For a complete decoupling to be possible these must be satisfied separately by each and every term of the expansion given by Eq. (16.1). Insertion of a zero displacement in the final reduced problem implies in fact a zero displacement fixed throughout all terms in the z direction by definition. Care must be taken not to treat the final matrix therefore as a simple reduced problem. Indeed, this is one of the limitations of the process described. When the loading is complicated and many Fourier components need to be considered the advantages of the approach outlined here reduce and the full three-dimensional solution sometimes becomes more efficient. Other permutations of the basic definitions of the type given by Eq. (16.1) are obviously possible. For instance, two independent sets of parameters ~e may be specified with each of the trigonometric terms. Indeed, on occasion use of other orthogonal functions may be possible. The appropriate functions are often related to a reduction of the differential equation directly using separation of variables. 2
Consider a prismatic bar, illustrated in Fig. 16.1, which is assumed to be held at z = 0 and z = a in a manner preventing all displacements in the xy plane but permitting unrestricted motion in the z direction (traction tz = 0). The problem is fully three dimensional and three components of displacement u, o, and w have to be considered. Subdividing into finite elements in the xy plane we can prescribe the/th displacement components as
sin /z 0
U1 - -
Ul Wl
=~-~Nb[~ b
/~
sin 7/Z 0
V/ COS 71Z
(16.14)
11)/
where 71 = 17r/a. In this, Nb are simply the (scalar) shape functions appropriate to the elements used in the xy plane and again 71 = 17r/a. If, as shown in Fig. 16.1, simple triangles are used then the shape functions are given in area coordinates by Nb = L b, but any other isoparametric elements would be equally suitable. The displacement expansion ensures zero u and v displacements at the ends and the zero tz traction condition can be imposed in a standard manner.
501
502 Semi-analytical finite element processes
X
Z
Fig. 16.1 A prismatic bar reduced to a series of two-dimensional finite element solutions.
As the problem is fully three dimensional, the appropriate expression for strain involving all six components needs to be considered. This expression is given in Eq. (2.9a) of Chapter 2. On substitution of the shape function given by Eq. (16.14) for a typical term of the B matrix we have
B/=
Nb,x sin 71z 0 0 Nb, y sin 7l Z
0 Nb,y sin 7lZ 0 Nb,xsin 7l Z
0 Nb7l COS 71Z .Nb'~l COS 71Z 0
0 0 --NbTl sin 7lZ 0
(16.15)
Nb,yCOS 71Z Nb,x COS ")/lZ
It is convenient to separate the above as (16.16)
B / = !~/ sin 7lZ + ~lb COS7lZ where
Nb,x 0 0
0 Nb,y 0
Nb,y Nb,x 0 0
0 0
0 0 --NbTl 0 0 0
and
=l B b --
0 0 0 0 0
NbTl
0 0 0 0
0 0 0 0
N671 Nb,y 0 Nb,x
Prismatic bar
In all of the above it is assumed that the parameters are listed in the usual order:
U~ta --[~.lla
~la 1)
(16.17)
-1 T Wo]
and that the axes are as shown in Fig. 16.1. The stiffness matrix can be computed in the usual manner, noting that (K//b)e -- f ~
(16.18)
B lT a D B lb dr2 e
On substitution of Eq. (16.16), multiplying out, and noting the value of the integrals from Eq. (16.9), this reduces to
(Klalb) e -" -~ a/fA(
d x dy 1 = 1, 2 . . . .
13/TD 1~/ "k- B=/T a D B= lb)
(16.19)
e
The integration is now simply carded out over the element area.* The contributions from distributed loads, initial stresses, etc., are found as the loading terms. To match the displacement expansions distributed body forces may be expanded in the Fourier series
bia =
f0 a
[sin 71z
Na [ 0
0
sinT/z0
o l{ x,xyz } by(x, y, z)
cos ~/l z
bz(x, y, z)
dz
(16.20)
Similarly, concentrated line loads can be expressed directly as nodal forces
fl
f0 a
[sin 7lZ
Na [ 0
0
sink'/z0
o]{ x xyz
cos 'ffl Z
fy(X, y,z) fz (X, y, Z)
dz
(16.21)
in which f/are intensities per unit length. The boundary conditions used here have been of a type ensuring simply supported conditions for the prism. Other conditions can be inserted by suitable expansions. The method of analysis outlined here can be applied to a range of practical problems - one of these being a popular type of box girder, concrete bridge, illustrated in Fig. 16.2. Here a particularly convenient type of element is the distorted, serendipity or Lagrangian quadratic or cubic element. 3 Finally, it should be mentioned that some restrictions placed on the general shapes defined by Eq. (16.1) or Eq. (16.14) can be removed by doubling the number of parameters and writing expansions in the form of two sums:
L
u = ~ l=l
L
lq(x, y) cos ~/l z ~1At "21-y ~ ff~(X, y) sin 'ffl z LIBl
(16.22)
l=l
Parameters II Al and II Bl are independent and for every component of displacement two values have to be found and two equations formed. * It should be noted that now, even for a single triangle, the integration is not trivial as some linear terms from Na will remain in 1] and ~.
503
504
Semi-analytical finite element processes
1.0
i L.~,,,.,, ~--..__ I~11111111111 o = 3o*~----~
!llllllllltltlllllllll
~5.0
''0
1.0 4.0 2.0
\
I
\
I -
r"
z
I I II
~y
4.0 -'
E = 1000.0 v = 0.25
~
6.5
Mesh (a)
Mesh (b)
Load amplitude = 12.73
(a) Mesh of isoparametric elements
0.0 __ -
\"
\
"
0:o..,,
.....
Mesh
(a)
oo Plci d o o Io o d d ~ 9 o op-~ I ~ - ~'- r 04 1 ,4 t ~ ~ ~ I ~,'1
oo ~176176 ooooo 000004 ~ 9
o. o O e~
y - stress at midspan
' --~"
-~'4"~
f,---i
o o
,
L/z
~x
p = 10
(Gy)l/2
~,
=Y 000
(b) Distribution of Gy stress on mid-span: computer stress plot. Point load on cantilevered span Fig. 16.2 A thick box bridge prism of straight or curved platform.
An alternative to the above process is to write the expansion as u-
~
IN(x, y ) e x p ( / % z ) ] ~e
and to observe that both N and ~ are then complex quantities. Complex algebra is available in standard programming languages and the identity of the above expression with Eq. (16.22) will be observed, noting that exp i0 = cos 0 + i sin 0 i~i~6~3i~ii ~iiiTE~| i~i~7~i~i~ii~iiiii~mii~i~i~iib{i~Fiiiii{aii~ni!iiiiei~i i{i iiii~iii{iiiiiiii!i~i1i~i~ii!~ii~iii~i!iiiii}}ii~!iiii~i~{~iii!iiiiiiii iii~i
ii ilil
iiiiiiib~ oi ixiiiiiiii~ ~ { ~i u{~~uT~~s~ 1 ~ 1 ~ } ~ ~ } ~ { ~ ~ { ~ ~ { ~ s t i~il
i ii
i
In the previous section a three-dimensional problem was reduced to that of two dimensions. Here we shall see how a somewhat similar problem can be reduced to onedimensional elements (Fig. 16.3).
Plates and boxes with flexure
////
/t /
///
///I
t
//
/t//
/I /
/
///
/tt /
/1
///
1t
//I
/ /1t/ /
i
//
/ /
, ,,/,,///r / /,,//"j''''/I
J
Fig. 16.3 A 'membrane' box with one-dimensional elements.
A box-type structure is made up of thin shell components capable of sustaining stresses only in its own plane. Now, just as in the previous case, three displacements have to be considered at every point and indeed similar variation can be prescribed for these. However, a typical element ab is 'one dimensional' in the sense that integrations have to be carried out only along the line ab and only stresses in that direction need be considered. Indeed, it will be found that the situation and solution are similar to that of a pin-jointed framework.
ii!ii•ii•iiiiiii•i•iiii•••!i!i••••iiii•iiii•i••ii•i•••ii••ii••i i i iiiiiii
i ii!ii iiiiiiii!iiiiiiiiiiiiiiii ilii
!i
!i ! i iiii i iii i !
i
!!
Consider now a rectangular plate simply supported at the ends and in which all strain energy is contained in flexure. Only one displacement, w, is needed to specify fully the state of strain (see Chapter 11). For consistency of notation with the plate and shell chapters where z is the thickness direction, the direction in which geometry and material properties do not change is now taken as y (see Fig. 16.4). To preserve slope continuity the functions now need to include a 'rotation' parameter 0 a . Use of simple beam functions (cubic Hermitian interpolations) is easy and for a typical element ab we can write the plate transverse displacement as (with 71 = 17r/a) w t = lq(x) sin ~ly(fll) e
(16.23)
ensuring simply supported end conditions. In this, the typical nodal parameters are ~I
Ua --
{} Wa
-I
Oa
(16.24)
The shape functions of the cubic type are easy to write and are in fact identical to the Hermitian polynomials given in Sec. 10.4.1 and also those used for the asymmetric thin shell problem [Chapter 14, Eq. (14.9)].
505
506
Semi-analytical finite element processes
1
2
Fig. 16.4 The 'strip' method in slabs.
Using all definitions of Chapter 11 the strains (curvatures) are found and the B matrices determined; now with C I continuity satisfied in a trivial manner, the problem of a two-dimensional kind has here been reduced to that of one dimension. This application has been developed by Cheung and others, 4-~8 named the 'finite strip' method, and used to solve many rectangular plate problems, box girders, shells, and various folded plates. It is illuminating to quote an example from the above papers here. This refers to a square, uniformly loaded plate with three sides simply supported and one clamped. Ten strips or elements in the x direction were used in the solution, and Table 16.1 gives the results corresponding to the first three harmonics. Not only is an accurate solution of each I term a simple one involving only some 19 unknowns but the importance of higher terms in the series is seen to decrease rapidly. Extension of the process to box structures in which both membrane and bending effects are present is almost obvious when this example is considered together with the one in the previous section. In the examples just quoted a thin plate theory using the single displacement variable w and enforcing C1 compatibility in the x direction was employed. Obviously, any of the independently interpolated slope and displacement elements of Chapter 12 could be used here, again employing either reduced integration or mixed methods.
Table 16.1 Square plate, uniform load q; three sides simply supported, one clamped (Poisson ratio = 0.3)
Term I 1 2 3 Z Series Multiplier
Central deflection
Central Mx
Maximum negative Mx
0.002832 -0.000050 0.002786 0.002786 0.0028
0.0409 -0.0016 0.0396 0.0396 0.039
-0.0858 0.0041 -0.0007 -0.0824 -0.084
qa 4 / D
qa 2
qa 2
Axisymmetric solids with non-symmetrical load 507 Parabolic-type elements with reduced integration are employed in references 14 and 15, and linear interpolation with a single integration point is shown to be effective in reference 16. Other applications for plate and box-type structures abound and additional information is given in the text of reference 18.
One of the most natural and indeed earliest applications of Fourier expansion occurs in axisymmetric bodies subject to non-axisymmetric loads. Now, not only the radial (u) and axial (w) displacement will have to be considered but also a tangential component (v) associated with the tangential angular direction 0 (Fig. 16.5). It is in this direction that the geometric and material properties do not vary and hence here that the elimination will be applied. To simplify matters we shall consider first components of load which are symmetric about the 0 = 0 axis and later include those which are antisymmetric. Describing now only the nodal loads (with similar expansion holding for body forces, boundary conditions, initial strains, etc.) we specify forces per unit of circumference as R = ~
~ l COS l0
l=l L
T = ~
~r,l sin lO
(16.25)
l=l L Z --" ~
ffTl COS l O
/=1
in the direction of the various coordinates for symmetric loads [Fig. 16.6(a)]. The apparently non-symmetric sine expansion is used for T, since to achieve symmetry the direction of T has to change for 0 > 7r.
IZ i
Fig. 16.5 An axisymmetric solid; coordinate displacement components in an axisymmetric body.
508
Semi-analytical finite element processes
I
I
(a)
(b)
Fig. 16.6 Load and displacement components in an axisymmetric body: (a) symmetric; (b) antisymmetric.
The displacement components are described again in terms of the two-dimensional (r, z) shape functions appropriate to the element subdivision, and, observing symmetry, we write, as in Eq. (16.14), U l ---
1)l
N a
wI
o] Illnat
(ColO O
sin l 0
0
Va
0
cos/O
Wal
(16.26)
To proceed further it is necessary to specify the general, three-dimensional expression for strains in cylindrical coordinates. These are given by 19 ~r
U,F
Ez
W,z
[u + o0]/r
"~rz
U,Z "~ ll),r
%o
V,z + w,o/r u,o/r Jr" U,r -- v/r
'ffOr
(16.27)
We have on substitution of Eq. (16.26) into Eq. (16.27), and grouping the variables as in Eq. (16.17):
Na,r cos lO 0 Na/r cos/0 Na,z cos l O 0 _-IUa/r sinl0
0 0 INa/r cos/0 0 Na,z sinl0 (Ua,r - Ua/r) sinl0
0 Na,z cos lO 0 N a , r COS l0 - I N a / r sinl0 0 .
(16.28)
Axisymmetric solids with non-symmetrical load 509 A purely axisymmetric problem may be described for the complete zero harmonic (l = 0) and a further simplification arises in that the strains split into two problems: the first involves the displacement components u and w which appear only in the first four components of strain; and the second involves only the v displacement component and appears only in the last two shearing strains. This second problem is associated with a torsion problem on the axisymmetric b o d y - with the first problem sometimes referred to as a torsionless problem. For an isotropic elastic material the stiffness matrix for these two problems completely decouples as a result of the structure of the D matrix, and they can be treated separately. However, for inelastic problems a coupling occurs whenever both torsionless and torsional loading are both applied as loading conditions on the same problem. Thus, it is often expedient to form the axisymmetric case including all three displacement components (as is necessary also for the other harmonics). For the elastic case the remaining steps of the formulation follow precisely the previous derivations and can be performed by the reader as an exercise. For the antisymmetric loading, of Fig. 16.6(b), we shall simply replace the sine by cosine and vice versa in Eqs (16.25) and (16.26). The load intensity terms in each harmonic are obtained by virtual work as
2lcos2lo l I
tl -- fJO
~'l sin 2 lO
21 cos 2 lO
dO =
71" ~r,l 21
when I = 1, 2 . . . .
27r
when 1 = 0
0
(16.29)
21 for the symmetric case. Similarly, for the antisymmetric case
2lsin2l~ l I
tl = fJO
~l COS2 lO 2 l sin 2 lO
71"
~-I
/~
when 1 = 1, 2 . . . .
(21
dO =
27r
~t 0
(16.30) when I = 0
The final nodal loads are then computed from the appropriate integrals f~ -- fA N~ fl dA
(16.31)
e
for body loads and
f fl = / _ ga f l dr dl"
(16.32)
e
for surface loads. We see from this and from an expansion of K e that, as expected, for 1 = 0 the problem reduces to only two variables and the axisymmetric case is retrieved when
510 Semi-analyticalfinite element processes
Fig, 16.7 Torsion of a variable section circular bar.
symmetric terms only are involved. Similarly, when I = 0 only one set of equations remains in the variable for v for the antisymmetric case. This corresponds to constant tangential traction and solves simply the torsion problem of shafts subject to known torques (Fig. 16.7). This problem is classically treated by the use of a stress function 2~ and indeed in this way has been solved by using a finite element formulation. 2~ Here, an alternative, more physical, approach is available. The first application of the above concepts to the analysis of axisymmetric solids was made by Wilson. 22 A simple example illustrating the effects of various harmonics is shown in Figs 16,8(a) and 16.8(b). i!!{,,{~!i!~,,~i~!~i~!,{i,~i~~!~~~~~ {iili i~ii{i iiiiii! ii ~~~iil!~~~'~~{ii{i{!i!i~'~~~~i!i!i ii!lili!ili!{i~'l~!iiii !ili !iii!'l~'~'~ili i i!i iili iili!i}ili i i il ili!i i!lilil!ii~iililili i i!i!i~i~~ii{!iii!i!ii~!li i i !iiiii!ili~~~ii~!~, {i!!{ii{il~'i'!~ilili!ili !ili!ili!ili~ii'~~i'~i i ii i lili i~!i!i i il~i !ii!i~ili i~ !ii'~l"~{iii!ii{!i i ~i{il!i!ii !i{i i{i!i i{i!i i{i ii!i{iii i!ii lili!l{iilil!i!i i ii i!ii !ii!i ii!i!ilil!ii{i ~ii ili!i!!i!i~ii!i~ii i i {iii~i~"i !{ii!i{ili!l!ili{ii il}ii'~l~'' ~i{iil'~~!~!~ i~~"i' i{ilili{li}{!i!ili{ili{i{ilili{ii ~~!i!i!ilii{iiii ilif!!i!i ilili!i!i iiil{{~,~,~! ili i {iii iili i i i~!i
16.6.1 Thin case- no shear deformation The extension of analysis of axisymmetric thin shells as described in Chapter 14 to the case of non-axisymmetric loads is simple and will again follow the standard pattern. It is, however, necessary to extend the definition of strains to include now all three displacements and force components (Fig. 16.9). Three membrane and three bending effects are now present and, extending Eq. (7.1) involving straight generators, we now define strains a s 23'24 *
~m
~S
U,s
G0
o,0/r + (~ cos ~b - ~ sin ~b)/r
%e
~t,o/r + ~),s - ~) cos ~ / r
Xo
--W,SS - f o eo/r 2 - fo ~ cos ~b/r + f),o sin ~b/r
Xso
2 (-CO,so/r + fo,o cos ~b/r 2 + f),s sin ~ / r - ~ sin ~cos ~ / r
2)
(16.33) * Various alternatives are available as a result of the multiplicity of shell theories. The one presented is quite commonly accepted.
Axisymmetric shells with non-symmetrical load
l
z
J ! ! I
i i i i
I
i i
I , . . . . , . - - - -- ! -- - - . ,
.
.
.
.
.
.
.
1~ ~
r
1 ~Actual IIIIIIIIIIIIIIII1~ IIIIIIIIIIIIIIIIIIt11~
j
.. _
Harmonics
n:o/
Approximation by 5
IIIIIIIIIIIIIIIIIIIIIIIIIlllllllln~',.. o
I
loading
i
..
= 4 1800"~
~ . ~ / n ~
3
360 ~
(a)
Fig. 16.8 (a) An axisymmetrir tower under non-symmetricload; four cubic elementsare used in the solution; the harmonicsof load expansionused in the analysisare shown.
511
512
Semi-analytical finite element processes
az(+Ve) .
.
.
.
.
.
.
.
.
.
(s nig~nnnrevl
.
.
.
.
.
.
.
.
)
/9 > ~/2
n=4 n=2
y
_
m
Combined stress (b) Fig. 16.8 Cont. (b) Distribution of O-z,the vertical stress on base arising from various harmonics and their combination (third harmonic identically zero), the first two harmonics give practically the complete answer.
The corresponding 'stress' matrix is
o'--
Ns No Nse Ms Me Mse
with the three membrane and bending stresses defined as in Fig. 16.9.
(16.34)
Axisymmetric shells with non-symmetrical load 513
Z
so
0s os
Y
11
No N~0
(a)
Ms
(b)
Fig. 16.9 Axisymmetricshell with non-symmetric load; (a) geometry and displacements, (b) stress resultants. Once again, symmetric and antisymmetric variation of loads and displacements can be assumed, as in the previous section. As the processes involved in executing this extension of the application are now obvious, no further description is needed here, but note again should be made of the more elaborate form of equations necessary when curved elements are involved [see Chapter 14, Eq. (14.23)]. The reader is referred to the original paper by Grafton and Strome 24 in which this problem is first treated and to the many later papers on the subject listed in Chapter 14.
16.6.2 Thick case- with shear deformation The displacement definition for a shell which includes the effects of transverse shearing deformation is specified using the forms given in Eqs (8.4) and (8.26). For a case of loading which is symmetric about 0 -- 0, the decomposition into global trigonometric components involves the three displacement components of the nth harmonic as Un
CO nO
{w" } - - Z N a I i Vn
0 sinn0 0
fi'a
~]ta
1
6"a
0 / ~a// ~ 1 l/ fo~na" / .~___~__LI-COS SO~3a '~)a Ol
cosnO
(16.35)
Ua,/l)a, ~)La
and stand for the displacements and rotation illustrated in Fig. 15.5, In this Va is a displacement of the middle surface node in the tangential (0) direction, and/3a is a rotation about the vector tangential to the mid-surface. Global strains are conveniently defined by the relationship ]9
Er ~z
U,r tO,z
~rz
U,z ~ W,r
7zO
V,z + w,o/r l),r -- P/r + U,O/r
~/Or
(16.36)
514 Semi-analytical finite element processes These strains are transformed to the local coordinates, and the component normal to r/ 07 = constant) is neglected. As in the axisymmetric case described in Chapter 15, the D matrix relating local stresses and strains takes a form identical to that defined by Eq.
(8.13).
A purely axisymmetric problem may again be described for the complete zero harmonic problem and again, as in the non-symmetric loading of solids, the strains split into two problems defining a torsionless and a torsional state. However, for inelastic problems a coupling again occurs whenever both torsionless and torsional loading are both applied as loading conditions on the same problem. Thus, it is often expedient to form the axisymmetric case including all three displacement components.
::iiliiiii!iliiiiiiiiiii!iil iii',i~,!iiiiiiii ili i',',ii',iiiiiiiiiiiiiiiiii~'i!ii~, ,~!'~i~iii iiiii ~,'~,~!~iiili~i',i~' i'il,~i~',i,'i~i,~i'~iiii'i'~,i!'i~,i'~iliiiiii' i~,i~i'~ii!iiiil~i,,i~,!'~,,~,:i':~iill ~iiii'i ,i',i',iii!iiii' iill ,iiiii~i',iiiii!iii' i~',i,i~,~i',iiiiiil ii~, ~,i~',i!,iii~,ilii'~~,'i'~,iiiiiiiiii!!iiiiiiiiiili!i!i!!iiiiiiii!!i!ii!~,!iiii i'~ii iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii!i iiiiiiiii!iiiiiiiiiiiiiiiiiiiiiil iiiiiiiiii!i!!i!!iiiiiii!~,iiiiiiiiiiiiiiiiiiiiii!ii i':iiiiiiiiiii!iiiiiiiiii!iiiiiiii'i':iiiiiiiii!iii i':iili!i!ii~i!iiiiiiiii,i':i',i'i:,i'i'i
A fairly general process combining some of the advantages of finite element analysis with the economy of expansion in terms of generally orthogonal functions has been illustrated in several applications. Certainly, these only touch on the possibilities offered, but it should be borne in mind that the economy is achieved only in certain geometrically constrained situations and those to which the number of terms requiring solution is limited. Similarly, other 'prismatic' situations can be dealt with in which only a segment of a body of revolution is developed (Fig. 16.10). Clearly, the expansion must now be taken in terms of the angle 17rO/o~, but otherwise the approach is identical to that described previously. 3 In the methods of this chapter it was assumed that material properties remain constant with one coordinate direction. This restriction can on occasion be lifted with the
Fig. 16.10 Other segmental, prismatic situations.
References 515 same general process maintained. An early example of this type was presented by Stricklin and DeAndrade. 25 Inclusion of inelastic behaviour has also been successfully treated. 26-29 All the problems we have described in this chapter could be derived in terms of semidiscretization in time. We would thus first semi-discretize, describing the problem in terms of an ordinary differential equation in z of the form d2a da K I ~ z 2 + K 2 ~ zz + K 3 a + f = 0 Second, the above equation system would be solved in the domain O < z < a by means of orthogonal functions that naturally enter the problem as solutions of ordinary differential equations with constant coefficients. This second solution step is most easily found by using a diagonalization process described in dynamic applications. 1 Clearly, the final result of such computations would turn out to be identical with the procedures here described, but on occasion the above formulation is more self-evident.
1. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann, Oxford, 6th edition, 2005. 2. P.M. Morse and H. Feshbach. Methods of Theoretical Physics. McGraw-Hill, New York, 1953. 3. O.C. Zienkiewicz and J.J.M. Too. The finite prism in analysis of thick simply supported bridge boxes. Proc. Inst. Civ. Eng., 53:147-172, 1972. 4. Y.K. Cheung. The finite strip method in the analysis of elastic plates with two opposite simply supported ends. Proc. Inst. Civ. Eng., 40:1-7, 1968. 5. Y.K. Cheung. Finite strip method of analysis of elastic slabs. Proc. Am. Soc. Civ. Eng., 94(EM6): 1365-1378, 1968. 6. Y.K. Cheung. Folded plates by the finite strip method. Proc. Am. Soc. Civ. Eng., 95(ST2):963979, 1969. 7. Y.K. Cheung. The analysis of cylindrical orthotropic curved bridge decks. Publ. Int. Ass. Struct. Eng., 29-11:41-52, 1969. 8. Y.K. Cheung, M.S. Cheung and A. Ghali. Analysis of slab and girder bridges by the finite strip method. Building Sci., 5:95-104, 1970. 9. Y.C. Loo and A.R. Cusens. Development of the finite strip method in the analysis of cellular bridge decks. In K. Rockey et al., editors, Conf. on Developments in Bridge Design and Construction. Crosby Lockwood, London, 1971. 10. Y.K. Cheung and M.S. Cheung. Static and dynamic behaviour of rectangular plates using higher order finite strips. Building Sci., 7:151-158, 1972. 11. G.S. Tadros and A. Ghali. Convergence of semi-analytical solution of plates. Proc. Am. Soc. Civ. Eng., 99(EM5):1023-1035, 1973. 12. A.R. Cusens and Y.C. Loo. Application of the finite strip method in the analysis of concrete box bridges. Proc. Inst. Civ. Eng., 57-11:251-273, 1974. 13. T.G. Brown and A. Ghali. Semi-analytic solution of skew plates in bending. Proc. Inst. Civ. Eng., 57-11:165-175, 1974. 14. A.S. Mawenya and J.D. Davies. Finite strip analysis of plate bending including transverse shear effects. Building Sci., 9:175-180, 1974. 15. P.R. Benson and E. Hinton. A thick finite strip solution for static, free vibration and stability problems. International Journal for Numerical Methods in Engineering, 10:665-678, 1976.
516 Semi-analytical finite element processes 16. E. Hinton and O.C. Zienkiewicz. A note on a simple thick finite strip. International Journal for Numerical Methods in Engineering, 11:905-909, 1977. 17. H.C. Chan and O. Foo. Buckling of multilayer plates by the finite strip method. Int. J. Mech. Sci., 19:447-456, 1977. 18. Y.K. Cheung. Finite Strip Method in Structural Analysis. Pergamon Press, Oxford, 1976. 19. I.S. Sokolnikoff. The Mathematical Theory of Elasticity. McGraw-Hill, New York, 2nd edition, 1956. 20. S.P. Timoshenko and J.N. Goodier. Theory of Elasticity. McGraw-Hill, New York, 3rd edition, 1969. 21. O.C. Zienkiewicz, P.L. Arlett and A.K. Bahrani. Solution of three-dimensional field problems by the finite element method. The Engineer, October 1967. 22. E.L. Wilson. Structural analysis of axi-symmetric solids. Journal of AIAA, 3:2269-2274, 1965. 23. V.V. Novozhilov. Theory of Thin Shells. Noordhoff, Dordrecht, 1959. (English translation). 24. P.E. Grafton and D.R. Strome. Analysis of axi-symmetric shells by the direct stiffness method. Journal of AIAA, 1:2342-2347, 1963. 25. J.A. Stricklin and J.C. de Andrade. Linear and non-linear analysis of shells of revolution with asymmetrical stiffness properties. In Proc. 2nd Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-68-150, Wright Patterson Air Force Base, Ohio, October 1968. 26. L.A. Winnicki and O.C. Zienkiewicz. Plastic or visco-plastic behaviour of axisymmetric bodies subject to non-symmetric loading; semi-analytical finite element solution. International Journal for Numerical Methods in Engineering, 14:1399-1412, 1979. 27. W. Wunderlich, H. Cramer and H. Obrecht. Application of ring elements in the nonlinear analysis of shells of revolution under nonaxisymmetric loading. Computer Methods in Applied Mechanics and Engineering, 51:259-275, 1985. 28. H. Obrecht, E Schnabel and W. Wunderlich. Elastic-plastic creep buckling of circular cylindrical shells under axial compression. Z. Angew. Math. Mech., 67:T118-T120, 1987. 29. W. Wunderlich and C. Seiler. Nonlinear treatment of liquid-filled storage tanks under earthquake excitation by a quasistatic approach. In B.H.V. Topping, editor, Advances in Computational Structural Mechanics, Proceedings 4th International Conference on Computational Structures, pages 283-291, August 1998.
i ii!iiiiiil
Non-linear structural problemslarge displacement and instability iiiii iii !iiiiiii i i iiiii iii iiii i iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii iii!i iiiil iii!iiiiii iiiii i i In the previous chapter the question of finite deformations and non-linear material behaviour was discussed and methods were developed to allow the standard linear forms to be used in an iterative way to obtain solutions. In the present chapter we consider the more specialized problem of large displacements but with strains restricted to be small. Generally, we shall assume that 'small strain' stress-strain relations are adequate but for accurate determination of the displacements geometric non-linearity needs to be considered. Here, for instance, stresses arising from membrane action, usually neglected in plate flexure, may cause a considerable decrease of displacements as compared with the linear solution discussed in Chapters 11 and 12, even though displacements remain quite small. Conversely, it may be found that a load is reached where indeed a state may be attained where load-carrying capacity decreases with continuing deformation. This classic problem is that of structural stability and obviously has many practical implications. The applications of such an analysis are clearly of considerable importance in aerospace and automotive engineering applications, design of telescopes, wind loading on cooling towers, box girder bridges with thin diaphragms and other relatively 'slender' structures. In this chapter we consider the above class of problems applied to beam, plate, and shell systems by examining the basic non-linear equilibrium equations. Such considerations lead also to the formulation of classical initial stability problems. These concepts are illustrated in detail by formulating the large deflection and initial stability problems for beams and flat plates. A Lagrangian approach is adopted throughout in which displacements are referred to the original (reference) configuration.
17.2.1 Geometrically exact formulation In Chapter 10 we described the behaviour for the bending of a beam for the small strain theory. Here we present a form for cases in which large displacements with finite rotations occur. We shall, however, assume that the strains which result are small.
518 Non-linearstructural problems- large displacement and instability
l
l
Z
Z
Y
//1
/
iit I
,/
/
//1
~
=y
iiI I
A
!,
I
I i
I/---J"
X
X
Fig. 17.1 Finite motion of three-dimensional beams.
A two-dimensional theory of beams (rods) was developed by Reissner 1 and was extended to a three-dimensional dynamic form by Simo. 2 In these developments the normal to the cross-section is followed, as contrasted to following the tangent to the beam axis, by an orthogonal frame. Here we consider an initially straight beam for which the orthogonal triad of the beam cross-section is denoted by the vectors ai (Fig. 17.1). The motion for the beam can then be written as (Di ~ xi = xO -~ A i l Z l (17.1) where the orthogonal matrix is related to the ai vectors as A = [al
a2
(17.2)
a3]
If we assume that the reference coordinate X1 (X) is the beam axis and X2, X3 (Y, Z) are the axes of the cross-section the above motion may be written in matrix form as
{x} {i} {x} {u} jill X2
X3
=
--
0 0
+
1)
+
W
21
A22
A23/
Y
31
A32
A33J
Z
(17.3)
where u(X), v(X), and w(X) are displacements of the beam reference axis and where A(X) is the rotation of the beam cross-section which does not necessarily remain normal to the beam axis and thus admits the possibility of transverse shearing deformations. The derivation of the deformation gradient for Eq. (17.3) requires computation of the derivatives of the displacements and the rotation matrix. The derivative of the rotation matrix is given by 2'3 A,x = O,xA (17.4) ^
where 0,x denotes a skew symmetric matrix for the derivatives of a rotation vector 0 and is expressed by
i O _Oz x Or, ] 0x = Oz,x O' -Ox,x L-O ,,x
Ox, x
(17.5)
Large displacement theory of beams 519
Z,z
.9~ ,
,
,
,,..-
~
".
X,x
Fig. 17.2 Deformedbeam configuration.
Here we consider in detail the two-dimensional case where the motion is restricted to the X - Z plane. The orthogonal matrix may then be represented as (Or =/3) A =
[Co/3 [-sinfl
0 sin/3/3] 1 0 0 cos
(17.6)
Inserting this in Eq. (17.3) and expanding, the deformed position then is described compactly by x = X + u(X) + Z sin/3(X) y = Y (17.7) z = w ( X ) + Z cos 3 ( x )
This results in the deformed configuration for a beam shown in Fig. 17.2. It is a twodimensional specialization of the theory presented by Simo and co-workers 2'4'5 and is called geometrically exact since no small-angle approximations are involved. The deformation gradient for this displacement is given by the relation
Fil
[[l+u,x+Z~,xCOS/3] --
0
0
1
[W,x - Z/3,x sin/3]
0
si 0/31
(17.8)
cos/3J
Using Eq.(10.15) and computing the Green-Lagrange strain tensor, two non-zero components are obtained which, ignoring a quadratic term in Z, are expressed by
E x x = U , x + ~ '(U2x , + wix) , + z A ~ ,x = E~ + z / ( b 2Exz = (1 + U,x) sin/3 + W x cos/3 = F
(17.9)
where E ~ and I" are strains which are constant on the cross-section and K b measures change in rotation (curvature) of the cross-sections and (17.10)
A -- (1 -t- U,x) cos/3 - W,x sin/3
A variational equation for the beam can be written now by introducing second Piola-Kirchhoff stresses as described in Chapter 5 to obtain
~I-I = L (~ExxSxx + 2~ExzSxz) dV
-
t~I'Iext
(17.11)
520
Non-linear structural problems- large displacement and instability where ~l-'Iex t denotes the terms from end forces and loading along the length. If we separate the volume integral into one along the length times an integral over the beam cross-sectional area A and define force resultants as
TP=LSxxdA
, SP-/]Sxzda
and
Mb-LSxxZda
(17.12)
the variational equation may be written compactly as cSFl = / u (~E~ p + ~FS p + c~KbMb) dX
- ~I'Iext
(17.13)
where virtual strains for the beam are given by c~E~ = (1 + U x)~SUx + W,x~SW,x d/F = sin ~SU,x + cos/3~SW,x + Ac5/3
(17.14)
~K b -- A~/~,x q- F~/~ + cos t~SU,x + sin 3~SW,x A finite element approximation for the displacements may be introduced in a manner identical to that used in Sec. 17.4 for axisymmetric shells. Accordingly, we can write
I nw/
9
a/ -- N~(X) / UgOa
(17.15)
L
where the shape functions for each variable are the same. Using this approximation the virtual work is computed as
Spb dX - ~l"lext (~II- [~Ua ~lT~ ~L] L BaT M
(17.16)
where Ba :
[(1 + U,x)No,x / sin flNa,x [/3,x cos/3Na,x
to,xNa,x
cos flNa,x -/3 x sin flNa,x
0 1 ANa J (ANa,x - Ffl, xNa)
(17.17)
Just as for the axisymmetric shell described in Sec. 17.4 this interpolation will lead to 'shear locking' and it is necessary to compute the integrals for stresses by using a 'reduced quadrature'. For a two-node beam element this implies use of one quadrature point for each element. Alternatively, a mixed formulation where F and S p are assumed constant in each element can be introduced as was done in Sec. 12.6 for the bending analysis of plates using the T6S3B3 element. The non-linear equilibrium equation for a quasi-static problem that is solved at each load level (or time) is given by TnP+l p
lien+1 --
f~+l - /
JL
Ba~
sP+l
b Mn+l
dX = 0
(17.18)
Large displacement theory of beams 521 For a Newton-type solution the tangent stiffness matrix is deduced by a linearization of Eq. (17.18). To give a specific relation for the derivation we assume, for simplicity, the strains are small and the constitution may be expressed by a linear elastic relation between the Green-Lagrange strains and the second Piola-Kirchhoff stresses. Accordingly, we take (17.19)
and Sxz = 2 G E x z
Sxx = E E x x
where E is a Young's modulus and G a shear modulus. Integrating Eq. (17.12) the elastic behaviour of the beam resultants becomes S p = t~GA F
T p -- EA E ~
and M b -- E I K b
in which A is the cross-sectional area, I is the moment of inertia about the centroid, and t~ is a shear correction factor to account for the fact that Sxz is not constant on the cross-section. Using these relations the linearization of Eq. (17.18) gives the tangent stiffness (KT)ab -- f
JL
BarDTBb dX 4- (KG)ob
(17.20)
where for the simple elastic relation Eq. (17.20)
Dx -
lEA
xoGA E1
1
(17.21)
and Kc is the geometric stiffness resulting from linearization of the non-linear expression for B. After some algebra the reader can verify that the geometric stiffness is given by
( [ ~P
(K~)ab=~ N~,x MboCOS/3 + Na,x
[!o oil 0 0
o ~p
-- M b sin/3
G2
Nb + Na
--Mbl -'
~sin~,~X+~a
[i 0~ ~ ~
[o o o] ) G1
0
0
G2
--MbF
Nb, x
0
G3
dX
(17.22) where G 1
-
-
S p cos/~ -
M / b3 , x
sin/3,
G2 _ _ Sp s i n fl --
and G3 - - S P F -
Mb/3,x A
Mb/3,XCOS/~,
522 Non-linearstructural problems-large displacement and instability
17.2.2 Large displacement formulation with small rotations In many applications the full non-linear displacement field with finite rotations is not needed; however, the behaviour is such that limitations of the small displacement theory are not appropriate. In such cases we can assume that rotations are small so that the trigonometric functions may be approximated as sin/3 ~ / 3
and
cos/3 ~ 1
In this case the displacement approximations become x = x + u(X)+ z~(x)
y = Y
(17.23)
z = w(X)
+ Z
which yield now the non-zero Green-Lagrange strain expressions Exx 2Exz
= U,x + ~~(u ,89+
w2x) + z/~,x = E ~ + Z K b
= W,x +/3 = F
(17.24)
where terms in Z 2 as well as products of/3 with derivatives of displacements are ignored. With this approximation and again using Eq. (17.15) for the finite element representation of the displacements in each element we obtain the set of non-linear equilibrium equations given by Eq. (17.18) in which now
Ba --
[(l "-I"U,x)Na,x w,xNa,x 0 ] 0 Na,x Na 0 0 No,X
(17.25)
This expression results in a much simpler geometric stiffness term in the tangent matrix given by Eq. (17.20) and may be written simply as
0 i]
(KG)ab -- fjL Na,x
Tp
Nb,x
dX
(17.26)
0
It is also possible to reduce the theory further by assuming shear deformations to be negligible so that from F = 0 we have fl -
(17.27)
- W,x
Taking the approximations now in the form
u=NU~a 1/3 -- N:/13 a -q--NaflL
(17.28)
in which ~a = - OJa,X at nodes. The equilibrium equation is now given by 'I'n+l = fn+l --
B~
b
M,+I
(17.29)
Elastic stability- energy interpretation
523
where the strain-displacement matrix is expressed as Ba = [ ( 1 +
U,x)NaUx w,xNaW,X w'xN~X]N fl I 0 -NWa,XX - a,XXJ
(17.30)
The tangent matrix is given by Eq. (17.20) where the elastic tangent moduli involve only the terms from T p and M b as
I)T = [EAo
0 ]
(17.31)
EI
and the geometric tangent is given by
INU,xTPN~,x 0 0 ] (KG)ab = Jl Na,x 0 NaW,,xTPN~,,xNWa,xTPN~x dX 0 N~xTPN~xa, Nt3a,XTPN~xJ
(17.32)
Example17.1 Aclamped-hingedarch
To illustrate the performance and limitations of the above formulations we consider the behaviour of a circular arch with one boundary clamped, the other boundary hinged and loaded by a single point load, as shown in Fig. 17.3(a). Here it is necessary to introduce a transformation between the axes used to define each beam element and the global axes used to define the arch. This follows standard procedures as used many times previously. The cross-section of the beam is a unit square with other properties as shown in the figure. An analytical solution to this problem has been obtained by da Deppo and Schmidt 6 and an early finite element solution by Wood and Zienkiewicz. 7 Here a solution is obtained using 40 two-node elements of the types presented in this section. The problem produces a complex load displacement history with 'softening' behaviour that is traced using the arc-length method described in Sec. 3.2.6 [Fig. 17.3(b)]. It is observed from Fig. 17.3(b) that the assumption of small rotation produces an accurate trace of the behaviour only during the early parts of loading and also produces a limit state which is far from reality. This emphasizes clearly the type of discrepancies that can occur by misusing a formulation in which assumptions are involved. Deformed configurations during the deformation history are shown for the load parameter fl = EI/PR 2 in Fig. 17.4. In Fig. 17.4(a) we show the deformed configuration for five loading levels - three before the limit load is reached and two after passing the limit load. It will be observed that continued loading would not lead to correct solutions unless a contact state is used between the support and the arch member. This aspect was considered by Simo et al. 8 and loading was applied much further into the deformation process. In Fig. 17.4(b) we show a comparison of the deformed shapes for/3 = 3.0 where the small-angle assumption is still valid. i!iiiiii!ii!~iiii~i~ii!i! ~~'~'~iiiii~ ~iiiiiiiiiillii~'~ii ~,~':~iii~ii!iiiii~iii~ ~il%!iiii!!!ii~iiii~iiiiiiiii~i'~'i!iiii!! ~ii~!~ii~ii!iiiiii!i~i!~i~i~iii~i~J!!~iiiiiii~iiii!~!~ii~i~iiiiii!iii~ii~iiiiiiiiiiiii~ii~i~iii~iiii~iii!i~iiiiiii!i~iii~iii~i~i~iii~ ~iiiiiiiiii~iiiiiiiiiiiii~i~iiii~ii~iii~iii!iiiiii~i~ii~i~ii~iiiii~iiii~i~ii~iiiiiiiiiiii!~i!i!~i~i~iii!iiiii!iiii~iii~iii!!ii~i
i iiiL|iii iiiiiii
i
i: .................ii ii i:, ........
...........iii
.......
i
..........
i i i: .......iiiiiHiiiiii,iiiiiiiii,iiiii,iiiiiiiiiiilJiiii,iii,
The energy expression given in Eq. (10.37) and the equilibrium behaviour deduced from the first variation given by Eq. (10.42) may also be used to assess the stability of
524
Non-linear structural problems-large displacement and instability
~
,.,i ~
~P
1V
t
',1 ,
~
\,l
.......... ..i~
...... J'r.~ "'~" . . . . .
",II
~
./. "t\ .
.,.,, .
,~1__~"'~ ~/ll/m'
"/ R= 100, ~ = 215", t= 1.0, El= 10 s
"" . . . . ""
.......
Deflected position at maximum load (true scale)
10
'/R
9 8 n- 7 O. --~ 6 LLI " 5
I
2 1
00 Fig. 17.3
....
Finite angle Small angle
i
0.5 1 Displacements: U/R, V/R
1.5
Clamped-hinged arch: (a) problem definition; (b)load deflection.
equilibrium. 9 For an equilibrium state we always have (~l"I -- _ ( ~ T ~
._ 0
(17.33)
that is, the total potential energy is stationary [which, ignoring inertia effects, is equivalent to Eq. (10.65)]. The second variation of rI is (17.34) The stability criterion is given by a positive value of this second variation and, conversely, instability by a negative value (as in the first case energy has to be added to the structure whereas in the second it contains surplus energy). In other words, if KT is positive definite, stability exists. This criterion is well known 9 and of considerable
Elastic s t a b i l i t y - energy interpretation
/
J
fl = 4.31 L21
J
~ "..
fl = 8.20 *"
..- .
.
.
.
.
.,. --- ""
~, - ' " ",fl=3.82 k
%
....
9
Finite Small angle
t
(a)
|
/
(b)
Fig. 17.4 Clamped-hingedarch: deformedshapes.(a) Finite-anglesolution;(b) finite-angleform compared with small-angleform. use when investigating stability during large deformation. 1~ An alternative test is to investigate the sign of the determinant of KT, a positive sign denoting stability. ~2 A limit on stability exists when the second variation is zero. We note from Eq. (10.66) that the stability test can then be written as (assuming KL is zero)
~fiTKM~fi + ~TKc6fi = 0
(17.35)
This may be written in the Rayleigh quotient form 13 = -A
(17.36)
where we have A
l
< 1, stable - 1, stability limit
(17.37)
> 1, unstable
The limit of stability is sometimes called neutral equilibrium since the configuration may be changed by a small amount without affecting the value of the second variation (i.e. equilibrium balance). Several options exist for implementing the above test and the simplest is to let A = 1 + AA and write the problem in the form of a generalized linear eigenproblem given by KT~U : AAKc6u (17.38) Here we seek the solution where AA is zero to define a stability limit. This form uses the usual tangent matrix directly and requires only a separate implementation for the geometric term and availability of a general eigensolution routine. To maintain numerical conditioning in the eigenproblem near a buckling or limit state where KT is singular a shift may be used as described for the vibration problem in Chapter 16 of reference 14.
Example 17.2
Euler buckling - propped cantilever
As an example of the stability test we consider the buckling of a straight beam with one end fixed and the other on a roller support. We can also use this example to show the usefulness of the small-angle beam theory.
525
526 Non-linearstructural problems- large displacement and instability Table 17.1 Linear buckling load estimates
Number of elements 20
100
500
20.36 61.14 124.79
20.19 59.67 118.85
20.18 59.61 118.62
An axial compressive load is applied to the roller end and the Euler buckling load computed. This is a problem in which the displacement prior to buckling is purely axial. The buckling load may be estimated relative to the small deformation theory by using the solution from the first tangent matrix computed. Alternatively, the buckling load can be computed by increasing the load until the tangent matrix becomes singular. In the case of a structure where the distribution of the internal forces does not change with load level and material is linear elastic there is no difference in the results obtained. Table 17.1 shows the results obtained for the propped cantilever using different numbers of elements. Here it is observed that accurate results for higher modes require use of more elements; however, both the finite rotation and small rotation formulations given above yield identical answers since no rotation is present prior to buckling. The properties used in the analysis are E = 12 x 10 6, A = 1, I = 1/12, and length L = 100. The classical Euler buckling load is given by Pcr = c e "
E1
L2
(17.39)
with the lowest buckling load given as a = 20.18. ~5
i!
!
ii!iiii!!ii!!ii!ii l i!,iiii!i'!!!!
17.4.1 Definitions The small rotation form for beams described in Sec. 17.2.2 may be used to consider problems associated with deformation of plates subject to 'in-plane' and 'lateral' forces, when displacements are not infinitesimal but also not excessively large (Fig. 17.5). In this situation the 'change-in-geometry' effect is less important than the relative magnitudes of the linear and non-linear strain-displacement terms, and in fact for 'stiffening' problems the non-linear displacements are always less than the corresponding linear ones (see Fig. 17.6). It is well known that in such situations the lateral displacements will be responsible for development of 'membrane'-type strains and now the two problems of 'in-plane' and 'lateral' deformation can no longer be dealt with separately but are coupled. Generally, for plates the rotation angles remain small unless in-plane strains also become large. To develop the equations for small rotations in which plate bending is modelled using the formulations discussed in Chapter 12 we generalize the displacement field given in Eq. (4.9) to include the effects of in-plane displacements.
Large displacement theory of thick plates 527 Ty
J
Tx = x(u)
y(v)
;x J
T~
z(w)
(a)
i~
dx--'-~
~X
(b) Fig. 17.5 (a)'In-plane' and bending resultants for a flat plate; (b) increase of middle surface length owing to lateral displacement. Accordingly, we write
u(X, Y)
Ul U --
U2
m
v(X, Y)
+Z
w(X, Y)
U3
4,x(X, Y) r r)
(17.40)
0
where q~are small rotations defined according to Fig. 11.3 and X, Y, Z denote positions in the reference configuration of the plate. Using these to compute the Green-Lagrange strains given by Eq. (5.15) we can write the non-zero terms as
l(w,x)2 V y + 89 2
Exx
U,x +
Eyy ~.Ex~
--
u , v Jr U,x .qt_ y.),xW,y
Cx,x Cy,y + Z
~ x , Y + ~Y,X
~Ex~
Cx + W,x
0
~Er~
~r + w,r
0
(17.41)
In these expressions we have used classical results ~6that ignore all square terms involving ~b and derivatives of u and v, as well as terms which contain quadratic powers of Z. Generally, the position of the in-plane reference coordinates X and Y change very little during deformations and we can replace them with the current coordinates x and y just as is implicitly done for the small strain case considered in Chapter 11. Thus,
528
Non-linearstructuralproblems-largedisplacementand instability //////A////////,~
2
300
~
/
/
P
f,
r ' ://///////k"//////Z I I2a---~
,r
v
'-4--!--i-.~: . !'!"'~-:~-'~ /
9
/
200
/
/
/
/
U/'-- Large deflection
study
100
f
fl ~eflection
f
1
theory
welt
Fig. 17.6 Central deflection Wc of a clamped square plate under uniform load p;12 u = v = 0 at edge.
we can represent the Green-Lagrange strains in terms of the middle surface strains and changes in curvature as 1 (W,x)2 E-
l),y --~- ~I(W ,y)2
-~- Z
: E p + ZK b
~y,y
U ,y -Ji- l),x ..3t- W,x W, y
(17.42)
(fix, y "]- (fiy,x
where E p denotes the in-plane membrane strains and K b the change in curvatures owing to bending. In addition we have the transverse shearing strains given by ~s={
(17.43)
} r "Jr W,y ~)x'Jl'Ul)'x
The variations of the strains are given by oVEP =
oeK b --
{,ux }Io x ~V,y (~U,y -t- (~V,x
,~4'x,x } (~r
+
k W, y
O]
W, y Wx
and 5 r s -
(~ll)' y
{5r5~y +.-.l-5W, x} 5W,y
(17.44)
(17.45)
(~~ x , y -I- (~q~y ,x
Using these expressions the variation of the plate equations may be expressed as ~1-I -- Ja (o"EP)rS d~ + f (OTs) r S s d~ + f~ ( ~ b ) T S Z d~2 - ~I'Iex t
(17.46)
Large displacement theory of thick plates 529 Defining the integrals through the thickness in terms of the 'in-plane' membrane forces
Ty Txy
Tp -
--
dZ
(17.47)
= i" t/2 S s dZ - I t / 2 { SXZ } dZ a -t /2 a -t /2 SY Z
(17.48)
a -t/2
S dZ -
Srr
,1-t/2
Sxy
transverse shears
} T s = { Txz Tyz and bending forces
Myy
M b -
--
a -t/2
Mxy
SZdZ -
Syy SX Y
,I -t/2
ZdZ
(17.49)
-- t~l-Iex t
(17.50)
we obtain the virtual work expression for the plate, given as ~l-I - JA[(O~EP)rTP + ~ ( r s ) r T s + ~(Kb)rMb] dA This may now be used to construct a finite element solution.
17.4.2 Finite element evaluation of strain-displacement matrices For further evaluation it is necessary to establish expressions for the finite element B and KT matrices. Introducing the finite element approximations, we have
{u} 1)
=
Na
w
~a N~a
(17.51)
eva
and (17.52)
I ~ ~ i -- N: I (~)x)a (~y)a}
The expressions for the strain-displacement matrices are deduced from Eqs (17.44) and (17.45) as
ot"EP :
BPt~fio -
Iax Ol,Oal IO 0] Na ,y [ (~l)a
0
Na,y
-+-
Na,x
W ,y G a [.lll),y W,x
t~( ~x ) a (~(~y)a
(17.53a)
--- Ba~(l a + BLt~'~,Va
5r
s -
BSt~wa
[NaW,x -- LNaW,y
-N~a 0
0 ] --N~a
(~(~x)a t~(~y)a
(17.53b)
530 Non-linear structural problems- large displacement and instability and o"Kb - Bba.a = where Ga =
0'
g~y
(~(~x)a
N~y
g~x
(~(~y)a
[Na x N~ L a,y
0: l 0
(17.53c)
(17.53d)
with nodal parameters defined by
8a-T__ [bla l)a Wa~ (r (~)y)a ]~ = [~T Wa~T] (r~ H~Ta "-- [fia Ua] and -Tw a = [~l)a (~x)a ~ We here immediately recognize an in-plane term which is identical to the small strain (linear) plane stress (membrane) form and a term which is identical to the small strain bending and transverse shear form. The added non-linear in-plane term results from the quadratic displacement terms in the membrane strains. Using the above strain-displacement matrices we can now write Eq. (17.50) as
al-I = afiTajfA(BPa)rTP dA + af.vTajfA(BSa)TTS dA + af,vTa fA(Bba)rMb dA -- 6I'Iext = O Grouping the force terms as O=
{Tp}
(17.54)
Ts Mb
(17.55)
and the strain matrices as B a ~"
BSa|
(17.56)
the virtual work expression may be written compactly as (17.57)
6l-I = a~ar f a I~T~ dA - 6FlCxt = 0 The non-linear problem to be solved is thus expressed as
~I/a ~" fa -- fA BT ~1"dA = 0
(17.58)
This may be solved by using a Newton process for which a tangent matrix is required.
17.4.3 Evaluation of tangent matrix A tangent matrix for the non-linear plate formulation may be computed by a linearization of Eq. (17.57). Formally, this may be written as
d(~n) = 6aa~ fA [d(Bar)er + Ba~d(er)] dA -
d(~I'Iext)
(17.59)
Large displacement theory of thick plates 531 We shall assume for simplicity that loading is conservative so that d (51-Iext) -- 0 and hence the only terms to be linearized are the strain-displacement matrix and the stressstrain relation. If we assume linear elastic behaviour, the relation between the plate forces and strains may be written as Ts Mb
o
__
Ds 0 Db
1-,s Kb
where for an isotropic homogeneous plate D p --
Et 1 - u2
1
0
o ]
,
DS
_
[1 o]
~Et
{
d(T s) d(M b)
2(i~u)
} [i 0 =
t2
_
d(EPs)
Ds 0
[~P
0
00]
0
Db
Ds
, and
Db
=]-~D p (17.61) Again, ~ is a shear correction factor which, for homogeneous plates, is usually taken as 5/6. Thus, the linearization of the constitution becomes d(O-)-
(l--u)/2
(17.60)
BI~ B~] ( B; /
B~]
)
(17.62)
d(ab) d(~Vb)
Using this result the material part of the tangent matrix is expressed as (KM)ab =
fA [(BaP)r 0 L(BaL)T (BSa)r
= fA BTI)T]~b dA =
0 (Bba)T
Ds 0 0 D b [ 00
BblB~/dA
(17.63)
"(K~)ab (Kr~)ab" L r (Kb)ab. .(KM)ab
where DT is the coefficient matrix from Eq. (17.60), and the individual parts of the tangent matrix are (KP)ab = fA(BaP)rDPBb da (KLM)ab -- fa (Bp)TDPB~ da
(17.64)
(K~)a~ = fA [~Bs~ D sBbs + (Bb)TDbBb] da We immediately recognize that the material part of the tangent matrix consists of the same result as that of the small displacement analysis except for the added term K~ which establishes coupling between membrane and bending behaviour.
532
Non-linearstructural problems-large displacement and instability The remainder of the computation for the tangent involves the linearization of the non-linear part of the strain-displacement matrix, B~. As in the continuum problem discussed in Chapter 5 it is easiest to rewrite this term as
d(BL)TTb = Gar
d(,x)
0
d(w,y)
O
d(ll),y
d(w,x
Zyp
TxPy (17.65)
It: rx"y]fd( ,x)
- G T LTxPy TyPJ L d ( w , y ) }
This may now be expressed in terms of finite element interpolations to obtain the geometric part of the tangent as -
fA
[TxP T~y]GbdA L J
(17.66)
which is inserted into the total geometric tangent as (17.67) This geometric matrix is also referred to in the literature as the initial stress matrix for plate bending. iiilili',i::iiil i!i~:i~:~'~:~il:i: ,~i,: ,i: ,;iiii~' : ,i: ~,i:i:'!i~i""'i~!ii~, iii'i ,iilili~',iii~,~~i',',~i'i'::iiill ,iii',i',iiiii',iii!iiiiii',i'~i'iil~iili'iii,i~,i'~,ii!!!U' iiii~i'ii,'i,!i~iiii!iiii' ~ ~i'i~i',iii',~i:,iiiiiiiiiii?:' ~ i"i~'i,~i~!',"i~'~iiii~iii i~i"i"i"i~' i~' !i"i"i'~i:'"~i"i',i'!i!ii,i',~i',iir :i~i~ii:i~i~ii~~i~i~i~i~i~!~::~':i:!irii!iiii','i::i',i':i~,i',iiiii:, i!ili'i~,i:i!'ii,iiiiil iiili'i,~i'iiii,: ii'i',i::,i;ii~iiiiiiiiiiiii'ii,', :'::iiiiiiiiiii~,i::i::::::::iiii i:~:'i~iii:i''i,;'i~iiiii , '~'~'i~'i,'i~ii'iiiii ,iii~i~~iiii i !~,iii~,i~:',:i~'i:i',ii'~~i,~ii~:'~i:,ii:,',:i!::i:~,:'~i, ~,',!i'!,i~,iif'!i,'i~i~,ii',;iil !, i'i,~ii~:i'iii:,, ii:ii:~iiiiiiii i ii'i,r il iiiii:~iii~i' !',i~'!iiiiii~'i' iiiii~i!i '~'iil,ii:~ii~,i~~i;iiiii~iiii'iilil,ii!',i!iiiii ','i,il',ilii~i:ii~i~i,i~i:,ili!iii! i~,i!i!ii!i;i;ii' ~,il :ii;ii!iii:
The above theory may be specialized to the thin plate formulation by neglecting the effects of transverse shearing strains as discussed in Chapter 11. Thus setting Exz -Erz = 0 in Eq. (17.41), this yields the result
~x -- - W,x
and
~br -- - w,r
(17.68)
The displacements of the plate middle surface may then be approximated as U ---
U2 U3
--"
u ( X , Y) w(X,Y)
- Z
w,r(X, Y) 0
(17.69)
Once again we can note that in-plane positions X and Y do not change significantly, thus permitting substitution of x and y in the strain expressions to obtain Green-Lagrange strains as
E-
U,x+~1 (tO,x)2 1 )2 V y + g(Wy U, y -JI- l),x "Jl- l13,x W, y
/ { w,x}x -- Z
lU,yy
= E p + ZK b
(17.70)
2w xy
where we have once again neglected square terms involving derivatives of the in-plane displacements and terms in Z 2. We note now that introduction of Eq. (17.68) modifies the expression for change in curvature to the same form as that used for thin plates in Chapter 11.
Large displacement theory of thin plates
17.5.1 Evaluation of strain-displacement matrices For further formulation it is again necessary to establish expressions for the !] and KT matrices. The finite element approximations to the displacements now involve only u, v, and w. Here we assume these to be expressed in the form
{U}-Na{~'la } 1)
l) a
(17.71)
N a tla
and 113
-- N aw 11)a
--]- N a~ -t~)a
(17.72)
where now the rotation parameters are defined as
(~f=
[(~x)a
(r
=-
[(ffO,x)a
(113,y)a]
(17.73)
The expressions for BP and B L are identical to those given previously except for the definition of G. Owing to the form of the interpolation for w, we now obtain G a -.
N a,x w Nw
a,y
N~ax Nr - a,y
l
N4,y |
(17.74)
a,y..I
The variation in curvature for the thin plate is given by
or'Kb __ _
I Na,x w x
g a ,Cx x X
N a,xx ~y |1
~Ul3a
w I Na'yy
Na,yy c~x
N a,yy ~y ||
((~)x)a
w L2No,xy
2Na,xy 4~x
2 N a,xY_l Cy |
((~@y)a
-- Bba(~'Wa
(17.75)
(17.76)
Grouping the force terms, now without the shears T s, as
or -and the strain matrices as
Ba --
Mb
["a "a l BbaJ
(17.77)
(17.78)
the virtual work expression may be written in matrix form as ~FI -- ~ ar f a Bs fir dA - ~Flext = 0 and once again a non-linear problem in the form of Eq. (17.58) is obtained.
(17.79)
533
534 Non-linearstructural problems-large displacement and instability
17.5.2 Evaluation of tangent matrix A tangent matrix for the non-linear plate formulation may be computed by a linearization of Eq. (17.57). If we again assume linear elastic behaviour, the relation between the plate forces and strains may be written as
=
)
where the elastic constants are given in Eq. (17.61). Thus, the linearization of the constitution becomes d(~r)--
~ re[B~, o]{a,
Db
B~
d(~)
Using this result the material part of the tangent matrix is expressed as
(KM)ab=Lr+,e'"="
[~
B~J dA
__ I (Kp)ab (KL)ab] L(KL)aT (Kb)abj
(17.82)
where K~ and K~ are given as in Eq. (17.64), and K~ simplifies to
(KD)ob -- L (Bba)TobBb d a
(17.83)
and now B~a is given by Eq. (17.76). Using Eq. (17.74) the geometric matrix has identical form to Eqs (17.66) and (17.67).
All the ingredients necessary for computing the 'large deflection' plate problem are now available. Here we may use results from either the thick or thin plate formulations described above. Below we describe the process for the thin plate formulation. As a first step displacements ~0 are found according to the small displacement uncoupled solution. This is used to determine the actual strains by considering the nonlinear relations for EP and the linear curvature relations for K b defined in Eq. (17.70). Corresponding stresses can be found by the elastic relations and a Newton iteration process set up to solve Eq. (17.58) [which is obtained from Eq. (17.79)]. A typical solution which shows the stiffening of the plate with increasing deformation arising from the development of 'membrane' stresses was shown in Fig. 17.6.12 The results show excellent agreement with an alternative analytical solution. The element properties were derived using for the in-plane deformation the simplest bilinear rectangle and for the bending deformation the non-conforming shape function for a rectangle (Sec. 11.3).
Solution of large deflection problems
16 S = aa2/EIt 2 P = qa4/E/t 4
/
14 //
/
f
/
/
12
r (9
~
Mid-side
10
8
~ 0
E
~ uJ
6
Centre
Analytical
FE 32 A elementsTM
0
4
8
12 Load P
16
20
Fig. 17.7 Clampedsquare plate: stresses.
Example 17.3 Clamped plate subjected to uniform load
An example of the stress variation with loads for a clamped square plate under uniform dead load is shown in Fig. 17.7.17 A quarter of the plate is analysed as above with 32 triangular elements, using the 'in-plane' triangular three-node linear element together with a modified version of the non-conforming plate bending element of Sec. 11.5. TM Many other examples of large plate deformation obtained by finite element methods are available in the literature. 17'19-24
17.6.1 Bifurcation instability In a few practical cases, as in the classical Euler problem, a bifurcation instability is possible similar to the case considered for straight beams in Sec. 17.3. Consider the situation of a plate loaded purely in its own plane. As lateral deflections, w, are not
535
536
Non-linearstructural problems-large displacement and instability produced, the small deflection theory gives an exact solution. However, even with zero lateral displacements, the geometric stiffness (initial stress) matrix can be found while B L remains zero. If the in-plane stresses are compressive this matrix will be such that real eigenvalues of the bending deformation can be found by solving the eigenproblem (17.84)
-
in which A denotes a multiplying factor on the in-plane stresses necessary to achieve neutral equilibrium (limit stability), and ~ is the eigenvector describing the shape that a 'buckling' mode may take. At such an increased load incipient buckling occurs and lateral deflections can occur without any lateral load. The problem is simply formulated by writing only the bending equations with K b determined as in Chapter 11 and with K~ found from Eq. (17.66). Points of such incipient stability (buckling) for a variety of plate problems have been determined using various element formulations. 25-3~ Some comparative results for a simple problem of a square, simply supported plate under a uniform compression Tx applied in one direction are given in Table 17.2. In this the buckling parameter is defined as C=
Txa 2 7r2 D
where a is the side length of a square plate and D the bending rigidity. The elements are all of the type described in Chapter 11 and it is of interest to note that all those that are slope compatible always overestimate the buckling factor. This result is obtained only for cases where the in-plane stresses TP are exact solutions to the differential equations; in cases where these are approximate solutions this bound property is not assured. The non-conforming elements in this case underestimate the load, although there is now no theoretical lower bound available. Figure 17.8 shows a buckling mode for a geometrically more complex case. 28 Here again the non-conforming triangle was used. Such incipient stability problems in plates are of limited practical importance. As soon as lateral deflection occurs a stiffening of the plate follows and additional loads can be carried. This stiffening was noted in the example of Fig. 17.6. Postbuckling behaviour thus should be studied by the large deformation process described in previous sections. 31,32
Table 17.2 Values of C for a simply supported square plate and compressed axially by Tx
Non-compatible Elements in quarter plate 2 • 2 4 • 4 8 • 8 Exact C = 4.00.15 d.o.f. = degrees of freedom.
Compatible
rectangle 27 12 d.o.f.
triangle 28 9 d.o.f.
rectangle 29 16 d.o.f.
3.77 3.93
3.22 3.72 3.90
4.015 4.001
quadrilateral 30 16 d.o.f. 4.029 4.002
Shells
~Y"'Rd2--------: = 26.12 :T2D
.
.
r,--1 . .
.
.
.
.
.
.
. ,
""
.
-~--t
. ~
"P
Flange dimensions L .t=4t ~ W= 3--0
Fig. 17.8 Buckling mode of a square plate under shear: clamped edges, central hole stiffened by flange.28
In shells, non-linear response and stability problems are much more relevant than in plates. Here, in general, the problem is one in which the tangential stiffness matrix KT should always be determined taking the actual displacements into account, as now the special case of uncoupled membrane and bending effects does not occur under load except in the most trivial cases. If the initial stability matrix K~ is determined for the elastic stresses it is, however, sometimes possible to obtain useful results concerning the stability factor A, and indeed in the classical work on the subject of shell buckling this initial stability often has been considered. The true collapse load may, however, be well below the initial stability load and it is important to determine at least approximately the deformation effects. If the shell is assumed to be built up of flat plate elements, the same transformations as given in Chapter 13 can be followed with the plate tangential stiffness matrix. 33 If curved shell elements are used it is important to revert to the equations of shell theory and to include in these the non-linear terms. 12'34-36 Alternatively, one may approach the problem from a degeneration of solids, as described in Chapter 15 for the small deformation case, suitably extended to the large deformation form. This approach was introduced by several authors and extensively developed in recent years. 37-48 A key to successful implementation of this approach is the treatment of finite rotations. For details on the complete formulation the reader is referred to the cited references.
537
538
Non-linearstructural problems-large displacement and instability
17.7.1 Axisymmetric shells Here we consider the extension for the beam presented above in Sec. 17.2 to treat axisymmetric shells. We limit our discussion to the extension of the small deformation case treated in Sec. 17.4 in which two-node straight conical elements (see Fig. 14.2) and reduced quadrature are employed. Local axes on the shell segment may be defined by /~ = cos ~(R - R0) - sin ~ ( Z - Z0) = sin ~(R - R0) + cos ~ ( Z - Z0) (17.85) where R0, Z0 are centred on the element as 1 $ ( R I + R2)
Ro =
(17.86)
1 Z 0 = ~ ( Z 1 -q- Z2)
with R i, Z t nodal coordinates of the element. The deformed position with respect to the local axes may be written in a form identical to Eq. (17.7). Accordingly, we have ? = / ~ + ~(k) + 2 sin/3(R) = ~ ( k ) + 2 cos ~(k)
(17.87)
To consider the axisymmetric shell it is necessary to integrate over the volume of the shell and to include the axisymmetric hoop strain effects. Accordingly, we now consider a segment of shell in the R-Z plane (i.e. X is replaced by the radius R). The volume of the shell in the reference configuration is obtained by multiplying the beam volume element by the factor 27rR. In axisymmetry the deformation gradient in the tangential (hoop) direction must be included. Accordingly, in the local coordinate frame the deformation gradient is given by
Fil
=
[1 + u,k + Z (COS/3)/3,k]
0
0
r/R
[t~k - Z(sin/3)/3,k]
sin/3~ 0/3J cos
0
(17.88)
Following the same procedures as indicated for the beam we obtain the expressions for Green-Lagrange strains as Ek~=~,k+~
u
, +
,
Ag, k = E k o k + Z K ~ k
l ( u ) 2 ( u+) 2s i n 13+=~E ~ R
(17.89)
E 77 = -~ + -~ -~
2Ek2 = (1 + ~,k) sin/3 + ~,k cos ~ = F where A = (1 + ~,k) cos/3 - fok sin/3, and the additional hoop strain results in two additional strain components, E~ and K~T. With the above modifications, the virtual work expression for the shell now becomes
5FI - f~ (6Ek}S}} + (~ETTSTT
+ 2~Ek2S}2) dV
-
(~I'Iex
t -
0
(17.90)
in which STT is the hoop stress in the cylindrical direction. The remainder of the development follows the procedures presented in Sec. 17.2.1 and is left as an exercise
Shells 539
for the reader. It is also possible to develop a small rotation theory following the methods described in Sec. 17.2.2. Here we demonstrate the use of the axisymmetric shell theory by considering a shallow spherical cap subjected to an axisymmetric vertical ring load (Fig. 17.9). The case where the ring load is concentrated at the crown has been examined analytically by Biezeno 49 and Reissner. 5~ Solutions using finite difference methods on the equations of Reissner are presented by Mescall. 51 Solutions by finite elements have been presented earlier by Zienkiewicz and co-workers. 7'52 Owing to the shallow nature of the shell, rotations remain small, and excellent agreement exists between the finite rotation and small rotation forms.
17.7.2 Shallow shells- co-rotational forms .
.
.
.
.
.
.
.
.
.
.
.
.
.
In the case of shallow shells the transformations of Chapter 13 may conveniently be avoided by adopting a formulation based on Marguerre shallow shell theory. 24'53'54 A simple extension to a shallow shell theory for the formulation presented for thin plates may be obtained by replacing the displacements by
{u}{uo+u} v w
--->
Vo+V Wo+W
(17.91)
in which Uo, Vo, and Wo describe the position of the shell reference configuration from the X - Y plane. Now the current configuration of the shell (where, often, u0 and v0 are taken as zero) may be described by X 1
(t) = X + uo(X, Y) + u ( X , Y, t) - Z [Wo,x ( X , Y) + w x ( X , Y, t)]
x2(t) = Y + vo(X, Y) + v ( X , Y, t) - Z [Wo,r(X, Y) + w , r ( X , Y, t)]
(17.92)
x3(t) = wo(X, Y) + w ( X , Y, t) where a time t is introduced to remind the reader that at time zero the reference configuration is described by x 1(0) = X -3I- uo(X , Y) - Z Wo,x ( X , Y)
x2(O)
--"
Y + vo(X, Y) - Z Wo,r(X, Y)
(17.93)
x3(0) = wo(X, Y) where u, v, w vanish. Using these expressions we can compute the deformation gradient for the deformed configuration and for the reference configuration. Denoting these by Fit and F ~ respectively, we can deduce the Green-Lagrange strains from 1
E I j -- ~[Fil Fij - F ~ F?j]
(17.94)
The remainder of the derivations are straightforward and left as an exercise for the reader. This approach may be generalized and also used to deduce the equations for deep shells. 44 Alternatively, we can note that as finite elements become small they are essentially shallow shells relative to a rotated plane. This observation led to the development of many general shells based on a concept named 'co-rotational'. Here the reader is referred to the literature for additional details. 55-69
540
Non-linear structural problems-large displacement and instability
80
I
I
I
I
I
I
70 60 50 A
..Q
~, 40 30 20 10 .... 0r 0
i
i
0.02
0.04
i
i
0.06
0.08 V(in)
i
0.1
Finite angle Small angle I
0.12
I
0.14
0.16
Load-deflection curves for vadous ring loads
t= 0.01576 in
I_r~r.~l
~--I-
-~
Total load = P 0.08598 in
0.9 in
_1_
0.9 in
_1
R = 4.758 in E= 10 x 10 slb in 2 v= 0.3 Fig. 17.9 Sphericalcap under vertical ring load: (a)load-deflection curvesfor various ring loads. Spherical cap under vertical ring load; (b) geometrydefinition and deflected shape.
Shells
17.7.3 Stability of shells It is extremely important to emphasize again that instability calculations are meaningful only in special cases and that they often overestimate the collapse loads considerably. For correct answers a full non-linear process has to be invoked. A progressive 'softening' of a shell under load is shown in Fig. 17.10 and the result is well below the one given by linearized buckling. 12 Figure 17.11 shows the progressive collapse of an arch at a load much below that given by the linear stability value. The solution from the finite rotation beam formulation is compared with an early solution obtained by Marcal 7~ who employed small-angle approximations. Here again it is evident that use of finite angles is important. The determination of the actual collapse load of a shell or other slender structure presents obvious difficulties (of a kind already discussed in Chapter 3 and encountered above for beams), as convergence of displacements cannot be obtained when load is 'increased' near the peak carrying capacity. In such cases one can proceed by prescribing displacement increments and computing the corresponding reactions if only one concentrated load is considered. By such processes, Argyris 71 and others 36'52 succeeded in following a complete snap-through behaviour of a shallow arch. Pian and Tong 72 show how the process can be generalized simply when a system of proportional loads is considered. This and other 'arc-length' methods are considered in Sec. 3.2.6.
Linear so
f 0.16
f
J
J
H
,
/ / f"
A 04 r
0.08
r /
I
100 in 0
0.2
0.4
10
wc
(in)
v = 0.3 E = 45 x 104 Ib/in 2
I
I
0.6
0.8
Fig. 17.10 Deflection of cylindrical shell at centre: all edges clamped.12
541
542
Non-linear structural problems- large displacement and instability
~P I-
70
34 in
I
601-
.............
I1"09 in
~ ~ ~ .
rI A = 0.188 in 2 I = 0.00055 in 4 E = 10 x 10s Ib/in 2
i arca'o
! ...............
Finite angle
! ......
~ ......
_ ..............
: ...............
i //Initial stability i
i ,,i i i -~ 40 |l~-.............~i........ .~ _/).J~. L.."II..............~.II.............. Rnite'aingle ............... o,'~176 30 ............. ' ~"7"- ........ _ : i ............. T .............. - "arc~'l ... ,.!...............
20
I- ........
10 00
l/,,~-.-'---':
,,4 ~, i
0.1
.......
i ..............
! i
-i- ..............
i i
0.2 0.3 Central deflection (in)
~ ...............
i =
0.4
0.5
Fig. 17.11 'Initial stability' and incremental solution for large deformation of an arch under central load p.70
iiiii~i!i~ii~i!i1!7~8~iiii~COnc~uid~| ~;~iiii~i~!iii~ii~i!i~iii~i~i~i~ii~ii~ma~ksii ~iiii!iiiii~iiiiiii~ii: ii ill ii!i!iiiiiii!ii!ii!iiiiiiiii!i
!!iiiii!iiii!iii!i!iiiiii!i!ii!!iiiiiiiiiiilii•iiii•ii•i•i•i•iiiiiiii•ii•iii•i
ii•iiiiiiii••iiiii••i!ii!iii!••ii••i!ii i iii ......!!ili This chapter presents a summary of approaches that can be used to solve problems in structures composed of beams (rods), plates, and shells. The various procedures follow the general theory presented in Chapter 5 combined with solution methods for non-linear algebraic systems as presented in Chapter 3. Again we find that solution of a non-linear large displacement problem is efficiently approached by using a Newtontype approach in which a residual and a tangent matrix are used. We remind the reader, however, that use of modified approaches, such as use of a constant tangent matrix, is often as, or even more, economical than use of the full Newton process. If a full load deformation study is required it has been common practice to proceed with small load increments and treat, for each such increment, the problem by a form of the Newton process. It is recommended that each solution step be accurately solved so as not to accumulate errors. We have observed that for problems which have a limit load, beyond which the system is stable, a full solution can be achieved only by use of an 'arc-length' method (except in the trivial case of one point load as noted above). Extension of the problem to dynamic situations is readily accomplished by adding the inertial terms. In the geometrically exact approach in three dimensions one may
References 543 encounter quite c o m p l e x forms for these terms and here the reader should consult literature on the subject before proceeding with detailed developments. 2-5 For the small-angle assumptions the treatment of rotations is identical to the small deformation p r o b l e m and no such difficulties arise.
1. E. Reissner. On one-dimensional finite strain beam theory: the plane problem. J. Appl. Math. Phys., 23:795-804, 1972. 2. J.C. Simo. A finite strain beam formulation. The three-dimensional dynamic problem: Part I. Computer Methods in Applied Mechanics and Engineering, 49:55-70, 1985. 3. A. Ibrahimbegovic and M. A1 Mikdad. Finite rotations in dynamics of beams and implicit timestepping schemes. International Journal for Numerical Methods in Engineering, 41:781-814, 1998. 4. J.C. Simo and L. Vu-Quoc. A three-dimensional finite strain rod model. Part II: Geometric and computational aspects. Computer Methods in Applied Mechanics and Engineering, 58:79-116, 1986. 5. J.C. Simo, N. Tarnow and M. Doblare. Non-linear dynamics of three-dimensional rods: exact energy and momentum conserving algorithms. International Journal for Numerical Methods in Engineering, 38:1431-1473, 1995. 6. D.A. da Deppo and R. Schmidt. Instability of clamped-hinged circular arches subjected to a point load. Trans. Am. Soc. Mech. Eng., pages 894-896, 1975. 7. R.D. Wood and O.C. Zienkiewicz. Geometrically non-linear finite element analysis of beamsframes-circles and axisymmetric shells. Computers and Structures, 7:725-735, 1977. 8. J.C. Simo, P. Wriggers, K.H. Schweizerhof and R.L. Taylor. Finite deformation post-buckling analysis involving inelasticity and contact constraints. International Journal for Numerical Methods in Engineering, 23:779-800, 1986. 9. H.L. Langhaar. Energy Methods in Applied Mechanics. John Wiley & Sons, New York, 1962. 10. K. Marguerre. Ober die Anwendung der energetishen Methode auf Stabilit~itsprobleme. Hohrb., DVL, pages 252-262, 1938. 11. B. Fraeijs de Veubeke. The second variation test with algebraic and differential constraints. In Advanced Problems and Methods for Space Flight Optimization. Pergamon Press, Oxford, 1969. 12. C.A. Brebbia and J. Connor. Geometrically non-linear finite element analysis. Proc. Am. Soc. Civ. Eng., 95(EM2):463-483, 1969. 13. B.N. Parlett. The Symmetric Eigenvalue Problem. Prentice-Hall, Englewood Cliffs, NJ, 1980. 14. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann, Oxford, 6th edition, 2005. 15. S.P. Timoshenko and J.M. Gere. Theory of Elastic Stability. McGraw-Hill, New York, 1961. 16. R. Szilard. Theory and Analysis of Plates. Prentice-Hall, Englewood Cliffs, NJ, 1974. 17. R.D. Wood. The application offinite element methods to geometrically non-linear analysis. PhD thesis, Department of Civil Engineering, University of Wales, Swansea, 1973. 18. A. Razzaque. Program for triangular bending element with derivative smoothing. International Journal for Numerical Methods in Engineering, 5:588-589, 1973. 19. M.J. Turner, E.H. Dill, H.C. Martin and R.J. Melosh. Large deflection of structures subjected to heating and external loads. J. Aero. Sci., 27:97-106, 1960. 20. L.A. Schmit, F.K. Bogner and R.L. Fox. Finite deflection structural analysis using plate and cylindrical shell discrete elements. Journal of AIAA, 5:1525-1527, 1968. 21. R.H. Mallett and P.V. Marqal. Finite element analysis of non-linear structures. Proc. Am. Soc. Civ. Eng., 94(ST9):2081-2105, 1968.
544
Non-linearstructural problems-large displacement and instability 22. D.W. Murray and E.L. Wilson. Finite element large deflection analysis of plates. Proc. Am. Soc. Civ. Eng., 94(EM1):143-165, 1968. 23. T. Kawai and N. Yoshimura. Analysis of large deflection of plates by finite element method. International Journal for Numerical Methods in Engineering, 1:123-133, 1969. 24. P.G. Bergan and R.W. Clough. Large deflection analysis of plates and shallow shells using the finite element method. International Journal for Numerical Methods in Engineering, 5:543-556, 1973. 25. R.H. Gallagher and J. Padlog. Discrete element approach to structural instability analysis. Journal of AIAA, 1:1537-1539, 1963. 26. H.C. Martin. On the derivation of stiffness matrices for the analysis of large deflection and stability problems. In Proc. 1st Conf. Matrix Methods in Structural Mechanics, volume AFFDLTR-66-80, Wright Patterson Air Force Base, Ohio, October 1966. 27. K.K. Kapur and B.J. Hartz. Stability of thin plates using the finite element method. Proc. Am. Soc. Civ. Eng., 92(EM2):177-195, 1966. 28. R.G. Anderson, B.M. Irons and O.C. Zienkiewicz. Vibration and stability of plates using finite elements. International Journal of Solids and Structures, 4:1033-1055, 1968. 29. W.G. Carson and R.E. Newton. Plate buckling analysis using a fully compatible finite element. Journal of AIAA, 8:527-529, 1969. 30. Y.K. Chan and A.P. Kabaila. A conforming quadrilateral element for analysis of stiffened plates. Technical Report UNICIV Report R-121, University of New South Wales, 1973. 31. D.W. Murray and E.L. Wilson. Finite element post buckling analysis of thin plates. In Proc. 2nd Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-68-150, Wright Patterson Air Force Base, Ohio, October 1968. 32. K.C. Rockey and D.K. Bagchi. Buckling of plate girder webs under partial edge loadings. Int. J. Mech. Sci., 12:61-76, 1970. 33. R.H. Gallagher, R.A. Gellately, R.H. Mallett and J. Padlog. A discrete element procedure for thin shell instability analysis. Journal of AIAA, 5:138-145, 1967. 34. R.H. Gallagher and H.T.Y. Yang. Elastic instability predictions for doubly curved shells. In Proc. 2nd Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-68-150, Wright Patterson Air Force Base, Ohio, October 1968. 35. J.L. Batoz, A. Chattapadhyay and G. Dhatt. Finite element large deflection analysis of shallow shells. International Journal for Numerical Methods in Engineering, 10:35-38, 1976. 36. T. Matsui and O. Matsuoka. A new finite element scheme for instability analysis of thin shells. International Journal for Numerical Methods in Engineering, 10:145-170, 1976. 37. E. Ramm. Geometrishe nichtlineare Elastostatik und Finite elemente. Technical Report Vol. 76-2, Institut for Baustatik, Universit~it Stuttgart, 1976. 38. H. Parisch. Efficient non-linear finite element shell formulation involving large strains. Engineering Computations, 3:121-128, 1986. 39. J.C. Simo and D.D. Fox. On a stress resultant geometrically exact shell model. Part I: Formulation and optimal parametrization. Computer Methods in Applied Mechanics and Engineering, 72:267-304, 1989. 40. J.C. Simo, D.D. Fox and M.S. Rifai. On a stress resultant geometrically exact shell model. Part II: The linear theory; computational aspects. Computer Methods in Applied Mechanics and Engineering, 73:53-92, 1989. 41. J.C. Simo, M.S. Rifai and D.D. Fox. On a stress resultant geometrically exact shell model. Part IV: Variable thickness shells with through-the-thickness stretching. Computer Methods in Applied Mechanics and Engineering, 81:91-126, 1990. 42. J.C. Simo and N. Tarnow. On a stress resultant geometrically exact shell model. Part VI: 5/6 dof treatments. International Journal for Numerical Methods in Engineering, 34:117-164, 1992. 43. H. Parisch. A continuum-based shell theory for nonlinear applications. International Journal for Numerical Methods in Engineering, 38:1855-1883, 1993.
References 545 44. K.-J. Bathe. Finite Element Procedures. Prentice Hall, Englewood Cliffs, NJ, 1996. 45. P. Betsch, E Gruttmann and E. Stein. A 4-node finite shell element for the implementation of general hyperelastic 3d-elasticity at finite strains. Computer Methods in Applied Mechanics and Engineering, 130:57-79, 1996. 46. M. Bischoff and E. Ramm. Shear deformable shell elements for large strains and rotations. International Journal for Numerical Methods in Engineering, 40:4427-4449, 1997. 47. E. Ramm. From Reissner plate theory to three dimensions in large deformation shell analysis. Z. Angew. Math. Mech., 79:1-8, 1999. 48. M. Bischoff, E. Ramm and D. Braess. A class of equivalent enhanced assumed strain and hybrid stress finite elements. Computational Mechanics, 22:443-449, 1999. 49. C.B. Biezeno. 0ber die Bestimmung der Durchschlagkraft einer schwach-gekrummten kneinformigen Platte. Zeitschrifi fttr Mathematik und Mechanik, 15:10, 1935. 50. E. Reissner. On axisymmetric deformation of thin shells of revolution. In Proc. Symp. in Appl. Math., page 32, 1950. 51. J.E Mescall. Large deflections of spherical caps under concentrated loads. Trans. ASME, J. Appl. Mech., 32:936--938, 1965. 52. O.C. Zienkiewicz and G.C. Nayak. A general approach to problems of plasticity and large deformation using isoparametric elements. In Proc. 3rd Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-71-160, Wright-Patterson Air Force Base, Ohio, 1972. 53. T.Y. Yang. A finite element procedure for the large deflection analysis of plates with initial imperfections. Journal of AIAA, 9:1468-1473, 1971. 54. T.M. Roberts and D.G. Ashwell. The use of finite element mid-increment stiffness matrices in the post-buckling analysis of imperfect structures. International Journal of Solids and Structures, 7:805-823, 1971. 55. T. Belytschko and R. Mullen. Stability of explicit-implicit time domain solution. International Journal for Numerical Methods in Engineering, 12:1575-1586, 1978. 56. T. Belytschko, J.I. Lin and C.-S. Tsay. Explicit algorithms for the nonlinear dynamics of shells. Computer Methods in Applied Mechanics and Engineering, 42:225-251, 1984. 57. C.C. Rankin and EA. Brogan. An element independent co-rotational procedure for the treatment of large rotations. ASME, J. Press. Vessel Tech., 108:165-174, 1986. 58. Kuo-Mo Hsiao and Hung-Chan Hung. Large-deflection analysis of shell structure by using co-rotational total Lagrangian formulation. Computer Methods in Applied Mechanics and Engineering, 73:209-225, 1989. 59. H. Stolarski, T. Belytschko and S.-H. Lee. A review of shell finite elements and co-rotational theories. Computational Mechanics Advances, 2:125-212, 1995. 60. E. Madenci and A. Barut. A free-formulation-based flat shell element for non-linear analysis of thin composite structures. International Journal for Numerical Methods in Engineering, 30:3825-3842, 1994. 61. E. Madenci and A. Barut. Dynamic response of thin composite shells experiencing non-linear elastic deformations coupled with large and rapid overall motions. International Journal for Numerical Methods in Engineering, 39:2695-2723, 1996. 62. T.M. Wasfy and A.K. Noor. Modeling and sensitivity analysis of multibody systems using new solid, shell and beam elements. Computer Methods in Applied Mechanics and Engineering, 138:187-211, 1996. 63. A. Barut, E. Madenci and A. Tessler. Nonlinear elastic deformations of moderately thick laminated shells subjected to large and rapid rigid-body motion. Finite Elements in Analysis and Design, 22:41-57, 1996. 64. M.A. Crisfield and G.E Moita. A unified co-rotational framework for solids, shells and beams. International Journal of Solids and Structures, 33:2969-2992, 1996. 65. M.A. Crisfield and J. Shi. An energy conserving co-rotational procedure for non-linear dynamics with finite elements. Nonlinear Dynamics, 9:37-52, 1996.
546
Non-linearstructural problems-large displacement and instability 66. A.A. Barut, E. Madenci and A. Tessler. Nonlinear analysis of laminates through a Mindlintype shear deformable shallow shell element. Computer Methods in Applied Mechanics and Engineering, 143:155-173, 1997. 67. J.L. Meek and S. Ristic. Large displacement analysis of thin plates and shells using a flat facet finite element formulation. Computer Methods in Applied Mechanics and Engineering, 145 (3-4):285-299, 1997. 68. H.G. Zhong and M.A. Crisfield. An energy-conserving co-rotational procedure for the dynamics of shell structures. Engineering Computations, 15:552-576, 1998. 69. C. Pacoste. Co-rotational flat facet triangular elements for shell instability analyses. Computer Methods in Applied Mechanics and Engineering, 156:75-110, 1998. 70. P.V. Marqal. Effect of initial displacement on problem of large deflection and stability. Technical Report ARPA E54, Brown University, 1967. 71. J.H. Argyris. Continua and discontinua. In Proc. 1st Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-66-80, pages 11-189, Wright Patterson Air Force Base, Ohio, October 1966. 72. T.H.H. Pian and P. Tong. Variational formulation of finite displacement analysis. In Symp. on High Speed Electronic Computation of Structures, Lirge, 1970.
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In the previous chapters we mainly have used single-scale models to study material and structural behaviour at a macroscopic level. However, many natural and man-made materials exhibit an internal structure at more than one length scale. These internal structures may be of a translational nature, where the structure is more or less invariant with respect to a translation corresponding to the smallest length scale. Materials with internal structure may show also multiscale features, i.e. they may be invariant with respect to scaling. Such materials can be considered to be fractal-like, but they are not true fractals since the exponent n remains finite and the volume fraction does not go to zero even for large n. These examples of scalable structures are by no means exhaustive and many other possibilities exist. Thus, in many cases the microstructure may not be scalable and may be different at each structural level. 1 Examples of different scales and their domain of application are shown in Table 18.1. Materials with internal structure have in common that each structural level plays its own role in the global response: the material behaviour is controlled by the physical phenomena which take place at the various scales and by the interaction of these phenomena across scales. Single-scale models, usually at a macro scale, make use of constitutive equations which should reflect the behaviour of the underlying finer scales. These constitutive equations are generally of a phenomenological type. An alternative to the use of constitutive equations at a single (macro) scale is provided by multiscale modelling, in which the relevant physics is explicitly captured on multiple spatial and temporal scales. 2 This does not exclude the use of phenomenological constitutive equations at lower scale, except when the lowest scale is pushed to the electronic structure and ab-initio calculations are performed. However, multiscale modelling permits one to reduce the uncertainty incorporated in many single-scale constitutive models. The output of multiscale material modelling are usually effective material properties which are used at the higher scale. This analysis may also span over several scales where the output from the preceding, lower level is used as input for the next higher level. Examples of materials with internal structure are: dense hierarchical materials, composites and polycrystals, polymers, hierarchical cellular materials, cellular solids, hierarchical honeycombs of second order, hierarchical composites comprising continuous * This chapter was contributed by Professor Bernhard A. Schrefler, University of Padova, Italy.
548 Multiscalemodelling carbon nanotube fibres in a nanotube-reinforced matrix; natural materials with structural hierarchy are biological materials such as bone or soft tissue, wood, etc.; further we have granular materials such as soils, clays and foams. The reason why man-made materials with structural hierarchy are common in many engineering applications is that they permit one to obtain special desired properties due to the behaviour of the individual components, their geometric structure and the interaction between the components. Examples of such properties are: materials with both high stiffness and high damping, improved strength and toughness, improved thermal and electrical conductivity, permeability, unusual physical properties such as negative Poisson ratio, negative stiffness inclusions, etc. Beyond material behaviour the overall structural behaviour is also of interest. It is at this level that external loads are usually specified and the structural response has to be determined. At the structural level a direct simulation of multiscale systems is usually rather complex and time consuming because the discretization has to be reduced to the lowest scale at which information is needed. This would allow one to obtain at the same time the macroscopic behaviour of the entire structure as well as all the information at the lower scales. But this is not necessary and in a case where we start from an atomic or a nano-scale level is not yet possible. Multiscale modelling which links the different single-scale models in a hierarchical way is an answer to the problem. In the case of material multiscale modelling it is usually of interest to proceed from the lower scales upward in order to obtain homogenized material properties. Alternatively, in the case of structural modelling it is important to be able to step down through the scales until the desired scale of the real, not homogenized, material is reached. This technique is often known as unsmearing or localization. Usually in a global analysis both aspects need to be pursued. As can be seen from Table 18.1, there are many possible scales and the techniques for scale bridging are different. At the lower end we have a linkage between the electronic
Table 18.1 Different length scales and domain of interest; reconstructed from NASA web site
Level
Length scale (m)
Quantum
10-12
Nano
10- 9
Micro
10 -6
Meso
10- 3
Macro
10~
Scientific domain
Subject of manipulation
Nature of prediction
Computational chemistry Computational material mechanics Computational material mechanics
Molecular assembly, nuclei
Qualitative predictions Qualitative predictions
Computational mechanicscomputational material mechanics Computational mechanicsstructural mechanics
Molecular fragments, molecular interactions Surface interactions, orientation, anisotropy, crystals, molecular weight, free volumes Different constituents, different phases, damage
Composed structures
Qualitative predictionsQuantitative predictions Quantitative predictions
Quantitative predictions
Asymptotic analysis structure and the atomistic one and there a discrete-to-discrete linkage is commonly used. At the upper end usually we have a continuum-to-continuum linkage. In between these scales there is a transition from discrete to continuum. The scale at which this transition is made, depends largely on the problem. For instance, continuum elasticity may be used down to the atomic scale, whereas classical plasticity breaks down at the micrometre scale, at which dislocation self-organization takes place. 2'3 Discreteto-continuum linkage may also take place between the micro- or mesoscopic and the macroscopic level when particle mechanics or discrete elements are used at the lower scale. Discrete-to-discrete linkage at the lower end of Table 18.1 requires the definition of atomic potentials and is still in the research stage, e.g. see references 2,4. We deal here more extensively with continuum-to-continuum linkages and with discrete-to-continuum linkages at the upper end of Table 18.1. However, one example of multiple-scale bridging will go down to the micro level. For linkages at the lowest scale the reader is referred to the references mentioned above, 2'4 the references therein and to specialized literature. The most common methods for scale bridging are bounds, self-consistent methods and asymptotic analysis. Voigt 5 was the first to give an evaluation of effective mechanical properties for heterogeneous solids, followed later by Reuss. 6 Voigt and Reuss provide lower and upper bounds for equivalent material properties. Self consistent methods and the Mori-Tanaka 7 approach give instead an estimate of overall mechanical properties for composite materials. Further work on bounds, self consistent and other estimation methods can be found in Eshelby, 8 Hashin and Strikmann, 9 Krrner 1~ and Willis. 12 An alternative approach to scale bridging is asymptotic analysis of media with periodic (or quasi-periodic) structure at the micro level, also called homogenization theory.
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Asymptotic analysis not only permits one to obtain equivalent material properties but also allows one to solve the full structural problem down to stresses in the constituent materials at a micro (or local) scale. It is mostly applied to linear two-scale problems, but it can be extended to non-linear analysis and to several scales as will be shown later in Sec. 18.9. We do not intend to give a full account of the underlying theory in this chapter. In references 13 and 14 the interested reader will find a rigorous formulation of the method, its application in many fields and further references. We will show in detail, however, its finite element solution methodology that is a basic ingredient of many multiscale analyses. For the moment we consider just two levels, the micro (or local) level and the macro (or global) level. These levels are shown in Fig. 18.1 for a structure assumed to be periodic and, thus, asymptotic analysis can be successfully applied. Periodicity means that if we consider a body f2 with periodic structure and a genetic mechanical or geometric property f (for example, the constitutive tensor), we have ifx~f2
and ( x + Y ) ~ f 2 = ~ f ( x + Y ) - - f ( x )
(18.1)
where Y is the (geometric) period of the structure. Hence the elements of f are Y-periodic functions of the position vector x. The characteristic size of a single cell
549
550 Multiscale modelling
Y2 X2
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Fig. 18.1 Example of a periodic structure with two levels: global on the left and local on the right.
of periodicity is assumed to be much smaller than the geometric dimensions of the structure under analysis which means that a clear scale separation is possible.
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One important assumption for asymptotic analysis is that it must be possible to distinguish two length scales associated with the macroscopic and microscopic phenomena. The ratio of these scales defines a small parameter e (Fig. 18.1). Two sets of coordinates related by 1
y = -x
(18.2)
E
formally express this separation of scales between macro and micro phenomena. The global coordinate vector x refers to the whole body f2 and the stretched local coordinate vector y is related to the single, repetitive cell of periodicity. In this way the single cell is mapped into the unitary domain Y [here and in the remainder of this chapter Y indicates the unitary domain occupied by the cell of periodicity and not the period of the composite material as in Eq. (18.1)]. In an asymptotic analysis the normalized cell of periodicity is mapped onto a sequence of finer and finer structures as e tends to 0. If, as defined below, equivalent material properties are employed, the fields considered (temperature, displacement, etc.) converge toward the homogeneous macroscopic solution as the micro-structural parameter e tends to 0. In this sense problems for the heterogeneous body and the homogenized one are equivalent. (For more details concerning the mathematical meaning see references 13 and 14.) We consider now a small strain problem of thermoelasticity in a heterogeneous body, such as that depicted in Fig. 18.1, defined by the following equations: Balance
equations
~ i , j (x) + j5 (x) - 0; qi~i -- r - - 0
cr/j - crj/
(18.3)
Statement of the problem and assumptions 551
Constitutive equations S S S O'ij(X) -- Cijkl(X) ekl (U s ( x ) ) -- OgijO
qT= - k i j O , Strain definition
j
,(u, j
eij (U s (X)) -- ~
+ u j,i
u (i,j)
(18.4)
(18.5)
Boundary and discontinuity conditions ti
(x)nj = 0 on ~t
and
u Ei (x) = 0 on Qu
q[ (x)ni -- 0 on ~~q
and
0~(x) = 0 on ~20
s
= ~ij
[u~(x)] = 0
and
[tzi~(x)nj] = 0 on Sj
[0s(x)] = 0
and
[q~ (x)ni] = 0 on S j
(18.6a)
(18.6b)
The superscript e is used to indicate that the variables of the problem depend on the cell dimensions related to the global length. Square parentheses denote the jump of the enclosed value. The other symbols have the usual meaning: u is the displacement vector, e(x) denotes the linearized strain tensor, crij(x) the stress tensor, Cijkl(X) the tensor of elastic moduli, kij(x) the tensor of thermal conductivity, c~ij(x) = Cijkl(X)13klwherel3kl(X) the tensor of thermal expansion coefficients, 0(x), qi(x) temperature and heat flux respectively, and r(x), j~ (x) stand for thermal sources and body forces, respectively. Since the components of the elasticity and thermal conductivity tensors are discontinuous, differentiation [in the above equations and in Eqs (18.10a) and (18.10b) below] should be understood in the weak (variational) sense. This is the main reason why most of the problems posed in the sequel will be presented in a variational formulation. We introduce now the second hypothesis of homogenization theory" we assume that the periodicity of the material characteristics imposes an analogous periodical perturbation on quantifies describing the mechanical behaviour of the body; hence we will use the following representation for displacements and temperatures US(X) --- U0(X) + ~olul(x, y) + E2u2(x, y) + . . .
+ ekuk(x, y)
0 e (X) -- 00 (X) + E 10 / (X, y) + E202 (X, y) + - . . + F~kok (X, y)
(18.7)
Using Eq. (18.7), similar expansion with respect to powers of e give expressions for stresses, strains and heat fluxes as O"e(X) --" U0(X) -[- E10"l (X, y) -+- ~720"2(X, y) + ' ' " - ~ - Eko"k(X, y) eS(x) = u~ qe(x) = u~
+ elel(x, y) + e2e2(x, y) + . . - + ekek(x, y) + elql(x, y) + ~o2qE(x, y) + . . . + ~okqk(x, y)
(18.8)
where u k, ~ , e k, Ok, qk, for k > 0 are Y-periodic, i.e. take the same values on the opposite sides of the cell of periodicity. The term scaled with the nth power of e in Eqs (18.7) and (18.8) is called a term of order n.
552 Multiscale modelling
The necessary mathematical tools are the chain rule of differentiation with respect to the micro variable and averaging over a cell of periodicity. We introduce the assumption (18.7) and (18.8) into equations of the heterogeneous problem (18.3)-(18.6b) and use the differential calculus rule (see reference 14)
df _ (Of
dxi
l Of)
~
+ -E ~
1
-- f ,i (x) -Jr -6 f ,i(y)
(18.9)
This equation also defines the notation used in the sequel for differentiation with respect to local and global independent variables. Because of Eq. (18.9) the equilibrium equations and the heat balance equation split into terms of different orders [the terms of the same power of e are equated to zero separately: e.g. Eqs (18.10a) and (18.10b) are of order l/e]. For the equilibrium equation we have (79.. tJ, J (y) (x ' y ) - 0 or~ 1 (x,y)+j~(x)=0 tj,j(x) (x ' y ) + O'ij,j(y)
(18.10a)
o1ij,j (x) (x, y) + O'ij,j 2 (y) (X, y) -- 0 We have a similar expression for the heat balance equation: 0 (x, y) = 0 qi,i(y) q0i,i(x) ( x , y ) + qi,i(y) 1 (x,y)-r(x)=0 2 (X, y) = 0 qli,i(x)(X, y) + qi,i(y) ,
o
.
From Eqs (18.5) and (18.9) it follows that the main term of depends not only on u ~ but also on u 1 e~
(18.lOb)
eij in expansions
0 1 -Jr- u(i,j)(y ) ~ eij(x) (U 0) -1" eij (y) (U 1) y) -- u(i,j)(x)
(18.82) (18.11)
The constitutive relationships (18.4) now assume the form or~
Y) - Cijkl(Y) (ekl(x)(U O) -Jr- ekl(y)(Ul)) -- O~ij(y) 0 0
o]j(x, y) -
Cijkl(Y) (ekl(x)(U 1) -q- ekl(y)(U2)) -- O~ij(y) 01
q~
00 -kk/(y) (,l(x)
y)-
q~(x, y)
-
-
--kkl (y)
01 (,l(x)
+ 01,'(y)) -JI-0 2,l(y))
(18.12a)
(18.12b)
It can be seen that the terms of order n in the asymptotic expansions for stresses (18.12a) and heat flux (18.12b) depend respectively on the displacement and temperature terms of order n and n + 1. In this way the influence of the local perturbation on the global quantifies is accounted for. This is the reason why for instance we need u 1(x, y) to define via the constitutive relationship the main term in expansion (18.8) for stresses [and u 2 (x, y) for the term of order 1, if needed, see below].
Global solution 553
i~i~i~i~iiiiiii!~!iiiiiii!i~i~iii!~iiii~iH~i~iii~i~iiiiiii~ii~iiiii~i~b~Ii~i~i~ii~ii~iii~iiiiiiiiii~i~i~i;iii~ii~i!i~;!ii;ii~i~i~iii~i~iii~iiii~Iiiiiiii!iiiiiii~i~iiiiiii!~iiiii!~i~ii~i;iiiiiiiii~i~i~ii~iiii!~iiiiiiiii Referring separately to the terms of the same powers of e leads to the following variational formulations for unknowns of successive order of the problem. Starting with the first order, using separation of variables it can be formally shown that u I (x, y) and 01 (x, y) can be represented by 14'15
U] (X, y) -- epq(x)(U 0 (X)))( pq (y) -+- Ci(x) O~(x' Y) __ oO ,p(x)(X) Op (y) + C(x)
(18.13)
We call X pq (y) and 0 p (y) the homogenization functions for displacements and temperature, respectively. The zero-order (often also referred to as first-order) component of the equation of equilibrium (18.10a) and of heat balance (18.10b) in the light of Eq. (18.13) yields the following boundary value problems for functions of homogenization: find X.pq E Vy such that: Vui ~ Vr
Pq (y) ) lfk,l(y) (y) dr2 - 0 f y Cijkl(Y) ((~ip(~jq "~ Xi,j(y)
(18.14a)
find 0 p E Vy such that: V~b ~ Vr
f y kij(Y) ((~ip AI- O,i(y)(y)) pq q~,j(y)(y)dr2 - - 0
(18.14b)
In the above equations Vr is the subset of the space of kinematically admissible functions that contains the functions with equal values on the opposite sides of the cell of periodicity Y. The tensor Xpq and the vector 0 p are functions that depend only on the geometry of the cell of periodicity and on the values of the jumps of material coefficients across Sj. Functions yi(y) and ~(y) are the usual arbitrary test functions having the meaning of Y-periodic displacements and temperature fields, respectively. They are used here to write explicitly the counterparts of the expressions (18.10a) and (18.10b), in which the prescribed differentiations again are understood in a weak (variational) sense. The solutions X pq and 0 p of the 'local' (i.e. defined for a single cell of periodicity) boundary value problems with periodic boundary condition (18.14a) and (18.14b) can be interpreted as obtained for the cell subject to a unitary average strain e pq and unitary average temperature gradient O,p(y), respectively. The true value of perturbations are obtained after by scaling X pq and 0 p with true global strains (gradient of global temperature), as prescribed by Eq. (18.13). In the asymptotic expansion for displacements and temperature given by Eq. (18.7) the dependence on x alone occurs only in the first term. The independence on y of these functions can be proved (see, for example, reference 14). The functions depending only on x define the macro behaviour of the structure and we will call these the global terms. To obtain the global behaviour of stresses and heat flux the following mean values over the cell of periodicity are defined 14
0.0j(X)_ [y[-1 [ a0j(x, y) dY dY
and
?/~
[y[-1 [ qO(x' y) dY dY
(18.15)
554 Multiscale modelling Averaging of Eqs (18.12al) and (18.12bl) results in the following, effective constitutive relationships
o'Oj(x) --- Chkl ekl (UO) -- OzhijO0
and
~0 = _ kh 00
(18.16)
where the effective material coefficients are computed according to
Ch'kl -- IYI -~ fy Cijpq(Y) ((~kp(~lq"~"X,k,l(y)(Y)) dY Pq kh = IYI-' fr
oLhj "--
kip(y)
IYI-1 fr OLij(y)
((~jp.2f_ L~,j(y) P
(18.17)
(y)) dY
dY
The macro behaviour can be defined now by averaging first-order terms in the equilibrium equations (18.10a2), flux balance equations (18.10b2) and boundary conditions (18.6a) and then substituting the averaged counterparts of stress and heat flux (18.15) [first-order perturbations vanish in averaging of Eqs (18.10a2), (18.10b2) because of periodicity]. Equation (18.4) is replaced by Eq. (18.16) and Eq. (18.6b) is no longer needed since we deal now with homogeneous uncoupled thermoelasticity. The heterogeneous structure can now be studied as a homogeneous one with effective material coefficients given by Eq. (18.17) from which the global displacements, strains and average stresses and heat fluxes can be computed. We then go back to Eq. (18.12a) to recover a local approximation of stresses. This last step corresponds to the unsmearing or localization mentioned in the introduction.
iiii iiiiiiiii iii i6i!iii
iiiii i ii iii
iiilii iiii iiiiiiii iiiiiiiiiiii!!ill ii
ii~i!~i~i~ii~i!i~i~@~!~i~i~i~!i~i~i~s~i!~i~i~!~!~i~i!~i~:~i~ii~@~i~i~!~i~!~i~!i~!~i)~i~i~ii~i~ iiiiiiiiiiiiiiii!iiii!i'~iiiii@',!i!iiii!iii~i'ii',iiiiiii iiillii',iilI l!i!ii!ii~,',i iiii',ii',',!ii ',',ii ~,i
iii iii~iiii iii~ii! iiiili
We note that the homogenization approach results in two different kinds of stress tensors. The first one is the average stress field defined by Eq. (18.16). It represents the stress tensor for the homogenized, equivalent but unreal macro body. Once the effective material coefficients are known, the stress field and the heat flux may be obtained from a standard finite element structural analysis and heat transfer program as described in the previous section. The other stress field is associated with a family of uniform states of strains epq(x) (u 0) over each cell of periodicity Y. This local stress is obtained by introducing Eq. (18.11) into Eq. (18.12a) and results in
a~
Y)=
Cijkl(Y) (r
--
Pq (U 0) Xk,l(y))epq(x)
-- O~/j(y) 0 0
(18.18)
Because of Eqs (18.10a) and (18.14a) this tensor satisfies the equations of equilibrium everywhere in Y. If needed, the stress description can be completed with a higher-order term in Eq. (18.8). This approach is presented in references 16 and 17. Finally the local approximation of heat flux is as follows: p ) (y)] 0 0,p(x) qj0 (y) = -kij(Y)[Sip + O,i(y
(18.19)
Finite element analysis applied to the local problem 555
For the finite element formulation it is convenient to introduce matrix notation for the quantities introduced above. Accordingly, the homogenization functions are ordered as defined by Eq. (18.20) [the numbers in the superscripts in Eq. (18.20) and subscripts in Eq. (18.21), refer to the reference coordinate axes 1, 2, 3]: XT(y)-- [xll(y)
X22(y) X33(y) X12(y) X23(y) X31(y)]3•
TT(y) -- [OI(y)
02(Y)
(18.20)
03(y)]1•
This is in accordance with the ordering of strains and fluxes e - - [ell q=
[ql
e22 e33 el2 q2
e23
e31]T-- {epq}6xl
q3]T'-- {qp}3xl
(18.21)
In the following a tilde again denotes a nodal value in the finite element mesh. We have the usual representations for each element: X(y) - N(y)R;
T(y) = N ( y ) l ~
(18.22)
where N contains the values of standard shape functions. It is easy to show that the variational formulation (18.14a) can be rewritten as follows: find X ~ Vr such that: u
~ Vr
r e r (v(y))D(y) (1 + EX(y)) dY = 0
(18.23)
In the above E denotes the matrix of differential operators, and D contains the material in the repetitive domain. Matrix X which contains the values of coefficients homogenization functions at the nodes of the mesh is obtained as a finite element solution of Eq. (18.23). The equation to solve is the following:
aijkt
KxX - f = 0
(18.24)
where X is Y-periodic, with zero mean value over the cell, and f = frBrD(y)dY;
Kx =
~BTD(y)BdY;
B = EN(y)
(18.25)
D contains the material coefficients aijkl. It can be shown that X in Eq. (18.23) and thus Eq. (18.24) is a solution of a boundary value problem, for which the loading consists of unitary average strains over the cell. This is seen in the form of the right-hand side of Eq. (18.24) which forms a matrix. We solve thus six equations for six functions of homogenization. The variational formulation (18.14b) can be represented in a form similar to Eq. (18.24), T being Y-periodic, with given mean zero value over the cell Kr T + f = 0
(18.26)
556
Multiscalemodelling where
Kr=frB~ko(y)BodY; B 0 = s ko contains the conductivities kij of materials in the repetitive domain. f=frB~k0(y)dY;
(18.27)
Differential operators in s are ordered suitably for the thermal problem. The periodicity conditions can be taken into account using Lagrange multipliers in the construction of a finite element code. The requirements of the zero mean value also has to be included in the program. Having computed X, I' and by consequence ll 1 and 01 one can derive the effective material coefficients, according to: O h = IY1-1 ~ O(y) (1 + BX) OY kh =
IYI-/i
k0(y) (1 + B01') d r
(18.28)
CI~h - - I Y I - 1 J I c~(y)dY With the homogenized material coefficients (18.28) any thermoelastic finite element program can be used to obtain global displacements and temperatures. For the unsmearing procedure we need the gradients of temperatures and strains in the regions of interest, see Eqs (18.18) and (18.19). Strains and temperature gradients are directly obtained from the finite element interpolations. To present the plots of stresses and heat fluxes over the single cell nodal projection can be used. To assure continuity of tangential stresses, this projection should be extended to patches of cells, for example, using SPR. 18
18.7.1 Example As an example of the procedure described above, the temperature effects on a superconducting coil are analysed. The structure has an overall shape of a large D, and is constructed by winding of a superconducting cable with a rectangular cross-section (Fig. 18.2). The whole D-shaped frame is immersed in liquid helium at a temperature of about 4K to assure superconductivity. The supports are designed to eliminate only the rigid motion of the structure. This is a typical example where asymptotic analysis can be successfully applied and the structure is clearly periodic: the macro scale is defined by a typical dimension of the coil cross-section, while the micro scale is given by the height of the section of a single cable (cell of periodicity). In the following a beam-like kinematics with deformable cross-section is used, instead of a full three-dimensional analysis. Homogenization is carried out in the cross-section plane (1,2) only, while in the axial direction (3) there is no periodic structure. 16'17 First effective elastic coefficients and six vectors of homogenization functions for unsmearing are computed. Then the thermal effective characteristics, i.e. effective conductivity and effective thermal expansion are calculated. Three scalar functions of
Finite element analysis applied to the local problem 557 Coil cross-section Yl
Cable cross-section Y2
Xl
=
21
=
[mm] Fig. 18.2 A superconducting coil cross-section and the single cell of periodicity (cable). The material properties are discontinuous, with discontinuities along regular surfaces 5j.
homogenization are further obtained for computations of the heat flux. These functions are shown in Figs 18.3-18.5. The macro analysis with the homogenized material (a standard thermoelastic finite element program) yields the global displacements and the temperature field. Unsmearing as described in the previous section then yields the local distribution of stresses in the region of interest in the coil. Stresses in each different material of a cell of periodicity can be recovered. In Fig. 18.6 the graphs of stress in the cross-section Plane homogenization functions
"
...
Z11,zee
~12 ~33
Fig. 18.3 Planefunctions of homogenization X 11, X ~2, )C22, X 33.
;;
....
558
Multiscale modelling
Y3
Homogenizationfunctions ~23
Y2~~
Y1
X,13 Y3
Fig. 18.4 Plots of the anti-plane functions of homogenization X 13,
X 23 on
four strands.
y~
~ .~
~:
Y2 2.
~,:j
81
82
83
Fig. 18.5 Three scalar functions of homogenization for temperature.
perpendicular to the fibres are shown for uniform cooling. Figure 18.7 shows the full complex state of stress in the neighbourhood of a single conductor for non-uniform cooling. Uniform cooling means here that the whole coil is immersed in liquid helium while non-uniform cooling indicates that the coil is only partially immersed in liquid helium.
18.7.2 Corrections for stresses and boundary effects A refined stress description over the cell of periodicity can be obtained by considering the second of Eq. (18.10a) (for simplicity, in the following we omit temperature effects).
Finite element analysis applied to the local problem . ~ . ~
Stress in steel
Zero level
Stress in epoxy Fig. 18.6 Graphs of stress o33 on the cross-section perpendicular to the fibres for uniform cooling (one cell of periodicity only).
This requires taking into account the second of Eq. (18.12a) and the solution for U 2 in the expansion for displacements (18.7). The rather lengthy procedure is fully described in reference 19 and has been applied to the example problem in reference 17. It also has to be extended to account for localization after the solution of the global problem. An advantage is that, over a single cell of periodicity, the equilibrium conditions are also fully satisfied in the presence of a non-uniform global state of strain. An alternative is to make the localization over a patch of neighbouring cells using the first-order homogenization described in Sec. 18.5.19 There is still another open problem: the periodicity condition used to find the local perturbation is strictly applicable only inside the body. We have hence the solution of a (thermo-)elasticity problem based on an assumed stress or displacement field which is valid nearly everywhere in the region occupied by the body under investigation, except
Stress in steel
Stress in epoxy
Zero level Fig. 18.7 Graphs of stress o33 on the cross-section perpendicular to fibres for non-uniform cooling (patch of four cells of periodicity).
559
560
Multiscale modelling on the boundary. The use of material coefficients based on the assumption of periodicity in the global solution (where the real boundary conditions are imposed) may implicitly impose some unrealistic constraints close to the boundary. This problem can be solved by some corrections which change the solution in the domain close to the boundary while away from the boundary the correction field should asymptotically decrease. 2~ Such a correction can be obtained by replacing the expansion (18.7) by
u=(x) = u~
+ e [u~(x, y) + b~(x, z)] + . . .
(18.29)
where z represents the coordinates of an additional system with the origin on the external surface, the axis z3 directed normal to the boundary surface with the other two axes oriented tangential to the surface and b is the boundary layer correction. Vector b should vanish exponentially when z3 goes to infinity. The procedure is described in detail in reference 21 and has been implemented by making use of one cell of periodicity and infinite elements with homogenized material properties around it. The use of infinite elements assures the desired exponential decay. Although the procedure is theoretically necessary near free edges, it is seldom applied in practical engineering problems.
i~ii~~i~i~ iiiiii~i~i~.~.!.~.~.~ii.~.i.~ii~i~ii~i~i~ii~i!~i!i~iii~i!i~i~i~i~iii~~i~i~i~~~i i i~i~iiiii~iii~i~i~i~i~i~!ii!ii~i~i!~ii!~i~i~i~i~!iiii~i~i~i~!~i~i~i~i~i~iii~ii~ii~i~!i~i~iiiii~~~i~!~i~ii~ ii~!~ii!i~ii!~iii~iii~i~i~ii~i~i!~i~@ii~ii~i~i~i!i~i~i~i~iiii~i~ii~!i~i~i~i~!1~iii!i~ii~i~i~ii~i~!~i~i!~ii~!ii~i~i~!i~ii~i~!~i~i~i~i~i~i~i~!~!i!i!i~i~i@~@i~i!iiiii~i~iii@ii
If applied iteratively, asymptotic theory of homogenization may also be used for nonlinear situations. Furthermore, it can obviously be used to bridge several scales. Here we deal with the case where three scales are bridged by applying in sequential manner the two-scale asymptotic analysis. The behaviour of the components is physically nonlinear. Again we refer to thermomechanical behaviour and introduce a micro, meso and macro level as shown in Fig. 18.8. At the stage of micro or meso modelling, some main features of the local structure can be extracted and used later in the macro analysis. The behaviour of the components, even if elastic-plastic, is assumed to be piecewise linear, so that the homogenization we perform is piecewise linear. Only monotonic loading and/or temperature increase (decrease) is considered, otherwise we need to store the whole history and use an incremental analysis.
X2
@mDB D
fi iS ll
X.
(a) Macro
z{
Yl (b) Meso
(c) Micro
Fig. 18.8 Example of a periodic structure with three levels: macro, meso and micro.
Asymptotic homogenization at three levels: micro, meso and macro
Because of the assumed form of material properties we deal with a sequence of problems of linear elasticity written for a non-homogeneous material domain and with coefficients that are functions of both temperature and stress level. At the top level of the hierarchy we consider an elastic body contained in the domain with a smooth boundary Of2 where on the part Of 2t of the boundary tractions are given and on the remaining part Of2, displacements are prescribed. The domain ~ is filled with repetitive cells of periodicity Y, shown in Fig. 18.8, where the material of the body is supposed to be piecewise homogeneous inside Y, as defined in Eq. (18.1). The governing equations are given by Eqs (18.3) to (18.5). For the lowest level all the formulations are formally the same with one difference: the boundary conditions are those of an infinite body. It is worth mentioning that all the macro fields at the micro level become the micro fields at the higher structural level. Effective material coefficients and mean fields obtained with the homogenization procedure at the lowest level enter as local perturbations at the next higher step. Before explaining the application of the homogenization procedure in sequential form to multi-level non-linear material behaviour we mention the solution by Terada and Kikuchi 22 who write a two-scale variational statement within the theory of homogenization. The solution of the microscopic problem at each Gauss point of the finite element mesh for the overall structure, and the deformation histories at time tn-~ must be stored until the macroscopic equilibrium state at current time tn is obtained. This procedure has not been applied to bridging of more than two scales. A three-scale asymptotic analysis is used by Fish and Yu 23 to analyse damage phenomena occurring at micro, meso, and macro scales in brittle composite materials (woven composites). These authors also retain the second-order term in the displacement expansion (18.7) and introduce a similar form for the expansion of the damage variable. We recall further that also stochastic aspects can be introduced in the homogenization procedure. 24
~!~~E~~~~i
~h~~~i
~
~i~~t~h~r~
~e~I~~l~i~ i ~r~i~'~:~
The two usual tools of homogenization of the previous section are used, i.e. volume averaging and total differentiation with respect to the global variable x that involves the local variable y. The homogenization functions are obtained similarly to Eqs (18.14a) and (18.14b). Only a factor A is introduced in Eq. (18.14a) to adapt the solution to the real strain level as explained below. find X pq ~_ Vy such that: Yui ~ Vr
fy
(A)
Cijkl(Y, ,~, O0)[(~ipt~jq "F ~i,j(y)] Pq I/k,l(y) dr2 -- O, or(A, X pq) E P
(18.30)
Material properties are assumed to depend on temperature and a set of representative temperatures is considered for the material input data where linear interpolation is used between the given values. P is the domain inside the surface of plasticity. The requirement that the stress belongs to the admissible region P [introduced in Eq. (18.30)] is verified using the classical unsmearing procedure described in the preceding section.
561
562
Multiscale modelling Table 18.2
Updating yield surface algorithm scheme i
1. 2. 3. 4. 5. 6.
Compute effective coefficients at micro level. Compute effective coefficients at meso level. Apply increment of forces and/or temperature at the macro level, solve global homogeneous problem. Compute global strain Eij" Eij -- eij (u 0) reminding that Eij -- e~j (x) [see Eq. (18.31)]. Apply Eij to meso-level cell by equivalent kinematic loading (displacement on the border). Solve the kinematic problem at the meso level for w(y), compute stress (unsmearing for meso level) and strain Eij" now Eij -- eij (w 0) and Eij -- ~ej (y). 7. Apply Eij from meso to micro level cell by equivalent kinematic loading (displacements on the border). 8. Solve the kinematic problem at the micro level for w I (z), compute stress (unsmearing for micro level). 9. Verify yielding of the material in the physically true situation at micro level. If 'yes' change mechanical parameter of the material and go to 1; else exit. ,,
,
,
,
,
It is to be noted that for 'solving the boundary value problem' mentioned in points 6 and 8 it is not always necessary to use the true finite element solution. If the cell of periodicity has not been changed before, this solution can be composed according to Eq. (18.13) (suitably rewritten).
The modification of the algorithm required by the non-linearity starts with the composite cell of periodicity with given elastic components. The uniform strain is increased step by step. Effective material coefficients are constant until the stress reaches the yield surface at some points of the cell. The yield surface in the space of stresses is different for each material component, being thus a function of position. The region where the material yield is of finite volume at the end of the step, thus it is easy to replace the material with the yielded one, with the elastic modulus equal to the hardening modulus of the elastic-plastic material and with the Poisson ratio tending to 0.5. The cell of periodicity is hence transformed into a form with one more material and the usual analysis procedure is restarted with a uniform strain, a new homogenization function and a new stress map over the cell. We identify each new region where further local yielding occurs, then redefine the cell and perform the analysis. The loop is repeated as many times as needed. In Eq. (18.30) the history of this replacement of materials at the micro level is marked by A, the level of the average stress, for which the micro yielding occurs each time. The algorithm is summarized in Table 18.2 where w(y) indicates displacement at meso level. At the end of each step we can compute also the mean stress over the cell having (generalized) homogenization functions [see Eq. (18.16)] and the effective coefficients can be computed using Eq. (18.17).
An important part of multiscale modelling is the recovery of stress and heat flux as well as strain, temperature and displacements at the level of the microstructure. This is obtained by Eqs (18.18) and (18.19) with the following procedure: first global (mean) fields are obtained from the homogeneous analysis where the material is characterized by the effective coefficients (18.17), then we go back to the original problem formulation, using homogenization functions. Thus we recover the main parts of the stress and heat flux. Because of the specified three-level hierarchical structure we are dealing with, the recovery process must be applied twice, also since material characteristics
Recovery of the micro description of the variables of the problem 563 are temperature dependent and non-linear the procedure must be applied for each representative temperature within the context of the correct stress state. We recall that the recovery process starts at the highest structural level while the homogenization begins at the lowest part of the structural hierarchy.
18.10.1 Example" the VAC strand analysis As an example of application, we consider a superconducting strand which is used to build the cable of Fig. 18.2. The structures and the three scales are shown in Figs 18.9 and 18.10, where the single filament (micro scale), groups of filaments (meso scale) and the superconducting strand (macro scale) are shown. The homogeneous effective properties will be defined for the inner part of the strand, shown on the left of Fig. 18.10. The diameter of the strand is about 0.80 mm. The application of the theory of homogenization is justified by the scale separation as clearly shown in Figs 18.9 and 18.10. As already indicated, periodic homogenization is applicable to structures obtained by a multiple translation of a representative volume element (RVE), called in this case the cell of periodicity. The considered strand shows two different levels of such a translational structure. On the meso level we have the repetitive pattern of the superconducting filament in the bronze matrix (micro scale RVE), filling the hexagonal region as illustrated in Fig. 18.11. The second translational structure is the net of the hexagonal filament groups (meso scale RVE) in the body of the single strand shown in Fig. 18.12. The homogenization splits thus into two steps, each one dealing with rather similar geometry and a comparable scale separation factor. Boundary conditions for the macro problem will be given in terms of interaction of the strand with the other strands in the cable e5 and will be of the type of Eq. (18.6). To form the superconducting alloy Nb3Sn the coil is kept for 175 hours at 923K. Afterward, to reach an operating temperature, it is cooled from 923K to 4.2K. In this example we analyse the effects of such cool-down, using the homogenization procedure to define the strain state of the strand at 4.2K which is caused by the different thermal :o ~ ~ , , ~ i ~ . ~. ~, .......... ~.~....... %
~,:~.~: .. ~.? ~:~?~:&.~:.
C~
.;:
~ ~.~ .~ :.!i~~i.:.~: ~.:"::,~-
-~L~: ~ ~io~ :~ ~.~.~~;~<~.....
i
... ~ ~
~~ ~?,~-~::~ ......
~ ~ ~ ~L. ~ ~ ~
~
~.~.~
~:; ~::~.
.~. :~.'~..~, ... :. :"~.:~......
~ ~ ~:: ~: ~:.~'~:: ........
~:~ ~ ~: ~L~'.::~
. .~
9
~ .~
~ : ~ ~
...~;" ~" ~..:~=~i
~" ::: IT ~ ~'~;~
~ ~ ~': ,:~ 9 i.~'~ ....
~
~r
~ " ~ .
~,~rr~:~
9
Fig. 18.9 A single Nb3Sn filament (left) and Nb3Sn filaments groups (right); the respective scales are also evidenced. Each filament group is made of 85 filaments.
564
Multiscale
modelling
Fig. 18.10 Three-level hierarchy in the VAC strand. The central part of the strand itself (left) consists of 55 groups of 85 filaments, embedded in tin-rich bronze matrix, while the outer region is made of high conductivity copper. contractions of the materials. This strain state is the initial condition for successive operations of the cable. It is recalled that superconductivity of the employed material is strain sensitive and hence a precise knowledge of these strains is of paramount importance. At the end of the cool-down in a reacted strand the filaments are in a compressive strain state while the bronze and copper matrices are in a tensile state. We assume that the strand components are stress free at 923K, since the strands remained at that temperature for 175 hours.
N
>e oco: oO, o 'c ~,~ .........,:, :'i~!!f :~
Q Q~
o,
.9
' ~ : ........ i. . . .
.
' .......
~-~
~::' .... ~.,
~...>.:~,.::
.%~
..% .
~ ~~
.~. ,~,,~:; ~:: ~>
.'~': :*~:~
,~ ,.. !!~i:~! ~ .
.
~ :
i!4:: i~!:
:~..,~ ~:~,,~ . . . . . .
:~):%!.
. ...
~.
,~,::
:~: ..~,,.
"
Fig. 18.11 Micro scale unit cell. Light elements: bronze material, dark elements: Nb3Sn alloy. The area of the cell is 90x 5.6 #m.
Material characteristics and homogenization results 565 ":~
~
~ ' ~
II ~:~e~
~<~
~1:~ ~ ~<~<~~,,
~:~ ~ ~-; ~ ~ ~ :
~~
-~@~ ....
INNNN
~==NNN~
Fig. 18.12 Meso scale unit cell. Dark element: bronze material, light elements: homogenized material at micro level. The area of the cell is 100.0 x 60.1 #m.
The Nb3Sn compound has a low thermal contraction but a relatively high elastic modulus and a very high yield strength. The bronze and copper reach their yield limits as soon as the temperature starts decreasing, so that they are plastically flowing for nearly the whole thermomechanical analysis. Material thermal characteristics are taken from the conductor database. 26 There are very few measurements of elastic-plastic properties of the strand components over the whole temperature range 4-923K, see references 27-30. Variation of the different materials' elastic moduli versus temperature is shown in Fig. 18.13. Due to their high yield limit the Nb3Sn filaments can be assumed to remain elastic over the whole temperature range, with a constant elastic modulus of 160 GPa. 31 Bronze and Nb3Sn thermal conductivity (W/mK) and thermal expansion (%) as a function of temperature (K) are illustrated in Figs 18.14 and 18.15, respectively. After the homogenization procedure, the resulting equivalent material has an orthotropic behaviour, depending upon the material characteristics and the geometrical configuration of the unit cell. Conceming the thermal behaviour, the Nb3Sn compound has a very low thermal conductivity, and is practically zero when compared to that of bronze. The bronze material has a thermal conductivity of less than 500 W/mK from reaction temperature until to about 100K, then, with decreasing temperature, thermal conductivity undergoes a strong increase of almost 2500 W/mK and then decreases again (Fig. 18.14). It is quite obvious that the resulting material will be guided by the bronze behaviour, both at meso level (Fig. 18.14, grey continuous lines) and on macro level (Fig. 18.14, grey dashed lines): kll, k22, k33 stand for the value of the conductivity referred to a Cartesian system of coordinates, where the third axis is parallel to the longitudinal axis of the strand.
566 Multiscalemodelling ................. ................................... i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ 180000~ ~
~:'. . . . . . . . . . . . . . .
v
160000
'
=
140000
0
E
120000 100000
W
80000 60000
,
,
i. . . . . . . . . . . . . . . . . ... .
,
20000
................... i. ................... i................... i ................... . . . i .......................................
.
. .
0
.
. .
. .
i 200
0
. .
i
i ...................
..................
.
~ . . . . . .. . . . . . . . . . . . .
......................................
40000
. .
i 400
i 600
i 1000
800
1200
T e m p e r a t u r e (K)
I -~-
Bronze
- A - Nb3sn
+
Copper I
Fig. 18.13 Variation of bronze (circles), Nb3Sn (triangles) and copper (squares) elastic modulus versus temperature.
~" 2500 E v
••i••i••i•••i••i••i••i••`• ~ 9 2000
o "O tO
o
1500
1000
" ! ............ ~............ ] ............ ~............. [ ............ .,"............ i ............ ~............. !
500
i ......
\~
----: .
0
0
100 I
= --u-
.
i
9
........
- .
200
~ ! i
: - -
i ..... ~!
! .........
i
:
i ............
i ............
: ............
:. . . .
i. . . .
!__--;
i
'.
:
,.
500
600
700
: i
: - - ~
.
300
400
............
Bronze
.-.o--K11 First level
~
K22 First level
K11 Second level
--z,.- K22 Second level
--o-
K33 Second level
:
u
'.
~
800 900 T e m p e r a t u r e (K)
-.-o-.-K33 First level • Nb3Sn
Fig. 1 8 . 1 4 Thermal conductivity (W/mK) of bronze (stars), Nb3Sn (crosses), meso and macro level homogenization results (grey continuous and grey dashed lines, respectively).
Multilevel procedures which use homogenization as an ingredient ~" 2.5000E-05 tO .m
t- 2.0000E-05 Q. X
E 1.5000E-05 t'-
I-
1.0000E-05
5.0000E-06
O.O000E+O0
0
,
0
~
0
200
i
400
Bronze - - ~-- - a l 1 S e c o n d
--o-.-
level
-- ~
600
a l 1 First level - a22 Second
.....
level
800 ~
1000
1200
Temperature (K) a 3 3 First level
a 2 2 First level
- - o-- - a 3 3 S e c o n d
i ........
level
•
Nb3Sn
Fig. 18.15 Thermal expansion [1/K] of bronze (stars), Nb3Sn (crosses),meso and macro level homogenization results (grey continuous and grey dashed lines, respectively).
Thermal expansion is almost linear with temperature, that of bronze being higher than that of Nb3Sn (Fig. 18.15). Resulting effective coefficients are illustrated in Fig. 18.15 for the meso level (grey continuous lines) and for the macro one (grey dashed lines): O~11, O~22, O~33 stand for the value of the expansion coefficients referred to the same Cartesian system of coordinates as for the conductivity values described above. Mechanical characteristics of the individual materials and homogeneous results are compared in Fig. 18.16, showing the diagonal terms of the elasticity tensors as a function of temperature. The peculiar disposition of the superconducting filaments gathered into groups results in an almost isotropic behaviour in the strand cross-section, while along the longitudinal direction of the strand the material behaviour is strongly influenced by the superconducting material. The procedure has been validated by comparing results of a homogenized group of filaments and those of a three-dimensional discretizationinvolving about 8000 elements - subjected to the cool-down analysis of a strand. 59
,,,~,,,!!~,~!~i!,~,,!~,,~,,,,~,,~,~,~,,a~,,an,m~u,|,e,n ~ ,~,,,~,~,,,~,,,~,,~,,,,,~,~,,
~,,~,,,,,,,,,~,~,,~,,,,,,~,~,,~,,,,,~,~,,~,,,,,,,,,,~,,~,~,,,~,~
,,
,,~,,,,,,,,~,,,~,,,
~,i
There exist several procedures for multilevel approaches which use asymptotic homogenization as an ingredient. We will consider here two of them. Feye132-35 introduced an integrated multiscale analysis, also called the FE 2 method which consists of the following ingredients typical for multiscale techniques:
567
568
Multiscalemodelling ~" 250000 ,
x
......
g 200000 ,m
150000 m W
100000 .
50000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~
0
0
200
400
Bronze m o-
- al 1 Second
~ level
--~-
600
al 1 First level - a22 Second
level
800 ~ m o-
1000 1200 Temperature(K)
a22 First level - a33 Second
level
--..o--x
a33 First level Nb3Sn
Fig. 18.16 Main diagonal elasticity terms for bronze (stars), Nb3Sn (crosses), meso and macro level homogenization results (grey continuous and grey dashed lines, respectively).
1. The mechanical behaviour is modelled on the lower scale, i.e. on the representative volume element (RVE). 2. A localization rule is chosen to determine the local solution inside RVE for any given overall strain. 3. A homogenization rule is selected which yields the macroscopic stress tensor, knowing the micro-mechanical stress state. 4. At each integration point of the macroscopic level an RVE is located where the above operations are carried out. 5. The overall strain and displacements are obtained at the macroscopic level. In principle any localization/homogenization rule can be used for steps 1 to 3, as indicated in the next section, but the most common method is the periodic homogenization described in the previous sections which is considered further in this section. Also in the integrated analysis two finite element meshes are needed as in the sequential application of the homogenization method described above, one for the cell of periodicity and the other for the macroscopic structure. The difference from that considered previously is that the computation is carried out simultaneously on both scales. In the cell of periodicity located at each Gauss point of the macroscopic structure the method computes the stress tensor at time t using the current strain, strain rate and mechanical history since the beginning of the analysis. In classical phenomenological models at macroscopic scale the mechanical history is taken into account through a set of internal variables. Here the internal variable set is constructed by assembling all microscopic data required by the finite element computation at a lower level. This includes of course microscopic internal variables needed to describe dissipative phenomena.
Multilevel procedures which use homogenization as an ingredient
The local analysis yields the macroscopic (average) stresses and strains, defined as
1 L O'ij d V ;
]~aiJ -- O'ij -- W
1 L eij d V
Eij -- O.ij -- V
(18.31)
also known as Bishop-Hill relations. 36 The integrated approach can be easily implemented in a classical finite element program based on a Newton algorithm to handle all non-linearities and in fact consists in a sequence of Newton algorithms at the global and local levels. For optimum performance the tangent stiffness matrix at the macroscopic level has to be computed. For this calculation we need the macroscopic (algorithmic) tangent matrix in the Gauss points which can be computed from the variation of the average strains and stresses of Eq. (18.31) as U macr~
-
OAE
(18.32)
0AE
where A denotes the increment of the quantity between time t and time t + At. This computation depends of course on the homogenization theory used and on its finite element implementation. In the case of the periodic homogenization theory it is convenient to add at the local level of each element in the cell of periodicity mesh some degrees of freedom corresponding to the average or macroscopic strain E. This involves at most 6 degrees of freedom shared by every element in the mesh. It is recalled that at a local level the unknown displacements are the periodic part U 1 of the total displacement u on the cell where u 1 is obtained using Eqs (18.24) and (18.13). These, together with the first of Eq. (18.31), are contained in the condensation procedure outlined below, proposed in reference 35. The deformation tensor is computed by Eq. (18.11) and may be written as e ~(x, y) - e(x)(u ~ + e(y)(u 1) - E + e(y)(u 1)
(18.33)
^
The B matrix needed in this case, called B, is similar to the usual one, except that a new part comes from degrees of freedom associated with E (these degrees of freedom may be inserted at the end of the finite element degree of freedom list): l ~ - [B
1]
(18.34)
where B denotes as usual the standard symmetric gradient of the shape functions. According to our choice, E has associated degrees of freedom and hence the associated reactions give the mean stress ~2, to be multiplied by the volume of the cell. The assembled tangent stiffness matrix at the cell scale can be written as K - fcen l]rD 1] dr2
(18.35)
Taking into account Eq. (18.34) this leads to GHr]
(18.36)
569
570 Multiscale modelling where D represents the tangent matrix given by all microscopic phenomenological constitutive equations. Recalling the meaning of the additional degrees of freedom and the associated reactions, the macroscopic tangent matrix ])macro of Eq. (18.32) is then nothing but a condensation of the previous matrix onto the degrees of freedom associated with E. That is O macr~
:
1
~
V
[H - G k - l G r ]
(18.37)
Conceptually this condensed matrix is then very easy to compute. However, for large structures the method requires a great computational effort to perform the reductions involving k and parallel computing is usually employed in this step. Another multiscale computational strategy which makes use of the homogenization theory is proposed by Ladevbze and co-workers. 37-39 In that case the structure is considered as an assembly of substructures and interfaces. The junction between the macro and micro scales takes place only at the interfaces.
The main characteristics of the methods of the previous section are: 1. The constitutive response at the macro scale is undetermined a p r i o r i and ensues from the solution of the micro scale boundary value problem. 2. The method can deal with large displacements in a straightforward way, provided that the micro structural constituents are modelled adequately. 3. The different constituents in the microstructure can be modelled with any desired non-linear constitutive model. 4. The micro scale problem is a classical boundary value problem; therefore also methods different from that of asymptotic homogenization can be used, i.e. any appropriate solution strategy is applicable. 5. The macroscopic constituent tangent operator can be obtained from the overall microscopic stiffness tensor by static condensation, as indicated above. This scale transition preserves consistency. 4~ The procedure is called a first-order computational homogenization since it includes only first-order gradients of the macroscopic displacement field. Other methods have been published by Ghosh, 41'42 Miehe, 43'44 Kouznetsova, 45 Suquet 46 and Wriggers and co-workers. 47'48 As already mentioned, they also make use of solution methods of the micro scale problem which differ from the asymptotic homogenization. First-order methods require, however, that the characteristic length of the spatial variation of the macroscopic loading must be very large with respect to the size of the micro structure and the macroscopic gradients must remain very small with respect to the micro scale. Possible size effects which may arise from the micro scale and localization problems cannot be dealt with properly. For this purpose the method has been extended by Geers et al. 4~ and Kouznetsova et al. 45 to higher-order continua.
General first-order and second-order procedures 571
18.13.1 Example As an example, we show micro scale computations including damage at interfaces on a single RVE under two different boundary conditions. 47 The investigation of a single RVE without embedding in a concurrent method as above may also be used for numerical material testing. 47 The considered unit cube of Fig. 18.17 contains 10 particles. Around the particles there is a cohesive zone as in reference 49 in which damage can occur. Since meshing of a complex three-dimensional structure is difficult and often leads to badly shaped element geometries, here the geometry is discretized and approximated with cube-shaped elements and subsequently refined at the boundaries of the particles and the cohesive zone to obtain a better geometry approximation, see Fig. 18.17. The matrix material, the particle material and the cohesive zone material are considered to be a simple neo-Hookean material with the strain energy function
' ( J-Z/3tr(C) - 3 ) + K ( J -
9 (C) - g #
1 - log(J)
)
; tr(C)
-
CII
where C - F r F and j2 __ det(C). The material parameters of the particle material and the matrix material are chosen to be #Matrix = 3, KMatrix -- 7, ~ p a r t i c l e = 30 and Kparticle "- 70. The damage law used for the cohesive zone is the one proposed by Zohdi and Wriggers. 48 The initial material parameters for the cohesive zone material are chosen to be the same as the parameters for the matrix material. When damage occurs they are weakened by a factor c~.
#(cz) _
o~ #o(cz) ~(cz) K(CZ) - - ~ "~0
where (.)(cz) means c o h e s i v e z o n e .
Fig. 18.17 Cut through the initial mesh.
0
' O < ce ~ 1
572 Multiscalemodelling The local constraint condition from which a can be computed is (a) = M(a) - K (a) < 0
where M (a) is a scalar valued term representing the stress state of the material point M(a)-
(g(O'deg(OZ)" g(O'deg(O~))1/2
g(~rdeg(a)) -- ~1 ~ tr(~rdeg) 1 + ~/2 (O.deg -- ~1 tr(~rdeg) 1) and ~7~ and r/2 are parameters scaling the isochofic and deviatofic parts of g(Ordeg(a)). K (a) is a threshold value which depends on the damage variable itself. K (a)
=
(I)li m -~- ( ( I ) c f i t -
(I)lim)o~
P
where ~cnt is the initial threshold value, and (I)lim is the threshold value in the limiting case that the material point has degraded completely (a - 0). Finally P is an exponent which controls the rate of degradation. The parameters chosen here are ~1 = 1, T]2 = 1, (I)li m - - 0.1, (I)crit = 1 and P = 0.1. The tests done here are a comparison between the results for pure Dirichlet boundary conditions and pure Neumann boundary conditions. For pure displacement boundary conditions a displacement gradient H-7
[1111 1
1
1 1
1 ,
7-0.1
1
has been enforced on the entire boundary of the RVE, 7 is a load factor. The resulting damage distribution and the first principal stresses are shown in Figs 18.18 and 18.19. For pure traction boundary conditions the Cauchy stress tensor ~r = q/
3.5
1
3.5
,
"7 -- 0.48
has been prescribed. The same quantities as above are shown in Figs 18.20 and 18.21. More details about the homogenization procedure and the statistical testing method chosen can be found in reference 47. As for the boundary conditions, it is recalled that linear deformations and uniform traction constraints yield upper and lower bounds of the sought material characteristics, respectively. On the contrary, in classical homogenization theory as shown in the previous sections periodicity conditions reflect exact results of the global stiffness for periodic structures. Hence, where applicable, periodicity conditions should be applied, see also the next section.
The methods described in the previous section in principle also can be applied to a discrete-to-continuum linkage in which the discrete body is contained in the RVE
Discrete-to-continuum linkage
.........................................................
Ni
..............,~:
....
N :2~: ....
.... , .................... .,.,.:,.,,.:,,,,:,. ............... .,,..,~,~,..~..,,,:,:,.,:.~.~,.
Fig. 18.18 Displacementboundary conditions: damage. while the macroscopic body represents the continuum, see, for example, Miehe and Dettmar, 5~ where a package of irregular discs is studied. In this case a periodic RVE is assumed but its size is rather large because of the irregularity of the disc size. As an alternative we show here a sequential method that can be used with any localization~omogenization rule. A numerical homogenized constitutive relation for the global behaviour is defined first by constructing subsequent yield surfaces by means
i Fig. 18.19 Displacementboundary conditions: first principal stresses.
iii~i~u~. ii ,.~
i
........ i
i
573
574 Multiscalemodelling
...................!!
i
i!
...........::2.,L.Z:
............................ i
Fig. 18.20 Traction boundary conditions: damage. of the solution of a series of local problems on a RVE. These permit the definition of so-called interpolation points in the global stress space, which are assumed to belong to the global yield surfaces. At a macroscopic level the method uses an elastic-plastic algorithm, where the necessary information (hardening law, flow rule) is extracted numerically from the interpolation points. This method is advantageous if several macroscopic loading conditions have to be considered, since the numerical constitutive relationship is established and stored once for all cases. In the following we select asymptotic homogenization for the solution of the local problems. We apply the procedure and the related stress recovery for the case
........IZ ..
i ii!i!iiiiiiiiiii~:~:~i......
ii
i..............................................................................
Fig. 18.21 Traction boundary conditions: first principal stresses.
Discrete-to-continuum linkage
( ( Cell of periodicity
Fig. 18.22 Periodicstructure composed of discs in contact. where in the RVE there are elastic-plastic discs in contact, as depicted in Fig. 18.22. This represents clearly a discrete structure, while the overall macroscopic behaviour is dealt with as a continuum. Plane situations, monotonic proportional loading and small strains are assumed. With enhanced interpolation procedures the method can be extended to more genetic situations, such as the discrete element methods described in Chapter 9, if they are used to obtain macroscopic constitutive behaviour. 5~ For an alternative approach see references 52 and 53.
18.14.1 Representative volume element and boundary conditions We take here advantage of the periodicity of the structure and hence the choice of the RVE is straightforward. The size of the RVE takes advantage of the regularity of the packing. As in the previous sections, microscopic quantifies are indicated with lower case letters while capital letters are used to indicate macroscopic quantifies. To derive the constitutive equations from micro mechanics and homogenization, it is necessary to relate the fields in the interior of the RVE to their volume and boundary averages. This can be obtained through the macroscopic stress and strain defined in Eq. (18.31) which in our case can also be expressed as ~-~ij :
-'~
E i j "-
1 2V
O'ij d V
E
'Ic
(uinj + u j n i ) d S - - ~
(18.38)
[ui]nj + [uj]ni dSc
where V* is the portion of RVE occupied by the solid phase, SRVEis the external surface of the RVE, Sc is the common surface of the discs in contact, [uj] is the jump of the displacement field across the surface Sc, and rti is a vector normal to the relevant surface, see Figs 18.23 and 18.26.15
575
576
Multiscalemodelling Su(2)
s#21 Su (~) *
~._
Fig. 18.23 Bodies in elastic or elastic-plastic contact.
...............................................................................
i!i ~, ...............................................................................
'~!~i~i~!~!~~?i~i~i~i~i~i~i:~i~i~i~:i~:i~i~i~i~i~i~i~:~:~i~i~:
i~iii~i!ii!ii~ii!i~i~i!!i!ii~iiii~i~i!i~!~!ii!i!~i~i'i~i!i!~ii~!~!i!ii!~!i~!i!i!ii!i!i!!iii!iii!i!ii!~i~!i~i!
Fig. 18.24 Periodic boundary conditions on a quadratic cell with circular inclusion: deformed configuration due to u1(on the right).
I
I
'
I
Fig. 18.25 Periodic boundary conditions on a quadratic cell with circular inclusion: periodic component of the deformation due to u 2 (on the right).
Discrete-to-continuum linkage
aY1
P2
ml aY2 Fig. 18.26 Single cell of periodicity.
We simulate a large number of different radial loading paths on the unit cell in such a way that any genetic monotonic proportional loading can be approximated by an interpolation between some paths previously simulated. One of the main problems in this approach is that the unit cell on which the different loading paths are simulated is constrained in a manner which cannot represent all the possible in situ conditions. We adopt here the following periodic boundary conditions on OV Ui -- E i j x j + u *" i, u * i periodic on OV
O'ij nj; anti-periodic on OV
(18.39)
where Ui = (u~, U2), for planar problems, is the displacement applied to the boundary of the cell and is the sum of two contributions: a linear part given by Eij y j and a periodic part u~ which gives no contribution to the global problem, u 7 can contain terms of the first order or also higher order of Eq. (18.7). As an example, in Fig. 18.24 a typical deformed configuration of the cell of periodicity shown in Fig. 18.1 is presented and in Fig. 18.25 we show the periodic component due to u2. 54 In the example only the linear part u 1 has been used. As for the anti-periodicity condition (18.392), the boundary of the unit cell is decomposed into two parts OV : OVI -+- OV2
where each point P1 :-- P1 E Fig. 18.26, such that
(18.40)
OV1 has a corresponding point P2 := P2 ~ OV2, see ( o i j n j ) p2 - - - - ( o ' i j n j ) p ,
(18.41)
From a numerical point of view periodic boundary conditions can be easily implemented in general-purpose finite element codes (see reference 18 for a discussion on periodic boundary conditions).
577
578 Multiscale modelling i~i~i~i~i~i~i~i~i~i~i~i~i~i!i~i~~~i!~i~1~i~i~i5i~i~i~!~i~i~i~i~~ii~i~i~i~!~i~~i~i~i~i~~!~i~i~i~~~ii~i~i~i~iC~| ~i~i~~i ~]~~~~~~~!~!~!~!~i~i~i~i~i~i~i~i~i~i~i~ Given a unit cell on which the periodic boundary conditions (18.39) are imposed, the problem to be solved is to find the constitutive law of the homogenized material described by macroscopic stress and strain (18.38). The unknown relation between these quantifies is numerically obtained by solving a large number of local problems given on the unit cell microscopic constitutive laws div ~r = 0; micro-equilibrium
Eij'-(c~0+!c~0m) E ~
(18.42)
, given
E ~ indicates the direction of the loading path in the strain space, c~0is a strain multiplier which allows the first yielding and/or slip in some points of the RVE to be reached and m = O, 1, 2 . . . . . mmax indicates the number of loading steps which follow the first elastic step; in the numerical examples shown below we choose m m a x = 25. The factor 1/c is the multiplier for plastic strain. For a description of contact formulations, see Chapter 5. A global strain tensor E ~ is hence imposed to the cell and it is monotonically increased to generate a kinematic loading path: in particular, an elastic step is applied so as to reach the global elastic frontier, which is defined as the set of points in stress space corresponding to the first yielding of any point of the unit cell or the first activation of relative displacements in the contact points for a fixed loading direction E ~ Then mmax 'small' load increments are applied to induce plastic deformations in the unit cell and/or slip in the contact points. Each 'small' load increment is equal to 1/c times the first (elastic) increment. The homogenized stress t e n s o r ]~ij is computed, by means of Eq. (18.38), for each step of the load history. Therefore we have one point in stress space for each loading step; all the points characterized by the same E ~ form a loading path in stress space. These points are called interpolation points: there the behaviour of the homogenized material (and precisely the value of the homogenized strains Eij and s t r e s s e s Eij) is known. Repeating the procedure for several different given tensors E ~ we know the behaviour of the homogenized material at a discrete number of points and for a discrete variety of load situations. Interpolation points and typical loading paths are shown in Fig. 18.27. 55
~ ~ 2~i~ ~ ~ ~~~ ~~~m~i~o~ :i~i~ini!|i~i~ ~ : ~ ~ : ~ : : ~ ~~~Di:~i ~ ~
~ ~i~i~i~i!iiiiiiiiiiiii!iiii~":~:i iiiiii ii! i ii!iiiiii!iiiiiiiiiiiiiiiii!ii
Equation (18.38) defines macroscopic stresses and strains, which are assumed to be linked by a macroscopic constitutive law. Such a law is constructed starting from the constitutive relations of the single components and the geometry of the unit cell; this step is referred to as homogenization procedure. The macroscopic behaviour is assumed
Homogenization procedure- definition of successive yield surfaces 579 60
. . . . . . .
Piasti'c surfaces . . . .
40 20
-20 -40 -6080
' -~50 -40 '
-20
0
Zll
2'0
40
60
80
Fig. 18.27 Grid of interpolation points and plastic surfaces.
to be of elastic-plastic type, even if at microscopic level the behaviour may be of a different type. 54,56There the behaviour should, however, follow the complementary rule as it does in the present case. We have hence to determine the following macroscopic quantities: elastic frontier, flow rule, hardening law and extremal surface of the global homogenized materials. In other words our aim is to determine the global elasticplastic constitutive law '~ij "- ~aij (Eij) (18.43) Given the nature of the components, the microscopic stress field is constrained by the usual relation tr(y) e P (y) (18.44) where P (y) is the set of stress states that the material can admit in the space of stresses. P (y) depends on the single material and on the contact conditions, and hence on the position y in the RVE. P (y) can be defined by means of a yield function f (y, or) P (y) = {tr If (Y, or, c p) < 0 }
(18.45)
Since local stress states cr must lie within the set given in Eq. (18.44), it seems reasonable that all physical global states ~ are contained in a macroscopic (or effective) domain peff whose frontier is called global (or effective) extremal yield surface E peff
(18.46)
The global constitutive behaviour of the homogenized material usually has an initial range with a linear relation between global strains and global stresses followed by a non-linear range, see Fig. 18.27. The limit of the initial range, the successive yield surfaces for different values of m (loading steps) and the extremal yield surfaces can be constructed with the numerical experiments on local problems described above. They are all shown in Fig. 18.27 for the case of only Ell and Eee. An example of a three-dimensional surface for the case of local damage behaviour is presented in reference 56.
580
Multiscale modelling
From the results obtained from the solution of many local problems we extract now the information necessary for a global elastic-plastic analysis. The behaviour of the material is linear within the most interior closed curve, see Fig. 18.27; a flow rule and hardening law have to be defined for the homogenized material outside the elastic zone. We consider here the simplest case where in the space of global stresses each interpolation point is characterized by the values of three variables" the r a t i o s E ~ 1 7 6 E~176 2 and m (number of steps). The constitutive law at the macro level can be written as
E E peff= {y] i f ( E , E p) < 0}
(18.47)
The consistency condition of Eq. (18.47) can be expressed as
Of Of Of (~, E p) - - ~ dN -Jr - ~ dE p - 0
(18.48)
The gradient of the yield function with respect to stress Of (E)/OE, for a genetic point lying within interpolation points, can be obtained easily by an interpolation technique which is the same as that generally used in post-processing to obtain contour lines or surfaces, see Fig. 18.27. The plastic strain increment d E p can be obtained from the local analysis with the following additive decomposition and relationship e
Eij -- E~ + Eij ~ij -- OiEklEE
(18.49)
where E/~, the macroscopic purely elastic strain, is defined as that part of the macro strain that is non-zero if and only if the macro stress is non-zero, and DiEkl is the elastic constitutive tensor obtained with elastic homogenization. If we assume an associative plastic flow for the macro level constitutive model, once the vector Of (E)/OE has been obtained the plastic flow direction is known at all points in the stress space for the constitutive law at the macro level. However, to implement the algorithm systematically, a hardening modulus has to be determined in advance. This requires an iterative procedure due to the unknown final state for one incremental step. We assume here that the mean hardening modulus can be expressed as H =
Of(~,E P)
(18.50)
0E e
Hence, from Eq. (18.48), the following iterative procedure can be established H (i) --
I
1
AE e 0
( (1 -
r/)
Of(i)+ r/ Of(i-1)) (gE
(gE
AN (i)
' ,
when AE P --/: 0 when AE e - 0 (18.51)
where 0 < r / < 1 is an integration parameter such as that used in a mid-point algorithm.
Global solution and stress-recovery procedure
Err
l
E22
Path n+l
~
Path n
1+1
T_.,//~ /~ J
k i+l
Currentstress ~k state
"
Interpolation points
~11
v
Fig. 18.28 Interpolation of the flow direction. When a non-associative plastic flow is used, the flow rule can be extracted from the radial return mapping algorithm written in a standard form as R "-- - - ~ a tr .qt_ ~a(i) "7!-k D m -
0,
k > 0
(18.52)
where D is the elastic constitutive tensor obtained with the elastic homogenization, k is the increment of the plastic flow, m is the flow direction and ~a tr, ~"]~(i) are the global stresses. The flow direction (which refers to an unknown plastic potential) can be calculated, at each interpolation point, in the following manner, see Fig. 18.28. Starting from an interpolation point corresponding to step i one multiplies the known strain increment E by the elastic effective tensor A, obtaining the trial stress ~a tr . Since there is plastic deformation or slip in some part of the unit cell, the strain AE actually generates the new stress ~a (i+l) , different from y]tr. The flow direction at the interpolation point for the level i + 1 can be computed as
m:cA-lI~tr
.-~-~-]~(i)]:
c IAE-A-I~(i) 1
(~8.53)
The value of k has been arbitrarily chosen to be 1/c [see local problem (18.42)], hence the quantity m not only gives information about the flow direction but also about the amount of the plastic flow. For stress points inside a patch, the flow direction is obtained again by interpolation. Once a consistency condition and a flow rule have been determined, the global constitutive law is fully defined and can be assumed as constitutive law of the homogenized material.
igi At this point a global problem can be solved. Given a periodic structure subjected to assigned external loads f and to assigned boundary conditions, the global displacements
581
582 Multiscalemodelling can be found solving the problem macroscopic constitutive law div 12 - f; macro-equilibrium global boundary conditions
(18.54)
The solution of the global problem (18.54) gives a reasonably good estimate of the displacements of the structure. Nevertheless, very often stresses are the most relevant mechanical quantities but they cannot be easily derived from global displacements. The global solution is used to evaluate the local distribution of micro stresses by solving a local problem (18.42). This is the third step of the homogenization procedure. If the stress distribution in a specific region is required, we take into consideration the integration points used in the global solution which are close to the examined region of the real structure. If the number of unit cells is high a single integration point corresponds to a group of cells, but usually one RVE corresponds to one integration point. The strains computed in the global solution are assumed as global strains of the investigated cell (or group of cells); in general the finite element computation gives a sequence of n values of strains if the load history is composed of n load steps. Such a sequence of values is taken as load history on a unit cell with periodic boundary conditions and the following local problem is solved: microscopic constitutive laws div tr - 0; micro-equilibrium Eij - Eij(t) given by the global problem
(18.55)
The solution of such a local problem gives a stress distribution which usually is a good approximation of the real one, as shown in the next example.
Numerical examples
We consider an assembly of discs shown in Fig. 18.22, under assumption of plane stress behaviour. The discs have the following mechanical properties E=128000MPa,
v=0.34,
crr=80MPa
The elastic-plastic constitutive law of the discs is of the von Mises type. The overall property of the structure will clearly be controlled by the behaviour of the discs around their contact points and by the elastic-plastic material behaviour. This character is captured by the analysis of three cases of different interface parameters. (a) Elastic-plastic material with purely frictional contact # = 0.25,
PnO --" Pto --O,
~* ~= 0
(b) Elastic-plastic material with cohesive frictional contact: #=0.25,
1 PnO0, Pto -- ~ 8 O
MPa,
~*r
(c) Elastic-plastic material with cohesive frictional contact: #-
0.10,
1 PnO0, PrO -- ---~_80MPa,
~/3
~* r 0
Global solution and stress-recovery procedure 583 ~-'11
Nt
~,'~f'~$' .....................................
(o.o)
.i ~'~;~~7~;"~ '~"ii!
~:11
E r . I,,
I
'
=",~, ~9'~I ~M l}'
~! 9
1
~,,.~.~,.I,.,.. ~..
~
~
~ (0.0)
~.,...................................... ~.,-:-~7~>,--...----f-. ~. D,li:"
~................. ~...
i'/
-+'
,
Initial
/~ ....~ yio,d~udace
~,,:
.....
~:~~ ~ ~ ~ ,7--,22
IX = 0.25
IX = 0.25
Z:11
~ 1~,~< ~ " ~.
i
Sll
! (0.0)
~r
i
~
.:,.
~,~-~~
i (o.o) .:~r
Initial
~:-. :.
~ / / /. . . . . . .
.~
i~:
...... ~:
~~/i: [J]
...... ,m -..,,,~! .. ~ : ~ . . , : .
~~l-iii/ttII
"
.:
. ,
yield surface
...~
o:~,~~..,..,, ~
.:,:!~22
.......................................
................................................. ! `7..`22
~=0.10
~
0.00
Fig. 18.29 Local numerical constitutive law for elastic-plastic bodies in contact with different material parameters: (a) purely frictional contact; (b) cohesive-frictional contact with # = 0.25; (c) cohesive-frictional contact with # = O.10; (d) purely cohesive contact.
(d) Elastic-plastic material with purely cohesive contact: # = 0.00,
PnO0,
PrO --"
1
-----~80MPa,
~/3
6* ~ 0
Pn, Pt are the limits of the normal and tangential contact forces respectively, and 6* is the gap, see Fig. 18.23. The contact algorithm used in the example is that of Zhang et al. 57 The cell of periodicity, indicated in Fig. 18.22, is discretized with four-node plane stress elements. Our interest is focused on the macro stress domain
E 11, E22 I Ell _ 0 , E22 ~ 0 } and the obtained interpolation points (yield surfaces) corresponding to the above four cases are presented in Figs 18.29 (a)-(d) in that quadrant. For the elastic case see reference 55. To verify the capability of the procedure a group of 140 discs arranged in a rectangle (10 x 14 discs) is analysed under uniform compression. For this purpose a uniform vertical displacement is assigned to the top nodes. The vertical displacements of the
584 Multiscale modelling 80 70 60 Z v
tO .D O
(D I:1:
heterog. homog.o homog.n
50 40 30
/-
20
/
f
J
/
10
o/
0.0 0.7
1.4 2.1
2.8 3.5 4.2 4.9 5.6 6.3 7.0
Top displacement (mm)
(a) 70 60
~" 50 v 40 C
o
g
30
w 20 10
0.0 0.7
1.4 2.1
2.8 3.5 4.2 4.9 5.6 6.3 7.0
Top displacement (mm)
(b)
70 60
~" 50 v
jl
= 40 o
g
30
/
w 20
o/
/
0.0 0.7 1.4 2.1
J
heterog. homog.o homog.n
2.8 3.5 4.2 4.9 5.6 6.3 7.0
Top displacement (mm) (c) Fig. 18.30 Comparisonbetween the reactions in the case of uniform top displacement. Homog.n indicates the results with the self-consistent procedure of this chapter,25 homog.othe results of reference 58.
Global solution and stress-recovery procedure VALUE OPTION:ACTUAL 8.01E@r .... ~ . , ~ tl
...... ..... ............ ,, . . . . . . . . . 9
.
.........i({~:::~c"
~,{~i"{.~
"i!.!i~:'~3i~:!i,,:.-!:~c.~. . . . . . . . .
;.;;::7~, ~.:" .:.:,..,:,: :,.~
..~,.,< 9
>>.@iS
....... ... ...... . < ~ , ~ . . . , ~ . . . .
.c<.!:~:!": :::i.#!:~,!~.::::~: <,:77,
...... 7.22E
!7ii~',.!i'.
~.~.:::;7,,~+~
. ......
~' .....
6.44E
5.65E
4.86E
4.07E
3.28E
2.49E
1.70E
9.14E
1.26E
(a) VALUE O P T I O N : A C T U A L 8.00E+01 9
7.21E+01
[
l:.~:. ~.:
.... :~..
.,
.,~.
s
1
6.42E+01
5.63E+01
I7
4.85E+01
4.06E+01
!: i'~:~~!
3.27E+01
,~i~..,, 9 ~ ..~:-,:
2.49E+01
1.70E+01
9.13E+00
.26E+00
(b) Fig. 18.31 Comparison between the stress distributions obtained for elastic-plastic bodies in contact: (a) stresses recovered from the homogeneous computation; (b) stresses recovered from the heterogeneous computation.
585
586 Multiscale modelling bottom line and the horizontal displacement of the right and left edge of the rectangular assembly are restrained to be zero. Associative behaviour of the homogenized material is assumed. The problem is solved using a rough discretization (43 nodes and 35 elements, called homogeneous) with the numerical constitutive law which takes into account the non-linear property due to the stick-slip behaviour between the discs. The vertical reactions at the middle point of the bottom line are compared to those of the equivalent model with a finite element discretization (called heterogeneous) which describes the real material distribution and the real mechanical characteristics of the single components. The comparisons between the vertical reactions in the homogeneous and heterogeneous models are shown in Figs 18.30(a)-(c) for the three different cases of contact parameters b, c, and d. The results obtained in reference 58 with a similar procedure, which is, however, not self-consistent, are also shown in the figure. The results obtained by the proposed method are in good agreement with the heterogeneous results. The case of a non-uniform displacement distribution on the top of the assembly was also investigated in reference 25. We show here the results of the stress recovery procedure, i.e. we consider a unit cell from a local point of view and impose to it a kinematic load represented by the displacement field computed with the homogenized solution. In the specific case of this example the comer displacements of the RVE were directly obtained from the homogenized solution and the displacements imposed on the other boundary nodes were obtained with a linear interpolation between the comer nodes. The solution of this local problem gives a stress distribution in the unit cell which can be compared to the stress distribution obtained with the heterogeneous (discrete) model. The results for one unit cell are shown in Fig. 18.31 where an excellent agreement can be observed. In conclusion, the approach requires the preliminary solution of many local problems and proceeds then with a macroscopic numerical analysis of the overall structure. Solution of local problems for obtaining the local stress distribution is needed then only at the Gauss points of interest.
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Discrete-continuum linkage becomes particularly important for nanoscale mechanics and materials. Continuum-based approaches are not applicable down to the nanoscale as non-continuum behaviour is observed at that scale, such as for instance in large deformation of carbon nanotubes or ion deposition processes. 4 Further, nanoscale components are generally used in conjunction with components that are larger and have a mechanical response at different length and time scale. Single-scale methods such as molecular dynamics or quantum mechanics are generally not applicable in this last case due to the disparity of the scales and scale bridging is a necessity. This is a new and rapidly developing area and the interested reader is referred, e.g., to the already mentioned special issue devoted to that topic. 4 Finally, the contributions to this chapter by D.P. Boso, M. Lefik, S. Loehnert and V. Salomoni are gratefully acknowledged.
References 587
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1. R. Lakes. Materials with structural hierarchy. Nature, 361:511-515, 1993. 2. R.C. Picu. Foreword to special issue on linking discrete and continuum models. Int. J. Multiscale Computational Engng., 1(1):vii-viii, 2003. 3. P. Ladev6ze and J. Fish. Preface to special issue on multiscale computational mechanics for materials and structure. Computer Methods in Applied Mechanics and Engineering, 192:28-30, 2003. 4. W.K. Liu, D. Qian and M.E Horstemeyer. Preface to special issue on multiple scale methods for nanoscale mechanics and materials. Computer Methods in Applied Mechanics and Engineering, 193:17-20, 2004. 5. W. Voigt. Lehrbuch der Kristallphysik. Teubner, Leipzig-Berlin, 1910. 6. A. Reuss. Berechnung der Fliessgrenze von Mischkristallen auf Grund der Plastizitatsdingung fiir Einkristalle. Z. Angew. Math. Mech., 9:49, 1926. 7. T. Moil and K. Tanaka. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallurgica, 21:571-574, 1973. 8. J.D. Eshelby. The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc. Roy. Soc.,A241:376-396, 1957. 9. Z. Hashin and S. Shtrikman. A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. Phys. Sol., 11(2):127-141, 1964. 10. E. Kr6ner. Bounds for effective elastic moduli of disordered materials. J. Mech. Phys. Sol., 25(2):137-155, 1977. 11. E. Kr6ner. Self-consistent scheme and graded disorder in polycristal elasticity. J. Phys., 8:22612267, 1978. 12. J.R. Willis. Bounds and self-consistent estimates for the overall properties of anisotropic composites. J. Mech. Phys. Sol., 25:185-202, 1977. 13. A. Bensoussan, J.L. Lions and G. Papanicolau. Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam, 1976. 14. E. Sanchez-Palencia. Non-Homogeneous Media and Vibration Theory. Springer Verlag, Berlin, 1980. 15. G.A. Francfort. Homogenisation and fast oscillations in linear thermoelasticity. In R. Lewis et al., editors, Numerical Methods for Transient and Coupled Problems, pages 382-392. Pineridge Press, Swansea, 1984. 16. M. Lefik and B.A. Schrefler. 3d finite element analysis of composite beams with parallel fibres based on the homogenisation theory. Computational Mechanics, 14:2-15, 1994. 17. M. Lefik and B.A. Schrefler. Application of the homogenisation method to the analysis of superconducting coils. Fusion Engineering and Design, 24:231-255, 1994. 18. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann, Oxford, 6th edition, 2005. 19. B.A. Schrefler, M. Lefik and U. Galvanetto. Correctors in a beam model for unidirectional composites. Mechanics of Composite Materials and Structures, 4:159-190, 1997. 20. P. Ladev6ze, editor. Local Effects in the Analysis of Structures, page 342. Elsevier, New York, 1985. 21. M. Lefik and B.A. Schrefler. Fe modelling of boundary layer correctors for composites using the homogenisation theory. Engineering Computations, 13:31-42, 1996. 22. K. Terada and N. Kikuchi. A class of general algorithms for multi-scale analyses of heterogeneous media. Computer Methods in Applied Mechanics and Engineering, 190:5427-5464, 2001. 23. J. Fish and Q. Yu. Multiscale damage modeling for composite materials: theory and computational framework. International Journal for Numerical Methods in Engineering, 52:161-192, 2001.
588
Multiscale modelling 24. M. Kaminski and B.A. Schrefler. Probabilistic effective characteristics of cables for superconducting coils. Computer Methods in Applied Mechanics and Engineering, 188:1-16, 2000. 25. H.W. Zhang, D.P. Boso and B.A. Schrefler. Homogeneous analysis of periodic assemblies of elastoplastic disks in contact. Int. J. of Multiscale Computational Engineering, 1:359-380, 2003. 26. Thermal, electrical and mechanical properties of materials at cryogenic temperatures. Conductor Database, Appendix C, Annex 11, 24 August 2000. 27. J. Ekin. Mechanical properties and strain effects in superconductors. In S. Foner and B. Schwartz, editors, Superconductor Materials Science: Metallurgy, Fabrication and Applications. Plenum Press, NATO Advanced Study Institute Series, 1980. 28. R.P. Reed and A.E Clark, editors. Materials at Low Temperature. American Society for Metals, Metals Park, Ohio, 1983. 29. S. Ochiai and K. Osamura. Prediction of variation of upper critical magnetic field of nb3sn superconducting composites as a function of applied stress at room temperature. Acta Metall., 37(9):2539-2549, 1989. 30. G. Rupp. The importance of being prestressed (nb3sn composite superconductor). In M. Suenaga and A.E Clark, editors, Filamentary A15 Superconductors, pages 155-170. Plenum Press, New York, 1980. 31. N. Mitchell. Analysis of the effect ofnb3sn strand bending on CICC superconductor performance. Cryogenics, 42:311-325, 2002. 32. E Feyel. Multiscale f e 2 elastoviscoplastic analysis of composite structures. Comp. Mat. Sci., 16:344-354, 1999. 33. E Feyel and J.L. Chaboche. f e 2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials. Computer Methods in Applied Mechanics and Engineering, 183:309-330, 2000. 34. E Feyel. Multiscale non-linear f e 2 analysis of composite structures: Fiber size effects. J. Phys. IV, France, 11:195-202, 2001. 35. E Feyel and J.L. Chaboche. Multi-scale non-linear f e 2 analysis of composite structures: damage and fiber size effects. In Khbmais Saanouni, editor, Revue Europ~enne des Elgments Finis, 'Numerical Modelling in Damage Mechanics '- NUMEDAM'O0, volume 10, pages 449472, 2001. 36. J.F. Bishop and R. Hill. A theory of the plastic distorsion of polycrystalline aggregate under combined stresses. Philos. Mag., 42:414--427, 1951. 37. P. Ladevrze, O. Loiseau and D. Dureisseix. A micro-macro and parallel computational strategy for highly heterogeneous structures. International Journal for Numerical Methods in Engineering, 52:121-138, 2001. 38. P. Ladevrze, A. Nouy and O. Loiseau. A multiscale computational approach for contact problems. Computer Methods in Applied Mechanics and Engineering, 191:4869-4891, 2002. 39. P. Ladevrze and A. Nouy. On a multiscale computational strategy with time and space homogenisation for structural mechanics. Computer Methods in Applied Mechanics and Engineering, 192:3061-3087, 2003. 40. M.G.D. Geers, V. Kouznetsova and W.A.M. Brekelmans. Gradient-enhanced computational homogenisation for the micro-macro scale transition. Journal de Physique IV, 11 (5):5145-5152, 2001. 41. S. Ghosh, K. Lee and S. Moorthy. Two scales analysis of heterogeneous elastic-plastic materials with asymptotic homogenisation and voronoi cell finite element model. Computer Methods in Applied Mechanics and Engineering, 132:63-116, 1996. 42. S. Ghosh, K. Lee and P. Raghavan. A multilevel computational model for multiscale damage analysis in composite and porous materials. International Journal of Solids and Structures, 38:2335-2385, 2001. 43. C. Miehe, J. Schroder and J. Schotte. Computational homogenisation analysis in finite plasticity. Simulation of texture development in polycrystalline materials. Computer Methods in Applied Mechanics and Engineering, 171:387-418, 1999.
References 589 44. C. Miehe, J. Schotte and J. Schroder. Computational micro-macro transitions and overall moduli in the analysis of polycrystals at large strains. Computational Materials Science, 16:372-382, 1999. 45. V. Kouznetsova, W.A.M. Brekelmans and EP.T. Baaijens. An approach to micro-macro modelling of heterogeneous materials. Computational Mechanics, 27:37-48, 2001. 46. P.M. Suquet. Plasticity today: modelling, methods and applications. In Local and GlobaIAspects in the Mathematical Theory of Plasticity, pages 279-310. Elsevier Applied Science Publishers, London, 1985. 47. S. Loehnert and P. Wriggers. Homogenisation of microheterogeneous materials considering interfacial delamination at finite strains. Technische Mechanik, 23(2-3): 167-177, 2003. 48. T.I. Zohdi and P. Wriggers. Computational micro-macro material testing. Archives of Computational Methods in Engineering, 8(2): 131-228, 2001. 49. A. Needleman. A continuum model for void nucleation by inclusion debonding. J. Applied Mechanics, ASME, 54:525-531, 1987. 50. C. Miehe and J. Dettmar. A framework for micro-macro transitions in periodic particle aggregates of granular materials. Computer Methods in Applied Mechanics and Engineering, 193:225-256, 2004. 51. A. Nardin, G. Zavarise and B.A. Schrefler. Modelling of cutting tool-soil interaction- Part I: Contact behaviour. Computational Mechanics, 31:327-339, 2003. 52. J.R. Wren and R.I. Borja. Micromechanics of granular media. Part I: Generation of overall constitutive equation for assemblies of circular disks. Computer Methods in Applied Mechanics and Engineering, 127:13-36, 1995. 53. J.R. Wren and R.I. Borja. Micromechanics of granular media Part II: Overall tangential moduli and localization model for periodic assemblies of circular disks. Computer Methods in Applied Mechanics and Engineering, 141:221-246, 1995. 54. C. Pellegrino, U. Galvanetto and B.A. Schrefler. Numerical homogenisation of periodic composite materials with non-linear material components. International Journal for Numerical Methods in Engineering, 46:1609-1637, 1999. 55. H.W. Zhang, U. Galvanetto and B.A. Schrefler. Local analysis and global non-linear behaviour of periodic assemblies of bodies in elastic contact. Computational Mechanics, 24:217-229, 1999. 56. D.P. Boso, C. Pellegrino, U. Galvanetto and B.A. Schrefler. Macroscopic damage in periodic composite materials. Comm. Numer. Meth. Engng, 16(9):615-623, 2000. 57. H.W. Zhang, W.X. Zhong and Y.X. Gu. A combined programming and iteration algorithm for finite element analysis of three-dimensional contact problems. Acta Mech. Sinica, 11:318-326, 1995. 58. H.W. Zhang and B.A. Schrefler. Global constitutive behaviour of periodic assemblies of inelastic bodies in contact. Mechanics of Composite Materials and Structures, 7:355-382, 2000. 59. D.P. Boso, M. Lefik and B.A. Schrefler. A multilevel homogenised model for superconducting strands thermomechanics. Cryogenics, 45:259-271, 2005.
i~i!i~i:i~i~i!i~i!i~i~i!ii!i!i!~i!~i!i~i~!~i~i:~i~!i!i!~!i!ii!~!~i~i~i
Computer procedures for finite element analysis iiii i Hii~i~ii~!~i~i~i~i~i~i~!ii~i~iiiii~i~i~iiiiiii~iiiiii~!~i~ii~i!iiiiii~i~iii!i~i~i~i~i~i~iiiiiiii~i~ii~i~ii~i~!~iii~i~iiiii~i!~i~i~i~i~iii~i~iii~iii~i~i~i~i~i~i~iTi!iiiiiii~ii~i~i~i~iiiiiiiii~i~i~i~ii~iii~iii~i~ii~i~!~ii~i~ d ii i iiiii iiiiiiiii iiHiHiiiii ii i iii iiiiiii i iiii ii In this chapter we describe some features of the companion computer program which may be used to perform numerical studies for many of the topics discussed in this book. The source program and manuals are available at no cost from the publisher's web page: http ://books.elsevier. com/c ompanion s or from the authors' web page: http://www.ce.berkeley.edu/-~rlt 9The computer program described in this volume is intended for use by those who are undertaking a study of the finite element method and wish to implement and test specific elements or specific solution steps. The program also includes a library of simple elements to permit solution to many of the topics discussed in this book. The program is called FEAPpv to emphasize the fact that it may be used as apersonal version system. With very few exceptions, the program is written using standard Fortran, hence it may be implemented on any personal computer, engineering workstation, or main frame computer which has access to a Fortran 77 or Fortran 90/95 compiler. It still may be necessary to modify some routines to avoid system-dependent difficulties. Non-standard routines are restricted to the graphical interfaces and file handling for temporary data storage. Users should consult their compiler manuals on alternative options when such problems arise. Users may also wish to add new features to the program. In order to accommodate a wide range of changes several options exist for users to write new modules without difficulty. There are options to add new mesh input routines through addition of routines named UNESHn or to include solution options through additions of routines named UMACRn. Finally, the addition of a user developed element module is accommodated by adding a single subprogram named ELMTnn. In adding new options the use of established algorithms as described in references 1-5 can be very helpful. The current chapter presents a brief discussion to describe aspects of the program that are related to solution of non-linear problems. The program FEAPpv includes capabilities to solve general non-linear finite element models for transient and steadystate (static) problems. The transient problem types include solution algorithms for
Solution of non-linear problems 591 both first-order (diffusion-type) and second-order (vibration/wave-type) ordinary differential equations in time. In addition an eigensolution system is included to compute eigenpairs of typical problems. A simultaneous vector iteration algorithm (subspace method) is used to extract the eigenpairs nearest to a specified shift of a symmetric tangent matrix. Hence, the eigensystem may be used with either linear or non-linear problems. Non-linear problems are often difficult to solve and time consuming in computer resources. In many applications the complete analysis may not be performed during one execution of the program; hence, techniques to stop the program at key points in the analysis for a later restart to continue the solution are available. The program described in this chapter has been developed and used in an educational and research environment over a period of nearly 35 years. The concept of the command language solution algorithm has permitted several studies that cover problems that differ widely in scope and concept, to be undertaken at the same time without need for different program systems. Unique features for each study may be provided as new solution commands. The ability to treat problems whose coefficient matrix may be either symmetric or unsymmetric often proves useful for testing the performance of algorithms that advocate substitution of a symmetric tangent matrix in place of an unsymmetric matrix resulting from a consistent linearization process. The element interface is quite straightforward and, once understood, permits users to test rapidly new types of finite elements. We believe that the program in this book provides a very powerful solution system to assist the interested reader in performing finite element analyses. The program FEAPpv is by no means a complete software system that can be used to solve any finite element problem, and readers are encouraged to modify the program in any way necessary to solve their particular problem. While the program has been tested on several sample problems, it is likely that errors and mistakes still exist within the program modules. The authors need to be informed about these errors so that the available system can be continuously updated. We also welcome readers' comments and suggestions conceming possible future improvements.
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The general methods described in this volume are directed toward the solution of nonlinear problems in solid and structural mechanics. The application of the finite element method to these problems leads to a set of non-linear algebraic equations. A solution to the non-linear algebraic problem by a Newton method, as described in Chapter 3, is given by 6
~,(u (~)) = p ( u (k)) + f(~) K(Tk)du (k) -- ~'(~) where K(T~) = U (k+l)
~_~ U (k) +
O~I, Ou i(k)
(19.1)
d u (k)
where f is a vector of applied loads and P is the non-linear internal force vector which is indicated as a function of the nodal parameters u. The vector 9 is the residual of
592
Computerprocedures for finite element analysis the problem, KT is the tangent matrix and a solution is defined as any set of nodal displacements, u, for which the residual is zero. In general, there may be more than one set of displacements which define a solution and it is the responsibility of a user to ensure that a proper solution is obtained. This may be achieved by starting from a state which satisfies physical arguments for a solution and then applying small increments to the loading vector, f. By taking small enough steps, a solution path may usually be traced. FEAPpv uses the basic Newton strategy defined in Chapter 3 to perform all solution steps. A key feature of the program is a command language which permits users to construct many different linear and non-linear solution algorithms as data statements of the analysis process. For example, a simple set of commands given by L O O P n e w t o n i0 TANGent FORM residual SOLVe NEXT newton
performs the necessary computations for a Newton algorithm. In the above LOOPNEXT defines the necessary commands to form 10 iterations of the Newton scheme. The command TANGent constructs the tangent matrix, FORM constructs the residual and S O L V e performs a solution of the linear equations. In FEAPpv only the first four characters of command words are processed and the remainder can be used to provide additional clarity, thus for emphasis the required data is shown above in upper case letters. The user manual describes the available commands which may be used to construct general linear and non-linear steady state or transient algorithms. Non-linear problems require use of many schemes to improve the convergence of the solution. Use of the BFGS algorithm described in Chapter 3 can lead to improved solution performance and/or reduced solution cost (see the necking example in Chapter 6). It is also particularly effective when no exact tangent matrix can be computed. In addition a linear line search is useful to limit the magnitude of du (k) during early iterations 7 and its use often allows use of larger load increments and still obtain rapid converge. A line search requires repeated computations of ~ ( u ) which may increase solution times. Thus, some assessment of the need of a line search should be made before proceeding with large numbers of solution steps. A modified Newton method also may be performed by removing the tangent computation from the loop structure given above (i.e. placing T A N G before LOOP). This points out the power of the command language scheme to efficiently include many solution algorithms. The solution of transient problems defined by the algorithms given in Chapter 2 may also be performed using FEAPpv. The program includes options to solve transient finite element problems which generate first- and second-order ordinary differential equations using the GN 11 and Newmark (GN22) algorithms. 8 Options also exist to use an explicit version of the GN22 algorithm.
The solution of a general linear eigenproblem is a useful feature included in the FEAPpv program. The program can compute a set of the smallest eigenvalues (in absolute value)
Eigensolutions 593 and their associated eigenvectors for the problem KTV -- M V A
(19.2)
In the above, KT is any symmetric tangent matrix which has been computed by using a TANG command statement; M is a mass or identity matrix computed using a r/ASS or IDEH command statement, respectively; the columns of V are the set of eigenvectors to be computed; and A is a diagonal matrix which contains the set of eigenvalues to be computed, For the second-order equations from solid or structural mechanics problems the eigenvalues A are the frequencies squared, w2. The tangent matrix can have zero eigenvalues and, for this case, the algorithm used requires the problem to be transformed to (KT -- a M ) V = MVA,~
(19.3a)
where c~ is a parameter called the shift, which can be selected to make the coefficient matrix on the left-hand side of Eq. (19.3a) nonsingular. A~ are the eigenvalues of the shift which are related to the desired values by A = A~ + a l
(19.3b)
The shift may also be used to compute the eigenpairs nearest to some specified value (e.g., a buckling load), The components of A are output as part of the eigenproblem solution. In addition, the vectors may be output as numerical values or presented graphically. The program uses a subspace algorithm T M to compute a small general eigenproblem defined as K*x = M*xA (19.4a) where V = Qx
(19.4b)
and K* -" QTMT (KT -- o~M)-IMQ M* = Qa'MQ
(19.4c)
Accordingly, after the projection, the A are reciprocals of A~ (i.e. AS l). An eigensolution of the small problem may be used to generate a sequence of iterates for Q which converge to the solution for the original problem (e.g. see reference 10). The solution of the projected small general problem is solved here using a transformation to a standard linear eigenproblem combined with a QL algorithm. 6 The transformation is performed by computing the Choleski factors of M* to define the standard linear eigenproblem Hy = yA (19.5a) where M* = LL T y -'LTx H = L-1K*L -T
(19.5b)
594
Computerprocedures for finite element analysis In the implementation described here scaling is introduced, which causes M* to converge to an identity matrix; hence the above transformation is numerically stable. Furthermore, use of a standard eigenproblem solution permits calculation of positive and negative eigenvalues. The subspace algorithm implemented provides a means to compute a few eigenpairs for problems with many degrees of freedom or all of the eigenpairs of small problems. A subspace algorithm is based upon a power method to compute the dominant eigenvalues. Thus, the effectiveness of the solution strategy depends on the ratio of the absolute value of the largest eigenvalue sought in the subspace to that of the first eigenvalue not contained in the subspace. This ratio may be reduced by adding additional vectors to the subspace. That is, if p pairs are sought, the subspace is taken as q vectors so that
/~p Aq+l
< 1
(19.6)
Of course, the magnitude of this ratio is unknown before the problem is solved and some analysis is necessary to estimate its value. The program tracks the magnitude of the shifted reciprocal eigenvalues A and computes the change in values between successive iterations. If the subspace is too small, convergence will be extremely slow owing to Eq. (19.6) having a ratio near unity. It may be desirable to increase the subspace size to speed the convergence. In some problems, characteristics of the eigenvalue magnitudes may be available to assist in the process. It should be especially noted that when p is specified as the total number of degrees of freedom in the problem (or q becomes this value), then Aq+~ is infinitely larger and the ratio given in Eq. (19.6) is zero. In this case subspace iteration converges in a single iteration, a fact which is noted by the program to limit the iterations to 1. Accordingly, it is usually more efficient to compute all the eigenpairs if q is very near the number of degrees of freedom.
The program FEAPpv permits a user to save a solution state and subsequently use it later to continue the analysis. This is called a restart option. To use the restart feature, the file names given at initiation of the program must be appropriately specified. The file name for the set of problem restart data files is specified at the time execution of FEAPpv is initiated. During a solution a restart file may be saved by using the command statement SAVE
<extender>
This saves the current solution data in a file that has the restart file name with an extension e x t e n d e r . For example, if the restart write file has the name 'Rprob', issuing the command SAVE
tiO
saves the data on a file named P p r o b . t i O. Alternatively, issuing the command as SAVE
References 595 saves the data on the file named P p r o b . For large problems the restart file can be quite large (especially if the elements use several history variables at each integration point) thus one should be cautious about use of too many files in these situations. To restore a file the command RESTart
is given to load the file without an extender, and the command RESTart
<extender>
to load the file with an extender.
In the discussion above we have presented a few of the ways the program FEAPpv may be used to solve non-linear finite element problems. The classes of non-linear problems which may be solved using this system is extensive and we cannot give a comprehensive summary here. The reader is encouraged to obtain a copy of the program source statements and companion documents from the publisher's website (http ://books.e lsevi er. com/companions). As noted in the introduction to this chapter the computer programs will undoubtedly contain some errors. We welcome being informed of these as well as comments and suggestions on how the programs may be improved. Although the programs available are written in Fortran it is quite easy to adapt these to permit program modules to be constructed in other languages. For example, an interface for element routines written in C has been developed by Govindjee. 12 The program system FEAPpv contains only basic commands to generate structured meshes as blocks of elements. For problems where graded meshes are needed (e.g. adaptive mesh refinements) more sophisticated mesh generation techniques are needed. There are many locations where generators may be obtained and two are given in references 13 and 14. The program GiD offers two- and three-dimensional options for fluid and structure applications.
!ii ii i !! ii!!
ii ! i i i!
ill
iii!i!i
i!
i
i
!
~ ~~~
~~~ ~~ ~~
~i~i~i~,~,~
1. W.H. Press et al., editors. Numerical Recipes in Fortran: The Art of Scientific Computing. Cambridge University Press, Cambridge, 2nd edition, 1992. 2. W.H. Press et al., editors. Numerical Recipes in Fortran 77 and 90: The Art of Scientific and Parallel Computing (Software). Cambridge University Press, Cambridge, 1997. 3. W.H. Press et al., editors. Numerical Recipes in Fortran 90: The Art of Parallel Scientific Computing, volume 2. Cambridge University Press, Cambridge, 1996. 4. G.H. Golub and C.F. Van Loan. Matrix Computations. The Johns Hopkins University Press, Baltimore MD, 3rd edition, 1996. 5. J. Demmel. Applied Numerical Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1997. 6. L. Collatz. The Numerical Treatment of Differential Equations. Springer, Berlin, 1966.
596
Computerprocedures for finite element analysis 7. H. Matthies and G. Strang. The solution of nonlinear finite element equations. International Journal for Numerical Methods in Engineering, 14:1613-1626, 1979. 8. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann, Oxford, 6th edition, 2005. 9. J.H. Wilkinson and C. Reinsch. Linear Algebra. Handbook for Automatic Computation, volume II. Springer-Verlag, Berlin, 1971. 10. K.-J. Bathe and E.L. Wilson. Numerical Methods in Finite Element Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1976. 11. K.-J. Bathe. Finite Element Procedures. Prentice-Hall, Englewood Cliffs, NJ, 1996. 12. S. Govindjee. Interface for c-language routines for feap programs. Private communication (see also at internet address: http://www.ce.berkeley.edu/~sanjay), 2000. 13. J. Shewchuk. Triangle. http://www.cs.berkeley.edu/~jrs). 14. GiD - The Personal Pre/Postprocesor. www.gidhome.com, 2004.
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T/I
Isoparametric finite element approximations ii~~~i~i~i~iiii~i~i~ii~ii~i~i~!i~!!!i!i~i~i~iiii!~i~ii~!!i~iii~!~!iii~!iiii!ii~i~iiiii!i!!~ii~i!~i~ii~iiiiii!iiiii~iiiii!i!ii~iiii!iiiiiiii~i~i~i!i~i!iiiiiiiiiiiii~iii!i!ii!!!ii!ii!iiiiii~i~ii!iiiiiiiii!iiiiiiii~iii~i~ii
An isoparametric formulation may be used for any problem in which the approximations are C Ocontinuous. In an isoparametric formulation a parent element is defined in terms of a set of natural coordinates. The shape functions are constructed on a parent element and used to compute the coordinates within each element using X "-- N a ( ~ ) X
(m.1)
a
where Na denotes the shape function, ~ are a set of natural coordinates and Xa are nodal coordinates. A dependent variable u is then approximated as U "~, U h =
ga(~)[l
(A.2)
a
The construction of shape functions requires the selection of an appropriate set of natural coordinates. Here we first summarize the form for quadrilateral and brick elements in two and three dimensions, respectively. We then consider triangular and tetrahedral elements. iiii',iii',i'~,ii~i~ iiiiiii i~i i~i~i:J~::~:~iiiii~ii::~Q:~:~i,'i:~,'ii~iiili~iiiii~:ii'ii'~,ii',:~iiiiii' ii'ii,~i',i~"'i,i:~':,ii~i'i~:~i~:ii~i ~i,i:!~i!':~~i':iii:, i'',~,iiliii'~iii~i~i~iiii~,i~iii:i~iii!ii!iiiiii! !~ililiiiiiiiiii"~'~i~!iiiiiii~iiiiiiiiiiiiiiiiiiiiii!iiiiiiiiiiiiiiiiiiiiiiiii' i!i~"":~:ii! ,~iiiiiiiii' ::~iiiiiiii!!iiii ,ili:~ i!i':iiiiiii!iii:,i:,ii:.ii iiiiiiiiiiliiiiiiiiiiliiiiiiiiiii!iiiiiiiiiiii!iii' iii':i!,iiii~:i:,!iiii!iiiiiiiiiiii',ii!iiiii!ii i!iiiiiiiiii iiiiiiii
ii
The natural coordinates for a quadrilateral element are given by =(~,~/);
-l~,r/~
1
as shown in Fig. A. 1. The simplest group of elements construct the shape functions from products of one dimensional Lagrangian interpolation functions given by 1~(~)-
(~-~o)(~-~1)""" (~a--~O)(~a--~l)''"
-II bCa
(~-~a-1)(~-~a+l)""" (~a--~a--1)(~a--~a+l)
(~--~n) " " " (~a--~n)
a = 1, 2, ..- , n - 1 '
598 Appendix A
1
Fig. A.1 Natural coordinatesfor a quadrilateral.
>!<
and la'(~)
=-
(~ - ~ 1 ) " " (~ - ~n) (~a - ~l)"" (G - ~n)'
In(~)
--
(~ -- ~ o ) " " (r -- G - l ) (~a -- ~ 0 ) ' ' " (~a -- ~n-1) '
a -- 0 a
=
n
which gives a unit value at ~a and passes through zero at n points specified by the points ~k, k -- 0, 2, .. 9, n. Using this form of interpolation we can construct the Lagrangian family of elements expressed as products of the one-dimensional functions given by
Na (~, ~7) -- lm (~) l: 07)
(A.3)
where m and n may be different orders in the natural coordinate directions. The simplest element uses linear interpolation in which ~0, % = - 1 and ~l, ~71 = 1 giving the four functions NI=
(~C_ l) ( r / _ l) (-2) (-2)
_L(l_~)(l_r/) -
4
N2 = (~ if- 1 ) ( T ] - 1) (2) (--2)
_-- i ( 1 i f - ~ ) ( 1 - r]) 4
N 3 - - (~ q- 1)(r/q--I) (2) (2)
_ i(1+~)(lq_77) -- 4
N4-
( ( - 1 ) ( r / + 1) (-2) (2)
(A.4)
_ i ( 1 _ ~)(1 4-n) --
4
where nodes are numbered as shown in Fig. A.2(a). We shall also often use elements constructed from quadratic functions in which ~0, % = - 1 , ~l, rh = 0, and ~2, r/e = 1 giving a nine-node element. For an element with nodes numbered as shown in Fig. A.2(b) the shape functions are given by
Quadrilateral elements q
4
3
(
4
(
)
7
)
1
(
2
0 5 (b)
1
(a)
9
()
3
)6 > ~
)
2
Fig. A.2 Node numberingfor four-node and nine-nodequadrilateralelements. N ~ - ~1 ( ~ -
1)r/(r/- 1)
N2--
1)r/(r/- 1)
88 1
N3 = ~ ~(~ + 1 ) r / 0 / + 1) 1
N4-
llr/07
~(~-
+ 11
1
N5 = ~ (1 - ~2)T](T] -- 1)
(A.5)
1
N6 = ~ ( ~ -I- 1)(1 - r/z) N7 -- $1 (1 - {2)~7(r/ + 11
N8 = ~1 ( ~ -
1)(1 - 7/2)
N9 = ( 1 - - ~21(1-- U2)
In problems using C O shape functions it is necessary to construct the derivatives of variables with respect to the global coordinates. For an isoparametric formulation these derivatives are computed using the chain rule given by
Ou
Ou Oxi ~Xi ~ On ~Xi ~Xi
Ou m
which may be written in matrix form as 0u o~
=
&l
Oxl
OX2
Ou
OXl
Ox2
Ou
O,7
oo
&l
599
600 Appendix A where J denotes the Jacobian matrix. The solution is given by Ou _ j_l Ou
--
~-~
(A.6)
and for the two-dimensional problem requires the inverse of a 2 • 2 Jacobian matrix.
Brick elements are generalization to three dimensions of the quadrilateral used in two dimensions. The global coordinates xi, i = 1, 2, 3 now require three natural coordinates which we give as (see Fig. A.3)
r
~, 0
The shape functions for the Lagrange family of elements are now given by
Na (~, rl, 0 = la (~) l~a(rl) lea(0
(A.7)
with the Jacobian transformation given by the 3 x 3 matrix
"Ox~ Ox= Ox3" J .m
Oxl Oxz Ox3
oo
(A.8)
oo
Ox~ Ox2 Ox3 _o;" o~ o~. 5
7
/ ..................!
IIW
r
//
/
//
11 "-~ f
k-
4
////~ / .
2
.
w
V
3
Fig. A.3 Natural coordinates and node order for an eight-node brick element.
Triangular elements 60 1
(A.10)
Higher-order elements may be constructedsimilarto that given for the two-dimensional form above.
For two-dimensional problems in which the elements are of triangular shape the natural coordinates are taken as the area coordinates, L a , a = 1 , 2 , 3 as shown in Fig. A.4(a), in which the constraint L1+ L2 L3 = 1 is used to limit the number of independent values to the same number as in the global XI,x2 Cartesian coordinates. The shape functions for the three-node triangular element are given by
+
Na = L a , u = 1 , 2 , 3
Derivatives with respect to xi for shape functions expressed in area coordinates may be computed using the chain rule aNa aNa aLb -=-axi aLb axi The derivatives aLb/axi may be computed by writing three equations
rl = x1 -
C N , Z ~ ,= o a
r2 = x2
Na.jtza = 0
a
r3 = 1 -
1L, = o a
602 AppendixA
I I
L2
,,2
,
,"
," /
O~
L1 -.
L3
2
i
",,,,...Xl
,... X l f
f
(b) Six-node triangle
(a) Three-node triangle and area coordinates
Fig. A.4 Natural coordinates and node order for triangular elements.
and using the chain rule
Orj
Orj OL b
Oxi
OLb Oxi
This gives 1
0
Ea ONa ~ ONa ~ OL----~Xla E a - ~ 2 X l a
~--
ONa YC2a Ea ONo ~ Ea O01Na j Ea-Z~---. OL 1 ~ x2a 3 x2a 1
ONa ~ "] EaoL3Xla I
1
"OL1 OL1 OXl OX2 01.,2 01-,2 OX1
OX2
OL3 01_,3 .Ox~ Ox2
which may be solved for the derivatives OZa/~X j . Higher-order triangles may also be constructed. We leave it to the reader to show that the shape functions for the six-node triangle shown in Fig. A.4(b) may be written as
No -- ~La(2LaNa+3 = 4 LaLb
1)
,
a=1,2,3;
b=2,3,1
Tetrahedral elements may be constructed in a manner similar to that used for triangular elements. In the case of tetrahedral elements we use volume coordinates La, a = 1, 2, 3, 4 with a constraint L1 + L2 + L3 -I- L4 = 1 The shape functions for a four-node tetrahedron are given by
Na - La;
a --
1,2,3,4
Tetrahedral elements 603
Derivatives are computed using the chain rule in the same way as for triangles, the difference being that arrays are of size 4 and 3 instead of 3 and 2. We leave this as an exercise for the reader to write appropriate relations for these steps.
Invariants of second-order tensors
Given any second-order Cartesian tensor a with components expressed as
a-
I
a11 a12 1a21 a22 /a31 a32
a13] a231 a33/
(B.1)
the p r i n c i p a l v a l u e s of a, denoted as al, a2, and a3, may be computed from the solution of the eigenproblem a q(m) _ a m q ( m ) (B.2) in which the (fight) eigenvectors q(m) denote p r i n c i p a l d i r e c t i o n s for the associated eigenvalue a m . Non-trivial solutions of Eq. (B.2) require (all a) a21 a31 m
det
a12 (a22 - a) a32
a13 a23 (a33 - a)
-
0
(B.3)
Expanding the determinant results in the cubic equation 3 -am
Ia a m2 --k I I a a m - Ilia -- 0
(B.4)
where:
Ia
-
all -~- a22 -q- a33
IIa
--
alia22 + a22a33 -+- a33a11 - a12a21 - a23a32 - a31a13
Ilia
m
a11a22a33 -
=
det a
a11a23a32 -
a22a31a13 -
(B.5)
a33a12a21 9 a12a23a31 9 a21a32a13
The quantities Ia, IIa, and IIIa are called the p r i n c i p a l i n v a r i a n t s o f a. The roots of Eq. (B.4) give the principal values am. The invariants for the deviator of a may be obtained by using a' -
a -
~1
(B.6)
where fi is the mean defined as -- ~1 (a 11 q- a22 q--a33) -- 3Ia
(B.7)
Moment invariants
Substitution of Eq. (B.6) into Eq. (B.2) gives
[a' 4- aI] q(m) =
amq(m)
(B.8)
fi)q(m) __ a~mq(m)
(B.9)
or
a'q (m) :
(am --
which yields a cubic equation for principal values of the deviator given as (a~) 3 4-
IiI'a - 0
II'aa'm -
(B.10)
where invariants of a' are denoted as I'a, II'a, and III'a . Since the deviators a' differ from the total a by a mean term only, we observe from Eq. (B.9) that the directions of their principal values coincide, and the three principal values are related through ! ai - - a i 4- ct; i = 1, 2, 3 (B.11) Moreover Eq. (B. 10) generally has a closed-form solution which may be constructed by using the Cardon formula, l'e The definition of a' given by Eq. (B.6) yields I a' - -
all' 4- a 2'2 4- a33' : 0
(B.12)
Using this result, the second invariant of the deviator may be shown to have the indicial fOrlTl 3 1
II'a = - - . ~ a i j a j i
t
(B.13)
t
The third invariant is again given by III'a - det a'
(B. 14)
however, we show in Sec. B.2 that this invariant may be written in a form which is easier to use in many applications (e.g. yield functions for elasto-plastic materials). iiiiiiiii!iiiiiiiiiil i~i~2i~i~i~i~i!i~ii!i~i~i i~iii~M iiii~Qi~i~mi~i!ie~iiNi~i~itii~i~i!ii!ii!i~!i!~i !i!iii!iiiii ii!i!iiiiii!i!ii|~iiiiii iRiiiViiliiiiiiiiiiiii!iiiiiiiiiiiii a F | a N ~i~iii!iiiiiiiiiii!i~ii :!si!iiiiiiiiiiiiiiiiiiiiiiii iiii!iliiiiiiiiiiiiiiiiiii i!iii!iili!iiiiiiiii!iiiiiiiiiiiliii!iiiliiiiiiiiiiii!iiiiii iiii!iii ii!iiiiiii! l iiiiiiiiiiiiiiiiiii iiiiiiiiiiiiiiiiiiiiiiii iiiiiiiii!iiiiiiiiiiiii!iiiiiiiiiiiii!i!iiiiiiiiiiiiiii iiiiiiiiiliili!iiiiiiiiiiiiiii iiiiiiiiiiiiiiiiiiiiiiiiii!iiiiiiiiiiiiiiiiii iliiiiiiiiiiiiiiiiiiiiiiiiii iiiiiiiiiiiiiiiiiiiii ii~iiiiii~iii~i~iiiiiii~i~!iiiiiii!iii!i~i!i~i~i!iii~iii~i~ii~!iii~i!i!i!i!i~i~ii~iiiiiii~i!i!!i~i~iii~i~i~i~i!ii~i~i~i~i~i~i!ii~i~i~iiii!i!i!!~iiiiiiii!iii!iiiiiiiiii!~ii~ii~!~!~i!i~z~r~i~!!!!~i!i~iiii!i!i!i~i~iii~iiiii
It is also possible to write the invariants in a form known as m o m e n t i n v a r i a n t s . 4 The moment invariants are denoted as ia, I-Ia, IIIa, and are defined by the indicial forms 1
Ia = aii,
(Ia -- ~ a i j a j i ,
IIIa
1
- - -gaijajkaki
(B. 15)
We observe that moment invariants are directly related to the t r a c e of products of a. The trace (tr) of a matrix is defined as the sum of its diagonal elements. Thus, the first three moment invariants may be written in matrix form (using a square matrix for a) as IIa -- 89 (aa),
Ia = tr (a),
Iila -- gtr (aaa)
(B.16)
The moment invariants may be related to the principal invariants as 4 Ia -- Ia, -
Ia -- Ia,
IIa = 89] -- IIa, 1-2
IIa -- ~ I a -- I-Ia,
1 3 IIIa -- Ilia - 5I a + IalIa -
1-3
IIIa -- IIIa + gI a - IaIIa
(B.17)
605
606 AppendixB Using Eq. (B.12) and the identities given in Eq. (B.17) we can immediately observe that the principal invariants and the moment invariants for a deviatoric second-order tensor are related through II'a = --fi'a
and
III'a = IiI'a -- det a'
(B. 18)
',i~,!',~,',!',!i!'i'',~""i,::,:'~:::::~i :::,,:,':i~,i',i~'~"'~'~::':::;iii!iii~',i,'i~i'l~il~""'~: '~~~ii'iiii', i~iiiiiiiiiiiiiiiiiiiii' ~ ~i:i iii',i'i,i',i'iii!iiii :ili~i~i',~,i!i~:ii::~::::::ili iil ili~:',i'i',!~, ~ i~,i'~,,i~~,i,iii'~i,!iiii:ilillil:,ii!iiii~, ~,i~',i',i~,i',i'i,i::'i i:',::iiii'~i:':ii'i~i'~',~iii::',!i!i i i!i~iiiiiiii!ili'iii'~~ii:,~i',!~, iii~,i'ii!'i,iii!,ii'i~,!~,~~iiiiiii~,i~,iii'~,,ii'i~: ,iii~:!i i',:i~i:::i~~iii',',i'i,i'i,i'i,i':',~,i'!',iiii, i',ili'ili',::i,iiiliii'i,i'i,li'i',i:iiii!~,ii!~,ii~,i!'i,iiiiiiii!!iliii'i,ii'~ii!ii',ilii','i'::,iii','iilili ~,iii',iii',iiiiiii!i'i,iiiiii'i'i',~,i'iii:i'',!iii~, iiiii::'~i i'iii'~i:,,ii'iiii,iiiii'~ii!ii'i:iliiiiiiii iiiiiiiiiiiiiiii',iliii!ili::iliiiiii',iliiiiiiiiiiliiiii!i':',iiiliiililil',iiii':'~!iii~,i:l:iilili~,i~iiiiii::iiii' ,i i'~ ii~:ii ,iii!!iiiii!iiiiiiiliiili~,iiiliiiiiiiiiiiiliiiiiiiiiiiiiiiiliiii'!i,iiiii',ii!i~i ! !i'i~i'~iiiiiiiiiii i~ii~:~iiiiiii~i~i~i~:~!~i~i!i~ii!i~iiiiiiiiii~i~i~iii~!ii~iii~i~!iii~iiii~ii~i~i~ii~i~i~i~i~i~i~iii~!i!~!~ii!iii~i~i~iiii ~!iiiiiiiiiiiiiiiii~!ii~i~i~iiiii~i~iiii~!~iii~iiii~i~i~ii!i!!i!!~iii~i~i~i~i~i~iii~i~iii~i~i~i~!ii!!iiiii!iiiiii~i!!iii~iii iiiiiiiiiiiiiiiii!!ii!iii!iiiiii iiii ii iii i
i
ii ii iiiiiiiii~,ii!i!iiiiiiiiii~i!iii iiii!ii~:~'~ii!',~,'~i~,~ii' ~ii ,',',i! ~'~i~:,~:,~::,~i~,~,~::::,ii~ ':,iii',i',ii~,ii!i',~i',',':~:,~i '
We often also need to compute the derivative of the invariants with respect to their components and this is only possible when all components are treated independentlythat is, we do not use any symmetry, if present. From the definitions of the principal and moment invariants given above, it is evident that derivatives of the moment invariants are the easiest to compute since they are given in concise indicial form. Derivatives of principal invariants can be computed from these by using the identities given in Eqs (B.17) and (B.18). The first derivatives of the principal invariants for symmetric second-order tensors may be expressed in a matrix form directly, as shown by Nayak and Zienkiewicz; 5,6 however, second derivatives from these are not easy to construct and we now prefer the methods given here.
B.3.1 First derivatives of invariants The first derivative of each moment invariant may be computed by using Eq. (B.15). For the first invariant we obtain Oi a
"-- t~ij
Oaij
(B. 19)
Similarly, for the second moment invariant we get
OIL
= aji
(B.20)
"-- ajkaki
(B.21)
~aij
and for the third moment invariant OIila Oaij
Using the identities, the derivative of the principal invariants may be written in indicial form as GQIa Oaij : r
OIIa Oaij = Ia~ij -- aji,
0IIIa
cOaij -- IIa~ij - Iaaji + ajtakilIIaa-j-i 1
(B.22)
The third invariant may also be shown to have the representation 3 cgIIIa
~aij
-- IIIaaj-i 1
(B.23)
Derivatives of invariants
where aj~-1 is the inverse (transposed) of the aij tensor. Thus, in matrix form we may write the derivatives as
OIIa = 1,1 - a T,
OIa = 1, Oa
0IIIa = iiiaa_ T Oa
0a
(B.24)
where here 1 denotes a 3 • 3 identity matrix. The expression for the derivative of the determinant of a second-order tensor is of particular use as we shall encounter this in dealing with volume change in finite deformation problems and in plasticity yield functions and flow rules. Performing the same steps for the invariants of the deviator yields 0I' Oa i~j
--
0I' Oa i~
-- O,
0II'
Off'
--
Oa i~j
Oa i~
0IIY
l
-- - - a j i ,
--
0IiI'
l
I
, -, -- a j k a k i Oa ij Oa ij
(B.25)
with only a sign change occurring in the second invariant to obtain the derivative of principal invariants from derivatives of moment invariants. Often the derivatives of the invariants of a deviator tensor are needed with respect to the tensor itself, and these may be computed as
o(.)' Oamn where
Oa,'j Oamn
--
o(.)' 0%
(B.26)
Oaij COamn
1
--- (~im(~jn -- ~(~ij(~mn
(B.27)
Combining the two expressions yields 0(.)'
_
Oamn
0(.)'
Oa tmn
1 (~mn 5ij
3
Oaij J
(B.28)
B.3.2 Second derivatives In developments of tangent tensors we need second derivatives of the invariants. These may be computed directly from Eqs (B. 19)-(B.21) by standard operations. The second derivatives of ia, I-Ia, IIIa yield
02Ia Oaij Oakl
= O,
02fia Oaij ~akl
= 5jkSil,
02IIIa
Oaij Oakl
= 5jkail + ajkSil
(B.29)
The computations for principal invariants follow directly from the above using the identities given in Eqs (B.17) and (B.18). Also, all results may be transformed to the vector form used extensively in this volume for the finite element constructions. These steps are by now a standard process and are left as an exercise for the reader.
607
608 AppendixB
1. H.M. Westergaard. Theory of Elasticity and Plasticity. Harvard University Press, Cambridge, 1952. 2. W.H. Press et al., editors. Numerical Recipes in Fortran: The Art of Scientific Computing. Cambridge University Press, Cambridge, 2nd edition, 1992. 3. I.H. Shames and EA. Cozzarelli. Elastic and Inelastic Stress Analysis. Taylor & Francis, Washington, DC, 1997. (Revised printing.) 4. J.L. Ericksen. Tensor fields. In S. Fltigge, editor, Encyclopedia of Physics, volume III/1. SpringerVerlag, Berlin, 1960. 5. G.C. Nayak and O.C. Zienkiewicz. Convenient forms of stress invariants for plasticity. Proc. Am. Soc. Civ. Eng., 98(ST4):949-953, 1972. 6. O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method, volume 2. McGraw-Hill, London, 4th edition, 1991.
Author index
Page numbers in bold are for pages at the end of chapters with names of author references. Abramowitz, M., 22, 44, 291,303,320, 374, 381,483, 496 Acary, V., 270, 277 Acheampong, C.B., 269, 276 Adams, M., 58, 61 Adini, A., 336, 378 Agelet de Saracibar, C., 212, 226 Ahmad, S., 387, 391,397, 421, 422, 427, 434, 451, 452, 454, 473, 475,479,483,484, 488,492, 495, 496 Ahmed, H., 117, 126 Ahtrikman, S., 549, 587 AI Mikdad, M., 233,244, 518, 543,543 Alart, P., 212, 226 Albasiny, E.L., 444, 453 Alfrey, T., 72, 120 Allman, D.J., 368, 369, 372, 381,436, 439, 452 Allwood, R.J., 368,380 Ambr6sio, J.A.C., 228, 243 Anandarajah, A., 269, 276 Andelfinger, U., 150, 157, 311,321 Anderson, C.A., 40, 45, 117, 119, 126 Anderson, R.G., 536, 544 Ando, Y., 344, 379 Andrade, J.C. de, 515, 516 Andrews, K.R.F., 252, 274 Areias, P.M.A., 127, 150, 156 Argyris, J.H., 233,244, 336, 340, 356, 365,377, 378, 427, 450, 541,546 Arlett, P.L., 510, 516 Armero, E, 115,125, 127, 150, 156, 179, 190 Armstrong, P.J., 77, 121 Arnold, D.N., 396, 406, 407, 418, 423 Arrow, K.J., 197, 226 Arruda, E.M., 166, 189 Arulanandank, K., 41, 45 Ashwell, D.G., 427, 451, 539, 545 Atluri, S.N., 228, 243
Auricchio, E, 78, 95, 121, 173, 189, 395, 396, 408, 409, 412, 415, 419, 423, 424 Ayoub, A., 297,321 Baaijens, EET., 570, 589 Babu~ka, I., 273, 277, 332, 356, 377, 392, 419, 423, 424 Bagchi, D.K., 536, 544 Bahrani, A.K., 510, 516 Bakhrebah, S.A., 492, 493, 496 Balan, T.A., 278, 320 Baldwin, J.T., 371,381 Ballesteros, E, 349, 379 Baraff, D., 252, 273 Barbosa, R., 26 l, 275 Barut, A., 539, 545, 546 Basant, Z.E, 113, 125 Basole, M.M., 269, 276 Bathe, K.-J., 52, 60, 191, 212, 224, 226, 349, 379, 392, 394, 395, 397, 399, 400, 405, 422, 423, 424, 494, 496, 537, 545, 593, 596 Batoz, J.L., 55, 60, 349,371,379, 381,405,424, 441, 452, 537, 544 Bauchau, O.A., 228, 243 Bauer, J., 467,469, 471,473, 474 Bayless, A., 113, 125 Ba2ant, Z.E, 113, 115, 116, 125, 267, 276 Bazeley, G.E, 335,336, 344, 360, 362, 377 Beer, EE Jnr., 278,320 Beisinger, Z.E., 427, 451 Bell, K., 336, 356, 365,377 Belytschko, T., 17, 24, 44, 65, 113, 115, 120, 125, 191, 225, 273, 277, 308, 321, 427, 451, 494, 496, 539, 545 Benson, D.J., 19 l, 201,224, 225, 228, 237, 242 Benson, ER., 506, 507, 515 Bensoussan, A., 549, 550, 587 Bergan, EG., 55, 56, 58, 60, 61, 336, 344, 378, 379, 436, 452, 535,539, 544 Besseling, J.E, 78, 121
610
Author index Best, B., 112, 125 Betsch, E, 236, 244, 319, 322, 537, 545 Bi6ani6, N., 26, 44, 48, 60, 78, 94, 108, 113,121,123, 124, 125, 260, 266, 275, 389.422 Biezeno, C.B., 539, 545 Bingham, E.C., 103, 123 Birkhoff, G., 365,380 Bischoff, M., 150, 157, 311, 319, 321,322, 428, 451, 537, 545 Bishop, J.F., 569, 588 Bittencourt, E., 191,225 Bjtirk, A., 48, 60 Blaauwendraad, J., 108, 124 Bogner, EK., 335,364, 377, 427, 450, 535,543 Bonet, J., 132, 134, 135, 146,156, 159, 169, 176,189, 230, 243, 252, 274 Bonnew, G., 427.450 Booker, J.R., 113,125 Boot, J.C., 367, 380 Boresi, A.E, 4, 16 Borja, R.I., 575,589 Boroomand, B., 18l, 190 Borri, M., 228, 243 Boso, D.E, 565,579, 588, 589 Bosshard, W., 336, 356, 365,377 Bottasso, C., 228, 243 Bouzinov, EA., 19l, 224 Boyce, M.C., 166, 189 Braess, D., 150, 157, 31 l, 321, 537,545 Braun, M., 319, 322 Brebbia, C., 427,450, 525,534, 537, 54 l, 543 Brekelmans, W.A.M., 570, 588, 589 Brekke, T., 112, 125, 260, 270, 275 Brezzi, E, 397,423, 424 Brogan, EA., 539, 545 Bron, J., 367, 380 Brown, T.G., 506, 515 Broyden, C.G., 58, 61 Bucalem, M.L., 397,424 Bucciarelly, L., 323,376 Btichter, N., 319, 322 Bugeda, G., 37 l, 381 Bushnell, D., 494, 497 Butlin, G.A., 336, 356, 365,377 Cantin, G., 427,450 Cardona, A., 228,243 Carlesimo, L., 297, 304, 320 Carnoy, E., 171,189 Carpenter, N., 308, 321, 427,451 Carr, A.J., 427,436, 450 Carson, W.G., 536, 544 Cassel, E., 40, 45, 58, 61 Cazzani, A., 228,243 Celigoj, C.C., 144, 156 Cervera, M., 115, 126
C6sar de Sfi, J.M.A., 127, 150, 156 Chaboche, J.L., 78, 121, 567,588 Chadwick, E, 128, 156 Chan, A.H.C., 41, 43, 45, 94, 105,123, 336, 340, 345, 378, 391,392, 406, 418,422 Chan, A.S.L., 454, 467,468, 469, 474 Chan, H.C., 506, 516 Chan, S.K., 19l, 224 Chan, Y.K., 536, 544 Chang, C.S., 269, 276 Chang, C.T., 78, 121 Chari, M.V.K, 117, 126 Chaterjee, A., 367, 380 Chattapadhyay, A., 537,544 Chaudhary, A.B., 191,212, 224, 226, 397, 399, 400, 423 Chawla, V., 155, 157, 191,225 Chen, A.J., 240, 244 Chen, G., 272, 277 Chen, W.E, 74, 78, 121 Cheung, M.S., 506, 515 Cheung, Y.K., 72, 101,120, 123, 335,336, 338, 340, 344, 348,360, 362, 377, 378, 427,441,450, 506, 507, 515, 516 Chi-Wang Shu, 18, 44 Chiang, M.Y.M., 397,423 Chinosi, C., 413, 424 Chong, K.E, 4, 16 Chou, EC., 4, 16 Christensen, R.M., 63, 120 Church, K., 368, 381 Ciampi, V., 297, 304, 320, 321 Ciarlet, EG., 426, 450 Cilia, M.A., 494, 497 Cimento, A.E, 52, 60 Clark, A.E, 565,588 Clark, H.T., 228, 243 Clausen, W.E., 278, 320 Clough, R.W., 335,336, 355,356, 360, 361,377,378, 388,395,422, 427,436,442,450, 452, 476, 488, 496, 535, 539, 544 Cockburn, B., 18, 44 Cohen, H., 3, 16, 228, 229, 242, 260, 272, 275, 493, 496 Cohen, M., 389, 394, 422 Collatz, L., 48, 60, 59 l, 593,595 Comini, G., 38, 45 Conceiqao Anto6nio, C.A., 127, 150, 156 Connor, J., 427, 450, 451, 525, 534, 537, 541,543 Cook, R.D., l, 15, 22, 24, 44, 368,380, 389,422, 436, 452, 493,496 Cormeau, I.C., 103, 104, 105, 115, 124 Comes, G.M.M, 368, 380 Cowper, G.R., 336, 356, 365,377, 427,451 Cozzarelli, EA., 4, 16, 83, 122, 128, 132, 146, 156, 158, 159, 160, 189, 430, 452, 605,606, 608
Author index 611 Cramer, H., 515,516 Creus, G.J., 191,225 Crisfield, M.A., 52, 53, 55, 56, 58, 60, 61, 155, 157, 304, 321,371,381,409, 419, 424, 539, 545, 546 Crook, A.J.L., 261,267, 275, 276 Cross, D.K., 371,381 Cundall, P.A., 246, 247, 249, 254, 255,273,273, 274, 277 Cuomo, M,, 191,225 Curnier, A., 191,197, 212, 225, 226 Cusens, A.R., 506, 515 Cyr, N.A., 101,123 Cyrus, N.J., 336, 378 D'Adetta, G.A., 267, 276 Dahlquist, G., 48, 60 Damilano, G., 228, 243 Darve, E, 92, 122 Davidon, W.C., 52, 60 Davies, J.D., 506, 507, 515 Dawe, D.J., 336, 340, 378, 427, 440, 441,451 De Borst, R., 108, 115, 124, 125, 181,190, 265,276 Defalias, Y., 93, 122 Del Guidice, S., 38, 45 Delpak, R., 454, 461,464, 468, 469, 470, 474 Demmel, J., 48, 60, 278,320, 590, 595 Dennis, J.E., 52, 60 Deppo, D.A. da, 523,543 Desai, C.S., 106, 124 Dettmar, J., 573, 589 Dhalla, A.K., 95, 123 Dhatt, G., 55, 60, 367, 371,379, 380, 381, 405, 424, 427, 450, 537, 544 Dhillon, B.S., 409, 424 Dill, E.H., 535,543 Dinno, K.S., 97, 99, 123 Doan, D.B., 228,243 Doblare, M., 519, 543,543 Doherty, W.P., 18, 44 Dong, S.B., 308,321 Doolin, D.M., 268, 269, 272, 276, 277 Draper, J.K., 355,356, 377 Drucker, D.C., 74, 86, 120, 121, 122 Duncan, W., 336, 364, 377 Dupuis, G., 427,451 Dureisseix, D., 570, 588 Duvaut, G., 103, 104, 124 Dvorkin, E.N., 171,189, 397,399, 400, 423, 494, 496 Dworsky, N., 323,376 Edwards, G., 368, 380, 427,451 Ekin, J., 565, 588 Elias, Z.M., 454, 474 Elsawaf, A.E, 58, 61 Ergatoudis, J.G., 336, 378 Ericksen, J.L., 605,608 Eshelby, J.D., 549, 587
Falk, R.S., 396, 406, 407, 418, 423 Favier, J.F., 259, 275 Felippa, C.A., 335,336, 344, 361,365,377, 379, 380, 436, 452 Feng, Y.T., 252, 258, 266, 269, 274, 276 Ferencz, R.M., 48, 60 Fernandes, A.A., 127, 150, 156 Feshbach, H., 501,515 Feyel, E, 567, 588 Filippou, EC., 297, 303,304, 321 Finnie, I., 104, 124 Firmin, A., 454, 467,468, 469, 474 Fish, J., 113,125, 547, 561,587 Fix, G.J., 19, 44 Fleming, M., 273, 277 Fletcher, R., 58, 61 Fltigge, W., 426, 450 Foo, O., 506, 516 Ford, R., 336, 356, 365,377 Fox, D.D., 428, 451,494, 496, 537, 544 Fox, R.L., 335, 364, 377, 427, 450, 535,543 Fraeijs de Veubeke, B., 147, 156, 308,321, 335,356, 361,362, 367, 368, 377, 380, 392, 422, 525,543 Franca, L.P., 436, 452 Francavilla, A., 191,224 Francofort, G.A., 553, 587 Frasier, G.A., 24, 44 Frederick, C.O., 77, 121 Frey, E, 427, 451 Fried, I., 336, 356, 365,377 Fulton, R.E., 336, 378 Galerkin, B.G., 17, 18, 44, 348, 379 Gallagher, R.H., 95, 106, 123, 124, 427, 450, 451, 454, 474, 536, 537, 544 Galvanetto, U., 155,157, 559, 577, 579, 587, 589 Garcfa Orden, J.C., 230, 243 Geers, M.G.D., 570, 588 Gelder, D., 117, 126 Gellately, R.A., 537, 544 Geradin, M., 52, 60, 228,243 Gere, J.M., 3, 16, 525,536, 543 Ghaboussi, J., 18, 44, 260, 261,269, 275, 276 Ghali, A., 506, 515 Ghionis, P., 58, 61 Ghosh, S., 570, 588 Gianni, M., 454, 465,467, 468, 469, 474 GiD-The Personal Pre/Postprocessor, 595,596 Gill, J.J., 253, 274 Gill, S.S., 97, 99, 123 Giroux, Y.M., 427,450 Glaser, S., 127, 150, 156 G0el, J.J., 365,380, 427, 451 Goicolea, J.M., 230, 243 Goldstein, H., 237, 244 Golub, G.H., 93, 122, 240, 244, 590, 595
612
Author index Gonz~ilez, O., 28, 45, 155, 157 Goodier, J.N., 4, 16, 193,225, 281,321), 510, 516 Goodman, R.E., 112, 125, 260, 26 l, 270, 275 Goudreau, G.L., 63, 69, 120, 19l, 201,224 Gould, EL., 454, 465,474 Govindjee, S., l, 16, 169, 171, 189, 191,225, 282, 320, 348, 379, 412, 424, 595,596 Grafton, EE., 454, 458,460, 461,473, 510, 513,516 Gray, N.G., 494, 497 Greenbaum, G.A., l01,123 Greene, B.E., 368, 380, 427, 450, 451 Greimann, L.E, 409, 424 Griffiths, D.W., 266, 276 Groger, T., 267,276 Gross, B., 63, 120 Gruttmann, E, 236, 244, 319, 322, 388,422, 537, 545 Gu, Y.X., 583,584, 589 Guex, L., 349, 369, 379 Gunderson, K., 371,379 Gurtin, M.E., 128, 132, 134, 156, 159, 189, 230, 243 Haisler, W.E., 95,123, 371,379, 465,474 Hallquist, J.O., 137, 191,201,224, 225, 228, 242 Hanssen, L., 336, 344, 378 Harou, M., 389, 422 Hart, R.D., 247,273 Harter, R.L., 436, 437,452 Hartz, B.J., 536, 544 Harvey, J.W., 367, 380 Hashin, Z., 549, 587 Haug, E.J., 228, 243 Heegaard, J.-H., 191, 197, 225 Heege, A., 212, 226 Hellan, K., 368, 380 Heller, W.R., 104, 124 Hellinger, E., 13, 16 Hencky, H., 348,379 Henshell, R.D., 336, 340, 368, 378, 380 Herrmann, L.R., 69, 120, 367, 368, 380, 427,451 Hestenes, M., 58, 61 Heyes, D., 267, 276 Hibbitt, H.D., 40, 45, 154, 155,157 Hildebrand, EB., 12, 16, 81,121 Hill, R., 74, 112, 12t), 569, 588 Hilton, H.H., 72, 120 Hinton, E., 58, 61, 78, 81,121, 122, 297, 388, 389, 395,397, 399,400, 422, 423, 494, 496, 506, 507, 515 Hirschberg, M.H., 101,123 Hjelmstad, K.D., 297, 321 Ho, L.W., 349, 379, 394, 395,397,405,423, 424 Hobbs, R.E., 40, 45, 58, 61 Hocking, G., 260, 265,275, 276 Hoffstetter, G., 108, 124 Hogge, M., 52, 60 Hogue, C., 253,254, 255,274
Holzapfel, G.A., 166, 189 Horstemeyer,, 549, 586, 587 Hrabok, M.M., 349, 379 Hrudey, T.M., 349, 379 Huang, E.C., 494, 496 Huang, H.C., 181,190, 397, 399, 400, 423 Huang, M., 181, 184, 190, 395,409, 423 Hughes, T.J.R., 1, 15, 17, 22, 24, 35, 44, 45, 48, 60, 65, 73, 76, 77, 80, 83, 90, 103, 104, 113, 120, 123, 135,156, 171,172, 173,176, 177, 179,189, 191,224,268,276, 336, 379, 388,389,392, 394, 395,397,409, 418,422, 423, 424,436, 439, 452, 471,474, 493,496 Htihlhaus, H.B., 115,125, 181,190 Hulbert, G.M., 228,243, 436, 452 Humpheson, C., 105, 124 Hung-Chan Hung, 539, 545 Hurwicz, L., 197, 226 Huston, R.L., 228, 243 Ibrahimbegovic, A., 233, 244, 436, 440, 452, 518, 543, 543 Idelsohn, S., 52, 60 Irons, B.M., 58, 61,335,336, 344, 355,356, 360, 362, 365,369, 371,377, 378, 381,387, 421,427,434, 440, 451,452, 454, 473, 475,483,484, 488,492, 495, 496, 536, 537, 544 Isenberg, J., 108, 124 Ishihera, K., 94, 123 Ito, T., 272, 277 Jean, M., 212, 226, 270, 276, 277 Jetteur, Ph., 427,436, 439, 451,452 Jirasek, M., 267,276 Jirousek, J., 349, 368, 369, 372, 379, 381 John, N.W.M., 260, 275 Johnson, C., 368,380 Johnson, C.E, 427,442, 450 Johnson, E.R., 278, 320 Johnson, K.W., 26, 44, 48, 60 Johnson, W., 74, 78, 120 Jones, R.E., 191,225, 368, 380, 427, 451,454, 473 Joo, T., 228, 243 Jordan, EE, 467,474 Jourdan, E, 212, 226 Ju, J.-W., 212, 226 Kabaila, A.E, 536, 544 Kalker, J.J., 191,224 Kaminski, M., 561,588 Kanchi, M.B., l0 l, 104, 123 Kane, C., 191,225, 27 l, 277 Kang, D.S., 228, 243 Kanoknukulchai, W., 388, 395,422, 47 l, 474 Kapur, K.K., 336, 349, 378, 536, 544 Karniadakis, G.E., 18, 44
Author index 613 Kasper, E.P., 151,154, 157 Kawai, T., 247, 273, 535,544 Ke, T.C., 262, 275 Kelsey, S., 367,380 Key, S.W., 427,451 Khachaturian, W., 108, 124 Khojasteh, Bakht, M., 454, 463,473 Kikuchi, F., 344, 379 Kikuchi, N., 191,224, 561,587 King, I.P., 63, 80, 96, 108, 120, 121, 123, 124, 336, 344, 378, 427,436, 441,442, 450 Kirchoff, G., 323, 376 Klapka, I., 228, 243 Klein, S., 454, 458,473 Klerck, P.A., 261,275 Klinkel, S., 169, 171,189, 282, 320 Koiter, W.T., 74, 80, 104, 120, 124 Kosko, E., 336, 356, 365,377 Kosloff, D., 24, 44 Kotanchik, J.J., 454, 474 Kouznetsova, V., 570, 588, 589 Krahl, N.W., 108, 124 Kraus, H., 467,474 Kremmer, M., 259, 275 Krieg, D.N., 81,122 Krieg, R.D., 81,122 Krtier, E., 549, 587 Kr/Sner, E., 549, 587 Krongauz, Y., 273,277 Krysl, P., 273, 277 Kuhn, M., 259, 274 Kun, E, 267,276 Kuo-Mo Hsiao, 539, 545 Ladev6ze, P., 547, 560, 570, 587, 588 Ladkany, S.G., 368,380 Lakes, R., 547, 587 Landers, J.A., 191,194.224 Langhaar, H.L., 524, 543 Lardeur, P., 405, 424 Larsson, R., 181,190 Latham, J.P., 260, 275 Laursen, T.A., 155,157, 191,197,201,212,213,218, 219, 225, 226, 227 Lawson, C.L., 365,380 le Tallec, P., 1, 16 Leckie, EA., 104, 124 Ledesma, A., 94, 123 Lee, E.H., 173, 189 Lee, K., 570, 588 Lee, S.-H., 539, 545 Lee, S.L., 349, 379 Lees, M., 38, 45 Lefebvre, D., 336, 337, 378, 391,392, 393,406, 407, 419, 422 Lefik, M., 554, 559, 560, 587
Leroy, Y., 113, 125, 181,190 Leung, K.H., 93, 122 Lewis, R.W., 38, 40, 45, 87, 105, 122, 124 Liepins, A., 467, 469, 470, 474 Lin, C., 254, 274 Lin, F.B., 113, 115, 116, 125 Lin, J.I., 539, 545 Lin, T.L., 1, 16 Lin, X., 254, 274 Lindberg, G.M., 336, 356, 365,377, 427,451 Ling, W., 212, 226 Lions, J.-L., 103, 104, 124, 549, 550, 587 Liu, D.T., 173, 189 Liu, W.K., 17, 24, 44, 394, 423, 493, 496, 549, 586, 587 Liu, Z.A., 454, 473 Lo, K.S., 444, 453, 491,496 Loden, W.A., 427, 450 Loehnert, S., 570, 571,572, 589 Ltihner, R., 58, 61 Loiseau, O., 570, 588 Loo, Y.C., 506, 515 Lovadina, C., 412, 413, 419, 424 Lubliner, J., 73, 77, 78, 95,120, 121 Luenberger, D.G., 194, 225 Lynch, E de S., 1, 15 Lyness, J.E, 117, 126 Lynn, P.P., 409, 424 Lyons, L.P.R., 371,381 Ma, M.Y., 263,275 McHenry, D., 72, 120 Mackerle, J., 191,225 McLaughlin, M.M., 262, 275 McLay, R.M., 368, 380 McMeeking, R.M., 95, 123 McNamara, S., 256, 274 MacNeal, R.H., 436, 437,452 Madenci, E., 539, 545, 546 Maenchen, G., 81,122 Mahalic, P.A., 1, 16 Malkus, D.S., 1, 15, 22, 24, 44, 392, 422 Mallett, R.H., 535,537, 543, 544 Malvern, L.E., 4, 16, 128, 156, 176, 189, 230, 243, 430, 452 Mandel, J., 77, 112, 121, 125, 173, 189 Mang, H., 108, 124 Mansfield, L., 365,380 Manson, S.S., 101,123 Manzoli, O., 115, 126 Marqal, P.V., 40, 45, 96, 101,123, 154, 155,157, 535, 541,542, 543, 546 Marcotte, L., 371,381 Marcus, H., 349, 379 Marguerre, K., 525,543 Mark, R., 447,448, 453
614 Author index Marketos, E., 96, 123 Marsden, J.E., 176, 189, 191,225, 271,277 Martin, D.W., 444, 453 Martin, H.C., 535, 536, 543, 544 Martin, J.B., 104, 124 Martins, J.A.C., 213,226 Martins, R.A.E, 371,381 Mason, W.E., 427, 451 Matsui, T., 537, 541,544 Matsuoka, O., 537, 541,544 Matte, Y., 371,381 Matthies, H., 52, 54, 55, 60, 592, 596 Mawenya, A.S., 506, 507, 515 Mazars, J., 113,125 Meek, J.L., 539, 546 Meguro, K., 270, 276 Melenk, J.M., 273, 277 Mellor, P.W., 74, 78, 120 Melosh, R.J., 336, 338,378, 535,543 Mendelson, A., 101,123 Mescall, J.F., 539, 545 Meschke, G., 108, 124 Miehe, C., 570, 573, 588, 589 Mier, J. van, 267, 276 Miles, G.A., 454, 465,467, 468, 469, 474 Milford, R.V., 494, 496 Mindlin, R.D., 323,377 Mira, P., 181,190 Mises, R. von, 74, 120 Mitchell, N., 565, 588 Moita, G.E, 539, 545 Monerie, Y., 270, 277 Mooney, M., 162, 189 Moorthy, S., 570, 588 Moran, B., 17, 24, 44 More, J., 52, 60 Moreau, J.J., 270, 276 Morgan, K., 40, 45, 184,190, 467,469, 471,473,474 Mori, T., 549, 587 Morikawa, H., 266, 276 Morley, L.S.D., 336, 345, 346, 367, 368, 378, 379, 417, 424 Morse, P.M., 501,515 Mr6z, Z., 78, 93, 112, 113, 121, 122, 125 Mullen, R., 539, 545 MUller, D., 254, 274 Muncaster, R.G., 3, 16, 228, 229, 242, 260, 272, 275 Munjiza, A., 247, 252, 260, 266, 267, 273, 274, 275, 276 Murray, D.W., 525,536, 544 Muscat, M., 117, 126 Mustoe, G.G.W., 247, 260, 265,266, 273, 275, 276 Naghdi, EM., 454, 455,471,474 Nagtegaal, J.C., 95, 123 Nakazawa, S., 93, 122, 336, 378, 393,422 Narasimhan, R., 392, 423
Narayanaswami, R., 336, 378 Nardin, A., 575, 589 Nath, E, 108, 110, 124 Navaratna, D.R., 454, 458,465,473 Nay, R.A., 37 l, 381 Nayak, G.C., 74, 78, 81, 85,121, 122, 539, 541,545, 606, 608 Naylor, D.J., 86, 87, 122 Neal, M.O., 191,225 Neale, B.K., 368, 380 Needleman, A., 113, 125, 181,190, 571,589 Neuenhofer, A., 297, 304, 321 Newland, D., 254, 274 Newmark, N., 25, 44 Newton, R.E., 536, 544 Ng, T.T., 254, 259, 274, 275 Nguyen, Q.A., 103, 124 Nithiarasu, E, 37, 45, 183, 190 Noll, W., 176, 189 Noor, A.K., 539, 545 Norris, V.A., 86, 87, 122 Nouy, A., 570, 588 Novozhilov, V.V., 454, 455,474, 510, 516 Nygard, M.K., 336, 344, 378 Oancea, V.G., 212, 225 Obrecht, H., 515, 516 Ochiai, S., 565,588 O'Connor, R., 252, 254, 274 Oden, J.T., 1, 16, 154, 157, 191,213,224, 226, 371, 381 Ogden, R.W., 166, 168, 189 Ohnishi, Y., 272, 277 Oliver, J., 115, 125, 126 Olson, M.D., 336, 356, 365,377, 427, 451 Ofiate, E., 371, 374, 381, 397, 400, 403, 423, 424, 467, 469, 471,473,474 Organ, D., 273,277 Ortiz, M., 113, 125, 18l, 190, 191,225, 271,277 Osamura, K., 565, 588 Otter, J.R.H., 40, 45, 58, 61 Ottosen, N.S., 181,190 Owen, D.R.J., 78, 81, 101,104, 108, 117, 121, 122, 123,124,126, 181,190, 252, 258,260, 261,266, 267, 269, 274, 275, 276, 371,381 Pacoste, C., 539, 546 Paczelt, I., 191,225 Padlog, J., 536, 537, 544 Pagano, N.J., 4, 16 Pamin, J., 115,125, 181,190 Pande, G.N., 86, 106, 112, 115,122, 124, 125 Pantuso, D., 171,189 Papadopoulos, P., 191, 218, 219, 225, 226, 228, 229, 242, 272, 277, 397, 400, 405,409, 420, 423, 425 Papadrakakis, M., 58, 61
Author index 615 Papanicolau, G., 549, 550, 587 Parekh, C.J., 427, 436, 441,442, 450, 452 Parisch, H., 191,225, 537, 544 Parks, D.M., 95, 123 Parlett, B.N., 525,543 Pastor, M., 41,43, 45, 93, 94, 105,122,123, 181,184, 190 Paul, D.K., 94, 123 Pawsey, S.E, 388, 395,422, 476, 488, 491,496 Peano, A.G., 365,380 Pellegrino, C., 577, 579, 589 Pentland, A.P., 254, 274 Penzien, J., 454, 473 Peraire, J., 181,184, 190, 252, 274 Percy, J.H., 454, 458, 473 Peric, D., 181,190, 261,275 Perkins, E., 252, 274 Perzyna, P., 103, 106, 123, 124 Peterson, EE., 69, 120 Petocz, E.G., 229, 243 Petrangeli, M., 297,321 Petrinic, N., 252, 265, 274 Phillips, D.V., 108, 124 Pian, T.H.H., 55, 60, 152, 157, 368, 380, 408, 424, 454, 465,473, 541,546 Pica, A., 58, 61 Picu, R.C., 41,547, 549, 587 Pietruszczak, S.T., 112, 113, 115,125 Pietrzak, G., 212, 226 Pijaudier-Cabot, G., 113, 115, 116, 125 Pinto, P.E., 297, 321 Pister, K.S., 63, 69, 120 Plesha, M.E., 1, 15, 22, 24, 44 Polak, E., 58, 61 Popov, E.P., 93, 122, 278, 320, 454, 473 Prager, W., 74, 76, 78, 86, 120, 121, 122 Prakash, A., 78, 121 Pramono, E., 113, 125 Pratt, C., 427,451 Preece, D.S., 251,252, 273 Press, W.H. et al (eds.), 590, 595, 605,608 Pugh, E.D.L., 388, 395,397, 422 Puso, M.A., 191,197, 212, 218, 219, 225, 226, 227 Qian, D., 549, 586, 587 Qu, S., 93, 122, 336, 378, 393,422 Raghavan, E, 570, 588 Rahier, C., 1, 16 Ralston, A., 48, 53, 60 Ramm, E., 55, 60, 150, 154, 155,157, 267, 276, 31 l, 319, 321,322, 428, 451, 537,544, 545 Randen, Y. van., 191,224 Ranjithan, S., 269, 276 Rankin, C.C., 539, 545 Rao, R.S., 397, 423
Razzaque, A., 336, 349, 362, 369, 371,378,379, 381, 441,445,452, 535,543 Reed, R.P., 565, 588 Reeves, C.M., 58, 61 Reinsch, C., 593,596 Reisman, W. von,, 95, 123 Reissner, E., 13, 16, 323, 377, 439, 452, 518, 539, 543, 545 Repetto, E., 171,189, 191,225, 271,277 Reuss, A., 549, 587 Rice, J.R., 95, 112, 113, 123, 125, 154, 155, 157 Riesa, J.D., 447, 448,453 Rifai, M.S., 19, 44, 311,321,428,451,494, 496, 537, 544 Riks, E., 55, 60 Ristic, S., 539, 546 Ritz, W., 348,379 Roberts, T.M., 539, 545 Robichaud, L.P.A., 427, 450 Rockey, K.C., 536, 544 Roehl, D., 150, 157, 311, 319, 321,322 Rots, J.G., 108, 124, 270, 276 Rubinstein, M.E, 101,123 Rubio, C., 181,190 Rudnicki, J.W., 112, 113, 125 Runesson, K., 181,190 Rupp, G., 565, 588 Russell, H.G., 72, 120 Rybicki, E.E, 95, 123 Sabir, A., 427, 451 Sackman, J.L., 191,224 Sacks, S., 81,122 Sakurai, T., 80, 121 Samuelsson, A., 336, 344, 379, 395,406, 408,423 Sanchez-Palencia, E., 549, 550, 552, 553, 587 Sander, G., 361,368, 379 Sanders, J.L. Jr., 467, 469, 470, 474 Saritas, A., 297, 303,321 Sawamoto, Y., 266, 276 Scapolla, T., 332, 377, 419, 424 Scharpf, D.W., 233,244, 336, 356, 365,377 Schlangen, E.J., 267, 276 Schmidt, L.A., 535, 543 Schmidt, R., 523,543 Schmit, L.A., 95,123, 335,364, 377, 427, 450 Schnabel, E, 515,516 Schnobrich, W.C., 492, 493,494, 496 Schotte, J., 570, 588, 589 Schrefler, B.A., 41, 43, 45, 94, 105, 123, 554, 559, 560, 561, 563, 575, 577, 579, 584, 586, 587, 588, 589 Schroder, J., 570, 588, 589 Schultz, M., 297,321 Schwarz, H.R., 48, 60 Schweizerhof, K.H., 101, 154, 155, 157, 191,224, 523,543
616 Author index Scordelis, A.C., 444, 453, 491,496 Scott, R.F., 41, 45 Secht, B., 336, 344, 379 Seiler, C., 515,516 Seiss, C.P., 108, 124 Sen, S.K., 454, 465,474 Setlur, A.V., 367, 380 Shabana, A.A., 228,242 Shames, I.H., 4, 16, 83, 122, 128, 132, 146, 156, 158, 159, 160, 189, 430, 452, 605, 606, 608 Shapiro, G.S., 376, 381 Sharma, K.G., 112, 125 Shewchuk, J., 595,596 Shi, G.(-)H., 171,229, 243, 261,272, 275, 277 Shi, J., 155,157, 539, 545 Shiomi, T., 41, 43, 45, 94, 105,123 Silvester, P., 117, 126 Simo, J.C., 19, 28, 35, 44, 45, 49, 60, 73, 76, 77, 80, 81, 83, 90, 91,101,104, 113, 115,120,121,122, 125, 127, 135, 145, 150, 154, 155,156,157, 158, 166, 168, 172, 173,177, 179, 189,190, 191,212, 218,224, 226, 228, 231,233,234, 236, 243,244, 311,321,336, 340, 345,378, 391,392, 406, 418, 422, 428,436, 439,451,452, 494, 496, 518, 519, 523,537, 543, 543, 544 Sitar, N., 262, 268, 269, 275, 276 Sloan, S.W., 113,125 Sluys, L.J., 115, 125, 181,190 Smith, I.M., 336, 364, 377 Sokolnikoff, I.S., 4, 7, 10, 11, 12, 16, 485, 496, 508, 513,516 Solberg, J.M., 191,225, 228, 229, 242, 272, 277 Southwell, R.V., 336, 348, 378 Souza Neto, E.A. de., 261,275 Spacone, E., 297, 304, 321 Stegun, I.A., 22, 44, 291, 303, 320, 374, 381, 483, 496 Stein, E., 212, 226, 236, 244, 319, 322, 537, 545 Steinmann, P., 181,190 Stiefel, E., 58, 61 Stolarski, H.K., 212, 226, 308, 321, 397, 423, 427, 451,494, 496, 539, 545 Strack, D.L., 249, 273 Strang, G., 19, 44, 48, 52, 54, 55, 60, 592, 596 Strickland, G.E., 427,450 Stricklin, J.A., 95,123, 371,379, 454, 463,465,473, 474,515,516 Strome, D.R., 368, 380, 427,450, 451,454, 458, 460, 461,473, 510, 513,516 Sture, S., 113, 125 SuArez, B., 397, 403,424 Sumihara, K., 152, 157, 408,424 Suneda, D.K., 117, 126 Suquet, P.M., 570, 589 Szabo, B.A., 191,225 Szabo, T., 191,225 Szilard, R., 526, 543
Tabbara, M., 115, 125 Taciroglu, E., 297, 321 Tadros, G.S., 506, 515 Tagel-Din, H., 270, 276 Tahar, M.B., 371,381, 441,452 Tanaka, K., 549, 587 Tarnow, N., 28, 44, 155,157, 519, 537,543,543, 544 Tatsueka, T., 94, 123 Taucer, EE, 297, 321 Taylor, L.M., 251,252, 273 Taylor, R.L., 1, 13, 14, 15, 17, 18, 19, 22, 24, 25, 26, 30, 36, 37, 38, 44, 45, 57, 61, 63, 65, 69, 78, 81, 83, 91,93, 95, 103, 112,120,121,122,123,125, 127, 129, 137, 145, 150, 151,152, 154,156,157, 166, 168, 173, 181,183,189,190, 191,194,212, 218, 219,224, 225,226, 227, 232,234,238,244, 260, 267,270, 275, 276, 290, 297, 303, 318,320, 321,324, 334, 336, 340, 341,345,346, 348,356, 377,378, 379, 388,389, 391,392, 393,395,396, 397,400, 403,405,406,407,408,409,410, 412, 415,418,420, 422, 423, 424, 425, 427,436, 439, 440, 441,450, 452, 458, 461,465,466, 471,474, 475,476, 479,480, 483,487,488,491,495, 496, 498,515, 523,525,543, 556, 587, 592,596, 606, 608
Terada, K., 561,587 Tessler, A., 308,321,409, 424, 539, 545, 546 Teter, R.D., 101,123 Tezduyar, T., 397, 418,423 Theocaris, P.S., 96, 123 Theron, N.J., 228,243 Thomas, G.R., 427,451 Timoshenko, S.P., 3, 4, 16, 193, 225, 281,284, 320, 323,325,333,376, 426, 450, 454, 455,467,474, 510, 516, 525,536, 543 Ting, J.M., 254, 274 Tisdale, P., 371,379 Tocher, J.L., 336, 349, 355, 356, 360, 361,377, 378, 450, 4427 Tong, P., 55, 60, 290, 320, 368,380, 541,546 Too, J., 388, 395, 422, 476, 479, 483,487,488, 491, 492, 496 Too, J.J.M., 503,514, 515 Torbe, I., 368,381 Toyoshima, S., 391,422 Truesdell, C., 176, 189 Tsay, C.-S., 539, 545 Tu, T., 407,424 Tuba, I.S., 191,224 Turner, M.J., 535,543 Tuzun, U., 267, 276 Utku, S., 371,381,427,450 Uzawa, H., 197,226 Vahdati, M., 184, 190 Valliappan, S., 80, 96, 108, 110, 121, 124
Author index 617 Van Loan, C.E, 93,122,240,244,590,595 Ventura, G.,191,225 Vilotte, J.E, 391,422 Visser, W.,233,244,336,356,365,367,377, 380
Vlachoutsis, S.,494,497 Voigt, W.,549,587 Volker, R.E.,l17,126 VuVan, Z, 212,226 Vu-Quoc, L.,49,60,254,274,519,543, 543 Wafy, T.M., 539, 545 Wagner, W., 388, 422 Wait, R., 254, 274 Waiters, D., 336, 340, 378 Walton, O.R., 254, 274 Walz, J.E., 336, 378 Warburton, G.B., 336, 340, 378 Washizu, K., 12, 14, 16, 454, 455,471,474 Watson, M., 63, 72, 120 Webster, J.J., 427, 451, 454, 463,473 Wehage, R.A., 228, 243 Weikel, R.C., 427, 450 Wempner, G., 278, 320, 371,381 Westergaard, H.M., 605,608 White, J.L., 63, 120 Wiberg, N.-E., 395,406, 408, 423 Wilkins, M.L., 81,122 Wilkinson, J.H., 593, 596 Will, G., 427, 451 Willam, K.J., 113, 125, 181,190 Williams, J.R., 247, 252, 254, 260, 265, 273, 274, 275, 276 Willis, J.R., 549, 587 Wilson, E.L., 18, 44, 153, 157, 361,379, 414, 424, 436, 440, 452, 459, 468,469,470, 474, 510, 516, 525,536, 544, 593, 596 Winnicki, L.A., 87, 122 Winslow, A.M., 117, 118, 126, 515,516 Withum, D., 365,368,379 Witmer, E.A., 454, 474 Witt, R.J., 1, 15, 22, 24, 44 Wittaker, E.T., 233, 244 Wohlmuth, B.I., 197, 218, 226, 227 Woinowski-Krieger, S., 284, 320, 323,325,333,376, 426, 450, 454, 455,467, 474 Wojtaszak, I.A., 348, 379 Wong, K., 228, 231,233,234, 243 Wood, R.D., 132, 134, 135, 146, 156, 159, 169, 176, 189, 230, 243, 336, 346, 378, 523,535,539, 543 Wren, J.R., 575, 589 Wriggers, P., 101,150, 154, 157, 191,197, 201,212, 213,218, 224, 226, 523,543, 570, 571,572, 589 Wu, J., 184, 190, 395,409, 423
Wu, S., 395,409, 423 Wunderlich, W., 515,516 Xie, Y.M., 94, 123 Xu, Z., 395,406, 407, 408,409, 415,423, 424 Yamada, Y.,80,121 Yang, H.ZY.,537,544 Yang, ZY.,539,545 Yeoh, O.H.,165,189 Yishimura, N.,80,121 Yoshimura, N.,535,544 Young, R.,256,274 Yu, J.,181,190,261,275 Yu, Q.,561,587 Zaman, M.,263,275 Zanisek, A.,365,380 Zfir~e, E, 371,374,381 Zarka, J.,103,124 Zavarise, G.,150,157,197,226,575,589 Zeng, L.E, 395,406,408,409,423 Zhang, H.W.,565,583,584,586,588,589 Zhang, X.,254,274 Zhong, H.G.,539,546 Zhong, W.X.,392,422,583,584,589 Zhong, Z.,191,225 Zhou, J.H.,263,275 Zhu, J.Z.,1,13,14,15,17,18,19,22,24,25,26,30, 36, 38, 44, 45, 57, 61, 65, 120, 127, 129, 137, 152,156,181,190,218,227,232,234,238,244, 267,276,290,318,320,324,334,340,341,346, 356,377,388,389,403,407,408,410,421,422, 425,427,440,441,450,458,461,465,466,471, 474,475,480,483,495,498,515,525,543,556, 587,592,596 Ziegler, H., 76, 121 Zienkiewicz, O.C., 1, 13, 14, 15, 17, 18, 19, 22, 24, 25, 26, 30, 36, 37, 38, 40, 41, 43, 44, 45, 55, 57, 58, 60, 61, 63, 65, 72, 74, 78, 80, 81, 85, 86, 87,93,94,96,101,103,104,105,106,108,112, 115,117,119,120, 121,122,123,124,125,126, 127,129,137,152, 156, 181,183, 184,190, 191, 218,224,227,232, 234, 238,244, 267,276, 290, 318,320,324,334, 335,336, 338,340, 341,344, 345,346, 348, 356, 360, 362,367,377, 378, 380, 387,388,389, 391, 392, 393,395,397, 400, 403, 406,407,408,409, 410, 418,419, 421,421,422, 423,424,425,427, 434, 436, 440, 441,442,450, 452,454,458,461, 465,466,467,469, 471,473, 473,474,475,476, 479, 480, 483,484, 488, 491, 492,495,496,498, 503,506, 507, 510, 514, 515, 515,516, 523,525,536, 537,539, 541,543,544, 545, 556, 587, 592, 596, 606, 608 Zohdi, T.I., 570, 571,589
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Subject index
Adaptive refinement and localization (slip line) capture, 180-8 based on norm error estimates, 181-3 using error indicators; discontinuity capture, 183-8 Almansi strain tensor, 131 Analogies, solutions by analogies, viscoelasticity, 72 Arches: clamped-hinged example, 523 dam shell example, 441-3 Arruda-Boyce model example, isotropic elasticity, 166 Asymptotic analysis/homogenization: about asymptotic analysis, 549-50 assumptions and problems, 550-1 balance equations, 550 boundary and discontinuity conditions, 551 bridging over several scales, 560-1 constitutive equations/relationships, 551,552 equilibrium equation, 552 finite element analysis applied to the local problem, 555-60 corrections for stresses and boundary effects, 558-60 example, 556-8 periodicity problems, 559-60 global solution, 553-4 heat balance equation, 552 homogenization formalization, 552 macroscopic and microscopic phenomena, distinguishing between, 550 non-linear case, 560-2 periodicity, 549-50 recovery of the micro description of the variables, 562-5 homogenization results, 565-7 Nb3Sn compound properties, 563-7 VAC strand analysis example, 563-5 strain definition, 551 stress vector, local approximation, 554 unsmearing technique, 548, 557 updating yield surface algorithm scheme, 562
Homogenization procedure; Multilevel procedures using homogenization Augmented Lagrangian form, node-node contact, 197 Axisymmetric problems, 10-12 Axisymmetric shells with non-symmetrical load, 510-14 Axisymmetric solids with non-symmetrical load, 507-10 see also
Balance mass principle, 132 Balance of momentum, equilibrium equations, 6 Barrel vault, shell example, 444-7 Beams, non-linear large displacements, 517-23 clamped-hinged arch example, 523 Euler-Bernoulli theory, 279, 284-5,287, 288, 306 formulation with small rotations, 522-3 non-zero Green-Lagrange strain expressions, 522 geometrically exact concept, 519 Piola-Kirchhoff stresses, 519-21 Timoshenko theory, 279, 289 two and three-dimensional theories, 518 Bergan method, with iterative techniques, 56-7 BFGS algorithm: wth computer procedures, 592 wth non-linear algebraic equations, 52 Bifurcation instability, thin plates, 535-7 Biharmonic equation of plate flexure, 331 Bishop-Hill relations, multilevel procedures, 569 Block deformability and the discrete element method: about block deformability, 260 block fracturing and fragmentation, 265-7 block interface constitutive law (Mohr-Coulomb), 262 bond failure, 266-7 combined finite/discrete element method (DEM/FEM), 247, 260-1 Cosserat continuum, 266 discontinuous deformation analysis (DDA), 261-5
620 Subjectindex Block deformability and the discrete element method - cont. discrete element deformation field, 260 interblock contact conditions of impenetrability, 264 Jaumann-Zaremba rate form, 261 Modal Expansion Discrete Element Method, 260 penalty format, 264 s e e a l s o Discrete element methods (DEMs) Boundary conditions: about boundary conditions, 6, 28 contact and the discrete element method, 256 displacement (essential) condition, 29-30 linear static problem, 29 finite deformation, geometrically non-linear problems, 133 with geometrically non-linear problems, 133 matrix notation, 9 mixed displacement/traction condition, 32 non-linear explicit problems, 30 non-linear implicit problems, 30 patch test for plate bending elements constraints, 395-6 rods, weak (Galerkin) forms, 286, 290 semi-analytical finite element processes using orthogonal functions and 'finite strip' methods, 498, 501 traction condition, 31 pressure loading, 31-2 Bounding surface plasticity models, 93 Boxes with flexure problems using 'finite strip' methods, 505-7 Brick elements, isoparametric formulations, 600-1 Lagrange family, 600 Brittle materials, 107-12 laminar material, 108-12 s e e a l s o No-tension material Buckling: about buckling, 1 Euler buckling-propped cantilever example, 525-6 plates, thin, large deflections, 536-7 Cartesian coordinate system, 5 Cartesian tensor indicial form, 4 Cauchy stress, 142, 146, 159-60, 176 Cauchy-Green deformation tensor, 131,158, 176 Clough and Tocher triangle 360, thin plates, 360 Combined discrete/finite element method (DEM/FEM), 247, 260-1 Computational modelling, 246 Computer procedures: about computer procedures, 590-1 s e e a l s o FEAPpv computer program Constitutive relations, 6-7 Constrained potential energy principle, 402
Constraint elimination approach, node-node contact, 195 Constraints, treatment of s e e Contact/contact problems; Tied interfaces Contact and the discrete element method: about contact constraints and boundary conditions, 256 about contact detection, 250-1 body based strategies, 251 body geometry characterization, 253-6 body-based search, 251 contact constraints between bodies, 256-9 contact constraints on model boundaries, 259-60 contact resolution, 252-6 corner-to-corner contact, 258-9 discrete functional representation (DFR), 254-6 finite wall (FW) method, 259-60 global neighbour or region search, 251-2 non-smooth contact, regularization of, 256-7 penalty function concept, 258 polygonal representation, 253-4 space-based search, 251 s e e a l s o Discrete element methods (DEMs) Contact/contact problems: about contact, 1,191-3 contact between disk and block example, 219-21 contact between two discs example, 219 frictional conditions, 192 frictional sliding of a flexible disk on a sloping block example, 221 frictionless condition, 192 impenetrability condition, 191 stick-slip-type response, 192 upsetting of a cylindrical billet example, 221-3 s e e a l s o Node-node contact: Hertzian contact; Node-surface contact; surface-surface contact Continuum-to-continuum linkage, 549 Convergence criteria, iterative techniques, 56-8 Cooling tower, shell example, 443-5 Coordinates and displacements, 5 Cosserat continuum, 266 Couplings s e e Multibody coupling by joints Crash analysis, 40, 42 Creep: basic problem formulation, 100-2 creep compliance functions, 64 fully explicit process with modified stiffness, 101-2 fully explicit solutions, 101-2 'initial strain' procedure, 101 of metals, 104-5 Norton-Soderberg creep law, 104 rate of creep strain, 100 s e e a l s o Viscoelasticity; Viscoplasticity Cylinder, edge loaded, example, 489
Subject index 621 d'Alembert principle, 138-9 rods, 282 Dams, shell examples: arched dams, 441-3 curved dams, 491-2 DEM/FEM (combined discrete/finite element method), 247, 260-1 s e e a l s o Discrete element methods (DEMs) Deviatoric and mean stress and strain components, 33 Dimension reduction for simplification s e e Semi-analytical finite element processes using orthogonal functions Dirac delta weighting, 398 Discontinuous deformation analysis (DDA), 229, 247, 261-5,267-8, 271-2 s e e a l s o Discrete element methods (DEMs) Discrete element methods (DEMs): about discrete element methods, 245-7, 272-3 associated discontinuous modelling methodologies, 270-1 computational modelling, 246 DEM algorithm, 249 discontinuous deformation analysis (DDA), 229, 247, 261-5, 267-8, 271-2 early formulations, 247-50 effective stiffness matrix, 268-9 modified distinct element method (MDEM), 270 non-smooth contact dynamics method (NSCDM), 270-1 predictor-corrector schemes, 269 time integration for, 267-70 unifying aspects, 271-2 s e e a l s o Block deformability and the discrete element method; Contact and the discrete element method Discrete functional representation (DFR), 254-6 Discrete Kirchhoff theory (DKT)/constraints: rods, 315-17 thick plates, 397, 405 thin plates, 369-72 Discrete-to-continuum linkage, 549 Displacement (irreducible) methods/models, 12, 13 Displacement-based finite element models, 142-3 Dome, spherical, under uniform pressure example, 488-9 Drilling degree of freedom, shells, 436 Drucker-Prager yield conditions, 86-8 Dummy index, indicial notation, 4-5 Earthquake response of soil, 40, 43 Effective stiffness matrix, 268-9 Eigensolutions, with FEAPpv computer program, 592-4 Elastic deviatoric stress, 172 Elastic moduli, 7
Elastic stability - energy interpretation, 523-6 Euler buckling-propped cantilever example, 525-6 neutral equilibrium, 525 stability criterion, 524 Elastic-plastic behaviour, 1 Elasticity: non-linear, 12-14 s e e a l s o Elastic stability- energy interpretation; Hyperelasticity; Isotropic elasticity; Non-linear and inelastic materials; Rods, basics; Viscoelasticity Elasto-plastic deformations, 112-16 concentrated discontinuity approach, 115 non-uniqueness and localization, 112-16 regularization process, 113, 115 Elasto-viscoplastic material, 102 Energy interpretation s e e Elastic stability - energy interpretation Energy-momentum conserving method, 28 Equilibrium equations: balance of momentum, 6 rods, 279-81 transient problems, 24 weak form, 21 Euler buckling-propped cantilever example, 525-6 Euler-Bernoulli theory: rods/beams, 279, 284-5,287, 288, 306 for thin plates, 324 Explicit methods, plasticity theory, 81-2 FE method, multilevel approach, 567-8 FEAPpv computer program: about computer procedures and the FEAPpv program, 590-1,595 eigensolutions, 592-4 non-linear problem solution, 591-2 restart option, 594-5 Finite deformation, about, 1 Finite deformation formulation, mixed-enhanced, 150-4 equations: matrix notation, 152-4 Pian-Sumihara plane elastic element, 152 Finite deformation formulation, three field, mixed, 145-9 hyperelastic problem, 145 matrix notation equations, 147-9 Newton scheme/linearization, 148 Finite deformation, geometrically non-linear problems: about geometrically non-linear problems, 127-8, 155-6 Almansi strain tensor, 131 balance of mass principle, 132 boundary conditions, 133 Cauchy stress, 142 Cauchy-Green deformation strain tensor, 131
622
Subjectindex Finite deformation, geometrically non-linear problems - c o n t . current configuration formulation, 140-3 current deformed configuration, 129 d'Alembert principle, 138-9 deformation gradient, 130, 138 displacement-based finite element models, 142-3 equilibrium equations, 132 finite element approximation, 137-8 finite element formulation, 141-3 forces dependent on deformation - pressure loads, 154-5 geometric stiffness, 139-41 Green strain/theorem, 130, 136, 137, 138 hyperelastic material, 134-5, 139 initial conditions, 133-4 kinematic relations, 128-31 Kronecker delta quantity, 129 load correction matrix, 154 matrix form, 136-7 Piola-Kirchhoff stress, 131,132, 134, 136-7 reference configuration formulation, 135-40 reference and deformed configurations, 127-8 Saint-Venant-Kirchhoff model, 134, 135, 164 stored energy function, 134 stress measures, 131 traction measures, 131-2 transient problems, 138-40 two dimensional forms: about two dimensional forms, 143 axisymmetric with torsion, 144-5 plain strain, 143-4 plain stress, 144 variational description, 135-43 Finite deformation, material constitution for: about material constitution for finite deformation, 158, 185 adaptive refinement and localization (slip line) capture, 180-8 based on energy norm error estimates, 181-3 adaptive refinement and localized (slip line) capture, using error indicators: discontinuity capture, 183-8 necking of a circular bar example, 178-80 s e e a l s o Hyperelasticity; Incremental formulations; Isotropic elasticity; Rate constitutive models 'Finite strip' methods s e e Semi-analytical finite element processes using orthogonal functions and 'finite strip' methods Finite wall (FW) method, 259-60 Flow rule (normality principle), plasticity theory, 74-6 Free index, indicial notation, 4-5 Friction: frictional materials, and plasticity, 92-4
frictional/frictionless conditions, 192 Node-surface contact, frictional case; Node-surface contact, frictionless case
see also
Galerkin method of approximation, 17-22 derivatives of displacements, 20 displacement approximation, 19-20 equation with node-surface contact, 215 incompatible approximations, 18-19 irreducible displacement method, 22 isoparametric form, 19-20 non-linear quasi-harmonic field problems, 37-8 strain-displacement equations, 20-1 stress divergence/stress force term, 21 variational crimes, 19 weak form, 18, 21 Gaussian quadrature formulae, shells, 483 General variational theorem, 12-14 Geometric modelling: node-node contact, 193-4 node-surface contact, 200-5 Geometric stiffness, 139-41 Geometrically exact concept, beams, 519 Geometrically non-linear problems s e e Finite deformation, geometrically non-linear problems GN22 method: explicit GN22 method, 25-6 implicit GN22 method, 26-7 strain energy density, 27 and the tangent modulus matrix, 27 and the tangent stiffness matrix, 27 Green strain, 130, 137, 138 Green-Lagrange strains: axisymmetric shells, 538 non-linear structures, 522 plates, thick, large displacement theory, 528 Green's theorem, 136 Guassian quadrature, 22-3 Hellinger-Reissner principle/form, 13 Rods, 285-6, 287, 289 thin plates, 367 Hermitian interpolations: axisymmetric shells, 457,463 beam displacements, 290-2 Hermitian rectangular shape function, thin plates, 363-4 Hertzian contact s e e Node-node contact: Hertzian contact Heterosis element, 389 Homogenization procedure: constitutive law at the macro level, 580 definition of successive yield surfaces, 578-9 elastic constitutive tensor, 581 global elastic-plastic analysis, 580-1 global solution and stress-recovery, 581-6
Subject index 623 numerical examples, elasto-plastic materials with various frictional type contact, 582-6 see also Asymptotic analysis/homogenization; Multilevel procedures using homogenization Hu-Washizu functional, thin plates, 371 Hu-Washizu theorem, 13 Huber-von Mises yield conditions, 86-8 Huber-von Mises-type material, thin plates, 376 Hyperelasticity: about hyperelasticity, 158 hyperelastic materials, 7, 134-5, 139, 145 see also Isotropic elasticity; Isotropic viscoelasticity Impenetrability condition, 191 Incremental formulations, 174-6 incremental deformation gradient, 174-5 incremental Saint-Venant-Kirchhoff model example, 175-6 Incremental and rate methods, non-linear equation solution, 58-9 Incremental-secant or quasi-Newton iterative methods, 51-3 BFGS update, 52 DFP update, 52 direct (Picard) iteration, 53 Indicial notation, 4-7 Inelastic material behaviour, thin plates, 374-6 Inelastic and non-linear materials: about inelastic materials, 62 see also Brittle material; Elasto-plastic deformations; No-tension material; Plasticity theory; Viscoelasticity; Viscoplasticity Initial conditions, 6 Initial stress matrix for plate bending, 532 Initial stress/stress transfer method, 50 Integral equation model, viscoelasticity, 67 Invariants of second-order tensors: derivatives of invariants: about derivatives, 606 first derivatives, 606-7 second derivatives, 607 moment invariants, 605-6 principal invariants, 604-5 Irreducible (displacement) models/methods, 12, 13, 22 see also Mixed or irreducible forms Isoparametric finite element approximations/formulations: about isoparametric formulations, 597 axisymmetric shells, 465 brick elements, 600-1 Galerkin method, 19-20 quadrilateral elements, 597-600 tetrahedral elements, 602-3 triangular elements, 601-2
Isotropic elasticity: An~da-Boyce model example, 166 compressible neo-Hookean material example, 162-3 formulation in invariants, 158-64 formulation in modified invariants, 165-6 formulation in principal stretches, 166-9 logarithmic principal stretch form example, 168-9 modified compressible neo-Hookean material example, 163-4 Newton-type solution, 160 plane stress applications, 169-71 plasticity models, 173-4 push forward transformation, 160-1 volumetric behaviour example, 161-2 Yeoh model example, 165-6 Isotropic hardening, 74, 76, 179 Isotropic materials, 7 Isotropic plasticity models, 85-91 isotropic yield surfaces, 85-7 Isotropic viscoelasticity, 172-3 elastic deviatoric stress, 172 Iterative techniques: about iterative techniques, 47-8 acceleration of convergence, 53-4 Bergan method, 56-7 convergence criteria, 56-8 incremental-secant or quasi-Newton methods, 51-3 line search procedures, 53-4 modified Newton's method, 49-51 Newton's (Newton-Raphson) method, 48-9 softening behaviour and displacement control, 55-6 JU2u model with isotropic and kinematic hardening, 87-90 JU2u plane stress, 90-1 Jacobian matrix: quadrilateral elements, 600 shells, 482 Jacobian transformation: Galerkin method, 20 and volume integrals, 23-4 Jaumann-Zaremba: rate form, 261 stress rate, 178 Joints: library of, 239-40 see also Multibody coupling by joints; Pseudo-rigid bodies; Rigid motions; Rigid-flexible bodies Kelvin model, viscoelasticity, 64 Kinematic hardening, 74, 76-8 Kirchhoff stress, 146
624
Subjectindex Kirchhoff-Love assumption, axisymmetric shells, 455-6 Koiter treatment for multi-surface plasticity, 104 Kronecker delta function/quantity, 7, 33, 129 Kuhn-Tucker form, 75 Lagrange family of elements, isoparametric formulations, 597-8, 600 Lagrange multiplier constraints: multibody coupling, 237-8 rigid to flexible bodies, 234-6 Lagrange multiplier form/approach: augmented Lagrangian form, 197 node-node contact, 194-5 tied interfaces, 198 Lam6 elastic parameters, 7 Laminar material, 108-12 Large displacements s e e Beams, non-linear large displacements; Plates, thick, large displacement theory; Plates, thin, large displacement theory Laurant transformation, 13 Lie derivative, 178 Line search procedures, iterative techniques, 53-4 Load bifurcations, 1 s e e a l s o Buckling Load correction matrix, 154 Localization, plasticity theory, 113 Localization technique, 548 McHenry-Alfrey analogies, Viscoelasticity, 72 Macroscopic phenomena s e e Asymptotic analysis/homogenization; Multiscale modelling Material constitution for finite deformation s e e Finite deformation, material constitution for Matrix notation, 7-9 for boundary conditions, 9 constitutive equations, 9 for coordinates, 7 for displacements, 8 for strains, 8 tensor/matrix index relations, 8 with transformations, 8 for transient problems, 9 Maxwell model, viscoelasticity, 63-4 'Membrane' stresses, thin plates, 534 Metals, creep, 104-5 Microscopic phenomena s e e Asymptotic analysis/homogenization; Multiscale modelling Mixed displacement/traction condition, boundary conditions, 32 Mixed or irreducible forms, 33-7 deviatoric and mean stress and strain components, 33 linear elastic tangent example, 36-7 three-field mixed method for general constitutive models, 34-7
Mixed models, 13 Modal Expansion Discrete Element Method, 260 Modified distinct element method (MDEM), 270 Mohr-Coulomb yield law/conditions/surface, 86-8, 106, 112-13 Mortar methods, surface-surface contact, 219 Multibody coupling by joints: about joints, 237 beam with attached mass example, 240-3 Lagrange multiplier constraint, 237-8 library of joints, 239-40 rotating disc example, 240-1 rotation constraints, 239 translational constraints, 237 s e e a l s o Joints; Rigid motions; Rigid-flexible bodies Multilevel procedures using homogenization, 567-70 Bishop-Hill relations, 569 discrete-to-continum linkage, 572-7 RVE and boundary conditions, 575-7 FE method, 567-8 general first and second-order procedures, 570-2 local analysis of a single cell, 578 representative volume element (RVE), 568, 571, 575-6 RVE under different boundary conditions example, 571-2 Multiscale modelling: about multiscale modelling, 547-9, 586 s e e a l s o Asymptotic analysis/homogenization; Homogenization procedure; Multilevel procedures using homogenization Necking of a circular bar example, 178-80 Neo-Hookean material, compressible, 162-3 modified, 163-4 Neutral equilibrium, elastic stability, 525 Newton scheme/linearization, 148 Newton-type solution/strategy: isotropic elasticity, 160 viscoelasticity, 66 Newton's (Newton-Raphson) iterative method for non-linear equations, 48-9 modified Newton's method, 49-51 stress transfer/initial stress method, 50 zone of attraction, 48, 50-1 No-tension material, 107-8 reinforced concrete example, 108, 110 underground power station example, 108-9 Node-node contact: Hertzian contact: augmented Lagrangian form, 197 constraint elimination approach, 195 geometric modelling, 193-4 Lagrange multiplier form/approach, 194-5 master nodes, 193 penalty function form, 197
Subject index 625 perturbed Lagrangian form, 195-6 perturbed tangent method, 196 slave nodes, 193 see also Contact/contact problems Node-surface contact, frictional case, 212-18 Galerkin equation, 215 incremental tangential slip, 214 residual and tangent for slip - normal to master surface example, 217-18 residual and tangent for stick - normal to master surface example, 216-17 sliding state, 215-16 stick condition, 213-15 see also Contact/contact problems Node-surface contact, frictionless case, 205-12 contact forces for normal to 2D master surface example, 207 contact forces for normal to 2D slave surface alternative form example, 208-9 contact forces for normal to 2D slave surface example, 207-8 contact residual, 206-7 contact tangent, 209-12 perturbed Lagrangian (penalty) form, 205-6 tangent for 2D linear master surface example, 210-11 tangent for normal to 2D slave surface alternative form example, 212 tangent for normal to 2D slave surface example, 211 see also Contact/contact problems Node-surface contact, geometric modelling, 200-5 master surface, 200 Newton solution method, 202 normal and tangent vector definitions, 202-4 normal vector to 2D linear master facets example, 204 normal vector to 2D linear slave facets example, 204-5 slave node, 200 tangent arrays, 204 see also Contact/contact problems Non-linear effects: crash analysis, 40, 42 earthquake response of soil, 40, 43 geometric non-linearity, 1 material non-linearity, 1 non-linear elasticity, 12-14 non-linear quasi-harmonic field problems, 37-8 transient calculation examples, 38-43 transient heat conduction, 38-41 Non-linear equation solution: about solution of algebraic equations, 46-7 direct integration procedure, 59 incremental and rate methods, 58-9 see also Iterative techniques
Non-linear geometrical problems see Finite deformation, geometrically non-linear problems Non-linear and inelastic materials: magnetic field in six-pole magnet example, 117-18 non-linear quasi-harmonic field problems, 116-18 spontaneous ignition example, 117-18, 119 see also Brittle material; Elasto-plastic deformations; No-tension material; Plasticity theory; Viscoelasticity; Viscoplasticity Non-linear large displacement problems: about non-linear problems, 517, 542-3 see also Beams, non-linear large displacements; Plates, thick, large displacement theory; Plates, thin, large deflection problems; Plates, thin, large displacement theory; Shells, non-linear response and stability problems Non-linear problems, solution with FEAPpv computer program, 591-2 BFGS algorithm, 592 Non-smooth contact dynamics method (NSCDM), 247,270-1 see also Discrete element methods (DEMs) Normality principle, plasticity theory, 75 Norton-Soderberg creep law, 104 Numerical integration see Quadrature Objective time derivative, 176 Orthogonal functions, use of see Semi-analytical finite element processes using orthogonal functions Parabolic type elements using reduced integration, 507 Patch test for plate bending elements: boundary constraints, 395-6 bubble mode approach, 396 design of some useful elements, 395-6 numerical patch test, 394-5 thin plates, 346-8 triangular mixed elements, 396 why elements fail, 392-6 Penalty function concept, and the discrete element method, 258 Penalty function form, node-node contact, 197 Periodicity, 549-50 Perturbed Lagrangian form, node-node contact, 195-6, 205-6 Perturbed tangent method, 196 Phenomenological constitutive equations, 547 Pian-Sumihara plane elastic element, 152 Piola-Kirchhoff stresses: beams, non-linear large displacements, 519-20 geometrically non-linear problems, 131,132, 134, 136-7 isotropic elasticity, 158-9
626
Subjectindex Piola-Kirchhoff stresses - cont. isotropic viscoelasticity, 172 mixed finite deformation formulation, 146, 150, 152 rate constitutive models, 177 Pipe penetration and spherical cap example, 492-5 Plane stress and plane strain, 9-10 traction vector for, 10 Plastic computation examples: about plastic computation, 95 perforated plate - plane strain solutions, 97, 98 perforated plate - plane stress solutions, 96-7 steel pressure vessel, 97-9 Plastic flow rule potential, 75-6 Drucker-Prager yield conditions, 86-8 Plasticity models, 173-4 Plasticity theory: about plasticity, 72-4 associative/non-associative plasticity, 76, 84, 92-5 bounding surface plasticity models, 93 computation of stress increments, 80-4 continum rate form, 87-9 discrete time integration algorithm, 80 elasto-plastic matrix, 79-80 explicit computational methods, 81-2 flow rule (normality principle), 74-6 frictional materials, 92-4 generalized plasticity, 92-5 hardening/softening rules, 76-8 Huber-von Mises yield conditions, 86-8 implicit computational methods, 82-3 incremental return map form, 89-90 isotropic hardening, 74, 76, 78 isotropic plasticity models, 85-91 isotropic yield surfaces, 85-7 JU2u generalized plasticity, 94-5 JU2u model with isotropic and kinematic hardening, 87-90 JU2u plane stress, 90-1 kinematic hardening, 74, 76-8, 79, 87 limit plastic state, 95 Mohr-Coulomb yield conditions, 86-8, 106, 112-13 normality principle, 75 overlay models, 78 plastic flow rule potential, 75-6 plastic stress-strain relations, 78-80 Prandtl-Reuss equations, 87-90 principle of maximum plastic dissipation, 77 return map algorithm, 82-3 Runge-Kutta process, 81 Sherman-Morrison-Woodbury formula, 93 softening behaviour, 80 subincrementation, 81 Tresca yield conditions, 86-8 yield functions, 74 see also Elasto-plastic deformations
Plate, folded, shell example, 447-50 Plates, basics/general: about plates, 2 about thin plate theory, 323-5,376 adaptive refinement, 420-1 basic assumptions/postulates on deformation and stresses, 325-7 biharmonic equation of plate flexure, 331 boundary conditions, 327-8, 331-2 composites, 330 discrete Kirchhoff constraints, 369-72 equilibrium equations, 326-7 general theory, 328-9 governing equations, 325-31 Hellinger-Reissner principle, 367 hybrid plate elements, 368-9 inelastic material behaviour, 374-6, 419-20 numeric integration through thickness, 374-6 resultant constitutive models, 376 irreducible, thin plate approximation, 332-4 mixed formulations, 366-8 moment components and displacement derivatives, 329-30 rotation-free elements, 371-4 Hu-Washizu functional, 371 shape function continuity requirements, 334-6 strain-displacement relations, 330-1 and thick plates/thick plate theory, 323-4 thin plate theory limitations, 415-19 Plates with flexure problems using 'finite strip' methods, 505-7 Plates, perforated: plane strain solutions, 97, 98 plane stress solutions, 96-7 Plates, thick: about thick Reissner-Mindlin plates, 382-5 basic equations, 382-5 clamped and simply supported, 415-19 discrete 'exact' thin plate limit, 413-15 elements with discrete collocation constraints: collocation constraints for triangular elements, 403-5 constrained potential energy principle, 402 Dirac delta weighting, 398 element matrices for discrete collocation constraints, 401-3 general possibilities - quadrilaterals, 397-401 patch count, degrees of freedom, 400 relation to the discrete Kirchhoff formulation, 403 elements with rotational bubble or enhanced modes, 405-8 linear triangular element T3S 1B 1,406-7, 409 quadratic element T6S3B3, 406-7 single bubble mode Q4S 1B 1,408, 409 hard and soft simple support, 415-19
Subject index 627 irreducible system/formulation- reduced integration, 385-9 heterosis element, 389 span to thickness variation effects, 387-9 total potential energy principle, 385 linked interpolation: about linked interpolation, 408-9 clamped plate by linked elements example, 413-14 derivation of linked function for three-node triangle, 410-11 linking function for linear triangles and quadrilaterals, 409-11 for quadratic elements, 411-13 Zienkiewicz-Lefebvre element, 412 mixed formulation, 390-2 approximation considerations, 390-1 continuity requirements, 391-2 equivalence of mixed forms with discontinuous S interpolation and reduced (selective) integration, 392 Reissner-Mindlin plate theory, 382, 409 special case of three dimensional analysis, 487 support condition considerations, 415-19 see also Patch test for plate bending elements; Plates, basics/general Plates, thick, large displacement theory, 526-32 definitions, 526-9 evaluation of tangent matrix, 530-2 finite element evaluation of strain-displacement matrices, 529-30 Green-Lagrange strains, 528 initial stress matrix for plate bending, 532 rotation considerations, 526-7 Plates, thin, conforming shape functions with additional degrees of freedom: 21 and 18 degree-of-freedom triangle, 364-6 Hermitian rectangular shape function, 363-4 Plates, thin, conforming shape functions with nodal singularities: 18 degree-of-freedom triangular element, 360 about conforming shape functions, 357 Clough and Tocher triangle, 360 compatible quadrilateral elements, 361-2 quasi-conforming elements, 362-3 singular shape functions for the simple triangular element, 357-60 Plates, thin, large deflection problems: about large deflections, 534 bifurcation instability, 535-7 buckling, 536-7 clamped plate with uniform load example, 535 'membrane' stresses, 534 Plates, thin, large displacement theory, 532-4 evaluation of strain-displacement matrices, 533 evaluation of tangent matrix, 534
Plates, thin, non-conforming shape functions: examples: comparison of convergence behaviour, 349, 352-6 deflections and moments for clamped square plate, 348-9 energy convergence in a skew plate, 349, 356 skewed slab bridge, 349, 350, 351 patch test, 346-8 quadrilateral and parallelogram elements, 340 rectangular element with comer nodes, 336-9 stiffness and load matrices, 339 uniform load on 12 DOF rectangle example, 339 triangular element with comer nodes, 340-6 triangular element of the simplest form, 345-6 Polar decomposition on the deformation tensor, 230 Potential energy principle, 12 Prandtl-Reuss equations, 87-90 Pressure vessel, steel, plastic computation example, 97-9 Prismatic bar problem, use of orthogonal functions, 504-5 Prony series for integral equation solution, 67-70 Pseudo-rigid bodies: about pseudo-rigid bodies, 228 deformation gradient, 229 discontinuous deformation analysis, 229 homogeneous motion, 229 pseudo-rigid motions, 228-9 St Venant-Kirchhoff relation, 229 see also Rigid motions; Rigid-flexible bodies Push forward transformation, isotropic elasticity, 160-1 Quadrature: Gaussian quadrature, 22-3 reduced quadrature, 24 selective reduced integration, 24 surface integrals, 24 and volume integrals, 23-4 Quadrilateral elements, isoparametric formulations, 597-600 Jacobian matrix, 600 Lagrangian interpolation functions/family, 597-8 natural coordinates, 597-8 node numbering, 599 Quasi-Newton or incremental-secant iterative methods, 51-3 BFGS update, 52 DFP update, 52 direct (Picard) iteration, 53 Rate constitutive models, 176-8 Jaumann-Zaremba stress rate, 178 Lie derivative, 178 objective time derivative, 176
628
Subjectindex Regularization process, elasto-plastic deformations, 113, 115 Reinforced concrete, no-tension material example, 108, 110 Reissner-Mindlin assumptions, shells, 475 Reissner-Mindlin theory, thick plates s e e Plates, thick Representative volume element (RVE), 568, 571, 575-6 Return map algorithm, plasticity theory, 82-3 Right Riemann (discontinuous deformation analysis), 267-8 Rigid block spring method (RBSM), 247 s e e a l s o Discrete element methods (DEMs) Rigid motions: about rigid motions, 230-1 construction from a finite element model, 232-3 equations of motion for a rigid body, 231-2 polar decomposition on the deformation tensor, 230 transient solutions, 233-4 s e e a l s o Multibody coupling by joints; Pseudo-rigid bodies; Rigid-flexible bodies Rigid-flexible bodies: about rigid-flexible bodies, 228 connecting rigid to flexible bodies, 234-6 Lagrange multiplier constraints, 234-6 s e e a l s o Pseudo-rigid bodies Rods, basics: about rods, 278-9 area inertia, 283 axial forces, 280 bending moment resultants, 281 correction factor, 284 d'Alembert principle, 282 elastic axial stiffness, 283 elastic bending stiffness, 283 elastic constitutive relations, 283-5 elastic torsional stiffness, 284 equilibrium equations, 279-81 Euler-Bernoulli theory, 279, 284 forms without rotational parameters, 317-19 kinematics, 281-2 length parameter considerations, 279 moment resisting frames, 319-20 polar inertia, 283 Timoshenko theory, 279 transient behaviour, 282-3 transverse shear forces, 280 Rods, curved s e e Shells, axisymmetric Rods, Euler-Bernoulli:finite element solution, 290-305 axial deformation- irreducible form example, 294-5 axial stiffness and load arrays example, 295 beam bending- irreducible form, 290-4
beam bending - mixed form, 296-9 beam bending element array example, 297-8 beam cantilever tapered bar example, 298-9 bending stiffness, mass and load matrices example, 294 Hermite interpolation for beam displacement, 290-3 inelastic behaviour of rods, 299-305 irreducible form, 301-2 mixed form, 302-4 simply supported beam with point load example, 304-5 stiffness and load arrays, 292-3 strain-displacement relations, 292 torsion deformation - irreducible form, 295-6 torsion stiffness and load arrays example, 296 Rods, Timoshenko form/theory: finite element solution, 305-17 beam bending - irreducible form, 305-11 cantilever tapered bar example, 313-15 constant strain form, 310 discrete Kirchhoff constraints, 315-17 one dimensional beam example, 315-17 enhanced assumed strain form, 311 equal order interpolation, 306 Euler-Bernoulli theory, 306 exact nodal solution form, 308-9 linear interpolation example, 306-7 mixed form, 311-15 mixed stiffness matrix example, 312-13 stiffness/load arrays for uniform beam example, 309-10 stiffness/load for uniform constant strain beam example, 310-11 Rods, weak (Galerkin) forms: about weak forms, 285 axial weak form, 285-6 bending weak forms, 287-90 boundary conditions, 286, 290 Euler-Bernoulli theory, 287, 288 Hellinger-Reissner form, 285,286, 287, 289 Timoshenko beam theory, 289 torsion weak form, 286-7 Rotation constraints, multibody coupling, 239 revolute joint example, 239 Runge-Kutta process, 81 St Venant-Kirchhoff relation, 229 Saint-Venant-Kirchhoff model, 134, 135, 164 incremental model example, 175-6 Self-consistent methods, 549 Semi-analytical finite element processes using orthogonal functions and 'finite strip' methods: about using 'finite strip' methods, 506 about using orthogonal functions, 498-501, 514-15
Subject index axisymmetric shells with non-symmetrical load: thick case - with shear deformation, 513-14 thin case- no shear deformation, 510-13 axisymmetrical solids with non-symmetrical load, 507-10 torsion/torsionless problem, 509 boundary condition considerations, 498, 501 linear elastic material requirements, 499 with parabolic type elements, 507 plates and boxes with flexure, 505-7 prismatic bar - reduce three dimensions to two, 501-4 boundary conditions, 503 stiffness matrix, 503 thin membrane box structures - reduce two dimensions to one, 504-5 Shape functions, non-conforming, thin plates, 334-6, 336-56 Shear relaxation modulus function, viscoelasticity, 67 Shells as an assembly of flat elements: about shells, 2, 426-8 arch dam analysis, 436 assembly of elements, 429-31 axisymmetrical shells, 427 centre displacement parameters, 439 curved element development, 428 drilling degree of freedom, 435-40 element choice, 440-1 elements with mid-side slope only, 440-1 flat element approximation, 427 local direction cosines, 431-5 local and global coordinates, 430 membrane error terms, 437 mixed patch test, 440 plane stress problem, 439-40 rectangular elements, 431-2 rotational stiffness, 435-40 'shallow' shell theory approach, 427 skew symmetric stress, 439-40 spherical problem, 437 stiffness matrix, 429-31 stiffness of a plane element in local coordinates, 428-9 transformation to global coordinates, 429-31 triangular elements arbitarily orientated, 432-5 Shells, axisymmetric: about axisymmetric shells, 454 accuracy examples, 459-61 curved elements, 461-70 additional nodeless variables, 465-8 branching shell, 470 Hermitian interpolations, 463 shape function, 462-4 spherical dome, 467-8 stiffness matrix, 465
strain expressions and properties, 464-5 toroidal shell, 468-9 element characteristics - axisymmetric loads, 456-8 enhanced strain mode added, 459, 466 Hermitian interpolations, 457-8 independent slope-displacement interpolation with penalty functions, 468-73 bending of a circular plate, 472-3 penalty functional, 471 isoparametric form, 465 Kirchhoff-Love assumption, 455-6 loading, displacements and stress resultants, 454-6 with non-symmetrical load, 510-14 see also Semi-analytical finite element processes using orthogonal functions and 'finite strip' methods Shells, element with displacement and rotation parameters: about omitting the third constraint of thin plate theory, 475-6 convergence, 487-9 curved thick shell elements, various, 477-8 definition of strains and stresses, 480-1 displacement field, 479-80 element properties, 481-3 Gaussian quadrature formulae, 483 geometric definition of an element, 477-9 Jacobian matrix, 482 local and global coordinates, 478 Reissner-Mindlin assumptions, 475 shell director, 478 stress presentation, 483-4 thick plate special case, 487 transformations required, 482-3 Shells, examples: arch dam shell, 441-3 barrel vault, 444-7 cooling tower, 443-5 curved dams, 491-2 cylindrical vault, 489-91 edge loaded cylinder, 489-90 folded plate structure, 447-50 pipe penetration and spherical cap, 492-5 spherical dome under uniform pressure, 488-9 Shells, non-linear response and stability problems: about non-linearity and stability, 537 axisymmetric shells, 538-40 Green-Lagrange strain expressions, 538 shallow shells - co-rotational forms, 539-40 stability, 541-2 Shells, as a special case of three-dimensional analysis: about three-dimensional analysis of shells, 475, 493-5 see also Shells, element with displacement and rotation parameters
629
630
Subjectindex Shells, thick, axisymmetric, curved, 484-6 axisymmetric loading, 485-6 global displacements, 485-6 inelastic behaviour, 488 Sherman-Morrison-Woodbury formula, 93 Soil mechanics, and viscoplasticity, 105-7 Solid mechanics, about solid mechanics problems, 1-4 Stability criterion, elastic stability, 524 Stability, elastic s e e Elastic stability - energy interpretation Stability of shells, 541-2 Stick-slip-type response, 192 Stiffness, linear stiffness matrix, 22 Stored energy function, 134 Strain: plane strain, 9-10 strain energy density, 27 Strain-displacement equations, Galerkin method, 20-1 Strain-displacement relations, 5 Strain-energy density function, 7 Strains, virtual, 14-15 Stress, plane stress, 9-10 Stress divergence/stress force term, Galerkin method, 21 Stress transfer/initial stress method, 50 Subincrementation, plasticity theory, 81 Surface integrals, 24 Surface-surface contact, 218-19 mortar methods, 219 s e e a l s o Contact/contact problems Tangent moduli, with viscoelasticity, 66 Tangent vectors/arrays, 199-200 Tensor/matrix index relations, 8 Tetrahedral elements, isoparametric formulations, 602-3 Thick plates/thick plate theory s e e Plates, basics/general; Plates, thick 'Thin' geometry, 1 Thin plates/thin plate theory s e e plates, basics/general; Plates, thin Three-field mixed method for general constitutive models, 34-7 Tied interfaces, 197-200 Lagrange multiplier functional, 198 two-dimensional using linear elements example, 199-200 Timoshenko theory: beams/rods, 279, 289 s e e a l s o Rods, Timoshenko form/theory Torsion/torsionless problems, 11 Total potential energy principle, 385 Traction condition, boundary conditions, 31-2 Traction vector for plane problems, 10
Transient problems: discrete approximation in time, 24 energy-momentum conserving method, 28 generalized mid-point implicit form, 28 non-linear calculation examples, 38-40 non-linear and steady state, 24-8 s e e a l s o Gn22 method transient heat conduction, 38-41 Translational constraints, multibody couplings, 237 Tresca yield conditions, 86-8 Triangular elements, isoparametric formulations, 601-2 Tyre analysis, 1-2 Unsmearing technique, 548, 557 Variational crimes, Galerkin method, 19 Variational forms for non-linear elasticity, 12-14 Virtual strains, 14-15 Viscoelasticity: aging effects, 66 creep compliance functions, 64 differential equation model, 65 history dependence, 63 integral equation model, 67 solution with Prony series, 67-70 isotropic models, 64-72 Kelvin model, 64 generalized, 64 linear models for, 63-4 McHenry-Alfrey analogies, 72 Maxwell model, 63-4 generalized, 63-4, 65, 68 non-linear effects, 66 Prony series for integral equation solution, 67-70 relaxation times, 63, 65 retardation time parameters, 64 shear relaxation modulus function, 67 solution by analogies, 72 strain-driven solution form, 65 tangent moduli, 66 thick-walled cylinder subject to internal pressure example, 70-2 s e e a l s o Creep; Plasticity theory Viscoplasticity: about viscoplasticity, 102-3 elasto-viscoplastic material, 102 iterative solution, 103-4 Koiter treatment for multi-surface plasticity, 104 soil mechanics applications, 105-7 viscoplastic (or creep) strain rate, 102 s e e a l s o Creep; Plasticity theory Volume integrals, 23-4 Volumetric behaviour example, isotropic elasticity, 161-2
Subject index 631 Weak forms: for equilibrium equation, 14-15 with Galerkin method of approximation, 21 of governing equations, 14-15
Yeoh model example, isotropic elasticity, 165-6 Yield functions, plasticity theory, 74 Zienkiewicz-Lefebvre element, 412 Zone of attraction, 48, 50-1
This Page Intentionally Left Blank
(a) Geometrical model of an arch dam
(b) Finite element discretization of an arch dam including the foundation
Plate 1 Three dimensional non-linear analysis of a dam
Courtesy of Prof. Miguel Cervera, CIMNE, Barcelona. Source: B. Suarez, M. Cervera and J. Miguel Canet, 'Safety assessment of the Suarna arch dam using non-linear damage model', Proc. Int. Sym. New Trends and Guidelines on Dam Safety, Barcelona, Spain, 1998. L. Berga (ed.) Balakema, Rotterdam.
(a) (a) Stamping Stamping die for an automotive structural component component
(b) Die (black) and sheet (green) discretization into 23522 rotation discretization triangles free BST shell triangles
(c) Thickness Thickness ratio ratio contours contours of of the the (c) sheet at at 17mm 17mm punch punch travel travel sheet
Plate ,ing analysis Plate 22 Non-linear Non-linear mete,, metal ..... forming analysis of of aa car car door door
Courtesy Courtesy of of Prof. Prof. E. E. OSate, Onate, CIMNE CIMNE and and DECAD DECAD S.A S.A Barcelona. Barcelona. Source: Source: E. E. OSate, Onate, F. F. Zarate, Zarate, J. J. Rojek, Rojek, G. G. Duffet, Duffet, L. L. Neamtui, Neamtui, 'Adventures 'Adventures in in rotation rotation free free elements elements for for sheet sheet stamping stamping analysis'. analysis'. 4th 4th Int. In!. Conf. Conf. Workshop Workshop on on Numerical Numerical Simulation of 3D Sheet Forming Processess (NUMISHEET' 99, Besancon, France, Sept 13-17, 1999) Simulation of 3D Sheet Forming Processess (NUMISHEET' 99, Besancon, France, Sept 13-17, 1999)
(d) Thickness ratio contours at 35mm punch travel
~
il,
y
(e) Thickness ratio contours at 54mm punch travel
(f) Thickness ratio contours at 72mm punch travel Plate 2 continued Non-linear metal forming analysis of a car door
Plate 3 Car crash analysis
Frontal crash of a Neon Car performed using LS-DYNA. Courtesy of Livermore Software Technology Corporation. Model developed by FHWA/NHTSA National Crash Analysis Center of the George Washington University
The Finite Element Method: Its Basis and Fundamentals Sixth edition
Professor O.C. Zienkiewicz, CBE, FRS, FREng is Professor Emeritus at the Civil and Computational Engineering Centre, University of Wales Swansea and previously Director of the Institute for Numerical Methods in Engineering at the University of Wales Swansea, UK. He holds the UNESCO Chair of Numerical Methods in Engineering at the Technical University of Catalunya, Barcelona, Spain. He was the head of the Civil Engineering Department at the University of Wales Swansea between 1961 and 1989. He established that department as one of the primary centres of finite element research. In 1968 he became the Founder Editor of the International Journal for Numerical Methods in Engineering which still remains today the major journal in this field. The recipient of 27 honorary degrees and many medals, Professor Zienkiewicz is also a member of five academies- an honour he has received for his many contributions to the fundamental developments of the finite element method. In 1978, he became a Fellow of the Royal Society and the Royal Academy of Engineering. This was followed by his election as a foreign member to the U.S. Academy of Engineering (1981), the Polish Academy of Science (1985), the Chinese Academy of Sciences (1998), and the National Academy of Science, Italy (Academia dei Lincei) (1999). He published the first edition of this book in 1967 and it remained the only book on the subject until 1971. Professor R.L. Taylor has more than 40 years' experience in the modelling and simulation of structures and solid continua including two years in industry. He is Professor in the Graduate School and the Emeritus T.Y. and Margaret Lin Professor of Engineering at the University of California at Berkeley. In 1991 he was elected to membership in the US National Academy of Engineering in recognition of his educational and research contributions to the field of computational mechanics. Professor Taylor is a Fellow of the US Association of Computational Mechanics - USACM (1996) and a Fellow of the International Association of Computational Mechanics - IACM (1998). He has received numerous awards including the Berkeley Citation, the highest honour awarded by the University of California at Berkeley, the USACM John von Neumann Medal, the IACM Gauss-Newton Congress Medal and a Dr.-Ingenieur ehrenhalber awarded by the Technical University of Hannover, Germany. Professor Taylor has written several computer programs for finite element analysis of structural and non-structural systems, one of which, FEAP, is used world-wide in education and research environments. A personal version, FEAPpv, available from the publisher's website, is incorporated into the book. Dr J.Z. Zhu has more than 20 years' experience in the development of finite element methods. During the last 12 years he has worked in industry where he has been developing commercial finite element software to solve multi-physics problems. Dr Zhu read for his Bachelor of Science degree at Harbin Engineering University and his Master of Science at Tianjin University, both in China. He was awarded his doctoral degree in 1987 from the University of Wales Swansea, working under the supervision of Professor Zienkiewicz. Dr Zhu is the author of more than 40 technical papers on finite element methods including several on error estimation and adaptive automatic mesh generation. These have resulted in his being named in 2000 as one of the highly cited researchers for engineering in the world and in 2001 as one of the top 20 most highly cited researchers for engineering in the United Kingdom.
The Finite Element Method: Its Basis and Fundamentals Sixth edition O.C. Zienkiewicz, CBE, FRS
UNESCO Professor of Numerical Methods in Engineering International Centre for Numerical Methods in Engineering, Barcelona Previously Director of the Institute for Numerical Methods in Engineering University of Wales, Swansea
R.L. Taylor
Professor in the Graduate School Department of Civil and Environmental Engineering University of California at Berkeley Berkeley, California
J.Z. Zhu
Senior Scientist ESI US R & D Inc. 5850 Waterloo Road, Suite 140 Columbia, Maryland
ELSEVIER BUTTERWORTH HEINEMANN
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Elsevier Butterworth-Heinemann Linacre House, Jordan Hill, Oxford OX2 8DP 30 Corporate Drive, Burlington, MA 01803 First published in 1967 by McGraw-Hill Fifth edition published by Butterworth-Heinemann 2000 Reprinted 2002 Sixth edition 2005 Copyright 9 2000, 2005. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. All rights reserved The rights of O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu to be identified as the authors of this work have been asserted in accordance with the Copyright, Designs and Patents Act 1988 No part of this publication may be reproduced in any material form (including photocopying or storing in any medium by electronic means and whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90, Tottenham Court Road, London, England W 1T 4LP. Applications for the copyright holder's written permission to reproduce any part of this publication should be addressed to the publisher. Permissions may be sought directly from Elsevier's Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: permissions @ elsevier.co.uk. You may also complete your request on-line via the Elsevier homepage (http://www.elsevier.com), by selecting 'Customer Support' and then 'Obtaining Permissions'
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Published with the cooperation of CIMNE, the International Centre for Numerical Methods in Engineering, Barcelona, Spain (www.cimne.upc.es) For information on all Elsevier Butterworth-Heinemann publications visit our website at http://books.elsevier.com Typeset by Kolam Information Services P Ltd, Pondicherry, India
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Dedication This book is dedicated to our wives Helen, Mary Lou and Song and our families for their support and patience during the preparation of this book, and also to all of our students and colleagues who over the years have contributed to our knowledge of the finite element method. In particular we would like to mention Professor Eugenio Oriate and his group at CIMNE for their help, encouragement and support during the preparation process.
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Contents
Preface 1 The standard discrete system and origins of the finite element method 1.1 Introduction 1.2 The structural element and the structural system 1.3 Assembly and analysis of a structure 1.4 The boundary conditions 1.5 Electrical and fluid networks 1.6 The general pattern 1.7 The standard discrete system 1.8 Transformation of coordinates 1.9 Problems 2
A direct physical approach to problems in elasticity: plane stress 2.1 Introduction 2.2 Direct formulation of finite element characteristics 2.3 Generalization to the whole region- internal nodal force concept abandoned 2.4 Displacement approach as a minimization of total potential energy 2.5 Convergence criteria 2.6 Discretization error and convergence rate 2.7 Displacement functions with discontinuity between elements non-conforming elements and the patch test 2.8 Finite element solution process 2.9 Numerical examples 2.10 Concluding remarks 2.11 Problems Generalization of the finite element concepts. Galerkin-weighted residual and variational approaches 3.1 Introduction 3.2 Integral or 'weak' statements equivalent to the differential equations 3.3 Approximation to integral formulations: the weighted residualGalerkin method
xiii 1 1
3 5 6 7 9 10 11 13 19 19 20 31 34 37 38 39 40 40 46 47 54 54 57 60
viii
Contents
3.4
Virtual work as the 'weak form' of equilibrium equations for analysis of solids or fluids 3.5 Partial discretization 3.6 Convergence 3.7 What are 'variational principles'? 3.8 'Natural' variational principles and their relation to governing differential equations 3.9 Establishment of natural variational principles for linear, self-adjoint, differential equations 3.10 Maximum, minimum, or a saddle point? 3.11 Constrained variational principles. Lagrange multipliers 3.12 Constrained variational principles. Penalty function and perturbed lagrangian methods 3.13 Least squares approximations 3.14 Concluding remarks - finite difference and boundary methods 3.15 Problems 4
5
'Standard' and 'hierarchical' element shape functions: some general families of Co continuity 4.1 Introduction 4.2 Standard and hierarchical concepts 4.3 Rectangular elements- some preliminary considerations 4.4 Completeness of polynomials 4.5 Rectangular elements- Lagrange family 4.6 Rectangular elements- 'serendipity' family 4.7 Triangular element family 4.8 Line elements 4.9 Rectangular prisms - Lagrange family 4.10 Rectangular prisms - 'serendipity' family 4.11 Tetrahedral elements 4.12 Other simple three-dimensional elements 4.13 Hierarchic polynomials in one dimension 4.14 Two- and three-dimensional, hierarchical elements of the 'rectangle' or 'brick' type 4.15 Triangle and tetrahedron family 4.16 Improvement of conditioning with hierarchical forms 4.17 Global and local finite element approximation 4.18 Elimination of internal parameters before assembly - substructures 4.19 Concluding remarks 4.20 Problems Mapped elements and numerical integration- 'infinite' and 'singularity elements' 5.1 Introduction 5.2 Use of 'shape functions' in the establishment of coordinate transformations 5.3 Geometrical conformity of elements 5.4 Variation of the unknown function within distorted, curvilinear elements. Continuity requirements
69 71 74 76 78 81 83 84 88 92 95 97 103 103 104 107 109 110 112 116 119 120 121 122 125 125 128 128 130 131 132 134 134 138 138 139 143 143
Contents
5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17
Evaluation of element matrices. Transformation in ~, 0, ff coordinates Evaluation of element matrices. Transformation in area and volume coordinates Order of convergence for mapped elements Shape functions by degeneration Numerical integration- one dimensional Numerical integration- rectangular (2D) or brick regions (3D) Numerical integration - triangular or tetrahedral regions Required order of numerical integration Generation of finite element meshes by mapping. Blending functions Infinite domains and infinite elements Singular elements by mapping - use in fracture mechanics, etc. Computational advantage of numerically integrated finite elements Problems
145 148 151 153 160 162 164 164 169 170 176 177 178
Problems in linear elasticity Introduction 6.1 Governing equations 6.2 Finite element approximation 6.3 Reporting of results: displacements, strains and stresses 6.4 Numerical examples 6.5 Problems 6.6
187 187 188 201 207 209 217
Field problems - heat conduction, electric and magnetic potential and fluid flow Introduction 7.1 General quasi-harmonic equation 7.2 Finite element solution process 7.3 Partial discretization- transient problems 7.4 Numerical examples- an assessment of accuracy 7.5 Concluding remarks 7.6 Problems 7.7
229 229 230 233 237 239 253 253
8 Automatic mesh generation Introduction 8.1 Two-dimensional mesh generation- advancing front method 8.2 Surface mesh generation 8.3 Three-dimensional mesh generation- Delaunay triangulation 8.4 Concluding remarks 8.5 Problems 8.6
264 264 266 286 303 323 323
9
329 329 330
6
The patch test, reduced integration, and non-conforming elements Introduction 9.1 Convergence requirements 9.2 The simple patch test (tests A and B) - a necessary condition for 9.3 convergence Generalized patch test (test C) and the single-element test 9.4 The generality of a numerical patch test 9.5 Higher order patch tests 9.6
332 334 336 336
ix
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Contents
9.7
Application of the patch test to plane elasticity elements with 'standard' and 'reduced' quadrature 9.8 Application of the patch test to an incompatible element 9.9 Higher order patch test- assessment of robustness 9.10 Concluding remarks 9.11 Problems 10
formulation and constraints- complete field methods Introduction Discretization of mixed forms - some general remarks Stability of mixed approximation. The patch test Two-field mixed formulation in elasticity Three-field mixed formulations in elasticity Complementary forms with direct constraint Concluding remarks - mixed formulation or a test of element 'robustness' Problems
356 356 358 360 363 370 375
Incompressible problems, mixed methods and other procedures of solution 11.1 Introduction 11.2 Deviatoric stress and strain, pressure and volume change 11.3 Two-field incompressible elasticity (u-p form) 11.4 Three-field nearly incompressible elasticity (u-p-~v form) 11.5 Reduced and selective integration and its equivalence to penalized mixed problems 11.6 A simple iterative solution process for mixed problems: Uzawa method 11.7 Stabilized methods for some mixed elements failing the incompressibility patch test 11.8 Concluding remarks 11.9 Problems
383 383 383 384 393
Mixed 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8
11
12
13
337 343 347 347 350
Multidomain mixed approximations- domain decomposition and 'frame' methods 12.1 Introduction 12.2 Linking of two or more subdomains by Lagrange multipliers 12.3 Linking of two or more subdomains by perturbed lagrangian and penalty methods 12.4 Interface displacement 'frame' 12.5 Linking of boundary (or Trefftz)-type solution by the 'frame' of specified displacements 12.6 Subdomains with 'standard' elements and global functions 12.7 Concluding remarks 12.8 Problems Errors, recovery processes and error estimates 13.1 Definition of errors 13.2 Superconvergence and optimal sampling points 13.3 Recovery of gradients and stresses
379 379
398 404 407 421 422 429 429 430 436 442 445 451 451 451 456 456 459 465
Contents
467 474 476 478
Superconvergent patch recovery - SPR Recovery by equilibration of patches - REP Error estimates by recovery Residual-based methods Asymptotic behaviour and robustness of error estimators - the Babu~ka patch test Bounds on quantities of interest 13.9 13.10 Which errors should concern us? 13.11 Problems
488 490 494 495
14
Adaptive finite element refinement 14.1 Introduction 14.2 Adaptive h-refinement 14.3 p-refinement and hp-refinement 14.4 Concluding remarks 14.5 Problems
500 500 503 514 518 520
15
Point-based and partition of unity approximations. Extended finite element methods 15.1 Introduction 15.2 Function approximation 15.3 Moving least squares approximations - restoration of continuity of approximation 15.4 Hierarchical enhancement of moving least squares expansions 15.5 Point collocation- finite point methods 15.6 Galerkin weighting and finite volume methods 15.7 Use of hierarchic and special functions based on standard finite elements satisfying the partition of unity requirement 15.8 Concluding remarks 15.9 Problems
13.4 13.5 13.6 13.7 13.8
16
17
The time dimension- semi-discretization of field and dynamic problems and analytical solution procedures 16.1 Introduction 16.2 Direct formulation of time-dependent problems with spatial finite element subdivision 16.3 General classification 16.4 Free response - eigenvalues for second-order problems and dynamic vibration 16.5 Free response - eigenvalues for first-order problems and heat conduction, etc. 16.6 Free response- damped dynamic eigenvalues 16.7 Forced periodic response 16.8 Transient response by analytical procedures 16.9 Symmetry and repeatability 16.10 Problems The time dimension- discrete approximation in time 17.1 Introduction
525 525 527 533 538 540 546 549 558 558 563 563 563 570 571 576 578 579 579 583 584 589 589
xi
xii
Contents
17.2 17.3 17.4 17.5 17.6 17.7 17.8 17.9 18
19
Simple time-step algorithms for the first-order equation General single-step algorithms for first- and second-order equations Stability of general algorithms Multistep recurrence algorithms Some remarks on general performance of numerical algorithms Time discontinuous Galerkin approximation Concluding remarks Problems
590 600 609 615 618 619 624 626
Coupled systems 18.1 Coupled problems- definition and classification 18.2 Fluid-structure interaction (Class I problems) 18.3 Soil-pore fluid interaction (Class II problems) 18.4 Partitioned single-phase systems - implicit-explicit partitions (Class I problems) 18.5 Staggered solution processes 18.6 Concluding remarks
631 631 634 645
Computer procedures for finite element analysis 19.1 Introduction 19.2 Pre-processing module: mesh creation 19.3 Solution module 19.4 Post-processor module 19.5 User modules
664 664 664 666 666 667
653 655 660
Appendix A: Matrix algebra
668
Appendix B: Tensor-indicial notation in the approximation of elasticity problems
674
Appendix C: Solution of simultaneous linear algebraic equations
683
Appendix D: Some integration formulae for a triangle
692
Appendix E: Some integration formulae for a tetrahedron
693
Appendix F: Some vector algebra
694
Appendix G: Integration by parts in two or three dimensions (Green's theorem)
699
Appendix H: Solutions exact at nodes
701
Appendix I: Matrix diagonalization or lumping
704
Author index
711
Subject index
719
Preface
It is thirty-eight years since the The Finite Element Method in Structural and Continuum Mechanics was first published. This book, which was the first dealing with the finite element method, provided the basis from which many further developments occurred. The expanding research and field of application of finite elements led to the second edition in 1971, the third in 1977, the fourth as two volumes in 1989 and 1991 and the fifth as three volumes in 2000. The size of each of these editions expanded geometrically (from 272 pages in 1967 to the fifth edition of 1482 pages). This was necessary to do justice to a rapidly expanding field of professional application and research. Even so, much filtering of the contents was necessary to keep these editions within reasonable bounds. In the present edition we have decided not to pursue the course of having three contiguous volumes but rather we treat the whole work as an assembly of three separate works, each one capable of being used without the others and each one appealing perhaps to a different audience. Though naturally we recommend the use of the whole ensemble to people wishing to devote much of their time and study to the finite element method. In particular the first volume which was entitled The Finite Element Method: The Basis is now renamed The Finite Element Method: Its Basis and Fundamentals. This volume has been considerably reorganized from the previous one and is now, we believe, better suited for teaching fundamentals of the finite element method. The sequence of chapters has been somewhat altered and several examples of worked problems have been added to the text. A set of problems to be worked out by students has also been provided. In addition to its previous content this book has been considerably enlarged by including more emphasis on use of higher order shape functions in formulation of problems and a new chapter devoted to the subject of automatic mesh generation. A beginner in the finite element field will find very rapidly that much of the work of solving problems consists of preparing a suitable mesh to deal with the whole problem and as the size of computers has seemed to increase without limits the size of problems capable of being dealt with is also increasing. Thus, meshes containing sometimes more than several million nodes have to be prepared with details of the material interfaces, boundaries and loads being well specified. There are many books devoted exclusively to the subject of mesh generation but we feel that the essence of dealing with this difficult problem should be included here for those who wish to have a complete 'encyclopedic' knowledge of the subject.
xiv
Preface
The chapter on computational methods is much reduced by transferring the computer source program and user instructions to a web site.t This has the very substantial advantage of not only eliminating errors in program and manual but also in ensuring that the readers have the benefit of the most recent version of the program available at all times. The two further volumes form again separate books and here we feel that a completely different audience will use them. The first of these is entitled The Finite Element Method in Solid and Structural Mechanics and the second is a text entitled The Finite Element Method in Fluid Dynamics. Each of these two volumes is a standalone text which provides the full knowledge of the subject for those who have acquired an introduction to the finite element method through other texts. Of course the viewpoint of the authors introduced in this volume will be continued but it is possible to start at a different point. We emphasize here the fact that all three books stress the importance of considering the finite element method as a unique and whole basis of approach and that it contains many of the other numerical analysis methods as special cases. Thus, imagination and knowledge should be combined by the readers in their endeavours. The authors are particularly indebted to the International Center of Numerical Methods in Engineering (CIMNE) in Barcelona who have allowed their pre- and post-processing code (GiD) to be accessed from the web site. This allows such difficult tasks as mesh generation and graphic output to be dealt with efficiently. The authors are also grateful to Professors Eric Kasper and Jose Luis Perez-Aparicio for their careful scrutiny of the entire text and Drs Joaquim Peir6 and C.K. Lee for their review of the new chapter on mesh generation.
Resources to accompany this book Worked solutions to selected problems in this book are available online for teachers and lecturers who either adopt or recommend the text. Please visit http://books.elsevier.com/ manuals and follow the registration and log in instructions on screen. OCZ, RLT and JZZ
t Complete source code and user manual for program FEAPpv may be obtained at no cost from the publisher's web page: http://books.elsevier.com/companionsor from the authors' web page: http://www.ce.berkeley.edu/~rlt
The standard discrete system and origins of the finite element method
The limitations of the human mind are such that it cannot grasp the behaviour of its complex surroundings and creations in one operation. Thus the process of subdividing all systems into their individual components or 'elements', whose behaviour is readily understood, and then rebuilding the original system from such components to study its behaviour is a natural way in which the engineer, the scientist, or even the economist proceeds. In many situations an adequate model is obtained using a finite number of well-defined components. We shall term such problems discrete. In others the subdivision is continued indefinitely and the problem can only be defined using the mathematical fiction of an infinitesimal. This leads to differential equations or equivalent statements which imply an infinite number of elements. We shall term such systems continuous. With the advent of digital computers, discrete problems can generally be solved readily even if the number of elements is very large. As the capacity of all computers is finite, continuous problems can only be solved exactly by mathematical manipulation. The available mathematical techniques for exact solutions usually limit the possibilities to oversimplified situations. To overcome the intractability of realistic types of continuous problems (continuum), various methods of discretization have from time to time been proposed by engineers, scientists and mathematicians. All involve an approximation which, hopefully, approaches in the limit the true continuum solution as the number of discrete variables increases. The discretization of continuous problems has been approached differently by mathematicians and engineers. Mathematicians have developed general techniques applicable directly to differential equations governing the problem, such as finite difference approximations, 1-3 various weighted residual procedures, 4' 5 or approximate techniques for determining the stationarity of properly defined 'functionals'. 6 The engineer, on the other hand, often approaches the problem more intuitively by creating an analogy between real discrete elements and finite portions of a continuum domain. For instance, in the field of solid mechanics McHenry, 7 Hrenikoff, 8 Newmark, 9 and Southwell 2 in the 1940s, showed that reasonably good solutions to an elastic continuum problem can be obtained by replacing small portions of the continuum by an arrangement of simple elastic bars. Later, in the same context, Turner et al. 1~ showed that a more direct, but no less intuitive, substitution of properties
2 The standard discrete system and origins of the finite element method
can be made much more effectively by considering that small portions or 'elements' in a continuum behave in a simplified manner. It is from the engineering 'direct analogy' view that the term 'finite element' was born. Clough 11 appears to be the first to use this term, which implies in it a direct use of a standard methodology applicable to discrete systems (see also reference 12 for a history on early developments). Both conceptually and from the computational viewpoint this is of the utmost importance. The first allows an improved understanding to be obtained; the second offers a unified approach to the variety of problems and the development of standard computational procedures. Since the early 1960s much progress has been made, and today the purely mathematical and 'direct analogy' approaches are fully reconciled. It is the object of this volume to present a view of the finite element method as a general discretization procedure of continuum
problems posed by mathematically defined statements. In the analysis of problems of a discrete nature, a standard methodology has been developed over the years. The civil engineer, dealing with structures, first calculates forcedisplacement relationships for each element of the structure and then proceeds to assemble the whole by following a well-defined procedure of establishing local equilibrium at each 'node' or connecting point of the structure. The resulting equations can be solved for the unknown displacements. Similarly, the electrical or hydraulic engineer, dealing with a network of electrical components (resistors, capacitances, etc.) or hydraulic conduits, first establishes a relationship between currents (fluxes) and potentials for individual elements and then proceeds to assemble the system by ensuring continuity of flows. All such analyses follow a standard pattern which is universally adaptable to discrete systems. It is thus possible to define a standard discrete system, and this chapter will be primarily concerned with establishing the processes applicable to such systems. Much of what is presented here will be known to engineers, but some reiteration at this stage is advisable. As the treatment of elastic solid structures has been the most developed area of activity this will be introduced first, followed by examples from other fields, before attempting a complete generalization. The existence of a unified treatment of 'standard discrete problems' leads us to the first definition of the finite element process as a method of approximation to continuum problems such that (a) the continuum is divided into a finite number of parts (elements), the behaviour of which is specified by a finite number of parameters, and (b) the solution of the complete system as an assembly of its elements follows precisely the same rules as those applicable to standard discrete problems. The development of the standard discrete system can be followed most closely through the work done in structural engineering during the nineteenth and twentieth centuries. It appears that the 'direct stiffness process' was first introduced by Navier in the early part of the nineteenth century and brought to its modern form by Clebsch 13 and others. In the twentieth century much use of this has been made and Southwell, 14 Cross 15 and others have revolutionized many aspects of structural engineering by introducing a relaxation iterative process. Just before the Second World War matrices began to play a larger part in casting the equations and it was convenient to restate the procedures in matrix form. The work of Duncan and Collar, 16-18Argyris, 19 Kron 20 and Turner 1~should be noted. A thorough study of direct stiffness and related methods was recently conducted by Samuelsson. 21
The structural element and the structural system 3 It will be found that most classical mathematical approximation procedures as well as the various direct approximations used in engineering fall into this category. It is thus difficult to determine the origins of the finite element method and the precise moment of its invention. Table 1.1 shows the process of evolution which led to the present-day concepts of finite 'element analysis. A historical development of the subject of finite element methods has been presented by the first author in references 34-36. Chapter 3 will give, in more detail, the mathematical basis which emerged from these classical ideas, l, 22-27, 29, 30, 32
To introduce the reader to the general concept of discrete systems we shall first consider a structural engineering example with linear elastic behaviour. Figure 1.1 represents a two-dimensional structure assembled from individual components and interconnected at the nodes numbered 1 to 6. The joints at the nodes, in this case, are pinned so that moments cannot be transmitted. As a starting point it will be assumed that by separate calculation, or for that matter from the results of an experiment, the characteristics of each element are precisely known. Thus, if a typical element labelled (1) and associated with nodes 1, 2, 3 is examined, the forces acting at the nodes are uniquely defined by the displacements of these nodes, the distributed loading acting on the element (p), and its initial strain. The last may be due to temperature, shrinkage, or simply an initial 'lack of fit'. The forces and the corresponding Table 1.1 Historyof approximate methods ENGINEERING
MATHEMATICS[
Trial functions]
I differences[ Finite
Rayleigh 187022 Ritz 190823
Variational methods [
Weighted I residuals I
Rayleigh 187022 Ritz 190823
Structural analogue substitution
Hrenikoff 19418 McHenry 194328 Newmark 19499
1
Richardson 19101 Liebman 191824 Southwell 19462
Gauss 179525 Galerkin 191526 Biezeno--Koch 192327
Piecewise continuous trialfunctions
Courant 194329 Prager-Synge 194730 Argyris 195519 Zienkiewicz 196431
Direct continuum elements 1 ~ Turner et al.
r
Variational finite differences
J
PRESENT-DAY FINITE ELEMENTMETHOD
Varga 196232 Wilkins 196433
4
The standard discrete system and origins of the finite element method
Fig. 1.1 A typical structure built up from interconnected elements.
displacements are defined by appropriate components (U, V and u, v) in a common coordinate system (x, y). Listing the forces acting on all the nodes (three in the case illustrated) of the element (1) as a matrixt we have qm=
q~
ql=
ql
U1 V1
'
Ul 131
,
etc.
(1 1)
etc.
(1.2)
and for the corresponding nodal displacements
u l ~.
ul
Ul __
Assuming linear elastic behaviour of the element, the characteristic relationship will always be of the form ql = Klu 1 + fl (1.3) in which f l represents the nodal forces required to balance any concentrated or distributed loads acting on the element. The first of the terms represents the forces induced by displacement of the nodes. The matrix K e is known as the stiffness matrix for the element (e). Equation (1.3) is illustrated by an example of an element with three nodes with the interconnection points capable of transmitting only two components of force. Clearly, the -~A limited knowledge of matrix algebra will be assumed throughout this book. This is necessary for reasonable conciseness and forms a convenient book-keeping form. For readers not familiar with the subject a brief appendix (Appendix A) is included in which sufficient principles of matrix algebra are given to follow the development intelligently. Matrices and vectors will be distinguished by bold print throughout.
Assembly and analysis of a structure
same arguments and definitions will apply generally 9 An element (2) of the hypothetical structure will possess only two points o f interconnection; others may have quite a large number of such points 9 Quite generally, therefore, Ul
qe .__
.
and
me - -
qe
U2
(1.4)
Um
with each ~ and U a possessing the same number of components or degrees offreedom. The stiffness matrices of the element will clearly always be square and of the form
[
K~I
K~2
K e -- ]K~I
""
/
LKel
...
K~m
(1 9
......
KernJ
in which K~I, K~2, etc., are submatrices which are again square and of the size l x l, where I is the number of force and displacement components to be considered at each node. The element properties were assumed to follow a simple linear relationship. In principle, similar relationships could be established for non-linear materials, but discussion of such problems will be postponed at this stage. In most cases considered in this volume the element matrices K e will be symmetric.
Consider again the hypothetical structure of Fig. 1.1. To obtain a complete solution the two conditions of (a) displacement compatibility and (b) equilibrium have to be satisfied throughout 9 Any system of nodal displacements
u:
u =
In1/ "
(1.6)
an
listed now for the whole structure in which all the elements participate, automatically satisfies the first condition. As the conditions of overall equilibrium have already been satisfied within an element, all that is necessary is to establish equilibrium conditions at the nodes (or assembly points) of the structure. The resulting equations will contain the displacements as unknowns, and once these have been solved the structural problem is determined. The internal forces in elements, or the stresses, can easily be found by using the characteristics established a priori for each element.
5
6 The standard discrete system and origins of the finite element method
If now the equilibrium conditions of a typical node, a, are to be established, the sum of the component forces contributed by the elements meeting at the node are simply accumulated. Thus, considering all the force components we have m
~-'~ q~ : q a 1 + q ~ + ' "
=0
(1.7)
e=l in which ql is the force contributed to node a by element 1, qa2 by element 2, etc. Clearly, only the elements which include point a will contribute non-zero forces, but for conciseness in notation all the elements are included in the summation. Substituting the forces contributing to node a from the definition (1.3) and noting that nodal variables Ua are common (thus omitting the superscript e), we have K e1 U l +
e=l
Ke2 u 2 + . . . +
fe=0
e=l
(1.8)
e=l
The summation again only concerns the elements which contribute to node a. If all such equations are assembled we have simply Ku + f = 0
(1.9)
in which the submatrices are rn
rn
Kab : ~ Keb e=l
and
fa - ~
fe
(1.10)
e=l
with summations including all elements. This simple rule for assembly is very convenient because as soon as a coefficient for a particular element is found it can be put immediately into the appropriate 'location' specified in the computer. This general assembly process can be found to be the common and fundamental feature of all finite element calculations and should be well understood by the reader. If different types of structural elements are used and are to be coupled it must be remembered that at any given node the rules of matrix summation permit this to be done only if these are of identical size. The individual submatrices to be added have therefore to be built up of the same number of individual components of force or displacement.
The system of equations resulting from Eq. (1.9) can be solved once the prescribed support displacements have been substituted. In the example of Fig. 1.1, where both components of displacement of nodes 1 and 6 are zero, this will mean the substitution of
Ul ---U6-- /00/ which is equivalent to reducing the number of equilibrium equations (in this instance 12) by deleting the first and last pairs and thus reducing the total number of unknown displacement
Electrical and fluid networks
components to eight. It is, nevertheless, often convenient to assemble the equation according to relation (1.9) so as to include all the nodes. Clearly, without substitution of a minimum number of prescribed displacements to prevent rigid body movements of the structure, it is impossible to solve this system, because the displacements cannot be uniquely determined by the forces in such a situation. This physically obvious fact will be interpreted mathematically as the matrix K being singular, i.e., not possessing an inverse. The prescription of appropriate displacements after the assembly stage will permit a unique solution to be obtained by deleting appropriate rows and columns of the various matrices. If all the equations of a system are assembled, their form is KllUl -+- K12u2 + " "
-+- fl =- 0
K21Ul @ K22u2 + " "
-F f2 -- 0
(1.11)
etc. and it will be noted that if any displacement, such as Ul = ill, is prescribed then the total 'force' fl cannot be simultaneously specified and remains unknown. The first equation could then be deleted and substitution of known values fil made in the remaining equations. When all the boundary conditions are inserted the equations of the system can be solved for the unknown nodal displacements and the internal forces in each element obtained.
Identical principles of deriving element characteristics and of assembly will be found in many non-structural fields. Consider, for instance, the assembly of electrical resistances shown in Fig. 1.2. If a typical resistance element, ab, is isolated from the system we can write, by Ohm's law, the relation between the currents (J) entering the element at the ends and the end voltages (V) as
je
1
= --~(Va-
and
Vb)
J{
1
= - ~ ( V b -- Va)
(1.12)
or in matrix form re
m
1 ll{
Vb
which in our standard form is simply je = Keve
(1.13)
This form clearly corresponds to the stiffness relationship (1.3); indeed if an external current were supplied along the length of the element the element 'force' terms could also be found. To assemble the whole network the continuity of the voltage (V) at the nodes is assumed and a current balance imposed there. With no external input of current at node a we must
7
8 The standard discrete system and origins of the finite element method
b
i
rO§ 4,%
Fig. 1.2 A network of electrical resistances.
have, with complete analogy to Eq. (1.8),
~ ~-~K~aVa
--
0
(1.14)
b=l e-1
where the second summation is over all 'elements', and once again for all the nodes
KV = 0
(1.15)
in which m
gab -- ~ geb e=l
Matrix notation in the latter has been dropped since the quantities such as voltage and current, and hence also the coefficients of the 'stiffness' matrix, are scalars. If the resistances were replaced by fluid-carrying pipes in which a laminar regime pertained, an identical formulation would once again result, with V standing for the hydraulic head and J for the flow. For pipe networks that are usually encountered, however, the linear laws are in general not valid and non-linear equations must be solved. Finally it is perhaps of interest to mention the more general form of an electrical network subject to an alternating current. It is customary to write the relationships between
The general pattern the current and voltage in complex arithmetic form with the resistance being replaced by complex impedance. Once again the standard forms of (1.13)-(1.15) will be obtained but with each quantity divided into real and imaginary parts. Identical solution procedures can be used if the equality of the real and imaginary quantities is considered at each stage. Indeed with modem digital computers it is possible to use standard programming practice, making use of facilities available for dealing with complex numbers. Reference to some problems of this class will be made in the sections dealing with vibration problems in Chapter 15.
An example will be considered to consolidate the concepts discussed in this chapter. This is shown in Fig. 1.3(a) where five discrete elements are interconnected. These may be of structural, electrical, or any other linear type. In the solution:
The first step is the determination of element properties from the geometric material and loading data. For each element the 'stiffness matrix' as well as the corresponding 'nodal
Fig. 1.3 The generalpattern.
9
10 The standard discrete system and origins of the finite element method loads' are found in the form of Eq. (1.3). Each element shown in Fig. 1.3(a) has its own identifying number and specified nodal connection. For example: element
1 2 3 4 5
connection
1 1 2 3 4
3 4 5 6 7
4 2 7 8
Assuming that properties are found in global coordinates we can enter each 'stiffness' or 'force' component in its position of the global matrix as shown in Fig. 1.3(b). Each shaded square represents a single coefficient or a submatrix of type Kab if more than one quantity is being considered at the nodes. Here the separate contribution of each element is shown and the reader can verify the position of the coefficients. Note that the various types of 'elements' considered here present no difficulty in specification. (All 'forces', including nodal ones, are here associated with elements for simplicity.)
The second step is the assembly of the final equations of the type given by Eq. (1.9). This is accomplished according to the rule of Eq. (1.10) by simple addition of all numbers in the appropriate space of the global matrix. The result is shown in Fig. 1.3(c) where the non-zero coefficients are indicated by shading. If the matrices are symmetric only the half above the diagonal shown needs, in fact, to be found. All the non-zero coefficients are confined within a band or profile which can be calculated a priori for the nodal connections. Thus in computer programs only the storage of the elements within the profile (or sparse structure) is necessary, as shown in Fig. 1.3(c). Indeed, if K is symmetric only the upper (or lower) half need be stored. The third step is the insertion of prescribed boundary conditions into the final assembled matrix, as discussed in Sec. 1.3. This is followed by the final step. The final step solves the resulting equation system. Here many different methods can be employed, some of which are summarized in Appendix C. The general subject of equation solving, though extremely important, is in general beyond the scope of this book. The final step discussed above can be followed by substitution to obtain stresses, currents, or other desired output quantities. All operations involved in structural or other network analysis are thus of an extremely simple and repetitive kind. We can now define the standard discrete system as one in which such conditions prevail.
In the standard discrete system, whether it is structural or of any other kind, we find that: 1. A set of discrete parameters, say Ua, can be identified which describes simultaneously the behaviour of each element, e, and of the whole system. We shall call these the system parameters.
Transformation of coordinates
2. For each element a set of quantities qe can be computed in terms of the system parameters Ua. The general function relationship can be non-linear, for example q~ = qe (u)
(1.16)
but in many cases a linear form exists giving q ; _.
e e K a l U l -+- Ka2U2 + " "
e + fa
(1.17)
3. The final system equations are obtained by a simple addition m ra --- E qa -'- 0 e=l
(1.18)
where ra are system quantities (often prescribed as zero). In the linear case this results in a system of equations Ku + f = 0 = 0
such that
m
(1.19) m
Kab "~-E Keb
and
e=l
fa = E
fa
(1.20)
e=l
from which the solution for the system variables u can be found after imposing necessary boundary conditions. The reader will observe that this definition includes the structural, hydraulic, and electrical examples already discussed. However, it is broader. In general neither linearity nor symmetry of matrices need exist - although in many problems this will arise naturally. Further, the narrowness of interconnections existing in usual elements is not essential. While much further detail could be discussed (we refer the reader to specific books for more exhaustive studies in the structural context 37' 38), we feel that the general expos6 given here should suffice for further study of this book. Only one further matter relating to the change of discrete parameters need be mentioned here. The process of so-called transformation of coordinates is vital in many contexts and must be fully understood.
It is often convenient to establish the characteristics of an individual element in a coordinate system which is different from that in which the external forces and displacements of the assembled structure or system will be measured. A different coordinate system may, in fact, be used for every element, to ease the computation. It is a simple matter to transform the coordinates of the displacement and force components of Eq. (1.3) to any other coordinate system. Clearly, it is necessary to do so before an assembly of the structure can be attempted. Let the local coordinate system in which the element properties have been evaluated be denoted by a prime suffix and the common coordinate system necessary for assembly have no embellishment. The displacement components can be transformed by a suitable matrix of direction cosines L as u ' = Lu
(1.21)
11
12
The standard discrete system and origins of the finite element method As the corresponding force components must perform the same amount of work in either systemt On inserting ( 1.21) we have
qTu=
q'Tu'
qTu=
qCrLu
(1.22)
or q = LXq '
(1.23)
The set of transformations given by (1.21) and (1.23) is called contravariant. To transform 'stiffnesses' which may be available in local coordinates to global ones note that if we write q' = K ' u '
(1.24)
then by (1.23), (1.24), and (1.21) q = LTK'Lu or in global coordinates K = LTK'L
(1.25)
In many complex problems an external constraint of some kind may be imagined, enforcing the requirement (1.21) with the number of degrees of freedom of u and u' being quite different. Even in such instances the relations (1.22) and (1.23) continue to be valid. An alternative and more general argument can be applied to many other situations of discrete analysis. We wish to replace a set of parameters u in which the system equations have been written by another one related to it by a transformation matrix T as u -- T v
(1.26)
In the linear case the system equations are of the form K u -- - f
(1.27)
KTv = - f
(1.28)
and on the substitution we have The new system can be premultiplied simply by T T, yielding ( T T K T ) v = T T - T Tf
(1.29)
which will preserve the symmetry of equations if the matrix K is symmetric. However, occasionally the matrix T is not square and expression (1.26) represents in fact an approximation in which a larger number of parameters u is constrained. Clearly the system of equations (1.28) gives more equations than are necessary for a solution of the reduced set of parameters v, and the final expression (1.29) presents a reduced system which in some sense approximates the original one. We have thus introduced the basic idea of approximation, which will be the subject of subsequent chapters where infinite sets of quantities are reduced to finite sets. t With ( )T standing for the transpose of the matrix.
Problems
1.1 A simple fluid network to transport water is shown in Fig. 1.4. Each 'element' of the network is modelled in terms of the flow, J, and head, V, which are approximated by the linear relation je __ _ K e V e where K e is the coefficient array for element (e). The individual terms in the flow vector denote the total amount of flow entering (+) or leaving ( - ) each end point. The properties of the elements are given by
K e
=
c e
I: ] 3
-2
-1
2
4
-2
1
-2
3
for elements 1 and 4, and for elements 2 and 3 by K e
-
.
Ce
where ce is an element related parameter. The system is operating with a known head of 100 m at node 1 and 30 m at node 6. At node 2, 30 cubic metres of water per hour are being used and at node 4, 10 cubic metres per hour. (a) For all c e = 1, assemble the total matrix from the individual elements to give J=KV N.B. J contains entries for the specified usage and connection points. (b) Impose boundary conditions by modifying J and K such that the known heads at nodes 1 and 6 are recovered. (c) Solve the equations for the heads at nodes 2 to 5. (Result at node 4 should be V4 = 30.8133 m.) (d) Determine the flow entering and leaving each element. 1.2 A plane truss may be described as a standard discrete problem by expressing the characteristics for each member in terms of end displacements and forces. The behaviour of the elastic member shown in Fig. 1.5 with modulus E, cross-section A and length L is given by q' -- K'e u' 3
lb,
(2)
4
A
1
(3) 2
Fig. 1.4 Fluid network for Problem 1.1.
5
6
13
14 The standard discrete system and origins of the finite element method where
q,= { UU{~ } ' 9u' = {'}ul'
and K e' = EA[
u2
L
1 -11]
-1
To obtain the final assembled matrices for a standard discrete problem it is necessary to transform the behaviour to a global frame using Eqs 1.23 and 1.25 where
L=
[o c
0
sinO 0
0 cos 0
0 sin 0
l
U1
V1 ; q --
Ul
and u =
U2
u2
V2
u2
(a) Compute relations for q and K in terms of L, q' and K'e. (b) If the numbering for the end nodes is reversed what is the final form for K compared to that given in (a)? Verify your answer when 0 = 30 ~ 1.3 A plane truss has nodes numbered as shown in Fig. 1.6(a). (a) Use the procedure shown in Fig. 1.3 to define the non-zero structure of the coefficient matrix K. Compute the maximum bandwidth. (b) Determine the non-zero structure of K for the numbering of nodes shown in 1.6(b). Compute the maximum bandwidth. Which order produces the smallest band? 1.4 Write a small computer program (e.g., using MATLAB 39) to solve the truss problem shown in Fig. 1.6(b). Let the total span of the truss be 2.5 m and the height 0.8 m and use steel as the property for each member with E -- 200 GPa and A = 0.001 m 2. Restrain node 1 in both the u and v directions and the right bottom node in the v direction only. Apply a vertical load of 100 N at the position of node 6 shown in Fig. 1.6(b). Determine the maximum vertical displacement at any node. Plot the undeformed and deformed position of the truss (increase the magnitude of displacements to make the shape visible on the plot). You can verify your result using the program FEAPpv available at the publisher's web site (see Chapter 18). 1.5 An axially loaded elastic bar has a variable cross-section and lengths as shown in Fig. 1.7(a). The problem is converted into a standard discrete system by considering each prismatic section as a separate member. The array for each member segment is given as qe __ K e u e
y(v, V)
'
y'
x'
2
1
>
(a) Trussmemberdescription Fig. 1.5 Trussmemberfor Problem1.2.
x (u,U)
U
(b) Displacements
Problems
where Ke
EAe[lh - 1
- 1}] qe 1= { qe+l qeee
and ue
=
{ }Ue+l ue
Equilibrium for the standard discrete problem at joint e is obtained by combining results from segment e - 1 and e as
qe-1 + qe + Ue
__ 0
where Ue is any external force applied to a joint. Boundary conditions are applied for any joint at which the value of Ue is known a priori. Solve the problem shown in Fig. 1.7(b) for the joint displacements using the data E1 = E2 = E3 = 200 GPa, A 1 = 25 cm 2, A2 = 20 cm 2, A3 -- 12 cm 2, L 1 = 37.5 cm, L2 = 25.0 cm, L3 = 12.5 cm, P2 = 10 kN, P3 = - 3 . 5 kN and P4 = 6 kN. 1.6 Solve Problem 1.5 for the boundary conditions and loading shown in Fig. 1.7(c). Let E1 = E2 = E3 = 200GPa, A1 = 30cm2, A2 = 20cm2, A3 = 10cm2, L1 = 37.5 cm, L2 = 30.0 cm, L3 = 25.0 cm, P2 = - 1 0 kN and P3 = 3.5 kN. 1.7 A tapered bar is loaded by an end load P and a uniform loading b as shown in Fig. 1.8(a). The area varies as A (x) = A x/L when the origin of coordinates is located as shown in the figure. The problem is converted into a standard discrete system by dividing it into equal length segments of constant area as shown in Fig. 1.8(b). The array for each segment is determined from qe __ Ken e + fe
1
7
8
9
10
2
3
4
5
~'
9
w
6
(a) 3
w
1
5
A
A
,IL
w
v
w
2
4
6
(b) Fig. 1.6 Truss for Problems 1.3 and 1.4.
9
v
8
v
10
15
16 The standard discrete system and origins of the finite element method E1 ,A1
E2'A2
E3,A3
~lJ
L2
L3
~1
(a) Bar geometry
+
~_ Ul=0
~
(b) Problem 1.5
+
" > Ul=0
>P3
>92
~ >u4=O
(c) Problem 1.6
Fig. 1.7 Elasticbars. Problems1.5 and 1.6.
Us=0
~ I ~-
2
x
A -'rl~-.
f
I~"
..rl
(a) Tapered bar geometry
rl
(b) Approximation by 4 segments
Fig. 1.8 Taperedbar. Problem1.7.
where K e and u e are defined in Problem 1.5 and
fe =-21bh{1}1
4
For the properties L = 100 cm, A = 2 cm 2, E = 10 kN/cm 2, P = 2 kN, b = - 0 . 2 5 kN/cm and u(2L) = 0, the displacement from the solution of the differential = - 0 . 0 3 1 4 2 5 1 3 cm. equation is Write a small computer program (e.g., using MATLAB 39) that solves the problem for the case where e = 1, 2, 4, 8, . . . segments. Continue the solution until the absolute error in the tip displacement is less than 10 -5 cm (let error be E = l u ( t ) - ull where Ul is the numerical solution at the end).
u(L)
References
1. L.E Richardson. The approximate arithmetical solution by finite differences of physical problems. Trans. Roy. Soc. (London), A210:307-357, 1910. 2. R.V. Southwell. Relaxation Methods in Theoretical Physics. Clarendon Press, Oxford, 1st edition, 1946. 3. D.N. de G. Allen. Relaxation Methods. McGraw-Hill, London, 1955. 4. S. Crandall. Engineering Analysis. McGraw-Hill, New York, 1956. 5. B.A. Finlayson. The Method of Weighted Residuals and Variational Principles. Academic Press, New York, 1972. 6. K. Washizu. Variational Methods in Elasticity and Plasticity. Pergamon Press, New York, 3rd edition, 1982. 7. D. McHenry. A lattice analogy for the solution of plane stress problems. J. Inst. Civ. Eng., 21:59-82, 1943. 8. A. Hrenikoff. Solution of problems in elasticity by the framework method. J. Appl. Mech., ASME, A8:169-175, 1941. 9. N.M. Newmark. Numerical methods of analysis in bars, plates and elastic bodies. In L.E. Grinter, editor, Numerical Methods in Analysis in Engineering. Macmillan, New York, 1949. 10. M.J. Turner, R.W. Clough, H.C. Martin, and L.J. Topp. Stiffness and deflection analysis of complex structures. J. Aero. Sci., 23:805-823, 1956. 11. R.W. Clough. The finite element method in plane stress analysis. In Proc. 2nd ASCE Conf. on Electronic Computation, Pittsburgh, Pa., Sept. 1960. 12. R.W. Clough. Early history of the finite element method from the view point of a pioneer. Int. J. Numer. Meth. Eng., 60:283-287, 2004. 13. R.E Clebsch. Thdorie de l'elasticitd des corps solides. Dunod, Pads, 1883. 14. R.V. Southwell. Stress calculation in frame works by the method of systematic relaxation of constraints, Part I & II. Proc. Roy. Soc. London (A), 151:56-95, 1935. 15. Hardy Cross. Continuous Frames of Reinforced Concrete. John Wiley & Sons, New York, 1932. 16. W.J. Duncan and A.R. Collar. A method for the solution of oscillation problems by matrices. Phil. Mag., 17:865, 1934. Series 7. 17. W.J. Duncan and A.R. Collar. Matrices applied to the motions of damped systems. Phil. Mag., 19:197, 1935. Series 7. 18. R.R. Frazer, W.J. Duncan, and A.R. Collar. Elementary Matrices. Cambridge University Press, London, 1960. 19. J.H. Argyris and S. Kelsey. Energy Theorems and Structural Analysis. Butterworths, London, 1960. Reprinted from a series of articles in Aircraft Eng., 1954-55. 20. G. Kron. Equivalent Circuits of Electrical Machinery. John Wiley & Sons, New York, 1951. 21. A. Samuelsson. Personal communication, 2003. 22. Lord Rayleigh (J.W. Strutt). On the theory of resonance. Trans. Roy. Soc. (London), A161:77-118, 1870. 23. W. Ritz. Uber eine neue Methode zur L/Ssung gewisser vadationsproblem der mathematischen physik. Journal ftlr die reine und angewandte Mathematik, 135:1-61, 1908. 24. H. Liebman. Die angen~iherte Ermittlung: harmonishen, functionen und konformer Abbildung. Sitzber. Math. Physik Kl. BayerAkad. Wiss. Mtlnchen, 3:65-75, 1918. 25. C.E Gauss. Werke. Dietrich, GtJttingen, 1863-1929. See: Theoretishe Astronomie, Bd. VII. 26. B.G. Galerkin. Series solution of some problems in elastic equilibrium of rods and plates. Vestn. Inzh. Tech., 19:897-908, 1915. 27. C.B. Biezeno and J.J. Koch. Over een Nieuwe Methode ter Berekening van Vlokke Platen. Ing. Grav., 38:25-36, 1923. 28. D. McHenry. A new aspect of creep in concrete and its application to design. Proc. ASTM, 43:1064, 1943.
17
18 The standard discrete system and origins of the finite element method 29. R. Courant. Variational methods for the solution of problems of equilibrium and vibration. Bull. Am. Math Soc., 49:1-61, 1943. 30. W. Prager and J.L. Synge. Approximation in elasticity based on the concept of function space. Quart. J. Appl. Math, 5:241-269, 1947. 31. O.C. Zienkiewicz and Y.K. Cheung. The finite element method for analysis of elastic isotropic and orthotropic slabs. Proc. Inst. Civ. Eng., 28:471-488, 1964. 32. R.S. Varga. Matrix Iterative Analysis. Prentice-Hall, Englewood Cliffs, N.J., 1962. 33. M.L. Wilkins. Calculation of elastic-plastic flow. In B. Alder, editor, Methods in Computational Physics, volume 3, pages 211-263. Academic Press, New York, 1964. 34. O.C. Zienkiewicz. Origins, milestones and directions of the finite element method. Arch. Comp. Meth. Eng., 2:1-48, 1995. 35. O.C. Zienkiewicz. Origins, milestones and directions of the finite element method. A personal view. In P.G. Ciarlet and J.L Lyons, editors, Handbook of NumericalAnalysis, volume IV, pages 3-65. North Holland, 1996. 36. O.C. Zienkiewicz. The birth of the finite element method and of computational mechanics. Int. J. Numer. Meth. Eng., 60:3-10, 2004. 37. J.S. Przemieniecki. Theory of Matrix Structural Analysis. McGraw-Hill, New York, 1968. 38. R.K. Livesley. Matrix Methods in Structural Analysis. Pergamon Press, New York, 2nd edition, 1975. 39. MATLAB. www.mathworks.com, 2003.
A direct physical approach to problems in elasticity: plane stress
The process of approximating the behaviour of a continuum by 'finite elements' which behave in a manner similar to the real, 'discrete', elements described in the previous chapter can be introduced through the medium of particular physical applications or as a general mathematical concept. We have chosen here to follow the first path, narrowing our view to a set of problems associated with structural mechanics which historically were the first to which the finite element method was applied. In Chapter 3 we shall generalize the concepts and show that the basic ideas are widely applicable. In many phases of engineering the solution of stress and strain distributions in elastic continua is required. Special cases of such problems may range from two-dimensional plane stress or strain distributions, axisymmetric solids, plate bending, and shells, to fully three-dimensional solids. In all cases the number of interconnections between any 'finite element' isolated by some imaginary boundaries and the neighbouring elements is continuous and therefore infinite. It is difficult to see at first glance how such problems may be discretized in the same manner as was described in the preceding chapter for simpler systems. The difficulty can be overcome (and the approximation made) in the following manner: 1. The continuum is separated by imaginary lines or surfaces into a number of 'finite elements'. 2. The elements are assumed to be interconnected at a discrete number of nodal points situated on their boundaries and occasionally in their interior. The displacements of these nodal points will be the basic unknown parameters of the problem, just as in simple, discrete, structural analysis. 3. A set of functions is chosen to define uniquely the state of displacement within each 'finite element' and on its boundaries in terms of its nodal displacements. 4. The displacement functions now define uniquely the state of strain within an element in terms of the nodal displacements. These strains, together with any initial strains and the constitutive properties of the material, define the state of stress throughout the element and, hence, also on its boundaries. 5. A system of 'equivalent forces' concentrated at the nodes and equilibrating the boundary stresses and any distributed loads is determined, resulting in a stiffness relationship
20 A direct physical approach to problems in elasticity: plane stress of the form of Eq. (1.3). The determination of these equivalent forces is done most conveniently and generally using the principle of virtual work which is a particular mathematical relation known as a weak form of the problem. Once this stage has been reached the solution procedure can follow the standard discrete system pattern described in Chapter 1. Clearly a series of approximations has been introduced. First, it is not always easy to ensure that the chosen displacement functions will satisfy the requirement of displacement continuity between adjacent elements. Thus, the compatibility condition on such lines may be violated (though within each element it is obviously satisfied due to the uniqueness of displacements implied in their continuous representation). Second, by concentrating the equivalent forces at the nodes, equilibrium conditions are satisfied in the overall sense only. Local violation of equilibrium conditions within each element and on its boundaries will usually arise. The choice of element shape and of the form of the displacement function for specific cases leaves many opportunities for the ingenuity and skill of the analyst to be employed, and obviously the degree of approximation which can be achieved will strongly depend on these factors. The approach outlined here is known as the displacement formulation. 1"2 The use of the principle of virtual work (weak form) is extremely convenient and powerful. Here it has only been justified intuitively though in the next chapter we shall see its mathematical origins. However, we will also show the determination of these equivalent forces can be done by minimizing the total potential energy. This is applicable to situations where elasticity predominates and the behaviour is reversible. While the virtual work form is always valid, the principle of minimum potential energy is not and care has to be taken. The recognition of the equivalence of the finite element method to a minimization process was late. 2'3 However, Courant 4 in 1943t and Prager and Synge 5 in 1947 proposed minimizing methods that are in essence identical. This broader basis of the finite element method allows it to be extended to other continuum problems where a variational formulation is possible. Indeed, general procedures are now available for a finite element discretization of any problem defined by a properly constituted set of differential equations. Such generalizations will be discussed in Chapter 3, and throughout the book application to structural and some non-structural problems will be made. It will be found that the process described in this chapter is essentially an application of trial-function and Galerkin-type approximations to the particular case of solid mechanics.
The 'prescriptions' for deriving the characteristics of a 'finite element' of a continuum, which were outlined in general terms, will now be presented in more detailed mathematical form. t It appears that Courant had anticipated the essence of the finite element method in general, and of a triangular element in particular, as earlyas 1923in a paperentitled 'On a convergenceprinciplein the calculus of variations.' Krn. Gesellschaftder Wissenschaften zu Grttingen, Nachrichten, Berlin, 1923. He states: 'We imagine a mesh of triangles covering the domain.., the convergenceprinciples remain valid for each triangular domain.'
Direct formulation of finite element characteristics
It is desirable to obtain results in a general form applicable to any situation, but to avoid introducing conceptual difficulties the general relations will be illustrated with a very simple example of plane stress analysis of a thin slice. In this a division of the region into triangular-shaped elements may be used as shown in Fig. 2.1. Alternatively, regions may be divided into rectangles or, indeed using a combination of triangles and rectangles. In later chapters we will show how many other shapes also may be used to define elements.
2.2.1 Displacement function A typical finite element, e, with a triangular shape is defined by local nodes 1, 2 and 3, and straight line boundaries between the nodes as shown in Fig. 2.2(a). Similarly, a rectangular element could be defined by local nodes 1, 2, 3 and 4 as shown in Fig. 2.2(b). The choice of displacement functions for each element is of paramount importance and in Chapters 4 and 5 we will show how they may be developed for a wide range of types; however, in the rest of this chapter we will consider only the 3-node triangular and 4-node rectangular element shapes 9 Let the displacements u at any point within the element be approximated as a column vector, fi: U ~ fi ---- Z
Naue ---- [N1,
N2 . . . .
a
Fig. 2.1 A plane stress region divided into finite elements.
]
lull/e U2
--
N~ e
(2.1)
21
22 A direct physical approach to problems in elasticity: plane stress
Fig. 2.2 Shapefunction N3 for one element. In the case of plane stress, for instance, u =
{u xy } v(x, y)
represents horizontal and vertical movements (see Fig. 2.1) of a typical point within the element and
the corresponding displacements of a node a. The functions Na, a = 1, 2 . . . . are called shape functions (or basis functions, and, occasionally interpolation functions) and must be chosen to give appropriate nodal displacements when coordinates of the corresponding nodes are inserted in Eq. (2.1). Clearly in general we have Na(xa, Ya) - - I (identity matrix) while
Na (Xb, Yb) = O, a ~ b If both components of displacement are specified in an identical manner then we can write
Na -- Na I
(2.2)
and obtain Na from Eq. (2.1) by noting that Na(xa, Ya) 1 but is zero at other vertices. The shape functions N will be seen later to play a paramount role in finite element analysis. -
-
Triangle with 3 nodes
The most obvious linear function in the case of a triangle will yield the shape of Na of the form shown in Fig. 2.2(a). Writing, the two displacements as U :
0/1 + 0/2 X -'[-"0/3 Y
(2.3)
V "--0/4 -'~"0/5 X -+- 0/6 Y
we may evaluate the six constants by solving two sets of three simultaneous equations which arise if the nodal coordinates are inserted and the displacements equated to the appropriate nodal values. For example, the u displacement gives fil =0/1 + 0/2Xl -+- 0/3 yl fi2 = 0/1 + 0/2 x2 + 0/3 Y2 fi3 = 0/1 + 0/2 X3 + 0/3 Y3
(2.4)
Direct formulation of finite element characteristics
We can easily solve for obtain finally
0/1, 0t 2
and
Ot3
in terms of the nodal displacements fi 1,
1
u2
and
u = 2A [(al + b l x + c l y ) Ul ~" (a2 -k- b2x -4c-c2y) U2 "q- (a3 -k- b3x -Jr c3y) fi3]
fi3
and
(2.5)
in which al = x2Y3
-
-
x3Y2
(2.6)
bl = Y2 - Y3 r
-
-
"
X3 m X2
with other coefficients obtained by cyclic permutation of the subscripts in the order 1, 2, 3, and where 1 Xl yl 2A = det 1 x2 y2 = 2. (area of triangle 123) (2.7) 1 x3 y3 From (2.5) we see that the shape functions are given by Na -- (aa + ba x -k- Ca y)/(2A);
a = 1, 2, 3
(2.8)
Since displacements with these shape functions vary linearly along any side of a triangle the interpolation (2.5) guarantees continuity between adjacent elements and, with identical nodal displacements imposed, the same displacement will clearly exist along an interface between elements. We note, however, that in general the derivatives will not be continuous between elements.
Rectangle with 4 nodes
An alternative subdivision can use rectangles of the form shown in Fig. 2.3. The rectangular element has side lengths of a and b in the x and y directions, respectively. For the derivation
y 4
f
()
3
|
119
I I
() 1
L.
C)__~.x' 2 ._1 x
Fig. 2.3 Rectangularelement geometry and local node numbers.
23
24 A direct physical approach to problems in elasticity: plane stress of the shape functions it is convenient to use a local cartesian system x', y' defined by X t -- X ~ Xl
Y~= Y-Yl
We now need four functions for each displacement component in order to uniquely define the shape functions. In addition these functions must have linear behaviour along each edge of the element to ensure interelement continuity. A suitable choice is given by U = Ot 1 + X t Ol2 +
yt or3 + x l y 1014
V = Ot5 + X ~ Ot6 + y t o t 7 +
The coefficients
O~a
(2.9)
x~y ~or8
may be obtained by expressing (2.9) at each node, giving for u Ul = 0 / 1
(2.10)
U2 = 0tl -+- a ot2
fi 3 = Otl + a or2 + b or3 + a b ~l 4 = Ol 1 +
We can again easily solve for
O~a
Ol4
b or3
in terms of the nodal displacements to obtain finally
1
u - ~--~[(a - x ' ) ( b - y') fil + x' (b - y') fi2 -+- x ' y ' fi3 + (a - x') y'
fi4]
(2.11)
An identical expression is obtained for v by replacing fia by ~a. From (2.11) we obtain the shape functions N1 = (a - x ' ) ( b - y ' ) / ( a b ) N2 -- x' (b - y ' ) / ( a b )
N3=
x' y' / ( a b )
(2.12)
N4 = (a - x') y' / ( a b )
2.2.2 Strains With displacements known at all points within the element the 'strains' at any point can be determined. These will always result in a relationship that can be written in matrix notation ast e = ,Su (2.13) where ,9 is a suitable linear differential operator. Using Eq. (2.1), the above equation can be approximated by r ,~ ~ - - B ~ e (2.14) with B = ,SN
(2.15)
t i t is known that strain is a second rank tensor by its transformation properties; however, in this book we will normally represent quantities using matrix (Voigt) notation. The interested reader is encouraged to consult Appendix B for the relations between tensor forms and the matrix quantities.
Direct formulation of finite element characteristics For the plane stress case the relevant strains of interest are those occurring in the plane and are defined in terms of the displacements by well-known relations 6 which define the operator ,S Ou
a
Ox e =
ey
=
Yxy
Oy ~U
-
o =
O,
OU
~
-~y + -~x
Oy
-~y
v
Ox _
With the shape functions N1, N2 and N3 already determined for a triangular element, the matrix B can easily be obtained using (2.15). If the linear form of the shape functions is adopted then, in fact, the strains are constant throughout the element (i.e., the B matrix is constant). A similar result may be obtained for the rectangular element by adding the results for N4; however, in this case the strains are not constant but have linear terms in x and y.
2,2,3 Stresses In general, the material within the element boundaries may be subjected to initial strains such as those due to temperature changes, shrinkage, crystal growth, and so on. If such strains are denoted by e0 then the stresses will be caused by the difference between the actual and initial strains. In addition it is convenient to assume that at the outset of the analysis the body is stressed by some known system of initial residual stresses tr0 which, for instance, could be measured, but the prediction of which is impossible without the full knowledge of the material's history. These stresses can simply be added on to the general definition. Thus, assuming linear elastic behaviour, the relationship between stresses and strains will be linear and of the form tr = D(e - So) + t r o (2.16) where D is an elasticity matrix containing the appropriate material properties. Again for the particular case of plane stress three components of stress corresponding to the strains already defined have to be considered. These are, in familiar notation, o'--
Cry 72xy
and for an isotropic material the D matrix may be simply obtained from the usual stressstrain relationship 6 ~x - ~xO =
1 1) -E (ax -,~xO) - -~ (ay - a y o )
v 1 F,y -- F,y0 = ----E (t7x --O'xO ) -1t- ~ (fly --tYyO) ~xy -- ~xyO =
2(1 + v) E
(72xy -- rxyO)
25
26 A direct physical approach to problems in elasticity: plane stress i.e., on solving, D
1 -
v2
ElY 0v)/2] 1
0
(1 -
0
2.2.4 Equivalent nodal forces Let qe __
q~
define the nodal forces which are statically equivalent to the boundary stresses and distributed body forces acting on the element. Each of the forces ~ must contain the same number of components as the corresponding nodal displacement Ua and be ordered in the appropriate, corresponding directions. The distributed body forces b are defined as those acting on a unit volume of material within the element with directions corresponding to those of the displacements u at that point. In the particular case of plane stress the nodal forces are, for instance, ve
with components U and V corresponding to the directions of u and v, respectively (viz. Fig. 2.1), and the distributed body forces are
b~lbybxI bx
in which and by are the 'body force' components per unit of volume. In the absence of body forces equivalent nodal forces for the 3-node triangular element can be computed directly from equilibrium considerations. In Fig. 2.4(a) we show a triangular element together with the geometric properties which are obtained by the linear interpolation of the displacements using (2.1) to (2.8). In particular we note from the figure [and (2.6)] that bl+b2+b3=0 and c 1 + c 2 + c 3 = 0 The stresses in the element are given by (2.16) in which we assume that so and o'0 are constant in each element and strains are computed from (2.14) and, for the 3-node triangular element, are also constant in each element. To determine the nodal forces resulting from the stresses, the boundary tractions are first computed from
{tx}--t [oX 0 nY] { fix } ty ny nx ay (2.17) 75xy where t is a constant thickness of the plane strain slice and nx, n y arethe direction cosines t=
of the outward normal to the element boundary. For the triangular element the tractions
Direct formulation of finite element characteristics
Fig. 2.4 3-node triangle, geometry and constant stress state. are constant. The resultant for each side of the triangle is the product of the triangle side times the traction. Here is the length of the side opposite the triangle node a and we note from Fig. 2.4(a) that
length
(la)
la
la nx = - ba
la ny
and
:
--
(2.18)
c a
Therefore,
flatx I --ba lat = [late "- t 0
0
--Ca]{trxtry I 75xy
--Ca -ba
The resultant acts at the middle of each side of the triangle and, thus, by sum of forces and moments is equivalent to placing half at each end node. Thus, by static equivalence the nodal forces at node 1 are given by
ql:2
([--0b2
0 --C2
-b2
0 C1]t r = Cl bl
+
0
0 -c3
-b3
o" (2.19a)
B~r At
Similarly, the forces at nodes 2 and 3 are given by q2 -- B2tr A t
(2.19b)
q3 -" B3 tr A t Combining with the expression for stress and strain for each element we obtain q - - B T[D(Bfi e - e 0 ) + t r 0 ]
: Keu e -t- fe
At
(2.20a)
27
28
A direct physical approach to problems in elasticity: plane stress where K e = BTDB At
and
fe __ BT(o.0 _ D s o )
At
(2.20b)
The above gives a result which is now in the form of the standard discrete problem defined in Sec. 1.2. However, when body forces are present or we consider other element forms the above procedure fails and we need a more general approach. To make the nodal forces statically equivalent to the actual boundary stresses and distributed body forces, the simplest general procedure is to impose an arbitrary (virtual) nodal displacement and to equate the external and internal work done by the various forces and stresses during that displacement. Let such a virtual displacement be ~Ule at the nodes. This results, by Eqs (2.1) and (2.14), in virtual displacements and strains within the element equal to 8u = N ~ e
and 8s = B ~ e
(2.21)
respectively. The external work done by the nodal forces is equal to the sum of the products of the individual force components and corresponding displacements, i.e., in matrix form ~ eT e -, ~ eT e = ~eTqe 8 Ul ql -+- 0U2 q 2 " ' "
(2.22)
Similarly, the internal work per unit volume done by the stresses and distributed body forces subjected to a set of virtual strains and displacements is ~S TO" -- ~uTb
(2.23)
~UIeT (BTo " -- NTb)
(2.24)
or, after using (2.21),t Equating the external work with the total internal work obtained by integrating (2.24) over the volume of the element, f2e, we have ~ e T q e -- 6fieT ( ~ e
BTo" d ~ - s
NTb d ~ )
(2.25)
As this relation is valid for any value of the virtual displacement, the multipliers must be equal. Thus qe=f~
BTcrdf2_ f~ NTbdf2 e
(2.26)
e
This statement is valid quite generally for any stress-strain relation. With (2.14) and the linear law of Eq. (2.16) we can write Eq. (2.26) as qe __ Kefie at- fe
(2.27)
K e -- f~ BTD B dr2
(2.28a)
where 9 e
and
fe
NT. o e
.To 0dO e
.T 0 O e
t Note that by the rules of matrix algebra for the transpose of products (A B) T = BTAT.
(2.28b)
Direct formulation of finite element characteristics 29
For the plane stress problem f
(')df2=fa(')tdA e
e
where Ae is the area of the element. Here t now can be allowed to vary over the element. In the last equation the three terms represent forces due to body forces, initial strain, and initial stress respectively. The relations have the characteristics of the discrete structural elements described in Chapter 1. If the initial stress system is self-equilibrating, as must be the case with normal residual stresses, then the forces given by the initial stress term of Eq. (2.28b) are identically zero after assembly. Thus frequent evaluation of this force component is omitted. However, if for instance a machine part is manufactured out of a block in which residual stresses are present or if an excavation is made in rock where known tectonic stresses exist a removal of material will cause a force imbalance which results from the above term. For the particular example of the plane stress triangular element these characteristics will be obtained by appropriate substitution. It has already been noted that the B matrix in that example was not dependent on the coordinates; hence the integration will become particularly simple and, in the absence of body forces, K e and fe are identical to those given in (2.20b). The interconnection and solution of the whole assembly of elements follows the simple structural procedures outlined in Chapter 1. This gives r -- E
qe = 0
(2.29)
e
A note should be added here concerning elements near the boundary. If, at the boundary, displacements are specified, no special problem arises as these can be satisfied by specifying some of the nodal parameters ~. Consider, however, the boundary as subject to a distributed external loading, say t per unit area (traction). A loading term on the nodes of the element which has a boundary face Fe will now have to be added. By the virtual work consideration, this will simply result in /.t
fe ~
fe _ / _ NT~ dF ,/I
7
(2.30)
e
with integration taken over the boundary area of the element. It will be noted that t must have the same number of components as u for the above expression to be valid. Such a boundary element is shown again for the special case of plane stress in Fig. 2.1. Once the nodal displacements have been determined by solution of the overall 'structural'type equations, the stresses at any point of the element can be found from the relations in Eqs (2.14) and (2.16), giving O" = I) ( B u e - c0) + o'0
(2.31)
Example 2.1: Stiffness matrix for 3-node triangle. The stiffness matrix for an individual element is computed by evaluating Eq. (2.28a). For a 3-node triangle in which the moduli and thickness are constant over the element the solution for the stiffness becomes K e - B~ D B A t
(2.32)
30 A direct physical approach to problems in elasticity: plane stress
0]
where A is the area of the triangle computed from (2.7). Evaluating (2.15) using the shape functions in (2.8) gives Ba -2A
Lca
(2.33)
Ca ba
Thus, the expression for the stiffness of the triangular element is given by t K a b - - ~za~
[~ O baCa]
D12 LD31 D32
Ca
D13] [ ~
0]
D33
bb
LCb
where Dij : D j i are the elastic moduli. Example 2.2: Nodal forces for boundary traction. Let us consider a problem in which a traction boundary condition is to be imposed along a vertical surface located at x = Xb. A triangular element has one of its edges located along the boundary as shown in Fig. 2.5 and is loaded by a specified traction given by t
=
ty
~xy
The normal stress ax is given by a linearly varying stress in the y direction and the shearing stress is assumed zero, thus, to compute nodal forces we use the expressions
75xy
ax = kx y
and
rxy = 0
in which kx is a specified constant. Along the boundary the shape functions for either a triangular element or a rectangular element are linear functions in y and are given by N1 = (Y2 - Y ) / ( Y 2 - Yl)
and
N2 = (y - Y l ) / ( Y 2 -
(xb,~)
(xb,~)
Fig. 2.5 Traction on vertical face.
Yl)
Generalization to the whole region -internal nodal force concept abandoned 31
thus, the nodal forces for the element shown are computed from Eq. (2.30) and given by
fl -- -- fyY2t N1 { ax } d y 1
"gxy
f2 ~-fyY2lN2I~
75xy
-
- {kxt(2yl+Y2)(y2-Yl)/6} 0
and
kx t(yl + 2y2)(y2 - yl)/6"~ 0
f
In the preceding section the virtual work principle was applied to a single element and the concept of equivalent nodal force was retained. The assembly principle thus followed the conventional, direct equilibrium, approach. The idea of nodal forces contributed by elements replacing the continuous interaction of stresses between elements presents a conceptual difficulty. However, it has a considerable appeal to 'practical' engineers and does at times allow an interpretation which otherwise would not be obvious to the more rigorous mathematician. There is, however, no need to consider each element individually and the reasoning of the previous section may be applied directly to the whole continuum. Equation (2.1) can be interpreted as applying to the whole structure, that is, u=Iqfi
and 8 u = N 6 f i
(2.35)
in which fi and ~ list all the nodal points and
/~a -- Z Ne
(2.36)
e
when the point concerned is within a particular element e and a is a node point associated with the element. If a point does not occur within the element (see Fig. 2.6)
Na = 0
(2.37)
A matrix 1) can be similarly defined and we shall drop the bar, considering simply that the shape functions, etc., are always defined over the whole domain, f2. For any virtual displacement 6fi we can now write the sum of internal and external work for the whole region as
In the above equation, 3fi, 3u and 6e can be completely arbitrary, providing they stem from a continuous displacement assumption. If for convenience we assume they are simply variations linked by relations (2.35) and (2.14) we obtain, on substitution of the constitutive relation (2.16), a system of algebraic equations K fi + f = 0
(2.39)
32 A direct physical approach to problems in elasticity: plane stress
X
Fig. 2.6 Shapefunction ]~a for whole domain. where K = f a BTD B dr2
(2.40a)
and f - -s
NTbd~ - ~r NTt dF + ~a BT(cr0 -- De0) dr2
(2.40b)
The integrals are taken over the whole domain f2 and over the whole surface area F on which tractions are given. It is immediately obvious from the above that
Kab = Z
Keb and fa = Z
e
fae
(2.41)
e
by virtue of the property of definite integrals requiting that the total be the sum of the parts: /a(.)dfi=Z/a(.)dt2 e
e
and
fr(.)dF=Zj(r(.)dF e
(2.42)
e
The same is obviously true for the surface integrals in Eq. (2.40b). We thus see that the 'secret' of the approximation possessing the required behaviour of a 'standard discrete system' of Chapter 1 lies simply in the requirement of writing the relationships in integral form. The assembly rule as well as the whole derivation has been achieved without involving the concept of 'interelement forces' (i.e., qe). In the remainder of this book the element superscript will be dropped unless specifically needed. Also no differentiation between element and system shape functions will be made. However, an important point arises immediately. In considering the virtual work for the whole system [Eq. (2.38)] and equating this to the sum of the element contributions it is implicitly assumed that no discontinuity in displacement between adjacent elements develops. If such a discontinuity developed, a contribution equal to the work done by the stresses in the separations would have to be added.
Generalization to the whole region -internal nodal force concept abandoned ,
|
zone
du
dx
t t t
t
i I
i
d2u
dx2
|1
is is i | i i
| !
i ,
| i
i
! |
i i t
i
Fig. 2.7 Differentiation of function with sloped discontinuity (Co continuous).
Put in other words, we require that the terms integrated in Eq. (2.42) be finite. These terms arise from the shape functions Na used in defining the displacement u [by Eq. (2.35)] and its derivatives associated with the definition of strain [viz. Eq. (2.14)]. If, for instance, the 'strains' are defined by first derivatives of the functions N, the displacements must be continuous. In Fig. 2.7 we see how first derivatives of continuous functions may involve a 'jump' but are still finite, while second derivatives may become infinite. Such functions we call Co continuous. In some problems the 'strain' in a generalized sense may be defined by second derivatives. In such cases we shall obviously require that both the function N and its slope (first derivative) be continuous. Such functions are more difficult to derive but are used in the analysis of thin plate and shell problems (e.g., see volume on solid mechanics7). The continuity involved now is called C1.
33
34
A direct physical approach to problems in elasticity: plane stress
The principle of virtual displacements used in the previous sections ensured satisfaction of equilibrium conditions within the limits prescribed by the assumed displacement pattern. Only if the virtual work equality for all, arbitrary, variations of displacement was ensured would the equilibrium be complete. As the number of parameters of fi which prescribes the displacement increases without limit then ever closer approximation of all equilibrium conditions can be ensured. The virtual work principle as written in Eq. (2.38) can be restated in a different form if the virtual quantities 6fi, 8u, and 8s are considered as variations of the real quantities. 8' 9 Thus, for instance, we can write the first term of Eq. (2.38), for elastic materials, as ~U -- f a ~sT~ dr2
(2.43)
where U is the strain energy of the system. For the linear elastic material described by Eq. (2.16) the strain energy is given by U -- ~
sTD S d ~ +
S T (cr0
- D So) dr2
(2.44)
and will, after variation, yield the correct expression providing I) is a symmetric matrix (this is a necessary condition for a single-valued U to exist). 8' 9 The last two terms of Eq. (2.38) can be written as , W = - , ( f a uTbdf2 + f r uTtdF )
(2.45)
where W is the potential energy of the external loads. The above is certainly true if b and i are conservative (or independent of displacement) where we obtain simply W = -~
uTbdf2 - f r uTtdF
(2.46)
Thus, instead of Eq. (2.38), we can write the total potential energy, I-I, as FI = U + W
(2.47)
in which U is given by (2.44) and W by (2.46) and require 8rI = 8(u + w) = 0
(2.48)
In this form FI is known as a functional and (2.48) is a requirement which renders the functional stationary. The above statement means that for equilibrium to be ensured the total potential energy must be stationary for variations of the admissible displacements. The finite element equations for the total potential energy are obtained by substituting the approximation for displacements [viz. Eq. (2.35)] into Eqs (2.44) and (2.46) giving lI = I~TK ~ nt- ~Tf 2
(2.49)
Displacement approach as a minimization of total potential energy
in which K (where K = K T) and f are given by Eqs (2.40a) to (2.41). The variation with respect to displacements with the finite number of parameters ~ is now written as OH FI _
aa
~ I-I
7u2
= K fi + f = 0
(2.50)
It can also be shown that in stable elastic situations the totalpotential energy is not only stationary but is a minimum. 8 Thus the finite element process seeks such a minimum within the constraint of an assumed displacement pattern. The greater the number of degrees of freedom, the more closely the solution will approximate the true one, ensuring complete equilibrium, providing the true displacement can, in the limit, be represented. The necessary convergence conditions for the finite element process could thus be derived. Discussion of these will, however, be deferred to subsequent sections. It is of interest to note that if true equilibrium requires an absolute minimum of the total potential energy, l-I, a finite element solution by the displacement approach will always provide an approximate 1-I greater than the correct one. Thus a bound on the value of the total potential energy is always achieved. If the functional I7 could be specified, a priori, then the finite element equations could be derived directly by the differentiation specified by Eq. (2.50). The well-known Rayleighl~ 11 process of approximation frequently used in elastic analysis is based precisely on this approach. The total potential energy expression is formulated and the displacement pattern is assumed to vary with a finite set of undetermined parameters. A set of simultaneous equations minimizing the total potential energy with respect to these parameters is set up. Thus the finite element process as described so far can be considered to be the Rayleigh-Ritz procedure. The difference is only in the manner in which the assumed displacements are prescribed. In the traditionally used Ritz process the functions are usually given by expressions valid throughout the whole region, thus leading to simultaneous equations in which the coefficient matrix is full. In the finite element process this specification is usually piecewise, each nodal parameter influencing only adjacent elements, and thus a sparse and usually banded matrix of coefficients is found. By its nature the conventional Ritz process is limited to relatively simple geometrical shapes of the total region while this limitation only occurs in finite element analysis in the element itself. Thus complex, realistic, configurations can be assembled from relatively simple element shapes. A further difference is in the usual association of the undetermined parameter fia with a particular nodal displacement. This allows a simple physical interpretation invaluable to an engineer. Doubtless much of the early popularity of the finite element process is due to this fact.
2.4.1 Bound on strain energy in a displacement formulation While the approximation obtained by the finite element displacement approach always overestimates the true value of the total potential energy H (the absolute minimum corresponding
35
3t5 A direct physical approach to problems in elasticity: plane stress to the exact solution), this is not directly useful in practice. It is, however, possible to obtain a more useful limit in special cases. Consider the problem in which no initial strains ~0 or initial stresses ~r0 exist. Now by the principle of energy conservation the strain energy will be equal to the work done by the external loads which increase uniformly from zero. 12 This work done is equal to - W/2 where W is the potential energy of the loads. Thus, 1 u + :w 2
= o
(2.51)
or
FI = U + W = - U
(2.52)
whether an exact or approximate displacement field is assumed. If only one external concentrated load Ra is present, the strain energy bound immediately informs us that the finite element deflection under this load has been underestimated (as U -- -W/2 -- -Raua/2, where Ua is the deflection at the load point). In more complex loading cases the usefulness of this bound is limited as neither local displacements nor local stresses, i.e., the quantities of real engineering interest, can be bounded. It is also important to remember that this bound on strain energy is only valid in the absence of any initial stresses or strains. The expression for U in this case can be obtained from Eq. (2.44) as
1s
g = ~
sTD s dr2
(2.53)
which becomes by Eq. (2.14) simply U = ~1 fiT J~ B T D B d f l a = 1 fitTK
(2.54)
a quadratic matrix form in which K is the stiffness matrix previously discussed. When sufficient supports are provided to prevent rigid body motion and only linear elastic materials are considered, the above energy expression is always positive from physical considerations. It follows therefore that the matrix K occurring in all the finite element assemblies is not only symmetric but is positive definite (a property defined in fact by the requirement that the quadratic form should always be greater than zero). This feature is of importance when the numerical solution of the simultaneous equations is considered, as simplifications arise in the case of symmetric positive definite equations. 13
2.4.2 Direct minimization The fact that the finite element approximation reduces to the problem of minimizing the total potential energy FI defined in terms of a finite number of nodal parameters led us to the formulation of the simultaneous set of equations given symbolically by Eq. (2.50). This is the most usual and convenient approach, especially in linear solutions, but other search procedures, now well developed in the field of optimization, could be used to estimate the lowest value of Yl. In this text we shall continue with the simultaneous equation process but the interested reader could well bear the alternative possibilities in mind. 14,15
Convergence criteria 37
The assumed shape functions limit the infinite degrees of freedom of the real system, and the true minimum of the energy may never be reached, irrespective of the fineness of subdivision. To ensure convergence to the correct result certain simple requirements must be satisfied. Obviously, for instance, the displacement function should be able to represent the true displacement distribution as closely as desired. It will be found that this is not so if the chosen functions are such that straining is possible when the element is subjected to rigid body displacements at the nodes. Thus, the first criterion that the displacement function must obey is: Criterion 1. The displacement shape functions chosen should be such that they do not permit straining of an element to occur when the nodal displacements are caused by a rigid body motion.
This self-evident condition can be violated easily if certain types of function are used; care must therefore be taken in the choice of displacement functions. A second criterion stems from similar requirements. Clearly, as elements get smaller nearly constant strain conditions will prevail in them. If, in fact, constant strain conditions exist, it is most desirable for good accuracy that a finite size element is able to reproduce these exactly. It is possible to formulate functions that satisfy the first criterion but at the same time require a strain variation throughout the element when the nodal displacements are compatible with a constant strain solution. Such functions will, in general, not show good convergence to an accurate solution and cannot, even in the limit, represent the true strain distribution. The second criterion can therefore be formulated as follows: Criterion 2. The displacement shape functions have to be of such a form that if nodal displacements are compatible with a constant strain condition such constant strain will in fact be obtained.
It will be observed that Criterion 2 in fact incorporates the requirement of Criterion 1, as rigid body displacements are a particular case of constant strain - with a value of zero. This criterion was first stated by Bazeley et al. 16 in 1966. Strictly, both criteria need only be satisfied in the limit as the size o f the element tends to zero. However, the imposition of these criteria on elements of finite size leads to improved accuracy, although in certain situations (such as in axisymmetric analysis) the imposition of the second one is not possible or essential. Lastly, as already mentioned in Sec. 2.3, it is implicitly assumed in this derivation that no contribution to the virtual work arises at element interfaces. It therefore appears necessary that the following criterion be included: Criterion 3. The displacement shape functions should be chosen such that the strains at the interface between elements are finite (even though they may be discontinuous).
This criterion implies a certain continuity of displacements between elements. In the case of strains being defined by first derivatives, as in the plane stress example quoted here, the displacements only have to be continuous (Co continuity). If, however, the 'strains' are defined by second derivatives, first derivatives of these have also to be continuous (C1 continuity).:
38 A direct physical approach to problems in elasticity: plane stress The above criteria are included mathematically in a statement of 'functional completeness' and the reader is referred elsewhere for full mathematical discussion. 17-22The 'heuristic' proof of the convergence requirements given here is sufficient for practical purposes and we shall generalize all of the above criteria in Sec. 3.6 and more fully in Chapter 9. Indeed in the latter we shall show a universal test which justifies convergence even if some of the above criteria are violated.
In the foregoing sections we have assumed that the approximation to the displacement as represented by Eq. (2.1) will yield the exact solution in the limit as the size h of elements decreases. The arguments for this are simple: if the expansion is capable, in the limit, of exactly reproducing any displacement form conceivable in the continuum, then as the solution of each approximation is unique it must approach, in the limit of h ~ 0, the unique exact solution. In some cases the exact solution is indeed obtained with a finite number of subdivisions (or even with one element only) if the polynomial expansion used in that element fits the exact solution. Thus, for instance, if the exact solution is of the form of a quadratic polynomial and the shape functions include all the polynomials of that order, the approximation will yield the exact answer. The last argument helps in determining the order of convergence of the finite element procedure as the exact solution can always be expanded in a Taylor series in the vicinity of any point (or node) a as a polynomial: U --- U a +
-~X
( X - - X a ) "~ a
a
(Y
-- Ya) +''"
(2.55)
If within an element of 'size' h a polynomial expansion complete to degree p is employed, this can fit locally the Taylor expansion up to that degree and, as x - Xa and y - Ya are of the order of magnitude h, the error in u will be of the order O(h p+I). Thus, for instance, in the case of the plane elasticity problem discussed, we used a complete linear expansion and p = 1. We should therefore expect a convergence rate of order O(h2), i.e., the error in displacement being reduced to 1/4 for a halving of the mesh spacing. By a similar argument the strains (or stresses) which are given by the mth derivatives of displacement should converge with an error of O(hp+l-m),i.e., as O(h) in the plane stress example, where m = 1. The strain energy, being given by the square of the stresses, will show an error of O(h 2(p+I-m)) or O(h 2) in the plane stress example. The arguments given here are perhaps 'heuristic' from a mathematical viewpoint- they are, however, true 21' 22 and correctly give the orders of convergence, which can be expected to be achieved asymptotically as the element size tends to zero and if the exact solution does not contain singularities. Such singularities may result in infinite values of the coefficients in terms omitted in the Taylor expansion of Eq. (2.55) and invalidate the arguments. However, in many well-behaved problems the mere determination of the order of convergence often suffices to extrapolate the solution to the correct result. Thus, for instance, if the displacement converges at O(h 2) and we have two approximate solutions u I and u 2 obtained with meshes of size h and h/2, we can write, with u being the exact solution, u1- u
O(h 2)
u2 - u -
O(h/2) 2
,~ 4
(2.56)
Non-conforming elements and the patch test From the above an (almost) exact solution u can be predicted. This type of extrapolation was first introduced by Richardson e3 and is of use if convergence is monotonic and nearly asymptotic. We shall return to the important question of estimating errors due to the discretization process in Chapter 13 and will show that much more precise methods than those arising from convergence rate considerations are possible today. Indeed automatic mesh refinement processes can be introduced so that the specified accuracy can be achieved (viz. Chapters 8 and 14). Discretization error is not the only error possible in a finite element computation. In addition to obvious mistakes which can occur when introducing data into computers, errors due to round-offare always possible. With the computer operating on numbers rounded off to a finite number of digits, a reduction of accuracy occurs every time differences between 'like' numbers are being formed. In the process of equation solving many subtractions are necessary and accuracy decreases. Problems of matrix conditioning, etc., enter here and the user of the finite element method must at all times be aware of accuracy limitations which simply do not allow the exact solution ever to be obtained. Fortunately in many computations, by using modem machines which carry a large number of significant digits, these errors are often small. Another error that is often encountered occurs in approximation of curved boundaries by polynomials on faces of elements. For example, use of linear triangles to approximate a circular boundary causes an error of O(h 2) to be introduced.
In some cases considerable difficulty is experienced in finding displacement functions for an element which will automatically be continuous along the whole interface between adjacent elements. As already pointed out, the discontinuity of displacement will cause infinite strains at the interfaces, a factor ignored in this formulation because the energy contribution is limited to the elements themselves. However, if, in the limit, as the size of the subdivision decreases continuity is restored, then the formulation already obtained will still tend to the correct answer. This condition is always reached if (a) constant strain condition automatically ensures displacement continuity, and (b) the constant strain criteria of the previous section are satisfied. To test that such continuity is achieved for any mesh configuration when using such non-conforming elements it is necessary to impose, on an arbitrary patch of elements, nodal displacements corresponding to any state of constant strain. If nodal equilibrium is simultaneously achieved without the imposition of external nodal forces and if a state of constant stress is obtained, then clearly no external work has been lost through interelement discontinuity. Elements which pass such apatch test will converge, and indeed at times non-conforming elements will show a superior performance to conforming elements.
39
40 A direct physical approach to problems in elasticity: plane stress The patch test was first introduced by Irons 16 and has since been demonstrated to give a sufficient condition for convergence. 22' 24-28 The concept of the patch test can be generalized to give information on the rate of convergence which can be expected from a given element. We shall return to this problem in detail in Chapter 9 where the test will be fully discussed.
The finite element solution of a problem follows a standard methodology. The steps in any solution process are always performed by the following steps: 1. Define the problem to be solved in terms of differential equations. Construct the integral form for the problem as a virtual work, variational or weak formulation. 2. Select the type and order of finite elements to be used in the analysis. 3. Define the mesh for the problem. This involves the description of the node and element layout, as well as the specification of boundary conditions and parameters for the formulation used. The process for mesh generation will be described in more detail in Chapter 8. 4. Compute and assemble the element arrays. The particular virtual work, variational or weak form provide the basis for computing specific relationships of each element. 5. Solve the resulting set of linear algebraic equations for the unknown parameters. See Appendix C for a brief discussion on solution of linear algebraic equations. 6. Output the results for the nodal and element variables. Graphical outputs also are useful for this step. An accurate procedure to project element values to nodes is described in Chapter 6. Much of the discussion in the following chapters is concerned with the development of the theory needed to compute element arrays. For a steady-state problem the two arrays are a coefficient array K, which we refer to as a 'stiffness' matrix, and a force array f. In the next section, however, we first illustrate the solution steps for two problems for which the exact solution is available.
Let us now consider the solution to a set of problems for which an exact solution is known. This will enable us to see how the finite element results compare to the known solution and also to demonstrate the convergence properties for different element types. Of course, the power of the finite element method is primarily for use on problems for which no alternative solution is possible using results from classical books on elasticity and in later chapters we will include results for several such example problems.
2.9.1 Problems for accuracy assessment Example 2.3: Beam subjected to end shear. We consider a rectangular beam in a state of plane stress. The geometric properties are shown in Fig. 2.8(a). The solution to the problem is given in Timoshenko and Goodier based on use of a stress function solution. 6 The solution for stresses is given by
Numerical examples 41
5 C
/\/\/\/\ X
C
\/\/
\/\/
L
(a)
(b)
Fig. 2.8 End loaded beam' (a) Problem geometry and (b) coarse mesh.
Ox= %= 12xY =
3 Pxy 2 c3 0
3P [ 1 - (Y)21 4c
c
where P is the applied load and c the half-depth of the beam. For the displacement boundary conditions u ( L , O) = v ( L , O) = O and u ( L , c) = u ( L , - c ) = O As shown in Fig. 2.8(a), the solution for displacements is given by u=-
p(x 2 _ L2)y 2EI 2EI
6EI
P ( x 3 - L 3)
vPxy 2 V --
v P y ( y 2 _ c 2)
-
+
6EI
+
( PL 2
p y ( y 2 _ c 2) 6GI vPc 2
- \ 2EI + 6EI
pc 2 ) + ~
(x-L)
In the above E and v are the elastic modulus and Poisson ratio, G is the shear modulus given by E/[2(1 + v)] and I is the area moment of inertia which is equal to 2tc3/3 where t is a constant beam thickness. For this solution the tractions on the boundaries become
tx} {o} ,x} ,{.} ty
-- t
_ rxY
ty
=
Txy
forx--O; forx=L;
-c_
For the numerical solution we choose the properties c=10;
L=100;
t=l;
P=80;
E=1000andv--0.25
In order to perform a finite element solution to the problem we need to compute the nodal forces for the tractions using Eq. 2.30. When many elements are used in an analysis this
42 A direct physical approach to problems in elasticity: plane stress
Fig. 2.9 Convergencein energy error for 3-node triangles and 4-node rectangular elements. step can be quite tedious and it is best to write a small computer program to carry out the integrations (e.g., using MATLAB 29 or any other programming language).t The solution to the problem is carried out using a uniform mesh of (a) 3-node triangular elements and (b) 4-node rectangular elements and the results for the error in energy given by
OE=
[Eex-Efel 2 ~Ch Eex
is plotted versus the log in element size h in Fig. 2.9. Here Eex is the energy of the exact solution and Efe that of the finite element solution. Results for the energy are given in Table 2.1 and the exact value for the geometry and properties selected is 3296 (energy here is work done which is twice the stored elastic strain energy). The element size is normalized to that of the coarsest mesh [hi shown in Fig. 2.8(b)] and the energy error is computed using the exact value. The expected slope of 2 is achieved for both element types with the 4-node element giving a smaller constant C due to the presence of the xy term in each shape function. The stresses in each triangle are constant. The values for Crx, O'y and 'tSxy obtained in the elements at the fight end of the beam (where Crx is largest) are shown in Fig. 2.10(a). The distribution of Crx for x = 90 is shown in Fig. 2.10(b) where we also include values computed by a nodal averaging method. In Chapter 6 we will show how more accurate stresses may be obtained at nodes.
Example 2.4: Circular beam subjected to end shear. We consider a circular beam in a state of plane stress. The geometric properties are shown in Fig. 2.11. The solution to the problem is given in Timoshenko and Goodier based on use of a stress function. 6 The geometry and loading for the problem are shown in Fig. 2.11. The solution for stresses is t For the triangular elementsdiscussedin this chapter, the programFEAPpv availableas a companionto this book includes automatic computation of nodal forces for this type of loading.3~
Numerical examples Table 2.1 Mesh size and energy for end loaded beam (a) 3-node triangles
(b) 4-node rectangles
Nodes
Elmts
Energy
Elmts
Energy
55 189 697 2673 10465 41409
80 320 1280 5120 20480 81920
2438.020814633 3027.225730752 3223.959303515 3277.628191064 3291.381304522 3294.843527071
40 160 640 2560 10240 40960
2984.863896144 3212.088124234 3274.561666674 3290.607667261 3294.649630776 3295.662249951
-
3296.000000000
-
3296.000000000
Exact
(-78.28,5
.
~
w
(-47.92,-6.83,-10.86) ~4 ~)
(-1.64,-6.56, 7.56)
(1.64,6 . 5 6 ~ (44.55,-6.66, -24.62) ~10"86(78.28,-5.96, 119.1) (a)
(b)
Lr--
~ 108 --1 J
Fig. 2.10 End loaded beam: (a) Element stresses (~x, cry, rxy) and (b) o-x stress distribution for x = 90, element values and [] nodal average values.
C C C
c
i
i
(a)
(b)
Fig. 2..11 End loaded circular beam: (a) Problem geometry and (b) coarse mesh.
0
43
44 A direct physical approach to problems in elasticity: plane stress given by
P[
~rrr =
.N .
croo =
~
a2b2
. .r3 r-Jr
PI
a2b2 3r
P[ . "grO = -N. where N = a 2 - b 2 + (a 2 nt- b 2) log for displacements is given by Ur =
P { [2
-~
(1 - 3v)r 2 -
.
r.
b/a.
r3
a2b2 . r 3.
nt
a 2 + b 2]
r
sin 0
a2 -+ b2] -r sin 0
a2 -+-b 2 ] r
cos 0
For the restraints shown in Fig. 2.11 (a) the solution
a2b2(l+v) 2r 2
- (a 2 -+- b2)(1 - v) logr
1 sin0
-+-(a2 nt- b2)(0 - :rr) cos 0 } - K sin 0 J
Uo = ~
(5 + v)r 2 -- a2b2(1 + v) + (a 2 -k- b2 )[(1 - v ) l o g r - (1 + v)] cos0 2r 2
- ( a 2 + b2)(0 - 7r) sin 0 } - K cos 0 where for
Ur(a, zr/2) P
K = NE
= 0 we obtain [1 ~(1-3v)a
2-
b2(l+v)_(a2+b2)(l_v)loga] 2
In the above E and v are the elastic modulus and Poisson ratio; a and b are the inner and outer radii, respectively (see Fig. 2.11). For this solution the displacement Ur for 0 = 0 is constant and given by zrP Ur(r, 0) = - - ~ ( a
EN
2 -k- b 2) = uo
Thus, instead of computing the nodal forces for the traction on this boundary we merely set all the nodal displacements in the x direction to a constant value. For the numerical solution we choose the properties a -- 5; b = 10; t = 1; u0 = - 0 . 0 1 ; E = 10000 and v = 0.25 In addition the displacements on the boundaries are prescribed as
u(x, O) = uo
and u(O, y) = v(O, a) = 0
The finite element solution to the problem is carried out using a uniform mesh of 3-node triangular elements oriented as shown in Fig. 2.11 (b). Results for the energy are given in Table 2.2 and the 'exact' value is computed from
Eex ----1 [ l o g 2 - 0 . 6 ]
=0.02964966844238
Numerical examples 45 for the geometry and properties selected. The element size is normalized to that of the coarsest mesh [shown in Fig. 2.11 (b)] and the energy error given in Table 2.2 again has the expected slope of 2. Finally, in Fig. 2.13 we compare the Ur and uo displacements from the finite element solution of the coarsest mesh to the exact values. We observe that even with this coarse distribution of elements the solution is quite good. Unfortunately, the stress distribution is not as accurate and quite fine meshes are needed to obtain good values. In Chapter 6 we will show how use of higher order elements can significantly improve both the displacements and stresses obtained.
2.9.2 A practical application Obviously, the practical applications of the finite element method are limitless, and it has superseded experimental technique for plane problems because of its high accuracy, low cost, and versatility. The ease of treatment of general boundary shapes and conditions, material anisotropy, thermal stresses, or body force problems add to its practical advantages.
Stress flow around a reinforced opening
An example of an actual early application of the finite element method to complex problems of engineering practice is a steel pressure vessel or aircraft structure in which openings are introduced in the stressed skin. The penetrating duct itself provides some reinforcement round the edge and, in addition, the skin itself is increased in thickness to reduce the stresses due to concentration effects. Analysis of such problems treated as cases of plane stress present no difficulties. The elements are chosen so as to follow the thickness variation, and appropriate values of this are assigned. The narrow band of thick material near the edge can be represented either by special bar-type elements, or by very thin triangular elements of the usual type, to which appropriate thickness is assigned. The latter procedure was used in the problem shown in Fig. 2.14 which gives some of the resulting stresses near the opening itself. The fairly large extent of the region introduced in the analysis and the grading of the mesh should be noted. Table 2.2 Mesh size and energy for curved beam Nodes
3-node triangles Elmts Energy
35 117 425 1617 6305 24897 Exact
48 192 768 3072 12288 49152 -
0.04056964168222 0.03245261212845 0.03035760000738 0.02982725603614 0.02969411302439 0.02966078320581 0.02964966844238
Error (%) 36.830 9.454 2.388 0.598 0.150 0.037
46 A direct physical approach to problems in elasticity: plane stress
Fig. 2.13 End loaded circular beam: (a) Ur displacement and (b) uo displacement for r = a.
The 'displacement' approach to the analysis of elastic solids is still undoubtedly the most popular and easily understood procedure. In many of the following chapters we shall use the general formulae developed here in the context of linear elastic analysis (e.g., in Chapter 6). These are also applicable in the context of non-linear analysis, the main variants being the definitions of the stresses, generalized strains, and other associated quantities. 7 In Chapter 3 we shall show that the procedures developed here are but a particular case of finite element discretization applied to the governing differential equations written in terms of displacements. 31 Clearly, alternative starting points are possible. Some of these will be mentioned in Chapters 10 and 11.
Problems
Fig. 2.14 A reinforced opening in a plate. Uniform stress field at a distance from opening crx - 100, cry = 50. Thickness of plate regions A, B, and C is in the ratio of 1 : 3 : 23.
2.1 For the triangular element shown in Fig. 2.15(a), the dimensions are: a = 3 cm and b = 4 cm. Compute the shape functions N for the three nodes of the element. 2.2 For the rectangular element shown in Fig. 2.15(b), the dimensions are: a = 6 cm and b = 4 cm. Compute the shape functions N for the four nodes of the element. 2.3 Use the results from Problem 2.1 to compute the strain-displacement matrix B for the triangular element shown in Fig. 2.15(a). 2.4 Use the results from Problem 2.2 to compute the strain-displacement matrix B for the rectangular element shown in Fig. 2.15(b). The body force vector in a plane stress problem is given by bx = 5 and by -- 0. Using the shape functions determined in Problem 2.1 compute the body force vector for the triangular element shown in Fig. 2.15(a). --30. 2.5 Repeat Problem 2.4 using bx = 0 and by 2.6 The body force vector in a plane stress problem is given by bx = 5 and by = 0. Using the shape functions determined in Problem 2.2 compute the body force vector for the rectangular element shown in Fig. 2.15(b). 2.7 Repeat Problem 2.6 using bx = 0 and by = - 3 0 . 2.8 The edge of the triangular element defined by nodes 2-3 shown in Fig. 2.15(a) is to be assigned boundary conditions Un = 0 and ts = 0 where n is a direction normal to the edge and s tangential to the edge. Determine the transformation matrix L -
'
-
47
48
A direct physical approach to problems in elasticity: plane stress ;E---'
4
3
3
,J
b I r
j2 v
1
,r.-,
(a) Triangle
(b) Rectangle
Fig. 2.15 Elements for Problems 2.1 to 2.4. [viz. Eq. (1.21)] required to transform the nodal degrees of freedom at node 2 and 3 to be able to impose the boundary conditions. 2.9 A concentrated load, F, is applied to the edge of a two-dimensional plane strain problem as shown in Fig. 2.16(a). (a) Use equilibrium conditions to compute the statically equivalent forces acting at nodes 1 and 2. (b) Use virtual work to compute the equivalent forces acting on nodes 1 and 2. 2.10 A triangular traction load is applied to the edge of a two-dimensional plane strain problem as shown in Fig. 2.16(b). (a) Use equilibrium conditions to compute the statically equivalent forces acting at nodes 1 and 2. (b) Use virtual work to compute the equivalent forces acting on nodes 1 and 2. 2.11 For the rectangular and triangular element shown in Fig. 2.17, compute and assemble the stiffness matrices associated with nodes 2 and 5 (i.e., K22, K25 and K55). Let E = 1000, v = 0.25 for the rectangle and E = 1200, v = 0 for the triangle. The thickness for the assembly is constant with t = 0.2 cm.
Fig. 2.16 Traction loading on boundary for Problems 2.9 and 2. I0.
Problems
4
I
4 em
2
1
I|
3cm
6 cm
_<
J
Fig. 2.17 Element assembly for Problem 2.11.
2.12 The formulation given in Sec. 2.3 may be specialized to one dimension by using displacement functions Ue
= 0~1 "l" 0/2 X = N 1 (x) Ul -1- N 2 ( x ) u2
~u e =
N1(x) ~ 1
+ N 2 ( x ) ~u2
and simplifying the equation to: du
e=--=Bfi
Strain:
dx tr=Ee=EBfi
Stress:
where E is the modulus of elasticity and we assume initial stress tr0 and initial strain e0 are zero. (a) For a two-node element with coordinates located at x~ and x~ compute the shape functions Na which satisfy the linear approximation given above. (b) Compute the strain matrix B for the shape functions determined in (a). (c) Using Eqs (2.25) to (2.28b) compute the element stiffness and force vector. Assume the body force b in the x direction is constant in each element. (d) For the two element problem shown in Fig. 2.18 let each element have length a = 5, the end traction t = 4, the body force b -- 2 and the modulus of elasticity E -- 200. i. Generalize the element formulation given in (b) to form the whole problem. ii. Impose the boundary condition u (0) = fil = 0. iii. Determine the solution for fie and fi3. iv. Plot the computed finite element displacement u and stress cr vs x. 1
[]3
2
,,0
Fig. 2.18 One-dimensional elasticity. Problem 2.12.
3
0
t
49
50 A direct physical approach to problems in elasticity: plane stress (e) The exact solution to the problem satisfies the equilibrium equation da ~-Fb=0 dx
2.13
2.14 2.15 2.16
and boundary conditions u (0) = 0 and t (10) = cr (10) = 4. Compute and plot the exact solution for u (x) and cr (x). (f) What is the maximum error in the finite element solutions for u and tr ? (g) Subdivide the mesh into four elements and repeat the above solution steps. Download the program FEAPpv and user manual from a web site given in Chapter 19. Note that both source code for the program and an executable version for Windowsbased systems are available at the site. If source code is used it is necessary to compile the program to obtain an executable version. Use FEAPpv (or any available program) to solve the rectangular beam problem given in Example 2.3 - verify results shown in Table 2.1. Use FEAPpv (or any available program) to solve the curved beam problem given in Example 2 . 4 - verify results shown in Table 2.2. The uniformly loaded cantilever beam shown in Fig. 2.19 has properties L=2m;
h=0.4m;
t=0.05m
and q 0 = 1 0 0 N / m
Use FEAPpv (or any available program) to perform a plane stress analysis of the problem assuming linear isotropic elastic behaviour with E = 200 GPa and v = 0.3. In your analysis: (a) Use 3-node triangular elements with an initial mesh of two elements in the depth and ten elements in the length directions. (b) Compute consistent nodal forces for the uniform loading. (c) Compute nodal forces for a parabolically distributed shear traction at the restrained end which balances the uniform loading q0. (d) Report results for the centreline displacement in the vertical direction and the stored energy in the beam. (e) Repeat the analysis three additional times using meshes of 4 x 20, 8 x 40 and 16 x 80 elements. Tabulate the tip vertical displacement and stored energy for each solution.
y
%
r-x
Fig. 2.19 Uniformly loaded cantilever beam. Problem 2.16.
Problems
3
l
5
~ I i
X
J
2 6x= 2
r
Fig. 2.20 Patch test for triangles. Problem 2.17. (f) If the energy error is given by AE
= En - E n - 1 -- C h p
estimate C and p for your solution. (g) Repeat the above analysis using rectangular 4-node elements. 2.17 Program development project:t Write a MATLAB 29 programJ~ to solve plane stress problems. Your program system should have the following features" (a) Input module which describes" i. Nodal coordinate values, Xa; ii. Nodes connected to each element and material properties of each element; iii. Node and degree-of-freedom (dof) for each applied nodal forces; iv Node and dof for fixed (essential) boundary condition - also value if non-zero. (b) Module to compute the stiffness matrix for a 3-node triangular element [use Eq. (2.34)]. (c) Module to assemble element arrays into global arrays and specified nodal forces and displacements. (d) Module to solve Kfi + f = 0. (e) Module to output nodal displacements and element stress and strains. Use your program to solve the p a t c h test problem shown in Fig. 2.20. Use the properties: E = 2 . 1 0 5 , v = 0.3 and t = 1 (t is thickness of slab). You can verify the correctness of your answer by computing an exact solution to the problem. The correctness of computed arrays may be obtained using results from F E A P p v (or any available plane stress program). t If programming is included as a part of your study, it is recommended that this problem be solved. Several extensions will be suggested later to create a solution system capable of performing all steps of finite element analysis. J~Another programming language may be used; however, MATLAB offers many advantages to write simple programs and is also useful to easily complete later exercises.
51
52
A direct physical approach to problems in elasticity: plane stress 2.18 Program development project: Add a graphics capability to the program developed in Problem 2.17 to plot contours of the computed finite element displacements. (Hint: MATLAB has contour and surf options to easily perform this operation.) Solve the curved beam problem for the mesh shown in Fig. 2.11. Plot contours for u and v displacements. (Hint: Write a separate MATLAB program to generate the nodal coordinates and element connections for the simple geometry of the curved beam.) Refine the mesh by increasing the number of segments in each direction by a factor of 2 and repeat the solution of the curved beam problem.
1. R.W. Clough. The finite element method in plane stress analysis. In Proc. 2nd ASCE Conf. on Electronic Computation, Pittsburgh, Pa., Sept. 1960. 2. R.W. Clough. The finite element method in structural mechanics. In O.C. Zienkiewicz and G.S. Holister, editors, Stress Analysis, Chapter 7. John Wiley & Sons, Chichester, 1965. 3. J. Szmelter. The energy method of networks of arbitrary shape in problems of the theory of elasticity. In W. Olszak, editor, Proc. IUTAM Symposium on Non-Homogeneity in Elasticity and Plasticity. Pergamon Press, 1959. 4. R. Courant. Variational methods for the solution of problems of equilibrium and vibration. Bull. Am. Math Soc., 49:1-61, 1943. 5. W. Prager and J.L. Synge. Approximation in elasticity based on the concept of function space. Quart. J. Appl. Math., 5:241-269, 1947. 6. S.P. Timoshenko and J.N. Goodier. Theory of Elasticity. McGraw-Hill, New York, 3rd edition, 1969. 7. O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method for Solid and Structural Mechanics. Butterworth-Heinemann, Oxford, 6th edition, 2005. 8. K. Washizu. Variational Methods in Elasticity and Plasticity. Pergamon Press, New York, 3rd edition, 1982. 9. EB. Hildebrand. Methods of Applied Mathematics. Prentice-Hall (reprinted by Dover Publishers, 1992), 2nd edition, 1965. 10. Lord Rayleigh (J.W. Strutt). On the theory of resonance. Trans. Roy. Soc. (London), A161:77118, 1870. 11. W. Ritz./Ober eine neue Methode zur L6sung gewisser variationsproblem der mathematischen physik. Journal fttr die reine und angewandte Mathematik, 135:1-61, 1908. 12. B. Fraeijs de Veubeke. Displacement and equilibrium models in finite element method. In O.C. Zienkiewicz and G.S. Holister, editors, Stress Analysis, Chapter 9, pages 145-197. John Wiley & Sons, Chichester, 1965. 13. J. Demmel. Applied Numerical Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1997. 14. R.L. Fox and E.L. Stanton. Developments in structural analysis by direct energy minimization. J. AIAA, 6:1036-1044, 1968. 15. EK. Bogner, R.H. Mallett, M.D. Minich, and L.A. Schmit. Development and evaluation of energy search methods in non-linear structural analysis. In Proc. 1st Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-66-80, Wright Patterson Air Force Base, Ohio, Oct. 1966. 16. G.P. Bazeley, Y.K. Cheung, B.M. Irons, and O.C. Zienkiewicz. Triangular elements in bending -conforming and non-conforming solutions. In Proc. 1st Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-66-80, pages 547-576, Wright Patterson Air Force Base, Ohio, Oct. 1966.
References 17. S.C. Mikhlin. The Problem of the Minimum of a Quadratic Functional. Holden-Day, San Francisco, 1966. 18. M.W. Johnson and R.W. McLay. Convergence of the finite element method in the theory of elasticity. J. Appl. Mech., ASME, 274-278, 1968. 19. EG. Ciarlet. The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978. 20. T.H.H. Pian and E Tong. The convergence of finite element method in solving linear elastic problems. Int. J. Solids Struct., 4:865-880, 1967. 21. E.R. de Arantes e Oliveira. Theoretical foundations of the finite element method. Int. J. Solids Struct., 4:929-952, 1968. 22. G. Strang and G.J. Fix. An Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs, N.J., 1973. 23. L.E Richardson. The approximate arithmetical solution by finite differences of physical problems. Trans. Roy. Soc. (London), A210:307-357, 1910. 24. B.M. Irons and A. Razzaque. Experience with the patch test for convergence of finite elements. In A.K. Aziz, editor, The Mathematics of Finite Elements with Application to Partial Differential Equations, pages 557-587. Academic Press, New York, 1972. 25. B. Fraeijs de Veubeke. Variational principles and the patch test. Int. J. Numer. Meth. Eng., 8:783-801, 1974. 26. R.L. Taylor, O.C. Zienkiewicz, J.C. Simo, and A.H.C. Chan. The patch t e s t - a condition for assessing FEM convergence. Int. J. Numer. Meth. Eng., 22:39-62, 1986. 27. O.C. Zienkiewicz, S. Qu, R.L. Taylor, and S. Nakazawa. The patch test for mixed formulations. Int. J. Numer. Meth. Eng., 23:1873-1883, 1986. 28. O.C. Zienkiewicz and R.L. Taylor. The finite element patch test revisited: a computer test for convergence, validation and error estimates. Comp. Meth. Appl. Mech. Eng., 149:523-544, 1997. 29. MATLAB. www.mathworks.com, 2003. 30. R.L. Taylor. FEAP - A Finite Element Analysis Program: Personal version, User Manual. University of California, Berkeley. http://www.ce.berkeley.edu/~rlt/feappv. 31. O.C. Zienkiewicz and K. Morgan. Finite Elements and Approximation. John Wiley & Sons, London, 1983.
53
Generalization of the finite element concepts. Galerkin-weighted residual and variational approaches We have so far dealt with one possible approach to the approximate solution of the particular problem of linear elasticity. Many other continuum problems arise in engineering and physics and usually these problems are posed by appropriate differential equations and boundary conditions to be imposed on the unknown function or functions. It is the object of this chapter to show that all such problems can be dealt with by the finite element method. Posing the problem to be solved in its most general terms we find that we seek an unknown function u such that it satisfies a certain differential equation set .A(u) --
AI(U) A2(u)
=0
(3.1)
in a 'domain' (volume, area, etc.), f2, together with certain boundary conditions J~(u) :
B1 (u) / B2(U) = 0
(3.2)
on the boundaries, F, of the domain as shown in Fig. 3.1. The function sought may be a scalar quantity or may represent a vector of several variable~ Similarly, the differential equation may be a single one or a set of simultaneous equations and does not need to be linear. It is for this reason that we have resorted to matrix notation in the above. The finite element process, being one of approximation, will seek the solution in the approximate formt n U ,~ fl : Z Nafla -- Nfi (3.3) a=l
where Na are shape functions prescribed in terms of independent variables (such as the coordinates x, y, etc.) and all or most of the parameters Ua are unknown. t In the sequel we will also use summation convention for any repeated index. Thus NaUa ----~ a NaUa, etc.
Introduction
Fig. 3.1 Problemdomain f2 and boundaryY. In the previous chapter we have seen that precisely the same form of approximation was used in the displacement approach to elasticity problems. We also noted there that (a) the shape functions were usually defined locally for elements or subdomains and (b) the properties of discrete systems were recovered if the approximating equations were cast in an integral form [viz. Eqs (2.38)-(2.42)]. With this object in mind we shall seek to cast the equation from which the unknown parameters Ua are to be obtained in the integral form
J~ Gb(fl)df2 + fr gb(fl)dF = O
b = l to n
(3.4)
in which Gb and gb prescribe known functions or operators. These integral forms will permit the approximation to be obtained element by element and an assembly to be achieved by the use of the procedures developed for standard discrete systems in Chapter 1, since, providing the functions Gb and gb are integrable, we have
f
f
wm e=l
fF )
(f~
gbdF
e
e
where ~'2 e is the domain of each element and 1-'e its part of the boundary. Two distinct procedures are available for obtaining the approximation in such integral forms. The first is the method of weighted residuals (known alternatively as the Galerkin procedure); the second is the determination of variationalfunctionals for which stationarity is sought. We shall deal with both approaches in turn. If the differential equations are linear, i.e., if we can write (3.1) and (3.2) as .A(u) - s + b = 0 B(u) - .A4u + t = 0
in [2
on F
(3.6)
then the approximating equation system (3.4) will yield a set of linear equations of the form K fi + f = 0
with
m
e=l
(3.7) m
e=l
55
56 Generalizationof the finite element concepts The reader not used to abstraction may well now be confused about the meaning of the various terms. We shall introduce here some typical sets of differential equations for which we will seek solutions (and which will make the problems a little more definite). Example 3.1: Steady-state heat conduction equation in a two-dimensional domain. Here the equations are written as a(r
B(r
k-ff-x-x - ~ y
--
{k~+O=O
k-~y
+Q=0
in
(3.8)
on Fr
0r
on Fq
where u - r indicates temperature, k is the conductivity, Q is a heat source, r and ~/are the prescribed values of temperature and heat flow on the boundaries and n is the direction normal to F. In the context of this equation the boundary condition for Fr is called a Dirichlet condition and the one on 1-'q a Neumann one. In the above problem k and Q can be functions of position and, if the problem is nonlinear, of r or its derivatives.
Example 3.2: Steady-state heat conduction-convection equation in two dimensions. When convection effects are added the differential equation becomes A(r
= -0--xx k~x-x -
k
+
+
+ a = 0 in f2
(3.9)
with boundary conditions as in the first example. Here Ux and Uy are known functions of position and represent velocities of an incompressible fluid in which heat transfer occurs.
Example 3.3: A system of three first-order equations equivalent to Example 3.1. The steady-state heat equation in two dimensions may also be split into the three equations Oqx
A(u) -
Oqy
qx + k 0 r Ox
= 0 in f2
qy + k 0r Oy
and
B(u) = [ r
~ = 0
on PC
[ qn -- ?1 = 0
on F q
where qn is the flux normal to the boundary. Here the unknown function vector u corresponds to the set
u/:} l, qy
(3.10)
Integral or 'weak' statements equivalent to the differential equations
This last example is typical of a so-called mbced formulation. In such problems the number of dependent unknowns can always be reduced in the governing equations by suitable algebraic operation, still leaving a solvable problem [e.g., obtaining Eq. (3.8) from (3.10) by eliminating qx and qy]. If this cannot be done [viz. Eq.(3.8)] we have an irreducible formulation. Problems of mixed form present certain complexities in their solution which we shall discuss in Chapter 11. In Chapter 7 we shall return to detailed examples of the first problem and other examples will be introduced throughout the book. The above three sets of problems will, however, be useful in their full form or reduced to one dimension (by suppressing the y variable) to illustrate the various approaches used in this chapter.
Weighted residual methods
As the set of differential equations (3.1) has to be zero at each point of the domain f2, it follows that p ~ vTc4.(u)da = / [ V I A l ( U ) + v2A2(u)+ "" ]dr2 =- 0 (3.11) Ya where Vl V-"
V2
is a set of arbitrary functions equal in number to the number of equations (or components of u) involved. The statement is, however, more powerful. We can assert that if(3.11) is satisfied for all v then the differential equations (3.1) must be satisfied at all points oylhe domain. The proof of the validity of this statement is obvious if we consider the possibility that ,,4(u) y~ 0 at any point or part of the domain. Immediately, a function v can be found which makes the integral of (3.11) non-zero, and hence the point is proved. If the boundary conditions (3.10) are to be simultaneously satisfied, then we require that f r $'TB(u)dF = j(r[VlBl(U) + g'2B2(u) + ' "
]dI" = 0
(3.12)
for any set of arbitrary functions ~e. Indeed, the integral statement that
fa
v T.,4.(u)df2 + / r ~TB(u)dF - 0
(3.13)
is satisfied for all v and ~ is equivalent to the satisfaction of the differential equations (3.1) and their boundary conditions (3.2). In the above discussion it was implicitly assumed that integrals such as those in Eq. (3.13) are capable of being evaluated. This places certain restrictions on the possible families to
57
58 Generalizationof the finite element concepts -q A I~ i
one
du
i
dx
i
I
i
d2u dx 2
iI iI ! | i
| i
i
| | i | i
Fig. 3.2 Differentiation of function with slope discontinuity (Co continuity).
which the functions v or u must belong. In general we shall seek to avoid functions which result in any term in the integrals becoming infinite. Thus, in Eq. (3.13) we generally limit the choice of v and ~eto bounded functions without restricting the validity of previous statements. What restrictions need to be placed on the functions? The answer obviously depends on the order of differentiation implied in the equations ,A(u) [or B(u)]. Consider, for instance, a function u which is continuous but has a discontinuous slope in the x direction, as shown in Fig. 3.2 which is identical to Fig. 2.7 but is reproduced here for clarity. We imagine this discontinuity to be replaced by a continuous variation in a very small distance A (a process known as 'molification') and study the behaviour of the derivatives. It is easy to see that although the first derivativeis not defined here, it has finite value and can be integrated easily but the second derivative tends to infinity. This therefore presents difficulties if integrals are to be evaluated numerically by simple means, even though the integral is finite. If such derivatives are multiplied by each other the integral does not exist and the function is known
Integral or 'weak' statements equivalent to the differential equations as
non-square integrable. Such a function is said to be Co continuous.
In a similar way it is easy to see that if nth-order derivatives occur in any term of.A or/3 then the function has to be such that its n - 1 derivatives are continuous (Cn_ 1 continuity). On many occasions it is possible to perform an integration by parts on Eq. (3.13) and replace it by an alternative statement of the form (3.14)
c(v)T'D(u)d~ if" fF ~(~')T'~'(u)dl-' -" 0
In this the operators C to .~" usually contain lower order derivatives than those occurring in operators .,4 or/3. Now a lower order of continuity is required in the choice of the u function at a price of higher continuity for v and ~. The statement (3.14) is now more 'permissive' than the original problem posed by Eq. (3.1), (3.2), or (3.13) and is called a weak form of these equations. It is a somewhat surprising fact that often this weak form is more realistic physically than the original differential equation which implied an excessive 'smoothness' of the true solution. Integral statements of the form of (3.13) and (3.14) will form the basis of finite element approximations, and we shall discuss them later in fuller detail. Before doing so we shall apply the new formulation to an example.
Example 3.4: Weak form of the heat conduction equation- forced and natural boundary conditions. Consider now the integral form of Eq. (3.8). We can write the statement (3.13) as v
k~-~-
k
+Q
dxdy+
0 k~-,- +~/ d F = 0
<3.15)
q
noting that v and ~ are scalar functionst and presuming that one of the boundary conditions, i.e.,
~-~ =0 is automatically satisfied by the choice of the functions ~b on F,. This type of boundary condition is often called a 'forced' or an 'essential' one. Equation (3.15) can now be integrated by parts to obtain a weak form similar to Eq. (3.14). We shall make use here of general formulae for such integration (Green's formulae) which we derive in Appendix G and which on many occasions will be useful, i.e.,
k-z-- d x d y = -
s
k-=- dxdy+
v
\ Ox
nx
y)nyd
oy
(3.16)
We have thus in place of Eq. (3.15)
J2 ( Ovk O~ Ox + Oy Ovk Oy +vQ ) dxdy- J: vk ( ~dp_~x nx + O c~ ) dF oyny (3.17)
+
~ k-~d+O d r = O q
t Two functions are introduced such that simplificationsare possible at later stages of the development.
59
60
Generalizationof the finite element concepts Noting that the derivative along the normal is given as
On - Ox nx
+
-ff-fyny
(3.18)
and, further, making = v
on F
(3.19)
without loss of generality (as both functions are arbitrary), we can write Eq. (3.17) as
q
where the operator 17 is simply
On
dp
0
We note that (a) the variable ~p has disappeared from the integrals taken along the boundary 1-'q and that the boundary condition
B(~> =~a~ +0 =0
On on that boundary is automatically satisfied- such a condition is known as a natural boundary condition- and (b) if the choice of 0~is restricted so as to satisfy the forced boundary conditions O~- ~ = O, we can omit the last term of Eq. (3.20) by restricting the choice of v to functions which give v - 0 on F~. The form of Eq. (3.20) is the weak form of the heat conduction statement equivalent to Eq. (3.8). It admits discontinuous conductivity coefficients k and temperature 0~ which show discontinuous first derivatives, a real possibility not easily admitted in the differential form.
If the unknown function u is approximated by the expansion (3.3), i.e., U ,~ fl - - ~
Nafla --
Nfl
I
a=l
then it is clearly impossible to satisfy both the differential equation and the boundary conditions in the general case. The integral statements (3.13) or (3.14) allow an approximation to be made if, in place of any function v, we put a finite set of approximate functions
V,~~Wb~tlb b=l
and
V:~Wb~ll b=l
b
(3.21)
Approximation to integral formulations: the weighted residuaI-Galerkin method 61 in which ~Ub the relation
are
arbitrary parameters. Inserting the approximations into Eq. (3.13) we have
and since 8fib is arbitrary we have a set of equations which is sufficient to determine the parameters fi as ~w/,A(Nfi)df2+fr,~/3(Nfi)dl-'
=0;
b = 1,2 . . . . . n
(3.22)
Performing similar steps using Eq. (3.14) gives the set f
C T(wb)79(N fi) dr2 + f r eT(qeb)'~'(N fi) dF - 0; b = 1, 2, . . . , n
(3.23)
If we note that ,A(Nfi) represents the residual or error obtained by substitution of the approximation into the differential equation [and B(Nfi), the residual of the boundary conditions], then Eq. (3.22) is a weighted integral of such residuals. The approximation may thus be called the method of weighted residuals. In its classical sense it was first described by Crandall, 1 who points out the various forms used since the end of the nineteenth century. Later a very full expos6 of the method was given by Finlayson. e Clearly, almost any set of independent functions wb could be used for the purpose of weighting and, according to the choice of function, a different name can be attached to each process. Thus the various common choices are: 1. Point collocation. 3 Wb -- 6b in (3.22), where 6b is such that for x ~ Xb; y ~ Yb, Wb -- 0 but f~ Wbd~2 = 1 (unit matrix). This procedure is equivalent to simply making the residual zero at n points within the domain and integration is 'nominal' (incidentally although Wb defined here does not satisfy all the criteria of Sec. 3.2, it is nevertheless admissible in view of its properties). Finite difference methods are particular cases of this weighting. 2. Subdomain collocation. 4 Wb -- I in subdomain ~b and zero elsewhere. This essentially makes the integral of the error zero over the specified subdomains. When used with (3.23) this is one of the many finite volume methods. 5 3. The Galerkin method (Bubnov-Galerkin). 4' 6 Wb - - Nb. Here simply the original shape (or basis) functions are used as weighting. This method, as we shall see, frequently (but by no means always) leads to symmetric matrices and for this and other reasons will be adopted in this book almost exclusively.
The name of 'weighted residuals' is clearly much older than that of the 'finite element method'. The latter uses mainly locally based (element) functions in the expansion of Eq. (3.3) but the general procedures are identical. As the process always leads to equations which, being of integral form, can be obtained by summation of contributions from various subdomains, we choose to embrace all weighted residual approximations under the name of generalized finite element method. On occasion, simultaneous use of both local and 'global' trial functions will be found to be useful. In the literature the names of Petrov and Galerkin 6 are often associated with the use of weighting functions such that Wb ~ Nb. It is important to remark that the well-known finite difference method of approximation is a particular case of collocation with locally defined
62
Generalization of the finite element concepts
Fig. 3.3 Problem description and loading for 1-d heat conduction example. basis functions and is thus a case of a Petrov-Galerkin scheme. We shall return to such unorthodox definitions in more detail in Chapter 15. To illustrate the procedure of weighted residual approximation and its relation to the finite element process let us consider some specific examples.
Example 3.5: One-dimensional equation of heat conduction. The problem here will be a one-dimensional representation of the heat conduction equation [Eq. (3.8)] with unit conductivity. (This problem could equally well represent many other physical situations, e.g., deflection of a loaded string with unit tension.) Here we have (see Fig. 3.3) d2~p A(dp) =
dx 2 + Q(x) = O (O < x < L)
(3.24a)
with Q (x) given by
Q(x) =
0
O<x
- 2 Qo ( x / L - 1 / 2 )
L/2<x
(3.24b)
The boundary conditions assumed will be simply 4~ = 0 at x = 0 and x = L. The problem is solved by a Galerkin-weighted residual method in which the field ~b(x) is approximated by piecewise defined (locally based) functions. Here we use the equivalent of Eq. (3.14) which results from an integration by parts of Wb
-- ~X 2
~-~ Na qba + Q
dx=0
a
to obtain
L dx M 4""
i"
+
i
, /v2
I I
I I
1
",,
x x
2
Fig. 3.4 Linear locally based one-dimensional shape functions.
ax = 0
1
(3.25)
Approximation to integral formulations: the weighted residuaI-Galerkin method in which the boundary terms disappear if we require Wb = 0 at the two ends. For the Galerkin solution we use Wb = Nb; hence, the above equation can be written as K ~b + f = 0
(3.26)
where for each element of length h,
Kbe =
jfoh dNb dNa dx dx dx
f~ =
(3.27)
Ub Q(x) dx
with the usual rules of adding pertaining, i.e., gba=Zgb
e
fb=--~~f~,
and
e
e
The reader will observe that the matrix K is symmetric, i.e., Kba = gab. AS the shape functions need only be of Co continuity, a piecewise linear approximation is conveniently used, as shown in Fig. 3.4. Considering a typical element 1-2 shown, we can write (translating the cartesian origin of x to point 1)
N1 = 1 - x / h
and
N2=x/h
(3.28)
giving for a typical element
Ke = -hl[ -11 -1]1 h ( 2 Q Il ++2QQ221 6
fe
(3.29)
where Q1 and Q2 are load intensities at the Xl and x2 coordinates, respectively. Assembly of four equal size elements results, after inserting the boundary conditions ~1 = ~5 = 0, in the equation set
412_1
-
-1
L
2
0
-1
-1
r
2
~4
=
QoL
48
{0) 1
6
(3.30)
The solution is shown in Fig. 3.5 along with the exact solution to the problem. For comparison purposes we also show a finite difference solution in which simple collocation is used in a weighted residual equation together with the approximation for the second derivative given by a Taylor expansion d24~ I axe Xa
1 ~
-~
(~a--1- 2~a "31-~a-t-1)
(3.31)
which yields the approximation for each node point
1
h2 (--~a-1 + 2 ~ a - ~a+l) "~- Qo = 0
(3.32)
63
64
Generalization of the finite element concepts
Again after including the boundary conditions a set of three equations for the points 2, 3 and 4 is expressed as
16[ ,
L2
-1
0
2
--1
-1
2
~3
~4
=Q0
{0} 0
(3.33)
1/2
The reader will note that the coefficient matrix for the finite element and finite difference methods differ by only a constant multiplier (for the boundary conditions assumed in this one-dimensional problem); however, the fight sides differ significantly. We also plot the solution to (3.33) (and one for half the mesh spacing) in Fig. 3.5. Here we note that the nodal results for the finite element method are exact whereas those for the finite difference solution are all in error (although convergence can be observed for the finer subdivision). The nodal exactness is a property of the particular equation being solved and unfortunately does not carry over to general problems. 7 (See also Appendix H.) However, based on the above result and other experiences we can say that the finite element method always achieves (the same or) better results than classical finite difference methods. In addition, the finite element method permits an approximation of the solution at all points in the domain as indicated by the dashed lines in Fig. 3.5 for the one-dimensional problem. The problem is repeated using 4-quadratic order finite elements and results are shown in Fig. 3.6. It is evident that the use of quadratic order greatly increases the accuracy of the results obtained. Indeed, if cubic order elements were used results would be exact, since for linear varying Q the solution over the loaded portion will only contain polynomials up to cubic order. 0.03
0.025
0.02
~" 0.015 :5
cf
ool
0.005
0
0.2
0.4
0.6 x/L coordinate
n
0.8
1
Fig. 3.5 One-dimensional heat conduction. Solution by finite element method with linear elements and
h = LI4; finite difference method with h = LI4 and h = LI8.
Approximation to integral formulations: the weighted residuaI-Galerkin method 0.03
!
0.025 .i-., t"
m 0.02 E tl)
0 m
-~9 0.015
"0
cf
o.01
0.005
0
0.2
0.4 0.6 x/L coordinate
0.8
1
Fig. 3.6 One-dimensional heat conduction. Solution by finite element method with quadratic elements and
h = LI4.
Example 3.6: Steady-state heat conduction in two dimensions: Galerkin formulation with triangular elements. We have already introduced the problem in Sec. 3.1 and defined it by Eq. (3.8) with appropriate boundary conditions. The weak form has been obtained in Eq. (3.20). Approximating the weight by v = ~ Nb3(bb and solution by q~ = ~ Na(ba we have immediately that K~+f=0 where
Kbe =
s
~
k
ONa+ aNbk ONa)d a Ox
-~y
Oy
(3.34)
ff,=/~NbQdf2+frNbEtdF Once again the components of gba and fb can be evaluated for a typical element or subdomain and the system of equations built by standard methods. For instance, considering the set of nodes and elements shown shaded in Fig. 3.7(b), to compute the equation for node 1 it is only necessary to compute the K~,a for two element shapes as indicated in Fig. 3.7. For the Type 1 element (left element in Fig. 3.7(c)) the shape functions evaluated from (2.8) using (2.6) and (2.7) gives Ul=l-
Y" N2
h,
x
~, N 3 =
y-x h
65
66
Generalization of the finite element concepts thus, the derivatives are given by:
ON1 ON Ox =
ON2
-~
ON3
0
=
1
i
37 1_ ON
ands-
1
0~ 2
1
-i __
3
Similarly, for the Type 2 element the shape functions are expressed by
Fig. 3.7 Linear triangular elements for heat conduction example.
0 1
Approximation to integral formulations: the weighted residuaI-Galerkin method 67 and their derivatives by
ON
Ox
N1
1
N2
1
N3
0
07 ---ff and
0 1
=
1
0/~
--ft.
Evaluation of the matrix K[,a and f [ for Type 1 and Type 2 elements gives Ke~ e
lk
=2
[, 0 0
--1
1
--1
-1
~
2
~e3
and Ke~be=
2
k
-1 0
2 -1
-1 1
~. ~
'
respectively. Note that the stiffness matrix does not depend on the size h of the element. This is a property of all two-dimensional elements in which the B matrix depends only on first derivatives of Co shape functions. The force vector for a constant Q over each element is given by fe = _1 Qh 2 6
1 1
for both types of elements. Assembling the patch of elements shown in Fig. 3.7(b) gives the equation with non-zero coefficients for node 1 as
k[4
-1
-1
-1
-1]
~4
--
Qh 2
~6 The reader should note that the final stiffness for node 1 does not depend on nodes 3 and 7 of the patch, whereas there are non-zero stiffness coefficients in the individual elements. Thus, the final result is only true when the arrangement of the nodes is regular. If the location of any of the nodes lies on an irregular pattern then final stiffness coefficients will remain for these nodes also. Repeating the construction of the stiffness terms using a finite difference approximation [as given in Eq. (3.31)] directly in the differential equation (3.8) gives the approximation
k [4
h2
-1
-1
-1
-1]
~4 ~6
-- Q
and once again the assembled node is identical to the finite difference approximation to within a constant multiplier. If all the boundary conditions are forced (i.e., ~p = ~) no differences arise between a finite element and a finite difference solution for the regular mesh assumed. However, if any boundary conditions are of natural type or the mesh is
68
Generalization of the finite element concepts
irregular differences will arise. Indeed, no restrictions on shape of elements or assembly type are imposed by the finite element approach.
Example 3.7: Steady-state heat conduction-convection in two dimensions: Galerkin formulation. We have already introduced the problem in Sec. 3.1 and defined it by Eq. (3.9) with appropriate boundary conditions. The equation differs only in the convective terms from that of simple heat conduction for which the weak form has already been obtained in Eq. (3.20). We can write the weighted residual equation immediately from this, substituting v = Wb~CPband adding the convective terms. Thus we have (VWb)rkVd~d~2+
Wb
Ux +Uy
wbOd +
WbgldF=O q
(3.35)
with ~ = ~-~aga~)a being such that the prescribed values of ~ are given on the boundary 1-'e and that ~4~b = 0 on that boundary [ignoring that term in (3.35)]. Specializing to the Galerkin approximation, i.e., putting Wb = Nb, we have immediately a set of equations of the form KO + f = 0
(3.36t
with
gba =
('~TNb)Tk'~TNadff2+
NbUx---~x + NbUy -~v
dr2
j~fl ( ONbk ONa + ONbk ONa) dr2 + ffl ( NbUx Oga + Nbu ONa) dr2 Ox Ox Oy Oy ~ Y'--~y fb -- ~ NbOdS2+ Jrq Nb~ldr =
(3.37)
Once again the components Kba and fb can be evaluated for a typical element or subdomain and systems of equations built up by standard methods. At this point it is important to mention that to satisfy the boundary conditions some of the parameters Ca have to be prescribed and the number of approximation equations must be equal to the number of unknown parameters. It is nevertheless often convenient to form all equations for all parameters and prescribe the fixed values at the end using precisely the same techniques as we have described in Chapter 1 for the insertion of prescribed boundary conditions in standard discrete problems. A further point concerning the coefficients of the matrix K should be noted here. The first part, corresponding to the pure heat conduction equation, is symmetric (Kab -- Kba) but the second is not and thus a system of non-symmetric equations needs to be solved. There is a basic reason for such non-symmetries which will be discussed in Sec. 3.9. To make the problem concrete consider the domain f2 to be divided into regular square elements of side h (Fig. 3.8(b)). To preserve Co continuity with nodes placed at comers, shape functions given as the product of the linear expansions can be written. For instance, for node 1, as shown in Fig. 3.8(a), xy N1 -hh and for node 2,
Virtual work as the 'weak form' of equilibrium equations for analysis of solids or fluids N2
~
Y
N1
s s ~~ ~ l ~ ~
,"
s S
1
~
I
1
h (a) Shape functions for square element
(b) 'Connected' equations for node 1
Fig. 3.8 Linear square elements for heat conduction-convection example.
(h-x) y etc. h h' With these shape functions the reader is invited to evaluate typical element contributions and to assemble the equations for point 1 of the mesh numbered as shown in Fig. 3.8(b). If Q is assumed to be constant, the result will be N2 =
8k~ ( k uxh) (b2- ( ~ uxh uyh) ~3_ ( k uyh) ~b4 3 3 12 12 3 3 (k uxh uyh)~5_ (k ~_~) (k uxh uyh~ - 3"~ 12 12 ~-'1"~6- ~'-[---~---~-12 /1~7 (k ~_~)~ 8 - (~ uxh uyh) - Q h 2 -- ~ + 12 ~- - - ~
--~-~bl --
(3.38)
~9 - "
This equation is similar to those that would be obtained by using finite difference approximations to the same equations in a fairly standard manner. 8, 9 In the example discussed some difficulties arise when the convective terms are large. In such cases the Galerkin weighting is not acceptable and other forms have to be used. For problems dealing with fluid dynamics this is discussed in detail in reference 10.
In Chapter 2 we introduced a finite element by way of an application to the solid mechanics problem of linear elasticity. The integral statement necessary for formulation in terms of the finite element approximation was supplied via the principle of virtual work, which was assumed to be so basic as not to merit proof. Indeed, to many this is so, and the virtual work principle is considered by some as a statement of mechanics more fundamental than the traditional equilibrium conditions of Newton's laws of motion. Others will argue with this view and will point out that all work statements are derived from the classical laws pertaining to the equilibrium of the particle. We shall therefore show in this section that the virtual work statement is simply a 'weak form' of equilibrium equations. In a general three-dimensional continuum the equilibrium equations of an elementary volume can be written in terms of the components of the symmetric cartesian stress tensor
69
70
Generalizationof the finite element concepts asll, 12
Offx { al } Ae
012xy
012xz
O~xyO~yOTy
=-
+ bX
z
--~x "JI--~-y -l- ---~-z -JI-by
A3
Orxz
Oryz
= 0
(3.39)
aCrz
-g-x + -03--y+
+ bz
where b = [bx by bz] T stands for the body forces acting per unit volume (which may well include acceleration effects by the d'Alembert principle). In solid mechanics the six stress components will be some general functions of the six components of strain (s) which are computed from the displacement u=
[u
v
w] T
(3.40)
and in fluid mechanics of the velocity vector u, which has identically named components. Thus Eq. (3.39) can be considered as a general equation of the form Eq. (3.1), i.e., ,A(u) = 0. To obtain a weak form we shall proceed as before, introducing an arbitrary weighting function vector defined as v=_6u=[$u,
6v,
(3.41)
6w] w
We can now write the integral statement of Eq. (3.11) as
If2 gu T.A(u) d ~
= -
~[(
O0"x + ~-OTJxy ~ v + _-OTJxz ~_
~u
+ bx
+ 6v(A2) + 8w(A3) ] dr2
=0 (3.42) where the volume, ~2, is the problem domain. Integrating each term by parts and rearranging we can write this as
-~-x O'x -Jr- -~y -1" ~ - Jr [(~u tx -I- ($v ty -I- ~w where t =
Z'xy +
....
~ u b x - ($v b y - 3 w b z dr2
(3.43)
tz] dr" =
0
ty tz
nxrxy -k- nyCry -+- nz'Cy z nxrxz -k- ny'Cyz -+- nz(rz
{}{nxx+nyxy+nzz}
(3.44)
are tractions acting per unit area of external boundary surface F of the solid [in (3.43)
Green's formulae of Appendix G are again used]. In the first set of bracketed terms in (3.43) we can recognize immediately the small strain operators acting on 6u, which can be termed a virtual displacement (or virtual velocity).
Partial discretization
We can therefore introduce a virtual strain (or strain rate) defined as 08u 0x 08v Oy (3.45) 08u
Oz
03v
where the strain operator is defined as in Chapter 2 [Eqs (2.13)-(2.15)]. Arranging the six stress components in a vector cr in an order corresponding to that used for 8e, we can write Eq. (3.43) simply as f
s~To'df2 -- f
, u T b d ~ - f r ~uTtdF = 0
(3.46)
which is the three-dimensional equivalent virtual work statement used in Eqs (2.25) and (2.38) of Chapter 2. We see from the above that the virtual work statement is precisely the weak form of equilibrium equations and is valid for non-linear as well as linear stress-strain (or stressstrain rate) relations. The finite element approximation which we have derived in Chapter 2 is infact a Galerkin formulation of the weighted residual process applied to the equilibrium equation. Thus if we take 6u as the shape function times arbitrary parameters ~u = ~
NbSfJb
(3.47)
b
where the displacement field is discretized, i.e., u
Nafla
=
(3.48)
a
together with the strain-displacement relations
Z S Nalla = ~ natla a
(3.49)
a
and constitutive relation of Eq. (2.16), we shall determine once again all the basic expressions of Chapter 2 which are so essential to the solution of elasticity problems. We shall consider this class of problems further in Chapter 6. Similar expressions are vital to the formulation of equivalent fluid mechanics problems as discussed in reference 10.
In the approximation to the problem of solving the differential equation (3.1) by an expression of the standard form of Eq. (3.3), we have assumed that the shape functions N include
71
72
Generalizationof the finite element concepts
all independent coordinates of the problem and that ~ was simply a set of constants. The final approximation equations were thus always of an algebraic form, from which a unique set of parameters could be determined. In some problems it is convenient to proceed differently. Thus, for instance, if the independent variables are x, y and z we could allow the parameters ~ to be functions of z and do the approximate expansion only in the domain of x, y, say f2. Thus, in place of Eq. (3.3) we would have u = N(x, y)~(z) (3.50) Clearly the derivatives of fi with respect to z will remain in the final discretization and the result will be a set of ordinary differential equations with z as the independent variable. In linear problems such a set will have the appearance K~+
C~ + . . .
+ f - 0
(3.51)
where ~ - dfl/dz, etc. Such a partial discretization can obviously be used in different ways, but is particularly useful when the domain f2 is not dependent on z, i.e., when the problem is prismatic. In such a case the coefficient matrices of the ordinary differential equations, (3.51), are independent of z and the solution of the system can frequently be carried out efficiently by standard analytical methods. This type of partial discretization has been applied extensively by Kantorovich 13 and is frequently known by his name. Semi-analytical treatments are presented in reference 14 for prismatic solids where the final solution is obtained in terms of Fourier (or other) series. However, the most frequently encountered 'prismatic' problem is one involving the time variable, where the space domain ~ is not subject to change. We shall address such problems in Chapter 16 of this volume. It is convenient by way of illustration to consider here heat conduction in a two-dimensional equation in its transient state. This is obtained from Eq. (3.8) by addition of the heat storage term c(OO/Ot), where c is the specific heat per unit volume. We now have a problem posed in a domain f2 (x, y, t) in which the following equation holds"
a( ~ ) -- --~x k-~x
- -~y k-~y
+ Q + c o--7 0
with boundary conditions identical to those of Eq. (3.8) and the temperature taken as zero at time zero. Taking ~p ~ qb = ~ Na(x, y)(ha(t) (3.53) a
and using the Galerkin weighting procedure we follow precisely the steps outlined in Eqs (3.35)-(3.37) and arrive at a system of ordinary differential equations
db
K(b + C-dT + f = 0
(3.54)
Here the expression for Kba and fb are identical with that of Eq. (3.34) and the reader can verify that the matrix C is defined by
Cab -- f NaCNb dx dy J~
(3.55)
Partial discretization Once again the matrix C can be assembled from its element contributions. Various analytical and numerical procedures can be applied simply to the solution of such transient, ordinary, differential equations which, again, we shall discuss in detail in Chapters 16 and 17. However, to illustrate the detail and the possible advantage of the process of partial discretization, we shall consider a very simple problem.
Example 3.8: Heat equation with heat generation. Consider a long bar with a square cross-section of size L • L in which the transient heat conduction equation (3.52) applies and assume that the rate of heat generation varies with time as Q-
(3.56)
Qoe -~t
(this might approximate a problem of heat development due to hydration of concrete). We assume that at t - 0, 4> = 0 throughout. Further, we shall take 4> = 0 on all boundaries for all times. An approximation for the solution is taken: M
N
~--ZZNmn(X,Y)~)mn(t) m=l n=l
gmn = cos
mrc x
L
nrc y
cos ~ ; L
(3.57)
m, n = 1, 3, 5 , . - .
with x and y measured from the centre (Fig. 3.9). The even components of the Fourier series are omitted due to the required symmetry of solution. Evaluating the coefficients (only diagonal terms exist in K), we have Kmn --
Cmn -'-
finn
ILl2 ILl2 [k (Ogmn) 2 + k (Ogmn)21 dx d y
a-L~2 a-L~2
fL/2 ;L/2 a-L~2 a-L~2
Ox
c N ~ n dx d y =
Oy
-
zr2k ~. (m 2 -k- n 2)
L2c
(3.58)
4
~[L/2[L/2 Nmn Qoe -~t dx d y a-L/2 a-L/2
=
4Qo L2
mnTr2
(_l)(m+3)/2(_l)(n+3)/2e-Ott
This leads to an ordinary differential equation with parameters
Kmnf~mn+ Cmn~d(bmn -Jr-fmn -- 0 dt
~mn"
(3.59)
with ~mn - - 0 when t = 0. The exact solution of this is easy to obtain, as is shown in Fig. 3.9 for specific values of the parameters M, N, ot and k / L 2 c . The remarkable accuracy of the approximation with M -- N -- 3 in this example should be noted. In this example we have used trigonometric functions in place of the more standard polynomials used in the finite element method. In Chapter 7 we recalculate the solution using a standard finite element method in which the solution to the time problem is computed using a finite difference method.
73
74 Generalization of the finite element concepts
Fig. 3.9 Two-dimensional transient heat development in a square prism - plot of temperature at centre.
In the previous sections we have discussed how approximate solutions can be obtained by use of an expansion of the unknown function in terms of trial or shape functions. Further, we have stated the necessary conditions that such functions have to fulfil in order that the various integrals can be evaluated over the domain. Thus if various integrals contain only the values of N and its first derivatives then N has to be Co continuous. If second derivatives are involved, C1 continuity is needed, etc. The problem which we have not yet addressed ourselves consists of the questions of just how good the approximation is and how it can be systematically improved to approach the exact answer. The first question is more difficult to answer and presumes knowledge of the exact solution (see Chapter 13). The second is more rational and can be answered if we consider some systematic way in which the number of parameters ~ in the standard expansion of Eq. (3.3),
tl = ~-~ Nafla a=l
is presumed to increase. In some examples we have assumed, in effect, a trigonometric Fourier-type series limited to a finite number of terms with a single form of trial function assumed over the whole domain. Here addition of new terms would be simply an extension of the number of terms in the series included in the analysis, and as the Fourier series is known to be able to represent any function within any accuracy desired as the number of terms increases, we can talk about convergence of the approximation to the true solution as the number of terms increases. In other examples of this chapter we have used locally based polynomial functions which are fundamental in the finite element analysis. Here we have tacitly assumed that
Convergence 75 convergence occurs as the size of elements decreases and, hence, the number of fi parameters specified at nodes increases. It is with such convergence that we need to be concerned and we have already discussed this in the context of the analysis of elastic solids in Chapter 2 (Sec. 2.6). We have now to determine (a) that, as the number of elements increases, the unknown functions can be approximated as closely as required, and (b) how the error decreases with the size, h, of the element subdivisions (h is here some typical dimension of an element). The first problem is that of completeness of the expansion and we shall here assume that all trial functions are polynomials (or at least include certain terms of a polynomial expansion). Clearly, as the approximation discussed here is to the weak, integral form typified by Eq. (3.11) or (3.14) it is necessary that every term occurring under the integral be in the limit capable of being approximated as nearly as possible and, in particular, giving a constant value over any arbitrary infinitesimal part of the domain ~2. If a derivative of order m exists in any such term, then it is obviously necessary for the local polynomial to be at least of the order m so that, in the limit, such a constant value can be obtained. We will thus state that a necessary condition for the expansion to be covergent is the criterion of completeness: that, if mth derivatives occur in the integral form, a constant value of all derivatives up to order m be attainable in the element domain when the size of any element tends to zero. This criterion is automatically ensured if the polynomials used in the shape function N are complete to mth order. This criterion is also equivalent to the one of constant strain postulated in Chapter 2 (Sec. 2.5). This, however, has to be satisfied only in the limit h --+ 0. If the actual order of a complete polynomial used in the finite element expansion is p > m, then the order of convergence can be ascertained by seeing how closely such a polynomial can follow the local Taylor expansion of the unknown u. Clearly the order of error will be simply O (h p+I) since only terms of order p can be rendered correctly. Knowledge of the order of convergence helps in ascertaining how good the approximation is if studies on several decreasing mesh sizes are conducted. Though, in Chapter 14, we shall see the asymptotic convergence rate is seldom reached if singularities occur in the problem. Once again we have re-established some of the conditions discussed in Chapter 2. We shall not discuss, at this stage, approximations which do not satisfy the postulated continuity requirements except to remark that once again, in many cases, convergence and indeed improved results can be obtained (see Chapter 9). In the above we have referred to the convergence of a given element type as its size is reduced. This is sometimes referred to as h convergence. On the other hand, it is possible to consider a subdivision into elements of a given size and to obtain convergence to the exact solution by increasing the polynomial order p of each element. This is referred to as p convergence, which is obviously assured. In general p convergence is more rapid per degree of freedom introduced. We shall discuss both types further in Chapter 14; although we have already noted in some examples how improved accuracy occurs with higher term polynomials being added at each element level.
76
Generalizationof the finite element concepts
Variational principles What are variational principles and how can they be useful in the approximation to continuum problems? It is to these questions that the following sections are addressed. First a definition: a 'variational principle' specifies a scalar quantity (functional) 17, which is defined by an integral form 17 =
F
0u ...
df~+
u, Ox'
E u,~,.., Ox
dF
(3.60)
in which u is the unknown function and F and E are specified differential operators. The solution to the continuum problem is a function u which makes 17 stationary with respect to arbitrary changes 6u. Thus, for a solution to the continuum problem, the 'variation' is 817 = 0
(3.61)
for any 8u, which defines the condition of stationarity. 15 If a 'variational principle' can be found, then means are immediately established for obtaining approximate solutions in the standard, integral form suitable for finite element analysis. Assuming a trial function expansion in the usual form [Eq. (3.3)] U ~ fl --" ~
Nail a
a=l we can insert this into Eq. (3.60) and write OH OH 017 ~l"I "-- O------~Ul+ ~u2~B2 .ql_....11_ ~Un ~ n ~. 0
(3.62)
This being true for any variations ~fi yields a set of equations 017 an _ 0~
.
= 0
(3.63)
317
from which parameters Ha are found. The equations are of an integral form necessary for the finite element approximation as the original specification of I7 was given in terms of domain and boundary integrals. The process of finding stationarity with respect to trial function parameters fi is an old one and is associated with the names of Rayleigh 16 and Ritz. 17 It has become extremely important in finite element analysis which, to many investigators, is typified as a 'variational process'.
What are 'variational principles'?
If the functional I7 is 'quadratic', i.e., if the function u and its derivatives occur in powers not exceeding 2, then Eq. (3.63) reduces to a standard linear form similar to Eq. (3.7), i.e., 017 0~
-- K~ + f = 0
(3.64)
It is easy to show that the matrix K will now always be symmetric. To do this let us consider a linearization of the vector 01-I/0~. This we can write as
zx
(0I-I) ~
=
O--~Ul ~Ul A~I
-- KT A~
(3.65)
in which KT is generally known as the tangent matrix, of significance in non-linear analysis, and A~ are small incremental changes to ~. Now it is easy to see that
02l"I
(3.66)
KTab -- O~laO~lb "- KTba
Hence KT is symmetric. For a quadratic functional we have, from Eq. (3.64), ....
A
-0--flu -- K A ~
with
K = KT
(3.67)
and hence symmetry must exist. The fact that symmetric matrices will arise whenever a variational principle exists is one of the most important merits of variational approaches for discretization. However, symmetric forms will frequently arise directly from the Galerkin process. In such cases we simply conclude that the variational principle exists but we shall not need to use it directly. Further, the discovery of symmetry from a weighted residual process leads directly to known (or previously unknown) variational principles. 18 How then do 'variational principles' arise and is it always possible to construct these for continuous problems? To answer the first part of the question we note that frequently the physical aspects of the problem can be stated directly in a variational principle form. Theorems such as minimization of total potential energy to achieve equilibrium in mechanical systems, least energy dissipation principles in viscous flow, etc., may be known to the reader and are considered by many as the basis of the formulation. We have already referred to the first of these in Sec. 2.4 of Chapter 2. Variational principles of this kind are 'natural' ones but unfortunately they do not exist for all continuum problems for which well-defined differential equations may be formulated. However, there is another category of variational principles which we may call 'contrived'. Such contrived principles can always be constructed for any differentially specified problem, either by extending the number of unknown functions u by additional variables known as Lagrange multipliers, or by procedures imposing a higher degree of continuity requirements such as in least squares problems. In subsequent sections we shall discuss, respectively, such 'natural' and 'contrived' variational principles. Before proceeding further it is worth noting that, in addition to symmetry occurring in equations derived by variational means, sometimes further motivation arises. When
77
78 Generalizationof the finite element concepts
'natural' variational principles exist the quantity FI may be of specific interest itself. If this arises a variational approach possesses the merit of easy evaluation of this functional. The reader will observe that if the functional is 'quadratic' and yields Eq. (3.64), then we can write the approximate 'functional' 1-I simply as n = I~TKfi + ~Tf
(3.68)
By simple differentiation
~l"I = ~1 (fiT)K fi + 21fiTK 8fi + 8fiTf __ 0 As K is symmetric, 3fiTKfi ~ fiTK3fi Hence FI = ~fiT (Kfi + f) = 0 which is true for all 6fi and hence Kfi+f=O when inserted into (3.68) we obtain 1 1 FI = - fiTf = _ _ fiTK 2 2
If we consider the definitions of Eqs (3.60) and (3.61) we observe that for stationarity we can write, after performing some differentiations and integrations by parts, ~n = f
~uT,A(u)d~ + f r 8uTB(u)dr - 0
(3.69)
As the above has to be true for any variations ~u, we must have ,At(u) = 0
in f2
and
B(u) = 0
on F
(3.70)
If ,At corresponds precisely to the differential equations governing the problem of interest and/3 to its boundary conditions, then the variational principle is a natural one. Equations (3.70) are known as the Euler differential equations corresponding to the variational principle requiting the stationarity of FI. It is easy to show that for any variational principle a corresponding set of Euler equations can be established. The reverse is unfortunately not true, i.e., only certain forms of differential equations are Euler equations of a variational functional. In the next section we shall consider the conditions necessary for the existence of variational principles and give a prescription for the establishing FI from a set of suitable linear differential equations. In this section we shall continue to assume that the form of the variational principle is known.
'Natural' variational principles and their relation to governing differential equations To illustrate the process let us now consider a specific example. Suppose we specify a problem by requiring the stationarity of a functional + gk
+ Q~b dr2 +
q~b dF
(3.71)
q
in which k and Q depend only on position and we assume r = ~b is satisfied on F~. We now perform the variation. 15 This can be written following the rules of differentiation
as
8I-1 =
k -3xx8 -~x + k ?y 8 -~y + a s ~
As
ar
I
dr2 +
(~/8~b)dF = 0
(3.72)
q
a
8 -~x =ff--xx(8r
(3.73)
we can integrate by parts (as in Sec. 3.3) and, since 8q~ = 0 on F4, obtain
3 ( k ~ y ) + QI dr2 +
i (
ox
&b k-~n + q
q
)
-
(3.74a)
dr=0
This is of the form of Eq. (3.69) and we immediately observe that the Euler equations are A(~b)=-~xxk
Oy
-~
k
+a=0
B(r = k a4) On + q = 0
inf2
(3.74b)
on Fq
If q~is prescribed so that q~ = ~) on F4 and 8q~ = 0 on that boundary, then the problem is precisely the one we have already discussed in Sec. 3.2 and the functional (3.71) specifies the two-dimensional heat conduction problem in an alternative way. In this case we have 'guessed' the functional but the reader will observe that the variation operation could have been carried out for any functional specified and corresponding Euler equations could have been established. Let us continue the process to obtain an approximate solution of the linear heat conduction problem. Taking, as usual,
qb~, ~ -~-~ ga~)a = N6
a
(3.75)
we substitute this approximation into the expression for the functional FI [Eq. (3.71)] and obtain
n-f
~2 (Z--~x :d f 2 + f ~ 2 !~ ( ~ - - ~ aNa~ aaNa)a - y C a ) d: r 2 (3.76)
"~ ~ O ~Na~)a d~-~"~-fF q ~a Na~)ad~ a q
79
80
Generalization of the finite element concepts
On differentiation with respect to a typical parameter ~b we have
01-1 s ( ONa )ONbd~_+_s O~)b-- k ~ ---~x ~)a OX
ONa~)ONb d~ ~a --~-ydt)a Or
(3.77)
q and a system of equations for the solution of the problem is ~
Kr
(3.78)
=0
with
Kab=Kba - s
ONaONb dr a + s Ox Ox fb -- s NbQ dS2 + J; Nb~/dF
8NaSNb dr2 Oy Oy
(3.79)
q
The reader will observe that the approximation equations are here identical with those obtained in Sec. 3.5 for the same problem using the Galerkin process. No special advantage accrues to the variational formulation here, and indeed we can predict now that Galerkin
and variationalproceduresmustgive the sameanswerfor caseswherenaturalvariational principles exist. 3.8.2 Relation of the Galerkin method to approximation via
variational principles
In the preceding example we have observed that the approximation obtained by the use of a natural variational principle and by the use of the Galerkin weighting process proved identical. That this is the case follows directly from Eq. (3.69), in which the variation was derived in terms of the original differential equations and the associated boundary conditions. If we consider the usual trial function expansion [Eq. (3.3)] u,~
fi = N ~
we can write the variation of this approximation as ~fi - N ~fi
(3.80)
and inserting the above into (3.69) yields
-
[o N
+
N
dr = 0
3.81
The above form, being true for all 6fi, requires that the expression under the integrals should be zero. The reader will immediately recognize this as simply the Galerkin form of the weighted residual statement discussed earlier [Eq. (3.22)], and identity is hereby proved. We need to underline, however, that this is only true if the Euler equations of the variational principle coincide with the governing equations of the original problem. The Galerkin process thus retains its greater range of applicability.
Establishment of natural variational principles for linear, self-adjoint, differential equations
General rules for deriving natural variational principles from non-linear differential equations are complicated and even the tests necessary to establish the existence of such variational principles are not simple. Much mathematical work has been done in this context by Vainberg, 19 Tonti, 18 Oden, 2~ 21 and others. For linear differential equations the situation is much simpler and a thorough study is available in the works of Mikhlin, 22' 23 and in this section a brief presentation of such rules is given. We shall consider here only the establishment of variational principles for a linear system of equations with forced boundary conditions, implying only variation of functions which yield 6u = 0 on their boundaries. The extension to include natural boundary conditions is simple and will be omitted. Writing a linear system of differential equations as .A(u) - L u + b = 0
(3.82)
in which E is a linear differential operator it can be shown that natural variational principles require that the operator E be such that L ~T (Z~'7) dr2 = f
7T(z2~.,) dr2 + b.t.
(3.83)
for any two function sets @ and "7. In the above, 'b.t.' stands for boundary terms which we disregard in the present context. The property required in the above operator is called that of self-adjointness or symmetry. If the operator Z2 is self-adjoint, the variational principle can be written immediately as 1-I = ~
[ 89T (Cu) + uTb] dr2 + b.t.
(3.84)
To prove the veracity of the last statement a variation needs to be considered. We thus write (omitting boundary terms)
~I'/ = f [ 8 9~uT~u -{- 1UT~ (~U) "-]-3uTb] dff2 = 0
(3.85)
Noting that for any linear operator 8(Z;u) = Z; ~u
(3.86)
and that u and 3u can be treated as any two independent functions, by identity (3.83) we can write Eq. (3.85) as ~l-I = J~ ~uT[/~U + b] dr2 - 0
(3.87)
We observe immediately that the term in the brackets, i.e., the Euler equation of the functional, is identical with the original equation postulated, and therefore the variational principle is verified.
81
82
Generalizationof the finite element concepts
The above gives a very simple test and a prescription for the establishment of natural variational principles for differential equations of the problem. Example 3.9: Helmholz problem in two dimensions. A Helmholz problem is governed by a differential equation similar to the heat conduction equation, e.g., V2~b -4- C ~ "4- Q = 0
(3.88)
with c and Q being dependent on position only. The above can be written in the general form of Eq. (3.82), with
s
Tx2 + ~ y 2 + c ;
b=Q
and
u-4~
(3.89)
Verifying that self-adjointness applies (which we leave to the reader as an exercise), we immediately have a variational principle I-I--f~ [ ~ q ~ ( ~-Yx 02q~ 02~b + c ~ )/ + q~Q ] d x d y 2 + ff~y2
(3.90)
with 4~ satisfying the forced boundary condition, i.e., 4~ - 4~ on FO. Integrating by parts of the first two terms results in b ) 2 + -~1 (O~b)2 1 c ~p2 - cp Q 1 dx d y 1-I = - f ~ [ ~ ( O ~-~x ~y - -~
(3.91)
on noting that boundary terms with prescribed ~p do not alter the principle. Example 3.10: First-order form of heat equation. This problem concerns the onedimensional heat conduction equation (Example 3.5, Sec. 3.3) written in first order form as
d~ - q - Ux
,A(u) =
dq
~+a
or, using Eq. (3.82), as s
l,
-dxx
d
.
b=
dxx' 0 1'
{0} Q
=0
and u =
{:}
Again self-adjointness of the operator can be tested and found to be satisfied. We now write the functional as
I{}T
d
Ux' dq~
{:} §
0
dx (3.92)
QI dx
Maximum, minimum, or a saddle point?
n j~
--
d21-[ t > 0
,
;o d2nl
_\
"',
~',-a
r
Fig. 3.10 Maximum, minimum and a 'saddle' point for a functional l I of one variable
The verification of the correctness of the above, by executing a variation, is left to the reader. These two examples illustrate the simplicity of application of the general expressions. The reader will observe that self-adjointness of the operator will generally exist if even orders of differentiation are present. For odd orders self-adjointness is only possible if the operator is a 'skew'-symmetric matrix such as occurs in the second example.
In discussing variational principles so far we have assumed simply that at the solution point g I-I = 0, that is the functional is stationary. It is often desirable to know whether 1-I is at a maximum, minimum, or simply at a 'saddle point'. If a maximum or a minimum is involved, then the approximation to FI will always be 'bounded', i.e., will provide approximate values of FI which are either smaller or larger than the correct ones.t The bound in itself may be of practical significance in some problems. When, in elementary calculus, we consider a stationary point of a function 1-I of one variable u, we investigate the rate of change of dFI with du and write
Ol-I d u )
d(dFl) = dk,-~u
02I'I
= -~u2 (du)2
(3.93)
The sign of the second derivative determines whether I7 is a minimum, maximum, or simply stationary (saddle point), as shown in Fig. 3.10. By analogy in the calculus of variations we shall consider changes of g I7. Noting the general form of this quantity given by Eq. (3.62) and the notion of the second derivative of Eq. (3.65) we can write, in terms of discrete parameters, g(gl"[) ~ g
~
gO -" goTg
~
-- gO T
oqUloq~lgUl
-- gOTKT glUl
(3.94)
If, in the above, g (g FI) is always negative then FI is obviously reaching a maximum, if it is always positive then 1-I is a minimum, but if the sign is indeterminate this shows only the existence of a saddle point. t Provided all integrals are exactly evaluated.
83
84
Generalizationof the finite element concepts
As ~ is an arbitrary vector this statement is equivalent to requiring the matrix KT to be negative definite for a maximum or positive definite for a minimum. The form of the matrix Kx (or in linear problems of K which is identical to it) is thus of great importance in the solution of variational problems.
Consider the problem of making a functional I-I stationary, subject to the unknown u obeying some set of additional differential relationships C(u) = 0
in f2
(3.95)
We can introduce this constraint by forming another functional l=l(u, ~k) = I-l(u) + f~ ~xTC(u)dg2
(3.96)
in which ,k is some set of functions of the independent coordinates in the domain f2 known as Lagrange multipliers. The variation of the new functional is now 81=I = 8Fl + f
)~TSC(u) d r q- f
8)~TC(u) d r = 0
(3.97)
which immediately gives C(u) = 0 and, simultaneously, an added contribution to the original 3 H involving ,,k. In a similar way, constraints can be introduced at some points or over boundaries of the domain. For instance, if we require that u obey E(u) = 0
on r'
(3.98)
we would add to the original functional the term r ,XTE(u) dF
(3.99)
with ,k now being an unknown function defined only on F. Alternatively, if the constraint C is applicable only at one or more points of the system, then the simple addition of ,xTC(u) at these points to the general functional 1-I will introduce a discrete number of constraints. It appears, therefore, possible to always introduce additional functions ,~ and modify a functional to include any prescribed constraints. In the 'discretization' process we shall now have to use trial functions to describe both u and ~k. Writing, for instance, Nafla -- Nf~
fl-- ~ a
~, = ~
Nb~b = N ~ b
(3.100)
Constrained variational principles. Lagrange multipliers we shall obtain a set of equations 01-I _
=0
0w
where
w = (~A}
(3.101)
from which both the sets of parameters fi and ,,~can be obtained. It is somewhat paradoxical that the 'constrained' problem has resulted in a larger number of unknown parameters than the original one and, indeed, has complicated the solution. We shall, nevertheless, find practical use for Lagrange multipliers in formulating some physical variational principles, and will make use of these in a more general context in Chapters 10 and 11. Before proceeding further it is of interest to investigate the form of equations resulting from the modified functional FI of Eq. (3.96). If the original functional I-I gave as its Euler equations a system ,A(u) - 0 (3.102) then we have (omitting the boundary terms) ~1=I = f~'uT.A(u) dr2 + f ~cTA dr2 + f 'ATC(u) dr2 = O
(3.103)
Substituting the trial functions (3.100) we can write for a linear set of constraints
C(u) -~ A~lU-if-C1 that
8l=I = BuT [J~ NT.,4.(fi)dff2+ J~ (ff-,1N)T~dff2] (3.104) + ~xT f a l~V(Z~lu + C1) dr2 = 0 As this has to be true for all variations 3fi and 8,~, we have a system of equations NTA(fi) dr2 + ~(~IN)T,Xdf2 = 0
f I~T(~Ifi -t" C1) dr2 =
(3.105) 0
For linear equations .,4, the first term of the first equation is precisely the ordinary, unconstrained, variational approximation Kuufi + fu
(3.106)
and inserting again the trial functions (3.100) we can write the approximated Eq. (3.105) as a linear system: =
[Ku\
'
+
fz
with KuT = ~ I~T (LIN)dr2;
fz = f I~Tc1 dr2
(3.108)
85
86 Generalizationof the finite element concepts Clearly the system of equations is symmetric but now possesses zeros on the diagonal, and therefore the variational principle H is merely stationary. Further, computational difficulties may be encountered unless the solution process allows for zero diagonal terms.
Example 3.11: Constraint enforcement using L a g r a n g e multiplier. The point about increasing the number of parameters to introduce a constraint may perhaps be best illustrated in a simple algebraic situation in which we require a stationary value of a quadratic function of two variables u 1 and u2: H = 2u 2 - 2u 1u2 "q- U2 + 18u 1 "q- 6u2
(3.109)
Ul -- /'12 = 0
(3.110)
subject to a constraint
The obvious way to proceed would be to insert directly the equality 'constraint' and obtain FI = u 2 + 24Ul
(3.111)
and write, for stationarity, OH
OUl
= 0 = 2 U l d- 2 4
Ul - - u2 = - - 1 2
(3.112)
Introducing a Lagrange multiplier )~ we can alternatively find the stationarity of (I = 2u 2 - 2 U l U 2 -~- u 2 -at- l g U l --[- 6 u 2 --[- ~,(Ul - u 2 )
(3.113)
and write three simultaneous equations
al=i = 4Ul - 2u2 + ~, -k- 18 = 0 OUl ~f-I = - - 2 U l 4- 2 u 2 -- ~, q- 6 = 0 OU2
(3.114)
=Ul--U2=0
The solution of the above system again yields the correct answer Ul - u2 - - 1 2
)~ = 6
but at considerably more effort. Unfortunately, in most continuum problems direct elimination of constraints cannot be so simply accomplished.t t In the finite element context, Szabo and Kassos 24 use such direct elimination; however, this involves considerable algebraic manipulation.
Constrained variational principles. Lagrange multipliers 87
3.11.2 Identification of Lagrange multipliers. Forced boundary conditions and modified variational principles Although the Lagrange multipliers were introduced as a mathematical concept necessary for the enforcement of certain external constraints required to satisfy the original variational principle, we shall find that in many situations they can be identified with certain physical quantities of importance to the original mathematical model. Such an identification will follow immediately from the definition of the variational principle established in Eq. (3.96) and through the first of the Euler equations in (3.105) corresponding to it. The variation l=I, written in Eq. (3.97), supplies through its third term the constraint equation. The first two terms can always be rewritten as ~ ~C(u)T)~ dr2 + f~ 6uT.gt(U)d~ = 0
(3.115a)
j~r 8E(u)Tzx dF + j~r ~uTB(u)dF = 0
(3.115b)
or
This supplies the identification of )~. In the literature of variational calculation such identification arises frequently and the reader is referred to the excellent text by Washizu e5 for numerous examples.
Example 3.12: Identification of Lagrange multiplier for boundary condition. Here we shall introduce this identification by means of the example considered in Sec. 3.8.1. As we have noted, the variational principle of Eq. (3.71) established the governing equation and the natural boundary conditions of the heat conduction problem providing the forced boundary condition E(r = r - r = 0 (3.116) was satisfied on Fr in the choice of the trial function for r The above forced boundary condition can, however, be considered as a constraint on the original problem. We can write the constrained variational principle as lYl = I7 + f
dF
)~(~b- ~) dF
(3.117)
where H is given by Eq. (3.71). Performing the variation we have
,~fI=~rI + f ~r dr" r
+f
.JFr
8X(r
~)dr =0
(3.118)
H is now given by the expression (3.74a) augmented by an integral
fr~ ar 0_r dr On which was previously disregarded (as we had assumed that 6tp = 0 on Fr the conditions of Eq. (3.74b), we now require that fr '1'(~b-~)dF+fr
'q~( k + k 0 q ~ d l - ' - 0
(3.119) In addition to
(3.120)
88
Generalizationof the finite element concepts
which must be true for all variations 3X and 3q~. The first simply reiterates the constraint (p - (p = 0
on
ro
(3.121)
The second defines X as X = -k~ On Noting that k(Ocp/On) is the negative to the flux qn on the boundary identification of the multiplier has been achieved- that is, ~. _-- qn.
(3.122) the physical
The identification of the Lagrange variable leads to the possible establishment of a modified variational principle in which ~, is replaced by the identification. We could thus write a new principle for the above example: l=I -
-n (~ fr k ~04
n -
~) dr
(3.123)
in which once again YI is given by the expression (3.71) but 4~ is not constrained to satisfy any boundary conditions. Use of such modified variational principles can be made to restore interelement continuity and appears to have been first introduced for that purpose by Kikuchi and Ando. 26 In general these present interesting new procedures for establishing useful variational principles. A further extension of such principles has been made use of by Chen and Mei 27 and Zienkiewicz et al. 28 Washizu 25 discusses many such applications in the context of structural mechanics. The reader can verify that the variational principle expressed in Eq. (3.123) leads to automatic satisfaction of all the necessary boundary conditions in the example considered. The use of modified variational principles restores the problem to the original number of unknown functions or parameters and is often computationally advantageous.
In the previous section we have seen how the process of introducing Lagrange multipliers allows constrained variational principles to be obtained at the expense of increasing the total number of unknowns. Further, we have shown that even in linear problems the algebraic equations which have to be solved are now complicated by having zero diagonal terms. In this section we shall consider alternative procedures of introducing constraints which do not possess these drawbacks.
3.12.1 Penalty functions Considering once again the problem of obtaining stationarity of FI with a set of constraint equations C(u) = 0 in domain ~, we note that the product =
+
+...
(3.124)
Constrained variational principles. Penalty function and perturbed lagrangian methods 89 where C x = [C1, C2, ...] must always be a quantity which is positive or zero. Clearly, the latter value is found when the constraints are satisfied and also clearly the variation ~ (CTC) = 0
(3.125)
as the product reaches that minimum. We can now write a new functional
fl
c T ( u ) c ( u ) d~
= n + ~ c~
(3.126)
in which ot is a 'penalty number' and then require the stationarity for the constrained solution. If FI is itself a minimum of the solution then c~ should be a positive number. The solution obtained by the stationarity of the functional 1-I will satisfy the constraints only approximately. The larger the value of c~ the better will be the constraints achieved. Further, it seems obvious that the process is best suited to cases where I-I is a minimum (or maximum) principle, but success can be obtained even with purely saddle point problems. The process is equally applicable to constraints applied on boundaries or simple discrete constraints. In this latter case integration is dropped.
3.12.2
Perturbed lagrangian ~
We consider once again the problem of obtaining stationarity of FI with a set of constraint equations C(u) = 0 in domain f2. The Lagrange multiplier form to embed the constraint is given in Eq. (3.96). Here we modify the expression by appending a quadratic term of the form ,~T~ scaled by a parameter c~. The form of the final equation is given by
~(u, ~,) = rI(u) + f~ ;~TC(u)dS2
1 ~ ,kT,,kdf2 2-~
(3.127)
We note that as the parameter c~ tends toward infinity the form approaches a Lagrange multiplier form. Accordingly, this form is called aperturbed lagrangianfunctional. Taking the variation we obtain the result
If the constraints are a linear form given by C(u) =
C0u
we can introduce the approximations (3.100) into (3.128) to obtain the set of equations
Kuu Kzu
where
1Kuz] fi -~KxxJ /,~}:
{f0}
Kuu is the coefficient array from ~l-I and Ku~ = f I~ITc0 dr2
and
Kzz = J~ I~TI~ dr2
(3.129)
90 Generalizationof the finite element concepts The second equation of (3.129) may be solved for )~ in terms of ~ and substituted into the first equation to obtain
Kuufl - [Kuu + ot KuzKx-lKxu]U = f It is now apparent that the perturbed lagrangian and penalty forms are closely related. The perturbed lagrangian uses Kux K~-~Kzu to impose the constraint whereas the penalty approach uses f
c ~ c 0 dr2
When the constraint is a simple scalar relation the two methods are identical; however, when any other form is considered the methods will yield different approximations unless the shape functions for )~ include all the terms contained in ~C(u). Example 3.13: Constraint enforcement by penalty method. To clarify ideas let us once again consider the algebraic problem of Sec. 3.11.1, in which the stationarity of a functional given by Eq. (3.109) was sought subject to a constraint. With the penalty function approach we now seek the minimum of a functional
1~I - 2u~ - 2Ul//2
+ U2 "[- 1 8 U l -~- 6u2 q- ~1 ot (Ul - - U 2 ) 2
with respect to the variation of both parameters equations
OUl we find
=
I
0,
Ul
OU2
and
U2.
(3.130)
Writing the two simultaneous
=0
] u2
3131
and note as ot is increased we approach the correct solution. In Table 3.1 the results are set out demonstrating the convergence. The reader will observe that in a problem formulated in the above manner the constraint introduces no additional unknown parameters - but neither does it decrease their original number. The process will always result in strongly positive definite matrices if the original variational principle is one of a minimum and, similarly, negative definite matrices are obtained for a maximum principle if c~ is negative. In practical applications the method of penalty functions has proved to be quite effective, 29 and indeed is often introduced intuitively. T a b l e 3.1 C o n v e r g e n c e of t w o - t e r m solution ot
1/2
1
3
5
50
u1
- 12.000
- 12.000
u2
-13.500
-13.000
500
- 12.000
- 12.000
- 12.000
- 12.000
-12.429
-12.273
-12.030
-12.003
Constrained variational principles. Penalty function and perturbed lagrangian methods 91 In the example presented next the forced boundary conditions are not introduced a priori and the problem gives, on assembly, a singular system of equations Kfi + f = 0
(3.132)
which can be obtained from the functional (providing K is symmetric) I-[ = ~1 0 T K 0 + 0Tf
Introducing a prescribed value of
U l,
(3.133)
i.e., writing Ul -- Ul -- 0
(3.134)
1 (U 1 - Ul) 2 fl _. l-I-+- ~or
(3.135)
the functional can be modified to
yielding /~11 = Kll + or
f l = fl --OtUl
(3.136)
and giving no change in any of the other matrix coefficients. Many applications of such a 'discrete' kind are discussed by Campbell. 3~ It is easy to show in another context 29'31 that the use of a high Poisson's ratio (v -+ 0.5) for the study of incompressible solids or fluids is in fact equivalent to the introduction of a penalty term to suppress any compressibility allowed by an arbitrary displacement variation. The use of the penalty function in the finite element context presents certain difficulties. First, the constrained functional of Eq. (3.126) leads to equations of the form (K1 + otK2)u + ~ = 0
(3.137)
where K1 derives from the original functional and K2 from the constraints. As o~ increases the above equation degenerates to: K2u--f/ct
--+ 0
and fi = 0 unless the matrix K2 is singular. The phenomenon where ~ ::~ 0 is known as locking and has often been encountered by researchers who failed to recognize its source. This singularity in the equations does not always arise and we shall discuss means of its introduction in Chapters 10 and 11. Second, with large but finite values ofct numerical difficulties will be encountered. Noting that discrefization errors can be of comparable magnitude to those due to not satisfying the constraint, we can make ot = constant(l/h) n ensuring a limiting convergence to the correct answer. Fried 32, 33 discusses this problem in detail. A more general discussion of the whole topic is given in reference 34 and in Chapter 11 where the relationship between Lagrange constraints and penalty forms is made clear.
92
Generalizationof the finite element concepts
A general variational principle also may be constructed if the constraints described in the previous section are simply the governing equations of the problem C(u) - A(u)
(3.138)
Obviously the same procedure can be used in the context of the penalty function approach by setting H = 0 in Eq. (3.126). We can thus write a 'variational principle' = 1 L (A 2 + A 2 + . . . ) d ~ -
1 j ~ .AT (u),A(u) d~2
(3.139)
for any set of differential equations. In the above equation the boundary conditions are assumed to be satisfied by u (forced boundary condition) and the parameter ct is dropped as it becomes a multiplier. Clearly, the above statement is a requirement that the sum of the squares of the residuals of the differential equations should be a minimum at the correct solution. This minimum is obviously zero at that point, and the process is simply the well-known least squares method of approximation. It is equally obvious that we could obtain the correct solution by minimizing any functional of the form = ~l f ~ (pl A2 + p2A 2 2 +...)dr2 -
l f~ .AT (u) p,A(u) dr2
(3.140)
in which Pl, P2 . . . . . etc., are positive valued weighting functions or constants and p is a diagonal matrix: 0 P --
P2
P3
(3.141) ".o
The above alternative form is sometimes convenient as it puts different importance on the satisfaction of individual components of the equation set and allows additional freedom in the choice of the approximate solution. Once again this weighting function could be chosen so as to ensure a constant ratio of terms contributed by various equations. A least squares method of the kind shown above is a very powerful alternative procedure for obtaining integral forms from which an approximate solution can be started, and has been used with considerable success. 35' 36 As a least squares variational principle can be written for any set of differential equations without introducing additional variables, we may well enquire what is the difference between these and the natural variational principles discussed previously. On performing a variation in a specific case the reader will find that the Euler equations which are obtained no longer give the original differential equations but give higher order derivatives of these. This introduces the possibility of spurious solutions if incorrect boundary conditions are used. Further, higher order continuity of trial functions is now generally needed. This may be a serious drawback but frequently can be by-passed by stating the original problem as a set of lower order equations.
Least squares approximations 93
We shall now consider the general form of discretized equations resulting from the least squares approximation for linear equation sets (again neglecting boundary conditions which are assumed forced). Thus, if we take ,,Zt(u) = s
+ b
(3.142)
and take the usual trial function approximation fi = Nfi
(3.143)
we can write, substituting into (3.140), lI = ~ -
1
/2
[(Z~N)fi+ b]Tp[(Z~N)fi + b] dr2
(3.144)
and obtain
= lf~
~I-I = ~
lf~ [(s163
6fiT(/2N)Tp[(/2N)fi+b]dg2+ ~
= 0 (3.145)
or, as p is symmetric, ~I~I = 6,T { [ ~ (Z~N)Tp(Z~N)dr21, + ~ (Z~N)Tpb dr2 } = 0
(3.146)
This immediately yields the approximation equation in the usual form: Kfi + f = 0
(3.147)
and the reader can observe that the matrix K is symmetric and positive definite. Example 3.14: Least squares solution for Helmholz equation. To illustrate an actual example, consider the Helmholz problem governed by Eq. (3.88) for which we have already obtained a natural variational principle [Eq. (3.91)] in which only first derivatives were involved requiting Co continuity for u. Now, if we use the operator/2 and term b defined by Eq. (3.89), we have a set of approximating equations with
gab -- f (V2Na "q-cNa)(V2Nb -k- cNb) dx dy J/.S2
(3.148)
fa = J(V2Na + cNa)Q dx dy
The reader will observe that due to the presence of second derivatives C1 continuity is now needed for the trial functions N. Example 3.15: Least squares solution for Helmholz equation in first-order form. An alternative, avoiding the requirement of C1 functions, is to write Eq. (3.88) as a first-order system. This can be written as
a(u)
=
Oqx Oqy --~-x + --~-y + cqb + Q aO ~ qx Ox a4~ Oy qY
=0
(3.149)
94
Generalization of the finite element concepts
or, introducing the vector u, (3.150)
u = [~, qx, qy]T _._ Nfl
as the unknown we can write an approximation as
0 0]
U ~, fi --
Nq 0
0 Nq
Clx
- N~
(]y
(3.151)
where N o and Nq are Co shape functions for the ~b and qx, qy variables, respectively. The least squares approximation is now given by ~l~i = 6fit f~ (Z~N)T [(Z~N)fi + b] dr2 = 0 where
0Nq cN4" s
0Nq-
Ox '
Oy
-Nq,
0
1. 8y '
(3.152a)
. {!}
(3.152b)
-Nq
O,
The reader can now perform the final steps to obtain the K and f matrices. The approximation equations in a form requiring only Co continuity are obtained, however, at the expense of additional variables. Use of such forms has been made extensively in the finite element context. 35-41
3.13.1 Galerkin least squares, stabilization It is interesting to note that the concept of penalty formulation introduced in the previous section was anticipated as early as 1943 by Courant 42 in a somewhat different manner. He used the original variational principle augmented by the differential equations of the problem employed as least squares constraints. In this manner he claimed, though never proved, that the convergence rate could be accelerated. The suggestion put forward by Courant has been used effectively by others though in a somewhat different manner. Noting that the Galerkin process is, for self-adjoint equations, equivalent to that of minimizing a functional, the least squares formulation using the original equation is simply added to the Galerkin form. Here it allows non-self-adjoint operators to be used, for instance, and this feature has been exploited with success. Consider, for instance, an equation of the form
d2O
dO
x----d 2 + ot ~
+ Q = 0
The first order term multiplying ot is a convective term and, due to its presence, no natural variational equation is available as the differential equation is non-self-adjoint. However, Galerkin methods have been successfully used in its solution providing the convection term (c~d~b/dx) remains relatively small compared to the second derivative term (the diffusion
Concluding remarks- finite difference and boundary methods 95 term). However, it is found that as the convection term increases the solution becomes highly oscillatory. Here we only consider the problem in a preliminary manner and refer the reader to references on fluid dynamics for further study (e.g., see reference 10). Suppose in a Galerkin form given by dx dx
v ot
dx
+ Q
dx = 0
(3 153)
we add a multiple of the minimization of the least squares of the total equation. The result is dx dx +
v ot
//
+Q
k,dx 2 + o t ~ -x
r
dx
)
~xZ+Ot~--X-X+ Q d x = O
(3.154)
and we see immediately that an additional diffusive term has been added which depends on the parameter r, though at the expense of having higher derivatives appearing in the integrals. If only linear elements are used and the discontinuities ignored at element interfaces, the process of adding the diffusive terms can stabilize the oscillations which would otherwise occur. The idea appears to have first been used by Hughes 43--45 and later studied by Codina. 46 This process in the view of the authors is somewhat unorthodox as discontinuity of derivatives is ignored, and alternatives to this are discussed at length in reference 10. It is interesting to note also that another application of the same Galerkin least squares process can be made to the mixed formulation with two variables u and p for incompressible problems. We shall discuss such problems in Chapter 11 of this volume and show how this process can be made applicable there. Finally, it is of interest to note that the simple procedure introduced by Courant can also be effective in the prevention of locking of other problems. The treatment for beams has been studied by Freund and Salonen 47 and it appears that quite an effective process can be reached.
This very extensive chapter presents the general possibilities of using the finite element process in almost any mathematical or mathematically modelled physical problem. The essential approximation processes have been given in as simple a form as possible, at the same time presenting a fully comprehensive picture which should allow the reader to understand much of the literature and indeed to experiment with new permutations. In the chapters that follow we shall apply to various physical problems a limited selection of the methods to which allusion has been made. In some we shall show, however, that certain extensions of the process are possible (Chapters 11 and 15) and in another (Chapter 9) how a violation of some of the rules here expounded can be accepted. The numerous approximation procedures discussed fall into several categories. To remind the reader of these, we present in Table 3.2 a comprehensive catalogue of the methods used here and in Chapter 2. The only aspect of the finite element process mentioned in
96
Generalizationof the finite element concepts
Table 3.2 Finiteelementapproximation Integral forms of continuum problems trial functions u
Direct physical model
= Y ~ NaSa a
1 Variational principles
Weighted integrals of partial differential equation governing (weak formulations)
Global physical statements (e.g. virtual work)
Meaningful physical principles Miscellaneous weight functions
Constrained lagrangian forms
--"4-1
Collocation (point or subdomain)
Penalty function forms
1 I (wb= Nb) I I Galerkin
Least square forms .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
i
this table that has not been discussed here is that of a direct physical method. In such models an 'atomic' rather than continuum concept is the starting point. While much interest exists in the possibilities offered by such models, their discussion is outside the scope of this book. In all the continuum processes discussed the first step is always the choice of suitable shape or trial functions. A few simple forms of such functions have been introduced as the need demanded and many new forms will be introduced in the next two chapters. Indeed, the reader who has mastered the essence of the present chapter will have little difficulty in applying the finite element method to any suitably defined physical problem. For further reading references 48-52 could be consulted. The methods listed do not include specifically two well-known techniques, i.e., finite difference methods and boundary solution methods (sometimes known as boundary elements). In the general sense these belong under the category of the generalized finite element method discussed here. 48 1. Boundary solution methods choose the trial functions such that the governing equation is automatically satisfied in the domain f2. Thus starting from the general approximation equation (3.22), we note that only boundary terms remain to be satisfied. We shall return to such approximations in Chapter 12. 2. Finite difference procedures can be interpreted as an approximation based on local, discontinuous, shape functions with collocation weighting applied (although usually the derivation of the approximation algorithm is based on a Taylor expansion). As Galerkin or variational approaches give, in the energy sense, the best approximation, this method has only the merit of computational simplicity and occasionally a loss of accuracy. To illustrate this process we recall the approximation carded out for the one-dimensional equation (3.24a) (viz. p. 62). We now represent a localized approximation through equally spaced nodal points by
Problems
I,..,....
i-2
i-1
9
i+ 1
i+ 2
~,,~X
Fig. 3.11 A local, discontinuous shape function by parabolic segments used to obtain a finite difference approximation 9
X--Xa)h X
' ( l - (X-Xa)2)h 2
' "~1( (X-xa)2h2
~a ~a+l
+ X--Xa)]]~..
(3.155)
where h = Xa+l -- Xa (shown in Fig. 3.11). It is now clear that adjacent parabolic approximations in this case are discontinuous between the nodes. Values of the function and its first two derivatives at a typical node i are given by
~)(Xa) = ~a 0~b] --~X X=Xa
O2~l
1 ~ =
~-~(q~a+l -- ~a-1)
(3.156)
1 x=x, = h2 (~a+~ -- 2~a 4- ~a-~)
If we insert these into the governing equation at node i, we note immediately that the approximating equation at the node becomes 1
h2 (~a-1 - 2~a + ~a+l) +
Qa - 0
(3.157)
This is identical to the result based on Taylor expansion given by Eq. (3.31). This is indeed one of the cases in which the finite difference approximation is identical to the finite element one rather than different. In Chapter 15 we shall be discussing such finite difference and point approximations in more detail. However, the reader will note the present exercise is simply given to underline the similarity of finite element and finite difference processes. Many textbooks deal exclusively with these types of approximations. References 8, 53-55 discuss finite difference approximation and references 56-59 relate to boundary methods.
3.1 Write weak forms for the following differential equations and boundary conditions. For each form state appropriate continuity conditions for approximations to the dependent
97
98
Generalization of the finite element concepts
variable u and the weighting function v. The domain for each one-dimensional differential equation is 0 < x < 1. du (a) a -~x + cu + q - O; u ( 0 ) = d(du) (b) ~--Xx a ~ xx (c) - ~
d(du) a~
d(d:u)
(d) dxx a ~ x 2
+q=0;
du
du u(0)=~, &a-~x+ku=~atx=l
+bdxx + q = 0 ; +f-0;
u(0)-g0;
u(0)-g0;
(e) - 1 7 T ( k v u ) + c b T ( 1 7 u ) + q = 0
duI
u(1)-g,2
dxx x=0
--h0 & u(1)--gl
inf2; u = ~ , o n F
The differential equations for bending of a beam are given by dV (1) ~ x + q = 0
dM (2) ~ + V = 0
dO M dw V =0 (4) -0 =0 dx EI dx GA in which V is shear force, M is moment, 0 is section rotation, w is displacement, E I is bending stiffness, GA is shear stiffness and q is load as shown in Fig. 3.12. Boundary conditions are given by (3)
(1) V = 1 2
or
w--C0
(2) M = &
or
0--0
3.2 Construct a weak form for the beam equations by multiplying (1) by 4-6 w, (2) by 4-30, (3) by 3M and (4) by 3V. Choose the correct sign for 3w and 30 to give symmetry. 3.3 Add all boundary conditions to the weak form obtained in Problem 3.2. 3.4 Construct a variational theorem which gives the weak form obtained in Problems 3.2 and 3.3 as the first variation. 3.5 For G A - c~ (no shear deformation) deduce the irreducible differential equation in terms of w. Express all boundary conditions in terms of w.
I Z
qz
ttttt V+AV
M+AM
z~
Fig. 3.12 Beam bending description.
X
Problems 99
3.6 Construct a weak form for Problem 3.5. What is the required continuity of the dependent variable needed for approximation by a finite element method? What are the natural and essential boundary conditions for the weak form? 3.7 Construct a variational theorem which has Problem 3.6 as its first variation. 3.8 For G A = c~ (no shear deformation) deduce the differential equations in terms of w and M. Express all boundary conditions in terms of these variables. 3.9 Deduce a weak form for Problem 3.8 that permits approximation using Co functions to approximate w and M. Let 2
W =
ZNaffOa and
2
ZNa]l/'Ia
M=
a=l
a--1
where Na are given by (3.28). Ensure your weak form gives a symmetric coefficient matrix for these approximations. Compute typical element matrices K and f for an element of length h with constant E l and q in the element. 3.10 For a simply supported beam of length 10 and constant cross-section E I = 3 compute the solution for a uniform load of q = 1. The boundary conditions at each end of the beam for a simple support are w = M = 0. Obtain a solution using 2, 4, and 8 elements. It is recommended that a small computer program be written using a high level language, e.g. MATLAB, 6~ to perform the numerical calculations. Compare your results to an exact solution. 3.11 Solve the one-dimensional heat equation given in Example 3.5 by enforcing the boundary conditions by the penalty formulation described in Sec. 3.12.1. How large must each penalty parameter be taken to make the boundary error less than lO-61qbmaxl? 3.12 Deduce the Euler differential equation and boundary conditions for the variational principle expressed as 1-I(u) --
E I (-~x ) - P u
dx - ug
x--b
;
u(a) = O
Classify rI as a minimum, maximum or saddle point form. 3.13 Deduce the Euler differential equation and boundary conditions for the variational principle expressed as 1-I(u) --
E A (-d--~x) 4- ku 2 - 2 q u
dx + ot [(u(a)) 2 + (u(b)) 2]
where EA and k are constant parameters and c~ is a penalty parameter. 3.14 Deduce the Euler equations and boundary conditions for the variational principle expressed as rI(u, )~a, ~.b) =
E A (~-X-x)2+ ku 2 - 2 q u
dx + 1.aU(a) + ~+u(b)
where EA, k and q are constant parameters and )~a, ~.b are Lagrange multipliers. 3.15 The transient heat equation in one dimension is given by 8x
-~x
+Q+co-7
100 Generalizationof the finite element concepts where q~ is temperature, k thermal conductivity, Q heat generation per unit length and c specific heat. Boundary conditions may be given as 4~-~
on F1 or q = -
k ~O~ =q0x
on F2
where q is the heat flux and ~, ~ are specified values. Initial conditions are given as
~(x, 0) = ~0(x). (a) Construct a weak form for the problem. (b) Using the shape functions given in Eq. (3.28) and the approximation u e = Nl(X)fil(t) + N2(x)ft2(t) 3u e = Nl(X)rfil + N2(x)rfi2
construct the semi-discrete form for a typical element of length h. (c) Consider a region of length 10, with properties k = 5, c = 1, Q = 0. Divide the region into four equal length elements and establish the set of global semi-discrete equations. (d) Consider a set of discrete times tn. Approximate time derivatives of nodal values by ddp/dt(tn) ~ ( ~ ) n - qbn-1)/At where t~n is the approximation to dp(tn) and A t = tn - t~-I and write the fully discrete equations. Write a computer program (e.g., using MATLAB) to solve the problem. Assume the initial temperature of the region is zero and boundary conditions q~(0) = 0 and 4~(10) = 1 are applied at time zero and held constant. Solve the problem using 10 steps with At = 0.01, followed by 9 steps with At = 0.1 and finally 9 steps with At = 1. Plot the finite element solution for q~ vs x at times 0.01, 0.1, 1.0 and 10.0. Replace the element matrix associated with c by a diagonal (lumped) form with c h / 2 on each diagonal (h = x~ - x~). Repeat the above solution and compare results with the consistent form for the matrix.
1. S. Crandall. Engineering Analysis. McGraw-Hill, New York, 1956. 2. B.A. Finlayson. The Method of Weighted Residuals and Variational Principles. Academic Press, New York, 1972. 3. R.A. Frazer, W.P. Jones, and S.W. Sken. Approximations to functions and to the solution of differential equations. Technical Report 1799, Aero. Research Committee Report, 1937. 4. C'B. Biezeno and R. Grammel. Technishe Dynamik. Springer-Verlag, Berlin, 1933, p. 142. 5. O.C. Zienkiewicz and E. Ofiate. Finite elements versus finite volumes. Is there a choice? In P. Wriggers and W. Wagner, editors, Nonlinear Computational Mechanics. State of the Art. Springer, Berlin, 1991. 6. B.G. Galerkin. Series solution of some problems in elastic equilibrium of rods and plates. Vestn. Inzh. Tech., 19:897-908, 1915. 7. P. Tong. Exact solution of certain problems by the finite element method. J. AIAA, 7:179-180, 1969. 8. R.V. Southwell. Relaxation Methods in Theoretical Physics. Clarendon Press, Oxford, 1st edition, 1946.
References 9. R.S. Varga. Matrix lterative Analysis. Prentice-Hall, Englewood Cliffs, N.J., 1962. 10. O.C. Zienkiewicz, R.L. Taylor, and P. Nithiarasu. The Finite Element Method for Fluid Dynamics. Butterworth-Heinemann, Oxford, 6th edition, 2005. 11. S.P. Timoshenko and J.N. Goodier. Theory of Elasticity. McGraw-Hill, New York, 3rd edition, 1969. 12. I.S. Sokolnikoff. The Mathematical Theory of Elasticity. McGraw-Hill, New York, 2nd edition, 1956. 13. L.V. Kantorovich and V.I. Krylov. Approximate Methods of HigherAnalysis. John Wiley & Sons (International), New York, 1964. English translation by Curtis D. Benster. 14. O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method for Solid and Structural Mechanics. Butterworth-Heinemann, Oxford, 6th edition, 2005. 15. EB. Hildebrand. Methods of Applied Mathematics. Prentice-Hall (reprinted by Dover Publishers, 1992), 2nd edition, 1965. 16. Lord Rayleigh (J.W. Strutt). On the theory of resonance. Trans. Roy. Soc. (London),A161:77118, 1870. 17. W. Ritz. Uber eine neue Methode zur L6sung gewisser variationsproblem der mathematischen physik. J. Reine angew. Math., 135:1-61, 1908. 18. E. Tonti. Variational formulation of non-linear differential equations. Bull. Acad. Roy. Belg. Classe Sci., 55:137-165 & 263-278, 1969. 19. M.M. Vainberg. Variational Methods for the Study of Nonlinear Operators. Holden-Day Inc., San Francisco, CA, 1964. 20. J.T. Oden. A general theory of finite elements. Part I. Topological considerations. Int. J. Numer. Meth. Eng., 1:205-246, 1969. 21. J.T. Oden. A general theory of finite elements. Part II. Applications. Int. J. Numer. Meth. Eng., 1:247-254, 1969. 22. S.C. Mikhlin. Variational Methods in Mathematical Physics. Macmillan, New York, 1964. 23. S.C. Mikhlin. The Problem of the Minimum of a Quadratic Functional. Holden-Day, San Francisco, 1966. 24. B.A. Szabo and T. Kassos. Linear equation constraints in finite element approximations. Int. J. Numer. Meth. Eng., 9:563-580, 1975. 25. K. Washizu. Variational Methods in Elasticity and Plasticity. Pergamon Press, New York, 3rd edition, 1982. 26. E Kikuchi and Y. Ando. A new variational functional for the finite element method and its application to plate and shell problems. Nucl. Eng. Des., 21(1):95-113, 1972. 27. H.S. Chen and C.C. Mei. Oscillations and water forces in an offshore harbour. Technical Report 190, Ralph M. Parsons Laboratory for Water Resources and Hydrodynamics, Massachusetts Institute of Technology, Cambridge, MA, 1974. 28. O.C. Zienkiewicz, D.W. Kelley, and P. Bettess. The coupling of the finite element and boundary solution procedures. Int. J. Numer. Meth. Eng., 11:355-375, 1977. 29. O.C. Zienkiewicz. Constrained variational principles and penalty function methods in finite element analysis. In Lecture Notes in Mathematics, No. 363, pages 207-214, Springer-Verlag, Berlin, 1974. 30. J. Campbell. A finite element system for analysis and design. Ph.D. thesis, Department of Civil Engineering, University of Wales, Swansea, 1974. 31. D.J. Naylor. Stresses in nearly incompressible materials for finite elements with application to the calculation of excess pore pressures. Int. J. Numer. Meth. Eng., 8:443-460, 1974. 32. I. Fried. Shear in c o and c 1 bending finite elements. Int. J. Solids Struct., 9:449-460, 1973. 33. I. Fried. Finite element analysis of incompressible materials by residual energy balancing. Int. J. Solids Struct., 10:993-1002, 1974. 34. O.C. Zienkiewicz and E. Hinton. Reduced integration, function smoothing and non-conformity in finite element analysis. J. Franklin Inst., 302:443-461, 1976.
101
102 Generalizationof the finite element concepts 35. EE Lynn and S.K. Arya. Finite elements formulation by the weighted discrete least squares method. Int. J. Numer. Meth. Eng., 8:71-90, 1974. 36. O.C. Zienkiewicz, D.R.J. Owen, and K.N. Lee. Least square finite element for elasto-static problems - use of reduced integration. Int. J. Numer. Meth. Eng., 8:341-358, 1974. 37. B.-N. Jiang. Optimal least-squares finite element method for elliptic problems. Comp. Meth. Appl. Mech. Eng., 102:199-212, 1993. 38. B.-N. Jiang. On the least-squares method. Comp. Meth. Appl. Mech. and Eng., 152:239-257, 1998. 39. B.-N. Jiang. Least Squares Finite Element Method: Theory and Applications in Computational Fluid Dynamics and Electromagnetics. Springer, New York, 1998. 40. B.-N. Jiang. The least-squares finite element method in elasticity. I. Plane stress or strain with drilling degrees of freedom. Int. J. Numer. Meth. Eng., 53:621-636, 2002. 41. B.-N. Jiang. The least-squares finite element method in elasticity. II. Bending of thin plates. Int. J. Numer. Meth. Eng., 54:1459-1475, 2002. 42. R. Courant. Variational methods for the solution of problems of equilibrium and vibration. Bull. Am. Math Soc., 49:1-61, 1943. 43. T.J.R. Hughes, L.P. Franca, and M. Balestra. A new finite element formulation for computational fluid dynamics: V. Circumventing the Babu~ka-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Comp. Meth. Appl. Mech. Eng., 59:85-99, 1986. 44. T.J.R. Hughes and L.P. Franca. A new finite element formulation for computational fluid dynamics: VII. The Stokes problem with various well-posed boundary conditions: symmetric formulation that converge for all velocity/pressure spaces. Comp. Meth. Appl. Mech. Eng., 65:85-96, 1987. 45. T.J.R. Hughes, L.P. Franca, and G.M. Hulbert. A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective-diffusive equations. Comp. Meth. Appl. Mech. Eng., 73:173-189, 1989. 46. R. Codina, M. V~izquez, and O.C. Zienkiewicz. General algorithm for compressible and incompressible flows, Part I I I - a semi-implicit form. Int. J. Numer. Meth. Fluids, 27:13-32, 1998. 47. Jouni Freund and Eero-Matti Salonen. Sensitizing according to Courant the Timoshenko beam finite element solution. Int. J. Numer. Meth. Eng., x:129-160, 1999. 48. O.C. Zienkiewicz and K. Morgan. Finite Elements and Approximation. John Wiley & Sons, London, 1983. 49. E.B. Becker, G.E Carey, and J.T. Oden. Finite Elements: An Introduction, volume 1. PrenticeHall, Englewood Cliffs, N.J., 1981. 50. B. Szabo and I. Babu~ka. Finite Element Analysis. John Wiley & Sons, New York, 1991. 51. T.J.R. Hughes. The Finite Element Method: Linear Static and Dynamic Analysis. Dover Publications, New York, 2000. 52. C.A.T. Fletcher. Computational Galerkin Methods. Springer-Verlag, Berlin, 1984. 53. D.N. de G. Allen. Relaxation Methods. McGraw-Hill, London, 1955. 54. EB. Hildebrand. Introduction to Numerical Analysis. Dover Publishers, 2nd edition, 1987. 55. A.R. Mitchell and D. Griffiths. The Finite Difference Method in Partial Differential Equations. John Wiley & Sons, London, 1980. 56. P.K. Banerjee. The Boundary Element Methods in Engineering. McGraw-Hill, London, 1994. 57. Prem K. Kythe. An Introduction to Boundary Element Methods. CRC Press, 1995. 58. G. Beer and J.O. Watson. Programming the Boundary Element Method: An Introduction for Engineers. John Wiley & Sons, Chichester, 2001. 59. L. Gaul. Boundary Element Methods for Engineers and Scientists. Springer, Berlin, 2003. 60. MATLAB. www.mathworks.com, 2003.
In Chapters 2 and 3 the reader was shown in some detail how linear elasticity and other problems could be formulated and solved using very simple element forms. Although the detailed algebra was only concerned with shape functions which arose from triangular or rectangular shapes, it should by now be obvious that other element forms could equally well be used. Indeed, once the element and the corresponding shape functions are determined, subsequent operations follow a standard, well-defined path. It will be seen later that it is possible to program a computer to deal with wide classes of problems by specifying the shape functions only. The choice of these is, however, a matter to which intelligence has to be applied and in which the human factor remains paramount. In this chapter some rules for the generation of several families of one-, two-, and three-dimensional elements will be presented. In the problems of elasticity illustrated in Chapters 2 and 3 the displacement variable was a vector with two or three components and the shape functions were written in matrix form. They were, however, derived for each component separately and the matrix expressions in these were derived by multiplying a scalar function by an identity matrix [e.g., Eq. (2.2)]. In this chapter we shall concentrate on the scalar shape function forms, calling these simply Na. The shape functions used in the displacement formulation of elasticity problems were such that they satisfy the convergence criteria of Chapter 2: 1. The continuity of the unknown only had to occur between elements (i.e., slope continuity is not required), or, in mathematical notation, Co continuity was needed; 2. The function has to allow any arbitrary linear form to be taken so that the constant strain (constant first derivative) criterion could be observed in each element. The shape functions described in this chapter will require the satisfaction of these two criteria. They will thus be applicable to all the problems requiring Co continuity (i.e., all problems governed by first or second order differential equations). Indeed they are applicable to any situation where the functional H or 81-I (see Chapter 3) is defined by derivatives of first order only.
104 'Standard' and 'hierarchical' element shape functions The element families discussed will progressively have an increasing number of degrees of freedom. The question may well be asked as to whether any economic or other advantage is gained by increasing the complexity of an element. The answer here is not an easy one although it can be stated as a general rule that as the order of an element increases so the total number of unknowns in a problem can be reduced for a given accuracy of representation. Economic advantage requires, however, a reduction of total computation and data preparation effort, and this does not follow automatically for a reduced number of total variables. However, an overwhelming economic advantage in the case of three-dimensional analyses occurs. The same kind of advantage arises on occasion in other problems but in general the optimum element may have to be determined from case to case. In Sec. 2.6 of Chapter 2 we have shown that the order of error in the approximation to the unknown function is O(hP+l), where h is the element 'size' and p is the degree of the complete polynomial present in the expansion. Clearly, as the element shape functions increase in degree so will the order of error increase, and convergence to the exact solution becomes more rapid. While this says nothing about the magnitude of error at a particular subdivision, it is clear that we should seek element shape functions with the highest complete polynomial for a given number of degrees of freedom.
The essence of the finite element method already stated in Chapters 2 and 3 is in approximating the unknown (displacement) by an expansion given in Eqs (2.1) and (3.3). This, for a scalar variable u, can be written as
u ~ ~t - ~
Naua -- Nfi e
(4.1)
a=l
where n is the total number of functions used and fia are the unknown parameters to be determined. We have explicitly chosen to identify such variables with the values of the unknown function at element nodes, thus making ~la = 1-l(Xa)
(4.2)
The shape functions so defined will be referred to as 'standard' ones and are the basis of most finite element programs. If polynomial expansions are used and the element satisfies Criterion 1 of Chapter 2 (which specifies that rigid body displacements cause no strain), it is clear that a constant value of fia specified at all nodes must result in a constant value of fi" o
when
fia
--" /'/0-
It follows that
(4.3)
n
y~'Na = 1 a--1
(4.4)
Standard and hierarchical concepts 105 at all points of the domain. This important property is known as a partition of unity I which we will make extensive use of here and in Chapter 15. The first part of this chapter will deal with such standard shape functions. A serious drawback exists, however, with 'standard' functions, since when element refinement is made totally new shape functions have to be generated and hence all calculations repeated. It would be of advantage to avoid this difficulty by considering the expression (4.1) as a series in which the shape function Na does not depend on the number of nodes in the mesh n. This indeed is achieved with hierarchic shape functions to which the second part of this chapter is devoted. The hierarchic concept is well illustrated by the one-dimensional (elastic bar) problem of Fig. 4.1. Here for simplicity elastic properties are taken as constant (D = E) and the body force b is assumed to vary in such a manner as to produce the exact solution shown on the figure (with zero displacements at both ends). Two meshes are shown and a linear interpolation between nodal points assumed. For both standard and hierarchic forms the coarse mesh gives C
"~C
K11u I
--
(4.5)
fl
For a fine mesh two additional nodes are added and with the standard shape function the equations requiring solution are
K~
Kg
K~I
0
1{0} U2
KFJ
"-
fi3
f2
(4.6)
f3
In this form the zero matrices have been automatically inserted due to element interconnection which is here obvious, and we note that as no coefficients are the same, the new equations have to be resolved [Eq. (2.28a) shows how these coefficients are calculated and the reader is encouraged to work these out in detail]. With the 'hierarchic' form using the shape functions shown, a similar form of equation arises and an identical approximation is achieved (being simply given by a series of straight segments). The final solution is identical but the meaning of the parameters fi] is now different, as shown in Fig. 4.1. Quite generally, K F -- K~I (4.7) as an identical shape function is used for the first variable. Further, in this particular case the off-diagonal coefficients are zero and the final equations become, for the fine mesh,
[
K~I 0 0
0 Kf2 0
~ K
~;
=
f2 f3
(4.8)
The 'diagonality' feature is only true in the one-dimensional problem, but in general it will be found that the matrices obtained using hierarchic shape functions are more nearly diagonal and hence usually imply better conditioning than those with standard shape functions. Although the variables are now not subject to the obvious interpretation (as local displacement values), they can be easily transformed to those if desired. Though it is not usual
106
'Standard' and 'hierarchical' element shape functions Coarse
Fine Exact proximate
1
J
J
2
1
3
N2
N1
N3
2
1
3
N2
N1
N3
jJ
(a)
1
...""
9 ss
s
..
"',
~
ii ~
s
,2-"
,"
s ~.;r s x
~
",,"
~
"-,," s s
9
, "-~
(b) Fig. 4.1 A one-dimensional problem of stretching of a uniform elastic bar by prescribed body forces.
to use hierarchic forms in linearly interpolated elements their derivation in polynomial form is simple and very advantageous. The reader should note that with hierarchic forms it is convenient to consider the finer mesh as still using the same, coarse, elements but now adding additional refining functions. Hierarchic forms provide a link with other approximate (orthogonal) series solutions. Many problems solved in classical literature by trigonometric, Fourier series, expansion are indeed particular examples of this approach. In the next sections of this chapter we shall consider the development of shape functions for high order elements with many boundary and internal degree of freedoms. Such development will generally be made on simple geometric forms and the reader may well question the wisdom of using increased accuracy for such simple shaped domains - having already observed the advantage of generalized finite element methods in fitting arbitrary domain shapes. This concern is well founded, but in the next chapter we shall show a general method to map high order elements into quite complex shapes.
Rectangular elements-some preliminary considerations 107
Part 1. 'Standard' shape functions Two-dimensional elements
Conceptually (especially if the reader is conditioned by education to thinking in the cartesian coordinate system) the simplest element form of a two-dimensional kind is that of a rectangle with sides parallel to the x and y axes. Consider, for instance, the rectangle shown in Fig. 4.2 with nodal points numbered 1 to 8, located as shown, and at which the values of an unknown function u (here representing, for instance, one of the components of displacement) form the element parameters. How can suitable Co continuous shape functions for this element be determined? Let us first assume that u is expressed in polynomial form in x and y. To ensure interelement continuity of u along the top and bottom sides the variation must be linear. Two points at which the function is common between elements lying above or below exist, and as two values uniquely determine a linear function, its identity all along these sides is ensured with that given by adjacent elements. Use of this fact was already made in specifying linear expansions on edges for a triangle and a rectangle. Similarly, if a cubic variation along the vertical sides is assumed, continuity will be preserved there as four values determine a unique cubic polynomial. Conditions for satisfying the first criterion are now obtained. To ensure the existence of constant values of the first derivative it is necessary that all the linear polynomial terms of the expansion be retained. Finally, as eight points are to determine uniquely the variation of the function, only eight coefficients of the expansion can be retained and thus we could write
,~Y
Fig. 4.2 A rectangular element.
1
8
2
7
3
6
4
5
108 'Standard' and 'hierarchical' element shape functions U
=
Ol 1 n t- O l 2 X
-Jr"ot3y nt- Ot4Xy nt- ot5y 2 nt- Ot6Xy 2 nt- o~7y3 -k- Ot8xy 3
(4.9)
The choice can in general be made unique by retaining the lowest possible expansion terms, though in this case apparently no such choice arises.t The reader will easily verify that all the requirements have now been satisfied. Substituting coordinates of the various nodes a set of simultaneous equations will be obtained. This can be written in exactly the same manner as was done for a triangle in Eq. (2.4) as
/01/Ii Xl yl Xlyl X fi8
,
x8,
Y8,
..
.
.
.
Xlyl ]/ l/
.
x8y3J
(4.10)
or8
or simply as ~e = Cog.
(4.11)
og : c - l ~ e
(4.12)
u = P(x, y)og = P(x, y ) C - l u e
(4.13)
P(x, y) = [1, x, y, x y , y2, x y 2 , y3, xy3]
(4.14)
Formally, and we could write Eq. (4.9) as
in which Thus the shape functions for the element defined by u = Nfi e = [N1, N2 . . . . . N8] a e
(4.15)
N(x, y) - P(x, y)C -1
(4.16)
can be found as This process has, however, some considerable disadvantages. Occasionally an inverse of C may not exist 2' 3 and always considerable algebraic difficulty is experienced in obtaining an expression for the inverse in general terms suitable for all element geometries. It is therefore worthwhile to consider whether shape functions Na (x, y) can be written down directly. Before doing this some general properties of these functions have to be mentioned. Inspection of the defining relation, Eq. (4.15), reveals immediately some important characteristics. First, as this expression is valid for all components of fie, Na (Xb, Yb )
1;
-- ~ab "-
0;
a=b a~b
where ~ a b is known as the Kronecker delta. Further, the basic type of variation along boundaries defined for continuity purposes (e.g., linear in x and cubic in y in the above example) must be retained. The typical form of the shape functions for the elements considered is illustrated isometrically for two typical nodes in Fig. 4.3. It is clear that these could have been written down directly as a product of a suitable linear function in x with a t Retention of a higher order term of expansion, ignoring one of lower order, will usually lead to a poorer approximation though still retaining convergence,2 providing the linear terms are always included.
Completeness of polynomials 109
N~
/
N~
st ,,,%s
1
Fig. 4.3 Shape functions for elements of Fig. 4.2. cubic function in y. The easy solution of this example is not always as obvious but given sufficient ingenuity, a direct derivation of shape functions is always preferable. It will be convenient to use normalized coordinates in our further investigation. Such normalized coordinates are shown in Fig. 4.4 and are chosen so that their values are + 1 on the faces of the rectangle:t ~-
x -
Xc
a
r/=
Y -
b
Yc
dx
d~----
a dy dr/= m b
(4.17)
Once the shape functions are known in the normalized coordinates, translation into actual coordinates or transformation of the various expressions occurring, for instance, in the stiffness derivation is trivial for rectangular shapes. Consideration of other more convenient 'mapping' methods will be addressed in Chapter 5.
The shape function derived in the previous section was of a rather special form [viz. Eq. (4.9)]. Only a linear variation with the coordinate x was permitted, while in y a full cubic was available. The complete polynomial contained in it was thus of order 1. In general use, a convergence order corresponding to a linear variation would occur despite an increase of the total number of variables. Only in situations where the linear variation t In Chapter 5 we will show that this is convenient for purposes of numerical integration.
110 'Standard' and 'hierarchical' element shape functions ~Y
,=1
/
q= 1 ~q c
Yc
q=-I xc
/
\ ~=1 ~.
r X
Fig. 4.4 Normal coordinates for a rectangle.
in x corresponded closely to the exact solution would a higher order of convergence occur, and for this reason elements with such 'preferential' directions should be restricted to special use, e.g., in narrow beams or strips. Usually, we will seek element expansions which possess the highest order of a complete polynomial for a minimum of degrees of freedom. In this context it is useful to recall the Pascal triangle (Fig. 4.5) from which the number of terms occurring in a polynomial in two variables x, y can be readily ascertained. For instance, first-order polynomials require three terms, second order require six terms, third order require ten terms, etc.
Consider the element shown in Fig. 4.6 in which a series of nodes, external and internal, is placed on a regular grid. It is required to determine a shape function for the point indicated
Fig. 4.5 The Pascal triangle. (Cubic expansion shaded - 10 terms.)
Rectangular elements- Lagrange family (o,rn) (I, d) ce ,~u ......
(n, rn) "P
)
c)
.~
.;
.;
c
)
()
")
,i)
..')
.."
f
f
/ ' . / l l i ~ ' , ~ l l ~ - " , / l i ~ l
(0, O)
(n, O)
1
1
Fig. 4.6
A typical shape function for a lagrangian element (n = 5, m = 4, / = 1, J = 4).
by the heavy circle. Clearly the product of a fifth-order polynomial in ~ which has a value of unity at points of the second column of nodes and zero at the other nodal columns and that of a fourth-order polynomial in r/having unity on the coordinate corresponding to the top row of nodes and zero at other nodal rows satisfies all the interelement continuity conditions and gives unity at the nodal point concerned. Polynomials in one coordinate having this property are known as Lagrange polynomials and can be written down directly as
l~(~)
=
(~: -- ~:0)(~: -- ~:1)""" (~ -- ~:k-1)(~:
-- ~:k+l)"""
(~: -- ~:n)
(~k --~0)(~k - - ~ l ) ' ' ' ( ~ k --~k-1)(~k - - ~ k + l ) ' ' " (~k -- ~n)
--II i=0
ir
(4.18t giving unity at ~k and passing through zero at the remaining n points. An easy and systematic method of generating shape functions of any order now can be achieved by simple products of Lagrange polynomials in the two coordinates. 4-6 Thus, in two dimensions, if we label the node by its column and row number, I, J, we have Na ~ N i j
-
l~ (~)l7 (rl)
(4.19)
where n and m stand for the number of subdivisions in each direction. Figure 4.7 shows a few members of this unlimited family where m - n. For m -- n = 1 we obtain the simple result
111
112 'Standard' and 'hierarchical' element shape functions
0 (b)
(a) 0
0
0
0
0
0
0
(c)
0
Fig. 4.7 Three elements of the Lagrange family: (a)linear, (b) quadratic, (c) cubic.
Na
--
1 ~(1 -F ~a~)(1 -F/']aT])
(4.20)
in which ~a, Tla are the normalized coordinates at node a. Indeed, if we examine the polynomial terms present in a situation where n = m we observe in Fig. 4.8, based on the Pascal triangle, that a large number of polynomial terms is present above those needed for a complete expansion. 7 However, when mapping of shape functions is considered (viz. Chapter 5) some advantages occur for this family.
It is often more efficient to make the functions dependent on nodal values placed on the element boundary. Consider, for instance, the first three elements of Fig. 4.9. In each a progressively increasing and equal number of nodes are placed on the element boundary.
Fig. 4.8 Terms generated by a lagrangian expansion of order 3 x 3 (or m x n). Complete polynomials of order 3 (or n).
Rectangular elements-'serendipity' family 113 q=l / "--I
\ q= -I
(a) 0
0
0
0
(b) '
(c)
0
0
0
0
0
0
(d)
Fig. 4.9 Rectangles of boundary node (serendipity) family: (a)linear, (b) quadratic, (c) cubic, (d) quartic. The variation of the function on the edges to ensure continuity is linear, parabolic, and cubic in increasing element order. To achieve the shape function for the first element it is obvious that a product of linear lagrangian polynomials of the form
+ ~:)(1 + r/) 1(1 4
(4.21)
gives unity at the top right comer where ~ = r/ = 1 and zero at all the other comers. Further, a linear variation of the shape function of all sides exists and hence continuity is satisfied. Indeed this element is identical to the lagrangian one with n -- 1 and again all the shape functions may be written as one expression: 1
Na = z(1 + ~:a~:)(1 -+-?Ta/'])
As a linear combination of these shape functions yields any arbitrary linear variation of u, the second convergence criterion is satisfied. The reader can verify that the following functions satisfy all the necessary criteria for quadratic and cubic members of the family.
"Quadratic" element
Comer nodes:
i (1 + ~a~)(1 + ~a~)(~a~ "3t- ~a T] -- 1) Na = -~
(4.22a)
Mid-side nodes:
~a=0
Na = ~1 ( 1 - ~2)(1 + Fla/'])
/']a = 0
1 + ~:a~)(1Na = ~(1
(4.22b)
/72)
"Cubic" element
Comer nodes:
Na = ~2(1 + ~:a~)(1 "l- r/ar/)[9(~ :2 + 0 2)
--
10]
(4.23a)
114
'Standard' and 'hierarchical' element shape functions
Mid-side nodes: ~a -- +1
and
9(1 +
Na =
1
0a = -}-3
~a~)(1 - 02)(1 -Jr-9000)
(4.23b)
and ~a = +~1
Na -'-
and
9(1
:t::1
0a =
- ~e2)(1 + 9~'a~')(1 -I- 0a0)
(4.23c)
which all satisfy the requirement
Na (~b, 0b)
~ab --"
:
1;a=b 0; a # b
(4.23d)
The above functions were originally derived by inspection, and progression to yet higher members is difficult and requires some ingenuity. 4, 5 It was therefore appropriate to name this family 'serendipity' after the famous princes of Serendip noted for their chance discoveries (Horace Walpole, 1754). However, a quite systematic way of generating the 'serendipity' shape functions can be devised, which becomes apparent from Fig. 4.10 where the generation of a quadratic shape function is presented. 7, 8 4
1
7
5
3
2
(a) N 5 = V 2 ( 1 - ~ 2 ) ( l - q )
0.5
Step 1 1
(b) N 8 = V2(1 - ~) (1 - 1]2)
0
.
0
~
1
= (1 - ~) (1 - q)/4
I '-' 0.5
Step 2
(c)
N~ = ~ - ~ N s - Y ~ N 8 Fig. 4.10 Systematic generation of 'serendipity' shape functions.
Rectangular elements-'serendipity' family 115 As a starting point we observe that for mid-side nodes a lagrangian interpolation of a quadratic x linear type suffices to determine Na at nodes 5 to 8. N5 and N8 are shown in Fig. 4.10(a) and (b). For a comer node, such as Fig. 4.10(c), we start with a bilinear lagrangian family ~/1 and note immediately that while b~l = 1 at node 1, it is not zero at nodes 5 or 8 (step 1). Successive subtraction of 1/2 N5 (step 2) and 1/2 N8 (step 3) ensures that a zero value is obtained at these nodes. The reader can verify that the final expressions obtained coincide with those of Eqs (4.22a) and (4.22b). Indeed, it should now be obvious that for all higher order elements the mid-side and comer shape functions can be generated by an identical process. For the former a simple multiplication of mth-order and first-order lagrangian interpolations suffices. For the latter a combination of bilinear comer functions, together with appropriate fractions of mid-side shape functions to ensure zero at appropriate nodes, is necessary. It also is quite easy to generate shape functions for elements with different numbers of nodes along each side by a similar systematic algorithm. This may be very desirable if a transition between elements of different order is to be achieved, enabling a different order of accuracy in separate sections of a large problem to be studied. Figure 4.11 illustrates the necessary shape functions for a cubic/linear transition. Use of such special elements was first introduced in reference 8, but the simpler formulation used here is that of reference 7. With the mode of generating shape functions for this class of elements available it is immediately obvious that fewer degrees of freedom are now necessary for a given complete
Fig. 4.11 Shape functions for a transition 'serendipity' element, cubic/linear.
116 'Standard' and 'hierarchical' element shape functions
Fig. 4.12 Terms generated by edge shape functions in serendipity-type elements (3 x 3 and rn x m).
polynomial expansion. Figure 4.12 shows this for a cubic element where only two surplus terms arise (as compared with six surplus terms in a lagrangian of the same degree). However, when mapping to general quadrilateral shape is introduced (Chapter 5) some of these advantages are lost rendering the lagrangian form of interpolation advantageous. It is immediately evident, however, that the functions generated by nodes placed only along the edges will not generate complete polynomials beyond cubic order. For higher order ones it is necessary to supplement the expansion by internal nodes or by the use of 'nodeless' variables which contain appropriate polynomial terms. For example, in the next, quartic, member 9 of this family a central node is added [viz. Fig. 4.9(d)] so that all terms of a complete fourth-order expansion will be available. This central node adds a shape function (1 - ~2)(1 - 02) which is zero on all outer boundaries and coincides with the internal function used in the quadratic lagrangian element. Once interior nodes are added it is necessary to modify the comer and mid-side shape functions to preserve the Kronnecker delta property (4.23d).
The advantage of an arbitrary triangular shape in approximating to any boundary configuration has been amply demonstrated in earlier chapters. Its apparent superiority here over rectangular shapes needs no further discussion. However, the question of generating more elaborate higher order elements needs to be further developed. Consider a series of triangles generated on a pattern indicated in Fig. 4.13. The number of nodes in each member of the family is now such that a complete polynomial expansion, of the order needed for interelement compatibility, is ensured. This follows by comparison with the Pascal triangle of Fig. 4.5 in which we see the number of nodes coincides exactly with the number of polynomial terms required. This particular feature puts the triangle family in a special, privileged position, in which the inverse of the C matrices of Eq. (4.11) will always exist. 3 However, once again a direct generation of shape functions will be preferred- and indeed will be shown to be particularly easy. Before proceeding further it is useful to define a special set of normalized coordinates for a triangle.
.Triangularelement
family
3 3 (a)
3
1
(b)
4
8
2
(c)
1
4
7
5
2
Fig. 4.13 Triangular element family: (a)linear, (b) quadratic, (c) cubic.
4.7.1 Area coordinates ........................
.....
,.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
...................................
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.......................................
. ...................................
...............
While cartesian directions parallel to the sides of a rectangle were a natural choice for that shape, in the triangle these are not convenient. A new set of coordinates, L1, L2, and L3 for a triangle 1, 2, 3 (Fig. 4.14), is defined by the following linear relation between these and the cartesian system: X -- LlXl + L2x2 + L3x3 y = L l Y l Jr- L2Y2 Jr- L3Y3 1 =
(4.24)
L1 Jr- L2 --F L3
To every set, L 1, L2, L 3 (which are not independent, but are related by the third equation), there corresponds a unique set of cartesian coordinates. At point 1, L 1 = 1 and L2 -- L3 -0, etc. A linear relation between the new and cartesian coordinates implies that contours of L1 are equally placed straight lines parallel to side 2-3 on which L1 = 0, etc.
Fig. 4.14 Area coordinates.
117
118 'Standard' and 'hierarchical' element shape functions Indeed it is easy to see that an alternative definition of the coordinate L1 of a point P is by a ratio of the area of the shaded triangle to that of the total triangle: L1 --
area P23
(4.25)
area 123
Hence the name a r e a c o o r d i n a t e s . Solving Eq. (4.24) gives Za
aa + b a x -+- Cay
--
2A
; k = 1, 2, 3
(4.26)
in which A
--
x2 Y2 1det Ei Xl yl1 x3 Y3
~
area 123
(4.27)
and al = x2Y3 - x3Y2
bl = Y2 - Y3
Cl -- x3 - x2
(4.28)
etc., with cyclic rotation of indices 1, 2, and 3. The identity of expressions with those derived in Chapter 2 [Eqs (2.6) and (2.7)] is worth noting.
4.7.2 Shape functions For the first element of the series [Fig. 4.13(a)], the shape functions are simply the area coordinates. Thus N1 -- L1 N2 -- L2 N3 = L3 (4.29) This is obvious as each individually gives unity at one node, zero at others, and varies linearly everywhere. To derive shape functions for other elements a simple recurrence relation can be derived. 3 However, it is very simple to write functions for an arbitrary triangle of order M in a manner similar to that used for the lagrangian element of Sec. 4.5. Denoting a typical node a by three numbers I, J, and K corresponding to the position of coordinates L la, L2a, and L3a we can write the shape function in terms of three lagrangian interpolations as [see Eq. (4.18)] (4.30)
Na -- l l I ( L 1 ) I J ( L 2 ) I K ( L 3 )
In the above l ], etc., are given by expression (4.18), with L1 taking the place of ~, etc. It is easy to verify that the above expression gives Na
= 1
at L1 = L I I ,
L2 -- L 2 j ,
L3 :
L3K
and zero at all other nodes. The highest term occurring in the expansion is L~ L2J L~ and as I + J + K -- M for all points the polynomial is also of order M. Expression (4.30) is valid for quite arbitrary distributions of nodes of the pattern given in Fig. 4.15 and simplifies if the spacing of the nodal lines is equal (i.e., 1/ m). The formula was first obtained by Argyris et al. 1~ and formalized in a different manner by others. 7' 11 The reader can verify the shape functions for the second- and third-order elements as given below and indeed derive ones of any higher order easily.
Line elements
.•,
O, M)
0 01/
\
Fig. 4.15 A general triangular element.
Quadratic triangle [Fig. 4.13(b)1 Comer nodes:
Na :
a=
(2La - 1)La,
1,2,3
Mid-side nodes: N4 :
4L1L2,
N5 - - 4 L 2 L 3 ,
N6 :
4L3L1
Cubic triangle [Fig. 4.13(c)1 Comer nodes:
Na :
1
-~(3La - 1 ) ( 3 L a - 2 ) L a ,
a=1,2,3
Mid-side nodes:
N 4 - 9 L I L e ( 3 L 1 - 1),
9
N5 -- ~ L 1 L 2 ( 3 L 2 -
1),
etc.
and for the internal node: N10 - - 2 7 L 1 L 2 L 3
The last shape again is a 'bubble' function giving zero contribution along boundaries and this will be found to be useful in other contexts (see the mixed forms in Chapter 11). The quadratic triangle was first derived by Veubeke 12 and used later in the context of plane stress analysis by Argyris. 13 When element matrices have to be evaluated it will follow that we are faced with integration of quantities defined in terms of area coordinates over the triangular region. It is useful to note in this context the following exact integration expression:
dx d y = (a + a! b+ a b! cc!+ 2)!2A f f A L1LbL~
(4.31)
One-dimensional elements
So far in this book the continuum was considered generally in two or three dimensions. 'One-dimensional' members, being of a kind for which exact solutions are generally available, were treated only as trivial examples in Chapter 3 and in Sec. 4.2. In many practical two- or three-dimensional problems such elements do in fact appear in conjunction with
119
120 'Standard' and 'hierarchical' element shape functions
the more usual continuum elements- and a unified treatment is desirable. In the context of elastic analysis these elements may represent lines of reinforcement (plane and threedimensional problems) or sheets of thin lining material in axisymmetric bodies. In the context of heat conduction and other field problems similar effects occur. Once the shape of such a function as displacement is chosen for an element of this kind, its properties can be determined, noting, however, that derived quantities such as strain, etc., have to be considered only in one dimension. Figure 4.16 shows such an element sandwiched between two adjacent quadratic-type elements. Clearly for continuity of the function a quadratic variation of the unknown with the one variable ~ is all that is required. Thus the shape functions are given directly by the Lagrange polynomial as defined in Eq. (4.18).
Three-dimensional elements
In a precisely analogous way to that given in previous sections equivalent lagrangian family elements of three-dimensional type can be described. Shape functions for such elements will be generated by a direct product of three Lagrange polynomials. Extending the notation of Eq. (4.19) we now have Na -~ NI j K
- - 17
(~)17 (o)l P (()
(4.32)
for n, m, and p subdivisions along each side and x -- Xc
~ = ~ ; a
~--
Y -- Yc
b
and ( =
z - - Zc
c
This element again is suggested by Zienkiewicz e t a l . 5 and elaborated upon by Argyris All the remarks about internal nodes and the properties of the formulation with mappings (to be described in the next chapter) are applicable here. The first three members of the three-dimensional Lagrange family are shown in Fig. 4.17(a). et al. 6
0
0 0
0
Fig. 4.16 A line element sandwiched between two-dimensional elements.
Rectangular prisms - 'serendipity' family
Fig. 4.17 Linear, quadratic and cubic right prisms with corresponding sheet and line elements. (Extra shading on 64-node element to show node location more clearly.)
For interelement continuity the simple rules given previously have to be modified. What is necessary to achieve such continuity is that along a whole face of an element the nodal values define a unique variation of the unknown function. It is obvious on a face that one of the l) will be unity and the remaining product defines the two-dimensional form given by (4.19), thus ensuring continuity.
The serendipity family of elements shown in Fig. 4.17(b) is precisely equivalent to that of Fig. 4.9 for the two-dimensional case. 4, 8,14 Using now three coordinates and otherwise following the terminology of Sec. 4.6 we have the following shape functions:
"Linear" element (8 nodes) Na
--
1 ~(1 + ~a~)(1 + r/at/)(1 + (a()
which is identical with the linear lagrangian element.
"Quadratic" element (20 nodes)
Comer nodes:
1 Na = ~(1 + ~a~)(1 + qaO)(1 + (a()(~a~ + qaq + (a( -- 2)
121
122
'Standard' and 'hierarchical' element shape functions
Typical mid-side node: ~a = 4-1
~a = 0
(a = +1
1 ~2 )(1 + ~Ta0)(1 + ~'a~) Na = ~ ( 1 -
"Cubic"elements (32 nodes)
Comer node:
1 (1 -+- ~a~)(1 + rlar])(1 -~- (a()[9(~ 2 + r]2 -~- ~2) _ 19] Na - -~ Typical mid-side node:
~a ---~'~1 Na - 9 ( 1 -
Oa = 4 - 1
(a - + 1
~2)(1 + 9~a~)(1 + 0a0)(1 + (a()
When (a( = (2 = 1 the above expressions reduce to those of Eqs (4.20)-(4.23c). Indeed such elements of three-dimensional type can be joined in a compatible manner to sheet or line elements of the appropriate type as shown in Fig. 4.17. Once again the procedure for generating the shape functions follows that described in Figs 4.10 and 4.11 and once again elements with varying degrees of freedom along the edges can be derived following the same steps. The equivalent of a Pascal triangle is now a tetrahedron and again we can observe the small number of surplus degrees of freedom- a situation of even greater magnitude than in two-dimensional analysis.
The tetrahedral family shown in Fig. 4.18 not surprisingly exhibits properties similar to those of the triangle family. First, once again complete polynomials in three coordinates are achieved at each stage. Second, as faces are divided in a manner identical with that of the previous triangles, the same order of polynomial in two coordinates in the plane of the face is achieved and element compatibility ensured. No surplus terms in the polynomial occur.
4.11.1 Volume coordinates Once again special coordinates are introduced defined by (Fig. 4.19): X = LlXl + L2x2 -]" L3x3 + Lnx4
y = LlYl + Ley2 + L3Y3 + L4Y4 Z = LlZl + L2Z2 + L3z3 + L4Z4
1 = L1 + Le + L3 + L4
Solving Eq. (4.33) gives Lk=
ak + bkx + ck y + dkz
6V
; k=1,2,3,4
(4.33)
Tetrahedral elements 123
Fig. 4.18 The tetrahedral family: (a)linear, (b) quadratic, (c) cubic.
with 1 1 6V = det 1 1
Xl x2 X3 x4
Yl Y2 Y3 Y4
Zl z2 Z3 z4
(4.34a)
in which, incidentally, the value V represents the volume of the tetrahedron. By expanding the other relevant determinants into their cofactors we have
al =
det
Cl = - d e t
ix2y2z21 x3 x4
x3 x4
Y3 y4
1 1
Z3
bl = - det
Z4
z3 z4
dl=
- det
[1 y2z2] 1 1
x3 x4
y3 y4
y3 y4
z3
Z4
11
(4.34b)
1 1
with the other constants defined by cyclic interchange of the subscripts in the order 1, 2, 3, 4. Again the physical nature of the coordinates can be identified as the ratio of volumes of tetrahedra based on an internal point P in the total volume, e.g., as shown in Fig. 4.19:
124 'Standard' and 'hierarchical' element shape functions volume P234 L1 -- volume 1234'
etc.
(4.35)
4.11.2 Shape functions As the volume coordinates vary linearly with the cartesian ones from unity at one node to zero at the opposite face then shape functions for the linear element [Fig. 4.18(a)] are simply Na = La a = 1, 2, 3, 4 (4.36) Formulae for shape functions of higher order tetrahedra are derived in precisely the same manner as for the triangles by establishing appropriate Lagrange-type formulae similar to Eq. (4.30). The reader may verify the following shape functions for the quadratic and cubic order cases.
"Quadratic" tetrahedron [Fig. 4.18(b)1
For comer nodes:
Na = (2Lo - 1)La
a=1,2,3,4
For mid-edge nodes: N5 - 4LIL2,
etc.
"Cubic" tetrahedron
Comer nodes:
N1-
Fig. 4.19 Volume coordinates.
1
~(3Lo - 1 ) ( 3 L o - 2)La
a = 1,2,3,4
Other simple three-dimensional elements 125 Mid-edge nodes:
N5 -- 9LIL2(3L1-
1),
etc.
Mid-face nodes: N17 -'-
27L1L2L3,
etc.
A useful integration formula may again be quoted here:
f /fvo 1LTL
L L 4
a y az =
a! b! c!+ ddl +
(a + b + c
3)!
6V
(4.37)
The possibilities of simple shapes in three dimensions are greater, for obvious reasons, than in two dimensions. A quite useful series of elements can, for instance, be based on triangular prisms (wedges) (Fig. 4.20). Here again variants of the product, Lagrange, approach or of the 'serendipity' type can be distinguished. The first element of both families, shown in Fig. 4.20(a), is identical and the shape functions are
Na --
1 =La(1 +(a() z
a=
1,2 . . . . . 6
For the 'quadratic' element illustrated in Fig. 4.20(b) the shape functions are Comer nodes
Na__l~La(2La-
1)(1 + ( a ( )
_ ~La(1 1 - ~2)
a=l,2
..... 6
Mid-edge of rectangle: N7 - L1 (1 - (2),
etc.
Mid-edge of triangles: N10 = 2LIL2(1 + (),
etc.
Such elements are not purely esoteric but have a practical application as 'fillers' in conjunction with 20-noded serendipity elements.
Part 2. Hierarchical shape functions
The general ideas of hierarchic approximation were introduced in Sec.4.2 in the context of simple, linear, elements. The idea of generating higher order hierarchic forms is again simple. We shall start from a one-dimensional expansion as this has been shown to provide a basis for the generation of two- and three-dimensional forms in previous sections. To generate a polynomial of order p along an element side we do not need to introduce nodes but can instead use parameters without any obvious physical meaning. We could use here a linear expansion specified by 'standard' functions N1 and N2 and add to this a
126 'Standard' and 'hierarchical' element shape functions
Fig. 4.20 Triangular prism elements (serendipity) family: (a)linear, (b) quadratic, (c) cubic. series of polynomials always designed so as to have zero values at the ends of the range (i.e., points 1 and 2). Thus for a quadratic approximation, we would write over the typical one-dimensional element, for instance, = Nlfil + g2u2 + N3u3
(4.38)
1 N2 = z(1 + ~) 2;
(4.39)
where 1 NI = z(1 - ~:) Z
N3 = (1 - ~2)
using in the above the normalized x coordinate [viz. Eq. (4.17)].
Hierarchic polynomials in one dimension
We note that the parameter fi3 does in fact have a meaning in this case as it is the magnitude of the departure from linearity of the approximation fi at the element centre, since N3 has been chosen here to have the value of unity at that point. In a similar manner, for a cubic element we simply have to add N4u4 tO the quadratic expansion of Eq. (4.39), where N4 is any cubic of the form (4.40)
N4 = (9/0 -+- ~IYl + ~:20t2 nI- ~:30g3
and which has zero values at ~ = 4-1 (i.e., at nodes 1 and 2). Again an infinity of choices exists, and we could select a cubic of a simple form which has a zero value at the centre of the element and for which dNn/d~ = 1 at the same point. Immediately we can write N4 = ~ ( 1
_~:2)
(4.41)
as the cubic function with the desired properties. Now the parameter U4 denotes the departure of the slope at the centre of the element from that of the linear approximation. We note that we could proceed in a similar manner and define the fourth-order hierarchical element shape function as N5 -- ~2(1 _~:2)
(4.42)
but a physical identification of the parameter associated with this now becomes more difficult (even though it is not strictly necessary). As we have already noted, the above set is not unique and many other possibilities exist. An alternative convenient form for the hierarchical functions is defined by 1 -~(~P - 1) p even
G+I (~)
//,
(4.43)
1 --7(~ p - ~) p odd p~
where p (> 2) is the degree of the introduced polynomial. 16 This yields the set of shape functions: 1
2
1
N3 = ~(~ - 1) N5 = 1 ( ~ 4 1)
N4 = ~ ( ~ 3 _ ~) 1 N 6 - 1-N(~ 5 - ~),
(4.44) etc.
We observe that all derivatives of Np+l of second or higher order have the value zero at = 0, apart from d p Np+l/d~p, which equals unity at that point, and hence, when shape functions of the form given by Eq. (4.44) are used, we can identify the parameters in the approximation as fip+l
=
d'~
~
=0
p > 2
(4.45)
This identification gives a general physical significance but is by no means necessary. In two- and three-dimensional elements a simple identification of the hierarchic parameters on interfaces will automatically ensure Co continuity of the approximation.
127
128 'Standard' and 'hierarchical' element shape functions
In deriving 'standard' finite element approximations we have shown that all shape functions for the Lagrange family could be obtained by a simple multiplication of one-dimensional ones and those for serendipity elements by a combination of such multiplications. The situation is even simpler for hierarchic elements. Here all the shape functions can be obtained by a simple multiplication process. Thus, for instance, in Fig. 4.21 we show the shape functions for a lagrangian nine-noded element and the corresponding hierarchical functions. The latter not only have simpler shapes but are more easily calculated, being simple products of linear and quadratic terms of Eq. (4.43) or (4.44). Using products of lagrangian polynomials the three functions illustrated are simply N1 -----(1 --~)(1 + r/)/4 N2 = (1 - ~ ) ( 1 - r/2)/2
(4.46)
N3 = (1 - ~2)(1 -- r] 2) The distinction between lagrangian and serendipity forms now disappears as for the latter in the present case the last shape function (N3) is simply omitted. Indeed, it is now easy to introduce interpolation for elements of the type illustrated in Fig. 4.11 in which a different expansion is used along different sides. This essential characteristic of hierarchical elements is exploited in adaptive refinement (viz. Chapter 14) where new degrees of freedom (or polynomial order increase) is made only when required by the magnitude of the error. A similar process clearly applies to the three-dimensional family of hierarchical bricktype elements.
Once again the concepts of multiplication can be introduced in terms of area or volume coordinates to define the triangle and tetrahedron family of elements. 15' 16 Starting from the linear shape functions for the comer nodes Na = Za
hierarchical functions for mid-side and interior nodes can be added. For the triangle shown in Fig. 4.14 we note that along the side 1-2, L3 is identically zero, and therefore we have (L1 + L2)1-2 = 1 (4.47) If ~, measured along side 1-2, is the usual non-dimensional local element coordinate of the type we have used in deriving hierarchical functions for one-dimensional elements, we can write 1 1 Ll11-2 -- ~(1 - ~) L211-2= ~(1 + ~) (4.48) from which it follows that we have = (L2 - L1)l-2
(4.49)
Triangle and tetrahedron family
:1 ~o3 1
:1 1
o
I
~,~%:=.L/o/"
.,~'.e
~ I x . , ".t. - ~ ,
'
-
. T r L.,< ,.,7".. L.,/.. / (a) Standard
(b) Hierarchical
Fig. 4.21 Standard and hierarchical shape functions corresponding to a lagrangian quadratic element.
This suggests that we could generate hierarchical shape functions over the triangle by generalizing the one-dimensional shape function forms produced earlier. For example, using the expressions of Eq. (4.43), we associate with the side 1-2 the polynomial of degree p (> 2) defined by Up(l-2) --
&[(L2 /]! ~.v [(L2 -
L1) p -
(L1 + L2) p]
p even
L1) p -
(L2 - L1)(L1 + L2) p-l]
p odd
(4.50)
It follows from Eq. (4.48) that these shape functions are zero at nodes 1 and 2. In addition, it can easily be shown that Np(1-2) will be zero all along the sides 3-1 and 3-2 of the triangle, and so Co continuity of the approximation fi is assured.
129
130 'Standard' and 'hierarchical' element shape functions It should be noted that in this case for p > 3 the number of hierarchical functions arising from the element sides in this manner is insufficient to define a complete polynomial of degree p, and internal hierarchical functions, which are identically zero on the boundaries, need to be introduced; for example, for p = 3 the function L1LeL3could be used, while for p - 4 the three additional functions L~L2L3,L1L~L3,LILzL~ could be adopted. In Fig. 4.22 typical, hierarchical, linear, quadratic, and cubic trial functions for a triangular element are shown. Identical procedures are obvious in the context of tetrahedra. Hierarchical functions of other forms can be found in reference 23.
We have already mentioned that hierarchic element forms give a much improved equation conditioning for steady-state (static) problems due to their form which is more nearly diagonal. In Fig. 4.23 we show the 'condition number' (which is a measure of such diagonality and is defined in standard texts on linear algebra; see Appendix A) for a single cubic element and for an assembly of four cubic elements, using standard and hierarchic forms in their
Fig. 4.22 Triangular elements and associated hierarchical shape functions of (a)linear, (b) quadratic, (c) cubic form.
Global and local finite element approximation Single element (Reduction of condition number = 10.7)
|
o
o
o
o
|
II
'11
~max/~min = 390
~max/~nin = 36
Four element assembly (Reduction of condition number = 13.2)
|
|
II
~,max/~min = 124
~max/~min- 1643
Cubic order elements ( ~ Standard shape function ( ~ Hierarchic shape function Fig. 4.23 Improvement of condition number (ratio of maximum to minimum eigenvalue of the stiffness matrix) by use of hierarchical form (isotropic elasticity, v = O.15).
formulation. The improvement of the conditioning is a distinct advantage of such forms and allows the use of iterative solution techniques to be more easily adopted. 17 Unfortunately much of this advantage disappears for transient analysis as the approximation must contain specific modes (see Chapter 16).
The very concept of hierarchic approximations (in which the shape functions are not affected by the refinement) means that it is possible to include in the expansion
U -- ~
Nabta
(4.51)
a=l
where functions N are not local in nature. Such functions may, for instance, be the exact solutions of an analytical problem which in some way resembles the problem dealt with, but do not satisfy some boundary or inhomogeneity conditions. The 'finite element', local, expansions would here be a device for correcting this solution to satisfy the real conditions,
131
132
'Standard' and 'hierarchical' element shape functions
,
(a)
/ / / /
r
(b)
/ / / / / / /
Fig. 4.24 Some possible uses of global-local approximation. (a) Rotating slotted disc. (b) Perforated beam. This use of the global-local approximation was first suggested by Mote TM in a problem where the coefficients of this function were fixed. The example involved here is that of a rotating disc with cutouts (Fig. 4.24). The global known solution is the analytical one corresponding to a disc without cutout, and finite elements are added locally to modify the solution. Other examples of such 'fixed' solutions may well be those associated with point loads, where the use of the global approximation serves to eliminate the singularity modelled badly by the discretization. In some problems the singularity itself is unknown and the appropriate function can be added with an unknown coefficient. Some aspects of this are mentioned in Chapter 15 and, for waves, in the context of fluid dynamics, in reference 19.
Internal nodes or nodeless internal parameters yield in the usual way the element properties ~I-Ie = Ke~e _+_fe a~ e
(4.52)
As ~e can be subdivided into parts which are common with other elements, ~ , and others which occur in the particular element only, ~ , we can immediately write aI-I
aI-I e
=0
Elimination of internal parameters before assembly- substructures 133 and eliminate fi~ from further consideration. Writing Eq. (4.52) in a partitioned form we have On e
O~le :
0I'I e --
1 / } /ff} _ "K~I K~2 fi~ + .K~.1 K~.2J ul l, f~
0U~
0 r[ e
N
=
{01"Ie}
Ofi--~-I
(4.53)
0
From the second set of equations given above we can write
U2 -- -- (K22) ~e
which on substitution yields
e
8I-1 e
-1
(K21u 1 -t- f~) e
~e
- e ~e : KllUl -~- fl- e
(4.54)
(4.55)
in which - e e e -1K~ Kll -- Klle - K12(K22) 1
f~ = f~ -- K~2(K~2)-lf~
(4.56)
This process of partial solution is also known in the literature as 'static condensation' .20 Assembly of the total region then follows, by considering only the element boundary variables, thus giving a saving in the equation-solving effort at the expense of a few additional manipulations carded out at the element stage. 2~ Perhaps a structural interpretation of this elimination is desirable. What in fact is involved is the separation of a part of the structure from its surroundings and determination of its solution separately for any prescribed displacements at the interconnecting boundaries. I~ *e is now simply the overall stiffness of the separated structure and f,e the equivalent set of nodal forces. If the triangulation of Fig. 4.25 is interpreted as an assembly of pin-jointed bars the reader will recognize immediately the well-known device of 'substructures' used frequently in structural engineering. Such a substructure is in fact simply a complex element from which the internal degrees of freedom have been eliminated. Immediately a new possibility for devising more elaborate, and presumably more accurate, elements is presented. Figure 4.25(a) can be interpreted as a continuum field subdivided into linear triangular elements. The substructure results in fact in one complex element shown in Fig. 4.25(b) with a number of boundary nodes. The only difference from elements derived in previous sections is the fact that the unknown u now is not approximated internally by one set of smooth shape functions but by a series of piecewise approximations. This presumably results in a slightly poorer approximation but an economic advantage may arise if the total computation time for such an assembly is saved. Substructuring is an important device in complex problems, particularly where a repetition of complicated components arises. In simple, small-scale finite element analysis, much improved use of simple triangular elements was found by the use of simple subassemblies of the triangles (or indeed tetrahedra). For instance, a quadrilateral based on four triangles from which the central node is eliminated was found to give an economic advantage over direct use of simple triangles (Fig. 4.26). This and other subassemblies based on triangles are discussed by Doherty e t al. 21 and used by Nagtegaal e t al. 22 and others.
134 'Standard' and 'hierarchical' element shape functions
I/._ i
,I (a)
I
,o
D
~3
1
i
'_....... ~._". . . . . _)'_...... ;4 ...... T
(b)
Fig. 4.25 Substructureof a complexelement.
An unlimited selection of element types has been presented here to the reader- and indeed equally unlimited alternative possibilities exist. 4' 8 What is the use of such complex elements in practice? As presented so far the triangular and tetrahedral elements are limited to situations where the real region is of a suitable shape which can be represented as an assembly of fiat facets and all other elements are limited to situations represented by an assembly of fight prisms. Such a limitation would be so severe that little practical purpose would have been served by the derivation of such shape functions unless some way could be found of distorting these elements to fit realistic curved boundaries. In fact, methods for doing this are available and will be described in the next chapter.
4.1 Develop an explicit form of the standard shape functions at nodes 1, 3 and 6 for the element shown in Fig. 4.27(a). Using a Pascal triangle in ~ and r/show the polynomials included in the element. 4.2 Develop an explicit form of the standard shape functions at nodes 2, 3 and 9 for the element shown in Fig. 4.27(b). Using a Pascal triangle in ~ and 17show the polynomials included in the element. 4.3 Develop an explicit form of the standard shape functions at nodes 1, 2 and 5 for the element shown in Fig. 4.27(c). Using a Pascal triangle in ~ and r/show the polynomials included in the element.
Fig. 4.26 A composite quadrilateral made from four simple triangles.
Problems 3
4
4
9
8
1
#
= 5
~6 8
(a)
=
2
3
~
v
@
=
5
3
7 I -
9o
1
4
v
80
(b)
=
2
=
1
=
5
(c)
_
2
Fig. 4.27 Quadrilateral element for Problems 4.1 to 4.4.
4.4 Develop an explicit expression in hierarchical form for all nodes of the element shown in Fig. 4.27(c). 4.5 Develop an explicit form of the standard shape functions at nodes 1, 2 and 5 for the element shown in Fig. 4.28(a). Using a Pascal triangle in ~ and 0 show the polynomials included in the element. 4.6 Develop an explicit form of the standard shape functions at nodes 1, 5 and 7 for the element shown in Fig. 4.28(b). Using a Pascal triangle in ~ and r/show the polynomials included in the element. 4.7 The mesh for a problem contains an 8-node quadratic serendipity rectangle adjacent to a 6-node quadratic triangle as shown in Fig. 4.29. Show that the coordinates computed from each element satisfy C Ocontinuity along the edge 3-7-11. 4.8 Determine an explicit expression for the shape function of node 1 of the linear triangular prism shown in Fig. 4.20(a). 4.9 Determine an explicit expression for the hierarchical shape function of nodes 1, 7 and 10 of the quadratic triangular prism shown in Fig. 4.20(b). 4.10 Determine an explicit expression for the shape function of nodes 1, 7, 13 and 25 of the cubic triangular prism shown in Fig. 4.20(c). 4.11 On a sketch show the location of the nodes for the quartic member of the tetrahedron family. Construct an explicit expression for the shape function of the vertex node located at (L1, L2, L3, L4) -- (1, 0, 0, 0) and the mid-edge node located at (0.25, 0.75, 0, 0). 4.12 On a sketch show the location of the nodes for the quartic member of the serendipity family. Construct an explicit expression for the shape function of the vertex node located at (~, 0, ~) = (1, 1, 1) and the mid-edge node located at (0.75, 1, 1). 4.13 On a sketch show the location of the nodes for the quartic member of the triangular prism family shown in Fig. 4.20. Construct an explicit expression for the hierarchical shape function of a vertex node, an edge node of a triangular face and an edge node of a rectangular face. 4.14 On a sketch show the location of the nodes for the quadratic member of the triangular prism family in which lagrangian interpolation is used on rectangular faces (see Fig. 4.20). Construct an explicit expression for the shape function of a vertex node, an edge node of a triangular face and an edge node of a rectangular face.
135
136
'Standard' and 'hierarchical' element shape functions
7
3 4
7o
-
=
1
5
=
=
t)'a" 6
2 1
9
(b)
v
6
2
Fig. 4.28 Quadrilateral element for Problems 4.5 and 4.6.
4 cm
9
10
I,
,qw
7q
6
q~
1 Jig 'qW
11
2
3 ~
8
4
i W
6 cm
3 cm
Fig. 4.29 Quadratic rectangle and triangle for Problem 4.7.
4.15 On a sketch show the location of the nodes for the cubic member of the triangular prism family in which lagrangian interpolation is used on rectangular faces (see Fig. 4.20). Construct an explicit expression for the shape function of a vertex node, an edge node of a triangular face, an edge node of a rectangular face, a mid-face node of a triangular face, a mid-face node of a rectangular face, and for any internal nodes.
1. W. Rudin. Principles of Mathematical Analysis. McGraw-Hill, 3rd edition, 1976. 2. P.C. Dunne. Complete polynomial displacement fields for finite element methods. Trans. Roy. Aero. Soc., 72:245, 1968. 3. B.M. Irons, J.G. Ergatoudis, and O.C. Zienkiewicz. Comments on 'complete polynomial displacement fields for finite element method' (by P.C. Dunne). Trans. Roy. Aeronaut. Soc., 72:709, 1968. 4. J.G. Ergatoudis, B.M. Irons, and O.C. Zienkiewicz. Curved, isoparametric, 'quadrilateral' elements for finite element analysis. Int. J. Solids Struct, 4:31-42, 1968.
References 137 5. O.C. Zienkiewicz, B.M. Irons, J.G. Ergatoudis, S. Ahmad, and EC. Scott. Isoparametric and associated elements families for two and three dimensional analysis. In Finite Element Methods in Stress Analysis, Chapter 13. Tapir Press, Trondheim, 1969. 6. J.H. Argyris, K.E. Buck, H.M. Hilber, G. Mareczek, and D.W. Scharpf. Some new elements for matrix displacement methods. In Proc. 2nd Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-68-150, Wright Patterson Air Force Base, Ohio, Oct. 1968. 7. R.L. Taylor. On completeness of shape functions for finite element analysis. Int. J. Numer. Meth. Eng., 4:17-22, 1972. 8. O.C. Zienkiewicz, B.M. Irons, J. Campbell, and EC. Scott. Three dimensional stress analysis. In IUTAM Symposium on High Speed Computing in Elasticity, Li'ege, 1970. 9. EC. Scott. A quartic, two dimensional isoparametric element. Undergraduate Project, University of Wales, 1968. 10. J.H. Argyris, I. Fried, and D.W. Scharpf. The TET 20 and TEA 8 elements for the matrix displacement method. Aero. J., 72:618-625, 1968. 11. P. Silvester. Higher order polynomial triangular finite elements for potential problems. Int. J. Eng. Sci., 7:849-861, 1969. 12. B. Fraeijs de Veubeke. Displacement and equilibrium models in finite element method. In O.C. Zienkiewicz and G.S. Holister, editors, Stress Analysis, Chapter 9, pages 145-197. John Wiley & Sons, Chichester, 1965. 13. J.H. Argyris. Triangular elements with linearly varying strain for the matrix displacement method. J. Roy. Aero. Soc. Tech. Note, 69:711-713, 1965. 14. J.G. Ergatoudis, B.M. Irons, and O.C. Zienkiewicz. Three dimensional analysis of arch dams and their foundations. In Proc. Symp. Arch Dams, Inst. Civ. Eng., London, 1968. 15. A.G. Peano. Hierarchics of conforming finite elements for elasticity and plate bending. Comp. Math. and Applications, 2:3-4, 1976. 16. J.E de S.R. Gago. A posteri error analysis and adaptivity for the finite element method. Ph.D. thesis, Department of Civil Engineering, University of Wales, Swansea, 1982. 17. O.C. Zienkiewicz, J.E De S.R. Gago, and D.W. Kelly. The hierarchical concept in finite element analysis. Comp. Struct., 16:53-65, 1983. 18. C.D. Mote. Global-local finite element. Int. J. Numer. Meth. Eng., 3:565-574, 1971. 19. O.C. Zienkiewicz, R.L. Taylor, and E Nithiarasu. The Finite Element Method for Fluid Dynamics. Butterworth-Heinemann, Oxford, 6th edition, 2005. 20. E.L. Wilson. The static condensation algorithm. Int. J. Numer. Meth. Eng., 8:199-203, 1974. 21. W.E Doherty, E.L. Wilson, and R.L. Taylor. Stress analysis of axisymmetric solids utilizing higher-order quadrilateral finite elements. Technical Report 69-3, Structural Engineering Laboratory, Univ. of California, Berkeley, Jan. 1969. 22. J.C. Nagtegaal, D.M. Parks, and J.R. Rice. On numerical accurate finite element solutions in the fully plastic range. Comp. Meth. Appl. Mech. Eng., 4:153-177, 1974. 23. S.J. Sherwin and G.E. Karniadakis. A new triangular and tetrahedral basis for high-order (hp) finite element methods. Int. J. Numer. Meth. Eng. 38:3775-3802, 1995.
In the previous chapter we have shown how some general families of finite elements can be obtained for Co interpolations. A progressively increasing number of nodes and hence improved accuracy characterizes each new member of the family and presumably the number of such elements required to obtain an adequate solution decreases rapidly. To ensure that a small number of elements can represent a relatively complex form of the type that is liable to occur in real, rather than academic, problems, simple rectangles and triangles no longer suffice. This chapter is therefore concerned with the subject of distorting such simple forms into others of more arbitrary shape. Elements of the basic one-, two-, or three-dimensional types will be 'mapped' into distorted forms in the manner indicated in Figs 5.1 and 5.2. In these figures it is shown that the 'parent' ~, 77, (, or L1, L2, L3, L4 coordinates can be distorted to a new, curvilinear set when plotted in cartesian x, y, z space. Not only can two-dimensional elements be distorted into others in two dimensions but the mapping of these can be taken into three dimensions as indicated by the flat sheet elements of Fig. 5.2 distorting into a three-dimensional space. This principle applies generally, providing a one-to-one correspondence between cartesian and curvilinear coordinates can be established, i.e., once the mapping relations of the type
x)
fx(~, 17, ()
YI = z
fy(~, 77, () fz(~, rl, ()
fx(L1, L2, L3, L4) or
fy(L1, L2, L3, L4) fz(L1, L2, L3, L4)
(5.1)
can be established. Once such coordinate relationships are known, shape functions can be specified in local (parent) coordinates and by suitable transformations the element properties established in the global coordinate system.
Use of 'shape functions' in the establishment of coordinate transformations
44-+-
/-~,-~/*l*
L2=O
3
L2 = 0
3 L1 = 0
1
J
L1 = 0
L 2
1
Local coordinates
2 L3=O
Cartesian map
x
Fig. 5.1 Two-dimensional 'mapping' of some elements.
In what follows we shall first discuss the so-called isoparametric form of relationship (5.1) which has found a great deal of practical application. Full details of this formulation will be given, including the establishment of element matrices by numerical integration. In later sections we shall show that many other coordinate transformations also can be used effectively.
Parametric curvilinear coordinates ....5.2 Use of 'shape functions'in the establishment of coordinate transformations A most convenient method of establishing the coordinate transformations is to use the 'standard' type of Co shape functions we have already derived to represent the variation of the unknown function.
139
140 Mappedelements and numerical integration
=
=1
1]
71'
1] "---1
4
2 2
Local coordinates
Cartesian map
x
Fig. 5.2 Three-dimensional'mapping' of some elements.
If we write, for instance, for each element
' x = N lxl @ N~x2 + . . . .
,
/Xl/ N'
,
Y = N l Yl + N~Y2 + "
Z -- N~Zl -q- N~Z2 Jr- . . . .
N'
x2
= N'x
/Zl/ Y2
-- W y
z2
-
(5.2)
N'z
in which N' are standard shape functions given in terms of the local (parent) coordinates, then a relationship of the required form is immediately available 9 Further, the points with coordinates Xl, yl, Zl, etc., will lie at appropriate points of the element boundary or interior (as from the general definitions of the standard shape functions we know that these have a value of unity at the point in question and zero elsewhere). These points can establish nodes a p r i o r i . To each set of local coordinates there will correspond a set of global cartesian coordinates and in general only one such set. We shall see, however, that a non-uniqueness may arise if the nodal coordinates are placed such that a violent distortion occurs.
Use of 'shape functions' in the establishment of coordinate transformations
The concept of using such element shape functions for establishing curvilinear coordinates in the context of finite element analysis appears to have been introduced first by Taig. 1 In his first application basic linear quadrilateral relations were used. Irons generalized the idea for other elements. 2' 3 Quite independently the exercises of devising various practical methods of generating curved surfaces for purposes of engineering design led to the establishment of similar definitions by Coons 4 and Forrest, 5 and indeed today the subjects of surface definitions and analysis are drawing closer together due to this activity. In Fig. 5.3 an actual distortion of elements based on the quadratic and cubic members of the two-dimensional 'serendipity' family is shown. It is seen here that a one-to-one relationship exists between the local (~, 7) and global (x, y) coordinates. If the fixed (nodal) points are such that a violent distortion occurs then a non-uniqueness can occur in the manner indicated for two situations in Fig. 5.4. Here at internal points of the distorted element two or more local coordinates correspond to the same cartesian coordinate and in addition to some internal points being mapped outside the element. Care must be taken in practice to avoid such gross distortion. Figure 5.5 shows two examples of a two-dimensional (~, r/) element mapped into a three-dimensional (x, y, z) space. We shall often refer to the basic element in undistorted, local, coordinates as a 'parent' element. In Sec. 5.5 we shall define a quantity known as thejacobian determinant. The well-known condition for a o n e - t o - o n e mapping (such as exists in Fig. 5.3 and does not in Fig. 5.4) is that the sign of this quantity should remain unchanged at all the points of the mapped element. It can be shown that with a parametric transformation based on bilinear shape functions, the necessary condition is that no internal angle [such as ot in Fig. 5.6(a)] be equal or greater than 180~ 6 In transformations based on quadratic 'serendipity' or 'lagrangian' functions,
11
l
1
~=-1
Fig. 5.3 Plots of curvilinear coordinates for quadratic and cubic elements (reasonable distortion).
141
142
Mapped elements and numerical integration
Fig. 5.4 Unreasonableelement distortion leading to non-unique mapping and 'overspill'. Quadraticand cubic elements.
a
f-- t<" ~t ~..< ~ . . . y ~,~ "~ ~.~/ -~/ --/...
i <" < x / < . - ' > " "
'<"
i
,.,./
'<
\/
~x
--.~
"~
<
x/x/
">
>'
.../
x/x/-,,,1/'"
~/
\
~
\/
\
/
\~
.
Y~
.Z
\ /\.
/\
/
--.~/
(. > ... >1
x \< .~,~ 1
Fig. 5.5 Flat elements (of quadratic type) mapped into three dimensions. it is necessary in addition to this requirement to ensure that the mid-side nodes are in the 'middle half' of the distance between adjacent comers but a 'middle third' shown in Fig. 5.6 is safer. For cubic functions such general rules are impractical and numerical checks on the sign of the jacobian determinant are necessary. In practice a quadratic distortion is usually sufficient.
Geometrical conformity of elements
,
> o~< 180 ~
(a) Linear element o~ < 180 ~
0
0
(b) Quadratic element
zone
for midpoint Fig. 5.6 Rules for uniqueness of mapping (a) and (b).
While it was shown that by the use of the shape function transformation each parent element maps uniquely a part of the real object, it is important that the subdivision of this into the new, curved, elements should leave no gaps. The possibility of such gaps is indicated by dotted lines in Fig. 5.7. Theorem 1. If two adjacent elements are generated from 'parents' in which the shape
functions satisfy Co continuity requirements then the distorted elements will be continuous (compatible). This statement is obvious, as in such cases uniqueness of any function u required by continuity is simply replaced by that of uniqueness of the x, y, or z coordinate. As adjacent elements are given the same sets of coordinates at nodes, continuity is implied.
With the shape of the element now defined by the shape functions N' the variation of the unknown, u, has to be specified before we can establish element properties. This is most conveniently given in terms of local, curvilinear coordinates by the usual expression u = Nfi e where fie lists the nodal values.
(5.3)
143
144 Mappedelements and numerical integration
)
()
) ~-0---()
(a)
(b)
Fig. 5.7 Compatibility requirements in a real subdivision of space.
(a)
(b)
(c) Fig. 5.8 Various element specifications: o point at which coordinate specified; [] point at which function parameter specified. (a)Isoparametric. (b) Superparametric. (c) Subparametric. T h e o r e m 2. If the shape functions N used in (5.3) are such that Co continuity of u is preserved in the parent coordinates then Co continuity requirements will be satisfied in distorted elements. The proof of this statement follows the same lines as that in the previous section. The nodal values may or may not be associated with the same nodes as used to specify the element geometry. For example, in Fig. 5.8 the points marked with a circle are used to define the element geometry. We could use the values of the function defined at nodes marked with a square to define the variation of the unknown.
Evaluation of element matrices 145
In Fig. 5.8(a) the same points define the geometry and the finite element analysis points. If then N = N' (5.4) i.e., the shape functions defining the geometry and the function are the same, the elements will be called isoparametric. We could, however, use only the four corner points to define the variation ofu [Fig. 5.8(b)]. We shall refer to such an element as superparametric, noting that the variation of geometry is more general than that of the actual unknown. Similarly, if for instance we introduce more nodes to define u than are used to define the geometry, subparametric elements will result [Fig. 5.8(c)]. While for mapping it is convenient to use 'standard' forms of shape functions the interpolation of the unknown can, of course, use hierarchic forms defined in the previous chapter. Once again the definitions of sub- and superparametric variations are applicable.
Transformations
To perform finite element analysis the matrices defining element properties, e.g., stiffness, etc., have to be found. These will be of the form f
G dr2
(5.5)
in which the matrix G depends on N or its derivatives with respect to global coordinates. As an example of this we have the stiffness matrix K = f~ B'rDB dr2
(5.6a)
f = 3f~ NTb dr2
(5.6b)
and associated body force vectors
For elastic problems the matrix for B is given explicitly by components [see the general form of Eq. (2.15)]. For plane stress problems we have
-~Na Ox '
ONa 8y
O,
Ba "-
ONa .
Oy
ONa '
~X
_
(5.7)
146 Mappedelements and numerical integration In elasticity problems the matrix G is thus a function of the first derivatives of N and this situation will arise in many other classes of problems. In all, Co continuity is needed and, as we have already noted, this is readily satisfied by the functions of Chapter 4, written now in terms of curvilinear coordinates. To evaluate such matrices we note that two transformations are necessary. In the first place, as Na is defined in terms of local (curvilinear) coordinates, it is necessary to devise some means of expressing the global derivatives of the type occurring in Eq. (5.7) in terms of local derivatives. In the second place the element of volume (or surface) over which the integration has to be carded out needs to be expressed in terms of the local coordinates with an appropriate change in limits of integration.
5.5.1 Computation of global derivatives Consider, for instance, the set of local coordinates ~, r/, ( and a corresponding set of global coordinates x, y, z. By the usual rules of partial differentiation we can write, for instance, the ~ derivative as
ONa 0~
=
ONa Ox 0x a~
~
ONa Oy ay 0~
~
ONa OZ
(5.8)
0z a~
Performing the same differentiation with respect to the other two coordinates and writing in matrix form we have
ONa
Ox
Oy
Oz
ONa
ONa
Ox
ay
Oz
ONa
ONa
Ox
Oy
Oz
ONa
W
ff'
ONa
=s
ONa
-fly
(5.9)
ONa
In the above, the left-hand side can be evaluated as the functions Na are specified in local coordinates. Further, as x, y, z are explicitly given by the relation defining the curvilinear coordinates [Eq. (5.2)], the matrix J can be found explicitly in terms of the local coordinates. The array J is known as the jacobian matrix for the transformation. To find now the global derivatives we invert J and write
aj
a
01
/~a
a
a
__
j-1
~a
(5.10)
/~a
In terms of the shape function defining the coordinate transformation N' (which as we have seen are only identical with the shape functions N when the isoparametric formulation is used) we have
Evaluation of element matrices 147
J
~_
2o-ON"Xa, o-ON'yo, 2a-ON"Zo ON; ON; ON; EaWXa, 2oWYa, 2aWZa ON;
Xa,
ON;
ON;
Yo,
ON[
o~' ON~ Or/ '
X1,
Or#
Za
i
,
Yl,
Zl
Y2,
Z2
9
~
(5.11) For two-dimensional problems we drop all the terms containing z and/or ( in Eqs (5.8) to (5.11).
5.5.2 Volume integrals To transform the variables and the domain with respect to which the integration is made, a standard process will be used which involves the determinant of J. Thus, for instance, a volume element becomes dx dy dz = J (~, r/, () d~ dr/d(
(5.12)
where J (~, r/, () = det J. This type of transformation is valid irrespective of the number of coordinates used. For its justification the reader is referred to standard mathematical texts.t (See also Appendix F.) Assuming that the inverse of J can be found we now have reduced the evaluation of the element properties to that of finding integrals of the form of Eq. (5.5). More explicitly we can write this as L
G(x, y, z) dr2 =
/1/1/1 1
1
G(~, r/, () J(se, 17, () d~" dr/d(
(5.13)
1
where G(x, y, z) = G(x(~, 17, (), y(~, r/, (), z(~, r/, ()) = G(~, 17, () and the curvilinear coordinates are of the normalized type based on the right prism. Indeed the integration is carried out within such a prism and not in the complicated distorted shape, thus accounting for the simple integration limits. One- and two-dimensional problems will similarly result in integrals with respect to one or two coordinates within simple limits. While the limits of integration are simple in the above case, unfortunately the explicit form of G is not. Apart from the simplest elements, algebraic integration usually defies our mathematical skill, and numerical integration has to be used. This, as will be seen from later sections, is not a severe penalty and has the advantage that algebraic errors are more easily avoided and that general programs, not tied to a particular element, can be written for various classes of problems. t The determinant of the jacobian matrix is known in the literature simply as 'the jacobian' and is often written as
J(~, ~, ~) -
0(x, y, z)
148
Mapped elements and numerical integration
5.5.3 Surface integrals In elasticity and other applications, surface integrals frequently occur. Typical here are the expressions for evaluating the contributions of surface tractions [see Chapter 2, Eq. (2.40b)]" f = -- f r NTi d F The element dF will generally lie on a surface where one of the coordinates (say () is constant. The most convenient process of dealing with the above is to consider dA as a vector oriented in the direction normal to the surface (see Appendix F). For three-dimensional problems we form the vector product
OJi ndA =dA=
~
OX ' •
i
Oi,
~
oy
dedr/
Oz
and on substitution integrate within the domain - 1 _< ~, r/_< 1. For two dimensions a line length dS arises and here the magnitude is simply
Ox ndF=dF-
Oy
-~ 0
Oy x
0 1
d~ -
Ox
d~
- -~ 0
on constant r/surfaces. This may now be reduced to two components for the two-dimensional problem.
The general relationship (5.2) for coordinate mapping and indeed all the subsequent statements are equally valid for any set of local coordinates and could relate the local L 1, L2 . . . . coordinates used for triangles and tetrahedra in the previous chapter, to the global cartesian ones. Indeed most of the discussion of the previous sections is valid if we simply rename the local coordinates suitably. However, two important differences arise. The first concerns the fact that the local coordinates are not independent and in fact number one more than the cartesian system. The matrix J would apparently therefore become rectangular and would not possess an inverse. The second is simply the difference of integration limits which have to correspond with a triangular or tetrahedral 'parent' element.
Evaluation of element matrices. Transformation in area and volume coordinates 149
A simple, though perhaps not the most elegant, way out of the first difficulty is to consider one variable as a dependent one. Thus, for example, we can introduce formally, in the case of the tetrahedra, ~--L1 r/=L2
(5.14)
( =L3 1-~-r/-(
=L4
(by definition in the previous chapter) and thus preserve without change Eq. (5.8) and all the equations up to Eq. (5.12). As the functions N a are given in terms of L1, L2, etc., we must observe that ONa
0~
=
ONa OL1
0L1 a~
t
ONa OL2
0L2 0~
t
ONa OL3
0L3 0~
~
ONa OL4
0L4 0~
(5.15)
On using Eq. (5.14) this becomes simply
aN~
aNo
aN~
O~
OL1
OL4
with the other derivatives obtainable by similar expressions. The integration limits of Eq. (5.13) now change, however, to correspond with the tetrahedron limits, typically
s
= j0'/0'
CJ(~:, r/, ( ) d ~ dr/d(
(5.16)
The same procedure will clearly apply in the case of triangular coordinates. It must be noted that once again the expression G will necessitate numerical integration which, however, is carried out over the simple, undistorted, parent region whether this be triangular or tetrahedral. An alternative to the above is to express the coordinates and constraint as rx - - x - Xl N~ - x z N ~ - x 3 N ~ . . . . . ry = y - y l N ~ - yEN~ - y 3 N ~ . . . . rz -- z rl -
-
Zl
U~ - z 2 N ~ - z 3 N ~ . . . . .
0 = 0
(5.17)
0
1 - L1 - L 2 - L 3 - L 4 = 0
where N a' = N a'(L1, L2, L3, L4), etc. Now derivatives of the above with respect to x, y
150 Mappedelements and numerical integration and z may be written directly as
"Orx Ox
Orx Oy
Orx Oz
Ox Orz Ox
Oy Orz Oy
Oz Orz Oz
Orl
Orl
Orl
Ox
Oy
Oz
a~
1
Ory Ory Ory
.
0
0
1 - . - -
m
Ea Xa-~2
o~
a~
Ea Ya~ 1 Ea Ya-~2 a~
0 0
a~
a~
Ea Xa-~3 o~
Ea Ya-~3 o~
a~-
Ea Xa-~4 o~
Ea ~-'~4 a~
Ea Za-~l
Ea Za-~2
Ea Za-~3
Ea Za-'~4
1
1
1
1
OLl
OL1
OL1-
Ox
Oy
Oz
l
a~
Ea Xa-~l
OL2 OL2 OL2
I -~x
Oy
Oz
x OL3 OL3 OL3
I
-fix
Oy
=0
Oz
OL4 OL4 OL4 k ~x Oy Oz.
(5.18)
The above may be solved for the partial derivatives of La with respect to the x, y, z coordinates and used directly with the chain rule written as
ONa ONa OL1 ONa OL2 ONa OL3 ONa OL4 = ! ~ ~ Ox OL10x OL2 0X OL3 0x OL4 0x
(5.19)
The above has advantages when the coordinates are written using mapping functions as the computation can still be more easily carded out. Also, the calculation of integrals will normally be performed numerically (as described in Sec. 5.11) where the points for integration are defined directly in terms of the volume coordinates. Finally it should be remarked that any of the elements given in the previous chapter are capable of being mapped. In some, such as the triangular prism, both area and rectangular coordinates are used (Fig. 5.9). The remarks regarding the dependence of coordinates apply once again with regard to the former but the processes of the present section should make procedures clear.
Fig. 5.9 A distorted triangular prism.
Order of convergence for mapped elements
If the shape functions are chosen in curvilinear coordinate space so as to observe the usual rules of convergence (continuity and presence of complete first-order polynomials in these coordinates), then convergence will occur. In the case of isoparametric (or subparametric) elements a complete linear field is always reproduced (i.e., 1, x, y) by the curvilinear coordinate expansion, and thus the lowest order patch test will be passed in the standard manner on such elements. The proof of this is simple. Consider a standard isoparametric expansion U "-- ~
Nail a ~ Nfi
N = N(~, 0, ~')
(5.20)
a=l
with coordinates of nodes defining the transformation as x--
y~gax
a
y---~-~Nay
a
z=~gaza
(5.21)
The question is under what circumstances is it possible for expression (5.20) to define a linear expansion in cartesian coordinates: U = 0/1 + O/2X -'['-ot3y -+- O~4Z
~ Otl -[-ot2 ~"~ Naxa +Ot3 y ~ NaYa .-[-.ot4 ~
NaZa
(5.22)
If we take bta --" Oll "[- Ol2X a "[- o[3 y a "Jr"Ol4 Za
and compare expression (5.20) with (5.22) we note that identity is obtained between these providing ~-~Na = 1 As this is the usual requirement of standard element shape functions [see Eq. (4.4)] we can conclude that the following theorem is valid. Theorem 3. The constant derivative condition will be satisfied for all isoparametric elements. As subparametric elements can always be expressed as specific cases of an isoparametric transformation this theorem is obviously valid here also. It is of interest to pursue the argument and to see under what circumstances higher polynomial expansions in cartesian coordinates can be achieved under various transformations. The simple linear case in which we 'guessed' the solution has now to be replaced by considering in detail the polynomial terms occurring in expressions such as (5.20) and (5.22) and establishing conditions for equating appropriate coefficients. Consider a specific problem: the circumstances under which the bilinear mapped quadrilateral of Fig. 5.10 can fully represent any quadratic cartesian expansion. We now have 4
X --
Z1 '
Nax a
4
y --
Z1 NtaYa
(5.23)
151
152 Mappedelements and numerical integration and we wish to be able to reproduce U = O/1 -]--O/2X + ot3y + 0/4x2 -+- otsxy + ot6y 2
(5.24)
Noting that the bilinear form of N a' contains terms such as 1, ~,/7 and ~/7, the above can be written as U = /71 q- /72~ "1- /73/7 -Jr"/74~2 + /75~/7 + /76/72 + /77~/72 + /78~2/7 + /79~2/72
(5.25)
where/71 to/79 depend on the values of or1 to or6. We shall now try to match the terms arising from the quadratic expansions of the serendipity kind shown in Fig. 5.10(b) where the interplation is 8 U -- ~ gaua a=l
(5.26)
where the appropriate shape functions are of the kind defined in the previous chapter. We also can write (5.26) directly using polynomial coefficients ba, a -- 1. . . . . 8, in place of the nodal variables fia (noting the terms occurring in the Pascal triangle) as u -- bl -+- b2~ -k- b3/7 + b4~ 2 -+- b5~/7 + b6/72 -k- b7~/72 + b8~2/7
(5.27)
It is immediately evident that for arbitrary values of/71 to/79 it is impossible to match the coefficients bl to b8 due to the absence of the term ~2/72 in Eq. (5.27). [However, if higher order (quartic, etc.) expansions of the serendipity kind were used such matching would evidently be possible and we could conclude that for linearly distorted elements the serendipity family of order four or greater will always represent quadratic polynomials in
x,y.] For the 9-node, lagrangian, element [Fig. 5.10(c)] the expansion similar to (5.28) gives 9 U -- ~ Naua a=l
(5.28)
which when expressed directly in polynomial coefficents ba, 1 -- 1. . . . . 9 yields u -- bl -k- b2~ + b3/7 q- b4~ 2 -k-'""-+- b8~2/7 -+- b9~2/72
(5.29)
and the matching of the coefficients of Eqs (5.29) and (5.25) can be made directly. We can conclude therefore that 9-node elements better represent cartesian polynomials (when distorted linearly) and therefore are generally preferable in modelling smooth solutions. This matter was first presented by Wachspress 7 but the simple proof presented above is due to Crochet. 8 An example of this is given in Fig. 5.11 where we consider the results of a finite element calculation with 8- and 9-node elements respectively used to reproduce a simple beam solution in which we know that the exact answers are quadratic. With no distortion both elements give exact results but when distorted only the 9-node element does so, with the 8-node element giving quite wild stress fluctuation. Similar arguments will lead to the conclusion that in three dimensions again only the lagrangian 27-node element is capable of reproducing fully a quadratic function in cartesian coordinates when trilinearly distorted (i.e., using the mapping for N a' for the 8-node hexahedron).
Shape functions by degeneration 153 Lee and Bathe 9 investigate the problem for cubic and quartic serendipity and lagrangian quadrilateral elements and show that under bilinear distortions the full order cartesian polynomial terms remain in lagrangian elements but not in serendipity ones. They also consider edge distortion and show that this polynomial order is always lost. Additional discussion of such problems is also given by Wachspress. 7
In the previous sections we have discussed the construction of shape functions for mapped elements of lagrangian and serendipity type, as well as those for triangular and tetrahedral type. We have also shown how mixtures of interpolation forms may be used to construct elements of prism type. One may ask what happens if we distort elements such that nodes for the lagrangian or serendipity type are coalesced - that is, they are assigned the same node number in the mesh. We call the approach where two or more nodes are common a degenerate form. In a degenerate form the shape function for a coalesced set of two or more nodes is obtained by adding together the shape functions of each individual node (in a hierarchic form, any mid-side and/or face functions are omitted). Example 5.1: Quadrilateral degenerated into a triangular element. As a simple example we consider the degeneration of a 4-node quadrilateral in which nodes 3 and 4 are coalesced to form the third node of a triangular element as shown in Fig. 5.12. For an isoparametric form given in ~, r/coordinates, the shape functions for the degenerate triangular element are given by r/) N1 -- ~1(1 --,~)(1 N : = ~ 1(1 -b ~)(1 - r/) 1 + r/) N3 ~(1 -
(5.30)
-
where the last function results from adding together the standard shape functions for nodes 3 and 4 of the quadrilateral element. Computing now the global derivatives for the above functions we obtain [using (5.10)]
aNa
ba(1 -- O) Ox = 2 A(1 - r/)'
(b)
aNa
Ca(1 -- O)
Oy
2 A(1 -- r/)
(c)
Fig. 5.10 Bilinear mapping of subparametric quadratic 8- and 9-node element.
(5.31)
154
Mapped elements and numerical integration
lar (R) mesh
.....
5.o
-I
AL
A v
~
M
8- and 9-noded elements
(a)
Exact
~_ deflection
[] R = 8/9 l = Exact o 1=9 J z~ 1=8
(b) /
Exact
zx
- 1000
z~z~ z~
/k z~
z~
m
900 800
Ox stress 700 on AA (Gauss point) 600
500 (c)
zx
- 400
Fig. 5.11 Quadratic serendipity and lagrangian 8- and 9-node elements in regular and distorted form. Elastic deflection of a beam under constant moment. Note poor results of 8-node element.
where ba and C a coincide with results for the standard 3-node triangular element shape functions given in (2.6) and A is the area of the triangle as given in (2.7). Except for the point 0 = 1 (the point where the nodes are coalesced) the shape function derivatives are constant and identical to those obtained using area coordinates L], L2, L3. Thus, for the
Shape functions by degeneration degeneration we have the identities 1 N1 = ~(1 - ~ ' ) ( 1 - r/)
= L1
N 2 - - ~ 1(1 --{- b~)(1 -- r])
= L2
N3 = 89 + r/)
= L3
(5.32)
and, provided we do not consider the point r/ = 1, we may compute the derivatives and integrals for 3-node triangular elements using the degeneration process. A similar form to the above example holds when an 8-node brick element is degenerated into a 4-node tetrahedron. In addition, however, we can compute shape functions for other degenerate forms as indicated in Fig. 5.13. In all cases, the computation of derivatives gives a 0/0 form at any point where nodes are coalesced. In addition, however, any faces which degenerate into an edge will also contain a 0/0 in the derivative along that edge. The behaviour on any remaining face of a degenerate element is either the original quadrilateral one or a triangular one in which the shape functions are identical to the results given in
(5.32).
5.8.1 Higher order degenerate elements When nodes for higher order quadrilateral and hexahedral elements are coalesced to give a degenerate form it is necessary to modify the shape functions for some of the non-coalesced nodes in order to produce results which are consistent with those computed using area or volume coordinates, respectively. This aspect was first studied by Newton 1~ and Irons 11 for serendipity-type elements. Here we extend the work reported in these references to include the lagrangian-type elements. Using lagrangian elements has a distinct advantage since all the degenerate elements preserve the properties of higher order approximation
3=4
3
,T,
A w
2 Fig. 5.12 Degeneration of a quadrilateral into a triangle.
1
2
155
156
Mapped elements and numerical integration
in global coordinates when the element is mapped according to the trilinear form (i.e., a subparametric form using the 8-node hexahedron). Example 5.2" Quadratic quadrilateral degenerated to a triangular element. As an example we consider the degeneration of a quadratic order quadrilateral to form a quadratic order triangular element. Expressing the shape functions in hierarchical form we have for
--/]
'
5
6j
7
I I I
, ii SSS
1~ ....
I I I I
1
1;
4
/
S
7=8
s
s s
s s
2
2
3
(b) Degenerate
(a) Brick 5=6
5=6=7=8
7=8
4
2
(c) Prism 5=6
3
2
(d) Pyramid
I
I
l
s
I
l
I
s
l
3
5=6=7=8
7=8 I
2
/ w
/!
l
1
(e) Chisel
3=4 I
2
I
(f) Tetrahedron
Fig. 5.13 Somedegenerate element forms for an 8-node brick element.
3=4
Shape functions by degeneration 157 3=4=7 11
9
1
5
w
2
,iw
1
5
lw,
2
Fig. 5.14 Degeneration of a 8- or 9-node quadrilateral into a 6-node triangle.
the 8- or 9-node quadfilateralt
NaQ=~1(1
+ ~a~)(1 +/7a/7);
a=1,2,3,4
NQ=~I(1
+ ~a~)(1 -- 172) ",
a =6,8
NQ=~I(1
+/7a/7)(1 - ~2).
a-5,7
N Q = (1 -se2)(1 -/72);
(5.33)
a=9
for which a hierarchical lagrangian interpolation of any function is given by 4
f -- ~
8
NQ(~,/7) fa + ~
a=l
NQ(~,/7) Afa + N ~ (~,/7) AAf9
(5.34)
a=5
where fa are nodal values, Afa are departures from linear interpolation for mid-side nodes, and A A f9 is the departure from the 8-node serendipity interpolation at the centre node. Thus, omitting the ninth function gives the serendipity form. If we now coalesce the nodes 3, 4, 7 and use the above hierarchic form, the shape functions for the vertex nodes again are given by 1 N Q = ~(1 -se)(1 -/7) = L1 = Nir
N Q - - ~ 1(1 + se)(1
/7) = L2 = N T
1
/~3Q = ~ ( l --I--/7)
(5.35)
~-- L3 ~. m ~
[note that A f7 = 0 in any interpolation and, thus, N f - 0]. Also, for the 6-node form we omit the interior node 9 and thus, for the degenerate element, N f = 0. If the resulting degenerate element is to be identical with the 6-node triangular element we require N 5r = 4L 1L 2 N { = 4L2L3
(5.36)
N T = 4L3L1 'f We use a superscript 'Q' for shape functions associated with the quadrilateral form and, later, 'T' to denote those for a triangular form.
158 Mappedelements and numerical integration Substituting the definitions for area coordinates given by (5.35) into (5.36) we find 1 N f = ~(1 - ~2)(1 - r/) 2
1 N~" = ~(I + ~ ' 1 ( I - 021
(5.37)
1
N f = ~(1 - ~1(1 - r/21 and, thus, comparing the forms given by (5.33) and (5.37) we obtain the result
Uf e Ug; U: = U6; Of=N8 Q
(5.38)
Thus, it only remains to correct the shape function for node 5. This is accomplished by noting N f = ~ ( 11 -
~2 ) ( 1 - 2 r / + 0
z)
-
1(1 - ~ 2 ) ( 2 - 2 0 - 1 +/'12)
--
1(1 - ~2)(1 2
--4
-
r]) -
1 -~(1
~2)(1
- - T] 2 )
giving the 'corrected' degenerate function for node 5 as
I NQ
(5.39)
The hierarchical forms now can be converted to standard isoparametric form using the process given in Sec. 4.6. E x a m p l e 5.3: D e g e n e r a t e f o r m s f o r a quadratic 2 7 - n o d e h e x a h e d r o n .
The construction for quadratic degenerate three-dimensional forms foUows a similar process and, when using the hierarchical form, the mid-side node opposite each coalesced node on a 'face' must be modified using a form similar to (5.39). Again, all the shapes shown in Fig. 5.13 are possible and permit the construction of meshes which use a mix of bricks, tetrahedra and degenerate transition forms. In addition to the 8 vertex nodes it is necessary to add 12 mid-edge nodes, 6 mid-face nodes and one internal node to form a lagrangian quadratic order hexahedron. For node numbers as given in Fig. 5.15 and using hierarchical interpolation, the shape functions are given by the following: 5
8
16
13
9..........................:i::..............
I 2
! .......
_ 3
,w
10
Fig. 5.15 Numbering for 27-node quadratic lagrangian hexagon. (Node 27 at origin of ~, r/, ~" coordinates).
Shape functions by degeneration 159 Table 5.1
Degeneration modifications for 27-node hexahedron
Coalesced nodes
Modified nodes
1 and 2 and 3 and 4 and
2 omit 3 omit 4 omit 1 omit
9 10 11 12
11 by 12 by 9 by 10 by
25 25 25 25
omit omit omit omit
25 25 25 25
13 by 14 by 15 by 16 by
23 22 24 21
omit omit omit omit
23 22 24 21
5 and 6 and 7 and 8 and
6 omit 7 omit 8 omit 5 omit
13 14 15 16
15 by 16 by 13 by 14 by
26 26 26 26
omit omit omit omit
26 26 26 26
9 by 10 by 11 by 12 by
23 22 24 21
omit omit omit omit
23 22 24 21
1 and 2 and 3 and 4 and
5 omit 6 omit 7 omit 8 omit
17 18 19 20
18 by 19 by 20 by 17 by
23 22 24 21
omit omit omit omit
23 22 24 21
20 by 17 by 18 by 19 by
21 23 22 24
omit omit omit omit
21 23 22 24
1. For vertex nodes 1 Na = ~(1 + ~a~')(1 -+-0a0)(1 "1-(a();
a=l,2
..... 8
(5.40a)
2. For mid-edge nodes 1
/
Na = ~
(
(1 - ~:2)(1 + 0 a 0 ) ( 1 -'l- ~'a~'); a = 9, 11, 13, 15 (1 + ~ase)(1 - 02)(1 q- ~'a~'); a = 10, 12, 14, 16
(5.40b)
(1 + ~a~')(1 + OaO)(1 - (2); a = 17, 18, 19, 20
3. For mid-face nodes
Na -- 1
(1 -~- ~ a ~ ) ( 1 -- 02)(1 --
~2); a = 21, 22
(1 -- ~ 2 ) ( 1 -'1- 0 a 0 ) ( 1 --
if2); a = 23, 24
(1 -- ~ Z ) ( 1 -- 0 2 ) ( 1 + ( a ( ) ;
a =
(5.40c)
25, 26
4. For interior node N a - - (1 -
~2)(1 - 02)(1 -
(2); a = 27
(5.40d)
Table 5.1 indicates which shape functions are modified when vertex nodes are coalesced. The hierarchical shape functions to be omitted are also indicated. Note that shape functions should only be omitted (set zero) after all coalesced node pairs are considered. Also if a tetrahedral element is formed then all mid-face nodes are deleted and the interior node may also be omitted, giving the final tetrahedron as a 10-node element. Again, if any of the element forms is mapped using the degenerate subparametric form of the 8-node hexahedron for N a' full quadratic behaviour in global coordinates is attained- showing the advantage of starting from lagrangian form elements. Consideration of cubic and higher order forms are also possible and are left as an exercise for the interested reader.
160 Mappedelements and numerical integration
In the most obvious procedure, points at which the function is to be found are determined a p r i o r i - usually at equal intervals - and a polynomial passed through the values of the function at these points and exactly integrated [Fig. 5.16(a)]. As n values of the function define a polynomial of degree n - 1, the errors will be of the order O(h n) where h is the element size. The well-known Newton-Cotes 'quadrature' formulae can be written as I =
/1
f ( ~ ) d~
f(~i) Wi
"-
1
(5.41)
i=1
for the range of integration between - 1 and + 1 [Fig. 5.16(a)]. For example, if n = 2, we have the well-known trapezoidal rule: I = f(-1) + f(1)
(5.42a)
for n = 3, the Simpson 'one-third' rule: 1
I = 5 [ f ( - 1 ) + 4 f ( O ) + f(1)]
(5.42b)
and for n = 4: 1
1
I = ~ [ f ( - 1 ) + 3 f ( - 5) + 3 f ( 1 ) + f(1)]
(5.42c)
Formulae for higher values of n are given in reference 12.
5.9.2 Gauss quadrature If in place of specifying the position of sampling points a priori we allow these to be located at points to be determined so as to aim for best accuracy, then for a given number of sampling points increased accuracy can be obtained. Indeed, if we again consider I =
f ( ~ ) d~ 1
f(~i) Wi
= i=1
t ' Q u a d r a t u r e ' is a n a l t e r n a t i v e t e r m to ' n u m e r i c a l i n t e g r a t i o n ' .
(5.43)
Numerical integration - one dimensional
Af
f(~8) i
f(~1) u
-1
~'
T
(a)
u
L.
A = 72
f
~L
(b)
~_1
0.86114
Fig. 5.16 (a) Newton-Cotes and (b) Gauss integrations. Each integrates exactly a seventh-order polynomial [i.e., error O(h8)].
and again assume a polynomial expression, it is easy to see that for n sampling points we have 2n unknowns (wi and ~i ) and hence a polynomial of degree 2n - 1 could be constructed and exactly integrated [Fig. 5.16(b)]. The error is thus of order O(h2n). The simultaneous equations involved are difficult to solve, but some mathematical manipulation will show that the solution can be obtained explicitly in terms of Legendre polynomials. Thus this particular process is frequently known as Gauss-Legendre quadrature. 12 Table 5.2 shows the positions and weighting coefficients for gaussian integration. For purposes of finite element analysis complex calculations are involved in determining the values of f , the function to be integrated. Thus the Gauss-type processes, requiting the least number of such evaluations, are ideally suited and from now on will be used exclusively.
161
162
Mappedelements and numerical integration
The most obvious way of obtaining the integral
I --
flf, 1
(5.44)
f (~, rl)d~ do 1
is first to e v a l u a t e t h e i n n e r i n t e g r a l k e e p i n g r / c o n s t a n t , i.e.,
1
f (~, ~) d~ =
f (~j, ~) w j - - ~ (/7) j=l
Table 5.2 Gaussian quadrature abscissae and weights for
f l 1 f (x)dx = ~ = 1 f (~J) wj. .4-~j
wj n=l
0
2.000 000 000 000 000 n=2
1/ ~
1.000 000 000 000 000 n=3 5/9 8/9
0.000 000 000 000 000 n=4
0.347 854 845 137 454 0.652 145 154 862 546
0.861 136 311 594053 0.339 981 043 584 856 n=5
0.236 926 885 056 189 0.478 628 670 499 366 0.568 888 888 888 889
0.906 179 845 938 664 0.538 469 310 105 683 0.000 000 000 000 000 n=6
0.171 324 492 379 170 0.360 761 573 048 139 0.467 913 934 572 691
0.932 469 514 203 152 0.661 209 386 466 265 0.238 619 186 083 197 n=7 0.949 0.741 0.405 0.000
107 912 342 531 185 599 845 151 377 000 000 000
759 394 397 000
0.960 0.796 0.525 0.183
289 856 497 666 477 413 532 409 916 434 642 495
536 627 329 650
0.129 0.279 0.381 0.417
484 966 168 870 705 391 489 277 830 050 505 119 959 183 673 469
0.101 0.222 0.313 0.362
228 381 706 683
536 290 034 453 645 877 783 378
376 374 887 362
0.081 0.180 0.260 0.312 0.330
274 648 610 347 239
388 361 160 694 696 402 077 040 355 001
574 857 935 003 260
0.066 0.149 0.219 0.269 0.295
671 451 086 266 524
344 308 688 349 150 581 362 515 982 719 309 996 224 714 753
n=8
n=9 0.968 160 239 507 626 0.836 031 107 326 636 0.613 371 432 700 590 0.324 253 423 403 809 0.000 000 000 000 000 n=10 0.973 0.865 0.679 0.433 0.148
906 063 409 395 874
528 517 172 366 688 985 568 299 024 394 129 247 338 981 631
(5.45)
Numerical integration - rectangular (2D) or brick regions (3D)
Evaluating the outer integral in a similar manner, we have
I =
~t(O) do
=
1
l~r(l~i) w i i=1
~Wi ~f(~j, rli)Wj i=1
(5.46)
j=l
--~~ ~ f (~j, rli) WiWj i=1 j = l
For a brick we have similarly ..
/_'/_'/_'
I --
1
1
1
f ( ~ , #7, ( ) d ~ d77 d( (5.4"7)
/=1 j = l i=1
In the above, the number of integrating points in each direction was assumed to be the same. Clearly this is not necessary and on occasion it may be an advantage to use different numbers in each direction of integration. It is of interest to note that in fact the double summation can be readily interpreted as a single one over (n x n) points for a rectangle (or n 3 points for a cube). Thus in Fig. 5.17 we show the nine sampling points that result in exact integrals of order 5 in each direction. In the sequel when numerical integration is used we will denote the summation as a single sum over unique points, thus we will write
I =
f ( ~ , O)d~ do = 1
f(~l, l~l) Wl
(5.48)
/=1
for two dimensions and . I =
fl/lfl 1
1
1
f ( ~ , 0, ( ) d ~ do d( =
/=1
f(~l, 11l, (1) Wl
(5.49)
for three dimensions. Here the weight WI denotes the product of the appropriate onedimensional weights. We can also approach the problem directly and require an exact integration of a fifthorder polynomial in two dimensions. At any sampling point two coordinates and a value of f have to be determined in a weighting formula of type .LI/1
I =
1
1
f (~, rl) d~ do =
/=1
f (~i, Oi) Wi
(5.50)
It would appear that only seven points would suffice to obtain the same order of accuracy. Some formulae for three-dimensional bricks have been derived by Irons 13 and used successfully. 14
163
164
Mapped elements and numerical integration
o
o
o 4
5
7
-1
1
o
o
8
9
P,
2
o
o
6
3
o
-1 Fig. 5.17 Integrating points for n = 3 in a square region. (Exact for polynomial of fifth order in each direction).
For a triangle, in terms of the area coordinates the integrals are of the form
I = f01f0 l-L1 f(L1, L2, L3) dL2 dL1
L3 = 1 - L1 - L2
(5.51)
Once again we could use n Gauss points and arrive at a summation expression of the type used in the previous section. However, the limits of integration now involve the variable itself and it is convenient to use alternative sampling points for the second integration by use of a special Gauss expression for integrals in which the integrand is multiplied by a linear function. These have been devised by Radau 15 and used successfully in the finite element context. 16 It is, however, much more desirable (and aesthetically pleasing) to use special formulae in which no bias is given to any of the natural coordinates La. Such formulae were first derived by Hammer et al. 17 and Felippa 18 and a series of necessary sampling points and weights is given in Table 5.3.19 (A more comprehensive list of higher formulae derived by Cowper is given in reference 19.) A similar extension for tetrahedra can obviously be made. Table 5.4 presents some formulae based on reference 17.
With numerical integration used in place of exact integration, an additional error is introduced into the calculation and the first impression is that this should be reduced as much as possible. Clearly the cost of numerical integration can be quite significant, and indeed in some early programs numerical formulation of element characteristics used a comparable amount of computer time as in the subsequent solution of the equations. It is of interest, therefore, to determine (a) the minimum integration requirement permitting convergence and (b) the integration requirements necessary to preserve the rate of convergence which would result if exact integration were used. It will be found later (Chapters 9 and 11) that it is in fact often a disadvantage to use higher orders of integration than those actually needed under (b) as, for very good
Required order of numerical integration Table 5.3 Numerical integration formulae for triangles Order
Figure
Linear
Quadratic
Error
Points
Triangular coordinates
R = O (h2)
a
1 3'
1 3,
a
1
1
b
1
C
0,
R = O(h 3)
Cubic
R = O (h4)
Quintic
R -- O(h 6)
1
3
1
3,
1
3
0.6,
0.2,
0.2
0.2,
0.6,
0.2
0.2,
0.2,
0.6
/31,
d
1
3,
0~2,
1
1
3
0.22500O0000
~1,
0.1323941527
r
0.1323941527
/31,
e
1
1
0r
r
1
1
3'
3,
b
1
1
1,
1
a
Weights
0tl
~2, ~2,
0.1323941527 0.1259391805
~2
0.1259391805
Or2
0.1259391805
with O t l = 0.059 715 8717 /31 = 0.470 142 064 1 ct2 = 0.797 426 985 3 /32 = 0.101 286 507 3
reasons, a 'cancellation of errors' due to discretization and due to inexact integration can occur.
5.12.1 Minimum order of integration for convergence .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
~
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
In problems where the energy functional (or equivalent Galerkin integral statements) defines the approximation we have already stated that convergence can occur providing any arbitrary constant value of mth derivatives can be reproduced. In the present case m = 1 and we thus require that in integrals of the form (5.5) a constant value of G be correctly integrated. Thus the volume of the element fa dr2 needs to be evaluated correctly
165
166 Mappedelements and numerical integration Table
5.4 Numerical integration formulae for tetrahedra
Order
Figure
Linear
Quadratic
~
Error
Points
Tetrahedral coordinates
R = O (h 2)
a
1 ~,
1 ~,
a b
ct, /3,
/3, ct,
1
c d
fl, fl,
fl, fl,
I
R = O (h 3)
Weights
1 ~,
1
1
fl,
~
1
a = 0.585 410 20 /3 = 0.13819660 a b
Cubic
~
R = O(h 4)
C
d e
1
~'
1
~'
1
~'
1 ~' 1 6' 1
1 6' 1 2' 1
1 6' 1 6' 1
1
1 6'
6'
~'
6'
~'
~' 1
1
~
1 6 1 6 1
~ 1
2
4
-5
9 2"-'0 9 2--0 9
2-6
9 2"0
f o r convergence to occur. In curvilinear coordinates we can thus argue that f J d ( d r / d ~ has to be evaluated exactly. 3' 6
5.12.2 Order of integration for no loss of convergence rate In a general problem we have already found that the finite element approximate evaluation of energy (and indeed all the other integrals in a Galerkin-type approximation, see Chapter 3) was exact to the order 2(p - m ) , where p was the degree of the complete polynomial present and m the order of differential occurring in the appropriate expressions. Pn ~ ,ding the integration is exact to the order 2(p-m), or shows an error of O (h 2(p-m)+ 1), or less, then no loss of convergence order will occur.t If in curvilinear coordinates we take a curvilinear dimension h of an element, the same rule applies. For Co problems (i.e., m = 1) the integration formulae should be as follows: p = 1,
linear elements
O(h)
p = 2,
quadratic e l e m e n t s
O ( h 3)
p = 3,
cubic elements
O (h 5)
t For an energy principle use of quadrature may result in loss of a bound for H (fi).
Required order of numerical integration We shall make use of these results in practice, as will be seen later, but it should be noted that for a linear quadrilateral or triangle a single-point integration is adequate. For parabolic quadrilaterals (or bricks) 2 x 2 (or 2 x 2 x 2), Gauss point integration is adequate and for parabolic triangles (or tetrahedra) three-point (and four-point) formulae of Tables 5.3 and 5.4 are needed. The basic theorems of this section have been introduced and proved numerically in published work. 2~
5.12.3 Matrix singularity due to numerical integration The final outcome of a finite element approximation in linear problems is an equation system K~ + f = 0 (5.52) in which the boundary conditions have been inserted and which should, on solution for the parameter fi, give an approximate solution for the physical situation. If a solution is unique, as is the case with well-posed physical problems, the equation matrix K should be non-singular. We have a priori assumed that this was the case with exact integration and in general have not been disappointed. With numerical integration, singularities may arise for low integration orders, and this may make such orders impractical. It is easy to show how, in some circumstances, a singularity of K must arise, but it is more difficult to prove that it will not. We shall, therefore, concentrate on the former case. With numerical integration we replace the integrals by a weighted sum of independent linear relations between the nodal parameters ~. These linear relations supply the only information from which the matrix K is constructed. If the number of unknowns fl exceeds the number of independent relations supplied at all the integrating points, then the matrix K must be singular. To illustrate this point we shall consider two-dimensional elasticity problems using linear and parabolic serendipity quadrilateral elements with one- and four-point quadrature respectively. Here at each integrating point three independent 'strain relations' are used and the total number of independent relations possible equals 3 x (number of integration points). The number of unknowns ~ is simply 2 x (number of nodes) less restrained degrees of freedom. In Fig. 5.18(a) and (b) we show a single element and an assembly of two elements supported by a minimum number of specified displacements eliminating rigid body motion. The simple calculation shows that only in the assembly of the quadratic elements is elimination of singularities possible, all the other cases remaining strictly singular. In Fig. 5.18(c) a well-supported block of both kinds of elements is considered and here for both element types non-singular matrices may arise although local, near singularity may still lead to unsatisfactory results (see Chapter 9). The reader may well consider the same assembly but supported again by the minimum restraint of three degrees of freedom. The assembly of linear elements with a single integrating point will be singular while the quadratic ones will, in fact, usually be well behaved. For the reason just indicated, linear single-point integrated elements are used infrequently in static solutions, although they do find wide use in 'explicit' dynamics codes - but needing
167
168 Mappedelements and numerical integration x o ~/.~///x
Integratingpoint (3 independent relations) Nodalpoint with 2 degrees of freedom l
(a)
~ / ~
~/(~//~ Both d.o.f. suppressed R
i xI
One d.o.f. suppressed
(b)
x
x
x
x
x
x
x
x
x
x
x
x
X
X
X
X
(c)
Degree of freedom
Linear Quadratic Independent Independent Degree of freedom relation relation
(a)
4x2-3=5>1x3=3 singular
2x8-3=
13 > 4 x 3 = 12 singular
(b)
6x2-3=9>2x3=6 singular
13x2-3=23<8x3=24
(c)
25 x 2-18 = 32< 16 x 3 = 48 48 x2 = 96 < 64x 3= 192
Fig. 5.18 Checkon matrix singularity in two-dimensional elasticity problems (a), (b), and (c).
certain remedial additions (e.g., hourglass control 23'24) - while four-point quadrature is often used for quadratic serendipity elements.t In Chapter 9 we shall return to the problem of convergence and will indicate dangers arising from local element singularities. However, it is of interest to mention that in Chapter 11 we shall in fact seek matrix singularities for special purposes (e.g., incompressibility) using similar arguments.
~fRepeating the test for quadratic lagrangianelements indicates a singularityfor 2 x 2 quadrature.
Generation of finite element meshes by mapping. Blendingfunctions 169
It will be observed that it is an easy matter to obtain a coarse subdivision of the analysis domain with a small number of isoparametric elements. If second- or third-degree elements are used, the fit of these to quite complex boundaries is reasonable, as shown in Fig. 5.19(a) where four parabolic elements specify a sectorial region. This number of elements would be too small for analysis purposes but a simple subdivision into finer elements can be done automatically by, say, assigning new positions for the nodes at the mid-points of the curvilinear coordinates and thus deriving a larger number of similar elements, as shown in Fig. 5.19(b). Indeed, automatic subdivision could be carried out further to generate a field of triangular elements. The process thus allows us, with a small amount of original input data, to derive a finite element mesh of any refinement desirable. In reference 25 this type of mesh generation is developed for two- and three-dimensional solids and surfaces and is reasonably efficient. However, elements of predetermined size and/or gradation cannot be easily generated. The main drawback of the mapping and generation suggested is the fact that the originally circular boundaries in Fig. 5.19(a) are approximated by simple parabolas and a geometric error can be developed there. To overcome this difficulty another form of mapping, originally developed for the representation of complex motor-car body shapes, can be adopted for this purpose. 26 In this mapping blending functions interpolate the unknown u in such a way as to satisfy exactly its variations along the edges of a square ~, r/domain. If the coordinates x and y are used in a parametric expression of the type given in Eq. (5.1), then any complex shape can be mapped by a single element. In reference 26 the region of Fig. 5.19 is in fact so mapped and a mesh subdivision obtained directly without any geometric error on the boundary. The blending processes are of considerable importance and have been used to construct some interesting element families 27 (which in fact include the standard serendipity elements as a subclass). To explain the process we shall show how a function with prescribed variations along the boundaries can be interpolated. Consider a region - 1 < ~, r / < 1, shown in Fig. 5.20, on the edges of which an arbitrary function q~ is specified [i.e., 4~(-1, O), q~(1, r/), 4~(~, - 1 ) , 4~(~, 1) are given]. The problem presented is that of interpolating a function 4~(~, r/) so that a smooth surface reproducing precisely the boundary values is obtained. Writing NI(~) = ~1 ( 1 - ~)
N2(~:)-- 1 ( 1 + ~ )
N1 (r/) = ~1 (1 - O)
N2 (r/) = 1( 1 "~-0)
(5.53)
for our usual one-dimensional linear interpolating functions, we note that Poq~ ---- Nl(0)~b(~,-1) + N2(0)q~(~, 1)
(5.54)
interpolates linearly between the specified functions in the 0 direction, as shown in Fig. 5.20(b). Similarly, P~q~ -----N1 (~)q~ (r/, - 1 ) + N2(~)q~(r/, 1)
(5.55)
interpolates linearly in the ~ direction [Fig. 5.20(c)]. Constructing a third function which is a standard bilinear interpolation of the kind we have already encountered [Fig. 5.20(d)],
170 Mappedelements and numerical integration (a) (b)
Fig. 5.19 Automatic mesh generation by quadratic isoparametric elements. (a) Specified mesh points. (b) Automatic subdivision into a small number of isoparametric elements. (c) Automatic subdivision into linear triangles. i.e.,
P~Pock = NI(~)NI(O)4~(-1,-1) + NI(~)Nz(O)q~(-1, 1) + Nz(~)NI(r/)4~(1,-1) + Nz(~)Nz(rl)dp(1, 1)
(5.56)
we note by inspection that ~(~, ~) = P0~ + P ~ - P~ P0~
(5.57)
is a smooth surface interpolating exactly the boundary functions. Extension to functions with higher order blending is almost evident, and immediately the method of mapping the quadrilateral region - 1 < ~, r/ < 1 to any arbitrary shape is obvious. Though the above mesh generation method derives from mapping and indeed has been widely applied in two and three dimensions, we shall see in the chapter devoted to adaptivity (Chapter 14) that the optimal solution or specification of mesh density or size should guide the mesh generation. In Chapter 8 we will discuss in much more detail how meshes with prescribed density can be generated.
In many problems of engineering and physics infinite or semi-infinite domains exist. A typical example from structural mechanics may, for instance, be that of three-dimensional (or axisymmetric) excavation, illustrated in Fig. 5.21. Here the problem is one of determining the deformations in a semi-infinite half-space due to the removal of loads with the specification of zero displacements at infinity. Similar problems abound in electromagnetics and fluid mechanics but the situation illustrated is typical. The question arises as to how such
Infinite domains and infinite elements
(1,1)
(a) (-1, -1)
@
(b)
@ (c)
@ (d) Fig. 5.20 Stagesof construction of a blending interpolation (a), (b), (c), and (d).
problems can be dealt with by a method of approximation in which elements of decreasing size are used in the modelling process. The first intuitive answer is the one illustrated in Fig. 5.21(a) where the infinite boundary condition is specified at a finite boundary placed at a large distance from the object. This, however, begs the question of what is a 'large distance' and obviously substantial errors may arise if this boundary is not placed far enough away. On the other hand, pushing this out excessively far necessitates the introduction of a large number of elements to model regions of relatively little interest to the analyst. To overcome such 'infinite' difficulties many methods have been proposed. In some a sequence of nesting grids is used and a recurrence relation derived. 28' 29 In others a boundarytype exact solution is used and coupled to the finite element domain. 3~ 31 However, without doubt, the most effective and efficient treatment is the use of 'infinite elements '32-35
171
172 Mapped elements and numerical integration
Fig. 5.21 A semi-infinite domain. Deformations of a foundation due to removal of load following an excavation. (a) Conventional treatment and (b) use of infinite elements. pioneered originally by Bettess. 36 In this process the conventional, finite elements are coupled to elements of the type shown in Fig. 5.21 (b) which model in a reasonable manner the material stretching to infinity. The shape of such two-dimensional elements and their treatment is best accomplished by mapping 34-36 these onto a finite square (or a finite line in one dimension or cube in three dimensions). However, it is essential that the sequence of trial functions introduced in the mapped domain be such that it is complete and capable of modelling the true behaviour as the radial distance r increases. Here it would be advantageous if the mapped shape functions could approximate a sequence of the decaying form
c--L1+ C2 + C3 + . . . r
(5.58)
~
where Ca are arbitrary constants and r is the radial distance from the 'focus' of the problem. In the next subsection we introduce a mapping function capable of doing just this.
5.14.2 The mapping function ......... :--:
:-:--:--::
--- 7 - : : : -
:::
::::
: ---:
:
:
. . . . . . . . . . . . . . .
:
: ----------------: . . . . . . . . .
::
:
- ::
: : :
::::
:
:
: :::::
::
:
:..................................................
:
..:
.-:-
:
:
~
- ......................
:::
:
:::::
:::
Figure 5.22 illustrates the principles for generation of the derived mapping function. We shall start with a one-dimensional mapping along a line CPQ coinciding with the x
Infinite domains and infinite elements
direction. Consider the following function:
~)XQ'--~[CXC'~'-NQXQ
~ xc+(l+
(5.59a)
and we immediately observe that XQ -]-" x C
= -1
corresponds to x =
= 0
corresponds to x = XQ
= 1
corresponds to x = cx~
2
-- xp
where xp is a point midway between Q and C. Alternatively the above mapping could be written directly in terms of the Q and P coordinates by simple elimination of xc. This gives, using our previous notation: X -- NQXQ + N p x p
=
(
2~ ) 1+ 1-~ XQ
2~ 1 - xp
(5.59b)
Both forms give a mapping that is independent of the origin of the x coordinate as NQ + Np = 1 = Nc + A/Q
(5.60)
The significance of the point C is, however, of great importance. It represents the centre from which the 'disturbance' originates and, as we shall now show, allows the expansion of the form of Eq. (5.58) to be achieved on the assumption that r is measured from C. Thus r = x - xc
(5.61)
If, for instance, the unknown function u is approximated by a polynomial function using, say, hierarchical shape functions and giving bt - - O/0 -+- Otl~ + 0/2~ 2 + 13/3~ 3 "~- 9 9 9
(5.62)
we can easily solve Eqs (5.59a) for ~, obtaining =l--XQ--xc
=l--XQ--Xc
x -- x C
(5.63)
r
Substitution into Eq. (5.62) shows that a series of the form given by Eq. (5.58) is obtained with the linear shape function in ~ corresponding to 1/r terms, quadratic to 1/r 2, etc. In one dimension the objectives specified have thus been achieved and the element will yield convergence as the degree of the polynomial expansion, p, increases. Now a generalization to two or three dimensions is necessary. It is easy to see that this can be achieved by simple products of the one-dimensional infinite mapping with a 'standard' type of shape function in r/(and ~') directions in the manner indicated in Fig. 5.22. First we generalize the interpolation of Eqs (5.59a) and (5.59b) for any straight line in x, y, z space and write (for such a line as C1PIQ1 in Fig. 5.22) x---1--~xr
1+
1+ 1-~
Y=-l-~Ycl+
1+1_
z=
1+ 1-~
l_~Zcl+
xQ1 (5.64)
~ YQ1 ZQ1
(in three dimensions)
173
174 Mappedelements and numerical integration
Fig. 5.22 Infinite line and element map. Linear r/interpolation. Second we complete the interpolation and map the whole ~r/(() domain by adding a 'standard' interpolation in the r/(() directions. Thus for the linear interpolation shown we can write for elements PP1QQ1RR1 of Fig. 5.22, as x -- Nl(r/) +No(r/) with
E
( 0)xQ]
- 1 - - ~ x c -+- 1 -+- 1 - l_--~xc, +
1 Nl(r/) = ~(1 + r/)
1+ 1-~:
XQ~
(5.65) ,
etc.
1 N0(r/) -- ~(1 - r/)
and map the points as shown. In a similar manner we could use quadratic interpolations and map an element as shown in Fig. 5.23 by using quadratic functions in 17. Thus it is an easy matter to create infinite elements and join these to a standard element mesh as shown in Fig. 5.21(b). In the generation of such element properties only the transformation jacobian matrix differs from standard forms, hence only this has to be altered in conventional programs. Moreover, integration is again over the usual 'parent' element. The 'origin' or 'pole' of the coordinates C can be fixed arbitrarily for each radial line, as shown in Fig. 5.22. This will be done by taking account of the knowledge of the physical solution expected. In Fig. 5.24 we show a solution of the Boussinesq problem (a point load on an elastic half-space). Here results of using a fixed displacement or infinite elements are compared
Infinite domains and infinite elements
Fig. 5.23 Infinite element map. Quadratic r/interpolation.
and the big changes in the solution noted. In this example the pole of each element was taken at the load point for obvious reasons. 35 Figure 5.25 shows how similar infinite elements (of the linear kind) can give excellent results, even when combined with very few standard elements. In this example where a solution of the Laplace equation is used (see Chapter 3) for an irrotational fluid flow, the poles of the infinite elements are chosen at arbitrary points of the aerofoil centre-line. In concluding this section it should be remarked that the use of infinite elements (as indeed of any other finite elements) must be tempered by background analytical knowledge and 'miracles' should not be expected. Thus the user should not expect, for instance, such excellent results as those shown in Fig. 5.24 for the displacement of a plane elasticity problem. It is 'well known' that in this case the displacements under any load which is not self-equilibrated will be infinite everywhere and the numbers obtained from the computation will not be, whereas for the three-dimensional or axisymmetric case it is infinite only at a point load. Further use of infinite elements is made in the context of the solution of wave problems in fluids in reference 37.
1
176 Mappedelements and numerical integration
Fig. 5.24 A point load on an elastic half-space (Boussinesq problem). Standard linear elements and infinite line elements (E -- 1, v = 0.1, p = 1).
In the study of fracture mechanics interest is often focused on the singularity point where quantities such as stress become (mathematically, but not physically) infinite. Near such singularities normal, polynomial-based, finite element approximations perform badly and attempts have frequently been made here to include special functions within an element which can model the analytically known singularity. References 38-53 give an extensive literature survey of the problem and finite element solution techniques. An alternative to the introduction of special functions within an element- which frequently poses problems of enforcing continuity requirements with adjacent, standard, elements - lies in the use of special mapping techniques. An element of this kind, shown in Fig. 5.26(a), was introduced almost simultaneously by Henshell and Shaw 49 and Barsoum 5~ 51 for quadrilaterals by a simple shift of the mid-side node to the quarter point in quadratic, isoparametfic elements. It can now be shown (and we leave this exercise to the curious reader) that along the element edges the derivatives Ou/Ox (or strains) vary as 1/~/7 where r is the distance from
Computational advantage of numerically integrated finite elements
Fig. 5.25 Irrotational flow around NACA 0018 wing section.31 (a) Mesh of bilinear isoparametric and infinite elements. (b) Computed o and analytical m results for velocity parallel to surface. the comer node at which the singularity develops. Although good results are achievable with such elements the singularity is, in fact, not well modelled on lines other than element edges. A development suggested by Hibbitt 52 achieves a better result by using triangular second order elements for this purpose [Fig. 5.26(b)]. Indeed, the use of distorted or degenerate isoparametric elements is not confined to elastic singularities. Rice 43 shows that in the case of plasticity a shear strain singularity of 1/r type develops and Levy et al. 54 use an isoparametric, linear quadrilateral to generate such a singularity by the simple device of coalescing two nodes but treating these displacements independently. A variant of this is developed by Rice and Tracey. 39 The elements just described are simple to implement without any changes in a standard finite element program. However, in Chapter 15 we introduce a method whereby any singularity (or other function) can be modelled directly. We believe the methods to be described there supersede the above described techniques.
One considerable gain that is possible in numerically integrated finite elements is the versatility that can be achieved in a single computer program. 55 It will be observed that for a
177
178
Mapped elements and numerical integration
)
~-o )
.-
(a)
m
)
0----"~
>_
i
a
=
o
.>--
(b)
Fig. 5.26 Singular elements from degenerate isoparametric elements (a), (b), and (c).
given class ofproblems the general matrices are always of the same form [see the example of Eq. (5.7)] in terms of the shape function and its derivatives. To proceed to evaluation of the element properties it is necessary first to specify the shape function and its derivatives and, second, to specify the order of integration. The computation of element properties is thus composed of three distinct parts as shown in Fig. 5.27. For a given class ofproblems it is only necessary to change the prescription of the shape functions to achieve a variety of possible elements. Conversely, the same shape function routines can be used in many different classes of problem. Use of different elements, testing the efficiency of a new element in a given context, or extension of programs to deal with new situations can thus be readily achieved, and considerable algebra avoided (with its inherent possibilities of mistakes). The computer is thus placed in the position it deserves, i.e., of being the obedient slave capable of saving routine work. The greatest practical advantage of the use of universal shape function routines is that they can be checked decisively for errors by a simple program with the patch test playing the crucial role (viz. Chapter 9). The incorporation of simple, exactly integrable, elements in such a system is, incidentally, not penalized as the time of exact and numerical integration in many cases is almost identical.
5.1 A quadratic one-dimensional element is shown in Fig. 5.28 in parent form and in the mapped configuration. Let a + b = h the total length of the mapped element. (a) Determine the shape functions Na (~) for the three nodes. (b) Plot ~ vs x for values of a ranging from 0.2h to 0.8h in increments of 0. lh. (c) Plot Navs x for the range of a given in part (b). (d) Plot dNa/dx vs x for the range of a given in part (b).
Problems
Shape function specification
r
General formulation for a particular type of element matrix
Order of integration
r
Fig, 5,27 Computationalschemefor numericallyintegrated elements.
IN
1
o
L
3
,
2
,
o
J
r~--~
I' 1
L
o
a
(a) Parent element
~
3
2
o
b
~-x
J
(b) Mapped element
Fig. 5.28 Quadraticelementfor Problem 5.1. 5.2 Consider the one-dimensional problem for 0 < x < 1 which is defined by the weak form
f0
1 [d~u du L dx dx
~uq] dx - ~ucr x=l
- 0 with u (0) = 0
with q = tr = 1. (a) Deduce the Euler differential equation and boundary conditions for the problem. (b) Construct an exact solution to the differential equation. (c) Solve the weak form using a single quadratic order element with nodes placed at x = 0, 5/16 and 1 and shape functions Na defined by: i. Lagrange interpolation in x directly. ii. Isoparametric interpolation for Na (~) with x = Na (~).Xa. Evaluate all integrals using two-point gaussian quadrature. (d) Plot u and du/dx for the two solutions. Comment on differences in quality of the two solutions. 5.3 It is proposed to create transition elements for use with 4-node quadrilateral element meshes as shown in Fig. 5.29. (a) Devise the shape functions for the transition element labelled A. The shape functions must maintain compatibility along all boundaries. (Hint: The element can be a composite form combining more than one 4-node element.) (b) Devise the shape functions for the transition element labelled B. (c) On a sketch show the location of quadrature points necessary to integrate each element form.
179
180 Mappedelements and numerical integration
Fig. 5.29 Transition elements for use with 4-node quadrilaterals.
(d) As an alternative to transition elements, 4-node elements may be used for all elements and constraints imposed to maintain compatibility. For the mesh shown in the figure, number all nodes and write the constraint equations necessary to maintain compatibility. The interior node of element B is not needed and can be ignored. 5.4 Determine the hierarchical interpolation functions in ~, 77coordinates for the 16-node cubic order quadrilateral shown in Fig. 5.30(a). Express your hierarchic shape functions in a form such that interpolation is given by 4
12
16
f (~, ~7)-- ~ Na(~, rl)fa -'b ~ Na(~, o)A fa -Jr-~ Na(~, rl)AA fa a=l
a=5
a=13
5.5 Determine the hierarchical interpolation functions in L1, L2, L3 area coordinates for the 10-node cubic order triangle shown in Fig. 5.30(b). Express your hierarchic shape functions in a form such that interpolation is given by 3
f (L1, L2, L 3 )
8
Na(L1, L2, L3) fa -k- ~ Na(L1, L2, L3)A fa
-- ~ a=l
a=5 12
+ ~ Na(L1, L2, L3)Afa -t" N 1 3 ( L 1 ,
L2,
L3)AAfl3
a=ll
5.6 Using the shape functions developed in Problem 5.4, determine the modified shape functions to degenerate the cubic 16-node quadrilateral into the cubic 10-node triangular element using numbering as shown in Fig. 5.30. The final element must be completely consistent with the shape functions developed in Problem 5.5. 5.7 Degenerate an 8-node hexahedral element to form a pyramid form with a rectangular base. Write the resulting shape functions for the remaining 5 nodes. 5.8 For the triangular element shown in Fig. 5.31 show that the global coordinates may be expressed in local coordinates as 6
x- ~ a=l
Na(Lb) Xa
"
-
12L2 + 18L3
Problems
5.9 For the triangular element shown in Fig. 5.31 compute the integrals fA N2N3 dA and f zx N2 N4 dA using: (a) Eq. (4.31) and (b) an appropriate numerical integration using Table 5.3. 5.10 For the triangular element shown in Fig. 5.31 compute the integrals fA Na dA; a = 1, 2 , - . . , 6 using: (a) Eq. (4.31) and (b) an appropriate numerical integration using Table 5.3. 5.11 The 4-node quadrilateral element shown in Fig. 5.32 is used in the solution of a problem in which the dependent variable is a scalar, u. (a) Write the expression for an isoparametric mapping of coordinates in the element. (b) Determine the location of the natural coordinates ~ and r/which define the centroid of the element. (c) Compute the expression for thejacobian transformation J of the element. Evaluate the jacobian at the centroid. (d) Compute the derivatives of the shape function N3 at the centroid. 5.12 A triangular element is formed by degenerating a 4-node quadrilateral element as shown in Fig. 5.33. If node 1 is located at (x, y) = (10, 8) and the sides are a = 20 and b = 30: (a) Write the expressions for x and y in terms of ~ and r/. (b) Compute the jacobian matrix J (~, r/) for the element. (c) Compute the jacobian J (~, r/). (d) For a one-point quadrature formula given by
I =
fill 1
f (~, rl) d~ do -- f (~i, Tli)Wi
1
determine the values of Wi, ~i and 0i which exactly integrate the jacobian J (and thus also any integral which is a constant times the jacobian). (e) Is this the same point in the element as that using triangular coordinates La and the one-point formula from Table 5.3? If not, why? 10 A v
11
12
9
3
16
15
13
14
--
5
3=4
w
t
6
8
7
11
/
,3
k
v
2
(a) 16-node quadrilateral Fig. 5.30 Degeneration of cubic triangle for Problem 5.5.
1
5
6
(b) 10-node triangle
2
181
182
Mapped elements and numerical integration
3
6
1
3
)
5
4
2
Fig. 5.31 Quadratic triangle. Problems 5.8 to 5.10. )k
y
4 )
X I
f
6
.._]2
Fig. 5.32 Quadrilateral for Problem 5.11. 5.13 In some instances it is desirable to perform numerical integration in which quadrature points are located at the end points as well as at interior points. One such formula is the Newton-Cotes type shown in Fig. 5.16(a); however, a more accurate formula (known as Gauss-Lobatto quadrature) may be developed as
l 1f(se) dse 1
f(~i)Wi
[ f ( - 1 ) 4- f(1)]Wo + i=1
Determine the location of the points ~i and the value of the weights Wi which exactly integrate the highest polynomial of f possible. Consider: (a) The three-point formula (n = 1). (b) The four-point formula (n = 2). 5.14 Write the blending function mapping for a two-dimensional quadrilateral region which has one circular edge and three straight linear edges. Make a clear sketch of the region defined by the function and a 3 • 3 division into 4-node quadrilateral elements.
Problems
~,Y
b
! !
_~L()
1L
a
..]2 x
Fig. 5.33 Degeneratetriangle for Problem5.12. 5.15 Consider a 6-node triangular element with straight edges in which two of the midside nodes are placed at the quarter point. Show that the interpolation along the edge produces a derivative which varies as 1/~/7 where r is the distance measured from the vertex. 5.16 Compute the x and y derivatives for the shape function of nodes 1, 7 and 10 of the quadratic triangular prism shown in Fig. 4.20(b). 5.17 Program development project: Extend the program system started in Problem 2.17 to permit mesh generation using as input a 4-node isoparametric block and mapping as described in Sec. 5.13. The input data should be the coordinates of the block vertices and the number of subdivisions in each direction. Include as an option generation of coordinates in r, 0 coordinates that are then transformed to x, y cartesian form. (Hint: Once coordinates for all node points are specified, MATLAB can generate a node connection list for 3-node triangles using DELAUNAY.t A plot of the mesh may be produced using TRIMESH.) Use your program to generate a mesh for the rectangular beam described in Example 2.3 and the curved beam described in Example 2.4. Note the random orientation of diagonals which is associated with degeneracy in the Delaunay algorithm (viz. Chap. 8). 5.18 Program development project: Extend the mesh generation scheme developed in Problem 5.17 to permit specification of the block as a blending function. Only allow two cases: (i) Lagrange interpolation which is linear or quadratic; (ii) circular arcs with specified radius and end points. Test your program for the beam problems described in Examples 2.3 and 2.4.
t In Chapter 8 we discuss mesh generation and some of the difficulties encounteredwith the Delaunaymethod.
183
184 Mappedelements and numerical integration
1. I.C. Taig. Structural analysis by the matrix displacement method. Technical Report No. S.0.17, English Electric Aviation Ltd, April 1962. Based on work performed 1957-58. 2. B.M. Irons. Numerical integration applied to finite element methods. In Proc. Conf. on Use of Digital Computers in Structural Engineering, University of Newcastle, 1966. 3. B.M. Irons. Engineering applications of numerical integration in stiffness methods. J. AIAA, 4:2035-2037, 1966. 4. S.A. Coons. Surfaces for computer aided design of space forms. Technical Report MAC-TR-41, MIT Project MAC, 1967. 5. A.R. Forrest. Curves and surfaces for computer aided design. Technical Report, Computer Aided Design Group, Cambridge, England, 1968. 6. G. Strang and G.J. Fix. An Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs, N.J., 1973. 7. E.L. Wachspress. High order curved finite elements. Int. J. Numer. Meth. Eng., 17:735-745, 1981. 8. M. Crochet. Personal communication, 1988. 9. Nam-Sua Lee and K.-J. Bathe. Effects of element distortion on the performance of isoparametric elements. Int. J. Numer. Meth. Eng., 36:3553-3576, 1993. 10. R.E. Newton. Degeneration of brick-type isoparametric elements. Int. J. Numer. Meth. Eng., 7:579-581, 1974. 11. B.M. Irons. A technique for degenerating brick type isoparametric elements using hierarchical midside nodes. Int. J. Numer. Meth. Eng., 8:209-211, 1974. 12. M. Abramowitz and I.A. Stegun, editors. Handbook of Mathematical Functions. Dover Publications, New York, 1965. 13. B.M. Irons. Quadrature rules for brick based finite elements. Int. J. Numer. Meth. Eng., 3:293-294, 1971. 14. T.K. Hellen. Effective quadrature rules for quadratic solid isoparametric finite elements. Int. J. Numer. Meth. Eng., 4:597-600, 1972. 15. R. Radau. l~tudes sur les formules d' approximation qui servent h calculer la valeur d'une intrgrale drfinie. Journ. de math., 6:283-336, 1880. 16. R.G. Anderson, B.M. Irons, and O.C. Zienkiewicz. Vibration and stability of plates using finite elements. Int. J. Solids Struct., 4:1033-1055, 1968. 17. P.C. Hammer, O.P. Marlowe, and A.H. Stroud. Numerical integraion over symplexes and cones. Math. Tables Aids Comp., 10:130-137, 1956. 18. C.A. Felippa. Refined finite element analysis of linear and non-linear two-dimensional structures. Ph.D. dissertation, Department of Civil Engineering, SEMM, University of California, Berkeley, 1966. Also: SEL Report 66-22, Structures Materials Research Laboratory. 19. G.R. Cowper. Gaussian quadrature formulas for triangles. Int. J. Numer. Meth. Eng., 7:405-408, 1973. 20. I. Fried. Accuracy and condition of curved (isoparametric) finite elements. J. Sound Vibration, 31:345-355, 1973. 21. I. Fried. Numerical integration in the finite element method. Comp. Struct., 4:921-932, 1974. 22. M. Zlamal. Curved elements in the finite element method. SlAM J. Num. Anal., 11:347-362, 1974. 23. D. Kosloff and G.A. Frasier. Treatment of hour glass patterns in low order finite element codes. Int. J. Numer. Anal. Meth. Geomech., 2:57-72, 1978. 24. T. Belytschko and W.E. Bachrach. The efficient implementation of quadrilaterals with high coarse mesh accuracy. Comp. Meth. Applied Mech. Eng., 54:276-301, 1986. 25. O.C. Zienkiewicz and D.V. Phillips. An automatic mesh generation scheme for plane and curved surfaces by isoparametric coordinates. Int. J. Numer. Meth. Eng., 3:519-528, 1971.
References 185 26. W.J. Gordon and C.A. Hall. Construction of curvilinear co-ordinate systems and application to mesh generation. Int. J. Numer. Meth. Eng., 3:461-477, 1973. 27. W.J. Gordon and C.A. Hall. Transfinite element methods - blending-function interpolation over arbitrary curved element domains. Numer. Math., 21:109-129, 1973. 28. R.W. Thatcher. On the finite element method for unbounded regions. SIAMM J. Num. Anal., 15(3):466-476, 1978. 29. P. Silvester, D.A. Lowther, C.J. Carpenter, and E.A. Wyatt. Exterior finite elements for 2-dimensional field problems with open boundaries. Proc. IEEE, 123(12), Dec. 1977. 30. S.E Shen. An aerodynamicist looks at the finite element method. In R.H. Gallagher et al., editor, Finite Elements in Fluids, volume 2, pages 179-204. John Wiley & Sons, New York, 1975. 31. O.C. Zienkiewicz, D.W. Kelley, and P. Bettess. The coupling of the finite element and boundary solution procedures. Int. J. Numer. Meth. Eng., 11:355-375, 1977. 32. P. Bettess. Infinite Elements. Penshaw Press, Cleadon, U.K., 1992. 33. P. Bettess and O.C. Zienkiewicz. Diffraction and refraction of surface waves using finite and infinite elements. Int. J. Numer. Meth. Eng., 11:1271-1290, 1977. 34. G. Beer and J.L. Meek. Infinite domain elements. Int. J. Numer. Meth. Eng., 17:43-52, 1981. 35. O.C. Zienkiewicz, C. Emson, and P. Bettess. A novel boundary infinite element. Int. J. Numer. Meth. Eng., 19:393-404, 1983. 36. P. Bettess. Infinite elements. Int. J. Numer. Meth. Eng., 11:53-64, 1977. 37. O.C. Zienkiewicz, R.L. Taylor, and P. Nithiarasu. The Finite Element Method for Fluid Dynamics. Butterworth-Heinemann, Oxford, 6th edition, 2005. 38. J.J. Oglesby and O. Lomacky. An evaluation of finite element methods for the computation of elastic stress intensity factors. J. Eng. Ind., 95:177-183, 1973. 39. J.R. Rice and D.M. Tracey. Computational fracture mechanics. In S.J. Fenves et al., editor, Numerical and Computer Methods in Structural Mechanics, pages 555-624. Academic Press, New York, 1973. 40. A.A. Griffiths. The phenomena of flow and rupture in solids. Phil. Trans. Roy. Soc. (London), A221:163-198, 1920. 41. D.M. Parks. A stiffness derivative finite element technique for determination of elastic crack tip stress intensity factors. Int. J. Fract., 10:487-502, 1974. 42. T.K. Hellen. On the method of virtual crack extensions. Int. J. Numer. Meth. Eng., 9:187-208, 1975. 43. J.R. Rice. A path-independent integral and the approximate analysis of strain concentration by notches and cracks. J. Applied Mech., ASME, 35:379-386, 1968. 44. P. Tong and T.H.H. Pian. On the convergence of the finite element method for problems with singularity. Int. J. Solids Struct., 9:313-321, 1972. 45. T.A. Cruse and W. Vanburen. Three dimensional elastic stress analysis of a fracture specimen with edge crack. Int. J. Fract. Mech., 7:1-15, 1971. 46. P.E Walsh. The computation of stress intensity factors by a special finite element technique. Int. J. Solids Struct., 7:1333-1342, 1971. 47. P.E Walsh. Numerical analysis in orthotropic linear fracture mechanics. Inst. Eng. Australia, Civ. Eng. Trans., 15:115-119, 1973. 48. D.M. Tracey. Finite elements for determination of crack tip elastic stress intensity factors. Eng. Fract. Mech., 3:255-265, 1971. 49. R.D. Henshell and K.G. Shaw. Crack tip elements are unnecessary. Int. J. Numer. Meth. Eng., 9:495-509, 1975. 50. R.S. Barsoum. On the use of isoparametric finite elements in linear fracture mechanics. Int. J. Numer. Meth. Eng., 10:25-38, 1976. 51. R.S. Barsoum. Triangular quarter point elements as elastic and perfectly elastic crack tip elements. Int. J. Numer. Meth. Eng., 11:85-98, 1977.
186 Mappedelements and numerical integration 52. H.D. Hibbitt. Some properties of singular isoparametric elements. Int. J. Numer. Meth. Eng., 11:180-184, 1977. 53. S.E. Benzley. Representation of singularities with isoparametric finite elements. Int. J. Numer. Meth. Eng., 8:537-545, 1974. 54. N. Levy, P.V. Marqal, W.J. Ostergren, and J.R. Rice. Small scale yielding near a crack in plane strain: a finite element analysis. Int. J. Fract. Mech., 7:143-157, 1967. 55. B.M. Irons. Economical computer techniques for numerically integrated finite elements. Int. J. Numer. Meth. Eng., 1:201-203, 1969.
In this and the next chapter we deal with the set of problems in elasticity and fields which are common in various engineering applications and will serve well to introduce practical examples of application of the various element forms discussed in the previous chapters. Specifically, in this chapter we again consider the problem of stress analysis for linear elastic solids which was introduced in Chapter 2 and briefly discussed in Sec. 3.4. The simplest two-dimensional continuum element is a triangle. In three dimensions its equivalent is a tetrahedron, an element with four nodal corners.t Two-dimensional elastic problems were the first successful examples of the application of the finite element method, l' 2 Indeed, we have already used this situation to illustrate the basis of the finite element formulation in Chapter 2 where the general relationships were derived. The first suggestions for the use of the simple tetrahedral element appear to be those of Gallagher et al. 3 and Melosh. 4 Argyris 5, 6 elaborated further on the theme and Rashid and Rockenhauser 7 were the first to apply three-dimensional analysis to realistic problems. It is immediately obvious, however, that the number of simple tetrahedral elements which has to be used to achieve a given degree of accuracy has to be very large and this will result in very large numbers of simultaneous equations in practical problems. This leads to large compute times when direct solution schemes based on Gauss elimination are used. Thus, in recent times there is increased interest in use of iterative solution methods. To realize the order of magnitude of the problems presented let us assume that the accuracy of a triangle in two-dimensional analysis is comparable to that of a tetrahedron in three dimensions. If an adequate stress analysis of a square, two-dimensional region requires a mesh of some 20 x 20 - 400 nodes, the total number of simultaneous equations is around 800 given two displacement variables at a node (this is a fairly realistic figure). The bandwidth of the matrix involves 20 nodes, i.e., some 40 variables. An equivalent three-dimensional region is that of a cube with 20 • 20 • 20 = 8000 nodes. The total number of simultaneous equations is now some 24000 as three displacement variables have to be specified. Further, the bandwidth now involves an interconnection of some 20 • 20 = 400 nodes or 1200 variables. Given that with direct solution techniques the computation effort is roughly proportional to the number of equations and to the square of the bandwidth, the magnitude of the problems can be appreciated. It is not surprising therefore that efforts to improve accuracy by use t The simplestpolygonal shape which permits the approximation of the domain is known as the simplex. Thus a triangular and tetrahedral element constitutes the simplex forms in two and three dimensions, respectively.
188
Problems in linear elasticity
Fig. 6.1 Two-dimensional analysis types for plane stress, plane strain and axisymmetry. of higher order elements was strongest in the area of three-dimensional analysis. 8-12 The development and practical application of such elements will be described in this chapter. We shall deal with elasticity problems for general three-dimensional applications, as well as its simplification to some special two-dimensional situations. What we mean by two dimensions is that the total field should be capable of being defined by two components of displacement in the manner similar to that illustrated in Chapter 2. The two-dimensional problems we consider are of three types: (a) The plane stress case dealt with in Chapter 2 and shown in Fig. 6.1 (a). In this problem the only non-zero stresses are those in the plane of the problem and normal to the lamina we have no stresses. (b) The second case where again two displacement components exist is that ofplane strain in which all straining normal to the plane considered is prevented. Such a situation may arise in a long prism which is being loaded in the manner shown in Fig. 6.1(b). (c) The third and final case of two-dimensional analysis is that in which the situation is axisymmetric. Here the plane considered is one at constant 0 in a cylindrical coordinate system r, z, 0 [Fig. 6.1(c)] and again two displacements define the state of strain. We assume the reader is familiar with the theory of linear elasticity; however, for completeness we will summarize the basic equations for the different problem classes to be considered. For a more general discussion the reader is referred to standard references on the subject (e.g., see references 13-18).
Although in some situations the use of indicial notation is advantageous as discussed in Appendix B, for simplicity we choose to continue here the matrix form of definitions.
6.2.1 Displacement function For the three-dimensional problem the displacement field is given by
Governing equations 189
u(x,y,z) } u =
(6.1)
v(x, y, z)
w(x, y,z) where positions are denoted by the cartesian coordinates x, y, z. For the two-dimensional cases considered the displacement field is given by
u(x, y) } u =
(6.2)
v(x, y)
for plane stress and plane strain problems; and by
u(r,z)} v(r,z)
u =
(6.3)
for problems with axisymmetric deformation. The only difference in the latter two is the coordinates used: x, y for cartesian coordinates and r, z for cylindrical coordinates. One may notice that in plane stress problems, changes of thickness occur; however, no explicit displacement assumption is given and the result will be included directly within the strain approximation.
6.2.2 Strain matrix The strains for a problem undergoing small deformations are computed from the displacements. The form of the strain was given in Eq. (2.13) as e = Su
(6.4)
where ,$ is a differential operator and u the displacement field. We write the six independent components of strain in e where
0
~'X ~y Sz
Yxy Yyz Yzx
0 0 0 Oy 0 0 _Oz
o
0 ~ Oy 0 m
Ox az 0
o
0 0 Oz 0
{u} V
(6.5)
1/3
a ay 0 Ox
Note that in matrix form shear strain components are twice that given in tensor form in Appendix B (e.g., Yxy = 2 exy).
190
Problems in linear elasticity
~z
,....-
i
"..'%L..
Ez (~z)
Fig. 6.2 Strains and stress involved in the analysis of axisymmetric solids. For convenience in considering all three classes of two-dimensional problems in a unified manner, we include four components of strain in e and write them as
"0~ Ox
O-
8y
o
5-;y
8z
0
0
0 Oy
O Ox
~X
gxy
a
(u)+0 0
V
8z
= S u + ez
(6.6)
0
for plane problems (where ez is zero for plane strain but not for plane stress) and
"0~ Or
o Oz
~r ~_._
8z
--
8O
Yrz
O-
1 r
0
a
a
_Oz
Or_
u / = Su /3
(6.7)
for the axisymmetric case (see Fig. 6.2). The three problem types differ only by the presence of ez in the plane stress problem and the e0 component in the ,S operator of the axisymmetric case.
6.2.3 Equilibrium equations The equilibrium equations for the three-dimensional behaviour of a solid were presented in Sec. 3.4. They may be written in a matrix form as ,STir + b = 0
(6.8)
Governing equations 191 where ,S is the same differential operator as that given for strains in (6.5); or is the array of stresses which are ordered as or =
EtTx ,
CTy ,
O'z ,
"gx y ,
"gy z ,
"gzx
IT
(6.9)
and b is the vector of body forces given as b=[bx,
by,
(6.10)
bz] T
In two-dimensional plane problems we omit ryz, rzx and bz. For axisymmetric problems the stress is replaced by Or "--[or r ,
O"z ,
O"0 ,
b=
[br,
bz] T
75rz]T
(6.11)
and body force by (6.12)
6.2.4 Boundaryconditions Boundary conditions must be specified for each point on the surface of the solid. Here we consider two types of boundary conditions: (a) displacement boundary conditions and (b) traction boundary conditions. Thus we assume that the boundary may be divided into two parts, Fu for the displacement conditions and l-'t for the traction conditions. Displacement boundary conditions are specified at each point of the boundary Fu as
u=~
(6.13)
where fi are known values. Traction boundary conditions are specified for each point of the boundary Ft and are given in terms of stresses by t = GTor = t
(6.14)
in which G T is the matrix G T --
IOx
0 ny 0
0 0 nz
ny nx 0
0 nz ny
nz] 0 nx
(6.15)
where nx, ny, nz are the direction cosines for the outward pointing normal to the boundary Ft. It should be noticed that the matrices G and ,S have identical non-zero structure. In two dimensions G reduces to GT
[oX
0
ny
0
0
ny] nx
(6.16)
with nx, ny the components of an outward pointing unit normal vector of the boundary. In axisymmetry nx = nr and ny nz. "
-
-
192
Problems in linear elasticity
Fig. 6.3 Repeatabilitysegmentsand analysis domain (shaded).
Boundary conditions on inclined coordinates
Each of the two boundary condition types are vectors and, at any point on the boundary, it is possible to specify some components by displacement conditions and others by traction ones. It is also possible that the conditions are given with respect to coordinates x', y', z' which are oriented with respect to the global axes x, y, z by x' = Tx
(6.17)
where T is an orthogonal matrix of direction cosines given by T=
cosx cos x cosx 1 cos(y',y) cos(y',z)
|cos(y',x)
Lcos(z',x)
cos(z',y)
=
cos(z',z)
t21 t31
t22 t32
t13] t23 t33
(6.18)
in which cos(x', x) is the cosine of the angle between the x' direction and the x direction. For two dimensions cos(x', z) = cos(z', x) = 0
cos(y', z) -- cos(z', y) -- 0 cos(z', z) = 1 The displacement and traction vectors transform to the prime system in exactly the same way as the coordinates; hence, we have T u = fi' t' - Tt = TGTcr = t'
u' =
Symmetry and repeatability
(6.19)
In many problems, the advantage of symmetry in loading and geometry can be considered when imposing the boundary conditions, thus reducing the whole problem to more manageable proportions. The use of symmetry conditions is so well known to the engineer and physicist that no statement needs to be made about it explicitly. Less known, however, appears to be the use of repeatability 19 when an identical structure (and) loading is continuously repeated, as shown in Fig. 6.3 for an infinite blade cascade. Here it is evident
Governing equations 193
Fig. 6.4 Repeatable sector in analysis of an impeller.
that a typical segment shown shaded behaves identically to the next one, and thus functions such as velocities and displacements at corresponding points of AA and BB are simply identified, i.e., UI -- UII
Similar repeatability, in radial coordinates, occurs very frequently in problems involving turbine or pump impellers. Figure 6.4 shows a typical three-dimensional analysis of such a repeatable segment. Proper use of symmetry and repeatability can reduce the required compute effort significantly. Similar conditions can obviously be imposed to enforce conditions of 'asymmetry' also.
Normal pressure loading
When a pressure loading is applied normal to a surface the traction may be specified as t = pn
(6.20)
where p is the magnitude of the traction and n again is the unit outward normal to the boundary. This is a condition which is often encountered in practical situations in which p arises from a fluid or gas loading in which tangential components are zero.
194
Problems in linear elasticity
6.2.5 Transformation of stress and strain The transformation of coordinates to a prime system may also be used to define transformations for stresses and strains. Expressing the stress in the cartesian tensor form
tYxx tYxy tYxz1
cr-I%x
(6.21)
O'yy Cryz
LOzx Ozy fizz
these transformations are given by t r ' = TtrT T
(6.22)
In a matrix form, however, we must transform the quantities using the forms identified in Appendix B. Using this we obtain the relations -tlltll
t12t12
tl3tl3
2 tllt12
2 t12t13
2 t13tll
ay,
t21 t21
/22t22
t23 t23
2 t21 t22
2 t22t23
2 t23 t21
az,
"
ay
t31 t31
t32t32
t33ta3
2 t31 t32
2 t32t33
2 t33t31
[ "Cxty,
tllt21
t12t22
t13t23
(tllt22 d- tlEt21)
(t12t23 at- t13t22)
(t13t21 -~- tllt23)
"Cytzt
t21t31
t22t32
t23t33
(t21t32 -~- t22t31 )
(t22t33 71- t23t32 )
(t23t31 -11-t21t33 )
-y]
7~Zt X t
.t31tll
t32t12
t33t13
(t31t12 -~- t32tll)
(t32t13 ~- t33tl2)
(t33tll -~- t31t13)_
rzx
r
az
ryz
= T~
(6.23) for stresses and Ext
Eyt
" tlltll
t12t12
t13t13
t11t12
t12t13
tl3tll
t21 t21
t22 t22
t23 t23
t21 t22
t22 t23
t23 t21
E E
Ezt
t31/31
ta2t32
t33t33
t31t32
t32t33
taat31
Yx'y'
2tllt21
2 tlEt22
2 t13t23
(tllt22 -~- t12t21)
(tlEt23 -~- tlat22)
(tlat21 -~- tllt23)
y y t zt
2t21t31 2 t22t32
2 t23t33
(t21/32 -~- t22t31)
(t22t33 -11-t23t32)
(t23t31 71- t21t33)
.2 t31tll
2 t33t13
(t31/12 -~- t32tll)
(t32t13 -~- t33t12)
(ta3tll -~- t31t13).
~ZI X !
2 t32t12
e' = Tee
(6.24) for strains. The differences between T~ and TE occur from the use of the engineering definition of shearing strain where we have introduced
Yxy -- 2 Exy,
etc.
If the principal material axes are oriented at angle of/3 with respect to the coordinate axes of the problem (Fig. 6.5), the two-dimensional representation of T~ is given by
T~=
COS2 fl sin2/3 0
sin 2/3 cos2/~ 0
0 0 1
sin/3 cos/3 ] - sin 0flcos/~ ]
- 2 sin/3 cos/3
2 sin/3 cos/3
0
cos 2/3 - sin 2/33
(6.25t
Governing equations 195 Table 6.1
Relations between isotropic elastic parameters
Parameters
E, v
K, G
Z,/x
9KG/(3K
+ 2G) (3K - 2 G ) / ( 6 K + 2G) -
/x(3~. + 2/z)/(~. + / x ) ~./(~. + / 2 ) / 2 ~. + 2/z/3
G K - 2G/3
tz -
E =
-
v =
-
K =
E/(1
G = Ix = L =
E/(1 + v)/2 r E ~ ( 1 + v)/(1
-
2v)/3 - 2v)
6.2.6 Elasticity matrix The stress-strain equations, also known as constitutive relations, for a linearly elastic material may be expressed by Eq. (2.16) or = D (e - Co) + oro
(6.26a)
e = D - 1 (or - - orO) + e0
(6.26b)
or by
The D matrix is known as the elasticity matrix o f m o d u l i and the D -1 matrix as the elasticity matrix o f compliances. 17 Without loss in generality, in the sequel we ignore the eo and oro terms. After obtaining final results they may be again added by replacing e and or by e - eo and or - oro, respectively.
Is o trop ic materials
We write a general expression for isotropic materials in terms of the six stress and strain terms. We may use any two independent elastic constants for an isotropic material. 15' 17 Here we use Young's modulus of elasticity, E, and Poisson's ratio, v. In Table 6.1 we indicate relationships between E, v and other parameters frequently encountered in the literature. Using cartesian coordinates, for example, the expression is given by ex ~,y ez Yx y
--"
Yyz ?'zx
1 -E
1 --U -v 0 0
-v 1
-v -v
-v 0 0 0
1 0 0 0
0 0 0
0 0 0
0 0 0
ax cry az
2(1 -1-1)) 0 0
0
0
"Yx y
2(1 + V) 0
0 2(1 + V)
"gyz rzx
(6.27)
Inverting to obtain the appropriate D matrix yields the result ax ay az rxy "gyz rzx
E =d-
(1 - v)( v l-v) v v v 0 0 0 0 0 0
v 0 0 v 0 0 (1 - v) 0 0 0 (1-2v)/2 0 0 0 (1 -- 2V)/2 0 0 0 (1 -
0 0 0 0 0 2v)/2
~'X F,y F,z
Yxy ?% Yzx
(6.28)
196
Problems in linear elasticity where d = (1 + v)(1 - 2v). The form of d places restrictions on the admissible values of v to keep the material parameters positive; thus, we have 1
-l
The limiting values of the parameters (namely - 1 and 1/2) are permitted by rewriting the material model with Lagrange multipliers replacing the terms which are indefinite. The case where v = 1/2 is associated with materials which are incompressible and we shall devote special attention to this problem in Chapter 11 since it has relevance for many applications in solid and fluid mechanics. Generally, of course, no material can be incompressible and we are only interested in the case where v --+ 1/2. However, even for this case we will find that care must be used when developing a finite element form. For isotropic materials the expression for two-dimensional problems is written in terms of the four stress and strain terms by omitting the last two rows and columns in (6.27). Using cartesian coordinates the expression then becomes ex ey ez ~xy
1 1 = -E
-v
-;
1 -v 0
-v
0
ax
-v
0 0 2(1 + v)
Cry ~z
1 0
0
(6.29)
"Cxy
For the plane stress case we must set ~z zero to compute the appropriate D matrix. This yields the result 1J
ez -
(6.30)
E (~x + ay)
and including this in the inverse of (6.29) we obtain ax Cry
E
=
az
(1
-
1)2)
"Yxy
1 v
v 1
0 0
0 0
ex ey
0
0
0
0
8z
0
0
0
(1 -- V)/2
Yxy
(6.31)
We have filled in the third column and row of the D array so that correct stresses are obtained. Indeed, if we deleted these we would once again get the result given in Chapter 2 and one may ask why we include the extra terms. The main reason is to permit a single form to be used for plane stress, plane strain and axisymmetric problems and therefore minimize the amount of programming needed to implement these in a computer program. For the plane strain and axisymmetric problems the inverse may be performed directly from (6.29), as all components of stress can exist. Accordingly, for these two cases we obtain (again writing for the cartesian coordinate form) ax ay az Txy
=
E
(1 + v)(1 - 2v)
(1 - v) v
v (1 - v)
v v
0 0
ex ey
v 0
v 0
(1 - v) 0
0
ez
(1 -- 2V)/2
)/xy
.32)
which is identical to the three-dimensional problem if the last two rows and columns of each array in (6.28) are omitted. Here we observe that the case where ez is zero is treated merely by inserting that value when computing stresses; however, az will always exist unless the Poisson ratio, v, is zero.
Governing equations 197
Anisotropic materials
We may write a general relationship for anisotropic linearly elastic materials as Crx
ay Crz rxy
=
ryz rzx
Dll D21 D31 D41 D51 D61
D12 D22 D32 D42 D52 D62
D13 D23 D33 D43 D53 D63
D14 D24 D34 D44 D54 D64
D15
D25 D35 D45 D55 D65
D16D26 D36 D46 D56 D66
~X
gy /?z
(6.33)
Yxy
Yyz Yzx
For an elastic material the D matrix must be symmetric and, hence, Dij : Dji. This results in a possibility of 21 elastic constants for the general problem. 16,20, 21 An important class of anisotropic materials is one for which three planes of symmetry exist and is called an orthotropic material. Here the principal axes are also in rectangular cartesian coordinates. For orthotropic materials it is common to define the elastic material parameters in terms of their Young's moduli, Poisson ratios and shear moduli. If we let x', y' and z' be the three axes of material symmetry, the elastic strain-stress relations may be expressed as "
~X t
~y, 8z~ Yx' y' Yyfzt yZt X t
1 Ex, Py'x' Ex, Vz'x' Ex,
Px'y' Ey,
Vx'z' Ez,
0
0
0
1 Ey, _ Vz,y, Ey,
VY'Z' Ex, 1 Ez,
0
0
0
0
0
0
0
0
crx,
0
0
0
1 Gx'y'
0
0
0
0
0
0
0
0
1
a y,z, 0
0
Cry, Crz'
"gx'y'
(6.34)
"gy'z' rz,x,
1 Gz,x,
where Ex,, Ey,,Ez, are elastic moduli; Px'y', Px'z', etc. are Poisson ratios; and Gx,y,, Gy,z,, Gz,x, are elastic shear modulus. Again symmetry of the D matrix results in 1)i,j, Vj, i, = (6.35) Ej, Ei, thus reducing the number of independent components for the three-dimensional cases considered to nine parameters (three direct moduli, three Poisson ratios, and three shear moduli). The elastic moduli for the D' matrix are computed by inverting the square matrix appearing in Eq. (6.34). The inverse may be written as o " = D' e' (6.36) If the principal material axes are expressed by the directions given in (6.17), it is necessary to transform Eq. (6.36) to the form (6.26a) before proceeding with an analysis. This is most easily performed by noting an equality of work given by o.tT~,l __ o.T~. ~D
I ~I = ~T D
(6.37)
198
Problems in linear elasticity
and using the expressions for transformation given by Eq. (6.24) in Eq. (6.37) gives D = T~D'T~
(6.38)
Such transformation also has been used in a slightly different context in Chapter 1 [viz. Eq. (1.25)] to transform a stiffness matrix. For treatment of anisotropic materials in the two-dimensional problems, it is necessary for the direction normal to the plane of deformation (i.e., the z direction for plane problems or 0 direction for axisymmetric problems) to be a direction of material symmetry. For this case we may write a general relationship for linearly elastic deformation as (using the cartesian form) tYx
O'y O'z Txy
i
Dll D21 -" O31 LD41
o41
D12 D22 D32 D42
D13 D23 D24 D33 034 D43 D44J
~y ez
(6.39)
Yxy
For an elastic material the D matrix must be symmetric and, hence, Dij -- Dji. This results in a possibility of ten elastic constants for the plane or axisymmetric problem. In order to consider the plane stress case it is again necessary to impose the constraint az = 0. If we solve for ez using the third row of (6.39) we have (6.40)
ez : -[D31 ex d- D32 ey + D34 Yxy]/D33 which may be substituted into the remaining three equations in (6.39) to give ax
[Dll
D12
0
D14
OY ' o.z :
[O21 b D 2 2 20 04 0 0
Txy
LD41 D42 0
D44
~X
6y
(6.41)
Ez Yxy
in which Dij : D i j -
(6.42)
Di3D331D3j
are reduced elastic moduli. For the plane stress and strain problems of an orthotropic material the principal axes are also in rectangular cartesian coordinates; however, for the axisymmetric problem the principal axes also must be axisymmetric (i.e., cylindrical orthotropy which is similar to rings of a tree). For orthotropic materials it is common to define their properties in terms of their Young's moduli, Poisson ratios and shear moduli. If we let x', y' (or r', z') be the two axes of material symmetry in the plane of deformation, the elastic strain-stress relations may be expressed by (6.34), after omitting the last two rows and columns of each array. Accordingly, 1 ~,Xt
~'Y' F,Zp Yx'y'
:
Ex, 1)y,x, Ex,
1)x,yt
1)x, z,
Ey, 1 Ey,
l)y' zt
O'X
EX t
Cry,
Ezt
1)ZtXt
VZ~yt
1
az,
EX
Ey,
E z,
Zx,y,
0
0
Gx,y,
(6.43)
Governing equations 199
Fig. 6.5 Coordinate definition for transformation of material axes.
After considering symmetry the number of independent components for the two-dimensional cases considered reduces to seven parameters (three direct moduli, three Poisson ratios, and one shear modulus). The elastic moduli for the D' matrix are computed by inverting the square matrix appearing in Eq. (6.43). If plane stress is considered, it is necessary to ensure crz, is zero. Inserting expression (6.24) into Eq. (6.37) again gives W ! D = T6D TE
(6.44)
Example 6.1" Anisotropic, stratified, material. With the z axis representing the normal to the planes of stratification, as shown in Fig. 6.6, we can rewrite (6.43) (again ignoring the initial strains and stresses for convenience) as O"r E1 P2ffr
E1 ~0 - -
P20"z
PlO'0
E2 ffz
E1 P2ff0
t
E2
E1
Plffr
P2~z
if0
El
E2
E1
"Crz
(6.45)
G Writing the parameters as E1
E2
~n~
o
G E2
=m
and
d = (1 + Vl)(1
-
Pl -
2nv 2)
200
Problems in linear elasticity
~ z
>,r
Fig. 6.6 Axisymmetrically stratified material. we have on solving for the stresses that I n ( 1 - nv2), D = E2 |nvz(1 + l)l), -d- | n ( v l + n v ] ) ,
L
o,
nv2(1 + Pl), 1 - v 2, nv2(1 + v , ) ,
n(vl + nv2), nv2(1 + 1)1), n(1-nv]),
0
o,
md
o,
0
(6.46)
Initial s t r a i n - thermal effects
Initial strains may be due to many causes. Shrinkage, crystal growth, or temperature change will, in general, result in an initial strain vector
eo--[SxO
8yO F~zO YxyO YyzO }/ZX0]T
(6.47)
The initial strain will usually depend on position which may be included by interpolation using shape functions. As an example, consider the effects of change in temperature in an isotropic material. The initial strain for a temperature change A T = T - To (with To a temperature where no straining is caused) with a linear coefficient of thermal expansion ot is given by e0 = ot AT m
(6.48)
where m-[1
1
1
0
0
0] T
(6.49)
For an isotropic material normal strains ex, ey, e z are all equal and no shear strains are caused by a temperature change.
Finite element approximation
Anisotropic materials present no special problems with the coefficients of thermal expansion varying with direction in the material. For example, in an orthotropic material no shearing strains are caused for the principal material directions and we may replace (6.48) by S'o--AT [Otx, Oly, Olz, 0 0 0] T (6.50) It is now again necessary to use the transformation between principal material directions and those used for coordinates of the analysis using S0
= T~-1 s o'
(6.51)
where Ts is given by (6.24). Now, in general, the ~xyO, YyzO, YzxOcomponents are no longer equal to zero.
The above describes the governing equations for three-dimensional behaviour of solids. Except for the elastic constitutive equations, all the remaining equations are valid for general materials undergoing small deformations. A finite element solution process for the equations may be established by: 1. 2. 3. 4. 5. 6. 7. 8. 9.
Using the virtual work (or weak form), equations for equilibrium given in Sec. 3.4. Introducing an approximation for the displacement field u in terms of shape functions. Computing strains from (6.5). Computing stresses from (6.26a) where for linear elastic behaviour we use one of the forms given in Sec. 6.2.6 for the D matrix. Performing the integrations over each element (usually by quadrature). Assembling the element contributions to form the global stiffness and load arrays. Imposing the known traction and displacement boundary conditions. Solving the resulting stiffness and load matrices. Reporting desired parts of the solution.
The above steps describe a general solution framework which we shall follow in all subsequent developments for solutions of problems in solid mechanics. Differences will occur later in the types of weak forms used and in the expressions for strains and constitutive equations. Otherwise the steps are standard. To illustrate the process we first consider the general three-dimensional problem in which the virtual work expression is given by [see (3.46) in Sec. 3.4] ~ , s T , df~ - ~ ,uTb dr2 - / ;
,uZt dF = 0
To simplify the solution process we split the boundary into Fu and l"t and introduce the known traction boundary condition. We shall also impose the displacement boundary conditions in the approximations for u and assume the virtual displacement ~u vanishes on Fu. For linear elastic behaviour we also may introduce the constitutive equation given by (6.26a) and thus our virtual work equation simplifies to
f,s~r[o'o+D(s-so)]
dr2-f
,uTbdf2-fr
,uZtdF = 0 t
with the constraint u = ~ on Fu.
(6.52)
201
202
Problems in linear elasticity
I
e= CO(h 2)
>,
/ Fig. 6.7 Approximation of curved surface by linear element.
In the finite element solution we divide all integrals to be sums over individual elements and approximate the weak form by
~/n 'eT[~176176 e
'uTbd~-e~fr'uTtdF=O
(6.53)
e
e
e
where ~'2e and F t denote element domains and parts of boundaries of any element where tractions are specified, respectively. The 'approximation' in this step is associated with the fact that for curved boundary surfaces the sum of element domains ~,~eis not always exactly equal to f2, nor is the sum of F t equal to Ft. This is easily observed for approximations using linear elements as shown in Fig. 6.7. We observe that the error is O(h 2) which is exactly the same as the error in displacement from shape functions using linear polynomials. Thus, the order of error in our solution is not increased by the boundary approximation.
Displacement and strain approximation
At this point we can introduce the finite element shape function expressions to define displacements. Accordingly, we have
U ,'~ ll :
{0} 0a}No { ~3
~)a
~13
t13a
-- ~
Nafla
(6.54)
a
where f i a , V a , Wa are nodal values of the displacement. Any of the three-dimensional interpolations given in Chapter 4 may be used to define the shape functions Na. Inserting
Finite element approximation 203
Fig. 6.8 8-node brick element. Local node numbering. the interpolation into Eq. (6.5) gives
FaNa
E--
Yxy Vy~ Yzx
o
ONa
0
Oy
~x
Ey Ez
o
0
~=~ a
|aNa
I ONa L Oz
aNa Oz
Oga
0
ONa
ONa
Oz
Oy
0
ONa
{OaI ~3a ~)a
-" ~
Batla
(6.55)
a
Ox
A similar expression may be written for virtual strains. Example 6.2: Strains for 8-node brick. As an example we consider the 8-node brick element shown in Fig. 6.8. The shape functions are given by
Na = g1 (1 + ~'a~')(1 + qar/)(1 + (a() for which the derivatives with respect to ~, 1/, ( are given by
204
Problems in linear elasticity
ONa
1
ONa
1
ONa
1
O~ = 8 ~a (1 + r/at/)(1 -+- (a() 0/1 = ~ /~a (1 + (a()(1 + ~a~)
For an 8-node brick element, the jacobian matrix in (5.11) may be expressed as 0b(1 +(b()(1 + ~b~)
[Xb Yb Zb]
The shape function derivatives are now obtained from (5.10) as
ONo'
ONa
~a(1 + 0a0)(1 + (a()
1
= ~ J-~
r/a(1 + (a()(1 + ~a~)
ONa
(a(1 + ~a~)(1 + /70/7)
These may be used directly to define the Ba strain matrix given in Eq. (6.55). For two-dimensional problems the finite element shape function expressions to define the displacements are given by U~fi:
(U/~ "- Z a
Na (~'la a
-:
~-~ Natla
(6.56)
a
Inserting the interpolation into Eqs (6.6) and (6.7) gives
-aNa Ox ~x
8z
za
Yxy
0
aNa Oy
0
0
aNa Oy
aNa Ox
0
{blal Ua 7t-
0 Ez 0
= ZBa~la_+_~ z
(6.57)
a
for the plane problems and
-aNa Or ~r 8O
Yrz
o
Ea
r -
0 aNa fia o
aNa
aNa
Oz
Or
-
Zo "a~
(6.58)
Finite element approximation for the axisymmetric problem, respectively. In the form for axisymmetry the radius is computed from the parametric form given by Eq. (5.2). Accordingly, for this case we have r - ~
N~ rb
(6.59)
b where rb are locations of the node points defining the N~ functions.
Stiffness and load matrices
Introducing the above approximations into the weak form (6.53) results in
e
,fT
f~ B T[O.0q_D(nbfb_~0)] d ~ - f ~ e
e
Nabds
Natdl-'] : 0
(6.60)
which after summing the element integrals and noting that ~fa is arbitrary gives the system of linear equations Kab fib + fa : 0 (6.61) where Kab = ~ / ~ e e
BTDBb dr2
fa : ~ ~ e e
(6.62)
InT(~176176
e
-
The integration over each element domain may be computed by quadrature in which
L3 f~ (.)d~'2 : fD (.)Jd[-] ~,~ ~ e l=1 L2
(.)l Jl Wl
fF (') d F -
(')l jl Wl
fD (.) jdE] ~ ~ l=1
(6.63)
where J = det J is the determinant of the jacobian transformation between the global and local volume coordinate frames, j - det j is the determinant of the jacobian transformation between the global and local surface coordinate frames, and subscript I is associated with each quadrature point with weight WI. The points and weights are taken from the tables given in Chapter 5.
Example 6.3: Quadrature for 8-node brick element. For an 8-node brick element it is sufficient to perform volume integrals using a 2 • 2 • 2 formula. Thus L3 in (6.63) is equal to 8 and the points and weights may be given ast 1 ~I Ol ~l WI
1 --C --C --C 1
2 C --C --C 1
3 --C C --C 1
4 C C --C 1
5 --C --C C 1
6 C --C C 1
7 --C C C 1
where c = 1/~/-3.
t Note that ordering is unimportant and any other permutation for I is permissible.
8 C C C 1
205
206
Problems in linear elasticity Similarly for the surface integrals a 2 • 2 formula may be used and give L2 equal 4 with points and weights ordered as l ~1 /71
Wl
1
2
3
4
--C --C 1
C --C 1
--C C 1
C C 1
in which again c = 1/~/~. These formulae always have error equal to or less than that in the approximation of domain or in the shape functions. Hence, it is never necessary to use higher order quadrature than the above. 22 A similar form holds for all the two-dimensional cases; however, the volume and surface elements are different for the plane and axisymmetric problems. For plane stress dr2 - t dx dy
and dF = t ds
(6.64)
where t is the thickness of the slab and may vary over the two-dimensional domain; for plane strain dS2 = dx dy and dF = ds (6.65) where a unit thickness is considered; and for axisymmetry t dr2 = 2zrr dr dz
and dF = 2zrr ds
(6.66)
In the above (6.67)
ds - (dx 2 -k- dy 2) 1/2
for the plane problem with a similar expression for axisymmetry. The finite element arrays may now be computed from (6.60) for quadrilateral-shaped elements of lagrangian or serendipity type; the stiffness and load matrices defined in (6.62) are computed using gaussian quadrature as Keb =
/if, 1
Z
1
BaT (~, r/) DBb (~, r/) J (~, r/) d~ dr/
BT(~I' r/l) O l b (~l, r/l) J (~l, r/l)
(6.68)
WI
l
and
f'f'
fae =
1
f_l
-
1
1
[BaT (r 17)(cro - Deo) - Na (r r/)b] J (r r/)d~ dr/
-
N~(~) t j(~) d~
(6.69)
[BT(~I' r/l)(or0 -- De0) - Na(~1, r//)b] J(~l, 77l) WI
Z l
- ~
N~ (~l) i(~l) j (~1) wt I
f Some programs omit the 2zr in the definition of d~ and dF and compute matrices for one radian of arc.
Reporting of results: displacements, strains and stresses 207
in which J=tdetJ; J = detJ;
j=t j-
[(8~)2
+
(8y)2] 1/2 ~-~
; Plane stress
[(Ox)2 8y 21 1/2 ~-~ + (~-~) ; Plane strain
J = 2zrr det J; j - 2zrr
(6.70)
[(8r)2 ( 8 z ) 2 ] 1/2 ~-~ + ~ ; Axisymmetric
for the domain and boundary. The quadrature points are denoted as ~l and/7l and weights as In (6.68) and (6.70) t, D, b, i and initial stress and strain may vary in space in an arbitrary manner and det J is computed as indicated in Sec. 5.5. The simplicity of computation using shape functions and numerical integration should be especially noted. This permits easy consideration of different types of interpolations for the shape functions and different quadrature orders for numerical integration to be assessed.
WI.
The reader will now have observed that, while the finite element representation of displacements is in a sense optimal as it is the primary variable, both the strains and the stresses are not realistic. In particular, in ordinary engineering problems both strains and stresses tend to be continuous within a single material. The answers which are obtained by the finite element calculation result in discontinuities of both strains and stresses between adjacent elements. Thus if the direct calculation of these quantities were presented the answers would be deemed unrealistic. For this reason, from the beginning of the finite element method it was sought to establish these rather important quantities in a more realistic, and possibly more accurate, way. In the very early days of finite element calculation with simple Co continuity elements an averaging of element strains and stresses, which are constant in triangular elements, was made at each node. This of course gave improved results at most points -except at those which were on the boundary. Since the simple days of averaging further attention was given to this subject and other methods were developed. The first of these methodologies was developed by Brauchli and Oden 23 in 1971 and consisted of assuming that a continuous representation of either strain or stress using the same Co functions as for displacements could be found by solving a least squares sense representation of the corresponding discontinuous (finite element) one. This method proved quite expensive but often gave results which were superior to the simple averaging - at least for some sets of problems. However, higher accuracy was not achieved despite the additional cost of solving a full set of algebraic equations. An alternative local procedure to improve results was proposed by Hinton and Campbell 24 and was once quite widely used. The methods of recovery of strain and stress have progressed much further in recent years and in Chapter 13 we discuss these fully. We find that currently an optimal procedure, which generally gives higher order accuracy and has similar cost to simple averaging, is the patch recovery method. In this the process of determining values of recovered strains or stresses assumes that:
208
Problems in linear elasticity 1. At some points of the domain or each element, the strains and stresses calculated by the direct differentiation of the shape functions are more accurate than elsewhere 9 Indeed on many occasions at such points 'superconvergence' is demonstrated which can make the accuracy at least one order higher than that of the finite element values computed from derivatives of shape functions 9 2. A continuous representation of such strains and stresses can be given by finding nodal values which in the least squares sense approximate those computed by the optimal points. Now the increased accuracy will exist over the entire domain 9 The discussion of the existence of such points at which higher order may exist is deferred to Chapter 13 but here we show how this can be easily incorporated into standard programs dealing with elasticity 9 Basically in the procedure we will assume a strain exists for an element and can be expressed by e* = ~ Nb~b (6.71) b where now e is any component of strain. A similar expression may be written for a stress component. The goal is to find appropriate values for ~b which give improved results. To do this we use a least squares method in which the strain in a patch surrounding a vertex node a on elements may be expressed in global coordinates by a polynomial expression of higher order, suitable for the number of unknown parameters in the strain expression. This polynomial expression is given by
e**--[1,
(X-Xa),
(Y--Ya),
"'']
ea o~1 C~2 = P a ( x ) C~a
(6.72)
For 3-node triangles or 4-node quadrilaterals in two dimensions and 4-node tetrahedrons or 8-node brick elements in three dimensions a linear interpolation is used. For the quadratic order elements the polynomial in Pa is also raised to quadratic order. Thus, for 6-node triangular and 8- or 9-node quadrilateral elements in two dimensions we use
Pa = [1,
(X--Xa),
(Y--Ya),
( X - - X a ) 2,
(X--Xa)(y--ya),
( y - - Y a ) 2]
The parameters in e** are determined using the least squares problem given by 1 na
1-I --- ~ Z Z [Po(X~)O~o -- 8(x~)] 2 -- min e=l l
(6.73)
where na is the number of elements attached to node a and x~ are locations where strains are computed. The minimization condition results in
(6.74)
Maoza -- fa
where
na
na
Ma "~ Z Z PT(x~)Pa(x~) e=l 1
and
fa - Z Z PaT(X~)~(X~) e=l l
(6.75)
Numerical examples 209 The values for the remaining nodes (e.g., at mid-side and boundary locations) may be computed by averaging the extrapolated values computed from (6.72). For example, from a patch, the result at node b (b r a) is given by e** (Xb) --- Pa (Xb)Ota
and averaging the result from all patches which contain node b gives the final result for ~b. An identical process may be used to compute stress values. We recommend that a patch recovery method be used to report all strain or stress values. In addition, the method serves as the basis for error assessment and methods to efficiently construct adaptive solutions to a specified accuracy as we shall present in Chapter 14.
To illustrate the application of the theory presented above we consider some example problems. Some of the problems we include can be solved by other analytic methods and thus serve to illustrate the accuracy of results obtained. Others, however, are from more practical situations where either no alternative solution method exists or the method is otherwise cumbersome to obtain thus rendering the finite element approach most useful. Here we first solve again the two-dimensional plane stress problems considered in Chapter 2 to illustrate the advantages of using 4-, 9-, and 16-node quadrilateral isoparametric elements of lagrangian type.
Example 6.4: Beam subjected to end shear. The rectangular beam considered in Sec. 2.9.1 is solved again using lagrangian rectangular elements with 4 nodes (bilinear), 9 nodes (biquadratic) and 16 nodes (bicubic). The mesh for the bilinear model initially has six elements in the depth direction and 12 along the length for a total of 72 elements and 91 nodes. This is subsequently subdivided to form meshes with 12 • 24, 24 x 48, 48 x 96 and 96 x 192 elements. All other data are as defined in Sec. 2.9.1. The analysis is repeated using 9-node biquadratic elements with an inital mesh of 3 • 6 elements, which gives the same number of nodes. Finally, the problem is solved with a mesh of 2 • 3 16-node bicubic elements which again gives a mesh with 91 nodes. Since the exact solution for displacements given in Sec. 2.9.1 contains all polynomial terms of degree 3 or less the solution with this coarse mesh is exact and no refinement is needed. In Table 6.2 we present the results for the energy obtained from each mesh and in Fig. 6.9 we show the convergence behaviour for the 4-node and 9-node element forms. Again, the expected rates of convergence are attained as indicated by the slopes of 2 and 4 in the figure. Example 6.5: Circular beam subjected to end shear. We consider next the circular beam problem described in Sec. 2.9.1. The solution to the problem is performed using isoparametric 4-node bilinear quadrilaterals, 9-node lagrangian quadrilaterals and 16-node lagrangian quadrilaterals. The geometric and material data for the problem is as given in Example 2.4. The initial mesh for all element types uses a regular subdivision of the domain that produces initial element patterns with 6 x 12 4-node elements, 3 x 6 9-node elements and 2 • 4 16-node elements. The mesh for each element form is shown in Fig. 6.10.
210
Problems in linear elasticity 10 0
9
9
i
o 4-node quads J a 9-node quads
10 -2
0 t_
>, 10-4 fLU
10 - 6
.iiiiiiiiiiiiiiiiiiill
10 -8
,_
10-1
10 0
h/h 1
Fig. 6.9 Convergencein energy error for 4-node and 9-node rectangular elements.
Table 6.2
Mesh size and energy for end loaded beam
Nodes
Elmts
4-node rectangles
91
325 1225 4753 18721 Exact
72
288 1152 4608 18432 -
Energy
3077.4986
3238.2915 3281.3465 3292.3206 3295.0790 3296.0000
9-node rectangles Elmts
18
72 288 1152 4608 -
Energy
3294.7512
3295.9174 3295.9947 3295.9997 3296.0000 3296.0000
16-node rectangles Elmts
8
Energy
3296.0000
3296.0000
Results for the energy are given in Table 6.3 and compared to the exact value computed from
Eex -- 0.02964966844238 using the geometry and properties selected. The element size is normalized to that of the coarsest mesh [shown in Fig. 6.10(a)] and the energy error computed from Table 6.3 has the expected slope for 4-node elements, for 9-node elements and for cubic elements (viz. Fig. 6.11). We now consider some practical examples for problems which have been solved using the finite element method. Some simple typical examples are given which use both tetrahedral and isoparametric brick-type elements. The isoparametric examples are all performed using Gauss quadrature to approximate the necessary integrals.
Numerical examples
(a)
(b)
(c) .
.
.
.
.
.
Fig. 6.10 End loaded circular beam' Coarsemeshfor 4-node, 9-node and 16-nodelagrangianelements. Table 6.3 Mesh size and energy for curved beam
4-node quadrilateral Nodes Elmts Energy 91 72 0.03042038175071 325 288 0.02984351371323 1225 1152 0.02969820784232 4753 4608 0.02966180825828 18721 18432 0.02965270370808 Exact - 0.02964966844238
9-node quadrilateral Elmts Energy 18 0.02970101373401 72 0.02965318188484 288 0.02964989418870 1152 0.02964968266120 4608 0.02964966933301 - 0.02964966844238
16-node quadrilateral Elmts Energy 8 0.02965327376971 32 0.02964975296446 128 0.02964966996157 512 0.02964966846707 2048 0.02964966844276 - 0.02964966844238
6.5.1 A dam subject to external and internal water pressures A buttress dam on a somewhat complex rock foundation is shown in Fig. 6.12 and analysed.25, 26 This dam (completed in 1964) is of particular interest as it is the first to which the finite element method was applied during the design stage. The heterogeneous foundation region is subject to plane strain conditions while the dam itself is considered in a state of plane stress of variable thickness. With external and gravity loading no special problems of analysis arise. When pore pressures are considered, the situation, however, requires perhaps some explanation. It is well known that in a porous material the water pressure is transmitted to the structure as a body force of magnitude
bx --
0p
Ox
by =
0p
Oy
(6.76)
and that now the external pressure need not be considered. The pore pressure p is, in fact, now a body force potential which may be determined by solving a 'field problem' as described in the next chapter. Figure 6.12 shows the element subdivision of the region and the outline of the dam. Figure 6.13(a) and (b) shows the stresses resulting from gravity (applied to the dam only) and due to water pressure assumed to be acting as an external load or, alternatively, as an internal pore pressure. Both solutions indicate large tensile regions, but the increase of stresses due to the second assumption is important.
211
212
Problems in linear elasticity 10 0
10 - 2
10 .-4
i
o [] o
9
9
4 node 9 node 16node
I
9
i
9
i
!
i
: !
>,,
t-
LU 10 .-6
10 -8 9
10-10
i
10 -1
10 0 h/h 1
Fig. 6.11 Curved beam Convergence in energy error for quadrilateral elements.
The stresses calculated here are the so-called 'effective' stresses. These represent the forces transmitted between the solid particles and are defined in terms of the total stresses o" and the pore pressures p by cr' = ,7 + m p
m T = [1, 1, 0]
(6.77)
i.e., simply by removing the hydrostatic pressure component from the total stress. 27' 28 The effective stress is of particular importance in the mechanics of porous media such as those that occur in the study of soils, rocks, or concrete. The basic assumption in deriving the body forces of Eq. (6.76) is that only the effective stress is of any importance in deforming the solid phase. This leads immediately to another possibility of formulation. 29 If we examine the equilibrium conditions of Eq. (6.8) we note that this is written in terms of total stresses. Writing the constitutive relation, Eq. (6.26a), in terms of effective stresses, i.e., ~r' -- D'(s - s0) + o'~
(6.78)
and substituting into the weak form we find that the stiffness matrix is given in terms of the matrix D' and the force terms are augmented by an additional force -
[ BTm p d ~ d~2
(6.79)
e
or, if p is interpolated by shape functions N~, the force becomes - [ BTmN ' dr2 ~e df~
(6.80)
e
This alternative form of introducing pore pressure effects allows a discontinuous interpolation of p to be used [as in Eq. (6.79) no derivatives occur] and this is now frequently used in practice.
Numerical examples 213
Here only 18 cubic serendipity elements are needed to obtain an adequate solution, arranged as shown in Fig. 6.14. It is of interest to observe that all mid-side nodes of the cubic elements may be generated within the computer program and need not be specified. Also, the problem requires the specification of body forces caused by the centrifugal effects of the rotating disk. Here, br = - p
r (.0 2
where p is the mass density of the material and co is the angular velocity.
u
i,_. Q. i,... t~
C
E
0 u
0
"0 c~
~
0
v~ if1 ~,m !,_
c ~
Lt~ ~
0 4.-J
m ~ c~ L~ Q-O 9~ Q.
~0 .m 0-0
E ~0
t ~ v~ -0 f~ Lt~ ~
c~ ~ m0
_Q
~
n t~ ~n
0
U
~ C~
o~
~
~E ~ C~
rV~ --C~
tim LL
Numerical examples 215
Fig. 6.15 Conical water tank.
6.5.3 Conical water tank In this problem cubic serendipity elements are again used as shown in Fig. 6.15. It is worth noting that single-element thickness throughout is adequate to represent the bending effects in both the thick and thin parts of the container. With simple 3-node triangular elements, several layers of elements would have been needed to give an adequate solution.
216
Problems in linear elasticity E = 10 7 Ib/in 2 v=0.2 t = 0.5 in
600 z~
400 5
M e Ib/in 200
Exact
I
15
-200
I
I
r0
~L
20
/t varies from
degrees
1to 20
Typical element Fig. 6.16 Encastr~, thin hemispherical shell. Solution with 15 and 24 cubic serendipity elements.
6.5.4 A hemispherical dome The possibilities of dealing with shells approached in the previous example are here further exploited to show how a limited number of elements can adequately solve a thin shell problem as illustrated in Fig. 6.16. This type of solution can be further improved upon from the economy viewpoint by making use of the well-known shell assumptions involving a linear variation of displacements across the thickness. Thus the number of degrees of freedom can be reduced (e.g., see reference 30).
6.5.5 Arch dam in a rigid valley This problem, perhaps a little unrealistic from the engineer's viewpoint, was the subject of a study carried out by a committee of the Institution of Civil Engineers and provided an excellent test for a convergence evaluation of three-dimensional analysis. 1~ In Fig. 6.17 two subdivisions into quadratic and two into cubic elements are shown. In Fig. 6.18 the
Problems 217
convergence of displacements in the centre-line section is shown, indicating that quite remarkable accuracy can be achieved with even one element. The comparison of stresses in Fig. 6.19 is again quite remarkable, though showing a greater 'oscillation' with coarse subdivision. The finest subdivision results can be taken as 'exact' from checks by models and alternative methods of analysis.
6.5.6 Pressure vessel problem A more ambitious problem treated with simple tetrahedra is given in reference 7. Figure 6.20 illustrates an analysis of a complex pressure vessel. Some 10000 degrees of freedom are involved in this analysis. A similar problem using higher order isoparametric elements permits a sufficiently accurate analysis for a very similar problem to be performed with only 2121 degrees of freedom (Fig. 6.21).
6.1 Use the transformation array given by
T
.__
cos0 - sin 0 0
sin0 cos 0 0
i]
with 0 -- 45 ~ to transform stress and strain components from their x, y, z components to their x t, yt, z' components. Let the material be linearly elastic with material parameters given by E and v. Show that G = E/[2(1 + v)]. 6.2 For an isotropic material expressed in E and v compute the mean stress p = (Crx + cry + crz). If the bulk modulus is given by p=Kev
where ev = ex + ey --1-e z is the volume strain, show that K = E/[3(1 - 2v)]. 6.3 The strain displacement equations for a one-dimensional problem in plane polar coordinates are given by OUr
--
{}r0 Err EO0
"--
1 OUo
Ur
r O0 ! r 1 OUr OUo (-:= uo) + r T;-r
D
The displacements are expanded in a Fourier series as Ur -- Z
un (r) COSnO and uo = Z
v" (r) sin nO
218
Problems in linear elasticity
Fig. 6.17 Arch dam in a rigid valley- various element subdivisions.
Problems 219
Level 6
120 (a) (b) (c) (d)
100
o z~ x [~i
32 El 9A El 9B El 1(96D)EI
,,, i::!.
80
r
60
,'..!....,
40
j
L
,~
y""
,///F f , ~
Level4
S
S
Level 2
Y
20
0
10
20
30
40
50
60
mm
Fig. 6.18 Archdam in a rigid valley- centre-linedisplacements.
3 ~
//Air fa!e
o 32 El z~ 9A El x 9B El Ei:i 1(96D)EI
(a) (b) (c) (d)
Water face
-80
-60 -40 -20 Compression
0
20
40
60 80 Tension
Fig. 6.19 Archdam in a rigid valley- vertical stresseson centre-line.
100 kg/cm2
220
Problems in linear elasticity
Fig. 6.20 A nuclear pressure vessel analysis using simple tetrahedral elements. 7 Geometry, subdivision, and some stress results. N.B. Not all edges are shown.
(a) Express u n ( r ) and v n ( r ) in terms of shape functions and parameter fia and Va, respectively, and determine the strain displacement matrix for each harmonic n. (b) For a linear elastic material show that the stiffness matrix for each harmonic is independent of other terms in the Fourier series 9 (Hint: Perform integrals in 0 analytically.) 6.4 Cartesian coordinates may be expressed in terms of spherical components r, 0, and ~p as
x=rcos0
sincp; y - - r
sin0 sin 4) and z = r c o s ~ p
This form permits the solution of spherically symmetric problems for which displacements depend only on r and the strain-displacement equations are expressed as 8rr
-
-
Yre~-
OUr Or '
Ou 4, Or
Ur eoo = eep4, = -7-
u4, " Ouo , YrO--" r Or
uo ; r
Y04~ = 0
(a) How many rigid body modes exist for this problem? (b) Express the displacement components Ur, uo and u, in finite element form using one-dimensional shape functions in r. (c) Determine the form of the strain-displacement matrix Ba for each shape function
Na.
Problems
Fig. 6.21 Three-dimensional analysis of a pressure vessel. (d) For a linear elastic isotropic material write the form of the stiffness matrix for the nodal pair a and b. Show that the problem decomposes into three separate problems in terms of each displacement component. (e) For linear shape functions obtain an expression for the stiffness components corresponding to the Ur displacements using a one-point quadrature formula. Check if the resulting stiffness matrix has correct rank. 6.5 For a linear elastic isotropic material the stiffness matrix may be computed by numerical integration using Eq. (6.68). Alternatively, the stiffness matrix may be computed in indicial form as indicated in Appendix B.
221
222
Problems in linear elasticity
Consider a plane strain problem which is modelled by 4-node quadrilateral elements. Assume the stiffness matrix is computed using a 2 x 2 gaussian quadrature formula. (a) Compute separately the number of additions/subtractions and multiplications necessary to evaluate the stiffness using Eq. (6.68). Count only operations involving non-zero values in B or D. (b) Repeat the above calculation using the method of Appendix B given by Eqs (B.52) and (B.54). 6.6 In the classical plane strain problem the strain normal to the plane of deformation (i.e., ez) was assumed to be zero. The problem may be 'generalized' by assuming ez is constant over the entire analysis domain. The constant strain may then be related to a resultant force Fz applied normal to the deformation plane. (a) Following the steps given in Sec. 6.3, develop the virtual work expression (weak form) for the generalized plane strain problem. (b) Write finite element approximations for all the terms in the weak form. (c) Write the expression for an element stiffness in terms of nodal parameters and the strain ez. (d) Show how the resultant force Fz is related to the constant strain ez.
Fig. 6.22 Traction loading on boundary for Problems 6.7 and 6.8.
6.7 A concentrated load, F, is applied to the edge of a two-dimensional plane strain problem which is modelled using quadratic order finite elements as shown in Fig. 6.22(a). Compute the equivalent forces acting on nodes 1, 2 and 3. 6.8 A triangular traction load is applied to the edge of a two-dimensional plane strain problem as shown in Fig. 6.22(b). (a) Compute the equivalent forces acting on nodes 1, 2 and 3 by performing the integrals exactly. (b) Use numerical integration to compute the integrals which define the equivalent forces. Use the minimum number of points that integrate the integral exactly. What is the result if one-order lower is used? 6.9 An arc of 20 for a circular boundary of radius R is approximated by the quadratic isoparametric interpolation as shown in Fig. 6.23. For this case h = R sin 0 and c - R(1 - cos0). A concentrated load, F, is applied normal to the boundary at the point labelled a (~). Let F -- 100 N, R -- 10 cm and 0 = 15 ~ For ~ - 0, 0.25, 0.50, 0.75, 1.0 determine:
Problems 223 y cL
Fig. 6.23 Concentratednormalloadon a curvedboundary.Problem6.9. ()
4
()8
()
1
L
0 7
t
3
Y
0 7
4
3
Y
b x
X
b
18
(a) Serendipityelement
2
5
5
0
J
6(
L
0 a
a
(b) Lagrangianelement
r
Fig. 6.24 Quadrilateral8- and 9-nodeelements. Problems6.10 and 6.11.
(a) The equivalent forces acting on nodes 1, 2 and 3 for the case when the normal is computed from the quadratic interpolation. (b) The equivalent nodal forces using the normal to the circular boundary. (c) The error between the two forms. Show on a sketch. 6.10 A mesh for a plane strain problem contains the quadratic order rectangular elements shown in Figs 6.24(a) and (b). The elements are subjected to a constant body force b = (0, -19 g)r where p is mass density and g is acceleration of gravity. For each element type: (a) Use standard shape functions for Na and develop a closed form expression for the nodal forces in terms of a, b and p g. (b) Use hierarchical shape functions for Na and develop a closed form expression for the nodal forces in terms of a, b and p g. 6.11 A mesh for a plane strain problem contains the quadratic order rectangular elements shown in Figs 6.24(a) and (b). The elements are subjected to a constant temperature change AT. Each element is made from an isotropic elastic material with constant properties E, v and ct. For each element type:
224 Problemsin linear elasticity
IY
qo
,~--x
L
L
r
Fig. 6.25 Uniformlyloadedcantileverbeam. Problem6.12. (a) Use standard shape functions for Na and develop a closed form expression for the nodal forces in terms of a, b and the elastic properties. (b) Use hierarchical shape functions for Na and develop a closed form expression for the nodal forces in terms of a, b and the elastic properties. 6.12 Use the program FEAPpv (or any other available program) to solve the rectangular beam problem given in Example 6.4 and verify the results shown in Table 6.2. 6.13 Use the program FEAPpv (or any other available program) to solve the curved beam problem given in Example 6.5 and verify the results shown in Table 6.3. 6.14 The uniformly loaded cantilever beam shown in Fig. 6.25 has properties L=2m;
h=0.4m;
t=0.05m
and q 0 = 1 0 0 N / m
Use FEAPpv or any other available program to perform a plane stress analysis of the problem assuming linear isotropic elastic behaviour with E = 200 GPa and v = 0.3. In your analysis: (a) Use quadratic lagrangian elements with an initial mesh of 1 element in the depth and 5 elements in the length directions. (b) Compute consistent nodal forces for the uniform loading. (c) Compute nodal forces for a parabolically distributed shear traction at the restrained end which balances the uniform loading q0. (d) Report results for the centre-line displacement in the vertical direction and the stored energy in the beam. (e) Repeat the analysis three additional times using meshes of 2 • 10, 4 x 20 and 8 • 40 elements. Tabulate the tip vertical displacement and stored energy for each solution. (f) If the energy error is given by A E = En - En-1
= Ch q
estimate C and q for your solution. Is the convergence rate as expected? Explain your answer. 6.15 A circular composite disk is restrained at its inner radius and free at the outer radius. The disk is spinning at a constant angular velocity co as shown in Fig. 6.26. The disk is manufactured by bonding a steel layer on top of an aluminium layer as shown in Fig. 6.26(b).
Problems 225
Fig. 6.26 Spinning composite disk. Problem 6.15. When spinning at an angular velocity of 50 rpm it is desired that the top surface be flat. This will be accomplished by milling the initial shape of the top to a specified level. Your task is to determine the profile for milling. To accomplish this (a) Perform an analysis for an initially flat top surface using the dimensions given in the figure (lengths given in mm). The elastic properties for steel are E = 200 GPa, v = 0.3 and p = 7.8/zg/mm3; those for aluminium are E = 70 GPa, v = 0.35 and p = 2.6/zg/mm 3 (where/z = 10-6). Be sure to use consistent units (say, mm, sec, and/zg). The inner radius of the disk is to be restrained in the radial direction (i.e., u(15, z) = 0). Axial restraint is only applied at the centre of the disk (i.e., v(15, O) - 0 ) . (b) Using the results for the vertical displacements computed in (a) reposition the top nodes to new values for which a reanalysis should give improved results. (c) Reanalyse the problem for the new coordinates. How accurate does this analysis predict the desired result? What would you do to improve your answer? 6.16 A rectangular region with a circular hole is shown in Fig. 6.27. The traction on the circular hole is zero. The region is to be used for the solution of an infinitely extending
Fig. 6.27 Rectangularregion with circular hole. Problem 6.16.
226 Problems in linear elasticity plane stress problem in which the stress at infinity is given by a uniformly distributed normal stress tr0 acting in the x direction. The stress distribution in polar coordinates for the problem is given by
o"0 --
([ 1 - (a )2] + [ 1 + 3 ( a)4 - 4 ( - )a2 ] cos 20 } r r r lif o {[1-~-(a) 2] -- [| + 3 ( a ) 4] cos20}
rr0=
go'0 1 -
ar=
1 ~ao
{
a} r
+2(-)2
r
sin 20
and the displacements by Ur
= -a~o r { [ 1 + (a )2] -t- [ 1 - (a)4 + 4( a )2 ] COS20 r r
(4,2]
o0r {[
uo=~
(a,4] cos20}
] [ a ]} l + ( a ) 4 r + 2(a)zr + v 1 + ( 4 ) 4 - 2 ( )2 sin20
In order for the region to satisfy the above solution it is necessary to: (a) Enforce symmetry conditions along the boundaries AB and DE and (b) Apply the tractions of the exact solution on the boundary BCD. Program development project: Write a program that uses numerical integration to compute the consistent nodal forces on the boundary BCD. (Hint: This may be done by adding an element to FEAPpv which computes only the nodal forces for line elements defined on the boundary BCD or by writing a MATLAB program which, given the location of nodal coordinates on BCD, computes the nodal forces.) Your program should also compute
E =t f
JB
CD
[Utxd-Vty]
dr
which is twice the stored energy in a slice of thickness t. When accurately computed (e.g., to 9 or 10 digit accuracy) this may be used as the 'exact' solution for the region. Use your program and FEAPpv (or any other available program) to solve a plane stress problem. Let the hole radius be R = 10 cm, the thickness of the slice be t = 0.1R and take E = 200 GPa and v - 0.3 for the elastic properties. The boundary BC should be placed at about 3R and the boundary CD at 2 to 3R. Assume a unit value for the stress or0. (a) Use 4-node quadrilateral elements to solve the problem on a sequence of meshes in which element sizes are reduced in half for each succeeding mesh. (b) Plot the displacement at the hole boundary and compare to the exact solution. (c) Compute the work done by your finite element program. (Note: In FEAPpvthis will be the 'energy' reported by the solver.) (d) Compute the rate of convergence for your solution and plot on a figure similar to that given in Fig. 6.9. (e) Repeat the solution using 8-node serendipity elements. (f) Repeat the solution using 9-node lagrangian elements.
References 227 Write a short report discussing your findings. 6.17 Program development project: Extend the program developed in Problem 2.17 to consider plane strain and axisymmetric geometry. 6.18 Program development project: Extend the program developed in Problem 2.17 to compute nodal forces for specified boundary tractions which are normal or tangential to the element edge. Assume tractions can vary up to quadratic order (i.e., constant, linear and parabolic distributions) and use numerical integration to compute values. Test your program for an edge with constant normal stress. Then test for linear normal and finally quadratic tangential values. Compare results with those computed by FEAPpv (or any available program). 6.19 Program development project: Extend the program developed in Problem 2.17 to compute nodal values of stress and strain. Follow the procedure given in Sec. 6.4 to project element values to nodes. Test your program using (a) the patch test of Problem 2.17 and (b) the curved beam problem shown in Fig. 2.11. 6.20 Program development project: Add a module to the program developed in Problem 2.17 to plot contours of stress and strain components for plane stress, plane strain and axisymmetric solids. Use the capability developed in Problem 6.19 to obtain nodal values and the contour routine developed in Problem 2.18. Test your program system by plotting contours of stress components for the curved beam meshes described in Problem 2.18. 6.21 Program development project: Add a 4-node quadrilateral element to the program system developed in Problem 2.17. Use shape functions and numerical integration to compute the element stiffness matrix. Also include the force vector from a constant element body force (you may need to add b to your input module). Test your program on the curved beam problems described in Problem 2.18. Compare the accuracy to that obtained using triangular elements.
1. M.J. Turner, R.W. Clough, H.C. Martin, and L.J. Topp. Stiffness and deflection analysis of complex structures. J. Aero. Sci., 23:805-823, 1956. 2. R.W. Clough. The finite element method in plane stress analysis. In Proc. 2nd ASCE Conf. on Electronic Computation, Pittsburgh, Pa., Sept. 1960. 3. R.H. Gallagher, J. Padlog, and P.P. Bijlaard. Stress analysis of heated complex shapes. ARS J., 29:700-707, 1962. 4. R.J. Melosh. Structural analysis of solids. J. Struct. Eng., ASCE, 4:205-223, Aug. 1963. 5. J.H. Argyris. Matrix analysis of three-dimensional elastic media- small and large displacements. J. AIAA, 3:45-51, Jan. 1965. 6. J.H. Argyris. Three-dimensional anisotropic and inhomogeneous media- matrix analysis for small and large displacements. Ingenieur Archiv, 34:33-55, 1965. 7. Y.R. Rashid and W. Rockenhauser. Pressure vessel analysis by finite element techniques. In Proc. Conf. Prestressed Concrete Pressure Vessels, Institute of Civil Engineering, 1968. 8. J.H. Argyris. Continua and discontinua. In Proc. 1st Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-66-80, pages 11-189, Wright Patterson Air Force Base, Ohio, Oct. 1966.
228
Problems in linear elasticity 9. B.M. Irons. Engineering applications of numerical integration in stiffness methods. J. AIAA, 4:2035-2037, 1966. 10. J.G. Ergatoudis, B.M. Irons, and O.C. Zienkiewicz. Three dimensional analysis of arch dams and their foundations. In Proc. Symp. Arch Dams, Inst. Civ. Eng., London, 1968. 11. J.H. Argyris and J.C. Redshaw. Three dimensional analysis of two arch dams by a finite element method. In Proc. Symp. Arch Dams, Inst. Civ. Eng., 1968. 12. S. Fjeld. Three dimensional theory of elastics. In I. Holand and K. Bell, editors, Finite Element Methods in Stress Analysis, Trondheim, 1969. Tech. Univ. of Norway, Tapir Press. 13. A.E.H. Love. A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press, Cambridge, 4th edition, 1927. 14. N.I. Muskhelishvili. Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, Groningen, 3rd edition, 1953. English translation by J.R.M. Radok. 15. S.P. Timoshenko and J.N. Goodier. Theory of Elasticity. McGraw-Hill, New York, 3rd edition, 1969. 16. S.G. Lekhnitskii. Theory of Elasticity of an Anisotropic Elastic Body. Holden Day, San Francisco, 1963. (Translation from Russian by P. Fern.) 17. I.S. Sokolnikoff. The Mathematical Theory of Elasticity. McGraw-Hill, New York, 2rd edition, 1956. 18. P.G. Ciarlet. Mathematical Elasticity. Volume 1: Three-dimensional Elasticity. North-Holland, Amsterdam, 1988. 19. O.C. Zienkiewicz and EC. Scott. On the principle of repeatability and its application in analysis of turbine and pump impellers. Int. J. Numer. Meth. Eng., 9:445-452, 1972. 20. R.ES. Hearmon. An Introduction to Applied Anisotropic Elasticity. Oxford University Press, Oxford, 1961. 21. T.C.-T Ting. Anisotropic Elasticity: Theory and Applications. Oxford University Press, New York, 1996. 22. G. Strang and G.J. Fix. An Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs, N.J., 1973. 23. H.J. Brauchli and J.T. Oden. On the calculation bf consistent stress distributions in finite element applications. Int. J. Numer. Meth. Eng., 3:317-325, 1971. 24. E. Hinton and J. Campbell. Local and global smoothing of discontinuous finite element function using a least squares method. Int. J. Numer. Meth. Eng., 8:461-480, 1974. 25. O.C. Zienkiewicz and Y.K. Cheung. Buttress dams on complex rock foundations. Water Power, 16:193, 1964. 26. O.C. Zienkiewicz and Y.K. Cheung. Finite element procedures in the solution of plate and shell problems. In O.C. Zienkiewicz and G.S. Holister, editors, Stress Analysis, Chapter 8. John Wiley & Sons, Chichester, 1965. 27. O.C. Zienkiewicz, A.H.C. Chan, M. Pastor, B.A. Schrefler, and T. Shiomi. Computational Geomechanics: With Special Reference to Earthquake Engineering. John Wiley & Sons, Chichester, 1999. 28. K. Terzhagi. Theoretical Soil Mechanics. John Wiley & Sons, New York, 1943. 29. O.C. Zienkiewicz, C. Humpheson, and R.W. Lewis. A unified approach to soil mechanics problems, including plasticity and visco-plasticity. In Int. Symp. on Numerical Methods in Soil and Rock Mechanics, Karlsruhe, 1975. See also Chapter 4 of Finite Elements in Geomechanics (ed. G. Gudehus), pages 151-78. Wiley, 1977. 30. O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method for Solid and Structural Mechanics. Butterworth-Heinemann, Oxford, 6th edition, 2005.
Field problems- heat conduction, electric and magnetic potential and fluid flow The general procedures discussed in the previous chapters can be applied to a variety of physical problems. Indeed, some such possibilities have been indicated in Chapter 3 and here more detailed attention will be given to a particular, but wide class, of such situations. Primarily we shall deal with situations governed by the general 'quasi-harmonic' equation, the particular cases of which are the well-known Laplace and Poisson equations. 1-6 The range of physical problems falling into this category is large. To list but a few frequently encountered in engineering practice we have: 9 9 9 9 9 9
Heat conduction Seepage through porous media Irrotational flow of ideal fluids Distribution of electrical (or magnetic) potential Torsion of prismatic shafts Lubrication of pad beatings, etc.
The formulation developed in this chapter is equally applicable to all, and hence only limited reference will be made to the actual physical quantities. In all the above classes of problems, the behaviour can be represented in terms of a scalar variable for which we will generally use the symbol 4~. In the applications to specific problems, however, we shall generally introduce the physical variable describing the behaviour. For instance, in discussing heat conduction applications we use the symbol T to denote the temperature. In Chapter 3 we indicated both the 'weak form' and a variational principle applicable to the Poisson and Laplace equations (see Secs 3.2 and 3.8.1). In the following sections we shall apply these approaches to a general, quasi-harmonic equation and indicate the ranges of applicability of a single, unified, approach by which one computer program can solve a large variety of physical problems. It will be observed that the Co 'shape functions' presented in Chapters 4 and 5 can be directly applied and that both isotropic and anisotropic behaviour can be treated with equal ease.
230
Field problems
In many physical situations we are concerned with the diffusion or flow of some quantity such as heat, mass, concentration, etc. In such problems, the rate of transfer per unit area (flux), q, can be written in terms of its cartesian components as
q = [qx,
qy,
qz]T
(7.1)
If the rate at which the relevant quantity is generated (or removed) per unit volume is Q, then for steady-state flow the balance or continuity requirement gives
Oqx Oqy Oqz Ox +--~-y +--~-z + Q - 0
(7.2)
Introducing the gradient operator
V=
I0Ox' OyO Oz0] T
(7.3)
VTq --I- Q = 0
(7.4)
we can write (7.2) as
Generally the rates of flow will be related to the gradient of some potential quantity q~. This may be temperature in the case of heat flow, etc. A very general linear relationship will be of the form
q =
qx i x xy xzl
qy qz
-- -
kyx,
kyy,
ky z
L~zx,
kzy,
kzzJ
= - k V~b
(7.5)
where k is a symmetric form due to energy arguments (i.e., kxy = kyx, etc.) and is variously referred to as Fourier's, Fick's, or Darcy's law depending on the physical problem. The final governing differential equation for the 'potential' q~ is obtained by substitution of Eq. (7.5) into (7.4), leading to - - V T (k Vq~) + Q = 0
(7.6)
which has to be solved in a domain f2. On the boundaries of such a domain we shall usually encounter one of the following conditions:
General quasi-harmonicequation 231 1. O n F , , (7.7a) i.e., the potential is specified (Dirichlet condition). 2. On 1-'q the normal component of flow (or flux), qn, is given as (Neumann condition) q n --" q - - H (dp -
qbo)
where H is a transfer or radiation coefficient, 4~0 is a known equilibrium value and E/is a specified value. Here q n is defined as qn
= nTq with n = [n x
, ny , n z
]T
where n is a vector of direction cosines of the normal to the boundary surface. Accordingly, we may write the second boundary condition ? / + n T (k X74~) + H (~b - q~0) = 0
(7.7b)
which holds on 1-'q.
7.2.2 Anisotropic and isotropic forms for k If we consider the general statement of Eq. (7.5) as being determined for an arbitrary set of coordinate axes x, y, z we shall find that it is always possible to determine locally another set of axes x', y', z' with respect to which the matrix k' becomes diagonal, as shown in
Fig. 7.1 Anisotropic material. Local coordinates coincide with the principal directions of stratification.
232
Field
problems
Fig. 7.1. With respect to such axes we have k'
--"
0 00]
ky,y, 0
(7.8)
kz'z'
Thus, the general form of the k has only three components which are associated with three orthogonal axes. Such materials are called anisotropic or orthotropic. The governing differential equation (7.6) for these axes can be written
---~x ~ kx,x,~ox, + ~
ky,y,~
+~Oz' kz,z,~ -- (V')T
+Q=0
(7.9)
(k' V'4~) + Q = 0
where
Ox"
Oy"
Oz'
defines the gradient operator for the 'prime' coordinate system. Alternatively, knowing k' and the orientation of the axes x', y', z' a transformation of coordinates is given by x' = Tx in which T are direction cosines defined as T -
rcos(x',x), |cos(y', x), Lcos(z', x),
cos(x',y), cos(y', y), cos(z', y),
cos(x',z)] cos(y', z) cos(z', z)
where cos(x', x) is the cosine of the angle between the x' direction and the x direction. The inverse of T is equal to its transpose; hence X = TTx /
In addition we may write the gradient with respect to the prime axes as V'(.) = T V (.) or alternatively V(.) = T T V'(.)
Using the above we obtain the expression ( v ' ) T (k'V'~b) = (V) T T T (k'TVq~)
(7.10a)
or
k=TTk'T
or k ' = T k T T
(7.10b)
Lastly for an isotropic material we can write k = kI
(7.11)
where I is an identity matrix. In two dimensions this leads to the simple form of Eq. (3.8) as discussed in Chapter 3.
Finite element solution process 233
7.2.3 Weak form and variational principle for the general
quasi-harmonic equation
Following the principles of Chapter 3, Sec. 3.2, we can obtain the weak form of Eqs (7.6) and (7.7b) by writing (using v = 8~p) f 8 4 ~ [ - V T (k Vq~) + Q] dr2 + f r 8q~ [~ + n T (k Vq~) + H (r - ~bo)] dF = o q
(7.12) for arbitrary functions &p. Integration by parts (see Appendix G) will result in the following weak statement
L [(VS~b)T(k Vq~)+ 6~b Q] J~2
/.
/.
/,
dr2 + L
to + H
~o>] dF
+ L 8q~qndF
Jl" q
=0
,]l
(7.13) Generally, the last term is omitted by requiting &p = 0 and imposing the forced (Dirichlet) boundary condition (7.7a) on 1-'q. It is also possible to express an integral form for the quasi-harmonic equation as a variational principle. The functional rI
---
j [1
~ (~7~) T
]
(k V4~) + 4~ Q dr2 +
~b0 + H (~
~2 _
~0)
]
dF
(7.14)
q
gives on minimization [subject to the constraint of Eq. (7.7a)] the original problem in Eqs (7.6) and (7.7b). The algebraic manipulations required to verify the above principle follow precisely the lines of Sec. 3.8. Clearly material properties defined by the k matrix can vary from element to element in a discontinuous manner. This is implied in both the weak and variational statements of the problem.
The finite element solution process follows the standard solution methodology and for the quasi-harmonic equation approximates the trial function using any of the Co shape function expressions given in Chapters 4 and 5. Accordingly, we use
"~ ~ -- Z
Na ~ a
-- N r
(7.15)
a
in either the weak formulation of Eq. (7.13) or the variational statement of Eq. (7.14). If, in the weak statement, we take
8cp~8~=ZWaS~a=WS(6 d
with W = N
(7.16)
234 Field problems according to the Galerkin principle, an identical form will arise with that obtained from the minimization of the variational principle. The gradient of 4~ is now given by the approximation V ~ --
Z(VNa)
a
[
~)a
ONa ONe]v
ON.
= ~a
-~X '
-~y '
OZ
(7.17)
~a = Z b a ~ a a
gradientmatrix
where now ba denotes the of shape functions. Substituting Eqs (7.15) to (7.17) into (7.13), we have a typical statement for an arbitrary 6~a giving, for each a (assuming summation convention for b),
{ IL bTkb9dK2+ Lq NaHNbdF1 ~b+fNaQd~'2+frq Na(~I-H~)o)dF} ----0 (7.18/ Evaluating the integrals for all elements leads to the set of standard discrete equations of the form
H~+f
-0
(7.19)
with
Hab=fbTakbbd~+fr NaHNbdF
q
and
fa-fNaQd~2+fr Na(~I-HcPo)dF
q
(7.20) to which prescribed values of ~ have to be imposed on boundaries 1-'~. We note that an additional 'stiffness' is contributed on boundaries for which a radiation constant H is specified. Indeed, standard operations are followed to evaluate the above arrays using quadrature. In the general three-dimensional case using Lagrange or serendipity-type 'brick' elements, use of Gauss quadrature results in
L3
nab : Z ba (~l, Ol, (l)Tkbb(~I, OI, (l) J (~l, Ol, (l) Wl /=1 L2
+ Z Na (~l, Ol) H Nb (~l, 17l)j (~l, 17l)Wl /=1
fa.
with a similar expression for Indeed in a computer program the same standard operations are followed to evaluate the fluxes using
q --- - k V4~ ~ - k Z bb ~b
(7.21)
b
The fluxes may be computed within the elements; however, it is often desirable to obtain their values at nodes. This is best accomplished by the procedure summarized in Sec. 6.4 and discussed in more detail later in Chapter 13.
Finite element solution process 235
7.3.2 Two-dimensional plane and axisymmetric problem The two-dimensional plane case is obtained by taking the gradient in the form
V= and taking the flux as q =
Ox'
(7.22)
Oy
{qx} [,xx,xy -- -
qy
kyx
kyy.
04~
(7.23)
_ff_fy
On discretization by Eqs (7.15) to (7.17) a slightly simplified form of the matrices will now be found with Da in Eq. (7.17) replaced by
ba --
Ox '
Oy
(7.24)
and the volume element by d~2 = t dx dy where t is the slab thickness. Alternatively the formulation may be specialized to cylindrical coordinates and used for the solution of axisymmetric situations by introducing the gradient 0
V=
-~r'
0] T
Oz
(7.25)
where r, z replace x, y to describe both the gradient and ba. With the flux now given by
q =
{qr} rrr 1 = -
qz
Lkzr kzzJ
(7.26)
Oqb
the discretization of Eq. (7.18) is now performed with the volume element expressed by dr2 = 2zrr dr dz and integration carried out using quadrature as described above.
Example 7.1: Plane triangular element with 3 nodes. We particularize here to the simplest triangular element (Fig. 7.2). With shape functions written in the alternative forms
aa -[- bax "Jr-Cay
Na=La
2A
in which A is defined in (4.26) and aa, ba, Ca in (4.28), we can compute the derivatives as
ONa OX
OLa OX
ba 2A
ONa Oy
OLa Oy
Ca 2A
236
Field problems
qn
r
Fig. 7.2 Division of a two-dimensional region into triangular elements.
giving the gradient matrix
1 [ba Ca]T
ba = ~-S
Since the gradient matrix is constant the element 'stiffness' matrix (ignoring the H boundary term) is given by
c1c2 C2C3 cic3] bib2 bzb3 bab3 -t- --~ kyyt [Cc~Ccl C2C2 bzb2 4A Lb3bI b3b2 b3b3 LC3C1 C3C2 C3C3 b1c2 blc3 kyx, [:;~: Clb2 c2b2 Clb31 c2b3/ kxyt rblCl /bzc I b2c2 b2c3 -t- - ~
He = kxxt rblbl |bzbl
+ - ~ Lb3cl
LC3bl c3b2 c3b3J
b3c2 b3c3
The load matrices follow a similar simple pattern and thus, for instance, due to constant Q and using a 1-point quadrature from Table 5.3 we have La : 1/3 so that
1
fe : -LaQtA = - ~ Q t A This is a very simple (almost 'obvious') result.
Example 7.2: 'Stiffness' matrix for axisymmetric triangular element with 3 nodes. The computation of the arrays for an axisymmetric problem may be performed using area coordinates as described in Sec. 4.7.1 and quadrature in Sec. 5.11. Since the integrals for
Partial discretization-transient problems 237 the 'stiffness' matrix only involve a linear function in r (from the volume element) a 1-point integration from Table 5.3 still is exact and results in
He ._
§ wlaere ~ =
{
]
bib2 bib3 kzz krr rb,bl /b2b I b2b2 b2b3 -Jr-- ~ 4-A Lb3bl b3b2 b3b3
FblCl blc2 kzr krz ib2c 1 b2c2 b2c3/ --kLb3c1 b3c2 b3c31
CLC2CC3]
[~12~11 C2C2 C2C3 LC3Cl c3c2 c3c3
I; ~11 Lc3bl
c2b2Clb2 c2b3Clb3]} 2rr ? c3b2 c3b3
(rl -k-r2 -k-r3)/3.
Example 7.3" Load matrix for axisymmetric triangular element with 3 nodes. The nodal forces from a constant source term Q are computed from
fa = fA La Q2rcrbLb dr dz
sum on b
and thus now has quadratic terms. From Table 5.3 use of a 3-point formula is adequate to obtain an exact result. For node 1 this gives fl e = 89
[1 (rl -+- r2) -f- ~(rl 1 -+- r3) 1 A g1 = l(2rl + r 2 + r3)zrQA
with results for f~ and f3e obtained by cyclic permutation. The use of a 1-point formula gives results which are of the same accuracy as that of the basic linear functions in the approximation of 4), namely, O(h 2) where h is the diameter of an element. Using this we obtain the force array fe~,~1 2zr?QA = g2 rr~QA
The above developments have assumed that the solution to the problem is independent of time. Many problems, however, require the solution to depend explicitly on time, both in the loading and in the differential equation. An example of a problem which is time dependent is a heat conduction problem in which the loading varies with time. The solution for the temperature now requires use of the differential equation given by c
OT 0t
V T (k V T) + Q = 0
(7.27)
where T is temperature (which now replaces (p), k the thermal conductivity, c the specific heat per unit volume and Q a heat source term. In addition to boundary conditions of the form given in (7.7a) and (7.7b) it is now necessary to provide the distribution of temperature at the initial time T(x, y, z, O) = To(x, y, z) (7.28)
238 Field problems Extending the method used to develop (7.13), a weak form t of the time dependent problem is given by 3T c 0----~+ (V3T)T (k V T ) + 3TQ
dr2 +
6T (71 + H ( T - To)) d r - o q
(7.29)
where we require ~ T = 0 and T = 7' on FT.
7.4.1 Finite element discretizations A finite element solution of (7.29) is constructed using an approximation of the type given in Sec. 3.5 where now we assume the separable form T(x, y, z, y) ,~ T ( x , y, z, t) - Na(x, y, z) Ta(t)
(7.30)
With this form the spatial derivatives are associated with the shape functions Na and the time derivative with the parameters 1"a. Substituting (7.30) into (7.29) yields the semi-discrete set of ordinary differential equations
dT C - : - + H'I~ + f = 0 at or for node a
(7.31)
dTb
Cab---~ + nab Tb -1I- L - - 0
where Hab and fa are given by (7.20) and Cab = f ~ NacNb dg2
In Chapters 16 and 17 we shall discuss in more detail methods of solution for large sets of equations of the form (7.31). Here, however, we consider a simple procedure in which the time dependence is given by a finite difference approximation. We will approximate the nodal temperatures at a time tn by
~r ( tn ) .~ qr. and the time derivative by
dT
~ ( T n - Tn-1) At dt t=t~ where At = tn - tn-1. An approximate solution to the semi-discrete equations at each time tn is obtained by solving the set of equation 1 c + H 1 T" = 1AtC S7
t,,-1 - f
(7.32)
If the initial condition is approximated as T(x, 0) ~ N(x) ~i'(0) with ~i'(0) = I'0 a solution for'i'l is immediately available from (7.32) by solving a set of algebraic equations. For each subsequent time step the solution process is identicalto the time independent t Note that no variational principle of the type (7.14) exists.
Numerical examples- an assessment of accuracy 239 /
2
(a)
(b)
S /
(c)
Fig. 7.3 'Regular' and 'irregular' subdivision patterns. problem except for the modified force vector and a need to use a coefficient matrix which has a term inversely proportional on the size of the time increment.
In Sec. 3.3, Example 3.6, we showed that by assembling explicitly worked-out 'stiffnesses' of triangular elements for the 'regular' mesh pattern shown in Fig. 7.3(a) the discretized equations are identical with those that are derived by well-known finite difference methods. The same result holds for the mesh pattern shown in Fig. 7.3(b). 7 For cases where all boundary conditions are given as prescribed values 4~=~
onF~
the solutions obtained by the two methods obviously will be identical, and so also will be the orders of approximation. However, if the mesh shown in Fig. 7.3(c) which is also based on a square arrangement of nodes but with 'irregular' element pattern is used a difference between the two approaches for the 'load' vector fe will be evident. The assembled equations will have the same 'stiffness' matrix as in Fig. 7.3(a) but will show 'loads' which differ by small amounts from node to node, but the sum of which is still the same as that due to the finite difference expressions. The solutions therefore differ only locally and will represent the same averages. Further advantages of the finite element process are:
240 Field problems 1. It can deal simply with non-homogeneous and anisotropic situations (particularly when the direction of anisotropy is variable). 2. The elements can be graded in shape and size to follow arbitrary boundaries and to allow for regions of rapid variation of the function sought, thus controlling the errors in a most efficient way (viz. Chapters 13 and 14). 3. Specified gradient or 'radiation' boundary conditions are introduced naturally and with a better accuracy than in standard finite difference procedures. 4. Higher order elements can be readily used to improve accuracy without complicating boundary conditions- a difficulty always arising with finite difference approximations of a higher order. 5. Finally, but of considerable importance in the computer age, standard programs may be used for assembly and solution.
7.5.1 Torsion of prismatic bars The torsion of prismatic elastic bars may be solved using a quasi-harmonic equation formulation. Here either a warping function or a stress function approach may be used. In Fig. 7.4(a) we show a rectangular bar loaded by an end torque Mt. The analysis is performed on the cross-section as shown in Fig. 7.4(b). The use of a warping function is governed by the formulation in which displacements are given as u = -yzO;
v = xzO
(7.33)
and w = ~ ( x , y ) O
where x, y are coordinates in the cross-section and z is a coordinate of the bar axis; 0 is the rate of twist and gr the warping function. The non-zero strain components resulting from these displacements are given by z
Mt I
b
-"""~
I
y r
(a) Fig. 7.4 Torsion of rectangular prismatic bar.
(b)
X
Numerical examples - an assessment of accuracy
yxz = 0
( 0--~x ~) -y
and
yyz -- O
( 0-~y ~ ) -t-- x
(7.34)
giving, for an isotropic elastic material, the stresses rxz = G yxz and ryz = G yyz
(7.35)
Inserting the stresses into the equilibrium equation gives the governing differential equation 0--~
~x
+ ~y
G~y
=0
(7.36)
and for stress-free boundary conditions (7.37)
rnz = nx rxz + ny ry z -- 0
in which nx and ny are the direction cosines for the outward normal to the boundary of the rectangular section. At least one value of the warping function must be specified to have a unique solution. The total torque acting on a cross-section is given by Mt = fA [--rxz Y + ryz X] dA =
(7.38)
/A [
G xZ+y2-Y-~x
+x
dAO=GJ~O
where G Jo is the effective torsional stiffness. A stress function formulation is deduced using the representation for stresses rxz =
0r
0y
0~
and ryz = 0--~-
(7.39)
Combining (7.36) and (7.37) with (7.39) and eliminating the warping function ~ gives the differential equation
~x ~~x +~y
+20=0
(7.40)
with 4~(s) = Constant on 1-'q
(7.41)
representing a stress-free boundary condition. The total torque acting on a cross-section is now given by Mr=
G X-~x + y
dAO=GJepO
(7.42)
where G J~ is the effective torsional stiffness. The two solutions provide a bound on the torsional stiffness with the warping function solution giving an upper bound, G J~, and the stress function a lower bound, G J~.
241
242
Field problems
42
,
~o)'
/
/
47931 (
/
5317
/
1
2
(a)
/
(b)
Fig. 7.5 Torsion of a rectangular shaft. Numbers in parenthesesshow a more accurate solution due to Southwell using a 12 x 16 mesh (values of ~/GOL=).
Example 7.4: Torsion of rectangular shaft. In Fig. 7.5 a test comparing the results obtained on an 'irregular' mesh of 3-node triangular elements with a relaxation solution of the lowest order finite difference approximation is shown. Both give results of similar accuracy, as indeed would be anticipated. In general superior accuracy is available with the finite element discretization. Furthermore, it is possible to get bounds on the torsional stiffness, as indicated above. To illustrate this latter aspect we consider a square bar which is solved using 4-node rectangular elements and a range of n x n meshes in which n is the number of spaces between nodes on each side. The results for the computed torsional stiffness values are plotted in Fig. 7.6. The improvement in the rate of convergence for higher order elements may also be illustrated by comparing the total error using 4-node and 9-node elements of lagrangian type. A very accurate solution is computed from the series solution given in reference 8 and used to compute the error in the finite element solution (see Fig. 7.7).
Example 7.5: Torsion of hollow bimetallic shaft. The pure torsion of a non-homogeneous rectangular shaft with a circular hole is illustrated in Fig. 7.8. In the finite element solution presented, the hollow section is represented by a material for which G has a value of the order of 10 -3 compared with the other materials.t The results compare well with the contours derived from an accurate finite difference solution. 9
7.5.2 Transient heat conduction Example 7.6: Transient heat conduction of a rectangular bar. In this example we consider the transient heat conduction in a long square prism with sides L x L and subjected to a rate of heat generation "~This was done to avoid difficulties due to the 'multiple connection' of the region and to permit the use of a standard program.
Numerical examples- an assessmentof accuracy 243
Fig. 7.7 Rate of convergence for square bar. 4- and 9-node lagrangian elements.
Q = Q0 e -at The problem is identical to the one considered in Sec. 3.5 where shape functions are assumed in a cosine form given by Eq. (3.57). Here, however, we use a standard finite element solution with 4-node square elements. The transient solution is performed using
244
Field problems
Fig. 7.8 Torsion of a hollow bimetallic shaft. ~p/GOL2 x 10 4.
the procedure given in Sec. 7.4.1. For the analysis we assume the following parameters: L=c=Q0=ot=l
and k =
0.75 7g 2
Using symmetry conditions, a mesh of 20 • 20 4-node elements is used to approximate one quadrant of the domain. A constant increment in time, At -- 0.01, is used to perform the solution. Results for the temperature at the centre of the prism are given in Fig. 7.9 and compared to the series solutions computed in Sec. 3.5, Fig. 3.9.
Transient heat conduction o f a rotor blade
In Fig. 7.10 we show some results for the transient temperature distributrion in a turbine rotor blade. The blade is subjected to a hot gas at 1145C ~ applied to the outer boundary in which a variable radiation constant H = ct is employed. Cooling is introduced in the internal ducts. The analysis is performed using cubic elements of serendipity type which permit the representation of the boundaries using very few elements.
7.5.3 Anisotropic seepage The next problem is concerned with the flow through highly non-homogeneous, anisotropic, and contorted strata. The basic governing equation is
Ox'
kx' x' -~x'
+ -~y' ky, y,
=0
(7.43)
in which H is the hydraulic head and kx,x, and ky,y, represent the permeability coefficients in the direction of the (inclined) principal axes. However, a special feature has to be incorporated to allow for changes of x' and y' principal directions from element to element.
Numerical examples - an assessment of accuracy 0.35
I
o M,N=I
[
[] M , N = 3
0.3 [- ........... !.... j . ~ ~ i
/
0.25
~
"~ ~
ii
<
~,
i ~ ~:
...........................
~
:
.../ . . ..................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.15
o.,
'/
............ ! ............ ! ............ i............. i............ i ...........
i-/f
................................
; ............
: ...........
..............
0.05 0c
0
0.5
1
1.5
2
2.5
3
3.5
ttime Fig. 7.9 Transient heat development in a square prism - plot of temperature at centre.
No difficulties are encountered in computation, and the problem together with its solution is given in Fig. 7.11. 3
7.5.4 Electrostatic and magnetostatic problems In this area of activity frequent need arises to determine appropriate field strengths and the governing equations are usually of the standard quasi-harmonic type discussed here. Thus the formulations are directly transferable. One of the first applications made as early as 19674 was to fully three-dimensional electrostatic field distributions governed by simple Laplace equations (Fig. 7.12). In Fig. 7.13 a similar use of triangular elements was made in the context of magnetic two-dimensional fields by Winslow 6 in 1966. These early works stimulated considerable activity in this area and much additional work has been published. 11-14 The magnetic problem is of particular interest as its formulation usually involves the with three components which leads to a formulation difintroduction of a v e c t o r p o t e n t i a l ferent from those discussed in this chapter. It is, therefore, worthwhile introducing a variant which allows the standard programs of this section to be utilized for this problem. 15-17 In electromagnetic theory for steady-state fields the problem is governed by Maxwell's equations which are
V• B =/zH VTB = 0
(7.44)
245
246
Field problems
Fig. 7.10 Temperature distribution in a cooled rotor blade, initially at zero temperature.
Numerical examples- an assessment of accuracy 247
Fig. 7.11 Flow under a dam through a highly non-homogeneousand contorted foundation.
with the boundary condition specified at an infinite distance from the disturbance, requiring H and B to tend to zero there. In the above J is a prescribed electric current density confined to conductors, H and B are vector quantities with three components denoting the magnetic field strength and flux density respectively,/z is the magnetic permeability which varies (in an absolute set of units) from unity in v a c u o to several thousand in magnetizing materials and x denotes the vector (cross) product, defined in Appendix F. The formulation presented here depends on the fact that it is a relatively simple matter to determine the field Hs which exactly solves Eq. (7.44) when/z -- 1 everywhere. This is given at any point defined by a vector coordinate r by an integral:
1 ~ J • ( r - r~)
Hs = ~
~,,~ I r - r'l 3
dr2;
I r - r'l -- v / ( r - r ' ) T ( r - r')
(7.45)
In the above, r' refers to the coordinates of dr2 and obviously the integration domain only involves the electric conductors where J # 0. With Hs known we can write H = H s -4-Hm and, on substitution into Eq. (7.44), we have a system
W• B = / z (Hs + Hm) '~7TB = 0
(7.46)
248
Field problems
J
90
90 7o
70
50
30 10
f
Fig. 7.12 A three-dimensional distribution of electrostatic potential around a porcelain insulator in an earthed
trough. 1~
If we now introduce a
scalarpotential q~, defining I-Im as H m ~ ~rt~
(7.47)
we find the first of Eqs (7.46) to be automatically satisfied and, on eliminating B in the other two, the governing equation becomes ,~rT (~'~7~) "Jr- '~TT (/xHs) = 0
(7.48)
with 4~ --+ 0 at infinity. This is precisely of the standard form discussed in this chapter [Eq. (7.6)] with the second term, which is now specified, replacing Q. An apparent difficulty exists, however, if/z has a discontinuity, as indeed we would expect it to do on the interfaces of two materials. Here the term Q is now undefined and, in the standard discretization of Eq. (7.18)or (7.19), the term (for node a)
f Na Q dr2 =_-f~ Na ~7T (/zl-ls)
dr2
(7.49)
Numerical examples-an assessment of accuracy 249
Fig. 7.13 Field near a magnet (afterWinslow6).
apparently has no meaning. Integration by parts comes once again to the rescue and we note that
f~ NaV T (/zHs)dS2 ~
- f ~ (VNa) T/zHs dS2 -t- f r
Nan T ( p H s ) d r
(7.50)
In subregions of constant/z, VTHs ----0, the only contribution to the forcing terms comes as a line integral of the second term at discontinuity interfaces. Introduction of the scalar potential makes both two- and three-dimensional magnetostatic problems solvable by a standard program used for all the problems in this chapter. Figure 7.14 shows a typical three-dimensional solution for a transformer. Here isoparametric quadratic brick elements of the type which were described in Chapter 5 were used. 15 In typical magnetostatic problems a high non-linearity exists with /z-/z(IHI)
where I e l - v/nx2 + By2 + nz2
(7.51)
The treatment of such non-linearities is outside the scope of this volume; however, the solution of such problems generally uses an iterative approach in which a sequence of linearized problems is solved. 18
250
Field problems
Fig. 7.14 Three-dimensional transformer. (a) Field strength H. (b) Scalar potential on plane z = 4.0 cm.
Considerable economy in this and other problems of infinite extent can be achieved by the use of the infinite elements discussed in Chapter 5. Many examples of practical applications of computing magnetic and electric field solutions have been given by Binns et al. and some are included in his recent book. 19 Plate 3 given in the front of this book presents one such example.
Numerical examples- an assessment of accuracy
Contours of ph~/6~VL L
2.625 in
/~
,.~1_.,
,.j
2.375 in
S[
r-
I
"
p=O
.E
---~
I///////////~I//////; ~ i / / / / / / / / / / / / / / / / / / / / / / A //////////////l~//////~v////////////////////////// h = ~ in
~ in
Fig. 7.15 A stepped pad bearing. Pressure distribution.
7.5.5 Lubrication problems Once again a standard Poisson type of equation is encountered in the two-dimensional domain of a bearing pad. In the simplest case of constant lubricant density and viscosity the equation to be solved is the Reynolds equation
~9 ( h3-~y ~)p) =6/zVm0x Oh Oi)x (h3OP) ~ + -~y
(7.52)
where h is the film thickness, p the pressure developed,/z the viscosity and V the velocity of the pad in the x direction. Figure 7.15 shows the pressure distribution in a typical finite width stepped pad. 2~ The boundary condition is simply that of zero pressure and it is of interest to note that the step causes an equivalent of a 'line load' on integration by parts of the right-hand side of Eq. (7.52), just as in the case of magnetic discontinuity mentioned above. More general cases of lubrication problems, including vertical pad movements (squeeze films) and compressibility, can obviously be dealt with, and much work has been done here. 21-28
7.5.6 Irrotational and free surface flows .
.
.
.
.
The basic Laplace equation which governs the flow of viscous fluid in seepage problems is also applicable in the problem of irrotational fluid flow outside the boundary layer created by viscous effects. The seepage example given above is adequate to illustrate the general
251
252
Field problems applicability in this context. Further examples for this class of problems are cited by Martin 29 and others. 3~ If no viscous effects exist, then it can be shown that for a fluid starting at rest the motion must be irrotational, i.e., 3u 3v =0 (7.53) C~ 3y 3x where u and v are appropriate velocity components. This implies the existence of a velocity potential, giving u -
0~ 3x
v=
0~ 3y
(7.54a)
or
u = -Vq~
(7.54b)
If, further, the flow is incompressible the continuity equation [which is similar to Eq. (7.4)] has to be satisfied, i.e., V T u "- 0 (7.55) and therefore V T ( V ~ ) "- V 2 ~ -
0
(7.56)
Alternatively, for two-dimensional flow a stream function may be introduced defining the velocities as 3 ~p 3~p u = v= (7.57) 3y 3x and this identically satisfies the continuity equation. The irrotationality condition must now ensure that V T(Vl/r) -- V2~/r = 0 (7.58) and thus problems of ideal fluid flow can be posed in either form. As the standard formulation is again applicable, there is little more that needs to be added, and for examples the reader can well consult the literature cited. We also discuss this problem in more detail in reference 37. The similarity with problems of seepage flow, which has already been discussed, is obvious.38, 39 A particular class of fluid flow deserves mention. This is the case when a free surface limits the extent of the flow and this surface is not known a priori. The class of problem is typified by two examples - that of a freely overflowing jet [Fig. 7.16(a)] and that of flow through an earth dam [Fig. 7.16(b)]. In both, the free surface represents a streamline and in both the position of the free surface is unknown a priori but has to be determined so that an additional condition on this surface is satisfied. For instance, in the second problem, if formulated in terms of the potential for the hydraulic head H, Eq. (7.43) governs the problem. The free surface, being a streamline, imposes the condition
OH On
= 0
(7.59)
be satisfied there. In addition, however, the pressure must be zero on the surface as this is exposed to atmosphere. As n = p + y Y
(7.60)
Concluding remarks 253 0
(a)
/ / ~ - p = 0
~
~
N
p =O
(b) Fig. 7.16 Typicalfree surface problems with a streamline also satisfyingan additional condition of pressure = O. (a) Jet overflow. (b) Seepage through an earth dam. where y is the fluid specific weight, p is the fluid pressure, and y the elevation above some (horizontal) datum, we must have on the surface H = y
(7.61)
The solution may be approached iteratively. Starting with a prescribed free surface streamline the standard problem is solved. A check is carried out to see if Eq. (7.61) is satisfied and, if not, an adjustment of the surface is carried out to make the new y equal to the H just found. A few iterations of this kind show that convergence is reasonably rapid. Taylor and Brown 4~ show such a process. Alternative methods including special variational principles for dealing with this problem have been devised over the years and interested readers can consult references 41-49.
7.1 The anisotropic properties for k are kx, = 0.4, ky, -- 2.1 and kz, = 1.0. The axes are oriented as shown in Fig. 7.17. For 0 = 30 ~ compute the terms in the matrix k (e.g., kxx, kxy, etc.) with respect to the axes x, y, z.
254
Field problems
J~
Y
x'
S J
x f Z=
Zt
Fig. 7.17 Orientation of axes for Problem 7.1. 7.2 A two-dimensional heat equation has its surface located in the x-y plane. The problem is allowed to convect heat from the surface of the surrounding region according to
Q(x, y) = -/3[q~(x, y) -q~0] where/3 is a convection parameter and 4~0 the temperature of the surrounding medium. Construct a weak form for the problem by modifying Eq. (7.13). For a finite element approximation to q~ and 64~ deduce the form of the matrices which result from the modified weak form. 7.3 For the quasi-harmonic equation consider a square 8-node serendipity element with unit side lengths in the x and y directions. Using FEAPpv (or any other available program) determine the rank of the element matrix H for the case where k = I (i.e., isotropic with k = 1)and H = 0 using 1 • 1 Gaussian quadrature. Repeat the calculation using 2 • 2, 3 • 3 and 4 • 4 quadrature. (a) What is the lowest order quadrature that gives a matrix H with full rank? (b) What is the lowest order quadrature that evaluates the matrix H exactly? [Hint: The rank of H may be determined from the eigenproblem given by: H
Vi =
)~i Vi
with
vf Vj ~-- (~ij
where ~ij is the Kronnecker delta. The rank of H is the number of non-zero eigenvalues (a zero is any value below the round-off limit).] 7.4 Solve Problem 7.3 for a 9-node lagrangian element. Using the eigenvector for the zero eigenvalue of the fully integrated element array H determine and sketch the shape of eigenvectors from any additional (spurious) zero eigenvalues. (Note: The fully integrated element has one zero eigenvalue, )~0.) (Hint: For the case where two zero eigenvectors Vl and v2 exist they may be expressed in terms of v0 and another orthogonal unit vector w0 as: ~,i
V0 m O/1V 1 _.[_ O/2V 2
where
O~i --" v T v i
W 0 - - 0~2V 1 - - 0 t l V 2
number of non-zero eigenvalues/~'i (a zero being a value below the computer round-off). The vectors v0, w0 and the vectors Vl, w2 are both eigenvectors of the same subspace.
Problems 255
-t-
.-I-
(a)
(b)
-I-
(c)
Fig. 7.18 Warping function for torsion of rectangular bar. Problem 7.5. 7.5 Consider the torsion of a rectangular bar by the warping function formulation discussed in Sec. 7.5.1. Let a and b be the side lengths in the x and y directions, respectively. For a homogeneous section with shear modulus G the warping function has the behaviour shown in Fig. 7.18 for a/b ratios of 1, 1.25 and 2. Note that the behaviour transitions from eight to four regions of -4- variation. Estimate the a/b ratio where this transition just occurs. To make your estimate use FEAPpv (or any other available program) with a fine mesh of quadratic lagrangian elements. Set the boundary conditions to make the warping function zero along the x and y axes. The transition will occur at the smallest a/b for which all the values on the perimeter of one quadrant of the cross-section have the same sign or are 'numerically' zero. 7.6 A cross-section of a long prismatic section is shown in Fig. 7.19 and subjected to constant uniform temperatures 370 C ~ on the left boundary and 66 C ~ on the fight boundary. The top and bottom edges are assumed to be insulated so that qn O. The cross-section is a composite of fir (A), concrete (B), glass wool (C) and yellow pine (D). The thermal conductivity for each of the parts is: ka = 0.11, k8 = 0.78, kc = 0.04 and ko = 0.147 in consistent units for the geometry of the section shown. (a) Estimate the heat flow through the cross-section assuming qy - 0 and qx constant -
Fig. 7.19 Thermal analysis of composite section. Problem 7.6.
-
256
Field problems in each part. Let the temperatures at each junction be T (0) = 370, T (0.025) = T1, T(0.10) = T2 and T(0.15) = 66. (Hint: Assume T is a function o f x only.) (b) Use FEAPpv (or any other available program) to compute a finite element solution using 4-node and 9-node quadrilateral elements. First perform a solution on a coarse mesh and use this to design a mesh using a finer discretization. Let your final 4-node element mesh have nodal locations which coincide with those used for the corresponding 9-node element mesh. Plot a distribution of heat flow q, across each of the internal boundaries. 7.7 The cross-section of two tubular sections is shown in Fig. 7.20. The parts are to be assembled by heating the outer part until it just passes over the inner part as shown in the figure. Let ri = 1 0 c m , t ---- 5 cm and h = 10 cm and take elastic properties as E = 200 Gpa, v = 0.3 and ot = 12 x 10 -6 per C ~ The parts are stress free at room temperature 20C ~ The parts just fit when the outer bar is heated to 220C ~ (while the inner part is maintained at room temperature). (a) What is the correct inner radius of the outer part at room temperature? (b) Solve the problem using FEAPpv (or any other available program). Use a mesh of 4-node quadrilateral elements to compute the final solution for the assembled part at room temperature assuming complete contact at the mating surface and no slip during cooling. Plot the radial displacement at the interaction surface. (c) Compute an estimate of the traction components at the interaction surface. Do you think there will be slip? Why? 7.8 Company X&Y plans to produce a rectangular block which needs to be processed by a thermal quench in a medium which is 100~ above room temperature. The block shown in Fig. 7.21 (a) has a = 10 and b = 20 (i.e., the block is 10 x 10 x 20). It has been determined that the thermal properties of the block may be specified by an isotropic Fourier model in which k = 1 and c = 1. The surface convection constant H is 0.05. The quench must be maintained until the minimum temperature in the block reaches 99~ above room temperature. Use FEAPpv (or any other available program) to perform a transient analysis to estimate the required quench time.
Fig. 7.20 Thermal assembly of tubular sections. Problem 7.7.
Problems
qn = H (T-Too)
/
~X
%=0
qn = H ( T- Too)
x
z
f
~
%=0
(a)
(b)
Fig. 7.21 Thermalquench in 2 and 3 dimensions. Problem 7.8.
(a) First perform a two-dimensional plane analysis on a 10 x 10 cross-section using a uniform mesh of 4-node quadrilateral elements. Use symmetry to reduce the size of the domain analysed. The surface convection will be modelled by 2-node line elements along the outer perimeter. The analysis region is shown in Fig. 7.21(b) with the boundary conditions to be imposed. Locate the node where the minimum temperature occurs and plot the behaviour vs time (a good option is to use MATLAB to perform plots). Estimate the duration of time needed for the minimum temperature to reach the desired value. (Hint: One approach to selecting time increments is to select a very small value, e.g. At = 10 -8 and perform 10 steps of the solution. Multiply the time increment by 10 and perform 9 more steps. Repeat the multiplication until the desired time is reached.) (b) Using the time duration estimated in (a) perform a three-dimensional analysis using a uniform mesh of 8-node hexahedral elements. Use symmetry to reduce the size of the region analysed. (Note: The convection condition applies to all outer surfaces.) Estimate the duration of quench time needed for the minimum temperature to reach the desired value. (c) What analyses would you perform if the block was 10 x 10 x 5? (d) Comment on use of a two-dimensional solution to estimate the required quench times for other shaped parts. 7.9 The distribution of shear stresses on the cross-section of a cantilever beam shown in Fig. 7.22(a) may be determined by solving the quasi-harmonic equation 5~
Oq2~ a2~ 07 + ~
=0
with boundary condition =~
P [f
y2
dx-
v y3 3(1+v)
257
258
Field problems
Fig. 7.22 End loaded cantilever beam. Problem 7.9.
where P is the end load, I is the moment of inertia of the cross-section, v is the Poisson ratio of an isotropic elastic material and 4) is a stress function. The shear stresses are determined from ~XZ
04) and
Oy
_ 04) + P
"gyz- OX
V
X 2 _ y2]
l+v
J
See reference 50 for details on the formulation. (a) Show that the stress function satisfies the equilibrium equation when the bending stress is computed from e (L - z) y O'z=n I and Crx = Cry = 75xy - - O. L is the length of the beam. (b) Develop a weak form for the problem in terms of the stress function 4). (c) For a finite element formulation develop the relation to compute the boundary condition for the case when either 3-node triangular or 4-node quadrilateral elements are used. (d) Write a program to determine the boundary values for the cross-section shown in Fig. 7.22(b). Let w = 2 and h = 3. Use the quasi-harmonic thermal element in FEAPpv (or any other available program) to solve for the stress function 4). Plot the distribution for 4) on the cross section. (e) Modify the expressions in FEAPpv (or any other program for which source code is available) to compute the stress distribution on the cross-section. Solve and plot their distribution. Compare your results to those computed from the classical strength of materials approach. (Hint: Normalize your solution by the factor P/2I to simpify expressions.) 7.10 A long sheet pile is placed in soil as shown in Fig. 7.23. The anisotropic properties of the soil are oriented so that x - x', and y - y'. The governing differential equation is given in Sec. 7.5.3. The soil has the properties kx = 2 and ky - 3. Use FEAPpv (or any other available program) to determine the distribution of head and the flow in the region shown. Solve the problem using a mesh of 4-node, 8-node, and 9-node
Problems 259 kY
H=20
Pile H=0
50
qn=O qn=0
50
30
qn=0
x r
90
,..__1~.,
110
Fig. 7.23 Seepageunder a sheet pile. Problem 7.10.
~, z
H=0
Pile H=40
50
qn=0
50 30
qn=O
r r
/
Fig. 7.24 Seepage under an axisymmetric sheet pile. Problem 7.11.
quadrilateral elements. Model the problem so that there are about four times as many 4-node elements as used for the 8- and 9-node models (and thus approximately an equal number of nodes for each model). Compare total flow obtained from each analysis. 7.11 An axisymmetric sheet pile is placed in soil as shown in Fig. 7.24. The anisotropic properties of the soil are oriented so that r = r' and z = z'. The governing equation for plane flow is given in Sec. 7.5.3. Deduce the Euler differential equation for the axisymmetric problem from the weak form given in Secs 7.2.3 and 7.3.2 suitably modified for the seepage problem.
260
Field problems Assuming isotropic properties with k -- 3, use FEAPpv (or any other available program) to determine the distribution of head and the flow in the region shown. Solve the problem using a mesh of 4-node, 8-node, and 9-node quadrilateral elements. Model the problem so that there are about four times as many 4-node elements as used for the 8- and 9-node models (and thus approximately an equal number of nodes for each model). Compare total flow obtained from each analysis. 7.12 A membrane occupies a region in the x - y plane and is stretched by a uniform tension T. When subjected to a transient load q (x, y, t) acting normal to the surface the goveming differential equation is given by
-r
1
LOx2 + 8Y 2 J + m - - ~ = q(x, y, t)
(a) Construct a weak form for the differential equation for the case when boundary conditions are given by u (s, t) = 0 for s on P. (b) Show that the solution by a finite element method may be constructed using C O functions. (c) Approximate the u and 3u by C O shape functions Na(x, y) and determine the semi-discrete form of the equations. (d) For the case of steady harmonic motion, u may be replaced by
u(x, y, t) = w(x, y) expioot where i = ~ and o) is the frequency of excitation. Using this approximation, deduce the governing equation for w. Construct a weak form for this equation. Using C Oapproximations for w determine the form of the discretized problem. 7.13 Program development project: Modify the program developed for solution of linear elasticity problems to solve problems described by the quasi-harmonic equation for heat conduction. Include capability to solve both plane and axisymmetric geometry. Specify the material properties by anisotropic values k'x, k'y and/3 (where/3 is the angle x' makes with the x axis). Use your program to solve the problem described in Problem 7.6. Plot contours for temperature and heat flows qx and qy. 7.14 Program development project: Extend the program developed in Problem 7.13 to solve transient problems. Include an input module to specify the initial temperatures. Also add a capability to consider time-dependent source terms for Q. Test your program by solving the problem described in Example 7.6 of Sec. 7.5.2. 7.15 Program development project: Extend the program developed in Problem 2.17 to compute nodal values of fluxes from the quasi-harmonic equation. Follow the procedure given in Sec. 6.4 to project element values to nodes. Test your program using (a) a patch test of your design and (b) the problem described in Example 6.6.
References 261
1. O.C. Zienkiewicz and Y.K. Cheung. Finite elements in the solution of field problems. The Engineer, pages 507-510, Sept. 1965. 2. W. Visser. A finite element method for the determination of non-stationary temperature distribution and thermal deformation. In Proc. 1st Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-66-80, Wright Patterson Air Force Base, Ohio, Oct. 1966. 3. O.C. Zienkiewicz, P. Mayer, and Y.K. Cheung. Solution of anisotropic seepage problems by finite elements. J. Eng. Mech., ASCE, 92(EM1):l 11-120, 1966. 4. O.C. Zienkiewicz, P.L. Arlett, and A.K. Bahrani. Solution of three-dimensional field problems by the finite element method. The Engineer, Oct. 1967. 5. L.R. Herrmann. Elastic torsion analysis of irregular shapes. J. Eng. Mech., ASCE, 91(EM6):I 1-19, 1965. 6. A.M. Winslow. Numerical solution of the quasi-linear Poisson equation in a non-uniform triangle 'mesh'. J. Comp. Phys., 1:149-172, 1966. 7. D.N. de G. Allen. Relaxation Methods. McGraw-Hill, London, 1955. 8. S.P. Timoshenko and J.N. Goodier. Theory of Elasticity. McGraw-Hill, New York, 3rd edition, 1969. 9. J.F. Ely and O.C. Zienkiewicz. Torsion of compound bars - a relaxation solution. Int. J. Mech. Sci., 1:356-365, 1960. 10. O.C. Zienkiewicz and Y.K. Cheung. The Finite Element Method in Structural Mechanics. McGraw-Hill, London, 1967. 11. P. Silvester and M.V.K. Chaff. Non-linear magnetic field analysis of D.C. machines. Trans. IEEE, No. 7:5-89, 1970. 12. P. Silvester and M.S. Hsieh. Finite element solution of two dimensional exterior field problems. Proc. IEEE, 118, 1971. 13. B.H. McDonald and A. Wexler. Finite element solution of unbounded field problems. Proc. IEEE, MTT-20(12), 1972. 14. E. Munro. Computer design of electron lenses by the finite element method. In Image Processing and Computer Aided Design in Electron Optics, page 284. Academic Press, New York, 1973. 15. O.C. Zienkiewicz, J.F. Lyness, and D.R.J. Owen. Three-dimensional magnetic field determination using a scalar potential. A finite element solution. IEEE Trans. Mag., MAG-13(5): 16491656, 1977. 16. J. Simkin and C.W. Trowbridge. On the use of the total scalar potential in the numerical solution of field problems in electromagnets. Int. J. Numer. Meth. Eng., 14:423-440, 1979. 17. J. Simkin and C.W. Trowbridge. Three-dimensional non-linear electromagnetic field computations using scalar potentials. Proc. Inst. Elec. Eng., 127(B(6)):423-440, 1980. 18. O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method for Solid and Structural Mechanics. Butterworth-Heinemann, Oxford, 6th edition, 2005. 19. K.J. Binns, P.J. Lawrenson, and C.W. Trowbridge. The Analytical and Numerical Solution of Electric and Magnetic Fields. John Wiley & Sons, Chichester, 1992. 20. D.V. Tanesa and I.C. Rao. Student project report on lubrication. Royal Naval College, 1966. 21. M.M. Reddi. Finite element solution of the incompressible lubrication problem. Trans. Am. Soc. Mech. Eng., 91(Ser. F):524, 1969. 22. M.M. Reddi and T.Y. Chu. Finite element solution of the steady state compressible lubrication problem. Trans. Am. Soc. Mech. Eng., 92(Ser. F):495, 1970. 23. J.H. Argyris and D.W. Scharpf. The incompressible lubrication problem. J. Roy. Aero. Soc., 73:1044-1046, 1969. 24. J.E Booker and K.H. Huebner. Application of finite element methods to lubrication: an engineering approach. J. Lubr. Techn., Trans, ASME, 14((Ser. F)):313, 1972.
262 Field problems 25. K.H. Huebner. Application of finite element methods to thermohydrodynamic lubrication. Int. J. Numer. Meth. Eng., 8:139-168, 1974. 26. S.M. Rohde and K.P. Oh. Higher order finite element methods for the solution of compressible porous bearing problems. Int. J. Numer. Meth. Eng., 9:903-912, 1975. 27. A.K. Tieu. Oil film temperature distributions in an infinitely wide glider bearing: an application of the finite element method. J. Mech. Eng. Sci., 15:311, 1973. 28. K.H. Huebner. Finite element analysis of fluid film lubrication- a survey. In R.H. Gallagher, J.T. Oden, C. Taylor, and O.C. Zienkiewicz, editors, Finite Elements in Fluids, volume II, pages 225-254. John Wiley & Sons, New York, 1975. 29. H.C. Martin. Finite element analysis of fluid flows. In Proc. 2nd Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-68-150, Wright Patterson Air Force Base, Ohio, Oct. 1968. 30. G. de Vries and D.H. Norrie. Application of the finite element technique to potential flow problems. Technical Reports 7 and 8, Dept. Mech. Eng., Univ. of Calgary, Alberta, Canada, 1969. 31. J.H. Argyris, G. Mareczek, and D.W. Scharpf. Two and three dimensional flow using finite elements. J. Roy. Aero. Soc., 73:961-964, 1969. 32. L.J. Doctors. An application of finite element technique to boundary value problems of potential flow. Int. J. Numer. Meth. Eng., 2:243-252, 1970. 33. G. de Vries and D.H. Norrie. The application of the finite element technique to potential flow problems. J. Appl. Mech., ASME, 38:978-802, 1971. 34. S.T.K. Chan, B.E. Larock, and L.R. Herrmann. Free surface ideal fluid flows by finite elements. J. Hydraulics Division, ASCE, 99(HY6), 1973. 35. B.E. Larock. Jets from two dimensional symmetric nozzles of arbitrary shape. J. Fluid Mech., 37:479-483, 1969. 36. A. Curnier and R.L. Taylor. A thermomechanical formulation and solution of lubricated contacts between deformable solids. J. Lub. Tech., ASME, 104:109-117, 1982. 37. O.C. Zienkiewicz, R.L. Taylor, and P. Nithiarasu. The Finite Element Method for Fluid Dynamics. Butterworth-Heinemann, Oxford, 6th edition, 2005. 38. C.S. Desai. Finite element methods for flow in porous media. In J.T. Oden, O.C. Zienkiewicz, R.H. Gallagher, and C. Taylor, editors, Finite Elements in Fluids, volume 1, pages 157-182. John Wiley & Sons, New York, 1976. 39. I. Javandel and P.A. Witherspoon. Applications of the finite element method to transient flow in porous media. Trans. Soc. Petrol. Eng., 243:241-251, 1968. 40. R.L. Taylor and C.B. Brown. Darcy flow solutions with a free surface. J. Hydraulics Division, ASCE, 93(HY2):25-33, 1967. 41. J.C. Luke. A variational principle for a fluid with a free surface. J. Fluid Mech., 27:395-397, 1957. 42. K. Washizu. Variational Methods in Elasticity and Plasticity. Pergamon Press, New York, 3rd edition, 1982. 43. J.C. Bruch. A survey of free-boundary value problems in the theory of fluid flow through porous media. Adv. Water Res., 3:65-80, 1980. 44. C. Baiocchi, V. Comincioli, and V. Maione. Unconfined flow through porous media. Meccanice, Ital. Ass. Theor. Appl. Mech., 10:51-60, 1975. 45. J.M. Sloss and J.C. Bruch. Free surface seepage problem. J. Eng. Mech., ASCE, 108(EM5): 10991111, 1978. 46. N. Kikuchi. Seepage flow problems by variational inequalities. Int. J. Numer. Anal. Meth. Geomech., 1:283-290, 1977. 47. C.S. Desai. Finite element residual schemes for unconfined flow. Int. J. Numer. Meth. Eng., 10:1415-1418, 1976.
References 263 48. C.S. Desai and G.C. Li. A residual flow procedure and application for free surface, and porous media. Adv. Water Res., 6:27-40, 1983. 49. K.J. Bathe and M. Koshgoftar. Finite elements from surface seepage analysis without mesh iteration. Int. J. Numer. Anal. Meth. Geomech., 3:13-22, 1979. 50. S.P. Timoshenko and J.N. Goodier. Theory of Elasticity. McGraw-Hill, New York, 2nd edition, 1951.
Automatic mesh generation
In the previous chapters we have introduced various forms of elements and the procedures of using these elements in the computation of approximate solutions of a wide range of engineering problems. It is now obvious that the first step in the finite element computation is to discretize the problem domain into a union of elements. These elements could be of any one type or a combination of different types of those described in Chapters 4 and 5. The union of these elements is the so-called finite element mesh. The process of creating a finite element mesh is often termed as mesh generation. Mesh generation has always been a time-consuming and error-prone process. This is especially true in the practical science and engineering computations, where meshes have to be generated for three-dimensional geometries of various levels of complexity. The attempt to create a fully automatic mesh generator, which is a particular mesh generation algorithm that is capable of generating valid finite element meshes over arbitrary domains and needs only the information of the specified geometric boundary of the domain and the required distribution of the element size, started from the work of Zienkiewicz and Phillips 1 in the early 1970s. Since then many methodologies have been proposed and different algorithms have been devised in the development of automatic mesh generators. The early proposed mesh generation methods, such as the isoparametric mapping method by Zienkiewicz and Phillips, 1 the transfinite mapping method by Gordon and Hall 2 discussed in Sec. 5.13, and the method of generating a mesh by solving various types of partial differential equations as described by Thompson et al., 3' 4 are often regarded as semi-automatic mesh generation methods. This is because in the mesh generation process the model domain has to be subdivided manually into simple subregions, i.e., multi-blocks, which are then mapped onto regular grids to produce a mesh. This manual process is tedious and occasionally difficult, particularly in the case of three-dimensional complex geometries. Such mesh generation procedures are complicated further by the requirement of varying mesh size distributions since the element sizes are controlled by the subdivision of the simple subregions. Thus, more subregions are needed to generate a mesh which can accommodate changes in the desired element sizes from region to region. However, one of the main features of these mapping techniques is that, once the domain is decomposed into mappable subregions, the generation of the elements is much easier than any other methods. In addition, the elements generated by mapping methods usually have good shape and regular orientation. Mapping methods are often used to generate quadrilateral elements
Introduction
in two dimensions and hexahedral elements in three dimensions. Triangles, tetrahedra and all other types of elements can be obtained by dividing quadrilaterals and hexahedra accordingly. These generated meshes are sometimes called structured meshes. Over the years, continuous efforts have been made to automate the mapping methods, 5-1~ although automatic decomposition of a complex domain into subregions seems to be a non-trivial task. Today, no fully automatic mesh generator using a mapping method has been achieved. In contrast with mapping methods, in recent years concrete achievements have been made in the development of various algorithms for the automatic generation of the so-called unstructured meshes. Most of the unstructured mesh generation methods are designed for generating triangular elements in two dimensions and tetrahedral elements in three dimensions (known as simplex forms). These simplex forms lead to the simplest discretization of two- and three-dimensional domains of any shape, especially when meshes with varying element sizes in different regions of the domain are requested. A large number of automatic unstructured mesh generation algorithms have been proposed in the literature, but the most widely used algorithms are based on one or some kind of combination of the three fundamentally distinctive methods, which are the Delaunay triangulation method, 11-20 the advancing front method 21-24, 26-28,43 and tree methods 29-31 (the finite quadtree method in two dimensions and the finite octree method in three dimensions). By observing the fact that a quadrilateral can be formed by two triangles which share a common edge, the abovementioned methods can be extended to automatically generate unstructured quadrilateral meshes in two dimensions. However, automatically generating a hexahedral mesh 1~ 32-35 encounters the almost identical difficulties as that in the mapping methods, much research is still needed in this direction. The automatic mesh generation process has been an active research subject since the early 1970s. The research literature on the subject is vast and different methodologies and techniques have been proposed. In this chapter, we are mainly concerned with the automatic mesh generation methods based on the advancing front method and the Delaunay triangulation method. These are the basis of many existing mesh generation programs and the basis of current research. We shall discuss the algorithmic procedures of the advancing front method in two dimensions and the Delaunay triangulation method in three dimensions. We shall also discuss curve and surface mesh generations. The reader is referred to Thompson et al. 5' 36 for discussions on the development of semi-automatic multi-block mesh generation methods. Before proceeding further on mesh generation schemes it is necessary to specify the kind of mesh we desire. Here we should give the following information: 1. The type of element and the number of nodes required on each; 2. The size of the desired element, here the minimum size of each element generally is specified; 3. Specification of regions of different material types or characteristics to be attached to a given element; and 4. In some cases, the so-called stretch ratio if we wish to present elements which are elongated in some preferential direction. This is often needed for problems in fluid mechanics in regions where boundary layers and shocks are encountered. Any and all of the above information has to be available at all points of the space in which elements are to be generated. It is often convenient to present this information as numbers attached to a background mesh, consisting say of elements of a linear kind, from
265
266 Automatic mesh generation
which these values can be interpolated to any point in space. The procedure is important particularly if the use of adaptive refinement is considered- and here all mesh generation schemes must ensure that the input data contains this information. In adaptive refinement, in fact in general analysis, the background mesh will simply be the last mesh used for analysis of the problem and the refinement will proceed from there as this is the starting point for any new mesh to be developed.
Conceptually, the advancing front method is one of the simplest generation processes. The element generation algorithm, starting from an initial 'front' formed from the specified boundary of the domain, generates elements, one by one, as the front advances into the region to be discretized until the whole domain is completely covered by elements. The representative element generation algorithms of the advancing front method include the procedure introduced by Lo, 21 which constructs a triangulation over a set of a priori generated points inside of the domain, and the methodology developed by Peraire et al., 22 which generates points and triangular elements at the same time. One of the main distinctions of the mesh generation algorithm of Peraire et al. is that the geometrical characteristics of the mesh, such as the location of the newly generated point, the shape of the element and the size of the element, can be controlled during the mesh generation process, due to the fact that individual points and elements are generated simultaneously. With the assistance of a background mesh, which is utilized to define the geometrical characteristics of the mesh, non-uniform distribution of element sizes, often required in highly graded meshes, can be achieved throughout the domain according to particular specifications. Any directional orientation of the elements can also be realized by introducing stretches in certain specified directions. These features are particularly desirable for the nearly optimal mesh design in adaptive analysis (viz. Chapter 14) and adaptive computations of fluid dynamics as discussed in reference 37. The mesh generation procedure includes three main steps: 9 Node generation along boundary edges to form a discretized boundary of the domain. 9 Element (and node) generation within the discretized boundary. 9 Element shape enhancement to improve the quality of the mesh. Before we proceed to the discussion of mesh generation procedures, the geometrical representation of the two-dimensional domain is introduced.
8.2.1 Geometrical characteristics of the mesh The geometrical characteristics of the mesh such as element size, element shape and element orientation are represented by means of mesh parameters which are spatial functions. The mesh parameters include two orthogonal directions defined by unit vectors O~i (i "-" 1, 2 ) and the associated element sizes hi (i -- 1, 2) as illustrated in Fig. 8.1. The orthogonal directions O~i (i "-- l , 2 ) describe the directions of element stretching. A mesh with stretched elements in certain directions isonly necessary when a non-isotropic mesh is desired, otherwise the
Two-dimensional mesh generation- advancing front method 267
Y
O~1 x
Fig. 8.1 Mesh parameters in two dimensions. stretching directions are set to be constant unit vectors in the coordinate directions and the related element sizes are set to be equal, i.e., h l = h2. In this case, the generated elements will not be stretched in any direction and an isotropic mesh type will be generated.
Background mesh
The background mesh may be represented by simple triangular elements and is employed to accurately control the distribution of the geometrical characteristics on the new mesh. A piecewise linear distribution of the mesh parameters (mainly element sizes and stretching as discussed above) is represented by data assigned to nodes of the background mesh. Values of the mesh parameters at any point inside the domain or on the boundary of the domain can be obtained by linear interpolation. There is no requirement that the background mesh precisely represent the geometry, but it should completely cover the domain to be meshed. The number of the elements and the position of the nodes in the background mesh are chosen so that the mesh parameters can be approximated in a satisfactory manner. One or two background elements will be sufficient if a uniform (isotropic) distribution of the element sizes hi (i = 1, 2) is required. Examples of background meshes for a given domain are illustrated in Fig. 8.2.
8.2.2 Geometrical representation of the domain A general two-dimensional domain, which is covered by a background mesh (viz. Fig. 8.2), is defined by its boundary which consists of a closed loop of curved boundary segments (viz. Fig. 8.3).
Boundary curve representation
The curvilinear boundary segments are in general represented by composite parametric spline curves. A curved boundary segment in two dimensions, can be expressed by a vector valued function, using a parameter t, as x(t) = {x(t) y(t)},
0 _< t ~ 1
(8.1)
268
Automatic
mesh generation
O,
9 xxxxxx/xx
/
/
/ t
I i/
/ I II
"~" .~. ~
:-"-Y--4," x
x
x
x
, ' I
@,
ix I x I \
I I
x
:I, I I
\
\
\
i
".
'-, -,, 't, ..,..,.. ~
,"
~
\
/
/"
,,-\
,...,,
.
%
(a)
I
,
/ \ _ _
.
%
I
"~
......... :: ....-',.' I I
x%
.,.
(b)
Fig. 8.2 Background meshes for a typical domain. (a) Two triangles are used in the background mesh to represent a uniform distribution of the mesh parameters. (b) Eleven triangles are used in the background mesh to generate a graded mesh. Boundary edge
/
J
/
w
Boundary data point Fig. 8.3 Boundary segments, boundary data points and orientation of a typical domain.
In general, a composite spline curve is required to be at least C 1 continuous to preserve the smoothness of the boundary curve and to satisfy the continuity conditions required by mesh generation algorithms. A Hermite cubic spline is used in the following; however, there are many other types of parametric spline curves that can be used to represent the curved boundary edges. 38' 39 The parametric description of a Hermite cubic spline is given by the form
x(0) x(1) x ( t ) - {no(t) e l ( t ) Go(t) Gm(t)} x,t(0)
,
0 _
(8.2)
X,t(1)
in which x(0), x(1) are the coordinates of the end points and x,t (0), x,t (1) are their respective tangent vectors defined as dx(t) x,(t) =
dt
(8.3)
Two-dimensional mesh generation- advancing front method 269 Cubic Hermite polynomials are expressed as H1 (t) -- 3t 2 - 2t 3 G1 (t) -- - t 2 -+- t 3
Ho(t) = 1 - 3t 2 -k- 2t3; Go(t) = t - 2t 2 + t3;
(8.4)
and depicted in Fig. 8.4. It is easy to verify that Ha (b) = G'a (b) = ~ab
and
Ga(b) - O; a, b -- O, 1
H~ (b) -
(8.5)
Substituting (8.4) into (8.2) we obtain
x(0) x(t) = {1
x(1) x,t(0)
t t 2 t 3} M
,
0 ~t
_<< 1
(8.6)
x,t(1) where M =
1
0
0
0
0 -3
0 3
1 -2
0 -1
1
1
2
-2
(8.7)
All boundary edges are transformed to their spline representations. For each boundary edge, a set of ordered data points xi (i = 0, 1 . . . . . n) are located. An interpolation by piecewise cubic Hermite polynomials through pairs of these points forms a composite parametric spline curve. The number and distribution of the points should be chosen such that the resulting piecewise cubic spline accurately represents the geometry of the boundary. For a curved edge segment [ui-1, ui] of length A i = U i - - U i _ 1 ( i ~-- 1, 2 . . . . . n) of the interpolated cubic composite curve, the cubic spline has the form of x(u) =
Ho(t)x(ui-1)-Jr- Hl(t)x(ui)-4-
AiGo(t)x,t(ui-1)-+- A i G l ( t ) x , t ( u i )
X(Ui-1)
--{1
t t2 t3}M
X(Ui)
Aix,t(Ui_l
)
0 < t < 1 __
__
AiX,t(Ui)
=t G1
Fig. 8.4 Cubic Hermite interpolation functions.
(8.8)
270 Automatic mesh generation where t = (u - U i _ l ) / A i is the local parameter of the segment [Ui_I, Ui] and u 6 [uo, u,]. The unknown tangents of the composite curve can be computed by the standard cubic spline interpolation procedures, provided that the final parametric curve is C 2 continuous, with proper end conditions, as, 40 The implementation of the spline interpolation can be simplified when the parametric coordinates of ui are taken to be ui = i (i - 1, 2 . . . . . n). For segment [b/i_l, Ui] = [i -- 1, /], the global parametric coordinates u is then related to the local coordinate t according to U'--Ui-1
+t
=i-
(8.9)
1 +t
The global mapping of the region u 6 [u0, Un ] in parametric space and the cubic composite curve provided by x(u) is depicted in Fig. 8.5. The collection of all boundary edges, following a specific sequence that is convenient for mesh generation, forms the complete boundary of the domain. For the advancing front method, the sequence of exterior boundary edges is usually in a counterclockwise order, but, for interior boundary edges, is set in a clockwise order, i.e., the domain to be discretized always has an interior area situated to the left of the boundary edges. Figure 8.3 shows the direction of the boundary edges together with the boundary data points, of a typical domain.
8.2.3 Triangular mesh generation Among all the steps in the mesh generation process, we are particularly concerned with the procedure for element generation, which include node and side generation on the boundary curve and triangular element generation inside of the two-dimensional domain. 22
x~
X
Xi+l
I X(u) ~X
A w
UO
Fig. 8.5
A w
A w
-,w
Ui
Compositecubicsplineinterpolationof a planarcurve.
A w
Ui+l
..L w
Un
U
Two-dimensional mesh generation - advancing front method
Geometrical transformation of the mesh
In order to simplify the mesh generation process, a symmetric transformation matrix T, defined by the mesh parameters, is introduced and has the form
21
T(x) = ~ - - ~ i c ~ T ; ~'~.= h i
Ore "--
~Otli} I,~
(8.10)
It is easy to verify that the local transformation x ' = Tx
(8.11)
is in fact imposing two scaling operations with factor 1/hi in each of the corresponding directions c~i. Figure 8.6 illustrates the effect of the transformation T on a triangle formed by nodes abc in the coordinate system (x, y) to form the triangle a'b'c' in the coordinate system (x', y'). This demonstrates that, at a particular point, the transformation T maps triangle with element size hi(i = 1, 2) formed in the neighbourhood of the point into a normalized space (x', y'), in which the triangular elements are approximately equilateral.
Example 8.1: Transformation of a triangle. As an example, the details of the coordinates transformation shown in Fig. 8.6 are given as follows. From the coordinates of nodes a, b and c, we can easily find that at node b
and
(
with the associated element sizes h l -- 4~/2 and h2 -- V/2. The transformation matrix T at node b is computed, using Eq. (8.10), as
T=-i-g
1 1
+q--
-1
1)
-1
Applying T to nodes a, b, c results in
Xa _.Txa - - ~/2 (71) 8
,
X b,
~/2{1) = TXb= -~ 1
and Xc, = Txc = -~-
3
These are the nodal positions for the triangle a'b'c' in the coordinate system (x', y').
Boundary node generation
The boundary of the domain will be discretized into a polygon which will form the element edges, herein called 'sides'. The sides are defined by nodes generated on the composite spline curves that represent the boundary edges. The nodes will be generated along the curve edge and expressed by their parametric positions. The coordinates of the nodes in the two-dimensional domain are determined using Eq. (8.8). The algorithmic procedure of the boundary node generation is described in the following. 1. For a curved edge with typical length L, a set of sampling points X I - - X ( U / ) (1 = 0, 1, 2 , . . . , m) is first placed along the curve with parameters Ul uniformly
271
272
Automatic mesh generation
c(6,6) a(2,4)
5(2,2) '
,
2
-
4
,,
6
X
x' = Tx
yl
1.5
_'"'"'"'---"-~ c'(~, ~)
a'(- ~'7@) 1 ~.0 0.5
b,~,/~ ,,,a x4 4 ~./ ~ - .5
I
0
I
0.5
I
1.0
IP
1.5
X t
Fig. 8.6 An irregular triangle abc in coordinate system (x, y) mapped to a regular triangle a'b'c' in coordinate system (x; y'). distributed as shown in Fig. 8.7. The unit tangent vector of the curve is determined at each of the sampling points as t~ = (tit , t2l ) (8.12) where tl, =
X Ul
'
v/X,ul + y, l
and
t2, =
Y, Ul
v/X,ul + y,ul
(8.13)
and X,u, = (X,u, , Y,u, ) ;
x,ul =
dx Iu, -~u
(8.14)
Two-dimensional mesh generation - advancing front method
hl 2 a/2
• x
Y~
hl~al, ~X
uo
ul
Urn
Fig. 8.7 Sampling points and mesh parameters on a composite curve segment.
2. The mesh parameters Otli and hli (i = 1, 2) are computed at each sampling point by interpolation from their values assigned on the background mesh. The transformation matrix TI is formed accordingly at each of the sampling point. 3. In order to find the position of the new nodes on the curve, an element size distribution function needs to be determined along the curve edge. Let hsl denote the element size at the arc length st corresponding to the sampling point Xl, a vector of length h sl in the tangent direction is defined as
(8.15)
1"l = hsltl
Applying transformation T1 and assuming that Tl maps malized space with unit length, i.e.,
1" l
to a vector T r l in the nor-
(8.16)
Tl'rl = T l ( h s l t l ) -- hslTltl
with T(T/T/) = hsl v/(Tltl) T(T/t/) -- 1
r
(8.17)
Thus, the curvilinear element size hsl (l = 0, 1, 2 . . . . , m) at the sampling points along the curve in the direction of the tangent is hsl =
1 r
where C-
=
1
v/tTC/t/
2 1 TTT =/~1= ~/20tiOtT
(8.18)
(8.19)
is the two-dimensional matrix of E u c l i d e a n m e t r i c t e n s o r in the normalized space. 4. Assume that TI is a constant matrix in the neighbourhood of Xl, it is observed, from Eq. (8.13), that hsl is a function of the parameter u. A continuous element size distribution
273
274
Automaticmesh generation function may be achieved by a piecewise linear interpolation of the nodal values hsl using (lagrangian) finite element shape functions along the arc length of the curve
m h(u) = ~
hsiNl(u )
(8.20)
/=0 The element density function, i.e., the number of elements per length scale along the curve, is defined as 1/h(u). 5. The total number of the sides N to be generated along the curved edge needs to be consistent with the specified element size, which is now represented by the element density function. Therefore, N is taken to be the nearest integer to A -- ~0 L h ~1
ds = fUl tm h -1- ~ v / ( x , ) 2 + (yu) 2 du
(8.21)
where A is the ideal number of the sides that should have been created on the boundary curve edge. However, A is in general not an integer. To measure how close N is to A, a consistency index is defined as 0 = ~A -- ~1 fUl !m h -1~ V/(Xu)2 + (Yu )2 du
(8.22)
Because the position of the nodes at the end of the edge x0 = x(u0) and Xm = X(Um) are already known, there will be (N - 1) new nodes to be generated. 6. Assume that every node on the boundary edge is generated with the same consistency index 0, the position of a particular new node nk(k = 1, 2 . . . . . ( N - 1)), represented by its parametric position uk on the boundary curve can be computed as 0 -- Ok -- ~1 ful 'k h -1~ v/(Xu)2 + (yu )2 du
(8.23)
and similarly, the position of new node nk+l is given by 1 fuk+~ 1 V/(Xu) 2 --[- (Yu) 2 du 0 = Ok+l = k + 1 ~uo h(u)
(8.24)
From Ok = 0k+l = 0, we obtain the parametric position of node nk+l computed consecutively as uk+l 1 0 -" g/(Xu) 2 -k- (yu) 2 du (8.25) ~uk h(u) where k = 0, 1, 2 . . . . . (N - 2). In general, Eq. (8.25) can be solved iteratively for Uk+l. For example, writing Eq. (8.25) in the form of F(Uk+l) =
f
uk+~ h(u----~ 1 V/(Xu) 2 -[- (Yu) 2 du - 0 -- 0
,Auk
A Newton iterative process results in uj+~
j
k+l "-- U k + l for j
---
O, 1 , 2 , . . .
h --
J
v/(Xu~+l)2 -[-- (yu~+l)2
with initial value U ko+
1 -'- Uk"
9
(8.26)
Two-dimensional mesh generation - advancing front method
7. Finally, the position of the new points is mapped onto the boundary curve using Eq. (8.8). The generation of the boundary nodes will be performed edge by edge following the above procedure. The boundary of the domain is finally discretized and transformed to a union of straight line sides formed by connecting the consecutive boundary nodes. Example 8.2: To verify the above procedure, the node generation technique is applied to a curve edge representing a quarter of a circle shown in Fig. 8.8(a). To simplify the presentation and also to demonstrate that the node generation procedure is independent of the curve representation of the boundary edge, we chose not to use Hermite spline to represent the curve, but use its parametric expression in the form of x = 4cos(u),
7t"
y = 4 sin(u);
0 < u < --
-
(8.28)
2
The mesh parameters a i and hi (i = 1, 2) are chosen to be independent of their spatial positions, the background mesh is therefore not required 9 We shall follow each step of the boundary node generation procedure to create nodes in accordance with the specified element size.
X(U/) (1 =
1. Three sampling points
0, 1, 2) are located with
u0 = 0;
7/"
u l
equally distributed at
7/" /12 - - 2-
Ul = 4 '
The unit tangent vector to the curve is, by noting Eq. (8.13), t = (-sin(u), cos(u)) T and t(u2) -- (1, 0) T
;
2. The stretching directions are chosen as depicted in Fig. 8.8(a), i.e., Ot 1 - -
(1,
Y
0)T;
Ot 2 - - ( 0 ,
1) T
Y (}{2; h 2 = 2
T
New node r- Or,l; h I = 2
J
.... ~ /
D i s c r e t i z e d b o u n d a r y side
Sampling point x]
'
(2)
(a)
~ X
u-~
~
"x
(b)
Fig. 8.8 Node generation on a curve. (a) Description of the curve and mesh parameters. (b) Nodes generated on the curve and sides formed by nodes.
275
276
Automaticmesh generation The corresponding element sizes are set to be constant and have the values of h l = h2 -2. That is, we are looking for a uniform discretization on the edge. As a result of our choice of mesh parameters, the transformation matrix T is a constant matrix and at all sampling points T=~
0)
0
0
+2
0
0) 1
=2
0
0) 1
3. Applying T to tangent vector t(uo), we have 1
1
0
Tt(u0) = ~ ( 0
1)(
0~ )
(~)
The mesh size at x(u0) along the curve is, using Eq. (8.18), hso = 2. Similarly, as expected, we have
Tt(ul) =
and Tt(u2) -
4
(1) -2 0
;
hs2 = 2
4. Each element size is obtained by linear interpolation 2
h(u) = Z
hslNl(U)
-- 2
/=0
because hsl
(l = 0, 1,
2) are constant. The element density function is 1
1
h(u)
2
5. The integration of the element density function, using Eq. (8.21), gives the ideal number of sides, as A=
~
16sinE(u) + 16cosZ(u) du = Jr
The nearest integer N to rr is 3, i.e., there should be 3 sides being generated along the curve. In addition to the nodes at the end of the curve, 2 new nodes are required. The consistency index has the value of A
Jr
N
3
6. The parametric position of the first new node is computed using Eq. (8.25) 01 = ~ =
~Ul ~1 / 16sin2(u) +
16cos2(u)du =
foul 2du
= 2Ul
which gives ux = zr/6. Here Eq. (8.25) can be solved exactly, no iterative scheme needs to be invoked. Using the above result, the parametric position u2 of the second node is calculated as 02=~= We have u2 = rr/3.
2du=2
u2-
Two-dimensional mesh generation- advancing front method 277 7. Finally, the coordinates of the new nodes are obtained from the parametric equations of the circle. Substituting Ul and u2 into Eq. (8.28), we have for node nl, Xl = 4 cos
(6)
= 2V~;
Yl
--
4 sin
~-
= 2
and for node n2 X2 =
4 cos
- 2;
Y2 = 4 sin
~
= 2~/-3
The curved edge is discretized by three sides after linking each of the nodes as shown in Fig. 8.8(b).
Generation front
A generation front is established prior to starting the triangular element generation. The initial generation front is a collection of all of the sides which form the discretized boundary edges of the domain. Thus, it consists of a set of closed loops of boundary sides. If the domain is composed of multiple connected regions, such as regions with different material properties, an initial generation front will be formed for each of the regions. Each side in the generation front is defined by its two end points. The sequence of the sides is also arranged such that the regions to be meshed are always situated to the left of the generation front. The initial generation front for a simple rectangular domain is shown in Fig. 8.9(a). At any stage of the element generation process, the generation front always forms the boundary of ttie region to be discretized as depicted in Fig. 8.9(b). In the process of element generation, a side from the generation front is chosen as a base to form a new element with either a newly generated node or an existing node from the generation front. Once a new element is formed, the generation front is updated. Any side that has been used to create a new element is removed from the generation front and the newly created side is added, as illustrated in Fig. 8.9(c) and Fig. 8.9(d). The updating procedure ensures that the generation front always forms the boundary of the region to be meshed. The sides and the nodes in the generation front are referred to as active sides and active nodes respectively. Processing of the generation front continues until the entire region is filled with elements and nodes.
Element generation
The process of generating a triangular element is illustrated in Fig. 8.10 and includes the following steps: 1. An active side a b connecting nodes a and b is selected from the generation front as a base to form the new element. To produce a mesh with smooth transition of the element size, the smallest side is considered first. The sides in the generation front are sorted and updated according to their length during the element generation process to increase the efficiency of the mesh generation algorithm. 2. At the middle point m of side a b compute the local mesh parameters Otmi and hmi (i -1, 2) for the new element by interpolating from the background mesh. 3. The element creation process can be significantly simplified when point m and all the nodes in the generation front with respect to coordinates (x, y) are mapped to the normalized coordinate system (x', y') by x' = Tx. The element generation process will be conducted in the coordinate system (x', y') to construct a triangle that is as regular as possible.
278 Automatic mesh generation 1
:
13
2
12
11
7
..4
,oo
8
glnlnlOImmlqll~lWIDIggwn~ gnlmlmOIHIOgalglnammlD
initial
front
(a) 13
1
11 =
12 =
=4
190
5 6
14
2"
7
8
3
11416 12 17 18 13 19 [1014 Ill h21131 16 12 17 18 13 19 11014 Il11121131141
current front
(b)
1
13
11 =
12
~
5
oO'~176
10
6 2 updated front
1=
4
8
11 =
12
.,,
=
6 2"
-3
.
..4 190
5
9 7
13
14 ~"
8
3
I 1612171813191101411111zll~l~ql~ I IzlelS131911~ 61
11416 12 17 18 la 19 11014 Illl I 1121 161217181al9110141111121 I 1141
(d)
(c)
Fig. 8.9 Generation front and its updating during the element generation process: (a) The initial generation front. (b) Generation front at a certain stage. (c) and (d) Updated generation front after the creation of a new element. 4. Determine the ideal position of new node c' to form the new triangle. Node c' is constructed in the direction normal to side a'b' and located at a distance h~ from node a' and node b' as shown in Fig. 8.10. Here the normal of side a'b' is pointing to the region to be meshed and h~ is chosen as
hi=
0.55la,b,
0.551a,b,
1 <
1
0.551a,b, < 1 < 21a'b' 2la'b' < 1
21a'b'
(8.29)
to ensure that an element with excessive distortion is not created. The constants appearing in the expression are empirical but have been shown to work well in practice. 5. Additional points are located to create a list of the potential node to generate the new element. These include
Two-dimensional mesh generation- advancing front method 279
!
r c = h~
Fig. 8.10 The ideal position of the new node and locations of the potential forming nodes. 9 all the nodes from the generation front that fall into the circle centred at c' with radius rc, - - N h~
(8.30)
' with These nodes are ordered by their distance from c' and denoted by n 1' n 2~. . . . . nM n~ being the closest to #. The value of constant N is given here as N = 1. The inclusion of such nodes creates the opportunity for the new element being formed from existing nodes on the generation front. 9 A collection of L points are generated along the straight line between points c' and m'. These points, denoted by p], p~ . . . . . p~ are also ordered according to how close they are to node c'. Their addition ensures that a new element can always be generated. Here L = 5 is commonly used. It is noted that the empirical values given to N and L can be automatically modified during the mesh generation process to include a sufficient number of nodes. ! 6. Nodes n j (j = 1, 2 . . . . . M) from the potential node list and node c' are considered sequentially to form the new triangle with side a ' b ' . Such ordering allows the existing nodes to be considered first when they are not too far away from c'. The point that forms the new triangle with side a ' b f is taken to be the first node that satisfies the criterion (8.29) such that the newly formed sides of the triangle do not intersect any of the existing sides in the generation front. ! If neither nodes n j (j - 1, 2 . . . . . M) nor node c' can form the new triangle, nodes p j' (j = 1, 2 . . . . . L) are tested. The first p j' that verifies the criterion is taken to be the new node.
280 Automatic mesh generation 7. New element is formed. If the new node c' is adopted to generate the new triangle, its coordinates are transformed back to the original coordinate system by Xc = T -1 x c
(8.31)
8. The generation front list is updated after each new element is added and any new node is added to the node list. The element generation process continues until the number of the sides in the generation front reduces to zero. The domain is then discretized completely by triangular elements.
8.2.4 Mesh quality enhancement for triangles Mesh quality enhancement is indispensable to all mesh generation algorithms, because the shape of the triangles generated directly is not always optimal, particularly for a strongly graded mesh with element size varying rapidly. To improve the shape of the elements, at the final stage of the mesh generation, various mesh quality enhancement techniques, such as mesh smoothing and mesh modification, are employed.
Mesh smoothing
In the process of mesh smoothing, the topological structure of the mesh is fixed, i.e., the nodal connections of the elements will not be altered, but the interior nodes are repositioned to produce triangles with somewhat improved shapes. The computationally most efficient smoothing algorithm is the well-known Laplacian smoothing 41 which repositions the internal node at the centroid of the polygon formed by its neighbouring nodes. The new position of an internal node i is computed as 1
N
xi - -~ Z
1
N
xj and Yi = -~ Z
j=l
Yj
(8.32)
j=l
where N is the number of the nodes linked to node i. The mesh smoothing process consists of several (usually three to five) iterations. The technique has proved to be effective and generally adjusts the mesh into one with better shaped elements as shown in Fig. 8.11. However, the algorithm may fail if some of the neighbouring nodes of interior node i are boundary nodes and the polygon formed by these nodes is concave. The following example demonstrates this possibility.
Fig. 8.11 Laplacian smoothing. Node i is repositioned.
eA
Ad
a w
wb
Two-dimensional mesh generation- advancing front method 281 Example 8.3- A simple triangulated concave domain with boundary nodes a, b, c, d, e, f and their coordinates is shown in Fig. 8.12. The new position of the only interior node i, calculated by Eq. (8.32), is at X i,
-
-
1 ~(-1 +4+4+0+01
1)- 1
Yi'-- ~ ( - - 1 - - 1 + 0 + 0 + 4 + 4 ) =
1
which clearly is outside the domain. To prevent such failure, various constraints can be added to Laplacian smoothing. One such constraint is to reposition node i only if the maximum interior angle of all the elements linked by node i is decreasing. This constraint is in fact only necessary for those interior nodes which have neighbouring nodes being boundary nodes.
Mesh modification
The topological structures of the mesh, such as the node-element relation, the side-element relation and the node-node relation, are established once the process of element generation is completed. These relations, to some extent, reflect the regularity of the mesh. For instance, the optimal value of node-element adjacency number NE, which shows how many elements are connected to a node in a triangular mesh, is defined as
NEop -
6
max(L~ + 89 1)
for interior nodes for boundary nodes
(8.33)
where 0 is the internal angle formed by boundary edges joined at the boundary node and LcJ is the integer part of the value c. When NEop is attained for all the nodes, most of the elements in the resulting mesh are approximately equilateral. However, if an interior node has a node-element adjacency number far bigger or smaller than NEop, the surrounding elements of the node may be very distorted. Y (-1,4)
~
A
i(0, 4)
i
l
0(1, 1)
(4, o) a
(-1,-1) Fig. 8.12 A pathological case of Laplacian smoothing.
(4, -1)
X
282
Automatic mesh generation
Distorted elements are, in general, inevitable for a mesh with varying element sizes as less regular transition elements are created during the element generation process. Distorted elements are also produced because the element size h is considered only locally when each element is formed. The distortion of the element caused by non-optimal topological structure of the mesh cannot be corrected by smoothing alone. In the following, techniques that alter the topology of the mesh in order to reduce the element distortion to a minimum are described. Node elimination. A node elimination process consists of a loop over all the interior nodes. A node i will be eliminated if:
9 i is linked with three elements, i.e., N Ei = 3. i is removed together with the three elements connected to it and replaced by a single element e~ as illustrated in Fig. 8.13. 9 i is shared by four elements, i.e., N Ei -" 4. i is deleted from the mesh and its four related elements are reduced to two. The possibility of such operations are depicted in Fig. 8.14. It is noted that the element size distribution is almost unchanged in the process of node elimination. Diagonal swapping. The process of diagonal swapping examines all the element sides com-
mon to two elements. Element sides that are part of a material interface should not be altered. Considering a side shared by two triangles el and e2 shown in Fig. 8.15, the edge ac will be replaced by edge bd together with elements el and e2 and be substituted by elements e] and e~ when one of the following frequently used criteria is satisfied: 9 The maximum internal angle of the new elements e '1 and e 2' should be smaller than that of elements el and e2. or 9 The node-element adjacency number improves to be closer to the optimal value after swapping. Both criteria work well in practice. Figure 8.16 shows two diagonal swapping steps that satisfy the criterion of reducing the maximum angle in a region with seven elements. The quality of the elements is obviously improved. Swapping of diagonals is not allowed if it results in a negative area for one of the newly created elements. C
C
ii,
a"
(a)
Wb
Fig. 8.13 Elimination of node/with N E i = 3.
aw
(b)
Wb
Two-dimensional mesh generation - advancing front method d A
A c
e~ (b) dA
AC
'~1
a-
w
or d_
a"
(a)
"b
A
(3
"~
e~_
(c)
a" Fig. 8.14 Elimination of node e with NE e = 4.
The mesh quality, such as the smoothness of the mesh and the regularity of the elements, can be significantly improved after combined applications of mesh quality enhancement techniques. Example 8.4: Two triangular meshes generated by a mesh generator using the advancing front method are shown in Fig. 8.17 and Fig. 8.18. The mesh plotted in Fig. 8.17 is used for finite element analysis of a dam. The mesh parameters are arbitrarily given. The mesh of Fig. 8.18 was generated in the adaptive analysis of fluid dynamics. The mesh reflects the distribution of the specified mesh parameters which are computed from an a posteriori error estimator. 42
8.2.5 Higher order elements Higher order elements can be created easily by adding additional intermediate nodes to each element edge. For an interior edge, the position of an intermediate node is determined directly by interpolation using the positions of the nodes at each end of the edge. For a boundary edge, however, the parametric position of the intermediate node between nodes n~ and nk+l may be computed either by Eq. (8.25) or by interpolating the parametric position of nk and nk+l directly, so that the position of the intermediate node can be mapped onto the d_
.c
d-
-c
ID
(a)
(b)
F!g. 8.15 Diagonal swapping. Diagonal ac replaced by diagonal bd. Elements e 1 and e 2 changed to elements e 1 and e 2.
283
284
Automatic mesh generation e
d
b
f e
d J
J 11~"
~ g
1
b
f
e
A A
/I
A
1
"
d
b ",,~,JIII/IIIIIIII. C
3
f
"'"" h
g Fig. 8.16 Diagonal swapping for elements in a boundary region.
Fig. 8.17 Triangular mesh for a dam.
curvilinear boundary. These nodes are generated at the boundary node generation stage and placed on the boundary curve after the completion of the element generation process. The position of any interior nodes can be interpolated by the position of the element perimeter nodes. Figure 8.19 shows the locations of vertex nodes, edge nodes and interior nodes for some quadratic elements.
Two-dimensional mesh generation - advancing front method
Before we proceed to the discussion of surface mesh generation, several remarks are given on issues related to the topics discussed in this section. Remark 8.1. The algorithm for the advancing front method has been shown to be robust in two-dimensional mesh generation of triangles and, although not discussed here, can easily be extended to generate quadrilaterals. 87 However, in the process of generating triangles, several empirical constants have been adopted, e.g., those used in Eqs (8.29) and (8.30). The optimal values of these constants are still unknown. In addition, what kind of correlation between the value of these constants and the structure of the resulting mesh is also an open question. Because of the lack of the mathematical rigour, a robust element generation process for the advancing front method in three dimensions is much more complicated than that described here for two dimensions. 23-26, 28 Remark 8.2. In order to implement the search algorithms required in the element generation process efficiently, use of special data structures such as those proposed in references 44--46 for the advancing front method are advantageous.
285
286 Automaticmesh generation R e m a r k 8.3. Other methodologies for generating quadrilateral mesh can be found in refer-
ences 30, 47-52 and 88. The various algorithms that convert an existing triangular mesh to a quadrilateral mesh 48-5~ 52 are all robust, although the element size and element orientation of the resulting mesh are influenced by the pre-existing triangular mesh. Among these algorithms, the one proposed by Owen et al., 52 uses the advancing front method in the process of converting triangles to quadrilaterals.
Surface mesh generation is a prerequisite for many three-dimensional mesh generation algorithms, such as those based on the advancing front method and the Delaunay triangulation method, but it is not required for the finite octree method. However, generating a surface mesh prior to three-dimensional mesh generation has its advantages: (a) The quality of three-dimensional meshes is strongly dependent on the quality of the surface mesh. A distorted triangle in the surface mesh will almost certainly result in a tetrahedron with poor quality. (b) For many applications, an accurate description of the surface of a three-dimensional domain is essential. This can be readily realized by increasing the accuracy of the parametric representation of the surface and by assigning proper surface mesh element size distribution during the generation of the surface mesh. (c) In modem engineering design, nearly all three-dimensional geometries are created by computer aided design (CAD) systems. The boundary representation (B-Rep) of the three-dimensional geometry exported by CAD systems often contains defects, e.g., gaps between connecting surfaces and discontinuities of boundary edges of the surface. These defects can be corrected before and during the surface mesh generation process. Consequently, accurate surface mesh generation prevents three-dimensional mesh generation algorithms from failing due to presence of defects. Although essential to many three-dimensional mesh generation algorithms, surface mesh generation also has its own application in finite element analysis, this is especially true in the solution of shell problems. In the process of generating a surface mesh for a threedimensional geometry, each face of the geometry is discretized individually, the complete surface mesh of the geometry will be formed by a final assembly of the faces. Since we are mainly concerned with the algorithmic procedure of the surface mesh generation, we shall discuss the mesh generation algorithm for an individual face. The basic idea of the algorithm described below, proposed by Peraire et al. 53 and Peir6, 54 is to perform mesh generation, according to the prescribed element size distribution, in the two-dimensional parametric plane and map the two-dimensional mesh onto the three-dimensional surface. From a computational standpoint, generating triangles (or quadrilaterals) on a plane is much simpler than that on a three-dimensional surface. The two-dimensional element generation algorithm described in the previous section can be readily applied. Nevertheless, in order to obtain a surface mesh that respects the prescribed geometrical characteristics, such as element size and element shape, the mesh parameters given to the three-dimensional surface mesh need to be transformed to the parametric plane. The mesh generation procedure includes four major steps:
Surface mesh generation 287
Fig. 8.20 (a) A machine part. (b) Boundary faces and boundary edges of the machine part. 9 Perform node generation along the curved boundary edge to form the discretized boundary of the surface. 9 Transform boundary nodes, therefore the discretized surface boundary, to the parametric plane. 9 Perform element generation in the parametric space within the discretized boundary. 9 Map the mesh in parametric space onto the surface using its parametric representation. In the following we will mainly be concerned with the processes of boundary node generation and element generation in the parametric space, starting with the parametric representation of three-dimensional curves and surfaces.
8.3.1 Geometrical representation In the boundary representation of CAD systems for three-dimensional solids, the surfaces and curves are usually given in parametric forms represented by a variety of composite spline surfaces and curves [e.g., in the form of B6zier, B-spline or NURBS (Non-Uniform Rational B-Splines)]. The faces of the solid are sections of the surfaces on which they are defined. The edges that connect the faces are portions of the spline curves and are the boundary of the faces. Figure 8.20 illustrates a machine part, its boundary faces and their connecting edges. Figure 8.21 shows two of the faces of the same part and the composite spline surfaces that include the faces as sections. Although the surface mesh generation algorithm described in this section is independent of the parametric forms of the spline function that represents the curves and surfaces, we choose for simplicity to have boundary edges and surfaces expressed by the interpolatory cubic Hermite parametric spline curves and surfaces, respectively. It is worth noting that
288 Automatic mesh generation
Fig. 8.21 Surfaces of the machine part. (a) A flat surface. (b) A curved surface. Darker lines are the boundaries of the machine part surfaces.
transforming various curve and surface representations created by CAD systems to a single convenient form can simplify the development of a mesh generation program.
Curve
representation
The same curve representation described in Sec. 8.2.2 can be used for curves in three dimensions except that the vector valued function is now in the form of
x ( t ) - {x(t) y(t) z(t)},
0 < t < 1
(8.34)
with t again as the parameter. An interpolation by piecewise cubic Hermite polynomials through a set of ordered data points xi (i = 0, 1 . . . . . n) located on the curve generated by CAD systems for each boundary edge constructs a composite parametric spline curve in the same format as described by Eq. (8.8), except that now x(u) represents a position in three-dimensional space. Similarly, the number of the interpolation points and the distribution of the points should be chosen such that the surface edge can be accurately represented. The global mapping between the region u 6 [u0, un] in parametric space and a three-dimensional cubic composite curve provided by x(u) is depicted in Fig. 8.22.
Surface representation
Surfaces are represented by composite Hermite surfaces. A parametric bicubic Hermite surface patch can be obtained from the tensor product of two cubic Hermite parametric segments. It is represented by the four cubic curves that form its boundary and the twist vectors at its four comer points. In parametric form, the surface patch is expressed as
x(s, t) -- {H0(s)
Hi(s) Go(s) Gl(s)}Bc
Ho(t) Hi(t) Go(t) Gl(t)
(8.35)
1
-{1
t s s 2 s 3 } M B c M T t2 t3
,
0<s,t<
1
Surfacemeshgeneration289 Xi+l
x
T x(u)
/
x
UO
Ui
Un
Ui+l
= U
Fig. 8.22 Three-dimensional composite cubic spline interpolation. where M is the matrix defined in Eq. (8.7) and
Bc =
x(0, 0) x(1, 0)
x(0, 1) x(1, 1)
X,s (0, O)
X,s (0, 1)
X,s(1,0)
X,s(1,1)
x,t (0, 0) x,t(1, 0) X,st (0, 0) X,st(1,0)
x,t (0, 1) x,t(1, 1) X,st(0, 1) X,st(1,1)
(8.36)
is often called the boundary condition matrix, in which x(0, 0), x(0, 1), x(1, 0), x(1, 1) are the four comers of the patch. The tangents x,s and x t, and the twists X,st at the comers are given by 8x 8x 82x X,s = Oss' x,t = Ot X,st = 8 s 8 t (8.37) A piecewise composite surface is obtained by interpolation through a topologically rectangular set of data points xij (i - 0 . . . . . m; j - 0 . . . . . n) and their corresponding parameter values (vi, w j) (i = 0 . . . . . m; j = 0 . . . . . n) generated from the CAD representation of the surfaces. It consists of a network of m • n quadrilateral surface patches. The surface patch [Ui_I, Ui] X [//)j-i, //3j] (i --" 1, 2 . . . . . m; j = 1 . . . . . n) in the network is described by 1
x(v, w) = { 1 s s 2 S 3 } MBcM T
t t2
0 < s, t < 1
(8.38)
t3 Here s = ( v - Vi_l)/Ai and t = (w - W j _ l ) / A j are local coordinates of the corresponding intervals [vi-1, vi] and [ w j - 1 , wj], A i -" Ui -- l)i_ 1 and A j : Wj -- Wj_ 1 are the respective lengths of the intervals, and the boundary condition matrix is of the form
BC "--
X('Oi-1, Wj-1)
X(Ui-1, Wj)
AjX, w(Vi-1, Wj-1)
AjX, w(1)i_l, wj)
X(Vi, Wj_I)
X(Vi, //)j)
AjX, w(Vi, Wj-1)
Ajx, w(vi, wj)
Aix, v(Vi_l, Wi_l)
AiX, v(Vi_l, Wj)
Ai AjX, vw(l)i-1, Wj-1)
AiAjX,vw(1)i_l, wj)
AiX, v(Vi, //)j-l)
AiX, v(Vi, Wj)
mi mjX, vw(Ui, wj-1)
A i Ajx, vw(Ui, Wj)
(8.39)
290 Automatic mesh generation The unknown tangents x,v and x,w and twist X,vw of the composite surface can be determined by following the standard procedure for piecewise parametric surface interpolation requiring the resulting composite surface to be at least C 1 continuous 39' 40 (a C 2 continuous composite surface is usually adopted in CAD surface mesh generation). Similar to that used in the implementation of curve interpolation described in Sec. 8.2.2, the parametric coordinates of vi and wj are usually taken as vi = i and wj = j respectively. For surface patch [vi-1, vi] x [wj-1, wj] = [i - 1, i] x [ j - 1, j], the global parametric coordinates v and w are related to the local coordinates s and t through u--l)i_
1
--~-s = i -
IlO'--Wj_I + t = j -
1 +s,
1 +t,
i -- 1,2 . . . . . n; j=l,2
..... m
(8.4o)
The global one-to-one mapping between the parametric plane and the composite surface provided by x(v, w) is illustrated in Fig. 8.23. Indeed, a region with curved boundary, as depicted in Fig. 8.24, within the parametric plane [v0, On] x [w0, wn] constitutes a portion of the composite spline surface in a three-dimensional space.
8.3.2 Geometrical characteristics of the surface mesh A surface mesh generally is three dimensional, thus, the spatial distribution of the shape and size of the surface elements needs to be specified in three dimensions. Because the mesh generation is performed in the two-dimensional parametric plane, proper spatial distribution of the element shape and size in the parametric space also needs to be defined.
Mesh control function in three dimensions
As was adopted in two-dimensional mesh generation, the geometrical characteristics of the surface mesh such as the distribution of the element shapes and sizes on the surface are controlled by mesh parameters. For surface mesh, mesh parameters include a set of three mutually orthogonal directions Ot i (i --- 1, 2, 3), and their associated element sizes hi (i = 1, 2, 3) as shown in Fig. 8.25. Mesh parameters are defined at the nodes of a three-dimensional background mesh, which usually consists of a small number of tetrahedral elements. The background mesh can be constructed to cover the entire surface of the three-dimensional geometry or to cover each face of the geometry individually. In either case, the background mesh will be created automatically by dividing one or several hexahedra (tetrahedra or prisms) into tetrahedra. Figure 8.26 shows a tetrahedral background mesh generated from a single hexahedron for a single surface. The spatial distribution of the mesh parameters is furnished by the background mesh. At a particular point on the surface, the mesh parameters are computed by a linear interpolation of the values assigned at the nodes of the backgroun d mesh. The geometrical characteristics at a point of the surface mesh is therefore attained. For instance, when all three element sizes are found to be equal at the point, the tetrahedral elements in the surrounding area of the point will be approximately equilateral. Since the faces of the tetrahedral elements form the surface elements they are therefore approximately equilateral triangles.
Surface mesh generation 291
Z
X
/ I x(v,w) W
w~+l w~
Vi
Vi+l
Parametricplane Fig. 8.2.3 Compositecubicsurfaceinterpolation 9 A three-dimensional geometrical transformation matrix T, similar to that defined in twodimensional mesh generation, is in the form of 3 1 T(x) =/~1"= ~c~'~T
(8.41)
and constitutes a scaling factor 1/hi in each of the Ot i directions (i = 1, 2, 3). When it is applied to tetrahedra with element sizes hi in directions Ot i a t a given point in the coordinate system (x, y, z), the tetrahedra will be mapped to equilateral tetrahedra in the normalized space (x', y', z'). Consequently, the surface element will be mapped to equilateral triangles in the normalized space. Indeed, transformation T also provides a mapping relationship
292 Automaticmesh generation Surface Spline
Boundary curve
,y x
Tx(v,w) wi+l
/ ( \
/
\
3
( f
Vi
Vi+l
~ ~ -
V
Surface on parametric plane Fig. 8.24A region in parametric plane and its image on spline surface.
between parametric coordinates and the three-dimensional normalized coordinates x'(v, w) = Tx(v, w)
(8.42)
As indicated, surface mesh generation will be performed in the parametric space (v, w) and mapped onto the surface in three dimensions. In order to accomplish the planar mesh generation on the parametric plane, appropriate planar mesh parameters have to be assigned. These mesh parameters must be given in such a way that, after being mapped onto the surface, the final surface mesh respects the specified geometrical characteristics. Such
Surface mesh generation
293
i
z
.,y x
Fig. 8.25 Mesh parameters defined in three dimensions.
h
g
.........zoo~176176176176176176176176176176176176176176176176176176176 e
o
a
b
Fig. 8.26 A six tetrahedra background mesh derived from a hexahedron. The six tetrahedra are (e, f, h, a), (a, b, f, d), (f, h, d, a), (b, f, d, c), (d, g, h, f), (d, c, g, f).
a requirement can be achieved by deriving the planar mesh parameters from the threedimensional surface mesh parameters.
Mesh parameters in parametric plane
We start by examining a curve in the parametric plane expressed by parameter ~ and illustrated in Fig. 8.27, i.e., v = v(~), w = w(~) (8.43) or in vector form u(~) = (v(~), w(~))
(8.44)
294
Automaticmesh generation
At a particular point U(~p), the square of arc length element ( along the curve is expressed
as (d()2 = (dr)2 + (dw)2 = ( d~U d )~T / d ut ~
d ~)' = tfft~ (l~d~ )2 -- (l~d~) 2
(8.45)
where l~ is the length of the tangent vector ~ l~=
du ~r (d--~)(-~)
and t~ is the unit tangent vector
1 du
t~ = l~ d~
(8.46)
(8.47)
This shows that the arc length element in the direction of a unit tangent along the curve in the parametric plane can be expressed by d( = l~ d~
(8.48)
We now consider the image of the planar curve at point x(V(~p), 1/3(~p)) on the surface represented by x(v, w) = (x(v, w), y(v, w), z(v, w)) (8.49) The square of the arc length element in the direction of the unit tangent t along the curve on the surface is given by
(ds) 2 = (dx) 2 + (dy) 2 -'l-(dz) 2 = (dx)T(dx) where
dx dx=
dr
=
dz
Ox Ox O---v Ow __OY Or Ov Ow Oz Oz Ov Ow
X~dv~ ,, = [x,o, X,w]du [ dw J
(8.50)
(8.51)
Assume that the transformation matrix Tds, correlated to ds at x(v(~p), W(~p)), maps dx to the normalized space with a unit length, i.e.,
where
(ds') 2 - (dx') 2 + (dy') 2 + (dz') 2 = (dx)TCds(dx) -" 1
(8.52)
3 1 Cds = TTs Tds =/El= h2s----~ CltiOfT
(8.53)
corresponding to the Euclidean metric tensor in the normalized space, and hds, can be viewed as element size associated with ds in the direction of c~i. Substitute dx into Eq. (8.52), we have
x, l
x, l d,,) =
(8.54)
Surface mesh generation
dst( v(~p), W(~p))
x(v,w)
Z 9
x
w
I x(v, w)
u(~)
m
~
V
Parametric plane
Fig. 8.27 Arc length elementd~ in parametricplane and its image on the surface. From Eqs (8.47) and (8.48), we know that du = It d~t~ = dft~
(8.55)
which when substituted into Eq. (8.54) gives
([x,~, x,~] tq) T Cds ([X,v, X,w] tr (dff)2 _ 1
(8.56)
This shows that for an arc length element ds along a curve on the surface, the arc length
295
296
Automaticmesh generation
along its curve in the parametric plane is d( =
1
(8.57)
V/([x,o, x,w]tt)TCds ([x,o, x,~]tt) Replacing ds by the curvilinear element size hs and Tds by T on the surface. Using he, the element size along the planar curve on the parametric plane, in place of d~', we have ([x~, X,w]tr or
h~ =
x,w]tr
= 1
1
= ?([x,o, x,~l t~)Tc ([X,o, X,~] t t ) v / t ~ : G
(8.58) 1
t~
(8.59)
The matrix of the metric tensor is now expressed as 3
1
C = TTT =/~1": ~/2C~icX/T
(8.60)
G = [x,o, X,w]TC[x,o, X,w]
(8.61)
and is the matrix of the metric tensor in the parametric space. If we assume that T is a constant matrix in the neighbourhood of point x(V(~p), W(~p)), substitute Eq. (8.60) into Eq. (8.59) and note the mapping relationship of Eq. (8.42), we obtain h~ in a somewhat different form h~ =
(8.62)
v/tffgt; where [gvv gvw ] g = g~v gww
(8.63)
is the first fundamental matrix of the surface in the normalized space, 38 and =(0xI)2 gvv
(
(0y,)2
(aZ,)2
Oz'Oz')
\Or + -~v + Ox' Ox' Oy' Oy' + ~ - g~--2 Ov Ow ~ Ov Ow Ov Ow (0x,)2 (0y,)2 (Oz,)2 g ww = -~w + -~w + -~w
= gwo
(8.64)
Consequently, we have established the relationship between G and g. When transformation T is a constant matrix, G=g This shows that the matrix of the metric tensor in parametric space is the same as the first fundamental matrix of the surface in the normalized space.
Surface mesh generation
The two-dimensional mesh parameters Oti(V(~p) , 113(~p)) and hi(v(~p), llO(~p)) for the planar mesh on the parametric plane are computed from the directions in which h~ attains an extremum. To this end, Eq. (8.59) is rewritten in the form of 1
Let
tffGt~ = h--~
(8.65)
[Gll G12] G "- ]_G21 G22_]
(8.66)
where Gll : XT,vcx,v, G12 -- xTCx, w,,
G22 -- x w T C ,x ,
(8.67)
and G21 -- G12. From Eq. (8.65), we know that finding the direction in which 1/h~ reaches an extremum involves the solution of an eigenproblem for the symmetric matrix G. To this end we let ~1, )~2 with ~.1 >__~2 denote the eigenvalues and al, a2 the eigenvectors of G. The mesh parameters in the parametric plane at point (V(~p), W(~p)) are given as 1
hi (v, w) = ~ / ~ ,
h2(v, w) -- ~
1
(8.68)
and C~l(v, w) = al,
Ot2(V, W) -- a2
(8.69)
where the eigenvalues are computed as
~.1--
Gll + G22 v / ( G l l - G22)2 4- G22, 2 + 4
~,2 --
Gll + G22 ~/(Gll -- G22)2 2 - 7 4 4- G22
(8.7o)
the eigenvectors are al = (cos0, sin0), with
1 0 = ~ tan-1
a2 = ( - sin0, cos0) (2G12) G 11 - G22
(8.71)
(8.72)
8.3.3 Discretization of three-dimensional curves In order to perform mesh generation on the parametric plane, the boundary curves of the surface will be discretized in three-dimensional space and then projected to the parametric plane by inverse mapping.
Node generation on the curves
The algorithmic procedure for the node generation on three-dimensional curves is identical to that described in Sec. 8.2.2 for the two-dimensional curve. The curves are of course now represented by Eq. (8.34). The procedure listed below is the same as that discussed in Sec. 8.2.2 but now expressed in its three-dimensional form.
297
298 Automaticmesh generation 1. A set of sampling points X l = X ( U / ) (l = 0, 1, 2 . . . . . m) is first placed along the curve with parameters Ul uniformly distributed as shown in Fig. 8.28. The unit tangent vector for the three-dimensional curve is computed at the sampling point as (8.73)
tl = (tl l , t2,, t3 l)
where tll =
X'ul ~ lu I ,
t2l
--
Y,u, . lu--~' t3l--
(8.74)
Z,U l lul
2 -+- Y,ul 2 -~- Z21, and X, ul = (X,u I ' Y, Ut , Z ,ul ) the tangent vector. with lul -- ~ X,ul 2. Using the interpolated values from the background mesh, the mesh parameters ali, hli (i = 1, 2, 3) and the associated transformation matrix TI are computed at each sampling point. 3. The element size hsl a t the sampling point Xl (l = 0, 1, 2 . . . . . m) in the tangent direction is calculated by hsl =
1
=
v/(Tltl)T(Tltl)
1
(8.75)
v/tTCltl
where the metric tensor C is defined in Eq. (8.60). 4. A continuous element size distribution function is obtained by linear interpolation of the nodal values h sl a t sampling points using finite element shape functions m
h(u) = Z
(8.76)
hs, Nl(U)
/=0
The element density function is set to be 1/h(u). 5. The total number of sides N to be generated along the curve is taken to be the nearest integer to A =
/oL
h (1u )
ds=
~ ' x
/uourn1 V/(Xu) a -I- (yu) 2 -I- (Zu) 2 du ~ x,
x
h3~
(8.77)
/2'
h2or,2
I x(u) x
i/
-Y uo
u/
Urn
=U
Fig. 8.28 Sampling points and mesh parameters on a three-dimensional composite curve segment.
Surface mesh generation 299 where L is the length of the curve. The consistency index is calculated as A 0 = -(8.78) N In addition to the already known end points x(u0) and X(Um) of the curve, there will be (N - 1) new nodes created along the curve. 6. The parametric positions Uk+l (k = 0, 1, 2 . . . . . (N - 2)) of the new nodes are computed consecutively from Uk+l 1 0 -V/(Xu)2 + (yu) 2 + (Zu) 2 du (8.79) .,uk h(u)
f
by an iterative method, such as a Newton method. 7. Finally, the position of all the new points are mapped onto the boundary curve using Eq. (8.34).
Place boundary nodes to parametric plane
In order to perform surface mesh generation in the parametric plane the discretized boundary curves must be placed to the parametric plane to form the boundary of the region to be meshed. However, the inverse mapping
u(x) = (v(x), w(x))
(8.80)
is in general not expressed explicitly. When a composite cubic spline surface is used to represent the surface of the geometry, the above inverse mapping is clearly non-linear. In addition, in the boundary representation of CAD systems and in our discussion of the geometrical representation of curves and surfaces, the composite parametric spline curve that represents the boundary is in fact an approximation to the edge of the surface, which is often formed by the intersection of two or more surfaces. Similarly, the composite spline surface is also an approximation to the surface of the geometry. As a result of such approximations, the boundary curve and its nodes are not exactly located on the surface. The parametric position of a boundary node on the surface can be found by assuming that its parametric coordinates are the same as its closest point on the surface, which is to find a point x(v, w) on the surface that is the closest point to boundary node X(Uk). The problem of finding the closest point can be formulated as: find the parametric coordinates (v, w) of a surface point such that D = IIx(o, w) - x(uk)ll : minimum
(8.81)
where I1" II denotes the Euclidean norm. This problem is usually non-linear and may be solved by various iterative methods. 55, 56 The initial approximation of the parametric coordinates of a boundary node is taken as the computed position of a previous boundary point. After all the boundary nodes are placed on the parametric plane, they are linked by straight lines to form the discretized boundary of the region to be meshed.
8.3.4 Element generation in parametric plane The element generation procedure in the parametric plane is the same as that described in Sec. 8.2.3 except that when creating a new element, the mesh parameters c~i(v, w) and
300 Automatic mesh generation
Fig. 8.30 Finer meshes of different element sizes. (a) View one. (b) View two.
hi(l), 119) computed from the specified mesh parameters for the surface mesh have to be utilized. After the completion of the element generation quality enhancement techniques need to be applied which include a constraint of preserving the curvature of the surface to improve the quality of the mesh. 54, 57 The final mesh is then mapped onto the surface by x (v, w) to obtain the required surface mesh. The complete surface of a three-dimensional solid can be achieved after assembling the surface mesh of all the faces. Such surface mesh may be used as the discretized boundary for three-dimensional mesh generation, which we shall discuss in the next section. Example 8.5: Some of the faces of the machine part shown in Fig. 8.20 are discretized by a surface mesh generator using the algorithms discussed in this section. Figure 8.29(a) shows the boundary representation of two of the surfaces. Coarse meshes of the surfaces are illustrated in Fig. 8.29(b). Finer meshes of the surfaces with different element size distribution are shown in Fig. 8.30(a) and Fig. 8.30(b) with a view from a different angle. The surface mesh of the part with side faces removed is shown in Fig. 8.31. Example 8.6: The boundary edges of the surface representation of a gearbox part are shown in Fig. 8.32(a). The geometry of the part is realistic and hence somewhat complex.
Surface mesh gene,
Fig. 8.31 Surfacemesh of the machine part with side faces removed.
A complete surface mesh of the part generated automatically using the algorithms described in this section is illustrated in Fig. 8.32(b).
8.3.5 Higher order surface elements One of the advantages of generating a surface mesh in the parametric plane is that it can produce higher order elements without additional difficulties. When a mesh of higher order elements is generated in the parametric plane, it will be mapped onto the surface to form a boundary fit surface mesh with all the nodes on the surface. To preserve the boundary curves, it is important to generate the intermediate boundary edge nodes by following the node generation procedure for curves. Figure 8.33 demonstrates a surface mesh of quadratic elements for a mechanical part.
8.3.6 Remarks Several remarks on issues related to the surface mesh generation are given below. Remark 8.5. Once the discretized boundary of the region in the parametric plane is available, any two-dimensional element generation algorithm could be used to generate a valid
302 Automatic mesh generation
Fig. 8.33 Surfacemeshof quadraticelements.
Three-dimensional mesh generation- Delaunay triangulation 303 mesh on the parametric plane. For convenience, we only mentioned the two-dimensional advancing front method described in Sec. 8.2. Remark 8.6. Equations (8.59) and (8.62) have revealed that the mesh parameters in the parametric plane is a function of the parameterization of the surface. In order to satisfy the specified geometrical characteristics of the surface, the two-dimensional mesh generation algorithm is required to be capable of generating the mesh strictly following the computed mesh parameters for the parametric plane. Otherwise, serious distortion in the surface mesh may occur, when the planar mesh is mapped onto the surface. We refer to references 58-63 for additional discussion on issues of surface mesh generation. Remark 8.7. Besides the parametric representation, the surface of the three-dimensional geometry can also be represented in a discrete form. Surface mesh generation algorithms for surfaces represented in such form are topics of active research. Several surface mesh generation algorithms based on discrete representation of surfaces can be found in references 64-68. Remark 8.8. The surface mesh generation algorithms discussed in this section assume that the geometrical representation and topological representation of the surfaces are correct, i.e., there are no defects in the surface representation of the three-dimensional geometry. In practical computations, this is often not the case. The boundary representation of the geometry provided by the CAD systems sometimes contains errors or undesirable features that will either cause the mesh generation algorithm to fail or the quality of the surface mesh become unacceptable for three-dimensional mesh generation algorithms. Although methodologies that automatically remove defects and detrimental features from the boundary representations of the surface are not in the scope of surface mesh generation, they critically affect the success of automatic surface mesh generation and therefore deserve further research. Remark 8.9. Finally, surface mesh generation is often used as the boundary discretization of a three-dimensional geometry. The quality of the surface mesh not only affects the quality of the three-dimensional mesh, it also affects the robustness of any three-dimensional mesh generation algorithm. This is particularly true for the three-dimensional advancing front mesh generation method. Although the robustness of the three-dimensional Delaunay triangulation method is less dependent on the quality of the surface mesh, the quality of the final three-dimensional mesh is certainly affected. To have a successful three-dimensional mesh generation algorithm, the quality of the surface mesh must be insured before the interior mesh generation process starts.
Many practical finite element computations are carried out on complex three-dimensional domains. The level of difficulty to automatically generate valid meshes for arbitrary threedimensional domains is much greater than in two dimensions. In principle, a Delaunay triangulation, advancing front and finite octree method are all applicable to three-dimensional
304 Automaticmesh generation
mesh generation. However, the Delaunay triangulation method has attracted most of the attention in theoretical research and software development, due to its conceptual simplicity, mathematical rigour and algorithmic robustness. In this section, we shall be concerned with the Delaunay triangulation method and its application to three-dimensional mesh generation. We shall also introduce mesh quality enhancement methods which are crucial to ensure the final mesh can be used in the finite element computations.
8.4.1 Voronoi diagram and Delaunay triangulation Delaunay triangulation 69 is the dual of the Voronoi diagram. 7~ The properties of the Voronoi diagram and Delaunay triangulation provide the theoretical foundation for all the mesh generation methods based on the Delaunay method. In order to facilitate the description of the Delaunay triangulation method for mesh generation, a brief review of the basic properties of the Voronoi diagram and Delaunay triangulation is presented in a two-dimensional setting for visualization convenience, but these properties are equally valid in three dimensions. Let P = {Pi, i = 1, 2 . . . . . N} be a set of distinct points in the two-dimensional Euclidean plane R 2. They are referred to as the forming points in the mesh generation literature. The Voronoi region V (Pi) is defined as the set of points x 6 R 2 that are at least as close to pi as to any other forming point, i.e.,
V(pi)
=
{X E R2" I l x - Pi II ~ IIx-
Pill,
Vj 5/=i}
(8.82)
Figure 8.34(a) depicts ten Voronoi regions, with two interior regions bounded by eight others, the total being defined by an equal number of forming points. It follows that the Voronoi region V (Pi) represents a convex polygonal region, possibly unbounded; and any point x inside V (Pi) i s nearer to Pi than any other forming point in P. The points that belong to more than one region form the edges of the Voronoi regions and the edges of the Voronoi region V (Pi) are portions of the perpendicular bisectors separating the segment joining forming points Pi and pj when V (Pi) and V (pj) are contiguous. The union of the Voronoi regions is called the Voronoi diagram of the forming point set P. The dual graph of the Voronoi diagram is produced by connecting the forming points of the neighbouring Voronoi regions sharing a common edge with straight lines. It forms the Delaunay triangulation D(P) of the Voronoi forming points P. Figure 8.34(b) illustrates the Delaunay triangulation and its corresponding Voronoi diagram. In addition to those already mentioned, several properties of the Delaunay triangulations and Voronoi diagrams that are most relevant to the mesh generation algorithms of Delaunay triangulation are listed below: 71, 72 i. Delaunay triangulation is formed by triangles if no four points of the forming points P are co-circular. These triangles are called Delaunay triangles. ii. Each Delaunay triangle corresponds to a Voronoi vertex, which is the centre of the circumcircle of the triangle, as depicted in Fig. 8.35. iii. The interior of the circumcircle contains no forming points of P. iv. The boundary of the Delaunay triangulation is the convex hull of the forming points. In the Delaunay triangulation-based mesh generation algorithm, property (i) is used to avoid the degeneracy; property (ii) is often used to construct data structures; property
Three-dimensional mesh generation- Delaunay triangulation 305 (iii) forms the well-known Delaunay criterion, the empty circle criterion, or the in-circle criterion when verifying whether it is violated by the new point introduced to the Delaunay triangulation; property (iv) is the theoretical origin of using a convex hull, which contains all the mesh points, in mesh generation. Example 8.7: Each Voronoi diagram corresponds to a set of forming points which forms Delaunay triangulation. Adding a new forming point will inevitably result in a modification of the Voronoi diagram and the Delaunay triangulation. The process of constructing a new Voronoi diagram and Delaunay triangulation after the insertion of a new node is frequently used in automatic mesh generation and is illustrated here in the same two-dimensional setting shown in Fig. 8.34(b). Let the new forming point n be inserted in the Delaunay triangulation shown in Fig. 8.36(a). It falls into the circumcircles of Delaunay triangles afg, abf and bef , therefore
I
9
II
9
/
I
9
, i
9
I
!
/
,,
/
.(~ "(~r I
/
~.o
I
iI
I
i
"-(2) - -~
9
9 x
x
x
, I I
9 Forming points Pi 0 Voronoi vertex Edges of the Voronoi region V(Pi) (a) Voronoi diagram
,
Delaunay triangulation
- - - Voronoi diagram (b) Delaunay triangulation
Fig. 8.34 Voronoi and Delaunay diagrams for 10 forming points.
C) Voronoi vertex and centre of the circumcircle Fig. 8.35 Circumcirclesof the Delaunay triangles. Only three are shown.
306
Automatic mesh generation
violating property iii. This causes the removal of the three Voronoi vertices which are the centres of the circumcircles and their corresponding Delaunay triangles, as illustrated in Fig. 8.36(b). The new Delaunay triangulation is constructed by linking the new forming point n and its contiguous forming points that form a face of the neighbouring triangle followed by the construction of the new Voronoi diagram as shown in Fig. 8.36(c). As we have indicated previously, the process used in the last example is applicable to three dimensions.
8.4.2 Three-dimensionalmesh generation by Delaunay
triangulation
Although by definition Delaunay triangulation decomposes the convex hull of the forming points into triangles in two dimensions and tetrahedra in three dimensions, it does not address the issues of how Delaunay triangulation can be formed effectively; how to generate those points that will be inserted in the triangulation; and how to preserve the boundary of a region when the forming points are from the boundary of a concave region. These issues J
h
J
b
(a)
,.,
(b) ,
j
b
i
(c) Fig. 8.36 (a) Insertion of new forming point n into Delaunay triangulation. (b) Removal of Delaunay triangles, deleted Voronoi vertices are not shown. (c) New Delaunay triangulation and Voronoi diagram.
Three-dimensional mesh generation- Delaunay triangulation 307 are the three most important components of the automatic mesh generation algorithms of Delaunay type. A large body of literature exists on research of these three subjects. The most representative ones are the work of Bowyer 11 and Watson12 on the efficient Delaunay triangulation algorithms, which were introduced to mesh generation by Cavendish et al., 13 Weatherill, 14 Schroeder and Shephard, 15 Baker 16 and George et a/.; 17 the early work of Rebay, 18 Weatherill and Hassan, 19 Marcum and Weatherill 2~ on automatic point generation algorithms; and the work of Weatherill, 73 George et al., 17 Weatherill and Hassan 19 on preserving the integrity of the domain boundary. In the following, we shall introduce the three-dimensional mesh generation procedure of Weatherill and Hassan, which is one of the first Delaunay mesh generation procedures that contains all three necessary components for a robust three-dimensional Delaunay mesh generation algorithm. It includes a Delaunay triangulation algorithm; a node generation algorithm based on specified mesh size distribution; and a surface mesh recovery procedure that ensures the integrity of the boundary surface. The global procedure of the three-dimensional mesh generation algorithm is as follows: 1. Input the triangular surface mesh and derive the topological data of the surface mesh, such as edges of surface elements and node-element connections. (Figure 8.37 shows the surface mesh of a simple three-dimensional geometry.) 2. Build a convex hull that contains all the mesh points. (An eight node convex hull is shown in Fig. 8.38.) 3. Perform Delaunay triangulation using nodes of the surface mesh to form tetrahedra. (Figure 8.39(a) illustrates the Delaunay triangulation of the surface nodes. A crosssection of the triangulation is shown in Fig. 8.39(b).) 4. Create interior points, following the specified element size distribution function, and perform Delaunay triangulation to form tetrahedra. (The results are shown in Fig. 8.40(a) and Fig. 8.40(b).) 5. Recover any missing edges and triangular faces of the surface mesh to ensure the input surface triangulation being contained in the volume triangulation.
Fig. 8.37 Surface mesh of a simple three-dimensional geometry.
308 Automatic mesh generation
Fig. 8.38 Convex hull and surface mesh.
6. Identify and remove all the tetrahedra outside the domain of interest to give the final three-dimensional mesh. (Figure 8.41(a) shows the recovered surface mesh, which is identical to the input mesh, and Fig. 8.41 (b) demonstrates the interior of the tetrahedral mesh at a cross-section of the geometry.) The mesh generation procedure has been shown to be computationally efficient and, probably more importantly, very robust. Indeed, the robustness of the algorithm is independent on the complexity of the three-dimensional geometry. We shall, in the following, describe in more detail the three main components of the procedure, i.e., the Delaunay triangulation algorithm, the node creation algorithm and the surface mesh recovery methods.
Delaunay triangulation algorithm
The Delaunay triangulation algorithm discussed below is based on the algorithm proposed by Bowyer, 11 but it can be readily replaced by the similar algorithm of Watson 12 which differs only in its data structure. The process of generating Delaunay triangulation is sequential. Each point is introduced into an existing structure of the Voronoi diagram and the Delaunay triangulation, which will be reformulated based on the in-circle criterion to form a new Delaunay triangulation. The process is similar to that described in Example 8.7, but of course now is in three dimensions. The main steps of the procedure are as follows: 1. Define a set of points which form a convex hull that encloses all the points to be used in the tetrahedral mesh. 2. Introduce a new point into the convex hull. 3. Determine all vertices of the Voronoi diagram to be deleted. A vertex will be deleted if the circumsphere, centred at the vertex, of four forming points contains the new point. This follows from the in-circle criterion.
Three-dimensional mesh generation - Delaunay triangulation
Fig. 8.39 (a) De!aunay triangulation of the surface nodes. (b) A cross-section of the surface nodes Delaunay triangulation. Six additional points are inserted before surface nodes for efficiency.
4. Find the forming points of all the deleted Voronoi vertices. These form the contiguous points to the new point. 5. Determine the neighbouring Voronoi vertices to the deleted vertices which have not themselves been deleted. This data provides the necessary information to enable valid combinations of the contiguous points to be constructed. 6. Determine the forming points of the new Voronoi vertices. The forming points of new vertices include the new point together with three points which are contiguous to the new point and form a face of a neighbouring tetrahedra. This forms the new Delaunay triangulation. 7. Determine the neighbouring Voronoi vertices to the new vertices. In Step 6, the forming points of all new vertices have been computed. For each new vertex, perform a search through the forming points of the neighbouring vertices, as found in Step 5, to identify common triples of forming points. When a common combination
309
310 Automatic mesh generation
Fig. 8.41 (a) Recovered surface. (b) Tetrahedral mesh shown at a cross-section.
occurs, the neighbour of the new Voronoi vertex has been found. This forms the new Voronoi diagram. 8. Reorder the Voronoi diagram data structure and replace (overwrite) the entries of the deleted vertices. 9. Repeat Steps 2-8 until all the points have been inserted.
Three-dimensional mesh generation- Delaunaytriangulation 311 In the mesh generation process, the new points to be inserted into the Delaunay triangulation and Voronoi diagram are the surface mesh nodes and the mesh points generated automatically following the adopted node generation algorithm.
Automatic node generation
The detailed step-by-step node generation algorithm described below creates points based on the element size distribution of the surface mesh:
1. The node spacing function for each surface mesh node a of position x a is taken as the average of the surface element edge length,
ha :
M 1 ~ ~ IlXb -- Xa II b=l
(8.83)
where Xb(b = 1, 2 . . . . . M) are positions of the surface nodes connected to node a. 2. Perform Delaunay triangulation using surface mesh nodes. 3. Initialize the number of new interior points to be created, set N = 0. 4. For each tetrahedron within the domain: (a) Locate a prospective point c at the centroid of the tetrahedron. (b) Derive the node spacing function hc, for point c, by interpolating the node spacing function hm (m -- 1, 2, 3, 4) from the nodes of the tetrahedron. (c) Compute the distances dm (m = 1, 2, 3, 4) from the prospective point c to each of the four nodes of the tetrahedron. If {dm < othc} for any m = 1, 2, 3, 4 then reject the point and return to the beginning of Step 4 for the next tetrahedron. Else compute the distance dj from the prospective point c to other already created nodes pj (j = 1, 2 . . . . . N). If {dj < flhc } then reject the point and return to the beginning of Step 4 for the next tetrahedron. Else accept the point c and add it to the interior node list p j ( j -= 1, 2 . . . . . N) and update N. (d) Assign point distribution function hc to new node c. (e) Go to the next tetrahedron. 5. If N = 0, i.e., no new point is created, exit the node generation process. 6. Perform the Delaunay triangulation of the derived points pj (j = 1, 2 . . . . . N). Then, go to Step 3. The parameter ot controls the element density by changing the allowable shape of the formed tetrahedra, while/3 has an influence on the regularity of the triangulation by not allowing points within a specified distance of each other. Both parameters can be adjusted to control the mesh density. In practical computations, ct can be chosen in the range of 0.85-1.1, and/3 in the range of 0.6-1.0 for an isotropic mesh. An effort to find the optimal value of these parameters has been made in reference 74.
312
Automatic mesh generation
It is noted that, with minor modification, the node generation algorithm can also create nodes-based element size distribution defined by a three-dimensional background mesh.
Surface mesh recovery
The property (iv) of the Voronoi diagram and Delaunay triangulation presented in Sec. 8.4. l implies that the surface mesh and the surface boundary of a general three-dimensional geometry, which is seldom convex, will not be respected during the mesh generation process. Very often, some of the surface triangles and their edges are not present in the resulting Delaunay triangulation due to penetrations by other tetrahedra. The loss of completeness of the original surface mesh causes the loss of integrity of the surface boundary of the geometry. In order to derive a valid three-dimensional mesh for the given geometry, the integrity of the surface boundary of the geometry must be respected, which can be realized by recovering the original surface mesh. In the surface mesh recovery procedure, the surface triangles and surface edges that are missing from the Delaunay triangulation are first identified and then restored by the following procedure:
Edge swapping. Edge swapping is illustrated in Fig. 8.42. If faces abd and bed appear in the Delaunay triangulation, but faces abe and acd exist in the surface mesh, replacing edge bd by edge ac recovers two surface triangles. This process is attempted for each surface edge because it is the most efficient method to recover the missing edges and triangular faces. Boundary edge recovery. Consider the case when surface edge joining points a and b are missing from the Delaunay triangulation, a line ab is formed and its intersection with the faces, edges and points of the Delaunay triangulation are identified, as shown in Fig. 8.43, with all the possible types of intersections depicted in Fig. 8.44. Local transformations with the newly added nodes at the intersection as shown in Fig. 8.45 are performed to all the involved tetrahedra to recover the edge, segment by segment. This process is executed for every missing edge. When the combined intersection involves a node-to-face type and a face-to-node type, edge ab can be recovered by directly linking nodes a and b, with the two involved tetrahedra transformed to three tetrahedra as shown in Fig. 8.46. d
0
0 a
a
b Faces appear in Delaunay triangulation Fig. 8.42 Edge swapping to recover the surface edge and faces.
b Faces exist in surface mesh
Three-dimensional mesh generation - Delaunay triangulation
Fig. 8.43 Edgeab of a surface triangle missing in the Delaunay triangulation.
w
v
Node-node-type intersection
l
Node-edge-type intersection and edge-node-type intersection
,iw
lw
Node-face-type intersection and face-node-type intersection
Edge-edge-type intersection
Edge-face-type intersection and face-edge-type intersection
Face-face-type intersection
Fig. 8.44 Edge-tetrahedron intersections for missing surface edges.
Boundary face recovery. The recovery of the surface triangles is conducted after the completion of the edge recovery. A surface triangle may still be missing from the tetrahedral mesh even though all its edges are present, because the interior of the triangle face is penetrated by other tetrahedra. There are a total of four possibilities that a face can be intersected by a tetrahedra as illustrated in Fig. 8.47. Every missing face can be recovered after all the intersecting tetrahedra are determined and transformed, with newly added points, according to their intersection type. The intersecting tetrahedron shown in Fig. 8.47 is transformed to a combined shape of a tetrahedron, a pyramid or a prism which can be further divided into tetrahedra as illustrated in Fig. 8.48 and Fig. 8.49.
313
314 Automatic mesh generation Created tetrahedra
Transformation type
4
3 (1,2,3,n) (1,3,4,n)
3
(1,4,2,n) (2,3,4,n) (1,3,4,n)
1w
-2
3 (1,2,3,p) (2,3,p,n) (3,p,n,4)
4
3
(1,2,3,p) (2,3,p,n) (4,2,p,n)
1w
(3,4,p,n)
-2
3
(4,3,n,p) (1 ,n,p,4) (1,3,p,n)
Iw
-2
(1,4,2,n) (1,2,3,n)
Fig. 8.45 Tetrahedral transformation to recover a segment of the missing edges.
Three-dimensional mesh generation - Delaunay triangulation a
a
e
e
c
c
b
b
Fig. 8.46 Recoveryof boundary edge ab by the deletion of face cde. 1
1 c
a
8
1 c
a
b
b 4
2 2
3
(a) One edge intersects face abc 1
2
3
(c) Three edges intersect face abc
1
2
1
2
t
r
4
3
r a
a
a
~,
b
3 (b) Two edges intersect face abc
4 q~----~ 3
(d) Four edges intersect face abc
Fig. 8.47 Transformationsfor the recovery of surface faces. Shaded area shows the recovered portion of the boundary face.
When a face is intersected by only one edge and the edge is common to three tetrahedra, the face can be recovered directly by deleting the edge, with three tetrahedra transformed to two as shown in Fig. 8.50.
Removal of added points. The points that are added in the process of recovering the boundary edges and faces will be removed one by one together with the connected tetrahedra. The empty polyhedron left after the deletion of each added point and its connected tetrahedra will be triangulated directly, often with additional interior points. It is noted, with the reference to the global mesh generation procedure, that once Step 5 of the global procedure for surface recovery starts, the tetrahedral mesh, in general, does
315
3115
A u t o m a t i c mesh g e n e r a t i o n
e
c
a
e
a
c
v
e
a,.
c
,,.
Fig. 8.48 Different patterns of dividing a pyramid to tetrahedra.
not continue to be a Delaunay triangulation, which may seriously damage the quality of the mesh near the boundary. Therefore, mesh quality enhancement becomes indispensable after the completion of the mesh generation procedure.
8.4.3 Mesh quality enhancement In three-dimensional mesh generation, mesh quality enhancement is almost as important as element generation itself. This is because some poorly shaped tetrahedral elements are created either in the Delaunay triangulation process, due to the position of the inserting points, or in the surface mesh recovery process. Without applying certain mesh quality enhancement procedures to improve the element quality, these poorly shaped elements may render the three-dimensional mesh unusable in finite element computation. Unlike in twodimensional mesh generation, the process of improving the quality of a three-dimensional mesh is much more complicated and tedious. The quality of a tetrahedral element may be evaluated by different measures. A wide range of measures for the quality of tetrahedral elements are presented in references 75-78, and any one of these measures can be employed as criterion in the mesh quality enhancement procedure. Here we shall not get into the details of a particular quality measure, but are mainly concerned with the methodologies that can be used to improve the quality of the tetrahedral elements, under any specified quality measure. Several effective element quality enhancement methods are described below.
Element transformation
Modifying the topological structure of the mesh is probably the most effective way to improve the quality of the mesh in three dimensions and is realized by performing element transformations of the following form:
Three-dimensional mesh generation - Delaunay triangulation a
c
d
f e
a
c
aA
d
f
d
d~
f e
e
a,,,,
Ac
.-c
c
f
f
e
e
a
c
a
c
d
f
d
f
e
e
Fig. 8.49 Different patterns of dividing a prism to tetrahedra.
Two elements transformation. Two elements common to a face can be transformed to three elements as shown in Fig. 8.51, if one of the elements does not satisfy the quality criterion. To ensure that the new elements are valid, the new edge ab must intersect the removed face cde. Three elements transformation. As an inverse of the two element transformation, three elements common to an edge are transformed to two elements as illustrated in Fig. 8.51, if one of the elements does not meet the quality criterion.
317
318 Automatic mesh generation a
8
c . / ..... i ....
e
i
a
b Fig. 8.50 Recoveryof boundary face cde by the deletion of edge ab. a
a e
c
e
c
b (c,d,e,a) (c,e,d,b)
b (c,d,b,a) (d,e,b,a) (e,c,b,a)
Fig. 8.51 Two elements transformed to three elements and three elements transformed to two elements.
Four elements transformation. Four elements common to an edge can be transformed to two topologically different patterns of four elements common to an edge as depicted in Fig. 8.52, when one of the elements fails to satisfy the quality criterion. Five or more elements transformation. A split-collapse procedure is used for element transformation when an edge is common to five or more elements. For an element that fails the quality criterion, e.g., element abcd as shown in Fig. 8.53(a), find its edge ab and all the elements common to the edge (five elements are shown in Fig. 8.53(a)). A node n is added to the middle of the edge ab and splitting the elements as illustrated in Fig. 8.53(b). The new node n is then collapsed to one of its connecting nodes, except nodes a and b, to form new elements as demonstrated in Fig. 8.53(c) (six new elements are formed). This procedure can be used for an edge common to any number of elements and is attempted for each edge of the element.
All the transformations are carded out under the condition that the worst quality measure improves after the transformation.
Three-dimensional mesh generation - Delaunay triangulation
a
b
(a,d,c,f) (d,b,c,f) (a,c,d,e) (d,c,b,e) a
b
(a,d,b,f) (a,b,c,f) (a,c,b,e) (a,b,d,e)
b e
(a,d,e,f) (d,b,e,f) (b,c,e,f) (c,a,e,f)
Fig. 8.52 Two patterns of four elements transformed to four elements. (1) Edge ab changed to edge cd. (2) Edgeab changedto edge ef.
Node addition and node elimination
Unlike element transformations which will only change the topological structure of the mesh, node addition and node elimination will locally change the node density of the mesh. Node addition. A node is added to an edge of the element if the edge is deemed too long. This requires all the elements common to the edge being split, the process is shown in Fig. 8.53(a) and (b). Node elimination. A node of the element that fails the quality criterion and all the elements connected to it are shown in Fig. 8.54(a). The node is collapsed to one of its connecting
319
320
Automatic mesh generation b c g
f e
b g
(c,d,a,b) (d,e,a,b)
(c,d,n,b) (d,e,n,b)
(d,c,n,a) (e,d,n,a)
(e,f,n,b)
(f,e,n,a)
(f,g,n,b)
(g,f,n,a)
(g,c,n,b)
(c,g,n,a)
f
cq
(e,f,a,b) (f,g,a,b) (g,c,a,b) (c,d,e,b)
a
e
c
(c,e,f,b) (c,f,g,b) (c,e,d,a) (c,f,e,a) (c,g,f,a)
a
Fig. 8.53 Five or more elements transformation. (a) Edge ab and all the elements sharing it. (b) Node n is added to edge ab, all the common elements are split. (c) New node n is collapsed to node c.
nodes, as illustrated in Fig. 8.54(b), so that the quality of the resulting elements improves. The procedure is attempted for each node of the element. The quality of the elements can be further improved if the positions of the interior nodes are repositioned, which leads to the so-called mesh smoothing algorithm.
Mesh smoothing
The standard Laplacian smoothing cannot be applied directly to a tetrahedral mesh. It in fact reduces the quality of the mesh. The procedure is modified to move a node incrementally and iteratively towards each of its connecting nodes and is placed at the position that will increase the quality of the worst element. The procedure stops when quality of the connected elements does not improve. Several combined applications of the quality enhancement method usually results in a mesh with much improved quality. It is noted that the condition attached to all the quality enhancement methods, which requires the worst element quality to improve according to an element quality criterion,
Three-dimensional mesh generation - Delaunay triangulation
Fig. 8.54 Node collapsing. (a) Node b and its connecting elements. (b) Collapse node b to node d.
corresponds to an optimization problem. Its application guarantees improvement of the element quality, but could also be computationally expensive. For additional information on mesh enhancement methods using a specific quality measure as the objective function in the optimization process we refer to references 79-81.
8.4.4 Higher order elements With higher order surface mesh available, higher order tetrahedral elements can be readily obtained by finding the positions of the intermediate nodes using linear interpolation.
8.4.5 Numerical examples Tetrahedral meshes are generated using the mesh generation procedure of Delaunay triangulation described in this section. Figure 8.55(a) shows the mesh for a flask body casting. 89 A cross-section of the tetrahedral mesh is illustrated in Fig. 8.55(b) to demonstrate the regularity of the mesh. Figure 8.56 presents a tetrahedral mesh for a complete V8 engine block.
8.4.6 Remarks Remark 8.10. The automatic node generation procedure described in this section was intro-
duced by Weatherill and Hassan, many other node generation methods also exist. Indeed, the procedure by which new nodes are generated is the main difference between various Delaunay mesh generation algorithms reported in the literature. From our discussion, it is clear that once the points are available, a mesh can always be generated following a Delaunay triangulation algorithm. Therefore, it is important to adopt a suitable node generation method that can meet the specific requirements for a particular application, so that
321
322 Automatic mesh generation
Fig. 8.56 Tetrahedral mesh of a V8 engine block.
an optimal mesh can be obtained for finite element computation. We refer to references 20, 82-85 for additional information. Remark 8.11. As we have mentioned at the beginning of the chapter, it is still a demanding and challenging task to generate structured or unstructured hexahedral meshes automatically. In Chapter 11, we will observe that some hexahedral elements have advantages in the finite element computation for incompressible materials, in addition they generally have better accuracy compared to tetrahedral elements of the same order. Developing an
Concluding remarks 323 automatic mesh generation algorithm for hexahedral elements certainly deserves further research.
We have shown in this chapter how to generate mesh on curves, in arbitrary two-dimensional domains, on curved surfaces and for realistic three-dimensional geometries. We presented detailed discussion as well as algorithmic procedure for curve and surface mesh generation. We also described the advancing front method in two-dimensional mesh generation and the Delaunay triangulation method in three-dimensional mesh generation. The algorithms and methodologies presented in this chapter are not only robust and have been implemented for science and engineering applications, but also provide a basis for further research in the development of various aspects of automatic mesh generation methods. Additional applications of the automatic mesh generation methods discussed in this chapter will be presented again in Chapter 14 for adaptive finite element analysis and also appear in reference 37 for fluid dynamics applications.
8.1 Write expressions for two-dimensional boundary curves in which the Hermite parametric segments are replaced by a cubic Brzier spline. 8.2 Write a MATLAB program 86 to implement the boundary node generation procedure described in Sec. 8.2.3. 8.3 Develop an algorithm to update the generation front for a two-dimensional advancing front method. 8.4 For triangular meshing in two dimensions formulate a diagonal swapping criterion such that the node-element adjacency number NEop is closer to the optimal value after each swap. 8.5 When using the advancing front method to generate a quadrilateral mesh, prove that a necessary condition is that initial front must contain an even number of sides. 8.6 Devise a quadrilateral mesh generation algorithm for the advancing front method that forms each quadrilateral element from two neighbouring triangular elements with a common edge. Assume the triangular element mesh already exists. 8.7 Devise a quadrilateral mesh generation algorithm for the advancing front method that forms quadrilateral elements by subdividing a triangle as shown in Fig. 8.57. Assume the triangular element mesh already exists. 8.8 Define the optimal value of node-element adjacency number NEop for a quadrilateral mesh. 8.9 Write expressions for three-dimensional boundary curves in which the Hermite parametric segments are replaced by a cubic Brzier spline. 8.10 Write expressions for three-dimensional boundary surfaces in which the bi-Hermite parametric surface segments are replaced by bicubic Brzier splines. 8.11 Devise an algorithm to generate a quadrilateral surface mesh using the advancing front method.
324 Automatic mesh generation
Fig. 8.57 Triangle subdivided into three quadrilaterals. Problem 8.7. 8.12 When a new point is inserted into a Delaunay triangulation in three-dimensional mesh generation, the new point is sometimes found to lie on a co-sphere with four other forming points and, thus, violates the Delaunay triangulation property i of Sec. 8.4.1. Devise an algorithm to avoid such violation. 8.13 Devise an automatic numbering algorithm to generate nodes in an advancing front method for a Delaunay triangulation mesh generation procedure. 8.14 Show that the three elements transformation and four elements transformation used to improve the quality of the tetrahedral mesh in Sec. 8.4.3 are special cases of the split-collapse procedure used for the five or more elements transformation. 8.15 In element transformations in a tetrahedral mesh show that, after applying the splitcollapse procedure when an edge is common to six or more elements, the mesh quality can be further improved by performing edge swapping to the newly formed edges.
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326 Automaticmesh generation 32. T.D. Blacker and R.J. Meyers. Seams and wedges in plastering: a 3D hexahedral mesh generation algorithm. Eng. Comp., 2(9):83-93, 1993. 33. T.J. Tautges, T.D. Blacker, and S.A. Mitchell. The Whisker-Weaving algorithm: a connectivitybased method for constructing all-hexahedral finite element meshes. Int. J. Numer. Meth. Eng., 39:3327-3349, 1996. 34. N.A. Calvo and S.R. Idelsohn. All-hexahedral element meshing: generation of the dual mesh by recurrent subdivision. Comp. Meth. Appl. Mech. Eng., 182:371-378, 2000. 35. G. Dhondt. A new automatic hexahedral mesher based on cutting. Int. J. Numer. Meth. Eng., 50:2109-2126, 2001. 36. J.E Thompson, B.K. Soni, and N.P. Weatherill, editors. Handbook of Grid Generation. CRC Press, Jan. 1999. 37. O.C. Zienkiewicz, R.L. Taylor, and P. Nithiarasu. The Finite Element Method for Fluid Dynamics. Butterworth-Heinemann, Oxford, 6th edition, 2005. 38. I.D. Faux and M.J. Pratt. Computational Geometry for Design and Manufacture. Ellis Horwood, 1985. 39. G. Farin. Curves and Surfaces for Computer Aided Geometric Design. Academic Press, 1990. 40. E Yamaguchi. Curves and Surfaces in Computer Aided Geometric Design. Springer-Verlag, Berlin, 1988. 41. J.C. Cavendish. Automatic triangulation of arbitrary planar domains for the finite element method. Int. J. Numer. Meth. Eng., 8:679-696, 1974. 42. J. Wu, J.Z. Zhu, J. Szmelter, and O.C. Zienkiewicz. Error estimation and adaptivity in NavierStokes incompressible flows. Comp. Mech., 6:259-270, 1990. 43. S.H. Lo. Volume discretization into tetrahedra- II. 3D triangulation by advancing front approach. Comp. Struct., 39(5):501-511, 1991. 44. R. Lrhner. Some useful data structures for the generation of unstructured grids. Comm. Appl. Numer. Meth., 4:123-135, 1988. 45. J. Bonet and J. Peraire. An alternating digit tree (ADT) algorithm for 3d geometric search and intersection problems. Int. J. Numer. Meth. Eng., 31:1-17, 1990. 46. W. Kwok. An efficient data structure for the advancing front triangular mesh generation technique. Comm. Num. Meth. Eng., 11:465-473, 1995. 47. J.A. Talbert and A.R. Parkinson. Development of an automatic, two-dimensional finite element mesh generator using quadrilateral elements and Bezier curve boundary definitions. Int. J. Numer. Meth. Eng., 29:1551-1567, 1990. 48. B.P. Johnston, J.M. Sullivan, and A. Kwasnik. Automatic conversion of triangular finite element meshes to quadrilateral elements. Int. J. Numer. Meth. Eng., 31:67-84, 1991. 49. E. Rank, M. Schweingruber, and M. Sommer. Adaptive mesh generation. Comm. Numer. Meth. Eng., 9:121-129, 1993. 50. C.K. Lee and S.H. Lo. A new scheme for the generation of a graded quadrilateral mesh. Comp. Struct., 52:847-857, 1994. 51. B. Joe. Quadrilateral mesh generation in polygonal regions. Comp. Aided Design, 27:209-222, 1995. 52. S.J. Owen, M.L. Staten, S.A. Canann, and S. Saigal. Q-morph: an indirect approach to advancing front quad meshing. Int. J. Numer. Meth. Eng., 44:1317-1340, 1999. 53. J. Peraire, J. Peir6, and K. Morgan. Adaptive remeshing for 3-dimensional compressible flow computations. J. Comp. Phys., 103:269-285, 1992. 54. J. Peir6. Surface grid generation. In Handbook of Grid Generation, Chapter 19, pages 19.1 -19.20. CRC Press, 1999. 55. W.H. Press et al., editor. Numerical Recipes in Fortran: The Art of Scientific Computing. Cambridge University Press, Cambridge, 2nd edition, 1992. 56. O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method for Solid and Structural Mechanics. Butterworth-Heinemann, Oxford, 6th edition, 2005.
References 327 57. E Hansbo. Generalized Laplacian smoothing of unstructured grids. Comm. Num. Meth. Eng., 11:455-464, 1995. 58. Y. Zheng, R.W. Lewis, and D.T. Gethin. Three-dimensional unstructured mesh generation: Part 2. Surface meshes. Comp. Meth. Appl. Mech. Eng., 134:269-284, 1996. 59. M. Suzuki. Surface grid generation with linkage to geometric generation. Int. J. Numer. Meth. Eng., 34:163-176, 1993. 60. C.K. Lee. Automatic metric advancing front triangulation over curved surfaces. Eng. Comp., 17(1):48-74, 2000. 61. H. Borouchaki, P. Laug, and P.L. George. Parametric surface meshing using combined advancing-front generalized Delaunay approach. Int. J. Numer. Meth. Eng., 49:233-259, 2000. 62. S.J. Sherwin and J. Peir6. Mesh generation in curvilinear domain using high-order elements. Int. J. Numer. Meth. Eng., 53:207-223, 2002. 63. Y.K. Lee and C.K. Lee. Automatic generation of anisotropic quadrilateral meshes on threedimensional surfaces using metric specifications. Int. J. Numer. Meth. Eng., 53:2673-2700, 2002. 64. R. Lrhner. Regridding surface triangulations. J. Comp. Phys., 126:1-10, 1996. 65. A. Rassineux, P. Villon, J.-M. Savignat, and O. Stab. Surface remeshing by local Hermite diffuse interpolation. Int. J. Numer. Meth. Eng., 49:31-49, 2000. 66. Y. Ito and K. Nakahashi. Surface triangulation for polygonal models based on CAD data. Int. J. Numer. Meth. Fluids, 39:75-96, 2002. 67. C.K. Lee. Automatic metric 3D surface mesh generation using subdivision surface geometrical model. Part 1: Construction of underlying geometrical model. Int. J. Numer. Meth. Eng., 56:1593-1614, 2003. 68. C.K. Lee. Automatics metric 3D surface mesh generation using subdivision surface geometrical model. Part 2: Mesh generation algorithm and examples. Int. J. Numer. Meth. Eng., 56:16151646, 2003. 69. B. Delaunay. Sur la sphere vide. Izv. Akad. Nauk SSSR, Otdelenie Matematicheskii i Estestvennyka Nauk, 7:793-800, 1934. 70. G. Voronoi. Nouvelles applications des param~tres continus ~ la throrie des formes quadratiques. J. Reine Angew. Math, 133:97-178, 1907. 71. F.P. Preparata and M.I. Shamos. Computational Geometry. Springer-Verlag, New York, 1988. 72. J. O'Rourke. Computational Geometry in C. Cambridge University Press, 2nd edition, 2001. 73. N.P. Weatherill. The integrity of geometrical boundaries in the two-dimensional Delaunay triangulation. Comm. Appl. Numer. Meth., 6:101-109, 1990. 74. Y. Zheng, R.W. Lewis, and D.T. Gethin. Three-dimensional unstructured mesh generation: Part 1. Fundamental aspects of triangulation and point creation. Comp. Meth. Appl. Mech. Eng., 134:249-268, 1996. 75. V.N. Parthasarathy, C.M. Graichen, and A.E Hathaway. A comparision of tetrahedron quality measures. Fin. Elem. Anal. Design, 15:255-261, 1993. 76. N.P. Weatherill, P.R. Eiseman, J. Hause, and J.E Thompson. Numerical Grid Generation in Computational Fluid Dynamics and Related Fields. Pineridge Press, Swansea, 1994. 77. A. Liu and B. Joe. Relationship between tetrahedron shape measures. BIT, 34:268-287, 1994. 78. R.W. Lewis, Y. Zheng, and D.T. Gethin. Three-dimensional unstructured mesh generation: Part 3. Volume meshes. Comp. Meth. Appl. Mech. Eng., 134:285-310, 1996. 79. L.A. Freitag and P.M. Knupp. Tetrahedral mesh improvement via optimization of element condition number. Int. J. Numer. Meth. Eng., 53:1377-1391, 2002. 80. J.M. Escobar, E. Rodriguez, R. Montenegro, G. Montero, and J.M. Gonz~ilez-Yuste. Simultaneous untangling and smoothing of tetrahedral meshes. Comp. Meth. Appl. Mech. Eng., 192:27722787, 2003. 81. C.L. Bottasso. Anisotropic mesh adaption by metric-driven optimization. Int. J. Numer. Meth. Eng., 60:597-639, 2004.
328 Automaticmesh generation 82. D.L. Marcum. Adaptive unstructured grid generation for viscous flow applications. AIAA jr., 34:2440, 1996. 83. D.J. Mavriplis. An advancing front Delaunay triangulation algorithm designed for robustness. J. Comp. Phys., 117:90-101, 1995. 84. H. Borouchaki, P.L. George, F. Hecht, P. Laug, and E. Saltel. Delaunay mesh generation governed by metric specifications. Part I. Algorithms. Fin. Elem. Anal. Design, 25:61-83, 1997. 85. H. Borouchaki, P.L. George, and B. Mohammadi. Delaunay mesh generation governed by metric specifications. Part II. Applications. Fin. Elem. Anal. Design, 25:85-109, 1997. 86. MATLAB. www.mathworks.com, 2003. 87. J.Z. Zhu, O.C. Zienkiewicz, E. Hinton and J. Wu. A new approach to the development of automatic quadrilateral mesh generation. Int. J. Numer. Meth. Eng. 32:849-866, 1991. 88. T.D. Blacker and M.B. Stephenson. Paving: A new approach to automatic quadrilateral mesh generation. Int. J. Numer. Meth. Eng. 32:811-847, 1991. 89. J.Z. Zhu, O.C. Zienkiewicz. 'A posteriori' error estimation and three dimensional automatic mesh generation. Fin. Elem. Anal. Design. 25:167-184, 1997.
The patch test, reduced integration, and non-conforming elements
We have briefly referred in Chapter 2 to the patch test as a means of assessing convergence of displacement-type elements for elasticity problems in which the shape functions violate continuity requirements. In this chapter we shall deal in more detail with this test which is applicable to all finite element forms and will show that (a) it is a necessary condition for assessing the convergence of any finite element approximation and further that, if properly extended and interpreted, it can provide (b) a sufficient requirement for convergence, (c) an assessment of the (asymptotic) convergence rate of the element tested, (d) a check on the robustness of the algorithm, and (e) a means of developing new finite element forms which can violate compatibility (continuity) requirements. While for elements which a priori satisfy all the continuity requirements, have correct polynomial expansions, and are exactly integrated such a test is superfluous in principle, but it is nevertheless useful as it gives (f) a check that correct programming was achieved. For all the reasons cited above the patch test has been, since its inception, and continues to be the most important check for practical finite element codes. The original test was introduced by Irons et al. 1-3 in a physical way and could be interpreted as a check which ascertained whether a patch of elements (Fig. 9.1) subject to a constant strain reproduced exactly the constitutive behaviour of the material and resulted in correct stresses when it became infinitesimally small. If it did, it could then be argued that the finite element model represented the real material behaviour and, in the limit, as the size of the elements decreased would therefore reproduce exactly the behaviour of the real structure. Clearly, although this test would only have to be passed when the size of the element patch became infinitesimal, for most elements in which polynomials are used the patch size did not in fact enter the consideration and the requirement that the patch test be passed for any element size became standard.
330 The patch test, reduced integration, and non-conforming elements 0---
(~ constant
~xdX
d
Fig. 9.1 A patch of element and a volume of continuum subject to constant strain ex. A physical interpretation of the constant strain or linear displacement field patch test.
Quite obviously a rigid body displacement of the patch would cause no strain, and if the proper constitutive laws were reproduced no stress changes would result. The patch test thus guarantees that no rigid body motion straining will occur. When curvilinear coordinates are used the patch test is still required to be passed in the limit but generally will not do so for a finite size of the patch. (An exception here is the isoparametric coordinate system in problems discussed in Chapter 5 since it is guaranteed to contain linear polynomials in the global coordinates.) Thus for many problems such as shells, where local curvilinear coordinates are used, this test has to be restricted to infinitesimal patch sizes and, on physical grounds alone, appears to be a necessary and sufficient condition for convergence. Numerous publications on the theory and practice of the test have followed the original publications cited 4-6 and mathematical respectability was added to those by Strang. 7' 8 Although some authors have cast doubts on its validity 9' 10 these have been fully refuted 11-13 and if the test is used as described here it fulfils the requirements (a)-(f) stated above. In the present chapter we consider the patch test applied to irreducible forms (see Chapter 3) but an extension to mixed forms is more important. This has been studied in references 13, 14 and 15 and made use of in many subsequent publications. The matter of mixed form patch tests will be fully discussed in the next chapter; however, the consistency and stability tests developed in the present chapter are always required. One additional use of the patch test was suggested by Babu~ka et al. 16 with a shorter description given by Boroomand and Zienkiewicz. 17 This test can establish the efficiency of gradient (stress) recovery processes which are so important in error estimation as will be discussed in Chapter 13.
We shall consider in the following the patch test as applied to a finite element solution of a set of differential equations .A.(u) - Z~u + b = 0
(9.1)
in the domain ~ together with the conditions /~(u) = 0
on the boundary of the domain, F.
(9.2)
Convergencerequirements The finite element approximation is given in the form u ~ fi -
Nfi
(9.3)
where N are shape functions defined in each element, ~'2e, and fi are unknown parameters. By applying standard procedures of finite element approximation the problem reduces in a linear case to a set of algebraic equations Kfi = f
(9.4)
which when solved give an approximation to the differential equation and its boundary conditions. What is meant by 'convergence' in the approximation sense is that the approximate solution, fi, should tend to the exact solution u when the size of the elements h approaches zero (with some specified subdivision pattern). Stated mathematically we must find that the error at any point becomes (when h is sufficiently small) lu--fil : 0 ( h q )
<-- Chq
(9.5)
where q > 0 and C is a positive constant, depending on the position. This must also be true for all the derivatives of u defined in the approximation. By the order of convergence in the variable u we mean the value of the index q in the above definition. To ensure convergence it is necessary that the approximation fulfil both consistency and stability conditions. TM The consistency requirement ensures that as the size of the elements h tends to zero, the approximation equation (9.4) will represent the exact differential equation (9.1) and the boundary conditions (9.2) (at least in the weak sense). The stability condition is simply translated as a requirement that the solution of the discrete equation system (9.4) be unique and avoid spurious mechanisms which may pollute the solution for all sizes of elements. For linear problems in which we solve the system of algebraic equations (9.4) as = K-if (9.6) this means simply that the matrix K must be non-singular for all possible element assemblies (subject to imposing minimum stable boundary conditions). The patch test traditionally has been used as a procedure for verifying the consistency requirement; the stability was checked independently by ensuring non-singularity of matrices. 19 Further, it generally tested only the consistency in satisfaction of the differential equation (9.1) but not of its natural boundary conditions. In what follows we shall show how all the necessary requirements of convergence can be tested by a properly conceived patch test. A 'weak' singularity of a single element may on occasion be permissible and some elements exhibiting it have been, and still are, successfully used in practice. One such case is given by the eight-node isoparametric element with a 2 • 2 Gauss quadrature, to which we shall refer later here. This element is on occasion observed to show peculiar behaviour (though its use has advantages as discussed in Chapter 10). An element that occasionally fails is termed non-robust and the patch test provides a means of assessing the degree of
robustness.
331
332
The patch test, reduced integration, and non-conforming elements
We shall first consider the consistency condition which requires that in the limit (as h tends to zero) the finite element approximation of Eq. (9.4) should model exactly the differential equation (9.1) and the boundary conditions (9.2). If we consider a 'small' region of the domain (of size 2h) we can expand the unknown function u and the essential derivatives entering the weak approximation in a Taylor series. From this we conclude that for convergence of the function and its first derivative in typical problems of a secondorder equation and two dimensions, we require that around a point i assumed to be at the coordinate origin, U-Ua+
Ox --
~
-~x
x q-
y + . . . + O ( h p)
+ ' ' " -q- O(hp-1 )
(9.7)
Ou -- (0U)a OY
~YY
+ " " " -q- O(hp-1)
with p > 2. The finite element approximation should therefore reproduce exactly the problem posed for any linear f o r m s of u as h tends to zero. Similar conditions can obviously be written for higher order problems. This requirement is tested by the current interpretation of the patch test illustrated in Fig. 9.2. We refer to this as the base solution. For problems involving Co approximation we compute first an arbitrary solution of the differential equation using a linear polynomial as the base solution and set the corresponding parameters ~ [see Eq. (9.3)] at all 'nodes' of a p a t c h which assembles completely the nodal variable Ua (i.e., provides all the equation terms corresponding to it). In test A we simply insert the exact value of the parameters fi into the node a equations and verify that K a b f l b -- fa ~ 0
(9.8)
The simple patch test (tests A and B) - a necessary condition for convergence
where fa is a force which results from any 'body force' required to satisfy the differential equation (9.1) for the base solution. Generally in problems given in cartesian coordinates the required body force is zero; however, in curvilinear coordinates (e.g., axisymmetric elasticity problems) it can be non-zero. In test B only the values of ~ corresponding to the boundaries of the 'patch' are inserted and Ua is found as Ua - - K a 1
(fa
--
Kabflb)
b :fi a
(9.9)
and compared against the exact value. Both patch tests verify only the satisfaction of the basic differential equation and not of the boundary approximations, as these have been explicitly excluded here. We mentioned earlier that the test is, in principle, required only for an infinitesimally small patch of elements; however, for differential equations with constant coefficients the size of the patch is immaterial and the test can be carried out on a patch of arbitrary dimensions. Indeed, if the coefficients are not constant the same size independence exists providing that a constant set of such coefficients is used in the formulation of the test. This applies, for instance, in axisymmetric problems where coefficients of the type 1/r (radius) enter the equations and when the patch test is here applied, it is simply necessary to enter the computation with such quantities assumed constant. Alternatively, a body force can be computed which allows the base solution to satisfy the differential equation exactly. If mapped curvilinear elements are used it is not obvious that the patch test posed in global coordinates needs to be satisfied. Here, in general, convergence in the mapping coordinates may exist but a finite patch test may not be satisfied. However, once again if we specify the nature of the subdivision without changing the mapping function, in the limit the jacobian becomes locally constant and the previous remarks apply. To illustrate this point consider, for instance, a set of elements in which local coordinates are simply the polar coordinates as shown in Fig. 9.3. With shape functions using polynomial expansions in the r, 0 terms the patch test of the kind we have described above will not be satisfied with elements of finite size - nevertheless in the limit as the element size tends to zero it will become true. Thus it is evident that patch test satisfaction is a necessary condition which has always to be achieved providing the size of the patch is infinitesimal. This proviso which we shall call weak patch test satisfaction is not always simple to verify, particularly if the element coding does not easily permit the insertion of constant
Fig. 9.3 Polar coordinate mapping.
333
334
The patch test, reduced integration, and non-conforming elements coefficients or a constant jacobian. It is indeed fortunate that the standard isoparametric element form reproduces exactly the linear polynomial global coordinates (see Chapter 5) and for this reason does not require special treatment unless some other crime (such as selective or reduced integration) is introduced.
The patch test described in the preceding section was shown to be a necessary condition for convergence of the formulation but did not establish sufficient conditions for it. In particular, it omitted the testing of the boundary 'load' approximation for the case when the 'natural' (e.g., 'traction of elasticity') conditions are specified. Further it did not verify the stability of the approximation. A test including a check on both of the above conditions is easily constructed. We show this in Fig. 9.4 for a two-dimensional plane problem as test C. In this the patch of elements is assembled as before but subject to prescribed natural boundary conditions (or tractions around its perimeter) corresponding to the base function. The assembled matrix of the whole patch is written as Kfi=f
Fixing only the minimum number of parameters fi necessary to obtain a physically valid solution (e.g., eliminating the rigid body motion in an elasticity example or a single value of temperature in a heat conduction problem), a solution is sought for the remaining fi values and compared with the exact base solution assumed. Now any singularity of the K matrix will be immediately observed and, as the vector f includes all necessary source and boundary traction terms, the formulation will be completely tested (providing of course a sufficient number of test states is used). The test described is now not only necessary but sufficient for convergence. With boundary traction included it is of course possible to reduce the size of the patch to a single element and an alternative form of test C is illustrated in Fig. 9.4(b), which
,~
,,"
,,y" \
...... ~ .
Natural boundary conditions specified \ ........
Y
boundary conditions
(a)
x
Fig. 9.4 (a) Patch test of form C. (b) The single-elementtest.
Generalized patch test (test C) and the single-element test Gauss integration points o +
I
+
---+§
§
:,,'
-i
Nine-noded
elements only
Eight and nine nodes
(a)
(b)
Fig. 9.5 (a) Zero energy (singular) modes for 8- and 9-noded quadratic elements and (b) for a patch of bilinear elements with single integration points.
is termed the single-element test. 11 This test is indeed one requirement of a good finite element formulation as, on occasion, a larger patch may not reveal the inherent instabilities of a single element. This happens in the well-documented case of the plane strain-stress 8-noded isoparametric element with (reduced) four-point Gauss quadrature, i.e., where the singular deformation mode of a single element (see Fig. 9.5) disappears when several elements are assembled.t It should be noted, however, that satisfaction of a single element test is not a sufficient condition for convergence. For sufficiency we require at least one internal element boundary to test that consistency of a patch solution is maintained between elements.
t This figure also shows a similar singularity for a patch of four 4-noded (bilinear interpolation) elements with single-point quadrature, and we note the similar shape of zero energy modes (see Chapter 5, Sec. 5.12.3).
335
336
The patch test, reduced integration, and non-conforming elements
In the previous section we have defined in some detail the procedures for conducting a patch test. We have also asserted the fact that such tests if passed guarantee that convergence will occur. However, all the tests are numerical and it is impractical to test all possible combinations. In particular let us consider the base solutions used. These will invariably be a set of polynomials given in two dimensions as U -- ~
01,i e i ( x , y)
(9.10)
i
where Pi are a suitable set of low order polynomials (e.g., 1, x, y for Galerkin forms possessing only first order derivatives) and C~ i are parameters. It is fairly obvious that if patch tests are conducted on each of these polynomials individually any base function of the form given in Eq. (9.10) can be reproduced and the generality preserved for the particular combination of elements tested. This must always be done and is almost a standard procedure in engineering tests, necessitating only a limited number of combinations. However, as various possible patterns of elements can occur and it is possible to increase the size without limit the reader may well ask whether the test is complete from the geometrical point of view. We believe it is necessary in a numerical test to consider the possibility of several pathological arrangements of elements but that if the test is purely limited to a single element and a complete patch around a node we can be confident about the performance on more general geometric patterns. Indeed even mathematical assessments of convergence are subject to limits often imposed a posteriori. Such limits may arise if for instance a singular mapping is used. The procedures referred to in this section should satisfy most readers as to the validity and generality of the test. On some limited occasions it is possible to perform the test purely algebraically and then its validity cannot be doubted. Some such algebraic tests will be referred to later in connection with incompatible elements. In this chapter we have only considered linear differential equations and linear material behaviour; however, the patch test can well be used and extended to cover non-linear problems.
While the patch tests discussed in the last three sections ensure (when satisfied) that convergence will occur, they did not test the order of this convergence, beyond assuring us that in the case of Eq. (9.7) the errors were, at least, of order O (h 2) in u. It is an easy matter to determine the actual highest asymptotic rate of convergence of a given element by simply imposing, instead of a linear solution, exact higher order polynomial solutions. 6' 8 The highest value of such polynomials for which complete satisfaction of the patch test is achieved automatically evaluates the corresponding convergence rate. It goes without saying that for such exact solutions generally non-zero source (e.g., body force) terms in the original equation (9.1) will need to be involved.
The patch test for plane elasticity problems 337 In addition, test C in conjunction with a higher order patch test may be used to illustrate any tendency for 'locking' to occur (see Chapter 10). Accordingly, element robustness with regard to various parameters (e.g., Poisson's ratios near one-half for elasticity problems in plane strain) may be established. In such higher order patch tests it will of course first be assumed that the patch is subject to the base expansion solution as described. Thus, for higher order terms it will be necessary to start and investigate solutions of the type or3x2 W ot4xy + a s y 2 + . . . each of which should be applied individually or as linearly independent combinations and for each the solution should be appropriately tested. In particular, we shall expect higher order elements to exactly satisfy certain order solutions. However, in Chapter 13 we shall use this idea to find the error between the exact solution and the recovery using precisely the same type of formulation.
In the next few sections we consider several applications of the patch test in the evaluation of finite element models. In each case we consider only one of the necessary tests which need to be implemented. For a complete evaluation of a formulation it is necessary to consider all possible independent base polynomial solutions as well as a variety of patch configurations which test the effects of element distortion or alternative meshing interconnections which will be commonly used in analysis. As we shall emphasize, it is important that both consistency and stability be evaluated in a properly conducted test. In Chapter 5 (Sec. 5.12) we have discussed the minimum required order of numerical integration for various finite element problems which results in no loss of convergence rate. However, it was also shown that for some elements such a minimum integration order results in singular matrices. If we define the standard integration as one which evaluates the stiffness of an element exactlyt (at least in the undistorted form) then any lower order of integration is called reduced. Such reduced integration has some merits in certain problems for reasons which we shall discuss in Sec. 11.5, but it can cause singularities which should be discovered by a patch test (which supplements and verifies the arguments of Sec. 5.12.3). Application of the patch test to some typical problems will now be shown.
Example 9.1" Patch test for base solution. We consider first a plane stress problem on the patch shown in Fig. 9.6(a). The material is linear, isotropic elastic with properties E = 1000 and v = 0.3. The finite element procedure used is based on the displacement form using 4-noded isoparametric shape functions and numerical integration as described in Chapter 5. Since the stiffness computation includes only first derivatives of displacements, the formulation converges provided that the patch test is satisfied for all linear polynomial t An alternatedefinitionfor standardintegrationis the lowestorderof integrationfor whichthe rank of the stiffness matrix does not increase.
338
The patch test, reduced integration, and non-conforming elements
3
1
3
2
(a)
1
2
(b)
Fig. 9.6 Patch for evaluation of numerically integrated plane stress problems. (a) Five-element patch. (b) One-element patch. solutions of displacements in the base solution. Here we consider only one of the six independent linear polynomial solutions necessary to verify satisfaction of the patch test. The solution considered is u =
0.0020 x
(9.11a)
v = -0.0006 y which produces zero body forces and zero stresses except for Ox = 2
(9.11b)
The solution given in Table 9.1 is obtained for the nodal displacements and satisfies Eq. (9.11 a) exactly. The patch test is performed first using 2 x 2 gaussian 'standard' quadrature to compute each element stiffness and resulting reaction forces at nodes. For patch test A all nodes are restrained and nodal displacement values are specified according to Table 9.1. Stresses are computed at specified Gauss points (1 x 1, 2 • 2, and 3 • 3 Gauss points were sampled) and all are exact to within round-off error (double precision was used which produced round-off errors less than 10 -15 in the quantities computed). Reactions were also computed at all nodes and again produced the force values shown in Table 9.1 to within round-off limits. This approximation satisfies all conditions required for a finite element procedure (i.e., conforming shape functions and standard order quadrature). Accordingly, the patch test Table 9.1 Patch solution for Fig. 9.6 Coordinates
Computed displacements
Forces
Node a
Xa
Ya
Ua
1)a
Fxa
1
0.0 2.0 2.0 0.0 0.4 1.4 1.5 0.3
0.0 0.0 3.0 2.0 0.4 0.6 2.0 1.6
0.0 0.0040 0.0040 0.0 0.0008 0.0028 0.0030 0.0006
0.0 0.0 -0.00180 -0.00120 -0.00024 -0.00036 -0.00120 -0.00096
-2
2 3 4 5 6 7 8
3 2 -3 0 0 0 0
Fy a
0
0 0 0 0 0 0 0
The patch test for plane elasticity problems 339 merely verifies that the programming steps used contain no errors. Patch test A does not require explicit use of the stiffness matrix to compute results; consequently the above patch test was repeated using patch test B where only nodes 1 to 4 are restrained with their displacements specified according to Table 9.1. This tests the accuracy of the stiffness matrix and, as expected, exact results are once again recovered to within round-off errors. Finally, patch test C was performed with node 1 fully restrained and node 4 restrained only in the x direction. Nodal forces were applied to nodes 2 and 3 in accordance with the values generated through the boundary tractions by ax (i.e., nodal forces shown in Table 9.1). This test also produced exact solutions for all other nodal quantities in Table 9.1 and recovered ax of 2 at all Gauss points in each element. The above test was repeated for patch tests A, B, and C but using a 1 x 1 'reduced' Gauss quadrature to compute the element stiffness and nodal force quantities. Patch test C indicated that the global stiffness matrix contained two global 'zero energy modes' (i.e., the global stiffness matrix was rank deficient by 2), thus producing incorrect nodal displacements whose results depend solely on the round-off errors in the calculations. These in turn produced incorrect stresses except at the 1 • 1 Gauss point used in each element to compute the stiffness and forces. Thus, based upon stability considerations, the use of 1 • 1 quadrature on 4-noded elements produces a failure in the patch test. The element does satisfy consistency requirements, however, and provided a proper stabilization scheme is employed (e.g., stiffness or viscous methods are used in practice) this element may be used for practical calculations. 2~ 21 It should be noted that a one-element patch test may be performed using the mesh shown in Fig. 9.6(b). The results are given by nodes 1 to 4 in Table 9.1. For the one-element patch, patch tests A and B coincide and neither evaluates the accuracy or stability of the stiffness matrix. On the other hand, patch test C leads to the conclusions reached using the five-element patch: namely, 2 • 2 gaussian quadrature passes a patch test whereas 1 x 1 quadrature fails the stability part of the test (as indeed we would expect by the arguments of Chapter 5, Sec. 5.12). A simple test on cancellation of a diagonal during the triangular decomposition step is sufficient to warn of rank deficiencies in the stiffness matrix.
Example 9.2: Patch test for quadratic elements: quadrature effects. In Fig. 9.7 we show a two-element patch of quadratic isoparametric quadrilaterals. Both 8-noded serendipity and 9-noded lagrangian types are considered and a basic patch test type C is performed for load case 1. For the 8-noded element both 2 x 2 ('reduced') and 3 x 3 ('standard') gaussian quadrature satisfy the patch test, whereas for the 9-noded element only 3 x 3 quadrature is satisfactory, with 2 x 2 reduced quadrature leading to failure in rank of the stiffness matrix. However, if we perform a one-element test for the 8-noded and 2 x 2 quadrature element, we discover the spurious zero-energy mode shown in Fig. 9.5 and thus the one-element test has failed. We consider such elements suspect and to be used only with the greatest of care. To illustrate what can happen in practice we consider the simple problem shown in Fig. 9.8(a). In this example the 'structure' modelled by a single element is considered rigid and interest is centred on the 'foundation' response. Accordingly only one element is used to model the structure. Use of 2 x 2 quadrature throughout leads to answers shown in Fig. 9.8(b) while results for 3 x 3 quadrature are shown in Fig. 9.8(c). It should be noted that no zero-energy mode exists since more than one element is used. There is here, however, a spurious response due to the large modulus variation between
340
The patch test, reduced integration, and non-conforming elements
15
J q
10
L F"
5
15
Load 1
Load 2
._1 Vl
9 9
15
9 .. l.. dJ i~
=
-'1
A
2
..B
Load 1
Load 2
Fig. 9.7 Patch test for 8- and 9-noded isoparametric quadrilaterals.
structure and foundation. This suggests that problems in which non-linear response may lead to a large variation in material parameters could also induce such performance, and thus use of the 8-noded 2 x 2 integrated element should always be closely monitored to detect such anomalous behaviour. Indeed, support or loading conditions may themselves induce very suspect responses for elements in which near singularity occurs. Figure 9.9 shows some amusing peculiarities which can occur for reduced integration elements and which disappear entirely if full integration is used. 22 In all cases the assembly of elements is non-singular even though individual elements are rank deficient.
Example 9.3: Higher order patch test- assessment of order. In order to demonstrate a higher order patch test we consider the two-element plane stress problem shown in Fig. 9.7 and subjected to bending loading shown as Load 2. As above, two different types of element are considered: (a) an 8-noded serendipity quadrilateral element and (b) a 9-noded lagrangian quadrilateral element. In our test we wish to demonstrate a feature for nine-noded element mapping discussed in Chapter 5 (see Sec. 5.7) and first shown by Wachspress. 23 In particular we restrict the mapping into the xy plane to be that produced by the 4-noded isoparametric bilinear element, but permit the dependent variable to assume the full range of variations consistent with the 8- or 9-noded shape functions. In Chapter 5 we showed that the 9-noded element can approximate a complete quadratic displacement function in x, y whereas the eight-noded element cannot. Thus we expect that the nine-noded element when restricted to the isoparametric mappings of the 4-noded element will pass a higher order patch test for all arbitrary quadratic displacement fields. The pure bending solution in elasticity is composed of polynomial terms up to quadratic order. Furthermore, no body force loadings are necessary to satisfy the equilibrium equations. For the mesh considered the nodal loadings are equal and opposite on the top and bottom nodes as shown in Fig. 9.7. The results for the two elements are shown in Table 9.2 for the indicated quadratures with E = 100 and v = 0.3. From this test we observe that the 9-noded element does pass the higher order test performed. Indeed, provided the mapping is restricted to the 4-noded shape it will always
The patch test for plane elasticity problems 341
Fig. 9.8 A propagating spurious mode from a single unsatisfactory element. (a) Problem and mesh. (b) 2 x 2 integration. (c) 3 x 3 integration.
pass a patch test for displacements with terms no higher than quadratic. On the other hand, the 8-noded element passes the higher order patch test performed only for rectangular element (or constant jacobian) mappings. Moreover, the accuracy of the 8-noded element deteriorates very rapidly with increased distortions defined by the parameter d in Fig. 9.7. The use of 2 x 2 reduced quadrature improves results for the higher order patch test performed. Indeed, two of the points sampled give exact results and the third is only slightly in error. As noted previously, however, a single-element test for the 2 x 2 integrated 8-noded element will fail the stability part of the patch test and it should thus be used with great care.
342
The patch test, reduced integration, and non-conforming elements
i
(a)
......~.
r
_
\/
_
.__.._
--
i
_ _
_
_
_
//
,I
L ~
_
_
_
_
-
~
l
---
F--
(b)
IIIII
--!
r
I
(c)
Fig. 9.9 Peculiar response of near singular assemblies of elements.22 (a) A column of 9-noded elements with point load response of full 3 x 3 and 2 x 2 integration. The whole assembly is non-singular but singular element modes are apparent. (b) A fully constrained assembly of 9-noded elements with no singularity- first six eigenmodes with full (3x3) integration. (c) Same as (b) but with 2x2 integration. Note the appearance of 'wild' modes called 'Escher' modes named so in reference 22 after this graphic artist.
Application of the patch test to an incompatible element Table 9.2 Bending load case (E = 100, v = 0.3) Element
Quadrature
8-node 8-node 9-node 8-node 8-node 9-node 8-node 8-node 9-node Exact
3 2 3 3 2 3 3 2 3 -
x x x x x x x x x
3 2 3 3 2 3 3 2 3
d
) )
0
) )
1
) )
2 -
VA
UB
1)B
0.750 0.750 0.750 0.7448 0.750 0.750 0.6684 0.750 0.750 0.750
0.150 0.150 0.150 0.1490 0.150 0.150 0.1333 0.150 0.150 0.150
0.75225 0.75225 0.75225 0.74572 0.75100 0.75225 0.66364 0.75225 0.75225 0.75225
In order to demonstrate the use of the patch test for a finite element formulation which violates the usually stated requirements for shape function continuity, we consider the plane strain incompatible modes first introduced by Wilson et al. 24 and discussed by Taylor et al. 25 The specific incompatible formulation considered uses the element displacement approximations: fl = Natla + N~Otl + N~ot2 (9.12) where Na (a = 1. . . . . 4) are the usual conforming bilinear shape functions and the last two terms are incompatible modes of deformation defined by the hierarchical functions N ~ ' - 1 - ~2
and
N~-
1-/7
2
(9.13)
defined independently for each element. The shape functions used are illustrated in Fig. 9.10. The first, a set of standard bilinear type, gives a displacement pattern which, as shown in Fig. 9.10(b), introduces spurious shear strains in pure bending. The second, in which the parameters C~l and c~2 are strictly associated with a specific element, therefore introduces incompatibility but assures correct bending behaviour in an individual rectangular element. The excellent performance of this element in the bending situation is illustrated in Fig. 9.11. In reference 25 the finite element approximation is computed by summing the potential energies of each element and computing the nodal loads due to boundary tractions from the conforming part of the displacement field only. Thus for the purposes of conducting patch tests we compute the strains using all parts of the displacement field leading to a generalization of (9.4) which may be written as K21
12]{ o } {,1}
K22
ot
=
f2
(9.14)
Here Kll and fl are the stiffness and loads of the 4-noded (conforming) bilinear element, K12 and K21 (= KIT) are coupling stiffnesses between the conforming and non-conforming displacements, and K22 and f2 are the stiffness and loads of the non-conforming displacements. We note that, according to the algorithm of reference 24, f2 must vanish from the patch test solutions.
343
344 The patch test, reduced integration, and non-conforming elements
(a)
r-i It
(b)
(c)
Fig. 9.10 (a) Linear quadrilateral with auxiliary incompatible shape functions. (b) Pure bending and linear displacements causing shear. (c) Auxiliary 'bending' shape functions with internal variables. For a patch test in plane strain or plane stress, only linear polynomials need be considered for which all non-conforming displacements must vanish. Thus for a successful patch test we must have KllUI -- fl
(9.15a)
K21u = f2
(9.15b)
and
If we carry out a patch test for the mesh shown in Fig. 9.12(a) we find that all three forms (i.e., patch tests A, B, and C) satisfy these conditions and thus pass the patch test. If we consider the patch shown in Fig. 9.12(b), however, the patch test is not satisfied. The lack of satisfaction shows up in different ways for each form of the patch test. Patch test A produces non-zero t"2 values when oz is set to zero and fi according to the displacements considered. In form B the values of the nodal displacements u5 are in error and oz are non-zero, also leading to erroneous stresses in each element. In form C all unspecified displacements are in error as well as the stresses. It is interesting to note that when a patch is constructed according to Fig. 9.12(c) in which all elements are parallelograms all three forms of the patch test are once again satisfied.
Application of the patch test to an incompatible element
,ooo l,,o
T i
ooo Mesh 1
56.25
I"
10
"7 1000
187.50 Load B
b
a
Mesh 2
1000
Load A
Displacement at a
Beam theory
Mesh 1 (a)
Mesh 2
(b)
Mesh 2
Mesh 1
t 5o2s 187.50 56.25
T 2
l
Load B
Displacement at b
Load A
Load B
Load A
Load B
10.00 6.81 7.06 10.00 10.00
103.0 70.1 72.3 101.5 101.3
300.0 218.2 218.8 300.0 300.0
4050 2945 2954 4050 4050
Fig. 9.11 Performanceof the non-conforming quadrilateral in beam bending treated as plane stress. (a) Conforming linear quadrilateral. (b) Non-conforming quadrilateral.
Accordingly we can note that if any mesh is systematically refined by subdivision of each element into four elements whose sides are all along ~, r/lines in the original element with values of - 1 , 0, or 1 (i.e., by bisections) the mesh converges to constant jacobian approximations of the type shown in Fig. 9.12(c). Thus, in this special case the incompatible mode element satisfies a weak patch test and will converge. In general, however, it may be necessary to use a very fine discretization to achieve sufficient accuracy, and hence the element probably has no practical (or efficient) engineering use. A simple artifice to ensure that an element passes the patch test is to replace the derivatives of the incompatible modes by
a~
aN n -~ aN n
=
Jo
J(~, ~)
jo 1
-~
aNa
(9.16)
where J (~, O) is the determinant of the jacobian matrix J(~, 77) and J0 and J0 are the values of the inverse jacobian matrix and jacobian evaluated at the element centre (~ = 17 = 0). This ensures satisfaction of the patch test for all element shapes, and with this alteration of the algorithm the incompatible element proves convergent and quite accurate. 25
345
346 The patch test, reduced integration, and non-conforming elements 7
8
4
9
,L
9
(a)
1
(b)
1
(c)
9
5
,
,L
,
2
3
1
6
3
Fig. 9.12 Patch test for an incompatible element form. (a) Regular discretization. (b) Irregular discretization about node 5. (r Constant jacobian discretization about node 5.
An alternative approach which also passes the patch test constructs the derivatives used in the strains as
ON~
OX r
ON~
Oy ~
ON~ OX
ON~ Oy
1 fn ON~dn
~e
e
OX
1 In ON~d~2
~e
e
(9.17)
Oy
where ~e is the volume of the element. 26 Indeed, this form may also be used to deduce terms in the strain matrix of enhanced strain forms (e.g., see Sec. 10.5.3) and the justification of the modification follows from the mixed approach used there. When the shape functions (9.13) are inserted into (9.17) and the jacobian is constant (as it will be for any parallelogram shape
Higher order patch test- assessment of robustness 347 element) we immediately find that the integral term is zero. However, when the element has a non-constant jacobian which for the two-dimensional element has the form J(~, r / ) = Jo + J ~ + J,Tr/, where J0, J~, J~ are constants depending on nodal coordinates of the element, the integral term is non-zero. Thus, the effectiveness of the modification is clearly evident in producing elements which pass the constant stress patch test for all element shapes.
A higher order patch test may also be used to assess element 'robustness'. An element is termed robust if its performance is not sensitive to physical parameters of the differential equation. For example, the performance of many elements for solution of plane strain linear elasticity problems is sensitive to Poisson's ratio values near 0.5 (called 'near incompressibility'). Indeed, for Poisson ratios near 0.5 the energy stored by a unit volumetric strain is many orders larger than the energy stored by a unit deviatoric strain. Accordingly finite elements which exhibit a strong coupling between volumetric and deviatoric strains often produce poor results in the nearly incompressible range, a problem discussed further in Chapter 11. This may be observed using a four-noded element to solve a problem with a quadratic displacement field (i.e., a higher order patch test). If we again consider a pure bending example and an eight-element mesh shown in Fig. 9.13 we can clearly observe the deterioration of results as Poisson's ratio approaches a value of one-half. Also shown in Fig. 9.13 are results for the incompatible modes described in Sec. 9.8. It is evident that the response is considerably improved by adding these modes, especially if 2 x 2 quadrature is used. If we consider the regular mesh and 4-noded elements and further keep the domain constant and successively refine the problem using meshes of 8, 32, 128, and 512 elements, we observe that the answers do converge as guaranteed by the patch test. However, as shown in Fig. 9.14, the rate of convergence in energy for Poisson ratio values of 0.25 and 0.4999 is quite different. For 0.25 the rate of convergence is nearly a straight line for all meshes, whereas for 0.4999 the rate starts out quite low and approaches an asymptotic value of 2 as h tends towards zero. For v near 0.25 the element is called robust, whereas for v near 0.5 it is not. If we use selective reduced integration (which for the plane strain case passes strong patch tests) and repeat the experiment, both values of v produce a similar response and thus the element becomes robust for all values of Poisson's ratio less than 0.5. The use of higher order patch tests can thus be very important to separate robust elements from non-robust elements. For methods which seek to automatically refine a mesh adaptively in regions with high errors, as discussed in Chapter 14, it is extremely important to use robust elements.
In the preceding sections we have described the patch test and its use in practice by considering several example problems. The patch test described has two essential parts: (a) a consistency evaluation and (b) a stability check. In the consistencytest a set of linearly
348 The patch test, reduced integration, and non-conforming elements Incompatible-regular (2 x 2)(3 x 3)
1.0
~'~
Incompatible-distorted (3 x 3)
-,
0.8
Incompatibledistorted (2 x 2) Limit = 0.83
Bilinear-regular (2 x 2) ~ 90.6
#
(n x n) quadrature order
Bilinear-distorted (3 x 3) 0.4
0.2
0
0.1
0.2
0.3
0.4
0.5
V
l
5
)
,,
Regular mesh
Distorted mesh
,~
"-5
...-"-
x(u)
-----~ 5
Fig. 9.13 Plane strain 4-noded quadrilaterals with and without incompatible modes (higher order patch test for performance evaluation).
independent essential polynomials (i.e., all independent terms up to the order needed to describe the finite element model) is used as a solution to the differential equations and boundary conditions, and in the limit as the size of a patch tends to zero the finite element model must exactly satisfy each solution. We presented three forms to perform this portion of the test which we call forms A, B, and C. The use of form C, where all boundary conditions are the natural ones (e.g., tractions for elasticity) except for the minimum number of essential conditions needed to ensure a unique solution to the problem (e.g., rigid body modes for elasticity), is recommended to test consistency and stability simultaneously. Both one-element and more-than-one-
Concluding remarks 349 Element size In (h) 10-2
10 0
10-1
10 0
v = 0.4999 10-1 ss S
,-'" ~"
10-2
~
10_3
=
10-4
uJ
10 -5
~) SS
/"
,o v = 0.4999
ss SS
,~2 .o 9 1
s
v = 0.2500 v = 0.2500
9"
4 node (2 x 2) on all terms
4 node (2 x 2) on shear terms (1 x 1) on bulk terms
Fig. 9.14 Higher order patch test on element robustness (see Fig. 9.13)- convergence test under subdivision of elements.
element tests are necessary to ensure that the patch test is satisfied. With these conditions and assuming that the solution procedure used can detect any possible rank deficiencies the stability of solution is also tested. If no such condition is included in the program a stability test must be conducted independently. This can be performed by computing the number of zero eigenvalues in the coefficient matrix for methods that use a solution of linear equations to compute the finite element parameters, Q. Alternatively, the loading used for the patch solution may be perturbed at one point by a small value (say square root of the round-off limit, e.g., by 10 -8 for round-off of order 10 -15 ) and the solution tested to ensure that it does not change by a large amount. Once an element has been shown to pass all of the essential patch tests for both consistency and stability, convergence is assured as the size of elements tends to zero. However, in some situations (e.g., the nearly incompressible elastic problem) convergence may be very slow until a very large number of elements are used. Accordingly, we recommend that higher order patch tests be used to establish element robustness. Higher order patch tests involve the use of polynomial solutions of the differential equation and boundary conditions with the order of terms larger than the basic polynomials used in a patch test. Indeed, the order of polynomials used should be increased until the patch test is satisfied only in a weak sense (i.e., as h tends to zero). The advantage of using a higher order patch test, as opposed to other boundary value problems, is that the exact solution may be easily computed everywhere in the model. In some of the examples we have tested the use of incompatible function and inexact numerical integration procedures (reduced and selective integration). Some of these
350 The patch test, reduced integration, and non-conforming elements violations of the rules previously stipulated have proved justified not only by yielding improved performance but by providing methods for which convergence is guaranteed. We shall discuss in Chapter 11 some of the reasons for such improved performance.
9.1 A Type C patch test for a plane strain problem is to be performed using the single element shown in Fig. 9.15(a). Assume an element has the dimensions a = 15, b = 12, c = 10 with elastic properties E = 200, v = 0.25. Nodes 1 and 2 are placed on the x axis and Ul = Vl = v2 = 0 are applied as boundary restraints. (a) Compute all nodal forces necessary to compute the test for a stress state ax - 8 with all other stresses zero. (b) Compute the displacements u(x, y) and v(x, y) for the solution. (c) Use FEAPpv (or any other available program) to perform the test. Is it passed? 9.2 Solve Problem 9.1 for an axisymmetric geometryt with node 1 satisfying Vl = 0 and all other nodes free to displace. For err = 8 consider the cases: (a) Node 1 placed at r = 0. (b) Node 1 placed at r = 15. 9.3 Solve Problem 9.1 for a plane stress problem with an orthotropic material given by
LOt250 00 0]{ x /
-,50 "tSx,y,
100
0
0
75
Yx' y'
Let nodes 1 and 2 lie on the x axis with node 1 placed at the origin. (a) Compute the nodal forces acting on all nodes when the orthotropic axes are aligned as shown in Fig. 9.15(b) with 0 = 30 ~ and a single stress crx' = 5 is applied. (b) Compute the displacement field for the case Ul = Vl = u4 = 0. (c) Use FEAPpv (or any other available program) to perform the patch test. Is it passed? 9.4 The example described in Problem 9.1 is used to perform a patch test on an element with incompatible modes. The element matrix is given by
where
t Note: FEAPpv computes all axisymmetric arrays on a 1-radian sector in the 0 direction, thus avoiding the 2Jr factor in a complete ring sector.
Problems kJ
r-
()
4
q
3
j\y
I I X'
b Ii
_~(___()1 i
J r-
v-~
Z= Z ' I
(a) Element geometry
(b) Orthotropic axes
Fig. 9.15 One element patch test for 4-node quadrilateral. Problems9.1 to 9.4.
K
--
f~eBTDBu
d~2
C = f ~ BuTDBo~ dr2 e
v =
B~DB,~ dr2 e
The element passes the patch test for all constant stress states when a -- c; however, it fails when a # c. Suggest a correction which will ensure the patch test is satisfied. 9.5 Perform a patch test for the 8-node element shown in Fig. 9.16(a) for the assumed displacements u=0.1x
and
v=0
Let the origin be at the lower left comer of the element. The dimensions are a - b - 3 and c = 3.3. The material is linear isotropic elastic with E = 200 and v = 0.3 and plane strain conditions are assumed. (a) Use 2 • 2 gaussian quadrature to compute the element arrays and conduct a type A, B, and C patch test. (b) Repeat the calculation using 3 x 3 quadrature. (c) Consider a higher order displacement
u--O.2xy and and repeat (a) and (b). (d) Discuss any differences noted.
v-O
351
352
The patch test, reduced integration, and non-conforming elements
r
0
To
0
F
T0 -q
b {
0
b
L
0
2
(a)
J
O
O
0
{
a
.L
a
j
(b)
Fig. 9.16 One-element patch test for 8- and 9-node quadrilateral. Problems 9.5 to 9.8.
9.6 Perform a patch test for the 9-node element shown in Fig. 9.16(b) for the assumed displacements u - O . l x and v - 0 Let the origin be at the lower left comer of the element. The dimensions are a = b = e = 3, c = 3.3 and d = 3.15. Material is linear isotropic elastic with E = 200 and v -- 0.3 and plane strain conditions are assumed. (a) Use 2 x 2 gaussian quadrature to compute the element arrays and conduct a type A, B, and C patch test. (b) Repeat the calculation using 3 x 3 quadrature. (c) Set d = 3.4 and e = 2.9 and repeat (a) and (b). (d) Consider a higher order displacement
u=O.2xy and v = 0 and repeat (a) to (c). (e) Discuss any differences noted. 9.7 Solve Problem 9.5 for an axisymmetric geometry (replace x, y by r, z). (a) Let the inner radius be located at r = 0. (b) Let the inner radius be located at r = 3. 9.8 Solve Problem 9.6 for an axisymmetric geometry (replace x, y by r, z). (a) Let the inner radius be located at r = 0. (b) Let the inner radius be located at r = 3. 9.9 For the 4-element mesh configurations shown in Fig. 9.17 devise a set of patch.tests for a plane strain problem in which individual constant stress components are evaluated. Choose appropriate dimensions and isotropic elastic properties with v ~ 0. Use FEAPpv (or any available program) to perform Type A, B, and C tests for the arrays evaluated by (a) 1 x 1 quadrature and (b) 2 x 2 quadrature. Discuss your findings.
Problems 4
)3
4
)2
1
)3
(
8
5
1
(a) Regular
( (b) Distorted
Fig. 9.17 Multi-element patch test. Problems 9.9 to 9.13.
9.10 Each quadrilateral subregion in Fig. 9.17 is to be represented by an 8-node isoparametric serendipity element. Each side of the region has a length of 10 units. A higher order patch test of a plane strain problem with isotropic material with E = 200 and v = 0 has the displacements
u=-O.lxy and v=O.O5x 2
9.11 9.12 9.13 9.14
9.15
(a) Compute the state of stress for the given displacement field. (b) Select appropriate positions for nodes 5 to 8 for configurations (a) and (b). Specify appropriate nodal boundary conditions to prevent rigid body motion. (Hint: Place the origin of coordinates at the mid-point between nodes 1 and 4.) (c) Compute appropriate nodal forces and perform a Type C patch test for each configuration using 3 x 3 gaussian quadrature to compute arrays. Briefly discuss your findings. Solve Problem 9.10 using 9-node isoparametric lagrangian elements. Replace x by r and y by z and solve Problem 9.10 using 8-node isoparametric serendipity elements on an axisymmetric geometry. Replace x by r and y by z and solve Problem 9.12 using 9-node isoparametric lagrangian elements. Construct the generalization of the mesh configuration shown in Fig. 9.17 to a threedimensional problem. For E = 200, v = 0.25 and equal side lengths of 10 units use 8-node isoparametric hexagonal elements to perform a Type C patch test for the single stress crz = 5. Use both regular and distorted positions for the internal nodes. (Hint: Check that there are no negative jacobian determinants at the nodes of each element.) Select dimensions and use FEAPpv (or any available program) to verify the results shown in Fig. 9.8.
353
354
The patch test, reduced integration, and non-conforming elements 9.16 Select shown 9.17 Select shown
dimensions and use F E A P p v (or any available program) to verify the results in Fig. 9.9(a). dimensions and use F E A P p v (or any available program) to verify the results in Fig. 9.9(b) and (c).
1. B.M. Irons. Numerical integration applied to finite element methods. In Proc. Conf. on Use of Digital Computers in Structural Engineering, University of Newcastle, 1966. 2. G.P. Bazeley, Y.K. Cheung, B.M. Irons, and O.C. Zienkiewicz. Triangular elements in bending - conforming and non-conforming solutions. In Proc. 1st Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-66-80, pages 547-576, Wright Patterson Air Force Base, Ohio, Oct. 1966. 3. B.M. Irons and A. Razzaque. Experience with the patch test for convergence of finite elements. In A.K. Aziz, editor, The Mathematics of Finite Elements with Application to Partial Differential Equations, pages 557-587. Academic Press, New York, 1972. 4. B. Fraeijs de Veubeke. Variational principles and the patch test. Int. J. Numer. Meth. Eng., 8:783-801, 1974. 5. G. Sander. Bournes sup6rieures et in6rieures dans l'analyse matricielle des plates en flexiontorsion. Bull. Soc. Royale des Sci. de Lidge, 33:456-494, 1974. 6. E.R. de Mantes Oliveira. The patch test and the general convergence criteria of the finite element method. Int. J. Solids Struct., 13:159-178, 1977. 7. G. Strang. Variational crimes and the finite element method. In A.K. Aziz, editor, Proc. Foundations of the Finite Element Method, pages 689-710. Academic Press, New York, 1972. 8. G. Strang and G.J. Fix. An Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs, N.J., 1973. 9. F. Stummel. The limitations of the patch test. Int. J. Numer. Meth. Eng., 15:177-188, 1980. 10. J. Robinson et al. Correspondence on patch test. Fin. Elem~ News, 1:30-34, 1982. 11. R.L. Taylor, O.C. Zienkiewicz, J.C. Simo, and A.H.C. Chan. The patch test - a condition for assessing FEM convergence. Int. J. Numer. Meth. Eng., 22:39-62, 1986. 12. R.E. Griffiths and A.R. Mitchell. Non-conforming elements. In Mathematical Basis of Finite Element Methods, pages 41-69, Oxford, 1984. Clarendon Press. Inst. Math. and Appl. Conference series. 13. O.C. Zienkiewicz and R.L. Taylor. The finite element patch test revisited: a computer test for convergence, validation and error estimates. Comp. Meth. Appl. Mech. Eng., 149:523-544, 1997. 14. O.C. Zienkiewicz, S. Qu, R.L. Taylor, and S. Nakazawa. The patch test for mixed formulations. Int. J. Numer. Meth. Eng., 23:1873-1883, 1986. 15. W.X. Zhong. FEM patch test and its convergence. Technical Report 97-3001, Research Institute Engineering Mechanics, Dalian University of Technology, 1997 (in Chinese). 16. I. Babu~ka, T. Strouboulis, and C.S. Upadhyay. A model study of the quality of a posteriori error estimators for linear elliptic problems. Error estimation in the interior of patchwise uniform grids of triangles. Comp. Meth. Appl. Mech. Eng., 114:307-378, 1994. 17. B. Boroomand and O.C. Zienkiewicz. An improved REP recovery and the effectivity robustness test. Int. J. Numer. Meth. Eng., 40:3247-3277, 1997. 18. A. Ralston. A First Course in Numerical Analysis. McGraw-Hill, New York, 1965. 19. B.M. Irons and S. Ahmad. Techniques of Finite Elements. Horwood, Chichester, 1980. 20. D. Kosloff and G.A. Frasier. Treatment of hour glass patterns in low order finite element codes. Int. J. Numer. Anal. Meth. Geomech., 2:57-72, 1978.
References 355 21. T. Belytschko and W.E. Bachrach. The efficient implementation of quadrilaterals with high coarse mesh accuracy. Comp. Meth. Appl. Mech. Eng., 54:276-301, 1986. 22. N. Bi~ani~ and E. Hinton. Spurious modes in two dimensional isoparametric elements. Int. J. Numer. Meth. Eng., 14:1545-1557, 1979. 23. E.L. Wachspress. High order curved finite elements. Int. J. Numer. Meth. Eng., 17:735-745, 1981. 24. E.L. Wilson, R.L. Taylor, W.P. Doherty, and J. Ghaboussi. Incompatible displacement models. In S.T. Fenves et al., editor, Numerical and Computer Methods in Structural Mechanics, pages 43-57. Academic Press, New York, 1973. 25. R.L. Taylor, P.J. Beresford, and E.L. Wilson. A non-conforming element for stress analysis. Int. J. Numer. Meth. Eng., 10:1211-1219, 1976. 26. A. Ibrahimbegovic and E.L. Wilson. A modified method of incompatible modes. Comm. Numer. Meth. Eng., 7:187-194, 1991.
Mixed formulation and
constraints - complete field methods
The set of differential equations from which we start the discretization process will determine whether we refer to the formulation as mixed or irreducible. Thus if we consider an equation system with several dependent variables u written as [see Eqs (3.1) and (3.2)] ,A(u) = 0
in domain f2
(10.1 a)
on boundary F
(10. lb)
and B(u) = 0
in which none of the components of u can be eliminated still leaving a well-defined problem, then the formulation will be termed irreducible. If this is not the case the formulation will be called mixed. These definitions were given in Chapter 3 (p. 56). This definition is not the only one possible 1 but appears to the authors to be widely applicable 2, 3 if in the elimination process referred to we are allowed to introduce penalty functions. Further, for any given physical situation we shall find that more than one irreducible form is usually possible. As an example we shall consider the simple problem of heat conduction (or the quasiharmonic equation) to which we have referred in Chapters 3 and 7. In this we start with a physical constitutive relation defining the flux [see Eq. (7.5)] in terms of the potential (temperature) gradients, i.e., q = - k Vdp
q --
(qx) qy
The continuity equation can be written as [see Eq. (7.7)] (10.3)
Oqx + aqy VTq = -~x Oy ___ - Q
If the above equations are satisfied in f2 and the boundary conditions =~
onF~
are obeyed then the problem is solved.
or
qn=rtn
OnFq
(10.4)
Introduction
Clearly elimination of the vector q is possible and simple substitution of Eq. (10.2) into Eq. (10.3) leads to - V T(k Vtp) + Q = 0 in f2 (10.5) with appropriate boundary conditions expressed in terms of (p or its gradient. In Chapter 7 we showed discretized solutions starting from this point and clearly, as no further elimination of variables is possible, the formulation is irreducible. On the other hand, if we start the discretization from Eqs (10.2)-(10.4) the formulation would be mixed. An alternative irreducible form is also possible in terms of the variables q. Here we have to introduce a penalty form and write in place of Eq. (10.3) VTq + Q = 4)
(10.6)
O/
where ot is a penalty number which tends to infinity. Clearly in the limit both equations are the same and in general if ot is very large but finite the solutions should be approximately the same. Now substitution into Eq. (10.2) gives the single governing equation ~'(VTq) + --1 k_ 1q + V Q - 0
O/
(10.7)
which again could be used for the start of a discretization process as a possible irreducible form. 4 The reader should observe that, by the definition given, the formulations so far used in this book were irreducible. In subsequent sections we will show how elasticity problems can be dealt with in mixed form and indeed will show how such formulations are essential in certain problems typified by the incompressible elasticity example to which we have referred in Chapter 6. In Chapter 3 (Sec. 3.9) we have shown how discretization of a mixed problem can be accomplished. Before proceeding to a discussion of such discretization (which will reveal the advantages and disadvantages of mixed methods) it is important to observe that if the operator specifying the mixed form is symmetric or self-adjoint (see Sec. 3.9) the formulation can proceed from the basis of a variational principle which can be directly obtained for linear problems. We invite the reader to prove by using the methods of Chapter 3 that stationarity of the variational principle given below is equivalent to the differential equations (10.2) and (10.3) together with the boundary conditions (10.4):
lf~ qTk- lq dff2+ L qTvtp dr2 --J~ (pQ dff2 --fF (])OndF
H = ~
(10.8)
q
for (P = 4)
on F~
The establishment of such variational principles is a worthy academic pursuit and had led to many famous forms given in the classical work ofWashizu. 5 However, we also know (see Sec. 3.7) that if symmetry of weighted residual matrices is obtained in a linear problem then a variational principle exists and can be determined. As such symmetry can be established by inspection we shall, in what follows, proceed with such weighting directly and thus avoid some unwarranted complexity.
357
358
Mixed formulation and constraints - complete field methods
We shall demonstrate the discretization process on the basis of the mixed form of the heat conduction equations (10.2) and (10.3). Here we start by assuming that each of the unknowns is approximated in the usual manner by appropriate shape functions and corresponding unknown parameters. Thus, q m q] = Nqr
and
~b ~ ~ = N,(b
(10.9)
where r and ~b are the nodal (or element) parameters that have to be determined. Similarly the weighting functions are given by Vq "~ 'Vq --- Wq~Cl
and
v4, ~ O~ = Wo6qb
(10.10)
where 8r and 3~b are arbitrary parameters. Assuming that the boundary conditions for ~b = ~ are satisfied by the choice of the expansion, the weighted statement of the problem is, for Eq. (10.2) after elimination of the arbitrary parameters, WqT (k -1 q + Vt~) dr2 "-- 0 (10.11) and, for Eq. (10.3) and the 'natural' boundary conditions, - fn W~(vTq + Q) dr2 + f r W ~ ( q n -
qn)dr" - 0
(10.12)
q
The reason we have premultiplied Eq. (10.2) by k -1 is now evident as the choice Wq -- Nq
We = Nr
(10.13)
will yield symmetric equations [using Green's theorem to perform integration by parts on the gradient term in Eq. (10.12)] of the form [CA with A=
Jo
fl--O
NTk-INq dr2
CI { ~ } .= {ff12}
C-
(10.14)
Nq XTNr dr2 (10.15)
f2 -- ~ N ~ Q d Q + J ; N~/n dF q
This problem, which we shall consider as typifying a large number of mixed approximations, illustrates the main features of the mixed formulation, including its advantages and disadvantages. We note that: 1. The continuity requirements on the shape functions chosen are different. It is easily seen that those given for N~ can be Co continuous while those for Nq can be discontinuous in or between elements (C_ 1 continuity) as no derivatives of this are present. Alternatively,
Discretization of mixed forms- some general remarks 359
this discontinuity can be transferred to N~ (using Green's theorem on the integral in C) while maintaining Co continuity for Nq. This relaxation of continuity is of particular importance in plate and shell bending problems (see reference 6) and indeed many important early uses of mixed forms have been made in that context. 7-1~ 2. If interest is focused on the variable q rather than ~b, use of an improved approximation for this may result in higher accuracy than possible with the irreducible form previously discussed. However, we must note that if the approximation function for q is capable of reproducing precisely the same type of variation as that determinable from the irreducible form then no additional accuracy will result and, indeed, the two approximations will yield identical answers. Thus, for instance, if we consider the mixed approximation to the field problems discussed using a hnear triangle to determine N~ and piecewise constant Nq, as shown in Fig. 10.1, we will obtain precisely the same results as those obtained by the irreducible formulation with the same N~ applied directly to Eq. (10.5), providing k is constant within each element. This is evident as the second of Eqs (10.14) is precisely the weighted continuity statement used in deriving the irreducible formulation in which the first of the equations is identically satisfied. Indeed, should we choose to use a linear but discontinuous approximation form of Nq in the interior of such a triangle, we would still obtain precisely the same answers, with the additional coefficients becoming zero. This discovery was made by Fraeijs de Veubeke 11 and is called the principle of limitation, showing that under some circumstances no additional accuracy is to be expected from a mixed formulation. In a more general case where k is, for instance, discontinuous and variable within an element, the results of the mixed approximation will be different and on occasion superior. 2 Note that a Co continuous approximation for q does not fall into this category as it is not capable of reproducing the discontinuous ones.
OR
Constant q
+
Linear q
Linear
Linear Fig. 10.1 A mixed approximation to the heat conduction problem yielding identical results as the corresponding irreducible form (the constant k is assumed in each element).
360
Mixedformulation and constraints- complete field methods 3. The equations resulting from mixed formulations frequently have zero diagonal terms as indeed in the case of Eq. (10.14). We noted in Chapter 3 that this is a characteristic of problems constrained by a Lagrange multiplier variable. Indeed, this is the origin of the problem, which adds some difficulty to a standard gaussian elimination process used in equation solving. As the form of Eq. (10.14) is typical of many two-field problems we shall refer to the first variable (here el) as the primary variable and the second (here ~b) as the constraint variable. 4. The added number of variables means that generally larger size algebraic problems have to be dealt with. The characteristics so far discussed did not mention one vital point which we elaborate in the next section.
Despite the relaxation of shape function continuity requirements in the mixed approximation, for certain choices of the individual shape functions the mixed approximation will not yield meaningful results. This limitation is indeed much more severe than in an irreducible formulation where a very simple 'constant gradient' (or constant strain) condition sufficed to ensure a convergent form once continuity requirements were satisfied. The mathematical reasons for this difficulty are discussed by Babu~ka 12' 13 and Brezzi, 14 who formulated a mathematical criterion associated with their names. However, some sources of the difficulties (and hence ways of avoiding them) follow from quite simple reasoning. If we consider the equation system (10.14) to be typical of many mixed systems in which Clis the primary variable and q5 is the constraint variable (equivalent to a lagrangian multiplier), we note that the solution can proceed by eliminating Cl from the first equation and by substituting into the second to obtain (CTA-1C)~ -- --f2 + CTA-lfl
(10.16)
which requires the matrix A to be non-singular (or A~I ~ 0 for all ~ ~ 0). To calculate q5 it is necessary to ensure that the bracketed matrix, i.e.,
H
-
CTA-1C
(10.17)
is non-singular. Singularity of the H matrix will always occur if the number of unknowns in the vector el, which we call nq, is less than the number of unknowns n o in the vector (b. Thus for avoidance of singularity nq >_ nep (10.18) is necessary though not sufficient as we shall find later. The reason for this is evident as the rank of the matrix (10.17), which needs to be n 0, cannot be greater than n q , i.e., the rank of A -1.
Stability of mixed approximation. The patch test In some problems the matrix A may well be singular. It can normally be made nonsingular by addition of a multiple of the second equation, thus changing the first equation to A = A + FCC T fl
--
fl -+- vCf2
where V is an arbitrary number. We note that the solution to (10.14) is not changed by this modification. Although both the matrices A and CC T are singular their combination A~ should not be, providing we ensure that for all vectors (1 7(= 0 either A~I 7(= 0
or
CT~l~ 0
In mathematical terminology this means that A is non-singular in the null space of CC T. The requirement of Eq. (10.18) is a necessary but not sufficient condition for nonsingularity of the matrix H. An additional requirement evident from Eq. (10.16) is C(b 7(= 0
for all
(b :~ 0
If this is not the case the solution would not be unique. The above requirements are inherent in the Babugka-Brezzi condition previously mentioned, but can always be verified algebraically.
10.3.2 Locking The condition (10.18) ensures that non-zero answers for the variables fl are possible. If it is violated locking or non-convergent results will occur in the formulation, giving near-zero answers for fl [see Chapter 3, Eq. (3.137) ft.]. To show this, we shall replace Eq. (10.14) by its penalized form: [A Cx
-
~]{~} I
{fl} =
t"2
with ot ---~ c~ and I = identity matrix
(10.19)
Elimination of ~b leads to (A + c~CCT)~I = fl + otCf2
(10.20)
As c~ -+ c~ the above becomes simply (CCX)~l = Cf2
(10.21)
Non-zero answers for Cl should exist even when 1'2 is zero and hence the matrix CC T must be singular. This singularity will always exist if nq > nr but can exist also when nq = nep if the rank of C is less than nq. The stability conditions derived on the particular example of Eq. (10.14) are generally valid for any problem exhibiting the standard Lagrange multiplier form. In particular the necessary count condition will in many cases suffice to determine element acceptability; however, final conclusions for successful elements which pass all count conditions and the full test to ensure consistency must be evaluated by rank tests on the full matrix.
361
362 Mixedformulation and constraints- complete field methods In the example just quoted Cl denotes flux and (b temperature and perhaps the concept of locking was not clearly demonstrated. It is much more definite where the first primary variable is a displacement and the second constraining one is a stress or a pressure. There locking is more evident physically and simply means an occurrence of zero displacements throughout as the solution approaches a numerical instability limit. This unfortunately will happen on occasion.
10.3.3 The patch test The patch test for mixed elements can be carried out in exactly the way we have described in the previous chapter for irreducible elements. As consistency is easily assured by taking a polynomial approximation for each of the variables, only stability needs generally to be investigated. Most answers to this can be obtained by simply ensuring that count condition (10.18) is satisfied for any isolated patch on the boundaries of which we constrain the maximum number of primary variables and the minimum number of constraint variables. 15
Example 10.1: A single-element test.
In Fig. 10.2 we illustrate a single-element test for two possible formulations with Co continuous Nr (quadratic) and discontinuous Nq, assumed to be either constant or linear within an element of triangular form. As no values of (1 can here be specified on the boundaries, on the patch (which is here simply that of a single element) we shall fix a single value of ~ only, as is necessary to ensure uniqueness. A count shows that only one of the formulations, i.e., that with linear flux variation, satisfies condition (10.18) and therefore may be acceptable (but will always determine elements which fail!).
Example 10.2: Asingle-element test with Co, (1and ~5. In Fig. 10.3 we illustrate a similar patch test on the same element but with identical Co continuous shape functions specified for both (1and ~ variables. This example shows satisfaction of the basic condition of Eq. (10.18)
nq < n(h
+ (a)
Test failed n~=6- 1=5
nq= 2
/ +
(~
/ (b)
nq = 6
~
/
(~
/-
-~
0
\
~
/ /
/
Restrained nq>nr
Test passed (but results equivalent to irreducible form)
n~=6- 1=5
Fig. 10.2 Single-elementpatchtest for mixedapproximationsto the heatconductionproblemwith discontinuousflux q assumed. (a) QuadraticCo, ~; constantq. (b) QuadraticCo, ~; linearq.
Two-field mixed formulation in elasticity
nq> n~
nq= 12
~ n ~ = 6 - 1=5
w Restrained
Fig. 10.3 As Fig. 10.2 but with quadratic Co continuous q. and therefore is apparently a permissible formulation. The permissible formulation must always be subjected to a numerical rank test. Clearly condition (10.18) will need to be satisfied and many useful conclusions can be drawn from such counts. These eliminate elements which will not function and on many occasions will give guidance to elements which will. Even if the patch test is satisfied occasional difficulties can arise, and these are indicated mathematically by the Babu~ka-Brezzi condition already referred to. 16 These difficulties can be due to excessive continuity imposed on the problem by requiring, for instance, the flux condition to be of Co continuity class. In Fig. 10.4 we illustrate some cases in which the imposition of such continuity is physically incorrect and therefore can be expected to produce erroneous (and usually highly oscillating) results. In all such problems we recommend that the continuity be relaxed on all surfaces where a physical discontinuity can occur. We shall discuss this problem further in Sec. 10.4.3.
In all the previous formulations of elasticity problems in this book we have used an irreducible formulation, using the displacement u as the primary variable. In earlier chapters,
Fig. 10.4 Some situations for which Co continuity of flux q is inappropriate. (a) Discontinuous change of material properties. (b) Singularity.
363
364
Mixed formulation and constraints- complete field methods
the virtual work principle was used to establish the equilibrium conditions and was written as
f
'eTo.d~-
f
' u T b d f 2 - f r 6uTidF = 0
(10.22)
t
where t are the tractions prescribed on I"t and with o" = De
(10.23)
as the constitutive relation (omitting here initial strains and stresses for simplicity). We recall that statements such as Eq. (10.22) are equivalent to weighted residual forms (see Chapter 3) and in what follows we shall use these frequently. In the above the strains are related to displacement by the matrix operator ,S introduced in Chapter 2, giving e - - SU 6e - ,S 6u
(10.24)
with the displacement expansions constrained to satisfy the prescribed displacements on Fu. This is, of course, equivalent to Galerkin-type weighting. With the displacement u approximated as u ~ fi -- Nufi
(10.25)
the required stiffness equations were obtained in terms of the unknown displacement vector and the solution obtained. It is possible to use mixed forms in which either o" or e, or, indeed, both these variables, are approximated independently. We shall discuss such formulations below.
10.4.2 The u - o r mixed form In this we shall assume that Eq. (10.22) is valid but that we approximate or independently as
o" ~ & = No&
(10.26)
and approximately satisfy the constitutive relation o" = D,Su
(10.27)
which replaces (10.23) and (10.24). The approximate integral form is written as o~ C~O.T( S U - D -lo") dr2 - 0
(10.28)
where the expression in the brackets is simply Eq. (10.27) premultiplied by D- 1 to establish symmetry and 8o" is introduced as a weighting variable. Indeed, Eqs (10.22) and (10.28) which now define the problem are equivalent to the stationarity of the functional 1-Irm =
L
O.T,Sudr2 -
1LO.TD-lo.d~- f uTbd~"2-JF t
uTidF
(10.29)
Two-field mixed formulation in elasticity
where the boundary displacement UmU
is enforced on Fu, as the reader can readily verify. This is the well-known HellingerReissnerl7, 18 variational principle, but, as we have remarked earlier, it is unnecessary in deriving approximate equations. Using Nuafi inplace of ~u B~fi = ,SNu6fl
in place of ~
No.adr in place of 3o" we write the approximate equations (10.28) and (10.22) in the standard form [see Eq. (10.14)] [CA with
F A -- - L NO.D T -1 No.dS2
OC] { u } - {ff'2}
(10.30)
C -- £ N~TBdS2
(lO.31) fl -
0
f2 - - / 2 N,Vbd~ + f r N [ t d F t
In the form given above the Nu shape functions have still to be of Co continuity, though No. can be discontinuous. However, integration by parts of the expression for C allows a reduction of such continuity and indeed this form has been used by Herrmann 7' 19, 2o for problems of plates and shells.
10.4.3 Stability of two-field approximation in elasticity (u-~r) Before attempting to formulate practical mixed approach approximations in detail, identical stability problems to those discussed in Sec. 10.3 have to be considered. For the u-or forms it is clear that cr is the primary variable and u the constraint variable (see Sec. 10.2), and for the total problem as well as for element patches we must have as a necessary, though not sufficient, condition no- > n.
(10.32)
where no. and n, stand for numbers of degrees of freedom in appropriate variables. In Fig. 10.5 we consider a two-dimensional plane problem and show a series of elements in which No. is discontinuous while Nu has Co continuity. We note again, by invoking the Veubeke 'principle of limitation', that all the elements that pass the single-element test here will in fact yield identical results to those obtained by using the equivalent irreducible form, providing the D matrix and the determinant of the jacobian matrix are constant within each element. They are therefore of little interest. However, we note in passing that the Q 4/8, which fails in a single-element test, passes that patch test for assemblies of two or more elements, and performs well in many circumstances. We shall see later that this is equivalent to using four-point Gauss, reduced integration (see Sec. 11.5), and as we have mentioned in Chapter 9 such elements will not always be robust.
365
366
Mixed formulation and constraints- complete field methods T 3/3 I
Tl/3 I
no=3 nu=3X2-3=3 (pass)
1o '41 X
no=3 nu=4 X 2 - 3 = 5 (fail)
T 3/6 I
=3x3=9 =3x2-3=3 (pass)
Io
I
o
X
=3x3=9 =6x2-3=9 (pass)
X X
o =3x3=9 =8x2-3=13 (fail)
I Q 4/8 I o
I Q 4/9 I o
X
X
X
X
X
X
o =4 x 3 = 12 =8x2-3=13 (fail)
o
X X
o =4 x 3 = 12 =9x2-3=15 (fail)
Two-element Q 4/8 assembly patch test o o
X
X
X
X
X
X
X
X
o o no=8x3=24 nu= 13 x 2 - 3 = 23 (pass)
Fig. 10.5 Elasticity by the mixed o--u formulation. Discontinuous stress approximation. Single-element patch test. No restraint on ~ variables but three ~ degrees of freedom restrained on patch. Test condition no > n o [x denotes ~- (3 DOF)and o the 6 (2 DOF)variables].
It is of interest to note that if a higher order of interpolation is used for tr than for u the patch test is still satisfied, but in general the results will not be improved because of the principle of limitation. We do not show the similar patch test for the Co continuous No assumption but state simply that, similarly to the example of Fig. 10.3, identical interpolation of No and Nu is acceptable from the point of view of stability. However, as in Fig. 10.4, restriction of excessive continuity for stresses has to be avoided at singularities and at abrupt material property change interfaces, where only the normal and tangential tractions are continuous. The disconnection of stress variables at corner nodes can only be accomplished for all the stress variables. For this reason an alternative set of elements with continuous stress nodes at element interfaces can be introduced (see Fig. 10.6). 21 In such elements excessive continuity can easily be avoided by disconnecting only the direct stress components parallel to an interface at which material changes occur. It should be noted that even in the case when all stress components are connected at a mid-side node such elements do not ensure stress continuity along the whole interface. Indeed, the amount of such discontinuity can be useful as an error measure. However, we observe that for the linear element [Fig. 10.6(a)] the interelement stresses are continuous in the mean.
Two-field mixed formulation in elasticity
+
linear
(a)
u linear
_
T,
(~nt~'
/ T "r'x(~xx y Onn~ ~YY
(b)
/ ~tt
Fig. 10.6 Elasticity by the mixed cr-u formulation. Partially continuous ~r (continuity at nodes only). (a) ~r linear, u linear. (b) Possible transformation of interface stresses with ~rtt disconnected.
It is, of course, possible to derive elements that exhibit complete continuity of the appropriate components along interfaces and indeed this was achieved by Raviart and Thomas 22 in the case of the heat conduction problem discussed previously. Extension to the full stress problem is difficult 23 and as yet such elements have not been successfully noted.
Example 10.3: Pian-Sumihara rectangle. Today very few two-field elements based on interpolation of the full stress and displacement fields are used. One, however, deserves to be mentioned. We begin by first considering a rectangular element where interpolations may be given directly in terms of cartesian coordinates. A 4-node plane rectangular element with side lengths 2a in the x direction and 2b in the y direction, shown in Fig. 10.7, has ,~kY A
b
i i
:F-I
X
(Xo, yo)
b
.J_ "T Fig. 10.7 Geometry of rectangular ~ - u element.
367
368
Mixed formulation and constraints- complete field methods
displacement interpolation given by 4
U = ~ Na (x, y)fla a=l
The shape functions are given by
1(
Nl(X, y) = -~ 1 1(
N3(x, y) = -~ 1 +
X - - X o ) ( Y 1- - Y o ) a
x-xo)(
1+
a
1(
b
Y-Yo) b
" N2(x, y) = ~ 1+ ' 1(
" Nn(x y) -- -~ 1
'
X - - X o ) ( Y 1- - Y o ) a
x-xo)(
'
a
Id-
b
Y-Yo) b
in which x0 and Y0 are the cartesian coordinates at the element centre. The strains generated from this interpolation will be such that
8x - -
/71 d - t l 2 y ;
e y - - /73 +
/']4X;
Yxy --- 175+ 06x + O7Y
rlj are expressed in terms of ~. For isotropic linear elasticity problems these strains will lead to stresses which have a complete linear polynomial variation in each element (except for the special case when v = 0). Here the stress interpolation is restricted to each element individually and, thus, can be discontinuous between adjacent elements. The limitation principle restricts the possible choices which lead to different results from the standard displacement solution. Namely, the approximation must be less than a complete linear polynomial. To satisfy the stability condition given by Eq. (10.18) we need at least five stress parameters in each element. A viable choice for a five-term approximation is one which has the same variation in each element as the normal strains given above but only a constant shear stress. Accordingly,
where
{ax}
[i
tTy
"-"
"Cxy
0 0 1 0
0 1
Y-Yo
0
0
X -- X0
0
0
_
Otl
]
0l 2 0l 3
Ol4 015
Indeed, this approximation satisfies Eq. (10.18) and leads to excellent results for a rectangular element.
Example 10.4: Pian-Sumihara quadrilateral. We now rewrite the formulation given in Example 10.3 to permit a general quadrilateral shape to be used. The element coordinate and displacement field are given by a standard bilinear isoparametric expansion 4
x= ~ a=l
4
Na (~, r/)~a and fi = y ~ Na (~, 7)fia a=l
where now ma(~, ~) = 1(1 + ~a~)(1 +/~a~)
in which ~a and r/a are the values of the parent coordinates at node a.
Two-field mixed formulation in elasticity The problem remains to deduce an approximation for stresses for the general quadrilateral element. Here this is accomplished by first assuming stresses E on the parent element (for convenience in performing the coordinate transformation the tensor form is used, see Appendix B) in an analogous manner as the rectangle above:
~--](~: T]).__ F~]~ ]~r/] IO/1 "~'-0~4r] ' L][]r/~ ][]r/r/J -013
013 ] 13/2 + 0/5~
In the above the parent normal stresses again produce constant and bending terms while shear stress is only constant. These stresses are then transformed to cartesian space using o" = TTN(~, r/)T It remains now only to select an appropriate form for T. The transformation must 1. produce stresses in cartesian space which satisfy the patch test (i.e., can produce constant stresses and be stable); 2. be independent of the orientation of the initially chosen element coordinate system and numbering of element nodes (invariance requirement). Pian and Sumihara 24 use a constant array (to preserve constant stresses) deduced from the jacobian matrix at the centre of the element. Accordingly, with
Ox
1,O,ll ,o,11 Jo =
lJo,21
Jo,22J
=
Ox
o~
Oy Oy
~
q,o=o
the elements of the jacobian matrix at the centre are given by [see Eq. (5.11)] 1 1 J0,11 -" "~X a ~ a Jo,12 "- -~X a T]a 1 1 J0,21 -- -~Ya ~a J0,22 "- -~Ya Oa Using T = J0 gives the stresses (in matrix form)
O'y "rxy
--
I~2 (~3
"nt-
i
Jg, 12~
1
J02,21 r/
LJo, 12 Jo,21 t ]
Jo, 12 Jo,22~J
where the parameters (~i, i -- 1, 2, 3, replace the transformed quantities for the constant part of the stresses. This approximation satisfies the constant stress condition (Condition 1) and can also be shown to satisfy the invariance condition (Condition 2). The development is now complete and the arrays indicated in Eq. (10.31) may be computed. We note that the integrals are computed exactly for all quadrilateral elements (with constant D) using 2 x 2 gaussian quadrature. An alternative to the above definition for T is to use the transpose of the jacobian inverse at the centre of the element (i.e., T = JOT). This has also been suggested recently by several authors as an invariant transformation. However, as shown in Fig. 10.8, the sensitivity to element distortion is much greater for this form than the original one given by Pian and Sumihara for the above two-field approximation. The other two options (e.g., T = J~ and T = Jo 1) do not satisfy the frame invariance requirement, thus giving elements which depend on the orientation of the element with respect to the global coordinates.
369
370
Mixed formulation and constraints- complete field methods
21
\
i 0.5 ~1~, -9- i - ~
5 5 E=75, v=0
1.0 I
~- 0.5
~1 v I
0.8
.......-.-.-o 0.6
0.4 -
'i~ ,
o P-S:J [] P-S: J-inverse O Q-4
0.2 >-'0.., 0 "-0,.r ~-E3.......43 .... r .... O .......... 0 I I 0 1 2 a
~ I 3
. ........c] <> I 4
Fig. 10.8 Pian-Sumihara quadrilateral (P-S) compared with displacement quadrilateral (Q-4). Effect of element distortion (Exact -- 1.0).
It is, of course, possible to use an independent approximation to all the essential variables entering the elasticity problem. We can then write the three equations (10.22), (10.23), and (10.24) in their weak form as fl~r
-- t r ) d r 2 = 0
~~Or T(~II -- r dr2 = 0
(10.33)
t
where u - fi on Fu is enforced.t The variational principle equivalent to Eq. (10.33) is k n o w n by the name of H u - W a s h i z u 5 (see Problem 10.1). Introducing the approximations u ~ fi = Nufi
cr ~ & = N ~ #
and
e ~ @ - NE~
(10.34)
with corresponding 'variations' (i.e., the Galerkin form Wu = Nu, etc.) into Eq. (10.33), and writing the approximating equations in a similar fashion as we have in the previous t It is possible to include the displacement boundary conditions in Eq. (10.33) as a natural rather than imposed constraint; however, most finite element applications of the principle are in the form shown.
Three-field mixed formulations in elasticity
[A C
section yields an equation system of the following form: CT
0
o"
0
ET
u
{
--
(10.35)
f2 f3
where A = ~ NTDN~ dr2; fl = t " 2 - 0 ;
f3-
/.
E = f~ NTB dr2;
/.
C:
- f
NTNo d~2 (10.36)
LNuTbdf2+ /_ NuTidF d~l.
dl"
t
The reader will observe that in this section we have developed all the approximations directly without using a variational principle. In Problem 10.2 we suggest that the reader show the equivalence of a development from the variational principle.
10.5.2 Stability condition of three-field approximation (u-or-e) The stability condition derived in Sec. 10.3 [Eq. (10.18)] for two-field problems, which we later used in Eq. (10.32) for the simple mixed elasticity form, needs to be modified when three-field approximations of the form given in Eq. (10.35) are considered. Many other problems fall into a similar category (for instance, plate bending) and hence the conditions of stability are generally useful. The requirement now is that n~ + nu > n,r
-
(10.37)
n,r > nu
This was first stated in reference 25 and follows directly from the two-field criterion as shown below. The system of Eq. (10.35) can be 'regularized' by adding yE times the third equation to the second, with y being an arbitrary constant. We now have
c
yEE T ET
8" O
=
{ fl } t"2 + gEf3 f3
On elimination of r using the first of the above we have
[(~/EET-CTA-1C),ET, ~] (0"), -" (f2"~-~/Ef3-CTA-lfl/f3 From the two-field requirement [Eq. (10.18)] it follows that we require no > nu
for the equation system to have a solution. To establish the second condition we rearrange Eq. (10.35) as
(10.38)
371
372
Mixed formulation and constraints - complete field methods
This again can be regularized by adding multiples ?'C and y E T of the third of the above equations to the first and second respectively obtaining YETCT,
YETE E
C T,
fi ~
--
f3 + vETf2 f2
By partitioning as above it is evident that we require (10.39)
n~ + nu > no
We shall not discuss in detail any of the possible approximations to the 6-or-u formulation or their corresponding patch tests as the arguments are similar to those of two-field problems. In some practical applications of the three-field form the approximation of the second and third equations in (10.33) is used directly to eliminate all but the displacement terms. This leads to a special form of the displacement method which has been called a B (B-bar) form.26, 27 In the 171form the shape function derivatives are replaced by approximations resulting from the mixed form. We shall illustrate this concept with an example of a nearly incompressible material in Sec. 11.4.
10.5.3 The U--O'--Sen form. Enhanced strain formulation In the previous two sections the general form and stability conditions of the three-field formulation for elasticity problems are given in Eqs (10.32) and (10.37). Here we consider a special case of this form from which several useful elements may be deduced. In the special form considered the strain approximation is split into two parts: one the usual displacement-gradient term and, second, an added or enhanced strain part. Accordingly, we write : S U "4- ~en ~ --- ~ ( S U ) + ~ e n (10.40) Substitution into Eq. (10.33) yields the weak forms as
~
~ (SU) T
f
(D ( S u +
E:en) -- or)
dg2 = 0
3~eT (D (SU + ~en) -- or) dg2 -- 0 (10.41) / ~ ~iorT~en d ~ = 0
/, (Su)Toraft.-/,uTbd~-/r ,uT{dF= 0 t
where, as before, u = fi is enforced on Fu. We can directly discretize Eq. (10.41) by taking the following approximations u ~ fi - Nufi
or ,~ & = N~rr
~en ~'~ ~en :
Nen~en
(10.42)
with corresponding expressions for variations. Substituting the approximations into Eq. (10.41) yields the discrete equation system
[AoT T'1}oo i]{'en} "0 {
(10.43)
Three-field mixed formulations in elasticity where A =
NenD Nen d~2;
LT
C -- -
NenN~rdr2;
K--/fBTDBdff2;
G :
/T
NenD B d ~
f,-~NTbdff2+fr NTtdF
fl = t " 2 = 0 ;
(10.44)
t
In this form there is only one zero diagonal term and the stability condition reduces to the single condition nu +nen _> n~r (10.45) Further, the use of the strains deduced from the displacement interpolation leads to a matrix which is identical to that from the irreducible form and we have thus included this in Eq. (1 0.44) as K.
Example 10.5: Simo-Rifai quadrilateral. An enhanced strain formulation for application to problems in plain elasticity was introduced by Simo and Rifai. 28 The element has 4 nodes and employs isoparametric interpolation for the displacement field. The derivatives of the shape functions yield a form
~Na aNa
=
ax,a(Yb) -4;-bx,a(Yb)~ "[-Cx,a(Yb)rl) J(~, o)
ay,a(Xb) -J- by,a(Xb)~ + Cy,a(Xb)r]
where aa, ba and C a depend on the nodal coordinates, and the jacobian determinant for the 4-node quadrilateral is given byt d e t J = J(~, O) = Jo + J ~ + Jorl The enhanced strains are first assumed in the parent coordinate frame and transformed to the cartesian frame using a transformation similar to that used in developing the PianSumihara quadrilateral in Example 10.4. Due to the presence of the jacobian determinant in the strains computed from the displacements (as well as the requirement to later pass the patch test for constant stress states) the enhanced strains are computed from ~en --
I
J (~, O)
TTE(~, r/)T
where E-
F
LE.~
Er
E,,TJ
In matrix form this may be written as
•y
---
Yxy en
J(~i O)
T2 2TilT12
T2
T12T22
]{} , |
2T21T222 TilT22 + T12T21J
Eoo
2Er176
t In general, the determinantof the jacobian for the two-dimensionalLagrangefamilyof elementswill not contain the term with the product of the highest order polynomial, e.g., ~r/for the 4-node element, ~2172for the 9-node element, etc.
373
374
Mixed formulation and constraints-complete field methods The parent strains (strains with components in the parent element frame) are assumed as
Eo0
=
0
2E~0
0
0
772 773
~ 77 /74
0
The above is motivated by the fact that the derivatives of the shape functions with respect to parent coordinates yields
ONa = ~
ONo = oo
a~ + b~ 0
a,7 + b,7~
and these may be combined to form strains in the usual manner, but in the parent frame. Thus, by design, the above enhanced strains are specified to generate complete polynomials in the parent coordinates for each strain component. References 29 and 30 discuss the relationship between the design of assumed stress elements using the two-field form and the selection of enhanced strain modes so as to produce the same result.
Remarks
1. The above enhanced strains are defined so that the C array is identically zero for constant assumed stresses in each element. 2. Parent normal strains have linearly independent terms added. However, the assumed parent shear strains are linearly dependent. Due to this linear dependence the final sheafing strain will usually be nearly constant in each element. Accordingly, to be more explicit, normal strains are enhanced while sheafing strain is de-enhanced. Since the C array vanishes, the equation set to be solved becomes fl [GAT GI /Ee~n/~If3/ and in this form no additional count conditions are apparently needed. The solution may be accomplished partly at the element level by eliminating the equation associated with the enhanced strain parameters. Accordingly, K*fi = f ~ where K* = K - GTA-1G
and
f~ -- t"3 - GTA-lfl
The sensitivity of the enhanced strain element to geometric distortion is evaluated using the problem shown in Fig. 10.9. The transformation from the parent to the global frame is assessed using T = J0 and T = Jo T. These are the only options which maintain frame invafiance for the element. As observed in Fig. 10.9 the results are now better using the inverse transpose. Since the stress and strain are conjugates in an energy sense, this result could be anticipated from the equivalence relationship
1L tr Te dr2 -
E = ~
-2
~2TEd [7
Complementary forms with direct constraint
where E is energy and D denotes the domain of the element in the parent coordinate system (i.e., the bi-unit square for a quadrilateral element). The performance of the enhanced element is compared to the Pian-Sumihara element for a shear loading on the mesh shown in Fig. 10.10. In Fig. 10.11 the convergence results for various order meshes are shown for linear elastic, plane strain conditions with: (a) E = 70 and v = 1/3 and (b) for E = 70 and v = 0.499995. The results shown in Fig. 10.11 clearly show the strong dependence of the displacement formulation on Poisson's ratio - namely the tendency for the element to lock for values which approach the incompressibility limit of v = 1/2. On the other hand, the performance of both the enhanced strain and the Pian-Sumihara element are nearly insensitive to the value of Poisson's ratio selected, with somewhat better performance of the enhanced element on coarse meshing.
In the introduction to this chapter we defined the irreducible and mixed forms and indicated that on occasion it is possible to obtain more than one 'irreducible' form. To illustrate this in the problem of heat transfer given by Eqs (10.2) and (10.3) we introduced a penalty function ot in Eq. (10.6) and derived a corresponding single governing equation (10.7) given in terms of q. This penalty function here has no obvious physical meaning and served simply as a device to obtain a close enough approximation to the satisfaction of the continuity of flow equations. On occasion it is possible to solve the problem as an irreducible one assuming a priori that the choice of the variable satisfies one of the equations. We call such forms directly constrained and obviously the choice of the shape function becomes difficult. We shall consider two examples.
1.0 0.8 0 v
0.6 i
0.4 'X b"
0.2
~-~>..<>. " ~ , 0
I 1
o S-R:J [] S-R: J-inverse ~ Q-4
" " C , . . . . ~, . . . . . . . . . . I 2 a
C, I 3
I 4
Fig. 10.9 Simo-Rifai enhanced strain quadrilateral (S-R) compared with displacement quadrilateral (Q-4). Effect of element distortion (Exact - 1.0).
375
376
Mixed formulation and constraints- complete field methods
"l
48
L~
I
J~
16
44
!' t t t |
,,.__1
F
F/8 1=/4 F/4 FI4 FI8
,
(b) Mesh and nodal loads
(a) Geometry Fig. 10.10 Mesh with 4 x 4 elements for shear load.
35 30
f
35-
o
30-
25 . ~ , , , , , , , , , , - - - - - ~ O
25
w,"
.-- 20 ~" 15 ~, v
o
0
~
I 10
I I I I 20 30 40 50 n-elements/side (v = 1/3)
20
o o S-R o.-- - .----o P-S ............o Q-4
=" 15
0 ............~ Q-4
10 0
o S-R
D - - - - - - - o P-S
10
I 60
0
~ . - 0 .......~ ................0 ....................................
0
I
I
I
I
I
I
10 20 30 40 50 60 n- elements/side (v = 0.499995)
Fig. 10.11 Convergence behaviour for: (a) v = 1/3; (b) v = 0.499995.
The complementary heat transfer problem
In this we assume a priori that the choice of q is such that it satisfies Eq. (10.3) and the natural boundary conditions V T q -- - Q in f2
and
qn = q Tn : On o n ~ q
(10.46)
where n is the unit normal to the boundary. Thus we only have to satisfy the constitutive relation (10.2), i.e., k - l q + V~p = 0 in f2
with
~p = ~ on FO
(10.47)
A weak statement of the above is f
,~qT(k-aq + V ~ ) dr2 - f r 'qn(q~ - ~) dF - 0 r
(10.48)
Complementary forms with direct constraint 377
in which 3 q n - - 3qTn represents the variation of normal flux on the boundary. Use of Green's theorem transforms the above into ~SqTk-lqdf2-f
VTBq~pdf2+fr 8 q n @ d F + f r ' q n ~ p d F - - 0 r
(10.49)
q
If we further assume that ~7 T*q ~ 0 in if2 and 3qn = 0 on Fq, i.e., that the weighting functions are simply the variations of q, the equation reduces to
'qn~dF--O
L ,qTk-lqdf2 + f r
s
(10.50)
This is in fact the variation of a complementary flux principle
5q k
q da +
qn~ dr'
(10.51)
Numerical solutions can obviously be started from either of the above equations but the difficulty is the choice of the trial function satisfying the constraints. We shall return to this problem in Sec. 10.6.2.
The complementary elastic energy principle
In the elasticity problem specified in Sec. 10.4 we can proceed similarly, assuming stress fields which satisfy the equilibrium conditions both on the boundary 1-'tand in the domain f2. Thus in an analogous manner to that of the previous example we impose on the permissible stress field the constraints which we assume to be satisfied by the approximation identically, i.e., (10.52) ,sT0. + b -- 0 in f2 and t - t on I" t Thus only the constitutive relations and displacement boundary conditions remain to be satisfied, i.e., D-10. - ,Su - 0 in f2 and u - ~ on Fu (10.53) The weak statement of the above can be written as /_ 8o'T(D-lor -- Su)dr2 + f
8 t T ( u - fi)dF = 0
(10.54)
all' u
d~2
which on integration by Green's theorem gives f ,~rTD-IordQ + f ( S T , o - ) T u d Q - - f r , t T ~ d F - f r u
,tTudF = 0
(10.55)
t
Again assuming that the test functions are complete variations satisfying the homogeneous equilibrium equation, i.e., 8T30" =
0 in f2
and
3t -- 0 on Ft
(10.56)
we have as the weak statement L 30.TD-10. aft2 - ~
,tTfi dI" = 0
(10.57)
u
The corresponding complementary energy variational principle is FI = ~1 ~ o.TD-1 0. dr2 -
tT~ dF
(10.58)
u
Once again in practical use the difficulties connected with the choice of the approximating function arise but on occasion a direct choice is possible. 31
378
Mixedformulation and constraints-complete field methods
10.6.2 Solution using auxiliary functions Both the complementary forms can be solved using auxiliary functions to ensure the satisfaction of the constraints.
Example 10.6: Heat transfer solution by potential function. In the lem it is easy to verify that the homogeneous equation
heat transfer prob-
VTq -- -~x Oy = 0 Oqx + OqY
(10.59)
is automatically satisfied by defining a function ~p such that qx =
00
00
qy "-
Oy
(10.60)
Ox
Thus we define and
q= gTr+q0
6q = s
(10.61)
where q0 is any flux chosen so that VTq0 = - Q
(10.62)
and 12 =
0y
0x
(10.63)
the formulations of Eqs (10.50) and (10.51) can be used without any constraints and, for instance, the stationarity
I-[ =
falg(Z~0+ q0)Tk- 1(gO + q0) df2 - j~F (01~) -~s
~>dF
(10.64)
will suffice to so formulate the problem (here s is the tangential direction to the boundary). The above form will require shape functions for ~ satisfying Co continuity.
Example 10.7: Elasticity solution by Airy stress function. In the elasticity problem a two-dimensional form can be obtained by the use of the so-called Airy stress function ~.32 Now the equilibrium equations 0ax
~ T o" "Jr-
b-
+~oy +bx -fix OTxy -+---~y +by
/
= 0
(10.65)
are identically solved by choosing (10.66)
a = C~+~ro where 12 "--
2
02
Oy 2'
OX 2'
02 1 OX Oy
(10.67)
Concluding remarks - mixed formulation or a test of element 'robustness'
and tr0 is an arbitrary stress chosen so that ,STtr0 + b = 0
(10.68)
Again the substitution of (10.66) into the weak statement (10.57) or the complementary variational problem (10.58) will yield a direct formulation to which no additional constraints need be applied. However, use of the above forms does lead to further complexity in multiply connected regions where further conditions are needed. The reader will note that in Chapter 7 we encountered this in a similar problem in torsion and suggested a very simple procedure of avoidance (see Sec. 7.5). The use of this stress function formulation in the two-dimensional context was first made by de Veubeke and Zienkiewicz 33 and Elias, 34 but the reader should note that now with second order operators present, C1 continuity of shape functions is needed in a similar manner to the problems which we have to consider in plate bending (see reference 6). Incidentally, analogies with plate bending go further here and indeed it can be shown that some of these can be usefully employed for other problems. 35
The mixed form of finite element formulation outlined in this chapter opens a new range of possibilities, many with potentially higher accuracy and robustness than those offered by irreducible forms. However, an additional advantage arises even in situations where, by the principle of limitation, the irreducible and mixed forms yield identical results. Here the study of the behaviour of the mixed form can frequently reveal weaknesses or lack of 'robustness' in the irreducible form which otherwise would be difficult to determine. The mixed approximation, if properly understood, expands the potential of the finite element method and presents almost limitless possibilities of detailed improvement. Some of these will be discussed further in the next two chapters, and others in references 6 and 36.
10.1 Show that the stationarity of the variational principle given by
t
where u - fi on Fu is equivalent to Eq. (10.33). 10.2 Using the variational principle of Problem 10.1 with the approximations (10.34) show that the stationarity condition gives (10.35) and (10.36). 10.3 Show that the variational principle given by stationarity of
lien --- ~ 1 (SU+r
(SU-q-r
d~+fo'T~end~
-- ~ uTbd" - frt uT'dF with u = fi enforced on Fu is equivalent to Eq. (10.41).
379
380
Mixed formulation and constraints- complete field methods 10.4 For the rectangular element shown in Fig. 10.7 develop the expressions for T]i for the Pian-Sumihara element described in Sec. 10.4.3. For an isotropic elastic material and a plane stress problem compute the expressions for the stresses which result from the strains (these are those of the displacement model described in Chap. 6). How do these differ from those assumed for the mixed element? 10.5 For the enhanced strain formulation described in Sec. 10.5.3 use the constant stress patch test for a plane strain problem to show that ~en • O. Show that a necessary condition to satisfy this requirement is
f
Nend~-O e
10.6 Generalize the Simo-Rifai quadrilateral given as Example 10.5 in Sec. 10.5.3 for a three-dimensional solid modelled by 8-node hexahedral elements. 10.7 Generalize the Simo-Rifai quadrilateral given as Example 10.5 in Sec. 10.5.3 for an axisymmetric geometry. 10.8 A plane stress problem has the geometry shown in Fig. 10.11 and is loaded by a uniformly distributed shear traction (i.e., ty = const.). Use FEAPpv to solve the problem using a series of 3-node triangular meshes. The first mesh should be as shown with each quadrilateral divided into two triangles. Consider two values for the elastic properties: (a) E = 70, v = 1/3 and (b) E = 70, v = 0.499995. Let the thickness of the slab be one unit. Next, perform the solution using 4-node quadrilaterals based on (a) the displacement solution described in Chap. 6; (b) the Simo-Rifai enhanced element described in Sec. 10.5.3. Plot the displacement convergence for the top and bottom points at the loaded end. Plot contours for displacement and principal stresses. Repeat the calculations assuming plane strain conditions. Briefly discuss your findings.
1. S.N. Atluri, R.H. Gallagher, and O.C. Zienkiewicz, editors. Hybrid and Mixed Finite Element Methods. John Wiley & Sons, New York, 1983. 2. O.C. Zienkiewicz, R.L. Taylor, and J.A.W. Baynham. Mixed and irreducible formulations in finite element analysis. In S.N. Atluri, R.H. Gallagher, and O.C. Zienkiewicz, editors, Hybrid and Mixed Finite Element Methods, pages 405-431. John Wiley & Sons, 1983. 3. I. Babu~ka and J.E. Osborn. Generalized finite element methods and their relations to mixed problems. SlAM J. Num. Anal., 30:510-536, 1983. 4. R.L. Taylor and O.C. Zienkiewicz. Complementary energy with penalty function in finite element analysis. In R. Glowinski, E.Y. Rodin, and O.C. Zienkiewicz, editors, Energy Methods in Finite Element Analysis, Chapter 8. John Wiley & Sons, Chichester, 1979. 5. K. Washizu. Variational Methods in Elasticity and Plasticity. Pergamon Press, New York, 3rd edition, 1982. 6. O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method for Solid and Structural Mechanics. Butterworth-Heinemann, Oxford, 6th edition, 2005. 7. L.R. Herrmann. Finite element bending analysis of plates. In Proc. 1st Conf. Matrix Methods in Structural Mechanics, AFFDL-TR-66-80, pages 577-602, Wright-Patterson Air Force Base, Ohio, 1965.
References 381 8. K. Hellan. Analysis of elastic plates in flexure by a simplified finite element method. Technical Report Civ. Eng. Series 46, Acta Polytechnica Scandinavia, Trondheim, 1967. 9. R.S. Dunham and K.S. Pister. A finite element application of the Hellinger-Reissner variational theorem. In Proc. 1st Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-66-80, Wright Patterson Air Force Base, Ohio, Oct. 1966. 10. R.L. Taylor and O.C. Zienkiewicz. Mixed finite element solution of fluid flow problems. In R.H. Gallagher, G.E Carey, J.T. Oden, and O.C. Zienkiewicz, editors, Finite Elements in Fluids, volume 1, Chapter 4, pages 1-20. John Wiley & Sons, 1982. 11. B. Fraeijs de Veubeke. Displacement and equilibrium models in finite element method. In O.C. Zienkiewicz and G.S. Holister, editors, Stress Analysis, Chapter 9, pages 145-197. John Wiley & Sons, Chichester, 1965. 12. I. Babu~ka. Error bounds for finite element methods. Numer. Math., 16:322-333, 1971. 13. I. Babu~ka. The finite element method with lagrangian multipliers. Numer. Math., 20:179-192, 1973. 14. F. Brezzi. On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers. Rev. Fran9aise d'Automatique Inform. Rech. Opdr., Ser. Rouge Anal. Numgr., 8(R-2):129-151, 1974. 15. O.C. Zienkiewicz, S. Qu, R.L. Taylor, and S. Nakazawa. The patch test for mixed formulations. Int. J. Numer. Meth. Eng., 23:1873-1883, 1986. 16. J.T. Oden and N. Kikuchi. Finite element methods for constrained problems in elasticity. Int. J. Numer. Meth. Eng., 18:701-725, 1982. 17. E. Hellinger. Die allgemeine Aussetze der Mechanik der Kontinua. In E Klein and C. Muller, editors, Encyclopedia der Mathematishen Wissnschafien, volume 4. Tebner, Leipzig, 1914. 18. E. Reissner. On a variational theorem in elasticity. J. Math. Phys., 29(2):90-95, 1950. 19. L.R. Herrmann. Finite element bending analysis of plates. J. Eng. Mech., ASCE, 94(EM5): 13-25, 1968. 20. L.R. Herrmann and D.M. Campbell. Finite element analysis for thin shells. J. AIAA, 6:18421847, 1968. 21. O.C. Zienkiewicz and D. Lefebvre. Mixed methods for FEM and the patch test. Some recent developments. In E Murat and O. Pirenneau, editors, Analyse Mathematique of Application. Gauthier Villars, Paris, 1988. 22. P.A. Raviart and J.M. Thomas. A mixed finite element method for second order elliptic problems. In Lect. Notes in Math., no. 606, pages 292-315. Springer-Verlag, Berlin, 1977. 23. D.N. Arnold, E Brezzi, and J. Douglas. PEERS, a new mixed finite element for plane elasticity. Japan J. Appl. Math., 1:347-367, 1984. 24. T.H.H. Pian and K. Sumihara. Rational approach for assumed stress finite elements. Int. J. Numer. Meth. Eng., 20:1685-1695, 1985. 25. O.C. Zienkiewicz and D. Lefebvre. Three field mixed approximation and the plate bending problem. Comm. Appl. Num. Meth., 3:301-309, 1987. 26. T.J.R. Hughes. Generalization of selective integration procedures to anisotropic and non-linear media. Int. J. Numer. Meth. Eng., 15:1413-1418, 1980. 27. J.C. Simo, R.L. Taylor, and K.S. Pister. Variational and projection methods for the volume constraint in finite deformation plasticity. Comp. Meth. Appl. Mech. Eng., 51:177-208, 1985. 28. J.C. Simo and M.S. Rifai. A class of mixed assumed strain methods and the method of incompatible modes. Int. J. Numer. Meth. Eng., 29:1595-1638, 1990. 29. U. Andelfinger and E. Ramm. EAS-elements for two-dimensional, three-dimensional, plate and shell structures and their equivalence to HR-elements. Int. J. Numer. Meth. Eng., 36:1311-1337, 1993. 30. M. Bischoff, E. Ramm, and D. Braess. A class of equivalent enhanced assumed strain and hybrid stress finite elements. Comput. Mech., 22:443-449, 1999.
382
Mixed formulation and constraints-complete field methods 31. C. Loubignac, G. Cantin, and C. Touzot. Continuous stress fields in finite element analysis. J. AIAA, 15:1645-1647, 1978. 32. S.P. Timoshenko and J.N. Goodier. Theory of Elasticity. McGraw-Hill, New York, 3rd edition, 1969. 33. B. Fraeijs de Veubeke and O.C. Zienkiewicz. Strain energy bounds in finite element analysis by slab analogy. J. Strain Anal., 2:265-271, 1967. 34. Z.M. Elias. Duality in finite element methods. Proc. Am. Soc. Civ. Eng., 94(EM4):931-946, 1968. 35. R.V. Southwell. On the analogues relating flexure and displacement of flat plates. Quart. J. Mech. Appl. Math., 3:257-270, 1950. 36. O.C. Zienkiewicz, R.L. Taylor, and P. Nithiarasu. The Finite Element Method for Fluid Dynamics. Butterworth-Heinemann, Oxford, 6th edition, 2005.
Incompressible problems, mixed
methods and other procedures of solution
We have noted earlier that the standard displacement formulation of elastic problems fails when Poisson's ratio v becomes 0.5 or when the material becomes incompressible. Indeed, problems arise even when the material is nearly incompressible with v > 0.4 and the simple linear approximation with triangular elements gives highly oscillatory results in such cases. The application of a mixed formulation for such problems can avoid the difficulties and is of great practical interest as nearly incompressible behaviour is encountered in a variety of real engineering problems ranging from soil mechanics to aerospace engineering. Identical problems also arise when the flow of incompressible fluids is encountered. In this chapter we shall discuss fully the mixed approaches to incompressible problems, generally using a two-field manner where displacement (or fluid velocity) u and the pressure p are the variables. Such formulation will allow us to deal with full incompressibility as well as near incompressibility as it occurs. However, what we will find is that the interpolations used will be very much limited by the stability conditions of the mixed patch test. For this reason much interest has been focused on the development of so-called stabilized procedures in which the violation of the mixed patch test (or Babugka-Brezzi conditions) is artificially compensated. A part of this chapter will be devoted to such stabilized methods.
The main problem in the application of a 'standard' displacement formulation to incompressible or nearly incompressible problems lies in the determination of the mean stress or pressure which is related to the volumetric part of the strain (for isotropic materials). For this reason it is convenient to separate this from the total stress field and treat it as an independent variable. Using the 'vector' notation of stress, the mean stress or pressure is given by
p - - 5 (O'x + Cry + o-z)
~mTo 9
(11.1)
384 Incompressibleproblems, mixed methods and other procedures of solution where m for the general three-dimensional state of stress is given by
m--J1,
1,
1,
0,
0,
0] T
For isotropic behaviour the 'pressure' is related to the volumetric strain, ev, by the bulk modulus of the material, K. Thus, e v - - e x -at- e y + E z ---
mTe = P
(11.2)
For an incompressible material K = ~ (v = 0.5) and the volumetric strain is simply zero. The deviatoric strain e a is defined by 1 ~mev -- (I - l m m T ) e
E d --E-
=
Ide
(11.3)
where Id is a deviatoric projection matrix which also proves useful in problems with more general constitutive relations. 1 In isotropic elasticity the deviatoric strain is related to the deviatoric stress by the shear modulus G as tr d = Ido" - - 2 G I o e d
= 2G (Io - ~ m m l T) e
(11.4)
where the diagonal matrix -2 I0=
1
2
1 1 1
-
is introduced because of the vector notation. A deviatoric form for the elastic moduli of an isotropic material is written as Dd = 2G (I0 - l m m x)
(11.5)
for convenience in writing subsequent equations. The above relationships are but an alternate way of determining the stress-strain relations shown in Chapters 2 and 6, with the material parameters related through G= K=
E 2 (1 + v) E
(11.6)
3(1 - 2 v )
and indeed Eqs (11.4) and (11.2) can be used to define the standard D matrix in an alternative manner.
In the mixed form considered next we shall use as variables the displacement u and the pressure p.
Two-field incompressible elasticity (u-p form) Now the equilibrium equation (10.22) is rewritten using (11.4), treating p as an independent variable, as f
6eTDae dr2 + s 6eTmp d r 2 - s 6uTb d r 2 - f r 6uVidP = 0
(11.7)
t
and in addition we shall impose a weak form of Eq. (11.2), i.e., f
Sp
mTe-- P ] d~2 = 0
(11.8)
with e = Su. Independent approximation of u and p as and
u~fi=Nufi
p ~ ~ = Np[~
(11.9)
immediately gives the mixed approximation in the form EA
_v {o}= {ff,}
<,,.,o,
where
A V --
s
C s Np ~ N p dfa; fl =
s
NTb df~ +
J;
(11.11) NTi d r
f2 = 0
t
We note that for incompressible situations the equations are of the 'standard' form, see Eq. (10.14) with V = 0 (as K = oo), but the formulation is useful in practice when K has a high value (or v -+ 0.5). A formulation similar to that above and using the corresponding variational theorem was first proposed by Herrmann 2 and later generalized by Key 3 for anisotropic elasticity. The arguments concerning stability (or singularity) of the matrices which we presented in Sec. 10.3 are again of great importance in this problem. Clearly the mixed patch condition about the number of degrees of freedom now yields [see Eq. ( 10.18)] nu > n p (11.12) and is necessary for prevention of locking (or instability) with the pressure acting now as the constraint variable of the lagrangian multiplier enforcing zero volumetric strain. In the form of a patch test this condition is most critical and we show in Figs 11.1 and 11.2 a series of such patch tests on elements with Co continuous interpolation of u and either discontinuous or continuous interpolation of p. For each we have included all combinations of constant, linear and quadratic functions. In the test we prescribe all the displacements on the boundaries of the patch and one pressure variable as it is well known that in fully incompressible situations pressure will be indeterminate by a constant for the problem with all boundary displacements prescribed.t *Alternatively, it is possible to omit all boundary conditions on pressure if one displacement with a component normal to the boundary is allowed to exist.
385
386
Incompressible problems, mixed methods and other procedures of solution The single-element test is very stringent and eliminates most continuous pressure approximations whose performance is known to be acceptable in many situations. For this reason we attach more importance to the assembly test and it would appear that the following elements could be permissible according to the criteria of Eq. (11.12) (indeed all pass the B-B condition fully):
Triangles: T6/1; T10/3; T6/C3 Quadrilaterals" Q9/3; Q8/C4; Q9/C4 We note, however, that in practical applications quite adequate answers have been reported with Q4/1, Q8/3 and Q9/4 quadrilaterals, although severe oscillations of p may occur. If full robustness is sought the choice of the elements is limited. 4 It is unfortunate that in the present 'acceptable' list, the linear triangle and quadrilateral are missing. This appreciably restricts the use of these simplest elements. A possible and indeed effective procedure here is not to apply the pressure constraint at the level of a single element but on an assembly. This was done by Herrmann in his original presentation 2 where four elements were chosen for such a constraint as shown in Fig. 11.3(a). This composite 'element' passes the single-element (and multiple-element) patch tests but apparently so do several others fitting into this category. In Fig. 11.3(b) we show how a single triangle can be internally subdivided into three parts by the introduction of a central node. This coupled with constant pressure on the assembly allows the necessary count condition to be satisfied and a standard element procedure applies to the original triangle treating the central node as an internal variable. Indeed, the same effect could be achieved by the introduction of any other internal element function which gives zero value on the main triangle perimeter. Such a bubble function can simply be written in terms of the area coordinates (see Chapter 4) as
IT 3/1 I
IT 6/1 I
IT 6/3 I
IT 10/31
nu=O np=O
=0 =0 (pass)
=0 =2 (fail)
=2 =2 (pass)
(pass)
IQ
I
I Q 8/4 1 9
A
I Q 8/31 9
A
LQ 9/41 9
9 zx
nu=O np=O (a)
(pass)
A
o
=0 =3 (fail)
zx
zx
=0 =2 (fail)
zx
zx w
=2 =3 (fail)
I Q 9/3 I 9
A
o zx
=2 =2 (pass)
Fig. 11.1 Incompressible elasticity u-p formulation. Discontinuous pressure approximation. (a) Single-element patch tests, u variable (o restrained, o free) 2 DOF, p variable (A restrained,/x free) 1 DOE
Two-field incompressible elasticity (u-p form)
I T3/11
[T6/1 I
nu= 2 np=5
= 7 x 2 = 14 =5 (pass)
(fail)
IQ4/' I
w
[
w
w
~
IQ8/4 J
= 19x2=38 =17 (pass)
[ Q 9/3 ]
-9x2-18 -11
(fail)
(pass)
IQ9/41
IQ8/31 m,
14
=15
(fail)
d==
=7x2 =17 (fail)
= 5 x 2 = 10
nu= 2 np=3
IT ~o/31
IT 6/3 I
m,
9
9
A
"A )A
(b)
n u = 5 X 2 = 10 np=11 (fail)
w
= 9 x 2 = 18 =15 (pass)
,w
Fig. 11.1 (Cont.) Incompressible elasticity u-p formulation. Discontinuous pressure approximation. (b) Multiple-element patch tests.
L 1L2L3. However, as we have stated before, the degree of freedom count is a necessary but not sufficient condition for stability and a direct rank test is always required. In particular it can be verified by algebra that the conditions stated in Sec. 10.3 are not fulfilled for this triple subdivision of a linear triangle (or the case with the bubble function) and thus C~ = 0 for some non-zero values of indicating instability.
387
388 Incompressible problems, mixed methods and other procedures of solution
9 o u variable (restrained, free) 2 DOF A ~ p variable (restrained, free) 1 DOF I T3/03 I
I T6/06 1
i T6/03 I
nu=O np=2 (fail)
=0 =5 (fail)
=0 =2 (fail)
I Q4/04 I
I Q8/08 I
I Q8/04 I
I Q9/04.,.1
o
(a)
nu=O np=3 (fail)
=0 =7 (fail)
=0 =3 (fail)
=2 =3 (fail)
T3/c3 !
I To/co I
I T6/c3 I
nu=2 np=6 (fail)
=7x2= 14 =18 (fail)
=7x2= 14 =6 (pass)
I Q4/04 I
I Q 8/04 I
Z
I Q 9/04 ..I A
(b)
nu=2 np=8 (fail)
=5x2= 10 =8 (pass)
=9x2= 18 =8 (pass)
Fig. 11.2 Incompressibleelasticity u-p formulation. Continuous (Co) pressure approximation. (a) Singleelement patch tests. (b) Multiple-element patch tests.
Two-field incompressible elasticity (u-p form) 389
(a)
..i-
nu= 2
(b)
w
nu = 2
np=O
w
np=0 (Bubble function)
(c)
w
nu=2
w
np=2
OR (Bubble function) w
w
OR ubble function)+,
(d)
nu=2
~
np=2
Fig. 11.3 Somesimplecombinationsof lineartrianglesand quadrilateralsthat passthe necessarypatchtest counts. Combinations(a), (c),and (d) are successfulbut (b)is still singularand not usable.
390
Incompressibleproblems, mixed methods and other procedures of solution
~r
~
~Load Only vertical movement ,-o possiblefor ," ~ no volume ,," change
Triangle 1
9
I~~/2 / Z,/ / / ,/ / / ,/,/ / /~ t
a
s
t
s
t
s
t
t
s
)-~}~ Only horizontal movement possible for no volume change Triangle 2
(np
Fig. 11.4 Locking (zero displacements) of a simple assembly of linear triangles for which incompressibility is - n u - 24). fully required In Fig. 11.3(c) we show, however, that the same concept can be used with good effect for Co continuous p.5 Similar internal subdivision into quadrilaterals or the introduction of bubble functions in quadratic triangles can be used, as shown in Fig. 11.3(d), with success. The performance of all the elements mentioned above has been extensively discussed 6-11 but detailed comparative assessment of merit is difficult. As we have observed, it is essential to have nu > n p but if near equality is only obtained in a large problem no meaningful answers will result for u as we observe, for example, in Fig. 11.4 in which linear triangles for u are used with the element constant p. Here the only permissible answer is of course u -- 0 as the triangles have to preserve constant volumes. The ratio nu / np which occurs as the field of elements is enlarged gives some indication of the relative performance, and we show this in Fig. 11.5. This approximates to the behaviour of a very large element assembly, but of course for any practical problem such a ratio will depend on the boundary conditions imposed. We see that for the discontinuous pressure approximation this ratio for 'good' elements is 2-3 while for Co continuous pressure it is 6-8. All the elements shown in Fig. 11.5 perform very well, though two (Q4/1 and Q9/4) can on occasion lock when most boundary conditions are on u. Example 11.1: Simple triangle with bubble- MINI element. In Fig. 11.3(c) we indicate that the simple triangle with Co linear interpolation and an added bubble for the displacements u together with continuous Co linear interpolation for the pressure p satisfied the count test part of the mixed patch test and, verifying the consistency condition, can be used with success. 5 Here we consider this element further to develop some understanding about its performance at the incompressible limit. The displacement field with the bubble is written in hierarchical form as U ~, !1 -- ~
N a !1a Jr" Nbub tlbub a
(11.13)
Two-field incompressible elasticity (u-p form) (a)
T6/1
3,=4 2 "Y" / / / / / / / / ~ / / / / / / / / / ~
".m2"//////////////////~/////
"/////
(
A y=3
T10/3
Q9/3
"/~'/////~/////'/~'/////~'/////
~ t
y = 2.66
"/~////////7~////////2"Y'/////
A Q9/4
y= 2
T6B1/3
ZX
O ZX
ZX
0 t ZX
O
ZX
y=2
A
~Y////////~'/////////~'/////
~Y'///////,/~'////////2"~'/////
(b) T6/3C
h,
z~t:////////7~////////~L2,'////
T3B1/3C
y= 8
Q9/4C
y=8
,..,
~///////7~////////z/lYt2z////
y=6 zz~,'////////////////z-z~v////
Fig. 11.5 The freedom index or infinite patch ratio for various u - p elements for incompressible elasticity (y = nu/np). (a) Discontinuous pressure. (b) Continuous pressure. B-bubble, C-continuous.
where here Nbub -- L 1 L 2 L 3
(11.14)
Ua are nodal parameters of displacement and Llbub are parameters of the hierarchical bubble function. The pressures are similarly given by p ~/~ = ~ a
N~Pa
(11.15)
391
392
Incompressibleproblems, mixed methods and other procedures of solution
where Pa are nodal parameters of the pressure. In the above the shape functions are given by (e.g., see Eqs (4.26) and (4.29)) 1
Na -- La = 2A (aa + bax -!-Cay)
(11.16)
where ba = Yb -- Yc;
aa = XbYc -- XcYb;
Ca =
Xc -- Yb
b, c are cyclic permutations of a and 1 X1 Yl] 2A = det 1 X2 Y2 = al + a2 + a3 1 X3 Y3 The derivatives of the shape functions are thus given by ONa
ba
Ox
2A
ONa __ C__5_a 2A Oy
and
Similarly the derivatives of the bubble are given by ONbub 8x 8Nb~b Oy
1
2A 1
2A
(blL2L3 + b2L3L1 W b3L1L2) (ClL2L3 + c2L3L1 + c3L1L2)
r0a0]
0]
The strains may be expressed in terms of the above and the nodal parameters ast E -=. Za
"~
a
Ca f i a + ba
LCa
2A
Ca flb~b LCa ba
(11.17)
where again b, c are cyclic permutations of a. Substituting the above strains into Eq. (11.11) and evaluating the integrals give All A
A12
A13
A32
A33
0 0 0
0
0
Abb
A21 A22 A23
__.
A~I
(11.18)
where G [(4babb + 3CaCb) [(3baCb -- 2Cabb)
(3Cabb -- 2baCb)] (3babb + 4CaCb)J
Aab -- ~
G Abubbub = 2160A
[ (4bTb + 3cTc) b Tc
bTc ] (3bTb + 4cTc)J
and
b = [bl,
b2,
b3] T
and
c = [Cl,
C2,
C3] T
t At this point it is also possible to consider the term added to the derivatives to be enhancedmodes and delete the bubble mode from displacement terms.
Three-field nearly incompressible elasticity (u-p-ev form) 393 Note in the above that all terms except Abubbub are standard displacement stiffnesses for the deviatoric part. Similarly,
C --
Cll C21 C31
C12 C22 C32
C13 ] C23 / C33 |
(11.19)
Cbubl Cbub2 Cbub3J
'Ebb]
where
Cab = g r
120
In all the above arrays a and b have values from 1 to 3 and b u b denotes the bubble mode. We note that the bubble mode is decoupled from the other entries in the A array - it is precisely for this reason that the discontinuous constant pressure case shown in Fig. 11.3(b) cannot be improved by the addition of the internal parameters associated with fibub. Also, the parameters fibub are defined separately for each element. Consequently, we may perform a partial solution at the element levell2 to obtain the set of equations in the form Eq. (11.10) where now
rAil A12 A131 rCll C12 C13] A = |A21 A22 A23|; C= |C21 C22 C23|; ]A31 A32 A33J LC31 C32 C33J with Vob =
and
ba Ca]F11
2--A ~
L~'=I r=2
3 A2 [(3bTb-F- 4cTc) -bXc
"1" = l O G c L
rV l V= []/21
/bb/
V12 V13] V22 V23 V31 V32 V33
J
(11.20a)
r
--bTc ] (4bXb+ 3eTc )
(ll.20b)
in which c -- 12 (bTb)2 + 25 (bTb)(cTc) + 12 (cTc) 2 -- (bTc) 2 The reader may recognize the V array given above as that for the two-dimensional, steady heat equation with conductivity k = 7- and discretized by linear triangular elements. The direct reduction of the bubble matrix Abubbub as given above leads to a full matrix 7-. Some numerical experiments including the above formulation are presented in Sec. 11.7.
!iii ii
iiiiiiiiiiiiiiiiii!iii
A direct approximation of the three-field form leads to an important method in finite element solution procedures for nearly incompressible materials which has sometimes been called the B-bar method. The methodology can be illustrated for the nearly incompressible isotropic problem. For this problem the method often reduces to the same two-field form previously discussed. However, for more general anisotropic or inelastic materials and in
394 Incompressibleproblems, mixed methods and other procedures of solution finite deformation problems the method has distinct advantages as are discussed in reference 1. The usual irreducible form (displacement method) has been shown to 'lock' for the nearly incompressible problem. As shown in Sec. 11.3, the use of a two-field mixed method can avoid this locking phenomenon when properly implemented (e.g., using the Q9/3 two-field form). Below we present an alternative which leads to an efficient and accurate implementation in many situations. For the development shown we shall assume that the material is isotropic linear elastic but it may be extended easily to include anisotropic materials. Assuming an independent approximation to ev and p we can formulate the problem by use of Eq. (11.7) and the weak statement of relation (11.2) written as f6p[m TSu-ev] d~--0
(11.21)
f 3e,, [Keo - p] dr2 = 0
(11.22)
and If we approximate the u and p fields by Eq. (11.9) and eo ~ ~v = Noffo
(11.23)
we obtain a mixed approximation in the form of Sec. (10.5.3) but now only for p and eo
[
A C CT 0 0 .... E T
0 -E H
]{} {} fit P ~v
:
fl f2 f3
(11.24)
where A, C, fl, f2 are given by Eq. (11.11) and H = / ~ N~KNv dg2;
E = f N~Npdf2;
1'3=0
(11.25)
For completeness we give the variational theorem whose first variation gives Eqs (11.7), (11.21) and (11.22). First we define the strain deduced from the standard displacement approximation as Cu = S u ,~ Bfi (11.26) The variational theorem is then given as
rI=~l f~ (euTDdeu + evKev)d~2+
f
p (mTcu - e v ) d g 2 (11.27)
- ~ u T b d f 2 - ~r uTidF
Example 11.2: An enhanced strain triangle. In Example 11.1 we presented a two-field formulation using continuous u and p approximations together with an added hierarchical bubble mode to the displacements. For more general applications this form is not the most convenient. For example, if transient problems are considered the accelerations will also involve the bubble mode and affect the inertial terms. We will also find in the Sec. 11.7 that use of the above bubble is not fully effective in eliminating pressure oscillations in
Three-field nearly incompressible elasticity (u-p-~v form) 395 solutions. An alternative form is discussed in which we use a three-field approximation involving u, p and ev discussed above, together with an enhanced strain formulation as discussed in Sec. 10.5.3. The enhanced strains are added to those computed from displacements as (11.28)
-- ~u -~- ~e
in which 8e represents a set of enhanced strain terms. The internal strain energy is represented by = 1 (~TDa ~ + evKev) (11.29)
W(~, ev)
Using the above notation a Hu-Washizu-type variational theorem for the deviatoric-spherical split may be written as I-lnw = ~
[W(~, eo) + p (mT~ -- ev) + crT (eu -- ~)] dr2 + I-Iex t
(11.30)
where FIex t represents the terms associated with body and traction forces. After substitution for the mixed enhanced strain the last term simplifies to
/s2 crT (eu -- ~) d~2 = -- ~ o'T ee d~2
(11.31)
Taking variations with respect to u, p, ev, ~e and cr the principle yields the weak form ~Flnw = / ~ 8uTB T [Da~ + mp] dr2 + ~I"[ex t t/~G
q-~'~T[Dd~-q-mp--o'] d~2"q-~'o'T~ed~'-O Equal order interpolation with shape functions N are used to approximate u, p and to as u~fi-Nfi
p ~ p = N~
(11.33)
e~ ~ ~ = N ~ However, only approximations for u and p are Co continuous between elements. The approximation for ev may be discontinuous between elements. The stress ~r in each element is assumed constant. Thus, only the approximation for Se remains to be constructed in such a way that the third equation in (10.41) is satisfied. For the present we shall assume that this approximation may be represented by
Se ~ ge = Be&e
(11.34)
so that the terms involving cr and its variation in Eq. (11.32) are zero and thus do not appear in the final discrete equations. With the above approximations, Eq. (11.32) may be evaluated as
Aue Cu AeuAeeCeO
Auu Cux o
0
Cex
0
-E
0
-E x
H
fi
He P
~v
fl
=
0
t"2
f3
(11.35)
396 Incompressibleproblems, mixed methods and other proceduresof solution where Auu = A, Cu = C, fi, E and H are as defined in Eqs (11.11), (11.25) and /,
Aue = _/o BDdBe d~ -- AeT Aee - " / ~ BeDdne d~2
(1 1.36)
Ce = J~ BemN d~ Since the approximations for eo and ~e are discontinuous between elements we can again perform a partial solution for ev and &e using the second and fourth row of (11.35). After eliminating these variables from the first and third equation we again, as in Example 11.1, obtain a form identical to Eq. (11.10). As an example we consider again the 3-noded triangular element with linear approximations for N in terms of area coordinates Li. We will construct enhanced strain terms from the derivatives of an assumed function. Here we consider three enhanced functions given by
N~ --/~Li + L j L k
(11.37)
in which i, j, k is a cyclic permutation and/3 is a parameter to be determined. Note that this form only involves quadratic terms and thus gives linear strains which are fully consistent with the linear interpolations for p and ev. The derivatives of the enhanced function are given by DN~
1
cgx = 2A [/3bi 4- Ljbk + Lkbj] Oy
(11.38)
= 1. [gci + Z.jc + L cj] 2A
where
bi -- yj - Yk
and
Ci --" Xk --Xj
and A is the area of a triangular element. For constant p the requirement imposed by Eq. 11.21 gives fl = 1/3. The derivatives are inserted in the usual strain-displacement matrix
"aN~
o
o
aN~
Ox
B~=
ONie . Oy
Oy ONe
(11.39)
Ox.
While the use of added enhanced modes leads to increased cost (over use of a simple bubble mode, as in Example 11.1) in eliminating the ~o and Ore parameters in Eq. (11.35) the results obtained are improved considerably, as indicated in the numerical results presented in Sec. 11.7. Furthermore, this form leads to improved consistency between the pressure and strain.
Three-field nearly incompressible elasticity (u-p-ev form)
11.4.1 The B-bar method for nearly incompressible problems The second of (11.24) has the solution ~v - - E - 1 c T f i
"- W f i
(11.40)
In the above we assume that E may be inverted, which implies that Nv and N p have the same number of terms. Furthermore, the approximations for the volumetric strain and pressure are constructed for each element individually and are not continuous across element boundaries. Thus, the solution of Eq. (11.40) may be performed for each individual element. In practice No is normally assumed identical to N p so that E is symmetric positive definite. The solution of the third equation of (11.24) yields the pressure parameters in terms of the volumetric strain parameters and is given by = E-THffv
(11.41)
Substitution of (11.40) and (11.41) into the first of (11.24) gives a solution that is in terms of displacements only. Accordingly, Aft = fl (11.42) where for isotropy /.
A =/,.,
BTDd B dr2 + WTHW
(11.43)
= A + WTHW The solution of (11.42) yields the nodal parameters for the displacements. Use of (11.40) and (11.41) then gives the approximations for the volumetric strain and pressure. The result given by (11.43) may be further modified to obtain a form that is similar to the standard displacement method. Accordingly, we write = L BTD[I dr2
(1 1.44)
where the strain-displacement matrix is now 1 171= IdB + ~mNvW
(11.45)
For isotropy the modulus matrix is D = Dd + K m m T
(11.46)
We note that the above form is identical to a standard displacement model except that B is replaced by 1). The method has been discussed more extensively in references 13, 14 and 15. The equivalence of (11.43) and (11.44) can be verified by simple matrix multiplication. Extension to treat general small strain formulations can be simply performed by replacing the isotropic D matrix by an appropriate form for the general material model. The formulation shown above has been implemented into an element included as part of the program available on the web site. The elegance of the method is more fully utilized when
397
398 Incompressibleproblems, mixed methods and other procedures of solution considering non-linear problems, such as plasticity and finite deformation elasticity (see reference 1). We note that elimination starting with the third equation of (11.24) could be accomplished leading to a u - p two-field form using K as a penalty number. This is convenient for the case where p is continuous but ev remains discontinuous - as already discussed in Example 11.2. Such an elimination, however, points out that precisely the same stability criteria operate here as in the two-field approximation discussed earlier.
In Chapter 5 we mentioned the lowest order numerical integration rules that still preserve the required convergence order for various elements, but at the same time pointed out the possibility of a singularity in the resulting element matrices. In Chapter 9 we again referred to such low order integration rules, introducing the name 'reduced integration' for those that did not evaluate the stiffness exactly for simple elements and pointed out some dangers of its indiscriminate use due to resulting instability. Nevertheless, such reduced integration and selective integration (where the low order integration is only applied to certain parts of the matrix) has proved its worth in practice, often yielding much more accurate results than the use of more precise integration rules. This was particularly noticeable in nearly incompressible elasticity (or Stokes fluid flow which is similar) 16-18 and in problems of plate and shell flexure dealt with as a case of a degenerate solid 19' 20 (see reference 1 for more information on plate and shell problems). The success of these procedures derived initially by heuristic arguments proved quite spectacular- though some consider it somewhat verging on immorality to obtain improved results while doing less work! Obviously fuller justification of such processes is required. 21 The main reason for success is associated with the fact that it provides the necessary singularity of the constraint part of the matrix [viz. Eqs (10.19)-(10.21)] which avoids locking. Such singularity can be deduced from a count of integration points, 22, 23 but it is simpler to show that there is a complete equivalence between reduced (or selective) integration procedures and the mixed formulation already discussed in Sec. 11.3. This equivalence was first shown by Malkus and Hughes 24 and later in a general context by Zienkiewicz and Nakazawa. 25 We shall demonstrate this equivalence on the basis of the nearly incompressible elasticity problem for which the mixed weak Galerkin integral statement is given by Eqs (11.7) and (11.8). It should be noted, however, that equivalence holds only for the discontinuous pressure approximation. The corresponding irreducible form can be written by satisfying the second of Eq. (11.8) exactly, implying p = Km Te
and substituting above into (11.7) as
(11.47)
Reduced and selective integration and its equivalence to penalized mixed problems 399 ~ 8r
(Io -- ~l m T m ) e d S 2 + f ~ e T m K m T e d ~ (11.48) --~~uTbd~-
~r ~uTt, d r = 0 t
On substituting u ,~ fi = Nu~
and
e ,~ ~ = SNu~ = B~
(11.49)
we have (A +/i,) ~ = fl
(11.50)
where A and fl are exactly as given in Eq. (11.11) and /i = f~ BTmKmTB dr2
(11.51)
The solution of Eq. (11.50) for fi allows the pressures to be determined at all points by Eq. (11.47). In particular, if we have used an integration scheme for evaluating (11.51) which samples at points (~k) we can write
Npa(~k)tga
P(~k) = KmTe(~k) = KmTB(~k)fi = ~
(11.52)
a
Now if we turn our attention to the penalized mixed form of Eqs (11.7)-(11.11) we note that the second of Eq. (11.10) is explicitly Np mTBfi-- ~ N p ~
d~ = 0
(11.53)
If a numerical integration is applied to the above sampling at the pressure nodes located at coordinate (~l), previously defined in Eq. ( 11.52), we can write for each scalar component of Np Npa(~l)
(
1
mTB(~'I) ~1 -- -~-
p(~l)P
l
)
Wl : 0
(11.54)
in which the summation is over all integration points (~l) and Wl are the appropriate weights and jacobian determinants. Now as Npa(~l) = ~al
if ~l is located at the pressure node a and zero at other pressure nodes, Eq. (11.54) reduces simply to the requirement that at all pressure nodes mTB(~/) ~ = ~I N p(~l)P
(11.55)
This is precisely the same condition as that given by Eq. (11.52) and the equivalence of the procedures is proved, providing the integrating scheme usedfor evaluating A gives an
identical integral of the mixed form of Eq. (11.53).
This is true in many cases and for these the reduced integration-mixed equivalence is exact. In all other cases this equivalence exists for a mixed problem in which an inexact rule of integration has been used in evaluating equations such as (11.53).
400 Incompressibleproblems, mixed methods and other proceduresof solution For curved isoparametric elements the equivalence is in fact inexact, and slightly different results will be obtained using reduced integration and mixed forms. This is illustrated in examples given in reference 26. We can conclude without detailed proof that this type of equivalence is quite general and that with any problem of a similar type the application of numerical quadrature at n p points in evaluating the matrix ,~ within each element is equivalent to a mixed problem in which the variable p is interpolated element by element using as p-nodal values the same integrating points. The equivalence is only complete for the selective integration process, i.e., application of reduced numerical quadrature only to the matrix/~, and ensures that this matrix is singular, i.e., no locking occurs if we have satisfied the previously stated conditions (nu > n p). The full use of reduced integration on the remainder of the matrix determining ~, i.e., A, is only permissible if that remains non-singular- the case which we have discussed previously for the Q8/4 element. It can therefore be concluded that all the elements with discontinuous interpolation of p which we have verified as applicable to the mixed problem (viz. Fig. 11.1, for instance) can be implemented for nearly incompressible situations by a penalized irreducible form using corresponding selective integration.t In Fig. 11.6 we show an example which clearly indicates the improvement of displacements achieved by such reduced integration as the compressibility modulus K increases (or the Poisson ratio tends to 0.5). We note also in this example the dramatically improved performance of such points for stress sampling. For problems in which the p (constraint) variable is continuously interpolated (Co) the arguments given above fail as quantities such as rote are not interelement continuous in the irreducible form. A very interesting corollary of the equivalence just proved for (nearly) incompressible behaviour is observed if we note the rapid increase of order of integrating formulae with the number of quadrature points (viz. Chapter 5). For high order elements the number of quadrature points equivalent to the p constraint permissible for stability rapidly reaches that required for exact integration and hence their performance in nearly incompressible situations is excellent, even if exact integration is used. This was observed on many occasions 27-29 and Sloan and Randolf 3~ have shown good performance with the quintic triangle. Unfortunately such high order elements pose other difficulties and are seldom used in practice. A final remark concerns the use of 'reduced' integration in particular and of penalized, mixed, methods in general. As we have pointed out in Sec. 10.3.1 it is possible in such forms to obtain sensible results for the primary variable (u in the present example) even though the general stability conditions are violated, providing some of the constraint equations are linearly dependent. Now of course the constraint variable (p in the present example) is not determinate in the limit. This situation occurs with some elements that are occasionally used for the solution of incompressible problems but which do not pass our mixed patch test, such as Q8/4 and Q9/4 of Fig. 11.1. If we take the latter number to correspond to the integrating points these will yield acceptable u fields, though not p. t The Q9/3 element would involve three-point quadrature which is somewhatunnatural for quadrilaterals. It is therefore better to simplyuse the mixedformhere - and, indeed, in any problemwhichhas non-linearbehaviour between p and u (see reference 1).
~0 14row c~ in
0 r~
~0
CIJ
~m
~z o~
Ls~
O
im
0
1in
in C~
Ill
q~m Vm
~D
olm WR
E
im
O
C13 Im Q.
C~ tm Im CIJ
CIJ 1ram
Fig. 11.6 Sphere under internal pressure. Effect of numerical integration rules on results with different Poission ratios.
402
Incompressibleproblems, mixed methods and other proceduresof solution
Fig. 11.7 Steady-state, low Reynolds number flow through an orifice. Note that pressure variation for element Q8/4 is so large it cannot be plotted. Solution with u/p elements Q8/3, Q8/4, Q9/3, Q9/4.
Figure 11.7 illustrates the point on an application involving slow viscous flow through an orifice - a problem that obeys identical equations to those of incompressible elasticity. Here elements Q8/4, Q8/3, Q9/4 and Q9/3 are compared although only the last completely satisfies the stability requirements of the mixed patch test. All elements are found to give a reasonable velocity (u) field but pressures are acceptable only for the last one, with element Q8/4 failing to give results which can be plotted. 4
Reduced and selective integration and its equivalence to penalized mixed problems 403
Fig. 11.8 A quadrilateral with intersecting diagonals forming an assembly of four T311 elements. This allows displacements to be determined for nearly incompressible behaviour but does not yield pressure results. It is of passing interest to note that a similar situation develops if four triangles of the T3/1 type are assembled to form a quadrilateral in the manner of Fig. 11.8. Although the original element locks, as we have previously demonstrated, a linear dependence of the constraint equation allows the assembly to be used quite effectively in many incompressible situations, as shown in reference 31. Example 11.3: A weak patch test - selective integration.
In order to illustrate the performance of an element which only satisfies a weak patch test we consider an axisymmetric linear elastic problem modelled by 4-noded isoparametric elements. The material is assumed isotropic and the finite element stiffness and reaction force matrices are computed using a selective integration method where terms associated with the bulk modulus are evaluated by a single-point Gauss quadrature, whereas all other terms are computed using a 2 • 2 (standard) gaussian quadrature. It may be readily verified that the stiffness matrix is of proper rank and thus stability of solutions is not an issue. On the other hand, consistency must still be evaluated. In order to assess the performance of a selective reduced quadrature formulation we consider the patch of elements shown in Fig. 11.9. The patch is not as generally shaped as desirable and is only used to illustrate performance of an element that satisfies a weak patch test. The polynomial solution considered is u-2r
(11.56)
v=O
and material constants E = 1 and v -- 0 are used in the analysis. The resulting stress field is given by t7r
--"
O"0
"--
2
(11.57)
with other components identically zero. The exact solution for the nodal quantities of the mesh shown in Fig. 11.9 are summarized in Table 11.1. Patch tests have been performed for this problem using the selective reduced integration scheme described above and values of h of 0.8, 0.4, 0.2, 0.1, and 0.05. The result for the radial displacement at nodes 2 and 5 (reported to six digits) is given in Table 11.2. All other quantities (displacements, strains, and stresses) have a similar performance with convergence rates of at least O (h) or more. Based on this assessment we conclude the element passes a weak patch test. A similar result will be found for elements which are not rectangular and thus the element produces convergent results.
404
Incompressible problems, mixed methods and other procedures of solution tl
h
~b "I ~'
,..._I ~I
r=l
Fig. 11.9 Patchfor selective,reducedquadrature on axisymmetric4-noded elements. Table 11.1 Exact solution for patch Displacement
Force
Node a
Radius ra
tla
~)a
Era
1,4 2,5 3, 6
1 -h 1 1+ h
2(1 - h ) 2 2(1 + h)
0 0 0
-(1 - h ) h 0 (1 + h)h
Fza
Table 11.2 Radial displacement at nodes 2 and 5 h 0.8 0.4 0.2 0.1 0.05
2.01114 2.00049 2.00003 2.00000
2.00000
In the general remarks on the algebraic solution of mixed problems characterized by equations of the type [viz. Eq. (10.14)] A
fl
we have remarked on the difficulties posed by the zero diagonal and the increased number of unknowns (nx + ny) a s compared with the irreducible form (nx o r ny). A general iterative form of solution is possible, however, which substantially reduces the cost. 32 In this we solve successively y(k+l)
_
y(k) +
pr(k)
(11.59)
where r (k~ is the residual of the second equation computed as r (k~ = CTx (k~ - t"2
(11.60)
A simple iterative solution process for mixed problems: Uzawa method and follow with solution of the first equation, i.e., x(k+l) - -
A-1 (fl - CY(k+l))
(11.61)
In the above p is a 'convergence accelerator matrix' and is chosen to be efficient and simple to use. The algorithm is similar to that described initially by Uzawa 33 and has been widely applied in an optimization context. 28, 34--38 Its relative simplicity can best be grasped when a particular example is considered.
11.6.2 Iterative solution for incompressible elasticity In this case we start from Eq. (11.10) now written with V - 0, i.e., complete incompressibility is assumed. The various matrices are defined in (11.11), resulting in the form
fi
IC/]kT 0el {~) -- { l~'0 lj
(11.62)
Now, however, for three-dimensional problems the matrix A is singular (as volumetric changes are not restrained) and it is necessary to augment it to make it non-singular. We can do this in the manner described in Sec. 10.3.1, or equivalently by the addition of a fictitious compressibility matrix, thus replacing A by /~ = A + f~ B T()~GmmT)B dr2
(11.63)
If the second matrix uses an integration consistent with the number of discontinuous pressure parameters assumed, then this is precisely equivalent to writing /~ = A + )~GCC T
(11.64)
and is simpler to evaluate. Clearly this addition does not change the equation system. The iteration of the algorithm (11.59)-(11.61) is now conveniently taken with the 'convergence accelerator' being simply defined as p = ~.GI
(11.65)
We now have the iterative system given as
~(k+l) _ ~(k) ..1_XGr(k) where
r (k) --
cTfi (k)
(11.66)
(11.67)
the residual of the incompressible constraint, and fi(k+l) = ~ - l ( f 1 _ C~(k+l))
(11,68)
In this A can be interpreted as the stiffness matrix of a compressible material with bulk modulus K = ~.G and the process may be interpreted as the successive addition of volumetric 'initial' strains designed to reduce the volumetric strain to zero. Indeed this simple
405
406
Incompressible problems, mixed methods and other procedures of solution
approach led to the first realization of this algorithm. 39-41 Alternatively the process can be visualized as an amendment of the original equation (11.62) by subtracting the term p/(~.G) from each side of the second to give (this is often called an augmented lagrangian form)32,37, 38 [Ac"r
C} ] {1~ } =1 { _ f~ ~ XG E-G
and adopting the iteration [c/~T
C1 i ) { ; } ( k + l ) ( kG
fl~ - ~ G p(k) }
(11.69)
(11.70)
With this, on elimination, a sequence similar to Eqs (11.66)-(11.68) will be obtained provided A is defined by Eq. (11.64). Starting the iteration from 8(o) = 0
and
~(o) = 0
in Fig. 11.10 we show the convergence of the maximum div u computed at any of the integrating points used. We note that this convergence becomes quite rapid for large values of ~. = (103-104). 12.0 A
u=1t10.01 100
"" ,'~ <
~"~
20.0
~I
5.0
~,~10
\\, f
10_5
~'\, \'\~: ~o~ \ \ "\ "\,
.._x
> ,m
-o 10-1~ _
m
i
\
\~=1o~
\ "\, \
10-15 _
0
..
\.
\
g
\,
\
= lo4"\
v--~...Z.,
I 5 Number of iterations ~ - ~
Round-off limit I 10
Fig. 11.10 Convergenceof iterations in an extrusion problem for different values of the parameter ~..
Stabilized methods for some mixed elements failing the incompressibility patch test For smaller ~. values the process can be accelerated by using different 1032 but for practical purposes the simple algorithm shown above suffices for many problems, including applications in large strain. 42 Clearly much better satisfaction of the incompressibility constraint can now be obtained by the simple use of a 'large enough' bulk modulus or penalty parameter. With ~. = 104, for instance, in five iterations the initial div u is reduced from the value ~ 10 -4 to 10 -16, which is at the round-off limit of the particular computer used. Finally, we remind the reader that the above iterative process solves the equations of a mixed problem. Accordingly, it is fully effective only when the element used satisfies the stability and consistency conditions of the mixed patch test.
It has been observed earlier in this chapter that many of the two-field u - p elements do not pass the stability conditions imposed by the mixed patch test at the incompressible limit (or the Babugka-Brezzi conditions). Here in particular we have such methods in which the displacement and pressure are interpolated in an identical manner (for instance, linear triangles, linear quadrilaterals, quadratic triangles, etc.) and many attempts for stabilization of such elements have been introduced. Indeed one may view the bubble introduced in Example 11.1 and the enhanced strain treatment of Example 11.2 as stabilized methods. However, several alternative categories to these exist. The first category is the introduction of non-zero diagonal terms of the constraint equation by adding a least squares form to the Galerkin formulation. This was first suggested by Courant 43 as a means of improving accuracy in solutions. It appears that Brezzi and Pitkaranta in 198444 were the first to add terms to the Galerkin solution in an attempt to stabilize results. Numerous further suggestions have been proposed by Hughes et al. between 1986 and 1989 with the final form again a least squares approach called the Galerkin least squares method. 45--47 An alternative proposal of achieving similar answers has been proposed by Ofiate 48 which gains the addition of diagonal terms by the introduction of so-called finite increment calculus to the formulation. More recently, a very simple stabilization has been proposed by Dohrmann and Bochev 49 in which a stabilization involving the difference between the interpolated pressure and a direct projection of pressure is appended to the Galerkin equations in a least squares form. There is, however, an alternative possibility introduced by time integration of the full incompressible formulation. Here many of the algorithms will yield, when steady-state conditions are recovered, a stabilized form. A number of such algorithms have been discussed by Zienkiewicz and Wu in 199150 and a very efficient method has appeared as a by-product of a fluid mechanics algorithm named the characteristic-based split (CBS) procedure 51-55 (which is discussed at length in reference 56). In the latter algorithm there exists a free parameter. This parameter depends on the size of the time increment. In the other methods there is a weighting parameter applied to the additional terms introduced. We shall discuss each of these algorithms in the following subsections and compare the numerical results obtainable.
407
408
Incompressibleproblems, mixed methods and other procedures of solution One may question, perhaps, that resort to stabilization procedures is not worthwhile in view of the relative simplicity of the full mixed form. But this is a matter practice will decide and is clearly in the hands of the analyst applying the necessary solutions.
11.7.1 Laplacian pressure stabilization In the first part of this chapter we separated the stress into the deviatoric and pressure components as tr=cr d +rap Using the tensor form described in Appendix B this may be written in index form as
O'ij --- t7d .-~ t~ijP The deviatoric stresses are related to the deviatoric strains through the relation
___ (~Ui ~Uj cr~ = 2Geaij = G \ axj + OXi
2r 3
0u,
3xk J
(11.71)
The equilibrium equations (in the absence of inertial forces) are:
3xi + ~xj + bj = 0 Substituting the constitutive equations for the deviatoric part yields the equilibrium form (assuming G is constant) G
32uj 1 32ui 1 3p 3Xi 3Xi q- -3 3Xi 3Xj J + ~Xj + by
= 0
(11.72)
In vector form this is given as G[V2u + ~1 V(div u)] + V P + b = 0 where V 2 is the laplacian operator and V the gradient operator. The constitutive equation (11.2) is expressed in terms of the displacement as
ev
OUi 1 3xi = div u = K p
(11.73)
where div (.) is the divergence of the quantity. A single equation for pressure may be deduced from the divergence of the equilibrium equation. Accordingly, from Eq. (11.72) we obtain
4G V2 (div u) + V2p + div b = 0 3 where upon noting (11.73) we obtain finally
( 1+~-~
V2 p + d i v b = 0
(11.74)
(11.75)
Thus, in general, the pressure must satisfy a Poisson equation, or in the absence of body forces, a Laplace equation.
Stabilized methods for some mixed elements failing the incompressibility patch test
We have noted the dangers of artificially raising the order of the differential equation in introducing spurious solutions; however, in the context of constructing approximate solutions to the incompressible problem the above is useful in providing additional terms to the weak form which otherwise would be zero. Brezzi and Pitkaranta 44 suggested adding Eq. (11.75) to Eq. (11.7) and (on setting the body force to zero for simplicity) obtain ~p m X e - ~ p
~pV2pdf2=
d~ + Z/3 e
0
(11.76)
e
where/3 is a parameter introduced to control accuracy. The last term may be integrated by parts to yield a form which is more amenable to computation as ~
( 1 ) 3p m Te - ~ p dr2 - Z / 3
~ 06p Opdf2=O Oxi Oxi
e
(11.77)
e
in which the resulting boundary terms are ignored. Upon discretization using equal order linear interpolation on triangles for u and p we obtain a form identical to that for the bubble in Example 11.1 with the exception that 7- [viz. Eq. (11.20b)] is now given by 7- = / 3 I
(11.78)
On dimensional considerations with the first term in Eq. (11.77) the parameter/3 should have a value proportional to L4/F, where L is length and F is force. We defer discussion on the particular value until after presenting the Galerkin least squares method.
11.7.2 Galerkin least squares method The Galerkin least squares (GLS) approach is a general scheme for solving the differential equations (3.1) by a finite element method. We may write the GLS form as
fSu T.A(u) d~ + y~ f~ 8.,4(u) TT-.,4(u) dr2 = 0 e
(11.79)
e
where the first term represents the normal Galerkin form and the added terms are computed for each element individually including a weight 7- to provide dimensional balance and scaling. Generally, the 7- will involve parameters which have to be selected for good performance. Discontinuous terms on boundaries between elements that arise from higher order terms in ,A(u) are commonly omitted. The form given above has been used by Hughes 47 as a means of stabilizing the fluid flow equations, which for the case of the incompressible Stokes problem coincide with those for incompressible linear elasticity. For this problem only the momentum equation is used in the least squares terms. After substituting Eq. (11.73) into Eq. (11.72) the momentum equation may be written as (assuming that G and K are constant in each element and body forces are ignored)
02Uj(.+ Ox?
1+
~G)OP
= 0
(11.80)
A more convenient form results by using a single parameter defined as (~ =
G
1+ G/3K
(11.81)
409
410
Incompressible problems, mixed methods and other procedures of solution With this form the least squares term to be appended to each element may be written as
ff2 (
022
02Uj21-~OP) d~"~ + ~xiOaP) 15iJ( a~-~m
(11.82)
This leads to terms to be added to the standard Galerkin equations and is expressed as
[AS {o} CS, r
v s
where
ASb -- s
G2V2NaT-V2Nb dfa e
CSb= fa GV2NaT-VNbd~2
fo (VNa) T 7-VNbd~ e
VaSb--
e
and the operators on the shape functions are given in two dimensions by
02Na 02Na V2Na - Ox21-+- Ox2
and
[ ONa ONa] VNa= [-0--~11 OX2
Note again that all infinite terms between elements are ignored (i.e., those arising from second derivatives when Co functions are used). For linear triangular elements the second derivatives of the shape functions are identically zero within the element and only the V term remains and is now nearly identical to the Brezzi-Pitkaranta form if/3 coincides with the definition of 7-. In the work of Hughes et al., 7- is given by c~he 7- = - ~ I (11.83) 2G where ct is a parameter which is recommended to be of O(1) for linear triangles and quadrilaterals and h is the size of the element.
11.7.3 Direct pressure stabilization In the previous two sections we have discussed the procedures which needed certain disregard for consistency to be introduced. In particular, in both methodologies certain integrals were allowed over the individual elements with high order derivatives and interelement values being omitted especially if these reached infinity, as happens for instance in the GLS method when second derivatives on the interface between elements or boundary terms in the Brezzi-Pitkaranta method are ignored. In this section we introduce another process proposed in a recent paper by Dohrmann and Bochev 49 which seems to be totally correct and is arrived at without ignoring any terms in the overall integrals. In this procedure we try to ensure that the difference between the Co interpolated pressures gives answers consistent with those in which a lower order, discontinuous approximation is u s e d - i.e., one which is consistent with the general approximation
Stabilized methods for some mixed elements failing the incompressibility patch test for stresses. Thus, for instance, in triangular elements in which linear displacements are used the stresses are only allowed to be constant within any element and the assumption of any component being also linear is not consistent. For this reason the method looks at the difference between the interpolated pressure which is of the same order as the displacement and its projection onto one order lower expansion consistent with that of the stresses. The work of Dohrmann and Bochev considers a two-field mixed approximation given by f a 'sTDa6 dr2 + j a ' e T m p dr2 - f a 'uTb df2 - f r
8p
dr2-
t
~uT~dl-"= 0 (11.84)
(Sp - 6D) ~ e
in which the displacements u and pressure p are approximated by k order continuous polynomial shape functions,/~ is a discontinuous projection of p onto a polynomial space of order k - 1 and ot is a parameter to be selected for stability. When K --~ e~ the above form represents a stable approximation for the incompressible problem for all order of elements provided o~ is set at a nominal value. In the examples we use oe = 2.t We note that the form (11.84) requires no integration by parts in which terms are ignored. Thus, the method has considerable theoretical advantages over the previously discussed stabilization methods. The pressure stabilization is computed for each element individually using f a 8/~(p -
e
P) d ~ - 0
(11.85)
and, thus, has low additional cost. Due to this form, which also holds when the variation is interchanged on pressures, the stabilization term may also be written as
e
(Sp - 6D) ~ (p - P) d~2 -
e
-~ (~pp - ~/~/~) d~2
(11.86)
which is now in the form of a difference of two 'mass'-type arrays. If we approximate the pressure by
p~p=~NaPa-N~ a
in which Na contain the set of polynomials of order k and P = Z
h b (x)/~b = h(x)
/3
b
where ho (x) are the polynomials of order k - 1 the solution to (11.85) is determined from
f~e hWh d ~ ~ --- f~e hTN d~'~p H/~ = G~
Thus, the pressure projection is given by p - h(x)H-1G~
(11.87)
t Reference 49 uses a = 1. While this leads to convergence we find this value is somewhat small for our examples.
411
412 Incompressibleproblems, mixed methods and other procedures of solution Using the usual finite element approximation for the displacement u -- Z
Na f~a = Nfi a
the stabilized weak form may be written in matrix form as
[cK~ _ C ] { p } =
{f0}
(11.88)
where the arrays are given by C = ~ BXmNdf2
Kcl=J2BTDaBdf2;
V ---
/oN T
+
Ndf2 -
~
and f is the usual force due to boundary traction and body loads. It is clear from the definition of V that, when G/or is much smaller than K, the effect of the direct pressure stabilization is apenalty form on the difference of the interpolated and projected pressure. The patch test only requires pressures of order k - 1 (i.e., the order of the projected pressures) to satisfy the consistency condition. This partly explains why the above approach is successful. Further validation is provided by the numerical experiments given below.
Example 11.4: Direct stabilization for 3-node triangular element.
As an example consider the problem of the two-dimensional plane strain problem in which the solution is performed using linear triangles (k = 1) with shape functions given by Na = La; a = 1,2,3
Here the projection for/~ is given by a constant (k - 1 = 0) value 1
Note that numerical integration of the stabilizing term may not be performed using onepoint quadrature at the element baricentre as then no contribution to the stabilizing term would be found. Performing the integrations for the stabilizing term (11.86) gives the result
Vstab
--
"" [ili] ,.[,li]
12G
2
1
- ~
1
1
--
1 1
"" [i 1 ,]
18G
2
-1
-1
2
(11.89)
where A is the area of the triangular element. We recognize this result to have the same form as the deviatoric projection array which is positive semi-definite. The singular nature of this array permits the constant values of p to be unaffected by the stabilizing terms, thus, maintaining optimal accuracy for the method. If we assemble the stabilization array given in (11.89) for the four-element patch shown in Fig. 11.11 (a) we obtain an equation for node 0 Aot 4 Po 18G
1
Pi
Stabilized methods for some mixed elements failing the incompressibility patch test
3
4 )
4
3 )
5(
1 (a) 4-element patch
6
(
1 (b) 6-element patch
Fig. 11.11 Mesh patterns for pressure stabilization matrix evaluation.
which we recognize as a laplacian-type form. Similarly, for the six-element patch shown in Fig. 11.11 (b) we obtain Aot
18G
4 Po -
Pi
1
which has a similar form but is not the same as the laplacian operator for this mesh pattem. The simplicity of the direct pressure stabilization is one of its main advantages. However, it also permits applications on elements of other order and shape without significant complication. For example, if degenerate element forms for quadratic elements are used as discussed in Chapter 5 the direct approach provides a means of stabilizing computations for incompressible forms without the need to add any second derivative terms (as, for example, needed for the GLS form). We will show later on numerical examples of how well the direct stabilization approach works.
11.7.4 Incompressibilityby time stepping The fully incompressible case (i.e., K = oo) has been studied by Zienkiewicz and Wu 5~ using various time stepping procedures. Their applications concern the solution of fluid problems in which the rate effects for the Stokes equation appear as first derivatives of time. We can consider such a method here as a procedure to obtain the static solutions of elasticity problems in the limit as the rate terms become zero. Thus, this approach is considered here as a method for either the Stokes equation or the case of static incompressible elasticity. The governing equations for slightly compressible Stokes flow may be written as
413
414
Incompressible problems, mixed methods and other proceduresof solution OU i
Po-Ot
Or7d
Op
OXj
OX i
Op
1
= 0
(11.90)
Oui
pO c 2 0 t
=0
OXi
where P0 is density (taken as unity in subsequent developments), c = (K/po) 1/2 is the speed of compressible waves, p is the pressure (here taken as positive in tension), and u i is a velocity (or for elasticity interpretations a displacement) in the/-coordinate direction. Note that the above form assumes some compressibility in order to introduce the pressure rate term. At the steady limit this term is not involved, consequently, the solution will correspond to the incompressible case. Deviatoric stresses trd are related to deviatoric strains (or strain rates for fluids) as described by Eq. (11.71). Zienkiewicz and Wu consider many schemes for integrating the above equations in time. Here we introduce only one of the forms, which is widely used in the solution of the fluid equations which include transport effects (see reference 56). For the full fluid equations the algorithm is part of the characteristic-based split (CBS) method. 51' 52, 54-57 The equations are discretized in time using the approximations u(tn) ,~ u ~ and time derivatives OUi
Un+l -- U n
Ot
At
(11.91)
where At ----tn+l - tn. The time discretized equations are given by U n+l -- Un
Op"
O0 "d'n
At
=
Oxj
0 Ap
-+" ~ x i
+ 02 Ox------7
(11.92a)
and (11.92b)
1 p n + l _ pn OU n OAui -~ c At -- Oxi -]- O1 Oxi
U7 +1 -- UT; 01 can vary between 1/2 and 1; and 0 2 c a n where Ap pn+l _ pn; Aui vary between 0 and 1. In all that follows we shall use 01 = 1. The form to be considered uses a split of the equations by defining an intermediate approximate velocity u~' at time tn+l when integrating the equilibrium equation (11.92a). Accordingly, we consider =
:
(11.93a)
U* -- tin -- O(Tid'n
At
and
u~'+~ -u~' At
Oxj
--
Op" Oxi
~Ap
-'[-- 02 ~
(11.93b)
Oxi
Differentiating the second of these with respect to xi to get the divergence of U n+l and combining with the discrete pressure equation (11.92b) results in L
A__pp _
c2 At
02 At
O2Ap Oxi Oxi
= At
O2p n Oxi Oxi
~
8u* Oxi
(11.93c)
Thus, the original problem has been replaced by a set of three equations which need to be solved successively.
Stabilized methods for some mixed elements failing the incompressibility patch test
Equations (11.93a), (11.93b) and (11.93c) may be written in a weak form using as weighting functions 8u*, 8u and 8p, respectively (viz. Chapter 3). They are then discretized in space using the approximations U n ,~ fin ~__ Nuff'
and
6 u ~ 8fi = N u 3 f
fi* = Nuf* pn ,~ pn : Np~n
and and
8u* ,~ 8fi* =
u* ~
NuSf*
~p ~ ~ = NpS~
with similar expressions for Un+l and pn+l. The final discrete form is given by the three equation sets 1
Mu (fi* A--~
fn)
.+_fl
._ _ A f n
1
p + 02AtH A~ = - C f * -
At 1
AtH~ n + f2
(11.94)
A t M , (fn+a _ f , ) = _ C T (~n _31_0 2 A ~ ) In the above we have integrated by parts all the terms which involve derivatives on deviator stress (a/~), pressure (p) and displacements (velocities). In addition we consider only the case where u~'+1 = u* = fii on the boundary Fu (thus requiting 6ui = 6u* = 0 on Fu). Accordingly, the matrices are defined as
Mu - f. NuNu d.
Mp=
A = f~ BTDdB dg2
C =
f
1 NTNp d f 2 ~5
~ONp ~Nu
d~ (11.95)
H
=
f.
OXi
OXi
da
fl "- fF NT (i -- knpn) dI" t
t"2 = f r NTnTudF u
in which Dd are the deviatoric moduli defined previously. The parameter k denotes an option on alternative methods to split the boundary traction term and is taken as either zero or unity. We note that a choice of zero simplifies the computation of boundary contributions; however, some would argue that unity is more consistent with the integration by parts. The boundary pressure acting on 1-'t is computed from the specified surface tractions d,, (ti) and the 'best' estimate for the deviator stress at step (n + 1) which is given by oij . Accordingly, d,, ~n+l ~ ni-ti -- nio'ij nj
is imposed at each node on the boundary Ft. In general we require that At < Atcrit where the critical time step is h2/2G (in which h is the element size). Such a quantity is obviously calculated independently for each element and the lowest value occurring in any element governs the overall stability. It is possible and useful to use here the value of At calculated for each element separately when calculating incompressible stabilizing terms in the pressure calculation and the overall time step elsewhere (we shall label the time increments multiplying H in Eq. (11.94)3 as Atint).
415
416 Incompressibleproblems, mixed methods and other procedures of solution A ratio of F = Atint / At greater than unity improves considerably the stabilizing properties. As Eq. (11.94)3 has greater stability than the first two equations in (11.94), and for 02 > 1/2 is unconditionally stable, we recommend that the time step used in this equation be y Atcr for each node. Generally a value of 2 is good as we shall show in the examples (for additional details see reference 54). Equation (11.94) defines a value of ~* entirely in terms of known quantifies at the n-step. If the mass matrix Mu is made diagonal by lumping (see Chapter 16 and Appendix I) the solution is thus trivial. Such an equation is called explicit. The equation for A~, on the other hand, depends on both Mp and I-I and it is not possible to make the latter diagonal easily.t It is possible to make Mp diagonal using a similar method as that employed for Mu. Thus, if 02 is zero this equation will also be explicit, otherwise it is necessary to solve a set of algebraic equations and the method for this equation is called implicit. Once the value of A~ is known the solution for ~n+l is again explicit. In practice the above process is quite simple to implement; however, it is necessary to satisfy stability requirements by limiting the size of the time increment. This is discussed further in Chapter 17 and in reference 51. Here we only wish to show the limit result as the changes in time go to zero (i.e., for a constant in time load value) and when full incompressibility is imposed. At the steady limit the solutions become ~n = ~n+l __ ~
and
~n = ~n+l __ ~
(11.96)
Eliminating u* the discrete equations reduce to the mixed problem
{~~} + ( : } - 0
(11.97)
At the steady limit we again recover a term on the diagonal which stabilizes the solution. This term is again of a Laplace equation t y p e - indeed, it is now the difference between two discrete forms for the Laplace equation. The term CTMulC makes the bandwidth of the resulting equations larger- thus this form is different from all the previously discussed methods.
11.7.5 Numerical comparisons To provide some insight into the behaviour of the above methods we consider two example problems. The first is a problem often used to assess the performance of codes to solve steady-state Stokes flow problems - which is identical to the case for incompressible linear elasticity. The second example is a problem in nearly incompressible linear elasticity.
Example 11.5: Driven cavity. A two-dimensional plane (strain) case is considered for a square domain with unit side lengths. The material properties are assumed to be fully incompressible (v = 0.5) with unit viscosity (elastic shear modulus, G, of unity). All boundaries of the domain are restrained in the x and y directions with the top boundary having a unit tangential velocity (displacement) at all nodes except the comer ones. Since t It is possible to diagonalize the matrix by solving an eigenproblem as shown in Chapter 1 6 - for large problems this requires more effort than is practical.
Stabilized methods for some mixed elements failing the incompressibility patch test
Fig. 11.12 Mesh and GLS/Brezzi-Pitkaranta results.
the problem is incompressible it is necessary to prescribe the pressure at one point in the mesh - this is selected as the centre node along the bottom edge. The 10 • 10 element mesh of triangular elements (200 elements total) used for the comparison is shown in Fig. 11.12(a). The elements used for the analysis use linear velocity (displacement) and pressure on 3-noded triangles. Results are presented for the horizontal velocity along the vertical centre-line AA and for vertical velocity and pressure along the horizontal centre-line BB. Three forms of stabilization are considered: 1. Galerkin least squares (GLS)/Brezzi-Pitkaranta (BP) where the effect of ot on 7- is assessed. The results for the horizontal velocity are given in Fig. 11.12(b) and for the vertical velocity and pressure in Figs 11.12(c) and (d), respectively. From the analysis it
417
418 Incompressibleproblems, mixed methods and other procedures of solution
Fig. 11.13 Vertical velocity and pressure for driven cavity problem. is assessed that the stabilization parameter ot should be about 0.5 to 1 (as also indicated by Hughes et al.47). Use of lower values leads to excessive oscillation in pressure and use of higher values to strong dissipation of pressure results. 2. Cubic bubble (MINI) element stabilization. Results for vertical velocity are nearly indistinguishable from the GLS results as indicated in Fig. 11.13; however, those for pressure show oscillation. Such oscillation has also been observed by others along with some suggested boundary modifications. 58 No free parameters exist for this element (except possible modification of the bubble mode used), thus, no artificial 'tuning' is possible. Use of more refined meshes leads to a strong decrease in the oscillation. 3. Direct pressure stabilization (DB). Results for vertical velocity are again well captured as shown in Fig. 11.13; pressure results are also smooth and give good peak answers. We have not explored the range of c~ which may be used for the stabilization. 4. The CBS algorithm. Finally in Fig. 11.13 we present results using the CBS solution which may be compared with GLS, ct = 0.5. Once again the reader will observe that with y = 2, the results of CBS reproduce very closely those of GLS, a = 0.5. However,
Stabilized methods for some mixed elements failing the incompressibility patch test
Fig, 11.14 Region and mesh used for slotted tension strip. in results for ?, = 1 no oscillations are observed and they are quite reasonable. This ratio for y is where the algorithm gives excellent results in incompressible flow modelling as will be demonstrated further in results presented in reference 56.
Example 11.6: Tension strip with slot. As our next example we consider a plane strain linear problem on a square domain with a central slot. The domain is two units square
419
420
Incompressible problems, mixed methods and other procedures of solution 2.5
I
'
I
DB 0.9
GLS
CBS
0.8 0.7 o)
._~ 13
0.6 0.5
o
1.5
~
~, 0.4
.......................................
03 0.2 0.1
i
03
0.4
|
0.5
0
i
0.6
0.7
x-coordinate
0.8
0.9
0
|
|
0.2
0.4
(a) Pressure at x-axis
"--,~
3.5 "E
Q.
0.8
p-pressure
1
1.2
(b) Pressure at y-axis
x 10 -3
4
0.6
- - GLS .-., CBS
:
- .9:.
3 2.5
._~ "o
2 1.5 1
0
0.2
0.4 0.6 x coordinate
0.8
1
(c) Pressures and displacements for slot problems
Fig. 11.15 Pressuresand displacementsfor slot problems.
and the central slot has a total width of 0.4 units and a height of 0.1 units. The ends of the slot are semicircular. Lateral boundaries have specified normal displacement and zero tangential traction. The top and bottom boundaries are uniformly stretched by a uniform axial loading and lateral boundaries are maintained at zero horizontal displacement. We consider the linear elastic problem with elastic properties E = 24 and v = 0.5; thus, giving an incompressible situation. An unstructured mesh of triangles is constructed as shown in Fig. 11.14(b). The problem is solved using direct pressure (DB), Galerkin least squares (GLS) and characteristic-based split (CBS) stabilization methods. Results for pressure along the horizontal and vertical centre-lines (i.e., the x and y axes) are presented in Figs 11.15(a) and 11.15(b) and in Tables 11.3 and 11.4. The distribution of the vertical displacement is shown in Fig. 11.15(c). We note that the results for this problem cause very strong gradients in stress near the ends of the slot. The mesh used for the analysis is not highly refined in this region and hence results from different analyses can be expected to differ in this region. The results obtained elsewhere using all three formulations are indistinguishable on the plot. In general the results achieved with all forms are satisfactory and indicate that stabilized methods may be considered for use in problems where constraints, such as incompressibility, are encountered.
Concluding remarks 421 Table 11.3 Pressure for slot problem along x = 0 Pressure at x = 0 Coord. y
DB
GLS
CBS
0.0500 0.0693 0.0886 0.1079 0.1273 0.1466 0.1661 0.1860 0.2065 0.2275 0.2491 0.2714 0.2944 0.3182 0.3429 0.3686 0.3954 0.4235 0.4532 0.4846 0.5181 0.5543 0.5938 0.6380 0.6887 0.7505 0.8262 0.9090 1.0000
0.1841 0.2516 0.3563 0.4339 0.5091 0.5744 0.6311 0.6857 0.7243 0.7592 0.7943 0.8174 0.8503 0.8589 0.8918 0.8982 0.9163 0.9354 0.9360 0.9503 0.9605 0.9720 0.9813 0.9871 0.9976 1.0098 1.0222 1.0383 1.0534
0.2185 0.2593 0.3541 0.4337 0.5050 0.5693 0.6260 0.6770 0.7230 0.7589 0.7921 0.8187 0.8452 0.8636 0.8861 0.9016 0.9159 0.9301 0.9398 0.9506 0.9612 0.9712 0.9801 0.9886 0.9988 1.0096 1.0240 1.0390 1.0516
0.1964 0.2619 0.3537 0.4373 0.5083 0.5709 0.6261 0.6756 0.7226 0.7587 0.7936 0.8185 0.8470 0.8624 0.8886 0.9029 0.9157 0.9303 0.9386 0.9498 0.9610 0.9716 0.9803 0.9889 0.9974 1.0097 1.0222 1.0502 1.0275
In this chapter we have considered in some detail the application of mixed methods to incompressible problems and also we have indicated some alternative procedures. The extension to non-isotropic problems and non-linear problems is presented in reference 1, but will follow similar lines. Here we note how important the p r o b l e m is in the context of fluid mechanics and it is there that m u c h of the attention to it has been given. 56 In concluding this chapter we would like to point out three matters: 1. The mixed formulation discovers immediately the non-robustness of certain irreducible (displacement) elements and, indeed, helps us to isolate those which perform well from those that do not. Thus, it has merit which as a test is applicable to m a n y irreducible forms at all times. 2. In elasticity, certain mixed forms work quite well at the near incompressible limit without resort to splits into deviatoric and mean parts. These include the two-field quadrilateral element of P i a n - S u m i h a r a and the enhanced strain quadrilateral element of S i m o - R i f a i which were presented in the previous chapter. There we noted how well such elements work for Poisson's ratio approaching one-half as c o m p a r e d to the standard irreducible element of a similar type.
422 Incompressibleproblems, mixed methods and other procedures of solution Table 11.4 Pressure for slot problem along y = 0 Pressure at y = 0
Coord. x
DB
GLS
CBS
0.2000 0.2152 0.2313 0.2485 0.2668 0.2864 0.3062 0.3265 0.3474 0.3691 0.3915 0.4149 0.4392 0.4647 0.4916 0.5200 0.5502 0.5828 0.6183 0.6578 0.7027 0.7566 0.8268 0.9084 1.0000
2.4997 1.9449 1.6527 1.5073 1.4098 1.3392 1.2952 1.2532 1.2206 1.1964 1.1685 1.1514 1.1322 1.1195 1.1068 1.0988 1.0895 1.0793 1.0702 1.0645 1.0587 1.0525 1.0479 1.0444 1.0438
2.0775 1.8805 1.6682 1.5314 1.4291 1.3542 1.2957 1.2543 1.2209 1.1910 1.1702 1.1515 1.1348 1.1209 1.1082 1.0977 1.0879 1.0796 1.0722 1.0657 1.0596 1.0539 1.0489 1.0453 1.0443
2.1983 1.9405 1.6539 1.5219 1.4253 1.3510 1.2927 1.2523 1.2202 1.1902 1.1692 1.1508 1.1349 1.1205 1.1074 1.0975 1.0874 1.0795 1.0723 1.0655 1.0591 1.0536 1.0486 1.0450 1.0439
3. Use of stabilizing forms such as the direct pressure or time stepping form allows use of mixed u - p elements with equal order interpolation- a form which otherwise fails the mixed patch test (or Babugka-Brezzi condition).
11.1 Show that the variational theorem
FII-IR =
/~1
~e'rDaedf2-
f~
uTb dr2-
fF
u3:t dF
1 generates the problem given in Eq. (11.10) as its first variation. 11.2 Show that the variational theorem given in Eq. (11.27) generates the problem given
by (11.24). 11.3 If the approximation for p contains all the terms that are in the approximation to m Ts the limitation principle yields the result that the formulation will be identical to the standard displacement approximation given in Chapter 6. If the number of internal degrees of freedom for the displacement u in an element is equal to the number of parameters in the pressure p the mixed patch count condition is passed and the consistency condition is also passed an element will not lock. Consider the case where u is C o continuous and p is discontinuous.
Problems 423
(a) The first three members of the rectangular lagrangian family of two-dimensional plane strain elements is shown in Fig. 4.7. Consider the general member of this class in which the displacement u is of order n (i.e., has n + 1 nodes in each direction) and show on the Pascal triangle (viz. Fig. 4.8) the polynomial terms contained in the divergence term for volumetric strain m r e. (b) If the pressure p is approximated by an order m lagrangian interpolation determine the lowest order for m which will contain all the polynomial terms found in (a). (c) Determine the lowest order of n for which the limitation principle is satisfied and the element will not lock at the nearly incompressible limit. 11.4 Repeat Problem 11.3 for the triangular family of elements in plane strain (viz. Fig. 4.13). Let the displacement u be approximated by an order n polynomial (i.e., has n + 1 nodes on each edge). (a) Show on the Pascal triangle (viz. Fig. 4.8) the polynomial terms contained in the divergence term m r e. (b) What order approximation for p will contain all the polynomial terms in (a)? (c) Determine the lowest order ofn for which the limitation principle is satisfied. Note that this order of approximation will yield a displacement formulation which will not 'lock' near the incompressible limit. (d) Is the result valid for an axisymmetric geometry? Explain your answer. 11.5 For a plane strain problem consider a linear triangular element in which the displacement approximation is given by 3
U~ ~
Lafl
a=l
together with a constant approximation for the pressure p. Using Eq. (11.24) compute B. Can this formulation be used to model nearly incompressible problems? Justify your answer. 11.6 For an axisymmetric problem consider a linear triangular element in which the displacement approximation is given by 3
U .~ ~
Lafl
a=l
together with a constant approximation for the pressure p. Using Eq. (11.24) compute B. Can this formulation be used to model nearly incompressible problems? Justify your answer. 11.7 Consider a plane strain problem which is to be solved using a linear quadrilateral element with the displacement approximation 4
u~~
N~
a=l where N a - 1/4(1 + ~a~)(1 +/7a/7) together with a constant approximation for the pressure p. Let the element have a rectangular form with sides a and b in the x and y directions, respectively. Using Eq. (11.24) compute B. Can this formulation be used to model nearly incompressible problems? Justify your answer.
424 Incompressibleproblems, mixed methods and other procedures of solution 11.8 Consider an axisymmetric problem which is to be solved using a linear quadrilateral element with the displacement approximation 4
u~~
N~ a--1
where Na = 1/4(1 + ~a~)(1 + OAT}) together with a constant approximation for the pressure p. Let the element have a rectangular form with sides a and b in the r and z directions, respectively. Using Eq. (11.24) compute 1]. Can this formulation be used to model nearly incompressible problems? Justify your answer. 11.9 Consider a rectangular plane element with sides a and b in the x and y directions, respectively. (a) Compute the matrix V for GLS stabilization. Ignore second derivatives of u. (b) Consider four elements of equal size with a central node c and compute the assembled equation for the pressure p at this node. 11.10 Consider a rectangular axisymmetric element with sides a and b in the r and z directions, respectively. Let the inner radius of the element be located at ri > a/2. (a) Compute the matrix V for GLS stabilization. Ignore second derivatives of u. (b) Consider four elements of equal size with a central node c located at ri and compute the assembled equation for this node. 11.11 Consider a rectangular plane element with sides a and b in the x and y directions, respectively. (a) Compute the matrix V for direct pressure stabilization. (b) Consider four elements of equal size with a central node c and compute the assembled equation for this node. 11.12 Consider a rectangular axisymmetric element with sides a and b in the r and z directions, respectively. Let the inner radius of the element be located at ri > a/2. (a) Compute the matrix V for direct pressure stabilization. (b) Consider four elements of equal size with a central node c located at ri and compute the assembled equation for this node. 11.13 The steel-rubber composite bearing shown in Fig. 11.16(a) is used to support a heavy machine. The bearing is to have high vertical stiffness but be flexible in shear (similar beatings are also used to support structures in seismic regions). Consider a typical layer where tr = 1 cm and ts = 0.1 cm with a width to = 5 cm. Let the properties be Es = 200 GPa, Vs = 0.3 and Er -- 5 GPa, Vr = 0.495. For a state of plane strain use FEAPpv (or any appropriate available program) to compute the stiffness for a vertical and a horizontal applied loading. Use 4-node and 9-node mixed u - p - e v elements to compute the vertical and horizontal stiffness of a single layer [viz. Fig. 11.16(b)]. Compare your solution to answers from a standard displacement formulation in u. 11.14 Program development project: Extend your program system started in Problem 2.17 to permit solution of a stabilized method as described in Sec. 11.7. You may select a stabilization scheme from either the GLS method of Sec. 11.7.2 or the direct pressure stabilization method of Sec. 11.7.3.
References 425
Fig. 11.16 Support bearing. Problem 12.13.
Use your program to solve the driven cavity problem described in Example 11.5. Set the boundary velocity as shown in Fig. 11.12(a) and the nodal pressure to zero at the centre of the bottom. Plot values shown in Figs 11.12 and 11.13. Also plot contours for velocity components and pressure. Briefly discuss your findings.
1. O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method for Solid and Structural Mechanics. Butterworth-Heinemann, Oxford, 6th edition, 2005. 2. L.R. Herrmann. Finite element bending analysis of plates. In Proc. 1st Conf. Matrix Methods in Structural Mechanics, AFFDL-TR-66-80, pages 577-602, Wright-Patterson Air Force Base, Ohio, 1965. 3. S.W. Key. Variational principle for incompressible and nearly incompressible anisotropic elasticity. Int. J. Solids Struct., 5:951-964, 1969. 4. O.C. Zienkiewicz, R.L. Taylor, and J.A.W. Baynham. Mixed and irreducible formulations in finite element analysis. In S.N. Atluri, R.H. Gallagher, and O.C. Zienkiewicz, editors, Hybrid and Mixed Finite Element Methods, pages 405-431. John Wiley & Sons, 1983. 5. D.N. Arnold, E Brezzi, and M. Fortin. A stable finite element for the Stokes equations. Calcolo, 21:337-344, 1984. 6. M. Fortin and N. Fortin. Newer and newer elements for incompressible flow. In R.H. Gallagher, G.E Carey, J.T. Oden, and O.C. Zienkiewicz, editors, Finite Elements in Fluids, volume 6, Chapter 7, pages 171-188. John Wiley & Sons, 1985. 7. J.T. Oden. R.I.P. methods for Stokesian flow. In R.H. Gallagher, D.N. Norrie, J.T. Oden, and O.C. Zienkiewicz, editors, Finite Elements in Fluids, volume 4, Chapter 15, pages 305-318. John Wiley & Sons, 1982.
426 Incompressibleproblems, mixed methods and other proceduresof solution 8. M. Crouzcix and EA. Raviart. Conforming and non-conforming finite element methods for solving stationary Stokes equations. RAIRO, 7-R3:33-76, 1973. 9. D.S. Malkus. Eigenproblems associated with the discrete LBB condition for incompressible finite elements. Int. J. Eng. Sci., 19:1299-1370, 1981. 10. M. Fortin. Old and new finite elements for incompressible flow. Int. J. Numer. Meth. Fluids, 1:347-364, 1981. 11. C. Taylor and P. Hood. A numerical solution of the Navier-Stokes equations using the finite element technique. Comp. Fluids, 1:73-100, 1973. 12. R.E. Bank and B.D. Welfert. A comparison between the mini-element and the Petrov-Galerkin formulations for the generalized Stokes problem. Comp. Meth. Appl. Mech. Eng., 83:61-68, 1990. 13. T.J.R. Hughes. Generalization of selective integration procedures to anisotropic and non-linear media. Int. J. Numer. Meth. Eng., 15:1413-1418, 1980. 14. J.C. Simo, R.L. Taylor, and K.S. Pister. Variational and projection methods for the volume constraint in finite deformation plasticity. Comp. Meth. Appl. Mech. Eng., 51:177-208, 1985. 15. J.C. Simo and T.J.R. Hughes. Computational Inelasticity, volume 7 of Interdisciplinary Applied Mathematics. Springer-Verlag, Berlin, 1998. 16. D.J. Naylor. Stresses in nearly incompressible materials for finite elements with application to the calculation of excess pore pressures. Int. J. Numer. Meth. Eng., 8:443-460, 1974. 17. O.C. Zienkiewicz and P.N. Godbole. Viscous incompressible flow with special reference to nonNewtonian (plastic) flows. In R.H. Gallagher et al., editor, Finite Elements in Fluids, volume 1, Chapter 2, pages 25-55. John Wiley & Sons, 1975. 18. T.J.R. Hughes, R.L. Taylor, and J.F. Levy. High Reynolds number, steady, incompressible flows by a finite element method. In R.H. Gallagher et al., editor, Fin. Elem. Fluids, volume 3. John Wiley & Sons, 1978. 19. O.C. Zienkiewicz, J. Too, and R.L. Taylor. Reduced integration technique in general analysis of plates and shells. Int. J. Numer. Meth. Eng., 3:275-290, 1971. 20. S.E Pawsey and R.W. Clough. Improved numerical integration of thick slab finite elements. Int. J. Numer. Meth. Eng., 3:575-586, 1971. 21. O.C. Zienkiewicz and E. Hinton. Reduced integration, function smoothing and non-conformity in finite element analysis. J. Franklin Inst., 302:443-461, 1976. 22. O.C. Zienkiewicz and P. Bettess. Infinite elements in the study of fluid-structure interaction problems. In 2nd Int. Symp. on Computing Methods in Applied Science and Engineering, Versailles, France, Dec. 1975. 23. O.C. Zienkiewicz. The Finite Element Method. McGraw-Hill, London, 3rd edition, 1977. 24. D.S. Malkus and T.J.R. Hughes. Mixed finite element methods in reduced and selective integration techniques: a unification of concepts. Comp. Meth. Appl. Mech. Eng., 15:63-81, 1978. 25. O.C. Zienkiewicz and S. Nakazawa. On variational formulations and its modification for numerical solution. Comp. Struct., 19:303-313, 1984. 26. M.S. Engleman, R.L. Sani, P.M. Gresho, and H. Bercovier. Consistent vs. reduced integration penalty methods for incompressible media using several old and new elements. Int. J. Numer. Meth. Fluids, 2:25-42, 1982. 27. D.N. Arnold. Discretization by finite elements of a model parameter dependent problem. Num. Meth., 37:405-421, 1981. 28. M.J.D. Powell. A method for nonlinear constraints in minimization problems. In R. Fletcher, editor, Optimization. Academic Press, London, 1969. 29. M. Vogelius. An analysis of the p-version of the finite element method for nearly incompressible materials; uniformly optimal error estimates. Num. Math., 41:39-53, 1983. 30. S.W. Sloan and M.E Randolf. Numerical prediction of collapse loads using finite element methods. Int. J. Numer. Anal. Meth. Geomech., 6:47-76, 1982.
References 427 31. J.C. Nagtegaal, D.M. Parks, and J.R. Rice. On numerical accurate finite element solutions in the fully plastic range. Comp. Meth. Appl. Mech. Eng., 4:153-177, 1974. 32. O.C. Zienkiewicz, J.P. Vilotte, S. Toyoshima, and S. Nakazawa. Iterative method for constrained and mixed approximation. An inexpensive improvement of FEM performance. Comp. Meth. Appl. Mech. Eng., 51:3-29, 1985. 33. K.J. Arrow, L. Hurwicz, and H. Uzawa. Studies in Non-LinearProgramming. Stanford University Press, Stanford, CA, 1958. 34. M.R. Hestenes. Multiplier and gradient methods. J. Opt. Theory Appl., 4:303-320, 1969. 35. C.A. Felippa. Iterative procedure for improving penalty function solutions of algebraic systems. Int. J. Numer. Meth. Eng., 12:165-185, 1978. 36. M. Fortin and E Thomasset. Mixed finite element methods for incompressible flow problems. J. Comp. Phys., 31:113-145, 1973. 37. M. Fortin and R. Glowinski. Augmented Lagrangian Methods: Applications to Numerical Solution of Boundary- Value Problems. North-Holland, Amsterdam, 1983. 38. D.G. Luenberger. Linear and Nonlinear Programming. Addison-Wesley, Reading, Mass., 1984. 39. J.H. Argyris. Three-dimensional anisotropic and inhomogeneous m e d i a - matrix analysis for small and large displacements. Ingenieur Archiv, 34:33-55, 1965. 40. O.C. Zienkiewicz and S. Valliappan. Analysis of real structures for creep plasticity and other complex constitutive laws. In M. Te'eni, editor, Structure of Solid Mechanics and Engineering Design, volume, Part 1, pages 27-48, 1971. 41. O.C. Zienkiewicz. The Finite Element Method in Engineering Science. McGraw-Hill, London, 2nd edition, 1971. 42. J.C. Simo and R.L. Taylor. Quasi-incompressible finite elasticity in principal stretches: continuum basis and numerical algorithms. Comp. Meth. Appl. Mech. Eng., 85:273-310, 1991. 43. R. Courant. Variational methods for the solution of problems of equilibrium and vibration. Bull. Amer. Math Soc., 49:1-61, 1943. 44. E Brezzi and J. Pitk~anta. On the stabilization of finite element approximations of the Stokes problem. In W. Hackbusch, editor, Efficient Solution of Elliptic Problems, Notes on Numerical Fluid Mechanics, volume 10. Vieweg, Wiesbaden, 1984. 45. T.J.R. Hughes, L.P. Franca, and M. Balestra. A new finite element formulation for computational fluid dynamics: V. Circumventing the Babu~ka-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Comp. Meth. Appl. Mech. Eng., 59:85-99, 1986. 46. T.J.R. Hughes and L.P. Franca. A new finite element formulation for computational fluid dynamics: VII. The Stokes problem with various well-posed boundary conditions: symmetric formulation that converge for all velocity/pressure spaces. Comp. Meth. Appl. Mech. Eng., 65:85-96, 1987. 47. T.J.R. Hughes, L.P. Franca, and G.M. Hulbert. A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective-diffusive equations. Comp. Meth. Appl. Mech. Eng., 73:173-189, 1989. 48. E. Ofiate. Derivation of stabilized equations for numerical solution of advective-diffusive transport and fluid flow problems. Comp. Meth. Appl. Mech. Eng., 151:233-265, 1998. 49. C.R. Dohrmann and P.B. Bochev. A stabilized finite element method for the Stokes problem based on polynomial pressure projections. Int. J. Numer. Meth. Fluids, to appear, 2005. 50. O.C. Zienkiewicz and J. Wu. Incompressibility without tears! How to avoid restrictions of mixed formulations. Int. J. Numer. Meth. Eng., 32:1184-1203, 1991. 51. O.C. Zienkiewicz and R. Codina. A general algorithm for compressible and incompressible flow - Part I: The split, characteristic-based scheme. Int. J. Numer. Meth. Fluids, 20:869-885, 1995. 52. O.C. Zienkiewicz, P. Nithiarasu, R. Codina, M. Vasquez, and P. Ortiz. The characteristic-basedsplit (CBS) procedure: an efficient and accurate algorithm for fluid problems. Int. J. Numer. Meth. Fluids, 31:359-392, 1999.
428 Incompressibleproblems, mixed methods and other procedures of solution 53. O.C. Zienkiewicz, K. Morgan, B.V.K. Satya Sai, R. Codina, and M. Vasquez. A general algorithm for compressible and incompressible flow- Part II: Tests on the explicit form. Int. J. Numer. Meth. Fluids, 20:887-913, 1995. 54. P. Nithiarasu and O.C. Zienkiewicz. On stabilization of the CBS algorithm. Internal and external time steps. Int. J. Numer. Meth. Eng., 48:875-880, 2000. 55. P. Nithiarasu. An efficient artificial compressibility (AC) scheme based on the characteristic based split (CBS) method for incompressible flows. Int. J. Numer. Meth. Eng., 56:1815-1845, 2003. 56. O.C. Zienkiewicz, R.L. Taylor, and P. Nithiarasu. The Finite Element Method for Fluid Dynamics. Butterworth-Heinemann, Oxford, 6th edition, 2005. 57. O.C. Zienkiewicz. Origins, milestones and directions of the finite element method. A personal view. In P.G. Ciarlet and J.L. Lyons, editors, Handbook of NumericalAnalysis, volume IV, pages 3-65. North Holland, 1996. 58. R. Pierre. Simple C Oapproximations for the computation of incompressible flows. Comp. Meth. Appl. Mech. Eng., 68:205-227, 1988.
In the previous chapters we have assumed in the approximations that all the variables were defined in the same manner throughout the domain of the analysis. This process can, however, be conveniently abandoned on occasion with the same or different formulations adopted in different subdomains of the problem. In this case some variables are only approximated on surfaces joining such subdomains. There are two motivations for separating the whole domain into several subdomain regions. In the first of these the concept of parallel computation is paramount. Such parallel computation has become very important in many fields of engineering and allows us to use completely different methodologies for solving the problem in each individual part and even if this is not used allows us very much to increase the computer power by having separate operations going on simultaneously. In general the process we have just mentioned is referred to as domain decomposition and we shall devote the first part of this chapter to domain decomposition methodologies. As this volume is not concemed in detail with the process of calculation and therefore does not discuss the subject of parallel computing in extended form, we refer the reader to references on the subject. 1-4 Indeed the whole problem of domain decomposition associated with parallel computation is today an active field in which many conferences are held at frequent intervals and which seem to stir the imagination of many mathematicians and engineers. We shall discuss the problems of this kind in part one of this chapter. It is of interest to note, however, that the methodologies for connecting separate subdomains can have other outcomes and objectives. In particular here the so-called frame methods have proved successful and, for instance, the introduction of hybrid elements by Pian et al. pioneered this type of approximation in the 1960s. 5' 6 More recently, other forms of frame approximation have been introduced and of particular interest is one in which socalled boundary approximations are used within an element with standard displacements specified on a frame. This allows for the introduction of many complex elements, capable of dealing with interesting problems on their own, which can be linked with more standard finite element computations. 7-14
430 Multidomainmixed approximations- domain decomposition and 'frame' methods
Domain decompostion methods
In this section we deal with the problem of connecting two or more subdomains in which standard finite element approximations of one form or another have been used. In particular we shall give as examples the process in which 'irreducible' formulations are used but, of course, other approximations could be introduced. The linking of such subdomains can be easily accomplished by the introduction of Lagrange multiplier methods to which we already referred to in Chapter 3 and elsewhere. The Lagrange multipliers for this case are defined on the boundary interface of the connecting subdomains. In the present case we consider two subdomains, ~'~1 and f~2, which are to be joined together along an interface 1-'i. The generalization to multiple domains follows the same pattern. Independently approximated Lagrange multipliers (fluxes or tractions) are used on the interface to join the subdomains, as in Fig. 12.1. In the first problem considered we treat the quasi-harmonic equation expressed in terms of the scalar potential function 4~. This is followed by treatment for the elasticity problem.
12.2.1 Linkingsubdomainsfor quasi-harmonicequations In Chapter 7 we considered the general problem for steady-state field problems. This problem resulted in a weak form in terms of a potential function ~b. The approximation in a domain f21 may be expressed as [viz. see Eq. (7.13)] (we ignore ~b1 for simplicity)
q
i
q
i
(12.1) where the normal flux has been replaced by a Lagrange multiplier function k defined on the interface 1-'~. Similarly, for the domain f22 we have
n2
(12.2)
F 2 U Ft2
F~ O F t1 Fig. 12.1 Linking two (or more) domains by traction variables defined only on the interfaces. Variables in each domain are displacement u (irreducible form).
Linking of two or more subdomains by Lagrange multipliers
in which we have used q~ = -~. to satisfy flux continuity on the interface. The two subdomain equations are completed by a weak statement of continuity of the potential between the two subdomains. Thus we have f r 6 ~ . (4)1 - 4)2) dF = 0
(12.3)
I
Discretization of the potential in each domain and the Lagrange multiplier on the interface yields the final set of equations. Thus expressing the independent approximations as ~1 __ N I ~ 1., (])2 : N2~ 2 and ~. = Nz~,
we have
H1
0
Q1
~1
0
H2
Q2
(7)2
Q1T Q2T o
(12.4)
fl _+_
3,
f2
= 0
(12.5)
o
where H 1 - f~ (VN1) T(k 1VN1) dr2, 1
f + [
f H 2 = ] (VN2) T(k2vN2) dfl, J~ 2
NTH1N1 dF,
+ f r NTHZNzdl-"
q~
Jr 1
Q1 : f r NTNx dl-',
Q2
(12.6)
- fri N2TN~dr,
I
fl --
NTQ 1 dr2 -Jr-
NIT~/1dr,
q~
f2=f
NN T QT2 d~f 2/+ f2r d F 2
The formulation outlined above for two domains can obviously be extended to many subdomains and in many cases of practical analysis is useful in ensuring a better matrix conditioning and allowing the solution to be obtained with reduced computational effort. 15 The variables ~bI and ~b2 appear as intemal and boundary variables within each subdomain (or superelement) and can be eliminated locally providing the matrices H 1 and H 2 are non-singular. Such non-singularity presupposes, however, that each of the subdomains has enough prescribed values to prevent the singular modes. If this is not the case partial elimination is always possible, retaining the singular modes until the complete solution is achieved. We note that in the derivation of the matrices in Eq. (12.6) the shape function Nz and hence ~. itself are only specified along the interface surface. The choice of appropriate functions for the Nz must, of course, satisfy the mixed patch requirement with counts performed for the interface degree of freedoms. Here the count condition can be more difficult to satisfy when multiple subdomains are connected at a point or along a line due to presence of multiple ~. functions at these locations. One procedure to satisfy the condition is to use m o r t a r or d u a l m o r t a r methods. This matter is taken up later in this section. However, prior to this we consider the use of the above to include the Dirichlet boundary condition 4) - 4) = 0 as part of the weak solution to the problem.
431
432
Multidomain mixed approximations- domain decomposition and 'frame' methods
Treatment for forced boundary conditions
We note that the above form also may be used to satisfy the forced boundary condition r = r on Fr For this we let FI = Fr and from Eq. (12.1) (dropping the superscript '1') obtain
f~ [(V6r T (k Vr
t;r Q] d~2 + fr 6r (//+ He)dr + q
Similarly, from Eq. (12.3) with r
fr
~r Z d r r
=0
(12.7)
.__ t~ we have
fr ~Z(r
=o
(12.8)
The discrete form of the equations becomes
H
QT
Q
(12.9)
0
where
q
Q-
f r NTNzdF
(12.10)
r
_ _
f q
NzCdF r
in which r = N(b and )~ -- Nx.~
Mortar and dual mortar methods
(12.11)
The mortar m e t h o d is a procedure which is used to join multiple subdomains. 16 Consider as an example a two-dimensional problem in which we use 4-noded (bilinear) quadrilaterals in ~,'~1 and 9-noded (biquadratic) quadrilaterals in f22. To connect subdomains the Lagrange multiplier may be approximated as shown in Fig. 12.2 for a subdomain with five segments along the interface. The use of the constant part at an end is required if multiple subdomains exist at the end point, otherwise the interpolation may be continued with normal linear interpolation as shown for the left end in Fig. 12.2. Along the interface we may connect subdomains with a different number of segments as shown in Fig. 12.3(a). Thus, if we
Fig. 12.2 Mortar function for Lagrange multiplier. Formfor linear edges on ~1 elements.
Linking of two or more subdomains by Lagrange multipliers
O•D 2
t
,,
~2
1
0
~
,,:
T
~0
,,:
_U I
To'
0
i
,,.
T
(b) Subincrements for quadrature
(a) Interface for D1 and D2 Fig. 12.3 Two-dimensional mortar interface.
assume the Lagrange multiplier interpolation uses Nz = N1 (except at end points)t the interface term resulting from (12.3) yields Ql=fr
I
NTNldl -' and Q 2 = - f r
I
NTNldI '
(12.12)
The integral for Q1 may be evaluated for each element edge using quadrature described in Chapter 4; however, evaluation by quadrature of the integral for Q2 requires further subdivision into subincrements along the element edges as indicated in Fig. 12.3(b). The dual mortar method is an alternate form of the mortar method which has advantages for Lagrange multiplier and penalty forms. The dual shape functions are defined to satisfy Na Nb d~ = 3ab e
Nb d~
(12.13)
e
A
where Na denotes a dual shape function and (~ab is a Kronecker delta function. Figure 12.4 shows the dual functions computed for the standard linear functions shown in Fig. 12.2. The dual functions are discontinuous between elements, which is permitted since no derivatives appear for the Lagrange multipliers A. The dual functions may be computed for each element edge separately. For linear edges the result is shown in Fig. 12.5. The process may be repeated for higher order functions without difficulty; however, for higher order edges nodes appear between the ends and, thus, with arbitrary spacing the computation must be computed for each case separately. The advantage of the dual functions is evident from the definition of Q1 (assuming we start with N~ = N1). Here we observe that Q1 = fF N T i s ' d r
=
~1
(12.14)
I
where ~1 is diagonal by the properties of Eq. (12.13). t An alternativeto avoiding modification at end points is to use a stabilization method such as one equivalentto the direct pressure stabilizationpresented in Sec. 11.7.3.
433
434 Multidomainmixed approximations- domain decompositionand 'frame' methods
Fig. 12.5 Mortar and dual mortar shape functions for two-dimensional linear edge.
12.2.2 Linking subdomains for elasticity equations In this problem we formulate the approximation in domain ~'21 in terms of displacements U 1 resulting from an irreducible (displacement) form of the elasticity equations. The traction t 1 on the interface is denoted by ,,k. With the weak form using the standard virtual work expression [see Eq. (6.52)] ignoring tr0 and e0 we havet
1
I
1
t1
in which as usual we assume that the satisfaction of the prescribed displacement on F 1 is implied by the approximation for u 1. Similarly in domain ~"22 w e can write, now putting the interface traction as t 2 = -)~ to ensure equilibrium between the two domains,
2
I
2
2
The two subdomain equations are completed by a weak statement of displacement continuity on the interface between the two domains, i.e., t Here we use (6.4) to replace e and 8r by ,Su and ,SSu, respectively.
Linking of two or more subdomains by Lagrange multipliers
fr ~)~T(u2 -- u l ) d r -- 0 I
(12.17)
Discretization of displacements in each domain and of the Lagrange multipliers (tractions) )~ on the interface yields the final system of equations. Thus putting the independent approximations as U 1 -- N l u l ; U2 -- N2u2; ,~ - NxJ, (12.18) we have
K 0
0
K2 Q2T
Q1T
~176 {,1}
Q2 o
1~12
__
X
f2 o
(12.19)
where K1 =~f~ B1TD1B1 d ~ ' QI=_
fl
f r N1TNzdF' t
K2=s
B2TD2B2df2 (12.20)
Q 2 = J~r N~Nzdr I
o~~1 NITbl d~'~-~= jry i NT~I d r ,
f2
o~~1N2Tb2 d~"~-~-~2 NTt'2 d r
The process described here is very similar to that introduced by Kron 17 at a very early date and, more recently, used by Farhat et al. in the FETI (finite element tearing and interconnecting) method 18 which uses the process on many individual element partitions as a means of iteratively solving large problems. The formulation is, of course, subject to limitations imposed by the stability and consistency conditions of the mixed patch test for selection of appropriate number of X variables. The formulation just used can, of course, be applied to a single field displacement formulation in which we are required to specify the displacement on the boundaries in a weak sense - rather than imposing these directly on displacement shape functions. This problem can be approached directly or can be derived simply using (12.15) and (12.17) in which we put u 2 = ~, the specified displacement on FI -- Fu. Now the equation system is simply
[QKllT Q1] (,1 _ f,
fz=-fr
where
(12.21)
N~8dr
(12.22)
I This formulation is often convenient for imposing a prescribed displacement on an element field when the boundary values cannot fit the shape function form. We have approached the above formulation directly via weak or weighted residual forms. A variational principle could be given here simply as the minimization of total potential energy (see Chapter 2) subject to a Lagrange multiplier A imposing subdomain continuity. The stationarity of H =
._
-~
i
( S u i ) T D i S u i dr2 -
i
dr2 -
~
dr
-+-
i
)kT(u 2 - u
dr
(12.23)
435
436
Multidomain mixed approximations- domain decomposition and 'frame' methods
would result in the equation set (12.15) to (12.17). The formulation is, of course, subject to limitations imposed by the stability and consistency conditions of the mixed patch test for selection of the appropriate number of )~ variables.
Example 12.1: A mortar method for two-dimentional elasticity. Mortar and dual mortar methods may also be used in the solution of elasticity problems. The formulation follows that given for the quasi-harmonic equation with appropriate change in variables. To indicate the type of result which occurs using mortar or dual mortar forms we consider the problem of a strip loaded by a uniform pressure along a short segment. The problem is solved as a single region using a fine mesh over the whole domain and also by a two subdomain form in which fine elements are in the top layer only (see Fig. 12.6). Contours for the vertical displacement are presented in Fig. 12.7 for the two cases. It is evident that the mortar treatment produces excellent continuity in displacement. A comparison for the vertical stress, Cry,is not shown here again the results exhibit very small discontinuity.
In the previous section we have shown how linking can be achieved using Lagrange multipliers. A disadvantage of the Lagrange multiplier approach is the addition of extra unknowns (the Lagrange multipliers )~) and the creation of equations which have zero on the diagonal. As we have shown previously (viz. Chapter 3) it is possible to avoid both of these situations using a perturbed lagrangian or penalty form. The perturbed lagrangian form of the equations may be achieved by modifying Eq. (12.17) to f
~)kT(u2--ul)dl I
-"
1 f r 6)~x,,~dF = 0 13/
(a) No interface
Fig. 12.6 Mesh and nodal loading for vertically loaded strip.
l
(b) Mortar interface
(12.24)
Linking of two or more subdomains by perturbed lagrangian and penalty methods
Fig. 12.7 Vertical displacement for strip loaded over short segment of top.
in which ct is a large (penalty) parameter. Inserting the approximation (12.18) into (12.15), (12.16) and (12.24) results in the form
Fo 1 Ko2 LQ,T Q2T
ol]{ol} {fl}
Q2
~12
-~v
x
f.2
__
(12.25)
o
where in addition to the arrays defined in Eq. (12.20) V = f r N~Nz dl-'
(12.26)
I
Clearly, as the parameter ct tends to infinity the result becomes identical to the Lagrange multiplier form. Such approximation thus behaves as a penalty-type form. Formally, we can eliminate the Lagrange multiplier parameters from (12.25) to obtain otQ2V-1Q 1T
I{o'} {,'}
(K 2 4- otQ2V-1Q2T)]
fi2
=
f2
(12.27)
which we recognize as a penalty-type form [K1 + o~K2] u = f
[viz. Eq. (3.137)]. An alternative to the above solves (12.24)for each point on the boundary FI yielding ,,~ = ot (u 2 - u')
(12.28)
Substituting this into (12.15) and (12.16) then gives
f~ (~(Sui)TD1SulH~'~ mO~ f~ ~ulT(u2mul)dF-f~ ~ulWbd~~-f~ ~ulTidF --0 1 I 1 1
(12.29)
437
438 Multidomainmixed approximations- domain decomposition and 'frame' methods and
L '(SuZ)TDZSuZdf2 +afr 'u'T(uZ-u')dF-f.'uZTbdf2-fr'uZTidF= O 2
I
2
2
(12.30)
Introducing now the approximations for u 1 and u 2 produces the penalty form (K
where
c~KJ1) (K2..I._o/K22)j -c~KJ e ] UI1 {fi2 }--{f2 f' }
(12.31)
-Jlo/K21
K~= / f
(12.32)
N/TNgdF; i , j = 1,2
dF I
which we again recognize as a penalty form as given in Eq. (3.137). The differences between the penalty form (12.27) and that of (12.31) are significant: (a) The form given by (12.27) will not exhibit l o c k i n g provided the choice for the Nz satisfies the conditions for the mixed patch test. (b) The form given by (12.31) and (12.32) usually requires use of r e d u c e d q u a d r a t u r e on K~j in order to avoid locking for reasons we discussed in Chapter 11. (c) Using standard or dual mortar methods the form (12.27) satisfies consistency conditions (e.g., constant stress) across the interface. 19 Generally, the form (12.31) does not transmit a constant stress condition correctly at the interface unless perfect matching of meshes occurs on FI. The above remarks clearly favour the form (12.27); however, this form requires the inversion of the matrix V (or a solution process equivalent to such inversion) and this can present difficulties. For the dual mortar method discussed in Sec. 12.2.1, the Lagrange multiplier can be eliminated by a perturbed lagrangian approach using the discretized form of (12.26) approximated as V ,~ / dF
lq~Nx d r
(12.33)
i
In this case the matrix to be inverted is also diagonal and the Lagrange multiplier may be locally eliminated to give a penalty form.
12.3.1 Nitsche method and discontinuous Galerkin approximation An alternative to Lagrange multiplier and penalty methods for including the Dirichlet boundary condition was introduced by Nitsche. 2~ Here we consider the procedure to include the condition 4~ = ~ in the weak form of the quasi-harmonic equation. We first add together Eqs (12.7) and (12.8) to obtain
q
0
+ Jl ~x(O - ~) dr = 0 The normal flux qn on the boundary part FO is replaced by -- qn(dp) = -nT(kV~P)
6~ = qn(SCP) = - n T(kvSo)
(12.34)
Linking of two or more subdomains by perturbed lagrangian and penalty methods 439
Fig. 12.8 Solution of one-dimensional heat equation of Example 3.5 in Chapter 3.
thus eliminating the appearance of the Lagrange multiplier and giving a weak form expressed entirely in terms of ~. One now can note that these two terms on FO can be zero for ~ and g0 having constant values. Thus, to make the method stable Nitsche adds a penalty-like term
However, it is not required that ot be large to ensure a good satisfaction of the boundary condition. The value recommended by Nitsche for linear elements is
c~=c
Ikl
c = O(10)
h
where h is an element size and Ikl a norm of the diffusion matrix. The above steps give the weak form
j f [(V6q~) T ( k V O ) + 8~bQ] dr2 + f r 80 (~/+ H O ) d F + Jfr
8~qn(dp)dF
q
-+-fr qn(~b)[~b-- ~] d r Sym~netry
q- fF ~q~ff[~b - ~] dl-" ---0
(12.35)
Stab'qlity
Substituting the approximation for 0 given by Eq. (12.11) into (12.35) gives
H~=f
(12.36)
440
Multidomain mixed approximations - domain decomposition and 'frame' methods
where H = J~ (~rN)Tk(VN) dr2 + f r NTHN dF q
-fr
NT(nT[k(VN)])dF-fr
(nT[k(VN)])TNdF+fr
NTc~Ndl-"
q
- j; (nT[k(~7N)])T~ dF -I-fr NTot~dF The Nitsche method results in a form in terms of the original primary variables of the problem. We can easily extend this to consider the connection of multiple subdomains.
Example 12.2: Dirichlet boundary condition. To indicate the performance of the Nitsche method in satisfaction of the Dirichlet boundary condition, we consider the one-dimensional problem given in Chapter 3 as Example 3.5. There the differential equation was given as A(~) =
d2r
dx 2 + Q ( x ) = 0 0 < x <_ L
with boundary conditions ~p(0) = cp(L) = 0. We shall consider two domains: f21 for 0 < x < L/2 and f22 for L/2 < x < L. The loading on f22 is a linear continuous function and that on f21 is zero. Using the Lagrange multiplier solution for this problem results in exact satisfaction of the boundary conditions 4~(0) = ~p(L) = 0, and, consequently, the same solution as given for the standard finite element solution in Chapter 3, Fig. 3.5. Using the Nitsche method with c = 10 and h = L / 4 and L / 8 (k = 1) gives the solution shown in Fig. 12.8(a). For comparison we drop the terms on the boundary with q(40, and q(8~p) (i.e., use the penalty form alone) but keep the same value for c. This solution is shown in Fig. 12.8(b). Of course, increasing the size of c with either approach will improve the satisfaction of the boundary condition- but with an increased sensitivity in equation solution. The overall improvement of the Nitsche method is clearly evident and is accomplished without an increase in equation condition number. The results using quadratic results are even better as shown in Fig. 12.9.
Multiple subdomain problems
We again return to the problem of connecting two subdomains defined in f21 and ~'22 in which the common interface is FI. The weak form of the problem may be written now as 2 9__
i
iq
(12.37) I
I
+ fr [a4~ - ~4 ~2] ~ [4~1 - 4~2] dr' = 0 I
Linking of two or more subdomains by perturbed lagrangian and penalty methods
Fig. 12.10 Two subdomain solution using Nitsche method for one-dimensional heat equation of Example 3.5 in Chapter 3. Linear elements.
441
442 Multidomainmixed approximations-domain decomposition and 'frame' methods which results from adding Eqs (12.1) and (12.2) and setting the Lagrange multiplier to ~, : qn(~ 1, t~2) and 8)~ = qn(S~b 1, ~q~2)
(12.38)
which now becomes a function of the flux from both sides of the interface. This form is an extension of the concept of Nitsche and, of course, can be effectively used to consider multiple subdomains in an obvious manner. When extended to the case where each element becomes a subdomain the problem assumes a form known as the discontinuous Galerkin method. 21-25
The discontinuous Galerkin method was first introduced by Reed and Hill 26 for analysis of neutron transport problems. It was analysed by Lesaint and Raviart for its mathematical properties. 27 As shown in the paper by Zienkiewicz et al. the method is most effective in problems which have significant convection effects - and is less accurate than standard (continuous) finite elements for problems which possess only diffusion effects. 28 Here we are interested in the method primarily for connecting subdomains which contain either a large number of standard elements or have high order expansions with significant number of parameters not associated with the boundary.
Example 12.3: Two domain problem. To indicate the performance in the presence of multiple domains we again consider the one-dimensional of Example 12.2. We shall consider two domains: f21 for 0 < x < L / 2 and f22 for L / 2 < x < L. The loading on ~22 is a linear continuous function and that on ~1 is zero. The Nitsche method is used with four and eight elements (two and four in each subdomain, respectively) and a value of c = 10. The solution is shown in Fig. 12.10 and again indicates quite rapid convergence with increased number of elements. We also show results for the same problem with quadratic elements, Fig. 12.11, in which no discernible jump exists for the eight-element case.
Frame methods
In the preceding examples we have used traction as the Lagrange multiplier interface variable linking two or more subdomains of elasticity problems. Due to lack of rigid body constraints the elimination of local subdomain displacements has generally been impossible. For this and other reasons it is convenient to accomplish the linking of subdomains via a displacement field defined only on the interface [Fig. 12.12(a)] and to eliminate all the interior variables so that this linking can be accomplished via a standard stiffness matrix procedure using only the interface variables. A displacement frame can be made to surround the subdomain completely and if all internal variables are eliminated will yield a stiffness matrix of a new 'element' which can be used directly in coupling with any other element with similar displacement assumptions on the interface, irrespective of the procedure used for deriving such an element [Fig. 12.12(b)]. In all the examples of this section we shall approximate the frame displacements as v - Nv~
on
F/
(12.39)
Interface displacement 'frame'
0.03[
~;
]
~176.......................................... ~
~ ~176.................. ................. . /
~o.o,,[ /
~176
............1
'
l
~176..............................................~
1 '~ ~176
i ll ~~176 /
.................................. ~ , ........
................. ............... /
....... 1
..................................
\ "1
.~0o.o,i....../..... ..............................~.......~'T .......%.~0~ .....o.o, ... ............... ,~..................................................... ~............... ............................................ ....
V4
O
0
0.2
~
0.4 0.6 xlL coordinate
0.8
0
1
0
(a) 4-Elements: c = 10
0.2
................................i............. "1
0.4 0.6 xlL coordinate
0.8
1
(b) 8-Elements: c = 10
Fig. 12.11 Two subdomain solution using Nitsche method for one-dimensional heat equation of Example 3.5 in Chapter 3. Quadratic elements.
F 1
1-"I
Interface frame u = v Nv~
Nodes defining v
(a)
(b)
Fig. 12.12 Interface displacement field specified on a 'frame' linking subdomains. (a) Two-domain link. (b) A 'superelement' (hybrid) which can be linked to many other similar elements.
and consider the 'nodal forces' contributed by a single subdomain ~,-~1 to the 'nodes' on this frame. Using virtual work (or weak) statements we have with discretization J I N T t dF = ql
1
(12.40)
where t are the tractions the interior exerts on the imaginary frame and q l are the nodal forces developed. The balance of the nodal forces contributed by each subdomain now provides the weak condition for traction continuity. As finally the tractions t can be expressed in terms of the frame parameters r only, we shall arrive at ql = KI~ + fl (12.41)
443
444 Multidomain mixed approximations- domain decomposition and 'frame' methods
where K 1 is the stiffness matrix of the subdomain ~,'~1 and f~ its intemally contributed 'forces'. From this point onwards the standard assembly procedures are valid and the subdomain can be treated as a standard element which can be assembled with others by ensuring that Z qj = 0 J
(12.42)
where the sum includes all subdomains (or elements). We thus have only to consider a single subdomain ~"2e in what follows.
12.4.2 Linking displacement frame on equilibrating form subdomains In this form we shall assume a priori the stress field expansion is given by aT = ~r + tr0
(12.43)
and that the equilibrium equations are identically satisfied as ST o" ~ 0; S TO"0 ~ b in f2
and
Get : 0; Gtr0 : i on Fte
The equation '(Su)Tcrd~-~ e
'uTbd~-~r
8uTtdF-0
e
(12.44)
te
is identically satisfied and we write a weak form of the constitutive equation and interface condition as (see Chapter 10, Sec. 10.6) ~
~r rT(D-ltrr - S u ) dr2 + f e
~tT(u --V) dF le
e
le
On discretization, noting that the field u does not enter the problem tr=No#
and v=N~r162
we have, on including Eq. (12.40), [QAe; Qe] { O} ' } - - ~{ q e f ~--f~
(12.45)
where Ae - f~e NaD-1Na dr2
Qe = _ fr (GNo)TNo dF Ie
ff = - f~e NaD-ltr~ dQ
and f~ = / i
NvGtro dF Ie
Linking of boundary (or Trefftz)-type solution by the 'frame' of specified displacements 445 Here elimination of 6" is simple and we can write directly and
Ke~, __ qe _ f~ _~. Q e T ( A e ) - l f ~
K e = QeT(Ae)-IQe
(12.46)
In Secl0.6 we have discussed the possible equilibration fields and have indicated the difficulties in choosing such fields for a finite element, subdivided, field. In the present case the situation is quite simple as the parameters describing the equilibrating stresses inside the element can be chosen arbitrarily in a polynomial expression.
Example 12.4: Equilibrium field. If we use a simple polynomial expression in two dimensions: Crx = c~0 + C~lX + ot2y O'y : /~O -~- /~l X -lt- /~2 y rxy = YO + ylX -k- Y2Y
we note that to satisfy the equilibrium we require
~ T o " "-
0
Oy
Oy
Ox
0
0
dr =
Oil + ) " 2 /~2 -+- YI
=0
and this simply means )/2 - - m ~ l Yl = --/32
Thus a linear expansion in terms of 9 - 2 -- 7 independent parameters is easily achieved. Similar expansions can of course be used with higher order terms. It is interesting to observe that: 1. n,~ > nv - 3 is needed to preserve stability. 2. By the principle of limitation, the accuracy of this approximation cannot be better than that achieved by a simple displacement formulation with compatible expansion of v throughout the element, providing similar polynomial expressions arise in stress component variations. In practice two advantages of such elements, known as hybrid-stress elements, are obtained. In the first place it is not necessary to construct compatible displacement fields throughout the element (a point useful in their application to, say, a plate bending problem). In the second for distorted (isoparametric) elements it is easy to use stress fields varying with the global coordinates. The first use of such elements was made by Pian 5 and many successful variants are in use today. 6' 29-41
Boundary methods in which the chosen fields for both displacement and stress fields satisfy a priori the homogeneous equations of equilibrium and constitutive equations (and indeed
446 Multidomainmixed approximations-domain decomposition and 'frame' methods
on occasion some prescribed boundary traction or displacement conditions) have been considered by Trefftz. 42 Here such methods are called Trefftz-type solutions. Thus in Eqs (12.45) and (12.44) the subdomain (element e) ~2e integral terms disappear and, as the internal ~t and ~u variations are linked, we combine all into a single statement (in the absence of body force terms) as
~tT(u-v) dF-fr 8uT(t--i) dl'=O
-fr Ie
(12.47)
te
This coupled with the boundary statement (12.40) provides the means of devising stiffness matrix statements of such subdomains. For instance, if we express the approximate fields as u = Nfi
(12.48)
implying tr = D(SN)fi
and
t = Gcr = GD(SN)~
we can write [_QHe~ Qe] { ~ } = { ~ }
(12.49)
where
He -
frIe[GD(SN)]TNdF+frteNTGD(SN)dF
Qe__
f r [GD(SN)ITNvdF
(12.50)
Ie
f f - - - - f r NTidl-' te
In Eqs (12.49) and (12.50) we have omitted the domain integral of the particular solution tr0 corresponding to the body forces b but have allowed a portion of the boundary Fte to be subject to prescribed tractions. Full expressions including the particular solution can easily be derived. Equation (12.49) is immediately available for solution of a single boundary problem in which v and t. are described on portions of the boundary. More importantly, however, it results in a very simple stiffness matrix for a full element enclosed by the frame. We now have Ke~ = q + fe (12.51) in which K e = QeT(He)-IQe fe __ QeT(He)-lf~
(12.52)
This form is very similar to that of Eq. (12.46) except that now only integrals on the boundaries of the subdomain element need to be evaluated. Much has been written about so-called 'boundary elements' and their merits and disadvantages.9-11,13, 43-51 Very frequently singular Green's functions are used to satisfy the governing field equations in the domain. 46-5~ The singular function distributions used do
Linking of boundary (or Trefftz)-type solution by the 'frame' of specified displacements 447 not lend themselves readily to the derivation of symmetric coupling forms of the type given in Eq. (12.49). Zienkiewicz et al. 51-54 show that it is possible to obtain symmetry at a cost of two successive integrations. Further it should be noted that the singular distributions always involve difficult integration over a point of singularity and special procedures need to be used for numerical implementation. For this reason the use of generally non-singular Trefftz functions is preferable and it is possible to derive complete sets of functions satisfying the governing equations without introducing singularities, 51-54 and simple integration then suffices. While boundary solutions are confined to linear homogeneous domains these give very accurate solutions for a limited range of parameters, and their combination with 'standard' finite elements has been occasionally described. Several coupling procedures have been developed in the past, 51-54 but the form given here coincides with the work of Zielinski and Zienkiewicz, 55 Jirousek 7, 56-58 and Piltner.14 Jirousek et al. have developed very general two-dimensional elasticity and plate bending elements which can be enclosed by a many-sided polygonal domain (element) that can be directly coupled to standard elements providing that same-displacement interpolation along the edges is involved, as shown in Fig. 12.13. Here both interior elements with a frame enclosing an element volume and exterior elements satisfying tractions at free surface and infinity are illustrated. Rather than combining in a finite element mesh the standard and the Trefftz-type elements ('T-elements '13) it is often preferable to use the T-elements alone. This results in the whole domain being discretized by elements of the same nature and offering each about the same degree of accuracy. The subprogram of such elements can include an arsenal of homogeneous 'shape functions' N e [see Eq. (12.48)] which are exact solutions to different types of singularities as well as those which automatically satisfy traction boundary conditions on internal boundaries, e.g., circles or ellipses inscribed within large elements as shown in Fig. 12.14. Moreover, by completing the set of homogeneous shape functions by suitable 'load terms' representing the non-homogeneous differential equation solution, u0,
Fig. 12.13 Boundary-Trefftz-type elements (T) with complex-shaped 'frames' allowing combination with standard, displacement elements (D). (a) An interior element. (b) An exterior element.
448 Multidomainmixed approximations- domain decomposition and 'frame' methods
Fig. 12.14 Boundary-Trefftz-type elements. Some useful general forms,s8 one may account accurately for various discontinuous or concentrated loads without laborious adjustment of the finite element mesh. Clearly such elements can perform very well when compared with standard ones, as the nature of the analytical solution has been essentially included. Figure 12.15 shows excellent results which can be obtained using such complex elements. The number of degrees of freedom is here much smaller than with a standard displacement solution but, of course, the bandwidth is much larger. 58 Two points come out clearly in the general formulation of Eqs (12.47)-(12.50). First, the displacement field, u given by parameters ~, can only be determined by excluding any rigid body modes. These can only give strains ,$N identically equal to zero and hence make no contribution to the It matrix. Second, stability conditions require that (in two dimensions) nu > n v -
3
and thus the minimum nu can be readily found (viz. Chapter 10). Once again there is little point in increasing the number of internal parameters substantially above the minimum number as additional accuracy may not be gained. We have said earlier that the 'translation' of the formulation discussed to problems governed by the quasi-harmonic equations is almost evident. Now identical relations will hold if we replace u ---~ 4~ o" --+ q
(12.53)
t --~ qn S-+
For the Poisson equation
V2q~ ---
V
Q
(12.54)
a complete series of analytical solutions in two dimensions can be written as Re
(Z n ) =
1, x , X 2 - - y2, x 3 - 3 x y 3 . . . .
Im (z n) = y, 2 x y . . . .
forz=x+iy
(12.55)
Linking of boundary (or Trefftz)-type solution by the 'frame' of specified displacements 449
Fig. 12.15 Application of Trefftz-type elements to a problem of a plane stress tension bar with a circular hole. (a) Trefftz element solution. (b) Standard displacement element solution. (Numbers in parentheses indicate standard solution with 230 elements, 1600 DOE)
450
Multidomain mixed approximations- domain decomposition and 'frame' methods
Fig. 12.16 Boundary-Trefftz-type 'elements' linking two domains of different materials in an elliptic bar subject to torsion (Poisson equations), s5 (a) Stress function given by internal variables showing almost complete continuity. (b) x component of shear stress (gradient of stress function showing abrupt discontinuity of material junction).
With the above we get N e-
[1,
x,
y, x 2 - y 2, 2xy, x 3 - 3 x y 2, 3x2y . . . . ]
(12.56)
A simple solution involving two subdomains with constant but different values of Q and a linking on the boundary is shown in Fig. 12.16, indicating the accuracy of the linking procedures.
Subdomains with 'standard' elements and global functions 451
Fig. 12.17 'Superelements' built from assembly of standard displacement elements with global functions eliminating singularities confined to the assembly.
The procedure just described can be conveniently used with approximations made internally with standard (displacement) elements and global functions helping to deal with singularities or other internal problems. Now simply an additional term will arise inside nodes placed internally in the subdomain but the effect of global functions can be contained inside the subdomain. The formulation is somewhat simpler as complicated Trefftz-type functions need not be used. We leave details to the reader and in Fig. 12.17 show some possible, useful, subdomain assemblies. We shall return to this again in Chapter 15.
The possibilities of elements or 'superelements' constructed by the mixed-incomplete field methods of this chapter are numerous. Many have found practical use in existing computer codes as 'hybrid elements'; others are only now being made widely available. The use of a frame of specified displacements is only one of the possible methods for linking Trefftztype solutions. As an alternative, a frame of specified boundary tractions t has also been successfully investigated. 1~ 45 In addition, the so-called 'frameless formulation '9,11 has been found to be another efficient solution (for a review see reference 13) in the Trefftz-type element approach. All of the above-mentioned alternative approaches may be implemented into standard finite element computer codes. Much further research will elucidate the advantages of some of the forms discovered and we expect the use of such developments to continue to increase in the future.
12.1 Compute explicit relations for linear one-dimensional dual shape functions using Eq. (12.13). Verify the results shown in Fig. 12.5. 12.2 Compute explicit relations for quadratic one-dimentinal dual shape functions using Eq. (12.13). Assume the element side is straight and the interior node is at the centre of the edge.
452 Multidomainmixed approximations- domain decompositionand 'frame' methods 12.3 Compute an explicit relation at node a for 4-node dual shape functions. Use Eq. (12.13) and assume the surface mesh for elements is as shown in Fig. 12.18. Sketch the shape of the global dual function at node a (e.g., as shown for a twodimensional edge in Fig. 12.4). 12.4 The mesh segment shown in Fig. 12.19 occurs in a problem in which the two sides are to be joined using a standard mortar method. If node a is located at 0.4h from node b perform the integrals necessary to construct the contributions to the Qi arrays appearing in Eq. (12.12). 12.5 The mesh segment shown in Fig. 12.19 occurs in a problem in which the two sides are to be joined using a dual mortar method. If node a is located at 0.4h from node b perform the integrals necessary to construct the contributions to the Qi arrays appearing in Eq. (12.12). (Note: It is necessary to replace one N by/V for the dual approach.) 12.6 Write a MATLAB program to solve the one-dimensional problem of Example 3.5 in Chapter 3. Modify the program to enforce the boundary conditions using the Nitsche method described in Sec. 12.3.1. Verify your program by solving the example illustrated in Fig. 12.8(a). 12.7 Perform the derivations given in Sec. 12.5 which include the effects of a non-zero body force b to define tr0. 12.8 For the quasi-harmonic equation given by V2~b = Q construct the linking of Trefftztype solutions by a 'frame' of specified values for q~. (Hint: Follow the suggestions given in Eq. (12.53).)
h
a
H
h
h
Fig. 12.18 Surfacedescription for Problem 12.3. F"
"-1
bT
L.,
Fig. 12.19 Tied segment for Problems 12.4 and 12.5.
._1
References 453
1. Ch. Farhat and E-X. Roux. Optimal convergence properties of the FETI domain decomposition method. Comp. Meth. Appl. Mech. Eng., 115:365-385, 1994. 2. C. Farhat and J. Mandel. The two-level FETI method. I. An optimal iterative solver for biharmonic systems. Comp. Meth. Appl. Mech. Eng., 155:129-151, 1998. 3. C. Farhat, Chen Po-shu, J. Mandel, and EX. Roux. The two-level FETI method. II. Extension to shell problems, parallel implementation and performance results. Comp. Meth. Appl. Mech. Eng., 155:153-179, 1998. 4. C. Farhat, K. Pierson, and M. Lesoinne. The second generation FETI methods and their application to the parallel solution of large-scale linear and geometrically non-linear structural analysis problems. Comp. Meth. Appl. Mech. Eng., 184:333-374, 2000. 5. T.H.H. Pian. Derivation of element stiffness matrices by assumed stress distribution. J. AIAA, 2:1332-1336, 1964. 6. T.H.H. Pian and P. Tong. Basis of finite element methods for solid continua. Int. J. Numer. Meth. Eng., 1:3-28, 1969. 7. J. Jirousek and L. Guex. The hybrid-Trefftz finite element model and its application to plate bending. Int. J. Numer. Meth. Eng., 23:651-693, 1986. 8. J. Jirousek. Improvement of computational efficiency of the 9 dof triangular hybrid-Trefftz plate bending element. Int. J. Numer. Meth. Eng., 23:2167-2168, 1986. (Letter to Editor.) 9. J. Jirousek and A.P. Zieliriski. Study of two complementary hybrid-Trefftz p-element formulations. In Numerical Methods in Engineering 92, pages 583-590. Elsevier, 1992. 10. J. Jirousek and A.P. Zielifiski. Dual hybrid-Trefftz element formulation based on independent boundary traction frame. Int. J. Numer. Meth. Eng., 36:2955-2980, 1993. 11. J. Jirousek and A. Wr6blewski. Least-squares T-elements: equivalent FE and BE forms of a substructure-oriented boundary solution approach. Comm. Numer. Meth. Eng., 10:21-32, 1994. 12. J. Jirousek and A. Wr6blewski. T-elements: a finite element approach with advantages of boundary solution methods. Adv. Eng. Soft., 24:71-88, 1995. 13. J. Jirousek and A. Wr6blewski. T-elements: state of the art and future trends. Arch. Comput. Meth. Eng., 3(4), 1996. 14. R. Piltner. Special elements with holes and internal cracks. Int. J. Numer. Meth. Eng., 21:14711485, 1985. 15. N.-E. Wiberg. Matrix structural analysis with mixed variables. Int. J. Numer. Meth. Eng., 8:167-194, 1974. 16. B.I. Wohlmuth. Discretization Methods and Iterative Solvers Based on Domain Decomposition. Springer-Verlag, Heidelberg, 2001. 17. G. Kron. Tensor Analysis of Networks. John Wiley & Sons, New York, 1939. 18. Ch. Farhat and F.-X. Roux. A method of finite element tearing and interconnecting and its parallel solution algorithm. Int. J. Numer. Meth. Eng., 32:1205-1227, 1991. 19. M.A. Puso and T.A. Laursen. Mesh tying on curved interfaces in 3d. Eng. Comput., 20:305-319, 2003. 20. J.A. Nitsche. Uber ein Variationsprinzip zur Lrsung Dirichlet-Problemen bei Verwendung von Teilr~iumen, die keinen Randbedingungen uneworfen sind. Abh. Math. Sem. Univ. Hamburg, 36:9-15, 1971. 21. C.G. Makridakis and I. Babu~ka. On the stability of the discontinuous Galerkin method for the heat equation. SIAM J. Num. Anal., 34:389-401, 1997. 22. J.T. Oden, I. Babu~ka, and C.E. Baumann. A discontinuous hp finite element method for diffusion problems. J. Comp. Phys., 146(2):491-519, 1998. 23. T.J.R. Hughes, G. Engel, L. Mazzei, and M.G. Larson. A comparison of discontinuous and continuous Galerkin methods based on error estimates, conservation, robustness and efficiency.
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41. 42. 43. 44. 45. 46.
In Discontinuous Galerkin Methods: Theory, Computation and Applications, pages 135-146. Springer-Verlag, Berlin, 2000. B. Cockburn, G.E. Karniadakis, and Chi-Wang Shu. Discontinuous Galerkin Methods: Theory, Computation and Applications. Springer-Vedag, Berlin, 2000. D.N. Arnold, E Brezzi, B. Cockburn, and D. Marini. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal., 39:1749-1779, 2002. W.H. Reed and T.R. Hill. Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, 1973. E Lesaint and E-A. Raviart. On a finite element method for solving the neutron transport equation. In C. de Boor, editor, Mathematical Aspects of Finite Elements in Partial Differential Equations. Academic Press, New York, 1974. O.C. Zienkiewicz, R.L. Taylor, S.J. Sherwin, and J. Peir6. On discontinuous Galerkin methods. Int. J. Numer. Meth. Eng., 58:1119-1148, 2003. S.N. Atluri, R.H. Gallagher, and O.C. Zienkiewicz, editors. Hybrid and Mixed Finite Element Methods. John Wiley & Sons, New York, 1983. T.H.H. Pian. Element stiffness matrices for boundary compatibility and for prescribed boundary stresses. In Proc. 1st Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-66-80, pages 457-478, Wright Patterson Air Force Base, Ohio, Oct. 1966. R.D. Cook and J. At-Abdulla. Some plane quadrilateral 'hybrid' finite elements. J. AIAA, 7:2184-2185, 1969. S.N. Atluri. A new assumed stress hybrid finite element model for solid continua. J. AIAA, 9:1647-1649, 1971. R.D. Henshell. On hybrid finite elements. In J.R. Whiteman, editor, The Mathematics of Finite Elements and Applications, pages 299-312. Academic Press, London, 1973. R. Dungar and R.T. Severn. Triangular finite elements of variable thickness. J. Strain Anal., 4:10-21, 1969. R.J. Allwood and G.M.M. Comes. A polygonal finite element for plate bending problems using the assumed stress approach. Int. J. Numer. Meth. Eng., 1:135-160, 1969. T.H.H. Pian. Hybrid models. In S.J. Fenves et al., editors, Numerical and Computer Methods in Applied Mechanics. Academic Press, New York, 1971. Y. Yoshida. A hybrid stress element for thin shell analysis. In V. Pulmano and A. Kabailia, editors, Finite Element Methods in Engineering, pages 271-286. University of New South Wales, Australia, 1974. R.D. Cook and S.G. Ladkany. Observations regarding assumed-stress hybrid plate elements. Int. J. Numer. Meth. Eng., 8:513-520, 1974. J.A. Wolf. Generalized hybrid stress finite element models. J. AIAA, 11:385-388, 1973. EL. Gould and S.K. Sen. Refined mixed method finite elements for shells of revolution. In Proc. 3rd Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-71-160, WrightPatterson Air Force Base, Ohio, 1972. E Tong. New displacement hybrid models for solid continua. Int. J. Numer. Meth. Eng., 2:73-83, 1970. E. Trefftz. Ein Gegenstruck zum Ritz'schem Verfohren. In Proc. Int. Cong. Appl. Mech., Zurich, 1926. EK. Banerjee and R. Butterfield. The Boundary Element Methods in Engineering Science. McGraw-Hill, London, 1981. J.A. Ligget and EL-E Liu. The Boundary Integral Equation Method for Porous Media Flow. Allen and Unwin, London, 1983. C.A. Brebbia and S. Walker. Boundary Element Technique in Engineering. Newnes-Butterworth, London, 1980. I. Herrera. Boundary methods: a criteria for completeness. In Proc. Nat. Acad. Sci., volume 77, pages 4395-4398, USA, Aug. 1980.
References 455 47. I. Herrera. Boundary methods for fluids. In R.H. Gallagher, H.D. Norrie, J.T. Oden, and O.C. Zienkiewicz, editors, Finite Elements in Fluids, volume 4, Chapter 19. John Wiley & Sons, New York, 1982. 48. I. Herrera. Trefftz method. In C.A. Brebbia, editor, Progress in Boundary Element Methods, volume 3. John Wiley & Sons, New York, 1983. 49. I. Herrera and H. Gourgeon. Boundary methods, c-complete system for Stokes problems. Comp. Meth. Appl. Mech. Eng., 30:225-244, 1982. 50. I. Herrera and EJ. Sabina. Connectivity as an alternative to boundary integral equations construction of bases. In Proc. Nat. Acad. Sci., volume 75, pages 2059-2063, USA, May 1978. 51. O.C. Zienkiewicz, D.W. Kelley, and P. Bettess. The coupling of the finite element and boundary solution procedures. Int. J. Numer. Meth. Eng., 11:355-375, 1977. 52. O.C. Zienkiewicz, D.W. Kelly, and P. Bettess. Marriage a la mode - the best of both worlds (finite elements and boundary integrals). In R. Glowinski, E.Y. Rodin, and O.C. Zienkiewicz, editors, Energy Methods in Finite Element Analysis, Chapter 5, pages 81-107. John Wiley & Sons, London and New York, 1979. 53. O.C. Zienkiewicz and K. Morgan. Finite Elements and Approximation. John Wiley & Sons, London, 1983. 54. O.C. Zienkiewicz. The generalized finite element method- state of the art and future directions. J. Appl. Mech., ASME, 1983.50th anniversary issue. 55. A.P. Zielinski and O.C. Zienkiewicz. Generalized finite element analysis with t complete boundary solution functions. Int. J. Numer. Meth. Eng., 21:509-528, 1985. 56. J. Jirousek. A powerful finite element for plate bending. Comp. Meth. Appl. Mech. Eng., 12:7796, 1977. 57. J. Jirousek. Basis for development of large finite elements locally satisfying all field equations. Comp. Meth. Appl. Mech. Eng., 14:65-92, 1978. 58. J. Jirousek and P. Teodorescu. Large finite elements for the solution of problems in the theory of elasticity. Comp. Struct., 15:575-587, 1982.
We have stressed from the beginning of this book the approximate nature of the finite element method and on many occasions we have compared it with known exact solutions. Also, in reference to the 'accuracy' of the procedures, we suggested and discussed the manner by which this accuracy could be improved. Indeed one of the objectives of this chapter is concerned with the question of accuracy and a possible improvement on it by a posteriori treatments of the finite element data. We refer to such processes as recovery. We shall also consider the discretization error of the finite element approximation and a posteriori estimates of such error. In particular, we describe two distinct types ofaposteriori error estimators, recovery-based error estimators and residual-based error estimators. The importance of highly accurate recovery methods in the computation of the recovery-based error estimators is discussed. We also demonstrate how various recovery methods can be used in the construction of residual-based error estimators. Before proceeding further it is necessary to define what we mean by error. This we consider to be the difference between the exact solution and the approximate one. This can apply to the basic function, such as displacement which we have called u, and is given as e--u-fi
(13.1)
where, as before, fi denotes a finite element solution and u the exact solution. In a similar way, however, we could focus on the error in the strains (i.e., gradients in the solution), such as e or stresses or and describe the error in these quantities as e~ - e - ~ eo = or - 6"
(13.2)
The specification of local error in the manner given in Eqs (13.1) and (13.2) is generally not convenient and occasionally misleading. For instance, under a point load both errors in displacements and stresses will be locally infinite but the overall solution may well be acceptable. Similar situations will exist near re-entrant corners where, as is well known, stress singularities exist in elastic analysis and gradient singularities develop in field problems. For this reason various 'norms' representing some integral scalar quantity are often introduced to measure the error.
Definition of errors
13.1.1 Norms of errors If, for instance, we are concerned with a general linear equation of the form of Eq. (3.6) (cf. Chapter 3), i.e., Z~u + b = 0
we can define an
written for the error as
energy norm
Ilell -
(13.3)
L eTs
_= L ( u - fil)Ts (U-- ~) d ~
(13.4)
where I 9I denotes the absolute value of the argument. This scalar measure corresponds in fact to the square root of the quadratic functional such as we have discussed in Sec. 3.9 of Chapter 3 and where we sought its minimum in the case of a self-adjoint operator/2. For elasticity problems the energy norm is defined in the same manner and yields,
Ilell =
(,Se)TD,Se d f2
1
(13.5)
(with symbols as used in Chapters 2 and 6). Here e is given by Eq. (13.1), the operator ,S defines the strains as e = ,Su
and
g = ,9fi
(13.6a)
and D is the elasticity matrix (see Chapters 2 or 6), giving the stress as tr = De
and
0" = D~
(13.6b)
in which for simplicity we ignore initial stresses and strains. Using the above relations the energy norm of Eq. (13.5) can be written alternatively as
llell=
EL (~
_
~)T D (e -- ~) dr2
I1 I
(13.7) 1 -
(O" -- f~.)T D-1 (o" - 6") d Q
-
and its relation to strain energy is evident. Other scalar norms can easily be devised. For instance, the Le norm of displacement error can be written as 1
IlellL~
-
(u-
I1) T ( u -
u ) dr2
(13.8a)
and that for stresses error as
lle~IIL2 = IL (~ - &)T (~ -- 6") dg2
(13.8b)
457
458
Errors,recovery processes and error estimates
Such norms allow us to focus on a particular quantity of interest and indeed it is possible to evaluate 'root mean square' (RMS) values of its error. For instance, the RMS error in displacement, A u, becomes for the domain f2 ]Au] = ([[el'~2) 89 f2
(13.9)
Similarly, the RMS error in stress, Acr, becomes for the domain f2
IA~rl - ("e~'l~2) ~ f2
(13.10)
Any of the above norms can be evaluated over the whole domain, any subdomain, or even an individual element. We note that
Ilell =
Ilell~
(13.11)
m
where K refers to individual elements f2I< such that their sum (union) is f2. We note further that the energy norm given in terms of the stresses, the L2 norm of stress and the RMS stress error have a very similar structure and that these are similarly approximated.
Effect of a
singularity
At this stage it is of interest to invoke the discussion of Chapter 2 (Sec. 2.6) concerning the rates of convergence. We noted there that with trial functions in the displacement formulation of degree p, the errors in the stresses were of the order O (hP). This order of error should therefore apply to the energy norm error Ilell. While the arguments are correct for well-behaved problems with no singularity, it is of interest to see how the above rule is violated when singularities exist. To describe the behaviour of stress analysis problems we define the variation of the relative energy norm error (percentage) as r/=
Ilell • 100% Ilull
(13.12)
where Ilull -- [ j f 6TD~ dff~
(13.13)
is the energy norm of the solution. In Figs 13.1 and 13.2 we consider two similar stress analysis problems. In the first a strong singularity is present, however, in the second the singularity is removed by introducing a rounded corner. In both figures we show the relative energy norm error for an h refinement constructed by uniform subdivision of the initial mesh and for a p refinement in which polynomial order is increased throughout the original mesh. We note two interesting facts. First, the h convergence rate for various polynomial orders of the shape functions is nearly the same in the example with singularity (Fig. 13.1) and is well below the theoretically predicted optimal order O(hP), [or O(NDF)-p/2 as the NDF (number of degrees of freedom) is approximately inversely proportional to h 2 for a two-dimensional problem].
Superconvergenceand optimal sampling points
Fig. 13.1 Analysis of L-shaped domain with singularity. Second, in the case shown in Fig. 13.2, where the singularity is avoided by rounding the re-entrant comer, the h convergence rate improves for elements of higher order, although again the theoretical (asymptotic) rate is not quite achieved. The reason for this behaviour is clearly the singularity, and in general it can be shown that the rate of convergence for problems with singularity is O (NDF) -[minO''p)]/2
(13.14)
where ~. is a number associated with the intensity of the singularity. For elasticity problems ~, ranges from 0.5 for a nearly closed crack to 0.711 for a 90 ~ comer. The rate of convergence illustrated in Fig. 13.2 approaches the theoretically optimal order for all values of p used in the elements.
In this section we shall consider the location of points at which the stresses, or displacements, give their most accurate values in typical problems of a self-adjoint kind. We shall note that on many occasions the displacements, or the function itself, are most accurately sampled at the nodes defining an element and that the gradients or stresses are best sampled at some interior points. Indeed, in one dimension at least, we find that such points often exhibit the quality known as superconvergence (i.e., the values sampled at these points show an error which decreases more rapidly than elsewhere). Obviously, the user of finite element analysis should be encouraged to employ such points but at the same time note that the errors overall may be much larger. To clarify these ideas we start with a typical problem of second order in one dimension.
459
460
Errors, recovery processes and error estimates I
0 0
O0 O0 I
I
i
r
<11
I
I
I
I
,--
I
NDF
I
0.5 /
Quadrat ra e (1)./'/;-Av, g p / convergence
Subdivision 1 . . . .
i i
/
0 0
40 18
5.6
i
(a)
O0
Linea~ ~
i . . . .
0
,-- 0~ ~O 0
(/i/
<11
/
0 0
~
i
I //cu ic
v
o
1.8 uJ 1.0
///10.7T (1)
0.17
(b) Fig. 13.2 Analysisof L-shapeddomain without singularity.
13.2.1 A one-dimensional example Here we consider a problem of a second order equation such as we have discussed in Chapter 3 and which may be typical of either one-dimensional heat conduction or the displacements of an elastic bar with varying cross-section. This equation can readily be written as
d (kdU) dx +/3u +
d--~ \
Q = 0
(13.15)
with the boundary conditions either defining the values of the function u or of its gradients at the ends of the domain. Let us consider a typical problem as illustrated in Fig. 13.3. Here we show an exact for a span of several elements and indicate the type of solution solution for u and which will result from a finite element calculation using linear elements. We have already noted that on occasions we shall obtain exact solutions for u at nodes (see Figs 3.5 and 3.6). This will happen when the weighting function contains the exact solution of the homogeneous differential equation (Appendix H) - a situation which happens for Eq. (13.15) when /3 = 0, k is constant in each element and polynomial shape functions are used. In all cases, even when/3 is non-zero and linear shape functions are used, the nodal values generally will be much more accurate than those elsewhere, Fig. 13.3(a). For the gradients shown
du/dx
Superconvergence and optimal sampling points 461
Fig. 13.3 Optimal sampling points for the function (a) and its gradient (b) in one dimension (linear elements). in Fig. 13.3(b) we observe large discrepancies of the finite element solution from the exact solution but we note that somewhere within each element the results are exact. It would be useful to locate such points and indeed we have already remarked in the context of two-dimensional analysis that values obtained within the elements tend to be more accurate for gradients (strains and stresses) than those values calculated at nodes. Clearly, for the problem illustrated in Fig. 13.3(b) we should sample somewhere near the centre of each element. Pursuing this problem further in a heuristic manner we note that if higher order elements (e.g., quadratic elements) are used the solution still remains exact or nearly exact at the end nodes of an element but may depart from exactness at the interior nodes, as shown in Fig. 13.4(a). The stresses, or gradients, in this case will be optimal at points which correspond to the two Gauss quadrature points for each element as indicated in Fig. 13.4(b). This fact was observed experimentally by Barlow. 1 We shall now state in an axiomatic manner that: (a) the displacements are best sampled at the nodes of the element, whatever the order of element used, and (b) the best accuracy for gradients or stresses is obtained at the Gauss points corresponding to the order of polynomial used in the solution.
462
Errors,recovery processes and error estimates
Fig. 13.4 Optimal sampling points for the function (a) and its gradient (b) in one dimension (quadratic elements).
At such points the order of the convergence of the function or its gradients is at least one order higher than that which would be anticipated from the appropriate polynomial and thus such points are known as superconvergent. The reason for such superconvergence will be shown in the next section where we introduce the reader to a theorem developed by Herrmann. 2
13.2.2 The Herrmann theorem and optimal sampling points The concept of least squares fitting has additional justification in self-adjoint problems in which an energy functional is minimized. In such cases, typical of a displacement
Superconvergence and optimal sampling points formulation of elasticity, it can be readily shown that the minimization is equivalent to a least squares fit of the approximate stresses to the exact ones. Thus quite generally we can start from a theory given by the differential equation J~U --"
~.qT
(ASu) = p
(13.16)
In the above, /2 is a self-adjoint operator defined by S and A (symmetric) and p are prescribed matrices of position. The minimization of an energy functional FI defined as I - I = ~I f
(Su)TASu dS2 - /
uTp dr2
(13.17)
gives at an absolute minimum the exact solution u = O, is equivalent to minimization of another functional 1-I* defined as
l"[* = 1 f~
IS (u - a)]TAS (u -- a) dS2
(13.18)
The above quadratic functional [Eq. (13.17)] arises in all linear self-adjoint problems. For elasticity problems this theorem is given by Herrmann 2 and shows that the approximate solution for ,.qu approaches the exact one ,.qfi as a weighted least squares approximation. The proof of the Herrmann theorem is as follows. The variation of H defined in Eq. (13.17) gives, at u = fi (the exact solution),
t~I-I -- ~1 (St~U)T ASfi dr2 + ~1 /~ (Sfi)T AS'u dr2 - ~ 'uTp dr2 = 0 or as A is symmetric ~fl = ~ (,S~u)T A,.q. dr2 - ~ 8uTp dr2 = 0 in which ~u is any arbitrary variation. Thus we can select 6u=u and
L(
,Su)TASfi dr2 - ~ uTp dr2 = 0
Subtracting the above from Eq. (13.17) and noting the symmetry of the A matrix, we can write 1-I = ~lf~ [S (u - O)]T A S (u - ~)dr2 - ~1 f ( ~ f i ) T A S f i d ~ (1319) where the last term is not subject to variation. Thus H* = 1-I + constant
(13.20)
and its stationarity is equivalent to the stationarity of I-I. It follows directly from the Herrmann theorem that, for one dimension and by a wellknown property of the Gauss-Legendre quadrature points, if the approximate gradients
463
464
Errors,recovery processes and error estimates /9=1
-1
.J
J
o
1
/ p=2
p=3
_~_
"
\,
Fig. 13.5 The integration property of Gauss points: p - 1, p - 2, and p - 3 which guarantees superconvergence. are defined by a polynomial of degree p - 1, where p is the degree of the polynomial used for the unknown function u, then stresses taken at these quadrature points must be superconvergent. The single point at the centre of an element integrates precisely all linear functions passing through that point and, hence, if the stresses are exact to the linear form they will be exact at that point of integration. For any higher order polynomial of order p, the Gauss-Legendre points numbering p will also provide points of superconvergent sampling. We see this from Fig. 13.5 directly. Here we indicate one, two, and three point Gauss-Legendre quadrature showing why exact results are recovered there for gradients and stresses. For points based on rectangles and products of polynomial functions it is clear that the exact integration points will exist at the product points as shown in Fig. 13.6 for various rectangular elements assuming that the weighting matrix A is diagonal. In the same figure we show some triangles and what appear to be 'good' but are not necessarily superconvergent sampling points. Though we find that superconvergent points do not exist in triangles, the points shown in Fig. 13.6 are optimal. In Fig. 13.6 we contrast these points with the minimum number of quadrature points necessary for obtaining an accurate (though not always stable) stiffness representation and find these to be almost coincident at all times.
Recovery of gradients and stresses 465
p
Optimalerror 0 (h2(p-m)+2) O(h 2)
(•kb
Minimal quadrature 0
(h2(p--m)+l)
_>O(h2)
0 (h2)
I
A
I
O(h2)
0 (h 4)
0 (h2)
I. I
0 (h2)
:> 0 (h 3)
O(h 4) O(h 4) O(h 4)
G•
0 (h 3) O(h4 ) O(h4 )
Fig. 13.6 Optimalsuperconvergentsamplingand minimumintegrationpointsfor some Co elements. In Fig. 13.7 representing an analysis of a cantilever beam by four rectangular quadratic serendipity elements we see how well the stresses sampled at superconvergent points behave compared to the overall stress pattern computed in each element. The extension of the idea of superconvergent points from one-dimensional elements to two-dimensional rectangles is fairly obvious. However, the full order of superconvergence is lost when isoparametric distortion of elements occurs. We have shown, however, that results at the pth order Gauss-Legendre points still remain excellent and we suggest that superconvergent properties of the integration points continue to be used for sampling. In all of the above discussion we have assumed that the weighting matrix A is diagonal. If a diagonal structure does not exist the existence of superconvergent points is questionable. However, excellent results are still available through the sampling points defined as above. Finally, we refer readers to references 3-8 for surveys on the superconvergence phenomenon and its detailed analyses.
In the previous section we have shown that sampling of the gradients and stresses at certain points within an element is optimal and higher order accuracy can be achieved. However, we would also like to have similarly accurate quantities elsewhere within each element for general analysis purposes, and in particular we need such highly accurate displacements,
466
Errors, recovery processes and error estimates
40 i !
30
! | ! !
20
i
! i
I
i s
."
Exact average 9 shear stressl ',
'
; ~ Nodal values extrapolated E I" from Gauss points
10
i
-10
i
i
". ,'
'
I
l
values
9
Land 0.24 per unit 9
)
9
2
9
40
o
o
-I T
2 Gauss points
Fig. 13.7 Cantilever beam with four quadratic (Q8) elements. Stress sampling at cubic order (2 x 2) Gauss points with extrapolation to nodes. gradients and stresses when energy norm or other norms representing the particular quantity of interest have to be evaluated in error estimates. We have already shown how with some elements very large errors exist beyond the superconvergent point and attempts have been made from the earliest days to obtain a complete picture of stresses which is more accurate overall. Here attempts are generally made to recover the nodal values of stresses and gradients from those sampled internally and then to assume that throughout the element the recovered stresses tr* are obtained by interpolation in the same manner as the displacements o-* = N.8-*
(13.21)
We have already suggested a process used almost from the beginning of finite element calculations for triangular elements, where elements are sampled at the centroid (assuming linear shape functions have been used) and then the stresses are averaged at nodes. We have referred to such recovery in Chapter 6. However, this is not the best for triangles and for higher order elements such averaging is inadequate. Here other procedures were necessary, for instance Hinton and Campbell 9 suggested a method in which stresses at all nodes were calculated by extrapolating the Gauss point values. A method of a similar kind was suggested by Brauchli and Oden 1~ who used the stresses in the manner given by Eq. (13.21) and assumed that these stresses should represent in a least squares sense the actual finite element stresses. This is therefore an L2 projection. Although this has a
Superconvergent patch recovery- SPR 467
Fig. 13.8 Interior superconvergent patches for quadrilateral elements (linear, quadratic, and cubic) and triangles (linear and quadratic).
similarity with the ideas contained in the Herrmann theorem it reverses the order of least squares application and has not proved to be always stable and accurate, especially for even order elements. In the following presentation we will show that highly improved results can be obtained by direct polynomial 'smoothing' of the optimal values. Here the first method of importance is called superconvergent patch recovery. 11-13
We have noted above that the stresses sampled at certain points in an element possess a superconvergent property (i.e., converge at a rate comparable to that of displacement) and have errors of order O (h p+ 1). A fairly obvious procedure for utilizing such sampled values seems to the authors to be that of involving a smoothing of such values by a polynomial of order p within a patch of elements for which the number of sampling points can be taken as greater than the number of parameters in the polynomial. In Fig. 13.8 we show several such patches each assembled around an interior vertex (comer) node. The first four represent rectangular elements where the superconvergent points are well defined. The last two give patches of triangles where the 'optimal' sampling points used are not quite superconvergent. If we accept the superconvergence of t~ at certain points k in each element then it is a simple matter (which also turns out computationally much less expensive than the L2 projection) to compute tr* which is superconvergent at all points within the element. The procedure is illustrated for two dimensions in Fig. 13.8, where we shall consider interior patches (assembling all elements at interior nodes) as shown. At each superconvergent point the values of & are accurate to order p + 1 (not p as is true elsewhere). However, we can easily obtain an approximation tY* given by a polynomial of
468
Errors,recovery processes and error estimates
degree p, with identical order to those occurring in the shape function for displacement, which has superconvergent accuracy everywhere when this polynomial is made to fit the superconvergent points in a least squares manner. Thus we proceed for each component ~i of t~"as follows: writing the recovered solution as (13.22a)
c7/* = p(x, y ) a i in which p(x,y)=[1, ai --
2, al,
~,
a2,
..., "" ,
~P]
(13.22b)
am
with Yc - x - Xc, ~ - y - Yc where Xc, yc are the coordinates of the interior vertex node describing the patch. For each element patch we minimize a least squares functional with n sampling points, n
l"I -- -~ Z [ffi(Xk, k=l
Yk)
-
pkai] 2
(13.23)
where Pk : p(xk, Yk) [(Xk, Yk) correspond to the coordinates of the sampling superconvergent point k)] obtaining immediately the coefficient ai as ai -" A - 1bi (13.24)
where
n
A - ~ k=l
P~Pk
and
bi --
pgT ,cri , (Xk,
Yl,)
(13.25)
k=l
The availability of ~* allows superconvergent values of t~* to be determined at all nodes. For example, each component of the recovered solution at node a in the element patch is obtained by (O'?)a -- O'?(Xa, Ya) - - p ( X a , y a ) a i (13.26) It should be noted that on external boundaries or indeed on interfaces where stresses are discontinuous the nodal values should be calculated from interior patches and evaluated in the manner shown in Fig. 13.9. As some nodes belong to more than one patch, average values of #* are best obtained. The superconvergence of tr* throughout each element is established by Eq. (13.21). In Fig. 13.10 we show in a one-dimensional example how the superconvergent patch recovery reproduces e x a c t l y the stress (gradient) solutions of order p + 1 for linear or quadratic elements. Following the arguments of Chapter 9 on the patch test it is evident that superconvergent recovery is now achieved at all points. Indeed, the same figure shows why averaging (or L2 projection) is inferior (particularly on boundaries). Figure 13.11 shows experimentally determined convergence rates for a one-dimensional problem (stress distribution in a bar of length L = 1; 0 < x < 1 and prescribed body forces). A uniform subdivision is used here to form the elements, and the convergence rates for the stress error at x = 0.5 are shown using the direct stress approximation 8, the L2 recovery a L and cr* obtained by the SPR procedure using elements from order p = 1
Superconvergent patch recovery- SPR 469
Fig. 13.9 Recoveryof boundary or interface gradients.
to p = 6. It is immediately evident that ~r* is superconvergent with a rate of convergence being at least one order higher than that of 8. However, as anticipated, the L2 recovery gives much poorer answers, showing superconvergence only for odd values of p and almost no improvement for even values of p, while or* shows a two-order increase of convergence rate for even order elements (tests on higher order polynomials are reported in reference 14). This ultraconvergence has been verified mathematically. 15-17 Although it is not observed when elements of varying size are used, the important tests shown in Figs 13.12 and 13.13 indicate how well the recovery process works for problems in two dimensions. In the first of these, Fig. 13.12, a field problem is solved in two dimensions using a very irregular mesh for which the existence of superconvergent points is only inferred heuristically. The very small error in Crx*is compared with the error of 8x and the improvement is obvious. Here Crx = 8u/Ox where u is the field variable. In the second, i.e., Fig. 13.13, a problem of stress analysis, for which an exact solution is known, is solved using three different recovery methods. Once again the recovered solution cr* (SPR) shows much improved values compared with crL. It is clear that the SPR process should be included in all codes if simply to present improved stress values, to which we have already alluded in Chapters 6 and 7. The SPR procedure which we have just outlined has proved to be a very powerful tool leading to superconvergent results on regular meshes and much improved results (nearly superconvergent) on irregular meshes. It has been shown numerically that it produces superconvergent recovery even for triangular elements which do not have superconvergent points within the element. Recent mathematical proofs confirm these capabilities of SPR. 16-21 It is also found, for linear elements on irregular meshes, that SPR produces superconvergence of order O(h 1+~) with ot greater than zero. 22 The SPR procedure, introduced by Zienkiewicz and Zhu in 1992,11-13 is recommended as the best recovery procedure which is simple to use. However, the procedure has been modified by various investigators. 23-27 Some of the modifications have been shown to produce improved results in certain instances but with additional computational costs. One such modification appends satisfaction of discrete equilibrium equations and/or boundary conditions to the functional where the least squares fit is performed. While the satisfaction of known boundary tractions can on occasion
470
Errors, recovery processes and error estimates
S
l
(~
Exact (~ and (~* (5 h
"13 c-
Interior patch
()
O
9,' r t ' t
9
) rX
(a)
9Superconvergent values [] Nodal SPR values
l~
Quadratic exact solution
~ B o u n d a r y r-X
(b)
Fig. 13.10 Recovery of exact o- of degree p by linear elements (/9 - 1) and quadratic elements (p - 2).
be useful most of the additional constraints introduced have affected the superconvergent properties adversely and in general the modified versions of SPR such as those by Wiberg et al 23, 24 and by Blacker and Belytschko 25 have not proved to be effective. Example 13.1: SPR stress projection for rectangular element patch. As an example we consider the SPR projection for a stress component ai on the patch of rectangular elements shown in Fig. 13.14. The elements are 4-node rectangles in which shape functions are given by bilinear interpolations. Thus, the optimal sampling points are given by the points at the centre of each element. The recovered solution is given by a linear polynomial expressed as a* -- [1,
(x
-- Xl),
(Y - Yl)]
{01} a2 63
Superconvergent patch recovery- SPR 471
Fig. 13.12 Poisson equation in two dimensions solved using arbitrary-shaped quadratic quadrilaterals.
For this patch of elements, (13.23) is given by 1
4
H= ~ ~ k=l
[~i (xk, Yk) - Pka] 2
472
Errors,recovery processes and error estimates
Fig. 13.13 Plane stress analysis of stresses around a circular hole in a uniaxial field.
where [1 -a/2
Pk=
-b/2] [1 a/2 -b/2] b/2] [1 a/2 [1 -a/2 b/Z]
fork=l fork=2 fork=3 fork-4
Superconvergent patch recovery- SPR 473 8
(
(
() 1
2L.
(
3 a
~..~
6
5
4
a
Fig. 13.14 Patch of rectangular elements for SPR projection. Optimal points to sample stresses indicated by D.
Evaluating the minimum for FI and performing the sum gives the equations
[ioo]
Aa=b where
a2 0
A
and
b=
(F 1 ] -a/2
L-b/2J
(7il ~-
Ill
a/2
t~i2-~-
L-b/2J
0 b2
E1]
a/2 t~i3-4;- - a / 2 t~i4 b/2J L b/2J
The solution for the parameters is given by a l "-
1
~[
/
t~il + ~ i 2
+~i3
+~i4]
1 a2 - ~-s [-d'il + 6"/2 + ~i3 - 6"/4] 1
a3 -- =-7 [--tTil -- t7i2 -~- t7i3 -~- t~i4] 2/9
Inserting the parameters into the equation for the recovered stress gives , [~ O'i --'
(X -- X1)
2a
y_ym]{ 2b
--~i 1 "~- tTi 2 "nt- tTi 3 -- t7i4 - - ~ i l -- t7i2 nt- ~i3 "1-"~i4
)
474
Errors, recovery processes and error estimates
We note that the above yields SPR values at an internal node of a regular mesh which are the same as that obtained by averaging. Unfortunately, this is not the case when the mesh is irregular or boundary nodes are considered, as the reader can easily establish, where SPR will retain high accuracy but averaging will not.
13.4.2 SPR for displacements and stresses The superconvergent patch recovery can be extended to produce superconvergent displacements. The procedure for the displacements is quite simple if we assume the superconvergent points to be at nodes of the patch. However, as we have already observed it is always necessary to have more data than the number of coefficients in the particular polynomial to be able to execute a least squares minimization. Here of course we occasionally need a patch which extends further than before, particularly since the displacements will be given by a polynomial one order higher than that used for the shape functions. In Fig. 13.8, however, we show for most assemblies that an identical patch to that used for stresses will suffice. Larger element patches have also been suggested in reference 28 but it does not seem anything is gained. The recovered solution u* has on occasion been used in dynamic problems (e.g., Wiberg28, 29), since in this class of problems the displacements themselves are often important. We also find such recovery useful in problems of fluid dynamics. When both recovered displacements and stresses are desired, it is advantageous to compute the recovered stresses directly using the derivatives of the recovered displacements. The advantage of computing recovered stresses directly from displacements means that we have now obtained fully superconvergent results for all element types. Indeed, a recent study by Zhang and Naga, 3~ for field problems, has found that SPR using nodal field variable sampling produces better recovered gradients in certain instances. For example, although both SPR using gradient sampling and SPR using field variable sampling achieve ultraconvergence in the recovered gradient at vertex nodes of quadratic triangles, ultraconvergence of the recovered gradient at the mid-edge nodes can only be obtained by SPR using field variable sampling. A similar procedure to that studied in reference 30 has been used by Wiberg and Hager 31 in eigenfrequency computations. Thus, field variable recovery should probably always be used for triangular and tetrahedral elements, as well as for other element types when both superconvergent displacements and stresses or strains are required. The SPR recovery technique described in this section takes advantage of the superconvergence property of the finite element solutions and/or the availability of optimal sampling points. A recovery method which does not need such information has been devised and will be discussed in the next section.
Although SPR has proved to work well generally and much research has been devoted to its mathematical analyses, the reason behind its capability of producing an accurate recovered solution even when superconvergent points do not in fact exist remains an open question. We have therefore sought to determine viable recovery alternatives. One of these, known
Recovery by equilibration of patches- REP 475 by the acronym REP (recovery by equilibrium of patches), will be described next. This procedure was first presented in reference 32 and later improved in reference 33. To some extent the motivation is similar to that of Ladev~ze et al. 34' 35 who sought to establish (for somewhat different reasons) a fully equilibrating stress field which can replace that of the finite element approximation. However, we believe that the process presented here and in reference 33 is simpler although equilibrium is satisfied in an approximate manner. The starting point for REP is the governing equilibrium equation S TO" -~- b
= 0
(13.27)
In a finite element approximation this becomes fa BT~df2-fa p
NTbdfZ-fr p
NTtdF-0
(13.28)
p
where & are the stresses from the finite element solution. In the above f2p is the domain of a patch and the last term comes from the tractions on the boundary of the patch domain Fp. These can, of course, represent the whole problem, a patch of a few elements or a single element. As is well known the stresses dr which result from the finite element analysis will in general be discontinuous and we shall seek to replace them in every element patch by a recovered system which is smooth and continuous. To achieve the recovery we proceed in an analogous way to that used in the SPR procedure, first approximating the stress in each patch by a polynomial of appropriate order ~*, second using this approximation to obtain nodal values of #* andfinally interpolating these values by standard shape functions. The stress cr is taken as a vector of appropriate components, which for convenience we write as: crl if2 cr = . (13.29) Crn The above notation is general with, for instance, Crl -- Crx,or2 = Cry and Cr3 - - "Cxy describing a two-dimensional plane elastic analysis. We shall write each component of the above as a polynomial expansion of the form: c~* -- [1,
~:,
~,
...] ai :-- p(x, y)ai
(13.30)
where p is a vector of polynomials, ai is a set of unknown coefficients for the ith component of stress and 2, ~ are as described for (13.22b). For equilibrium we shall always attempt to ensure that the smoothed stress #* satisfies in a least squares sense the same patch equilibrium conditions as the finite element solution. Accordingly, ~
B T & d ~ 2 - ~ ~ BT&*d~2 p
(13.31)
p
where #* = Pa =
p
a2
0
a3
(13.32)
476
Errors,recovery processes and error estimates
written here again for the case of three stress components. Obvious modifications are made for more or fewer components. It has been found in practice that the constraints provided by Eq. ( 13.31) are not sufficient to always produce non-singular least squares minimization. Accordingly, the equilibrium constraints are split into an alternative form in which each component of stress is subjected to equilibrium requirements. This may be achieved by expressing the stress as
0"* =
i
~" = Z
li o"i ~
i
O"i (13.33)
lit~i -- Z t ~ i i i
in which
11--[1,
0,
0] T',
12 -- [0,
1,
0] T etc.
(13.34)
The equations are now obtained by imposing the set of constraints
3~~ BTO'id~'~ 3f~ BTO'*df2 = 3f~ B T l i p d ~ a i P
P
(13.35)
P
The imposition of the approximate equation (13.35) allows each set of coefficients ai to be solved independently reducing considerably the solution cost and here repeating a procedure used with success in SPR. A least squares minimization of Eq. (13.35) is expressed as
n=~1 (H/a/ where
Hi=fnBTlipdf2
- f;)T (H/a/- f;) and
f/P = / ~
P
BTr"i dr2
(13.36)
(13.37)
P
The minimization condition results in
ai = [HTI-Ii] -1 HTf;
(13.38)
Nodal values &* are obtained from Eq. (13.30) and the final recovered solution is given by Eq. (13.21). The REP procedure follows precisely the details of SPR near boundaries and gives overall an approximation which does not require knowledge of any superconvergent points. The accuracy of both processes is comparable.
One of the most important applications of the recovery methods is its use in the computation of error estimators. With the recovered solutions available, we can now evaluate errors simply by replacing the exact values of quantities such as u, or, etc., which are in general unknown, in Eqs (13.1) and (13.2), by the recovered values which are more accurate
aposteriori
Error estimates by recovery 477
than the direct finite element solution. We write the error estimators in various norms such
as
Ilell ~ I1~11 = I l u * - ~ l l (13.39)
IlellL= ~ II~IIL= - - I l u * - ~ I I L = Ile~ IlL= ~ IIG IlL= = liar* - &ILL=
For example, an error estimator of the energy norm for elasticity problems has the form 1
,,e,, = [f~ (o'*--o')TD-1 (o'*--~') d~"2]~
(13.40)
Similarly, estimates of the RMS error in displacement and stress can be obtained through Eqs (13.9)-(13.10). Error estimators formulated by replacing the exact solution with the recovered solution are sometimes called recovery-based error estimators. This type of error estimator was first introduced by Zienkiewicz and Zhu. 36 The accuracy or the quality of the error estimators is measured by the effectivity index O, which is defined as I1~11 0 = (13.41) Ilell A theorem presented by Zienkiewicz and Zhu 12 shows that for all estimators based on recovery we can establish the following bounds for the effectivity index: 1
Ile*ll
<0<
Ilell -
Ile*ll 1 + ~ Ilell
(13.42)
where e is the actual error and e* is the error of the recovered solution, e.g., Ile*ll = Ilu - u*ll
(13.43)
The proof of the above theorem is straightforward if we write Eq. (13.40) as II~ll = Ilu*-fill = II ( u - f i ) - ( u - u*)II = l i e - e*ll
(13.44)
Using now the triangle inequality we have Ilell-
Ile*ll ~ I1~11 ~ Ilell + Ile*ll
(13.45)
from which the inequality (13.42) follows after division by Ilell. Obviously, the theorem is also true for error estimators of other norms. Two important conclusions follow:
1. any recovery process which results in reduced error will give a reasonable error estimator and, more importantly, 2. if the recovered solution converges at a higher rate than the finite element solution we shall always have asymptotically exact estimation. To prove the second point we consider a typical finite element solution with shape functions of order p where we know that the error (in the energy norm) is Ilell-- O(hp)
(13.46)
478
Errors,recovery processes and error estimates
If the recovered solution gives an error of a higher order, e.g.,
Ile*ll-
ot > 0
O(h p+~)
(13.47)
then the bounds of the effectivity index are: 1 - O(h ~) < 0 < 1 + O(h ~)
(13.48)
and the error estimator is asymptotically exact, that is 0 ~
1
as
h ~ 0
(13.49)
This means that the error estimator converges to the true error. This is a very important property of error estimators based on recovery and is not generally shared by residual-based estimators which we discuss in the next section.
Other methods to obtain error estimators have been proposed by many investigators working in the field. 37--49Most of these make use of the residuals of the finite element approximation, either explicitly or implicitly. Error estimators based on these methods are often called residual error estimators. Those using residuals explicitly are termed explicit residual error estimators; the others are called implicit residual error estimators. In this section we are concerned with both explicit and implicit residual error estimators. To simplify the presentation, we use the quasi-harmonic equation in a two-dimensional domain as the model problem. The governing equation of the problem is given by - V T (k ~7~b) + O = 0 in f2
(13.50)
with boundary conditions 4~=~
onF~
q Tn ---=qn = g/ on 1-'q In the above q = -kV~
= [qx,
qy]T
(13.51)
is a flux, n is the unit outward normal to the boundary F and qn is the flux normal to the boundary (see Chapters 3 and 7). The error of the finite element solution ~ is written as e = 4~ - q~
(13.52)
The global energy norm error for domain ~ [viz. Eq. (13.11)] is I[e[I -
I[e[l~
(13.53)
where for each element K Ilell~ -- f a (Ve)TN V e dr2 (13.54)
K
-s 1
= fa
- t ] x ) 2 -t-- (qy - t]y) 2] d ~
K
In what follows, we shall first discuss the explicit residual error estimator.
Residual-based methods 479
13.7.1 Explicit residual error estimator The energy norm for an explicit residual error estimator has been derived by various authors48, 49 and has a general form 1
IlOll
I1~112
--
(13.55)
rK
m
with element contributions
--Clf~ r2d~+C2frj2dI"
IlOll2rK
K
(13.56)
K
where (13.57)
rK = -- V T (k V ~ ) + Q
is the element interior residual and J is the discontinuity in the normal flux q, at each edge of element K, which we call a jump discontinuity. For example, at an edge shared by element K and its neighbouring element I, we have J = 0,K + 0,,
(13.58)
where On~ = ~TnK
and
~,, = ~Tnl
are the finite element normal fluxes. The constants C1 and C2 that appear in (13.56) are mesh dependent parameters and generally are unknown. This renders the explicit residual error estimators in the form of Eq. (13.56) less useful in practical computations. For the particular case of constant k an explicit form for C1 and C2 has been obtained for a 4-node quadrilateral element. 38' 39 This element explicit residual error estimator has the form I1~[I 2 -
r 2d~ + dl-" (13.59) 24k K ~ K The derivation of Eq. (13.59) was achieved following some heuristic assumptions on the error distribution and manipulations of the element residuals. It was found that the major contribution to the error estimator is from the term involving the jump discontinuities and that the term for the element interior residual is of higher order. Therefore, in practice the following form rK
--
11~112~ =
h f r j2dl" (13.60) 24k x is often used. Indeed, this form of the explicit residual error estimator has been most widely used. In the following, we shall show, as an example, that the explicit residual error estimator of Eq. (13.60) can also be derived from a particular recovery-based error estimator.
Example 13.2: Deriving explicit residual error estimator. For simplicity we consider a square element f2X and its neighbouring elements as shown in Fig. 13.15. The element contribution of the recovery-based error estimator is in the form 1 II0112 = f a ~ ~[(qx* - qx )2 + (qy
-
Oy)2]dg2
(13.61)
480
Errors,recovery processes and error estimates Yi+ 1 -~YK+ I
DK
~-'~XK_ 1
~'-~XK+ 1
Yi-1 ~"~YK-1 ~X
Yi-2 Xi-2
Xi-1
Xi
Xi+l
Fig. 13.15 An element patch. Element~K and its neighbors. The main steps involved in the derivation of the residual error estimator of Eq. (13.60) are as follows" 1. Construct a recovered solution for element f2K from elements f2K, f2xK+l and ~"~YK+I and forming recovery-based error estimator IlOll~,. 2. Construct a recovered solution for element f2K from elements f2K, f2xK_~ and f2yK_1 and forming recovery-based error estimator I1~II2K2" 3. Average 11Oll2rland I1~11~=to obtain the final recovery-based error estimator which results in the explicit residual error estimator of Eq. (13.60). In the first step, consider f2r and its two neighbouring elements recovered solutions are expressed as
q*x~= ~lxK+ OglZx(x)
~"~XK+I and
~"2yg+l,
the
(13.62)
q Y l = ~lyK " 1 - / ~ I Z y ( Y )
where
Zx
and
Zy are linear functions
in x and y respectively, i.e.,
Zx(x)-l-2(
xi - x ) h
Zy(y)-l-2(
(13.63)
yi-y)h
with h the edge length of the square element
h --
x i -
xi-1
-~- Yi -
Yi-1
and Ctl, /31 are unknown parameters to be determined by a recovery process. A recovery process for q*Xl is shown in Fig. 13.16. A similar result holds for qyl The recovery method of averaging is used by requiring the recovered solution to be the average of the finite element solution at the boundary of the element, i.e., at the edge shared by f2K and f2xK+, 1
q* (xi) -- [~lxK(Xi)+ ~lx~+~(xi)]
(13.64a)
Residual-based methods 481
and at the edge shared by element ~2K and ~"2yK+1 1
(13.64b)
qy~ (yi) - '~ [OYK(Yi) + OYK+I(Yi)] Substituting Eq. (13.62) into Eqs (13.64a)-(13.64b), Ot1 and/31 have the solution 1 1 0[1 -- ~2Zx(Xi-----~(OXK+l (Xi) -- QxK(Xi)) --" ---~ J ( x i ) /~1 --
1
2Zy(Yi)
(QyK+, (Yi)
OyK(Yi))
(13.65)
1
-gJ(Yi) /_.,
where J (Xi) is the jump discontinuity along edge Xi (viz. Fig. 13.16) and J (Yi) is the jump discontinuity along edge yi. In the above, we have used the fact that at xi On K --- OXK
and
On K+l --- --OXK+I
Ong --" OYK
and
OnK+l -" --OYK+I
and at yi The determined recovered solutions are now in the form of q* -- glxK -
1
J(xi)Zx
1 qYl --- OYK -- -2 J
Error estimator Eq. (13.61)
(13.66)
(Yi ) Z y
11~112for element ~'2K is attained by substituting the a b o v e
q*l and qy~ into
1l~.12 1 ~ , , (J(xi ) 2 Zx2 + J (Yi )2 Z 2) dr2 K1 = 4k
(13.67)
Notice that J (Xi) and Zy are the only function of y and J (yi) and Zx are the only function of x, we have the first recovery-based error estimator for element ~K 1
l[~'ll21 = -~
(Jxi /yi Xi
-1
Z 2 dx
Yi
-1
J(xi )2 dy +
_--hl2k (fryi J ( x i ) 2 d F + f r x i
J;i /xi Yi
-1
2 dy Zy
-1
J (yi)2 dx
)
(13.68)
J(yi)2dr')
where FXi denotes the limits from Xi-1 to Xi and I"yi from Yi-1 to Yi. In the above we have used the following results fXi lfi Z 2 d x -- fyi yi Zy2 d y = -~ h -1 -1
Similarly, in the second step consider elements f2K, ~'2XK_1 and ~"~YK-1with the recovered solutions written as q*X2 = OxK + 012Zx qy2 -- OyK -Jr-/32Zy
(13.69)
482
Errors,recovery processes and error estimates y
J(xi) .,..,
... A
.,,..
,,,,.
r
xi_l
xi
x
xi+l
~t
~t
DE
DXK+ 1
Fig. 13.16 Recoveredsolution and jump for f2xx and S2xx+ 1 .
To determine the unknown parameters c~2 and/32 we again use the recovery method of averaging and require that at the edge shared by f2x and ~2xK_l 1
qx*2(Xi-1) -- "~(OXK-1(Xi--1) + qXK (Xi--1))
(13.70)
and at the edge shared by element ~K and ~"~YK-1 ,
1
(13.71)
qy2 (Yi-1) -'- 2(qYK-1 (Yi--1) -t- QYK(Yi--1)) Following the exact procedure used in step one, ot2 and/32 are solved as 1
1
1
1
Or2 --" 2Zx(xi-1) (qxx-1 (Xi-1) -- qXK (Xi--1)) --" - - ~ J ( X i - 1 )
(13.72)
/32 -- 2Zy(Yi-1) (qYK-1 (Yi--1) -- QYK(Yi--1)) = - - ~ J(Yi-1) Here J (Xi-1) and J (Yi-1) are jump discontinuities along edges Yi_ 1 and Yi-1 respectively. The recovered solutions are now written as 1
qx*2 = glxx - ~J(xi-1)Zx
(13.73)
1
qy: -= QYK -- -2 J (Yi--1) Zy Substituting Eq. (13.73) into Eq. (13.61) the second recovery-based error estimator can be obtained
III1~
--~--.~2
=
12k
x,_~
J(Yi_l) 2 dF +
y,_~
J(Xi_l) 2 dl-'
(13.74)
Residual-based methods 483
Finally, to include the influence of all the neighbouring elements, it is only natural to let the and i.e., recovery error estimator for element ~K be taken as the average of
I1~11~
1 II~ll~ = ~(ll~ll~:~ + 11~112=)
J(xi) 2 dF + Yi
_
(13.75)
I1~11~and I1~11~=,we have
Substituting the expressions of r = 24k
I1~11~=,
J(yi) 2 dF +
J(Yi_l) 2 dF +
xi
x i -- 1
J(xi-1
dF
Yi - 1
h f JZdF 24k JrK (13.76)
This is exactly the explicit residual error estimator of Eq. (13.60). We have demonstrated, by the above example, that the explicit residual error estimator for the bilinear element can be derived from a recovery-based error estimator using averaging as the recovery method. For more general discussions on the relationship between recoverybased error estimators and explicit residual error estimators we refer to references 50 and 51; for discussion on the equivalence of recovery-based error estimators with certain explicit residual estimators we refer to references 48 and 52; for using a recovery method in the computation of the explicit residual error estimator the reader is referred to reference 53. We shall now turn our attention to how to use a recovery method in the computation of implicit residual error estimators.
13.7.2 Implicit residual error estimators The computation of implicit residual error estimators requires solving an auxiliary boundary value problem with residuals as input data for the approximation error. Among all the existing implicit residual error estimators, the equilibrated residual estimator has been shown to be the most robust. 54-56 In what follows we restrict our discussion to the equilibrated residual error estimator for the model problem of Eq. (13.50). The construction of an equilibrated residual error estimator for other problems, such as elasticity problems, proceeds in an analogous manner. 57 We again consider an interior element K. Substituting the finite element solution ~ into Eq. (13.50) results in, for element K,
- V T (k V(b) + Q = rr
in f2K
(13.77)
and -(kV~)Tn=qn
onFK
Subtracting the above equations from Eq. (13.50) gives an element boundary value problem for error e as - V T (k ~re) + rx = 0 in ~X (13.78) with boundary condition - (k Ve) T n = qn
-
-
qn on FK
484
Errors,recovery processes and error estimates
We notice immediately that Eq. (13.78) is not solvable because the exact normal flux
qn on the element boundary is in general unknown. A natural strategy to overcome this
difficulty is to replace the exact solution by a recovered solution qn which can be computed from the finite element flux in element K and its surrounding elements (as we did in the computation of a recovery-based error estimator). We can now write the Neumann boundary value problem for the element error as
--~7T (k We) + rK = 0 in ~K
(13.79)
with boundary condition - (k Ve) T n - qn*
-
t]n on F K
An approximate solution to the above equation for E appearing in the energy norm, I1~IlK, defines an implicit element residual error estimator. _ , 34 42, 43 Various recovery techniques can be used to compute the normal flux tln. ' However, the Neumann problem of Eq. (13.79) will have a solution if qn* is computed such that the residuals satisfy the equilibrium condition
/~ Ncrx df2 + fr Nc (q~ - ~n) dF - 0 K
(13.80)
K
where Nc is the shape function for node c of element K. Although Nc can be a shape function of any order, a linear shape function seems to be the most practical in the following computation. The residuals which satisfy Eq. (13.80) are said to be equilibrated, thus the recovered solution qn* satisfying Eq. (13.80) is called the equilibrated flux. An error estimator which uses the solution of the element error problem of Eq. (13.80) with the equilibrated flux q* is termed an equilibrated residual error estimator. This type of residual error estimator was first introduced by Bank and Weiser 42 and later more rigorously pursued by Ainsworth and Oden. 46 It is apparent that the most important step in the computation of the equilibrated residual error estimator is to achieve the recovered normal flux q,~ which satisfies Eq. (13.80). Once q~* is determined, the error problem Eq. (13.79) can be readily solved for an element following a standard finite element procedure. Therefore we shall focus our attention on the recovery process. The technique of recovering normal flux by equilibrated residuals was first proposed by Ladev6ze et al., 34 Kelly 41 and followed by Ohtsubo and Mitamura. 58 A different version of this technique was later used by Ainsworth and Oden 49 where a detailed description of the application to various mesh patterns can be found. Here we shall consider a typical element patch of triangles as shown in Fig. 13.17. To determine q,~, we first substitute the residual r K of Eq. (13.77) into Eq. (13.80) and upon integrating by parts obtain
]~ NcQ df2 + ~ K
(VNc)T (k V(b) df2 + fr Ncq: dF - O K
(13.81)
K
Let the recovered interelement boundary normal flux have the form 9 1 )T qn = ~ ({IK +{11 ns + Zs
(13.82)
Residual-based methods 485
n2
S4 e.
Fig. 13.17 Typical patch with interior vertex node a showing a local numbering of elements e; and edges S;. where the first term on the right-hand side is the average of the normal flux of the finite element solution from element K and its neighbour element I as shown in Fig. 13.18; ns is the outward normal on the edge s of element K; and Zs is a linear function defined on the edge s, shared by elements K and I, with end nodes a and b and AS
S
AS
S
(13.83)
Zs -- Naa a 't- Nba b
where N~,/f/~ are the dual shape functions introduced in Sec. 12.2.1 and in the present case are given by /~ls =
2 Ihsl [2 N s - N{]
and
N~ = ~
2
[2 N{ - N s]
(13.84)
where N,~ and N[ are the linear shape functions defined for edge S and Ihsl is the length of the edge. The unknown parameters a s and ag are to be determined from the residual equilibrium equation ( 13.81). It is easy to verify that
fS N asAs N b dl-' = (~ab where
(13.85)
(~abis the Kronecker delta given by: (~ab=
1, 0,
a=b
a:/=b
(13.86)
Let a denote a typical interior vertex node. Choose Arc = Na in Eq. (13.81) and consider the element patch associated with the linear shape function Na as shown in Fig. 13.17. Assign element 1 as element K in the patch (i.e., el = eK). It is obvious that Na is zero at the exterior boundary of the element patch. A local numbering for the elements and edges connected to node a in the patch is given. The edge normals shown here are the result of a global edge orientation.
486
Errors,recovery processes and error estimates
a t9
Fig. 13.18 Element interface for equilibrated flux recovery. For element el in the patch, substituting Eq. (13.82) into Eq. (13.81) for each edge and observing that Na is non-zero only on the edges Sl and s2 and in the n l and n2 directions we have
f~ NaQd~"+-~ el
+
el
+ qes)Tnsl dF (VNa)T(kVqb)df2 " f ~Na(Cle~ 1
J~s2~ga(~lel 1
(13.87) +
qe2)Tns2 dr - ~s~ gaZsi
dr +
J~s2NaZs2 d r
= 0
where a boundary integral takes a negative sign if the edge normal shown in Figs 13.17 and 13.18 points inward to the element. Let fe~ denote the first four, computable, terms of the above equation and notice that [using Eq. (13.85)]
/saNaZs~
dr'
-
/s Na (N~,assa + N~,a sl b ) /-
and
f S NaZs2 d r -- /
Js2
dr
= aS~
(13.88a)
Na (Naa ^ S aS2 -4- N~,ab ^ S $2 ) dF
= a s2
(13.88b)
Equation (13.87) now becomes -aS~ + aS2 = - fel
(13.89)
Similar equations result for element e2 to e5 of the patch in Fig. 13.17 giving the equation set Aa = f (13.90) where
-1
A
1
0
0
0
0 0 0
-1 0 0
1 -1 0
0 1 -1
1
0
0
0
0 o 1 - 1
Residual-based methods 487 I t - - [ a a sl ,
a s2,
aas3,
a s4,
aS'] T
and
f---[--fel,
--fe2, --fe3, --fee,
--fe5] T
It is easy to verify that these equations are linearly dependent but have solutions determined up to an arbitrary constant. A procedure to obtain an optimal particular solution is described as follows. 35' 42, 49 First, a particular solution ao of Eq. (13.90) is found by choosing, for example, a s5 = 0. Second, the corresponding homogeneous equation
Ab=0
(13.91)
with b = [bl, be, b3, b4, bs] T is solved for a non-zero particular solution with the choice of, corresponding to asS, b5 = 1. It is easy to verify that bi is either 1 or - 1 due to the structure of A. In the element patch considered here b = [1, 1, 1, 1, 1]T. The final particular solution of Eq. (13.90) takes the form of a = a0 + gb
(13.92)
where the constant y is determined by the minimization of FI = a X a The minimization condition gives y =
bTa0 bT b
(13.93)
(13.94)
The solution gives the nodal value aa~` at node a for each connected edge of the element patch. Boundary nodes and their related element patches can be considered in the same fashion except that we can take q* = On, the known flux, for the element edge being part of I"q. For edges coincident with FO, we let the first term on the fight-hand side of Eq. (13.82) be zero. By considering each vertex node of the mesh and its associated element patch, we will be able to determine a s and a~ in Eq. (13.83) for every edge, thus the recovered normal flux qn* in the form defined by Eq. (13.82) on the element boundary is achieved. The procedure described above for recovering the normal flux is a recovery by element residuals. We note that the non-uniqueness of the solution of Eq. (13.90) represents the nonuniqueness of the equilibrium status of the element residuals. The choice of the arbitrary constant in solving Eq. (13.90) will certainly affect the accuracy of the recovered solution qn*, and therefore the accuracy of the error estimator. With q* determined, the local error problem Eq. (13.79) is usually solved by a higher order (e.g., p + 1 or even p + 2) approximation. The solution of the problem is then employed in the equilibrated residual error estimator IlOllr~. The global error estimator I1~11 is obtained through Eq. (13.55). The global error estimator has been shown to be an upper bound of the exact error,46 although it is not a trivial task to prove its convergence. We have shown here that a proper recovery method is the key to the computation of equilibrated residual error estimators. Indeed, carefully chosen recovery methods are very important in the computation of all the implicit residual error estimators. Numerical performance of residual-based error estimators was tested by Babugka et al. 54-56 and Carstensen et al. 59 and compared with that of recovery-based error estimators.
488
Errors,recovery processes and error estimates
Fig. 13.19 Repeating patch of irregular and quadrilateral elements.
It is well known that elements in which polynomials of order p are used to represent the unknown u will reproduce exactly any problem for which the exact solution is also defined by such a polynomial. Indeed the verification of this behaviour is an essential part of the 'patch test' which has to be satisfied by all elements to ensure convergence, as we have discussed in Chapter 9. Thus if we are attempting to determine the error in a general smooth solution we will find that this error is dominated by terms of order p + 1. The response of any patch to an exact solution of order p + 1 will therefore determine the asymptotic behaviour when both the size of the patch and of all the elements tends to zero. If the patch is assumed to be one of a repeatable kind, its behaviour when subjected to an exact solution of order p + 1 will give the exact asymptotic error of the finite element solution. Thus, any estimator can be compared with this exact value and the asymptotic effectivity index can be established. Figure 13.19 shows such a repeatable patch of quadrilateral elements which evaluate the performance of the error estimators for quite irregular meshes. We have indeed shown how true superconvergent behaviour reproduces exactly such higher order solutions and thus leads to an effectivity index of unity in the asymptotic limit. In the papers presented by Babu~ka et a/. 54-56 the procedure of dealing with such repeatable patches for various patterns of two-dimensional elements is developed. Thus, if we are interested in solving the differential equation /Zu + f = 0
(13.95)
where s is a linear differential operator of order 2p, we consider exact solutions (harmonic solutions) to the homogeneous equation ( f = 0) of the form Uex = ~ m
amxmy
n --
P(x, y)a;
n = p + 1- m
(13.96)
The boundary conditions are taken as
Uexlx+Zx--" uexlx
and
Uex[yq_Ly = Uex[y
(13.97)
where Lx and Ly are periodic distances in the x and y directions, respectively (viz. repeatability, Sec. 6.2.4, page 192). In general, the individual terms of Eq. (13.96) do not satisfy
Asymptotic behaviour and robustness of error estimators - the Babugka patch test
the differential equation and it is necessary to consider linear combinations in terms of the parameters in L; as a ' = Ta (13.98) This solution serves as the basis for conducting a patch test in which the boundary conditions are assigned to be periodic and to prevent constant changes to u.t The correct constant value may be computed from fpa
tch
(Nil + C) dr2 - foa
Uex
tch
d~2
(13.99)
To compute upper and lower bounds (Or and OL) on the possible effectivity indices of the error estimators, all possible combinations of the harmonic solution must be considered. This may be achieved by constructing an error norm of the solutions, for example the L2 norm of the flux (or stress) Ileq I1~ = fpatch (qex -- q)T (qex -- q) d~2 = (a')T TTEexTa,
(13.100)
(q* - q) T (q, _ q) dQ = (a')T TTE, Ta,
(13.101)
and II~-qI1~
fpa =
tch
and solving the eigenproblem TTE*Ta ' = 02TTEexTa '
(13.102)
to determine the minimum (lower bound) and maximum (upper bound) effectivity indices. Further details of the process summarized here are given in Boroomand and Zienkiewicz32, 33 and by Zienkiewicz et al. 6~ These bounds on the effectivity index are very useful for comparing various error estimators and their behaviour for different mesh and element patterns. However, a single parameter called the robustness index has also been devised 54 and is useful as a guide to the robustness of any particular estimator R = max
I1 - 0 L I + I1 - O v l ,
I1 - ~-/LI+ I1 - ~--~vl
(1.3.103)
A large value of this index obviously indicates a poor performance. Conversely the best behaviour is that in which OL = OU = 1 (13.104) and this gives R =0
(13.105)
In the series of tests reported in references 54-56 various estimators have been compared. Table 13.1 shows the highest robustness index value of an equilibrating residual-based error estimator, ERpB, and the SPR recovery error estimator for a set of particular patches of triangular elements. 54 t For elasticity-typeproblems the periodic boundary conditionspreventrigid rotations.
489
490
Errors,recovery processes and error estimates Table 13.1 Robustness index R for equilibrated residual ERpB and SPR (ZZ-discrete) estimators for a variety of anisotropic situations and element patterns, p = 2 Estimator form
Robustness i n d e x - R
ERpB SPR (ZZ-discrete)
10.21 0.02
This performance comparison is quite remarkable and it seems that in all the tests quoted by Babu~ka eta/. 54-56 and summarized in Babu~ka and Strouboulis 61 the recovery estimator using SPR performs best. Indeed we shall observe that in many cases of regular subdivision, when full superconvergence occurs the ideal, asymptotically exact solution characterized by R = 0 will be obtained. In Table 13.2 we show some results obtained for regular meshes of triangles and rectangles with linear and quadratic elements. In the rectangular elements used for problems of heat conduction type, superconvergent points are exact and the ideal result is obtained for both linear and quadratic elements. It is surprising that this also occurs in elasticity where the proof of superconvergent points is lacking [since for v > 0 A in (13.17) is not diagonal]. Further, the REP procedure also seems to yield superconvergence except for elasticity with quadratic elements. For regular meshes of quadratic triangles generally superconvergence is not expected and it does not occur for either heat conduction or elasticity problems. However, the robustness index has very small values (R < 0.10 for SPR and R < 0.12 for REP) and these estimators are therefore very accurate. In Fig. 13.20 and Table 13.3 very irregular meshes of triangular and quadrilateral elements are analysed in repeatable patterns. It is of course not possible to present here all tests conducted by the effectivity patch test. The results shown are, however, typical - others are given in reference 32. It is interesting to observe that the performance measured by the robustness index on quadrilateral elements is always superior to that measured on triangles.
Although we have shown that excellent estimators of errors exist today, many are striving to know that these estimators are not only close but that they are bounded. The strain energy was one of the first quantities in which bounds could be established. Here the classical work of Fraeijs de Veubeke in the mid-1960s is of vital importance. 62' 63 It was quickly realized by Fraeijs de Veubeke that the standard (displacement) procedures from which structural analysis usually started would provide a lower bound of the strain energy contained in the structure and thus always underestimated the value of strain energy. He therefore sought procedures which could solve the same structural problem by concentrating on so-called complementary energy which would allow to be established solutions in which the strain energy would always be overestimated. This process proved very difficult as equilibrating solutions have to be established at all stages. A possible way, useful for many two-dimensional problems, was suggested in reference 64 in which stress functions and the slab analogy were used. Nevertheless the methodology never succeeded as a practical way of providing the bounds of strain energy in an actual analysis.
Bounds on quantities of interest Table 13.2 Effectivity bounds and robustness of SPR and REP recovery estimator for regular meshes of triangles and rectangles with linear and quadratic shape function (applied to heat conduction and elasticity problems). Aspect ratio = length(L)/height(H) of elements in patch tested Linear triangles and rectangles (heat conduction/elasticity) SPR
REP
Aspect ratio L/H
OL
OU
R
OL
Ou
R
1/1 1/2 1/4 1/8 1/16 1/32 1/64
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Quadratic rectangles (heat conduction)
1/1 1/2 1/4 1/8 1/16 1/32 1/64
OL
OU
R
OL
OU
R
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Quadratic rectangles (elasticity)
1/1 1/2 1/4 1/8 1/16 1/32 1/64
OL
Ou
R
OL
Ou
R
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.9991 0.9991 0.9991 0.9991 0.9968 0.9950 0.9945
1.0102 1.0181 1.0136 1.0030 1.0001 1.0000 1.0000
0.0111 0.0189 0.0145 0.0039 0.0033 0.0050 0.0055
Quadratic triangles (elasticity)
1/1 1/2 1/4 1/8 1/16 1/32 1/64
OL
Ou
R
OL
OU
R
0.9966 0.9966 0.9967 0.9967 0.9966 0.9966 0.9965
1.0929 1.0931 1.0937 1.0943 1.0946 1.0947 1.0947
0.0963 0.0965 0.0970 0.0976 0.0980 0.0981 0.0982
0.9562 0.9559 0.9535 0.9522 0.9518 0.9517 0.9516
1.0503 1.0481 1.0455 1.0603 1.0666 1.0684 1.0688
0.0940 0.0923 0.0924 0.1081 0.1148 o. 1167 0.1172
M u c h later w h e n the residual m e t h o d w a s b e i n g applied to d e t e r m i n e error in structural analysis it w a s r e a l i z e d that o n c e again o p p o r t u n i t y existed for establishing b o u n d s . T h e residuals are n o t h i n g else but a m e a s u r e by w h i c h the n u m e r i c a l solution fails to satisfy the differential e q u a t i o n s o f the p r o b l e m . B y using a local solution, g e n e r a l l y b a s e d on a few e l e m e n t s or even a single e l e m e n t , the error can be e s t i m a t e d locally and the total error o b t a i n e d by c o m b i n i n g these estimates for all e l e m e n t s . T h e c o m p l e t e l y i n d e p e n d e n t solution for the d i s p l a c e m e n t , stresses, etc. e s t a b l i s h e d b y the residuals p r o v i d e s the m e a s u r e o f the error. This solution can be carried out in a n u m b e r o f w a y s and here a d e p a r t u r e f r o m
491
492
Errors, recovery processes and error estimates
(a)
(e)
(b)
(f)
(c)
(g)
(d)
(h)
Fig. 13.20 Repeating patch types. the original, say displacement, method can be made. As many local problems are solved, it is assumed the total error is a combination of local (patch) solutions. It seems that one of the first to extend the concept of establishing upper bounds is Kelly in 198441 and subsequent work. 65 He endeavoured to obtain solutions for the residual placed as a load with
Bounds on quantities of interest Table 13.3 Effectivity bounds and robustness of SPR and REP recovery estimator for irregular meshes of (a, b, c, d) and quadrilaterals (e, f, g, h) Linear element (heat conduction) SPR
REP
Mesh pattern
OL
OU
R
OL
OU
R
a b c d e f g h
0.9626 0.9715 0.9228 0.8341 0.9943 0.9969 0.9987 0.9991
1.0054 1.0156 1.4417 1.2027 1.0175 1.0152 1.0175 1.0068
0.0442 0.0447 0.5189 0.3685 0.0232 0.0183 0.0188 0.0077
0.9709 0.9838 0.8938 0.9463 0.9800 0.9849 0.9987 0.9979
1.0145 1.0167 1.8235 1.9272 1.0589 1.0582 1.0175 1.0062
0.0443 0.0329 0.9297 0.9810 0.0789 0.0733 0.0188 0.0083
Linear elements (elasticity) SPR
a b c d e f g h
REP
OL
OU
R
OL
OU
R
0.9404 0.8869 0.8550 0.7945 0.9946 1.0038 0.9959 0.9972
1.0109 1.0250 1.6966 1.2734 1.0247 1.0281 1.0300 1.0139
0.0741 0.1520 0.8415 0.4788 0.0301 0.0318 0.0341 0.0168
0.9468 0.9392 0.8037 0.7576 0.9579 0.9612 0.9960 0.9965
1.0148 1.0275 2.0522 1.9416 1.0508 1.0467 1.0298 1.0122
0.0707 0.0915 1.2486 1.1840 0.0928 0.0855 0.0338 0.0157
Quadratic elements (heat conduction)
a b c d e f g h
OL
OU
R
OL
OU
R
0.9443 0.8146 0.7640 0.8140 0.9762 0.9691 0.9692 0.9906
1.0295 1.0037 1.0486 1.0141 1.0053 1.0045 1.0004 1.0113
0.0877 0.2313 0.3000 0.2423 0.0296 0.0363 0.0322 0.0207
0.9339 0.9256 0.9559 0.9091 0.9901 0.9901 0.9833 1.0045
1.0098 1.0028 1.2229 1.2808 1.0177 1.0322 1.0024 1.0261
0.0805 0.0832 0.2670 0.3717 0.0276 0.0421 0.0195 0.0307
Quadratic elements (elasticity)
a b c d e f g h
OL
Ou
R
OL
OU
R
0.9144 0.7302 0.7556 0.7624 0.9702 0.9651 0.9457 0.9852
1.0353 1.0355 1.1024 1.0323 1.0102 1.0085 1.0115 1.0141
0.1277 0.4038 0.4163 0.3430 0.0408 0.0446 0.0688 0.0290
0.9197 0.8643 0.8387 0.8244 0.9682 0.9749 0.9807 0.9996
1.0244 1.0346 1.2422 1.2632 1.0058 1.0286 1.0125 1.0522
0.1111 0.1905 0.4035 0.4388 0.0386 0.0537 0.0321 0.0526
equilibrating methodologies. Two similar alternative approaches, though stemming from completely different origins, were proposed by Ladev~ze 34' 66 and almost simultaneously by Bank and Weiser. 42 These ideas were later adopted by Ainsworth and Oden forming much of the basis to their book. 49
493
494
Errors,recovery processes and error estimates
The methodologies so far produced are very effective when the basic quantity of interest is a simple one, such as strain energy or the energy norm. However, when the quantity of interest is more localized and if for instance it is the displacement at some part of the structure, rather than an overall measure of stresses as it is in the case of energy, different procedures arise and pure examination of energy errors does not suffice or it is not very selective in showing how to obtain the answers. For this reason much effort has been given in recent years in discussing the possibilities of bounds and the manner by which such localized goals of analysis can be solved. Much of the recent work in this field concentrates on such methodologies. The first to give attention to the possible extension of norms to other quantities of interest appeared in a series of papers presented by Babu~ka and Miller in 1984. 67-69 These papers laid the foundations for much of the work continued some ten years later and which today occupies much interest. Here it appears that the first full extension of the methodology is due to Peraire et al. first published in 1997 with many papers following. 7~ By extending the ideas introduced in the Babu~ka and Miller papers, Peraire et al. show that whatever the quantity of interest is, it is always possible to establish an adjoint problem which can be solved on the same mesh of the original problem but now with different loads for dealing with the accuracy. Such adjoint problems may be called differently and names such as extraction problems and dual problems are also used. Although many people are now entering the field and the methodology has been followed by Oden and Prudhomme in a series of papers, it appears that only Peraire so far has extended the approach to nonself-adjoint problems such as fluid dynamics. 73-75 The equilibrated methods have always provided upper bounds for such quantities as strain energy. A similar bounding occurs if we look at the energy in the adjoint approaches. However, some interest now goes back to placing satisfactorily the lower bounds and thus bracketing the solution. Of course, lower bounds of zero values have been used and this is rather defeatist because they do not provide tight estimates. It is our belief that more precise bounds are required and here the work of Dfez and Huerta seems to lead the way. 47' 76, 77
In this chapter we have shown how various recovery procedures can accurately estimate the overall error of the finite element approximation and thus provide a very accurate error estimating method. We have also shown that estimators based on SPR recovery are superior to those based on residual computation. The error estimation discussed here concerns, however, only the original solution and if the user takes advantage of the recovered values a much better solution is already available. In the next chapter we shall be concerned with adaptivity processes which are aimed at reduction of the original finite element error. Here again we shall show the excellent values of the effectivity index which can be obtained with SPR-type methods on examples for which an 'exact' solution is available from very fine mesh computations. What perhaps we should also be concerned with are the errors remaining in the recovered solutions, if indeed these are to be made use of. This problem is still unsolved and at the moment all the adaptive methods simply aim at the reduction of various norms of error in the finite element solution directly provided.
Problems 495
13.1 Let the assumed stress for Example 13.1 in Sec. 13.4.1 be given as
o"i = [ 1 ,
(X--X1),
(Y--Y1),
(x - Xl)(y - Yl)]
~2 fi3 a4
13.2
13.3
13.4
13.5
13.6
13.7 13.8
13.9
Compute the recovered stress and compare the result with that of Example 13.1. Is this result superconvergent? Show all the relations necessary to extend the SPR algorithm to three-dimensional elastic problems. What is the expression for or/* that should be used for 8-node hexahedral elements? Program development project: Implement the SPR procedure in the solver system developed in Problem 2.17 and subsequent chapters. Assume the problem is modelled by the quasi-harmonic equation using 4-node quadrilateral elements. (Hint: Extend the result from Problem 6.19.) Program development project: Implement the SPR procedure in the solver developed in Problem 2.17 and subsequent additions. Assume a linear elastic problem that is modelled using 4-node quadrilateral elements. (Hint: Extend the result from Problem 6.19.) The element size h appearing in the explicit residual error estimator given by Eq. (13.59) is often taken as a constant for a particular element of certain shape. Consider results from Example 13.2 and explain why the accuracy of the explicit residual error estimator will deteriorate when the aspect ratio of the element increases, i.e., when the mesh becomes more anisotropic. Extend the technique of recovering normal flux by equilibrated residuals described in Sec. 13.7.2 to two-dimensional elastic problems. Consider both plane and axisymmetric geometry. Extend the technique of recovering normalflux by equilibrated residuals described in Sec. 13.7.2 to three-dimensional elastic problems. Program development project: Implement a recovery-based error estimator or a residual-based error estimator in the solver system developed in Problem 2.17 and subsequent exercises. Program development project: Extend the program developed in Problem 2.17 to compute the SPR solution for displacements. Use the recovered displacements to compute strains and from these stresses. Follow the procedure given in Sec. 13.4.2 to project 3-node triangular and 4-node quadrilateral element values to nodes. Test your program using (a) the patch test of Problem 2.17 and (b) the curved beam problem shown in Fig. 2.11. Report results for both displacements and stresses.
496
Errors,recovery processes and error estimates
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In the previous chapter we have discussed at some length various methods of recovery by which the finite element solution results could be made more accurate and this led us to devise various procedures for error estimation. In this chapter we are concerned with methods which can be used to reduce the errors once a finite element solution has been obtained. As the process depends on previous results at all stages it is called adaptive. Such adaptive methods were first introduced to finite element calculations by Babu~ka and Rheinboldt in the late 1970s. 1' 2 Before proceeding further it is necessary to clarify the objectives of refinement and specify 'permissible error magnitudes' and here the engineer or user must have very clear aims. For instance the naive requirement that all displacements or all stresses should be given within a specified tolerance is not acceptable. The reasons for this are obvious as at singularities, for example, stresses will always be infinite and therefore no finite tolerance could be specified. The same difficulty is true for displacements if point or knife edge loads are considered. The most common criterion in general engineering use is that of prescribing a total limit of the error computed in the energy norm. Often this error is required not to exceed a specified percentage of the total energy norm of the solution and in the many examples presented later we shall use this simple criterion. However, using a recovery type of error estimator it is possible to adaptively refine the mesh so that the accuracy of a certain quantity of interest, such as the RMS error in displacement and/or RMS error in stress (see Chapter 13, Eqs (13.9) and (13.10)), satisfies some user-specified criterion. We should recognize that mesh refinement based on reducing the RMS error in displacement is in effect reducing the average displacement error in a user-specified region (e.g., in each element); similarly mesh refinement based on reducing the RMS error in stress is the same as reducing the average stress error in a user-specified region. Here we could, for instance, specify directly the permissible error in stresses or displacements at any location. Some investigators (e.g., Zienkiewicz and Zhu 3) have used RMS error in stress in the adaptive mesh refinement to obtain more accurate stress solutions. Others (e.g., Ofiate and Bugeda 4) have used the requirement of constant energy norm density in the adaptive analysis, which is in fact equivalent to specifying a uniform distribution of RMS error in stress in each element. We note that the recovery type of error estimators are particularly useful and convenient in designing adaptive analysis procedures for the quantities of interest. For
Introduction
other methodologies of designing adaptive analysis procedures based on error estimation of the quantities of interest, we refer to references 5-7. As we have already remarked in the previous chapter we will at all times consider the error in the actual finite element solution rather than the error in the recovered solution. It may indeed be possible in special problems for the error in the recovered solution to be zero, even if the error in the finite element solution itself is quite substantial. (Consider here for instance a problem with a linear stress distribution being solved by linear elements which result in constant element stresses. Obviously the element error will be quite large. But if recovered stresses are used, exact results can be obtained and no errors will exist.) The problem of which errors to consider still needs to be answered. At the present time we shall consider the question of recovery as that of providing a very substantial margin of safety in the definition of errors. Various procedures exist for the refinement of finite element solutions. Broadly these fall into two categories: 1. The h-refinement in which the same class of elements continue to be used but are changed in size, in some locations made larger and in others made smaller, to provide maximum economy in reaching the desired solution. 2. The p-refinement in which we continue to use the same element size and simply increase, generally hierarchically, the order of the polynomial used in their definition. It is occasionally useful to divide the above categories into subclasses, as the h-refinement can be applied and thought of in different ways. In Fig. 14.1 we illustrate three typical methods of h-refinement: 1. The first of these h-refinement methods is element subdivision (enrichment) [Fig. 14.1 (b)]. Here refinement can be conveniently implemented and existing elements, if they show too much error, are simply divided into smaller ones keeping the original element boundaries intact. Such a process is cumbersome as many hanging points are created where an element with mid-side nodes is joined to a linear element with no such nodes. On such occasions it is necessary to provide local constraints at the hanging points and the calculations become more involved. In addition, the implementation of de-refinement requires rather complex data management which may reduce the efficiency of the method. Nevertheless, the method of element subdivision is quite widely used. 2. The second method is that of a complete mesh regeneration or remeshing [Fig. 14.1(c)]. Here, on the basis of a given solution, a new element size is predicted in all the domains and a totally new mesh is generated. Thus a refinement and de-refinement are simultaneously allowed. This of course can be expensive, especially in three dimensions where mesh generation is difficult for certain types of elements, and it also presents a problem of transferring data from one mesh to another. However, the results are generally much superior and this method will be used in most of the examples shown in this chapter. For many practical engineering problems, particularly of those for which the element shape will be severely distorted during the analysis, adaptive mesh regeneration is a natural choice. 3. The final method, sometimes known as r-refinement [Fig. 14.1 (d)], keeps the total number of nodes constant and adjusts their position to obtain an optimal approximation. 5-7 While this procedure is theoretically of interest it is difficult to use in practice and there is little to recommend it. Further it is not a true refinement procedure as a prespecified accuracy cannot generally be reached.
501
502 Adaptive finite element refinement
(a) Original mesh
(b) Mesh enhancement by subdivision (enrichment).
j J
//
/
(c) Mesh enhancement by remeshing
(d) r-refinement of original mesh by reposition of nodes Fig. 14.1 Variousproceduresby h-refinement.
Adaptive h-refinement 503 We shall see that with energy norms specified as the criterion, it is a fairly simple matter to predict the element size required for a given degree of approximation. Thus very few re-solutions are generally necessary to reach the objective. With p-refinement the situation is different. Here two subclasses exist: 1. One in which the polynomial order is increased uniformly throughout the whole domain; 2. One in which the polynomial order is increased locally using hierarchical refinement. In neither of these has a direct procedure been developed which allows the prediction of the best refinement to be used to obtain a given error. Here the procedures generally require more resolutions and tend to be more costly. However, the convergence for a given number of variables is more rapid with p-refinement and it has much to recommend it. On occasion it is possible to combine efficiently the h- and p-refinements and call it the hp-refinement. In this procedure both the size of elements h and their degree of polynomial p are altered. Much work has been reported in the literature by Babugka, Oden and others and the interested reader is referred to the references. 8-18 In Secs 14.2 and 14.3 we shall discuss both the h- and the p-refinement methods. In Sec. 14.3 we also include some details of the very simple and yet efficient hp-refinement process introduced by Zienkiewicz, Zhu and Gong. 19 i!il!i!!','!i~,iE!iEi'i,I"ii' !~~i~,i'~i,i'~,i~ii il,il~,i,U~~~ii',liii,'ii,,ii i~,i~~iii ',i ~,i ii',ii i ii"i,i',i",i i'i ',i i',!i'~,i'i~~ii~ii i~i ii i!i'i,~i!ii'~i!'!,i~!''i,!'i'~,"',~i'~'i,~',',i',iiii~i 'i~i i~i~i,~' ,,i;i~:,~i'~,~ii i ~i i~:,,i ~'~i'i ,ii, i~,:,~,i~,,~i',i,',i',i',~,i',i',i'i~i, i',i',~i!'',i'!'!i'~'~!i'~,!i',!i!''~!i',~"i! i ~,',~,~,,ii;',i~,~,~i,i~,,,i~i'~,i,'~,,i,',~, ~ i!~
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14.2.1 Predicting the required element size in h adaptivity In the introduction to this chapter we have mentioned several alternative processes of hadaptivity and we suggested that the process in which the complete mesh is regenerated is in general the most efficient. Such a procedure allows elements to be de-refined (or enlarged) as well as refined (made smaller) and invariably starts at each stage of the analysis from a specification of the mesh size h defined at each nodal point of the previous mesh. Standard interpolation is used to find the size of elements required at any point in the domain. As the refinement process proceeds for each subsequent stage of analysis the computed mesh sizes h are based on a prescribed accuracy at the nodes of the previous mesh. The error estimators discussed in the previous chapter allow the global energy (or similar) norm of the error to be determined and the errors occurring locally (at the element level) are usually also well represented. If these errors are within the limits prescribed by the analyst then clearly the work is completed. More frequently these limits are exceeded and refinement is necessary. The question which this section addresses is how best to effect this refinement. Here obviously many strategies are possible and much depends on the objectives or goals to be achieved. In the simplest case we shall seek, for instance, to make the relative energy norm error 77 [viz. Eq. (13.12)] less than some specified value f7 (say 5% for many engineering applications). Thus r / < f/ (14.1) is to be achieved. In an 'optimal mesh' it is desirable that the distribution of element energy norm error (i.e., Ilellr) should be equal for all elements. Thus if the total permissible error is determined (assuming that it is given by the result of the approximate analysis) as
504 Adaptive finite element refinement Permissible error -- ~/llull = ~7 (11~112+ Ilell2) 1/2
(14.2)
Ilell2 --Ilull 2 - I1~112
(14.3)
here we have used 2~ We could pose a requirement that the error in any element k should be IlellK < ~7( ]lull2 1-q-/]]ell2) 2 m
~em
(14.4)
where m is the number of elements involved. Elements in which the above is not satisfied are obvious candidates for refinement. Thus if we define a refinement ratio by Ilellzc ~K = (14.5) em
we shall refine whenevert ~r > 1
(14.6)
The refinement ratio ~r can be approximated, of course, by replacing the true error in Eqs (14.4) and (14.5) with the error estimators. The refinement could be carried out progressively by refining only a certain number of elements in which ~ is higher than a specified limit. This type of element subdivision process is also known as mesh enrichment as depicted in Fig. 14.1 (b). This process of refinement though ultimately leading to a satisfactory solution being obtained with a relatively small number of total degrees of freedom, is in general not economical as the total number of trial solutions is usually excessive. It is more efficient to try to design a completely new mesh which satisfies the requirement that ~r < 1 (14.7) in all elements. One possibility here is to invoke the asymptotic convergence rate criteria to predict the element size distribution. For instance, if we assume IlellK o~ h/~
(14.8)
where hK is the current element size and p the polynomial order of approximation, then to satisfy the requirement of Eq. (14.4) the new generated element size should be no larger than hnew - (14.9)
~K1/PhK
Mesh generation programs in which the local element size can be specified are available now as we have already discussed in Chapter 8 and these can be used to design a new mesh for which the reanalysis is carded out. 21,22 In the figures we show how starting from a relatively coarse solution a single mesh prediction often allows a solution (almost) satisfying the specified accuracy requirement to be achieved. The reason for the success of the mesh regeneration based on the simple assumption of asymptotic convergence rate implied in Eq. (14.8) is the fact that with refinement the mesh t We can indeed 'de-refine' or use a larger element spacing where ~tr < 1 if computational economy is desired.
Adaptive h-refinement 505 tends to be 'optimal' and the localized singularity influence no longer affects the overall convergence. Of course the effects of any singularity will still remain present in the elements adjacent to it. An improved mesh results if in such elements we use the appropriate convergence and replace p by )~ in Eqs (14.8) and (14.9), to obtain hnew - -
~K1/ZhK
(14.10)
in which ~. is the singularity strength, see Chapter 13, Eq. (13.14). A conservative number to use here is ~. = 0.5 as most singularity parameters lie in the range 0.5-1.0. With this procedure, added to the refinement strategy, we frequently achieve accuracies better than the prescribed limit in one remeshing.
14.2.2 Numerical examples In the examples which follow we will show in general a process of refinement in which the total number of degrees of freedom increases with each stage, even though the mesh is redesigned. This need not necessarily be the case as a fine but badly structured mesh can show much greater error than a near-optimal one. To illustrate this point we show in Fig. 14.2 a refinement designed to reach 5% accuracy in one step starting from uniform mesh subdivisions. We note that now, in at least one refinement, a decrease of total error occurs with a reduction of total degrees of freedom (starting from a uniform 8 • 8 subdivision with 544 equations and r / = 9.8% to ~ = 3.1% with 460 equations). We shall now present further typical examples of h-refinement with mesh adaptivity. In all of these, full mesh regeneration is used at every step.
Example 14.1: Short cantilever beam. This problem refers to a short cantilever beam in which two very high singularities exist at the comers attached to a rigid wall. The beam is loaded by a uniform load along the top boundary as illustrated in Figs 14.3 and 14.4. In the refinement process we use both the mesh criteria of Eqs (14.9) and (14.10). 23 In Figs 14.3 and 14.4 we show three stages of an adaptive solution and in Fig. 14.5 we indicate how rapidly these converge, although all uniform refinements converge at a very slow rate (due to the singularities). The same problem is also solved by both mesh enrichment and mesh regeneration using linear quadrilateral elements to achieve 5% accuracy. The prescribed accuracy is obtained with optimal rate of convergence being reached by both adaptive refinement processes (Fig. 14.6). However, the mesh enrichment method requires seven refinements, as shown in Fig. 14.7, while mesh regeneration requires only three (see Fig. 14.8). Here the refinement criterion, Eq. (14.6), is used for the mesh enrichment process.
Example 14.2: Stressed cylinder. As we mentioned earlier, the value of the energy norm error is not necessarily the best criterion for practical refinement. Limits on the local stress error can be used effectively. Such errors are quite simply obtained by the recovery processes described in the previous chapter (SPR in Sec. 13.4 and REP in Sec. 13.5). In Fig. 14.9 we show a simple exercise recently conducted by Ofiate and Bugeda 4 in which a refinement of a stressed cylinder is made using various criteria as described in the caption
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Fig. 14.2 The influence of initial mesh to convergence rates in h version. Adaptive refinement using quadratic triangular elements. Problem of Fig. 14.3. Note that if initial mesh is finer than h = 1/8 adaptive refinement reducesthe number of equations.
Adaptive h-refinement 507
Fig. 14.4 Adaptive mesh of quadratic triangular elements for short cantilever beam. of Fig. 14.9. It will be observed that the stress tolerance method generally needs a much finer mesh.
Example 14.3: A Poisson equation in a square domain. This example is fairly straightforward and starts from a simple square domain in which suitable loading terms exist in a Poisson equation to give the solution shown in Fig. 14.10.12 In Fig. 14.11 we show the first subdivision of this domain into regular linear and quadratic elements and the subsequent refinements. The elements are of both triangular and quadrilateral shape and for the linear ones a target error of 10% in total energy has been set, while for quadratic elements the target error is 1% of total energy. In practically all cases three refinements suffice to reach a very accurate solution satisfying the requirements despite the fact that the original mesh cannot capture in any way the high intensity region illustrated in the previous figure. It is of interest to note that the effectivity indices in all cases are very close to one - this is true even for the original refinement. Figure 14.12 shows the convergence for various elements
508
Adaptive finite element refinement
10
30
I
50
I
100
I
Number of degrees of freedom 250 500 1000 2500 5000 10000 I
I
I
I
I
I
I
30000 I
50 40 30
-0.2 -0.4 -0.6 (1)1 m
o _J
~ ~~~,,,,,,_
-
6-node element /Uniform refinement
20~ o
- 6-node element ~ ' ~ " ~ i ~ . ""'t",ll,.~_ -1.0 , Adaptive refinement / ."~-, ,~~~ _ - ~ .........
......
10~
1.0
> ,4..,,
-1.2 -1.4
9 Average slope
9-node element . . . . . . . . . Uniform refinement
-1.6 I
I
1.0
I
1.5
I
2.0
I
2.5
I
I
3.0
I
4.0
3.5
Log N ;~/2 = 0.356, theoretical rate of convergence for uniform refinement
4.5
P/2 = 1.0, maximum rate of convergence Fig. 14.5 Experimentalrates of convergencefor short cantilever beam.
0.5
Number degrees of freedom 30 100 300 1000
10
I
I
I
I
3000 I
Adaptive mesh enrichment _
-
70
-40 v
0.5
-20
o
d.)
m
o'} o
"
-lo
-1.0
-5
Adaptive mesh regeneration
-1.5 -2.0
I
I
1.0
2.0
P/2 -
0.5,
I
Log N
3.0
-3
.
>~ m
m
r EE
-1 4.0
optimal rate of convergence
Fig. 14.6 Short cantilever beam. Mesh enrichment versus mesh regeneration using linear quadrilateral elements.
Adaptive h-refinement 509
Fig. 14.8 Short cantilever solved by mesh regeneration. Linear quadrilateral elements. with the error plotted against the total number of degrees of freedom. The reader should note that the asymptotic rate of convergence is exceeded when the refinement gets closer to its final objective. Example 14.4: An L-shaped domain. It is of interest to note the results in Fig. 14.13 which come from an analysis of a re-entrant comer using isoparametric quadratic quadrilaterals. Here two meshes are shown together with the convergence data of the solution. Example 14.5: A machine part. For this machine part problem plane strain conditions are assumed. A prescribed accuracy of 5% relative error is achieved in one adaptive refinement (see Fig. 14.14) with linear quadrilateral elements. The convergence of the shear stress ?Sxy is shown in Fig. 14.15.
510 Adaptive finite element refinement
Fig. 14.9 Sequence of adaptive mesh refinement strategies based on (a) equal distribution of the global energy error between all the elements, (b) equal distribution of the density of energy error, (c) equal distribution of the maximum error in stresses at each point, and (d) equal distribution of the maximum percentage of the error in stresses at each point. All final meshes have less than 5% energy norm error.
Fig. 14.10 Poisson equation 'exact' solutions. (a) au/ax contours. (b) au/ay contours.
Adaptive h-refinement 511
Fig. 14.11 Poisson problem of Fig. 14.10. Adaptive solutions for: (a)linear triangles; (b)linear quadrilaterals; (c) quadratic triangles; (d) quadratic quadrilaterals. ~ based on SPR,~L based on L2 projection. Target error 10% for linear elements and 1% for quadratic elements.
512 Adaptive finite element refinement 100~(0.436) ~" ~ ~r--. "~ (I.064) " ' - - . . 1 '.L.
. (0.506) A" "~,....,,..
"'--..'~,
~
~
-~_
(0.988) "'., 9
10
Theoretical convergence I ./P=
~ . _
~
I
=~\ ,....-'"~ "~.'.~
~
v.,.,
,, " - , ~ (1.155)',,~(1.024) ~ -, ~ LJ/,,(1.036) Theoretical~ "-, "~ convergence" ~ "".. "~ p=2 1 ~~ """ "~l~k(1.063)
"-
1 10
I 100
z~ Linear triangle [] Linear quads 9Quadratic triangle 9Quadratic quads
1
\
(1 078")'I " "1 ,~(1.044) 1000 (1.032) 10000
d.o.f.
Bracketed numbers show effectivity index achieved
l Aim 10%
!
} I Aim 1%
Fig. 14.12 Adaptive refinement for Poisson problem of Fig. 14.10.
p= 1.0~
m
9
50
q
,I ~
50
Mesh 1"27 elements (252 DOF) q = 8.3%, 0* = 1.110
-I
Mesh 2:101 elements (876 DOF) q = 3.1%, e* = 1.057
Fig. 14.13 Adaptive refinement of an L-shapeddomain in plane stress with prescribed error of 1%.
Adaptive h-refinement 513
Fig. 14.14 Adaptive refinement of machine part using linear quadrilateral elements. Target error 5%.
Fig. 14.15 Adaptive refinement of machine part. Contours of shear stress for original and final mesh.
Example 14.6: A perforated gravity dam. The final example of this section shows a more practical engineering problem of a perforated dam. This dam was analysed in the late 1960s during its construction. The problem was revisited to choose a suitable mesh of quadratic triangles. Figure 14.16(a) shows the mesh chosen. Despite the high order of elements the error is quite high, being around 17%. One stage of adaptive refinement reaches the specified value of 5% error in energy norm. As we have seen in previous examples such convergence is not always possible but it is achieved here. We believe this typical example shows the advantages of adaptivity and the ease with which a final good mesh can be arrived at automatically.
514 Adaptive finite element refinement
The use of non-uniform p-refinement is of course possible if done hierarchically and many attempts have been made to do this efficiently. Some of this was done as early as 1983. 24, 25 However, the general process is difficult and necessitates many assumptions about the decrease of error. Certainly, the desired accuracy can seldom be obtained in a single step and most of the work on this requires a sequence of steps. We illustrate such a refinement process in Fig. 14.17 for the perforated dam problem presented in the previous section. The same applies to hp-processes in which much work has been done during the last two decades. 8-18 We shall quote here only one particular attempt at hp-refinement which seems to be particularly efficient and where the number of resolutions is quite small. The
p-refinement and hp-refinement Percentage error in total energy I~. = 6.0 Exact IE=a.6 By estimates
!~. = 7.81 By ~corrective'
~ ~ .~,
Percentage error in total energy
I~ = 3.ol
Exact
I' = 2.9 j'By estimates
IE* = 4"61 estimates By 'cOrrective'
-
Quadratic d.o.f. (x direction) Cubic d.o.f. (x direction) \ ~ . ~
Cubic d.o.f. (x direction) Quartic d.o.f. (x direction) ox Centre quadratic d.o.f. (x direction)
~ T-ox\ o x ~
,-,,/-\ . ~ ~ _
~C(Ov~
o__-ox Quadraticson element boundaries not represented
-0-
Fig. 14.17 Adaptive solution of perforated dam by p-refinement. (a) Stage three, 206 d.o.f. (b) Stage five, 365 d.o.f.
methodology was introduced by Zienkiewicz et al. in 198919 and we shall quote here some of the procedures suggested. The first procedure is that of pursuing an h-refinement with lower order elements (e.g., linear or quadratic elements) to obtain, say, a 5% accuracy, at which stage the energy norm error is nearly uniformly distributed throughout all elements. From there a p-refinement is applied in a uniform manner (i.e., the same p is used in all elements). This has very
515
516 Adaptive finite element refinement substantial computational advantages as programming is easy and can be readily accomplished, especially if hierarchical functions are used. The uniform p-refinement also allows the global energy norm error to be approximately extrapolated by three consecutive solutions, e6 The convergence of the p-refinement finite element solution can be written as 27 Ilell ~ CN -3
(14.11)
where C and/3 are positive constants depending on the solution of the problem and N is the number of degrees of freedom. We assume that for each refinement the error is, observing Eq. (14.3), Ilull 2 -Ilfiq[I 2 = CNq 23
(14.12)
with q = p - 2, p - 1, p for the three solutions. Eliminating the two constants C and/3 from the above three equations, Ilull 2 can be solved by l~
1/Np)
Ilu[I 2 _ II~pll 2 = (llull 2 _ II~p_lll 2 ) ~o~<~_=/~_,> Ilull 2- II~p-lll 2 Ilull 2- II~p_2ll2
(14.13)
The global energy norm error for the final solution and indeed the error at any stage of the p-refinement can be determined using
Ilell 2 = Ilull 2q=l,2
II~qll 2
(14.14)
. . . . . p.
Example 14.7: h-p-refinement of L-shaped domain and short cantilever beam. Generally the high accuracy is gained rapidly by refinement, at least from examples performed to date. In Figs 14.18 and 14.19 we show two examples for which we have previously used an h-refinement. The first illustrates the L-shaped domain with one singularity and the second the short cantilever beam with two strong singularities. Both problems are solved first using h-refinement until target 5% accuracy is reached using quadratic triangles. At this stage the p is increased to third and fourth order so that three solutions are available. At the end of the third solution the error is less than 1%. In the same paper 19 an alternative procedure is suggested. This uses a very coarse mesh at the outset followed by p-refinement. In this case the error at the element level is estimated at the last stage of the p-refinement as the difference between the last two refinements (e.g., the third and fourth order when the maximum p is 4). The global error estimator is calculated by the extrapolation procedure used in the previous example. The element error estimator is for order p - 1 rather than the highest order p. It is, however, very accurate. The element error estimator is subsequently used to compute the optimal mesh size as described in Sec. 14.2.1. Nearly optimal rate of convergence is expected to be achieved because the optimal mesh is designed for p - 1 order elements. Details of this process will be found again in the reference and will not be discussed further. At no stage of the hp-refinements have we used here any of the estimators quoted in the previous chapter. However, their use would make the optimal mesh design at order p
p-refinement
and hp-refinement
Poisson's ratio, v = 0.3 Plane stress conditions m=l.0 ~
Mesh 2 (385 d.o.f.) 1"1= 4.67% p = 2 (1322 d.o.f.) 1"1= 0.97% p = 4 (b) Quadratic triangles for 5% error
Mesh 1 (120 d.o.f.) rl - 15% p = 2 (a) Original mesh
30
50
I
m
I
-0.8 -1.0 -
.
- 11.4 "2-
l
;- -1.6 o -- -1.8 _2.0 -2.2 -2.4
Number of degrees of freedom 250 1000 5000 100 500 2500 10000 I
I
~t
I
I
I
I
6-node element Uniform refinement
I
/
-I, 30 -q 20 /
~ ~ - element",~-B"l'~J_ Adaptive .... o ~ = z ~ - 1 , . ~ ,
-~ 10~176 o, 1.0 "<:z30.272 / ~ 3 ~, ~ ._h'refinement 1.~ =3 9.~nodeelement--[2 ~ ~N, Uniform refinement I 1 rr p-refinement 0=4 I I I I I I / 0.5 1.5 2.0 2.5 3.0 3.5 4.0 Log N
-6-node
(c)h-p refinement. 1% accuracy reached with 1322 d.o.f. Fig. 14.18 Solution of L-shaped domain by h-p-refinement (as defined in Example 14.4 of previous section) using procedure one of reference 19.
possible, because the element error can be accurately estimated at order p. It will result in an optimal hp-refinement. The two examples we have quoted above are reanalysed using the alternative process described above and presented in Figs 14.20 and 14.21. In both cases the final accuracy shows an error of less than 1% but it is noteworthy that the total number of degrees of freedom used with the second method is considerably less than that in the first and still achieves a nearly optimal rate of convergence. We can conclude this section on hp-refinement with a final example where a highly singular crack domain is studied. Once again the second procedure is used showing in Fig. 14.22 a remarkable rate of convergence.
517
518 Adaptive finite element refinement
The methods of estimating errors and adaptive refinement which are described in this and the previous chapter constitute a very important tool for practical application of finite element methods. The range of applications is large and we have only touched here upon the relatively simple range of linear elasticity and similar self-adjoint problems. A recent survey shows many more areas of application 28 and the reader is referred to this publication for interesting details. At this stage we would like to reiterate that many different norms or measures of error can be used and that for some problems the energy norm is not in fact 'natural'. A good example of this is given by problems of high-speed gas flow, where very
Concluding
YY
o.
yyy y0yf --
.b
50
.
.I
Mesh 1, for p = 4 d.o.f. = 4301"1 = 6.72%
Mesh 2, for p = 4 d.o.f. = 84611 = 0.76%
(a)
(b)
30
Number of degrees of freedom 100 500 2500 10 000 50 250 1000 5000 30 000
10 I I I I I I I I I I I h-refinement -0.4 p=l~.c~Pp=l --..e- 6-node element -0.6 _ ~ \ Uniform r e f i n e m e n t -0.8 ~\ h-p refinement -1.0 ~ --6-Initial mesh -1.2 2~'~~_ - - ~ Regenerated mesh 2"~k~"e"e.....~ 1.0 -1.4 o , ~ 4"-., ~ l ' . e ~ l ~ . 0.272 -- -1.6 1.62 INk \ ""., 1.0 ..... -1.8 ~ -% 1.0 "",,,~:7~]0.S44 -2.0 -2.2 -2.4 -4LJ I I I I I I I I1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Log N -
(c) Fig.
14.20
erence 19.
remarks
60.0 50.0 40.0 30.0 20.0
!._ O
10.0 e. (1) 5.0 4.0 3.0 2.0
r
._> -~ rr
1.0 0.7 0.5
Note: 1% accuracy reached with 846 d.o.f.
Solution of L-shaped domain by h-p-adaptive refinement using alternative procedure of ref-
steep gradients (shocks) can develop. The formulation of such problems is complex, but this is not necessary for the present argument. For problems in fluid mechanics discussed in reference 29 and similarly for problems of strain localization in plastic softening discussed in reference 30 no global norms can be used effectively. In such situations it is convenient to base the refinement on the value of the maximum curvatures developed by the solution of u. On occasion an elongation of the elements will be used to refine the mesh appropriately. Figure 14.23 shows a typical problem of shock capturing solved adaptively.
519
520 Adaptive finite element refinement
14.1 Program development project: Implement a mesh enrichment algorithm (as described in Sec. 14.2.1) in the solution system started in Problem 2.17 and extended in subsequent exercises. Assume the problem is given by the quasi-harmonic equation and modelled using 3-node triangular elements. (Hint: Adapt the mesh generation program developed in Problems 5.17 and 5.18 to generate the mesh using the new coordinates resulting from enrichment.)
Problems On all boundaries prescribed traction = analytic solution
~Y
x
a
x
L. r-
.1. ~1~
a
I~
d~
a
a
Mesh 1, for p = 4 d.o.f. = 21711 = 9.17% (a)
(b)
Crack A
._L. ._L. -i-'9 "-i-" Mesh 2, for p = 4 d.o.f. = 1245 q = 0.89%
(c)
10 i
Number of degrees of freedom 100 500 2500 10000 50 250 1000 5000 30000 i i i i i i i i i i .p =1 h-refinement z~ p =1 ~ 6-node element "'"~c:~l"~ - uniform refinement 30
-0.4 -0.6 -0.8 -1.0 "~ -1.2 c~ -1.4 - h-ore, nement o .a -1.6 Initial mesh -1.8 ---o-Regeneiated -2.0 mesh -2.2 _ -2.4
,o
_
I
1.0
(d)
I
1.5
I
2.0
60.0 50.0 40.0 30.0 20.0 10.0
\
\
,.oo
b ,
1.5o~ I
5.0 4.0 3.0 2.0
?---
4 I
2.5 3.0 Log N
I
3.5
I
4.0
I
.~ rr
1.0 0.7 0.5
4.5
Note: 1% accuracy reached with 1245 d.o.f.
Fig. 14.22 Adaptive h-p-refinement for a singular crack using alternative procedure of reference 19.
521
522
Adaptive finite element refinement
Fig. 14.23 Directional mesh refinement. Gas flow past a circular cylinder- Mach number 3. Third refinement mesh 709 nodes (1348 elements).
14.2 Program development project: Implement an adaptive mesh regeneration algorithm (as described in Sec. 14.2) in the solution system started in Problem 2.17 and extended in subsequent exercises. Assume the problem is given by the quasi-harmonic equation and modelled using 3-node triangular elements. 14.3 Solve Example 2.3 using linear (3-node) elements. Follow the mesh refinement procedure described in Sec. 14.2 for adaptive h refinement and show that the optimal rate of convergence of the finite element method can be attained when a prescribed accuracy is achieved. 14.4 Solve Problem 2.3 using quadratic (6-node) elements. Follow the mesh refinement procedure described in Sec. 14.2 for adaptive h refinement and show that the optimal rate of convergence of the finite element method can be attained when prescribed accuracy is achieved. 14.5 Program development project: Devise and implement in the solution system started in Problem 2.17 an hp refinement strategy (see Sec. 14.3) to attain a prescribed accuracy. Assume the problem is given by the quasi-harmonic equation and hierarchical triangular elements are used to define the finite element p-models.
1. I. Babu~ka and C. Rheinboldt. A-posteriori error estimates for the finite element method. Int. J. Numer. Meth. Eng., 12:1597-1615, 1978. 2. I. Babu~ka and C. Rheinboldt. Adaptive approaches and reliability estimates in finite element analysis. Comp. Meth. Appl. Mech. Eng., 17/18:519-540, 1979. 3. O.C. Zienkiewicz and J.Z. Zhu. A simple error estimator and adaptive procedure for practical engineering analysis. Int. J. Numer. Meth. Eng., 24:337-357, 1987.
References 523 4. E. Ofiate and G. Bugeda. A study of mesh optimality criteria in adaptive finite element analysis. Eng. Comput., 10:307-321, 1993. 5. E.R. de Arantes e Oliveira. Theoretical foundations of the finite element method. Int. J. Solids Struct., 4:929-952, 1968. 6. E.R. de Arantes e Oliveira. Optimization of finite element solutions. In Proc, 3rd Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-71-160, pages 423-446, WrightPatterson Air Force Base, Ohio, 1972. 7. R.L. Taylor and R. Iding. Applications of extended variational principles to finite element analysis. In C.A. Brebbia and H. Tottenham, editors, Proc. of the International Conference on Variational Methods in Engineering, volume II, pages 2/54-2/67. Southampton University Press, 1973. 8. W. Gui and I. Babu~ka. The h, p and h-p version of the finite element method in 1 dimension. Part 1: The error analysis of the p-version. Part 2: The error analysis of the h- and h-p version. Part 3: The adaptive h-p version. Numerische Math., 48:557-683, 1986. 9. B. Guo and I. Babu~ka. The h-p version of the finite element method. Part 1: The basic approximation results. Part 2: General results and applications. Comp. Mech., 1:21-41,203226, 1986. 10. I. Babu~ka and B. Guo. The h-p version of the finite element method for domains with curved boundaries. SIAM J. Numer. Anal., 25:837-861, 1988. 11. I. Babu~ka and B.Q. Guo. Approximation properties of the hp version of the finite element method. Comp. Meth. Appl. Mech. Eng., 133:319-349, 1996. 12. L. Demkowicz, J.T. Oden, W. Rachowicz, and O. Hardy. Toward a universal h-p adaptive finite element strategy. Part 1: Constrained approximation and data structure. Comp. Meth. Appl. Mech. Eng., 77:79-112, 1989. 13. W. Rachowicz, J.T. Oden, and L. Demkowicz. Toward a universal h-p adaptive finite element strategy. Part 3: Design of h-p meshes. Comp. Meth. Appl. Mech. Eng., 77:181-211, 1989. 14. K.S. Bey and J.T. Oden. hp-version discontinuous Galerkin methods for hyperbolic conservation laws. Comp. Meth. Appl. Mech. Eng., 133:259-286, 1996. 15. C.E. Baumann and J.T. Oden. A discontinuous hp finite element method for convection-diffusion problems. Comp. Meth. Appl. Mech. Eng., 175:311-341, 1999. 16. P. Monk. On the p and hp extension of Nedelecs curl-conforming elements. J. Comput. Appl. Math., 53:117-137, 1994. 17. L.K. Chilton and M. Suri. On the selection of a locking-free hp element for elasticity problems. Int. J. Numer. Meth. Eng., 40:2045-2062, 1997. 18. L. Vardapetyan and L. Demkowicz. hp-Adaptive finite elements in electromagnetics. Comp. Meth. Appl. Mech. Eng., 169:331-344, 1999. 19. O.C. Zienkiewicz, J.Z. Zhu, and N.G. Gong. Effective and practical h-p-version adaptive analysis procedures for the finite element method. Int. J. Numer. Meth. Eng., 28:879-891, 1989. 20. P.G. Ciarlet. The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978. 21. J. Peraire, M. Vahdati, K. Morgan, and O.C. Zienkiewicz. Adaptive remeshing for compressible flow computations. J. Comput. Phys., 72:449-466, 1987. 22. J.Z. Zhu, O.C. Zienkiewicz, E. Hinton, and J. Wu. A new approach to the development of automatic quadrilateral mesh generation. Int. J. Numer. Meth. Eng., 32:849-866, 1991. 23. J.Z. Zhu and O.C. Zienkiewicz. Adaptive techniques in the finite element method. Comm. Appl. Num. Math., 4:197-204, 1988. 24. D.W. Kelly, J.P. De S.R. Gago, O.C. Zienkiewicz, and I. Babu~ka. A posteriori error analysis and adaptive processes in the finite element method: Part I - Error analysis. Int. J. Numer. Meth. Eng., 19:1593-1619, 1983. 25. J.P. De S.R. Gago, D.W. Kelly, O.C. Zienkiewicz, and I. Babu~ka. A posteriori error analysis and adaptive processes in the finite element method: Part II - Adaptive mesh refinement. Int. J. Numer. Meth. Eng., 19:1621-1656, 1983.
524 Adaptive finite element refinement 26. B.A. Szabo. Mesh design for the p version of the finite element. Comp. Meth. Appl. Mech. Eng., 55:181-197, 1986. 27. I. Babu~ka, B.A. Szabo, and I.N. Katz. The p version of the finite element method. SIAM J. Numer. Anal., 18:512-545, 1981. 28. P. Ladevbze and J.T. Oden, editors. Advances in Adaptive Computational Methods in Mechanics. Studies in Applied Mechanics 47. Elsevier, 1998. 29. O.C. Zienkiewicz, R.L. Taylor, and P. Nithiarasu. The Finite Element Method for Fluid Dynamics. Butterworth-Heinemann, Oxford, 6th edition, 2005. 30. O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method for Solid and Structural Mechanics. Butterworth-Heinemann, Oxford, 6th edition, 2005.
In all of the preceding chapters, the finite element method was characterized by the subdivision of the total domain of the problem into a set of subdomains called elements. The union of such elements gave the total domain. The subdivision of the domain into such components is of course laborious and difficult necessitating mesh generation as discussed in Chapter 8. Further if adaptivity processes are used, generally large areas of the problem have to be remeshed. For this reason, much attention has been given to devising approximation methods which are based on points without necessity of forming elements. When we discussed the matter of generalized finite element processes in Chapter 3, we noted that point collocation or in general finite differences did in fact satisfy the requirement of the pointwise definition. However, the early finite differences were always based on a regular arrangement of nodes which severely limited their applications. To overcome this difficulty, since the late 1960s the proponents of the finite difference method have worked on establishing the possibility of finite difference calculus being based on an arbitrary disposition of collocation points. Here the work of Girault, 1 Pavlin and Perrone, 2 and Snell et al. 3 should be mentioned. However, a full realization of the possibilities was finally offered by Liszka and Orkisz, 4' 5 and Krok and Orkisz 6 who introduced the use of least squares methods to determine the appropriate shape functions. At this stage Orkisz and coworkers realized not only that collocation methods could be used but also the full finite element, weak formulation could be adopted by performing integration. Questions of course arose as to what areas such integration should be applied. Liszka and Orkisz 4 suggested determining a 'tributary area' to each node providing these nodes were triangulated as shown in Fig. 15.1(a). On the other hand in a somewhat different context Nay and Utku 7 also used the least squares approximation including triangular vertices and points of other triangles placed outside a triangular element thus simply returning to the finite element concept. We show this kind of approximation in Fig. 15.1 (b). Whichever form of tributary area was used the direct least squares approximation centred at each node will lead to discontinuities of the function between the chosen integration areas and thus will violate the rules which we have imposed on the finite element method. However, it turns out that such rules could be violated and here the patch test will show that convergence is still preserved.
526
Point-basedand partition of unity approximations. Extended finite element methods
(a)
(b)
Fig. 15.1 Patchesof triangular elements and tributary areas. However, the possibility of determining a completely compatible form of approximation existed. This compatible form in which continuity of the function and of its slope if required and even higher derivatives could be accomplished by the use of so-called moving least squares methods. Such methods were originated in another context (Shepard, 8 Lancaster and Salkauskas 9' 10). The use of such interpolation in the meshless approximation was first suggested by Nayroles et al. 11-13 This formulation was named by the authors as the diffuse finite element method. Belytschko and coworkers 14' 15 realized the advantages offered by such an approach especially when dealing with the development of cracks and other problems for which standard elements presented difficulties. His so-called 'element-free Galerkin' method led to many seminal publications. An alternative use of moving least squares procedures, called hp-cloud methods, was suggested by Duarte and Oden. 16' 17 They introduced at the same time a concept of hierarchical forms by noting that all shape functions derived by least squares possess the partition of unity property (viz. Chapter 4). Thus higher order interpolations could be added at each node rather than each element, and the procedures of element-free Galerkin or of the diffuse element method could be extended. The use of all the above methods still, necessitates integration. Now, however, this integration need not be carried out over complex areas. A background grid for integration purposes is introduced though internal boundaries are no longer required. Thus such numerical integration on regular grids is used by Belytschko 18' 19 and other approaches are being explored. However, another interesting possibility was suggested by Babu~ka and Melenk.20, 21 Babu~ka and Melenk use a partition of unity but now the first set of basic shape functions is derived on a standard finite element, say the linear triangle. Most of the approximations then arise through addition of hierarchical variables centred at nodes. We feel that this kind of approach which necessitates very few elements for integration purposes combines well the methodologies of 'element-free' and 'standard element' approximation procedures. We shall demonstrate a few examples later for the application of such methods which seem to present a very useful extension of the hierarchical approach.
Function approximation 527 Incidentally the procedures based on local elements also have the additional advantage that global functions can be introduced in addition to the basic ones to represent special phenomena, for instance the presence of a singularity or waves. Both of these are important and the idea presented by this can be exploited. This is especially useful in solution of certain wave phenomena 22 and Belytschko and coworkers have coined the term 'XFEM' (extended finite element method) and exploited the approach to insert cracks in domains. 23-34 This chapter will conclude with reference to other similar procedures which we do not have time to discuss.
We consider here a local set of n points in two (or three) dimensions defined by the coordinates xk, Yk, Zk; k = 1, 2 . . . . , n or simply Xk = [xk, Yk, Zk] at which a set of data values of the unknown function fik are given. It is desired to fit a specified function form to the data points. In order to make a fit it is necessary to: 1. Specify the form of the functions, p(x), to be used for the approximation. Here as in the standard finite element method, it is essential to include low order polynomials necessary to model all the derivatives contained in the differential equation or in the weak form approximation being used. Certainly a complete linear and sometimes quadratic polynomial will always be necessary. 2. Define the procedure for establishing the fit. Here we will consider some least s q u a r e s f i t methods as the basis for performing the fit. The functions will mostly be assumed to be polynomials; however, in addition other functions can be considered if these are known to model well the solution expected (e.g., see reference 22 on use of 'wave' functions).
15.2.1 Least squares fit We shall first consider a least squares fit scheme which minimizes the square of the distance between n data values fik defined at the points Xk and an approximating function evaluated at the same points fi(Xk). We assume the approximation function is given by a linearly independent set of m polynomials p j (x) m ~(X) --" E p j ( x ) o t j -~ p ( x ) ~ j=l
(15.1)
in which c~ is a set of parameters to be determined. A least squares scheme is introduced to perform the fit to a set of n data points and this is written as (see Chapter 13 for similar operations). Minimize j=~l
(~(Xk) -- ~k) 2 = min k=l
(15.2)
528
Point-basedand partition of unity approximations. Extended finite element methods where the minimization is to be performed with respect to the values of c~. Substituting the values of ~ at the points xk we obtain 8ftk . (~(xk) - ilk) = 0;
OJ _ ~
00/j
j = 1, 2 . . . . . m
k=l ~ j
where
(15.3)
m
~lk : ~ pj(Xk)Olj : Pk Ot j=l
in which Pk - p(xk). This set of equations may be written in a compact matrix form as OJ _ - L p ~ ( p k c ~ - ~ ) = 0
ac~
.
k=l
(15.4)
We can define the result of the sums as Pk Pk = pT
g =
k=l
in which
Pk Uk = pT
Pl P -
(15.5)
k=l
P2
and
Pn
~2
Un
The above process yields the set of linear algebraic equations Hc~=g which, provided H is non-singular, has the solution c~ -- H - l g
(15.6)
We can now write the approximation for the function as = p(x)H-1pTfi = N(x)~ where N(x) are the appropriate shape or basis functions. In general Hi (x j) 5~ ~ij as it always has been for standard finite element shape functions. However, the partition of unity [viz. Eq. (4.4)] is always preserved provided p(x) contains a constant. Example 15.1: Fit of a linear polynomial. To make the process clear we first consider a dataset, ilk, defined at four points, Xk, to which we desire to fit an approximation given by a linear polynomial ~(x) : al + xot2 + yc~3 : p(x)c~
Function approximation 529 If we consider the set of data defined by Xk = [--4.0
-1.0
0.0
6.0]
5.0
-5.0
0.0
3.0]
yk=[
~k = [--1.5
5.1
4.3]
3.5
we can write the arrays as 1
-4
p
!
5
0 6
and
u=
-1.5 5.1 3.5 4.3
Using Eq. (15.5) we obtain the values
H = pTp =
53
and
3
g = pT~ =
59
11.4} 26.7 --20.1
which from Eq. (15.6) has the solution c~ = [ 3 . 1 2 4 1 , 0 . 4 7 4 5 , - 0 . 5 2 3 7 ] T. The least squares fit for these data points together with the difference between the data points and the values of the fit at Xk is given in Table 15.1.
15.2.2 Weighted least squares fit L e t u s now assume that the point at the origin, x0 = 0 (k = 3 of Example 15.1), is the point about which we are making the expansion and, therefore, the one where we would like to have the best accuracy. Based on the linear approximation above we observe that the direct least squares fit yields at the point in question the largest discrepancy. In order to improve the fit we can modify our least squares fit for weighting the data in a way that emphasizes the effect of distance from a chosen point. We can write such a w e i g h t e d least squares fit as the minimization of J -- ~1
W(Xk -- x0) (~(Xk) -- ~k) 2 - - m i n k=l
Table 15.1 Data and least squares fit for Example 15.1 k
1
2
3
4
Xk Yk
--4 5
--1 --5
0 0
6 3 4.300 4.400 -0.100
k ~k Difference
-- 1.500 --1.392 -0.108
5.100 5.268 -0.168
3.500 3.124 0.376
(15.7)
530
Point-basedand partition of unity approximations. Extended finite element methods
1.0
0.8
v
0.6 t-
0.4
0.2
0
0
I 1
I 2
I 3
I 4 r distance
I"'~, 5
6
I 7
I 8
Fig. 15.2 Weighting function for Eq. (15.8): c = O. 125.
where w is the weighting function. Many choices may be made for the shape of the function w. If we assume that the weight function depends on a radial distance, r, from the chosen point we have w = w(r); r 2 -- (x -- x0)" (x -- x0) One functional form for w ( r ) is the exponential Gauss function: w(r) = exp(-cr2);
c > 0
and
r > 0
(15.8)
For c = 0.125 this function has the shape shown in Fig. 15.2 and when used with the previously given four data points yields the linear fit shown in Table 15.2.
15.2.3 Interpolation domains and shape functions In what follows we shall invariably use the least squares procedure to interpolate the unknown function in the vicinity of a particular node i. The first problem is that when approximating to the function it is necessary to include a number of nodes equal at least Table 15.2 Differencebetween weighted least squares fit and data xk Yk
-4 5
-1 --5
0 0
k -- 1.500 5 . 1 0 0 3.500 ~k - - 0 . 8 8 0 5 . 2 4 7 3.487 Error -0.620 -0.147 0.013
6 3
4.300 5.246 -0.946
Function approximation 531 to the number of parameters of cx sought to represent a given polynomial. This number, for instance, in two dimensions is three for linear polynomials and six for quadratic ones. As always the number of nodal points has to be greater than or equal to the bare minimum which is the number of parameters required. We should note in passing that it is always possible to develop a singularity in the equation used for solving cz, i.e., Eq. (15.6) if the data points lie for instance on a straight line in two or three dimensions. However, in general we shall try to avoid such difficulties by reasonable spacing of nodes. The domain of influence can well be defined by making sure that the weighting function is limited in extent so that any point lying beyond a certain distance rm is weighted by zero and therefore is not taken into account. Commonly used weighting functions are, for instance, in direction r, given by e x p ( - c r ) - e x p ( - c r 2) w(r)= 1 - e x p ( - c r 2) ; c>0 and O < _ r < r m (15.9)
r > rm
O;
which represents a truncated Gauss function. Another alternative is to use a Hermitian interpolation function as employed for the beam example in Sec. 2.9:
w(r) = I
1 - - 3 ( ~ 2mr)
3r ; O
(15.10)
r > rm
0;
or alternatively the function
w(r) --
; O
rm 0;
and n > 2
(15.11)
r>rm
is simple and has been effectively used. For circular domains, or spherical ones in three dimensions, a simple limitation of rm suffices as shown in Fig. 15.3(a). However, occasionally use of rectangular or hexahedral subdomains is useful as also shown in that figure and now of course the weighting function takes on a different form:
w(x, y) =
Xi(x)Yj(y); 0;
0 <_<X < Xm;
0
<_ y < Ym;
and i, j > 2
(15.12)
x > Xm, y > Ym
with
Xi(x)--
1-
x__ Xm
;
rj(y)=
1-
y Ym
The above two possibilities are shown in Fig. 15.3. Extensions to three dimensions using these methods is straightforward. Clearly the domains defined by the weighting functions will overlap and it is necessary if any of the integral procedures are used such as the Galerkin method to avoid such an overlap by defining the areas of integration. We have suggested a couple of possible ideas in Fig. 15.1 but other limitations are clearly possible. In Fig. 15.4, we show an approximation to a series of points sampled in one dimension. The weighting function here always
532
Point-based and partition of unity approximations. Extended finite element methods 6
--
6
- -
m
20-2
--2
--
-4 -6
-6
I -4
I -2
I 0
I 2
I 4
I 6
-6
(a)
-6
I -4
I -2
I 0
I 2
I 4
I 6
(b)
Fig. 15.3 Two-dimensional interpolation domains. (a) Circular. (b) Rectangular.
Piecewise least square approximation
ExacL~
o
s SS
Fig. 15.4 A one-dimensional approximation to a set of data points using parabolic interpolation and direct least squares fit to adjacent points.
embraces three or four nodes. Limiting, however, the domains of their validity to a distance which is close to each of the points provides a unique definition of interpolation. The reader will observe that this interpolation is discontinuous. We have already pointed out such a discontinuity in Chapter 3, but if strictly finite difference approximations are used this does not matter. It can, however, have serious consequences if integral procedures are used and for this reason it is convenient to introduce a modification to the definition of weighting and method of calculation of the shape function which is given in the next section.
Moving least squares approximations- restoration of continuity of approximation 533
The method of moving least squares was introduced in the late 1960s by Shepard 8 as a means of generating a smooth surface interpolating between various specified point values. The procedure was later extended for the same reasons by Lancaster and Salkauskas 9' 10 to deal with very general surface generation problems but again it was not at that time considered of importance in finite elements. Clearly in the present context the method of moving least squares could be used to replace the local least squares we have so far considered and make the approximation fully continuous. In the moving least squares methods, the weighted least squares approximation is applied in exactly the same manner as we have discussed in the preceding section but is established for every point at which the interpolation is to be evaluated. The result of course completely smooths the weighting functions used and it also presents smooth derivatives noting of course that such derivatives will depend on the locally specified polynomial. To describe the method, we again consider the problem of fitting an approximation to a set of data items fii, i = 1 . . . . . n defined at the n points xi. We again assume the approximating function is described by the relation m u(x)
~(X) -- S p j ( x ) o t j j=l
--
p(x)cz
(15.13)
where pj are a set of linearly independent (polynomial) functions and otj are unknown quantities to be determined by the fit algorithm. A generalization to the weighted least squares fit given by Eq. (15.7) may be defined for each point x in the domain by solving the problem Wx(Xk- x)[~k - p(x~)~] 2 -- rain
J(x) = g
(15.14)
k=l
In this form the weighting function is defined for every point in the domain and thus can be considered as translating or moving as shown in Fig. 15.5. This produces a continuous interpolation throughout the whole domain. Figure 15.6 illustrates the problem previously presented in Fig. 15.4 now showing continuous interpolation. We should note that it is now no longer necessary to specify 'domains of influence' as the shape functions are defined in the whole domain. The main difficulty with this form is the generation of a moving weight function which can change size continuously to match any given distribution of points xk with a limited number of points entering each calculation. One expedient method to accomplish this is to assume the function is symmetric so that Wx (xk -
x) =
W x
( x - xk)
and use a weighting function associated with each data point x~ as Wx (x~ - x ) = w k ( x - x k )
The function to be minimized now becomes J(x) - ~
W k ( X - Xk)[fik -- p(Xk)C~]2 = min k=l
(15.15)
534
Point-basedand partition of unity approximations. Extended finite element methods
X
| | i |
1.5
=
1.0
0.5
0
O/
u~
I
-2
', o
U/+I
~
/
-1
\
0
xi coordinates
1
2
I
3
Fig. 15.5 Moving weighting function approximation in MLS.
Piecewiseleast square approximation Moving least square
approximation (no jump)
Exact. :
o-
o.. ?
9
o
Fig. 15.6 The problem of Fig. 15.4 with moving least squares interpolation. In this form the weighting function is fixed at a data point x k and evaluated at the point x as shown in Fig. 15.7. Each weighting function may be defined such that
Wx(r) =
{fk(r), 0,
if Irl ~ rk otherwise
(15.16)
Moving least squares approximations - restoration of continuity of approximation
X
! i i i i
'
1.5
0
E ," 1.0
0.5
,
-2
.
-1
0 1 xi coordinates
2
3
Fig. 15.7 A 'fixed' weighting function approximation to the MLS method.
and the terms in the sum are zero whenever r 2 = (x - Xk) T ( x - - Xk) and Irl > rk. The parameter rk defines the radius of a ball around each point, Xk; inside the ball the weighting function is non-zero while outside the radius it is zero. Each point may have a different weighting function and/or radius of the ball around its defining point. The weighting function should be defined such that it is zero on the boundary of the ball. This class of function may be denoted as C O(rk), where the superscript denotes the boundary value and the subscript the highest derivative for which Co continuity is achieved. Other options for defining the weighting function are available as discussed in the previous section. The solution to the least squares problem now leads to _~n a ( x ) = H -l(x) ~
gj(x)hj = H -l(x)g(x)fi
(15.17)
j=l
where n
H(x) = E
Wk(X -- Xk)p(xk)Tp(xk)
(15.18a)
k=l
and gj (x) = w j (x - xj)p(xj)T
(15.18b)
In matrix form the arrays H(x) and g(x) may be written as H(x) = pTw(Ax)P g(x) - PTw(Ax)
(15.19)
535
536
Point-basedand partition of unity approximations. Extended finite element methods in which (x 0 xl) w(Ax) -Wl
0
......
.
-.
0
"'"
"'"
0
tOn(X -- Xn)
1/32(/1[ -- X2)
--
.
(15.20)
The moving least squares algorithm produces solutions for a which depend continuously on the point selected for each fit. The approximation for the function u (x) now may be written as n ~(X) -- Z j=l
Nj(x)uj
(15.21)
where Nj(x) -- p(x)H -1 (x)gj(x)
(15.22)
define interpolation functions for each data item fij. We note that in general these 'shape functions' do not possess the Kronecker delta property which we noted previously for finite element methods - that is Nj(xi) # c~ji (15.23) It must be emphasized that all least squares approximations generally have values at the defining points xj in which fij # fi(xj) (15.24) i.e., the local values of the approximating function do not fit the nodal unknown values. Indeed fi will be the approximation used in seeking solutions to differential equations and boundary conditions and fij are simply the unknown parameters defining this approximation. The main drawback of the least squares approach is that the approximation rapidly deteriorates if the number of points used, n, largely exceeds that of the m polynomial terms in p. This is reasonable since a least squares fit usually does not match the data points exactly. A moving least squares interpolation as defined by Eq. (15.21) can approximate globally all the functions used to define p(x). To show this we consider the set of approximations
u = ~ mj(x)Cdj
(15.25)
j=l
where U = [Ul(X)
u2(x)
...
Un(X)]T
(15.26a)
uj2
...
~'ljn]T
(15.26b)
and [Jj " - [ U j l
Next, assign to each fijk the value of the polynomial pk(xj) (i.e., the kth entry in p) so that [Jj = p(xj)
(15.27)
Using the definition of the interpolation functions given by Eqs (15.21) and (15.22) we have n
~
U = Z Nj(x)p(xj) = j=l
p(x)H-l(x)g/(x)p(xj) j=l
(15.28)
Moving least squares approximations - restoration of continuity of approximation
which after substitution of the definition of gj (x) yields U= ~
p(x)H -l(x)wj(x - xj)p(xj)Tp(xj)
j=l n
= P(x)H-1 E
wj(x - xj)p(xj)Tp(xj)
j=l
= p(x)H-1H(x)
= p(x)
Equation (15.29) shows that a moving least squares form can exactly interpolate any function included as part of the definition of p(x). If polynomials are used to define the functions, the interpolation always includes exact representations for each included polynomial. Inclusion of the zero order polynomial (i.e., 1) implies that n
E
Nj (x) = 1
(15.29)
j=l
This is called a partition of unity (provided it is true for all points, x, in the domain). 35 It is easy to recognize that this is the same requirement as applies to standard finite element shape functions. Derivatives of moving least squares interpolation functions may be constructed from the representation Nj(x) = p(x)vj(x) (15.30) where H(x)vj(x) = gj(x)
(15.31)
For example, the first derivatives with respect to x is given by ONj Op Ovj Ox = Ox vj + p Ox
and H
where
OVj
OH Ogj Ox + -fffx vj = Ox
OH = ~ Ox
0 w k ( x - xk) Ox p(xk)Tp(xk)
(15.32a) (15.32b)
(15.33a)
k=l
and
Ogj _ Owj(x - Xj)p(xj ) (15.33b) Ox Ox Higher derivatives may be computed by repeating the above process to define the higher derivatives of v j. An important finding from higher derivatives is the order at which the interpolation becomes discontinuous between the interpolation subdomains. This will be controlled by the continuity of the weight function only. For weight functions which are Cq0 continuous in each subdomain the interpolation will be continuous for all derivatives up to order q. For the truncated Gauss function given by Eq. (15.9) only the approximated function will be continuous in the domain, no matter how high the order used for the p basis functions. On the other hand, use of the Hermitian interpolation given by Eq. (15.10)
537
538 Point-basedand partition of unity approximations. Extended finite element methods produces C1 continuous interpolation and use of Eq. (15.11) produces Cn continuous interpolation. This generality can be utilized to construct approximations for high order differential equations. Nayroles et al. suggest that approximations ignoring the derivatives of c~ may be used to define the derivatives of the interpolation functions. 11-13While this approximation simplifies the construction of derivatives as it is no longer necessary to compute the derivatives for H and g j, there is little additional effort required to compute the derivatives of the weighting function. Furthermore, for a constant in p no derivatives are available. Consequently, there is little to recommend the use of this approximation.
The moving least squares approximation of the function u(x) was given in the previous section as n
fi(x) = ~
Nj(x)~j
(15.34)
j=l
where Nj (x) defined the interpolation or shape functions based on linearly independent functions prescribed by p(x) as given by Eq. (15.22). Here we shall restrict attention to one-dimensional forms and employ polynomial functions to describe p(x) up to degree k. Accordingly, we have p(x)=[1
x
x2
...
x I']
(15.35)
For this case we will denote the resulting interpolation functions using the notation N k (x), where j is associated with the location of the point where the parameter fij is given and k denotes the order of the polynomial approximating functions. Duarte and Oden suggest using Legendre polynomials instead of the form given above; 16 however, conceptually the two are equivalent and we use the above form for simplicity. A hierarchical construction based on N k (x) can be established which increases the order of the complete polynomial to degree p. The hierarchical interpolation is written as bjl
a(x) =
+ U)(x)
.
j=l
x p]
b j2
[~jq
n
= ~ N~(x)(fij j=l
N~(x)
+ q(x)b/) =
[1
j=l
where q = p - k and [gjm, m = 1. . . . . q, are additional parameters for the approximation. Derivatives of the interpolation function may be constructed using the method described by Eqs (15.30)-(15.33b). The advantage of the above method lies in the reduced cost of computing the interpolation function N k (x) compared to that required to compute the p-order interpolations N ff (x).
Hierarchical enhancement of moving least squares expansions 539
Shepard interpolation
For example, use of the functions N O(x), which are called Shepard interpolations, 8 leads to a scalar matrix H which is trivial to invert to define the N ~ Specifically, the Shepard interpolations are N~ = H-l(x)gj(x) (15.36) where H (x) -- ~ wk (x - Xk) k=l
(15.37a)
gj(x) = wj(x - xj)
(15.37b)
and The fact that the hierarchical interpolations include polynomials up to order p is easy to demonstrate. Based on previous results from standard moving least squares the interpolation with bj = 0 contains all the polynomials up to degree k. Higher degree polynomials may be constructed from bjl n
x k+2
...
x p]
b j2 .
(15.38)
j=l bjq
by setting all Uj to zero and for each interpolation term setting one of the bjk to unity with the remaining values set to zero. For example, setting bjl to unity results in the expansion
~l(X)-- ~
gk(x)x
k+l -- X k+l
(15.39)
j=l
This result requires only the partition of unity property
•
N~(x) = 1
(15.40)
j=l
The remaining polynomials are obtained by setting the other values of [~jk to unity one at a time. We note further that the same order approximation is obtained using k -- 0, 1 or p.16 The above hierarchical form has parameters which do not relate to approximate values of the interpolation function. For the case where k = 0 (i.e., Shepard interpolation), Babu~ka and Melenk 36 suggest an alternate expression be used in which q in Eq. (15.38) is taken as [1 x x 2 . . . x p] and the interpolation written as
o,x,
(15.41) j=l
k=0
In this form the lff (x) are Lagrange interpolation polynomials (e.g., see Sec. 4.5) and ftjk are parameters with dimensions of u for the jth term at point Xk of the Lagrange interpolation. The above result follows since Lagrange interpolation polynomials have the property Ik (xi ) -- (~ki --
1, if k = i 0, otherwise
(15.42)
540 Point-basedand partition of unity approximations. Extended finite element methods We should also note that options other than polynomials may be used for the q (x). Thus, for any function qi (x) we can set the a s s o c i a t e d to unity (with all others and Uj set to zero) and obtain
[gji
~t(x) -- ~ Nk(x)qi(x) = qi(x)
(15.43)
j=l
Again the only requirement is that
•
N ~ ( x ) -- 1
j=l
Thus, for any basic functions satisfying the partition of unity a hierarchical enrichment may be added using any type of functions. For example, if one knows that the structure of the solution involves exponential functions in x it is possible to include them as members of the q(x) functions and thus capture the essential part of the solution with just a few terms. This is especially important for problems which involve solutions with different length scales. A large length scale can be included in the basic functions, N k (x), while other smaller length scales may be included in the functions q(x). This will be illustrated further in Volume 3 in the chapter dealing with waves. The above discussion has been limited to functions in one space variable; however, extensions to two and three dimensions can be easily constructed. In the process of this extension we shall encounter some difficulties which we address in more detail in the section on partition-of-unity finite element methods. Before doing this we explore in the next section the direct use of least squares methods to solve differential equations using collocation methods.
Finite difference methods based on Taylor formula expansions on regular grids can, as explained in Chapter 3, Sec. 3.14, always be considered as point collocation methods applied to the differential equation. They have been used to solve partial differential equations for many decades. 37-39 Classical finite difference methods commonly restrict applications to regular grids. This limits their use in obtaining accurate solutions to general engineering problems which have curved (irregular) boundaries and/or multiple material interfaces. To overcome the boundary approximation and interface problem curvilinear mapping may be used to define the finite difference operators. 4~ The extension of the finite difference methods from regular grids to general arbitrary and irregular grids or sets of points has received considerable attention (Girault, 1 Pavlin and Perrone, 2 Snell et al.3). An excellent summary of the current state of the art may be found in a recent paper by Orkisz 4~ who himself has contributed very much to the subject since the late 1970s (Liszka and Orkisz4). More recently such finite difference approximations on irregular grids have been proposed by Batina 41 in the context of aerodynamics and by On~te et al. 42-44 who introduced the name 'finite point method'. Here both elasticity and fluid mechanics problems have been addressed. In point collocation methods the set of differential equations, which here is taken in the form described in Sec. 3.1, is used directly without the need to construct a weak form or
Point collocation- finite point methods 541 perform domain integrals. Accordingly, we consider .A(u) = 0
(15.44a)
as a set of governing differential equations in a domain f2 subject to boundary conditions B(u) - 0
(15.44b)
applied on the boundaries F. An approximation to the dependent variable u may be constructed using either a weighted or moving least squares approximation since at each collocation point the methods become identical. In this we must first describe the (collocation) p o i n t s and the w e i g h t i n g f u n c t i o n . The approximation is then constructed from Eq. (15.21) by assuming a sufficient order polynomial for p in Eq. (15.13) such that all derivatives appearing in Eqs (15.44a) and (15.44b) may be computed. Generally, it is advantageous to use the same order of interpolation to approximate both the differential and boundary conditions. 4~ The resulting discrete form for the differential equations at each collocation point becomes (15.45a) ,A(N(xi)ui) = 0; i = 1, 2 . . . . . ne and the discrete form for each boundary condition is B ( N ( x i ) u i ) : 0;
i = 1, 2 . . . . ,nb
(15.45b)
The total number of equations must equal the number of collocation points selected. Accordingly, ne -k- nb -- n (15.46) It would appear that little difference will exist between continuous approximations involving moving least squares and discontinuous ones as in both locally the same polynomial will be used. This may well account for the convergence of standard least squares approximations which we have observed in Chapter 3 for discontinuous least squares forms but in view of our previous remarks about differentiation, a slight difference will in fact exist if moving least squares are used and in the work of On~te et al. 42-44 which we mentioned before such moving least squares are adopted. In addition to the choice for p(x), a key step in the approximation is the choice of the weighting function for the least squares method and the domain over which the weighting function is applied. In the work of Orkisz 45 and Liszka 46 two methods are used: 1. A 'cross' criterion in which the domain at a point is divided into quadrants in a cartesian coordinate system originating at the 'point' where the equation is to be evaluated. The domain is selected such that each quadrant contains a fixed number of points, n q . The product of n q and the number of quadrants, q, must equal or exceed the number of polynomial terms in p less one (the central node point). An example is shown in Fig. 15.8(a) for a two-dimensional problem (q = 4 quadrants) and nq - - 2. 2. A 'Voronoi neighbour' criterion in which the closest nodes are selected as shown for a two-dimensional example in Fig. 15.8(b). There are advantages and disadvantages to both approaches - namely, the cross criterion leads to dependence on the orientation of the global coordinate axes while the Voronoi method gives results which are sometimes too few in number to get appropriate order
542
Point-based and partition of unity approximations. Extended finite element methods 0'
15 -'
10-
'-
15
0
lO
.
.
.
.
0
50
o
-5
-5
-10
-lO
-15 -, (a)
--'5
0 I
I
-10
-5
0
I
I
5
10
,15
-15 (b)
"5
, ~ ,
-10
-5
,,
0
,
5
,
10
,
15
Fig. 15.8 Methods for selecting points. (a) Cross. (b) Voronoi.
approximations. The Voronoi method is, however, effective for use in Galerkin solution methods or finite volume (subdomain collocation) methods in which only first derivatives are needed. The interested reader can consult reference 40 for examples of solutions obtained by this approach. Additional results for finite point solutions may be found in work by On~te et al. 42 and Batina. 41 One advantage of considering moving least squares approximations instead of simple fixed point weighted least squares is that approximations at points other than those used to write the differential equations and boundary conditions are also continuously available. Thus, it is possible to perform a full post-processing to obtain the contours of the solution and its derivatives. In the next part of this section we consider the application of the moving least squares method to solve a second-order ordinary differential equation using point collocation.
Example 15.2: Collocation (point) solution of ordinary differential equations. We consider the solution of ordinary differential equations using a point collocation method. The differential equation in our examples is taken as d2u -a d~
du
+ b -~x + cu - f (x) = 0
(15.47)
on the domain 0 < x < L with constant coefficients a, b, c, subject to the boundary conditions u(0) = gl and u ( L ) = ga. The domain is divided into an equally spaced set of points located at xi, i = 1 . . . . . n. The moving least squares approximation described in Sec. 15.3 is used to write difference equations at each of the interior points (i.e., i = 2 . . . . . n - 1). The boundary conditions are also written in terms of discrete approximations using the moving least squares approximation. Accordingly, for the approximate solution using p-order polynomials to define the p(x) in the interpolations
a(x) - ~ NiP(x)r~i j=l
(15.48)
Point collocation- finite point methods 543 we have the set of n equations in n unknowns" n
Z NiP(xl)fti -- gl
(15.49a)
i=1
~
(_a d2Nip
+ b-~-x + cNP
Ui - - f ( x j )
"-
j = 2..... n - 1
0;
(15.49b)
X -'Xj
and
n
(15.49c)
=
i=1
The above equations may be written compactly as: Kfi + f = 0
(15.50)
where K is a square coefficient matrix, f is a load vector consisting of the entries from gi and f (x j), and fi is the vector of unknown parameters defining the approximate solution fi (x). A unique solution to this set of equations requires K to be non-singular (i.e., rank (K) = n). The rank of K depends both on the weighting function used to construct the least squares approximation as well as the number of functions used to define the polynomials p. In order to keep the least squares matrices as well conditioned as possible, a different approximation is used at each node with p(J)(x) = [1
x
-- Xj
(X -- Xj) 2
...
(X -- Xj) p]
(15.51)
defining the interpolations associated with N f (x). The matrix K will be of correct rank provided the weighting function can generate linearly independent equations. The accurate approximation of second derivatives in the differential equation requires the use of quadratic or higher order polynomials in p(x). 40 In addition, the span of the weighting function must be sufficient to keep the least squares matrix It non-singular at every collocation point. Thus, the minimum span needed to define quadratic interpolations of p(x) (i.e., p = k = 2) must include at least three mesh points with non-zero contributions. At the problem boundaries only half of the weighting function span will be used (e.g., the fight half at the left boundary). Consequently, for weighting functions which go smoothly to zero at their boundary, a span larger than four mesh spaces is required. The span should not be made too large, however, since the sparse structure of K will then be lost and overdiffuse solutions may result. Use of hierarchical interpolations reduces the required span of the weighting function. For example, use of interpolations with k = 0 requires only a span at each point for which the domain is just covered (since any span will include its defining point, Xk, the H matrix will always be non-singular). For a uniformly spaced set of points this is any span greater than one mesh spacing. For the example we use the weighting function described by Eq. (15.11) with a weight span 4.4 (rm = 2.2h) times the largest adjacent mesh space for the quadratic interpolations with k = p = 2 and a weight 2.01 times the mesh space for the hierarchical quadratic interpolations with k = 0, p = 2.
544 Point-basedand partition of unity approximations. Extended finite element methods We consider the example of a string on an elastic foundation with the differential equation
d2u
- a d ~ + cu + f = 0;
0 < x < 1
(15.52)
with the boundary conditions u (0) = u (1) = 0. This is a special form of Eq. (15.47). The parameters for solution are selected as a = 0.01
c=
1
f =-1
The exact solution is given by sinh(mx)
(c)
u(x) = 1 - cosh(mx) - (1 - c o s h ( m ) ) - -
sinh(m) '
1/2
m =
The problem is solved using 27 points and k = p = 2 producing the results shown in Fig. 15.9. The process was repeated using the hierarchical interpolations with k = 0 and p = 2 using nine points (which results in 27 parameters, the same as for the first case). The results are shown in Fig. 15.10. The hierarchical interpolation permits the solution to be obtained using as few as two points. A solution with two points and interpolations with k = 0 and p = 3 and 5 is shown in Figs 15.11 and 15.12, respectively. Note, however, that with the hierarchical form additional collocation points have to be introduced to achieve a sufficient number of equations. We show such collocation points in Fig. 15.10. Collocation moving least square ODE solution 0.9 0.8 0.7 0.6
~o.5 0.4 0.3 0.2 0"10 0
0.1
0.2
0.3
0.4 0.5 0.6 x coordinate
0.7
0.8
0.9
1.0
Fig. 15.9 String on elastic foundation solution using MLS form based on nodes: 27 points, k = 2, p = 2.
Point collocation - finite point methods
Collocation moving least square ODE solution _'"
;.,
;'.
;'.
;'.
;'.
;'.
;'.
)..
-
0.9 0.8 0.7 0.6 0.5
o x
Exact Approximate Node " Col pt
l
0.4 0.3 0.2 0.1 0.1
0.2
0.3
0.4 0.5 0.6 x coordinate
0.7
0.8
0.9
1.0
Fig. 15.10 String on elastic foundation hierarchic solution: 9 nodal points, k - O, p = 2. Collocation moving least square ODE solution O.9
tsSS ~ 9~ ~ ~
0.8
~ ~~
,' #
i
0.7 0.6
~-o.5 0.4 0.3 0.2 0.1 0
i 0
0.1
0.2
0.3
0.4 0.5 0.6 x coordinate
0.7
0.8
Fig. 15.11 String on elastic foundation hierarchic solution' 2 points, k -- O, p = 3.
0.9
1.0
545
546
Point-basedand partition of unity approximations. Extended finite element methods
Fig. 15.12 String on elastic foundation solution: 2 points, k = O, p = 5.
Point collocation methods are straightforward and quite easy to implement, the main task being only the selection of the subdomain on which to perform the fit of the function from which the derivatives are computed. Disadvantages arise, however, in the need to use high order interpolations such that accurate derivatives of the order of the differential equation may be computed. Further the treatment of boundaries and material interfaces present difficulties. An alternative, as we have discussed in Chapter 3, is the use of 'weak' or 'variational' forms which are equivalent to the differential equation. Approximations then require functions which have lower order than in the differential equation. In addition, boundary conditions often appear as 'natural' conditions in the weak form - especially for flux (derivative or Neumann)-type boundary conditions. This advantage now is balanced by a need to perform integration over the whole domain. Here, we consider problems of the form given by (see Sec. 3.2)t f
C(v)TZ)(u)dr2 + f r ~(V)T'~'(U)dr = 0
(15.53)
in which the operators C, Z), E and .7r contain lower derivatives than those occurring in operators .,4 and B given in Eqs (15.44a) and (15.44b), respectively. For example, t We assume that the b o u n d a r y terms are described such that ~r = v.
Galerkin weighting and finite volume methods 547
the solution of second order differential equations (such as those occurring in the quasiharmonic or linear elasticity equation) have differential operators for C to .~" with derivatives no higher than first order. The approximate solution to forms given by Eq. (15.53) may be achieved using moving least squares and alternative methods for performing the domain integrals.
15.6.2 Subdomain collocation- finite volume method A simple extension of the point collocation method is to use subdomains (elements) defined by the Voronoi neighbour criterion. The integrals for each subdomain are approximated as a constant evaluated at the originating point as nd
nb C(vi)T~)(Ui)~'2i -~- ~ ~-,(vi)Tff~(Ui)I"i -- 0 i i
(15.54)
where nd + nb = n, the total number of unknown parameters appearing in the approximations of u and v. The validity of the above approximation form can be established using patch tests (see Chapter 9). This approach is often called subdomain collocation or the finite volume method. This approach has been used extensively in constructing approximations for fluid flow problems. 47-53 It has also been employed with some success in the solution of problems in structural mechanics. 54
15.6.3 Galerkin methods- diffuse elements Moving least squares approximations have been used with weak forms to construct Galerkintype approximations. The origin of this approach can be traced to the work of Liszka46 and Orkisz. 40 Additional work, originally called the diffuse element approximation, was presented in the early 1990s by Nayroles et al. 11-13 Beginning in the mid-1990s the method has been extensively developed and improved by Belytschko and coauthors under the name element-free Galerkin. 14" 15,55,56 A similar procedure, called 'hp-clouds', was also presented by Oden and Duarte. 16' 17,57 Each of the methods is also said to be 'meshless'; however, in order to implement a true Galerkin process it is necessary to carry out integrations over the domain. What distinguishes each of the above processes is the manner in which these integrations are carried out. In the element-free Galerkin method a background 'grid' is often used to define the integrals whereas in the hp-cloud method circular subdomains are employed. Differing weights are also used as a means of generating the moving least squares approximation. The interested reader is referred to the appropriate literature for more details. Another source to consult for implementation of the EFG method is reference 19. Here we present only a simple implementation for solution of an ordinary differential equation.
Example 15.3: Galerkin solution of ordinary differential equations. The moving least squares approximation described in Sec. 15.3 is now used as a Galerkin method to solve a second order ordinary differential equation. For an arbitrary function W(x) satisfying
548
Point-basedand partition of unity approximations. Extended finite element methods W(0) = W (L) = 0, a weak form for the differential equation may be deduced using the procedures presented in Chapter 3. Accordingly, we obtain
--d-ffxa-d--~x+ w \ dx + cu - f (x)
dx = 0
(15.55)
subject to the boundary conditions u(0) = gl and u(L) = g2. Using a hierarchical moving least squares form a p-order polynomial approximation to the dependent variable may be written as
gt(x ) = ~
N~ (x )~ljp (X)fi p
(15.56)
j=l
where qjp
"-"
[1
x -
xj
(x -
xj) 2
...
(x -
x j ) p]
(15.57)
Note that in the above form we have used the representation
~tjp(X)fip = fij + q(x)bj The approximation to the weight function is similarly taken as
re(x) -
~p
N~
(15.58)
j=l ~p
in which Wj are arbitrary parameters satisfying W(0) - W(L) = 0. The approximation yields the discrete problem
n
{jO L
"=
j=l
=
((,VP)w i=1
L
dx
a
dx
/o"
~l~pN~ (15.59)
Since W~ is arbitrary, the solution to the approximate weak form yields the set of equations E
j=l
d ~
a
= fOL~ITNO f (x) dx;
dx
+~ITN ~
b
dx
+c0j )] dx}oj
(15.60)
i = 1, 2 . . . . . n
The set of equations only needs to be modified to satisfy the essential boundary equations. This is accomplished by replacing the equations corresponding to W1 = Wn = 0 by fil = gl and gn "~ g2. The Galerkin form requires only first derivatives of the approximating functions as opposed to the second derivatives required for the point collocation method. This reduction, however, is accompanied by a need to perform integrals over the domain. For weighting functions given by Eq. (15.11) all functions entering the approximation are polynomial
Use of hierarchic and special functions 549
and rational polynomial expressions, thus, a closed form evaluation is impractical. Accordingly, we evaluate integrals using Gauss and Gauss-Lobatto quadrature over each interval generated by the basis points in the moving least squares representation (i.e., xj for j = 1, 2 . . . . . n). As an example of the type of solutions possible we consider the string on elastic foundation problem given in the previous section. For the parameters a = 0.001, c = 1 with loading f = - 1 and zero boundary conditions a Galerkin solution using 3- and 4-point Gauss quadrature and 4- and 5-point Gauss-Lobatto quadrature is shown in Figs 15.13-15.16. A mesh consisting of nine equally spaced points is used to define the intervals for the solution and quadrature. The weight function is generated for k = 0, p = 2 with a span of 2.1 mesh points. Based upon this elementary example it is evident that the answers for a 9-point mesh depend on accurate evaluation of integrals to produce high-quality answers.
In Sec. 15.4, we discussed the possibility of introducing hierarchical variables to shape functions based on moving least squares interpolations. However, a simpler approach to
Fig. 15.13 String on elastic foundation solution: 3-point Gauss quadrature.
550 Point-basedand partition of unity approximations. Extended finite element methods Galerkin moving least square ODE solution 0.9 0.8 0.7 0.6 0.5
O
Exact Approximate Node
0.4 0.3 0.2 0.1 0(
0
~
0.1
0.2
0.3
0.4 0.5 0.6 x coordinate
0.7
0.8
0.9
..(
1.0
Fig. 15.14 String on elastic foundation solution: 4-point Gauss quadrature. Galerkin moving least square ODE solution 0.9 0.8 0.7 ~.
0.6
e
.~ o.5 0.4 0.3 0.2 0.1 0
1 0
0.1
0.2
0.3
0,4 0.5 0.6 x coordinate
0.7
0.8
Fig. 15.15 String on elastic foundation solution: 4-point Gauss-Lobatto quadrature.
0.9
1.0
Use of hierarchic and special functions 551 Galerkin moving least square ODE solution
0.9 0.8 0.7 0.6 0.5
0.4 0.3 0.2 o.1 00
o
0.1
0.2
0.3
0.4 0.5 0.6 x coordinate
0.7
0.8
0.9
1.0
Fig. 15.16 String on elastic foundation solution: 5-point Gauss-Lobatto quadrature.
hierarchical forms and indeed to extensions by other functions can be based on simple finite element shape functions. One important application of the partition of unity method starts from a set of finite element basis functions, Ni (x). An approximation to u (x) is now given by
u(x) ~ f~(x) -- ~ Ni(x) [~zi+
ot
(15.61)
where Ni (x) is the conventional (possibly isoparametric) finite element shape function at node i, q(i) are global functions associated with node i, and ui, and bai are parameters associated with the added global hierarchical functions. We must note that as before ~/will not represent a local value of the function unless the function qi become zero at the node i. Here we assume that conventional shape functions which satisfy the partition of unity condition
~Ni-1 i
are used. Thus, the above form is a hierarchic finite element method based on the partition of unity.21, 58, 59 We note in particular that the function q(i) may be different for each node and thus the form may be effectively used in an adaptive finite element procedure as described in Chapter 14. Equation (15.61) provides options for a wide choice of functions for q(i). 1. Polynomial functions. In this case the method becomes an alternative hierarchical scheme to that presented in Part 2 of Chapter 4.
552
Point-based and partition of unity approximations. Extended finite element methods 2. Harmonic 'wave' functions. This is a multiscale method and will be discussed in detail in Volume 3. 3. Singular functions. These can be used to introduce re-entrant corner or singular load effects in elliptic problems (e.g., heat conduction or elasticity forms). Derivatives of Eq. (15.61) are computed directly as
OXk--.
LOxk~i+~~-~xkq=ot
-'~'Ni~xk
bai
(15.62)
The reader will note that the narrow band structure of the standard finite element method will always be maintained as it is determined by the connectivity of Ni. Note also that the standard element on which the shape functions Ni were generated can be used for all subsequent integrations. Such a formulation is very easy to fit into any finite element program.
15.7.2 Polynomial hierarchical method To give more details of the above hierarchical finite element method we first consider the one-dimensional approximation in a 2-noded element where -- N1 Jill q-q(1)bl] q- N2 [i2-k-q(2)b2] in which
X2 -- X N1 - - ~ ; X2 -- Xl
N2 - -
(15.63)
X -- Xl X2 -- Xl
and q(1) = q(2) = [x k, bi = W e r e c a l l t h a t N1 -k N2 - - 1 a n d
[bil,
xk+l,
""
"]
(15.64)
bi2, ...]T
Nix1 q- N2x2 = x.
Investigation of the term x k in the approximation -- N1 (x) [fil + xkbkl] + N2(x) [fi2 + xkbk2]
(15.65)
we observe that a linear dependence with the usual finite element approximation occurs when i i -- Xi[) 0 and k - 1 with b l l = b12 -- bl. In this case Eq. (15.65) becomes = [ N l X l -+- N 2 x 2 ] bo -+- [N1 -}- N2] x b l
= xbo + xbl In one dimension linear dependence can be avoided by setting k to 2 in Eqs (15.63) and (15.64). However, in two- and three-dimensional problems the linear dependence cannot be completely avoided, and we address this next. 58' 6o An approximation over two-dimensional triangles may be expressed as 3
u(x, y) ~ ~t(x, y) -- Z i=1
Li [ii @ q(i)bi]
(15.66)
Use of hierarchic and special functions 553
where Li are the area coordinates defined in Chapter 4. We consider the case where complete quadratic functions are added as
q ( i ) - - [ x 2,
y2]
xy,
(15.67)
to give a complete second-order polynomial approximation for u. Although this gives a complete second-order polynomial approximation there are two ways in which the cubic term x2y can be obtained. 1. The first sets tli - - bil = bi3
giving
= 0
and
bi2 - - xiol
3 ~l = ~
Li.
[ x y ] . xiol =
x2y~t
i=1 2. The second alternative to compute the same term sets ~li "-" bi2 - - b i 3
giving
--0
and
bil -= yiot
3
-- ~
L i " [x2]
9yiot
--
x2ygt
i=1 A similar construction may be made for the polynomial term xy 2. An alternative is to construct the interpolation to depend on each node as
q(i) = [(X --
Xi) 2
(X -- X i ) ( y -- Yi)
(y
_
yi)2]
(15.68)
This form, while conceptually the same as the original formulation, appears to be better conditioned and also avoids some of the problems of linear dependency. 6~ In Sec. 15.7.4 we will discuss in more detail a methodology to deal with the problem of linear dependency; however, before doing so we illustrate the use of the hierarchical finite element method by an application to two-dimensional problems in linear elasticity.
15.7.3 Application to linear elasticity In the previous section the form for polynomial interpolation in two dimensions was given. Here we consider the use of the interpolation to model the behaviour of problems in linear elasticity. For simplicity only the displacement model for plane strain as discussed in Chapters 2 and 6 is considered; however, the use of the hierarchic interpolations can easily be extended to other forms and to mixed models. For a displacement model the finite element arrays may be computed using the formulation given in Chapter 6. For two-dimensional plane strain problems, the strain-displacement relations may be written in matrix form as -
Ou
0x
8v 8y
e = 8u
(15.69)
8v
554
Point-basedand partition of unity approximations. Extended finite element methods Inserting the interpolations for u and v given by Eq. (15.61) and using Eq. (15.62) to compute derivatives, the strain-displacement relations become
SNik 8x
0 -
o
ON/k
i=l .
N
JY/k
[o/] Ui
8y 8x "( ON~ Oqki -~x q ki + N ~ ~x ) .
+Z i=l
0
(ONik k
Oqki
(0N/k
0q/~
--~y qi + N~.--~y )
(0N/k 0q/k _ - ~ y q/k + N/k-~-y )
[byjrl
\-~xqki+N~--~-xj.
where N is the number of nodes for an element. The first term is identical to the usual finite element strain-displacement matrices [see Eq. (6.57)] and the second term has identical structure to the usual arrays. Thus, the development of all element arrays follows standard procedures. Example 15.4: A quadratic triangular element. For a triangular element with linear
interpolation the shape functions and quadratic polynomial hierarchic terms are given by Ni -- Li and Eq. (15.68), respectively. Using isoparametric concepts the coordinates are given by 3
3
X = ~ Nilxi -- ~ Lixi i=l i=l
(15.70)
and are used to construct all polynomials appearing in hierarchical form (15.68). A set of patch tests is first performed to assess the stability and consistency of the above hierarchic form. The set consists of one-, two-, four-, and eight-element patches as shown in Fig. 15.17. First, we perform a stability assessment by determining the number of zero eigenvalues for each patch. The results for hierarchical interpolation are shown in Table 15.3. The eigenproblem assessment reveals that the hierarchic interpolation has excess zero eigenvalues (i.e., spurious zero energy modes) only for meshes consisting of one or two elements. Furthermore, only two element meshes in which one side is a straight line through both elements have excess zero values. Once the mesh has no straight intersections the number of zero modes becomes correct (e.g., contain only the three rigid body modes). Consistency tests verify that all meshes contain terms of up to quadratic polynomial order - thus also validating the correctness of the coding. As a simple test problem using the hierarchical finite element method we consider a finite width strip containing a circular hole with diameter half the width of the strip. The strip is subjected to axial extension in the vertical direction and, due to symmetry, of the loading and geometry, only one quadrant is discretized as shown in Figs 15.18 and 15.19. The meshes in Fig. 15.18 employ the hierarchical interpolation considered above; whereas those in Fig. 15.19 use standard 6-node isoparametric quadratic triangles with two degrees of freedom per node (i.e., u and v). The material is taken as linear elastic with E = 1000
Use of hierarchic and special functions 555
(a) One element
(b) Two element, a
(c) Two element, b
(d) Two element, c
(e) Four element
(f) Eight element
Fig. 15.17 Patchesfor eigenproblem assessment.
Table 15.3 Triangle element patch tests: number of eigenvalues, minimum non-zero value, and maximum value (k = 2) - quadratic hierarchical terms Mesh
No. zero
Min. value
Max. value
1 2a 2b 2c 4 8
7 5 5 3 3 3
4.7340E 4.0689E 4.1971E 1.5728E 1.0446E 9.5560E
2.0560E 2.1543E 2.2648E 2.3883E 2.9027E 3.4813E
+ 01 4- 01 4- 02 4- 02 4- 02 4- 01
+ 06 4- 05 4- 05 4- 06 4- 05 4- 05
and v = 0.25. The half-width of the strip is 10 units and the half-height is 18 units. The hole has radius 5. The problem size and computed energy (which indicates solution accuracy) are shown in Table 15.4 for the hierarchical method, in Table 15.5 for the 6-node isoparametric formulation and in Table 15.6 for 3-node linear triangular elements. The 6-node isoparametric method gives overall the best accuracy; however, the hierarchical element is considerably better than the 3-node triangular element and offers great advantages when used in adaptive analysis. 6~
556 Point-basedand partition of unity approximations. Extended finite element methods
(a) 28 elements
(b) 112 elements
Fig. 15.18 Hierarchic elements: tension strip.
(a) 28 elements Fig. 15.19 Isoparametric 6-noded elements: tension strip.
(b) 112 elements
Use of hierarchic and special functions 557 Table 15.4 Hierarchical element. Boundary segments straight
Nodes
Elements
Equations
Energy
30 85 279 1003
28 112 448 1792
156 537 1971 7527
131.7088 127.8260 126.7641 126.5908
Table 15.5 Isoparametric element. Boundary segments have curved sides
Nodes
Elements
Equations
Energy
30 279 1003 3795
28 112 448 1792
129 483 1863 7311
127.3350 126.6483 126.5661 126.5593
Table 15.6 Linear triangular element Nodes
Elements
Equations
Energy
30 85 279 1003 3795
28 112 448 1792 7168
36 129 483 1863 7311
137.652 131.065 128.008 126.958 126.662
15.7.4 Solution of forms with linearly dependent equations A typical problem for a steady-state analysis in which the algebraic equations are generated from the hierarchical finite element form described above, such as given by Eqs (15.66) and (15.67), produces algebraic equations in the standard form, i.e.,
Kfi+f =0
(15.71)
where the parameters fi include both nodal U i and hierarchical parameters bi. We assume that occasionally the 'stiffness matrix' K and 'force' vector f include equations which are linearly dependent with other equations in the system and, thus, K can be singular. If the system is solved by a direct elimination scheme (e.g., as described in Chapter 2 or in books on linear algebra such as references 61 or 62) it is possible to set a tolerance for the pivot below which an equation is assumed to be linearly dependent and can be omitted from the calculations (e.g., see references 63 and 64). An alternative to the above is to perturb Eq. (15.71) to [K + eDi,:] Aft ~ = f - Kfi k
(15.72)
where D K are diagonal entries of K, e is a specified value and ~lk+l = ~k + A~k
(15.73)
558 Point-basedand partition of unity approximations. Extended finite element methods is used to define an iterative strategy. An initial guess of zero may be used to start the solution process. Certainly a choice of a small value for e (e.g., 10 -6) leads to rapid convergence. 6~
In this chapter we have considered a number of methods which eliminate or reduce our dependence on meshing the total domain. There are a number of other approaches having the same aim which have been pursued with success. These include the s m o o t h p a r t i c l e hydrod y n a m i c s (SPH) method (Lucy, 65 Gingold and Monaghan, 66 Benz 67) and the r e p r o d u c i n g k e r n e l (RPK) method (Liu et al. 68' 69) applied to problems in solid and fluid mechanics. Bonet and coworkers 7~ improve the method of SPH and show its possibilities. Another approach has recently been introduced by Yagawa. 71' 72 These are not described here and the reader is referred to the literature for details.
15.1 Data for a distributed loading to be applied to an analysis is tabulated in Table 15.7. The data is to be fit using the least squares analysis described in Sec. 15.2.1 in which
p(x)--[1
x
x2
x3
x 4]
Determine and plot the solution obtained with this data. 15.2 Data for a distributed loading to be applied to an analysis is tabulated in Table 15.7. The data is to be fit using the moving least squares analysis described in Sec. 15.3. Use the weight function given by Eq. (15.11) with rm = 2 and a Shepard approximation, p(x) = 1
Write a MATLAB program to compute the fit at intervals on x of 0.1 units. Plot the solution obtained. 15.3 Data for a distributed loading to be applied to an analysis is tabulated in Table 15.7. The data is to be fit using the moving least squares analysis described in Sec. 15.3. Use the weight function given by Eq. (15.11) with rm - 2 and a linear approximation, p(x) = [1
for x j - rm < x < x j + rm
x - x j]
Write a MATLAB program to compute the fit at intervals on x of 0.1 units. Plot the solution obtained. 15.4 Solve the differential equation d2u dx 2
+u+f=0;
0<x<
1
Table 15.7 Datafor force in Problems 15.1 to 15.3 k
1
2
3
4
5
6
7
8
xk fk
0.0 6.0
1.0 3.0
2.0 1.6
3.5 -1.4
5.0 -1.3
6.5 0.3
8.3 0.9
10.0 0.0
References 559
where the boundary conditions are u (0) = u (1) = 1 and f is a concentrated unit load at x -- 0.4 units. Write a MATLAB program to solve the problem. Use a moving least square method with Shepard interpolation and the point collocation method described in Sec. 15.5. Let the points be located at 0.2 intervals (6 points). Repeat the solution using points spaced at 0.1 intervals (11 points). Plot the two solutions and comment on the behaviour obtained. 15.5 Solve the differential equation d2u dx 2 ~ - u + f = 0 ;
0<x
< 1
where the boundary conditions are u (0) = u (1) = 1 and f is a concentrated unit load at x - 0.4 units. Write a MATLAB program to solve the problem. Use a moving least square method with Shepard interpolation and the Galerkin method described in Sec. 15.6. Let the points be located at 0.2 intervals (6 points). Repeat the solution using points spaced at 0.1 intervals (11 points). Plot the two solutions and comment on the behaviour obtained. 15.6 Repeat Problem 15.4 using a hierarchical interpolation with q(J)(x) = X-
Xj
where x j is the coordinate of the point. 15.7 Repeat Problem 15.5 using a hierarchical interpolation with q(J) (x) -- x - x j
where x j is the coordinate of the point. 15.8 In the moving least squares approximation, the shape function N j (x) of Eq. (15.22) is derived by minimization of the function J (x) in Eq. (15.14). Derive a shape function for the moving least squares approximation by minimization of the function
1L w(x -
J(x) = ~
y)[u(y) - p(y)c~]2dy = min
15.9 In the least square fit and the construction of shape functions for the moving least squares approximation the matrix H in Eqs (15.5) and (15.18a) may be singular for a given set of points if the monomials used in the polynomial basis are not chosen properly. Devise an algorithm that can choose the terms used in the polynomial basis automatically so that matrix H is always non-singular.
1. V. Girault. Theory of a finite difference method on irregular networks. SlAM J. Num. Anal., 11:260-282, 1974. 2. V. Pavlin and N. Perrone. Finite difference energy techniques for arbitrary meshes. Comp. Struct., 5:45-58, 1975.
560 Point-basedand partition of unity approximations. Extended finite element methods 3. C. Snell, D.G. Vesey, and E Mullord. The application of a general finite difference method to some boundary value problems. Comp. Struct., 13:547-552, 1981. 4. T. Liszka and J. Orkisz. Finite difference methods of arbitrary irregular meshes in non-linear problems of applied mechanics. In Proceedings 4th Int. Conference on Structural Mechanics in Reactor Technology, San Francisco, California, 1977. 5. T. Liszka and J. Orkisz. The finite difference method at arbitrary irregular grids and its applications in applied mechanics. Comp. Struct., 11:83-95, 1980. 6. J. Krok and J. Orkisz. A unified approach to the FE generalized variational FD method in nonlinear mechanics. Concept and numerical approach. In Discretization Methods in Structural Mechanics, pages 353-362. Springer-Verlag, Berlin-Heidelberg, 1990. IUTAM/IACM Symposium, Vienna, 1989. 7. R.A. Nay and S. Utku. An alternative for the finite element method. Vari. Meth. Eng., 1, 1972. 8. D. Shepard. A two-dimensional function for irregularly spaced data. In ACM National Conference, pages 517-524, 1968. 9. E Lancaster and K. Salkauskas. Surfaces generated by moving least squares methods. Math. Comput., 37:141-158, 1981. 10. E Lancaster and K. Salkauskas. Curve and Surface Fitting. Academic Press, New York, 1990. 11. B. Nayroles, G. Touzot, and E Villon. La m6thode des 616ments diffuse. C.R. Acad. Sci. Paris, 313:133-138, 1991. 12. B. Nayroles, G. Touzot, and E Villon. L'approximation diffuse. C.R. Acad. Sci. Paris, 313: 293-296, 1991. 13. B. Nayroles, G. Touzot, and E Villon. Generalizing the FEM: diffuse approximation and diffuse elements. Comput. Mech., 10:307-318, 1992. 14. T. Belytschko, Y. Lu, and L. Gu. Element free Galerkin methods. Int. J. Numer. Meth. Eng., 37:397-414, 1994. 15. T. Belytschko, Y. Lu, and L. Gu. Crack propagation by element-free Galerkin methods. Eng. Fracture Mech., 51:295-315, 1995. 16. C.A. Duarte and J.T. Oden. h - p clouds - a meshless method to solve boundary-value problems. Technical Report TICAM Report 95-05, The University of Texas, May 1995. 17. C.A. Duarte and J.T. Oden. An h - p adaptive method using clouds. Comp. Meth. Appl. Mech. Eng., 139(1-4):237-262, 1996. 18. T. Belytschko, J. Fish, and A. Bayless. The spectral overlay on finite elements for problems with high gradients. Comp. Meth. Appl. Mech. Eng., 81:71-89, 1990. 19. J. Dolbow and T. Belytschko. An introduction to programming the meshless element free Galerkin method. Arch. Comput. Meth. Eng., 5(3):207-241, 1998. 20. I. Babu~ka and J.M. Melenk. The partition of unity finite element method. Technical Report Technical Note BN- 1185, Institute for Physical Science and Technology, University of Maryland, April 1995. 21. J.M. Melenk and I. Babu~ka. The partition of unity finite element method: basic theory and applications. Comp. Meth. AppL Mech. Eng., 139:289-314, 1996. 22. O.C. Zienkiewicz, R.L. Taylor, and P. Nithiarasu. The Finite Element Method for Fluid Dynamics. Butterworth-Heinemann, Oxford, 6th edition, 2005. 23. C. Daux, N. Moes, J. Dolbow, N. Sukumar, and T. Belytschko. Arbitrary branched and intersecting cracks with the extended finite element method. Int. J. Numer. Meth. Eng., 48:1741-1760, 2000. 24. N. Sukumar, N. Moes, B. Moran, and T. Belytschko. Extended finite element method for threedimensional crack modelling. Int. J. Numer. Meth. Eng., 48:1549-1570, 2000. 25. N. Sukumar, D.L. Chopp, N. Moes, and T. Belytschko. Modeling holes and inclusions by level sets in the extended finite element method. Comp. Meth. Appl. Mech. Eng., 190:6183-6846, 2001.
References 561 26. J. Dolbow, N. Moes, and T. Belytschko. An extended finite element method for modeling crack growth with frictional contact. Comp. Meth. Appl. Mech. Eng., 190:6825-6846, 2000. 27. N. Moes, A. Gravouil, and T. Belytschko. Non-planar 3D crack growth by the extended finite element and level sets. Part I: Mechanical model. Int. J. Numer. Meth. Eng., 53:2549-2568, 2002. 28. A. Gravouil, N. Moes, and T. Belytschko. Non-planar 3D crack growth by the extended finite element and level sets. Part II: Level set update. Int. J. Numer. Meth. Eng., 53:2569-2586, 2002. 29. N. Moes and T. Belytschko. Extended finite element method for cohesive crack growth. Eng. Frac. Mech., 69:813-833, 2002. 30. T. Belytschko, C. Parimi, N. Moes, N. Sukumar, and S. Usui. Structured extended finite element methods for solids defined by implicit surfaces. Int. J. Numer. Meth. Eng., 56:609-635, 2002. 31. J. Chessa, P. Smolinski, and T. Betytschko. The extended finite element method (XFEM) for solidification problems. Int. J. Numer. Meth. Eng., 53:1959-1977, 2002. 32. G. Zi and T. Belytschko. New crack-tip elements for XFEM and applications to cohesive cracks. Int. J. Numer. Meth. Eng., 57:2221-2240, 2003. 33. F.L. Stazi, E. Budyn, J. Chessa, and T. Belytschko. An extended finite element method with higher-order elements for curved cracks. Comput. Mech., 31:38-48, 2003. 34. J. Chessa, H. Wang, and T. Belytschko. On the construction of blending elements for local partition of unity enriched finite elements. Int. J. Numer. Meth. Eng., 57:1015-1038, 2003. 35. W. Rudin. Principles of Mathematical Analysis. McGraw-Hill, 3rd edition, 1976. 36. I. Babu~ka and J.M. Melenk. The partition of unity method. Int. J. Numer. Meth. Eng., 40:727758, 1997. 37. L. Collatz. The Numerical Treatment of Differential Equations. Springer, Berlin, 1966. 38. G.E. Forsythe and W.R. Wasow. Finite Difference Methods for Partial Differential Equations. John Wiley & Sons, New York, 1960. 39. R.D. Richtmyer and K.W. Morton. Difference Methods for Initial Value Problems. Wiley (Interscience), New York, 1967. 40. J. Orkisz. Finite difference method. In M. Kleiber, editor, Handbook of Computational Solid Mechanics. Springer-Verlag, Berlin, 1998. 41. J. Batina. A gridless Euler/Navier-Stokes solution algorithm for complex aircraft applications. In AIAA 93-0333, Reno, NV, Jan. 1993. 42. E. Ofiate, S. Idelsohn, and O.C. Zienkiewicz. Finite point methods in computational mechanics. Technical Report CIMNE Report 67, Int. Center for Num. Meth. Engr., Barcelona, July 1995. 43. E. Ofiate, S.R. Idelsohn, O.C. Zienkiewicz, R.L. Taylor, and C. Sacco. A stabilized finite point method for analysis of fluid mechanics problems. Comp. Meth. Appl. Mech. Eng., 139:315-346, 1996. 44. E. Ofiate, S.R. Idelsohn, O.C. Zienkiewicz, and R.L. Taylor. A finite point method in computational mechanics. Applications to convective transport and fluid flow. Int. J. Numer. Meth. Eng., 39:3839-3866, 1996. 45. J. Orkisz. Computer approach to the finite difference method (in Polish). Mech. i Komp., 2:7-69, 1979. 46. T. Liszka. An interpolation method for an irregular net of nodes. Int. J. Numer. Meth. Eng., 20:1599-1612, 1984. 47. R.B. Pelz and A. Jameson. Transonic flow calculations using triangular finite elements. J. AIAA, 23:569-576, 1985. 48. J.T. Batina. Vortex-dominated conical-flow computations using unstructured adaptively-refined meshes. J. AIAA, 28(11):1925-1932, 1990. 49. J.T. Batina. Unsteady Euler airfoil solutions using unstructured dynamic meshes. J. AIAA, 28(8):1381-1388, 1990. 50. D.J. Mavriplis and A. Jameson. Multigrid solution of the Navier-Stokes equations on triangular meshes. J. AIAA, 28:1415-1425, 1990.
562 Point-basedand partition of unity approximations. Extended finite element methods 51. R.D. Rausch, J.T. Batina, and H.T.Y. Yang. Spatial adaptation of unstructured meshes for unsteady aerodynamic flow computations. J. AIAA, 30(5):1243-1251, 1992. 52. R.D. Rausch, J.T. Batina, and H.T.Y. Yang. Three-dimensional time-marching aeroelastic analyses using an unstructured-grid Euler method. J. AIAA, 31 (9): 1626-1633, 1993. 53. K. Xu, L. Martinelli, and A. Jameson. Gas-kinetic finite volume methods. In S.M. Deshpande, S.S. Desai, and R. Narasimha, editors, Proc. 14th Int. Conf. Num. Meth. Fluid Dynamics, pages 106-111, 1995. 54. E. Ofiate, F. Zarate, and F. Flores. A simple triangular element for thick and thin plate and shell analysis. Int. J. Numer. Meth. Eng., 37:2569-2582, 1994. 55. Y. Lu, T. Belytschko, and L. Gu. A new implementation of the element-free Galerkin method. Comp. Meth. Appl. Mech. Eng., 113:397-414, 1994. 56. M. Tabbara, T. Blacker, and T. Belytschko. Finite element derivative recovery by moving least square interpolates. Comp. Meth. Appl. Mech. Eng., 117:211-223, 1994. 57. C.A. Duarte. A review of some meshless methods to solve partial differential equations. Technical Report TICAM Report 95-06, The University of Texas, May 1995. 58. R.L. Taylor, O.C. Zienkiewicz, and E. Ofiate. A hierarchical finite element method based on the partition of unity. Comp. Meth. Appl. Mech. Eng., 152:73-84, 1998. 59. J.T. Oden, C.A.M. Duarte, and O.C. Zienkiewicz. A new cloud-based hp finite element method. Comp. Meth. Appl. Mech. Eng., 153(1-2):117-126, 1998. 60. C.A. Duarte, I. Babu~ka, and J.T. Oden. Generalized finite element methods for three dimensional structural mechanics problems. Comp. Struct., 77:215-232, 2000. 61. G. Strang. Linear Algebra and its Application. Academic Press, New York, 1976. 62. J. Demmel. Applied Numerical LinearAlgebra. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1997. 63. I.S. Duff and J.K. Reid. Exploiting zeros on the diagonal in the direct solution of indefinite sparse linear systems. ACM Trans. Math. Soft., 22:227-257, 1996. 64. I.S. Duff and J.A. Scott. Ma62 - a frontal code for sparse positive-definite symmetric systems from finite element applications. In M. Papadrakakis and B.H.V. Topping, editors, Innovative Computational Methods for Structural Mechanics, pages 1-25, June 1999. 65. L.B. Lucy. A numerical approach to the testing of fusion process. The Astron. J., 88, 1977. 66. R.A. Gingold and J.J. Monaghan. Smoothed particle hydrodynamics: theory and application to non-spherical stars. Monthly Notices of Royal Astron. Sci., 181, 1977. 67. W. Benz. Smoothed particle hydrodynamics: a review. Preprint 2884, 1989. 68. W.K. Liu, S. Jun, S. Li, J. Adee, and T. Belytschko. Reproducing kernel particle methods for structural dynamics. Comp. Meth. Appl. Mech. Eng., 38:1655-1679, 1995. 69. W.K. Liu, S. Jun, and Y.E Zhang. Reproducing kernel particle methods. Int. J. Numer. Meth. Eng., 20:1081-1106, 1995. 70. J. Bonet and S. Kulasegaram. Correction and stabilization of smooth particle hydrodynamics methods with applications in metal forming simulations. Int. J. Numer. Meth. Eng., 47:11891214, 2000. 71. G. Yagawa and T. Yamada. Free mesh method. A kind of meshless finite element method. Comput. Mech., 18:383-386, 1996. 72. G. Yagawa, T. Yamada, and T. Furukawa. Parallel computing with free mesh method: virtually meshless FEM. In H.A. Mang and EG. Rammerstorfer, editors, IUTAM Sym., Solid Mechanics and its Applications, pages 165-172. Kluwer Acd. Pub., 1997.
In most of the problems considered so far in this text conditions that do not vary with time were generally assumed. There is little difficulty in extending the finite element idealization to situations that are time dependent as indicated briefly in Chapters 3 and 7. The'range of practical problems in which the time dimension has to be considered is great. Transient heat conduction, wave transmission in fluids and dynamic behaviour of structures are typical examples. While it is usual to consider these various problems separately sometimes classifying them according to the mathematical structure of the governing equations as 'parabolic' or 'hyperbolic '1 - we shall group them into one category to show that the formulation is identical. In the first part of this chapter we shall formulate, by a simple extension of the methods used so far, matrix differential equations governing such problems for a variety of physical situations. Here a finite element discretization in the space dimension only will be used and a semi-discretization process followed (see Chapter 3). In the remainder of this chapter various analytical procedures of the solution for the resulting ordinary linear differential equation system will be dealt with. These form the basic arsenal of steady-state and transient analysis. Chapter 17 will be devoted to the discretization of the time domain itself. -
In many physical problems the quasi-harmonic equation takes the form in which time derivatives of the unknown function ~b occur. In the three-dimensional case typically we might have [viz. Eq. (7.6)]
564 Semi-discretizationand analytical solution
Ox
~x
- -ff-fy k -~y
- -~z k -~z
+
Q + lz --~ + p Ot 2 j - O
(16.1)
In the above, quite generally, all the parameters may be prescribed functions of time, or in non-linear cases of 4~, as well as of space x, i.e., k = k(x, q~, t)
0 = Q(x, cp, t) etc.
(16.2)
If a situation at a particular instant of time is considered, the time derivatives of ~p and all the parameters can be treated as prescribed functions of space coordinates. Thus, at that instant the problem is precisely identified with those treated in Chapter 7 if the whole of the quantity in the last parentheses of Eq. (16.1) is identified as the source term Q. The finite element discretization of this in terms of space elements has already been fully discussed and we found that with the prescription
dp -- Z
NaSa
-"
Nfi with N - N(x, y, z) and fi - fi(t)
(16.3)
for each element, the standard form of assembled equationst
Kfi + f = 0
(16.4)
was obtained. Element contributions to the above matrices are defined in Chapter 7 and need not be repeated here except for that representing the 'load' term due to Q. This is given by / *
f -- ./o NT Q d~2
(16.5)
Replacing Q by the last bracketed term of Eq. (16.1) we have f=
N T
O+/x--~-+p--~-
dr2
(16.6)
However, from Eq. (16.3) it is noted that 4~is approximated in terms of the nodal parameters ft. On substitution of this approximation we have f =
NTQdf2 + ( f ~ NT/zNdfl ) d-f~i + ( ~ N T pNdf2 )d2fi dt 2
(16.7)
and on expanding Eq. (16.4) in its final assembled form we get the following matrix differential equation:
Mfi+ Cfi+Kfi+f = 0 dfi d2fi dt
(16.8)
dt 2
in which all the matrices are assembled from element submatrices in the standard manner with submatrices K e and fe still given by relations (7.20) in Chapter 7 and Ceb --" f
Ja
Nal3, Nb dr2 and Meb -- f Nap Nb dr2 J~
(16.9)
t We have replacedthe matrixH of Chapter 7 by K and ~ by fi to facilitate later comparisonwith other transient equations.
Direct formulation of time-dependent problems with spatial finite element subdivision 565 Once again these matrices are symmetric as seen from the above relations. Boundary conditions imposed at any time instant are treated in the standard manner. The variety of physical problems governed by Eq. (16.1) is so large that a comprehensive discussion of them is beyond the scope of this book. A few typical examples will, however, be quoted.
Equation (16.1) with p = 0
This is the standard transient heat conduction equation 1, 2 which has been discussed in the finite element context in Sec. 7.4 and by several authors. 3-6 This same equation is applicable in other physical situations - one of these being the soil consolidation equations 7 associated with transient seepage forms. 8
Equation (16.1) with i ~ - 0
Now the relationship becomes the famous Helmholz wave equation governing a wide range of physical phenomena. Electromagnetic waves, 9 fluid surface waves 1~ and compression waves 11 are but a few cases to which the finite element process has been applied.
Equation (16.1) with i~ ~ p ~ 0 This damped wave equation is of yet more general applicability and has particular significance in fluid mechanics (wave) problems. The reader will recognize that what we have done here is simply an application of the process of partial discretization described in Sec. 3.5. It is convenient, however, to perform the operations in the manner suggested above as all the matrices and discretization expressions obtained from steady-state analysis are immediately available.
16.2.2 Dynamicbehaviour of elastic structures with linear damping While in the previous section we have been concerned with, apparently, a purely mathematical problem, identical reasoning can be applied directly to the wide class of dynamic behaviour of elastic structures following precisely the general lines of Chapters 2 and 6. When displacements of an elastic body vary with time two sets of additional forces are called into play. The first is the inertia force, which for an acceleration characterized by ii can be replaced by its static equivalent - p i i using the well-known d'Alembert principle. This is a force with components in directions identical to those of the displacement u and (generally) given per unit of volume. In this context p is simply the mass per unit volume. The second force is that due to (frictional) resistances opposing the motion. These may be due to microstructure movements, air resistance, etc., and are often related in a non-linear way to the velocity/l. For simplicity of treatment, however, only a linear viscous-type resistance will be considered, resulting again in unit volume forces in an equivalent static problem of magnitude -/~/l. In the above/~ is a set of viscosity parameters which can presumably be given numerical values. 12 The equivalent static problem, at any instant of time, is now discretized precisely in the manner of Chapters 2 and 6, but replacing the distributed body force b by its equivalent
The element (nodal) forces given by Eq. (6.62) now become (excluding initial stress and strain contributions)
566
Semi-discretizationand analytical solution
fe _ f~e NYb d~ "-" - f~e NT6 d~'~+ ~e NTpu d~r~+ f~e NW~fi dK2
(16.10)
in which the first force is that due to an external distributed body load and need not be considered further. Substituting Eq. (16.10) into the general equilibrium equations we obtain finally, on assembly, the following matrix differential equation: Mfi + C~ + Kfi + f = 0
(16.11)
in which K and f are assembled stiffness and force matrices obtained by the usual addition of element stiffness coefficients and of element forces due to external specified loads, initial stresses, etc., in the manner fully described before. The new matrices C and M are assembled by the usual rule from element submatrices given by C~b = f
J~2 e
and
sT/~Sb d ~
(16.12a)
NTp Nb dr2
(16.12b)
/.
Meb = [
d~2e
The matrix M e is known as the element mass matrix and the assembled matrix M as the system mass matrix. Similarly, the matrix C e is known as the element damping matrix and the assembled matrix C as the system damping matrix. It is of interest to note that in early attempts to deal with dynamic problems of this nature the mass of the elements was usually arbitrarily 'lumped' at nodes, always resulting in a diagonal mass matrix even if no actual concentrated masses existed. The fact that such a procedure was, in fact, unnecessary and apparently inconsistent was simultaneously recognized by Archer 13 and independently by Leckie and Lindberg 14 in 1963. The general presentation of the results given in Eq. (16.12b) is due to Zienkiewicz and Cheung. 15 The name consistent mass matrix has been coined for the mass matrix defined here, a term which may be considered to be unnecessary since it is the logical and natural consequence of the discretization process. By analogy the matrices C e and C may be called consistent damping matrices, t For many computational processes the lumped mass matrix is, however, more convenient and economical. Many practitioners are today using such matrices exclusively- sometimes showing good accuracy. While with simple elements a physically obvious methodology of lumping is easy to devise, this is not the case with higher order elements and we shall return to the process of 'lumping' later. Determination of the damping matrix C is in practice difficult as knowledge of the viscous matrix/z is lacking. It is often assumed, therefore, that the damping matrix is a linear combination of stiffness and mass matrices, i.e., C = ctM + / 3 K
(16.13)
Here the parameters ot and/3 are determined experimentally. 12' ~6 Such damping is known as 'Rayleigh damping' and has certain mathematical advantages which we shall discuss later. On occasion C may be completely specified and such approximation devices are not necessary. t For simplicity we shall only consider distributed inertia- concentrated damping forces being a limiting case.
Direct formulation of time-dependent problems with spatial finite element subdivision 567
16.2.3 'Mass' or 'damping' matrices for some typical elements It is impractical to present in an explicit form all the mass matrices for the various elements discussed in previous chapters. Some selected examples only will be discussed here. Example 16.1: Plane stress and plane strain. Using triangular elements discussed in Chapter 2 the matrix N e is defined as
Ne-[N~
N~
N~] where N e - N a I ,
a--1,2,3
andI-[~
10]
Equation (2.8) gives the shape functions as Na--
aa + bax + Cay
, a=1,2,3 2A where A is the area of the triangular element. If the thickness of the element is h and this is assumed to be constant within the element, we have, for the mass matrix, Eq. (16.12b), Me=
phffNTNdxdy
//
or Meb =
phIJJ'NaNbdxdy
{1
If the relationships of Eq. (2.8) are substituted, it is easy to verify that NaNbdxdy=
~A when a = b ~ A when a -~ b
(16.14)
Thus taking the total mass of the element as
m=phA the (consistent) mass matrix becomes
Me = __m 12
2 0
0 1
1 0
1 0
0 2
1 0
1 0
0 1
2 0
(16.15)
If the mass is physically lumped at the nodes in three equal parts the 'lumped' mass matrix contributed by the element is
M e _ -
m
3
1 0
0 1
0 0
0 0
0 0
0 0
0 0
0 0
1 0
0 1
0 0
0 0
0 0
0 0
0 0
0 0
1 0
0 1
(16.16)
568 Semi-discretization and analytical solution Certainly both matrices differ considerably and yet in applications the results of the analysis can be almost identical.
Example 16.2: Mass for isoparametric elements. The mass matrix for an isoparametric element may be computed by numerical integration as described in Chapter 5. For example, for two-dimensional elements the mass is given by
Mea "~ E Na(~/, l~l)P Na (~l, l~l) Jl //)l
(16.17)
1
for plane problems and
Meb ~ ~ Na(~l, Ol)pNa(~l, 1"11)rl Jl Wl
(16.18)
l
for axisymmetric problems. Now since it is the shape functions which are integrated, the order of quadrature needs to be selected according to the requirements given in Sec. 5.12. Generally, the order used for standard integration will suffice to accurately compute the mass. Reduced quadrature should not be used since spurious results can be obtained due to loss in rank of the mass matrix.
16.2.4 Mass 'lumping' or diagonalization We have referred to the computational convenience of lumping of mass matrices and presenting these in diagonal form. On some occasions such lumping is physically obvious (see the linear triangle for instance), in others this is not the case and a 'rational' procedure is required. For matrices of the type given in Eq. (16.12b) several alternative approximations have been developed, as discussed in Appendix I. In all of these the essential requirement of mass preservation is satisfied, i.e.,
E ]~/laa ~---f~ p dr2
(16.19)
a
where ]~aa is the diagonal for a component of the lumped mass matrix lVI. Three main procedures exist (see Fig. 16.1)" 1. The row sum method in which ]~Iaa -- E
Mab
2. Diagonal scaling in which ]~/Iaa "-- C Maa
with c adjusted so that Eq. (16.19) is satisfied, 17' 18 and 3. Evaluation of M using a quadrature involving only the nodal points and thus automatically yielding a diagonal matrix for standard finite element shape functions 19' 20 in which Na = O f o r x - X b , b # a.
Direct formulation of time-dependent problems with spatial finite element subdivision
1/4
1/4
1/4I ( ~ O ( ~ I
1/4
1/3~,j
,J 1/3
1
7/24
7/24
4
1/36
8/36 0
0 8/36
v 5/16
,
0 k..)--''k.2 k.) 1/3 1/36
5/16v
3/57~ 1~1/36 -1112g
/
/
~1/36 -1/12~J
16/57
k_)3/57
1/3 0
| 0 1/3
318 3,8 Ok.3-----k..2 k,_) 1/3
-1112 1/36(i) 1/3
4/36(~
-1/12 1/36',...j
0
4/36 0
|
1/36
0 16/36
4/36
0 4/36
1/36
|
(~ Row sum procedure (~ Diagonalscalingprocedure (~ Quadratureusing nodal points Fig. 16.1 Masslumpingfor sometwo-dimensionalelements. It should be remarked that Eq. (16.19) does not hold for hierarchical shape functions where no lumping procedure appears satisfactory. The quadrature (numerical integration) process is mathematically most appealing but frequently leads to negative or zero lumped masses. Such a loss of positive definiteness is undesirable in some solution processes and cancels out the advantages of lumping. In Fig. 16.1 we show the effect of various lumping procedures on triangular and quadrilateral elements of linear and quadratic type. It is clear from these that the optimal choice to lump the mass is by no means unique. In general we would recommend the use of lumped matrices only as a convenient numerical device generally paid for by some loss of accuracy. An exception to this is for 'explicit' time integration of dynamics problems where the considerable efficiency of their use more than compensates for any loss in accuracy (see Chapter 17). However, we note
569
570
Semi-discretization and analytical solution that it has occasionally been shown that lumping can improve accuracy of some problem by error cancellation. It can be shown that in the transient approximation the lumping process introduces additional dissipation of the 'stiffness matrix' form and this can help in cancelling out numerical oscillation. To demonstrate the nature of lumped and consistent mass matrices it is convenient to consider a typical one-dimensional problem specified by the equation
at
ax
-a-7
=o
Semi-discretization here gives a typical nodal equation a as (Mab 4- n a b ) h b 4- gab~lb -~ 0
where
Mab =
Na Nb dx, Hab --
f
dNa dNb dx Kab-~ tt d----~ '
k
J; ddxga adx
dx
and it is observed that H and K have identical structure. With linear elements of constant size h the approximating equation at a typical node a (and surrounding nodes a - 1 and a 4- 1) can be written as follows h Mab~lb ~ -~
(~a-1 4- 4~a 4- Uo+I)
tt nab~lb ~ -~
(--Uo-1 4- 2~a- ~/a-t-1)
k Kabfib ~ -~
(--fia-1 4- 2fia
9
-
-
fia+l)
If a lumped approximation is used for M, that is If/l, we have, simply by adding coefficients using the row sum method, Mab~lb -- h~la The difference between the two expressions is h l~/la b ~lb -- Ma b ~lb ~ -~
.
(--no-1 4- 2~a -- Ua+l)
and is clearly identical to that which would be obtained by replacing tt by h 2/6. As tt in the above example can be considered as a viscous dissipation we note that the effect of using a lumped matrix is that of adding an extra amount of such viscosity and can often result in smoother (though possibly less accurate) solutions.
Eigenvalues and analytical solution procedures
We have seen that as a result of semi-discretization many time-dependent problems can be reduced to a system of ordinary differential equations of the characteristic form given by
Free response - eigenvalues for second-order problems and dynamic vibration
Mfi + Cd + Kfi + f = 0
(16.20)
In this, in general, all the matrices are symmetric. Cases involving non-symmetric matrices are also found in some fluid problems. 21 This second-order system often becomes first order if M is zero as, for instance, in transient heat conduction problems. We shall now discuss some methods of solution of such ordinary differential equation systems. In general, the above equations can be non-linear (if, for instance, stiffness matrices are dependent on nonlinear material properties or if large deformations are involved) but here we shall concentrate on linear cases only. Systems of ordinary linear differential equations can always in principle be solved analytically without the introduction of additional approximations. The remainder of this chapter will be concerned with such analytical processes. While such solutions are possible they may be so complex that further recourse has to be taken to the process of approximation; we shall deal with this matter in the next chapter. The analytical approach provides, however, an insight into the behaviour of the system which the authors always find helpful. Some of the matter in this chapter will be an extension of standard well-known procedures used for the solution of differential equations with constant coefficients that are encountered in most studies of dynamics or mathematics. In the following we shall deal successively with: 1. Determination of free response (f = 0) 2. Determination of steady-state periodic response (f(t) periodic) 3. Determination of transient response (f(t) arbitrary). In the first two, initial conditions of the system are not required and a general solution is simply sought. The transient response initial conditions are required and we will devote considerable attention to this type in Sec. 16.8.
If no damping or forcing terms exist in the dynamic problem of Eq. (16.20) it reduces to Mfi + Kfi = 0
(16.21)
A general solution of such an equation may be written as fi -- fi exp(iwt) the real part of which simply represents a harmonic response as exp(iwt) = cos wt + i sin cot. Then on substitution we find that co can be determined from (-co2M + K)fi = 0
(16.22)
This is a general linear eigenvalue or characteristic value problem and for non-zero solutions the determinant of the above coefficient matrix must be zero: I-co2M + KI - 0
(16.23)
571
572
Semi-discretization and analytical solution 2 j = 1, 2 . . . . . n) Such a determinant will in general give n positive values of o92 (or ogj, when the size of the matrices K and M is n x n, providing the matrices K and M are symmetric positive definite.t While the solution of Eq. (16.23) cannot determine the actual values of fi we can find n vectors fij that give the proportions for the various terms. Such vectors are known as the normal modes of the system or eigenvectors and are made unique by normalizing so that
fiTMfij -- 1;
j = 1, 2 . . . . . n
(16.24)
At this stage it is useful to note the property of modal orthogonality, i.e., that fiTMfij = 0 ;
(i#j)
and fiTKfij -- 0;
(i~-j)
(16.25)
The proof of the above statement is simple. As Eq. (16.22) is valid for any mode we can write o92Mui - Kui and o9~Mfij - Kfij Premultiplying the first by uj -x and the second by fi/T and noting the symmetry of M and K so that f i T M f i i - - fiTMfij a n d fiTKfi/ - - f i T K f i j the difference becomes
O9j)U i M U j _
-
-
:
0
and if ogi =/=wj ~ the orthogonality condition for the matrix M has been proved. From this the orthogonality of the vectors with K follows immediately. The final condition fiWKfi i - - 092
follows from Eq. (16.24) and a premultiplication of Eq. (16.22) for equation i by Ui.
16.4.2 Determination of eigenvalues To find the actual eigenvalues it is seldom practical to write the polynomial expanding the determinant given in Eq. (16.23) and alternative techniques have to be developed. Many extremely efficient procedures are available and the reader can find some interesting matter in references 22-28. In some processes the starting point is the standard eigenvalue problem given by Hx = )~x
(16.26)
in which H is a symmetric matrix and hence has real eigenvalues. Equation (16.22) can be written as M-1Kfi = o92fi (16.27) ~fA symmetric matrix is positive definite if all the diagonals of the triangular factors are positive, this is a usual case with structural problems - all roots of Eq. (16.23) are real positive numbers (for a proof see reference 1). These are known as the natural frequencies of the system. If only the M matrix is symmetric positive definite while K is symmetric positive semidefinite the roots are real and positive or zero. J; For any case where repeated frequencies occur we merely enforce the orthogonality by construction.
Free response- eigenvalues for second-order problems and dynamic vibration 573 on inverting M with ~. = 09 2, but symmetry is in general lost. If, however, we write in triangular form (i.e., the Cholesky factors) M = LL T
and
M -1 = L - T L -1
in which L is a lower triangular matrix (i.e., has all zero coefficients above the diagonal), Eq. (16.22) may now be written as Kfi = 092LLTfi Calling LTfi = x
(16.28)
and multiplying by L -1 we have finally Hx
=
092X
(16.29)
in which H = L - 1KL-T
(16.30)
which is of the standard form of Eq. (16.26), as H is now symmetric. Having determined 092 (all, or only a few, of the selected smallest values corresponding to fundamental periods) the modes of x are found, and hence by use of Eq. (16.28) the modes of ft. If the matrix M is diagonal - as it will be if the masses have been ' l u m p e d ' - the procedure of deriving the standard eigenvalue problem is simplified and here appears the first advantage of the diagonalization, which we have discussed in Sec. 16.2.4.
16.4.3 Free vibration with the singular K matrix In static problems we have always introduced a suitable number of support conditions to allow the stiffness matrix K to be inverted, or what is equivalent to solve the static equations uniquely. If such 'support' conditions are in fact not specified, as may well be the case with a rocket travelling in space, the arbitrary fixing of a minimum number of support conditions allows a static solution to be obtained without affecting the stresses. In dynamic situations such a fixing is not permissible and frequently one is faced with the problem of a free oscillation for which K is singular and therefore does not possess unique triangular factors or an inverse. To preserve the applicability of methods which require an inverse (e.g., methods based on inverse power iteration 27) a simple artifice is possible. Equation (16.22) is modified to [(K + otM) - (092 + ot)M]fi = 0
(16.31)
in which ot is an arbitrary constant of the same order as the typical o92 sought. The new matrix (K + c~M) is no longer singular and can be factored (or inverted) for use in the standard eigensolution procedure to find (o92 + or) and hence o92. This simple but effective avoidance of an otherwise serious difficulty was first suggested by Cox 29 and Jennings. 3~Alternative methods of dealing with the above problem are given in references 31 and 32.
574 Semi-discretization and analytical solution
16.4.4 Reduction of the eigenvalue system Independent of which technique is used to determine the eigenpairs of the system (16.22), the effort for n x n matrices is at least one order greater than that involved in an equivalent static situation. Further, while the number of eigenvalues of the real system is infinite, in practice, we are generally interested only in a relatively small number of the lower frequencies and it is possible to simplify the computation by reducing the size of the problem. To achieve a reduced problem we assume that the unknown fi can be expressed in terms of m (<< n) vectors tl, t2 . . . . . tm and corresponding participating factors X i . We now write = tlXl + t2x2 + " "
+
tmXm m_
Tx
(16.32)
Inserting Eq. (16.32) into Eq. (16.22) and premultiplying by T T we have a reduced problem with only m eigenpairs: (w*)2M*x = K*x
(16.33)
where M* = TTMT
K* = TTKT
and co* are now eigenvalues of the reduced system, which for the appropriate choice of the ti vectors can be good approximations to the eigenvalues of the original system. If by good fortune the trial vectors were to be chosen as eigenvectors of the original matrix the system would become diagonal and all eigenvalues (i.e., in this case w* = w) could be determined by a trivial calculation. This indeed is what some iterative eigenproblem strategies attempt (e.g., subspace or Lanczos methods 27, 33). It is also of course possible by physical insight to find vectors t that correspond closely to the principal modes of the movement (e.g., see reference 34).
16.4.5 Some examples There are a variety of problems for which practical solutions exist, so only a few simple examples will be shown.
Example 16.3: Vibration of a simply supported beam. Figure 16.2 shows the first three vibration modes of a simply supported beam with length 40 and rectangular cross-section of width 1 and depth 2 units. The elastic properties are E = 30 000, v = 0, and p = 0.1 units. The beam is modelled using 9-noded quadrilateral elements of lagrangian type with the central node at the left end restrained in the x and y direction and the central node at the right end restrained only in the y direction. The problem is also solved using a mesh with 1000 2-noded beam elements which include effects of transverse shearing deformation. In Table 16.1 we present the values for the first three frequencies obtained from the finite element analysis and compare to the value obtained from an exact solution for the EulerBernoulli beam without shear deformation. It is evident that transverse sheafing strains affect the frequencies computed for this problem and, thus, illustrate the importance of using a correct theory for calculations.
Free response- eigenvalues for second-order problems and dynamicvibration 575 Table 16.1 Frequencies for a simply supported beam Solution form
COl
0)2
093
9-noded element 2-noded element Beam theory
3.7785 3.7787 3.8050
59.2236 59.2338 60.8807
290.0804 290.1774 308.2080
i I
,
~
(a)
(b)
(c)
Fig. 16.2 Simplysupported beam. (a) col - 3.7785. (b) co2 - 59.2236. (c) co3 - 290.0804.
Example 16.4: Vibration of an earth dam. Figure 16.3 shows the vibration of a twodimensional earth dam resting on a rigid foundation. The earth dam is modelled by linear triangular elements and includes the effects of different material layers.
Example 16.5: Electromagnetic fields. The basic dynamic equation (16.8) can be derived for a variety of non-structural problems. The eigenvalue problem once again occurs with 'stiffness' and 'mass' matrices now having alternate physical meanings. A particular form of the more general equations discussed earlier is the wen-known Helmholz wave equation which, in two-dimensional form, is 02r
02~
1 02r
_ 0
(16.34)
OX 2 -~- ~ y 2 -+- ~2 Ot 2 --
If the boundary conditions do not force a response, an eigenvalue problem results which has significance in several fields of physical science. The first application is to electromagneticfields. Figure 16.4 shows a modal shape of a field for a waveguide problem. Simple linear triangular elements are used here. More complex three-dimensional oscillations are also discussed in reference 9.
Example 16.6: Waves in shallow water. A similar equation also describes to a reasonable approximation the behaviour of shallow water waves in a body of water:
576
Semi-discretization and analytical solution
Fig. 16.3 (a) Mesh showing layers considered. (b) Earth dam, Mode 1. (c) Earth dam, Mode 2. (d) Earth dam, Mode 3.
8 (hS~r) 8 (0~) 1 82~p Ox ~x + ~y h-~y + g Ot2 = 0
(16.35)
in which h is the average water depth, ~p the surface elevation above average and g the gravity acceleration. 21 Thus natural frequencies of bodies of water contained in harbours of varying depths may easily be found. 1~ Figure 16.5 shows the modal shape for a particular harbour.
If in Eq. (16.20) M = 0, we have a form typical of the transient heat conduction equation [see Eq. (16.1)]. For free response we seek a solution of the homogeneous equation Cu + Kfi = 0
(16.36)
Free response - eigenvalues for first-order problems and heat conduction, etc.
I
Fig. 16.4 A 'lunar' waveguide; 9 mode of vibration for electromagnetic field. Outer diameter -- d, 0 0 ' = 0.13d, r = 0.29d, S = 0.055d, ~ = 22 ~
0 I
Scale of feet 1000 2000 I
I
I
I
Harbour oscillation
Line of equal horizontal movement Limit of water surface
9, x Fig. 16.5 Oscillations of a natural harbour: contours of velocity amplitudes! ~
577
578
Semi-discretization and analytical solution
Once again an exponential form can be used: fi = fitexp(-~.t) Substituting we have (-~.C + K)fi = 0
(16.37)
which again gives an eigenvalue problem identical to that of Eq. (16.22). As C and K are usually positive definite, ~. will be positive and real. The solution therefore represents simply an exponential decay term and is not really steady state. Combination of such terms, however, can be useful in the solution of initial value transient problems but is of little value p e r se.
We shall now consider the full equation (16.20) for free response conditions. Writing M~ + Cd + Kfi = 0
(16.38)
fi = fi exp(ott)
(16.39)
(o~2M + c~C --t--K)fi = 0
(16.40)
and substituting we have the characteristic equation
where c~and fi will in general be found to be complex. The real part of the solution represents a decaying vibration. The eigenvalue problem involved in Eq. (16.39) is more difficult than that arising in the previous sections. In solutions to date the problem is usually solved by splitting Eq. (16.38) into two first-order equations. This is accomplished by defining 2,
and writing the split form as
[:
- M0] {_~}+ [C
K] { v } =
{00}
(16..41)
(00}
(16.42)
Now substituting fi = fi exp(c~t)
v = r162 exp(ott)
gives the general linear eigenproblem
0]+
K]) {~}=
This form has been studied by Chen et al. 35-37 Similar to the first-order problem, no steady-state solution exists and once more the concept of eigenvalues of the above kind is generally of importance only in modal analysis, as we shall see later.
Forced periodic response 579
If the forcing term in Eq. (16.20) is periodic or, more generally, if we can express it as f = f exp(c~t)
(16.43)
c~2
(16.44)
where c~ is complex, i.e. =
I~ 1 + i
then a general solution can once more be written as = fi exp(ott)
(16.45)
Substituting the above in Eq. (16.20) gives (a2M + a C + K) ~ - I ~ = - f
(16.46)
which is no longer an eigenvalue problem but can be solved formally by inverting the matrix I~ as = -l~-lf (16.47) The solution is thus precisely of the same form as that used for static problems but now, however, has to be determined in terms of complex quantities. With periodic input the solution after an initial transient is not sensitive to the initial conditions and the above solution represents the finally established response. It is valid for problems of dynamic structural and fluid-structure responses as well as for problems typical of heat conduction in which we simply put M = 0.
16.8.1 General In the previous sections we have been concerned with steady-state general solutions which took no account of the initial conditions of the system or of the non-periodic form of the forcing terms. The response taking these features into account is essential if we consider, for instance, the earthquake behaviour of structures or the transient behaviour of the heat conduction problem. The solution of such general cases requires either a full-time discretization, which we shall discuss in detail in the next chapter, or the use of special analytical procedures.
16.8.2 Frequency response procedures In Sec. 16.7 we have shown how the response of the system to any forcing terms of the general periodic type or in particular to a periodic forcing function f = f exp(i ogt)
(16.48)
580 Semi-discretization and analytical solution can be obtained by solving a simple equation system. As a completely arbitrary forcing function can be represented approximately by a Fourier series or in the limit, exactly, as a Fourier integral, the response to such an input can be obtained by a synthesis of a curve representing the response of any quantity of interest, e.g., the displacement at a particular point, etc., to all frequencies ranging from zero to infinity. In fact only a limited number of such forcing frequencies has to be considered and a result can be synthesized efficiently by fast Fourier transform techniques. 38 We shall not discuss the mathematical details for such procedures which can be found in standard texts on structural dynamics. 12' 16 The technique of frequency response is readily adapted to problems where the damping matrix C is of an arbitrary specified form. This is not the case with the more widely used modal decomposition procedures which are to be described in the next section. By way of illustration we show in Fig. 16.6 the frequency response of an artificial harbour [see Eq. (16.35)] to an input of waves with different frequencies and damping due to the radiation of reflected waves which imposes a very particular form on the damping matrix. Details of this problem are given elsewhere. 21' 39,40 Similar techniques are frequently used in the analysis for the foundation response of structures where radiation of energy occurs. 41
16.8.3 Modal decomposition analysis This procedure is probably the most important and widely used in practice. Further, it provides an insight into the behaviour of the whole system, which is of value where strictly numerical processes are used. We shall therefore describe it in detail in the context of the general problem of Eq. (16.20), i.e., M6+
(16.49)
Cu + Kfi + f = 0
where f is an arbitrary function of time. We have seen that the general solution for the free response is of the form (16.50)
~-- ~ Ui exp(ctit) i=1
where O~i are the (complex) eigenvalues and Ui a r e the (complex) eigenvectors (Sec. 16.6). For forced response we shall assume that the problem is linear such that the solution can be written as a linear combination of the modes n
tl --" Z
tli Yi (t) = [fil,
U2. . . .
] y(t)
(16.51)
i=1 where the scalar modal participation factor yi is now a function of time. This shows in a clear manner the proportions of each mode occurring. Such a decomposition of an arbitrary vector presents no restriction as all the modes are linearly independent vectors (with those for repeated frequencies being constructed to be linearly independent as mentioned in Sec. 16.4). If expression (16.51) is substituted into Eq. (16.49) and the result is premultiplied by the complex conjugate transposed, 0/T (i = 1 . . . . . n), then the result is simply a set of scalar, independent, equations mi ];i + ci Yi + ki Yi + 3~ = 0 (16.52)
Transient response by analytical procedures Outer domain stretches to infinity Series solution
/ 'Infinite' elements !
B'I
Idealization using triangular linear element
I ,
I,
~
Idealization using quadratic isoparametric elements
~ for ka = 5
I (shortest wavelength) I
(a) Geometric details and FEM idealization. Wave forcing frequency co= k ~ = ka, h = depth of water 7 6 l 5
Chen and Mei 39
II
o
Zienkiewicz and Bettess
1 0
0
1
2 3 4 Frequency ka
5
6
(b) Amplitude magnification response of mean depth in harbour for various frequencies Fig. 16.6 Frequencyresponse of an artificial harbour to an input of periodic wave.
581
582
Semi-discretization and analytical solution where mi = f i T M f i i , ci = f i T c f i i ,
ki = f T K f i
and 3'} = fiTf
as for true eigenvectors fii "
fTMfj - fTcfj = fTKfj - 0
when i # j (this result was proved in Sec. 16.4 for real eigenpairs but is valid generally for complex pairs, as could be verified by the reader). Each scalar equation of (16.52) can be solved by elementary procedures independently and the total vector of response obtained by superposition following Eq. (16.51). In the general case, as we have shown in Sec. 16.6, the eigenpairs are complex and their determination is not simple. 31 The more usual procedure is to use real eigenpairs corresponding to the solution of Eq. (16.21): K f = co2Mfi (16.53) Decoupled equations with real variables y exist only if fTcfij = O;
i # j
which generally does not occur as the eigenvectors now guarantee only orthogonality with M and K and not of the damping matrix. However, if the damping matrix C is of the form of Eq. (16.13), i.e., a linear combination of M and K, such orthogonality will obviously occur. Unless the damping is of a definite form which requires special treatment, an assumption of orthogonality is made and Eq. (16.52) is assumed valid in terms of such eigenvectors. From Eq. (16.53) we have K f i ---- co/2Mfiz (16.54a) and on premultiplying by fiT we obtain
ki -- co2mi
(16.54b)
Ci = 2co~i
(16.54c)
Writing the modal damping in the form
(where ~i represents the ratio of damping to its critical value) and assuming that the modes have been normalized so that mi -- 1 [see Eq. (16.24)], Eq. (16.52) can be rewritten in standard second order form:
Yi + 2Will Yi -+-c02yi q- f i --0
(16.54d)
A general solution is then obtained as
Yi -- exp(--~icoit) Yio -+-co~icoiYi~
sin&it -4- YioCOS&it]
(16.55) + --l
coi
in which
&i = coi V/1
for
e x p ( - - e s i c o i [ t - r ] ) s i n & i ( t - r)3~(r) dr
- ~/2 and Yio, Yio are initial conditions computed from Yio ----fiTMfi(0)
and
Yi0 =
fiTMfi(0)
(16,56)
Symmetry and repeatability 583 The solution of Eq. (16.55) can be carried out by assuming the forcing function is given by linear interpolation between discrete time points tk and then evaluating the resulting integrals exactly. Alternatively, a numerical solution can be carried out and the response obtained. In practice, often a single calculation is carried out for each mode to determine the maximum responses and a suitable addition of these results is used. Such processes are described in standard texts and are used as procedures to calculate the bounds on behaviour of structures subjected to seismic loading. 12' 16, 28
16.8.4 Damping and participation of modes The type of calculation implied in modal decomposition apparently necessitates the determination of all modes and eigenvalues, a task of considerable magnitude. In fact only a limited number of modes usually need to be taken into consideration as often the response to higher frequency is critically damped and insignificant. To show that this is true consider the form of the damping matrices. In Sec. 16.2 [Eq. (16.13)] we have indicated that the damping matrix is often assumed as C = aM +/3K
(16.57)
Indeed a form of this type is necessary for the use of modal decomposition, although other generalizations are possible. 42, 43 From the definition of ~i, the ratio to critical damping ratio in Eq. (16.54c), we see that this can now be written as !
1
(16.58)
Thus if the coefficient/3 is of greater importance, as is the case with most structural damping, ~i grows with COi and at high frequency an overdamped condition will arise. 12 This is indeed fortunate as, in general, an infinite number of high frequencies exist which are not modelled by any finite element discretization. We shall see in the next chapter that in the step-by-step recurrence computation the high frequencies often control the problem, and this effect needs to be 'filtered out' for realistic results.
In concluding this chapter it is worth remarking that in dynamic calculation we have once again encountered all the general principles of assembly, etc., that are applicable to static problems. However, some aspects of symmetry and repeatability which were used previously (see Sec. 6.2.4) need amending. It is obviously possible for symmetric structures to vibrate in an unsymmetrical manner, for instance, and similarly a repeatable structure contains modes which are themselves non-repeatable. However, even here considerable simplification can still be made; details of this are discussed by Williams, 44 Thomas 45 and Evensen. 46
584 Semi-discretizationand analytical solution
16.1 Specialize the problem given in Sec. 16.2 for the case where p = 0. Construct C e and K e for a typical 3-node triangular element and a 6-node hierarchical triangle in which coordinates are given by 3 X=
ZLaxa a=l
16.2 An axial bar under transient loading is governed by
0 (EAOU) 02u Ox \ Ox + q - pA igt2 and boundary conditions
u(x, t) = ~(t), x
on
On
I"1 or E A m
Ox
-- P(t), x on 1-'2
where ~,(t) and P (t) are specified displacement and force, respectively. (a) Construct a weak form for the problem. Is there a variational theorem for the problem? (b) Consider an isoparametric element interpolation
xe(~) ue(~, t)
-
-
--
NI(~) 2~ + N2(~) 2~ NI(~) fi~(t) + N2(~) fi~(t)
with N 1 - - ( 1 - ~)/2 and N2 = (1 + ~)/2 and show the stiffness and mass element arrays are given by =
A[1 111
h
-
1
and
=
~
,
respectively (where h - 2~ - 2~). (c) Let
U(X, t) = Z
i
fii(x) exp iwit
and determine the discrete eigenproblem resulting from the weak form developed in (a). Is there a variational theorem for the problem? (d) Consider a two-element problem shown in Fig. 16.7 and solve the eigenproblem developed in (c). Let u(0, t) = P(L, t) = 0 and use material properties
E=A=p=L=landq=O. 0
2
L
Fig. 16.7 Two-elementbar for Problem 16.2.
3 0
J
Problems (e) Write a MATLAB program to solve the discrete eigenproblem. Check your program using the two-element solution; then solve the problem using 4, 8, and 16 elements. Plot the first two eigenvalues vs the number of elements. (f) Obtain an exact value for the first two eigenvalues and plot the error for each vs the number of elements on a log-log plot. What is the rate of convergence? (g) Replace the element mass matrix by a lumped form given by
~/Ie = -~1 ph [~ 01] and repeat parts (d) and (e) 16.3 Compute the lumped mass matrix by row sum (Method I) for a cubic order serendipity element which is a square with side length a. Note it is only necessary to compute one vertex and one mid-side value. 16.4 Compute the lumped mass matrix by diagonal scaling (Method II) for a cubic order serendipity element which is a square with side length a. Note it is only necessary to compute one vertex and one mid-side value. 16.5 Compute the lumped mass matrix by row sum (Method I) for a cubic order lagrangian element which is a square with side length a. Note it is only necessary to compute one vertex, one mid-side value and the interior node value. 16.6 For a three-dimensional cube with side lengths a in each direction compute the lumped mass matrix by row sum (Method I) for a quadratic order lagrangian element. Note it is only necessary to compute one vertex, one mid-side value and the interior node value. 16.7 Show that the degeneration of a 9-node lagrangian quadrilateral in ~, r/coordinates into a 6-node triangle as described in Example 5.2 of Sec. 5.8.1 yields the same mass matrix as that derived for the triangle using L1, L2, L3 area coordinates. Note it is only necessary to compute one vertex and one mid-side value for the triangle. 16.8 The bar shown in Fig. 16.8 is divided into four elements and has the fight end attached to a damper. A weak form for the problem may be written as M~ + C ~ + K f i + f = 0
where M and K are obtained using the element arrays given in Problem 16.1. For
a'-" In2 U3 U4 U5]T"
(a) Construct M and K for the problem. (b) Construct C for the problem. (c) Use MATLAB to compute the eigensolution for the problem. Plot real and imaginary parts for the problem. On a separate plot show as vectors the real and imaginary parts of the complex frequencies. Are they proportional? 16.9 Use FEAPpv to verify the results given for Example 16.3 in Table 16.1 and Fig. 16.2. Use a consistent mass matrix for the computation. Repeat the analysis using a lumped mass. If the mesh is refined several times, do you expect the results to converge? Why? 0
0 L
0
Fig. 16.8 Four-elementbar with end damperfor Problem 16.8.
585
586 Semi-discretizationand analytical solution 16.10 Use FEAPpv to compute the first three eigenpairs for the rectangular beam problem described in Example 2.3 of Sec.2.9. Use the same properties for E and v and let p = 0.001. Using the first mode and P = 1 for 0 < t < 2 obtain the solution for the first 3 seconds using At = 0.1. Plot the results for the vertical displacement at the tip where loading is applied. Repeat the solution using three modes. 16.11 The curved beam problem described in Example 2.4 of Sec. 2.9 is to be solved for the case where the boundary condition at y = 0 is specified by the shear stress of the exact solution. For the case of linear variation of displacements on the boundary edge write a MATLAB program to compute the consistent nodal loads for a unit force P and four equally spaced segments. For the mesh shown in Fig. 2.1 l(b) use FEAPpv to compute the first eigenpair (the one where oJi is smallest) for the problem assuming the same properties for E and v and take p = 1. Using the first mode and u (x, 0) = sin e t for 0 < t < n: obtain the solution for the first 5 seconds using At = 0.1. Plot the results for the vertical displacement at the tip where loading is applied. Repeat the solution using three modes. 16.12 Program development project: Extend the program system started in Problem 2.17 to compute a lumped and a consistent mass matrix for 3-node triangular and 4-node quadrilateral elements. Use the generalized eigenproblem [V, D] = EIG(K, M) from MATLAB to compute the eigenvectors (V = ui) and eigenvalues (D = wi). Use your program to determine the eigenvalues and eigenvectors for the curved beam analysed in Problem 16.11. Results may be checked using FEAPpv. 16.13 Program development project: Extend the program system developed for Problem 16.12 to perform mode superposition as described in Sec. 16.8.3. You may omit the modal damping factors ~i for simplicity. For the rectangular beam considered as Example 2.3 (using triangular elements as shown in Fig. 2.8), assume the end shear is applied suddenly at time zero and held constant for 2 seconds at which time it is suddenly removed. Perform a modal solution in which only the lowest eigenvalue mode is used. For a time increment of At = 0.001 determine and plot the first 5 seconds of response for the vertical displacement at the tip centre-line. Repeat the solution using the lowest three eigenvalue modes. Compare your solutions with that in which all modes are included (which is the exact solution for the semi-discrete equations). Comment on differences obtained.
1. S. Crandall. Eng. Analysis. McGraw-Hill, New York, 1956. 2. H.S. Carslaw and J.C. Jaeger. Conduction of Heat in Solids. Clarendon Press, Oxford, 2nd edition, 1959. 3. W. Visser. A finite element method for the determination of non-stationary temperature distribution and thermal deformation. In Proc. 1st Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-66-80, Wright Patterson Air Force Base, Ohio, Oct. 1966.
References 587 4. O.C. Zienkiewicz and Y.K. Cheung. The Finite Element Method in Structural Mechanics. McGraw-Hill, London, 1967. 5. E.L. Wilson and R.E. Nickell. Application of finite element method to heat conduction analysis. Nuclear Eng. Design, 4:1-11, 1966. 6. O.C. Zienkiewicz and C.J. Parekh. Transient field problems - two and three dimensional analysis by isoparametric finite elements. Int. J. Numer. Meth. Eng., 2:61-71, 1970. 7. K. Terzhagi and R.B. Peck. Soil Mechanics in Engineering. John Wiley & Sons, New York, 1948. 8. D.K. Todd. Ground Water Hydrology. John Wiley & Sons, New York, 1959. 9. P.L. Arlett, A.K. Bahrani, and O.C. Zienkiewicz. Application of finite elements to the solution of Helmholz's equation. Proc. lEE, 115:1762-1764, 1968. 10. C. Taylor, B.S. Patil, and O.C. Zienkiewicz. Harbour oscillation: a numerical treatment for undamped natural modes. Proc. Inst. Civ. Eng., 43:141-156, 1969. 11. O.C. Zienkiewicz and R.E. Newton. Coupled vibration of a structure submerged in a compressible fluid. In Proc. Int. Symp. on Finite Element Techniques, pages 359-371, Stuttgart, 1969. 12. A.K. Chopra. Dynamics of Structures. Prentice-Hall, Upper Saddle River, N.J., 1995. 13. J.S. Archer. Consistent mass matrix for distributed systems. Proc. Am. Soc. Civ. Eng., 89(ST4):161, 1963. 14. EA. Leckie and G.M. Lindberg. The effect of lumped parameters on beam frequencies. Aero. Q., 14:234, 1963. 15. O.C. Zienkiewicz and Y.K. Cheung. The finite element method for analysis of elastic isotropic and orthotropic slabs. Proc. Inst. Civ. Eng., 28:471-488, 1964. 16. R.W. Clough and J. Penzien. Dynamics of Structures. McGraw-Hill, New York, 2nd edition, 1993. 17. S.W. Key and Z.E. Beisinger. The transient dynamic analysis of thin shells in the finite element method. In Proc. 1st Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-66-80, pages 667-710, Wright Patterson Air Force Base, Ohio, Oct. 1966. 18. E. Hinton, T. Rock, and O.C. Zienkiewicz. A note on mass lumping and related processes in the finite element method. Earth. Eng. Struct. Dyn., 4:245-249, 1976. 19. P. Tong, T.H.H. Pian, and L.L. Bociovelli. Mode shapes and frequencies by the finite element method using consistent and lumped matrices. Comp. Struct., 1:623-638, 1971. 20. I. Fried and D.S. Malkus. Finite element mass matrix lumping by numerical integration with the convergence rate loss. Int. J. Solids Struct., 11:461-465, 1975. 21. O.C. Zienkiewicz, R.L. Taylor, and P. Nithiarasu. The Finite Element Method for Fluid Dynamics. Butterworth-Heinemann, Oxford, 6th edition, 2005. 22. J.H. Wilkinson. The Algebraic Eigenvalue Problem. Clarendon Press, Oxford, 1965. 23. I. Fried. Gradient methods for finite element eigen problems. J. AIAA, 7:739-741, 1969. 24. J.H. Wilkinson and C. Reinsch. Linear Algebra. Handbook for Automatic Computation, volume II. Springer-Verlag, Berlin, 1971. 25. K.K. Gupta. Solution of eigenvalue problems by Sturm sequence method. Int. J. Numer. Meth. Eng., 4:379-404, 1972. 26. A. Jennings. Mass condensation and similarity iterations for vibration problems. Int. J. Numer. Meth. Eng., 6:543-552, 1973. 27. B.N. Parlett. The Symmetric Eigenvalue Problem. Prentice Hall, Englewood Cliffs, N.J., 1980. 28. K.-J. Bathe. Finite Element Procedures. Prentice Hall, Englewood Cliffs, N.J., 1996. 29. H.L. Cox. Vibration of missiles. Aircraft Eng., 33:2-7 and 48-55, 1961. 30. A. Jennings. Natural vibration of a free structure. Aircraft Eng., 34:8, 1962. 31. W.C. Hurty and M.E Rubinstein. Dynamics of Structures. Prentice Hall, Englewood Cliffs, N.J., 1974. 32. A. Craig and M.C.C. Bampton. On the iterative solution of semi definite eigenvalue problems. Aero. J., 75:287-290, 1971.
588
Semi-discretization and analytical solution 33. J. Demmel. Applied Numerical Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1997. 34. O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method, volume 2. McGraw-Hill, London, 4th edition, 1991. 35. H.-C. Chen and R.L. Taylor. Using Lanczos vectors and Ritz vectors for computing dynamic responses. Eng. Comp., 6:151-157, 1989. 36. A. Ibrahimbegovic, H.-C. Chen, E.L. Wilson, and R.L. Taylor. Ritz method for dynamic analysis of large discrete linear systems with non-proportional damping. Earth. Eng. Struct. Dyn., 19:877-889, 1990. 37. H.-C. Chen and R.L. Taylor. Properties and solutions of the eigensystem of non-proportionally damped linear dynamic systems. Technical Report UCB/SEMM-86/10, University of California, Berkeley, Nov. 1986. 38. E.O. Brigham. The Fast Fourier Transform. Prentice Hall, Englewood Cliffs, N.J., 1974. 39. H.S. Chen and C.C. Mei. Hybrid-element method for water waves. In Proc. Modelling Techniques Conf. (Modelling 1975), volume 1, pages 63-81, San Francisco, 1975. 40. O.C. Zienkiewicz and P. Bettess. Infinite elements in the study of fluid-structure interaction problems. In 2nd Int. Symp. on Computing Methods in Applied Science and Engineering, Versailles, France, Dec. 1975. 41. J. Penzien. Frequency domain analysis including radiation damping and water load coupling. In O.C. Zienkiewicz, R.W. Lewis, and K.G. Stagg, editors, Numerical Methods in Offshore Engineering. John Wiley & Sons, 1978. 42. E.L. Wilson and J. Penzien. Evaluation of orthogonal damping matrices. Int. J. Numer. Meth. Eng., 4:5-10, 1972. 43. H.T. Thomson, T. Collins, and P. Caravani. A numerical study of damping. Earth. Eng. Struct. Dyn., 3:97-103, 1974. 44. F.W. Williams. Natural frequencies of repetitive structures. Q. J. Mech. Appl. Math., 24:285-310, 1971. 45. D.L. Thomas. Standing waves in rotationally periodic structures. J. Sound Vibr., 37:288-290, 1974. 46. D.A. Evensen. Vibration analysis of multi-symmetric structures. J. AIAA, 14:446-453, 1976.
In the last chapter we have shown how semi-discretization of dynamic or transient field problems leads, in linear cases, to sets of ordinary differential equations of the formt Mti + Cfi + K u + f = 0
where
du dt
= fi, etc.
(17.1)
subject to initial conditions u(0) = u0
and
fi(0) = rio
or for transient field problems (e.g., heat conduction) to Cfl + Ku + f = 0
(17.2)
subject to the initial condition u(0) = u0
In many practical situations non-linearities exist, typically altering the above equations by making M = M(u) C = C(u) Ku = P(u) (17.3) The analytical solutions previously discussed, while providing much insight into the behaviour patterns (and indispensable in establishing such properties as natural system frequencies), are in general not economical for the solution of transient problems in linear cases and not applicable when non-linearity exists. In this chapter we shall therefore revert to discretization processes applicable directly to the time domain. For such time discretization the finite element method, including in its definition the finite difference approximation, is of course widely applicable and provides the greatest possibilities, though much of the classical literature on the subject uses only the latter. 1-6 We shall demonstrate here how the finite element method provides a useful generalization unifying many existing algorithms and providing a variety of new ones. As the time domain is infinite we shall inevitably curtail it to a finite time increment At and relate the initial conditions at tn (and sometimes before) to those at time t n + l = tn + A t , obtaining so-called r e c u r r e n c e relations. In all of this chapter, the starting point will be that of the semi-discrete equations (17.1) or (17.2), though, of course, the full space-time t To simplifynotation, we omitthe 'tilde' on approximationsin time of the independentvariable, thus, fl(tn) "~ Un.
590 The time dimension- discrete approximation in time
domain discretization could be considered simultaneously. This, however, usually offers no advantage, for, with the regularity of the time domain, irregular space-time elements are not required. Indeed, if product-type shape functions are chosen, the process will be identical to that obtained by using first semi-discretization in space followed by time discretization. An exception here is provided in convection dominated problems where simultaneous discretization may be desirable (as is discussed in reference 7). The first concepts of space-time elements were introduced in 1969-708-11 and the development of processes involving semi-discretization is presented in references 12-21. Full space-time elements are described for convection-type equations in references 22, 23 and 24 and for elastodynamics in references 25, 26 and 27. The presentation of this chapter will be divided into three parts. In the first we shall derive a set of single-step recurrence relations for the linear first and second order problems of Eqs (17.2) and (17.1). Such schemes have a very general applicability and are preferable to multistep schemes described in the second part as the time step can be easily and adaptively varied. In the third part we briefly describe a discontinuous Galerkin scheme and show its application in some simple problems. When discussing stability problems we shall often revert to the concept of modally uncoupled equations introduced in the previous chapter. Here we recall that the equation systems (17.1) and (17.2) can be written as a set of scalar equations: m i Yi + r Yi 3I- ki Yi + f i -" 0
(17.4)
Ci Yi 21- ki Yi + J~ - " 0
(17.5)
or
in the respective eigenvalue participation factors Yi. We shall find that the stability requirements here are dependent on the eigenvalues associated with such equations, ~oi. It turns out, however, fortunately, that it is never necessary to obtain the system eigenvalues or eigenvectors due to a powerful theorem first stated for finite element problems by Irons and Treharne. 28 The theorem states simply that the system eigenvalues can be bounded by the eigenvalues of individual elements me . Thus m.in(~oj) 2 > min(af) 2 and j
e
max(~oj) 2 < max(oge) 2 j
e
(17.6)
The stability limits can thus (as will be shown later) be related to Eq. (17.4) or (17.5) written for a single element.
Single-step algorithms .
.
.
.
We shall now consider Eq. (17.2) which may represent a semi-discrete approximation to a particular physical problem or simply be itself a discrete system. The objective is to
Simple time-step algorithms for the first-order equation 591 obtain an approximation for Un-t-1 given the value of u n and the forcing vector f acting in the interval of time At. It is clear that in the first interval u,, is the initial condition u0, thus we have an initial value problem. In subsequent time intervals Un will always be a known quantity determined from the previous step. In each interval, in the manner used in all finite element approximations, we assume that u varies as a polynomial and take here the lowest (linear) expansion as shown in Fig. 17.1 writing
T
u ~ fi(t) = Un + -~-~ (Un+l - Un)
(17.7)
with r = t - t n . This can be translated to the standard finite element expansion giving fi(t) = ~
1
Niui - -
At-- Un "~-
-~
Un+l
(17.8)
in which the unknown parameter is Un+l. The equation by which this unknown parameter is provided will be a weighted residual approximation to Eq. (17.2). Accordingly, we write the variational problem f0 TM W(T)T [Cr + Ku + f] dr - 0
(17.9)
in which w(r) is an arbitrary weighting function. We write the approximate form
W(r)-- W(z')~Un+l
(17.10)
in which t~Un+l is an arbitrary parameter. With this approximation the weighted residual equation to be solved is given by
L
(17.11)
A, W(r)[Cfi + Kfi + f] dr = 0
Introducing 0 as a weighting parameter given by 0 --
1 f o t W(r)r dr
(17.12)
At f?t W(r) dr JU
Un+
I
(to be determined) I;
t.+1 At Fig. 17.1 Approximation to u in the time domain.
r
592 The time dimension- discrete approximation in time
we can immediately write 1
~C(Un+l
At
-- Un) + K[un + 0(Un+l -- Un)] + f
=
(17.13)
0
where f represents an average value of f given by :
f0At Wf dr
(17.14)
let Wdr or
= fn + O(fn+l -- fn)
(17.15)
if a linear variation of f is assumed within the time increment. Equation (17.13) is in fact almost identical to a finite difference approximation to the governing equation (17.2) at time tn + 0 At, and in this example little advantage is gained by introducing the finite element approximation. However, the averaging of the forcing term is important, as shown in Fig. 17.2, where a constant W (that is 0 = 1/2) is used and a finite difference approximation presents difficulties. Figure 17.3 shows how different weight functions can yield alternate values of the parameter 0. The solution of Eq. (17.13) yields Unq- 1 -- ( C -Jr- 0 A t K ) - I [ ( c - (1 -
O)AtK)un - Atf]
(17.16)
and it is evident that in general at each step of the computation a full equation system needs to be solved though of course a single inversion (or factorization using a Gauss-type solution process) is sufficient for linear problems in which the time increment At is held constant. Methods requiting such an inversion are called implicit. However, when 0 = 0 and the matrix C is approximated by its lumped equivalent CL the solution is called explicit and is exceedingly cheap for each time interval. We shall show later that explicit algorithms are conditionally stable (requiring the At to be less than some critical value Atcrit) whereas implicit methods may be made unconditionally stable for some choices of the parameters. 1 ~At
-~At
n
I
2 1 At
n+l
n
At
n+l
W /= 1.5
/=~
fn +1/2 indeterminate (a)
fn +1/3 indeterminate (b)
Fig. 17.2 'Averaging' of the forcing term in the finite-element-time approach.
Simple time-step algorithms for the first-order equation
n
n+l "~ ~
L r
0=0
t/At
At
J -I
W~
(a)
0=~1
(b)
0=1 (c)
0=~1
(d)
0=~2
/
J
(e)
1
0= 5
(f)
Fig. 17.3 Shape functions and weight functions for two-point recurrence formulae.
17.2.2 Taylor series collocation
..................
A frequently used alternative to the algorithm presented above is obtained by approximating separately Un+l and lin+l by truncated Taylor series. We can write, assuming that Un and ~!n a r e known: Un+l --Un "+"Atiln + flAt(lln+l --(.In) (17.17a) and use collocation to satisfy the governing equation at t.+l [or alternatively using the weight function shown in Fig. 17.3(c)] which gives C/ln+l + Ku.+I + fn+l = 0
(17.17b)
In the above fl is a parameter, 0
-[- 3AtK)-I[K(un + (1 -
3)Atu.) + f n + l ]
(17.18)
593
594 The time dimension- discrete approximation in time
where Un+l is now computed by substitution of Eq. (17.18) into Eq. (17.17a). We remark that: (a) the scheme is not self-startingt and requires the satisfaction of Eq. (17.2) at t = 0 [whereas the finite element in the time scheme given by (17.13) is self-starting]; (b) the computation requires, with identification of the parameters/~ = 0, an identical equation-solving problem to that in the finite element scheme of Eq. (17.16) and, finally, as we shall see later, stability considerations are identical. The procedure is introduced here as it has some advantages in non-linear computations.
17.2.3 Other single-step procedures As an alternative to the weighted residual process other possibilities of deriving finite element approximations exist, as discussed in Chapter 3. For instance, variational principles in time could be established and used for the purpose. This was indeed done in the early approaches to finite element approximation using Hamilton's or Gurtin's variational principle. 29-32 However, as expected, the final algorithms turn out to be identical. A variant on the above procedures is the use of a least square approximation for minimization of the equation residual. 13' 14 This is obtained by insertion of the approximation (17.7) into (17.2). The reader can verify that the recurrence relation becomes
I --~tcTc + 2
1 (KTC+CTK)+
-- [ ~tcTc-+- ~1 ( K T C 1 + ~-~
cT~0Atf dr
+
31AtKT K ] Un+l
CTK) - 1 gAtKTK] Un
~t K T f0 Atf r
(17.19)
dr = 0
requiring a more complex equation solution and always remaining 'implicit'. For this reason the algorithm is largely of purely theoretical interest, though as expected its accuracy is nearly exact for results shown in Fig. 17.4, in which a single degree of freedom equation (17.2) is used with K--+ K = 1 C--> C = 1 f---~ f = 0 with initial condition u0 = 1. Here, the various algorithms previously discussed are compared. Now we see from this example that the 0 = 1/2 algorithm performs almost as well as the least squares one. It is popular for this reason and is known as the CrankNicolson scheme after its originators. 33
17.2.4 Consistencyand approximation error For the convergence of any finite element approximation, it is necessary and sufficient that it be consistent and stable. We have discussed these two conditions in Chapter 9 t By 'self-starting' we mean an algorithmis directly applicable without solvingany subsidiaryequations. Other definitions are also in use.
Simple time-step algorithms for the first-order equation 1.2
. ~ . F_xact ,.., O=O
i e-" ID
E
o t'tl Q" "0
,~
0.8
i
i
i
i -o-0=2/3 ...... .-v- e= 1
1.21
:i t--~" o=oEXact I1 ......... i .......... :........... :........... : 0=112 '
o.8
..
0.6
~ 0.6
0.4
.~Q" 0.4
0.2
"~ 0.2
i
i
!
...-
.
o ...................'ii.. .i:::.,.i,.,.,.,...............................,............ -0.2
0
0.5
i 1
1.5
2
t time (a)
. 2.5
3
---0.2
0
0.5
,
1
1.5
t time (b)
2
2.5
Fig. 17.4 Comparison of various time-stepping schemes on a first-order initial value problem. and introduced appropriate requirements for boundary value problems. In the temporal approximation similar conditions apply though the stability problem is more delicate. Clearly the function u itself and its derivatives occurring in the equation have to be approximated with a truncation error of O(At~), where ot > 1 is needed for consistency to be satisfied. For the first-order equation (17.2) it is thus necessary to use an approximating polynomial of order p > 1 which is capable of approximating/I to at least O (At). The truncation error in the local approximation of u with such an approximation is O(At 2) and all the algorithms we have presented here using the p = 1 approximation of Eq. (17.7) will have at least that local accuracy, 34 as at a given time, t = n A t , the total error can be magnified n times and the final accuracy at a given time for schemes discussed here is of order O (At) in general. We shall see later that the arguments used here lead to p > 2 for the second-order equation (17.1) and that an increase of accuracy can generally be achieved by use of higher order approximating polynomials. It would of course be possible to apply such a polynomial increase to the approximating function (17.7) by adding higher order degrees of freedom. For instance, we could write in place of the original approximation a quadratic expansion: u ~ flU) = Un -~- - ~
(Un+I -- Un) "q- ~-~
1 --
Iln+ 1
(17.20)
where fi is a hierarchic internal variable. Obviously now both U,+l and fin+l are unknowns and will have to be solved for simultaneously. This is accomplished by using the weighting function W = W ( r ) ~ U n + l + Vl/r(~')~lln+l (17.21) where W (r) and ff'(r) are two independent weighting functions. This will obviously result in an increased size of the problem. It is of interest to consider the first of these obtained by using the weighting W alone in the manner of Eq. (17.11). It is easy to verify that we now have to add to Eq. (17.13) a term i n v o l v i n g Iln+ 1 which is
['
~-~(1 - 20)C + (0 - 0 ) K
]
Iln+ 1
(17.22)
595
596
The time dimension- discrete approximation in time where
0=
1 fot W'c2dz" f~t W dr
At 2
It is clear that the choice of 0 = 0 = 1/2 eliminates the quadratic term and regains the previous scheme, thus showing that the values so obtained have a local truncation error of O (At3). This explains why the Crank-Nicolson scheme possesses higher accuracy. In general the addition of higher order internal variables makes recurrence schemes too expensive and we shall later show how an increase of accuracy can be more economically achieved. In a later section of this chapter we shall refer to some currently popular schemes in which often sets of us have to be solved for simultaneously. In such schemes a discontinuity is assumed at the initial condition and additional parameters (~) can be introduced to keep the same linear conditions we assumed previously. In this case an additional equation appears as a weighted satisfaction of continuity in time. The procedure is therefore known as the discontinuous Galerkin process and was introduced initially by Reed and Hill 35 to solve neutron transport problems. An analysis of the method was given by Lesaint and Raviart. 36 It has subsequently been applied to solve problems in fluid mechanics and heat transfer 23' 37, 38 and to problems in structural dynamics. 25-27 As we have already stated, the introduction of additional variables is expensive, so somewhat limited use of the concept has so far been made. However, one interesting application is in error estimation and adaptive time stepping. 39
17.2.5 Stability If we consider any of the recurrence algorithms so far derived, we note that for the homogeneous form (i.e., with f = 0) all can be written in the form Un+l --
Aun
(17.23)
where A is known as the amplification matrix. The form of this matrix for the first algorithm derived is, for instance, evident from Eq. (17.16) as A = (C + 0 A t K ) -1 ( C - (1 - 0 ) A t K ) (17.24) Any errors present in the solution will of course be subject to amplification by precisely the same factor. A general solution of any recurrence scheme can be written as 40 U n + 1 - - ]-s
(17.25)
and by insertion into Eq. (17.23) we observe that/z is given by eigenvalues of the matrix as (A - / z l ) u, = 0
(17.26)
Clearly if any eigenvalue/z is such that I~1 > 1
(17.27)
Simple time-step algorithms for the first-order equation all initially small errors will increase without limit and the solution will be unstable. In the case of complex eigenvalues the above is modified to the requirement that the modulus of /z satisfies Eq. (17.27). As the determination of system eigenvalues is a large undertaking it is useful to consider only a scalar equation of the form (17.5) (representing, say, one-element performance). The bounding theorems of Irons and Treharne 28 will show why we do so and the results will provide general stability bounds if maximums are used. Thus for the case of the algorithm discussed in Eq. (17.26) we have for the scalar form of (17.24)
O)Atk c + 0 Atk
c - (1 -
A =
O)coAt 1 + OcoAt
1 - (1 -
=
=/z
(17.28)
where co = k/c and/z is evaluated from Eq. (17.26) simply as/z = A to allow non-trivial Un. (This is equivalent to making the determinant of A - / x l zero in the more general case.) In Fig. 17.5 we show how/z (or A) varies with coat for various 0 values. We observe immediately that: (a) for 0 > 1/2
(17.29)
I/~1 ~ ] and such algorithms are (b) for 0 < 1/2 we require
unconditionally stable; o9At < -
2
(17.30)
1--20
for stability. Such algorithms are therefore only the explicit form with 0 = 0 is typical.
conditionally stable.
Here of course
The critical value of At below which the scheme is stable with 0 < 1/2 needs the determination of the maximum value of/z from a typical element. For instance, in the case of the thermal conduction problem in which we have the coefficients Caa and kaa defined by expressions
Caa=.J~~'N2df2 1.0 0.8 t-- ~ ' ~ , .
e= 1
0.6 t-
t-
/
0.4
._o 0.2 o 0 ca. -0.2 E
< -0.4 -0.6 -0.8 -1"00
o=o
1
~
and
2
kaa=/VNafcVNadg2
8=0.878 /
'Exact' e -~
o=~1
3
Fig. 17.5 The amplification A for various versions of the 0 algorithm.
4
r
coAt
(17.31)
597
598
The time dimension-
in time
discrete approximation
we can presuppose uniaxial behaviour with a single degree of freedom and write for a linear element N
.._
c=
h
/0
, ~h
?N 2 d x =
Now
k O.)
This gives At < -
----
--
c
2 1-20
/o' k
k=
~
dx=-
h
3~ --"
?h 2
~-h2 - = Atcrit 3k
(17.32)
which of course means that the smallest element size, hmin, dictates overall stability. We note from the above that: (a) in first-order problems the critical time step is proportional to h 2 and thus decreases rapidly with element size making explicit computations difficult; (b) if mass lumping is assumed where Clump > Ccons the critical time step is larger than that obtained using a consistent mass. In Fig. 17.6 we show the performance of the scheme described in Sec. 17.2.1 for various values of 0 and At in the example we have already illustrated in Fig. 17.4, but now using larger values of At. We note now that the conditionally stable scheme with 0 = 0 and a stability limit of At = 2 shows oscillations as this limit is approached (At = 1.5) and diverges when exceeded. Stability computations which were presented for the algorithm of Sec. 17.2.1 can of course be repeated for the other algorithms which we have discussed.
/ 1.0 ~
~
Exact
1.0 ~ ; ,
0.8
0.4
0
t
i-'..~--
2 t 3 I t t---~
:u
i
4" 4 5
o
, 0
~1
...... i
,,
11
,
2 b..3-T-'4
I
~-
5
6
t ~ ~
~Oscillatory result
Unstable result
'
0.2 11 I I
l
~176 ',
-
0.2 0
~
~8 ~kO. 6
I ~,',
0.6
Exact
---~.---
x
~
~ i
] i
',]
----
o-----
e=OEuler e = ~1 Crank-Nicolson
----o----
t9 = g Galerkin
. . . . . 43 . . . . . .
e = 1 Backward difference
!
t i
X
Fig. 17.6 Performance of some 0 algorithms in the problem of Fig. 17.4 and larger time steps. Note oscillation and instability.
Simple time-step algorithms for the first-order equation 599 If identical procedures are used, for instance on the algorithm of Sec. 17.2.2, we shall find that the stability conditions, based on the determinant of the amplification matrix (A - / z l ) , are identical with the previous one providing we set 0 = /3. Algorithms that give such identical determinants will be called similar in the following presentations. In general, it is possible for different amplification matrices A to have identical determinants of (A - / z l ) and hence identical stability conditions, but differ otherwise. If in addition the amplification matrices are the same, the schemes are known as identical. In the two cases described here such an identity can be shown to exist despite different derivations.
17.2.6 Some further remarks. Initial conditions and examples The question of choosing an optimal value of 0 is not always obvious from theoretical accuracy considerations. In particular with 0 = 1/2 oscillations are sometimes present, 14 as we observe in Fig. 17.6 (At = 2.5), and for this reason some prefer to use 0 = 2/3, which is considerably 'smoother' (and which incidentally corresponds to a standard Galerkin approximation in time41). In Table 17.1 we show the results for a one-dimensional finite element problem where a bar at uniform initial temperature is subject to zero temperatures applied suddenly at the ends. Here 10 linear elements are used in the space dimension with L = 1. The oscillation errors occurring with 0 = 1/2 are much reduced for 0 = 2/3. The time step used here is much longer than that corresponding to the lowest eigenvalue period, but the main cause of the oscillation is in the abrupt discontinuity of the temperature change. For similar reasons Liniger 42 derives 0 which minimizes the error in the whole time domain and gives 0 = 0.878 for the simple one-dimensional case. We observe in Fig. 17.5 how well the amplification factor fits the exact solution with these values. Again this value will smooth out many oscillations. However, most oscillations are introduced by simply using a physically unrealistic initial condition. In part at least, the oscillations which for instance occur with 0 -- 1/2 and At = 2.5 (see Fig. 17.6) in the previous example are due to a sudden jump in the forcing term introduced at the start of the computation. This jump is evident if we consider this simple problem posed in the context of the whole time domain. We can take the problem as implying
f(t) = -1
for
t < 0
Table 17.1 Percentageerror for finite elements in time: 0 - 2/3 (Galerkin) and 0 = 1/2 (Crank-Nicolson) scheme; At -- 0.01 t/0
x --0.1 2/3 1/2
x =0.2 2/3 1/2
x =0.3 2/3 1/2
x =0.4 2/3 1/2
x =0.5 2/3 1/2
0.01 0.02 0.03 0.05 0.10 0.15 0.20 0.30
10.8 0.5 1.3 0.5 0.1 0.3 0.6 1.4
1.6 2.1 0.5 0.4 0.1 0.3 0.6 1.4
0.5 0.1 0.8 0.5 0.1 0.3 0.6 1.4
0.6 0.5 0.5 0.4 0.1 0.3 0.6 1.4
0.5 0.7 0.5 0.5 0.1 0.3 0.6 1.4
28.2 3.5 9.9 4.5 1.4 2.2 2.6 3.5
3.2 9.5 0.7 0.2 2.0 2.1 2.6 3.5
0.7 0.0 3.1 2.3 1.4 2.2 2.6 3.5
0.1 0.7 0.2 0.8 1.9 2.1 2.6 3.5
0.2 0.4 0.6 1.0 1.6 2.2 2.6 3.5
600
The time dimension- discrete approximation in time
giving the solution u - 1 with a sudden change at t -- 0, resulting in f(t)-O
for t>_O
As shown in Fig. 17.7 this represents a discontinuity of the loading function at t - 0. Although load discontinuities are permitted by the algorithm they lead to a sudden discontinuity of ti and hence induce undesirable oscillations. If in place of this discontinuity we assume that f varies linearly in the first time step At ( - A t / 2 _< t _< At/2) then smooth results are obtained with a much improved physical representation of the true solution, even for such a long time step as At = 2.5, as shown in Fig. 17.7. Similar use of smoothing is illustrated in a multidegree of freedom system (the representation of heat conduction in a wall) which is solved using two-dimensional finite elements 43 (Fig. 17.8). Here the problem corresponds to an instantaneous application of prescribed temperature (T -- 1) at the wall sides with zero initial conditions. Now again troublesome oscillations are almost eliminated for 0 = 1/2 and improved results are obtained for other values of 0 (2/3, 0.878) by assuming the step change to be replaced by a continuous one. Such smoothing is always advisable and a continuous representation of the forcing term is important.
We shall introduce in this section two general single-step algorithms applicable to Eq. (17.1): Mti + C~i + K u + f = 0 These algorithms will of course be applicable to the first-order problem of Eq. (17.2) simply by putting M = 0. An arbitrary degree polynomial p for approximating the unknown function u will be used and we must note immediately that for the second-order equations p > 2 is required for consistency as second-order derivatives have to be approximated. The first algorithm SSpj (single step with approximation of degree p for equations of order j - 1, 2) will be derived by use of the weighted residual process and we shall find that the algorithm of Sec. 17.2.1 is but a special case. The second algorithm GNpj (generalized Newmark 44 with degree p and order j) will follow the procedures using a truncated Taylor series approximation in a manner similar to that described in Sec. 17.2.2. In what follows we shall assume that at the start of the interval, i.e., at t -- tn, we know p-1
the values of the unknown function u and its derivatives, that is un,/In, lin up to Un and p-1
our objective will be to determine Un+l, fi~+l, ii~+l up to Un+l, where p is the order of the expansion used in the interval. This is indeed a rather strong presumption as for first-order problems we have already stated that only a single initial condition, u(0), is given and for second-order problems two conditions, u(0) and fi(0), are available (i.e., the initial displacement and velocity of the
General single-step algorithms for first- and second-order equations
Ul '
0
B
u0
0.---
rt
!
"
I
I
~
Smoothed (A) I
~.J
I
At
I
I
Standard (B)
I
I
At Fig. 17.7 Importance of 'smoothing' the force term in elimination of oscillations in the solution. A t = 2.5.
system). We can, however, argue that if the system starts from rest we could take u(0) to
p-1
u (0) as equal to zero and, providing that suitably continuous f o r c i n g o f the system occurs, the solution will remain smooth in the higher derivatives. Alternatively, we can differentiate the differential equation to obtain the necessary starting values.
17.3.2 The weighted residual finite element form SSpj In the SSpj algorithm the expansion of the unknown vector u is taken as a polynomial of p-1
degree p.19, 20 With the k n o w n values of u,,, ft,, ii,, up to u, at the beginning of the time step At, we write, as in Sec. 17.2.1, r = t - tn
At
--
tn+l
--
tn
(17.33)
and using a polynomial expansion of degree p, u ~ fi = Un + r/~n + l r 2 t i n + " "
+
1
(p-
1)!
.gp-1 p-1 1 rpc~ p Un +~..
(17.34)
601
602
The time dimension-
discrete approximation
in time
Interpolated
Discontinuous "e-
1.0
1.0
I
v t_
L_
E
E
I-.
I
0
10
I
Time
I
20
I 10
0
30
(a) Crank-Nicholson (e = 0.5) 1.0
1.0 "e-
m
v
Time
20
I 30
I 20
I 30
I 20
I 30
S
e e i
E
E
!-
F-
0
0
i
i
10
Time (b) Galerkin (0 = 0.667)
-e-
I 20
I 30
0
I 10
0
1.0
1.0
f
v
ss ~
n
E
E
F-
t-
0
0
Time
10
Time (c) Liniger (0 = 0.878)
10 quadratic elements
I 20
o=lll! i
,4
I 30
--x
0
I 10
0
!llilll 4
a~ =
0
Centre-line
Time
Exact . . . . Numerical At = 2
Fig. 17.8 Transient heating of a bar; comparison of discontinuous and interpolated (smoothed)initial conditions for single-step schemes.
where the only unknown is the vector a p, p dPll a np ~ u = dtP
(17.35)
which represents some average value of the pth derivative occurring in the interval At. The approximation to u for the case of p = 2 is shown in Fig. 17.9. We recall that in order to obtain a consistent approximation to all the derivatives that occur in the differential equations (17.1) and (17.2), p > 2 is necessary for the full
General single-step algorithms for first- and second-order equations 603 A
I
(Zn 2 At2/2
Un + ilnAt= On
Un
r
At
tn
tn+ l
Fig. 17.9 A second-order time approximation.
dynamic equation and p > 1 is necessary for the first-order equation. Indeed the lowest approximation, that is p = 1, is the basis of the algorithm derived in the previous section. The recurrence algorithm will now be obtained by inserting u, fi and ii [obtained by differentiating Eq. (17.34)] into Eq. (17.1) and satisfying the weighted residual equation with a single weighting function W(r). This gives (p _1 p) f0At W(r) IM ( On + r'l~l'n -'1-""" + 2)-------------~rP-2ot
1
+ C (fin -I- z'fin + " " + ~
( p - 1)!
+ K
z'P--10tp
)
( Iln + Z'l]ln-l-""" + -p1-! r
(17.36) =
0
as the basic equation for determining anP. Without explicitly specifying the weighting function used we can, as in Sec. 17.2.1, generalize its effects by writing
Ok = f~ Wr~ dr A t k fOt W dr = fot Wf dr f~t W d r
k=0, 1,...,p (17.37)
where we note 00 is always unity. Equation (17.36) can now be written more compactly as
Ac~p + M~n+l -t- C~n+l "-I-KUn+l + P -- 0
(17.38)
604
The time dimension - discrete approximation in time
where A
--
Op-2Atp-2
(p - 2)!
M+
Op-1 A t p-1 C -]- O p A t p K
( p - 1)!
p!
x-~P-10qAt q q
Un+l -- ~ q=0 9
Un+l
q!
Un
(17.39)
_ V TMp-10q-1 A t q-1 q
/---' q=l
( q - 1)*
Un
p-10q-zAt q-2 q fi~+l=~ ( q - 2 ) v u,~ q=2 " As an+l, ~n+l and Un+l can be computed directly from the initial values we can solve Eq. (17.38) to obtain O~p = - A
(17.40)
-1 [MUn+l + Can+l .qt..KUn+l + f]
It is important to observe that an+l, ~n+l and Un+l here represent some mean predicted values of Un+l, Lln+l and fin+l in the interval and satisfy the governing Eq. (17.1) in a weighted sense if C~nPis chosen as zero. The procedure is now complete as knowledge of the vector C~nPpermits the evaluation of p-1 Un+l to Un+l from the expansion originally used in Eq. (17.34) by putting r = At. This gives Un+l ---Un -~- Atfl~ + - . . + I]ln+1 --Ill n --[- Atii~ + . . .
+
At p
p i9 a p = fi"+l + A t p-1
D
.
At p
p.
i ap A t p-1
+ 1)--------5'~n (p= fin+l ( p -
D
1)! C~n
(17.41)
p-1 p-1 Un+l -- Un + Atc~nP
In the above fi, fi, etc., are again quantities that can be written down a priori (before solving for c~). These represent predicted values at the end of the interval with C~nP= 0. To summarize, the general algorithm necessitates the choice of values for 01 to Op and requires (a) computation of an+l, ~n-k-1 and fin-k-1 using the definitions of Eqs (17.39); (b) computation of a ff by solution of Eq. (17.40); p-1 (c) computation of Un+l to Un+l by Eqs (17.41). After completion of stage (c) a new time step can be started. In first-order problems the computation of ~ can obviously be omitted 9 If matrices C and M are diagonal the solution of Eq. (17.40) is trivial providing we choose Op = 0 (17.42)
General single-step algorithms for first- and second-order equations 605 With this choice the algorithms are explicit but, as we shall find later, only sometimes
conditionally stable. When Op > O, implicit algorithms of various kinds will be available and some of these will be found to be unconditionally stable. Indeed, it is such algorithms that are of great practical use. Important special cases of the general algorithm are the SS 11 and SS22 forms given below. Example 17.1: The S S l l algorithm. If we consider the first-order equation (that is j = 1) it is evident that only the value of Un is necessarily specified as the initial value for any computation. For this reason the choice of a linear expansion in the time interval is natural (p = 1) and the SS 11 algorithm is for that reason most widely used. Now the approximation of Eq. (17.34) is simply U --- U n -~- "CO:
(Og1
--
a = r)
(17.43)
and the approximation to the average satisfaction of Eq. (17.2) is simply Cc~ + K(fin+l + 0 A t a ) + f = 0 with
Uln+ 1 --" Un.
(17.44)
Solution of Eq. (17.44) determines c~ as -- - - ( C + OAtK) -1 (f
+ Nun)
(17.45)
and finally U n + l " - Un +
At c~
(17.46)
The reader will verify that this process is identical to that developed in Eqs (17.7)-(17.13) and hence will not be further discussed except perhaps for noting the more elegant computation form above. Example 17.2: The SS22 algorithm. With Eq. (17.1) we considered a second-order system (j = 2) in which the necessary initial conditions require the specification of two quantities, Un and fin- The simplest and most natural choice here is to specify the minimum value of p, that is p -- 2, as this does not require computation of additional derivatives at the start. This algorithm, SS22, is thus basic for dynamic equations and we present it here in full. From Eq. (17.34) the approximation is a quadratic 1 2r u = Un + rfin + ~r
~
(c~2 = c~ ~ ii)
(17.47)
The approximate form of the 'weighted' dynamic equation is now Ms +
C(Un+l
-~- 0 1 A t c ~ )
+ K(Un+l + 102At200 + f -- 0
(17.48)
with predicted 'mean' values fin+ l ~- On -+- O1Atfln O n + l - - tin
(17.49)
606 The time dimension- discrete approximation in time
After evaluation of c~ from Eq. (17.40), the values of Un+1 are found by Eqs (17.41) which become simply 1
Un+l "-- Un -~- Atfin + ~ At2ot
(17.50)
I~ln+l = fin -+- Atc~
This completes the algorithm which is of much practical value in the solution of dynamics problems. In many respects the previous example resembles the Newmark algorithm 44 which we shall discuss in the next section and which is widely used in practice with the forms an+ 1 "- U n -+- Atfi, + ( 89- / ~ ) a t 2 0 n I]ln+ 1 --" II n -~" (1
-
+/~At20n+l
+ y Atiin+l
y)Atfin
Indeed, the stability properties of the SS22 algorithm turn out to be identical with the Newmark algorithm if 01 --" y;
02 "-- 2/3;
1 01 _> 02 _> ~
(17.51)
for unconditional stability. In the above y and/3 are conventionally used Newmark parameters. For 02 = 0 the algorithm is 'explicit' (assuming both M and C to be diagonal) and can be made conditionally stable if 01 >__ 1/2. The algorithm is clearly applicable to first-order equations described as SS21 and we shall find that the stability conditions are identical. In this case, however, it is necessary to identify an initial condition for fi0 and I~lo -- - C -1 ( K u o -+- fo)
is one possibility.
17.3.3 TruncatedTaylor series collocation algorithm GNpj In the derivation using collocation, we consider the satisfaction of the governing equation (17.1) only at the end points of the interval At [which results from the weighting function shown in Fig. 17.3(c)] and write
Miin+1 +
Cl~ln+ 1 -[-
Ku~+1 "~- fn+ 1 :
0
(17.52)
with appropriate approximations for the values of Un+l, Iln+l and an+l- It will be shown that again as in Sec. 17.2.2 a non-self-starting process is obtained, which in most cases, however, gives an algorithm similar to the SSpj one we have derived. The classical Newmark method 44 will be recognized as a particular case together with its derivation process in a form presented generally in existing texts. 45 Because of this similarity we shall term the new algorithm generalized Newmark (GNpj).
General single-step algorithms for first- and second-order equations 607 If we consider a truncated Taylor series expansion similar to Eq. (17.17a) for the function u and its derivatives, we can write Un+l ---Un -+- Atfin -F""-Il[ln+l = Il n "~- A t fin + " "
AtPp AtP(p P) p V un -']-'/~PT.I n+l -- Un
Atp-I P AtP-1 (~1 P ) d- ( p - 1)! Un +/3p-1 ( p _ 1)----~ n + l - Un
(17.53)
p-1 p-1 (~ p) Un+l "-- Un -~- At ~_1n --/~1 At n+l -- Un
In Eqs ( 17.53) we have effectively allowed for a polynomial of degree p (i.e., by including terms up to A t p) plus a Taylor series remainder term in each of the expansions for the function and its derivatives with a parameter flj, j -- 1, 2 . . . . . p, which can be chosen to give good approximation properties to the algorithm. Insertion of the first three expressions of (17.53) into Eq. (17.52) gives a single equation p p-1 from which Un+l can be found. When this is determined, Unq-1 to un+l can be evaluated using Eqs (17.53). Satisfying Eq. (17.52) is almost a 'collocation' which could be obtained by inserting the expressions (17.53) into a weighted residual form (17.36) with W = 8 (t,,+l) (the Dirac delta function). However, the expansion does not correspond to a unique function u. In detail we can write the first three expansions of Eqs (17.53) as At p p Un+l = fin+l +/3p ~ U.+I 9 A t p-1 p Iln+l : Iln+l -~-tip--1 ( p _ 1)! Un+l
(17.54)
A t p-2 p Un+l : Iln+l -~-/~p-2 (p _ 2)! Un+l ..
where At p p Iln+ 1 "-" Un --~ Atiln + . . . + (1 - ~ p ) 7 . 1 U n .t-
AtP-1 P ( p - 1)! A t p-2 p Iln+l = fin -]- A t ll'n -+-"" + (1 - J ~ p - 2 ) ~ Un I~n+l
illn + Atiin +
.
+ (1
-
.
.
t/p-l) .
.
U
n
(17.55)
..
(p - 2)!
P Inserting the above into Eq. (17.52) and solving for an+ 1 gives P 1-- - A -1 [Man+l-+-Cl~n+l-+-Ki)n+l an+
where A "- " - 2 A t p - E M (p - 2)!
+
+ fn+l]
(17.56)
Atp-1
C-q- flpAt p K (p - 1)! p!
We note immediately that the above expression is formally identical to that of the SSpj algorithm, Eq. (17.40), if we make the substitutions
608 The time dimension- discrete approximation in time ~p = Op
/~p-1 --- Op-1
/~p-2 = Op-2
(17.57)
However, Un+l, U~+l, etc., in the generalized Newmark, GNpj, are not identical to Un+l, fin+l, etc., in the SSpj algorithms. In the SSpj algorithm these represent predicted mean values in the interval At while in the GNpj algorithms they represent predicted values at tn+l.
The computation procedure for the GN algorithms is very similar to that for the SS algorithms, starting now with known values of Un to Un. As before we have the given initial conditions and we can often arrange to use the differential equation and its derivatives to generate higher derivatives for u at t - 0. However, the GN algorithm requires use of u0 in the computation of the next time step. An important member of this family is the GN22 algorithm.
The Newmark algorithm (GN22)
We have already mentioned the classical Newmark algorithm as it is one of the most popular for dynamic analysis. It is indeed a special case of the general algorithm of the preceding section in which a quadratic (p = 2) expansion is used, this being the minimum required for second-order problems. We describe here the details in view of its widespread use. The expansion of Eq. (17.53) for p = 2 gives 1 1 Un+l -- Un "[- Atiln + 5(1 -/~2)At2iin -[- ~/~2At2iin+l -- Un+l "-{-1/~2At2Un+l
(17.58) lin+l --/In + (1 - ill)Atlln +/31Atfln+l -- I]n+l +/31Atlln+l and this together with the dynamic equation (17.52),
Miin+l + Clin+l --1-KUn+l + fn+l -- 0
(17.59)
allows the three unknowns Un+l, Iln+ 1 and fin+l to be determined. We now proceed as we have already indicated and solve first for iin+l by substituting (17.58) into (17.59). This yields as the first step
iin+l ~-- - A - 1 {fn+l -'[- C~n+l --[-Kiln+l}
(17.60)
1 A = M +/31AtC + ~/32AteK
(17.61)
where After this step the values of Un+l and lln+ 1 can be found using Eqs (17.58). As in the general case,/32 = 0 produces an explicit algorithm whose solution is very simple if M and C are assumed diagonal. It is of interest to remark that the accuracy can be slightly improved and yet the advantages of the explicit form preserved for SS/GN algorithms by a simple iterative process within each time increment. In this, for the GN algorithm, we predict Un+ l i , Un+l" i and Un+l..i using expressions (17.54) with (~ln+l) i - (~ln+l) i-1 and setting for i - 1
(Un+l) 0 __ 0
Stability of general algorithms 609 This is followed by rewriting the governing equation (17.52) as
M [/1 Un+l @ (p -- 2)!
IOn+llil
+ CIlin-+l "+- Kuin-+l -+- f,,+l = 0
(17.62)
( )
p i and solving for On+1 This predictor-corrector iteration has been successfully used for various algorithms, though of course the stability conditions remain unaltered from those of a simple explicit scheme. 46 For implicit schemes we note that in the general case, Eqs (17.58) have scalar coefficients while Eq. (17.59) has matrix coefficients. Thus, for the implicit case some users prefer a slightly more complicated procedure than indicated above in which the first unknown determined is Un+l. This may be achieved by expressing Eqs (17.58) in terms of the U~+l to obtain 2 Un+l -- Un+l -+" /32At 2 Un+l (17.63) 2/31 Iln+l = Un+l ~ Unq-1 /~2At where .~ Un+l --
2 2 ~1n /32At2 an -- /~2A--~
an+l----/~2A------~Un-+-
1 -/32 .. TUn
1----~-2jan-+-
1-~-~2
(17.64) Atiin
These are now substituted into Eq. (17.59) to give the result
Un+l : _A-1 (fn+l + C~n+l -'~ M~n+l)
(17.65)
where now
2/31 2 M+ C+K A = 32At-------~ 32At which again on using Eqs (17.63) and (17.64) gives fi and ii. The inversion is here identical to within a scalar multiplier and, thus, precludes use of the explicit form where/32 is zero.
Consistency of the general algorithms of SS and GN type is self-evident and assured by their formulation. In a similar manner to that used in Sec. 17.2.5 we can conclude from this that the local truncation error is O (At p+I) as the expansion contains all terms up to r p for SS and At p for GN algorithms. However, the total truncation error after n steps is only O (At P) for the first-order equation system and O (At p-l) for the second-order one. Details of accuracy discussions and reasons for this can be found in reference 6. The question of stability is paramount and in this section we shall discuss it in detail for the SS type of algorithms. The establishment of similar conditions for the GN algorithms follows precisely the same pattern and is left as an exercise to the reader. It is, however, important to remark here that it can be Shown that
610 The time dimension- discrete approximation in time
(a) the SS and GN algorithms are generally similar in performance; (b) their stability conditions are identical when Op - ~p. The proof of the last statement requires some elaborate algebra and is given in reference 6. The determination of stability requirements follows precisely the pattern outlined in Sec. 17.2.5. However, for practical reasons we shall (a) avoid writing explicitly the amplification matrix A; (b) immediately consider the scalar equation system implying modal decomposition and no forcing, i.e., mii + cit + ku = 0
(17.66)
Equations (17.37), (17.40) and (17.41) written in scalar terms define the recurrence algorithms. For the homogeneous case the general solution can be written down as Un+l ~ ~Un k n + l = /ZUn
(17.67) p-1 p-1 Un+ 1 --" /Z Un
and substitution of the above into the equations governing the recurrence can be written quite generally as SX,, = 0
(17.68)
where Un
Atitn X n ~-
(17.69)
p-1 A t p-1 Un A t p Pn
The matrix S is given below in a compact form which can be verified by the reader:
S __
bo
bl
1-/~
1
0
0 0
b2
"'"
bp-1
bp
2-~ "'"
(Pll)!
/]!
1 -/z
1
...
(p - 2)!
0 0
0 0
-....
1 1-/x
1
1
1
(p-
1)!
1
25 1
(17.70)
Stability of general algorithms where bo = OoAt2k bl -- OoAtc q-O1At2k
Oq-2 Oq-lAt OqAt 2 bq -- (q _ 2)!m + (q _ 1)-----------~c + q! k,
q - 2, 3 . . . . . p
and00 = 1. For non-trivial solutions for the vector Xn to exist it is necessary for detS = 0
(17.71)
This provides a characteristic polynomial of order p for/z which yields the eigenvalues of the amplification matrix 9 For stability it is sufficient and necessary that the moduli of all eigenvalues [see Eq. (17.27)] satisfy I~1 _< 1 (17.72) We remark that in the case of repeated roots the equality sign does not apply 9 The reader will have noticed that the direct derivation of the determinant of S is much simpler than writing down matrix A and finding the eigenvalues. The results are, of course, identical 9 The calculation of stability limits, even with the scalar (modal) equation system, is nontrivial. For this reason in what follows we shall only do it for p = 2 and p = 3. However, two general procedures will be introduced here. The first of these is the so-called z transformation. In this we use a change of variables in the polynomial putting l+z /z = (17.73) 1-z where z as well as/x are in general complex numbers. It is easy to show that the requirement of Eq. (17.72) is identical to that demanding the realpart ofz to be negative (see Fig. 17.10). The second procedure introduced is the well-known Routh-Hurwitz condition 47-49 which states that for a polynomial with
CoZ n -+" C lZ n - 1 "Jl- " ' " "~- C n - l Z nt- Cn = 0
[cc3c]
co > 0
(17.74)
the real part of all roots will be negative if, for Cl > 0, det cl
c3 > 0 ;
CO
det
C2
c2
ca
Cl
C3
>0
(17.75)
and generally
det
-Cl
C3
C5
C7
Co
C2
C4
C6
"'" "'"
0
Cl
c3
c5
...
0
0
C2
C4
"'"
.0
0
0
"'"
Cn_ 2
>0
(17.76)
Cn
With these tools in hand we can discuss in detail the stability of specific algorithms 9
611
612
The time dimension-discrete approximation in time
The recurrence relations for the algorithm given in Eqs (17.48) and (17.50) can be written after inserting (17.77) Un+l = ]J~Un; ~ln+l = ]J~{gn a n d f = 0 as
mot + c (Un -~- O1 mtot) + k (U n -~- 01Atitn + 102 At2ot) = 0 --~Un -~- Un + Ati~n + ~1 At2ot = 0 --lZi~n + ion + Atot = 0
Changing the variable according to Eq. (17.73) results in the characteristic polynomial coz 2 + ClZ + c2 = 0
(17.78)
with co = 4m + (401 - 2)Atc + 2(02 -- 0 1 ) A t 2 k (17.79)
cl = 2 A t c + (201 -- 1)AtZk c2 = A t Z k
The Routh-Hurwitz requirement for stability is simply that C0>0
el _>0
det IClco c201 > 0
or simply co > 0
Cl >__0
c2 > 0
(17.80)
These inequalities give for unconditional stability the condition that 02 >__01 >__
1
(17.81)
Stability of general algorithms This condition is also generally valid when m = 0, i.e., for the SS21 algorithm (the firstorder equation) though now 02 = 01 must be excluded. It is possible to satisfy the inequalities (17.80) only at some values of At yielding conditional stability. For the explicit process 02 = 0 with SS22/SS21 algorithms the inequalities (17.80) demand that 2m + (201 - 1 ) A t c - 01At2k > 0 2c + (201 - 1)Atk >_ 0
(17.82)
The second one is satisfied whenever
01 ~ ~1
(17.83)
and for 01 --- 1/2 the first supplies the requirement that At 2 < 4m - k
(17.84)
The last condition does not permit an explicit scheme for SS21, i.e., when m = 0. Here, however, if we take 01 > 1/2 we have from the first equation of Eq. (17.82) 201-1
At <
01
c
k
(17.85)
It is of interest for problems of structural dynamics to consider the nature of the bounds in an elastic situation. Here we can use the same process as that described in Sec. 17.2.5 for first-order problems of heat conduction. Looking at a single element with a single degree of freedom and consistent mass yields in place of condition (17.84) At <
2
h
-~/~C
where h is the element size and
= Atcrit
c:j
is the speed of elastic wave propagation. For lumped mass matrices the factor becomes ~/2. Once again the ratio of the smallest element size over wave speed governs the stability but it is interesting to note that in problems of dynamics the critical time step is proportional to h while, as shown in Eq. (17.32), for first-order problems it is proportional to h 2. Clearly for decreasing mesh size explicit schemes in dynamics are more efficient than in thermal analysis and are exceedingly popular in certain classes of problems.
17.4.2 Stability of various higher order schemes and equivalence with some known alternatives Identical stability considerations as those described in previous sections can be applied to SS32/SS31 and higher order approximations. We omit here the algebra and simply quote some results. 6
613
614
The time dimension- discrete approximation in time SS32/31. Here for zero damping (c = 0) in SS32 we require for unconditional stability that 1 01 > ~
02 > 01 -at- 1 02 > 1 _ ~ -- ~ 30102 - - 302 + 01 >__ 03
03 > 3 - g
(17.86)
For first-order problems (m = 0), i.e., SS31, the first requirements are as in dynamics but the last one becomes 3 0 1 0 2 - - 302 -k-01 >__ 03 - -
[601(01 -- 1) + 112 9 ( 2 0 1 - 1)
(17.87)
With 03 = 0, i.e., an explicit scheme when c = 0, At 2 < -
12(201 - 1) m -602- 1 k
(17.88)
and when m = 0, 02--01
C
At < - 602- 1 k
(17.89)
SS42/41. For this (and indeed higher orders) unconditional stability in dynamics problems m 7~ 0 does not exist. This is a consequence of a theorem by Dahlquist. 5~ The SS41 scheme can have unconditional stability but the general expressions for this are cumbersome. We quote one example that is unconditionally stable: 01 - - ~5
02 - - ~
03 - - T 25
04-
24
This set of values corresponds to a backward difference four-step algorithm of Gear. 51 It is of general interest to remark that certain members of the SS (or GN) families of algorithms are similar in performance and identical in the stability (and hence recurrence) properties to others published in the large literature on the subject. Each algorithm claims particular advantages and properties. In Tables 17.2-17.4 we show some members of this family.a1, 50-56 Clearly many more algorithms that are applicable are present in the general f o r / I l d 1 ae.
We remark here that identity of stability and recurrence always occurs with multistep algorithms, which we shall discuss briefly in the next section. Table 17.2
SS21 equivalents
Algorithms
Theta values
Zlama141
O1 -- 5, 02 = 2
Gear 51 Liniger 52 Liniger 52
01 ~-~ 3, 02 = 2 01 = 1.0848, 02 = 1 01 = 1.2184, 02 = 1.292
Multistep recurrence algorithms Table 17.3
SS31 equivalents
Algorithms
Theta values
Gear 51 Liniger 52 Liniger 52
01 = 2 , 0 2 = ~ , 0 3 = 6 01 : - 1.84, 02 = 3.07, 03 = 4.5 01 = 0.8, 02 = 1.03, 03 = 1.29
Table 17.4
SS32 equivalents
Algorithms
Theta values
Houbolt 53 Wilson {~)54 B o s s a k - N e w m a r k 55 (mfi + ku = 0, YB = 89- orB) B o s s a k - N e w m a r k 55 (mii + cil + ku = O, YB ~--- ~ -- OlB,
H i l b e r - H u g h e s - T a y l o r s6 (mfi + ku = O, •
=
89 -
2,02 = ~ , 0 3 = 6 01 = O, 02 = 0 2, 03 -- O 3 (| = 1.4 c o m m o n l y used) 01 = 1 - CrB
01 - -
,~,,)
o2 = 2 _ ,~8 + 2/~B 03 01 02 03
= = = =
6/3B 1 --O~B 1--2uB 1 -- 3C~B
01=1 02 = 2 + 2/3H -- 20t 2 03 = 6/3H(1 + C~H)
Multistep methods
In the previous sections we have been concerned with recurrence algorithms valid within a single time step and relating the values of Un+l, tin+l, fin+l to Un, fin, ii~, etc. It is possible to derive, using very similar procedures to those previously introduced, multistep algorithms in which we relate Un+l to the values u~, U~-l, Un-e, etc., without explicitly introducing the derivatives. Much classical work on stability and accuracy has been introduced on such multistep algorithms and hence they deserve mention here. We shall show in this section that a series of such algorithms may be simply derived using the weighted residual process. For constant time increments At, it can be shown that this set possesses identical stability and accuracy properties to the SSpj procedures.
17.5.2 The approximation procedure for a general multistep
algorithm
As in Sec. 17.3.2 we shall approximate the function u of the second-order equation M i i + Cli + K u + f = 0
(17.90)
615
616 The time dimension- discrete approximation in time
by a polynomial expansion of the order p, now containing a single unknown Un_t_1. This polynomial assumes knowledge of the value of Un, Un-1. . . . . U,-p+l at appropriate times tn, tn-1 . . . . . tn-p+l (Fig. 17.11). We can write this polynomial as 1
U(t)-- Z Nj(t)Un+j j=l-p
(17.91)
where Lagrange interpolation in time is given by (see Chapter 4) 1
Nj(t) = I I t - tn+k k=l-p tn+j -- tn+k k#j
(17.92)
Substituting this approximation into Eq. (17.91) gives 1
1
]Vj(t)lln+j and ii= Z
!1-- Z j=l-p
(17.93)
~[j(t)Un+j
j=l-p
where Nj and Nj denote the time derivatives of the shape functions. Insertion of u, li and ti into the weighted residual equation form yields ltn+l a tn
1 W(t) Z [(]VjM + NjC + NjK) Un+j + Njfn+j] d t = 0 j=l-p
(17.94)
with the forcing functions interpolated similarly from its nodal values. Using (17.92) and the definition for Ok given by (17.37) leads to a recurrence relation which may be used to compute u,,+a.
i
Un-p+l n-p+1
n-2 i~!:e::ig.na~!l~c~~t ~ .p~f .' [ 9
.>,(
At -F n
n-1
_
At
n+l
~ ~ ' ~~ I
11
...............
Approxi domaimnate Fig. 17.11 Multistep polynomial approximation.
Un+l
,,
Multistep recurrence algorithms Example 17.3: Two-point interpolation: p = 1. Evaluating Eq. (17.92) for the two points we obtain N1
--
No where At =
tn+ 1 -
tn
and
t -
tn
tn+l
-
tn
tn+l -
t
1
~(t At
-
r
tn) =
1
t n + l - - tn
z = t -
=
At
- t) = 1
= ~(tn+l At
(17.95a) At
Here the derivative is computed directly as
tn.
dN1 dt
dN0 dt
1 At
(17.95b)
Second derivatives are obviously zero, hence this form may only be used for first-order equations as 1 ~C(un+
1 - a n ) -~- K[(1 - O ) u n + 0Un+l] -~- f = 0 At which is, obviously, identical to the SS 11 result given previously.
(17.95c)
Example 17.4: Three-point interpolation: p = 2. Evaluating Eq. (17.92) for the three points gives N1
(t -- tn-1)(t
:
-- tn)
(tn+l - t n - 1 ) ( t n + l (t -
No =
tn-1)(t
-
(t -- tn)(t (tn-1
tn)
tn+l)
-- tn+l)
(tn - - t n - 1 ) ( t n
N_I =
-
(17.96a)
-- tn+l)
-- tn)(tn-1
tn+l)
--
The derivatives follow immediately from Eqs (17.92) and (17.93) as dN1
(t -
dt
dN0 dt
t n ) -t- ( t -
tn-1)
(tn+l -- t n - 1 ) ( t n + l
=
dN_l
(t (tn -
tn+l)
+ (t -
tn-1)(tn
(t - tn+l)
dt
-- tn)
--
tn-1)
tn+l)
+ (t -
(17.96b)
tn)
-- tn+l)
(tn--1 - - t n ) ( t n - - 1
This is the lowest order which can be used for second-order equations and has second derivatives d2N1
2
dt 2 d2No dt 2
(tn+l - - t n - 1 ) ( t n + l
=
2 (tn -
tn-1)(tn
(tn-1
-- tn)(tn-1
d2N_l dt 2
-- tn)
--
tn+l)
2 --
tn+l)
(17.96c)
617
618 The time dimension- discrete approximation in time
The recurrence relation for the two-step method with At constant is given by
(~-~M 1 + I
1 1 -Jr-~l(02+01)K]Un+l + ~-~(01 -~- ~)C
2 - ~sM-
2_
_
~ - 7 0 1 C -~- (1 - 0 2 ) K
Un'+"
(17.96d)
[ A 1t 2 M + -A-~(01 1 -- ~1 ) C + ~1(02 - 01)K] Un-1 + f = 0 where f is the effect of the integrated force resultant and Ok is computed using (17.92), but now has different values for stability than given for Ok in the SS22 form. The above form is identical to the form originally derived by Newmark 44 (however, the conventional parameters are usually/3 and 3/) and also corresponds to the SS22 and GN22 forms when parameters are related by: Y -- 01 + ~1 = 01 = fll
and
1 = ~1 132 /~ = ~1(02 ~-01) m_ ~02
The explicit form of this algorithm with 2/3 = 02 = 02 =/32 = 0 and V = 01 + 1/2 = 01 --/31 = 1/2 is frequently used as an alternative to the single-step explicit form. It is then known as the central difference approximation obtained by direct differencing. The reader can easily verify that the simplest finite difference approximation of Eq. (17.1) in fact corresponds to the above with 02 - 0 and 01 = 0. Higher order multistep forms follow the general pattern given above for the two- and three-point forms and need not be discussed more here. In general there are no added advantages using the multistep form and, quite generally, we recommend use of the onestep forms SSpj and GNpj given above.
In Secs 17.2.5 and 17.3.3 we have considered the exact solution of the approximate recurrence algorithm given in the form etc.
Un+ 1 - - ].ZUn,
(17.97)
for the modally decomposed, single degree of freedom systems typical of Eqs (17.4) and (17.5). The evaluation of/z was important to ensure that its modulus does not exceed unity so that stability is preserved. However, analytical solution of the linear homogeneous differential equations is also easy to obtain in the form fi -- fie x/ or
Un+ 1
Un e~.At
=
(17.98)
and comparison of/z with such a solution is always instructive to provide information on the performance of algorithms in the particular range of eigenvalues. In Fig. 17.5 we plotted the exact solution e -~~ and compared it with the values of/z for various 0 algorithms approximating the first-order equation, noting that here ~, -- -o9 -and is real.
k C
Time discontinuous Galerkin approximation 619
Immediately we see there that the performance error is very different for various values of At and obviously deteriorates at large values. Such values in a real multivariable problem correspond of course to the 'high frequency' responses which are often less important, and for smooth solutions we favour algorithms where/z tends to values much less than unity for such problems. However, response through the whole time range is important and attempts to choose an optimal value of 0 for various time ranges has been performed by Liniger. 52 Table 17.1 of Sec. 17.2.6 illustrates how an algorithm with 0 = 2/3 and a higher truncation error than that of 0 = 1/2 can perform better in a multidimensional system because of such properties. Similar analysis can be applied to the second-order equation. Here, to simplify matters, we consider 0nly the homogeneous undamped equation in the form
mii + ku = 0
(17.99)
in which the value of ~. is purely imaginary and corresponds to a simple oscillator. By examining/z we can find not only the amplitude ratio (which for high accuracy should be unity) but also the phase error. In Fig. 17.12(a) we show both the variation of the modulus of/z (which is called the spectral radius) and in Fig. 17.12(b) that of the relative period for the SS22/GN22 schemes, which of course are also applicable to the two-step equivalent. The results are plotted against At
2zr k where T = ~ ; o92=T co m In Fig. 17.13(a) and (b) similar curves are given for the SS23 and GN23 schemes frequently used in practice and discussed previously. Here as in the first-order problem we often wish to suppress (or damp out) the response to frequencies in which At / T is large (say greater than 0.1) in multidegree of freedom systems, as such a response will invariably be inaccurate. At the same time below this limit it is desirable to have amplitude ratios as close to unity as possible. It is clear that the stability limit with 01 - 02 = 1/2 giving unit response everywhere is often undesirable (unless physical damping is sufficient to damp high frequency modes) and that some algorithmic damping is necessary in these cases. The various schemes shown in Figs 17.12 and 17.13 can be judged accordingly and provide the reason for a search for an optimum algorithm. We have remarked frequently that although schemes can be identical with regard to stability their performances may differ slightly. In Fig. 17.14 we illustrate the application of SS22 and GN22 to a single degree of freedom system showing results and errors in each scheme.
A time discontinuous Galerkin formulation may be deduced from the finite element in the time approximation procedure considered in this chapter. This is achieved by assuming the weight function w and solution variables u are approximated within each time interval At as +
u-
u n + Au(t)
W =
W n
+
-~-
Aw(t)
tn < t < t~-+l t n < t < tn+ 1
(17.100)
where the time tn is the limit from times smaller than tn and tn+ is the limit from times larger than tn and, thus, admit a discontinuity in the approximation to occur at each discrete
620 The time d i m e n s i o n - discrete approximation 1.0
in time
~':-:4,-,~, ,,
02/2 = 13-- 0.25, 01 = 7 = 0.5
0.902/2 = 13= 0.5, 01 = 7 = 0.6
:=L m
02/2 = 13= 0.3025, 01 = 7 = 0.6
0.8-
0.7 10-2
I
I 11
1 0 -1
(a) Spectral radius I~tl
At/T
I 0
I 102
103
0.5 e e
01=7=0.6,02/2=13=0.5;
a .... n~ "1 - t ......
;
a /o_ 2 " - - 13 = 0 . 3 0 2 5
0.4
0.3~1... 0.2 -
0
0
,~
0.1
025
I
I
0.2 0.3 0.4 At/T (b) Relative period elongation, T = exact period, T= numerical period Fig. 17.12 SS22, GN22 (Newmark) or their two-step equivalent.
time location. The functions Au and Aw are defined to be zero at tn and continuous up to the time tn-+l where again a discontinuity can occur during the next time interval. The discrete form of the governing equations may be deduced starting from the time dependent partial differential equations where standard finite elements in space are combined with the time discontinuous Galerkin approximation and defining a weak form in a spacetime slab. Alternatively, we may begin with the semi-discrete form as done previously in this chapter for other finite element in time methods. In this second form, for the first-order case, we write
Time discontinuous Galerkin approximation 1.1
1.0
0.9
_~ 0.8
Wilson 54 e= 1.4
A 0.7
0.6
0.5 10-2
I 10-1
I 1
I 10
I 102
103
At/T
(a) Spectral radius 0.5 H 0.4 -
0.3 F... 0.2
Hilber et al. 56 0.1
/
/ /
0 S 0
_Wilson 54 01=1"4
0.1
(b) Relative period elongation
0.2
At/T
I 0.3
0.4
Fig. 17.13 SS23,GN23 or their two-step equivalent (see Problem 17.10 for description of o~).
621
622
The time dimension - discrete approximation in time
Fig. 17.14 Comparison of the 5S22 and GN22 (Newmark) algorithms: a single DOF dynamic equation with periodic forcing term, 01 -/31 - 1/2, 02 --/32 - 0.
I =
f
t;+l w x (Cfi + K u + f) d r = 0
(17.101)
Jt~-
Due to the discontinuity at tn it is necessary to split the integral into I =
J t~-
w T (Cfi + Ku + f) dr +
J t+
w T (Cfi + K u + f) dr = 0
(17.102)
which gives
] = (Wn+)T[C(un+ - an)]-[-(Wn+)
T
[ tn-bl (Cfi + K u + f ) dr Jt +
+
(17.103)
(Aw) T (Cli + K u + f) d r = 0 J t~+
in which now all integrals involve approximations to functions which are continuous. To apply the above process to a second-order equation it is necessary first to reduce the equation to a pair of first-order equations. This may be achieved by defining the momenta p = Mfi
(17.104)
Time discontinuous Galerkin approximation 623 and then writing the pair Mli-p
=0
(17.105)
+ Cfi+ Ku+ f = 0
The time discrete process may now be applied by introducing two weighting functions as described in reference 39. E x a m p l e 17.5: Solution of a scalar equation. To illustrate the process we consider the
simple first-order scalar equation cft + k u + f = 0
(17.106)
We consider the specific approximations u ( t ) = u ,+ + r AUn+ 1
(17.107)
+
w(t) = w n + rAw,+ 1
- 1 -where Au2+ 1 = Un+
+ , etc., and
U n
t -- tn 72
---
tn+l - tn
t -- t, ---
At
defines the time interval 0 < r < At. This approximation gives the integral form I
=
w
+ [c(u +
n
-
Un)] +
w
+At
cZXUn+I + ~(U+n +
n
+
+ At
foI +
Awn-+lr
[1
- - ~ c A u 2 + 1 + k (u + + rAU~+l) + f
~AUn+l) + I a~
1
dr = 0 (17.108)
Evaluation of the integrals gives the pair of equations
[~c + ~t,t) l~At 1 1 89 { ~n+ lkAt ~c+
-
C~n { 0 }
(17.109a)
where
Thus, with linear approximation of the variables the time discontinuous Galerkin method - 1. gives two equations to be solved for the two unknowns u,+ and u,+ To illustrate the performance of the above scheme we compare the amplification matrix for the discontinuous Galerkin and standard Galerkin method in Fig. 17.15. In addition we use the method to solve the example described in Fig. 17.4 and present the results in Fig. 17.16. It is possible to also perform the solution with c o n s t a n t approximation. Based on the above this is achieved by setting Au~-+l and Aw,+ 1 to zero yielding the single equation (17.110) (c + k A t ) u , + + A t f = cunand now since the approximation is constant over the entire time the u n+ also defines exactly the Un+ 1 value. This form will now be recognized as identical to the b a c k w a r d difference implicit scheme defined in Fig. 17.4 for 0 = 1.
624
The time dimension- discrete approximation in time
Fig. 17.15 The amplification A for standard and discontinuous Galerkin schemes. Finally, we compare the error in the amplification matrix for different step sizes. The error defined by
E =]~(At)-Aex(At) ^
where the A is the amplification for the approximate form and Aex -- e x p ( - A t ) , the exact value. In Fig. 17.17(a) we present the values for the single-step algorithms and in Fig. 17.17(b) those for the discontinuous Galerkin and two-step quadratic continuous Galerkin solution. We note that the 0 = 1/2 (Crank-Nicolson), discontinuous Galerkin and p = 2 continuous Galerkin solutions are all second-order accurate (slope zero for small At -- 0) while other values have finite slope and hence are only first-order accurate. It is also evident that the error at larger steps for the p -- 2 continuous Galerkin is more accurate than the discontinuous Galerkin. Thus, for the same computational effort the use of the continuous form is more appropriate in this class of problems. For this reason we will not pursue use of the discontinuous Galerkin time integration procedure further here.
The derivation and examples presented in this chapter cover, we believe, the necessary toolkit for efficient solution of many transient problems governed by Eqs (17.1) and (17.2). In the next chapter we shall elaborate further on the application of the procedures discussed here and show that they can be extended to solve coupled problems which frequently arise in practice and where simultaneous solution by time stepping is often needed. Finally, as we have indicated in Eq. (17.3), many problems have coefficient matrices or other variations which render the problem non-linear. This topic is addressed further for structural and solid mechanics problems in reference 57 and we note also that the issue of stability after many time steps is more involved than the procedures introduced here to investigate local stability.
Concluding
121
I--~xact
11%,,J,.
~ eI -=-o2 / 3D1G1
................................................................. ~ ~ ~
0.8 I ~ , , ,
:~..............
......
~
::...............
i ...............
::...............
121
I"
9
1
I--~xact I I
"-'.,.--.i ........................................................................ ~,,
0.6
9
E o 0.6
........................
.
-0.2
0
0.5
1
1.5 ttime
' 2.5
2
-0.2
3
i ,oG_.o. 311
...... i ................................................
0.8
::.............
.
.
0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0.5
.
.
.
.
.
1
.
.
.
.
1 ..................................................................
Exact DG
o
I i
I--.
1 ,
:
:
::...............
::..............
~ .............
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
i 2
1.5 ttime
.................................................................
i
1
:
1.2
1.2
remarks
i
i
.
.
.
.
.
.
.
.
.
2.5
3
___' Exact o
DG
- - - 0=2/31
I
Ij....
0.8 E 0.6
0.6
'
iliiiiI ii
n
.~ 0.4 0.2
'0.4 0.2 ___
0
-0.2
0
.........................................................................................
0
2
3
4
..................................
-0.2
5
0
!. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2
ttime
3
ttime
4
Fig. 17.16 Comparison of standard and discontinuous Galerkin schemes on a first-order initial value problem.
0.25 0.2 0.15
.
.
.
.
0.1
~ i ..................... .......... i ........... '
.
--;---.--..-. ..............
.......... i:.......... ~ ......... ~........... - i.........i :.. o=O ..... i .......... .......... ~
.
0.05
.
0
~-0.05 ILl -0.1 -0.15
.o--
..........
*. -. :: '. : - ....
.... ! .........
~
: ..........
" :
~
:
..........
......
......
! ...........
-= : :
, ......................
-~:
:
"~.
i .........
.....
0
i"-: i + :, '0.5
i i i 1
t
/ t ...................
o.o8 ~ o.o, 0.02
'~'~
:: "= :....
z .....
"
: ......
.::........... i i 1.5
::....... i :, 2
: .......
" .
.
.
9
.
.
i. :
~ ' . ~ .
.
.
.
:: i .
.........
"e..
.
:
,~
! ..........
:: "o.
: . . . . . . . . . .
.
:
i i
~m~'"
i
"
.,
0
,~=-0.02 ~
-0.04
1
-0.06
::. . . . . . . . . . i ; 3.5 4
0.08
. i
:-.......... !"a. : i -.L 2.5 3"-.
"....--o. ..........
k z~#c step size
(a)
! ..........
"e
::
: ...........
,= .....
:
...........
::
: ...........
P ""
:
e ~ . i
i": . ! ""m, . .!........... . ! "'n . .......... i "'.. ....... i ...........
-0.2 -0.25
/
-..~Z..~
, , ,-o-FE-p=2 - m. DG
0.08
"",,
Fig. 17.17 Error in amplification matrix for single steps.
-0.1
. 0
1
.
!iiiii,i.i,,iiill
. 2
.
"
3
4 t time
(b)
i'~" " 5
,~" D 6
7
8
625
626
The time dimension- discrete approximation in time
17.1 Verify the recurrence relation given in Eq. (17.19) using a least squares minimization process in which (17.7) is substituted into (17.2). 17.2 Determine the stability characteristics for a scalar form of the least squares recurrence relation given in Eq. (17.19). Plot the behaviour of the amplification matrix vs At. 17.3 Houbolt's method was originally developed as a multi-step method for the equation of motion written as
Miin+1 -+- C l h n +
1 "at-
Kun+1 3r- f n +
1 --" 0
with updates given by Ihn+l "-- ~
1 1
Un+l -- ~ ~
( l l U n + l -- 18Un + 9Un-1 -- 2Un-2) (2Un+l -- 5Un + 4Un-I -- Un-2)
(a) Following the approach given in Sec. 17.2.5 determine the amplification matrix in the form Xn+I=AXn
with X n + l = [Un+l
Un
Un_I]T
(b) For the undamped and unloaded case (i.e., c - f = 0) determine and plot the spectral radius I/~l and period elongation A T / T vs the time increment A t / T as shown in Fig. 17.13. T is the period of the undamped equation. 17.4 Consider the scalar first-order equation: cit + ku + f -
0
Construct the discrete form for transient solution using the SS 11 algorithm described in Sec. 17.3. For the data c = k = 1 and f = sin 2 t obtain, by hand, the solution for the first five steps using a time step of At = 0.1. Write a MATLAB program to solve the problem for 0 < t < 2. 17.5 Consider the scalar second-order equation: m i;t + cfa + ku + f = 0
Construct the discrete form for transient solution using the SS22 algorithm described in Sec. 17.3. (a) For the data m = k = 1, c = 0 and f = sin 2 t obtain, by hand, the solution for the first five steps using a time step of At -- 0.05. Write a MATLAB program to solve the problem for 0 < t < 2. (b) Repeat (a) for c = 0.05. 17.6 Consider the scalar second-order equation: mi2 + cfa + ku + f = 0
Construct the discrete form for transient solution using the GN22 algorithm described in Sec. 17.3.
Problems 627 (a) For the data m = k = 1, c = 0 and f = sin 2 t obtain, by hand, the solution for the first five steps using a time step of At = 0.05. Write a MATLAB program to solve the problem for 0 < t < 2. (b) Repeat (a) for c = 0.05. 17.7 For the general second-order equation
mfi + cit + ku + f = 0 develop the discrete form for the SS32 algorithm. 17.8 For the general second-order equation
mii +cit + k u + f = 0 develop the discrete form for the GN32 algorithm. 17.9 The general second-order equation may be split into the pair of first-order equations given by M~, + Cv + Ku + f = 0
u-v
=0
(a) Develop the discrete form of the equations using the SS 11 algorithm. (b) For the scalar form of the equations determine the amplification matrix and the stability characteristics of the method. (c) For the data m = k = 1, c = 0 and f = sin 2 t obtain, by hand, the solution for the first five steps using a time step of At -- 0.05. (d) Write a MATLAB program to solve the problem for 0 _< t _< 2. 17.10 The Hilber-Hughes-Taylor (HHT) algorithm 56 is given b y t Miin +~ + Cdn+~ + Kun +~ -+- fn+~ -- 0
where tn+~ -- (1 - an)tn + antn+l, Un+ot :
(1 - an)Un + anUn+l
fln+a :- (1 -- all)fin -t- aHtln+l fin +or :
fin+ 1
and fn+~ is the force at t,,+~. The algorithm is completed using the GN22 relations (17.58) with 3 1 /31 = 2 a n and /~2 "-- ~ ( 2 - a n ) 2 (a) For the scalar form of the equations determine the amplification matrix and the stability characteristics of the method. (If necessary, use MATLAB to determine the roots of the stability equation.) (b) For the data m -- k - 1, c = 0 and f -- sin 2 t obtain, by hand, the solution for the first five steps using a time step of At -- 0.05. (c) Write a MATLAB program to solve the problem for 0 < t < 2. t In the original publication a H = 1 + a and a had negative values. The definition used here is more consistent with other usage in this chapter and an is always positive.
628 The time dimension- discrete approximation in time
17.11 Using the Routh-Hurwitz criterion described in Sec. 17.4 perform the stability analysis for the first-order problem described by SS 11. 17.12 Using the Routh-Hurwitz criterion described in Sec. 17.4 perform the stability analysis for the second-order problem described by SS22. 17.13 Using the Routh-Hurwitz criterion, described in Sec. 17.4 perform the stability analysis for the second-order problem described by GN22. 17.14 Use FEAPpv to solve the rectangular beam problem described in Problem 16.13. Compare the solution with that computed by modal analysis. (Note: The comparison with the full modal solution gives the error between the discrete integration and an exact integration of the semi-discrete system.) 17.15 Use FEAPpv to solve the curved beam problem described in Problem 16.11. Compare the solution with that computed by modal analysis. 17.16 Program development project: Extend the program system started in Problem 2.17 to perform time integration using the single-step algorithms described in Sec. 17.3. Your implementation should include: (a) SS 11 to integrate a first-order system such as encountered for thermal analysis. (b) SS22 to integrate a second-order system for transient analysis of solids. (c) GN22 to integrate a second-order system for transient analysis of solids. Test your program by integrating a single degree of freedom problem for which you have a hand calculation for verification use. Solve the rectangular beam problem described in Problem 16.13. Compare the solution with that computed by modal analysis. (Note: The comparison with the full modal solution gives the error between the discrete integration and the exact integration of the semi-discrete system.)
1. R.D. Richtmyer and K.W. Morton. Difference Methods for Initial Value Problems. Wiley (Interscience), New York, 1967. 2. T.D. Lambert. Computational Methods in Ordinary Differential Equations. John Wiley & Sons, Chichester, 1973. 3. P. Henrici. Discrete Variable Methods in Ordinary Differential Equations. John Wiley & Sons, New York, 1962. 4. EB. Hildebrand. Finite Difference Equations and Simulations. Prentice-Hall, Englewood Cliffs, N.J., 1968. 5. G.W. Gear. Numerical Initial Value Problems in Ordinary Differential Equations. Prentice-Hall, Englewood Cliffs, N.J., 1971. 6. W.L. Wood. Practical Time Stepping Schemes. Clarendon Press, Oxford, 1990. 7. O.C. Zienkiewicz, R.L. Taylor, and P. Nithiarasu. The Finite Element Method for Fluid Dynamics. Butterworth-Heinemann, Oxford, 6th edition, 2005. 8. J.T. Oden. A general theory of finite elements. Part II. Applications. Int. J. Numer. Meth. Eng., 1:247-254, 1969. 9. I. Fried. Finite element analysis of time-dependent phenomena. J. AIAA, 7:1170-1173, 1969. 10. J.H. Argyris and D.W. Scharpf. Finite elements in time and space. Nucl. Eng. and Design, 10:456--469, 1969.
References 629 11. O.C. Zienkiewicz and C.J. Parekh. Transient field problems - two and three dimensional analysis by isoparametric finite elements. Int. J. Numer. Meth. Eng., 2:61-71, 1970. 12. O.C. Zienkiewicz. The Finite Element Method in Engineering Science. McGraw-Hill, London, 2nd edition, 1971. 13. O.C. Zienkiewicz and R.W. Lewis. An analysis of various time stepping schemes for initial value problems. Earth. Eng. Struct. Dyn., 1:407-408, 1973. 14. W.L. Wood and R.W. Lewis. A comparison of time marching schemes for the transient heat conduction equation. Int. J. Numer. Meth. Eng., 9:679-689, 1975. 15. O.C. Zienkiewicz. A new look at the Newmark, Houbolt and other time stepping formulas. A weighted residual approach. Earth. Eng. Struct. Dyn., 5:413-4 18, 1977. 16. W.L. Wood. On the Zienkiewicz four-time-level scheme for numerical integration of vibration problems. Int. J. Numer. Meth. Eng., 11:1519-1528, 1977. 17. O.C. Zienkiewicz, W.L. Wood, and R.L. Taylor. An alternative single-step algorithm for dynamic problems. Earth. Eng. Struct. Dyn., 8:31-40, 1980. 18. W.L. Wood. A further look at Newmark, Houbolt, etc. time-stepping formulae. Int. J. Numer. Meth. Eng., 20:1009-1017, 1984. 19. O.C. Zienkiewicz, W.L. Wood, N.W. Hine, and R.L. Taylor. A unified set of single-step algorithms. Part 1: General formulation and applications. Int. J. Numer. Meth. Eng., 20:1529-1552, 1984. 20. W.L. Wood. A unified set of single-step algorithms. Part 2: Theory. Int. J. Numer. Meth. Eng., 20:2302-2309, 1984. 21. M. Katona and O.C. Zienkiewicz. A unified set of single-step algorithms. Part 3: The beta-m method, a generalization of the Newmark scheme. Int. J. Numer. Meth. Eng., 21:1345-1359, 1985. 22. E. Varoglu and N.D.L. Finn. A finite element method for the diffusion convection equations with concurrent coefficients. Adv. Water Res., 1:337-341, 1973. 23. C. Johnson, U. N~ivert, and J. Pitk~anta. Finite element methods for linear hyperbolic problems. Comp. Meth. Appl. Mech. Eng., 45:285-312, 1984. 24. T.J.R. Hughes, L.P. Franca, and G.M. Hulbert. A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective-diffusive equations. Comp. Meth. Appl. Mech. Eng., 73:173-189, 1989. 25. T.J.R. Hughes and G.M. Hulbert. Space-time finite element methods in elastodynamics: formulation and error estimates. Comp. Meth. Appl. Mech. Eng., 66:339-363, 1988. 26. G.M. Hulbert and T.J.R. Hughes. Space-time finite element methods for second-order hyperbolic equations. Comp. Meth. Appl. Mech. Eng., 84:327-348, 1990. 27. G.M. Hulbert. Time finite element methods for structural dynamics. Int. J. Numer. Meth. Eng., 33:307-331, 1992. 28. B.M. Irons and C. Treharne. A bound theorem for eigenvalues and its practical application. In Proc. 3rd Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-71-160, pages 245-254, Wright-Patterson Air Force Base, Ohio, 1972. 29. K. Washizu. Variational Methods in Elasticity and Plasticity. Pergamon Press, New York, 3rd edition, 1982. 30. M.E. Gurtin. Variational principles for linear initial-value problems. Q. Appl. Math., 22: 252-256, 1964. 31. M.E. Gurtin. Variational principles for linear elastodynamics. Arch. Rat. Mech. Anal., 16:34-50, 1969. 32. E.L. Wilson and R.E. Nickell. Application of finite element method to heat conduction analysis. Nucl. Eng. Design, 4:1-11, 1966. 33. J. Crank and P. Nicolson. A practical method for numerical integration of solutions of partial differential equations of heat conduction type. Proc. Camb. Phil. Soc., 43:50, 1947.
630 The time dimension- discrete approximation in time 34. R.L. Taylor and O.C. Zienkiewicz. A note on the 'order of approximation'. Int. J. Solids Struct., 21:793-798, 1985. 35. W.H. Reed and T.R. Hill. Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, 1973. 36. P. Lesaint and P.-A. Raviart. On a finite element method for solving the neutron transport equation. In C. de Boor, editor, Mathematical Aspects of Finite Elements in Partial Differential Equations. Academic Press, New York, 1974. 37. C. Johnson. Numerical Solutions of Partial Differential Equations by the Finite Element Method. Cambridge University Press, Cambridge, 1987. 38. K. Eriksson and C. Johnson. Adaptive finite element methods for parabolic problems I: A linear model problem. SIAM J. Numer. Anal., 28:43-77, 1991. 39. X.D. Li and N.-E. Wiberg. Structural dynamic analysis by a time-discontinuous Galerkin finite element method. Int. J. Numer. Meth. Eng., 39:2131-2152, 1996. 40. M. Salvadori and M. Baron. Numerical Methods in Engineering. Prentice-Hall, New York, 1952. 41. M. Zlamal. Finite element methods in heat conduction problems. In J. Whiteman, editor, The Mathematics of Finite Elements and Applications, pages 85-104. Academic Press, London, 1977. 42. W. Liniger. Optimisation of a numerical integration method for stiff systems of ordinary differential equations. Technical Report RC2198, IBM Research, 1968. 43. J.M. Bettencourt, O.C. Zienkiewicz, and G. Cantin. Consistent use of finite elements in time and the performance of various recurrence schemes for heat diffusion equation. Int. J. Numer. Meth. Eng., 17:931-938, 1981. 44. N. Newmark. A method of computation for structural dynamics. J. Eng. Mech., ASCE, 85:67-94, 1959. 45. T. Belytschko and T.J.R. Hughes, editors. Computational Methods for Transient Analysis. North-Holland, Amsterdam, 1983. 46. I. Miranda, R.M. Ferencz, and T.J.R. Hughes. An improved implicit-explicit time integration method for structural dynamics. Earth. Eng. Struct. Dyn., 18:643-655, 1989. 47. E.J. Routh. A Treatise on the Stability of a Given State or Motion. Macmillan, London, 1977. 48. A. Hurwitz. Uber die Bedingungen, unter welchen eine Gleichung nur Wtirzeln mit negativen reellen teilen besitzt. Math. Ann., 46:273-284, 1895. 49. ER. Gantmacher. The Theory of Matrices. Chelsea, New York, 1959. 50. G.G. Dahlquist. A special stability problem for linear multistep methods. BIT, 3:27-43, 1963. 51. C.W. Gear. The automatic integration of stiff ordinary differential equations. In A.J.H. Morrell, editor, Information Processing 68. North-Holland, Dordrecht, 1969. 52. W. Liniger. Global accuracy and A-stability of one and two step integration formulae for stiff ordinary differential equations. In Proc. Conf. on Numerical Solution of Differential Equations, Dundee University, 1969. 53. J.C. Houbolt. A recurrence matrix solution for dynamic response of elastic aircraft. J. Aero. Sci., 17:540-550, 1950. 54. K.-J. Bathe and E.L. Wilson. Stability and accuracy analysis of direct integration methods. Earth. Eng. Struct. Dyn., 1:283-291, 1973. 55. W. Wood, M. Bossak, and O.C. Zienkiewicz. An alpha modification of Newmark's method. Int. J. Numer. Meth. Eng., 15:1562-1566, 1980. 56. H. Hilber, T.J.R. Hughes, and R.L. Taylor. Improved numerical dissipation for the time integration algorithms in structural dynamics. Earth. Eng. Struct. Dyn., 5:283-292, 1977. 57. O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method for Solid and Structural Mechanics. Butterworth-Heinemann, Oxford, 6th edition, 2005.
Frequently two or more physical systems interact with each other, with the independent solution of any one system being impossible without simultaneous solution of the others. Such systems are known as coupled and of course such coupling may be weak or strong depending on the degree of interaction. An obvious 'coupled' problem is that of dynamic fluid-structure interaction. Here neither the fluid nor the structural system can be solved independently of the other due to the unknown interface forces. A definition of coupled systems may be generalized to include a wide range of problems and their numerical discretization as: 1
Coupled systems and formulations are those applicable to multiple domains and dependent variables which usually (but not always) describe different physical phenomena and in which (a) neither domain can be solved while separated from the other; (b) neither set of dependent variables can be explicitly eliminated at the differential equation level. The reader may well contrast this with definitions of mixed and irreducible formulations introduced in Chapter 3 and discussed fully in Chapter 10 and find some similarities. Clearly 'mixed' and 'coupled' formulations are analogous, with the main difference being that in the former elimination of some dependent variables is possible at the governing differential equation level. In the coupled system a full analytical solution or inversion of a (discretized) single system is necessary before such elimination is possible. Indeed, a further distinction can be made. In coupled systems the solution of any single system is a well-posed problem and is possible when the variables corresponding to the other system are prescribed. This is not always the case in mixed formulations. It is convenient to classify coupled systems into two categories:
Class L This class contains problems in which coupling occurs on domain interfaces via the boundary conditions imposed there. Generally the domains describe different physical situations but it is possible to consider coupling between domains that are physically similar in which different discretization processes have been used.
632 Coupledsystems Class II. This class contains problems in which the various domains overlap (totally or
partially). Here the coupling occurs through the governing differential equations describing different physical phenomena. Typical of the first category are the problems of fluid-structure interaction illustrated in Fig. 18.1(a) where physically different problems interact and also structure-structure interactions of Fig. 18.1(b) where the interface simply divides arbitrarily chosen regions in which different numerical discretizations are used. The need for the use of different discretizations may arise from different causes. Here for instance:
Fig. 18.1 Class I problems with coupling via interfaces (shown as thick line).
Coupled problems-definition and classification 633 1. Different finite element meshes may be advantageous to describe the subdomains. 2. Different procedures such as the combination of boundary method and finite elements in respective regions may be computationally desirable. 3. Domains may simply be divided by the choice of different time-stepping procedures, e.g., of an implicit and explicit kind. In the second category, typical problems are illustrated in Fig. 18.2. One of these is that of metal extrusion where the plastic flow is strongly coupled with the temperature field while at the same time the latter is influenced by the heat generated in the plastic flow. This problem is included to illustrate a form of coupling that commonly occurs in analyses of solids. The other problem shown in Fig. 18.2 is that of soil dynamics (earthquake response of a dam) in which the seepage flow and pressures interact with the dynamic behaviour of the soil 'skeleton'. We observe that, in the examples illustrated, motion invariably occurs. Indeed, the vast majority of coupled problems involve such transient behaviour and for this reason the present chapter will only consider this area. It will thus follow and expand the analysis techniques presented in Chapters 16 and 17. As the problems encountered in coupled analysis of various kinds are similar, we shall focus the presentation on three examples: 1. fluid-structure interaction (confined to small amplitudes); 2. soil-fluid interaction;
Fig. 18.2 Class II problems with coupling in overlapping domains.
634
Coupledsystems 3. implicit-explicit dynamic analysis of a structure where the separation involves the process of temporal discretization. In these problems all the typical features of coupled analysis will be found and extension to others will normally follow similar lines. As a final remark, it is worthwhile mentioning that problems such as linear thermal stress analysis to which we have referred frequently in this volume are not coupled in the terms defined here. In this the stress analysis problem requires a knowledge of the temperature field but the temperature problem can be solved independently of the stress field.t Thus the problem decouples in one direction. Many examples of truly coupled problems will be found in available books. 3-5
The problem of fluid-structure interaction is a wide one and covers many forms of fluid which we do not discuss in this book. The consideration of problems in which the fluid is in substantial motion is considered in standard texts on fluid dynamics (e.g., see reference 6) and, thus, we exclude at this stage such problems as flutter where movement of an aerofoil influences the flow pattern and forces around it leading to possible instability. For the same reason we also exclude here the 'singing wire' problem in which the shedding of vortices reacts with the motion of the wire. However, in a very considerable range of problems the fluid displacement remains small while interaction is substantial. In this category fall the first two examples of Fig. 18.1 in which the structural motions influence and react with the generation of pressures in a reservoir or a container. A number of symposia have been entirely devoted to this class of problems which is of considerable engineering interest, and here fortunately considerable simplifications are possible in the description of the fluid phase. References 7-22 give some typical studies. In such problems it is possible to write the linearized dynamic equations of fluid behaviour about the hydrostatic state as 0(pv) at
Ov po~ = - Vp + b at
(18.1)
where v is the fluid velocity, p is the fluid density (with P0 the density in the hydrostatic state), p the pressure and b is a constant body force of gravity. In postulating the above we have assumed 1. that the density P0 varies by a small amount only so may be considered constant; 2. that velocities are small enough for convective effects to be omitted; 3. that viscous effects by which deviatoric stresses are introduced can be neglected in the fluid. t In a general settingthe temperaturefield does dependuponthe strainrate. However,these terms are not included in the form presented in this volumeand in many instances produce insignificantchanges to the solution.2
Fluid-structure interaction (Class I problems) The reader can in fact note that with the preceding assumption Eq. (18.1) is a special form of a more general relation (described in reference 6). The linearized continuity equation based on the same assumption is P0 div v - p0VTv =
Op Ot
(18.2)
and noting that
Op Po Op ~ Ot K Ot
(18.3)
where K is the bulk modulus of the fluid, we can write VT v =
1 Op K Ot
(18.4)
Elimination of v between (18.1) and (18.4) gives the well-known scalar wave equation governing the pressure p: 1 02p ~72p-- c 2 0 t 2 (18.5) where
c=( K P0
(18.6)
denotes the speed of sound in the fluid. The equations described above are the basis of acoustic problems.
18.2.2 Boundary conditions for the fluid. Coupling and radiation In Fig. 18.3 we focus on the Class I problem illustrated in Fig. 18.1 (a) and on the boundary conditions possible for the fluid part described by the governing equation (18.5). As we know well, either normal gradients or values of p now need to be specified.
Fig. 18.3 Boundary conditions for the fluid component of the fluid-structure interaction.
635
636
Coupledsystems
Interface with solid
On the boundaries (D and ~ in Fig. 18.3 the normal velocities (or their time derivatives) are prescribed. Considering the pressure gradient in the normal direction to the face n we can thus write, by Eq. (18.1), Op On
= - P o O n = - p0nTv
(18.7)
where n is the direction cosine vector for an outward pointing normal to the fluid region and 0,, is prescribed. Thus, for instance, on boundary (!)coupling with the motion of the structure described by displacement u occurs. Here we put U n --- iL n "--
nTii
(18.8)
while on boundary ~ where only horizontal motion exists we have Vz - 0
(18.9)
Coupling with the structure motion occurs only via boundary ~ .
Free surface
On the free surface (boundary Q in Fig. 18.3) the simplest assumption is that p =0
(18.10)
However, this does not allow for any possibility of surface gravity waves. These can be approximated by assuming the actual surface to be at an elevation 0 relative to the mean surface. Now p = pogrl (18.11) where g is the acceleration due to gravity. From Eq. (18.1) we have, on noting Vz = O0/Ot and assuming P0 to be constant, 0277 Op = (18.12) Po Ot 2
OZ
and on elimination of r/, using Eq. (18.11), we have a specified normal gradient condition Op
az
=
10Zp
1 = --~0 g 0t 2 g
(18.13)
This allows for gravity waves to be approximately incorporated in the analysis and is known as the linearized surface wave condition.
Radiation boundary
Boundary (~)physically terminates an infinite domain and some approximation to account for the effect of such a termination is necessary. The main dynamic effect is simply that the wave solution of the governing equation (18.5) must here be composed of outgoing waves only as no input from the infinite domain exists. If we consider only variations in x (the horizontal direction) we know that the general solution of Eq. (18.5) can be written as p = F ( x - ct) + G(x + ct)
(18.14)
Fluid-structure interaction (Class I problems)
where c is the wave velocity given by Eq. (18.6) and the two waves F and G travel in positive and negative directions of x, respectively. The absence of the incoming wave G means that on boundary (~) we have only p -- F ( x - ct)
(18.15)
Op = Op = F' On Ox
(18.16)
Thus and
Op Ot
=
(18.17)
-cF'
where F' denotes the derivative of F with respect to (x - ct). We can therefore eliminate the unknown function F' and write Op
1
(18.18) On c which is a condition very similar to that of Eq. (18.13). This boundary condition was first presented in reference 7 for radiating boundaries and has an analogy with a damping element placed there. More accurate forms are possible to represent far field radiation conditions. For example, use of so-called perfectly matched layers (PML) is reported in references 23, 24. --
=
--~b
18.2.3 Weak form for coupled systems A weak form for each part of the coupled system may be written as described in Chapter 3. Accordingly, for the fluid we can write the differential equation as 8I-lf --
6p ~5~0-
p
d~ = 0
(18.19)
f
which after integration by parts and substitution of the boundary conditions described above yields fa
f
[~P ~-5 1 )~ + (V~P)T(VP) ] dr2+ Jfr ~pp0nTiidF+ f r 8 p -1~ d I ' + 1 3 g
= 0 f r ~p-/~dF 1 4 C (18.20)
where ~ f is the fluid domain and Fi the integral over boundary part O. Similarly for the solid the weak form after integration by parts is given by guT[ps i] +/Z/It + s T I ) S u - b] dr2 - ~ ~uTtdF = 0
(18.21)
,/1 t
where for pressure defined positive in compression the surface traction is defined as i = -pns
= pn
(18.22)
since the outward normal to the solid is ns = - n . The traction integral in Eq. (18.21) is now expressed as f r ,uTidF = f r ,uTnpdF t
t
(18.23)
637
638
Coupledsystems (1) In complex physical situations, the interaction between compressibility and internal gravity waves (interaction between acoustic modes and sloshing modes) leads to a modified scalar wave equation. Equation (18.5) should then be replaced by a more complex equation: in a stratified medium for instance, the irrotationality condition for the fluid is not totally verified (the fluid is irrotational in a plane perpendicular to the stratification axis). 16 (2) The variational formulation defined by Eq. (18.20) is valid in the static case provided the following constraints conditions are added f~i p d~ + po c2 fs~2y nTu dl-'= 0 for a compressible fluid filling a cavity, frl nTu dF + fr2 P/Pog dF = 0, for an incompressible liquid with a free surface contained inside a reservoir. The static behaviour is important for the modal response of coupled systems when modal truncation needs static corrections in order to accelerate the convergence of the method. This static behaviour is also of prime importance for the construction of reduced matrix models when using dynamic substructuring methods for fluid structure interaction problems. 17' 18
18.2.4 The discrete coupled system We shall now consider the coupled problem discretized in the standard (displacement) manner with the displacement vector approximated as u ~ fi = NuR
(18.24)
and the fluid similarly approximated by p ~ ~b = Np~
(18.25)
where ~ and ~ are the nodal parameters of each field and Nu and N p are appropriate shape functions. The discrete structural problem thus becomes Mfi + C~ + K~ - Q~ + f = 0
(18.26)
where the coupling term arises due to the pressures (tractions) specified on the boundary as f r NuTtdl-' - f r NuTnNpdl-'P = QP t
(18.27)
t
The terms of the other matrices are already well known to the reader as mass, damping, stiffness and force. Standard Galerkin discretization applied to the weak form of the fluid equation (18.20) leads to (including the possibility of a source term, q) S~ + Cb + H~ + p0QTfi + q = 0
(18.28)
where T%N d~-+Npc2 P
S=
T1NpdF C-- f r Npc 4
I - I - f (~TNp)TVNp dr2 and Q is identical to that of Eq. (18.27).
3 NpgNpdl7 (18.29)
Fluid-structure interaction (Class I problems)
18.2.5 Free vibrations If we consider free vibrations and omit all force and damping terms (noting that in the fluid component the damping is strictly that due to radiation energy loss) we can write the two equations (18.26) and (18.28) as a set:
M
[o
0
{~
J = 0
(18.30)
and attempt to proceed to establish the eigenvalues corresponding to natural frequencies. However, we note immediately that the system is not symmetric (nor positive definite) and that standard eigenvalue computation methods are not directly applicable. Physically it is, however, clear that the eigenvalues are real and that free vibration modes exist. The above problem is similar to that arising in vibration of rotating solids and special solution methods are available, though costly. 25 It is possible by various manipulations to arrive at a symmetric form and reduce the problem to a standard eigenvalue one. 14-22' 25-28 A simple method proposed by Ohayon proceeds to achieve the symmetrization objective by putting ~ - lie it~ p : p e ic~ and rewriting Eq. (18.30) as Kfl - Q~ - co2M~ = 0
(18.31)
H~ - co2S~ - co2p0QTl~ -- 0 and an additional variable t) such that = co2q)
(18.32)
After some manipulation and substitution we can write the new system as
" ~ I:~ ~ ~
1p0S
0
-co 2
QT
0
•p0
0
1 ST
_1 n
l~
p0
= 0
(18.33)
which is a symmetric generalized eigenproblem. Further, the variable q can now be eliminated by static condensation and the final system becomes symmetric and now contains only the basic variables. The system (18.32), with static corrections, may lead to convenient reduced matrix models through appropriate dynamic substructuring methods. 19 An alternative that has frequently been used is to introduce a new symmetrizing variable at the governing equation level, but this is clearly not necessary. 14' 15 As an example of a simple problem in the present category we show an analysis of a three-dimensional flexible wall vibrating with a fluid encased in a 'rigid' container 29 (Fig. 18.4).
18.2.6 Forced vibrations and transient step-by-step algorithms The reader can easily verify that the steady-state, linear response to periodic input can be readily computed in the complex frequency domain by the procedures described in Chapter 16. Here no difficulties arise due to the non-symmetric nature of equations and standard procedures can be applied. Chopra and coworkers have, for instance, done many
639
640
Coupled
systems
":2~
."""
,,
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9
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9
O
iiii
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o
9
.
_[':2
0
TI o -"
..""
()
o
..( L..
9s. - 9
.e"
o
.(b.
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.-"
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Mode2 Frequency:43.6 Hz
9149149
9
[]
D 9149149149 D
9
Frequency:9"8 Hz
. ~ r 9149 ,
".~'.. ....O'"" "-....-" ]
"''''''()
Mode 1
9
.-'""" 9 9S
'
iiii
..Q,,
(a)
(:
9149149149
':)
.0.
()
Mode3
.-'""
---iii ..o-----
Fig. 18.4 Bodyof fluidwith a freesurfaceoscillatingwith a wall. Circlesshowpressureamplitudeandsquares indicateoppositesigns.Three-dimensionalapproachusingparabolicelements.
Fluid-structure interaction (Class I problems) studies of dam/reservoir interaction using such methods. 3~ 31 However, such methods are not generally economical for very large problems and fail in non-linear response studies. Here time-stepping procedures are required in the manner discussed in the previous chapter. However, simple application of methods developed there leads to an unsymmetric problem for the combined system (with fi and ~ as variables) due to the form of the matrices appearing in (18.30) and a modified approach is desirable. 32 In this, each of the equations (18.26) and (18.28) is first discretized in time separately using the general approaches of Chapter 17. Thus in the time interval At we can approximate 0 using, say, the general SS22 procedure as follows. First we write r2 UI = Un + an z" + O ~ 2
(18.34)
with a similar expression for p, T2 = Pn + 0n "g + ~ ~ -
(18.35)
where r = t - tn. Insertion of the above into Eqs (18.26) and (18.28) and weighting with two separate weighting f u n c t i o n s results in two relations in which cz and/3 are the unknowns. These are Mo~ + C(Un+l + 01Alex) -+- K(Un+l -+- 102A/2(x) -- Q(Pn+l + 102At2/~) + fn+l -- 0
(18.36a)
and 8/3 + QTcz + H (Pn+l + ~102At2/3) + qn+l -" 0
(18.36b)
where 1Uln+l = an + 01Attin ~n+l -- [In
(18.37)
Pn+l = Pn + 01Atpn
are the predictors for the n + 1 time step. In the above the parameters 0 i and 0i are similar to those of Eq. (18.49) and can be chosen by the user. It is interesting to note that the equation system can be put in symmetric form as
/. + where the second equation has been multiplied by - 1 , the unknown/3 has been replaced by /~ -- 102At2/3 (18.39) and the forces are given by F1 -- --fn+l -- C~n+l - KUn+l -+- Qpn+l
(18.40)
F2 - qn+l + H0n+l It is not necessary to go into detail about the computation steps as these follow the usual patterns of determining cz and/3 and then evaluation of the problem variables, that is Un+l,
641
642
Coupledsystems Pn+l, tln+l and Pn+l at tn+l before proceeding with the next time step. Non-linearity of structural behaviour can also be accommodated (e.g., see reference 33). It is, however, important to consider the stability of the linear system which will, of course, depend on the choice of Oi and Oi. Here we find, by using procedures described in Chapter 17, that unconditional stability is obtained when 1
02 >__O1 02 ~ O1
O1 >_ ~ O1 ~
--
--
(18.41)
1
It is instructive to note that precisely the same result would be obtained if GN22 approximations were used in Eqs (18.34) and (18.35). The derivation of such stability conditions is straightforward and follows precisely the lines of Sec.17.4 of the previous chapter. However, the algebra is sometimes tedious. Nevertheless, to allow the reader to repeat such calculations for any case encountered we shall outline the calculations for the present example.
Stability of the fluid-structure time-stepping scheme 32
For stability evaluations it is always advisable to consider the modally decomposed system with scalar variables. We thus rewrite Eqs (18.36a) and (18.36b) omitting the forcing terms and putting Oi = Oi as mot + C([l n
-+- 01Atot) + k(un + O1AtUn -k- 102 At2ot)
-- q ( p . + 01Atpn + 102At2/3) = 0
and
1 s/3 + q a + h(pn -~- 01Atp + ~02At2/~)
-- 0
(18.42a)
(18.42b)
To complete the recurrence relations we have 1
Un+ 1 ~" Un -+- A t { t n Jr ~ AtZot
/~n+l = i/n -~- Atot Pn+l = Pn + A t p n -t-
1 At2/3
(18.42c)
ign+l = Pn + At/3
The exact solution of the above system will always be of the form Un+l ~ ll~Un
/~/n+l --" /Z/~n
(18.43)
Pn+l -'-/zp, /)n+l = /ZPn and immediately we put /z=
l+z
1-z knowing that for stability we require the real part of z to be negative. Eliminating all n + 1 values from Eqs (18.42c) and (18.43) leads to 2z
~n -- - - ~ Ign
4Z2 ot = (1 - z) At 2 Un
2z
19n = - ~ pn
/~
4Z2 (1 - z) At 2 Pn
(18.44)
Fluid-structure interaction (Class I problems) Inserting (18.44) into the system (18.42a) and (18.42b) gives (allZ 2 -+- bllZ --1-k)un + (al2z 2 q- bl2z - q)Pn = 0 4qZ2Un -]- (a22z 2 + b22z -+- h ' ) p n - 0
(18.45)
where all
--
4 m ' - 2(1
-
a12 =
2q(01 - 02)
a22 - -
4s
-
201)c t-
2k(01
-02)
2(01 - O 2 ) h '
(18.46)
bll = 2 c ' - k(1 - 201) b12 = (1 -
201)q
b22 - - - ( 1
-
201)h'
m At 2
ct =
in which mt =
c At
ht _ A t 2 h
For non-trivial solutions to exist the determinant of the coefficient matrix Eq. (18.45) has to be zero. This determinant provides the characteristic equation for z which, in the present case, is a polynomial of fourth order of the form aoz 4 -Jr a l z 3 + a2z 2 + a3z -t- a4 -- 0
Thus use of the Routh-Hurwitz conditions given in Sec. 17.4 ensures stability requirements are satisfied, i.e., that the roots of z have negative real parts. For the present case the requirements are the following a0 > 0
and
ai > 0,
i = 1,2,3,4
The inequality a l i a 2 2 - - 8 q 2 ( 0 1 - - 02) > 0
(18.47)
is satisfied for m', c', k, s, h' > 0 if 1
O1 ~> ~
02 ~ O1
The inequality al = all [ - h ' ( 1 - 201)] + [ 2 c ' - k(1 - 201)] a22 - 4qb12 > 0
(18.48)
is also satisfied if 1
O1 >__ ~
02 >__O1
The inequalities all h p q-bllb22 -+ a22k -+-4q 2 > 0 a3 -- bll h t + b:z2k > 0 a2
=
(18.49)
are satisfied if (18.47) and (18.48) are satisfied. The inequality a4 -- k h ' > 0
(18.50)
643
644
Coupledsystems is automatically satisfied. Finally the two inequalities a l a 2 -- aoa3 >__ 0
(18.51)
a l a 2 a 3 -- a o a 2 -- a 4 a 2 > 0
are also satisfied if (18.47) and (18.48) are satisfied. If all the equalities hold then m ' s > 0 has to be satisfied. In case then 02 > 01 must be enforced.
m's
= 0 and c' = 0
18.2.7 Special case of incompressible fluids If the fluid is incompressible as well as being inviscid, its behaviour is described by a simple laplacian equation VZp = 0
(18.52)
obtained by putting c = cx~ in Eq. (18.5). In the absence of surface wave effects and of non-zero prescribed pressures the discrete equation (18.28) becomes simply I-I~ = _QT~
(18.53)
as wave radiation disappears. It is now simple to obtain = --H-1QTfi
(18.54)
and substitution of the above into the structure equation (18.26) results in (M + QH-1QT)~ + C~ + K~ + f = 0
(18.55)
This is now a standard structural system in which the mass matrix has been augmented by an a d d e d m a s s m a t r i x as Mu = QH-1QT
(18.56)
and its solution follows the standard procedures of previous chapters. We have to remark that: 1. In general the complete inverse of H is not required as pressures at interface nodes only are needed. 2. In general the question of when compressibility effects can be ignored is a difficult one and will depend much on the frequencies that have to be considered in the analysis. For instance, in the analysis of the reservoir-dam interaction much debate on the subject has been recorded. 34 Here the fundamental compressible period may be of order H / c where H is a typical dimension (such as height of the dam). If this period is of the same order as that of, say, earthquake forcing motion then, of course, compressibility must be taken into account. If it is much shorter then its neglect can be justified.
Soil-pore fluid interaction (Class II problems)
18.2.8 Cavitation effects in fluids In fluids such as water the linear behaviour under volumetric strain ceases when pressures fall below a certain threshold. This is the vapour pressure limit. When this is reached cavities or distributed bubbles form and the pressure remains almost constant. To follow such behaviour a non-linear constitutive law has to be introduced. Although this book is primarily devoted to linear problems we here indicate some of the steps which are necessary to extend analyses to account for non-linear behaviour. A convenient variable useful in cavitation analysis was defined by Newton 35 s = div(pu)
(18.57)
-- V T (pu)
where u is the fluid displacement. The non-linearity now is such that p = - K divu P = Pa -
=
c2s,
if s < (Pa -- P v ) / c 2 if s > (Pa -- P v ) / c 2
Pv,
(18.58)
Here Pa is the atmospheric pressure (at which u = 0 is assumed), pv is the vapour pressure and c is the sound velocity in the fluid. Clearly monitoring strains is a difficult problem in the formulation using the velocity and pressure variables [Eq. (18.1) and (18.5)]. Here it is convenient to introduce a displacement potential ~ such that pu = - V ~ (18.59) From the momentum equation (18.1) we see that =
=
-re
and thus ~) = p
(18.60)
The continuity equation (18.2) now gives s-
1
1 ..
p d i v u - - - V 2 1 / / " = ~ - - ~ p - - ~--~lp
(18.61)
in the linear case [with an appropriate change according to conditions (18.58) during cavitation]. Details of boundary conditions, discretization and coupling are fully described in reference 36 and follow the standard methodology previously given. Figure 18.5, taken from that reference, illustrates the results of a non-linear analysis showing the development of cavity zones in a reservoir.
It is well known that the behaviour of soils (and indeed other geomaterials) is strongly influenced by the pressures of the fluid present in the pores of the material. Indeed, the
645
646
Coupledsystems
Fig. 18.5 The Bhakra dam-reservoir system36 Interaction during the first second of earthquake motion showing the development of cavitation. concept of effective stress is here of paramount importance. Thus if ~r describes the total stress (positive in tension) acting on the total area of the soil and the pores, and p is the pressure of the fluid (positive in compression) in the pores (generally of water), the effective stress is defined as cr' = ~r + m p
(18.62)
Here m T = [1, 1, 1, 0, 0, 0] if we use the notation in Chapter 11. Now it is well known that it is only the stress o" which is responsible for the deformations (or failure) of the solid skeleton of the soil (excluding here a very small volumetric grain compression which has to be included in some cases). Assuming for the development given here that the soil can be represented by a linear elastic model we have o - ' = De
(18.63)
Soil-pore fluid interaction (Class II problems) Immediately the total discrete equilibrium equations for the soil-fluid mixture can be written in exactly the same form as is done for all problems of solid mechanics: Mfi + Cfi + f~ BTtr dr2 + f = 0
(18.64)
where fi are the displacement discrefization parameters, i.e., u ~
N~
fi -
(18.65)
B is the strain-displacement matrix and M, C, f have the usual meaning of mass, damping and force matrices, respectively. Now, however, the term involving the stress must be split as f BTtr dr2 = ~ BTtr' dr2 - ~ BTmp dr2
(18.66)
to allow the direct relationship between effective stresses and strains (and hence displacements) to be incorporated. For a linear elastic soil skeleton we immediately have Mfi + Cd + Kfi - Q~ + f = 0
(18.67)
where K is the standard stiffness matrix written as j ( BTtr' dr2 = ( f
BTDBdf2) fi = Kfi
(18.68)
and Q couples the field of pressures in the equilibrium equations assuming these are discretized as p ~ ~b = Np~ (18.69) Thus Q - f~ BTmNp dr2
(18.70)
In the above discretization conventionally the same element shapes are used for the fi and variables, though not necessarily identical interpolations. With the dynamic equations coupled to the pressure field an additional equation is clearly needed from which the pressure field can be derived. This is provided by the transient seepage equation of the form 1
- V T ( k V p ) + ~ b + ev = 0
(18.71)
where Q is related to the compressibility of the fluid, k is the permeability and ev is the volumetric strain in the soil skeleton, which on discretization of displacements is given by ev = m Te = mTBfi
(18.72)
The equation of seepage can now be discretized in the standard Galerkin manner as QTfi + S~ + n ~ + q = 0
(18.73)
where Q is precisely that of Eq. (18.70), and S = f~ N pT~1 N p dr2
H = f~ ( V N p ) T k V N p dr2
(18.74)
647
648
Coupledsystems with q containing the forcing and boundary terms. The derivation of coupled flow-soil equations was first introduced by Biot 37 but the present formulation is elaborated upon in references 32, 34--45 where various approximations, as well as the effect of various non-linear constitutive relations, are discussed. We shall not comment in detail on any of the boundary conditions as these are of standard type and are well documented in previous chapters.
18.3.2 The format of the coupled equations The solution of coupled equations often involves non-linear behaviour, as noted previously in the cavitation problem. However, it is instructive to consider the linear version of Eqs (18.67) and (18.73). This can be written as
[o :1
C
:]
[:
d
{o} {q} fi
f
(18.75)
Once again, like in the fluid-structure interaction problem, overall asymmetry occurs despite the inherent symmetry of the M, C, K, S and H matrices. As the free vibration problem is of no great interest here, we shall not discuss its symmetrization. In the transient solution algorithm we shall proceed in a similar manner to that described in Sec. 18.2.6 and again symmetry will be observed.
18.3.3 Transient step-by-step algorithm Time-stepping procedures can be derived in a manner analogous to that presented in Sec. 18.2.6. Here we choose to use the GNpj algorithm of lowest order to approximate each variable. Thus for fi we shall use GN22, writing Un+l -m- Un +
1
Atfin + ~ Ataiin
1
+ ~/~2At2AUn+l
P + 21/~2 At2 Afin+l Un+l
(18.76)
I]ln+l = tlln + A t tl n - t - / 3 1 A t A O n + l .p Un+ 1 + / 3 1 A t A O n + l
For the variables p that occur in first-order form we shall use GN11, as Pn+l - - P n + A t p n - + - O A t A p n + l p = Pn+ 1 -+- 0 At Apn +1
(18.77)
In the above u.+ p 1, etc., denote values that can be 'predicted' from known parameters at time tn and AiJn+l = Un+l -- iin APn+I = 0n+l -- Pn (18.78) are the unknowns. To complete the recurrence algorithm it is necessary to insert the above into the coupled governing equations [(18.64) and (18.73)] written at time tn+l. Thus we require the following equalities
Soil-pore fluid interaction (Class II problems) MUn+l at- Ciin+l at-
B O'n+ 1 - Qpn+l + fn+l --" 0
(18.79)
QTlin+l @ Spn+l + Hpn+l + qn+l : 0 in which o"+1 is evaluated using the constitutive equation (18.63) in incremental form and knowledge of trnf as I ! O'n4_1 -- or n +
DA6n+I
I = or n -'~ DBAu,,+I
(18.80)
In general the above system may be non-linear and indeed on many occasions the H matrix itself may be dependent on the values of u due to permeability variations with strain. It is of interest to look at the linear form as the non-linear system usually solves a similar form iteratively. 33 Here insertion of Eqs (18.76), (18.77) and (18.80) into (18.79) results in the equation system [ (M d-/31AtC lfl2At2K)_QT d-
+ ~-~S)] (AU. }nPn+l 'n+I } _ {F1 F2
_ (-Q1H
(18.81)
where F1 and F2 are vectors that can be evaluated from loads and solution values at Symmetry in the above is obtained by multiplying Eq. (18.36b) by - 1 and defining
tn.
(18.82)
Apn+I -- /~1AtA0n+l
The solution of Eq. (18.81) and the use of Eqs (18.76) and (18.77) complete the recurrence relation. The stability of the linear scheme can be found by following identical procedures to those used in Sec. 18.2.6 and the result is that stability is unconditional when 27
/32 >_ /~1
/~1 >_ ~1
0 >_ ~1
(18.83)
18.3.4 Special cases and robustness requirements Frequently the compressibility of the fluid phase, which forms the matrix S, is such that S~0 compared with other terms. Further, the permeability k may on occasion also be very small (as, say, in clays) and H~0 leading to so-called 'undrained' behaviour. Now the coefficient matrix in (18.81) becomes of the lagrangian constrained form (see Chapter 10), i.e., _QT
-Q01 { A_i_i A n +Pln}+ l
{F:}
and is solvable only if nu > n p
where
nu
and n p denote the number of fi and ~ parameters, respectively.
(18.84)
649
650 Coupledsystems
OU
Dp
0
Fig. 18.6 'Robust' interpolations for the coupled soil-fluid problem. The problem is indeed identical to that encountered in incompressible behaviour and the interpolations used for the u and p variables have to satisfy identical criteria. As Co interpolation for both variables is necessary for the general case, suitable element forms are shown in Fig. 18.6 and can be used with confidence. Alternatively, equal order interpolation may be used for u and p in conjunction with stabilized forms discussed in Sec.11.7. The formulation can of course be used for steady-state solutions but it must be remarked that in such cases an uncoupling occurs as the seepage equation can be solved independently. Finally, it is worth remarking that the formulation also solves the well-known soil consolidation problem where the phenomena are so slow that the dynamic term Mfi tends to 0. However, no special modifications are necessary and the algorithm form is again applicable.
18.3.5 Examples - soil liquefaction As we have already mentioned, the most interesting applications of the coupled soil-fluid behaviour is when non-linear soil properties are taken into account. In particular, it is a well-known fact that repeated straining of a granular, soil-like material in the absence of the pore fluid results in a decrease of volume (densification) due to particle rearrangement. Constitutive equations which include this effect are available; 33 however, here we only represent a typical result which they can achieve when used in a coupled soil-fluid solution. When a pore fluid is present, densification will (via the coupling terms) tend to increase the fluid pressures and hence reduce the soil strength. This, as is well known, decreases with the compressive mean effective stress. It is not surprising therefore that under dynamic action the soil frequently loses all of its strength (i.e., liquefies) and behaves almost like a fluid, leading occasionally to catastrophic failures of structural foundations in earthquakes. The reproduction of such phenomena with computational models is not easy as a complete constitutive behaviour description for soils is imperfect. However, much effort devoted to the subject has produced good results 38-45
Soil-pore fluid interaction (Class II problems)
Fig. 18.7 Soil-pressure water interaction. Computation and centrifuge model results compared on a problem of a dyke foundation subject to a simulated earthquake.
651
652 Coupledsystems
Fig. 18.7 (Cont.) and a reasonable confidence in predictions achieved by comparison with experimental studies exists. One such study is illustrated in Fig. 18.7 where a comparison with tests carried out in a centrifuge is made. 44, 45 In particular the close correlation between computed pressure and displacement with experiments should be noted.
18.3.6 Biomechanics, oil recovery and other applications The interaction between a porous medium and interstitial fluid is not confined to soils. The same equations describe, for instance, the biomechanics problem of bone-fluid interaction in vivo. Applications in this field have been documented. 46, 47 On occasion two (or more) fluids are present in the pores and here similar equations can again be written 46' 47 to describe the interaction. Problems of ground settlement in oil fields
Partitioned single-phase systems -implicit-explicit partitions (Class I problems) 653 due to oil extraction, or flow of water/oil mixtures in oil recovery are good examples of application of techniques described here.
In Fig. 18.1 (b), describing problems coupled by an interface, we have already indicated the possibility of a structure being partitioned into substructures and linked along an interface only. Here the substructures will in general be of a similar kind but may differ in the manner (or simply size) of discretization used in each or even in the transient recurrence algorithms employed. In Chapter 12 we have described special kinds of mixed formulations allowing the linking of domains in which, say, boundary-type approximations are used in one and standard finite elements in the other. We shall not return to this phase and will simply assume that the total system can be described using such procedures by a single set of equations in time. Here we shall only consider a first-order problem (but a similar approach can be extended to the second-order dynamic system): Cd + Kfi + f -- 0
(18.85)
which can be partitioned into two (or more) components, writing
Cll C12] ~1 C21 C22J / ~ 2 ) "l- IK121 K12]
(18.86)
Now for various reasons it may be desirable to use in each partition a different time-step algorithm. Here we shall assume the same structure of the algorithm (SS 11) and the same time step (At) but simply a different parameter 0 in each. Proceeding thus as in the other coupled analyses we write
UI.1= Uln "+""~'0~1 1HI2~ U2n -I'-T'O~2
(18.87)
Inserting the above into each of the partitions and using different weight functions, we obtain
CllO~l -+- C120~2 -+- Kll(Uln + 0AtCzl) + K12(U2n -+- 0Ato~2) + f l = 0 C210~1 + C220~2 + K21 (Uln -+- 0At~l) -+- K22(U2n -+-0Atcz2) + f2 = 0
(18.88)
This system may be solved in the usual manner for O~ 1 and O~2 and recurrence relations obtained even if 0 and 0 differ. The remaining details of the time-step calculations follow the obvious pattern but the question of coupling stability must be addressed. Details of such stability evaluation in this case are given elsewhere 48 but the result is interesting. 1. Unconditional stability of the whole system occurs if 0>~ 1 2. Conditional stability requires that
0 >_1~
654
Coupledsystems At _< Atcrit where the Atcrit condition is that pertaining to each partitioned system considered without its coupling terms.
Indeed, similar results will be obtained for the second-order systems Mfi + Cfi + Kfi + f - 0
(18.89)
partitioned in a similar manner with SS22 or GN22 used in each. The reader may well ask why different schemes should be used in each partition of the domain. The answer in the case of implicit-implicit schemes may be simply the desire to introduce different degrees of algorithmic damping. However, much more important is the use of implicit-explicit partitions. As we have shown in both 'thermal' and dynamictype problems the critical time step is inversely proportional to h e and h (the element size), respectively. Clearly if a single explicit scheme were to be used with very small elements (or very large material property differences) occurring in one partition, this time step may become too short for economy to be preserved in its use. In such cases it may be advantageous to use an explicit scheme (with 0 - 0 in first-order problems, 02 = 0 in dynamics) for a part of the domain with larger elements while maintaining unconditional stability with the same time step in the partition in which elements are small or otherwise very 'stiff'. For this reason such implicit-explicit partitions are frequently used in practice. Indeed, with a lumped representation of matrices C or M such schemes are in effect staggered as the explicit part can be advanced independently of the implicit part and immediately provides the boundary values for the implicit partition. We shall return to such staggered solutions in the next section. The use of explicit-implicit partitions was first recorded in 1978. 49-51 In the first reference the process is given in an identical manner as presented here; in the second, a different algorithm is given based on an element split (instead of the implied nodal split above) as described next.
partition
Implicit-explicit solution - element
We again consider the first-order problem given in Eq. (18.85) and split as
Ci~I -JI--CEUE ']- Klfll + Kefie + f = 0
(18.90)
where the subscript I denotes an implicit partition and subscript E an explicit one. An iteration process may be used in which one or more iterations per time step are used. The recurrence relation for u at iteration j is written using GN11 as i.t(j) n+l
with
. n( +j -l 1) + -- U
0
9( j ) AtUn+l
a(0)
n+ 1 -- Un + (1 -- O) Ati~n
(18.91) (18.92)
Using the iteration process an approximation for the implicit-explicit split is now taken as ~ Hi
. (J) -- Un+ 1
~ BE
(j-l) -- Un+ 1
~I
=
~E
=
"U(nJ +) 1
thus yielding the system of equations at iteration j as
Staggered solution processes 655 (C +
OAtKI)fi~J)+l + F (j)
(18.93)
= 0
where F (j) contains the loading terms which depend on known values at tn and possibly previous iterate values (j - 1). The above algorithm has stability properties which depend on the choice of 0. For a linear system with 0 >_ 0.5 the implicit part is unconditionally stable and stability depends onthe Atcr,t of the explicit elements. 5~ 51 Performing only one iteration in each time step is normally used; however, improved accuracy in the explicit partition can occur if additional iterations are used, although the cost of each time step is obviously increased.
We have observed in the previous section that in the nodal-based implicit-explicit partitioning of time stepping it was possible to proceed in a staggered fashion, achieving a complete solution of the explicit scheme independently of the implicit one and then using the results to progress with the implicit partition. It is tempting to examine the possibility of such staggered procedures generally even if each uses an independent algorithm. In such procedures the first equation would be solved with some assumed (predicted) values for thevariable of the other. Once the solution for the first system was obtained its values could be substituted in the second system, again allowing its independent treatment. If such procedures can be made stable and reasonably accurate many possibilities are immediately open, for instance: 1. Completely different methodologies could be used in each part of the coupled system. 2. Independently developed codes dealing efficiently with single systems could be combined. 3. Parallel computation with its inherent advantages could be used. 4. Finally, in systems of the same physics, efficient iterative solvers could easily be developed. The problems of such staggered solutions have been frequently discussed 36' 52-55 and on occasion unconditional stability could not be achieved without substantial modification. In the following we shall indicate some options available.
18.5.2 Staggered process of solution in single-phase systems We shall look at this possibility first, having already mentioned it as a special form arising naturally in the implicit-explicit processes of Sec~l8.4. We return here to consider the problem of Eq. (18.85) and the partitioning given in Eq. (18.86). Further, for simplicity we shall assume a diagonal form of the C matrix, i.e., that the problem is posed as
1 c 2]
{:}
1894
656 Coupledsystems As we have already remarked, the use of 0 = 0 in the first equation and 0 > 0.5 in the second [see Eq. (18.88)] allowed the explicit part to be solved independently of the implicit. Now, however, we shall use the same 0 in both equations but in the first of the approximations, analogous to Eq. (18.88), we shall insert a predicted value for the second variable: UI2 -" U~ "-- UZn
(18.95)
This is similar to the treatment of the explicit part in the element split of the implicit-explicit scheme and gives in place of Eq. (18.88) (18.96)
CllOtl -~- Kll(Uln -~-0AtOtl) = - f l - K121LI2n
allowing direct solution for c~1. Following this step, the second equation can be solved for ot2 with the previous value of c~1 inserted, i.e., (18.97)
C220t2 -+- K22(u2 n -+-0Atot2) = - f 2 - K21 (Uln -+-0AtOtl)
This scheme is unconditionally stable if 0 > 0.5, i.e., the total system is stable provided each stagger is unconditionally stable. A similar condition holds for linear second-order dynamic problems. Obviously, however, some accuracy will be lost as the approximation of Eq. (18.96) is that of the predicted value of u2. The approximation is consistent and hence convergence will occur as At ~ 0. The advantage of using the staggered process in the above is clear as the equation solving, even though not explicit, is now confined to the magnitude of each partition and computational economy occurs. Further, it is obvious that precisely the same procedures can be used for any number of partitions and that again the same stability conditions will apply. Define the arrays "ell
C22 ~
C
__.
(18.98a)
Cii Ckk
"Kll
K21
0
~
1
4
9
1
4
0
9
K22
+
.
K
m
K12
o
0
..
.
Klk
.
~
~
Kii .
Kkl
""
Kk,k-1
0
Kkk.
Kk- l,k 0
.
o
.
0
"-'KL + K u (18.98b) and consider the partition
Staggered solution processes
0.8
I
m
0.6 0.4 0.2
m
Implicit = 1 At/Atcrit = 4
y/
~..'"
'
~s S
###"
-
J
~
iExplicit'-sPlit
,""
At/Atcrit = 4
_-......"~_ _ . I.. -'~176176176I
0
20
40
T-, 1
I
I
I
I
60
80
100
120
t/Atcrit
~
140
I
# To=0
I
critical time step for standard explicit form Tc = temperature on centre-line
Atcrit =
Fig. 18.8 Accuracyof an explicit-split procedure compared with a standard implicit process for heat conduction of a bar.
C~ + KL~ + Ku~ + f = 0
(18.99)
Introducing now the approximation Ui --Uin
Jr- 150t i
(18.100)
and using Eq. (18.95) gives the discrete form (C + 0AtKL) c~ + Kuun 4- t' - 0
(18.101)
where f contains the load and effects from u,,. In approximating the first equation set it is necessary to use predicted values for u2, u3, . . . , uk, writing in place of Eq. (18.96),
CllO~l -q- Kll(Uln -+-0AtOll) -+- K12u2n "q- K13u3n "~-""" "~-fl -" 0
(18.102)
and continue similarly to (18.97), with the predicted values now continually being replaced by better approximations as the solution progresses. The partitioning of Eq. (18.98a) can be continued until only a single equation set is obtained. Then at each step the equation that requires solving for Ot i is of the form ( C i i -3t- O A t K i i )
ot i - - Fi
(18.103)
where Fi contains the effects of the load and all the previously computed u i . For partitions where each submatrix is a scalar Eq. (18.103) is a scalar equation and computation is thus fully explicit and yet preserves unconditional stability for 0 > 0.5. This type of partitioning and the derivation of an unconditionally stable explicit scheme was first proposed by Zienkiewicz et al. 56 An alternative and somewhat more limited scheme of a similar kind was given by Trujillo. 57
657
658 Coupledsystems 9
lw
9 9
=A~, ~ 5,,
~,
lw
i
9
9
3,, 4
,
Z,
~
8.,
9
lw
.~,, 9
9r
9
~w
9F
2. ~, 4,,~. ~,
9
~,~
9
9
; 9,;,10,, 11(, 12,
13,,; 14,, , 15,,) 1~)
13, 14 15, 1~ ',27.
~~"
""
~"
:
Fig. 18.9 Partitionscorrespondingto the well-known ADI (alternating direction implicit) finite difference scheme. Clearly the error in the approximation in the time step decreases as the solution sweeps through the partitions and hence it is advisable to alter the sweep directions during the computation. For instance, in Fig. 18.8 we show quite reasonable accuracy for a onedimensional heat-conduction problem in which the explicit-split process was used with alternating direction of sweeps. Of course the accuracy is much inferior to that exhibited by a standard implicit scheme with the same time step, though the process could be used quite effectively as an iteration to obtain steady-state solutions. Here many other options are also possible. It is, for instance, of interest to consider the system given in Eqs (18.98a), (18.98b) and (18.99) as originating from a simple finite difference approximation to, say, a heatconduction equation on the rectangular mesh of Fig. 18.9. Here it is well known that the so-called alternating direction implicit (ADI) scheme 58 presents an efficient solution for both transient and steady-state problems. It is fairly obvious that the scheme simply represents the procedure just outlined with partitions representing lines of nodes such as (1, 5, 9, 13), (2, 6, 10, 14), etc., ofFig. 18.9 alternating with partitions (1, 2, 3, 4), (5, 6, 7, 8), etc. Obviously the bigger the partition, the more accurate the scheme becomes, though of course at the expense of computational costs. The concept of the staggered partition clearly allows easy adoption of such procedures in the finite element context. Here irregular partitions arbitrarily chosen could be made but so far applications have only been recorded in regular mesh subdivisions. 58 The field of possibilities is obviously large. Use in parallel computation is obvious for such procedures. A further possibility which has many advantages is to use hierarchical variables based on, say, linear, quadratic and higher expansions and to consider each set of these variables as a partition. 59 Such procedures are particularly efficient in iteration if coupled with suitable preconditioning 60 and form a basis of multigridprocedures. 61-63
18.5.3 Staggered schemes in fluid-structure systems and
stabilization processes
The application of staggered solution methods in coupled problems representing different phenomena is more obvious, though, as it turns out, more difficult.
Staggered solution processes 659 For instance, let us consider the linear discrete fluid-structure equations with Po = 1 and damping omitted, written as [see Eqs (18.26) and 18.28)]
sl {;}+
(18.104)
where we have omitted the tilde superscript for simplicity. For illustration purposes we shall use the GN22 type of approximation for both variables and write using Eq. (18.76) p 1 Un_l_1 -~ Un+ 1 -~- ~/32AtZAfin+l .p I]ln+l = Un+l "]"/~1 At Aiin+l p 1 P n + l = P n + l -+- ~ / ~ 2 A t 2 A l J n + l 1J-n+l -" P"np+ l + / ~ 1
(18.105)
At A P n + l
which together with Eq. (18.104) written at t -- tn+l completes the system of equations requiring simultaneous solution for A n n + 1 and AlJn+l. Now a staggered solution of a fairly obvious kind would be to write the first set of equations (18.104) corresponding to the structural behaviour with a predicted (approximate) p value of Pn+l "- Pn+l, as this would allow an independent solution for Aiin+l writing Miin+l -+- KUn+l
-
-
P - f + QPn+I
(18.106)
This would then be followed by the solution of the fluid problem for AlJn+ 1 writing Sii~+l + Hp~+I = - q - QTiin+I
(18.107)
This scheme turns out, however, to be only conditionally stable, 48 even if/3/and/3i are chosen so that unconditional stability of a simultaneous solution is achieved. (The stability limit is indeed the same as if a fully explicit scheme were chosen for the f l u i d phase.) Various stabilization schemes can be used here. 27' 48 One of these is given below. In this Eq. (18.106) is augmented to MUn+l -I-- (K -~- QS-1QT)un+I = - f -t- QpnP+I -I- Q s -1QTun+lp
(18.108)
before solving for Aiin+l. It turns out that this scheme is now unconditionally stable provided the usual conditions
/~2 >__/~1
~1 >__
1
are satisfied. Such stabilization involves the inverse of S but again it should be noted that this needs to be obtained only for the coupling nodes on the interface. Another stable scheme involves a similar inversion of H and is useful as incompressible behaviour is automatically given. Similar stabilization processes have been applied with success to the soil-fluid system.64, 65
660
Coupled systems
iii ig77:7 Sq!iiiiiiii !{iiSiiiii!:!ii!{ili;iii;!:i i!!!iiiiii iiiiiiii iiiiii !i :;==:::i : i :;i:~71: : ~:17i77i{77i 7!!!: 7!17 :iT::iiii! i777i7~;ili!~i!i~i77i1i1iii~7i:!ii!i!iiTi=~:Ti ii~;'=i!:~iT=iii!~::i:i77i1T1ii!i:i~i~i !i~:iii!i:T ~i~illT ~ii~!~:~::i~~!i ~i!:i~~i ::=ii77 !i=i717 ~~ili:iii7i!7= ~=i?i!i?i~!i:i!iili;iliii; !i:i,:7i~!iiil ;~i;i:7~=:i;7~~7i!!i7:?i;:i~!ii;ii i!~~:?1T1i7!7i?!!~i i:~i!!?; i!!7i ii;i; i!iiii7:i:Tiii~!i!!7~i !iiiTiiTii!;ilii i!i)i~!iTiiii!il i7ii7ii7~i??ii7iiiii ii~Ti 7i!iiii! iii~7i!~iiii;:i!iT;i~!!17~~77:?!?i ~i??i~?iT?i{!i i?ii?iii;7iiiii!!~i :iTiiiiT7!ii!:!i~if:!i;~iiiilii; :,ii:7iii! ~i;i;!;ii?i;1;;i7ii;!?7~:71;?=:~:?i:i:!::~I.ii:!~!?; T h e r a n g e o f p r o b l e m s w h i c h m a y b e c o n s i d e r e d as c o u p l e d is v e r y large a n d f o r m s studies w h i c h are n o w o f t e n r e f e r r e d to as ' m u l t i - p h y s i c s ' p r o b l e m s . T h e r a n g e o f p o s s i b l e algor i t h m s to s o l v e s u c h p r o b l e m s has b e e n s u m m a r i z e d above; h o w e v e r , n e w m e t h o d s often are p r o p o s e d (e.g., see r e f e r e n c e 66). A n o t h e r class of p r o b l e m s w h i c h m a y be c o n s i d e r e d as c o u p l e d c o n s i d e r s ' m u l t i - s c a l e ' effects. T h e s e a t t e m p t to b r i d g e the b e h a v i o u r o f m a t e r i a l s from, for e x a m p l e , a m i c r o to a m a c r o scale. T h i s topic is v e r y p o p u l a r t o d a y a n d is d i s c u s s e d f u r t h e r in r e f e r e n c e 33.
1. O.C. Zienkiewicz. Coupled problems and their numerical solution. In R.W. Lewis, R Bettess, and E. Hinton, editors, Numerical Methods in Coupled Systems, Chapter 1, pages 35-68. John Wiley & Sons, Chichester, 1984. 2. B.A. Boley and J.H. Weiner. Theory of Thermal Stresses. Dover Publications, Mineola, New York, 1997. 3. R.W. Lewis, E Bettess, and E. Hinton, editors. Numerical Methods in Coupled Systems. John Wiley & Sons, Chichester, 1984. 4. R.W. Lewis, E. Hinton, E Bettess, and B.A. Schrefler, editors. Numerical Methods in Coupled Systems. John Wiley & Sons, Chichester, 1987. 5. J.C. Simo and T.J.R. Hughes. Computational Inelasticity, volume 7 of Interdisciplinary Applied Mathematics. Springer-Verlag, Berlin, 1998. 6. O.C. Zienkiewicz, R.L. Taylor, and E Nithiarasu. The Finite Element Method for Fluid Dynamics. Butterworth-Heinemann, Oxford, 6th edition, 2005. 7. O.C. Zienkiewicz and R.E. Newton. Coupled vibration of a structure submerged in a compressible fluid. In Proc. Int. Symp. on Finite Element Techniques, pages 359-371, Stuttgart, 1969. 8. R Bettess and O.C. Zienkiewicz. Diffraction and refraction of surface waves using finite and infinite elements. Int. J. Numer. Meth. Eng., 11:1271-1290, 1977. 9. O.C. Zienkiewicz, D.W. Kelly, and R Bettess. The Sommerfield (radiation) condition on infinite domains and its modelling in numerical procedures. In Proc. IRIA 3rd Int. Symp. on Computing Methods in Applied Science and Engineering, Versailles, Dec. 1977. 10. O.C. Zienkiewicz, R Bettess, and D.W. Kelly. The finite element method for determining fluid loadings on rigid structures. Two- and three-dimensional formulations. In O.C. Zienkiewicz, R.W. Lewis, and K.G. Stagg, editors, Numerical Methods in Offshore Engineering, pages 141183. John Wiley & Sons, Chichester, 1978. 11. O.C. Zienkiewicz and R Bettess. Dynamic fluid-structure interaction. Numerical modelling of the coupled problem. In O.C. Zienkiewicz, R.W. Lewis, and K.G. Stagg, editors, Numerical Methods in Offshore Engineering, pages 185-193. John Wiley & Sons, Chichester, 1978. 12. O.C. Zienkiewicz and R Bettess. Fluid-structure dynamic interaction and wave forces. An introduction to numerical treatment. Int. J. Numer. Meth. Eng., 13:1-16, 1978. 13. O.C. Zienkiewicz and E Bettess. Fluid-structure dynamic interaction and some 'unified' approximation processes. In Proc. 5th Int. Symp. on Unification of Finite Elements, Finite Differences and Calculus of Variations. University of Connecticut, May 1980. 14. R. Ohayon. Symmetric variational formulations for harmonic vibration problems coupling primal and dual v a r i a b l e s - applications to fluid-structure coupled systems. La Rechereche Aerospatiale, 3:69-77, 1979.
% :7
References 661 15. R. Ohayon. True symmetric formulation of free vibrations for fluid-structure interaction in bounded media. In R.W. Lewis, P. Bettess, and E. Hinton, editors, Numerical Methods in Coupled Systems. John Wiley & Sons, Chichester, 1984. 16. R. Ohayon. Fluid-structure interaction. In Proc. of the ECCM'99 Conference IACM/ECCM'99, Munich, Germany, Sept. 1999 (on CD-ROM). 17. H. Morand and R. Ohayon. Fluid-Structure Interaction. John Wiley & Sons, London, 1995. 18. M.P. Paidoussis and P.P. Friedmann, editors. 4th International Symposium on Fluid-Structure Interactions, Aeroelasticity, Flow-Induced Vibration and Noise, vol. 1, 2, 3, volume AD-vo. 52-3, Dallas, Texas, Nov. 1997. ASME/Winter Annual Meeting. 19. T. Kvamsdal et al., editors. Computational Methods for Fluid-Structure Interaction. Tapir Publishers, Trondheim, 1997. 20. R. Ohayon and C.A. Felippa (editors). Computational Methods for Fluid-Structure Interaction and Coupled Problems. Comp. Meth. Appl. Mech. Eng., 190:2977-3292 (Special issue). 21. M. Geradin, G. Roberts, and J. Huck. Eigenvalue analysis and transient response of fluid structure interaction problems. Eng. Comput., 1:152-160, 1984. 22. G. Sandberg and P. Gorensson. A symmetric finite element formation of acoustic fluid-structure interaction analysis. J. Sound Vibr., 123:507-515, 1988. 23. U. Basu and A.K. Chopra. Perfectly matched layers for time-harmonic elastodynamics of unbounded domains: theory and finite-element implementation. Comp. Meth. Appl. Mech. Eng., 192:1337-1375, 2003. 24. U. Basu and A.K. Chopra. Perfectly matched layers for transient elastodynamics of unbounded domains. Int. J. Numer. Meth. Eng., 59:1039-1074, 2004. 25. K.K. Gupta. On a numerical solution of the supersonic panel flutter eigenproblem. Int. J. Numer. Meth. Eng., 10:637-645, 1976. 26. B.M. Irons. The role of part inversion in fluid-structure problems with mixed variables. J. AIAA, 7:568, 1970. 27. W.J.T. Daniel. Modal methods in finite element fluid-structure eigenvalue problems. Int. J. Numer. Meth. Eng., 15:1161-1175, 1980. 28. C.A. Felippa. Symmetrization of coupled eigenproblems by eigenvector augmentation. Comm. Appl. Numer. Meth., 4:561-563, 1988. 29. J. Holbeche. Ph.D. thesis, Department of Civil Engineering, University of Wales, Swansea, 1971. 30. A.K. Chopra and S. Gupta. Hydrodynamic and foundation interaction effects in earthquake response of a concrete gravity dam. J. Struct. Div. Am. Soc. Civ. Eng., 578:1399-1412, 1981. 31. J.E Hall and A.K. Chopra. Hydrodynamic effects in the dynamic response of concrete gravity dams. Earth. Eng. Struct. Dyn., 10:333-395, 1982. 32. O.C. Zienkiewicz and R.L. Taylor. Coupled problems- a simple time-stepping procedure. Comm. Appl. Numer. Meth., 1:233-239, 1985. 33. O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method for Solid and Structural Mechanics. Butterworth-Heinemann, Oxford, 6th edition, 2005. 34. O.C. Zienkiewicz, R.W. Clough, and H.B. Seed. Earthquake analysis procedures for concrete and earth dams - state of the art. Technical Report Bulletin 32, Int. Commission on Large Dams, Paris, 1986. 35. R.E. Newton. Finite element study of shock induced cavitation. In ASCE Spring Convention, Portland, Oregon, 1980. 36. O.C. Zienkiewicz, D.K. Paul, and E. Hinton. Cavitation in fluid-structure response (with particular reference to dams under earthquake loading). Earth. Eng. Struct. Dyn., 11:463-481, 1983. 37. M.A. Biot. Theory of propagation of elastic waves in a fluid saturated porous medium, Part I: Low frequency range; Part II: High frequency range. J. Acoust. Soc. Am., 28:168-191, 1956.
662
Coupledsystems 38. O.C. Zienkiewicz, C.T. Chang, and E. Hinton. Non-linear seismic responses and liquefaction. Int. J. Numer. Anal. Meth. Geomech., 2:381-404, 1978. 39. O.C. Zienkiewicz and T. Shiomi. Dynamic behaviour of saturated porous media, the generalized Biot formulation and its numerical solution. Int. J. Numer. Anal. Meth. Geomech., 8:71-96, 1984. 40. O.C. Zienkiewicz, K.H. Leung, and M. Pastor. Simple model for transient soil loading in earthquake analysis: Part I - Basic model and its application. Int. J. Numer. Anal. Meth. Geomech., 9:453-476, 1985. 41. O.C. Zienkiewicz, K.H. Leung, and M. Pastor. Simple model for transient soil loading in earthquake analysis: Part I I - non-associative models for sands. Int. J. Numer. Anal. Meth. Geomech., 9:477-498, 1985. 42. O.C. Zienkiewicz, A.H.C. Chan, M. Pastor, and T. Shiomi. Computational approach to soil dynamics. In A.S. Czamak, editor, Soil Dynamics and Liquefaction, volume Developments in Geotechnical Engineering 42. Elsevier, Amsterdam, 1987. 43. O.C. Zienkiewicz, A.H.C. Chan, M. Pastor, D.K. Paul, and T. Shiomi. Static and dynamic behaviour of soils: a rational approach to quantitative solutions, I. Proc. R. Soc. London, 429:285-309, 1990. 44. O.C. Zienkiewicz, Y.M. Xie, B.A. Schrefler, A. Ledesma, and N. Bi~ani~. Static and dynamic behaviour of soils: a rational approach to quantitative solutions, II. Proc. R. Soc. London, 429:311-321, 1990. 45. O.C. Zienkiewicz, A.H.C. Chan, M. Pastor, B.A. Schrefler, and T. Shiomi. Computational Geomechanics: With Special Reference to Earthquake Engineering. John Wiley & Sons, Chichester, 1999. 46. B.R. Simon, J. S-S. Wu, M.W. Carlton, L.E. Kazarian, E/P. France, J.H. Evans, and O.C. Zienkiewicz. Poroelastic dynamic structural models of rhesus spinal motion segments. Spine, 10(6):494-507, 1985. 47. B.R. Simon, J. S-S. Wu, and O.C. Zienkiewicz. Higher order mixed and Hermitian finite element procedures for dynamic analysis of saturated porous media. Int. J. Numer. Meth. Eng., 10:483499, 1986. 48. O.C. Zienkiewicz and A.H.C. Chan. Coupled problems and their numerical solution. In I.S. Doltsinis, editor, Advances in Computational Non-linear Mechanics, Chapter 3, pages 109-176. Springer-Vedag, Berlin, 1988. 49. T. Belytschko and R. Mullen. Stability of explicit-implicit time domain solution. Int. J. Numer. Meth. Eng., 12:1575-1586, 1978. 50. T.J.R. Hughes and W.K. Liu. Implicit-explicit finite elements in transient analyses. Part I and Part II. J. Appl. Mech., 45:371-378, 1978. 51. T. Belytschko and T.J.R. Hughes, editors. Computational Methods for Transient Analysis. NorthHolland, Amsterdam, 1983. 52. C.A. Felippa and K.C. Park. Staggered transient analysis procedures for coupled mechanical systems: formulation. Comp. Meth. Appl. Mech. Eng., 24:61-111, 1980. 53. K.C. Park. Partitioned transient analysis procedures for coupled field problems: stability analysis. J. Appl. Mech., 47:370-376, 1980. 54. K.C. Park and C.A. Felippa. Partitioned transient analysis procedures for coupled field problems: accuracy analysis. J. Appl. Mech., 47:919-926, 1980. 55. O.C. Zienkiewicz, E. Hinton, K.H. Leung, and R.L. Taylor. Staggered time marching schemes in dynamic soil analysis and selective explicit extrapolation algorithms. In R. Shaw et al., editors, Proc. Conf. on Innovative Numerical Analysis for the Engineering Sciences, University of Virginia Press, 1980. 56. O.C. Zienkiewicz, C.T. Chang, and P. Bettess. Drained, undrained, consolidating dynamic behaviour assumptions in soils. Geotechnique, 30:385-395, 1980.
References 663 57. D.M. Trujillo. An unconditionally stable explicit scheme of structural dynamics. Int. J. Numer. Meth. Eng., 11:1579-1592, 1977. 58. L.J. Hayes. Implementation of finite element alternating-direction methods on non-rectangular regions. Int. J. Numer. Meth. Eng., 16:35-49, 1980. 59. A.W. Craig and O.C. Zienkiewicz. A multigrid algorithm using a hierarchical finite element basis. In D.J. Pedolon and H. Holstein, editors, Multigrid Methods in Integral and Differential Equations, pages 310-312. Clarendon Press, Oxford, 1985. 60. I. Babu~ka, A.W. Craig, J. Mandel, and J. Pitk~anta. Efficient preconditioning for the p-inversion finite element method in two dimensions. SlAM J. Num. Anal., 28:624-661,1991. 61. R. Lthner and K. Morgan. An unstructured multigrid method for elliptic problems. Int. J. Numer. Meth. Eng., 24:101-115, 1987. 62. M. Adams. Heuristics for automatic construction of coarse grids in multigrid solvers for finite element matrices. Technical Report UCB//CSD-98-994, University of California, Berkeley, 1998. 63. M. Adams. Parallel multigrid algorithms for unstructured 3D large deformation elasticity and plasticity finite element problems. Technical Report UCB//CSD-99-1036, University of California, Berkeley, 1999. 64. K.C. Park. Stabilization of partitioned solution procedures for pore fluid-soil interaction analysis. Int. J. Numer. Meth. Eng., 19:1669-1673, 1983. 65. O.C. Zienkiewicz, D.K. Paul, and A.H.C. Chan. Unconditionally stable staggered solution procedures for soil-pore fluid interaction problems. Int. J. Numer. Meth. Eng., 26:1039-1055, 1988. 66. J.Y. Kim, N.R. Aluru, and D.A. Tortorelli. Improved multi-level Newton solvers for fully coupled multi-physics problems. Int. J. Numer. Meth. Eng., 58:463-480, 2003.
Computer procedures for finite element analysis
A companion program to this book is available which can carry out analyses for most of the theory presented in previous chapters. In particular the computer program discussed here may be used to solve any one-, two-, or three-dimensional linear steady-state or transient problem. The program also has capabilities to perform non-linear analysis for the type of problems discussed in reference 1. Source listings and a user manual may be obtained at no charge from the author's intemet web site (http://www.ce.berkeley.edu/~rlt) or the publisher's internet web site (http:// books.elsevier.com/companions). The program is written mostly in Fortran with some routines in C (see author's web site for more information on using C for user modules). Any errors reported by readers will be corrected so that up-to-date versions are available. The version available for download is called FEAPpv which is an acronym for Finite Element Analysis Program-personal version. It is intended mainly for use in learning finite element programming methodologies and in solving small to moderate size problems on single processor computers. A simple management scheme is employed to permit efficient use of main memory with limited need to read and write information to disk. Finite element programs can be separated into three basic parts: 1. Data input module and pre-processor 2. Solution module 3. Results module and post-processor.
FEAPpv is mainly a solution module but provides simple data input and pre-processor capabilitites which permit generation of meshes using the multiblock schemes of Zienkiewicz and Phillips 2 and Gordon and Hall. 3 Alternatively the data may be input from neutral files written by other pre-processing systems (e.g., GiD4). Data input for the program consists of specification (or generation) of: (1) the coordinates for each node; (2) the element form and the nodal connection list for each element;
Pre-processing module: mesh creation 665 (3) boundary conditions and loads to be applied; and (4) material property data. The user manual describes the format for specifying the data to be used by FEAPpv.
19.2.1 Element library As part of the input data it is necessary to describe the element formulation to be used in forming the 'stiffness' matrix and 'load' vector of each problem. This may be provided either by user written modules (see below) or using the element library provided with the program. Currently, the element library in FEAPpv includes:
1. Solid elements for two-dimensional linear elasticity. Forms are provided for the irreducible formulation described in Chapters 2 and 6; the three-field mixed form described in Sec. 11.3 and the enhanced strain form described in Sec. 10.5.3. The elements permit consideration of elastic models which are isotropic or orthotropic as described in Chapter 6. (a) For the irreducible form the element shape may range from a 3-node triangle to a 9-node lagrangian quadrilateral. (b) For the three-field mixed form the element shape may be a 4-node, 8-node or 9-node quadrilateral form. (c) For the enhanced strain model the element is restricted to a 4-node quadrilateral form. 2. Solid elements for three-dimensional linear elasticity. Only the irreducible form for a 4-node tetrahedron or an 8-node brick may be used. The 8-node brick may be degenerated into other forms by giving the same node number to nodes used to perform the degenerate shape (see Sec.5.8). The elastic material model may be isotropic or orthotropic as described in Chapter 6. 3. Frame (rod) elements for two- and three-dimensional elasticity. Conventional structural elements are provided to perform analysis of elastic two- and three-dimensional frame structures. While these forms have not been discussed in this text, except as suggested problems for solution, they are useful for use in general analysis. The theory is contained in standard references for structural analysis and also in reference 1. 4. Truss elements for two- and three-dimensional elasticity. Similar to frame elements, the FEAPpv system includes conventional truss elements which may be used to analyse plane and space truss structures. 5. Plate element for linear elasticity. A plate bending element for use in the analysis of plates which include the primary effects of transverse shear (so-called Reissner-Mindlin theory 1) is provided. The element form may be either a 3-node triangle or a 4-node quadrilateral. The theory is described in references 1, 5, 6. 6. Shell element for three dimensions with linear elasticity. A 4-node quadrilateral element form for use in modelling general shell forms is provided. The element includes membrane and bending effects only and, thus, may be used only for analysis of 'thin' shells. The theory for the element is given in reference 7. The element form should be a 4-node quadrilateral.
666 Computerprocedures for finite element analysis
7. Membrane element for linear elasticity. A general elastic membrane form is provided which is the same as the shell element but without the bending terms. The element form should be a 4-node quadrilateral. 8. Thermal elements for two- and three-dimensional Fourier heat conduction. The theory described in Chapter 7 for transient heat conduction is provided in elements which solve two- and three-dimensional problems. The Fourier model may be isotropic or orthotropic. 9. User developed elements. Users may develop and add element modules for any problem which can be formed by the finite element approach described in this book. Details for writing modules will be found in the Programers Manual available at the web sites.
The main part of FEAPpv is a solution module which permits users to analyse a large range of problems formulated by the finite element method. Specific solution methods are prepared by the user using a unique command language, which is a sequence of statements which describe each algorithm. The current version of FEAPpv permits both 'batch' and 'interactive' problem solution. The commands provided permit specification of problems with either symmetric or unsymmetric 'stiffness' matrices, selection of direct or iterative solution of the linear algebraic equation system, selection of different transient solution algorithms, and output of solution results in either a text or graphics format. Commands which permit solution of a symmetric generalized linear eigenproblem (see Chapter 16) using a 'subspace' method 8, 9 are also available as well as a feature to compute the eigenvalues and vectors for an element stiffness. While the main thrust of this book is the solution of linear problems, the system FEAPpv is capable of solving both linear and non-linear problems. The use of special 'loop' commands permits the construction of algorithms which require iteration or time stepping. In addition features to solve problems in which load following is needed are provided in the form of 'arc-length'-type methods. 1~ The solution of problems for which it is not possible to deduce an accurate 'stiffness' matrix may be attempted using a quasi-Newton method based on the BFGS method. TM 14 The user manual available at the web site provides examples for several algorithms as well as a list of all available commands.
As noted above the FEAPpv system contains capabilities to report results as text data written to an output file or in graphical form which may be displayed on the screen or written to files for processing by other systems. Files are written in PostScript format (in an encapsulated form which may be used by many programs -e.g., TeX or LaTeX). The general features of graphical post-processing are limited to displaying two-dimensional objects. More complex forms require an interface to a separate pre-/post-processing system (e.g., GiD.4). The two-dimensional capabilities in FEAPpv include display of the mesh including node and element numbers, boundary conditions and loads. Contour plots for each degree of freedom of the solution system may be displayed as well as contours of
User modules 667
element values such as stress or strain components. The user manual provides a list of all commands for constructing graphical outputs. The available version for graphics is limited to X-window applications and compilers compatible with the current HP Fortran 95 compiler for Windows-based systems. 15
A key ingredient of the F E A P p v system is the ability of a user to add their own modules to extend the capabilities of the program to other classes of problems, material models, or solution strategies. Some user developed modules are available at the authors' web site given above and include element modules for other problem forms, an interface to other linear equation solvers, etc. Experienced programmers should be able to easily adapt these routines to include additional features. Programming additions to the system may be performed following descriptions in the Programmer Manual available at the web sites.
1. O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method for Solid and Structural Mechanics. Butterworth-Heinemann, Oxford, 6th edition, 2005. 2. O.C. Zienkiewicz and D.V. Phillips. An automatic mesh generation scheme for plane and curved surfaces by isoparametric coordinates. Int. J. Numer. Meth. Eng., 3:519-528, 1971. 3. W.J. Gordon and C.A. Hall. Transfinite element methods - blending-function interpolation over arbitrary curved element domains. Numer. Math., 21:109-129, 1973. 4. GiD - The Personal Pre/Postprocessor. www.gidhome.com, 2004. 5. E Auricchio and R.L. Taylor. A shear deformable plate element with an exact thin limit. Comp. Meth. AppL Mech. Eng., 118:393-412, 1994. 6. E Auricchio and R.L. Taylor. A triangular thick plate finite element with an exact thin limit. Finite Elements in Analysis and Design, 19:57-68, 1995. 7. R.L. Taylor. Finite element analysis of linear shell problems. In J.R. Whiteman, editor, The Mathematics of Finite Elements and Applications VI, pages 191-203. Academic Press, London, 1988. 8. J.H. Wilkinson and C. Reinsch. Linear Algebra. Handbook for Automatic Computation, volume II. Springer-Verlag, Berlin, 1971. 9. K.-J. Bathe. Finite Element Procedures. Prentice Hall, Englewood Cliffs, N.J., 1996. 10. E. Riks. An incremental approach to the solution of snapping and buckling problems. Int. J. Solids Struct., 15:529-551, 1979. 11. K. Schweizerhof. Nitchlineare Berechnung von Tragwerken unter verformungsabhangiger belastung mitfiniten Elementen. Doctoral dissertation, U. Stuttgart, Stuttgart, Germany, 1982. 12. J.C. Simo, P. Wriggers, K.H. Schweizerhof, and R.L Taylor. Finite deformation post-buckling analysis involving inelasticity and contact constraints. Int. J. Numer. Meth. Eng., 23:779-800, 1986. 13. J.E. Dennis and J. More. Quasi-Newton methods- motivation and theory. SlAM Rev., 19:46-89, 1977. 14. H. Matthies and G. Strang. The solution of nonlinear finite element equations. Int. J. Numer. Meth. Eng., 14:1613-1626, 1979. 15. HP Fortran home page. http://h18009.wwl.hp.com/fortran, 2004.
Matrix algebra
The mystique surrounding matrix algebra is perhaps due to the texts on the subject requiring a student to 'swallow too much' at one time. It will be found that in order to follow the present text and carry out the necessary computation only a limited knowledge of a few basic definitions is required.
The linear relationship between a set of variables x and b allXl -~- a12x2 -~- a13x3 q- a14x4 :- bl a21x1 d- a22x2 q-- a23x3 -]- a24x4 -- b2
(A.1)
a31xl 9 a32x2 9 a33x3 9 a34x4 "- b3 can be written, in a short-hand way, as [A] {x} = {b}
(A.2)
Ax - b
(A.3)
or where all
A -- [A] -
a12
a13
a14]
a21 a22 a23 a24 a31
a32
a33
a34
x1 X ~ {X} --
X2 X3 X4
b -- {b} -
b2 b3
(A.4)
The above notation contains within it the definition of both a matrix and the process of multiplication of two matrices. Matrices are defined as 'arrays of number' of the
Matrix addition or subtraction
type shown in Eq. (A.4). The particular form listing a single column of numbers is often referred to as a vector or column matrix, whereas a matrix with multiple columns and rows is called a rectangular matrix. The multiplication of a matrix by a column vector is defined by the equivalence of the left and fight sides of Eqs (A.1) and (A.2). The use of bold characters to define both vectors and matrices will be followed throughout the t e x t - generally lower case letters denoting vectors and capital letters matrices. If another relationship, using the same a constants, but a different set of x and b, exists and is written as l
I
l
!
!
l
!
l
I
l
t
/
!
I
/
allX 1 -~- al2x 2 -~- al3x 3 + al4x 4 -- b 1 a21x 1 -[- a22x 2 -t- a23x 3 -t- a24x 4 "- b 2
(A.5)
a31x 1 nt- a32x 2 + a33x 3 -[- a34x 4 --- b 3
then we could write [A] [ X ] = [ B ]
xa Xl
in which
or A X = B
(A.6)
I
X--[X]=
x2,
x2,
X3, X4,
x3, X4
B--[B]=
b2, b3,
b; b~
(A.7)
implying both the statements (A. 1) and (A.5) arranged simultaneously as
[
allXl+''',
allX 1 + " "
a21xl + . . . ,
azlx 1 + . . .
a31xl + ' '
]
I
",
/
= B-
[B] -
a31x 1 + " "
I
bl,
b~l
b2,
b; b~
b3,
]
(A.8)
It is seen, incidentally, that matrices can be equal only if each of the individual terms is equal. The multiplication of full matrices is defined above, and it is obvious that it has a meaning only if the number of columns in A is equal to the number of rows in X for a relation of the type (A.6). One property that distinguishes matrix multiplication is that, in general, AX # XA i.e., multiplication of matrices is not commutative as in ordinary algebra.
If relations of the form from (A. 1) and (A.5) are added then we have a l l (Xl + X~l) -+- alE(X2 + x;) + a13(x3 nt- x ; ) + a14(x4 + x 4) = bl -t- b'~ a21 (Xl --[- Xtl) 71--a22(x2 + x ; ) -[- a23(x3 nt- x ; ) -~- a24(x4 -~- x ; ) -- b2 +
b;
a31 (Xl + Xtl) nt- a32(x2 --[- x ; ) -[- a33(x3 -q- x ; ) -[-- a34(x4 -[- x ; ) -- b3 nt- b;
which will also follow from Ax + Ax' = b + b'
(A.9)
669
670
Matrix algebra
9
if we define the addition of matrices by a simple addition of the individual terms of the array. Clearly this can be done only if the size of the matrices is identical, i.e., for example,
[al1 a121 [bll b12] Jail+b11a12+b121 a21 a31
a22 a32
--1--
b21 b31
b22 b32
-
a21 + b21 a31 -+- b31
a22 -+- b22 a32 q-- b32
or
A+B:C
(A.10)
implies that every term of C is equal to the sum of the appropriate terms of A and B. Subtraction obviously follows similar rules.
This is simply a definition for reordering the terms in an array in the following manner: all a21
a12 a22
a13 a23
--
Iall a21] a12 a13
a22 a23
(A.11)
and will be indicated by the symbol T as shown. Its use is not immediately obvious but will be indicated later and can be treated here as a simple prescribed operation.
If in the relationship (A.3) the matrix A is 'square', i.e., it represents the coefficients of simultaneous equations of type (A. 1) equal in number to the number of unknowns x, then in general it is possible to solve for the unknowns in terms of the known coefficients b. This solution can be written as x = A-lb
(A.12)
in which the matrix A -1 is known as the 'inverse' of the square matrix A. Clearly A -1 is also square and of the same size as A. We could obtain (A.12) by multiplying both sides of (A.3) by A -1 and hence A-1A = I = AA -1
(A.13)
where I is an 'identity' matrix having zero on all off-diagonal positions and unity on each of the diagonal positions. If the equations are 'singular' and have no solution then clearly an inverse does not exist.
A s u m of products
In problems of mechanics we often encounter a number of quantifies such as force that can be listed as a matrix 'vector':
fl f2
f=
(A. 14)
fn These, in turn, are often associated with the same number of displacements given by another vector, say, Ul U2 .
a --
(A. 15)
Un
It is known that the work is represented as a sum of products of force and displacement n W
=
k=l
Clearly the transpose becomes useful here as we can write, by the rule of matrix multiplication, b/1 W
--
[fl
f2...
U2 .
f.]
--
fTu -- uTf
(A.16)
bl n
Use of this fact is made frequently in this book.
An operation that sometimes occurs is that of taking the transpose of a matrix product. It can be left to the reader to prove from previous definitions that (A B) T = BTA T
(A.17)
In structural problems symmetric matrices are often encountered. If a term of a matrix A is defined as a i j , then for a symmetric matrix aij -- aji
or A = A T
A symmetric matrix must be square. It can be shown that the inverse of a symmetric matrix is also symmetric A -1 _
(A-l)
T = A -T
671
672
Matrix algebra
It is easy to verify that a matrix product AB in which, for example,
A
all a21
a12 a13 a14 a15 a22 a23 a24 a25
a31
a32 a33 a34 a35 - bll
b12
-
b22 b31 b32
b21
B
b41 b42 . b51 b52
_
could be obtained by dividing each matrix into submatrices, indicated by the lines, and applying the rules of matrix multiplication first to each of such submatrices as if it were a scalar number and then carrying out further multiplication in the usual way. Thus, if we write I All A12 I [ B1 ] A -- A21 A22 BB2 then AB -
A11B1 A12B2 ] A21B1 A22B2
can be verified as representing the complete product by further multiplication. The essential feature of partitioning is that the size of subdivisions has to be such as to make the products of the type AllB1 meaningful, i.e., the number of columns in All must be equal to the number of rows in B1, etc. If the above definition holds, then all further operations can be conducted on partitioned matrices, treating each partition as if it were a scalar. It should be noted that any matrix can be multiplied by a scalar (number). Here, obviously, the requirements of equality of appropriate rows and columns no longer apply. If a symmetric matrix is divided into an equal number of submatfices Aq in rows and columns then Aij - AjT
An eigenvalue of a symmetric matrix A of size n • n is a scalar ,~i which allows the solution of ( A - ,~i I)qSi = 0 and det I A - ,~i I I= 0 (A.18) where r
is called the eigenvector.
The eigenvalue problem 673 There are, of course, n such eigenvalues ~i to each of which corresponds an eigenvector q5i. Such vectors can be shown to be orthonormal and we write
lfori--j 0fori # j The full set of eigenvalues and eigenvectors can be written as
A-
I
A1 0
"..
0 1
q' = [q~l,
A~
Using these the matrix A may be written in its spectral form by noting from the orthonormality conditions on the eigenvectors that
~I~-1 ---CI~T then from Aq, = q,A it follows immediately that A
=
~A~
T
(A.19)
The condition number tc (which is related to equation solution round-off) is defined as X=
I Amax I I Amin I
(A.20)
Tensor-indicial notation in the approximation of elasticity
problems
The matrix type of notation used in this volume for description of tensor quantities such as stresses and strains is compact and we believe easy to understand. However, in a computer program each quantity often will still have to be identified by appropriate indices and the conciseness of matrix notation does not always carry over to the programming steps. Further, many readers are accustomed to the use of indicial-tensor notation which is a standard tool in the study of solid mechanics. For this reason we summarize here the formulation of the finite element arrays in an indicial form. Some advantages of such reformulation from the matrix setting become apparent when evaluation of stiffness arrays for isotropic materials is considered. Here some multiplication operations previously necessary become redundant and the element module programs can be written more economically. When finite deformation problems in solid mechanics have to be considered the use of indicial notation is almost essential to form many of the arrays needed for the residual and tangent terms. This appendix adds little new to the discretization ideas - it merely repeats in a different language the results already presented.
A point P in three-dimensional space may be represented in terms of its cartesian coordinates xi, i - 1, 2, 3. The limits that i can take define its range. To define these components we must first establish an oriented orthogonal set of coordinate directions as shown in Fig. B. 1. The distance from the origin of the coordinate axes to the point define a position vector x. If along each of the coordinate axes we define the set of unit orthonormal base vectors, ii, i = 1, 2, 3 which have the property l
[i 9 ij - ~o -
~ 1 for i - j Ofori # j
(
(B.1)
Indicial notation: summation convention
x2
x2(P)
>xl Fig. B.10rthogonal axes and a point: Cartesian coordinates.
where ( ) 9( ) denotes the vector dot product. The components of the position vector are constructed from the vector dot product Xi
:
ii 9 X;
i -- 1, 2, 3
(B.2)
From this construction it is easy to observe that the vector x may be represented as 3 X -- ~
X i ii
(B.3)
i--1
In dealing with vectors, and later tensors, the form x is called the intrinsic notation of the coordinates and xi ii the indicialform.t An intrinsic form is a physical entity which is independent of the coordinate system selected, whereas an indicial form depends on a particular coordinate system. To simplify notation we adopt the common convention that any index which is repeated in any given term implies a summation over the range of the index. Thus, our short-hand notation for Eq. (B.3) is X - - X i ii - - X1 il d- X2 i2 "if- X3 i3
(B.4)
For two-dimensional problems, unless otherwise stated, it will be understood that the range of the index is two. Similarly, we can define the components of the displacement vector u as u =
ui ii
(B.5)
Note that the components (u 1, u2, u3) replace the components (u, v, w) used throughout most of this volume. To avoid confusion with nodal quantities to which we previously also attached subscripts we shall simply change their position to a superscript. Thus, ~ has the same meaning as ~a used previously, etc. t Often in an indicial form of equations the base vectors are omitted from the final equation.
675
676 Tensor-indicial notation in the approximation of elasticity problems
In indicial notation the derivative of any quantity with respect to a coordinate component xi is written compactly as a = ( ),i (B.6) Oxi Thus we can write the gradient of the displacement vector as Oui OXj ~" Ui'j; i, j = 1, 2, 3
(B.7)
In a cartesian coordinate system the base vectors do not change their magnitude or direction along any coordinate direction. Accordingly their derivatives with respect to any coordinate is zero as indicated in Eq. B.8 Oii OXj
(B.8)
= ii,j -- 0
Thus, in cartesian coordinates the derivative of the intrinsic displacement u is given by U,j -- gi,ji i at- Uiii,j -- Ui,jii
(B.9)
The collection of all the derivatives defines the displacement gradient which we write in intrinsic notation as ~Tu -- Ui,j ii | ij (B. 10) The symbol | denotes the tensor product between two base vectors and since only two vectors are involved the gradient of the displacement is called second rank. The notation used to define a tensor product follows that used in reference 1. Any second rank intrinsic quantity can be split into symmetric and a skew symmetric (anti-symmetric) parts as
1
A = ~1 [A + AT] + ~ [A - AT] = A (~) + A(a'
(B.11)
where A and its transpose have cartesian components
A-Aijii|
AT-Ayiii|
(B.12)
The symmetric part of the displacement gradient defines the (small) strain~
= Vu,S _ ! [Vu + (Vu T] 2 -- ! [Ui,j "[" Uj,i] ii (~ __
2
Eij
ii | ij
- - Eji
ij
(B.13)
ii | ij
t Note that this definition is slightly different from that occurring in Chapters 2 to 6. Now the shearing strain is given by eij -- 1/2 Yij when i # j.
Coordinate transformation 677
and the skew symmetric part gives the (small) rotation w -
Vu
1 [Vu-(Vu)
~'~ -
__ 1__ [Ui, j __ U j ' i ]
2
O)ij ii @ i: = - o ) j i
:
(B.14)
ii (~)ij ii @ i j
The strain expression is analogous to Eq. (2.13). The components/3ij and wij may be represented by a matrix as /3ij
--
/321 /331
/322 /332
/323 /333
wij
=
c021 0 0923 0931 0932 0
/312 /322 /323 /313 /323 /333
:
--
-0912 0 -0913 -o923
(B.15)
0923 0
(B.16)
Consider now the representation of the intrinsic coordinates in a system which has different orientation than that given in Fig. B.1. We represent the components in the new system by I *I x - x i, l i, (B. 17) Using Eq. (B.2) we can relate the components in the prime system to those in the original system as (B.18) X i,' - - I"'i 9X = I" i. i j x j - - A i , j x j where " A i , j --- li,
9ij = cos(x~,, x j )
(B 19)
define the direction cosines of the coordinate in a manner similar to that of Eq. (6.18). Equation (B.18) defines how the cartesian coordinate components transform from one coordinate frame to another. Recall that summation convention implies X i,'
-- Ai'IX1 --[- Ai,2x2 q- Ai,3x3
i' -- 1, 2, 3
(B.20)
In Eq. (B. 18) i' is called a f r e e i n d e x whereas j is called a d u m m y i n d e x since it may be replaced by any other unique index without changing the meaning of the term (note that the notation used does not permit an index to appear more than twice in any term). Summation convention will be employed throughout the remainder of this discussion and the reader should ensure that the concept is fully understood before proceeding. Some examples will be given occasionally to illustrate its use. Using the notion of the direction cosines, Eq. (B.18) may be used to transform any vector with three components. Thus, transformation of the components of the displacement vector is given by !
u i, :
Ai,juj
i', j = 1, 2, 3
(B.21)
678
Tensor-indicial notation in the approximation of elasticity problems Indeed we can also use the above to express the transformation for the base vectors since i I, -- ( i l , - i j ) ij -- A i , j i j (B.22) Similarly, by interchanging the role of the base vectors we obtain
(ij
ij --"
. l"') i,
I" i, -- Ai,jl "'i,
B23)
which indicates that the inverse of the direction cosine coefficient array is the same as its transpose. The strain transformation follows from the intrinsic form written as !
9
9
e = ei,j,1 i, Q lj, -- eklik Q il
(B.24)
Substitution of the base vectors from Eq. (B.23) into Eq. (B.24) gives e -- A i , k e k l A j , l i i ,
Q ij,
(B.25)
Comparing Eq. (B.25) with Eq. (B.24) the components of the strain transform according to the relation E~,j, - Ai,kEkll~j,l (B.26) Variables that transform according to Eq. (B.21) are called first rank cartesian tensors whereas quantities that transform according to Eq. (B.26) are called second rank cartesian tensors. The use of indicial notation in the context of cartesian coordinates will lead naturally to each mechanics variable being defined in terms of a cartesian tensor of an appropriate rank. Stress may be written in terms of its components crij which may be written in a matrix form similar to Eq. (B. 15) O'ij --
O'11 O'12 O'13 ] O'21 0"22 0"23 ; i, j = 1, 2, 3 O"31 0"32 0"33
J
(B.27)
In intrinsic form stress is given by tr -- ffijii |
ij
(B.28)
and, using similar logic as used for strain, can be shown to transform as a second rank cartesian tensor. The symmetry of the components of stress may be established by summing moments (angular momentum balance) about each of the coordinate axes to obtain O'ij : O'ji (B.29)
Introducing a body force vector b-
biii
(B.30)
Elastic constitutive equations 679
we can write the static equilibrium equations (linear momentum balance) for a differential element as div tr + b - (0"ji,j -3t- 0 (B.31)
bi)i/
where the repeated index again implies summation over the range of the index, i.e., 3
0"ji,j ~ ~_~ 0"ji,j -- 0"1i,1 -JI- 0"2i,2 + 0"3i,3 j=l
Note that the free index i must appear in each term for the equation to be meaningful. As a further example of the summation convention consider an internal energy term W - 0"ij8ij
(B.32)
This expression implies a double summation; hence summing first on i gives W = 0"1j81j '}- 0"2j82j + 0"3j83j
and then summing on j gives finally W - 0"11811 -~0"12812 +0"13813 "-['- 0"21 821 + 0"22 822 + 0"23 E23 ~- 0-31 831 "+" 0-32 832 + 0"33 833
We may use symmetry conditions on 0"ij and Eij to reduce the nine terms to six terms. Accordingly, W - 0"11811 + 0"22822 -+-0"33833 "[- 2(0"12812 -+" 0"23823 + 0"31831) -- 0"11811 -'l'- 0"22822 -~" 0"33E33 -+- 0"12}/12 + 0"23Y23 -1- 0"31}/31
(B.33)
Following a similar expansion we can also show the result 0-ijO)ij ~ 0
(B.34)
For an elastic material the most general linear relationship we can write for components of the stress-strain characterization is 0
0-ij~ Dijkl (Ekl ~ 8kl) 9 (70
(B.35)
Equation (B.35) is the equivalent of Eq. (2.16) but now written in indicial notation. We note that the elastic moduli which appear in Eq. (B.35)are components of the fourth rank tensor D : Dijklii ~ ij Q ik | it (B.36) The elastic moduli possess the following symmetry conditions Dijkl -- Djikl -- Dijlk --" Dklij
(B.37)
680
Tensor-indicial notation in the approximation of elasticity problems the latter arising from the existence of an internal energy density in the form z
1
W ( ~ ) -- -~EijOijkiEkl "+" Eij [(7.0 __ OijkiEkO]
(B.38)
which yields the stress from OW
(B.39)
O'ij = OEiJ
By writing the constitutive equation with respect to x~, and using properties of the base vectors we can deduce the transformation equation for moduli as D~,j,k,l, -- A i , m A j , n A k , pAl,qDmnpq
(B.40)
A common notation for the intrinsic form of Eq. (B.35) is (B.41) in which : denotes the double summation (contraction) between the elastic moduli and the strains. The elastic moduli for an isotropic elastic material may be written in indicial form as Oijkl "- /~ij ~kl -Jr- ].L((~ik(~jl -Jr- (~il~jk ) (B.42) where A, /z are the Lam6 constants. An isotropic linear elastic material is always characterized by two independent elastic constants. Instead of the Lam6 constants we can use Young's modulus, E, and Poisson's ratio, v, to characterize the material. The Lam6 constants may be deduced from E
/z = 2(1 + v)
and
A=
vE
(1 + v ) ( 1 - 2v)
(B.43)
If we now introduce the finite element displacement approximation given by Eq. (2.1), using indicial notation we may write for a single element U i ~ ~li : Na~-I a i -- 1,2,3; a -- 1,2 . . . . ,n
(B.44)
where n is the total number of nodes on an element. The strain approximation in each element is given by the definition of Eq. (B. 13) as /
,, -- -~1 [Na, jUi ~a .3t_ Na , i~lj ] Eij
(B.45)
The internal virtual work for an element is given as
(~ul - f~ 8r erdfl-f~ e
3eqaq d~ e
(B.46)
Finite element approximation 681 Using Eqs (B.45) and (B.46) and noting symmetries in Dijkl w e may write the internal virtual work for a linear elastic material as
8U I - ~
Na,jDijklNb,l d~2 uk (B.47)
e
P
+ ~ft a /.. Na,j(ff 0 - DijklSkO) d a e
which replaces in indicial notation the matrix form presented in Chapters 2 to 6. In describing a stiffness coefficient two subscripts have been used previously and the submatrix Kab implied 2 x 2 or 3 x 3 entries for the ab nodal pair, depending on whether two- or three-dimension displacement components were involved. Now the scalar components K~ b i , j 1,2,3; a , b - 1,2 . . . . . n (B.48) define completely the appropriate stiffness coefficient with ij indicating the relative submatrix position (in this case for a three-dimensional displacement). Note that for a symmetric matrix we have previously required that
Kab
-- K L
(B.49)
In indicial notation the same symmetry is implied if
K~jb -- K jb'~
(B.50)
The stiffness tensor is now defined from Eq. (B.47) as
Kiakb -- ~_ Na,j Dijkl Nb,l dr2
(B.51)
,.IS2 e
When the elastic properties are constant over the element we may separate the integration from the material constants by defining
Wij b ' - f~ Na,iNb,jd~
(B.52)
e
and then perform the summations with the material moduli as
Ki? - w;b nijkl
(B.53)
In the case of isotropy a particularly simple result is obtained
giakb -- )~W~kb -3I- ~b[W:b "3I- r W; b ]
(B.54)
which allows the construction of the stiffness to be carried out using fewer arithmetic operations as compared with the use of matrix form. 3 Using indicial notation the final equilibrium equations of the system are written as
gab~b ikuk + fi a - - 0
i--1,2,3
(B.55)
and in this scalar form every coefficient is simply identified. The reader can, as a simple exercise, complete the derivation of the force terms due to the initial strain e ~ stress cr~ body force bi and external traction ti. Indicial notation is at times useful in clarifying individual terms, and this introduction should be helpful as a key to reading some of the current literature.
682
Tensor-indicial notation in the approximation of elasticity problems Table B.1 Mapping between matrix and tensor indices for second rank symmetric tensors Form
Index number
Matrix
1
2
3
4
5
6
Tensor
11 xx
22 yy
33 zz
12 & 21 xy & yx
23 & 32 yz & zy
31 & 13 zx & xz
The matrix form used throughout most of this volume can be deduced from the indicial form by a simple transformation between the indices. The relationship between the indices of the second rank tensors and their corresponding matrix form can be performed by an inspection of the ordering in the matrix for stress and its representation shown in Eq. (B.27). In the matrix form the stress was given in Chapter 6 as O'=
If ill
0 " 2 2 033
O"12 0 " 2 3 O"31 ]T
(B.56)
This form includes use of symmetry of stress components. The mapping of the indices follows that shown in Table B. 1. Table B. 1 may also be used to perform the map of the material moduli by noting that the components in the energy are associated with the index pairs from the stress and the strain. Accordingly, the moduli transform as Dllll -+ Dll;
D2233-+ D23; D1231~ D46; etc.
(B.57)
The symmetry of the stress and strain is imbedded in Table B.1 and existence of an energy function yields symmetry of the modulus matrix, i.e., Dab -- Dba.
1. E Chadwick. Continuum Mechanics. John Wiley & Sons, New York, 1976. 2. I.S. Sokolnikoff. The Mathematical Theory of Elasticity. McGraw-Hill, New York, 2nd edition, 1956. 3. A.K. Gupta and B. Mohraz. A method of computing numerically integrated stiffness matrices. Int. J. Numer. Meth. Eng., 5:83-89, 1972.
Solution of simultaneous linear algebraic equations A finite element problem leads to a large set of simultaneous linear algebraic equations whose solution provides the nodal and element parameters in the formulation. For example, in the analysis of linear steady-state problems the direct assembly of the element coefficient matrices and load vectors leads to a set of linear algebraic equations. In this section methods to solve the simultaneous algebraic equations are summarized. We consider both direct methods where an a priori calculation of the number of numerical operations can be made, and indirect or iterative methods where no such estimate can be made.
Consider first the general problem of direct solution of a set of algebraic equations given by Ka=f (C.1) where K is a square coefficient matrix, ~ is a vector of unknown parameters and f is a vector of known values. The reader can associate these with the quantities described previously" namely, the stiffness matrix, the nodal unknowns, and the specified forces or residuals. In the discussion to follow it is assumed that the coefficient matrix has properties such that row and/or column interchanges are unnecessary to achieve an accurate solution. This is true in cases where K is symmetric positive (or negative) definite.t Pivoting may or may not be required with unsymmetric, or indefinite, conditions which can occur when the finite element formulation is based on some weighted residual methods. In these cases some checks or modifications may be necessary to ensure that the equations can be solved accurately. 1-3 For the moment consider that the coefficient matrix can be written as the product of a lower triangular matrix with unit diagonals and an upper triangular matrix. Accordingly, K = LU
(C.2)
t For mixed methods which lead to forms of the type given in Eq. (10.14) the solution is given in terms of a positive definite part for (1 followed by a negative definite part for t~.
684 Solutionof simultaneous linear algebraic equations where
I
1 L21
L -
01
.
LL~I and
Ull i
U -
Ln2
... 9 0i .. -.
"'"
U12 ''U22 '".' ' 0
(C.3)
..i
Uln U2n .
(C.4)
Un n
This form is called a triangular decomposition of K. The solution to the equations can now be obtained by solving the pair of equations Ly = f
(C.5)
Ufi = y
(C.6)
and where y is introduced to facilitate the separation, e.g., see references 1-5 for additional details. The reader can easily observe that the solution to these equations is trivial. In terms of the individual equations the solution is given by
Y l - fl
i-1
Yi -- f i - Z
j=l
Lij yj
(C.7)
i -2,3 ..... n
and
1(
Yn bl n - - -
~li ---
Unn
~ii
Yi-
j=i+l
Uijuj
)
i-n-l,n-2
..... 1
(c.8)
Equation (C.7) is commonly called forward elimination while Eq. (C.8) is called back substitution. The problem remains to construct the triangular decomposition of the coefficient matrix. This step is accomplished using variations on gaussian elimination. In practice, the operations necessary for the triangular decomposition are performed directly in the coefficient array; however, to make the steps clear the basic steps are shown in Fig. C. 1 using separate arrays. The decomposition is performed in the same way as that used in the subprogram DATRI contained in the FEAPpv program; thus, the reader can easily grasp the details of the subprograms included once the steps in Fig. C. 1 are mastered. Additional details on this step may be found in references 3-5. In DATRI a Crout form of gaussian elimination is used to successively reduce the original coefficient array to upper triangular form. The lower portion of the array is
Direct solution Active zone
Step 1. Active zone. First row and column to principal diagonal. rK21
Reduced zone i,/-- Active zone K22 K23
K31 K32 K3a
L21 = K21/Ull
L22 = 1
0
U22 = K22- L21 U12
Step 2. Active zone. Second row and column to principal diagonal. Use first row of K to eliminate L21 Ull. The active zone uses only values of K from the active zone and values of L and U which have already been computed in steps 1 and 2. ~/-- Reduced zone ~/-- Active zone
K31
K13
1
0
0
Ull
K23
L21 1
0
0
K32 K33
L31 L32 L33 = 1 0
U12 U13-- K13 U22 U23 = K23- L21U13 0
U33 = K33- L31U13- L32U23
L31 = K31/U11 L32 = (K32-L31U12)/U22
Step 3. Active zone. Third row and column to principal diagonal. Use first row to
eliminate L31 Ul1" use second row of reduced terms to eliminate L32 U22 (reduced coefficient K32). Reduce column 3 to reflect eliminations below diagonal.
Fig. C.1 Triangular decomposition of K. used to store L - I as shown in Fig. C.1. With this form, the unit diagonals for L are not stored. Based on the organization of Fig. C. 1 it is convenient to consider the coefficient array to be divided into three parts: part one being the region that is fully reduced; part two the region that is currently being reduced (called the active zone); and part three the region that contains the original unreduced coefficients. These regions are shown in Fig. C.2 where the jth column above the diagonal and the jth row to the left of the diagonal constitute the active zone. The algorithm for the triangular decomposition of an n • n square matrix can be deduced from Fig. C. 1 and Fig. C.3 as follows: Ull = K l l ;
Lll-
(C.9)
1
For each active zone j from 2 to n,
Kjl Ljl : ~ ;
(C.10)
Ulj -- Klj
U11 i-1
1
Lji : Uii (Kji - ~
LjmUmi)
m=l
(C.11)
i-1
Uij -- Kij - ~ m--1
Limgmj
i = 2, 3 .....
j -
1
685
686
Solution of simultaneous linear algebraic equations jth column active zone
U
K~E "x
Reduced zone
jth row active zone - - ~
Kjl, KE2
9
9
9
~'jj
",,,
Unreduced zone
Fig. C.2 Reduced,active and unreduced parts.
U1i
Ulj
uj_~,j
Ui_~ Lil Li2
9
9
9
Li, i-I
"_j
Lj~ Lj~
9
9
9
Lj,,_, k
%
%
% %
%.
%
Fig. C.3 Terms used to construct Uij and Lji.
and finally Ljj -- 1 j-1
Ujj
-- Kjj - y ~ L jmUmj m=l
(C.12)
Direct solution Table C.1 Example: triangular decomposition of 3 x 3 matrix
I'
K
Step 1.
I' I'
[ 1
U23
0.5
--
0.5 0.25
3
1 0.5
3
1
L31 = 1 = 0.25, U13 -- 1, L32 -2 - 0.5 x 1 = 1.5, L33
0.5 0.25
Step 4.
1
L21 -- 2 __ 0.5, U12 -- 2, U22 -- 1, U22 = 4 - 0.5 x 2 = 3
2
Step 3.
] ][421]
L l l -- 1, U l l -- 4
4 2
Step 2.
] [4
L
1 0.5
-'-
2 - 0.25 x 2 3
1.5 3
1.5 -----0.5 3
1, U33 = 4 - 0.25 x 1 - 0.5 x 1.5 = 3
3 1
1.5 3
=
4 2
Check
The ordering of the reduction process and the terms used are shown in Fig. C.3. The results from Fig. C.1 and Eqs (C.9)-(C.12) can be verified using the matrix given in the example shown in Table C. 1. Once the triangular decomposition of the coefficient matrix is computed, several solutions for different fight-hand sides f can be computed using Eqs (C.7) and (C.8). This process is often called a resolution since it is not necessary to recompute the L and U arrays. For large size coefficient matrices the triangular decomposition step is very costly while a resolution is relatively cheap; consequently, a resolution capability is necessary in any finite element solution system using a direct method. The above discussion considered the general case of equation solving (without row or column interchanges). In coefficient matrices resulting from a finite element formulation some special properties are usually present. Often the coefficient matrix is symmetric (Kij "-- K j i ) and it is easy to verify in this case that Uij -
L j i Uii
(no sum)
(C.13)
For this problem class it is not necessary to store the entire coefficient matrix. It is sufficient to store only the coefficients above (or below) the principal diagonal and the diagonal coefficients. Equation (C.13) may be used to construct the missing part. This reduces by almost half the required storage for the coefficient array as well as the computational effort to compute the triangular decomposition. The required storage can be further reduced by storing only those rows and columns which lie within the region of non-zero entries of the coefficient array. Problems formulated by the finite element method and the Galerkin process normally have a
687
688 Solutionof simultaneous linear algebraic equations symmetric profile which further simplifies the storage form. Storing the upper and lower parts in separate arrays and the diagonal entries of U in a third array is used in DATRI. Figure C.4 shows a typical profile matrix and the storage order adopted for the upper array AU, the lower array AL and the diagonal array AD. An integer array JD is used to locate the start and end of entries in each column. With this scheme it is necessary to store and compute only withiJa the non-zero profile of the equations. This form of storage does not severely penalize the presence of a few large columns/rows and is also an easy form to program a resolution process (e.g., see subprogram DASOL in FEAPpv and reference 4). The routines included in FEAPpv are restricted to problems for which the coefficient matrix can fit within the space allocated in the main storage array. In two-dimensional formulations, problems with several thousand degrees of freedom can be solved on today's personal computers. In three-dimensional cases, however, problems are restricted to a few thousand equations. To solve larger size problems there are several options. The first is to retain only part of the coefficient matrix in the main array with the rest saved on backing store (e.g., hard disk). This can be quite easily achieved but the size of problem is not greatly increased due to the very large solve times required and the rapid growth in the size of the profile-stored coefficient matrix in three-dimensional problems. A second option is to use sparse solution schemes. These lead to significant program complexity over the procedure discussed above but can lead to significant savings in storage demands and compute time- especially for problems in three dimensions. Nevertheless, capacity in terms of storage and compute time is again rapidly encountered and alternatives are needed.
One of the main problems in direct solutions is that terms within the coefficient matrix which are zero from a finite element formulation become non-zero during the triangular decomposition step. While sparse methods are better at limiting this fill than profile methods they still lead to a very large increase in the number of non-zero terms in the factored coefficient matrix. To be more specific consider the case of a three-dimensional linear elastic problem solved using 8-node isoparametric hexahedron elements. In a regular mesh each interior node is associated with 26 other nodes, thus, the equation of such a node has 81 non-zero coefficients - three for each of the 27 associated nodes. On the other hand, for a rectangular block of elements with n nodes on each of the sides the typical column height is approximately proportional to n 2 and the number of equations t o n 3. In Table C.2 we show the size and approximate number of non-zero terms in K from an element finite formulation for linear elasticity (i.e., with three degrees of freedom per node). The table also indicates the size growth with column height and storage requirements for a direct solution based on a profile solution method. From the table it can be observed that the demands for a direct solution are growing very rapidly (storage is approximately proportional to n 5) while at the same time the demands for storing the non-zero terms in the stiffness matrix grow proportional to the number of equations (i.e., proportional to n 3 for the block).
Iterative solution ~A
\
Half band width ~
Half band width ._
Profile
v
Kll K12 K13 K14
Kll
i
___/_:
K22 K23 K24 K33 K34 K35 K44 K45 K46
iK44 K45 K46 , . . . .
K55 K56
Symmetric' K55 K56 |
!
. . . . . . . . .
! . . . .
1
i
K77 K78
K77 K78
. . . . . i
i /
K66 K67 K68
K66 K67 K68
i i . . . .
i
i
i
,
i
K88
K88' i
Banded storage array ADi
AUi
ALi
JDi
K11 K22 K33
K12
I(21
0
K13 K23
K31 %2
3
K14 K24 K34
K41 K42 K43
K35 K45
Ks3 Ks4
10
K46 K56
K64 K6~
11
K67
K7~
12
/(18
K81
18
K7~
K8;
K44 Kss K66 K77 K88 Diagonals
_•
9
Profile
-- !~:~;iK~ K~4i
i K,,iK,~ K4~i
i _ _
r _ _ .
I
i
Symmetric iK55:K56:
iK~8I K,8 I
K581
!K881 Fig. C.4 Profile storage for coefficient matrix.
Storage of arrays
1 6 8 10 11 18
689
690
Solutionof simultaneous linear algebraic equations Table C.2 Partial list of solutions commands
Non-zeros in K
Profile storage data
Side nodes
Number of equations
Words (x 10 -6)
Mbytes
Col. Ht.
Words (x 10 -6)
Mbytes
5 10 20 40 80
375 3000 24000 192000 1536000
0.02 0.12 0.96 7.68 61.44
0.12 0.96 7.68 61.44 491.52
90 330 1260 4920 18440
0.03 0.99 30.24 944.64 28323.84
0.27 7.92 241.82 7557.12 226584.72
Iterative solution methods use the terms in the stiffness matrix directly and thus for large problems have the potential to be very efficient for large three-dimensional problems. On the other hand, iterative methods require the resolution of a set of equations until the residual of the linear equations, given by r (i) - -
f
- K f i (i)
(C.
14)
becomes less than a specified tolerance. In order to be effective the number of iterations i to achieve a solution must be quite small - generally no larger than a few hundred. Otherwise, excessive solution costs will result. At the time of writing this book the subject of iterative solution for general finite element problems remains a topic of intense research. There are some impressive results available for the case where K is symmetric positive (or negative) definite; however, those for other classes (e.g., unsymmetric or indefinite forms) are generally not efficient enough for reliable use in the solution of general problems. For the symmetric positive definite case methods based on a preconditioned conjugate gradient method have been particularly effective. 6-8 The convergence of the method depends on the condition number of the matrix K - the larger the condition number, the slower the convergence (see reference 3 for more discussion). The condition number for a finite element problem with a symmetric positive definite stiffness matrix K is defined as A, tc = - (C.15) )~1 where ,km and )kn a r e the smallest and largest eigenvalue from the solution of the eigenproblem (viz. Chapter 16) KcI, = ~ A
(C.16)
in which A is a diagonal matrix containing the individual eigenvalues/~i and the columns of 9 are the eigenvectors r associated with each of the eigenvalues. Usually, the condition number for an elasticity problem modelled by the finite element method is too large to achieve rapid convergence and a preconditionedconjugate gradient (PCG) is used. 6 A symmetric form of preconditioned system is written as Kpz --
pKpTz - Pf
(C.17)
where Prz - fi
(C.18)
References 691
Now the convergence of the PCG algorithm depends on the condition number of Kp. The problem remains to construct a preconditioner which adequately reduces the condition number. In FEAPpv the diagonal of K is used; however, more efficient schemes incorporating also multigrid methods are discussed in references 7 and 8.
1. A. Ralston. A First Course in Numerical Analysis. McGraw-Hill, New York, 1965. 2. J.H. Wilkinson and C. Reinsch. LinearAlgebra. HandbookforAutomatic Computation, volume II. Springer-Verlag, Berlin, 1971. 3. J. Demmel. Applied Numerical Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1997. 4. R.L. Taylor. Solution of linear equations by a profile solver. Eng. Comput., 2:344-350, 1985. 5. G. Strang. Linear Algebra and its Application. Academic Press, New York, 1976. 6. R.M. Ferencz. Element-by-element preconditioning techniques for large-scale, vectorized finite element analysis in nonlinear solid and structural mechanics. Ph.D. thesis, Department of Mechanical Engineering, Stanford University, Stanford, California, 1989. 7. M. Adams. A parallel maximal independent set algorithm. In Proc. 5th Copper Mountain Conference on Iterative Methods, 1998. 8. M. Adams. Parallel multigrid solver algorithms and implementations for 3D unstructured finite element problems. In Supercomputing '99: High Performance Networking and Computing, volume http://www.sc99.org/proceedings, Portland, Oregon, Nov. 1999.
Some integration formulae for a triangle Let a triangle be defined in the xy plane by three points (Xl, yj), (x2, Y2), (x3, Y3) with the origin of the coordinates taken at the centroid (or baricentre), i.e., Xl "~-X2 -~-X3
3
--
Yl "+- Y2 nt- Y3
3
=0
Then integrating over the triangle area we obtain:
1 dx dy - -~
xdxdy = f
1
Xl
Yl
1
X2
Y2
1
x 3
Y3
--
A - area of triangle
y d x d y =O
A x 2dxdy = -~ (x~ +x~ +x~) A y2 dx dy - -i2 (y~ 4- y22 + y32) f
A (Xlyl q- x2Y2 + x3Y3) x y dx dy - -i2
Some integration formulae for a tetrahedron Let a tetrahedron be defined in the x y z coordinate system by four points (Xl, yl, z 1), (x2, y2, z2), (x3, Y3, z3) (x4, Y4, z4) with the origin of the coordinates taken at the centroid, i.e., Xl "q- X2 @ X3 "~- X4
Yl -+- Y2 -+- Y3 q- Y4 ~"
4
Zl + Z2 "q" Z3 @ Z4 "-'-
4
4
~"
0
Then integrating over the tetrahedron volume 1 dx dy dz - -~
1
X1
Yl
Zl
1
x2
Y2
z2
1
X3
Y3
Z3
1
X4
Y4
Z4
- V - tetrahedron volume
Provided the order of numbering the nodes is as indicated on Fig. 4.18(a) then also"
/x xdydz=/ y x y z=/zdx ydz=O x 2 dx dy dz = ~ (x~ + x~ + x~ + x24) y2 dx dy dz = ~
f
z 2 dx dy dz --
(y~ + y~ + y~ + y~)
v (z~+ z~+ z~+ zl)
x y dx dy dz = 2-0 (XlYl nt- x2Y2 -t- x3Y3 -t- X424) y z dx dy dz - - ~ (ylzl + Y2Z2 + y3z3 + 2424) z x dx dy dz = ~ (ZlXl + ZzX2 + z3x3 + Z4X4)
Some vector algebra
Some knowledge and understanding of basic vector algebra is needed in dealing with complexities of elements oriented in space as occur in beams, shells, etc. Some of the operations are summarized here. Vectors (in the geometric sense) can be described by their components along the directions of the x, y, z axes. Thus, the vector V01 shown in Fig. E 1 can be written as Vol
(El)
xli q- Ylj q- zlk
-
in which i, j, k are unit vectors in the direction of the x, y, z axes. Alternatively, the same vector could be written as
Vol
:
{xl}
(F.2)
Yl Zl
(now a 'vector' in the matrix sense) in which the components are distinguished by positions in the column.
Addition and subtraction is defined by addition and subtraction of components. Thus, for example, V02 -- V01 -- (x2 -
xl)i +
(Y2 - -
Yl)j +
(Z2 - -
zl)k
(E3)
The same result is achieved by the definitions of matrix algebra; thus
V02 -
V0I
-'- V 2 1 =
I X2~X1 I Y2 - - Yl Z2 - - Z l
(F.4)
'Scalar' products z
2
.'"""
Z2
, i
V01
i
'
Y .
.
.
.
.
.
.
.
.
x2 .
.
.
X
.
.
.
.
.
.
.
.
.
........
r
""
~'x'=~i
!
"""
~'X
Fig. R1 Vector addition.
A scalar product of two vectors is defined as 3 A . B - B. A = Z
akbk
(ES)
k=l
If A
=
axi + ayj + azk
B
=
bxi + byj -+- bzk
(E6)
then A 9B = axbx + ayby + azbz
Using the matrix notation A =
{ax} Bay az
bx
by }
(E7)
(E8)
bz
the scalar product becomes A . B = ATB = BTA
(E9)
The length of the vector V21 is given, purely geometrically, as 121 -- r
-- Xl) 2 "[" (Y2 -- Yl) 2 + (Z2 -- Zl) 2
(El0)
695
696
Some vector algebra
Fig. R2 Vector multiplication (cross product). or in terms of matrix algebra as 12,- 4V21 " V21-
(Ell)
4VT1V21
Direction cosines of a vector are simply, from the definition of the projected component of lengths, given as (Fig. F. 1) cos Otx :
X2 -- X1 121
=
V21 9i
(F. 12)
121
The scalar product may also be written as (Fig. E2) A . B = B . A = lalb
(F.13)
COS y
where y is the angle between the two vectors A and B and respectively.
la
and
lb a r e
their lengths,
Another product of vectors is defined as a vector oriented normally to the plane given by two vectors and equal in magnitude to the product of the length of the two vectors multiplied by the sine of the angle between them. Further, the direction of the normal vector follows the fight-hand rule as shown in Fig. F.2 in which A • B - C
(El4)
is shown. Thus, from the fight-hand rule, we have A x B - -B x A
(F.15)
Elements of area and volume
It is worth noting that the magnitude (or length) of C is equal to the area of the parallelogram shown in Fig. E2. Using the definition of Eq. (E6) and noting that ixi
--
jxj-kxk-0
i•
=
k,j•
(F.16)
we have AxB
i ax bx
j ay by
k az bz
-
det
=
(aybz - azby)i + (azbx - axbz)j + (axby - a y b x ) k
In matrix algebra this does not find a simple counterpart but we can use the above to define the vector C t
{ ayz-azy}
C - A x B -
(F. 17)
azbx - axbz axby - aybx
The vector product will be found particularly useful when the problem of erecting a normal direction to a surface is considered.
If~ and r/are some curvilinear coordinates, then the following vectors in two-dimensional plane ax ax d~ --
0-~
d~
Oy
drl =
~
Oy
d77
(F. 18)
defined from the relationship between the cartesian and curvilinear coordinates, are vectors directed tangentially to the ~ and 0 equal constant contours, respectively. As the length or the vector resulting from a cross product of d~ x dr/is equal to the area of the elementary parallelogram we can write
d(area) -- det
0x
0x
0-~
~
Oy
Oy
d~ dr/
by Eq. (F.17). t If we rewrite A as a skewsymmetricmatrix A,=
az
0
ay
ax
aox]
then an altemativerepresentationof the vectorproductin matrixform is C = .~B.
(E19)
697
698 Somevector algebra
Similarly, if we have three curvilinear coordinates ~, r/, ( in the cartesian space, the 'triple' or box product defines a differential volume " Ox
d(vol) = (d~ x dr/). d~ = det
Ox
Ox-
Oy
Oy
Oy
Oz
Oz
Oz
d~ dr/d(
(F.20)
this follows simply from the geometry. The bracketed product, by definition, forms a vector whose length is equal to the parallelogram area with sides tangent to two of the coordinates. The second scalar multiplication by a length and the cosine of the angle between that length and the normal to the parallelogram establishes a differential volume element. The above equations serve in changing the variables in surface and volume integrals.
Integration by parts in two or three
dimensions (Green's theorem)
Consider the integration by parts of the following two-dimensional expression / 4~ 0x dx dy
(G.1)
~2
Integrating first with respect to x and using the well-known relation for integration by parts in one dimension
u dv - -
/xl
v du + (UV)x=x~ - (UV)x=xz,
(G.2)
we have, using the symbols of Fig. G. 1,
ff
iOx gOdxdy--
ff
-~x O~ 0 dx dy + fyyl'~ [(~b gr)x=xR - (qb gr)x=xL] dy (G.3)
If now we consider a direct segment of the boundary dl-" on the fight-hand boundary, we note that dy = nx dl-" (G.4) where nx is the direction cosine between the outward normal and the x direction. Similarly on the left-hand section we have
dy - - nx dI"
(G.5)
The final term of Eq. (G.3) can thus be expressed as the integral taken around an anticlockwise direction of the complete closed boundary:
/ dp~/nx dI"
(G.6)
r If several closed contours are encountered this integration has to be taken around each such contour. The general expression in all cases is
Ox ~2
-~x ~ dx dy + ~2
dp~nx dr" F
(6.7)
700
Integration by parts in two or three dimensions (Green's theorem)
'
YT
f~
F
~x Fig. G.1 Definitions for integrations in two dimensions.
Similarly, if differentiation in the y direction arises we can write f/
Oy~ d x d y - - f f O
(G.8)
-~y Ocp ~ dx dy + f dp~ny dF g2
F
n y is the direction cosine between the outward normal and the y axis. In three dimensions by identical procedure we can write
where
f/j dp O~p ~ gr dx dy dz + / 6q/ny Oy dx dy dz - - JjJ~-~y ~2
~
dl-"
(G.9)
F
where dl-' becomes the element of the surface area and the last integral is taken over the whole surface.
Solutions exact at nodes
The finite element solution of ordinary differential equations may be made exact at the interelement nodes by a proper choice of the weighting function in the weak (Galerkin) form. To be more specific, let us consider the set of ordinary differential equations given by A(u) + f(x) = 0 (H.1) where u is the set of dependent variables which are functions of the single independent variable 'X' and f is a vector of specified load functions. The weak form of this set of differential equations is given by (H.2)
fxl RV T [A(u) + f] dx = 0
The weak form may be integrated by parts to remove all the derivatives from u and place them on v. The result of this step may be expressed as ~xl=R [uTA*(v)-+-vTf] dx -4- [B* (v)] T B(u)
I
XR
- 0
(H.3)
XL
where A*(v) is the adjoint differential equation and B* (v) and B(u) are terms on the boundary resulting from integration by parts. If we can find the general integral to the homogeneous adjoint differential equation A* (v) = 0
(H.4)
then the weak form of the problem reduces to XR
fxl ~RvTf dx + [B* (v)] T B(u)
= 0
(H.5)
XL
The first term is merely an expression to generate equivalent forces from the solution to the adjoint equation and the last term is used to construct the residual equation for the problem. If the differential equation is linear these lead to a residual which depends linearly on the values of u at the ends XL and X R. If we now let these be the location of the end nodes of a typical element we immediately find an expression to generate a stiffness matrix. Since in this process we have never had to construct
702
Solutions exact at nodes
an approximation for the dependent variables u it is immediately evident that at the end points the discrete values of the exact solution must coincide with any admissible approximation we choose. Thus, we always obtain exact solutions at these points. If we consider that all values of the forcing function are contained in f (i.e., no point loads at nodes), the terms in B(u) must be continuous between adjacent elements. At the boundaries the terms in B(u) include a flux term as well as displacements. As an example problem, consider the single differential equation d2u du ---5 dx + P dx + f - 0
(H.6)
with the associated weak form
]xl
du 1
v l dx2+Pdxx + f
dx-0
(H,7)
After integration by parts the weak form becomes
/x?" (
-
dv) ] [ (dUxU) + Pu dx + v f d x + v
dv ] xR - -~x u =0
(H.8)
XL
The adjoint differential equation is given by a*(v) =
d2v dx 2
and the boundary terms by B*(v) --
P
dv dx
=0
{ v} dv dx
(H.9)
(H.10)
and a(u) =
dx + P u
(H.11)
u
For the above example two cases may be identified: 1. P z e r o - where the adjoint differential equation is identical to the homogeneous equation in which case the problem is called self-adjoint. 2. P non-zero - where we then have the non-self-adjoint problem. The finite element solution for these two cases is often quite different. In the first case an equivalent variational theorem exists, whereas for the second case no such theorem exists.t In the first case the solution to the adjoint equation is given by v = Ax + B
(H.12)
t An integrating factor often may be introduced to make the weak form generate a self-adjoint problem; however, the approximation problem will remain the same. See Sec. 3.11.2.
References 703
which may be written as conventional linear shape functions in each element as X R ~
Nt~ = ~
X
;
X R ~
X L
X
NR -- ~
XR
~
;
X L
~
(H.13)
X L
Thus, for linear shape functions in each element used as the weighting function the interelement nodal displacements for u will always be exact (e.g., see Fig. 3.4) irrespective of the interpolation used for u. For the second case the exact solution to the adjoint equation is v - - A e Px + B - - A z + B
(H.14)
This yields the shape functions for the weighting function ZR
--
N L = ~ ; ZR
--
Z ZL
Z --
NR--~;
ZR
--
ZL
(H. 15)
ZL
which when used in the weak form again yield exact answers at the interelement nodes. After constructing exact nodal solutions for u, exact solutions for the flux at the interelement nodes can also be obtained from the weak form for each element. The above process was first given by Tong for self-adjoint differential equations. 1
1. P. Tong. Exact solution of certain problems by the finite element method. J. A I A A , 7:179-180, 1969.
Matrix diagonalization or lumping
Some of the algorithms discussed in this volume become more efficient if one of the global matrices can be diagonalized (also called 'lumped' by many engineers). For example, the solution of some mixed and transient problems are more efficient if a global matrix to be inverted (or equations solved) is diagonal [viz. Chapter 11, Eq. (11.94) and Chapter 16, Secs 16.2.4 and 16.4.2]. Engineers have persisted with purely physical concepts of lumping; however, there is clearly a need for devising a systematic and mathematically acceptable procedure for such lumping. We shall define the matrix to be considered as
A = f~ N a'cN d~2
(I. 1)
where c is a matrix with small dimension. Often c is a diagonal matrix (e.g., in mass or simple least square problems e is an identity matrix times some scalar). When A is computed exactly it has full rank and is not diagonal - this is called the consistent form of A since it is computed consistently with the other terms in the finite element model. The diagonalized form is defined with respect to 'nodes' or the shape functions, e.g., Na = N~I; hence, the matrix will have small diagonal blocks, each with the maximum dimension of e. Only when e is diagonal can the matrix A be completely diagonalized. Four basic lines of argument may be followed in constructing a diagonal form. The first procedure is to use different shape functions to approximate each term in the finite element discretization. For the A matrix we use substitute shape functions Na for the lumping process. No derivatives exist in the definition of A; hence, for this term the shape functions may be piecewise continuous within and between elements and still lead to acceptable approximation. If the shape functions used to define A are piecewise constants, such that l~la is a certain part of the element surrounding the node a and zero elsewhere, and such parts are not overlapping or disjoint, then clearly the matrix of Eq. (I.1) becomes nodally diagonal as fa
{ faa e dr2 -
0
a=b
a 7~ b
(I.2)
Such an approximation with different shape functions is permissible since the usual finite element criteria of integrability and completeness are satisfied. We can verify this using a patch test to show that consistency is still maintained in the approximation.
Matrixdiagonalization or lumping
N~
N~
a
a
(a)
(b) ~
Fig. 1.1 (a) Linear and (b) piecewise constant shape functions for a triangle.
The functions selected need only satisfy the condition l~a
-
-
/VaI with ~ / V a
- 1
(I.3)
a
for all points in the element and this also maintains a partition of unity property in all of f2. In Fig. I. 1 we show the functions Na and Na for a triangular element. The second method to diagonalize a matrix is to note that condition (I. 1) is simply a requirement that ensures conservation of the quantity c over the element. For structural dynamics applications this is the conservation of mass at the element level. Accordingly, it has been noted that any lumping that preserves the integral of c on the element will lead to convergent results, although the rate of convergence may be lower than with use of a consistent A. Many alternatives have been proposed based upon this method. The earliest procedures performed the diagonalization using physical intuition only. Later alternative algorithms were proposed. One suggestion, often called a 'row sum' method, is to compute the diagonal matrix from Aab --
{ ~Cf~aNTacNcdf2 a-- b 0 a :/: b
This simplifies to Aab --
( f~aNTacdf2 a - b 0
a =/: b
(1.4)
(I.5)
since the sum of the shape functions is unity. This algorithm makes sense only when the degrees of freedom of the problem all have the same physical interpretation. An alternative is to scale the diagonals of the consistent mass to satisfy the conservation requirement. In this case the diagonal matrix is deduced from A a b __
~ m ~-'~aNaTCNbdr2 L
0
a -- b
a~b
(1.6)
where m is selected so that
a
Aaa
-- f~ C d ~
(I.7)
The third procedure uses numerical integration to obtain a diagonal array without apparently introducing additional shape functions. Use of numerical integration to
705
706
Matrixdiagonalization or lumping
evaluate the A matrix of Eq. (I. 1) yields a typical term in a summation form (following Chapter 8) / ' 8
(NTcNb)~q Jq Wq
A a b - /.. NaTCNbdr2 - ~
(1.8)
q
~q refers to the quadrature point at which the integrand is evaluated, J is the jacobian volume transformation at the same point and Wq gives the appropriate quadrature weight. If the quadrature points for the numerical integration are located at nodes then, for standard shape functions (viz. Chapter 4), by Eq. (I.3) the diagonal matrix is given where
Aab --
s 0
Wa
a - b a =/: b
(1.9)
where Ja is the jacobian and Wa is the quadrature weight at node a. Appropriate weighting values may be deduced by requiting the quadrature formula to exactly integrate particular polynomials in the natural coordinate system. In general the quadrature should integrate a polynomial of the highest complete order in the shape functions. Thus, for 4-noded quadrilateral elements, linear functions should be exactly integrated. Integrating additional terms may lead to improved accuracy but is not required. Indeed, only conservation of c is required. For low order elements, symmetry arguments may be used to lump the matrix. It is, for instance, obvious that in a simple triangular element little improvement can be obtained by any other lumping than the simple one in which the total c is distributed in three equal parts. For an 8-noded two-dimensional isoparametric element no such obvious procedure is available. In Fig. 1.2 we show the case of rectangular elements of 4-, 8-, and 9-noded type and lumping by Eqs (I.5), (1.6) and (1.9). It is noted that for the 8-noded element some of the lumped quantities are negative when Eq. (1.5) or Eq. (1.9) is used. These will have some adverse effects in certain algorithms (e.g., time-stepping schemes to integrate transient problems) and preclude their use. In Fig. 1.3 we show some lumped matrices for triangular elements computed by quadrature [i.e., Eq. (1.9)]. It is noted here that the cubic element has negative terms while the quadratic element has zero terms. The zero terms are particularly difficult to handle as the resulting diagonal A matrix no longer has full rank and thus may not be inverted. Another aspect of lumping is the performance of the element when distorted from its parent element shape. For example, as a rectangular element is distorted and approaches a triangular shape it is desirable to have the limit triangular shape case behave appropriately. In the case of a 4-noded rectangular element the lumped matrix for all three procedures gives the same answer. However, if the element is distorted by a transformation defined by one parameter f as shown in Fig. 1.4 then the three lumping procedures discussed so far give different answers. The jacobian transformation is given by J = ab(1 - f ) (I. 10) and c is here taken as the identity matrix. The form (I.5) gives Aaa -
ab (1 - f / 3 ) ab (1 + f /3)
at top nodes at bottom nodes
(I.11)
Matrixdiagonalization or lumping 1
1
8 36
1
-
-
1
-
-
1
4
1
-
0 8 3-6
3-6
1
"~
l
4 ""
I -36
" 16
~
o~
I 36
04 3-6
8-node (I.6)
4-node All methods
1
O
I
36
9-node
All methods
I
1 ~
1
12
1 g
12
8-node (I.5) and (I.9) Fig. 1.2 Diagonalization of rectangular elements by three methods.
A-~M 1
0M
g1 MO
og1 M
p=l O(h)
o
o p=2 O(h 2)
o0 M
p=3 O(h 5)
Fig. 1.3 Diagonalization of rectangular elements by three methods. L fa.l. (1-f)a ,_
VT
J
r
J
b X
,iw
Fig. 1.4 Distorted 4-noded element.
w
M
707
708
Matrix diagonalization or lumping
the form (1.6) gives Aaa
ab(1 - f/2) ab(1 + f /2)
--
at top nodes at bottom nodes
(I.12)
and the quadrature form (1.9) yields Aaa
ab(1-f) ab(1 + f )
--
at top nodes at bottom nodes
(I.13)
The 4-noded element has the property that a triangle may be defined by coalescing 2 nodes and assigning them to the same global node in the mesh. Thus, the quadrilateral is identical to a 3-noded triangle when the parameter f is unity. The limit value for the row sum method will give equal lumped terms at the 3 nodes while method (1.6) yields a lumped value for the coalesced node which is two-thirds the value at the other nodes and the quadrature method (1.9) yields a zero lumped value at the coalesced node. Thus, methods (1.6) and (1.9) give limit cases which depend on how the nodes are numbered to form each triangular element. This lack of invariance is not desirable in computer programs; hence for the 4-noded quadrilateral, method (1.5) appears to be superior to the other two. On the other hand, we have observed above that the row sum method (1.5) leads to negative diagonal elements for the 8-noded element; hence there is no universal method for diagonalizing a matrix. A fourth but not widely used method is available which may be explored to deduce a consistent matrix that is diagonal. This consists of making a mixed representation for the term creating the A matrix. Consider a functional given by
uTcudf2
(1.14)
I-I 1 -- f ~ t~uTcu dr2
(I. 15)
I-I1 -- ~
The first variation of 1-II yields
Approximation using the standard form U ,~, f l - - N a f l a
yields
- Nu
(1.16)
P
31-I1 = 3fiT / o NTeN dS2fi
(1.17)
This yields exactly the form for A given by Eq. (I. 1). We can construct an alternative mixed form by introducing a momenta-type variable given by p = cu (I.18) A Hellinger-Reissner-type mixed form may then be expressed as 1-I2 -- f~ uTp dr2 - ~1 f
pTc_ 1p dr2
(I.19)
Matrix diagonalization or lumping
and has the first variation ~FI2- f ~uTp dr2 q - f ~pr ( u - c-lp)dr2
(1.20)
The term with variation on u will combine with other terms so is not set to zero; however, the other term will not appear elsewhere so can be solved separately. If we now introduce an approximation for p as p "~ ~
--
n bPb
-
-
n~
(1.21)
then the variational equation becomes ~H2 -- ~;~r f Nrn d ~ + ~ r ( f
nrN d ~ - fa nrc-ln df2~)
(1.22)
If we now define the matrices G =
J~ nN r dr2
H -
f nrc-lndf2
(1.23)
then the weak form is ~I-I2--[~fi T ~ r ]
([G
0
Gr
_HI(
~
~ }=(
0
O })
(1.24)
Eliminating ~ using the second row of Eq. (I.24) gives A = GTH-1G
(1.25)
for which diagonal forms may now be sought. This form has again the same options as discussed above but, in addition, forms for the shape functions n can be sought which also render the matrix diagonal.
709
This Page Intentionally Left Blank
Author index
Page numbers in bold me for pages at the end of chapters with names of author references. Abdulwahab, E, 469, 470, 497 Abramowitz, M., 161,184 Adams, M., 658, 663, 690, 691 Adee, J., 558, 562 Ahmad, S., 111,114, 120, 136, 331,354 Ainsworth, M., 478, 479, 484, 487, 492, 498 Alberty, J., 487, 498 Allwood, R.J., 445, 454 Alturi, S.N., 356, 380, 445,454 Alum, N.R., 660, 663 Andelfinger, U., 374, 381 Anderson, R.(2, 164, 184 Ando, Y., 88, 101 Archer, J.S., 566, 587 Argyris, J.H., 3,17, 111,118, 119, 120,137, 187, 188, 227, 228, 251,252, 261,262, 406, 427, 590, 628 Arlett, P.L., 229, 245, 261, 565, 575, 577, 587 Armstrong, C.(2, 265, 324, 325 Arnold, D.N., 367, 381, 390, 400, 425, 426, 442, 454 Arrow, K.J., 405, 427 Arya, S.K., 92, 94, 102 At-Abdulla, J., 445, 454 Atluri, S.N., 445, 454 Auricchio, E, 665, 667 Babu~ka, I., 96, 102, 330, 354, 356, 360, 380, 381, 442, 453, 478,479, 483,487,488,489,490, 494, 497, 498, 499, 500, 503, 514, 516, 522, 523, 524, 526, 527,539, 551,552, 553,555,558,560, 561, 562, 658, 663 Bachrach, W.E., 168, 184, 339, 355 Baehmann, P.L., 265, 286, 325 Bahrani, 229, 245,261 Baiocchi, C., 253, 262 Baker, T.J., 265, 307, 325 Balestra, M., 95, 102, 407, 427 Bampton, M.C.C., 573,587 Banerjee, P.K., 97, 102, 446, 454 Bank, R.E., 393,426, 478, 484, 487, 492, 497 Barlow, J., 461,496
Barsoum, R.S., 176, 185 Basu, U., 637, 661 Bathe, K.-J., 153, 184, 253, 263, 572, 583, 587, 614, 630, 666, 667 Batina, J., 540, 542, 543,561 Batina, J.T., 547, 561, 562 Baumann, C.E., 442, 453, 503, 514, 523 Bayless, A., 526, 560 Baynham, J.A.W., 356, 359, 380, 386, 402, 425 Bazeley, (2E, 37, 40, 52, 329, 354 Becker, E.B., 96, 102 Beer, (2, 97, 102, 171,172, 185 Beisinger, Z.E., 568,587 Belytschko, T., 168, 184, 339, 355, 469, 470, 497, 526, 527, 547, 558, 560, 561, 562, 606, 630, 654, 655, 662 Benz, W., 558, 562 Benzley, S.E.176, 186 Bercovier, H., 400, 426 Bercovier, M., 265,325 Beresford, EJ., 343, 345,355 Bettencourt, J.M., 600, 630 Bettess, P., 88,101, 171,172, 175,185, 398,426, 447, 455, 580, 588, 634, 657, 660, 662 Bey, K.S., 503, 514, 523 Biqaniq, N., 340, 342, 355, 648, 650, 652, 662 Biezeno, C.B., 3, 17, 61,100 Bijlaard, P.P., 187, 227 Binns, K.J., 250, 261 Biot, M.A., 648, 661 Bischoff, M., 374, 381 Blacker, T., 547, 562 Blacker, T.D., 265, 326, 469, 470, 497 Bochev, P.B., 407, 410, 411,427 Bociovelli, L.L., 568,587 Bogner, EK., 36, 52 Boley, B.A., 635,660 Bonet, J., 285,326, 558, 562 Booker, J.E, 251,261 Boroomand, B., 330, 354, 475,489, 497, 498
712
Author index Borouchaki, H., 303, 322, 327, 328 Bossak, M., 614, 630 Bottasso, C.L., 321,327 Boyer, A., 265,307, 308, 325 Braess, D., 374, 381 Brauchli, H.J., 207, 228, 466, 496 Brebbia, C.A., 446, 451,454 Brezzi, E, 360, 367, 381, 390, 407, 409, 425, 427, 442, 454 Brigham, E.O., 580, 588 Brown, C.B., 253, 262 Bruch, J.C., 253,262 Buck, K.E., 111,120, 137 Budyn, E., 527, 561 Bugeda, (2, 500, 505,523 Butterfield, 446, 454 Calvo, N.A., 265, 326 Campbell, D.M., 365, 381 Campbell, J., 91,101, 114, 121,134, 137, 207, 228, 466, 496 Canann, S.A., 286, 326 Cantin, (2,377, 382, 600, 630 Caravani, E, 583,588 Carey, (2E, 96, 102 Carlton, M.W., 652, 662 Carpenter, C.J., 171,185 Carslaw, H.S., 565, 586 Carstensen, C., 487, 498 Cavendish, J.C., 265, 280, 325, 326 Chadwick, P., 676, 682 Chan, A.H.C., 40, 53, 211,228, 330, 335, 354, 648, 650, 652, 653, 659, 662, 663 Chan, S.T.K., 252, 262 Chang, C.T., 648, 650, 657,661, 662 Chad, M.V.K., 245 Chen, C.M., 465, 496 Chen, H.-C., 578, 588 Chen, H.S., 88, 101 Chen Po-shu, 429, 453 Chessa, J., 527, 561 Cheung, B.M., 37, 40, 52, 329, 354 Cheung, Y.K., 3, 18, 210, 228, 229, 245, 248, 261, 565, 566, 587 Chi-Wang Shu, 442, 454 Chiba, N., 265,325 Chilton, L.K., 503, 514, 523 Chopp, D.L., 527, 560 Chopra, A.K., 565,566, 580, 583,587, 637, 641,661 Chu, T.Y., 251,261 Ciarlet, P.G., 38, 53, 188, 228, 504, 523 Clesbsch, R.F., 2, 17 Clough, R.W., 1, 2, 3, 17, 20, 37, 52, 187, 227, 398, 426, 566, 580, 583, 587, 644, 648, 661 Cockburn, B., 442, 454 Codina, R., 95, 102, 407, 414, 427, 428
Coffignal, G., 493,498 Collar, A.R., 3, 17 Collatz, L., 540, 561 Collins, T., 583, 588 Cook, R.D., 445,454 Coons, S.A., 141,184 Copps, K., 483, 487, 488, 489, 490, 498 Comes, G.M.M., 445,454 Courant, R., 3, 18, 20, 52, 94, 102, 407, 427 Cowper, (2R., 164, 184 Cox, H.L., 573,587 Craig, A., 573,587 Craig, A.W., 658,663 Crandall, S., 1, 17, 61,100, 563, 565, 586 Crank, J., 594, 629 Crochet, M., 152, 184 Crouzcix, M., 390, 426 Cruse, T.A., 176, 185 Curnier, A., 252, 262 Dahlquist, (2(2, 614, 630 Daniel, W.J.T., 639, 649, 659, 661 Daux, C., 527,560 de Arantese Oliveira, E.R., 38, 53, 330, 336, 354, 501, 523 de (2 Allen, D.N., 1, 17, 97, 102, 239, 261 de Vries, (2,252, 262 Delaunay, B., 304, 327 Demkowicz, L., 478,484, 497, 503, 514, 523 Demmel, J., 36, 52, 557, 562, 574, 588, 684, 691 Dennis, J.E., 666, 667 Desai, C.S., 252, 253, 262, 263 Dhondt, G., 265,326 Dfez, E, 478, 494, 498, 499 Doctors, L.J., 252, 262 Doherty, W.E, 133, 137, 343,355 Dohrmann, C.R., 407, 410, 411,427 Dolbow, J., 526, 527, 547,560, 561 Douglas, J., 367,381 Duarte, C.A., 526, 538, 539, 547, 552, 553,555,558, 560, 562 Duarte, C.A.M., 551,562 Duff, I.S., 557, 562 Duncan, W.J., 3, 17 Dungar, R., 445,454 Dunham, R.S., 359, 381 Dunne, EC., 108, 136 Egozcue, J.J., 478, 494, 498 Eiseman, ER., 316, 327 Elias, Z.M., 379, 382 Ely, J.E, 242, 261 Emson, C., 171,172, 175, 185 Engel, (2,442, 453 Engleman, M.S., 400, 426
Author index 713 Ergatoudis, J.G., 108, 111, 114, 116, 118, 120, 121, 134, 136, 137, 188, 217, 228 Eriksson, K., 596, 630 Escobar, J.M., 321,327 Evans, J.H., 652, 662 Evenson, D.A., 583,588 Farhat, Ch., 429, 435,453 Farin, 268, 290, 326 Faux, I.D., 268, 270, 296, 326 Felippa, C.A., 164,184, 405,427, 634, 639, 655,661, 662 Ferencz, I., 609, 630 Ferencz, R.M., 690, 691 Field, D.A., 265, 325 Finlayson, B.A., 1, 17, 61,100 Finn, N.D.L., 590, 629 Fish, J., 526, 560 Fix, G.J., 38, 40, 53, 141, 166, 184, 206, 228, 330, 336, 354 Fjeld, S., 188, 228 Fletcher, C.A.T., 96, 102 Flores, E, 547, 562 Formaggio, L., 265, 285,325 Forrest, A.R., 141,184 Forsythe, CtE., 540, 561 Fortin, M., 390, 405,406, 425, 426, 427 Fortin, N., 390, 425 Fox, R.L., 36, 52 Fraeijs de Veubeke, B., 36, 40, 52, 53, 119, 137, 330, 354, 359, 379, 381, 382, 490, 498 Franca, L.E, 95, 102, 407, 427, 590, 629 France, E/P., 652, 662 Frasier, G.A., 168, 184 Frazer, R.A., 61,100 Frazer, R.R., 3, 17 Freitag, L.A., 321,327 Freund, J., 95, 102 Frey, W.H., 265, 325 Fried, I., 91,101, 118, 137, 167, 184, 568, 572, 587, 590, 628 Friedmann, EE, 634, 638, 639, 661 Fujishiro, K., 265,325 Furukawa, T., 558, 562 Gago, J.P De S.R., 127, 128, 131,137, 478, 479, 497, 514, 523 Galerkin, B.Ct, 3, 17, 61, 62, 100 Gallagher, R.H., 187, 227, 356, 380, 445,454 Gangaraj, S.K., 483,487, 488, 489, 490, 498 Gantmacher, ER., 611,630 Gaul, L., 97, 102 Gauss, C.E, 3, 17 Gear, C.W., 614, 630 Gear, G.W., 589, 628 George, EL., 265, 303, 307, 322, 325, 327, 328
Georges, M.K., 265, 286, 325 Geradin, M., 634, 639, 661 Gethin, D.T., 303, 311,327 Ghaboussi, J., 343,355 GiD, 664, 666, 667 Gingold, R.A., 558, 562 Girault, V., 525, 559 Glowinski, R., 405,406, 427 Godbole, EN., 398, 426 Gong, N.G., 503, 515, 516, 517, 518, 519, 520, 523 Gonz~ilez-Yuste,J.M., 321,327 Goodier, J.N., 40, 42, 52, 101, 188, 195, 228, 242, 257, 261, 263, 378, 382 Gordon, W.J., 169, 185, 264, 324, 664, 667 Gorensson, E, 634, 639, 661 Gould, EL., 445,454 Gourgeon, H., 446, 455 Graichen, C.M., 316, 327 Grammel, R., 61,100 Gravouil, A., 527, 561 Gresho, EM., 400, 426 (;rice, K.R., 265, 286, 325 Griffiths, A.A., 176, 185 Griffiths, D., 97, 102 Griffiths, R.E., 330, 354 Gu, L., 526, 547, 560, 562 Guex, L., 429, 453 Gui, W., 503, 514, 523 Guo, B., 503, 507, 514, 523 Guo, B.Q., 503, 514, 523 Gupta, A.K., 681,682 Gupta, K.K., 572, 587, 639, 661 Gupta, S., 641,661 Gurtin, M.E., 594, 629 Hager, E, 474, 497 Hall, C.A., 169, 185, 264, 324, 664, 667 Hall, J.E, 641,661 Hammer, EC., 164, 184 Hansbo, E, 265, 285, 300, 325, 327, 478, 497 Hardy Cross, 2, 17 Hardy, O, 503, 514, 523 Hassan, O., 265, 307, 325 Hathaway, A.E, 316, 327 Hause, J., 316, 327 Hayes, L.J., 658, 662 Hearmon, R.ES., 197, 228 Hecht, E, 265, 307, 322, 325, 328 Heimsund, B., 469, 497 Hellan, K., 359, 381 Hellen, T.K., 163, 176, 184, 185 Hellinger, E., 365,381 Henrici, E, 589, 628 Henshell, R.D., 176, 185, 445, 454 Herrera, I., 446, 454, 455
714 Author index Herrmann, L.R., 229, 252, 261, 262, 359, 365, 38t), 381, 385, 386, 425, 462, 496 Hestenes, M.R., 405,427 Hibbitt, H.D., 176, 177, 186 Hilber, H., 614, 627, 630 Hilber, H.M., 111,120, 137 Hildebrand, EB., 34, 52, 76, 79, 97, 101, 102, 589, 628 Hill, T.R., 442, 454, 596, 630 Hine, N.W., 590, 629 Hinton, E., 91,101,207,228, 340, 342, 355, 398,426, 466, 496, 504, 523, 568,587, 634, 645,646, 648, 650, 655,660, 661, 662 Holbeche, J., 639, 661 Hood, P., 390, 426 Houbolt, J.C., 614, 630 HP Fortran home page, 667, 667 Hrenikoff, A., 1, 3, 17 Hsieh, M.S., 245,261 Huang, Y., 465,496 Huck, J., 634, 639, 661 Huebner, K.H., 251,261,262 Huerta, A., 478,494, 498, 499 Hughes, T.R.J., 95, 96, 1t)2, 372, 381, 397, 398,407, 426, 427, 442, 453, 590, 606, 609, 614, 627,629, 630, 634, 654, 655,660, 662 Hulbert, G.M., 95, 102, 407, 409, 427, 590, 629 Humpheson, C., 212, 228 Hurty, W.C., 573, 582, 587 Hurwicz, L., 405,427 Hurwitz, A., 611,630 Ibrahimbegovic, A., 346, 355, 578, 588 Idelsohn, S., 540, 541,542, 561 Idelsohn, S.R., 265,326, 540, 541,561 Iding, R., 501,523 Irons, B.M., 37, 40, 52, 53, 108, 111, 114, 116, 118, 120, 121,134,136,137, 141,155, 163, 164, 166, 177,184,186, 188,217,228,319,329,331,354, 590, 597, 629, 639, 661 Ito, Y., 303,327 Jaeger, J.C., 565,586 Jameson, A., 547, 561, 562 Javandel, I., 252, 262 Jennings, A., 572, 573,587 Jian, B.-N., 94, 102 Jin, H., 265, 285,325 Jirousek, J., 429, 446, 447, 448, 451,453, 455 Joe, B., 286, 316, 326, 327 Johnson, C., 478,497, 590, 596, 629, 630 Johnson, M.W., 38, 53 Johnston, B.E, 286, 326 Jones, W.E, 61,100 Jun, S., 558,562
Kantorovich, L.V., 72, 101 Karniadakis, GE., 442, 454 Kassos, T., 86, 101 Katona, M., 590, 629 Katz, I.N., 516, 524 kazarian, L.E., 652, 662 Kelley, D.W., 88, 101, 131,137, 171,185 Kelly, D.W., 447, 455, 478, 479, 484, 492, 497, 498, 514, 523, 634, 660 Kelsey, S., 3, 17 Key, S.W., 385,425, 568, 587 Kikuchi, E, 88, 101, 363,381 Kim, J.Y., 660, 663 Kitamura, M., 484, 498 Knupp, EM., 321,327 Koch, J.J., 3, 17 Kolsoff, D., 339, 355 Koshgoftar, M., 253, 263 Kosloff, D., 168, 184 Krizek, M., 465,496 Krok, J., 525,560 Kron, G., 3, 17, 435,453 Krylov, 72, 101 Kulasegaram, S., 558,562 Kvamsdal, T., 634, 639, 661 Kwasnik, A., 286, 326 Kwok, W., 285,326 Kythe, Prem K., 97, 102 Ladev~ze, E, 475, 484, 487, 493,497, 498, 518, 524 Ladkany, S.G., 445,454 Lambert, T.D., 589, 628 Lancaster, E, 526, 533,560 Larock, B.E., 252, 262 Larson, M.G., 442, 453 Laug, E, 303, 322, 327, 328 Laursen, T.A., 438, 453 Lawrenson, EJ., 250, 261 Leckie, EA., 566, 587 Ledesma, A., 648, 650, 652, 662 Lee, C.K., 286, 303,326, 327 Lee, K.N., 92, 94, 102 Lee, S.W., 469, 497 Lee, T., 469, 497 Lee, Y.K., 303,327 Lefebvre, D., 366, 371,381 Leguillon, D., 475,484, 493,497 Lekhnitskii, S.Ct, 188, 197, 228 Lesaint, E, 442, 454, 596, 630 Lesoinne, M., 429, 453 Leung, K.H., 648, 650, 655, 662 Levy, J.E, 398,426 Levy, N.177, 186 Lewis, R.W., 212, 228, 303,311,327, 590, 594, 599, 629, 634, 660 Li, B., 469, 496
Author index 715 Li, G.C., 253,263 Li, S., 558, 562 Li, X.D., 474, 497, 596, 623, 630 Liebman, H., 3, 17 Ligget, J.A., 446, 454 Lin, Q., 465,496 Lin, R., 469, 496 Lindberg, CtM., 566, 587 Liniger, W., 599, 614, 630 Liszka, T., 525, 541,547,560, 561 Liu, A., 316, 327 Liu, EL-E, 446, 454 Liu, W.K., 558, 562, 654, 655,662 Livesley, R.K., 11, 18 Lo, S.H., 265,266, 285, 286, 325, 326 Lohner, R., 265, 285, 303,325, 326, 327, 658, 663 Lomacky, O., 176, 185 Loubignac, C., 377, 382 Love, A.E.H., 188, 228 Lowther, C.J., 171,185 Lu, Y., 526, 547, 560, 562 Lucy, L.B., 558, 562 Luenberger, D.G., 405,406, 427 Luke, J.C., 253,262 Lyness, J.E, 245, 249, 261 Lynn, P.P., 92, 94, 102 Machiels, L., 494, 499 Maday, Y., 494, 499 Makridakis, C.G., 442, 453 Malkus, D.S., 390, 398, 426, 568, 587 Mallet, R.H., 36, 52 Mandel, J., 429, 453, 658, 663 Marqal, P.V., 177, 186 Marcum, D.L., 265, 307, 322, 325, 328 Mareczek, G., 111,120, 137, 252, 262 Marini, D., 442, 454 Marlowe, O.P., 164, 184 Martin, C.W., 265,324 Martin, H.C., 1, 3, 17, 187, 227, 252, 262 Martinelli, L., 547, 562 Mastin, C.W., 264, 324 MATLAB, 16, 18, 42, 53, 99, 102, 323,328 Matthies, H., 666, 667 Mavriplis, D.J., 322, 328, 547, 561 Mayer, P., 229, 245,261 Mazzei, L., 442, 453 McDonald, B.H., 245,261 McHenry, D., 1, 3, 17 McLay, R.W., 38, 53 Meek, J.L., 171,172, 185 Mei, C.C., 88, 101 Melenk, J.M., 526, 539, 551,560, 561 Melosh, R.J., 187, 227 Meyers, R.J., 265,326 Mikhlin, S.C., 38, 53, 81,101
Miller, A., 494, 499 Minich, M.D., 36, 52 Miranda, I., 609, 630 Mitchell, A.R., 97, 102, 330, 354 Mitchell, S.A, 265,326 Moes, N., 527, 560, 561 Mohammadi, B., 322, 328 Mohraz, B., 681,682 MOiler, P., 265, 285,325 Monaghan, 558, 562 Monk, P., 503, 514, 523 Montenegro, R., 321,327 Montero, G., 321,327 Moran, B., 527, 560 Morand, H., 634, 638, 639, 661 More, J., 666, 667 Morgan, K., 46, 53, 96, 102, 265,266, 270, 285,286, 325, 326, 407, 428, 447, 455, 504, 523, 658, 663 Morton, K.W., 540, 561, 589, 628 Mote, C.D., 131,137 Mullen, R., 654, 662 Mullord, P., 525, 560 Munro, E., 245, 261 Muskhelish, N.I., 188,228 Naga, A., 474, 497 Nagtegaal, J.C., 133, 137, 403, 427 Nakahashi, K., 303, 327 Nakazawa, S., 40, 53, 330, 354, 362, 381, 398, 404, 406, 426, 427 Nam-Sua Lee, 153, 184 Navert, U., 590, 596, 629 Nay, R.A., 525,560 Naylor, D.J., 91,101, 398, 426 Nayroles, B., 526, 538, 547, 560 Neitaanmaki, P., 465, 496 Newmark, N., 600, 606, 618, 630 Newmark, N.M., 1, 3, 17 Newton, R.E., 155,184, 565,587, 634, 645,648,660, 661 Nickell, R.E., 565,587, 594, 629 Nicolson, P., 594, 629 Nikuchi, N., 253, 262 Nishigaki, I., 265,325 Nithiarasu, P., 69, 71,101, 132, 137, 175, 185, 252, 262, 266, 326, 379, 382, 407, 414, 419, 421,427, 428, 519, 524, 527,560, 571,576, 580, 587, 590, 628, 634, 635, 660 Nitsche, J.A., 438, 453 Norrie, D.H., 252, 262 Oden, J.T., 81, 96, 101, 102, 207, 228, 363,381, 390, 425, 442, 453, 466, 478,479,484, 487,496, 497, 498, 503, 514, 518,523, 524, 526, 538,539, 547, 551,552, 553, 555, 558, 560, 562, 590, 628 Oglesby, J.J., 176, 185
716
Author index Oh, K.E, 251,262 Ohayon, R., 634, 639, 660, 661 Ohnimus, S., 483,498 Ohtsubo, H., 484, 498 Ofiate, E., 61,100, 500, 407, 427, 505,523, 540, 541, 542, 547, 551,552, 561, 562 Orkisz, J., 525, 540, 541,547, 560, 561 O'Rourke, J., 304, 327 Ortiz, E, 407, 414, 427 Osborn, J.E., 356, 380 Ostergren, W.J., 177, 186 Owen, D.R.J., 92, 94, 102, 245, 249, 261 Owen, S.J., 286, 326 Padlog, J., 187, 227 Paidoussis, M.E, 634, 638, 639, 661 Paraschivoiu, M., 494, 499 Parekh, C.J., 565, 587, 590, 629 Par6s, N., 494, 499 Parikh, E, 265,285,325 Parimi, C., 527, 561 Park, H.C., 469, 497 Park, K.C., 655, 659, 662, 663 Parkinson, A.R., 286, 326 Parks, D.M., 133, 137, 176, 185, 403, 427 Parlett, B.N., 547, 572, 587 Parthasarathy, V.N., 316, 327 Pastor, M., 211,228, 648, 650, 652, 662 Patera, A.T., 494, 499 Patil, B.S., 565, 576, 577, 587 Paul, D.K., 645, 646, 648, 650, 659, 661, 662, 663 Pavlin, V., 525, 559 Pawsey, S.E, 398, 426 Peano, A.G., 128, 137 Peck, R.B., 565,587 Peir6, J., 265, 285,286, 300, 303, 325, 326, 327, 442, 454 Pelle, J.E, 487, 493,497, 498 Pelz, R.B., 547, 561 Penzien, J., 566, 580, 583,587, 588 Peraire, J., 265, 266, 270, 285, 286, 325, 326, 494, 499, 504, 523 Perrone, N., 525,559 Phillips, D.V., 169, 184, 264, 324, 664, 667 Pian, T.H.H., 38, 53, 176, 185, 369, 381, 429, 445, 453, 454, 568, 587 Picasso, M., 483,498 Pierre, R., 418, 428 Pierson, K., 429, 453 Pilmer, R., 447 Pister, K.S., 359, 372, 381, 397, 426 Pitkaranta, J., 407, 409, 427, 590, 596, 629, 658, 663 Powell, M.J.D., 400, 405,426 Prager, W., 3, 18, 20, 52 Pratt, M.J, 268, 270, 296, 326 Preparata, EE, 304, 327
Press, W.H., 299, 326 Price, M.A., 265,324, 325 Przemieniecki, J.S., 11, 18 Puso, M.A., 438, 453 Qu, S., 40, 53, 362, 381 Rachowicz, W., 478,484, 497, 503, 514, 523 Radau, R., 164, 184 Ralston, A., 331,354, 684, 691 Ramm, E., 374, 381 Randolf, M.E, 400, 426 Rank, E., 286, 326 Rao, D.V., 251,261 Rashid, Y.R., 227 Rassineux, A., 303,327 Rausch, R.D., 547, 562 Raviart, E-A., 367, 381, 390, 426, 442, 454, 596, 630 Rayleigh, Lord, 3, 17, 35, 52, 76, 101 Razzaque, A., 40, 53, 329, 354 Rebay, S., 265, 307, 325 Reddi, M.M., 251,261 Redshaw, J.C., 188, 228 Reed, W.H., 442, 454, 596, 630 Reid, J.K., 557, 562 Reinsch, C., 572, 587, 666, 667, 684, 691 Reissner, E., 365,381 Rheinboldt, C., 478,479, 497, 500, 522 Rice, J.R., 133, 137, 176, 177, 185, 186, 403, 427 Richardson, L.E, 1, 3, 17, 39, 53 Richtmyer, R.D., 540, 561, 589, 628 Rifai, M.S., 373,381 Riks, E., 666, 667 Ritz, W., 3, 17, 35, 52, 76, 101 Roberts, (2, 634, 639, 661 Robinson, J. et al., 330, 354 Rock, T., 568, 587 Rodr~guez, E., 321,327, 483,498 Rohde, S.M., 251,262 Rougeot, E, 487, 497 Routh, E.J., 611,630 Roux, E-X., 429, 435,453 Rubinstein, M.E, 573, 582, 587 Rudin, W., 105, 136, 561 Sabin, M.A., 265,324, 325 Sabina, EJ., 446, 455 Saigal, S., 286, 326 Salkauskas, K., 526, 533,560 Salonen, E-M., 95, 102 Saltel, E., 265, 307, 322, 325, 328 Salvadori, M., 596, 630 Samuelsson, A., 3, 17 Sandberg, (2, 634, 639, 661 Sander, (2,330, 354 Sani, R.L., 400, 426
Author index 717 Satya, B.V.K.,407,428 Savign~,J.-M.,303,327 Scharp~ D.W.,lll, l18,120,137,251,252,261,262, 590,628 Schmit, L.A.,36,52 Schrefle~ B.A., 211,228, 634, 648, 650, 652, 660, 662 Schroede~W.J.,265,307,325 Schweingrube~M.,286,326 Schweizerho~ K.,666,667 Scoa, EC., 111, 114, 116, 120, 121, 134, 136, 137, 192,228 Scott, J.A.,557,562 Seed, H.B.,644,648,661 Sen, S.K.,445,454 Seveno, E.,265,325 Severn, R.T.,445,454 Shamos, M.I.,304,327 Shaw, K.G.,176,185 Sheffe~ A.,265,325 Shen, S.E, 171,185 Shepard, D.,526,533,539,560 Shephard, M.S.,265,286,307,325 Sherwin, S.J., 303,327, 442, 454 Shiomi, T.,211,228,648,650,652,662 Silves~ E, 118,137,171,185,245 Simkin, J.,245,261 Simo, J.C.,40,53,330,335,354,372,373,381,397, 407,426,427,634,660,666,667 Simon, B.R.,652,662 Sken, S.W.,61,100 Sloan, S.,492,498 Sloan, S.W.,400,426 Sloss, J.M.,253,262 Smolinski, E, 527,561 Snell, C.,525,560 Sokolnikoff, I.S.,101,188,195,228,680,682 Somme~M.,286,326 Soni, B.K.,265,326 Southwell, R.V.,1,2,3,17,69,97,100,379,382 Stab, 0.,303,327 Stanton, E.L., 36, 52 St~en, M.L., 286, 326 Stazi, EL., 527, 561 Stegun, I.A., 161,184 S~in, E.,483,498 Stenberg, R.,465,496 S~ang, G, 38,40,53, 141, 166, 184,206,228,330, 336,354,557,562,666,667,684,691 S~ouboulis, T.,483,487,488,489,490,498 Stroud, A.H.,164,184 Stummel, E, 330,354 Sukumar, N.,527,560,561 Sullivan, J.M.,286,326 Sumihara, K.,369,381 Sun, M.,503,514,523
Suzuki, M., 303,327 Synge, J.L., 3, 18 Szabo, B., 96, 102 Szabo, B.A., 86, 101, 516, 524 Szmelter, J., 20, 52, 283,326 Tabbara, M.,547,562 Tai, X.,469,497 Taig, I.C.,141,184 Takizawa, C.,265,325 Talbe~,J.A.,286,326 Tam, T.K.H.,265,324 Tanesa, D.V.,251,261 Tanne~ R.I.,265,285,325 Tautges, T.J.,265,326 Taylo~ C.,390,425,565,576,577,587 Taylo~ R.L.,33,40,42,52,53, 69,71,72,101,112, 114,118,132,133,137,175,185,216,228,249, 252,253,261, 262,266,299,326,330,335,343, 345,354,355, 356,357,359,362,372,379,380, 381,382,384, 386,397,398,402,407,414,419, 421,425,426, 427,428,442,454,501,519,523, 524,527,540, 541,551,552,560,561,562,571, 574,576,578, 580,587,588,590,595,614,624, 627,628,629, 630,634,635,641,642,648,649, 650,655,660, 661,662,664,665,666,667,684, 688,691 Teodorescu, E, 447,448, 455 Terzhagi, K., 211,228, 565, 587 Thames, EC., 264, 324 Thatcher, R.W., 171,185 Thomas, D.L., 583, 588 Thomasset, E, 405,427 Thompson, J.E, 264, 265, 316, 324, 326, 327 Thomson, H.T., 583,588 Tieu, A.K., 251,262 Timoshenko, S.E, 40, 42,52,101, 188, 195,228,242, 257, 261, 263, 378, 382 Ting, T.C.-T., 197, 228 Todd, D.K., 565, 587 Tong, E, 38, 53, 64, 100, 176, 185, 429, 445, 453, 454, 568, 587, 703,703 Tonti, E., 77, 81,101 Too, J., 398, 426 Topp, L.J., 1, 3, 17, 187, 227 Tortorelli, D.A., 660, 663 Touzot, C., 377, 382 Touzot, G., 526, 538, 547, 56t) Toyoshima, S., 404, 406, 427 Tracey, D.M., 176, 177, 185 Trefftz, E., 446, 454 Treharne, C., 590, 597, 629 Trowbridge, C.W., 245, 250, 261 Trujillo, D.M., 657, 662 Turner, M.J., 1, 3, 17, 187, 227
718 Author index Upadhyay, C.S., 483,487,488, 489, 490, 498 Usui, S., 527, 561 Utku, S., 525, 560 Uzawa, H., 405, 427 Vahdati, M., 265, 266, 270, 325, 504, 523 Vainberg, M.M., 81,101 Valliappan, S., 406, 427 Vanburen, W., 176, 185 Vardapetyan, L., 503, 514, 523 Varga, R.S., 3, 18, 69, 101 Varoglu, E., 590, 629 Vazquez, M., 95, 102, 407, 414, 427, 428 Verfurth, R., 478, 479, 483,497, 498 Vesey, D.G., 525, 560 Victory, H.D., 469, 496 Villon, E, 303, 327, 526, 538, 547, 560 Vilotte, J.E, 404, 406, 407, 427 Visser, W., 229, 261, 565, 586 Vogelius, M., 400, 426 Voronoi, G., 304, 327 Wachspress, E.L., 152, 153, 184, 340, 355 Wahlbin, L.B., 465,496 Walhom, E., 483,498 Walker, S., 446, 451,454 Walsh, EE, 176, 185 Wang, H., 527, 561 Wang, J., 469, 497 Wang, S., 492, 498 Warsi, Z.U.A., 264, 265,324 Washizu, K., 1, 17, 34, 35, 52, 88, 101,253,262, 357, 380, 594, 629 Wasow, W.R., 540, 561 Watson, D.E, 265, 307, 308, 325 Watson, J.O., 97, 102 Weatherill, N.E, 265, 307, 316, 322, 325, 326, 327 Weiner, J.H., 634, 660 Weiser, A., 478,484, 487, 492, 497 Welfert, B.D., 393, 426 Westerman, T.A., 478, 484, 497 Wexler, A., 245,261 Wiberg, N.-E., 431,453, 469, 470, 474, 497, 596, 623, 630 Wilkins, M.L., 3, 18 Wilkinson, J.H., 572, 587, 666, 667, 684, 691 Williams, EW., 583,588 Wilson, E.L., 133,137, 343, 345,346, 355, 565, 578, 583, 587, 588, 594, 614, 629, 630 Winslow, A.M., 229, 245,261 Witherspoon, EA., 252, 262 Wittchen, S.L., 265,286, 325 Wohlmuth, B.I., 432, 453 Wolf, J.A., 445,454
Wood, W., 614, 630 Wood, W.L., 589, 590, 594, 599, 609, 613, 628, 629 Wriggers, E, 666, 667 Wr6blewski, A.E, 429, 446, 45 l, 453 Wu, J., 283,326, 407, 413, 427, 469, 496, 504, 523 Wu, J.S-S., 652, 662 Wyatt, E.A., 171,185 Xie, Y.M., 648, 650, 652, 662 Xu, J., 469, 496 Xu, K., 547, 562 Yagawa, (2, 558, 562 Yamada, T., 558, 562 Yamaguchi, E, 270, 290, 326 Yamashita, Y., 265,325 Yan, N., 465, 496 Yang, H.T.Y., 547,562 Yerry, M.A., 265, 286, 325 Yoshida, Y., 445,454 Zarate, E, 547, 562 Zhang, Y.E, 558, 562 Zhang, Z., 469, 474, 496 Zhao, Q., 469, 496 Zheng, Y., 303, 311,327 Zhong, W.X., 330, 354 Zhu, J.Z., 283,326, 467,469, 477,483,489, 496, 498, 500, 503,504, 505,515,516, 517, 518,519,520, 522, 523 Zhu, Q.D., 465,469, 496 Zi, (2,527, 561 Zielinski, A.E, 429, 446, 447,450, 451,453, 455 Zienkiewicz, O.C., 3, 17, 18, 33, 37, 40, 46, 52, 53, 61, 69, 71, 72, 88, 90, 91, 92, 94, 95, 96, 100, 101,102, 108, 111,114, 116, 118, 120, 121,131, 132, 134,136,137, 164, 169, 171,172, 175,184, 185, 188, 192,210,211,212,216,217,228,229, 242, 245,248,249, 252, 261,262, 264, 265,266, 270, 283,285,299,324, 325, 326, 329, 330, 335, 354, 356, 357, 359, 362, 366, 371,379,380, 381, 382, 384, 386, 398,402, 404, 406, 407, 413, 414, 419, 421,425, 426, 427, 428, 442, 445,446, 447, 450, 454, 455, 467,469,475,477,479, 483,489, 496, 497, 498, 500, 503,504, 505, 514, 515, 516, 517, 518, 519,520, 521,522, 523, 524, 527,540, 541,542, 55 l, 552, 560, 561,562, 565,566, 568, 571,574, 576, 577,580, 587, 588, 590, 595,600, 614, 624, 628, 629, 630, 631,634, 635,641,642, 644, 645,646, 648,649, 650, 652, 653,655,657, 658, 659, 660, 661, 662, 663, 664, 665, 667 Ziukas, S., 469, 470, 497 Zlamal, M., 167, 184, 599, 614, 630
Subject index
Abstraction, 55 Accuracy assessment, numerical examples, 239-53 Acoustic problems, equations for, 635 Adaptive finite element refinement: about adaptive refinement, 500-3, 518-20 asymptotic convergence rate, 504-5 h-refinement, 501-3, 503-14 element subdivision, 501 hanging points, 501 L-shaped domain example, 509 machine part example, 509, 513 mesh regeneration/remeshing, 501 perforated gravity dam example, 513, 515 Poisson equation in a square domain example, 506-11 predicting element size, 503-5 r-refinement, 501-2 short cantilever beam example, 505, 507-9 stressed cylinder example, 505-6, 519 mesh enrichment, 504 p and hp-refinement, 501-3, 514-18 about p and hp-refinement, 51 4-16 L-shaped domain and short cantilever beam example, 51 6-18 permissible error magnitudes, 500 ADI (alternating direction implicit) scheme, 658 Adjoint differential equations: non-self-adjoint, 702-3 self-adjoint, 702-3 Advancing front method of mesh generation s e e Mesh generation, two dimensional, advancing front method Airy stress function, 378-9 Algorithm stability, 609-15 Algorithmic damping, 619 Alternating direction implicit (ADI) scheme, 658 Amplification matrix, 596 Anisotropic and isotropic forms for k, 231-2 Anisotropic materials, 394 elasticity equations, 197-200 Anisotropic seepage problem, 244-5, 247
Approximations: about approximations, 1 and displacement continuity, 20 history of approximate methods, 3 and transformation of coordinates, 12 s e e a l s o Convergence of approximations; Elasticity finite element approximations for small deformations; Function approximation; Least squares approximations; Moving least squares approximations/expansions; Tensor-indicial notation in the approximation of elasticity problems Arch dam in a rigid valley example, 216-17 Area coordinates, 117-18 Assembly and analysis of structures: boundary conditions, 6-7 electrical networks, 7-9 fluid networks, 7-9 general process, 5-6 step one, determination of element properties, 9-10 step two, assembly of final equations, 10 step three, insertion of boundary conditions, 10 step four, solving the equation system, 10 Assessment of accuracy, numerical examples, 239-53 Asymptotic behaviour and robustness of error estimators, 488-90 Asymptotic convergence rate, 504-5 Augmented lagrangian form, 406 Automatic mesh and node generation s e e Mesh generation Auxiliary functions, with complementary forms, 378-9 Axisymmetric deformation problems, 188-9, 235-7 B-bar method for nearly incompressible problems, 397-8 Babu~ka patch test, 490 Babu~ka-Brezzi condition, 363 Back substitution, simultaneous equations, 684 Base solution, patch test, 332 Basis functions s e e Shape functions
720
Subjectindex Beam, circular, subjected to end shear example, 209-10 Beam, rectangular, subjected to end shear example, 209 Bearings, stepped pad, 251 Biomechanics problem of bone-fluid interaction, 652 Blending functions, 169-70 Body forces, distributed, 26 Boundary conditions: about boundary conditions, 6-7, 191 Dirichlet, 59 and equivalent nodal forces, 29 errors from approximation of curved boundaries, 39 forced, 59 forced with natural variational principles, 81 and identification of Lagrange multipliers, 87-8 linear elasticity equations: on inclined coordinates, 192 normal pressure loading, 193 symmetry and repeatability, 192-3 natural (Neumann condition), 60 nodal forces for boundary traction example, 30-1 Boundary value problems: element, 483 Neumann, 484 Bounded estimators, 490--4 equilibrated methods, 494 Boussinesq problem, 174-6 CAD, with surface mesh generation, 286 Cavitation effects in fluids, 645-6 CBS (characteristic-based split) procedure, 407 Central difference approximation, multistep recurrence algorithms, 618 Characteristic-based split (CBS) procedure, 407 Collocation: collocation methods, 525 subdomain/finite volume mehod, 61,547 Taylor series collocation, 593-4 see also Point collocation Complementary forms see u n d e r Mixed formulations Completeness of expansions, 75 Computer procedures: about computer procedures, 664 see also FEAPpv (Finite Element Analysis Program personal version) Conical water tank example, 215-16 Consistency index, mesh generation, 274, 299 Consistent damping matrices, 566 Consistent mass matrix, 566 Constant stress state, 3-node triangle, 26-7 Constitutive relations, 195 Constrained parameters, 12 Constrained variational principles: discretization process, 84-6 -
enforcement with Lagrange multiplier example, 86 locking, 91 penalty function method, 88-9, 90-1 penalty method for constraint enforcement example, 90 perturbed lagrangian functional, 89-91 see also Lagrange multipliers Constraint and primary variables, 360 Continuity requirements: mapped elements, 143-5 mixed formulations, 358-9 Continuous and discrete problems, 1 Contravariant sets of transformations, 12 Contrived variational principles, 77 Convergence of approximations, 74-5 and completeness of expansions, 75 criterion of completeness, 75 h convergence, 75 p convergence, 75 ultraconvergence, 469 see also Superconvergence Convergence criteria and displacement shape functions: and constant strain conditions, 37 and functional completeness statements, 38 and rigid body motion, 37 for standard and hierarchical element shapes, 103 strains to be finite, 37 Convergence rate and discretization error rate, 38-9 Convergence requirements, patch test, 330-1 Coordinates: coordinate transformation, 11-12, 677-8 global, 12 local, 12 Coupled systems: about coupled systems, 631-4, 660 classes, 631-2 definitions, 631 different discretizations, need for, 632-3 partitioned single-phase systems- implicit-explicit partitions (Class I problems), 653-5 see also Fluid-structure interaction (Class 1 problem); Soil-pore fluid interaction (Class II problems); Staggered solution processes Crank-Nicholson scheme, 594, 596 Cubic elements, serendipity family, 113-14 conical water tank example, 215-16 rotating disc analysis example, 212-15 Cubic Hermite polynomials, 269 Cubic triangle, triangular elements family, 119 Curvilinear coordinates, patch test, 330 D'Alembert principle, linear damping, 565 Dam subject to external and internal water pressure example, 210-14 Darn/reservoir interaction, 641
Subject index 721 Damping: algorithmic damping, 619 and participation of modes, 583 s e e a l s o Dynamic behaviour of elastic structures with linear damping; Time dependence; Transient response by analytical procedures Darcy's law, 230 DATRI (FEAPpv sub program), 684-8 Degeneration/degenerate forms: about degeneration, 153-9 degenerate forms for a quadratic 27-node hexahedron example, 158-9 higher order degenerate elements, 155-9 quadratic quadrilateral degenerated triangular element example, 156-8 quadrilateral degenerated into a triangular element example, 153 Degrees of freedom: and matrices, 5 and total potential energy, 35 Delaunay triangulation s e e Mesh generation, three-dimensional, Delaunay triangulation Deviatoric stress and strain, 383-4 deviatoric form for elastic moduli of an isotropic material, 384 laplacian pressure stabilization, 408-9 pressure change, 384 Diagonality with shape functions, 105 Diagonalization or mass lumping, 568-70, 704-9 Diffuse finite element method, 526 Diffusion or flow problems, 230 Direct minimization, 36 Direct pressure stabilization approach to incompressible problems, 41 0-13 Direction cosines, 232 Dirichlet boundary conditions, 59, 231 Nitsche method example, 440 Discontinuity of displacement problems, 39-40 between elements, 32-3 Discontinuous Galerkin method, 442, 596 Discrete and continuous problems, 1 Discrete systems, standard methodology for, 2 Discretization: about discretization procedures, 2 constrained variational principles, 84-6 discretization error and convergence rate, 38-9 singularities problems, 38 finite element process, 233-4 of mixed forms, 358-60 partial, 71-4 three-dimensional curves, 297-9 Displacement, virtual, 28 Displacement approach: and bound on strain energy, 35-6 direct minimization, 36 and minimizing total potential energy, 34-6
Displacement discontinuity between elements problems, 32-3, 39-40 and patch tests, 39-40 Displacement formulation, 20 Displacement functions: about displacement functions, 21-4 rectangle with 4 nodes, 22-4 shape functions, 22 triangle with 3 nodes, 22-3 Displacement gradient, 676 Displacements, result reporting, 207-9 Distributed body forces, 26 Domain decomposition methods s e e Subdomain linking by Lagrange multipliers; Subdomain linking by perturbed lagrangian and penalty methods Driven cavity incompressibility example, 41 6-19 Dual mortar method, 433-4 Dummy and free index, 677 Dynamic behaviour of elastic structures with linear damping: about dynamic behaviour, 565--6 consistent damping matrices, 566 consistent mass matrix, 566 d'Alembert principle, 565 element damping matrix, 566 element mass matrix, 566 mass for isoparametric elements example, 568 plane stress and plane strain example, 567-8 s e e a l s o Eigenvalues and time dependent problems; Time dependence Effective stress concept, 646 Effective stresses with pore pressure, 211-14 Effectivity (error recovery) index 0,477 Eigenproblem assessment, 554 Eigenvalues and time dependent problems: about time dependent problems, 570-1 eigenvalues determination, 572-3 eigenvectors, 572 electromagnetic fields example, 575, 577 forced periodic response, 579 free dynamic vibration- real eigenvalues, 571-2 free responses- damped dynamic eigenvalues, 578 free responses - for first-order problems, 576-8 free vibration with singular K matrix, 573 general linear eigenvalue/characteristic value problem, 572 matrix algebra, 672-3 and modal orthogonality, 572 reduction of the eigenvalue system, 574 standard eigenvalue problem, 572-3 vibration of an earth dam example, 575-6 vibration of a simple supported beam example, 574-5
722 Subjectindex Eigenvalues and time dependent problems - c o n t . waves in shallow water example, 575-7 s e e a l s o Transient response by analytical procedures Elastic constitutive equations, 679-80 Elastic continua, stress and strain in: approximations, 20 and principle of virtual work, 20 solution principle, 19, 24-6 'weak form of the problem', 20 Elastic solution by Airy stress function, 378-9 Elastic structures s e e Dynamic behaviour of elastic structures with linear damping Elasticity finite element approximations for small deformations: approximate weak form, 202 constitutive equation, 201 displacement and strain approximation, 202-5 strains for 8-node brick example, 203-4 equation solution process, 201-2 stiffness and load matrices, 205 quadrature for 8-node brick element example, 205-7 virtual work expression, 201 Elasticity (linear) equations: anisotropic materials, 197-9 example, 199-200 boundary conditions: about boundary conditions, 191 on inclined coordinates, 192 normal pressure loading, 193 symmetry and repeatability, 192-3 constitutive relations, 195 displacement function: axisymmetric deformation, 189 plane stress and plain strain, 189 three-dimensions, 188-9 two-dimensions, 189 elasticity matrix of compliances, 195 elasticity matrix of moduli, 195 equilibrium equations, 190-1 initial strain, 200 isotropic materials, 195-6 material symmetry, 198 orthotropic materials, 197, 198 strain matrix, 189-90 thermal effects, 200-1 transformation of stress and strain, 194-5 Elasticity (linear) problems: about direct physical approaches, 19-20, 46-7 about linear elasticity problems, 187-8 accuracy assessment: beam subjected to end shear example, 40-2 circular beam subjected to end shear example, 42-5 convergence criteria, 37-8
convergence rate considerations, 38-9 displacement approach, 34-6 displacement function, 21-4 finite element solution process, 40 formulation of finite element characteristics, 20-31 generalization to whole region, 31-3 nodal forces for boundary traction example, 30-1 result reporting, 207-9 stiffness matrix for 3-node triangle example, 29-30 stress flow around a reinforced opening application, 45-7 two-dimensional: axisymmetric, 188 plane strain, 188 plane stress, 188 Elasticity (linear) problem examples: arch dam in a rigid valley, 216-17 beam subjected to end shear, 209 circular beam subjected to end shear, 209-10 conical water tank, 215-16 dam subject to external and internal water pressure, 210-14 hemispherical dome, 216 pressure vessel problem, 217 rotating disc analysis, 212-15 Elasticity matrix: compliances, 195 moduli, 195 Electrical networks, assembly, 7-9 Electrostatic field problems, 245-51 Element boundary value problem, 483 Element damping matrix, 566 Element mass matrix, 566 Element matrices, evaluation of, 148-50 Element properties determination, 9-10 Element shape functions s e e Shape functions Element subdivision, h-refinement methods, 501 Energy: and equilibrium, 678-9 minimization of an energy functional, 463 Equation systems: assembly, 10 solving, 10 Equilibrated methods, bounded estimators, 494 Equilibrated residual estimators, 483-4 Equilibrating form subdomains, 444-5 Equilibrium: and energy, 678-9 and total potential energy, 35 Equilibrium equations, linear elasticity, 190-1 'Equivalent forces' concept, 19-20 Errors: about errors/error definitions, 456, 494 bounds on quantifies of interest, 490--4 effectivity index 0,477
Subject index 723 error estimators: asymptotic behaviour and robustness, 488-90 explicit residual error estimator, 479-83 implicit residual error estimator, 483-7 recovery-based, 476-8 residual-based, 478-87 from approximation of curved boundaries, 39 from round-off, 39 Herrmann theorem, 462-5 and irregular scalar quantities, 456 local errors, 456 norms of errors, 457-9 optimal sampling points, 459-65 permissible error magnitudes, 500 recovery of gradients and stresses, 465-7 relative energy norm error, 458 RMS error, 500 singularity effects, 457-8 s e e a l s o Adaptive finite element refinement; Discretization error and convergence rate; Recovery by equilibrium of patches (REP); Residual-based error estimators; Superconvergence Euclidean metric tensor, 294 Euler equations, 78-80 and constrained variational principles, 85 Explicit methods, with time discretization, 592 Explicit residual error estimators, 479-83 Extended finite element method (XFEM), 527 External loads, potential energy, 34 FEAPpv (Finite Element Analysis Program- personal version): about FEAPpv, 664 DATRI sub program, 684-8 element library, 665-6 post-processor module, 666-7 pre-processing module: mesh creation, 664-6 solution module, 666 user modules, 667 Fick's law, 230 Field, electrostatic and magnetostatic, problems, 245-51 Finite element approximations s e e Elasticity finite element approximations for small deformations Finite element characteristics: direct formulation, 20-31 s e e a l s o Displacement functions; Nodal forces; Strain in elastic continua; Stress in elastic continua Finite element discretization, 233-4 Finite element mesh generation by mapping, 169-70 Finite element method/concept: displacement approach, 34-6 history of approximate methods, 3 with indicial notation, 680-2
scalar and vector quantities, 54 solution process, 40 s e e a l s o Generalized finite element method/concept Finite volume method/subdomain collocation, 61,547 Flow or diffusion problems, 230 Fluid flow problems, 251-3 Fluid networks, assembly, 7-9 Fluid-structure interaction (Class 1 problem): about fluid behaviour equations, 634-5 acoustic problems, 635 boundary conditions for the fluid, 635-7 free surface, 636 interface with solid, 636 linearized surface wave condition, 636 perfectly matched layers (PML), 637 radiation boundary, 636--7 cavitation effects in fluids, 645-6 discrete coupled system, 638 forced vibrations and transient step-by-step algorithms, 639-44 dam/reservoir interaction, 641 Routh-Hurwitz conditions, 643 stability of the fluid-structure time-stepping scheme, 642-4 free vibrations, 639-40 linearized dynamic equations, 634 special case of incompressible fluids, 644 added mass matrix, 644 standard Galerkin discretization, 638 weak form for coupled systems, 637-8 s e e a l s o Soil-pore fluid interaction (Class II problems) Fluid-structure systems, staggered schemes, 658-9 Forced boundary conditions, 59 with natural variational principles, 81 Forced periodic response, 579 Forming points, Delaunay triangulation, 304 Formulations s e e Irreducible formulations; Mixed formulations Forward elimination, simultaneous equations, 684 Fourier's law, 230 Fracture mechanics, solutions with mapping, 176-7 Frame methods of linking displacement frames: about frame methods, 442-4 hybrid-stress elements, 445 interior and exterior elements, 447 linking on equilibrating form subdomains, 444-5 equilibrium field example, 445 subdomains with standard elements and global functions, 451 Trefftz-type solutions, 445-51 using virtual work, 443-4 Free and dummy index, 677 Free surface flow and irrotational problems, 251-3
724 Subjectindex Function approximation: about function approximation, 527 interpolation domains and shape functions, 530-2 least squares fit scheme, 527-9 weighted least squares fit, 527-9 Functionals, 34 stationary, 34 Galerkin method/procedure/principle: diffuse elements, 547 solution of ordinary differential equations example, 547-9 discontinuous, 442, 596 and finite element discretization, 233-4 least squares (GLS) stabilization method, 94-5, 409-10 and variational principles, 80 s e e a l s o Weighted residual-Galerkin method Galerkin standard discretization, 638 Galerkin time discontinuous approximation, 619-25 Gauss quadrature, 160, 234 Gauss-Legendre quadrature points, 161,463-5 General linear eigenvalue/characteristic value problem, 572 Generalized finite element method/concept: about finite element generalization, 54-6, 95-7, 525 convergence, 74-5 partial discetization, 71-4 virtual work as 'weak' form of equilibrium equations, 69-71 s e e a l s o Constrained variational principles; Lagrange multipliers; Variational principles; Weighted residual-Galerkin method Generalized Newmark (GN) algorithms: GN22 algorithm, 608-9 GNpj algorithm, 606-9 stability, 609-12 Global coordinates, 12 Global derivatives, computation of, 146-7 GLS (Galerkin least squares) stabilization method, 94-5,409-10 GN algorithms s e e Generalized Newmark (GN) algorithms Gradient, with rates of flow, 230 Gradient matrix, 234 Green's theorem (integration by parts in two or three dimensions), 699-700 Gurtin's variational principle, 594 h convergence, 75 h-refinement s e e Adaptive finite element refinement Hamilton's variational principle, 594 Hanging points, h-refinement methods, 501 Heat conduction:
steady-state, equation in two-dimensions example, 55-6 steady-state Galerkin formulation with triangular elements example, 65-8 time problems, 237-8 weak form-forced and boundary conditions example, 59 Heat conduction-convection: steady state, equation in two-dimensions example, 56 steady-state Galerkin formulation in two-dimensions example, 68-9 Heat equation: in first-order form example, 82-3 with heat generation example, 73-4 Heat transfer solution by potential function example, 378 Hellinger-Reissner variation principle, 365 Helrnholz equation, least squares solution example, 93-4 Helmholz problem in two-dimensions example, 82 Helmholz wave equation, 565 Hemispherical dome example, 216 Hermite cubic spline/Hermite polynomials, 268-9 Hermitian interpolation function, 531 Herrmann theorem and optimal sampling points, 462-5 Hierarchic finite element method based on the partition of unity: about hierarchical forms, 549-52 and global functions, 551 and harmonic wave functions, 552 linear elasticity application, 553-7 polynomial hierarchial method, 552-3 quadratic triangular element example, 554-7 and singular functions, 552 solution of forms with linearly dependent equations, 557-8 Hierarchical shape functions: concepts, standard and hierarchical, 104--6 diagonality, 105 global and local finite element approximation, 131-2 improving of conditioning with hierarchical forms, 130-1 one-dimensional (elastic bar) problem, 105-6 one-dimensional hierarchic polynomials, 125-7 polynomial form, 106 triangle and tetrahedron family, 128-30 two- and three-dimensional elements of the brick type, 128 Hu-Washizu variational principle/theorem, 370, 395 Hybrid-stress elements, 445 Identical and similar algorithms, 599 Identity matrix, 232
Subject index 725 Implicit equations, 416 Implicit methods, with time discretization, 592 Implicit residual error estimators, 483-7 equilibrated residual estimator, 483-4 Incompressible elasticity, three-field (u-p-ev form), 393-8 anisotropic materials, 394 B-bar method for nearly incompressible problems, 397-8 enhanced strain triangle example, 394-6 Hu-Washizu-type variational theorem, 395 variational theorem, 394 Incompressible elasticity, two-field (u-p form), 384-93 bubble function, 386-7 locking (instability) prevention, 385,390 patch tests, multiple-element, 386-9 patch tests, single-element, 386-9 simple triangle with bubble- MINI element example, 390-3 driven cavity example, 418 stability (or singularity) of the matrices, 385 Incompressible problems: about incompressible problems, 383, 421-2 and deviatoric stress and strain, 383-4 driven cavity example, 41 6-19 reduced and selective integration and its equivalence to penalized mixed problems, 398-404 slow viscous flow application, 402-3 with weak patch test example, 403-4 simple iterative solution for mixed problems: Uzawa method, 404-7 tension strip with slot example, 419-21 Incompressible problems for some mixed elements failing the incompressibility patch test, 407-21 about the stability conditions, 407-8 characteristic-based split (CBS) procedure, 407 direct pressure stabilization, 410-13 3-node triangular element example, 412-13 driven cavity example, 418 implicit equations, 416 Galerkin least squares method, 409-10 driven cavity example, 417-18 incompressibility by time stepping, 413-16 Stokes flow equation, 413-14 laplacian pressure stabilization, 408-9 deviatoric stresses and strains, 408 Indicial notation: summation convention, 674-5 see also Tensor-indicial notation in the approximation of elasticity problems Infinite domains and elements: about infinite domains and elements, 170-2 Boussinesq problem, 174--6 convergence considerations, 173 electrostatic and magnetostatic problems, 250
mapping function, 172-6 quadratic interpolations, 174-5 Initial strain, and elasticity equations, 200 Integral/'weak' statements, 57-60 Integration see Numerical integration Integration by parts in two or three dimensions (Green's theorem), 699-700 Integration formulae: tetrahedron, 693 triangles, 692 Interpolating functions, 169 see also Shape functions Interpolation domains, 530-2 circular/spherical domains, 531 discontinuous interpolation, 532 Hermitian interpolation function, 531 and weighting functions, 531 Irreducible formulations, 56, 356-7, 359, 360 see also Mixed formulations Irregular scalar quantities, and errors, 456 Irrotational and free surface flow problems, 251-3 Isoparametric concepts, 554 Isoparametric expansions/elements, 145, 151-2 transformer example, 249-50 Isotropic and anisotropic forms for k, 231-2 Isotropic materials, elasticity equations, 195-6 Iterative solution, simultaneous equations, 688-91 Jacobian matrix, mapped elements, 146, 174 Jump discontinuites, 479-83 Kantorovich partial discretization, 72 Kronecker delta function/property, 108-9, 116, 433, 485 Lagrange multipliers: boundary conditions identification example, 87-8 and constrained variational principles, 84-6 constraint enforcement example, 86 identification of, 87-8 see also Subdomain linking by Lagrange multipliers Lagrange polynomials, 110-12 Lam6 constants, 680 Least squares approximations, 92-5 Galerkin least squares, stabilization, 94-5 and the Herrmann theorem, 462-3 interpolation domains and shape functions, 530-2 solution for Helmholz equation example, 93-4 and variational principles, 92 see also Moving least squares approximations/ expansions Least squares fit scheme, 527-8 fit of a linear polynomial example, 528-9 weighted least squares fit scheme, 529-30 Line (one-dimensional) elements, 119-20
726 Subjectindex Linear damping s e e Dynamic behaviour of elastic structures with linear damping Linear differential operators, 24-5 Linear elasticity s e e Elastic continua, stress and strain in; Elasticity (linear) equations; Elasticity (linear) problems Linear and non-linear relationships, 11 Linearization of vectors, 77 Linearized surface wave condition, fluid-structure interaction, 636 Linking subdomains by Lagrange multipliers s e e Subdomain linking by Lagrange multipliers Load matrix for axisymmetric triangular element with 3 nodes example, 237 Local coordinates, 12 Local errors, 456 Locking, constrained variational principles, 91 Lubrication problems, 251 Magnetostatic field problems, 245-51 Mapped elements: about mapping, 138-9 blending functions, 169-70 Boussinesq problem, 174-5 continuity requirements, 143-5 evaluation of element matrices, 148-50 finite element mesh generation, 169-70 fracture mechanics application, 176-7 geometric conformity of elements, 143 global derivatives, computation of, 146-7 infinite domains and elements, 170-6 interpolating functions, 169 isoparametric elements, 145 jacobian matrix, 146 one-to-one mapping, 141 order of convergence, 151-3 parametric curvilinear coordinates, 139-45 parent elements, 141 quadratic distortion, 142-3 shape functions for coordinate transformations, 139-43 singular elements by mapping, 176-7 subparametric elements, 145 surface integrals, 148 transformations, 145-50 uniqueness rules, 142-3 unreasonable element distortion problems, 141-2 variation of the unknown function problems, 143-5 volume integrals, 147 s e e a l s o Degeneration; Numerical integration Mass lumping or diagonalization, 568-70, 704-9 Material symmetry, 198 Matrices/matrix notation: about matrices, 54 and degrees of freedom, 5 evaluation of element matrices, 148-50
stiffness matrix, 4-5 transformation matrix, 12 Matrix algebra: addition, 669-70 definitions, 668-9 arrays, 668-9 columns, 669 eigenvalue problem, 672-3 inversion, 670 partitioning, 672 spectral form of a matrix, 673 subtraction, 669-70 sum of products, 671 symmetric matrices, 671 transpose of a product, 671 transposing, 670 Matrix singularity due to numerical integration, 167-8 Maxwell's equations, 245-9 Mesh enrichment, 504 Mesh generation: about manual, semi-automatic and automatic mesh generation, 264-6 adaptive refinement, 266 background mesh, 267-8 background meshes, 265-6 boundary curve representation, 267-70 geometrical characteristics of meshes, 266-7 Hermite cubic spline, 268 with mapping, 169-70 structured and unstructured meshes, 265 using quadratic isoparametric elements, 169-70 Mesh generation, surface meshes: about surface mesh generation, 286-7, 301-3 CAD applications, 286 composite cubic surface interpolation, 290-1 curve representation, 288 discretization of three-dimensional curves: node generation on the curves, 297-9 place boundary nodes to parametric plane, 299 element generation in parametric plane, 299-300 examples, 300-1 geometrical characteristics, 290-7 geometrical representation, 287-90 higher order surface elements, 301-2 major steps, 287 mesh control function in three dimensions, 290-3 parametric plane parameters, 293-7 spline surface, 290, 292 surface representation, 288-90 Mesh generation, three-dimensional, Delaunay triangulation: about Delaunay triangulation, 303-4, 306-7, 321-3 automatic node generation procedure, 311-12 Delaunay triangulation algorithm, procedure for, 308-11
Subject index 727 element transformation: two elements, 317, 318 three elements, 317, 318 four elements, 318, 319 five or more elements, 318, 320 forming points, 304 global procedure, 307-8 higher order elements, 321 mesh quality enhancement, 316-21 mesh smoothing, 320-1 node addition and elimination, 319-20 numerical examples, 321,322 surface mesh recovery procedure: boundary edge recovery, 312, 313, 314 boundary face recovery, 313, 315, 316, 317 edge swapping, 312 removal of added points, 315-16 Voronoi diagram, 304-6, 308-10 example, 305-6 properties, 304 Voronoi vertices, 309-10 Mesh generation, two dimensional, advancing front method: about the advancing front method, 266, 285-6 active sides and active nodes, 277 boundary node generation: algorithmic procedure, 271-5 procedure verification example, 275-7 consistency index, 274 diagonal swapping, 282-4 distorted elements, 282 element generation steps, 277-80 Euclidean metric tensor, 273-4 generation front, 277-8 geometrical transformation of the mesh, 271 transformation of a triangle example, 271 higher order elements, 283-5 mesh modification, 281-3 mesh quality enhancement for triangles, 280-3 mesh smoothing, 280 example, 281 node elimination, 282 triangular mesh generation, 270-80 Method of weighted residuals s e e Weighted residual-Galerkin method Minimization of an energy functional, 463 Mixed formulations: about mixed and irreducible formulations, 56, 356-7, 379 complementary forms with direct constraint: about directly constrained forms, 375 auxiliary function solutions, 378-9 complementary elastic energy principle, 377 complementary heat transfer problem, 376-7 elastic solution by Airy stress function, 378-9
heat transfer solution by potential function example, 378 continuity requirements, 358-9 count condition satisfying, 362 discretization of, 358-60 Hu-Washizu variational principle, 370 locking, 361-2 patch test, 362-3 physical discontinuity problems, 363 single-element test examples, 362-3 primary and constraint variables, 360 principle of limitation, 359 singular and non-singular matrices, 361 solvability requirement, 360-1 stability of mixed approximation, 360-3 and the variational principle, 357 Mixed formulations in elasticity, three-field: stability condition, 371-2 u - a - e mixed form, 370-1 u-o'-een form- enhanced strain formulation, 372-5 enhanced strain aspects, 374-5 sheafing strain effects, 374 Simo-Rifai quadrilateral example, 373-4, 375 Mixed formulations in elasticity, two-field: about two-field formulations, 363-4 Hellinger-Reissner variation principle, 365 Pian-Sumihara quadrilateral example, 368-70 Pian-Sumihara rectangle example, 367-8 u-a mixed form, 364-5 u-a mixed form stability, 365-70 Modal decomposition analysis, 580-3 Modal orthogonality, 572 Mortar/dual mortar methods, 432-4 for two-dimensional elasticity example, 436 Moving least squares approximations/expansions, 533-8 hierarchical enhancement, 538-40 partition of unity, 537 Shepard interpolation, 539-40 and symmetric functions, 533-5 and weighting functions, 533-5 Multidomain mixed approximations: about multidomain mixed approximations, 429, 451 s e e a l s o Frame methods of linking displacement frames; Subdomain linking by Lagrange multipliers; Subdomain linking by perturbed lagrangian and penalty methods Multigrid procedures, staggered solution processes, 658 Multistep methods/multistep recurrence algorithms s e e Time discretization, multistep recurrence algorithms Natural variational principles principles
see
Variational
728 Subjectindex Neumann boundary value problem, 484 Neumann (natural) boundary condition, 60 Newton-Cotes quadrature, 160 Nitsche method for subdomain linking, 438-40 Dirichlet boundary condition example, 440 Nodal forces equivalent to boundary stresses and forces, 26-31 3-node triangle, 26-7 boundary considerations, 29 external and internal work done, 28 internal force concept abandoned for generalization, 31-3 nodal forces for boundary traction example, 30-1 plane stress problem, 29 stiffness matrix for 3-node triangle example, 29-30 s e e a l s o Whole region generalization Norms of errors, 457-9 Numerical algorithms, 618-19 Numerical integration: computational advantages, 177-8 Gauss quadrature, 160-1 and matrix singularity, 167-8 minimum order for convergence, 165-6 Newton-Cotes quadrature, 160 one dimensional, 160-1 order for no loss of convergence rate, 166-7 rectangular (2D) or brick regions (3D), 162-4 required order, 164-8 triangular or tetrahedral regions, 164 s e e a l s o Mapped elements Oil fields: ground settlement, 652-3 oil recovery, 653 One dimensional elements, line elements, 119-20 One-to-one mapping, mapped elements, 141 Optimal sampling points and the Herrmann theorem, 462-5 Orthotropic materials, elasticity equations, 197, 198 p convergence, 75 p and ph-refinement s e e Adaptive finite element refinement Parametric curvilinear coordinates, 139-45 Parent elements, mapped elements, 141 Partial discretization, 71-4 finite element discretizations, 238 heat equation with heat generation example, 73-4 Kantorovich, 72 transient problems, 237-9 Partition of unity: and shape functions, 105, 537 s e e a l s o Hierarchic finite element method based on the partition of unity Partitioned single-phase systems - implicit-explicit partitions (Class I problems), 653-5
Pascal triangle, 110, 116 Patch recovery, superconvergent (SPR), 467-74, 490 Patch test: about the patch test, 329-30, 347-50 application to an incompatible element, 343-7 application to elasticity elements with 'standard' and 'reduced' quadrature, 337-43 for base solution example, 337-9 higher order test-assessment of order example, 340-3 for quadratic elements: quadrature effects example, 339-40 Babu~ka patch test, 490 base solution, 332 consistency requirement, 331 convergence requirements, 330-1,332, 334-5 curvilinear coordinates, 330 degree of robustness, 331,488-9 and discontinuity of displacement, 39-40 generality of a numerical patch test, 336 higher order patch tests, 336-7 assessment of robustness, 347 mapped curvilinear elements, 333 mixed formulations, 362-3 non-robust elements, 331 single element tests, 335 size of patch, 333 stability condition, 331 tests A and B, simple tests, 332-4 test C, generalized test, 334-5 weak patch test satisfaction, 333-4 PCG (preconditioned conjugate gradient), with iterative solutions, 690-1 Penalized mixed problems, and reduced and selective integration, 398-404 Penalty functions, constrained variational principles, 88-9 Penalty methods s e e Subdomain linking by perturbed lagrangian and penalty methods Perfectly matched layers (PML), 637 Periodic response, forced, 579 Permissible error magnitudes, 500 Perturbed lagrangian functional, constrained variational principles, 89-91 Perturbed lagrangian method s e e Subdomain linking by perturbed lagrangian and penalty methods Plane stress problem, 29 Plane triangular element with 3 nodes example, 235-6 PML (perfectly matched layers), 637 Point collocation: about point collocation, 61,540-2, 546-7 cross criterion method, 541 Galerkin weighting and finite volume methods, 546-9 with hierarchical interpolations, 543-6
Subject index 729 solution of ordinary differential equations example, 542-6 subdomain collocation/finite element method, 61, 547 Voronoi neighbour criterion method, 541-2 s e e a l s o Least squares approximations; Moving least squares approximations/expansions Polynomial hierarchial method based on the partition of unity, 552-3 Porous material, pore pressure effects, 211-14 Potential energy: external loads, 34 total, 34 Preconditioned conjugate gradient (PCG), with iterative solutions, 690-1 Prescribed functions of space coordinates, 564 Pressure vessel problem example, 217 Primary and constraint variables, 360 Principle of limitation, mixed formulations, 359 Principle of virtual work, 20 Prismatic problems, 72 Quadratic distortion, mapped elements, 142-3 Quadratic elements, serendipity family, 113 Quadratic interpolations, 174-5 Quadratic isoparametric elements, 169-70, 249 Quadratic triangle, triangular elements family, 119 Quasi-harmonic equations: about quasi-harmonic equations, 229 anisotropic and isotropic forms for k, 231-2 axisymmetric problem, 235-7 governing equations, 230-1 with time differential, 563-5 and torsion of prismatic bars, 240-2 two-dimensional plane, 235-6 weak form and variational principal, 233 r-refinement s e e Adaptive finite element refinement Rayleigh-Ritz process/procedure, 35 Recovery, definition, 456 Recovery based error estimators, 476-8 s e e a l s o Errors Recovery by equilibrium of patches (REP), 474--6, 490 Rectangle with 4 nodes, displacement function, 22-4 Rectangular (square) bar, transient heat conduction example, 242-4, 246 Rectangular (three-dimensional) prisms: Lagrange family, 120-1 serendipity family, 121-2 Rectangular (two-dimensional) elements: concepts, 107-9 Lagrange family, 110-12 serendipity family, 112-16 s e e a l s o Standard shape functions Recurrence algorithm, 603
Recurrence relations, 589 Reduced and selective integration and its equivalence to penalized mixed problems, 398-404 Relative energy norm error, 458 REP (Recovery by equilibrium of patches), 474-6, 490 Reproducing kernel (RPK) method, 558 Residual-based error estimators: about residual error estimators, 478 explicit residual error estimators, 479 deriving, example, 479-83 implicit residual error estimators, 483-7 equilibrated residual estimator, 483-4 jump discontinuites, 479-83 recovery processes, 480-1 Result reporting, displacements, strains and stresses, 207-9 Ritz process, 35 RMS error, 500 Robustness index, 488-90 Rotating disc analysis example, 212-15 Rotor blade, transient heat conduction example, 242, 246 Round-off errors, 39 Routh-Hurwitz conditions, fluid-structure interaction, 643 Routh-Hurwitz stability requirements for SS22/SS21 algorithms, 612 Scalar and vector quantities, 54 Seepage: anisotropic, 244-5, 247 fluid flow, 252 soil-pore fluid interaction equation, 647 transient seepage, 565 Self-adjointness/symmetry properties, variational principles, 81,357 Semi-discretization s e e Time dependence Serendipity family, cubic elements: conical water tank example, 215-16 rotating disc analysis, 212-15 Serendipity family, rectangular elements, 112-16, 121-2 corner shape functions, 115 cubic elements, 113-14 mid-side functions, 114-15 quadratic elements, 113 shape function generation, 114 Shape functions: about shape functions, 22, 103-4 and convergence criteria, 103 for coordinate transformations, 139-43 diagonality, 105 elimination of internal parameters before assembly, 132-3 and partition of unity, 105, 537
730
Subjectindex Shape functions - cont. standard and hierarchical concepts, 104-6 substructuring, 133-4 tetrahedral elements, 124-5 and the triangular element family, 118-19 s e e a l s o Displacement functions; Hierarchical shape functions; Standard shape functions Shepard interpolation, moving least squares expansions, 539-40 Similar and identical algorithms, 599 Simo and Rifai enhanced strain formulation, 373-4, 375 Simultaneous discretization, 590 Simultaneous linear equations: back substitution, 684 DATRI (FEAPpv sub program), 684-8 direct methods/solutions, 683-8 forward elimination, 684 iterative solution, 688-91 preconditioned conjugate gradient (PCG), 690-1 resolution process, 687 triangular decomposition, 684, 685 Single-step (SS) algorithms: SS 11 algorithm: example, 605 stability, 612-13 SS22 algorithm: example, 605 stability, 612-13 SS32/SS31 algorithms, stability of, 613-15 SS42/SS41 algorithms, stability of, 614 stability, 609-15 weighted residual finite element form SSpj, 601-6 s e e a l s o Time discretization, single-step algorithms, first and second order equations Singular elements by mapping, 176-7 Singularities, effects on errors, 458-9 Singularities problems, and convergence rate, 38 Smooth particle hydrodynamics (SPH) method, 558 Soil consolidation equations, 565 Soil-pore fluid interaction (Class II problems): about soil-pore fluid interaction, 645-8 biomechanics problem of bone-fluid interaction, 652 coupled equations format, 648 effective stress concept, 646 oil fields, ground settlement, 652-3 robustness requirements, 650 soil liquefaction examples, 650-2 special cases, 649-50 transient step-by-step algorithm, 648-9 Solutions exact at nodes, 701-3 Spectral radius, 619, 620 SPR (superconvergent patch recovery), 467-74, 490 SS algorithms see Single-step (SS) algorithms
Stability/stabilization: algorithm stability, 609-15 generalized Newmark (GN) algorithms, 609-12 incompressible problems, direct pressure stabilization, 410-13 laplacian pressure stabilization, 408-9 least squares (GLS) stabilization method, 94-5, 409-10 patch test stability condition, 331 staggered schemes, 658-9 s e e a l s o Incompressible problems for some mixed elements failing the incompressibility patch test; Single-step (SS) algorithms; Time discretization Staggered solution processes: about staggered solutions, 655 alternating direction implicit (ADI) scheme, 658 in fluid-structure systems and stabilization processes, 658-9 multigrid procedures, 658 in single phase systems, 655-8 Standard discrete systems: about, 1-3, 55 definition and unified treatment, 2, 10-11 linear and non-linear relationships, 11 system equations, 11 system parameters, 10-11 transformation of coordinates, 11-12 s e e a l s o Assembly and analysis of structures Standard shape functions: Kronecker delta, 108-9 standard and hierarchical concepts, 104-6 one-dimensional (line)elements, 119-20 two-dimensional elements, 107-19 completeness of polynomials, 109-10 Lagrange family, 110-12 rectangular element concepts, 107-9 rectangular element families, 110-16 serendipity family, 112-16 triangular element family, 116-19 three-dimensional elements, 120-5 rectangular prisms, Lagrange family, 120-1 rectangular prisms, serendipity family, 121-2 tetrahedral elements, 122-5 Stepped pad bearings, 251 Stiffness, direct stiffness process, 2-3 Stiffness matrix, 4-5 for axisymmetric triangular element with 3 nodes example, 236-7 Stokes flow equation, 413-14 Strain in elastic continua, 19, 24-5 and relationship with stress, 25 s e e a l s o elastic continua, stress and strain in; Elasticity (linear) problems Strain energy: strain energy bound, 36
Subject index 731 of a system, 34 Strain matrix equations, 189-90 Strain rate (virtual strain), 71 Strains, result reporting, 207-9 Stress in elastic continua, 19, 25-6 initial residual stresses, 25 and relationship with strain, 25 s e e a l s o elastic continua, stress and strain in; Elasticity (linear) problems Stress function, and tension of prismatic bars, 240-2 Stresses, result reporting, 207-9 Structural element and system, 3-5 Structure assembly, general process, 5-6 Structured and unstructured meshes, 265 Subdomain collocation/finite volume method, 61,547 Subdomain linking by Lagrange multipliers, 430-6 for elasticity equations, 434-6 mortar method for two-dimensional elasticity example, 431,436 for quasi-harmonic equations, 430-4 mortar/dual mortar methods, 432-4 treatment for forced boundary conditions, 432 Subdomain linking by perturbed lagrangian and penalty methods: about, 436-8 discontinuous Galerkin method, 442 multiple subdomain problems, 440-2 Nitsche method, 438-41 two domain problem example, 442 Subparametric elements, 145 Substructuring, 133-4 Superconvergence, 208, 459-65 about superconvergence, 459 Herrmann theorem and optimal sampling points, 462-5 one-dimensional example, 460-2 superconvergent patch recovery (SPR), 467-74, 490 for displacement and stresses, 474 SPR stress projection for rectangular element patch example, 470-4 Surface integrals, 148 Surface mesh generation s e e Mesh generation, surface meshes Symmetric operators, 357 Symmetry properties/self-adjointness, variational principles, 81 Symmetry and repeatability, with time dependence, 583 System equations, 11 System parameters, 10-11 Taylor series collocation, 593-4 Tension strip with slot incompressibility example, 419-21
Tensor-indicial notation in the approximation of elasticity problems: about the tensor-indicial notation, 674 coordinate transformation, 677-8 free and dummy index, 677 derivatives, 676--7 displacement gradient, 676 elastic constitutive equations, 679-80 equilibrium and energy, 678-9 finite element displacement approximation, 680-2 stiffness coefficient/tensor, 681 first and second rank cartesian tensors, 678 indicial and matrix notation relation, 682 indicial notation: summation convention, 674-5 indicial form, 675 intrinsic notation, 675 Lam6 constants, 680 tensor products, 676 tensorial relations, 676-7 Tetrahedral (three-dimensional) elements, 122-5 cubic shape functions, 124-5 quadratic shape functions, 124 volume coordinates, 122-4 Tetrahedron, integration formulae, 693 Thermal effects, elasticity equations, 200-1 Three-dimensional elements: about three-dimensional elements, 120, 125 rectangular prisms, Lagrange family, 120-1 rectangular prisms, serendipity family, 121-2 tetrahedral elements, 122-5 Time dependence: about time dependence, 563 and boundary conditions, 565 damped wave equation, 565 direct formulation of with spatial finite element subdivision, 563-70 Helmholz wave equation, 565 mass lumping or diagonalization, 568-70 and partial discretization, 237-9 prescribed functions of space coordinates, 564 quasi-harmonic equation with time differential, 563-5 soil consolidation equations, 565 symmetry and repeatability, 583 transient heat conduction equation, 565 s e e a l s o Dynamic behaviour of elastic structures with linear damping; Eigenvalues and time dependent problems; Transient response by analytical procedures Time discontinuous Galerkin approximation, 619-24 solution of a scalar equation example, 623-5 Time discretization: about discrete approximation in time, 589-90 general performance of numerical algorithms, 618-19
732 Subjectindex Time discretization, multistep recurrence algorithms: about multistep recurrence algorithms, 615 approximation procedures, 615-18 central difference approximation, 618 and recurrence relations, 589 three-point interpolation example, 617-18 two-point interpolation example, 617 Time discretization, single-step algorithms, first order equations, 590-600 amplification matrix, 596 conditionally stable/unconditionally stable algorithms, 592 consistency and approximation error, 594-6 Crank-Nicholson scheme, 594, 596 different weight functions problems, 592 discontinuous Galerkin process, 596 explicit/implicit solutions, 592 Gurtin's variational principle, 594 Hamilton's variational principle, 594 identical and similar algorithms, 599 initial value problems, 591 load discontinuities, 600 optimal value of 0,599 smoothing usage, 600, 601,602 stability, 596-9, 609-15 conditional/unconditional stability, 597-8 Taylor series collocation, 593-4 starting/non-starting schemes, 594 weighted residual finite element approach, 590-3 Time discretization, single-step algorithms, first and second order equations: about general single-step algorithms, 600--1 GN22 Newmark algorithm, 608-9 GNpj truncated Taylor series collocation algorithm, 606-9 mean predicted values, 60 predictor-corrector iteration, 609 recurrence algorithm, 603 Routh-Hurwitz stability requirements, 612 SS 11 algorithm: example, 605 stability, 612-13 SS22 algorithm: example, 605-6 stability, 612-13 SS32/SS31 algorithms, stability, 613-15 SS42/SS41 algorithms, stability, 614 stability, conditional/unconditional, 605 stability of general algorithms, 609-15 weighted residual finite element form SSpj, 601-6 Time-stepping procedures, 641 Torsion of prismatic bars, 240-2 hollow bimetallic shaft example, 242 rectangular shaft example, 242 stress function approach, 241 warping function approach, 240-1
Total potential energy: and equilibrium, 35 minimization by displacement approach, 34-6 Tractions, and virtual work, 70 Transformation of coordinates, 11-12 and approximations, 12 and constrained parameters, 12 contravariant sets, 12 stress and strain for linear equations, 194-5 Transformation matrix, 12 Transformations, 145-50 Transient heat conduction: rectangular bar example, 242-4 rotor blade example, 244, 246 Transient response by analytical procedures: about transient response, 579 damping and participation of modes, 583 frequency response procedures, 579-80 modal decomposition analysis, 580-3 s e e a l s o Time dependence Trefftz-type solutions for boundary linking, 445-51 Triangle with 3 nodes, displacement function, 22-3 Triangles, integration formulae, 692 Triangular decomposition, simultaneous equations, 684, 685 Triangular (two-dimensional) element family, 116-19 area coordinates, 117-18 cubic triangle, 119 quadratic triangle, 119 shape functions, 118-19 Truncated Taylor series expansion algorithm GNpj, 606-9 Two-dimensional elements s e e Rectangular (two-dimensional) elements; Triangular (two-dimensional) element family Two-dimensional plane problem, 235-7 load matrix for axisymmetric triangular element with 3 nodes example, 237 plane triangular element with 3 nodes example, 235-6 stiffness matrix for axisymmetric triangular element with 3 nodes example, 236-7 u-a-e mixed forms s e e u n d e r Mixed formulations Ultraconvergence, 469 Uzawa method, iterative solution process for mixed problems, 404-7 Variational principles: about variational principles, 76-8 contrived variational principles, 77 Euler equations, 78-80 forced boundary condition equations, 81 and the Galerkin method/process, 80 heat equation in first-order form example, 82-3 Helmholz problem in two-dimensions example, 82
Subject index 733 least squares approximations, 92-5 maximum, minimum, or saddle point?, 83-4 natural variational principles, 78-80, 81-3 self-adjointness/symmetry properties, 81,357 see also Constrained variational principles; Lagrange multipliers 9Variational theorem, 394 Vector algebra: about vector algebra, 694 addition, 694-5 direction cosines, 696 elements of area and volume, 697-8 length of a vector, 695-6 scalar products, 695 subtraction, 694-5 vector or cross product, 696-7 Vector linearization, 77 Vector potential, 245 Vibration: of an earth dam example, 575-6 free vibration with singular K matrix, 573 of a simple supported beam example, 574-5 also see u n d e r Fluid-structure interaction (Class 1 problem); Eigenvalues and time dependent problems Virtual displacement, 28 Virtual strain (strain rate), 71 Virtual work: principle, 20, 34 and tractions, 70 as 'weak form' of equilibrium equations, 69-71 Viscous flow problems, 251-3 Volume integrals, 147 Voronoi diagram, 304-6, 308-10 see also Mesh generation, three-dimensional, Delaunay triangulation Voronoi neighbour criterion point collocation, 541-2
Weak form: coupled systems, 637-8 integral/'weak' statements, 57-60 quasi-harmonic equations, 233 small elastic deformations, 202 and virtual work, 69-71 'weak form of the problem', 20 'Weak'/integral statements, 57-60 Weighted least squares approximation, 463 Weighted least squares fit scheme, 529-30 Weighted residual-Galerkin method: about the weighted residual method, 55, 60-2 approximation to integral formulations, 60-9 convergence, 74-5 Galerkin formulation with triangular elements example, 65-8 and integral/'weak' statements, 57-60 one-dimensional equation of heat conduction example, 62-5 and partial discretization, 72 partial discretization, 71-4 and point collocation, 61 residuals, 61 restrictions needed, 58 steady-state heat conduction in two-dimensions example, 65-8 steady-state heat conduction-convection in two-dimensions example, 68-9 and subdomain collocation, 61 virtual work as the 'weak form' of equilibrium, 69-71 weak form of the heat conduction equation example, 59-60 Weighting function choice, 701 Whole region generalization, 31-3 Work done principle/concept, 28 virtual work, 34
Warping function, and tension of prismatic bars, 240-2
XFEM (extended finite element method), 527
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Plate 2 Analysis of an arch dam in China
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The Finite Element Method for Fluid Dynamics Sixth edition
Professor O.C. Zienkiewicz, CBE, FRS, FREng is Professor Emeritus at the Civil and Computational Engineering Centre, University of Wales Swansea and previously Director of the Institute for Numerical Methods in Engineering at the University of Wales Swansea, UK. He holds the UNESCO Chair of Numerical Methods in Engineering at the Technical University of Catalunya, Barcelona, Spain. He was the head of the Civil Engineering Department at the University of Wales Swansea between 1961 and 1989. He established that department as one of the primary centres of finite element research. In 1968 he became the Founder Editor of the International Journal for Numerical Methods in Engineering which still remains today the major journal in this field. The recipient of 27 honorary degrees and many medals, Professor Zienkiewicz is also a member of five academies - an honour he has received for his many contributions to the fundamental developments of the finite element method. In 1978, he became a Fellow of the Royal Society and the Royal Academy of Engineering. This was followed by his election as a foreign member to the US Academy of Engineering (1981), the Polish Academy of Science (1985), the Chinese Academy of Sciences (1998), and the National Academy of Science, Italy (Academia dei Lincei) (1999). He published the first edition of this book in 1967 and it remained the only book on the subject until 1971. Professor R.L. Taylor has more than 40 years' experience in the modelling and simulation of structures and solid continua including two years in industry. He is Professor in the Graduate School and the Emeritus T.Y. and Margaret Lin Professor of Engineering at the University of California at Berkeley. In 1991 he was elected to membership in the US National Academy of Engineering in recognition of his educational and research contributions to the field of computational mechanics. Professor Taylor is a Fellow of the US Association of Computational Mechanics - USACM (1996) and a Fellow of the International Association of Computational Mechanics- IACM (1998). He has received numerous awards including the Berkeley Citation, the highest honour awarded by the University of California at Berkeley, the USACM John von Neumann Medal, the IACM Gauss-Newton Congress Medal and a Dr.-Ingenieur ehrenhalber awarded by the Technical University of Hannover, Germany. Professor Taylor has written several computer programs for finite element analysis of structural and non-structural systems, one of which, FEAP, is used world-wide in education and research environments. A personal version, FEAPpv, available from the publisher's website, is incorporated into the book. Dr P. Nithiarasu, Senior Lecturer at the School of Engineering, University of Wales Swansea, has over ten years' experience in finite element based computational fluid dynamics research. He moved to Swansea in 1996 after completing his PhD research at IIT Madras. He was awarded the Zienkiewicz silver medal and prize of the Institution of Civil Engineers, UK in 2002. In 2004 he was selected to receive the European Community on Computational Methods in Applied Sciences (ECCOMAS) award for young scientists in computational engineering sciences. Dr Nithiarasu is the author of several articles in the area of fluid dynamics, porous medium flows and the finite element method.
The Finite Element Method for Fluid Dynamics Sixth edition O.C. Zienkiewicz, CBE, FRS
Professor Emeritus, Civil and Computational Engineering Centre University of Wales Swansea UNESCO Professor of Numerical Methods in Engineering International Centre for Numerical Methods in Engineering, Barcelona
R.L. Taylor
Professor in the Graduate School Department of Civil and Environmental Engineering University of California at Berkeley Berkeley, California
P. Nithiarasu
Civil and Computational Engineering Centre School of Engineering University of Wales Swansea
ELSEVIER B~RWORTH HEINEMANN AMSTERDAM
9 BOSTON
PARIS 9 SAN DIEGO
9 HEIDELBERG
9 SAN FRANCISCO
9 LONDON
9 NEW YORK
9 SINGAPORE
9 SYDNEY
9 OXFORD 9 TOKYO
Elsevier Butterworth-Heinemann Linacre House, Jordan Hill, Oxford OX2 8DP 30 Corporate Drive, Burlington, MA 01803 First published in 1967 by McGraw-Hill Fifth edition published by Butterworth-Heinemann 2000 Reprinted 2002 Sixth edition 2005 Reprinted 2006 Copyright 9 2000, 2005, O.C. Zienkiewicz, R.L. Taylor and P. Nithiarasu. All rights reserved The right of O.C. Zienkiewicz, R.L. Taylor and P. Nithiarasu to be identified as the authors of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988 No part of this publication may be reproduced in any material form (including photocopying or storing in any medium by electronic means and whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London, England W 1T 4LP. Applications for the copyright holder's written permission to reproduce any part of this publication should be addressed to the publisher Permissions may be sought directly from Elsevier's Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: [email protected]. You may also complete your request on-line via the Elsevier homepage (http://www.elsevier.com), by selecting 'Customer Support' and then 'Obtaining Permissions' British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication Data A catalogue record for this book is available from the Library of Congress
ISBN 0 7506 6322 7 Published with the cooperation of CIMNE, the International Centre for Numerical Methods in Engineering, Barcelona, Spain (www.cimne.upc.es)
For information on all Elsevier Butterworth-Heinemann publications visit our website at books.elsevier.com
Printed and bound in Great Britain by MPG Books Ltd., Bodmin, Cornwall
Dedication This book is dedicated to our wives Helen, Mary Lou and Sujatha and our families for their support and patience during the preparation of this book, and also to all of our students and colleagues who over the years have contributed to our knowledge of the finite element method. In particular we would like to mention Professor Eugenio Ofiate and his group at CIMNE for their help, encouragement and support during the preparation process.
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Contents
P refa c e Acknowledgements
1
2
3
xi xiii
Introduction to the equations of fluid dynamics and the finite element approximation 1.1 General remarks and classification of fluid dynamics problems discussed in this book 1.2 The governing equations of fluid dynamics 1.3 Inviscid, incompressible flow 1.4 Incompressible (or nearly incompressible) flows 1.5 Numerical solutions: weak forms, weighted residual and finite element approximation 1.6 Concluding remarks References
14 26 27
Convection dominated problems- finite element approximations to the convection-diffusion-reaction equation 2.1 Introduction 2.2 The steady-state problem in one dimension 2.3 The steady-state problem in two (or three) dimensions 2.4 Steady state - concluding remarks 2.5 Transients - introductory remarks 2.6 Characteristic-based methods 2.7 Taylor-Galerkin procedures for scalar variables 2.8 Steady-state condition 2.9 Non-linear waves and shocks 2.10 Treatment of pure convection 2.11 Boundary conditions for convection-diffusion 2.12 Summary and concluding remarks References
28 28 31 45 49 50 53 65 66 66 70 72 73 74
The characteristic-based split (CBS) algorithm. A general procedure for compressible and incompressible flow 3.1 Introduction 3.2 Non-dimensional form of the governing equations 3.3 Characteristic-based split (CBS) algorithm
79 79 81 82
1
4 11 13
viii
Contents
3.4 3.5 3.6 3.7 3.8 3.9 3.10
Explicit, semi-implicit and nearly implicit forms Artificial compressibility and dual time stepping 'Circumvention' of the Babu~ka-Brezzi (BB)restrictions A single-step version Boundary conditions The performance of two-step and one-step algorithms on an inviscid problem Concluding remarks References
92 95 97 98 100 103 104 105
4
Incompressible Newtonian laminar flows 4.1 Introduction and the basic equations 4.2 Use of the CBS algorithm for incompressible flows 4.3 Adaptive mesh refinement 4.4 Adaptive mesh generation for transient problems 4.5 Slow flows - mixed and penalty formulations 4.6 Concluding remarks References
110 110 112 123 131 131 136 136
5
Incompressible non-Newtonian flows 5.1 Introduction 5.2 Non-Newtonian flows - metal and polymer forming 5.3 Viscoelastic flows 5.4 Direct displacement approach to transient metal forming 5.5 Concluding remarks References
141 141 141 154 163 165 166
Free surface and buoyancy driven flows 6.1 Introduction 6.2 Free surface flows 6.3 Buoyancy driven flows 6.4 Concluding remarks References
170 170 170 189 191 193
Compressible high-speed gas flow 7.1 Introduction 7.2 The governing equations 7.3 Boundary conditions - subsonic and supersonic flow 7.4 Numerical approximations and the CBS algorithm 7.5 Shock capture 7.6 Variable smoothing 7.7 Some preliminary examples for the Euler equation 7.8 Adaptive refinement and shock capture in Euler problems 7.9 Three-dimensional inviscid examples in steady state 7.10 Transient two- and three-dimensional problems 7.11 Viscous problems in two dimensions 7.12 Three-dimensional viscous problems
197 197 198 199 202 203 205 206 212 217 226 227 240
Contents
7.13 7.14
Boundary layer-inviscid Euler solution coupling Concluding remarks References
241 242 242
8
Turbulent flows 8.1 Introduction 8.2 Treatment of incompressible turbulent flows Treatment of compressible flows 8.3 8.4 Large eddy simulation 8.5 Detached Eddy Simulation (DES) 8.6 Direct Numerical Simulation (DNS) 8.7 Concluding remarks References
248 248 251 264 267 270 270 271 271
9
Generalized flow through porous media Introduction 9.1 9.2 A generalized porous medium flow approach 9.3 Discretization procedure 9.4 Non-isothermal flows 9.5 Forced convection 9.6 Natural convection Concluding remarks 9.7 References
274 274 275 279 282 282 284 288 289
10 Shallow water problems 10.1 Introduction 10.2 The basis of the shallow water equations 10.3 Numerical approximation 10.4 Examples of application 10.5 Drying areas 10.6 Shallow water transport 10.7 Concluding remarks References
292 292 293 297 298 310 311 313 314
11 Long and medium waves 11.1 Introduction and equations 11.2 Waves in closed domains - finite element models 11.3 Difficulties in modelling surface waves 11.4 Bed friction and other effects 11.5 The short-wave problem 11.6 Waves in unbounded domains (exterior surface wave problems) 11.7 Unbounded problems 11.8 Local Non-Reflecting Boundary Conditions (NRBCs) 11.9 Infinite elements 11.10 Mapped periodic (unconjugated) infinite elements 11.11 Ellipsoidal type infinite elements of Burnett and Holford 11.12 Wave envelope (or conjugated) infinite elements 11.13 Accuracy of infinite elements
317 317 318 320 320 320 321 324 324 327 327 328 330 332
ix
x
Contents
11.14 11.15 11.16 11.17 11.18 11.19
Trefftz type infinite elements Convection and wave refraction Transient problems Linking to exterior solutions (or DtN mapping) Three-dimensional effects in surface waves Concluding remarks References
332 333 335 336 338 344 344
12 Shortwaves 12.1 Introduction 12.2 Background 12.3 Errors in wave modelling 12.4 Recent developments in short wave modelling 12.5 Transient solution of electromagnetic scattering problems 12.6 Finite elements incorporating wave shapes 12.7 Refraction 12.8 Spectral finite elements for waves 12.9 Discontinuous Galerkin finite elements (DGFE) 12.10 Concluding remarks References
349 349 349 351 351 352 352 364 372 374 378 378
13 Computer implementation of the CBS algorithm 13.1 Introduction 13.2 The data input module 13.3 Solution module 13.4 Output module References
382 382 383 384 387 387
Non-conservative form of Navier-Stokes equations Self-adjoint differential equations Postprocessing Integration formulae
389 391 392 395
Appendix E
Convection-diffusion equations: vector-valued variables
397
Appendix F
Edge-based finite element formulation
405
Appendix G
Multigrid method
407
Appendix H
Boundary layer-inviscid flow coupling
409
Appendix I
Mass-weighted averaged turbulence transport equations
413
Appendix Appendix Appendix Appendix
A B C D
Author index
417
Subject index
427
Preface
The major part of this book has been derived by updating the third volume of the fifth edition. However, it now contains three new chapters and also major improvements in the existing ones. Its objective is to separate the fluid dynamics formulations and applications from those of solid mechanics and thus to reach perhaps a different interest group. It is our intention that the present text could be used by investigators familiar with the finite element method in general terms and introduce them to the subject of fluid dynamics. It can thus in many ways stand alone. Although the finite element discretization is briefly covered here, many of the general finite element procedures may not be familiar to a reader introduced to the finite element method through different texts and therefore we advise that this volume be used in conjunction with the text on 'The Finite Element Method: Its Basis and Fundamentals' by Zienkiewicz, Taylor and Zhu to which we make frequent reference. In fluid dynamics, several difficulties arise. The first is that of dealing with incompressible or almost incompressible situations. These as we already know present special difficulties in formulation even in solids. The second difficulty is introduced by the convection which requires rather specialized treatment and stabilization. Here, particularly in the field of compressible, high speed, gas flow many alternative finite element approaches are possible and often different algorithms for different ranges of flow have been suggested. Although slow creeping flows may well be dealt with by procedures almost identical to those of solid mechanics, the high speed range of supersonic and hypersonic kind will require a very particular treatment. In this text we shall use the so-called Characteristic-Based Split (CBS) introduced a few years ago by the authors. It turns out that this algorithm is applicable to all ranges of flow and indeed gives results which are at least equal to those of specialized methods. We organized the text into 13 individual chapters. The first chapter introduces the topic of fluid dynamics and summarizes all relevant partial differential equations together with appropriate constitutive relations. Chapter 1 also provides a brief summary of the finite element formulation. In Chapter 2 we discuss convection stabilization procedures for convection-diffusion-reaction equations. Here, we make reference to methods available for steady and transient state equations and also one and multidimensional equations. We also discuss the similarity between various stabilization procedures. From Chapter 3 onwards the discussion is centred around the numerical
xii
Preface
solution of fluid dynamic equations. In Chapter 3, the CBS scheme is introduced and discussed in detail in its various forms. Its simplicity and universality makes it highly desirable for the study of incompressible and compressible flows and in the later chapters we shall indicate its widely applicable use. Though not all problems are necessarily solved using this method in this book, as work of several decades are reported here, the reader shall find the CBS method in general at least as accurate as other methods and that its performance is very good. For this reason we do not describe any other alternatives to make the reader's life simple. The topic of incompressible fluid dynamics is covered in Chapters 4, 5 and 6. Chapter 4 discusses the general Newtonian incompressible flows without reference to any special problems. This chapter could be used as a validating part of any fluid dynamics code development for incompressible flows. Chapter 5 discusses the non-Newtonian flows in general and metal forming and visco-elastic flows in particular. In Chapter 6 we discuss the special topics of gravity assisted incompressible flows which include treatment of free surfaces and buoyancy driven flows. Chapter 7 is devoted to compressible gas flows. Here, we discuss several special requirements for solving Navier-Stokes equations including phenomena such as shock capturing and adaptivity. Chapters 8 and 9 are new additions to the book. In Chapter 8 we discuss various basic turbulence modelling options available for both compressible and incompressible flows and in Chapter 9 we provide a brief description of flow through porous media. Chapter 10 discusses the shallow water flow and here application of the CBS scheme to a different incompressible flow approximation is considered. Although the flow is incompressible the approximations and variables involved produce a set of differential equations similar to those of compressible flows. Thus, the use of methods already derived for the solution of compressible flow is obvious for dealing with shallow water problems. Chapters 11 and 12 provide a detailed overview on the numerical treatment of long and short waves. Chapter 12 is a new chapter and both these chapters on waves are contributed by Professor Peter Bettess, University of Durham. The last chapter of this book is a brief outline on computer implementation. Further details, including source codes, are available from the author's personal home pages www.nithiarasu.co.uk and www.elsevier.com. We hope that the book will be useful in introducing the reader to the complex subject of computational fluid dynamics (CFD) and its many facets. Further, we hope it will also be of use to the experienced practitioner of CFD who may find the new presentation of interest to practical application.
Acknowledgements
The authors would like to thank Professor Peter Bettess for largely contributing the chapters on waves (Chapters 11 and 12), in which he has made so many achievements, and Dr Pablo Ortiz who with the main author was first to apply the CBS algorithm to shallow water equations and Chapter 10 of this text is partly contributed by him. Several other colleagues contributed to this text either directly or indirectly. Professors K. Morgan, N.P. Weatherill and O. Hassan, all from the University of Wales Swansea, Professors E. On~te and R. Codina, both from CIMNE, Barcelona, Professor J. Peraire from MIT and Professor R. Lrhner from George Mason University, USA, are a few to name. The third author thanks Professor P.G. Tucker, University of Wales, Swansea, and Dr S. Vengadesan, liT, Madras, for their constructive comments on the chapter on turbulence. The third author also thanks his graduate students Ray Hickey and Chun-Bin Liu for their assistance.
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Introduction to the equations of fluid dynamics and the finite
element approximation
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The problems of solid and fluid behaviour are in many respects similar. In both media stresses occur and in both the material is displaced. There is, however, one major difference. Fluids cannot support any deviatoric stresses when at rest. Thus only a pressure or a mean compressive stress can be carried. As we know, in solids deviatoric stresses can exist and a solid material can support general forms of structural forces. In addition to pressure, deviatoric stresses can develop when the fluid is in motion and such motion of the fluid will always be of primary interest in fluid dynamics. We shall therefore concentrate on problems in which displacement is continuously changing and in which velocity is the main characteristic of the flow. The deviatoric stresses which can now occur will be characterized by a quantity that has great resemblance to the shear modulus of solid mechanics and which is known as dynamic viscosity (molecular viscosity). Up to this point the equations governing fluid flow and solid mechanics appear to be similar with the velocity vector u replacing the displacement which often uses the same symbol. However, there is one further difference, even when the flow has a constant velocity (steady state), convective acceleration effects add terms which make the fluid dynamics equations non-self-adjoint. Therefore, in most cases, unless the velocities are very small so that the convective acceleration is negligible, the treatment has to be somewhat different from that of solid mechanics. The reader should note that for self-adjoint forms, approximating the equations by the Galerkin method gives the minimum error in the energy norm and thus such approximations are in a sense optimal. In general, this is no longer true in fluid mechanics, though for slow flows (creeping flows) where the convective acceleration terms are negligible the situation is somewhat similar. With a fluid which is in motion, conservation of mass is always necessary and, unless the fluid is highly compressible, we require that the divergence of the velocity vector be zero. Similar problems are encountered in the context of incompressible elasticity
2 Introductionto the equations of fluid dynamics and the finite element approximation and the incompressibility constraint can introduce difficulties in the formulation (viz. reference 1). In fluid mechanics the same difficulty again arises and all fluid mechanics approximations have to be such that, even if compressibility is possible, the limit of incompressibility can be modelled. This precludes the use of many elements which are otherwise acceptable. In this book we shall introduce the reader to finite element treatment of the equations of motion for various problems of fluid mechanics. Much of the activity in fluid mechanics has, however, pursued a finite difference formulation and more recently a derivative of this known as the finite volume technique. Competition between finite element methods and techniques of finite differences have appeared and led to a much slower adoption of the finite element process in fluid dynamics than in structures. The reasons for this are perhaps simple. In solid mechanics or structural problems, the treatment of continua often arises in combination with other structural forms, e.g. trusses, beams, plates and shells. The engineer often dealing with structures composed of structural elements does not need to solve continuum problems. In addition when continuum problems are encountered, the system can lead to use of many different material models which are easily treated using a finite element formulation. In fluid mechanics, practically all situations of flow require a two- or three-dimensional treatment and here approximation is required. This accounts for the early use of finite differences in the 1950s before the finite element process was made available. However, as pointed out in reference 1, there are many advantages of using the finite element process. This not only allows a fully unstructured and arbitrary domain subdivision to be used but also provides an approximation which in self-adjoint problems is always superior to or at least equal to that provided by finite differences. A methodology which appears to have gained an intermediate position is that of finite volumes, which were initially derived as a subclass of finite difference methods. As shown later in this chapter, these are simply another kind of finite element form in which subdomain collocation is used. We do not see much advantage in using this form of approximation; however, there is one point which seems to appeal to some investigators. That is the fact the finite volume approximation satisfies conservation conditions for each finite volume. This does not carry over to the full finite element analysis where generally satisfaction of conservation conditions is achieved only in an assembly region of elements surrounding each node. Satisfaction of the conservation conditions on an individual element is not an advantage if the general finite element approximation gives results which are superior. In this book we will discuss various classes of problems, each of which has a certain behaviour in the numerical solution. Here we start with incompressible flows or flows where the only change of volume is elastic and associated with transient changes of pressure (Chapters 4 and 5). For such flows full incompressible constraints must be available. Further, with very slow speeds, convective acceleration effects are often negligible and the solution can on occasion be reached using identical programs to those derived for linear incompressible elasticity. This indeed was the first venture of finite element developers into the field of fluid mechanics thus transferring the direct knowledge from solid mechanics to fluids. In particular the so-called linear Stokes flow is the case where fully incompressible but elastic behaviour occurs. A particular variant of Stokes flow is that used in metal forming where the material can no longer be described by a constant
General remarks for fluid mechanics problems 3 viscosity but possesses a viscosity which is non-Newtonian and depends on the strain rates. Chapter 5 is partly devoted to such problems. Here the fluid formulation (flow formulation) can be applied directly to problems such as the forming of metals or plastics and we shall discuss this extreme situation in Chapter 5. However, even in incompressible flows, when the speed increases convective acceleration terms become important. Here often steady-state solutions do not exist or at least are extremely unstable. This leads us to such problems as vortex shedding. Vortex shedding indicates the start of instability which becomes very irregular and indeed random when high speed flow occurs in viscous fluids. This introduces the subject of turbulence, which occurs frequently in fluid dynamics. In turbulent flows random fluctuation of velocity occurs at all points and the problem is highly time dependent. With such turbulent motion, it is possible to obtain an averaged solution using time averaged equations. Details of some available time averaged models are summarized in Chapter 8. Chapter 6 deals with incompressible flow in which free surface and other gravity controlled effects occur. In particular we show three different approaches for dealing with free surface flows and explain the necessary modifications to the general formulation. The next area of fluid dynamics to which much practical interest is devoted is of course that of flow of gases for which the compressibility effects are much larger. Here compressibility is problem dependent and generally obeys gas laws which relate the pressure to temperature and density. It is now necessary to add the energy conservation equation to the system goveming the motion so that the temperature can be evaluated. Such an energy equation can of course be written for incompressible flows but this shows only a weak or no coupling with the dynamics of the flow. This is not the case in compressible flows where coupling between all equations is very strong. In such compressible flows the flow speed may exceed the speed of sound and this may lead to shock development. This subject is of major importance in the field of aerodynamics and we shall devote a part of Chapter 7 to this particular problem. In a real fluid, viscosity is always present but at high speeds such viscous effects are confined to a narrow zone in the vicinity of solid boundaries (the so-called boundary layer). In such cases, the remainder of the fluid can be considered to be inviscid. There we can return to the fiction of an ideal fluid in which viscosity is not present and here various simplifications are again possible. Such simplifications have been used since the early days of aerodynamics and date back to the work of Prandtl and Schlichting. 2 One simplification is the introduction of potential flow and we shall mention this later in this chapter. Potential flows are indeed the topic of many finite element investigators, but unfortunately such solutions are not easily extendible to realistic problems. A particular form of viscous flow problem occurs in the modelling of flow in porous media. This important field is discussed in Chapter 9. In the topic of flow through porous media, two extreme situations are often encountered. In the first, the porous medium is stationary and the fluid flow occurs only in the narrow passages between solid grains. Such an extreme is the basis of porous medium flow modelling in applications such as geo-fluid dynamics where the flow of water or oil through porous rocks occurs. The other extreme of porous media flow is the one in which the solid occupies only a small part of the total volume (for example, representing thermal insulation systems, heat exchangers etc.). In such problems flow is almost the same as that occurring in fluids without the solid phase which only applies an added, distributed, resistance to flow. Both extremes are discussed in Chapter 9.
4 Introduction to the equations of fluid dynamics and the finite element approximation Another major field of fluid mechanics of interest to us is that of shallow water flows that occur in coastal estuaries or elsewhere. In this class of problems the depth dimension of flow is very much less than the horizontal ones. Chapter 10 will deal with such problems in which essentially the distribution of pressure in the vertical direction is almost hydrostatic. For such shallow water problems a free surface also occurs and this dominates the flow characteristics and here we note that shallow water flow problems result in a formulation which is closely related to gas flow. Whenever a free surface occurs it is possible for transient phenomena to happen, generating waves such as those occurring in oceans or other bodies of water. We have introduced in this book two chapters (Chapters 11 and 12) dealing with this particular aspect of fluid dynamics. Such wave phenomena are also typical of some other physical problems. For instance, acoustic and electromagnetic waves can be solved using similar approaches. Indeed, one can show that the treatment for this class of problems is very similar to that of surface wave problems. In what remains of this chapter we shall introduce the general equations of fluid dynamics valid for most compressible or incompressible flows showing how the particular simplification occurs in some categories of problems mentioned above. However, before proceeding with the recommended discretization procedures, which we present in Chapter 3, we must introduce the treatment of problems in which convection and diffusion occur simultaneously. This we shall do in Chapter 2 using the scalar convection-diffusion-reaction equation. Based on concepts given in Chapter 2, Chapter 3 will introduce a general algorithm capable of solving most of the fluid mechanics problems encountered in this book. There are many possible algorithms and very often specialized ones are used in different areas of applications. However, the general algorithm of Chapter 3 produces results which are at least as good as others achieved by more specialized means. We feel that this will give a certain unification to the whole subject of fluid dynamics and, without apology, we will omit reference to many other methods or discuss them only in passing. For completeness we shall show in the present chapter some detail of the finite element process to avoid the repetition of basic finite element presentations which we assume are known to the reader either from reference 1 or from any of the numerous texts available. i~!~!i~i~!!~!~!~i~2ii~!!!ii~~ i ii i i2i ililili!li~~i i ili i i~~i ~~iililii!ii i~i ~~ ~~~i i ii!i ii !iii i i i!i!ii ii ii~ii iii i i i iii!ii i!iiiii~i ii i ~i i i iii~i !i i i!i i i!i i ~i iliJ~i!iii ii~i i i!!~ii iii~i!i!i!ii~i ii iii!i i!i i! i~~ii i !~ii~ilii!i i ii i i!i ii ii i i ii i!ii i!i!i i i i~i~i i i i i ~~, ~~ii iilii ii~iil!ii i~ii ii i!i i!iii~~ii iiii iii i!i il i!ii ililimiiili!ii|ii!ii~!ii~ii i i i!i i!iiliii i~i,i,ii~i!~!!~!~i~i!~ i i !i !i ili~~,,~i~ i!i !i!iilililiii i!i i !iili i i !~i ili!~!!i li i i i!i!~ii ii!ii~!i !i~i~iiilili !iiii i i
1.2.1 Stresses in fluids As noted above, the essential characteristic of a fluid is its inability to sustain deviatoric stresses when at rest. Here only hydrostatic' stress' or pressure is possible. Any analysis must therefore concentrate on the motion, and the essential independent variable is the velocity u or, if we adopt indicial notation (with the coordinate axes referred to as xi, i = 1, 2, 3), ui,
i--1,2,3
or
u=
EUl,
u2,
u3
l
(1.1)
This replaces the displacement variable which is of primary importance in solid mechanics.
The governing equations of fluid dynamics 5 The rates of strain are the primary cause of the general stresses, oij , and these are defined in a manner analogous to that of infinitesimal strain in solid mechanics as
9
I(OUi
,ij =
+
OUj)
This is a well-known tensorial definition of strain rates but for use later in variational forms is written as a vector which is more convenient in finite element analysis. Details of such matrix forms are given fully in reference 1 but for completeness we summarize them here. Thus, the strain rate is written as a vector (e) and is given by the following form -- [~11, ~22, 2~12] T = [~11, ~22, ~/12] T (1.3a) in two dimensions with a similar form in three dimensions -" [~ll, ~22, ~33, 2~12, 2~23, 2~31] T
(1.3b)
When such vector forms are used we can write the strain rate vector in the form = ,$u
(1.4)
where ,.q is known as the strain rate operator and u is the velocity given in Eq. (1.1). The stress-strain rate relations for a linear (Newtonian) isotropic fluid require the definition of two constants. The first of these links the deviatoric stresses ~-ij to the deviatoric strain rates:
1
Tij ~ O'ij -- ~(~ij O'kk --
2# ( ~ij -- l(~ij~kk)
(~.5)
In the above equation the quantity in brackets is known as the deviatoric strain rate, (~ij is the Kronecker delta, and a repeated index implies summation over the range of the index; thus
O'kk ~ fill -+" 0"22 + 0"33
and
~kk ~ ~ll + ~22 + ~33
(1.6)
The coefficient/z is known as the dynamic (shear) viscosity or simply viscosity and is analogous to the shear modulus G in linear elasticity. The second relation is that between the mean stress changes and the volumetric strain rate. This defines the pressure as 1
P = -- gO'kk = --I~kk-4- Po
(1.7)
where ~ is a volumetric viscosity coefficient analogous to the bulk modulus K in linear elasticity and P0 is the initial hydrostatic pressure independent of the strain rate (note that p and P0 are invariably defined as positive when compressive). We can immediately write the 'constitutive' relation for fluids from Eqs (1.5) and (1.7)
O'ij = Tij -- (~ij P
2#(~:ij - lt~ij~kk) + I~(~ij ~kk - t~ijPo k
(1.8a)
6 Introductionto the equations of fluid dynamics and the finite element approximation or 2 O'ij -- 2#~ij -t" t~ij (l'~ -- -~ #) ~kk -- t~ij PO
(1.8b)
The Lam6 notation is occasionally used, putting - g2 # = A
(1.9)
but this has little to recommend it and the relation (1.8a) is basic. There is little evidence about the existence of volumetric viscosity and, in what follows, we shall take n~kk = 0 (1.10) giving the essential constitutive relation as (now dropping the suffix on P0)
O'ij = 2# ( ~ij -- ~l(~ij~kk ) -- t~ij P ~ 7-ij - t~ij P
(1 l l a )
without necessarily implying incompressibility ~kk = 0. In the above,
Tij __ 2# (~ij- l(~ij~kk)= # [(OUi
OUj 2 On k "-[- -~Xi ) -- -~(~ij ~Xk]
(1.11b)
The above relationships are identical to those of isotropic linear elasticity as we will note again later for incompressible flow. However, in solid mechanics we often consider anisotropic materials where a larger number ofparameters (i.e. more than 2) are required to define the stress-strain relations. In fluid mechanics use of such anisotropy is rare and in this book we will limit ourselves to purely isotropic behaviour. Non-linearity of some fluid flows is observed with a coefficient # depending on strain rates. We shall term such flows 'non-Newtonian'. We now consider the basic conservation principles used to write the equations of fluid dynamics. These are: mass conservation, momentum conservation and energy
conservation.
1.2.2 Mass conservation If p is the fluid density then the balance of mass flow pui entering and leaving an infinitesimal control volume (Fig. 1.1) is equal to the rate of change in density as expressed by the relation
0p
0
Op
--~ + ~xi (pui) =-- -~ +
VT
(pu) = 0
(1.12)
where V T - [O/Oxl, O/Ox2, O/Ox3] is known as the gradient operator. It should be noted that in this section, and indeed in all subsequent ones, the control volume remains fixed in space. This is known as the 'Eulerian form' and displacements of a particle are ignored. This is in contrast to the usual treatment in solid mechanics where displacement is a primary dependent variable.
The governing equations of fluid dynamics 7 It is possible to recast the above equations in relation to a moving frame of reference and, if the motion follows the particle, the equations will be named 'Lagrangian'. Such Lagrangian frame of reference is occasionally used in fluid dynamics and briefly discussed in Chapter 6.
1.2.3 Momentum conservation" dynamic equilibrium .
.
.
.
--~ ...........................
.. . . . . . . . . . . . . . . . . .
: ...........................................
-
..............................
-
: : = =
...............
~
..................................
= : =
................................................
__.
=
=
...........
: . . . . . . . . . . . . . . . . . . . . . . . . . . . .
In the j th direction the balance of linear momentum leaving and entering the control volume (Fig. 1.1) is to be in dynamic equilibrium with the stresses aij and body forces p g j . This gives a typical component equation
-O(puj) -f-
0
0
-~- -'~'~-_ [(puj)ui]-
-'-~'~-_ (O'ij ) -- pgj -- 0 uxi
uxi
(1.13)
or using Eq. (1.11 a),
O(puj)
T - t -
0
Orij
Op
=0
- -pgj
-'~-~iXi[ (pU j )Ui ] - ~ X i + "~Xj
(1.14)
with Eq. (1.11 b) implied. The conservation of angular momentum merely requires the stress to be symmetric, i.e. CYij -- O'j i
or
In the sequel we will use the term m o m e n t u m angular forms.
Ti j -- 7"j i conservation
to imply both linear and
1.2.4 Energy conservation and equation of state We note that in the equations of Secs 1.2.2 and 1.2.3 the dependent variables are U i (the velocity components), p (the pressure) and p (the density). The deviatoric stresses,
•
J
(y)
dXl
=
x~.(x)
x3; (z)
Fig. 1.1 Coordinate direction and the infinitesimal control volume.
8
Introduction to the equations of fluid dynamics and the finite element approximation of course, are defined by Eq. (1.1 l b) in terms of velocities and hence are dependent variables. Obviously, there is one variable too many for this equation system to be capable of solution. However, if the density is assumed constant (as in incompressible fluids) or if a single relationship linking pressure and density can be established (as in isothermal flow with small compressibility) the system becomes complete and solvable. More generally, the pressure (p), density (p) and absolute temperature (T) are related by an equation of state of the form (1.15a)
p = p(p, T)
For an ideal gas this takes the form p =
P RT
(1.15b)
where R is the universal gas constant. In such a general case, it is necessary to supplement the governing equation system by the equation of energy conservation. This equation is of interest even if it is not coupled with the mass and momentum conservation, as it provides additional information about the behaviour of the system. Before proceeding with the derivation of the energy conservation equation we must define some further quantifies. Thus we introduce e, the intrinsic energy per unit mass. This is dependent on the state of the fluid, i.e. its pressure and temperature or (1.16)
e=e(T,p)
The total energy per unit mass, E, includes of course the kinetic energy per unit mass and thus E -- e + ~1 UiU i (1.17) Finally, we can define the enthalpy as h = e+ p P
or
H = h + ~ UiU i = E
+P P
(1.18)
and these variables are found to be convenient to express the conservation of energy relation. Energy transfer can take place by convection and by conduction (radiation generally being confined to boundaries). The conductive heat flux qi for an isotropic material is defined as OT qi - - - k (1.19) OXi
where k is thermal conductivity. To complete the relationship it is necessary to determine heat source terms. These can be specified per unit volume as qH due to chemical reaction (if any) and must include the energy dissipation due to internal stresses, i.e. using Eqs (1.1 l a) and (1.11 b), 0
0
OX i ( O ' i j U j ) - - -'~Xi ( T i j U j )
-
0 -~xj ( p u j )
(1.20)
The governing equations of fluid dynamics The balance of energy in a infinitesimal control volume can now be written as O(pE)
t-
0
O / OT'x ~, +
(pui E ) -
0
(pui)
--
0
(7"ijU
J )-
pgiui -- qH = 0
(1.21a) or more simply O(pE)
~-
0 (pui H )
-
0(0k
)
0 (7"ijUj) -- pgiui
-
-- qH -- 0
(1.21b) Here, the penultimate term represents the rate of work done by body forces.
1.2.5 Boundary conditions On the boundary of a typical fluid dynamics problem b o u n d a r y conditions need to be specified to make the solution unique. These are given simply as: (a) The velocities can be described as on Fu
Ui - - Ui
or traction as ti - - n j o i j
--
-ti
on 1-'t
(1.22a) (1.22b)
where l-'u ~ 1-'t - 1-'. Generally traction is resolved into normal and tangential components to the boundary. (b) In problems for which consideration of energy is important the temperature on the boundary is expressed as T-7' onFr (1.23a) or thermal flux qn -- -nik
OT = - k OT
Ox----~
~ - -- g/" on 1-'q
(1.23b)
where Fr U Fq - - F . (c) For problems of compressible flow the density is specified as p=~
onFp
(1.24)
1.2.6 Navier-Stokes and Euler equations The governing equations derived in the preceding sections can be written in a general conservative form as 0~ 0Fi 0Gi --~ + ~ x / + ~ + Q - 0 (1.25) in which Eq. (1.12), (1.14) or (1.21b) provides the particular entries to the vectors.
9
10 Introduction to the equations of fluid dynamics and the finite element approximation Thus, using indicial notation the vector of independent unknowns is P pul
--
(1.26a)
pu2 pu3 pE
the convective flux is expressed as
Fi "-
pui pu 1ui -4- pt~li pu2ui -Jr-pt~2i pu3ui q- pt~3i pHui
(1.26b)
,
similarly, the diffusive flux is expressed as 0
Gi
--
and the source terms as
Q --
--,'i-li --T2i --'i-3i
(1.26c) OT
0 pgl Pg2 Pg3
(1.26d)
pgiui -- qu with ( Ou i
Ou j
2
Ou k
"l'i' -- #[~,OXj "qt- ~ x i ) -- "3tSiJ "~Xk ] The complete set of Eq. (1.25) is known as the Navier-Stokes equation. A particular case when viscosity is assumed to be zero and no heat conduction exists is known as the 'Euler equation' (where 7"ij = 0 and qi = 0). The above equations are the basis from which all fluid mechanics studies start and it is not surprising that many alternative forms are given in the literature obtained by combinations of the various equations. 5 The above set is, however, convenient and physically meaningful, defining the conservation of important quantities. It should be noted that only equations written in conservation form will yield the correct, physically meaningful, results in problems where shock discontinuities are present. In Appendix A, we show a particular set of non-conservative equations which are frequently used. The reader is cautioned not to extend the use of non-conservative equations to problems of high speed flow.
Inviscid, incompressible flow In many actual situations one or another feature of the flow is predominant. For instance, frequently the viscosity is only of importance close to the boundaries at which velocities are specified. In such cases the problem can be considered separately in two parts: one as a boundary layer near such boundaries and another as inviscidflow outside the boundary layer. Further, in many cases a steady-state solution is not available with the fluid exhibiting turbulence, i.e. a random fluctuation of velocity. Here it is still possible to use the general Navier-Stokes equations now written in terms of the mean flow with an additional Reynolds stress term. Turbulent instability is inherent in the Navier-Stokes equations. It is in principle always possible to obtain the transient, turbulent, solution modelling of the flow, providing the mesh size is capable of reproducing the small eddies which develop in the problem. Such computations are extremely costly and often not possible at high Reynolds numbers. Hence the Reynolds averaging approach is of practical importance. Two further points have to be made concerning inviscidflow (ideal fluid flow as it is sometimes known). First, the Euler equations are of a purely convective form: (~tI~ c~F i - - ~ -~ ~ X / "q- o "- D
F i -- F i ( t I ))
(1.27)
and hence very special methods for their solutions will be necessary. These methods are applicable and useful mainly in compressible flow, as we shall discuss in Chapter 7. Second, for incompressible (or nearly incompressible) flows it is of interest to introduce a potential that converts the Euler equations to a simple self-adjoint form. We shall discuss this potential approximation in Sec. 1.3. Although potential forms are also applicable to compressible flows we shall not use them as they fail in complex situations.
In the absence of viscosity and compressibility, p is constant and Eq. (1.12) can be written as Oui = 0 (1.28)
cgxi
and Eq. (1.14) as
0U i
(~
+ ~--(ujui) -~ oxj
10p P (~Xi
gi -- 0
(1.29)
1.3.1 Velocity potential solution The Euler equations given above are not convenient for numerical solution, and it is of interest to introduce a potential, qS, defining velocities as
or Ul --
OOX1
U2 w
or OOX2
/g3 --
or OOX3
or g i ---
Oxi
(1.30)
11
12
Introductionto the equations of fluid dynamics and the finite element approximation If such a potential exists then insertion of Eq. (1.30) into Eq. (1.28) gives a single governing equation ~r
OXiOXi
= V2r
0
(1.31)
which, with appropriate boundary conditions, can be readily solved. Equation (1.31) is a classical Laplacian equation. For contained flow we can of course impose the normal velocity Un on the boundaries"
Un =
On
"-" Un
(1.32)
and, as we shall see later, this provides a natural boundary condition for a weighted residual or finite element solution. Of course we must be assured that the potential function ~bexists, and indeed determine what conditions are necessary for its existence. Here we observe that so far in the definition of the problem we have not used the momentum conservation equations (1.29), to which we shall now return. However, we first note that a single-valued potential function implies that
OXj OXi
=
(1.33)
OXi OXj
Defining vorticity as rotation rate per unit area
l (Oui OJiJ --" -2 OXj
C~Uj) OXi
(1.34)
we note that the use of the velocity potential in Eq. (1.34) gives
OJij :
0
(1.35)
and the flow is therefore named irrotational. Inserting the definition of potential into the first term of Eq. (1.29) and using Eqs (1.28) and (1.34) we can rewrite Eq. (1.29) as
o(o ) o[:
OXi - ~
Jr- ~
UjUj + -- + P P
]
= 0
(1.36)
in which P is a potential of the body forces given by
gi --
OP
OXi
(1.37)
In problems involving constant gravity forces in the x2 direction the body force potential is simply P = g x2 (1.38) Equation (1.36) is alternatively written as -~- + H +
-
(1.39)
Incompressible (or nearly incompressible) flows where H is the enthalpy, given as H lgiUi/2 + p/p. If isothermal conditions pertain, the specific energy is constant and Eq. (1.39) implies that 0@ 1 p - - ~ -F -~ uiui + -- + P = constant (1.40) P over the whole domain. This can be taken as a corollary of the existence of the potential and indeed is a condition for its existence. In steady-state flows it provides the wellknown Bernoulli equation that allows the pressures to be determined throughout the whole potential field once the value of the constant is established. We note that the governing potential equation (1.31) is self-adjoint (see Appendix B) and that the introduction of the potential has side-stepped the difficulties of dealing with convective terms. It is also of interest to note that the Laplacian equation, which is obeyed by the velocity potential, occurs in other contexts. For instance, in twodimensional flow it is convenient to introduce a stream function the contours of which lie along the streamlines. The stream function, ~, defines the velocities as =
and
U1 = OX2
u2 =
OqXl
(1.41)
which satisfy the incompressibility condition (1.28)
Obli 0 0 OXi--OX1 (0-~22)-I---~Xl (--0-~11)--0
(1.42)
For an existence of a unique potential for irrotational flow we note that co12 - 0 [Eq. (1.34)] gives a Laplacian equation 02~D
-- V 2 ~ = 0
OxiOxi
(1.43)
The stream function is very useful in getting a pictorial representation of flow. In Appendix C we show how the stream function can be readily computed from a known distribution of velocities.
We observed earlier that the Navier-Stokes equations are completed by the existence of a state relationship giving [Eq. (1.15a)]
p - p(p, T) In (nearly) incompressible relations we shall frequently assume that: (a) The problem is isothermal. (b) The variation of p with p is very small, i.e. such that in product terms of velocity and density the latter can be assumed constant. The first assumption will be relaxed, as we shall see later, allowing some thermal coupling via the dependence of the fluid properties on temperature. In such cases we
13
14 Introduction to the equations of fluid dynamics and the finite element approximation shall introduce the coupling iteratively. For such cases the problem of density-induced currents or temperature-dependent viscosity will be typical (see Chapters 5 and 6). If the assumptions introduced above are used we can still allow for small compressibility, noting that density changes are, as a consequence of elastic deformability, related to pressure changes. Thus we can write P d p = ~ dp
(1.44a)
where K is the elastic bulk modulus. This also can be written as 1 dp -- ~ dp
(1.44b)
cOp lap - ~ - c2 0 t
(1.44c)
or
with c = ~/K/p being the acoustic wave velocity. Equations (1.25) and (1.26a)-(1.26d) can now be rewritten omitting the energy transport equation (and condensing the general form) as 10p /~Ui ~C -~- + P~X/ = 0
Ouj 0 10p Ot ~t_ ~Oxi ( Uj Ui ) .3t_ P Ox j
1 07"ji
p OXi
gj -- 0
(1.45a)
(1.45b)
In three dimensions j = 1, 2, 3 and the above represents a system of four equations in which the variables are u j and p. Here
1 l,,(OUi OUj -p 7"ij = \ OXj "~ OXi
2t~ij OUk) 3
-~xk :
where u = #/p is the kinematic viscosity. The reader will note that the above equations, with the exception of the convective acceleration terms, are identical to those governing the problem of incompressible (or slightly compressible) elasticity (e.g. see Chapter 11 of reference 1).
Numer ! o!u !o.si
................... l is .............
eS dua!......................
1.5.1 Strong and weak forms We assume the reader is already familiar with basic ideas of finite element and finite difference methods. However, to avoid a constant cross-reference to other texts (e.g. reference 1), we provide here a brief introduction to weighted residual andfinite element
methods.
Numerical solutions: weak forms, weighted residual and finite element approximation The Laplace equation, which we introduced in Sec. 1.3, is a very convenient example for the start of numerical approximations. We shall generalize slightly and discuss in some detail the quasi-harmonic (Poisson) equation k
0X i
+Q-0
(1.46)
where k and Q are specified functions. These equations together with appropriate boundary conditions define the problem uniquely. The boundary conditions can be of Dirichlet type q5 = 4) on Fr (1.47a) or that of Neumann type
0r
qn = - k ~
=?/n
on l-'q
(1.47b)
where a bar denotes a specified quantity. Equations (1.46) to (1.47b) are known as the strong f o r m of the problem.
Weak form of equations
We note that direct use of Eq. (1.46) requires computation of second derivatives to solve a problem using approximate techniques. This requirement may be weakened by considering an integral expression for Eq. (1.46) written as v
-~x/
k~x~
+ Q dr2 = 0
(1.48)
in which v is an arbitrary function. A proof that Eq. (1.48) is equivalent to Eq. (1.46) is simple. If we assume Eq. (1.46) is not zero at some point xi in f2 then we can also let v be a positive parameter times the same value resulting in a positive result for the integral Eq. (1.48). Since this violates the equality we conclude that Eq. (1.46) must be zero for every xi in ~ hence proving its equality with Eq. (1.48). We may integrate by parts the second derivative terms in Eq. (1.48) to obtain k~-~x/
d~+
v Q d S2 -
v n i
k -~xi
dF=0
(1.49)
We now split the boundary into two parts, F~ and Fq, with F = F~ U 1-'q, and use Eq. (1.47b) in Eq. (1.49) to give ~x/
k~-x/
dr2 +
v Q dr2 +
v qn dF - 0
(1.50)
q
which is valid only if v vanishes on Fr Hence we must impose Eq. (1.47a) for equivalence. Equation (1.50) is known as the weak form of the problem since only first derivatives are necessary in constructing a solution. Such forms are the basis for the finite element solutions we use throughout this book.
15
16
Introductionto the equations of fluid dynamics and the finite element approximation
Weighted residual approximation
In a weighted residual scheme an approximation to the independent variable ~bis written as a sum of known trial functions (basis functions) Na (xi) and unknown parameters ~ba. Thus we can always write
(9 ~, ~ -
Nl(Xi)~91 nt- N2(xi)@ 2 -Jr'''' : ~ Na(xi)dpa -- N(xi)dp a=l
(1.51)
where and
N - - [ N I , Nz,...Nn]
(1.52a)
t~--- [@1, q~2, ...@n] T
(1.52b)
In a similar way we can express the arbitrary variable v as
U ~ U -- Wl (xi)~31 .2f_W2(xi)~)2 -4-... -- ~ Wa(xi)l) a -- W ( x i ) v a=l
(1.53)
in which Wa are test functions and ~a arbitrary parameters. Using this form of approximation will convert Eq. (1.50) to a set of algebraic equations. In the finite element method and indeed in all other numerical procedures for which a computer-based solution can be used, the test and trial functions will generally be defined in a local manner. It is convenient to consider each of the test and basis functions to be defined in partitions ~'2e of the total domain f2. This division is denoted by
(1.54)
~ ~h -- U ~"~e
and in a finite element method ~'2 e a r e known as elements. The very simplest uses lines in one dimension, triangles in two dimensions and tetrahedra in three dimensions in which the basis functions are usually linear polynomials in each element and the unknown parameters are nodal values of 0. In Fig. 1.2 we show a typical set of such linear functions defined in two dimensions. In a weighted residual procedure we first insert the approximate function q~into the governing differential equation creating a residual, R (xi), which of course should be zero at the exact solution. In the present case for the quasi-harmonic equation we obtain
R--
Ox i
k ~-~ -~Xi a
+ Q
(1.55)
and we now seek the best values of the parameter set @a which ensures that
]~ WbR df2 = O; b - l , 2 , . . . , n
(1.56)
Note that this is the term multiplying the arbitrary parameter ~b. As noted previously, integration by parts is used to avoid higher-order derivatives (i.e. those greater than or
Numerical solutions: weak forms, weighted residual and finite element approximation
Y
.....
,,,,
. .
.
.
,
.
.
.
.
.
.
.
.
,,,
i
.
.
.
.
.
.
.
.
.
.
~
:
Fig. 1.2 Basis function in linear polynomials for a patch of trianglular elements. equal to two) and therefore reduce the constraints on choosing the basis functions to permit integration over individual elements using Eq. (1.54). In the present case, for instance, the weighted residual after integration by parts and introducing the natural boundary condition becomes k a~-~x /
WbQdS2+
dS2+
q
WbflndF=O
(1.57)
The Galerkin, finite element, method
In the Galerkin method we simply take equations
Wb =
Nb which gives the assembled system of
n
g b a @ a -+- f b
- - O;
b -- 1, 2 , . . .
,n
-
r
(1.58)
a--1
where r is the number of nodes appearing in the approximation to the Dirichlet boundary condition [i.e. Eq. (1.47a)] and Kba is assembled from element contributions K~a with
Kb~a--
L ONa
ONa --~-x/k-~x/dr2
(1.59)
e
Similarly,
fb is computed from the element as
f[ -- L Nb QdS2+ f NbClndF e
(1.60)
eq
To impose the Dirichlet boundary condition we replace @aby @afor the r boundary nodes. It is evident in this example that the Galerkin method results in a symmetric set of algebraic equations (e.g. Kba = Kab). However, this only happens if the differential equations are self-adjoint (see Appendix B). Indeed the existence of symmetry provides a test for self-adjointness and also for existence of a variational principle whose stationarity is sought. ~
17
18 Introduction to the equations of fluid dynamics and the finite element approximation 1 N1 y
D 1
2
3
.~X
Xl
I
(b) Shape function for node 1.
(a) Three-node triangle.
Fig. 1.3 Triangular element and shape function for node 1.
It is necessary to remark here that if we were considering a pure convection equation
0~
u i -~ix i --F a -" o
(1.61)
symmetry would not exist and such equations can often become unstable if the Galerkin method is used. We will discuss this matter further in the next chapter.
Example 1.1 Shape functions for triangle with three nodes
A typical finite element with a triangular shape is defined by the local nodes 1, 2, 3 and straight line boundaries between nodes as shown in Fig. 1.3(a) and will yield the shape of Na of the form shown in Fig. 1.3(b). Writing a scalar variable as ~) - - 0~1 -~" OZ2 Xl "q- 0~3 X2
(1.62)
we may evaluate the three constants by solving a set of three simultaneous equations which arise if the nodal coordinates are inserted and the scalar variable equated to the appropriate nodal values. For example, nodal values may be written as
~;~ -
~, + ~
x~ + ~ 3 x 1
~2 _ al + a2 Xl2 + a3 x~
(1.63)
@3 = Ol1 _+_ Ol2 X~ -+- OL3 X3
We can easily solve for al, a2 and a3 in terms of the nodal values ~l, ~2 and ~3 and obtain finally q~ "- ~
1
[(al + blXl -+- ClX2)~ 1 ~- (a2 + b2Xl nt- c2x2)@ 2 --t- (a3 + b3xl Jr c3x2)(~ 3]
(1.64) in which
23 32 al -- x 1X2 m Xl x2 b l
=
-
c, = x? -
(1.65)
Numerical solutions: weak forms, weighted residual and finite element approximation 19 where x a is the i direction coordinate of node a and other coefficients are obtained by cyclic permutation of the subscripts in the order 1, 2, 3, and 1
x~
x21
2A = det 1
X2
x~ = 2. (area of triangle 123)
1
x~
x3
(1.66)
From Eq. (1.64) we see that the shape functions are given by Na = (aa "+" ba Xl -+- Ca X 2 ) / ( 2 A ) ;
(1.67)
a = 1, 2, 3
Since the unknown nodal quantities defined by these shape functions vary linearly along any side of a triangle the interpolation equation (1.64) guarantees continuity between adjacent elements and, with identical nodal values imposed, the same scalar variable value will clearly exist along an interface between elements. We note, however, that in general the derivatives will not be continuous between element assemblies. 1
Poisson equation in two dimensions: Galerkin formulation with triangular elements Example 1.2
The relations for a Galerkin finite elemlent solution have been given in Eqs (1.58) to (1.60). The components of Kba and fb can be evaluated for a typical element or subdomain and the system of equations built by standard methods. For instance, considering the set of nodes and elements shown shaded in Fig. 1.4(a), to compute the equation for node 1 in the assembled patch, it is only necessary to compute the K~a for two element shapes as indicated in Fig. 1.4(b). For the Type 1 element [left element in Fig. 1.4(b)] the shape functions evaluated from Eq. (1.67) using Eqs (1.65) and (1.66) gives N1
=
1 -
--x2"
h,
N2
m
X l.
---~, N3
__ x 2 n X l
h
thus, the derivatives are given by:
ON _ 0Xl
• ON
0
ON
1
-
~
"ON ON_
and
Ol(!
-
Ox2
1
ON
1
0U
0 1
Similarly, for the Type 2 element [fight element in Fig. 1.4(b)] the shape functions are expressed by N1-
1 -xl"
h' N2
Xl - - X2
X2
h
h
20
Introduction to the equations of fluid dynamics and the finite element approximation
|
3 il
4
5
/ /1 /
9
/
/
(a) 'Connected' equations for node 1.
3
2
>q
3
I
2
~.
h
d
(b) Type 1 and Type 2 element shapes in mesh.
Fig. 1.4 Lineartriangularelementsfor Poissonequationexample. and their derivatives by
ON1
1
ON1 '
0
-
ON
yx;-
ON2
~
ON3
-
-h 1
and
~ o
ON
~
OU2
-
~ ON3 ~
-
-
1
and
1
-~
Evaluation o f the matrix K~a and f~ for Type 1 and Type 2 elements gives (refer to Appendix D for integration formulae)
[
Ke,f.e --
2
1
0
0
-1
1
-1
--I -1 2
1 ~le ~2e
@3e
and
Ke@e: ~ kl I - 11 0
-1
0
2 -1 -1 1
] @2e ~le ~3e
Numerical solutions: weak forms, weighted residual and finite element approximation respectively. The force vector for a constant Q over each element is given by
fe - g Ql h2{ 11 } 1
for both types of elements. Assembling the patch of elements shown in Fig. 1.4(b) gives the equation with non-zero coefficients for node 1 as (refer to references 1 and 10 for assembly procedure)
k [4
-1
-1
-1] ~4
-1
-'l- Q h 2
~6
=0
Using a central difference finite difference approximation directly in the differential equation (1.46) gives the approximation
h2
-1
-1
-1
-1]
~4 ~6
+Q-0
and we note that the assembled node using the finite element method is identical to the finite difference approximation though presented slightly differently. If all the boundary conditions are forced (i.e. ~b = qS) no differences arise between a finite element and a finite difference solution for the regular mesh assumed. However, if any boundary conditions are of natural type or the mesh is irregular differences will arise, with the finite element solution generally giving superior answers. Indeed, no restrictions on shape of elements or assembly type are imposed by the finite element approach.
Example 1.3
In Fig. 1.5 an example of a typical potential solution as described in Sec. 1.3 is given. Here we show the finite element mesh and streamlines for a domain of flow around a symmetric aerofoil.
Example 1.4
Some problems of specific interest are those of flow with a free surface. 11'12'13Here the governing Laplace equation for the potential remains identical, but the free surface position has to be found iteratively. In Fig. 1.6 an example of such a free surface flow solution is given. 12 For gravity forces given by Eq. (1.38) the free surface condition in two dimensions (Xl, x2) requires
12(b/1/'/1~ b/2U2)~ g x 2
-- 0
21
22 Introduction to the equations of fluid dynamics and the finite element approximation
Fig. 1.5 Potential flow solution around an aerofoil. Mesh and streamline plots.
Numerical solutions: weak forms, weighted residual and finite element approximation
Moving
grid.,
7/ /
Vj
I
ro
Ro
I
I
,
R0/~0 =
~.6
s~ ro = 0.8
~ = 5o.5~
Fig. 1.6 Free surface potential flow, illustrating an axisymmetric jet impinging on a hemispherical thrust reverser (from Sarpkaya and Hiriart12).
Solution of such conditions involves an iterative, non-linear algorithm, as illustrated by examples of overflows in reference 11.
1.5.2 A finite v o l u m e a p p r o x i m a t i o n Many choices of basis and weight functions are available. A large number of procedures are discussed in reference 1. An approximation which is frequently used in fluid mechanics is the finite volume process which many consider to be a generalized finite difference form. Here the weighting function is often taken as unity over a specified subdomain f2b and two variants are used: (a) an element (cell) centred approach; and (b) a node (vertex) centred approach. Here we will consider only a node centred approach with basis functions as given in Eq. (1.51) for each triangular subdomain and the specified integration cell (dual cell) for each node as shown in Fig. 1.7. For a solution of the Poisson equation discussed above, integration by parts of Eq. (1.56) for a unit Wb gives ni ~xi dF - 0
Q dr2 b
b
(1.68)
23
24 Introduction to the equations of fluid dynamics and the finite element approximation
Fig. 1.7 Finite volume weighting. Vertex centred method (~b).
for each subdomain f2b with boundary I'b. In this form the integral of the first term gives f
a dr2
(1.69)
Q f2b
b
when Q is constant in the domain. Introduction of the basis functions into the second term gives
~ dr" ,~ 7xi ~ fFO~)fF(~(/)f~Na~a ^
ni
ni
b
dr
dr
ni
=
b
(1.70)
b
requiting now only boundary integrals of the shape functions. In order to make the process clearer we again consider the case for the patch of elements shown in Fig. 1.4(a).
Example 1.5 Poisson equation in two dimensions: finite volume formulation with triangular elements
The subdomain for the determination of equation for node 1 using the finite volume method is shown in Fig. 1.8(a). The shape functions and their derivatives for the Type 1 and Type 2 elements shown in Fig. 1.8(b) are given in Example 1.2. We note especially that the derivatives of the shape functions in each element type are constant. Thus, the boundary integral terms in Eq. (1.70) become
ONo
n i - - ~ xi
dr
-
~
~
b
e
n e dr
ON~ Ox---~,
e
on I-'e
where e denotes the elements surrounding node a. Each of the integrals will be an I1, 12, 13 for the Type 1 and Type 2 elements shown. It is simple to show that the integral Ill --
JFF n~ d r e
=
J~4c n~ d r
+
f6
n~ d r
- x6 - x4
Numerical solutions: weak forms, weighted residual and finite element approximation 5
4
/ 4
3 le
ii
7"
l
7
8
(a) 'Connected' area for node 1.
5
6/
13
4
(b) Type 1 and Type 2 element boundary integrals. Fi 9. 1.8 Finite volume domain and integrations for vertex centred method.
w h e r e x 4, x 6 are m i d - e d g e c o o r d i n a t e s of the triangle as s h o w n in Fig. 1.8(b). Similarly, for the x2 derivative w e obtain
12 :
fF
n ~ d F --
e
fc
f6
n~dr +
n~dr
- x4 - x6
Thus, the integral
ONai d F 11 : fr ni-~x e
--
ONa (x 6 x~) + aXl _
OUa (x? - x 6)
25
26
Introductionto the equations of fluid dynamics and the finite element approximation The results 12 and 13 are likewise obtained as =
ONa (x 4 _ x5 ) -F ONa
=
ONa (x 5 _ x6 ) +
- Xr
ONa (x 6 -
x~)
Using the above we may write the finite volume result for the subdomain shown in Fig. 1.8(a) as
k [4 - 1
-1
-1
-1]
@4 + Oh 2 _ 0 ~6
We note that for the regular mesh the result is identical to that obtained using the standard Galerkin approximation. This identity does not generally hold when irregular meshes are considered and we find that the result from the finite volume approach applied to the Poisson equation will not yield a symmetric coefficient matrix. As we know, the Galerkin method is optimal in terms of energy error and, thus, has more desirable properties than either the finite difference or the finite volume approaches. Using the integrals defined on 'elements', as shown in Fig. 1.8(b), it is possible to implement the finite volume method directly in a standard finite element program. The assembled matrix is computed element-wise by assembly for each node on an element. The unit weight will be 'discontinuous' in each element, but otherwise all steps are standard.
We have observed in this chapter that a full set of Navier-Stokes equations can be written incorporating both compressible and incompressible behaviour. At this stage it is worth remarking that 1. More specialized sets of equations such as those which govern shallow water flow or surface wave behaviour (Chapters 10, 11 and 12) will be of similar forms and will be discussed in the appropriate chapters later. 2. The essential difference from solid mechanics equations involves the non-selfadjoint convective terms. Before proceeding with discretization and indeed the finite element solution of the full fluid equations, it is important to discuss in more detail the finite element procedures which are necessary to deal with such non-self-adjoint convective transport terms. We shall do this in the next chapter where a standard scalar convective-diffusivereactive equation is discussed.
References
i i ii i i
~iiiiiiiiiiiiiii}iiiiiiiiiiiiiiiiii iiiiDiii!iiiiii! iiiiii!iiii!iii!iiiii!i!ii!ii!iiiiiiiiiiii iiiiiiiiii iiiiiiii !iiiiiii iiiii i iiiiiiiiiiiiiii!iiiiiiii iiii!iiiiiiiiiiiiiiiiiiiiiiii ii!ii
1. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method: Its Basis and Fundamentals. Elsevier, 6th edition, 2005. 2. H. Schlichting. Boundary Layer Theory. Pergamon Press, London, 1955. 3. C.K. Batchelor. An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge, 1967. 4. H. Lamb. Hydrodynamics. Cambridge University Press, Cambridge, 6th edition, 1932. 5. C. Hirsch. Numerical Computation oflnternal and External Flows, volumes 1 & 2. John Wiley & Sons, New York, 1988. 6. P.J. Roach. Computational Fluid Mechanics. Hermosa Press, Albuquerque, New Mexico, 1972. 7. L.D. Landau and E.M. Lifshitz. Fluid Mechanics. Pergamon Press, London, 1959. 8. R. Temam. The Navier-Stokes Equation. North-Holland, Dordrecht, 1977. 9. I.G. Currie. Fundamental Mechanics of Fluids. McGraw-Hill, New York, 1993. 10. R.W. Lewis, P. Nithiarasu and K.N. Seetharamu. Fundamentals of the Finite Element Method for Heat and Fluid Flow. Wiley, Chichester, 2004. 11. P. Bettess and J.A. Bettess. Analysis of free surface flows using isoparametric finite elements. International Journal for Numerical Methods in Engineering, 19:1675-1689, 1983. 12. T. Sarpkaya and G. Hiriart. Finite element analysis of jet impingement on axisymmetric curved deflectors. In J.T. Oden, O.C. Zienkiewicz, R.H. Gallagher and C. Taylor, editors, Finite Elements in Fluids, volume 2, pages 265-279. John Wiley & Sons, New York, 1976. 13. M.J. O'Carroll. A variational principle for ideal flow over a spillway. International Journal for Numerical Methods in Engineering, 15:767-789, 1980.
27
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•i!i•i!•i!••i!i!•!i
Convection dominated problemsfinite element approximations to the convection-diffusion-reaction
equation
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In this chapter we are concerned with steady-state and transient solutions for equations of the type 0tI )
OF/
OCT/
Ot + ~ x / + ~ x / + Q - 0
(2.1)
where in general 4I, is the basic dependent, vector-valued variable, Q is a source or reaction term vector and the flux matrices F and G are such that Fi -- F i ( @ ) Gi "- Gi ( k
(2.2a)
and in general
Q - Q(xi, ~ )
(2.2b)
In the above, X i and i refer, in the indicial manner, to Cartesian coordinates and quantities associated with these. A linear relationship between the source and the scalar variable in Eq. (2.2b) is frequently referred to as a reaction term. The general equation (2.1) can be termed the transport equation with F standing for the convective and G for the diffusive flux quantities. Equation (2.1) is a set of conservation laws arising from a balance of the quantity with its fluxes F and G entering and leaving a control volume. Such equations are typical of fluid mechanics which we have discussed in Chapter 1. As such equations may also arise in other physical situations this chapter is devoted to a general discussion of their approximate solution. The simplest form of Eqs (2.1), (2.2a) and (2.2b) is one in which the unknown is a scalar. Most of this chapter is devoted to the solution of such equations. Throughout this book we shall show that there is no need dealing with convection of vector type
Introduction
quantities. Thus, for simplicity we may consider the form above "~ ~
r
Q ~
F i -+ Fi = Ui (fi
Q(xi, q~)
G i -->" G i =
-
k
or
(2.3)
~
OXi
We now have in Cartesian coordinates a scalar equation of the form Ot I
OX i
OX i k
+ Q - 0
(2.4)
In the above equation, Ui is a known velocity field and q~is a scalar quantity being transported by this velocity. However, diffusion can also exist and here k is the diffusion coefficient. The term Q represents any external sources of the quantity q5 being admitted to the system and also the reaction loss or gain which itself is dependent on the function ~b. A simple relation for the reaction may be written as Q - c q5
(2.5)
where c is a scalar parameter. The equation can be rewritten in a slightly modified form in which the convective term has been differentiated as
oui --~ -]- Ui --~xi + (~ Ox----~t
o Ox i
--~xi ) + Q -- 0
(2.6)
We note that the above form of the problem is self-adjoint with the exception of a convective term which is underlined. The reader is referred to Appendix B for the definition of self-adjoint problems. The third term in Eq. (2.6) disappears if the flow itself is such that its divergence is zero, i.e. if
OU/ Oxi
= 0
(2.7)
In what follows we shall discuss the scalar equation in much more detail as many of the finite element remedies are only applicable to such scalar problems and are not directly transferable to the vector form. In the CBS scheme, which we shall introduce in Chapter 3, the equations of fluid dynamics will be split so that only scalar transport occurs, where the treatment considered here is sufficient. From Eqs (2.6) and (2.7) we have "-~ -~- Ui Ox-----~
Ox i k~x~
+ Q = 0
(2.8)
With the variable ~bapproximated in the usual way: (2.9)
29
30 Convectiondominated problems the problem may be presented following the usual (weighted residual) semi-discretization process as M& + H ~ + f -- O
(2.101
where
Mab -- f~ WaNbdf2 nab = fa -- f
Wa Ui -~xi + OXi "~-X/J d a Wa a da + fF Waqn df2 q
Now even with standard Galerkin weighting the matrix H will not be symmetric. However, this is a relatively minor computational problem compared with inaccuracies and instabilities in the solution which follow the arbitrary use of the weighting function. This chapter will discuss the manner in which these difficulties can be overcome and the approximation improved. We shall in the main address the problem of solving Eq. (2.8), i.e. the scalar form, and to simplify matters further we shall start with the idealized one-dimensional equation:
---~ -~- U ox
Ox k
+Q=O
(2.11t
The term 05OU/Ox has been removed for simplicity, which of course is true if U is constant. The above reduces in steady state to an ordinary differential equation: U
d~ dx
d (kd~b)+Q_O dx \ dx or
(2.12)
s162 + Q = o
in which we shall often assume U, k and Q to be constant. The basic concepts will be evident from the above and will later be extended to multidimensional problems, still treating ~bas a scalar variable. Indeed the methodology of dealing with the first space derivatives occurring in differential equations governing a problem, which lead to non-self-adjointness, opens the way for many new physical situations. 1 The present chapter will be divided into three parts. Part I deals with steady-state situations starting from Eq. (2.12), Part II with transient solutions starting from Eq. (2.11) and Part III with treatment of boundary conditions for convective-diffusive problems where use of 'weak forms' is shown to be desirable. Although the scalar problem will mainly be dealt with here in detail, the discussion of the procedures can indicate the choice of optimal ones which will have much beating on the solution of the general case of Eq. (2.1). The extension of some procedures to the vector case is presented in Appendix E.
The steady-state problem in one dimension
Part I" Steady-state problems
2.2.1 General remarks We consider the discretization of Eq. (2.12) with
-- N(~
~ ~) "-- Z Nasa
(2.13)
where Na are shape functions and q~represents a set of still unknown parameters. Here we shall take these to be the nodal values of qS. The weighted residual form of the one-dimensional problem is written as (see Chapter 1)
d (kaY) ~x
J~ Wa [U d~ dx
]
dx
+ Q dr2 - 0
(2.14)
Integrating the second term by parts gives
Wa U
---h-T-k~d~'~+
+ Q d~ +
q
Wa Ondr
=0
(2.15)
^
where
or On=-k--~
onr'q
and ~=q~
onFe
is assumed. For a typical intemal node 'a' the approximating equation becomes
KabOb+ fa
= 0
(2.16)
where Kab - -
fa =
fo L
/0
Wa
U
dNb dx
dx +
fo L d Wak dNb dx dx dx
W a Q d x + WaOn
IFq
(2.17)
and the domain of the problem is 0 < x _< L. For linear shape functions (Fig. 2.1), Galerkin weighting (Wa - Na) and elements of equal size h, we have for constant values of U, k and Q (see Appendix D)
Ke - - 2g [-1 1 11 I f e -1Q }h2{ - 1
k I - 11 -11 1
+h
31
32 Convectiondominated problems Na
a-1
~, h
a
J..,. h
Na+ l
I
a+l
J . , _ .h
a+2 .
Fig. 2.1 A linear shape function for a one-dimensional problem.
which yields a typical assembled equation (after multiplying by h / k ) for node 'a'
( - P e - 1)~)a_ 1 "q- 2q~a q-
(Pe-
1)~a+ 1 +
Qh 2
=0
(2.18)
where Pe =
Uh
(2.19)
2k
is the element Peclet number. Incidentally, for the case of constant Q the above is identical to the usual central finite difference approximation obtained by putting (for node a) dq~ ~)a+l- ~)a-1 ,~ (2.20a) dx 2h and
d2* dx 2
~a+l -- 2~a -~- ~a-1 h2
(2.20b)
The algebraic equations (2.18) are obviously non-symmetric and their accuracy deteriorates as the parameter Pe increases i.e. when convective terms are of primary importance. Indeed as Pe --+ c~, the solution is purely oscillatory and bears no relation to the underlying problem. This may be ascertained by considering Eq. (2.18) for different element Peclet numbers and it is easy to show that with the standard Galerkin procedure oscillations in ~a Occur when IPel > 1 (2.21) To illustrate this point we consider a simple example.
Example 2.1
One-dimensional convection diffusion (Q = 0)
The domain of the problem considered is 0 < x < L and the boundary conditions are both of Dirichlet type and given by dp(O) = 1
and
r
-- 0
We approximate the solution using nine equal size linear elements and the Galerkin form of Eq. (2.15)
[/0L(da
]
The s t e a d y - s t a t e
1
:
Convection-Diffusion: I
0.9
- - -' . . . . . .
o.8
: ......
0.7
: ......
Pe
i ...... '- .....
: '- . . . . .
! ..... ; . . . . .
". . . . . .
" .....
- - -: . . . . . .
',- . . . . .
0.4
'
,
- ....
0.2
0.6 ~
0.5 0.4
.....
0.3
' - ....
0.1
0.2 - -
0
0.2
0.4
~L
0.6
0.1
1
0.8
Pe
Convection-Diffusion: 1.5
0
0
s
,i .
I
"L:
1 s
.
.
.
.
, i
,
|
.
.
"
i
i
0.2
I
"
.
0
1
i
"
0
-
012
!
.
.
11
i .
i
- i i
"
.
1
, i
.
:
|
. . . . . '
!
.
I
' s
.
.
.
,,
.
i . . .
i ' ,i
1
1 l-
s
i
:;
,
I
1
.
.
. r ,
.1.. 1
--0.6
. i i
,.
,..~
,
,
x//.
.
' . . . . . .
;,...,,.
,
T 014
I..I 1
,' ....
.
.... !
. . . . . .
.
t
,..,:
-
,
i
i .
.
I . . . I .
"
- .
-i
I .
'...:..'..,..,
i
11
,
.i
',1 .
"1 . . . .
. . . . . ,. ,
0.8
. i
I
- - -"'-: . . -
0.1 0.6
.
1
,
' ,
s
i, .
.1..I l
,..
.....
1
I
1
x/L
.... . . 1
"
....
~ :
i
L
1
I
.
i .
i
-1-
i
. . . . . . . . . . . . . .
0.3
i
.
-i
, I
i
9
,
I
1
=
iI
"11
'1
,
...,
i
. ;.,
1
i, "
1
0.4
|
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.4
Pe
||
,
1
::
1 .1.
.
,
1
i, ~
......
0.6
I
i . . . . . . . . i
-~ - ' . . . . .
. . . .
,
~- 0 . 5 ,
.
. .....
0.8 0.7
0.2
0.8
i
0.9
al a
a
0
0.6
~L
Convection-Diffusion:
,
0
0.4
1
,
0.5
0.2
= 2.5 ,
1
o.7
" .....
--';
=
0.9
i ......
0.5
0
Pe
1
',. . . . . .
0.3
in one d i m e n s i o n
Convection-Diffusion:
= 0
:. . . . . .
- - -', . . . . . .
0.6
:
problem
1 1"
' ,i
.
1l .
.
.i1
. . . . . s
9 f
. . . . .
1
1 : -
....
,,,:
1
. . . . .
ii,
~.8
Fig. 2.2 Approximations to Uddpldx - kd 2 ~ / d x 2 = 0 for ~b(O) = 1 and (;b(L) = 0 for various Peclet numbers. Solid line - exact solution" dotted line with triangular symbol- standard Galerkin solution.
The solution shown in Fig. 2.2 with curves labelled with triangles gives results for element Peclet numbers [given by Eq. (2.19)] Pe = 0, 1, 2.5 and co (the solution for this problem with Pe = co is only possible for an odd number of elements). We see that as the Pe number increases above 1 the solution becomes oscillatory and progressively departs from the smooth exact solution (solid line in Fig. 2.2)
eUx/k _ 1-
eUL/k
e uL/g
Of course the above is partly a problem of boundary conditions. When diffusion is omitted (k = 0) only a single boundary condition can be imposed and when the diffusion is small we note that the downstream boundary condition (~b(L) = 0) is felt in only a very small region of a boundary layer. The above example clearly demonstrates that the Galerkin method cannot be used to solve problems in which convective terms are large compared with those of diffusion. Of course, one can consider replacing the weight functions Na by other more general
i
33
34 Convectiondominated problems ones Wa. Indeed for the linear one-dimensional steady-state problem we can always find weight functions which give exact solutions at the interelement nodes.
Example 2.2
Weight function for exact nodal solutions
Here we consider the problem of Example 2.1 where the weak form including Q is given by v
[
Udx
dx
1
~-
+ Q dx-0
where x] < x < x2 denotes the domain. After integration by parts for all derivatives on ~bwe obtain dx
/xl
-+in which
dv - U dx
Qdx+v
v
(
Uq~ - kdq~ -dx
)1
x2+ ~ dv k~ Xl
i
x, - 0 x~
d (kdV) dx ~xx
is the adjoint equation for the original problem. We note that the presence of the first derivative term makes the problem non-self-adjoint (see Appendix B). Motivated by the fact that the propagation of information is in the direction of the velocity U, finite difference practitioners were the first to overcome the bad approximation problem of the central difference method. They used one-sided finite differences to approximate the first derivative. 4-7 Thus in place of Eq. (2.20a) and with positive U, the approximation was put as
de
~a --~a--1
(2.22) dx h changing the central finite difference form of the governing equation approximation [as given by Eq. (2.18)] to
(-2Pe
-
1)qSa_1 q- (2 +
2Pe)()
a -
~a+l "+"
Qh 2
= 0
(2.23)
With this upwind difference approximation, non-oscillatory solutions are obtained through the whole range of Peclet numbers as shown in Fig. 2.3 by curves labelled c~ = 1. Now exact nodal solutions are obtained for pure diffusion (Pe = 0) and for pure convection (Pe = c~); however, results for other values of Pe are generally not accurate. How can such upwind differencing be introduced into a finite element scheme and generalized to more complex situations? This is the problem that we now address and indeed show that, for linear one-dimensional elements, this form of finite element solution can also result in exact nodal values for all Peclet numbers.
2.2.2 Petrov-Galerkin methods for upwinding in one dimension The first possibility is that of the use of a Petrov-Galerkin type of weighting in which Wa ~ Na. 8-11 Such weighting was first suggested by Zienkiewicz et al. 8 in 1976 and
The steady-state ....
,a, . . . .
- - o-
-
------c
problem in one dimension
S t a n d a r d Galerkin a = 0 Petrov--Galerkin a - 1.0 (full u p w i n d difference)
Petrov-Galerkin a = aopt Exact
Pe = UN2k = 0
1.0
Z~=O=O (Allexact) i n
r
Pe= 1.0
Pe = 2 . 5
(Exact)
:
;
;,,
L
;,:'
Exact
L
p
'',, .] rl
Fig. 2.3 Approximations to Udd21dx- k d 2~/dx 2 = 0 with ~(0) = 0 and ~(L ) = 1 for various Peclet numbers.
used by Christie et al. 9 In particular, again for elements with linear shape functions Na, shown in Fig. 2.1, we shall take weighting functions constructed as shown in Fig. 2.4 so that Wa - ga -~- ogW: (2.24) where Wa is such that (to obtain finite difference equivalent) f~ Wa dx e
h
(2.25t
35
36 Convectiondominated problems
! Na i I ~
L
~'ll'~,,i,'II"I"~I"I"'~,i,,,i,,
h
r'-
A
P]
!
! !
!
I
o
!
; or
i~ , i i !
~
**
os
i ! !
i
Fig. 2.4 Petrov-Galerkinweight function W
a "-
B
Na q- 0/. W a.
Continuous and discontinuous definitions.
the sign depending on whether U is a velocity directed toward or away from the node. With this approximation Eq. (2.15) becomes
U
+ Q
\ dx
- (Ua +
dx
d--x
(2.26)
W;)On I = 0 Fq
Various forms of Wa are possible, but the most convenient is the following simple definition which is, of course, a discontinuous function (continuity requirements are discussed below) hUdNa W2 = (2.27) 2 ]UI dx where [U] denotes absolute value. With the above weighting functions the approximation from Eq. (2.15) for a typical node a becomes
[ - P e ( a + 1)
-
1]~a_ 1 "~- [2 + 2a(Pe)]~a -+- [-Pe(ce - 1)
--
1]~a_F1 "~-
Qh 2
=0 (2.28)
where Q is assumed constant for the whole domain and equal length elements are used. Immediately we see that with a = 0 the standard Galerkin approximation is recovered [Eq. (2.18)] and that with a = 1 the full upwind form [Eq. (2.23)] is available, each giving exact nodal values for purely diffusive or purely convective cases respectively (Fig. 2.3). Now if the value of a is chosen as O~ - -
O~op t
--
coth
IPel
IPel
(2.29)
The steady-state problem in one dimension 1.0 0.8 0.6
~ o
~
I
I
2
!
3
I
4 Pe
I
I
5
6
7
Fig. 2.5 Critical (stable) and optimal values of the 'upwind' parameter oL for different values of Pe= Uh/2k.
then exact nodal values will be given for all values of Pe. The proof of this is given in references 9 and 12 for the present, one-dimensional, case where it is also shown that if OL >
OLcrit =
1
1
(2.30)
IPel
oscillatory solutions never arise. Figure 2.5 shows the variation of OLopt and
OLcrit with
Pe. Although the proof of optimality for the upwinding parameter was given for the case of constant coefficients and constant size elements, nodally exact values will also be given if c~ = C~optis chosen for each element individually. We show some typical solutions in Fig. 2.613 for a variable source term Q = Q(x), convection coefficients U = U(x) and element sizes. Each of these is compared with a standard Galerkin solution, showing that even when the latter does not result in oscillations the accuracy is improved. Of course in the above examples the Petrov-Galerkin weighting must be applied to all terms of the equation. When this is not done (as in simple finite difference upwinding) totally wrong results will be obtained, as shown in the finite difference results of Fig. 2.7, which was used in reference 14 to discredit upwinding methods. The effect of c~ on the source term is not apparent in Eq. (2.28) where Q is constant in the whole domain, but its influence is strong when Q = Q (x).
Continuity requirements for weighting f u n c t i o n s The weighting function W a (or Wa) introduced in Fig. 2.4 can of course be discontinuous as far as the contributions to the convective terms are concerned [see Eq. (2.17)], i.e.
L
Wa
~d(UNb) dx dx
or
WaU
dx
dx
Clearly no difficulty arises at the discontinuity in the evaluation of the above integrals. However, when evaluating the diffusion term, we generally introduce integration by
37
38
Convection dominated problems ~x 10 4
d2r
d~
-~x 2 +200--=x
dx
O<x<
2
~ %
1
,,
I I I I
~=O,x=O,x= 1 15
:|
!
10
_
e I i I
i e i l
I 1
9
x
(a) Exact
o ---o---
Optimal Petrov-Galerkin Standard Galerkin
1.00
0.95
0.90
-
d2,
60 d,
2
l:
-1<X<2
-
li
*=1,x=1
l:
0.85
(b)
1.00
I
I
2.0
x
Fig. 2.6 Application of standard Galerkin and Petrov-Galerkin (optimal) approximation: (a) variable source term equation with constants k and h; (b) variable source term with a variable U. parts and evaluate such terms as
~oLdWa
dNb --~x k dx
dx
in place of the form
-
~o"I~ d (kdNb Wa-d-XX -~'x) dx
The steady-state problem in one dimension _
..
Exact and P e t r o v - G a l e r k i n
-
Finite difference u p w i n d solution
/ I
I
5
10
......
1.
15
Fig. 2.7 A one-dimensional pure convective problem (k = O) with a variable source term Q and constant U. Petrov-Galerkin procedure results in an exact solution but simple finite difference upwinding gives substantial error.
Here a local infinity will occur with discontinuous Wa. To avoid this difficulty we mollify or smooth the discontinuity of the Wa so that this occurs within the element 2 and thus avoid the discontinuity at the node in the manner shown in Fig. 2.4. Now direct integration can be used, showing in the present case zero contributions to the diffusion term.
2.2.3 Balancing diffusion in one dimension The comparison of the nodal equations (2.18) and (2.28) obtained on a uniform mesh and for a constant Q shows that the effect of using the Petrov-Galerkin procedure is equivalent to the use of a standard Galerkin process with the addition of a diffusion 1 kb = ~c~Uh
(2.31)
to the original differential equation (2.12). The reader can easily verify that with this substituted into the original equation, thus writing now in place of Eq. (2.12)
Udx
dx (k + kb)-~x + Q -- 0
(2.32)
we obtain an identical expression to that of Eq. (2.28) providing Q is constant and a standard Galerkin procedure is used. Such balancing diffusion is easier to implement than Petrov-Galerkin weighting, particularly in two or three dimensions, and has some physical merit in the interpretation of the Petrov-Galerkin methods. However, it does not provide the required
39
40
Convectiondominated problems modification of source terms, and for instance in the example of Fig. 2.7 will give erroneous results identical with a simple finite difference, upwind, approximation. The concept of artificial diffusion introduced frequently in finite difference models suffers of course from the same drawbacks and in addition cannot be logically justified. It is of interest to observe that a central difference approximation, when applied to the original equations (or the use of the standard Galerkin process), fails by introducing a negative diffusion 15 into the equations. This 'negative' diffusion is countered by the present, balancing, one.
2.2.4 A variational principle in one dimension Equation (2.12), which we are considering here, is not self-adjoint and hence is not directly derivable from any variational principle. However, it was shown by Guymon et al. 16 that it is a simple matter to derive a variational principle (or ensure selfadjointness which is equivalent) if the operator is premultiplied by a suitable function p. Thus we write a weak form of Eq. (2.12) as
foI~ vp l Udd~ dx where p =
dx d ( kd~b-~x) + QI dx - O
(2.33)
p(x) is as yet undetermined. This gives, on integration by parts,
]o'[
Ql
L ~0
(2.34)
0
Immediately we see that the operator can be made self-adjoint and a symmetric approximation achieved if the first term in square brackets is made zero. 1 This requires that p be chosen so that
pU + k u___g_v= 0
(2.35a)
p = Ce -Ux/k = Ce-2Pex/h
(2.35b)
dx
or that
For such a form corresponding to the existence of a variational principle the 'best' approximation is that of the Galerkin method with
1)--- y~ Nal)a
and
~ -- y ~ Nolo
(2.36)
where Ua is an arbitrary parameter. Indeed, such a formulation will, in one dimension, yield answers exact at nodes, l' 3 It must therefore be equivalent to that obtained earlier by weighting in the PetrovGalerkin manner. Inserting the approximation of Eq. (2.36) into Eq. (2.34), with Eqs (2.35a) and (2.35b) defining p using an origin at x = Xa, we have for the ath equation of the uniform mesh
f hdlga(ke-2P~exh/dNbh
)'-~'x~b + Nae-2Pex/hQ]dx
(2.37)
The steady-state problem in one dimension
with b = a - 1, a, a + 1. This gives, after some algebra, a typical nodal equation: (1 - e2ee)~a_ 1 -Jr-(e 2ee -- e-2ee)~ a -- (1 - e-2ee)~a+ 1 Q h 2 (e Pe -- e - P e ) 2 = 0
(2.38)
+ 2Pek which can be shown to be identical with expression (2.28) into which a = OZopt given by Eq. (2.29) has been inserted. Here we have a somewhat more convincing proof of the optimality of the proposed Petrov-Galerkin weighting 17' 18 introduced in the previous subsection. However, serious drawbacks exist. The numerical evaluation of the integrals is difficult and the equation system, though symmetric overall, is not well conditioned if p is taken as a continuous function of x through the whole domain. The second point is easily overcome by taking p to be locally defined, for instance taking the origin of x at point a for all assemblies as we did in deriving Eq. (2.38). This is permissible and is equivalent to scaling the full equation system row by row. 17 Now of course the total equation system ceases to be symmetric. The numerical integration difficulties disappear, of course, if the simple weighting functions previously derived are used. However, the proof of equivalence is important as the problem of determining the optimal weighting is no longer necessary. Although the development of Petrov-Galerkin methods has reached maturity, further performance improvements have been the subject of recent research. 19-21
2.2.5 Galerkin least squares approximation (GLS) in one dimension In the preceding sections we have shown that many, apparently different, approaches have resulted in identical (or almost identical) approximations. Here another procedure is presented which again will produce similar results. In this a combination of the standard Galerkin and least squares approximation is made. 22' 23 If Eq. (2.12) is rewritten as s
-t- Q = 0
4) ,~ q$ - N(b
(2.39a)
with L = Udx
dx
-~
(2.39b)
the standard Galerkin approximation gives for the ath equation
No [c(;b) + o] dx = o with boundary conditions omitted for simplicity. Similarly, a least squares minimization of the residual R -- s 1
d
2 d~a f~
R 2 dx --
f~ dsd~)a
[/~(q~) + Q] dx
(2.40)
+ Q results in (2.41)
41
42 Convection dominated problems
or
dx dx \k--~x ]] (17,(~)q- Q)dx -
f--,(Na)[/~(~)+ Q] dx
(2.42)
If the final approximation is written as a linear combination of Eqs (2.40) and (2.42), we have
f~ Iga "-[-)~(fdgadX dxd( kdga~dx,]] (s
+ Q) dx = 0
or
(2.43)
f~ [Na-t-A/~(Na)] (/~(~) -q- Q)dx
- 0
If the second derivative term on Na is omitted (as could be done assuming linear a mollification as in Fig. 2.4) Eq. (2.43) is the same as the Petrov-Galerkin approximation with an undetermined parameter A. Indeed, if we take
Na and
A=
ah 21uI
(2.44)
the approximation is identical to that of the Petrov-Galerkin method with the weighting given by Eqs (2.24) and (2.25). Once again we see that a Petrov-Galerkin form written as
ffl (ga'3t"~2 If I dx ) ( fd~dx dxd ( k - ~d~x ) - J v Q ) d x : O
(2.45)
is a result that follows from diverse approaches, though only the variational form of Sec. 2.2.4 and that using an exact solution of the adjoint differential equation explicitly determine the value of a that should optimally be used. In all the other derivations this value is determined by an a posteriori analysis.
2.2.6 Sub-grid scale (SGS) approximation The SGS method was originally introduced by Hughes 24' 25 following the principles of turbulence modelling. More details on the method as applied to convection--diffusion problems are available in references 26-29. In this method the scalar quantity ~b is divided into two parts q5 and ~b', and Eq. (2.12) is written as
U~x(~+~')- ~
~(~+~')
+ a -0
or z;(4~) + z;(4~') + a = o
(2.46)
In the above split ~b is assumed to be the solution given by a finite element discretization (the so-called resolved scale) and q~' the unresolved, fine-scale part of the solution.
The steady-state problem in one dimension
If we construct a weak form for the problem, we have (2.47)
f a y [/2(~) +/2(q5') + Q] dx - 0
where f2 is the domain considered. After integration by parts and assuming boundary conditions for ~b' are zero at the boundaries of each element (i.e. assuming that $ has values at element boundaries which are accurate and oscillation free) we obtain
L[ v
U--~x+a
]
dx+
i
~_,*(v) d x + l ) q n I"q = 0
-~xk-d~xdX+
(2.48)
where dv /2*(0) = - U dx
d (kdV) dx ~xx
(2.49)
is the adjoint differential equation. As shown before, the solution using ~b alone with a standard Galerkin method leads to oscillatory results when convection effects are significant. H e r e we wish to take account of effects from the unresolved part when computing q~. From Eq. (2.46) we obtain
m
/2(05') = -(/2(~b) + Q)
(2.50)
for which the Green's function solution gives
(2.51) dp'(x) = - [ g(x, y) [/2(~(y)) + Q] dy J What we need here, however, is a simple approximation which does not require such a complex solution (and also the Green's function). Using the simple approximation (2.52) with/3 appropriately defined has been shown by Codina to give accurate results. 26 Substitution of Eq. (2.52) into Eq. (2.48) gives
f~ v [U-7-d~)fi+x Q 1 dx + f~ --;--k--z-dxfix(Ix dU d~)
f~'~*(0)/3 [J~--'(7)-t- a ] d x+Ovn l OI"q= (2.53)
If higher-order derivative terms are neglected in the above equation and fl = o~h/2U, the formulation is identical to that of the Petrov-Galerkin method and the Galerkin least squares method. However, if Q contains reaction terms such equivalence is lost and some differences occur. 26
2.2.7 The finite increment calculus (FIC) for stabilizing the convective-diffusion equation in one dimension
As mentioned in the previous sections, there are many procedures which give nearly identical results to those of the Petrov-Galerkin approximation. We shall also find a
43
44 Convectiondominated problems number of such procedures arising directly from the transient formulations discussed in Part II of this chapter; however, there is one further process that can be applied directly to the steady-state equation. This process was suggested by On~te in 199830 and we describe its basis below. We start at the stage where the conservation equation of the type given by Eq. (2.6) is derived. Now instead of considering an infinitesimal control volume of length 'dx', we consider a finite length 3. Expanding to one higher order by Taylor series (backwards), we obtain instead of Eq. (2.12) U
d4~ dx
d
(k dq~) + Q dx \ dx ]
d 2 dx
U
dx
dx
k
+ Q = 0
(2.54)
with 3 being the finite distance which is smaller than or equal to that of the element size h. Rearranging terms and substituting 3 = c~h we have dq5 U dx
d dx
k+
2
+ Q
2 dx
= 0
(2.55)
In the above equation we have omitted the higher-order expansion for the diffusion term as in the previous sections. The final equations are now obtained by applying the Galerkin method with Wa = Na. From the last equation we see immediately that a stabilizing term has been recovered and the additional term o~hU/2 is identical to that of the Petrov-Galerkin form [Eq. (2.31)]. There is no need to discuss further and we see how the finite increment procedure has again yielded exactly the same result by directly modifying the conservation differential equations. In reference 30 it is shown further that arguments can be brought to determine a as being precisely the optimal value we have already obtained by studying the Petrov-Galerkin method. For further details on the method readers are referred to references 31 and 32.
2.2.8 Higher-order approximations -
-
The derivation of accurate Petrov-Galerkin procedures for the convective diffusion equation is of course possible for any order of finite element expansion. In reference 11 Heinrich and Zienkiewicz show how the procedure of studying exact discrete solutions can yield optimal upwind parameters for quadratic shape functions. However, here the simplest approach involves the procedures of Sec. 2.2.4, which are available of course for any element expansion and, as shown before, will always give an optimal approximation at nodes. We thus recommend the reader to pursue the example discussed in that section and, by extending Eq. (2.37), to arrive at an appropriate equation linking the two quadratic elements of Fig. 2.8. For higher-order elements the solution gives exact values at interior nodes.
The steady-state problem in two (or three) dimensions 45
o... a-2
a-1
L
..
r
h
a
a+l
d.
a+2
h
7-
J
7
Fig. 2.8 Assemblyof one-dimensionalquadraticelements. For practical purposes it is possible to extend the Petrov-Galerkin weighting of the type given in Eqs (2.24) to (2.27) now using OLop t =
O~~
--
coth Pe
1
Pe
for the midside node and
(coth Pe - 1/Pe) - (cosh pe)2(coth 2Pe - 1/(2Pe)) 1 - (cosh p e ) 2 / 2
for side nodes (2.56)
A simplified procedure, though not as exact as that for linear elements, is very effective and has been used with success for solution of Navier-Stokes equations. 33 In recent years, the subject of optimal upwinding for higher-order approximations has been studied further and several references show the developments. ]2, 34, 35
i~i!i~i~i~!2~i3~i~ii~i~i~iTi~ihiei~i~isiteiia~i !iJi~!i~ii~ii~i~i!~~ii~ii~i!i~i~iii!~i~ii!i!i~!i~!ii~i!lii~iii~!i!i~i~i~i!~i!i~i~!ii!i~!i~~i!ii~i!pi~irob! i!i!i~!ii!~Ii!Qi~iim~i iii~i~i!i~i!i!!is~i!tia~i~!~i~ii~i~i~i~iii~i~i~i!i~i~iQii!~iii~%!ii!~iOi iIn!ri!~~i ~ !i!iiii~i~i~i~!i~ii~ii~i•~i•i•!i•i•i~!i!iii•!ii!i!ii~•!•i~!ii•i•ii•~iii~•i•i•i•~i•i%i!~i~ie)ii~ 2.3.1 General remarks It is clear that the application of standard Galerkin discretization to the steady-state scalar convection-diffusion equation in several space dimensions is similar to the problem discussed previously in Sec. 2.2.1 in one dimension and will again yield unsatisfactory answers with high oscillation for local Peclet numbers greater than unity. The equation now considered is the steady-state version of Eq. (2.8), i.e. Ui
OXiO~ OX ( k ~X; O(~ J + Q -- 0
(2.57)
Obviously the problem is now of greater practical interest than the one-dimensional case so far discussed, and a satisfactory solution is important. Again, all of the possible numerical approaches we have discussed are applicable.
2.3.2 Streamline (Upwind) Petrov-Galerkin (SUPG) weighting The most obvious procedure is to use again some form of Petrov-Galerkin method of the type introduced in Sec. 2.2.2 and Eqs (2.24) to (2.29), seeking optimality of
46
Convectiondominated problems a in some heuristic manner. Restricting attention here to two dimensions, we note immediately that the Peclet parameter Uh 2k
Pe-
{U1}
U--
U2
(2.58)
is now a 'vector' quantity and hence that upwinding needs to be 'directional'. The first reasonably satisfactory attempt to do this consisted of determining the optimal Petrov-Galerkin formulation using a W* based on components of U associated to the sides of elements and of obtaining the final weight functions by a blending procedure. 10, 11 A better method was soon realized when the analogy between balancing diffusion and upwinding was established, as shown in Sec. 2.2.3. In two (or three) dimensions the convection is only active in the direction of the resultant element velocity U, and hence the corrective, or balancing, diffusion introduced by upwinding should be anisotropic with a coefficient different from zero only in the direction of the velocity resultant. This innovation introduced simultaneously by Hughes and Brooks 36' 37 and Kelly et al. 13can be readily accomplished by taking the individual weighting functions as
Wa=Ua+ W; =---Na -~
ah Ui ONa 2 IUI OXi
(2.59)
this last form being applicable to two and three dimensions. Here a is determined for each element by the previously found expression (2.29) written as follows: OL - - OLopt =
coth
Pe
Pe
(2.60)
where
Pe -- [Ulh 2k
with
[U[ - v/Ui Ui
(2.61)
The above expressions presuppose that the velocity components Ui in a particular element are substantially constant and that the element size h can be reasonably defined. Figure 2.9 shows an assembly of linear triangles and bilinear quadrilaterals for each of which the mean resultant velocity U is indicated. Determination of the element size h to use in expression (2.61) is of course somewhat arbitrary. In Fig. 2.9 we show it simply as the size in the direction of the velocity vector. The form of Eq. (2.59) is such that the 'non-standard' weighting W* has a zero effect in the direction where the velocity component is zero. Thus the balancing diffusion is only introduced in the direction of the resultant (convective) velocity vector U. Introducing the approximation
,~ ~ - y ~ Na (Xi)~a
(2.62)
a
and using the weights given by Eq. (2.59) the SUPG method is computed from the weighted residual form
f~ [Na -t o~h IUI Ui Ox~ Is
+ Q]d~-O
(2.63)
The steady-state problem in two (or three) dimensions 47
Fig. 2.9 A two-dimensional, streamline assembly. Element size h and streamline directions.
or after integration by parts and introduction of the natural boundary condition on Fq
f~ [ ( O~ ) O g a 0~ ogh Ui Oga (f_.,(@).jI..a)] d~.2 ga oi --~xi --[-Q + -~x k -~xi -~ 2 [UI Oxi + fr NaqndF
= 0
(2.64)
q
where g/n = -kO~/On. In the discretized form the 'balancing diffusion' term becomes
ON.
[r ~ONb dr2
(2.65)
with
[r -- ~
h IUI
2
(2.66)
This indicates a zero coefficient normal to the convective velocity vector direction. It is therefore named the streamline balancing diffusion 13' 36,37 or streamline upwind Petrov-Galerkin process. The streamline diffusion should allow discontinuities inthe direction normal to the streamline to travel without appreciable distortion. However, with the standard finite element approximations actual discontinuities cannot be modelled and in practice some oscillations may develop. For this reason some investigators add a smoothing diffusion in the direction normal to the streamlines (cross-wind diffusion). 38-41 The mathematical validity of the procedure introduced in this section has been established by Johnson et al. 42 for a - 1, showing convergence improvement over the standard Galerkin process. However, the proof does not include any optimality in the selection of a values as shown by Eq. (2.60). Figure 2.10 shows a typical solution of Eq. (2.57), indicating the very small amount of 'cross-wind diffusion', i.e. allowing discontinuities to propagate in the direction of flow without substantial smearing. 43 A more convincing 'optimality' can be achieved by applying the exponential modifying function, making the problem self-adjoint. This of course follows precisely the
48
Convection dominated problems T (y)
T= 0
U T=O Y T=O
(a) Boundary conditions for test problem
x --y
(b) Solutions for 0 = 45 ~ (top) and 0 = 60 ~ (bottom)
Fig. 2.10 'Streamline' procedures in a two-dimensional problem of pure convection. Bilinear elements.43
procedures of Sec. 2.2.4 and is easily accomplished if the velocities are constant in the element assembly domain. If velocities vary from element to element, again the exponential functions p = e -U~'/k (2.67) with x' orientated in the velocity direction in each element can be taken. This appears to have been first implemented by Sampaio 43 but problems regarding the origin of coordinates, etc., have once again to be addressed. However, the results are similar to those achieved here by the Streamline Upwind Petrov-Galerkin procedure.
2.3.3 Galerkin least squares (GLS) and finite increment calculus
(FIC) in multidimensional problems
It is of interest to observe that the somewhat intuitive approach to the generation of the 'streamline' Petrov-Galerkin weight functions of Eq. (2.59) can be avoided if the
Steady state - concluding remarks
least squares Galerkin procedure of Sec. 2.2.5 is extended to deal with the multidimensional equation. Simple extension of the reasoning given in Eqs (2.39a) to (2.45) will immediately yield the weighting of Eq. (2.59). Extension of the GLS to two or three dimensions gives (again using indicial notation) f~
0~ (Na"~-A~-'(Na))(UJ~xxj
where
0 (k -'~xj) 00 -Jl"Q)df2 "-" 0 Oxj
ONa
0 (kONa "-'~Xi)
ff--,(Na) = Ui-~x i .-~ ~
(2.68)
(2.69)
In the above equation after integration by parts, higher-order derivative terms of two or more are omitted for the sake of simplicity. As in one dimension [Eq. (2.43)] we have an additional weighting term. Now assuming A=
c~h
(2.70)
21UI
we obtain an identical stabilizing term to that of the streamline Petrov-Galerkin procedure [Eq. (2.64)]. The finite increment calculus method in multidimensions can be written as 3~
(~i O O~ 20x i [Uj Oxj
O (k OqS) + Ql = O
Oxj
or
/2(~b) + Q
-
(2.71)
~-
Note that the value of (~i is now dependent on the coordinate direction. To obtain streamline-oriented stabilization, we simply assume that (~i is the projection oriented along the streamlines. Now 6i =
Ui
6~
IuI
(2.72)
With 3 = c~h and again omitting the higher-order derivative terms in k, the streamline Petrov-Galerkin form of stabilization is obtained [Eq. (2.64)]. The reader can verify that both the GLS and FIC produce the correct weighting for the source term Q as of course is required by the Petrov-Galerkin method. The extension of the SGS procedure to multidimensions is straightforward and follows the approach explained in Sec. 2.2.6. iliil Jiiiiiiiiiiiiii~ 211i~ii~i ~~~~~iiiiiiiiii~~iiiiiiiiiii~ iiiii~~~ ~~ii!~i~iiiiiiiiiiiiii ~ iiiiiiiiiiiiiiil il'~'~'~iiiiii iiiMii~~!iiii~iiii~~~~ iiiiiiiii~i ~iiiiiiiiiiiiiii~ i~iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii~'~iiiiiiiiiiiiiiiiii '~i i~'~'~iiiiiiiiiiii!iiiiiiiiiiiiiiiiiiii i ii!iiiii ~iiiiiiiiilii~iiiiiiiiiiiiiii~iiiiiiiili!iiiiii iiiiiiiil~~iiiiiiiiiiiiiiiiiiiiii!iii i iiiiiiiiiiiiiiiiiiiiiiii i iiiiiiiiiiiiiiiiiiiiiiii i iiiiiiiiiiiiiiiiii iiiiiiiiiiiii!iiiiiiiiiiiiiiiiiii iiiiiiiiiliiiiiiiiiiil iiiiiililiiiiiiiiiiiiiiii!iiiiiiliiiiiiiiiliiiiiiiiiiiii iiiiiii iiliiiiiiiiiiiiiiiiiiiii!iiiii!iiii iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii iiiiiiiiii ii i i8 !ii iiii i!i!iiiiii!iiiiiili!i~ii! iiii iiiiiiiiiiiiii!iiiiii iiliil iii i
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In Secs 2.2 and 2.3 we presented several currently used procedures for dealing with the steady-state convection-diffusion equation with a scalar variable. All of these translate essentially to the use of streamline Petrov-Galerkin discretization, though of course the modification of the basic equations to a self-adjoint form given in Sec. 2.2.4 provides
49
50
Convectiondominated problems the full justification of the special weighting. Which of the procedures is best used in practice is largely a matter of taste, as all can give excellent results. The generalized representation of all the stabilization methods discussed in this part may be written as
f~ ( ga "~-~i -~x C~Na) ONa~a "lt---~x C~NlkC~gb 1 i IU -'~x -~x ~ q- Q d f ~ - O
(2.73)
where 'ffi is a stabilizing parameter. In the second part of this chapter dealing with transient problems it will be found that similar stabilizing forms arise directly when steady state is reached or assumed. In this case the parameter ")'i is now replaced by another one involving the length of the time step At. We shall show at the end of the next section a comparison between various procedures for stabilization and will note essentially the same forms in the steady-state situation.
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2.5.1 Mathematical background The objective of this section is to develop procedures of general applicability for the solution by direct time-stepping methods of Eq. (2.1) written for scalar values of 0, Fi and Gi. Starting from the scalar form of Eq. (2.4)
O~ O(UiO) 0 (0~) (gt ] OXi
OXi
k-~x i
+Q=0
(2.74)
though consideration of the procedure for dealing with a vector-valued function is included in Appendix E. To allow a simple interpretation of the various methods and of behaviour patterns the scalar equation in one dimension in non-conservation form [see Eq. (2.11)], i.e. -~- + U 0x
0x k
+ Q - 0
(2.75)
will be considered. The problem so defined is non-linear unless U is independent of ~b. However, the non-conservative equation (2.75) admits a spatial variation of U and is quite general. In the general form (2.74) the main behaviour patterns can be determined by a change of the independent variable x to x' such that d x ' = dx - U dt
(2.76)
Transients - introductory remarks
,~(x- ut)
,l,(x); t= o
V
X
Fig. 2.11 The wave nature of a solution with no conduction. Constant wave velocity U.
Noting that for q~ = q~(x', t) we have
Ocb l - ~ - x c o n s t -"
O~ Ox '
Oc~ l
OX' Ot ]"--~ x'const
-- - U Od?
Oq51
(2.77)
~Xt " ~ " - ~ - x'const
The one-dimensional equation (2.75) now becomes simply
Ot
Ox'
~x'
+ Q(x') = 0
(2.78)
and equations of this type can be readily discretized with self-adjoint spatial operators and solved by standard Galerkin finite element procedures. ~ The coordinate system of Eq. (2.76) describes characteristic directions and the moving nature of the coordinates must be noted. A further corollary of the coordinate change is that with no conduction or source terms, i.e. when k = 0 and Q - 0, we have simply
0___~= 0
(2.79a)
Ot or, for the one-dimensional case,
c~(x') - r
- Ut) = constant
(2.79b)
along a characteristic (assuming U to be constant). This is a typical equation of a wave propagating with a velocity U in the x direction, as shown in Fig. 2.11. The wave nature is evident in the problem even if the conduction (diffusion) is not zero, and in this case we shall have solutions showing a wave that attenuates with the distance travelled.
2.5.2 Possible discretization procedures In Part I of this chapter we have concentrated on the essential methods applicable directly to the steady-state equations. These procedures started off from somewhat heuristic considerations. The Petrov-Galerkin method was perhaps the most rational but even here the amount and the nature of the weighting functions were a matter of guesswork which was subsequently justified by consideration of the numerical error at nodal points. The Galerkin least squares (GLS) method in the same way provided no absolute necessity for improving the answers though of course the least squares method would tend to increase the symmetry of the equations and thus could be proved useful. It was only by results which turned out to be remarkably similar to those obtained by
51
52
Convectiondominated problems the Petrov-Galerkin method that we have deemed this method to be a success. The same remark could be directed at the finite increment calculus (FIC) method and indeed to other methods suggested dealing with the problems of steady-state equations. For the transient solutions the obvious first approach would be to try again the same types of methods used in steady-state calculations and indeed much literature has been devoted to this. 38-55 Petrov-Galerkin methods have been used here quite extensively. However, it is obvious that the application of Petrov-Galerkin methods will lead to non-symmetric mass matrices and these will be difficult to use for any explicit method as mass lumping is not by any means obvious. Serious difficulty will also arise with the Galerkin least squares (GLS) procedure even if the temporal variation is generally included by considering space-time finite elements in the whole formulation. This approach to such problems was made by Nguen and Reynen, 44 Carey and Jieng, 45' 46 Johnson and coworkers 42' 47, 48 and others. 49'5~ However, the use of space-time elements is expensive as explicit procedures are not available. Which way, therefore, should we proceed? Is there any other obvious approach which has not been mentioned? The answer lies in the wave nature of the equations which indeed not only permits different methods of approach but in many senses is much more direct and fully justifies the numerical procedures which we shall use later for the full fluid dynamics equations. We shall therefore concentrate on such methods and we will show that they lead to stabilizing diffusions which in form are very similar to those obtained previously by the Petrov-Galerkin method but in a much more direct manner which is consistent with the equations. The following discussion will therefore be centred on two main directions: (1) the procedures based on the use of the characteristics and the wave nature directly, leading to so-called characteristic Galerkin methods which we shall discuss in Sec. 2.6; and then (2) we shall proceed to approach the problem through the use of higher-order time approximations called Taylor-Galerkin methods. Of the two approaches the first one based on the characteristics is in our view more important. However, for historical and other reasons we shall discuss both methods which for a scalar variable can be shown to give identical answers. The solutions of convective scalar equations can be given by both approaches very simply. This will form the basis of our treatment for the solution of the full fluid mechanics equations in Chapter 3, where both explicit (time integration) processes as well as implicit methods can be used. Many of the methods for solving the transient scalar equations of convective diffusion have been applied to the full fluid mechanics equations, i.e. solving the full vectorvalued convective-diffusive equations we have given at the beginning of the chapter (Eq. 2.1). This applies in particular to the Taylor-Galerkin method which has proved to be quite successful in the treatment of high speed compressible gas flow problems. Indeed this particular approach was the first one adopted to solve such problems. However, the simple wave concepts which are evident in the scalar form of the equations do not translate to such multivariant problems and make the procedures largely heuristic. The same can be said of the direct application of the SUPG and GLS methods to multivariant problems. The procedures such as GLS can provide a useful stabilization of difficulties encountered with incompressible behaviour. ~ This does not justify their
Characteristic-based methods
use on the full equations and we therefore recommend the alternatives to be discussed in Chapter 3.
2.6.1 Mesh updating and interpolation methods We have already observed that, if the spatial coordinate is 'convected' in the manner implied by Eq. (2.76), i.e. along the problem characteristics, then the convective, firstorder, terms disappear. The remaining problem is that of simple diffusion for which discretizadon procedures with the standard Galerkin spatial approximation are optimal (in an energy norm sense). The most obvious use of this in the finite element context is to update the position of the mesh points in an incremental Lagrangian manner. In Fig. 2.12(a) we show such an update for the one-dimensional problem of Eq. (2.75) occurring in an interval At. t
=
I ~
aracteristic
I..
/ '
(a)Forward
/ h
'l =l
/
/ " \
/
UPdated node position
Initial node position
x
t
ik
r X
t n-1 = t n - At
(b)Backward Fig.2.12 Meshupdatingandinterpolation:(a)forward;(b)backward.
53
54 Convectiondominated problems For a constant x' coordinate (2.80)
dx = Udt
and for a typical nodal point a, we have
xn+l
n
= xa +
I tn+l J tn
Udt
(2.81)
where in general the 'velocity' U may be dependent on x. For a constant U we have simply
Xa+l
= x an + U A t
(2.82)
for the updated mesh position. This is not always the case and updating generally has to be done with variable U. On the updated mesh only the time-dependent diffusion problem needs to be solved using the Galerkin method. 1 The process of continuously updating the mesh and solving the diffusion problem on the new mesh is, of course, impractical. When applied to two- or three-dimensional configurations very distorted elements would result and difficulties will always arise near the boundaries of the domain. For these reasons it seems obvious that after completion of a single step a return to the original mesh should be made by interpolating from the updated values, to the original mesh positions. This procedure can of course be reversed and characteristic origins traced backwards, as shown in Fig. 2.12(b) using appropriate interpolated starting values. The method described is somewhat intuitive but has been used with success by Adey and Brebbia 56 and others as early as 1974 for solution of transport equations. The procedure can be formalized and presented more generally and gives the basis of so-called characteristic-Galerkin methods. 57 The diffusion part of the computation is carried out either on the original or on the final mesh, each representing a certain approximation. Intuitively we imagine in the updating scheme that the o p e r a t o r is s p l i t with the diffusion changes occurring separately from those of convection. This idea is explained in the procedures of the next section.
2.6.2 Characteristic-Galerkin procedures We shall consider that the equation of convective diffusion in its one-dimensional form (2.75) is split into two parts such that 4~ = ~b* + ~b**
(2.83)
and separate the differential equation into two additive parts. Accordingly,
o~* Ot is a purely convective system while 0t
o~
+ U~x = 0
0x
Ox
+ Q = 0
(2.84a)
(2.84b)
Characteristic-based methods y= x + U M
t
r
x
Fig. 2.13 Distortion of convected shape function.
represents the remaining terms [here Q contains the source, reaction and term ( OU/Ox ) r ] . Both ~b* and qS**now are approximated by expansions
NadPa
A
r
NadPa
r
(2.85)
and in a single time step t n to t n + At = t "+~ we shall assume that the initial conditions
are
t = tn
q~,n _ 0
c~**" - r
(2.86)
Standard Galerkin discretization of the diffusion equation allows q~**"+~to be determined on the given fixed mesh by solving an equation of the form MAq~** = At [H(q~" + 0Aq~**)+ f]
(2.87)
with
~)**n+l- (~**...~_A(~** and Mab = ~ NaNb d ~ f I4a
ONa
fa--f
ONo
i
--
i d
NaQd~
In solving the convective problem we assume that 4)* remains unchanged along the characteristic. However, Fig. 2.13 shows how the initial value of q5*n interpolated by standard linear shape functions at time n [see Eq. (2.85)] becomes shifted and distorted. The new value is given by (~.n+l
__
Na(y)~gan
y = x + UAt
(2.88)
As we require q5*n+l to be approximated by standard shape functions, we shall write a projection for smoothing of these values as f
N T(N~b*n+l - N(y)~b*") dx = 0
(2.89)
55
56
Convection dominated problems
giving
/ b
Mt~ *n+l -- / [NTN(y) dx]t~ *n
(2.90)
The evaluation of the above integrals is of course still complex, especially if the procedure is extended to two or three dimensions. This is generally performed numerically and the stability of the formulation is dependent on the accuracy of such integration. 57 The scheme is stable and indeed exact as far as the convective terms are concerned if the integration is performed exactly (which of course is an unreachable goal). However, stability and indeed accuracy will even then be controlled by the diffusion terms where several approximations have been involved.
2.6.3
A simple explicit
characteristic-Galerkin procedure
Many variants of the schemes described in the previous section are possible and were introduced quite early. References 56-67 present some successful versions. However, all methods then proposed are somewhat complex in programming and are time consuming. For this reason a simpler alternative was developed in which the difficulties are avoided at the expense of conditional stability. This method was first published in 198468 and is fully described in numerous publications. 69-72 Its derivation involves a local Taylor expansion and we illustrate this in Fig. 2.14. We can write Eq. (2.75) along the characteristic as
OdP(x,(t) t) - 0 ( 0 ~ ) ' Ox----; k-~x'
+ Q(x') - 0
(2.91)
As we can see, in the moving coordinate x', the convective acceleration term disappears and source and diffusion terms are averaged quantifies along the characteristic. Now the equation is self-adjoint and the Galerkin spatial approximation is optimal. The time discretization of the above equation along the characteristic (Fig. 2.14) gives
1 At
)
0
0
k~x_ x -a (2.92)
-~-(1- 0)[~X (kox)0(~ __ Q]n where 0 is equal to zero for explicit forms and between zero and unity for semi- and fully implicit forms. As we know, the solution of the above equation in moving coordinates
t
,
0r +I s
s
s
s
dpn(x-~))r-('" X
r
Fig. 2.14 A simple characteristic-Galerkin procedure.
s
s
s
s
t
0 Cn
r- I
Characteristic-based methods 57
leads to mesh updating and presents difficulties, so we will suggest alternatives. From the Taylor expansion we have
a2 02~)n Ox + 20qX 2
(2.93)
-+- O ( A t 3)
and assuming 0 = 0.5
1 0 (k 0(]5)
1 0 (~x~) n
2& 1
a n
(~ 0
0
n
x)] 2 0 x [ - ~ x ( k ~O~
+ O(At2) (2.94)
~ oa n
Ql(x-6) ~ 2
20x
where 3 is the distance travelled by the particle in the x direction (Fig. 2.14) which is
6 - OAt
(2.95)
where 0 is an average value of U along the characteristic. Different approximations of 0 lead to different stabilizing terms. The following relation is commonly u s e d 73' 74
OUn
U = Un - Un a t -
(2.96)
Ox
Inserting Eqs (2.93)-(2.96) into Eq. (2.92) we have
(~n+l - ~n ----At { o(9~)n~ OXO(kO~)'~)x + At{ At
n+l/2
+ Q.+1/2 }
O~ _ _ _ A t U -~x2
0
-~x
At
V
OQ
-~x
"
(2.97a)
where
O(k~x)nq-1/2
Ox and
1 0 (kO~)n+l
-- 2 0 x
~x
1 0 (0(~) n
+ -~ ~
k --~x
(2.97b)
Qn+l/2__ l(Qn+l _~_Q.)
(2.97c)
--2
If n + 1/2 terms are replaced with n terms the above equations become explicit in time. This, as already mentioned, is of a similar form to those resulting from TaylorGalerkin procedures which will be discussed fully in the next section, and the additional terms add the stabilizing diffusion in the streamline direction. For multidimensional problems, Eq. (2.97a) can be written in indicial notation and approximating n + 1/2 terms with n terms (for the fully explicit form)
n+l_
0Xi
+ Q} 5(2.98)
58
Convectiondominated problems An alternative approximation for 0 is 73
0--" ~1 (un+l --I-U n [(x--~)
(2.99)
Using the Taylor expansion Unl(x_6) ~ U n
_
OU" A t U n - - ~x- + O(At 2)
_
(2.100)
from Eqs (2.92)-(2.95) and Eqs (2.99) and (2.100) with 0 equal to 0.5 we have
1 (~n+l__on)____un+l/20~)n AtunOUn 0(9n Atun+I/2un+I/202 ~ At
Ox + T
-s
Ox
T
At un +~/2 0
0 (k~x)n+l/2
0
n
At un +1/20Q - Q + -2-
where
Ox
(2.101)
U n+1/2+ U n) m !(un+l 2
(2.102)
We can further approximate, as mentioned earlier, n + 1/2 terms using n, to get the fully explicit version of the scheme. Thus we have
U n+l/2 - U n Jr- O(At)
(2.103)
and similarly the diffusion term is approximated. The final form of the explicit characteristic-Galerkin method in one dimension can be written as
A ~) -- dpn+l -- ~n -- -- A t IU n O~POx OxOq( k O~--~x ) -+-Q] n At2un 0
+T
O~
[ nox
0 (k O(~
Ox
Q]n
(2.104)
Generalization to multidimensions and conservation form is direct and can be written in indicial notation for equations of the form of Eq. (2.6)"
zX~--- At [O(Ujdp)Oxj OXiO\(k ~x iO~/~ -+-a]n At 2 O [O(Sj~) Av T U ; ~ x k OXj
O(-~ixi) In OXi k +Q
(2.105)
The reader will notice the difference in the stabilizing terms obtained by two different approximations for U [Eqs (2.98) and (2.105)]. However, as we can see the difference between them is small and when U is constant both approximations give identical stabilizing terms. In the rest of the book we shall follow the latter approximation and always use the conservative form of the equations [Eq. (2.105)].
Characteristic-based methods 59
As we proved earlier, the Galerkin spatial approximation is justified when the characteristic-Galerkin procedure is used. We can thus write the approximation ~b = N ~
(2.106)
and use the weighting N T in the integrated residual expression. Thus we obtain M((]) n+l -- ~Dn) ~- - A t [ ( C ~ b n -F- K(]) n -'F- fn) _
At(Ku~p ~ + fn)]
(2.107)
In the above equation C=fNT
M = . ~ NTN dr2
/9 ~x/(UiN) d~2 (2.108)
K-
f~ ONTk ON ~ OXi dr2
f =
f a NT Q d~2 + b.t.
and K, and fn come from the new term introduced by the discretization along the characteristics. After integration by parts, the expression of Ku and fs is K U "---
2l f
O o -~xi(UiNT)-~xj(UjN)df2 (2.109)
f ~ -- - -~
-~x ( UiN T) Q dr2 + b.t.
where b.t. stands for integrals along region boundaries. Note that the higher-order (third and above) derivatives are not included in the above equation. The boundary terms from the discretization of stabilizing terms are ignored because the original residual is zero. The approximation is valid for any scalar convected quantity even if that is the velocity component Ui itself, as is the case with momentum-conservation equations. For this reason we have elaborated above the full details of the spatial approximation as the matrices will be repeatedly used. It is of interest to note that the explicit form of Eq. (2.107) is only conditionally stable. For one-dimensional problems, the stability condition is given as (neglecting the effect of sources) h At _< Atcrit = (2.110)
IUI
for linear elements in the absence of diffusion. If the diffusion is present a new criticial time step has to be calculated as 73' 74 m tcrit - -
AtuAtk Atu + Atk
(2.111
)
where Atu is given by Eq. (2.110) and Atk = h2/2k is the diffusive limit for the critical one-dimensional time step. In two and three dimensions the nodal element size h is taken as the minimum of element sizes surrounding a node. Further, with At = Atcrit the steady-state solution results in an (almost) identical balancing diffusion change to that obtained by using the optimal streamline upwinding
60
Convection dominated problems 1.0
Stability limit C = , ~ /
Unstable
1 +1 pe 2
1 Pe
0.8
J::
0.6
0.4 -
C = %pt = coth P e - l l P e
0.2 0
0
1
2
I ...... 3
Pe
I 4
I 5
I 6
I 7
Fig. 2.15 Stability limit for lumped mass approximation and optimal upwind parameter. procedures discussed in Part I of this chapter. Thus if steady-state solutions are the main objective of the computation such a value of At should be used in connection with the Ku term. A fully implicit form of solution is an expensive one involving unsymmetric matrices. However, it is often convenient to apply 0 > 1/2 to the diffusive term only. We call this a nearly (or quasi) implicit form and if it is employed we return to the stability condition h Atcrit = (2.112) IUI which can present an appreciable benefit. Figure 2.15 shows the stability limit variation prescribed by Eq. (2.110) with a lumped mass matrix for linear elements. It is of considerable interest to examine the behaviour of the solution when the steady state is reached - for instance, if we use the time-stepping algorithm of Eq. (2.107) as an iterative process. Now the final solution is given by taking ~Dnd-I .~ (~n __ ~D
which gives [(C + K - AtKu](b + f - Atfs -- 0
(2.113)
Inspection of Secs 2.2 and 2.3 shows that the above is identical in form with the use of the Petrov-Galerkin approximation. In the latter the stabilization matrix is identical ] and the matrix Ku includes balancing diffusion of the amount given by ~c~Uh. However, if we take
1 U2At -aUh =
(2.114) 2 2 the identity of the two schemes results. This can be written as a requirement that a --
UAt h
= C
(2.115)
Characteristic-based methods 61 where C is the Courant number. The Ccritical for a convection-diffusion problem in one dimension may be written a s Ccritical -- 4 1 / Pe 2 + 1 -- 1/ P e. In Fig. 2.15 we therefore plot the optimal value of a as given in Eq. (2.29) against Pe. We note immediately that if the time-stepping scheme is operated at or near the critical stability limit of the lumped scheme the steady-state solution reached will be close to that resulting from the optimal Petrov-Galerkin process for the steady state. However, if smaller time steps than the critical ones are used, the final solution, though stable, will tend toward the standard Galerkin steady-state discretization and may show oscillations if boundary conditions are such that boundary layers are created. Nevertheless, such small time steps result in very accurate transients so we can conclude that it is unlikely that optimality for transients and steady state can be reached simultaneously. Examination of Eqs (2.107) shows that the characteristic-Galerkin algorithm could have been obtained by applying a Petrov-Galerkin weighting
N" + Y Ox---; to the various terms of the governing equation (2.74) excluding the time derivative 0~b/0t to which the standard Galerkin weighting ofN T is attached. Comparing the above with the steady-state problem and the weighting given in Eq. (2.59) the connection is obvious. A two-dimensional application of the charactefistic-Galerkin process is illustrated in Fig. 2.16 in which we show pure convection of a disturbance in a circulating flow. It is remarkable to note that almost no dispersion occurs after a complete revolution. The present scheme is here contrasted with the solution obtained by the finite difference scheme of Lax and Wendroff 75 which for a regular one-dimensional mesh gives a scheme identical to the charactefistic-Galerkin except for mass matrix, which is always diagonal (lumped) in the finite difference scheme. It seems that here the difference is entirely due to the proper form of the mass matrix M now used and we note that for transient response the importance of the consistent mass matrix is crucial. However, the numerical convenience of using the lumped form is overwhelming in an explicit scheme. It is easy to recover the performance of the consistent mass matrix by using a simple iteration. In this we write Eq. (2.107) as
MA~b n - - S n
(2.116)
with S n being the fight-hand side of Eq. (2.107) and
~n--I-1 __ ~/~n ..~_A~n Substituting a lumped mass matrix ML to ease the solution process we can iterate as follows:
(A~b)7 = MLI[S n -k- (ML -- ]14)( m~t~))/_lln
(2.117)
where l is the iteration number. The process converges very rapidly and in Fig. 2.17 we show the dramatic improvements of results in the solution of a one-dimensional wave propagation with three such iterations done at each time step. At this stage the results are identical to those obtained with the consistent mass matrix.
62
Convectiondominated problems
(a) Original form
(b) Form after one revolution using consistent M matrix
(c) Form after one revolution using lumped mass (Lax-Wendroff)
Fig. 2.16 Advection of a Gaussian cone in a rotating fluid by characteristic-Galerkin method: (a) original form; (b) form after one revolution using consistent M matrix; (c) form after one revolution using lumped mass (Lax-Wendroff).
2.6.4 Boundaryconditions- radiation As we have already indicated the convection-diffusion problem allows us to apply on the boundary one of the following conditions = ~ on Fu
(2.118a)
Characteristic-based methods Courant number = 0.1
Lumped/consistent M Courant number = 0.5 Lumped
Lumped
~
"~J " , , J ' x . /
C= 0.5, 1/T
C= 0.1, 1/T
9
v
v
v
C= 0.5, 2/T
C= 0.1, 2/T
Consistent
Consistent
"
"-/
X/
k/"
A
v
~
y -
v
9
C= 0.1, 3/T (b) Courant number = 0.1
C = 0.5, 3/T (a) Courant number = 0.5
Fig. 2.17 Characteristic-Galerkin method in the solution of a one-dimensional wave progression. Effect of using a lumped mass matrix and of consistent iteration.
or
~)
_~(o~
=q
_
OnFq
(2.118b)
(where F = Fu U Fq), providing the equation is of second order and diffusion is present. In the case of pure convection this is no longer the case as the differential equation is of first order. Indeed this was responsible for the difficulty of obtaining a solution in the example of Fig. 2.2 when Pe ~ c~ and an exit boundary condition of the type given by Eq. (2.118a) was imposed. In this one-dimensional case for pure convection with positive U only the inlet boundary condition can be given; at the exit no boundary condition needs to be prescribed. For multidimensional problems of pure convection the same wave specification depends on the value of the normal component of U. Thus if Uini > 0
(2.119)
where ni is the outward normal vector to the boundary, the wave is leaving the problem and there no boundary condition is specified. If the problem has some diffusion, the
63
64 Convectiondominated problems same specification of 'no boundary condition' is equivalent to putting -k
~-
-0
(2.120)
at the exit boundary. In Fig. 2.18 we illustrate, following the work of Peraire, 76 how cleanly the same wave as that specified in the problem of Fig. 2.15 leaves the domain in the uniform velocity field 72,76 when the correct boundary condition is imposed.
Initial configuration
Fig. 2.18 A Gaussian distribution advected in a constant velocity field. Boundary condition causes no reflection.
Taylor-Galerkin procedures for scalar variables 65 i{ii{i{iiiiiiii;~ ii~i~ ii;~iiii~i!i '~iiiilililiiiiiii i ~'~~~~'~~iiiiiiii iiiiiiiiiiii! iiiiiiiiii ~ ~'iili'i~i!ii{iiji{i{i{!iiiii!ii~{~~'i~{~'~i~iiii!iili'{~'{i~'{iii{i{iiiii!'~{i'iiiiiiiiii~iiiiiiii 'ii iiliiiiiiiiiiiii i{~i{lii iilii{~iiiii!iiiiiil li!i iiili!iiiiil i~iliiiiiiiiii !ili iiiiliiiiiiiiiii ~ iiii~'i~'~iiiiiiiiiiii 'iiiiiiiiiiiiii{iiii~iiii~i{"'!iiiiiiiiiiiii iii~ili!iiiiiii!ii{iiiiiiii ili~iiii"~'iiiiiiiiiiiiiiiiiiii i iiiiiiiiiiii iiii{i~i iliiiiii{i~ii}iii!ii~iiiii i}ii{ili~i{iiii'~}i'~"''"~i{iiiiii iiiiiliii{iiil{iiiiiiiiiiiii iiiiiii}{iiiiiiiiiiiiil {{i iiiii}iiiiiiiiiiiiiiiiii iii iiiiiiii!i!ii!iiiiiiii
In the Taylor-Galerkin process, the Taylor expansion in time precedes the Galerkin space discretization. First, the scalar variable ~b is expanded by the Taylor series in time69, 77 O~n At2 02t~ n 3 ~bn + l - r 2 0t 2 b O ( A t ) (2.121) From Eq. (2.75) we have
0r n
(9r
0 (kO~t~
Q]n
(2.122a)
and Orlq2~n
-U~x +
0
=g
bTxV bTx/
+ Q
(2.122b)
Substituting Eqs (2.122a) and (2.122b) into Eq. (2.121) we have
4)n+l _ d/)n-- _At [U Odl) Ox
O (k OdP) Q]n Ox -~x/ +
At2 O EUO~ 20t Ox
O (kOd/)
Q]n
Ox
(2.123) Assuming U and k to be constant we have
~)n.-I-l__r __AtIgOd/)
Ox
O (kO ~)
Ox
-~x/ +
Q]n
At 2 o [uOr 2 Ox
Ot
Ox -if;)+ (2.124)
Inserting Eq. (2.122a) into Eq. (2.124) and neglecting higher-order terms cn+l
__ ~n __ _At IUOd/)r L Ox 20x
OO(k~x)Or-al n
Ox
(2.125)
U 2 --~x -
U -'~X k
+ UQ
+ O(At 3)
As we can see the above equation, having assumed constant U and k, is identical to Eq. (2.98) derived from the characteristic approach. Clearly for scalar variables both characteristic and Taylor-Galerkin procedures give identical stabilizing terms. Thus selection of a method for a scalar variable is a matter of taste. However, the sound
mathematicaljustification of the characteristic-Galerkin method should be emphasized here and for this reason the characteristic Galerkin procedure forms the fundamental basis for the remainder of this text. The Taylor-Galerkin procedure for the convection-diffusion equation in multidimensions can be written as
qsn+l
_ r =-At
UJ Oxj 2 ox,
Oxi k~xi
+ Q
(2.126)
66
Convectiondominated problems again showing the complete similarity with the appropriate characteristic-Galerkin form and identity when Ui and k are constant. The Taylor-Galerkin method is the finite element equivalent of the Lax-Wendroff method developed in the finite difference context. 75 The Taylor-Galerkin process has one important feature. The idea can be used directly for dealing with the vector form of the convection-diffusion equation, such as we have mentioned at the beginning of this chapter [viz. Eq. (2.1)]. This method was used with reasonable success to solve problems of high speed gas flow. In today's treatment we will find these forms are unnecessary. For this reason we relegate the treatment of vector-valued functions to Appendix E. iiiiiiiiiiiiiiii~iiiiiililiiii~i~i~iiiiiiiiiii i i i i ~i~i i i~i~i i i i i i i i i !~ii ii~i iiiiiiiiiiii!ii!iiii~i~i~i~iii~iii~iiiiiiii~ii~ii~iii~i~iiii~i~iiiii~:ii~ii~i:ii:i~ii:i:~i~i:~i:~i:i:~i:~i!ii:i:~:i:i:i:!:i:!:i:i i:i~i~i ii~i~i:~:i:ii:i:~i:i~:i:iili~:!i:~i:i:~~ii:i:i~i:'i:i!~:!i:i:i!~:i::i:i:i:i:i:i:!:i i:i:i:~i:ii:~i~i~:i:~i~!i~ii~~ii:i:~ii:!:i:i:i:i:i:i:i:i:i:i:i:i:i~i:i:i:i:i::!:i:i:i:i:i:i:i i:~i:i:i:i~i:i:i:i:i:i:i:i:!~i:~Ii~i:i:i:~i:i:~i~:ii:i:i::i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i i~i:i:i ~i:i:~ii:i~!: ~:i:ii~:i~i!:~i~ii~:i~~i~ii~i~i:i:i:i:i:i~~!:~i!:i:i:!:i:i:i:i:~ii:i:i~i~:~ii~i:~i~i:i:i:!i: iiiiiiiiiii!iiiiiiiiii~::.:~iiiiiiiiiii: .:iiiiiiiiiiiiliiii~::..%:!:i::":'iii~i."::ii~....... !i"i~":i~:.!:!i!:! !iii:~i! !i':::i!iii,::ss!~!!~..........' .........i'...... ii! ~| .........
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Both the Taylor-Galerkin and characteristic-Galerkin methods give an answer which compares directly with SUPG and GLS giving additional streamline diffusion (higherorder derivatives are omitted) and sources
At 20[UiU j 0(~ ] 2 0xi
~xj J
and
At O [UiQ ] 2 0x,
(2.127)
with At replacing the coefficient ah. With the characteristic-Galerkin method being the only method that has a full mathematical justification, we feel that even for steadystate problems this should be considered as an appropriate solution technique. iis~ii~i~i~i~iiii~i~i~ii!ii~iiiiiii~iii!i~i~i~iiii~ii:iii~i~i ii!i:~ii~:i!i~ii~!ii~!i~i!:ii~iii~i~ii~ii:i~:~:i iii~i:i~i~i:ii~~ii:iiii:i~ii~i:i~:ii~i:ii~~i:ii:i:i~i:~i:i:i~':i~:i:i~:~i~:i:ili:i:i:i~i:i::i:iii~iiiiiiiiiiiiiii%iiiiiiiiiiiiiiiiiiiiiiiiiiiiii i~i~:~:~:i iii:i:i~:i:i:iiii i~i~i:iiii~iiii~iiiiiiiii~iii~iiii~iii~i~i~iiiiiiiiiiiiii~iiiiiiiiii!iiiiiisiiiiiiiiiii iiiiii~84iiiiiiiii~i~iii!ii!iiii!iiiiii84iiiiTiiii:i:! ii~iiiiiiiiii~ii iiiiiiiiiiiii!iii!iiiiiiiiiiiiiii~ ~ :~iiiiiiiii: :iiiiiiiiill ~i ii............~i...........i~ ii ~7
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!i iii
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The procedures developed in the previous sections are in principle of course available for both linear and non-linear problems (with explicit procedures of time stepping being particularly efficient for the latter). Quite generally the convective part of the equation, i.e.
O~ Ot
OFi Odp Oxi = Ot t-Ui-~x~ - - 0
(2.128)
will have the vector Ui dependent on 4). Thus
Ui -
OFi
= Ui(~)
(2.129)
In the one-dimensional case with a scalar variable we shall have equations of the type
OO OF OO oo Ot t Ox --- Ot 4- U ( O) -~x -- O
(2.130)
corresponding to waves moving with a non-uniform velocity U. A typical problem in this category is that due to Burger, which is defined by
Ot
+ ox-~(~q~2) _ ~ - + 4)--~-- -- 0
(2.131)
Non-linear waves and shocks 67 Creation of a shock
r
x
Fig. 2.19 Progression of a wave with velocity U = ~.
In Fig. 2.19 we illustrate qualitatively how different parts of the wave moving with velocities proportional to their amplitude cause it to steepen and finally develop into a shock form. This behaviour is typical of many non-linear systems and in Chapter 7 we shall see how shocks develop in compressible flow at transonic and supersonic speeds. To illustrate the necessity for the development of the shock, consider the propagation of a wave with an originally smooth profile illustrated in Fig. 2.20(a). Here as we know the characteristics along which 05 is constant are straight lines shown in Fig. 2.20(b). These show different propagation speeds intersecting at time t = 2 when a discontinuous shock appears. This shock propagates at a finite speed (which here is the average of the two extreme values). In such a shock the differential equation is no longer valid but the conservation integral is. We can thus write for a small length As around the discontinuity d?ds + F(s + A s ) - F(s) = 0
-~
(2.132a)
S
or
CAq~ + A F = 0
(2.132b)
where C = lim A s / A t is the speed of shock propagation and A~b and A F are the discontinuities in ~b and F respectively. Equation (2.132b) is known as the RankineHugoniot condition. We shall find that such shocks develop frequently in the context of compressible gas flow and shallow water flow (Chapters 7 and 10) and can often exist even in the presence of diffusive terms in the equation. Indeed, such shocks are not specific to transients but can persist in the steady state. Clearly, approximation of the finite element kind in which we have postulated in general a Co continuity to ~ can at best smear such a discontinuity over an element length, and generally oscillations near such a discontinuity arise even when the best algorithms of the preceding sections are used. Figure 2.21 illustrates the difficulties of modelling such steep waves occurring even in linear problems in which the physical dissipation contained in the equations is incapable of smoothing the solution out reasonably, and to overcome this problem artificial diffusivity is frequently used. This artificial diffusivity must have the following characteristics: 1. It must vanish as the element size tends to zero. 2. It must not affect substantially the smooth domain of the solution. A typical diffusivity often used is a finite element version of that introduced by Lapidus 78 for finite differences, but many other forms of local smoothing have been
68 Convectiondominated problems
t=O i | ! | I
A B C D (a) Profile at time t = 0
E
F
G
x
t i
A B C (b) Characteristics
D
E
Shock
F
v
G
x
t=l
(c) Profile at time t = 1
x
t=2
(d) Profile at time t = 2
v
x
Fig. 2.20 Developmentof a shock (Burger equation).
proposed.79, 8o The additional Lapidus type diffusivity is of the form
_. CLaph2 ]~xl Oq~
(2.133)
where the last term gives the maximum gradient. In the above equation CLapis a coefficient and h is the element size. In Fig. 2.22 we show a problem of discontinuous propagation in the Burger equation (2.131) and how a progressive increase of the CLap coefficient kills spurious oscillation, but at the expense of rounding of a steep wave.
Non-linear waves and shocks
,.3 t -~ 0.8 o
"1o
>
"~ 0.3
-0.3
(a)
I.
I
I
I
I
,1
I
I
0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 Normalized length
1.3 [] 0=1/2
,.. t~ -~ 0.8 o
OO=0
"1o o
"B 0 . 3 -
.
-0.3-
L !
I
I
I
I
I
I
I
0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70
Normalized length
(b)
Fig. 2.21 Propagation of a steep wave by Taylor-Galerkin process: (a) explicit methods C = 0.5, step wave at Pe = 12 500; (b) explicit methods C= 0.1, step wave at Pe = 12 500.
For a multidimensional problem, a degree of anisotropy can be introduced and a possible expression generalizing Eq. (2.133) is
[s =
CLap
where
v/=
h 2 IV/VjJ
Ivl
(2.134)
oe OXi
Other possibilities are open here and numerous papers have been devoted to the subject of 'shock capture'. We will return to this problem in Chapter 7 where its importance in the high speed flow of gases is paramount.
69
70
Convection dominated problems
A
~
.
_
A
V
v
c=o.~, c~=o
c=o.~,c~=~ Fig. 2.22 Propagationof a steep front in Burger's equation with solution obtained using different values of Lapidus C v = CLap.
Part II1" Boundary conditions
ii~iiii!ii~i!i}i}i~i~i~~~~~i i~i~}iii~!i~ii}!ii~i~i~i}i~~ii~iii}!ii~!~ii~ii!}!ii~ii!ii~i!iii~iiii~}i~iiii~ii~ii~ii~~i!i!i!ii!~ii~ii~ii~ii~ii~iii~i~c~ii~i~!i!i~!i!i~i~i}iii~iiii!i!~i!iiiiiii!ii!~~~i~i!iiii!i~i}i~i}iiii~!iiii!i!ii!iiii!iii!ii{iiiii In pure convection problems the equation reduces to
or ~_ui ~x/4Or -~-Q -- 0
(2.135)
It is clear that only the 'inlet' values of q~can be set. By inlet we mean those where
Ui ni
<0
(2.136)
for ni an outward pointing normal to the boundary. This is particularly obvious when we consider the one-dimensional form of Eq. (2.135) in steady state
dr
U~
+ a = 0
(2.137)
This ordinary differential equation can only have conditions imposed on 4~ at one end of the domain. Let us consider the example where U d~ dx
+O-o;
r
Treatment of pure convection 71
with a constant u and Q. The Galerkin solution to Eq. (2.137) is given by
Na
~X § Q dx = 0
(2.138)
As shown in Sec. 2.2.1 where this type of problem is solved as the limit case Pe = oo and for one-dimensional problems in which boundary conditions are imposed at both ends a purely oscillatory solution occurs. However, when properly solved with only one boundary condition imposed, i.e. 05=1
for
0<x
the correct result for the example shown in Fig. 2.2, is obtained. The same result will be obtained no matter what order finite element approximation is used. The reader should note the solution 'develops' from the inlet in an element-wise manner when Eq. (2.136) is considered. The element by element process eliminates the need to solve a large set of simultaneous equations (which is not a serious matter in one-dimension problems). For two- and three-dimensional problems this saving can be quite substantial if proper boundary conditions are imposed and the process was described by Lesaint and Raviart. 81 Let us consider a one-dimensional domain as shown in Fig. 2.23(a) with linear elements. Since the flow direction is from the left to fight, the solution at node a is influenced only by the node or element upstream. Thus the need for assembly is eliminated and the resulting discrete Galerkin finite element form of Eq. (2.138) at node a is ( ua-ua-1 ) Ql=o (2.139) u 2 +T This expression is identical to the upwind finite difference approximation discussed previously. In two-dimensional problems [Fig. 2.23(b)] a similar rule applies and the solution at node a is only affected by the solutions of nodes b and c as shown in Fig. 2.23(b). The calculation can proceed using small groups of elements as illustrated in Fig. 2.23(b). The problem of pure convection is a common one in neutron transport where reaction effects are included. 8~' 82 We note that for.compressible flow without viscosity we shall later use the so-called Euler equations (viz. Chapter 7). Once again the values
.I / •
-
a-2
I"---/--"1) a-1
a
-
a
a+ 1
(a) One-dimensional solution sequence.
/Z, /
Z / Z c (b) Two-dimensional solution sequence.
Fig. 2.23 Solution of pure convection in element by element manner: source term Q and constant u.
72
Convectiondominated problems of the unknown functions will only be specified at the inlet boundaries if supersonic conditions exist.
If diffusion occurs, boundary conditions must be imposed on all boundaries. Thus, in the one-dimensional case we return to the situation discussed at the beginning of this chapter. We see immediately that the standard Galerkin process breaks down near the outlet boundary when we try to specify the value of the unknown ~b in a 'strong' (Dirichlet) manner when convection effects are important (i.e. Pe > 1). In such a case we shall try to impose the Dirichlet boundary condition in a 'weak' manner to balance the solution between the interior and the boundary treatments. Such a weak imposition of the Dirichlet condition was presented by Nitsche in 197183 and is used frequently when discontinuous Galerkin methods are discussed. We again start with the Galerkin equation in which the diffusive term is integrated by parts yielding
- jlNani
k-~ixi
dV -- frr Nan i
k-~ixi
dI" -k- frq NaOn d["
(2.140)
where the natural boundary condition has been imposed on 1-'q. When the approximate solution is used in Eq. (2.140) a contribution to the coefficient matrix arises and is given by
Kab --- -
f
r
N a n ~ kONb ~ dF
(2.141)
This term is clearly unsymmetric and to restore symmetry Nitsche adds two terms
--j;~ i nik(~-~)dI~-~-l~,fF
ga (q~-~)dI"=0
where t~is a parameter. In the above equation the desired value of the Dirichlet condition appears. The second term is a least squares type term used to maintain stability of the solution. This gives the final form for the boundary integrals as
--
Nanik-~xi -b
We observe as t~ ~ that choosing
. nik (c~- ~b) dr' -k-t~
6
Na (q~- ~)dl-'
(2.142)
cxz we recover the penalty method. However, Nitsche shows = O(k/h)
where h is an element size of the mesh, an effective solution results without strong imposition of the boundary condition. In Fig. 2.24 we show results for the problem originally given in Fig. 2.2 for a large Peclet number. The overall good results are obtained without any modification of the standard Galerkin process in the interior of
the domain.
Summary and concludingremarks 73 Finite element: linear elements = ,
1.2
1.2
Finite element: quadratic elements , ,
,
,
,
,
,
,
1 -,-*--,---,~'.:":= : " ' . : = - ' ~
"-,,'.'
,='~,-
,
,
,
,
,
~
,
o.a -I ~ u = :;o I . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.8
.......................................
0.2
....................................... ,
,
.
.............................. .
.
.
~-0.6
~-0.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.4
I ~ u=2ol
0.4 0.2 , ~
o0
Fig. 2 . 2 4
2
4 6 8 x - Coordinate (a) One-dimensional linear element solution.
10
O0
Solution on one dimensional convection-diffusion
2
4 6 8 10 x - Coordinate (b) One-dimensional quadratic element solution.
problem.
From results shown in Fig. 2.24 using linear and quadratic elements we see immediately that the results obtained by weakening the Dirichlet condition at the outlet are quite acceptable. If we contrast these results with those obtained by using the procedures of Part I we observe that exact values are no longer obtained at nodes but the solution is accurate in the 'mean'. The oscillations are almost non-existent and yet the standard Galerkin procedure is used throughout the whole domain with the special treatment confined to the outlet boundary.
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The reader may well be confused by the variety of apparently unrelated approaches given in this chapter. This may be excused by the fact that optimality guaranteed by the finite element approaches in elliptic, self-adjoint problems does not automatically transfer to hyperbolic non-self-adjoint ones. The major part of this chapter is concerned with a scalar variable in the convectiondiffusion reaction equation. The several procedures presented for steady-state and transient equations yield almost identical results. However, the characteristic-Galerkin method is the most logical one for transient problems and gives identical stabilizing terms to that derived by the use of Petrov-Galerkin, GLS and other procedures when the time step used is near the stability limit. For such a problem the optimality is assured simply by splitting the problem into the self-adjoint part where the direct Galerkin approximation is optimal and an advective motion where the unknown variable remains fixed in the characteristic space. Extension of the various procedures presented to vector variables has been made in the past and here we relegate such procedures to Appendix E as they present special problems. For this reason we recommend that when dealing with equations such as those arising in the motion of a fluid an operator split is made in a manner separating several scalar convection-diffusion problems for which the treatment described is used. We shall do so in the next chapter when we introduce the CBS algorithm using the
characteristic-based split.
74
Convection dominated problems
iiiJjiiiiijJjii
ii iJ i iYiZji
i ijiiiii i i i i iifiiiiiiii Ji i i i i i i i i i i i i iiiii!i i i i iiii i i i i i il
~iiiililii:iililililili:ili' '~i!iiiiiiiiil '~~ ililifilliiill!ii!iiiii!i!~!i!ii i!iiiiiiiii!il',i~ii~i~,'~iili!iiiiiiii'~ili'~~,,!i!'i,!i i!~,~:~,!i!iii~ !i!i~iii!i!'~,i,ilil i ~,!gli'~i i~!'~,iii'~,ii'~i!iiiii' i'~i'~,~,i~,iiii' '~i!i ,ii!!!i!!~ i!!!ili''~, !~,i~~,i~{iiiii' :'~'~ !'~i,'~i ,~,~~'i~i,}~i!ii i !ii',ii','i!,ii'i ~ii}iii ! !ililiiii'iiiii'~,~i!iilli',!',ii!i!i!i ~,'~il,ii',~iiiiii','~i:~i''~:~ii'~i!i!i'~iiii!i' i ~iii'iiiiiiiii' ,if ~i''~!iiiii' i'~i iii~:~!~!~:!~i~ii~i~!~ii~iii!~!~!i~i~i~iiiiiii~i~i~ii , ii',iiiiii'~',','~i'~i',i','~,'~i'~ii~,~,iiiii!i' i!i'', ,',iiiiii~ ~',',~~,,',~,',~~,~,',':',':,,ii':~:~i~ii' '~ii ~'~i!iiiiii',~,i:,'ii'i,i',:ii~,~,' i',i ,~,i:,',i:,i':i',;i',',ii',~,~,i~:iiii'.i
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References 58. R.E. Ewing and T.E Russell. Multistep Galerkin methods along characteristics for convectiondiffusion problems. In R. Vichnevetsky and R.S. Stepleman, editors, Advances in Computation Methods for PDEs, volume IV, pages 28-36. IMACS, Rutgers University, Brunswick, NJ, 1981. 59. J. Douglas Jr and T.E Russell. Numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures. SlAM J. Num. Anal., 19:871-885, 1982. 60. O. Pironneau. On the transport diffusion algorithm and its application to the Navier-Stokes equation. Num. Math., 38:309-332, 1982. 61. M. Bercovier, O. Pironneau, Y. Harbani and E. Levine. Characteristics and finite element methods applied to equations of fluids. In J. Whiteman, editor, The Mathematics of Finite Elements and Applications, volume V, pages 471-478. Academic Press, London, 1982. 62. J. Goussebaile, E Hecht, C. Labadie and L. Reinhart. Finite element solution of the shallow water equations by a quasi-direct decomposition procedure. International Journal for Numerical Methods in Fluids, 4:1117-1136, 1984. 63. M. Bercovier, O. Pironneau and V. Sastri. Finite elements and characteristics for some parabolichyperbolic problems. Appl. Math. Modelling, 7:89-96, 1983. 64. J.P. Benque, J.P. Gregoire, A. Hauguel and M. Maxant. Application des m6thodes du d6composition aux calculs num6riques en hydraulique industrielle. In INRIA, 6th Coll. Inst. Mdthodes de Calcul Sci. et Techn., Versailles, 12-16 Dec. 1983. 65. A. Bermudez, J. Durany, M. Posse and C. Vazquez. An upwind method for solving transportdiffusion-reaction systems. International Journal for Numerical Methods in Engineering, 28:2021-2040, 1984. 66. P.X. Lin, K.W. Morton and E. Suli. Characteristic Galerkin schemes for scalar conservation laws in two and three space dimensions. SIAM J. Num. Anal., 34:779-796, 1997. 67. O. Pironneau, J. Liou and T.T.I. Tezduyar. Characteristic Galerkin and Galerkin least squares space-time formulations for the advection-diffusion equation with time dependent domain. Computer Methods in Applied Mechanics and Engineering, 100:117-141, 1992. 68. O.C. Zienkiewicz, R. Ltihner, K. Morgan and S. Nakazawa. Finite elements in fluid mechanics - a decade of progress. In R.H. Gallagher et al., editors, Finite Elements in Fluids, volume 5, pages 1-26. John Wiley & Sons, Chichester, 1984. 69. R. L/Shner, K. Morgan and O.C. Zienkiewicz. The solution of non-linear hyperbolic equation systems by the finite element method. International Journal for Numerical Methods in Fluids, 4:1043-1063, 1984. 70. O.C. Zienkiewicz, R. L/3hner, K. Morgan and J. Peraire. High speed compressible flow and other advection dominated problems of fluid mechanics. In R.H. Gallagher et al., editors, Finite Elements in Fluids, volume 6, pages 41-88. John Wiley & Sons, Chichester, 1986. 71. R. L/3hner, K. Morgan and O.C. Zienkiewicz. An adaptive finite element procedure for compressible high speed flows. Computer Methods in Applied Mechanics and Engineering, 51:441-465, 1985. 72. O.C. Zienkiewicz, R. L/3hner and K. Morgan. High speed inviscid compressive flow by the finite element method. In J. Whiteman, editor, The Mathematics of Finite Elements and Applications, volume VI, pages 1-25. Academic Press, London, 1985. 73. O.C. Zienkiewicz and R. Codina. A general algorithm for compressible and incompressible flow - Part I. The split, characteristic based scheme. International Journal for Numerical Methods in Fluids, 20:869-885, 1996. 74. O.C. Zienkiewicz, P. Nithiarasu, R. Codina, M. Vazquez and P. Ortiz. The characteristic-basedsplit (CBS) procedure: An efficient and accurate algorithm for fluid problems. International Journal for Numerical Methods in Fluids, 31:359-392, 1999. 75. P.D. Lax and B. Wendroff. Systems of conservative laws. Comm. Pure Appl. Math., 13:217-237, 1960.
77
78 Convectiondominated problems 76. J. Peraire. A finite method for convection dominated flows. Ph.D. thesis, University of Wales, Swansea, 1986. 77. J. Donea. A Taylor-Galerkin method for convective transport problems. International Journal for Numerical Methods in Engineering, 20:101-119, 1984. 78. A. Lapidus. A detached shock calculation by second order finite differences. J. Comp. Phys., 2:154-177, 1967. 79. J.P. Boris and D.L. Brook. Flux corrected transport. I: Shasta- a fluid transport algorithm that works. J. Comp. Phys., 11:38-69, 1973. 80. S.T. Zalesiak. Fully multidimensional flux corrected transport algorithm for fluids. J. Comp. Phys., 31:335-362, 1979. 81. P. Lesaint and P.-A. Raviart. On a finite element method for solving the neutron transport equation. In C. de Boor, editor, MathematicaI Aspects of Finite Elements in Partial Differential Equations. Academic Press, New York, 1974. 82. W.H. Reed and T.R. Hill. Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, 1973. 83. J.A. Nitsche. Uber ein Variationsprinzip zur L/Ssung Dirichlet-Problemen bei Verwendung von Teilr~iumen, die keinen Randbedingungen uneworfen sind. Abh. Math. Sem. Univ. Hamburg, 36:9-15, 1971.
ii!iiii iii!iiiiiii !iiiiiiiliLiiGi!Ei!iili!iiiii
The characteristic-based split (CBS) algorithm. A general procedure for compressible and incompressible flow !!!!i! i!!ii!
!!i!!iii!ii!!ii!iiii!! i !!i!!i!!i!!ii!!iiii!i!!!!i!!!!ii!i!!!ii!i!i!i!i!!!!!i !!i!!! !!!ii!!i!!i!!!!!i i!!!i!!! !!i!i!!i!!!!i!i!!! !!!ii!!i
In the first chapter we have written the fluid mechanics equations in a very general format applicable to both incompressible and compressible flows. The equations included that of energy which for compressible situations is fully coupled with equations for conservation of mass and momentum. However, of course, the equations, with small modifications, are applicable for specialized treatment such as that of incompressible flow where the energy coupling disappears and to the problems of shallow water equations where the variables describe a somewhat different flow regime. Chapters 4 to 6 and 10 deal with such specialized forms. The equations have been written in Chapter 1 in fully conservative, standard form [Eqs (1.25) and (1.26a)-(1.26d)] but all the essential features can be captured by writing the three sets of equations as below. Mass conservation Op -~
10p = C2 0 t
OU~ --
OXi
(3.1)
where c is the speed of sound and depends on E, p and p and, assuming constant entropy,
C2.~ Op = ~/p Op
p
(3.2)
where 7 is the ratio of specific heats equal to cp/co, where Cp is the specific heat at constant pressure and co is the specific heat at constant volume, and we define the mass flow flux as Ui = pui
(3.3)
80
CBS algorithm" compressibleand incompressible flow For a fluid with a small compressibility K c2 = - P
(3.4)
in which K is the elastic bulk modulus. Depending on the application we use an appropriate relation for c 2.
Momentum conservation
oui
o
~(ujUi)j Ot -- --OX
orij
Op
+ OXj
OXi
d- Pgi
(3.5)
Energy conservation O(pE) Ot
0 -~(uipE) OXi
o (or)
+ ~
\ ~
o
o
-- ~xi (Uip) + -~Xi (7"ijUj) "31-pgiui + ql-I (3.6)
In all of the above Ui are the velocity components, p is the density, E is the specific energy, p is the pressure, T is the absolute temperature, Pgi represents body forces and other source terms, k is the thermal conductivity, and Tij are the deviatoric stress components given by [Eq. (1.1 lb)] 7"ij = ~
IOn i "q Ouj ~ OXi
2 ~ij Ouk) 3
~
(3.7)
where (~ij is the Kronecker delta - 1, if i - j, and - 0 if i # j. In general, #, the dynamic viscosity, in the above equation is a function of temperature, #(T), and appropriate physical relations will be used if necessary. The equations are completed by the universal gas law when the flow is coupled and compressible:
p -- pRT
(3.8)
where R is the universal gas constant. The reader will observe that the major difference in the momentum-conservation equations (3.5) and the corresponding ones describing the behaviour of solids (see reference 1) is the presence of a convective acceleration term. This does not lend itself to the optimal Galerkin approximation as the equations are now non-self-adjoint in nature. However, it will be observed that if a certain operator split is made, the characteristic-Galerkin procedure can be applied to the part of the system which is not self-adjoint but has an identical form to the convectiondiffusion equation. We have shown in the previous chapter that the characteristicGalerkin procedure is excellent for dealing with such equations. However, we have also shown that many alternatives will lead to precisely the same convection stabilizing terms. The choice of the name CBS is thus somewhat subjective indicating simply that the characteristic process is, for us, the most basic justification. It is important to state again here that the equations given above are of the conservation forms. As it is possible for non-conservative equations (Appendix A) to yield multiple and/or inaccurate solutions, this fact is very important.
Non-dimensional form of the governing equations 81 We believe that the algorithm introduced in this chapter is currently the most general one available for fluids, as it can be directly applied to almost all physical situations. We shall show, in this book, such applications ranging from low Mach number viscous or indeed inviscid flow to the solution of hypersonic flows. In all applications the algorithm proves to be at least as good as other procedures developed. Further, in problems of very slow viscous flow we find that the treatment can be almost identical to that of incompressible elastic solids and here we shall often find it expedient to use higher-order approximations satisfying the incompressibility conditions (the so-called Babu~kaBrezzi restriction). 2' 3 Indeed on certain occasions the direct use of incompressibility stabilizing processes described in Chapter 11 of reference 1 may be useful.
The governing equations described above, Eqs (3.1)-(3.8), are often written in nondimensional form. The scales used to non-dimensionalize these equations vary depending on the nature of the flow. We describe below the scales frequently used in flow computations, though of course many alternatives are possible _
_
xi
p
L '
L'
Poo
Ui
E
tucr
ui = ~ ; lgcx~
E'---2
lg oo
-
Poou2O0 ' ~2 .._ c2
Tcp.
T-
;
p
u2oo '
(3.9)
U2oo
where an over-bar indicates a non-dimensional quantity, subscript oo represents a free stream quantity and L is a reference length. Applying the above scales to the governing equations and rearranging we have the following forms: Conservation of mass 019
1 0[9
"-~ = e 2 0 ~
O(]i =
(3.10)
O~fi
Conservation of momentum OOi C~
=
0 OXj
.
10(llTl-ij) .
Re
.
.
O.~j
.
013 ~- [ggi
(3.11)
0~r i
where uooL Re = ~ ; Vcr
/r
gi L u2 , cr
P--~
u /-/oo
(3.12)
are the Reynolds number, non-dimensional body forces and viscosity ratio, respectively. In the above equation uis the kinematic viscosity equal to # / p with # being the dynamic viscosity.
82
CBS algorithm" compressibleand incompressibleflow Conservation of energy O(pE) Oi
1
0 (~iPF,) -~ -OYc---~i gePr
1 0
O ( Ok T )
Off i
-
~x i
0 (~jP)
~x i
(3.13)
-~- --~e -~x i ( ~' Ti j bl j )
where Pr is the Prandtl number and k* is the conductivity ratio given by the relations Pr = #c~CP" k~ '
(3.14)
k = ~k k~
where k~ is a reference thermal conductivity. Equation of state
p =
i)R]" Cp
= ~kl' - ~
(-),- 1) '3/
T
(3.15)
In the above equation R - Cp -- Co and/~ = (Cp - cv)/Cp are used. The following forms of non-dimensional equations are useful to relate the speed of sound, temperature, pressure, energy, etc. / ~ _ ~' + 1 - 7 2u'u' 72 _ ( 7 - 1)I'
(3.16)
The above non-dimensional equations are convenient when coding the CBS algorithm. However, the dimensional form will be retained in this and other chapters for clarity.
iiilili!!!iiiiiiiiii::i,:ii!i iiiiii!!iiii;iiiiiiiiiiiii iii!i!ii~iiiiiii;iiii!iiiililiiii!ililill ii::iiiiiiiiiiiiiii::!ii!; iiiili!!i!ili!iill~i!i~i~i~i~i!iii~i~::~ii!i~iiii~i~i~iii~i::i~!ii~!i~ii~i!~!~!~ii iiii',!iii!i!i!;i'i!iiii' ,!ii:i::i,:i:/,i!;i~i~iil ii!iiiiiiiiiiii iii',iiil;;ii::i::i'~ii~?,L!;i',iiiiiiiiil !~;i ~i i i~i~!~ii~!~!~ii i~i~i~i ~i~i!!~:~!!!!~!~!~i~i!i~i~i~i~i~i !i!: ~i~!~ii!~, i!~!~!i ii!!ii!i:. i !~!~i ~i:i:~~!~i i !i!!~!~i :~/~!~i~i~i~:i~!~i:~i~i:~i i~i~i~i~i i~i~....................................................................... i i i~i~!ii~!;!!i~i~i:i !i ~ :,:,~:,:,:,:, ~, ~,:,:, :,:,:~:,:~:,:,~,~,:~:,~,~
3.3.1 The split - general remarks The split follows the process initially introduced by Chorin 4' 5 for incompressible flow problems in a finite difference context. A similar extension of the split to a finite element formulation for different applications of incompressible flows has been carfled out by many authors. 6-3~ However, in this chapter we extend the split to solve the fluid dynamics equations of both compressible and incompressible forms using the characteristic-Galerkin procedure. 31-56 The algorithm in its full form was first introduced in 1995 by Zienkiewicz and Codina 31, 32 and followed several years of preliminary research. 57-61
Characteristic-based split (CBS) algorithm Although the original Chorin split 4' 5 could never be used in a fully explicit algorithm, the new form is applicable for fully compressible flows in both explicit and semi-implicit forms. The split provides a fully explicit algorithm even in the incompressible case for steady-state problems now using an 'artificial' compressibility which does not affect the steady-state solution. Recently this has been extended to the nonsteady case. 53 When real compressibility exists, such as in gas flows, the computational advantages of the explicit form compare well with other currently used schemes and the additional cost due to splitting the operator is insignificant. Generally for an identical cost, results are considerably improved throughout a large range of aerodynamical problems. However, a further advantage is that both subsonic and supersonic problems can be solved by the same algorithm.
3.3.2 The split- temporal discretization At this stage we will only consider the solution of Eqs (3.1) and (3.5) with variables Ui and p. The extension to include energy and any other variables will be treated after these are available. We can discretize Eq. (3.5) in time using the characteristic-Galerkin process. Except for the pressure term this equation is similar to the convection-diffusion equation (2.12). This term can, however, be treated as a known (source type) quantity providing we have an independent way of evaluating the pressure. Before proceeding with the algorithm, we repeat Eq. (3.5) below for use with the characteristic-Galerkin method
OUi Ot
0
= -~(ujUi) Oxj
-~
07"ij
Op
Oxj
Oxi
(3.17)
{- Pgi
At this stage we have to introduce the 'split' in which we substitute a suitable approximation. In all procedures the values of the solution (U n+l , pn+l) at time t n+l must be determined from the known values ( U n , pn) at time t n . Two alternative approximations are useful and we shall describe these as Split A and Split B, respectively. In each we assume during the time increment At -- t n+l -- t n Un+' -- Un + AU/* + AU/**
(3.18)
We also discretize in time using the approximation for the time interval t n <_ t <_ t ~+1
OU i 0t
=
U? + 1 - Un At
=
AU; At
AU 7 + ~ At
(3.19)
Using Eq. (2.105) of the previous chapter and replacing Uj by uj and q5 by Ui, we can write
0 07ffij U7 +1 - U? -- At [--~xj(.jui)n -~- ~xj Jr- (Pgi)n3 -- At At2 n 0 0 -lI-Tblk--~Xk[-~xj(bljSi)
07ffij OXj
Op n+02
pgi] n _~-
OX-----~ A t 2 blk 0
T
~
( Op n+02 ) OXi
(3.20)
83
84
CBS algorithm" compressibleand incompressible flow In the above equation
Opn+O2 = (1 -- 02) Opn Opn+l OXi ~Xi + 02 OX'----T
(3.21)
or
Opn+O2 Oxi
Opn
--
OAp
-+- 0 2 ~
Oxi
(3.22)
Oxi
where
Ap -- pn+l _ pn
(3.23)
The reader should note that the velocity and deviatoric stress terms are evaluated at
t n, whereas the pressure is evaluated at t n+02. This is certainly permitted as the method
may be shown to be fully consistent. Using the auxiliary variables A U/* and A U/** we split Eq. (3.20) into two parts. In the first form we remove all the pressure gradient terms from Eq. (3.20); in the second we retain in that equation the pressure gradient corresponding to the beginning of the step, i.e. Op n/OX i . Though it appears that the second split might be more accurate, there are other reasons for the success of the first split which we shall refer to later. Indeed Split A is the one which we shall universally recommend for steady state problems. However, in transient problems Split B appears to give a slightly better answer. 62
Split A In this we introduce an auxiliary variable A U* such that (removing third-order terms)
0 AU 7 -- At [--~xj(UjUi) + ~07"ij + Pgi + -2-Uk~xkAt O (~.~j (ujUi) - Pgi )In (3.24) We note that this equation is solved by an explicit time step applied to the discretized form and a complete solution is now possible. The 'correction' given below is available once the pressure increment is evaluated:
Opn+O2 AU~* -- - A t
OXi
At2 n 02 pn -~- T U k OXkOXi
(3.25)
From Eq. (3.1) we have Ap --
OU +~
Ap -- - A t "i ~ OXi
[ OUn
OA Ui J
=-At ---~x/-+-01 OXi
(3.26)
Replacing A Ui by the known intermediate, auxiliary variable A U7, using Eqs (3.19) and (3.25) and rearranging after neglecting third- and higher-order terms we obtain
Ap --
(1) n -~
[0un
Ap = - A t [ oxi -{- 0 1OA ~OxiUi
AtO1 ~ -~ 02 OxiOxi OxiOxi (3.27)
Characteristic-based split (CBS) algorithm The above equation is fully self-adjoint in the variable Ap (or Ap) which is the unknown. Now, therefore, a standard Galerkin-type procedure can be optimally used for spatial approximation of Eq. (3.27). It is clear that the governing equations can be solved after spatial discretization in the following order: 1. Eq. (3.24) to obtain All[; 2. Eq. (3.27) to obtain Ap or Ap; 3. Eq. (3.25) to obtain AU/** thus establishing the values of Ui and p at t n+l After completing the calculation to establish A Ui and Ap (or Ap) the energy equation is dealt with independently and the value of (pE) n+l is obtained by the characteristicGalerkin process applied to Eq. (3.6). It is important to remark that this sequence allows us to solve the goveming equations (3.1), (3.5) and (3.6), in an efficient manner and with adequate convection stabilization. Note that these equations are written in conservation form. Therefore, this algorithm is well suited for dealing with supersonic and hypersonic problems, in which the conservation form ensures that shocks will be placed at the fight position and a unique solution achieved. However, we must remark that near the actual shocks, additional numerical damping will always be needed.
Split B In this split we also introduce an auxiliary variable A U/* now retaining the known values of Op ~/Oxi, i.e. zx u.; = A t
-
0
j ( Uj Ui ) +
@ OXj
"~- -~- link ~
OX i
+ pgi
( U j g i ) "~- -~X i
pgi)] n
(3.28)
It would appear that now U/* is a better approximation than that from Split A in Eq. (3.24). We can now write the correction as
OAp A UT* = -02 At Ox----(-
(3.29)
i.e. the correction to be applied is smaller than that assuming Split A, Eq. (3.25). Further, if we use the fully explicit form with 02 = 0, no mass velocity (A U/**) correction is necessary. We proceed to calculate the pressure changes as in Split A
Ap =
1 ()n --fi
Ap = - A t
[OU['
--~x i --Jl-O l
"
OAU i __ AtO102 Ox i
"'1 Ox 2
(3.30)
The solution stages follow the same steps as in Split A. Later we will see that Split B does not possess the self-pressure stabilizing properties of Split A when incompressibility (or near incompressibility) is encountered (Sec. 3.6).
85
86
CBS algorithm: compressibleand incompressible flow
3.3.3 Spatial discretization and solution procedure In all of the equations given below the standard Galerkin procedure is used for spatial discretization as this was fully justified for the characteristic-Galerkin method in Chapter 2. We now approximate spatially using standard finite element shape functions as
Ui : NuUi ui -- Nufli
AUi -- NuAUi
AU/* = NuAf.I~' p - Np~
p = Np~
A U 7 - NuAfJ7
(3.31)
In the above equation
Ui -- [Ui1 N-
[N 1 N 2
...
va
...
...
Na
...
(3.32)
N m]
where a is the node (or variable) identifying number (and varies between 1 and m). Split A We have the following weak form of Eq. (3.24) for the standard Galerkin approximation (weighting functions are the shape functions)
ua A uT d e = - A t
[L ~
ua - xj (Uj U i ) dr2 +
~
7"ij dr2 -
mt2IfO ( 0 ) i n +'--f-~xk(Uk Na) -'~xj(Ujgi) + Pgi +At
NurijnjdF
N a (Pgi) dr2
In
df2
In
(3.33)
Here, the viscous and stabilizing terms are integrated by parts and the last term is the boundary integral arising from integrating by parts the viscous contribution. Since the residual on the boundaries can be neglected, other boundary contributions from the stabilizing terms are negligible. Note from Eq. (2.105) that the whole residual appears in the stabilizing term. However, we have omitted diffusion terms in the above equation since they are of higher order. As mentioned in Chapter 1, it is convenient to use matrix notation when the finite element formulation is carried out. We start here from Eq. (1.6) of Chapter 1 and we repeat the deviatoric stress and strain rate relations below
7ij -- 2# (~ij -- 5l (~ij~kk)
(3.34)
where the quantity in brackets is the deviatoric strain rate. In the above
9 l(OqUi OUj~
~ij -- -~ ~ and
-'}- ~X i /
~ui
~ii-- ~X i
(3.35)
(3.36)
Characteristic-based split (CBS) algorithm We now define the strain rate in three dimensions by a six-component vector as given below* "-" [~11 ~22 ~33 2~12 2~23 2~31] T (3.37) With a matrix m defined as m=
[1
1
1
0
T
00]
(3.38)
we find that the volumetric strain rate is ~v
--
~ii
(3.39)
~11 "4- ~22 + ~33 -- m T e
---
The deviatoric strain rate can now be written simply as [see Eq. (3.33)] ea
= e--
i1m e ,
9
=
(I
--
3mmT) e = I d &
(3.40)
where ld = ( I and thus
ld =
1
2 -1 -1 0 0 0
-1 2 -1 0 0 0
(3.41)
3mmT)
-1 -1 2 0 0 0
0 0 0 3 0 0
0 0 0 0 3 0
0 0 0 0 0 3
(3.42)
If stresses are similarly written in vector form as tr=
[Crll crze tr33 Crl2 cr23 cr31]~
(3.43)
where of course 0.11 is equal to 7"11 -- p while O'12 is identical to 7"12, etc. Immediately we can assume that the deviatoric stresses are proportional to the deviatoric strain rates and write directly from Eq. (3.34) tr d -- IdO" = # I 0 &d -- #
(I0 - 2mmT)/~
(3.44)
where the diagonal matrix I0 is -2 (3.45)
I0 --
To complete the matrix derivation the velocities and strain rates have to be appropriately related and the reader can verify that using the matrix strain definitions we can write = ,flu (3.46) *In two dimensions we use a three-component vector ~ = [~ll
~22
2~12] T, noting that ~33 is zero.
87
88 CBS algorithm: compressibleand incompressible flow where
U--[Ul
U2 U3] T
(3.47)
and ,S is an appropriate strain rate matrix (operator) defined below
0
0 0
0
0 Ox2 o
S
m
O Ox2 C~X1 O 0
Ox3
0
0
0 (3.48)
0
O Ox2 0
Finally the reader will note that the direct link between the strain rates and velocities will involve a matrix B defined simply by (3.49)
B = ,SNu
Now from Eqs (3.31), (3.33) and (3.44), the solution for AU/* in matrix form is: Step 1
A[J* -- - M u 1At [(Cu[J + KT-fi - f ) - At (KufJ + fs)] n
(3.50)
where the quantities with a - indicate nodal values and all the discretization matrices are similar to those defined in Chapter 2 for convection-diffusion equations [Eqs (2.108) and (2.109)] and are given as Cu - fa NuT(v(uNu)) dr2
Mu = fa NTNu dr2 K-r = fa BT# (Io -- 2mmT) B dr2
f -- fa NTpg dr2 +
fr NTtd dl-"
(3.51)
where g is [gl ge g3] T and t d is the traction corresponding to the deviatoric stress components. In Eq. (3.50) Ku and fs come from the terms introduced by the discretization along the characteristics. After integration by parts [i.e. from Eq. (3.33)], the expressions for Ku and fs are Ku = - ~1 f [grT(uNu)]T[VT(UNu)] d ~
(3.52)
and fs =
_1~ f~ [ V T(uNu) ]Tpg d ~
(3.53)
Characteristic-based split (CBS) algorithm where V r = [O/Oxl, ~OX2, ~OX3]. The weak form of the density-pressure equation is
Np Ap df2 --
Np -~ Ap df2
O( Opn+O2) Np ~xi U? ~t_O1mU7 - 01At Oxi dr2
= -At = At
~
--AtO1
S n -~-01 A N ; - At Oxi
f l ( Opnk-02)l Np ~ Jr_O1 AU; -- At Oxi ni dr
(3.54)
In the above, all the RHS terms are integrated by parts. Further we shall discretize p directly only in problems of compressible gas flows and therefore below we retain p as the main variable. Spatial discretization of the above equation gives
Step 2 (Mp + At2OlO2H)Ap = At[GU n + 01GA~J* - AtO1Hp n --fp] which can be solved for A~. The new matrices arising here are H -- fa (VNp)T'~TNp dr2
G = fa(VNp)TNudf2
Mp = fa Np
(1)n -~
(3.55)
NpdQ
fp = At frNTpnT[f.J n + 01(AIJ* -- AtVpn+O2)ldF (3.56)
In the above fp contains boundary conditions as shown above. We shall discuss these forcing terms fully in a later section as this form is vital to the success of the solution process. The weak form of the correction step from Eq. (3.25) is
s
r
=s
i
= -At
_s
f~(Op n Omp) N a ~ '~ 02OxiOxi dr2
(3.57)
At 2 s 0 Op n 2 -~xj(UjNa)-~xidf2 The final stage of the computation of the mass flow vector Un+l is completed by the following matrix form.
89
90
CBS algorithm: compressibleand incompressibleflow
Step 3 At
nq
AU** - AIJ - AU* -- - M u' At G T(~n + 02 mp) --[-- ~ P p ] where
(3.58)
P
P - I (V(uNu))TVNp dr2
(3.59)
At the completion of this stage the values of Qn+l and on+l are fully determined but the computation of the energy (pE) n+l is needed for gas flow problems so that new values of cn+l, the speed of sound, can be determined. Once again the energy equation (3.6) is identical in form to that of the scalar problem of convection-diffusion if we observe that p, Ui, etc. are known. The weak form of the energy equation is written using the characteristic-Galerkin approximation of Eq. (2.105) as
NkEA(pE) n+l dg] -- At
~ NkE-~xi(ui(pE -}- p)) d~2
-
Oxi ( JuJ + +---~+At
]
--~xj(UjNw ~ (-ui(pE q- p)) dS2
[fFNw (
OZ)ni In
TijU j -I- k-~xi
dF
(3.60)
With the additional approximations
pE -- NeE
and
T -- N r T
(3.61)
we have
Step 4 AI~ - --ME 1At [CEE + Cp~ + Kr'i" + K~Efi + fe --
At(KueF, + Kup~ + fes)] n (3.62)
where E contains the nodal values of pE and again the matrices and forcing vectors are similar to those previously obtained and given as
Kr -
(VNE)TkVNr d~2
K~E --
BT#(Io -
Characteristic-based split (CBS) algorithm Kue = - ~1 f (VT(uNe))T(VNe) dr2
fe = f r N+enT(tdu + k V T ) dF
Kup _ --~1 f (X7T(uNe))T (VNp) dr2 -
(3.63)
-
Ja
The forcing term fes contains source terms. If no source terms are available this term is equal to zero. It is of interest to observe that the process of Step 4 can be extended to include in an identical manner the equations describing the transport of quantifies such as turbulence parameters, 39 chemical concentrations, etc., once the first essential Steps 1-3 have been completed.
Split B With Split B, the discretization and solution procedures have to be modified slightly. Leaving the details of the derivation to the reader and using identical discretization processes, the final steps can be summarized as:
Step I AU~* = - M u -1At
[
(CuU + K~fi + GT~ -- f) -- At
(
Ku[l + fs + - ~ P P
)in (3.64)
where all matrices are the same as in Split A except the forcing term f which is -
N upg dr2 +
N u i dF
(3.65)
since the pressure term has now been integrated by parts
Step 2 (Mp + At2OIO2H)A[;- At[G[J n -1-01GA[J**-
fpln
(3.66)
and
Step 3
= -Mu' At [o:G
(3.67)
Step 4, calculation of the energy, is unchanged. The reader can notice the differences in the above equations from those of Split A.
3.3.4 Mass diagonalization (lumping) In Steps 1 to 3 of the split algorithm the solution only requires the inversion (or solution) of mass matrix Mu and Mp. Such steps are called explicit and generally are
91
92 CBS algorithm: compressible and incompressible flow accomplished using approximation by a diagonal (lumped) form. Procedures for such diagonalization are described in standard text. 1 Here we quote one which is generally efficient. In this the matrix form Mu is replaced by M L computed as
MLb -- (~abf~ Na dr2
no sum
which the reader will recognize as the r o w s u m of the full (consistent) mass matrix. Such lumping for steady-state problems make Steps 1 to 3 trivial and the errors involved are of no consequence as terms involving time variation disappear at a converged (steady-state) solution. However, for transient problems, quite serious errors can occur and in such cases an additional iteration is used to obtain a consistent solution. This was already discussed for the scalar equation in Sec. 2.6.3 [viz. Eq. (2.117)].
J~4 ~ p l i C i ~ semi~impliCi| anu !ne'-.ariilyIm. ............plliC|l ~ ~ 5 The split algorithm A or B will contain an explicit portion in the first characteristicGalerkin step. However, the second step, i.e. that of the determination of the pressure increment, can be made either explicit or implicit depending on the choice of 02 and speed of sound. Now different stability criteria will apply (01 will always be taken to satisfy 1/2 < 01 < 1 and thus stability does not depend on it). We refer to schemes as being fully explicit or semi-implicit depending on the choice of the parameter 02 as zero or non-zero, respectively. It is necessary at this point to mention that the fully explicit form is possible only for compressible flows for which c # c~; however, later we will show that this restriction may be removed by introducing an 'artificial compressibility'. It is also possible to solve the first step in a partially implicit manner to avoid severe time step restrictions. Now the viscous term is the one for which an implicit solution is sought. We refer to such schemes as quasi- (nearly) implicit schemes.
3.4.1 Fully explicit form In fully explicit forms, 51 < 01 < 1 and 02 = 0. In general the time step limitations explained for the convection-diffusion equations are applicable, i.e. At < -
c +
lul
(3.68)
if the viscosity effects are negligible. In the above equation h is the element size, assuming linear elements only and one-dimensional behaviour. For two- and threedimensional flow problems determination of the element size is more difficult. We shall discuss some possibilities in Sec. 3.4.4. This particular form is very successful in compressible flow computations and has been widely used by the authors for solving many complex problems (see Chapter 7). More recently the algorithm has also been successfully used in conjunction with artificial compressibility to solve many other flow situations, including those in which incompressibility is involved (see Chapter 4).
Explicit, semi-implicit and nearly implicit forms
3.4.2 Semi-implicit form In semi-implicit form the following values apply 12 --
1
and
(3.69)
_~< 0 2 < 1
Again the CBS algorithm is conditionally stable. The permissible time step is governed by the critical step of the characteristic-Galerkin explicit relation solved in Step 1 of the algorithm. This is the standard convection-diffusion problem discussed in Chapter 2 and the same stability limits apply, i.e. A t <_ Atu -
h
(3.70a)
lul
and/or h2
At < A t ~ - ~
(3.70b)
where u is the kinematic viscosity. A convenient form incorporating both limits can be written as AtuAt~ At < (3.71) - At~ + At~ The reader can verify that the above relation will give appropriate time step limits with and without the domination of viscosity. For slightly compressible or incompressible problems in which Mp is small or zero the semi-implicit form is efficient and it should be noted that the matrix I-I of Eqs (3.55) and (3.66) does not vary during the computation process. Therefore I-I can be factored into its triangular parts once leading to an economical direct procedure. The implicit equation is usually solved either by direct solvers or by iterative (conjugate gradient) methods.
3.4.3 Quasi- (nearly) implicit form To overcome the severe time step restriction made by the diffusion terms (e.g. viscosity or thermal conductivity), these terms can be treated implicitly. This involves solving separately an implicit form connecting the viscous terms with A U/*. Here, at each step, simultaneous equations need to be solved and this procedure can be of great advantage in certain cases such as high-viscosity flows and low Mach number flows. 16' 18,26,43 Now the only time step limitation is At < h/lul which appears to be a very reasonable and physically meaningful restriction.
3.4.4 Evaluation of time step limits. Local and global time steps .......................................................................................................................................................................................................
~ ......
..........
--_._~-
--_-
~_~
-~
..................
-.
.....
-
.._:
. . . . . . . . . . : ............
..:.-_:
..............
_:__::::.
_ _ . _ _ . . .
Though they are defined in terms of element sizes the time step limits are calculated at nodes. In Fig. 3.1 the manner in which the size of the element is easily established at nodes is shown. In such cases, as seen, the element size is not
93
94 CBS algorithm: compressible and incompressible flow
1
2
Fig. 3.1 Element sizes at different nodes of a linear triangle.
unique for each node. In the calculation, we shall specify, if the scheme is conditionally stable, the time step limit at each node by assigning the minimum value for such nodes calculated from all elements connected to that node. When a problem is being solved in true time then obviously the smallest of all nodal values has to be adopted for the solution. However, in many problems it is possible to use local time stepping if steady-state solution is the only interest. In such problems local 'nodal' time step values are conveniently used. Local time stepping can generally be applied to problems in which (1) the mass matrix is lumped and (2) the steady-state solution does not itself depend on the mass matrix. Thus with local time stepping we will use the minimum time step found at each node. This of course is equivalent to assuming identical time steps for the whole problem and simply adjusting the magnitude of the lumped masses. Such a problem with adjusted lumped masses is still physically and mathematically meaningful and we find that convergence will be achieved if a steady state solution exists. If only convective problems are considered, the time step only needs the element size in the direction of the streamline. This may very well be different from minimum h. The procedure has been shown to be efficient by Thomas and Nithiarasu. 63 This is particularly important if the elements are elongated as we shall later mention in the case of supersonic flows with shocks. Many steady-state problems have been successfully solved using such localized time stepping in the calculations. In the context of local and global time stepping it is interesting to note that the stabilizing terms introduced by the characteristic-Galerkin process will not take on the optimal value for any element in which the time step differs from the critical one; that is of course if we use local time stepping we shall automatically achieve this optimal value often throughout all elements at least for steady-state problems. However, on other occasions it may be useful to make sure that (a) in all elements we introduce optimal damping and (b) that the progressive time step for all elements is identical. The latter of course is absolutely necessary if for instance we deal with transient problems where all time steps are real. For such cases it is possible to consider the At as being introduced in two stages: (1) as the Atext resulting from rate terms which has of course to preserve stability and must be left at a minimum At calculated from any element; and (2) to use in the calculation of each individual element a Atint from the characteristic Galerkin terms which is optimal for an element, as of course exceeding the stability limit there does not matter and we are simply adding
Artificial compressibility and dual time stepping 95
Atint
better damping characteristics. Here the is the only one occurring in convection stabilizing terms. This internal-external subdivision is of some importance when incompressibility effects are considered. As shown in the next section, the stabilizing diagonal term occurs in steady-state problems depending on the size of the time step. If the mesh is graded and very small elements dictate the time step over the whole domain we might find that the diagonal term introduced overall is not sufficient to preserve incompressibility. For such problems we recommend the use of internal and external time steps which differ. 5~ In this case the internal time step Atint is the one multiplying the second order pressure terms at Step 2.
3.5.1 Artificial compressibility When the real compressibility is small, it is often assumed that the fluid is incompressible with the speed of sound approaching infinity. Even if the speed of sound is finite, its value may be large and hence a very severe time step limitation arises. However, an artificial compressibility method can be employed to eliminate the restrictions posed by the speed of sound at Step 2 by assuming an artificial value for the speed of sound which is sufficiently low. This of course is only possible if steady-state conditions exist and therefore in the limit the transient term disappears. The Step 2 of the CBS scheme may be rewritten in its semi-discrete form as 53
(1) n ~-~
Ap~
)]
(1) n I C~Un OAU[ AtO1( C~pn (~Ap 7 Ap=-At-~x/-t-01~ OXi OXiOXi "~-02OXiC~Xi
(3.72) where/3 is an artificial parameter with the dimensions of speed. This parameter may be either given as a constant throughout the domain or determined based on the convective or diffusive time step restrictions. We recommend the latter option, as this results in manageable local and global time step sizes. The/3 value may be locally computed using the following relation so that the convective and viscous time steps are represented. /3 = max (e, u conv, U d i f f )
(3.73)
where e is a small constant that ensures/3 is not approaching zero for any circumstances. Uconv and Udiff are respectively the convection and diffusion velocities given as Uconv
--
lUl
=
// Udiff
=
--
h
(3.74)
where h is the element size and u is the kinematic viscosity. The three steps of the CBS scheme follow exactly the procedure discussed in the preceding sections. However, the difference here is that no coupling exists between
96 CBS algorithm" compressibleand incompressibleflow the energy and the other governing equations. The time step limitation for the artificial compressibility method may be written as At -
lul
(3.75)
+/3
The above relation includes the viscous effect via the artificial parameter. The artificial compressibility method described here is valid for steady flows. However, an appropriate dual time stepping method, explained in the following subsection, should be employed to recover true transient solution.
3.5.2 Artificial compressibility in transient problems (dual time stepping)
..................................................................................................................................................................
- .................
:_.:::::.::::.
..................................................................................................
. ..........
: ..............................................................................................
Recovering the true transient for the artificial compressibility method is easy to implement. Within each real time step, an iterative procedure is carded out to go from time n to time n + 1. The dual time stepping of all the three steps of the CBS algorithm may be summarized as 53' 55
Step 1 A~.J7
U* - U m
AT
AT
= - M u -~ [(Cu[J + K~0 + GT~ -- f) (3.76)
as noticed the time step introduced is AT, which is a pseudo time step. The pseudo time iterations m are used to reach a local instantaneous steady state. Steps 2 and 3 of the CBS scheme can be written as
Step 2
~
pm
m
+ A~-0102H zXOm = ICE + 0~GAU"- f~]m _ ~ M ~ ~
At
pn
(3.77)
and
Step 3
m~j m ~.jm _~.]n A r = - M u l [02GT m~m] -At
(3.78)
where AU m -- U m+l - U m and Ap m -- pm+l _ pm. From the above steps it is obvious that for each real time step At, a certain number of pseudo time iterations are needed to get to the local pseudo steady state. After each local steady state, the new value of the field variables at the m + 1 time level are obtained. The above dual time stepping procedure has no stability restriction on the real time step value At used. It should also be noted that the pseudo time step can be evaluated locally by imposing a local stability condition.
'Circumvention' of the Babu~ka-Brezzi (BB) restrictions 97
iiii~ii!i!i!i!iii~!!iii~iiiiiiiii~i~iii%iii~i~i~i~i!i~i~iii~i~i!i~i~ii~i~i~i~i~i~iii~i~ii!i~iiii!~iii~i~i~i!~i~iii~i~i~iiii~i~i~iii~i~ii~i~i~ii~iii~i%i ~iii~iii~iiiiiiiiiiiiiiiiiiiiiiiiiiii~ii~iiiiiiii~iii~iiii
ii!i!!!ii !!i!!!~! i !i!!!i!~~~~ ! !ii!!!iii!i!!!!!ii!i!!ii!!!!ii!!!iii!!!ii!!i ii!!ii!!i!i!!!i!!!i!!ii!!ii!i!!i!!i!i!ii!i!!i!!if!i!!l!i!i!!i!!i!!!i!!ii! !l!!~ii!!i!ii!!i!il !i i!!!iii! i!;!!!!i In the previous sections we have not restricted the nature of the interpolation functions for shape functions Nu and N p. If we choose these interpolations in a manner satisfying the mixed patch test conditions 64' 65 or the Babu~ka-Brezzi (BB) restriction for incompressibility 2' 3 (see also Chapter 4 for some permissible interpolations), then of course completely incompressible problems can be dealt with without any special difficulties by both Split A and Split B formulations. However, Split A introduces an important bonus which permits us to avoid any restrictions on the nature of the two shape functions used for velocity and pressure. Let us examine here the structure of the equations obtained in steady-state conditions. For simplicity we shall consider here only the Stokes form of the governing equations in which the convective terms disappear. Further we shall take the fluid as incompressible (e.g. c -- c~) and thus uncoupled from the energy equations. Now the three steps of Split A given in Eqs (3.50), (3.55) and (3.58) are written as A[J* = -AtMu~[K~fi ~ - f] 1 ~H-I[G/J
n + 01GAf.J* - AtO1H[~ n - fp]
A l l = AtOl02
(3.79)
AU**= A [ J - A[J* = - A t M u l G T ( ~ n + 02A~) ,..,
,,.,
..,
In steady state we have A~ -- AU = 0 and eliminating AU* and AU** we can write (dropping now the superscript n) from the first and third of Eqs (3.79) K~fi + GT~ -- f
(3.80)
G[J + 01AtGMu1GT~ - At01H~ - fp = 0
(3.81)
and
from the second and third of Eqs (3.79). We finally have a system which can be written in the form IK~p
G 'r At01[GMulGT-H]] {~}=
fl {f2}
(3.82)
here fl and t"2 arise from the forcing terms. The system is now always positive definite 37 and therefore leads to a non-singular solution for any interpolation f u n c t i o n s Nu, N p chosen. In most of the examples discussed in this book and elsewhere equal interpolation is used for both the U~ and p variables, i.e. Nu = Np. We must, however, stress that any other interpolation can be used without violating the stability. This is an important reason for the preferred use of the Split A form. If the pressure gradient term is retained as in Eq. (3.28) (i.e. if we use Split B), the lower diagonal term of Eq. (3.82) is identically zero and the BB conditions in the full
98
CBS algorithm: compressibleand incompressible flow scheme cannot be avoided. To show this we start from Eqs (3.64), (3.66) and (3.67). For incompressible Stokes flow we have AIJ7 = - M u ~At [K~fi + G T ~ -- f)]n Ap :
~
1
H
AtOlO2
-1 At
[G~J
n --~ 0 1 G A [ J *
fp]
-
(3.83)
A[J** -- - M u 1 A t [02G T A~]
At steady state A~ = A[J -- 0, which gives the following two equations" K~fi + GT~ = f
(3.84)
and =
(3.85)
Note that AU~* is zero from the third of Eq. (3.83). As in Split A we can write the following system
~} = {t'f:}
(3.86)
where f~ and f2 arise from the forcing terms as in the Split A form. Clearly here the B B restrictions are not circumvented. It is interesting to observe that the lower diagonal term which appeared in Eq. (3.82) is equivalent to the difference between the so-called fourth-order and second-order finite difference approximations of the Laplacian. This justifies the use of similar terms introduced into the computation by Jameson and Mavripolis. 66 iiiiiiiiiiiiiiil~iiiii~!!!i~.!iiiiiiii~ iii!!~i ii~!iii"i~:iiiiiiii" i~i!i~i!:!iiiii!............ i~!iiii:'"'i iiii~iii::i•!••i•~••••::ii~::' i•••i••i••~••••i••i••.!•!!!i: ii!i•i••i•••i••~•'••:•~•i•........... iii~iii~i~iiiiiii!iiiii i~ii84 iii~ii!ii!iii~i~iiiii~ii!iiiiiiiii!iiiiiiiii!i~iii~i~iii~i~iiiiiii!iii~iiiiiiiiii~iiiiiiiiii~iii!iiiiiiiiiii!~iii~iM~i ii!iii~ii~i!!iii~i~i~i!~ii~~i!i~ii~iii~ii~i!i!i!iiiiiiii!i!i~iiiii!!ii!i~iiii ~ ~iiiiiiiiiii ••••••••iU •••••••••••••:••"••":•'•F••':•"••":•••••••••••••••••••••••••••••••••••.....••.. ili i iilli i i i:i:i:i i i iii i i i!i!i::iii::::::ii:i:i i ~!ilii Fiililili i i iilli:i:i i i ::ii::::::i:::.::ii ::i::ii i::ii:i:i::~i::i::i::ii ::::::iii!ii!:::i: ::iii::i:i!ii ...:.:~i:::i:iili i il::i:i:ii ::i::i:i~:ii!ii!ii iliii i i i:i:ii::iii::::ii:::i: :::i::::~::::~::~::::iilP : i i i:i:i:i:i i : i~::i:::::::::::::::::::::: i ::::i::i:i:i::!:i:il::ii ::::::ii ::iilii::ii::::::::ii ::i::ii:i:i: :ii i:i: :::::!::i~i:!::::i::!i::ii ::iiiiii!i::iiii i i :: il::~i::i::i::ii!:i:i:i!:ilili::ii::ilii ::i::i::ii::i::i::ii !!i i ~i i:i:ili:::::::::::::::::::::: iiii::iiii i:i:i:ii:i::i!::i::!!i::ii ii::::i:i: :ii i:iil:ii::::::::::i:i:i:i:ili: i::!ii ::i::::::iil:::!:i::iliii ::i::i::i!i:::::::::::::::::::::::::::::: i i i:i:i::i::ii i::ii ::i::i:i:iiiiiii:: :: i::i::ii::i:::i:i:ii:~i: ::i~:i::ii ::ii!ii::iiii::iiii!i ::i::i::iliiiiiiiiiiii::i::i::i::i::ii::ii::~ii:i:i:i:i i i i i::i::ii::ii i i :-i::!::i::ii::ii ::i::ii::il l::!ii i!i::ii i i ::i::iiiiiii::iii!iii::iiiiii::iiiii:.i:.i::il ::ii
If the AU 7 term in Eq. (3.27) is omitted, the intermediate variable U/* need not be determined. Instead we can directly calculate p (or p), Ui and pE. This of course introduces an additional approximation. The use of the approximation of Eq. (3.1) is not necessary in any expected fully explicit scheme as the density increment is directly obtained if we note that M p A ~ - MuA~
(3.87)
With the above simplifications and Split A we can return to the original equations and using the Galerkin approximation. We can therefore write directly
m~__Mulmt[fNT(OFi OC~i) l f~ +
N I)d
in
(3.88)
omitting the source terms for clarity [Fi and Gi are explained in Chapter 1, Eqs (1.26b) and (1.26c)] and noting that now ,I~ denotes all the variables. The added stabilizing terms D are defined below and have to be integrated by parts in the usual manner.
A single-stepversion
Op2 2010xiOxi U i -~iXi
( u j p u l ) "Jt- -'~X1
Ui-'~iXi
(UjpU2) "[-
Ui-'~X i
(UjpU3) "Jf-
Ui~xi
(ujpE + ujp)
(3.89)
The added 'diffusions' are simple and are streamline oriented, and thus do not mask the true effects of viscosity as happens in some schemes (e.g. the Taylor-Galerkin process). If only steady-state results are sought it would appear that At multiplying the matrix D should be set at its optimal value of Atcrit ~ h/lul and we generally recommend, providing the viscosity is small, this value for the full scheme. 33 The oversimplified scheme of Eq. (3.88) can lose some accuracy and even when steady state is reached will give slightly different results than those obtained using the full sequential updating. 33 However, at low Mach numbers the difference is negligible as we shall show later in Sec. 3.9. The small additional cost involved in computing the two-step sequence A[J* ~ Ap ~ A[J ~ AI~ will have to be balanced against the accuracy increase. In general, we have found that the two-step version is preferable. However, it is interesting to consider once again the performance of the single-step scheme in the case of Stokes equations as we did for the other schemes in the previous section. After discretization we have, omitting convective terms, only one additional diffusion term which arises [Eq. (3.82)] in the mass conservation equation. After discretization, in steady state
[K~p
01GASH] { ~ ) - ( f 2 }
fl
(3.90)
Clearly the single-step algorithm retains the capacity of dealing with full incompressibility without stability problems. We should remark here that this formulation now achieves precisely the same stabilization as that suggested by Brezzi and Pitkaranta, 67 see Chapter 11 in reference 1. We shall note the performance of single- and two-step algorithms in Sec. 3.9 of this chapter.
99
100 CBS algorithm: compressible and incompressible flow
3.8.1 Fictitious boundaries In a large number of fluid mechanics problems the flow in open domains is considered. A typical open domain describing flow past an aircraft wing is shown in Fig. 3.2. In such problems the boundaries are simply limits of computation and they are therefore fictitious. With suitable values specified at such boundaries, accurate solution for the flow inside the isolated domain can be achieved. Generally as the distance from the object grows, the boundary values tend to those encountered in the free domain flow or the flow at infinity. This is particularly true at the entry and side boundaries shown in Fig. 3.2. At the exit, however, the conditions are different and here the effect of the introduced disturbance can continue for a very large distance denoting the wake of the problem. We shall from time to time discuss problems of this nature but now we shall simply make the following remarks. 1. If the flow is subsonic the specification of all quantifies excepting the density (or pressure) can be made on both the side and entry boundaries. 2. Whereas for supersonic flows all the variables can be prescribed at the inlet or side boundaries, at the exit boundary, however, no conditions can be imposed simply because by definition no disturbance in a supersonic stream can travel upstream. With subsonic exit conditions the situation is somewhat more complex and here various possibilities exist. We again illustrate such conditions in Fig. 3.2. Condition A Denoting the most obvious assumptions with regard to the traction and velocities.
SIDE (Free-stream)
/
EXIT u2=0
r
INLET
(Free-stream)
tl=0 (parallel flow)
SLIP or NO-SLIP
(B)
/
eb
X2
l
SIDE (Free-stream)
"-Xl
r
Fig. 3.2 Fictitious and real boundaries.
(A)
Boundary conditions 101 Condition B A more sophisticated condition of zero gradient of traction and stresses existing there. Such conditions will of course always apply to the exit domains for incompressible flow. Condition B was first introduced by Zienkiewicz et al. 57 and is discussed fully by Papanastasiou et al. 68 This condition is of some importance as it gives remarkably good answers. Of considerable importance, especially in view of the CBS scheme, are, however, conditions which we will encounter on real boundaries.
3.8.2 Real boundaries By real boundaries we mean limits of fluid domains which are physically defined and here three different possibilities exist.
1. Solid boundaries with no slip conditions: On such boundaries the fluid is assumed to stick or attach itself to the boundary and thus all velocity components become zero. Obviously this condition is only possible for viscous flows. 2. Solid boundaries in inviscid flow (slip conditions): When the flow is inviscid we will always encounter slipping boundary conditions where only the normal velocity component is specified. This will generally be zero if the boundaries are stationary. Such boundary conditions will invariably be imposed for problems of Euler flow whether it is compressible or incompressible. 3. Prescribed traction boundary conditions: The last category is that of boundaries on which tractions are prescribed. This includes zero traction in the case of free surfaces of fluids or any prescribed tractions such as those caused by wind being imposed on the surface. These three basic kinds of boundary conditions have to be imposed on the fluid and special consideration has to be given to these when split operator schemes are used.
3.8.3 Ap.plication of real boundary conditions in the discretization using the CBS algorithm We shall first consider the treatment of boundaries described under (1) or (2) of the previous section. On such boundaries normal velocity zero
(3.91)
tangential traction zero for inviscid flow
(3.92a)
b/n = 0,
and either ts = 0,
102 CBS algorithm: compressible and incompressible flow
or Us = 0,
tangential velocity zero for viscous flow
(3.92b)
In early applications of the CBS algorithm it appeared correct that when computing A fJ f no velocity boundary conditions be imposed and to use instead the value of boundary tractions which corresponds to the deviatoric stresses and pressures computed at time tn. We note that if the pressure is removed as in Split A these pressures could also be removed from the boundary traction component. However, in Split B no such pressure removal is necessary. This requires, in viscous problems, evaluation of the boundary rij and this point is explained later. When computing Ap or Ap we integrate by parts obtaining [Eq. (3.54)]
L gkpl-~ A p d~2 -
- At
-- At
L Nkp ~O( U n + O1A U f -
~
O1At
OpOXi n+02) d~
ONkp Opn+O2 ~ [LUn + 01 (\AU/* - At Oxl ~]]jni dl-" L
Opn+02
--At NkRi[UT-+-OI(AU;-AtOxi
(3.93)
)lap
Here ni is the outward drawn normal. The last term in the above equation is identically equal to zero on boundaries in which slip and no-slip conditions are applied from the condition of Eq. (3.25):
[
Un - niUi - ni U~ + O1 A U . ~ - A t
Oxi
-0
(3.94)
For non-zero normal velocity this would simply become the specified normal velocity. This point seems to have baffled some investigators who simply assume
Op
Op
- ~ = ni-~x i = 0
(3.95)
on solid boundaries although this is not exactly true as shown by Codina et al. 41 Returning to the traction on the boundaries, the traction on the surface can be defined
as
ti = 7ijnj
--
pni
(3.96)
Prescribing the above traction using Split A, we replace the stress components in Step 1 [last term in Eq. (3.33)] as follows
Nu Tijnj dF -
ji
--Pt
N u T i j n j dF
"Jr-
/
N u (ti + pni ) dI"
(3.97)
t
where 1-'t represents the part of the boundary where the traction is prescribed. The above calculation may involve a substantial error in 'projecting' deviatoric stresses onto the boundary.
The performance of two-step and one-step algorithms on an inviscid problem The last step requires the solution for the velocity correction terms to obtain finally the g/n+l. Clearly correct velocity boundary values must always be imposed in this step. Although the above described procedure is theoretically correct and instructive, it is complex to implement. For this reason we will recommend the use of a much simpler procedure which relies on the fact that when summation is made of the CBS components correct velocity and traction conditions should be satisfied by the sum. Thus in the recommended procedure we apply all the specified tractions and velocity conditions at Step 1. Further, the need of calculating any boundary tractions from internal stress is now avoided even if the tangential velocity is taken as zero (no-slip condition).
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In this section we demonstrate the performance of the single- and two-step algorithms via an inviscid problem of subsonic and supersonic flow past an NACA0012 aerofoil. The problem domain and finite element mesh used are shown in Fig. 3.3(a) and (b). The discretization near the aerofoil surface is finer than that of other places and a total number of 969 nodes and 1824 elements are used in the mesh. The inlet Mach number is assumed to be equal to 0.5 and all variables except the density are prescribed at the inlet. The density is imposed at the exit of the domain. Both the top and bottom sides are assumed to be symmetric with normal component of velocity equal to zero. A slipping boundary is assumed on the surface of the aerofoil. No real or additional viscosity in any form is used in this problem when we use the CBS algorithm. However, other schemes do need additional diffusions to get a reasonable solution. Figure 3.3(c) shows the comparison of the density evolution at the stagnation point of the aerofoil. It is observed that the difference between the single- and two-step schemes is negligibly small. Further tests on these schemes were carded out at a higher inlet Mach number of 1.2 with the flow being supersonic, and a different mesh with a higher number of nodes (3753) and elements (7351). Here all the variables at the inlet were specified and the exit was free. As we can see from Fig. 3.3(d) and (e), the single-step scheme gives spurious oscillations in density values at the stagnation point. Therefore we conclude that here the two-step algorithm is valid for any range of Mach number and the single-step algorithm is limited to low Mach number flows with small compressibility. In Fig. 3.4 we compare the two-step algorithm results of the subsonic inviscid (M = 0.5) results with those obtained by the Taylor-Galerkin scheme for the same mesh. It is observed that the CBS algorithm gives a smooth solution near the stagnation point even though no additional artificial diffusion is introduced. However, the TaylorGalerkin scheme gives spurious solutions and a reasonable solution is obtained from this scheme only with a considerable amount of additional diffusion. Comparison of density distribution along the stagnation line shows [Fig. 3.4(d)] that the TaylorGalerkin scheme gives an incorrect solution even with additional diffusion. However,
103
104
CBS algorithm: compressibleand incompressibleflow
(a)
(b)
1.500 f 1.450 u.i 1.400 _i 1.350 1.300
Multi-step ........... Single step
1.250
"~ 1.200
1.15o
o 1.100 1.050 1.000
0
250 500 750 1000 1250 1500 No. of iterations (C) M = 0.5
1.500
1.500
Multi-step
uJ 1.250 J
1.250
Single step
..i .~ 1.o0o
.~ 1.ooo \ 8
a 0.750
0.500
L
a 0.750 t 0
I 500
I I 1 0 0 0 1500 Iterations (d)
I 2000
2500
0.5001 0 M = 1.2
I 500
I
I
1000 1500 Iterations
I
2000
I
2500
(o)
Fig. 3.3 Inviscid flow past an NACA0012 aerofoil, angle of attack = 0: (a) unstructured mesh 1824 elements and 969 nodes; (b) details of mesh near stagnation point; (c) convergence for M = 0.5 with two- and singlestep, fully explicit form; (d) convergence for M = 1.2 for two-step scheme; (e) convergence for M = 1.2 for single-step scheme.
the CBS algorithm again gives an accurate solution without the use of any additional artificial diffusion.*
ii~iii!i!ii~'i:':!:'~~:~iii~i~~'::"::~i:~~'Hii~iiiiiiiiiiiiiiiiiiiii~i~i~!~i~!i~!i~i~i~ii~iiii!iiiiiiiiii~i iii!~~iii ~ i ~ i ~i!;i i~~;iiiii~i~ii~i!~iiii~!iii~iiii~iiiii~!~ii!~i!l!iil~ii!i~~!ii~i~i~ii~i~i!i~i~i~i~i~!i~i~i~iiiiii~iiiii!!iiiiii!~i~i~ii~il~ii!~i ~i !i~i~i~i~i~!!ii~iii~i~i~ii~!iii~ii~i~i!i~iii~ii~iii~i~i;i!ii!ii~i~i;iii!ii~ii!!~!~ii~iiiiiiiiiii The general CBS algorithm is discussed in detail in this chapter for the equations of fluid dynamics in their conservation form. Comparison between the single- and twostep algorithms in the last section shows that the latter scheme is valid for all ranges
*Note that a variable smoothing operation is often necessary at low Mach numbers as explained in Chapter 7.
References 105
(a)
(b)
1.250 1.000
...'7.-.
d . . . . . . .
'-'. . . . . . . . . . . . . . . . .
"
8 0.750 O
0.500 -
. . . . . . . T - G scheme (Cs = 0.0) "............ T - G scheme (Cs = 0.5) Present schemes (Cs = 0.0)
~~176
1
| | | i i m
0.250 I I I I -50.00 --40.00 -30.00 -20.00 -10.00 Distance X (c)
M = 0.5
(d)
Fig. 3.4 Subsonic inviscid flow past an NACA0012 aerofoil with angle of attack = 0 and M = 0.5: (a) density contours with TG scheme with no additional viscosity; (b) density contours with TG scheme with additional viscosity; (c) density contours with CBS scheme with no additional viscosity; (d) comparison of density along the stagnation line.
of flow. In later chapters, we generally apply the two-step algorithm for different flow applications. Another important conclusion made from this chapter is about the accuracy of the present scheme. As observed in the last section, the present CBS algorithm gives excellent performance when the flow is slightly compressible compared to the Taylor-Galerkin algorithm. In the following chapters we show further tests on the algorithm for a variety of problems including general compressible and incompressible flow problems, shallow water problems, etc.
~ii
~~iii
i ii ili iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii iiiiiiiiiiiiiiiiiiiiliiiiiiiiiiiiiiiiiiiiiliiii !iiii i iiiiiiiiiiiiiiiiiiiiiii ii!i ii
iiiiiiiiiiiii~i~iiii~iiiii~ii~i~iii!i~iiiiii~!i~i~i~iii~ii~iii~iii~iiii!iii~i~!!iii!i~iiiiiii~iiiii!i!i~i!i~i~!i~i~!ii~i~i~i!i~ii~iii!~iiiiii~!~ii!ii~i~i!~i!~!i!iiii~iii!!iiiiiiii~iii~i!i!iiii!!iiiiiii!iiiii!i!iii~iiiii 1. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann, Oxford, 6th edition, 2005. 2. I. Babu~ka. The finite element method with Lagrangian multipliers. Numer. Math., 20:179-192, 1973.
106 CBS algorithm: compressibleand incompressibleflow 3. E Brezzi. On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers. Rev. Fran~aise d'Automatique Inform. Rech. Opdr., Ser. Rouge Anal. Numdr., 8(R-2):129-151, 1974. 4. A.J. Chorin. Numerical solution of Navier-Stokes equations. Math. Comput., 22:745-762, 1968. 5. A.J. Chorin. On the convergence of discrete approximation to the Navier-Stokes equations. Math. Comput., 23:341-353, 1969. 6. G. Comini and S. Del Guidice. Finite element solution of incompressible Navier-Stokes equations. Num. Heat Transfer, Part A, 5:463-478, 1972. 7. G.E. Schneider, G.D. Raithby and M.M. Yovanovich. Finite element analysis of incompressible flow incorporating equal order pressure and velocity interpolation. In C. Taylor, K. Morgan and C.A. Brebbia, editors, Numerical Methods in Laminar and Turbulent Flow. Plymouth, Pentech Press, 1978. 8. J. Donea, S. Giuliani, H. Laval and L. Quartapelle. Finite element solution of unsteady NavierStokes equations by a fractional step method. Computer Methods in Applied Mechanics and Engineering, 33:53-73, 1982. 9. P.M. Gresho, S.T. Chan, R.L. Lee and C.D. Upson. A modified finite element method for solving incompressible Navier-Stokes equations. Part I theory. International Journal for Numerical Methods in Engineering, 4:557-598, 1984. 10. M. Kawahara and K. Ohmiya. Finite element analysis of density flow using the velocity correction method. International Journal for Numerical Methods in Engineering, 5:981-993, 1985. 11. J.G. Rice and R.J. Schnipke. An equal-order velocity-pressure formulation that does not exhibit spurious pressure modes. Computer Methods in Applied Mechanics and Engineering, 58:135149, 1986. 12. B. Ramaswamy, M. Kawahara and T. Nakayama. Lagrangian finite element method for the analysis of two dimensional sloshing problems. International Journal for Numerical Methods in Fluids, 6:659-670, 1986. 13. B. Ramaswamy. Finite element solution for advection and natural convection flows. Comp. Fluids, 16:349-388, 1988. 14. M. Shimura and M. Kawahara. Two dimensional finite element flow analysis using velocity correction procedure. Earthquake Engineering and Structural Dynamics, 5:255-263, 1988. 15. S.G.R. Kumar, P.A.A. Narayana, K.N. Seetharamu and B. Ramaswamy. Laminar flow and heat transfer over a two dimensional triangular step. International Journal for Numerical Methods in Fluids, 9:1165-1177, 1989. 16. B. Ramaswamy, T.C. Jue and J.E. Akin. Semi-implicit and explicit finite element schemes for coupled fluid thermal problems. International Journal for Numerical Methods in Engineering, 34:675-696, 1992. 17. R. Rannacher. On chorin projection method for the incompressible Navier-Stokes equations. Lecture Notes in Mathematics, 1530:167-183, 1993. 18. B. Ramaswamy. Theory and implementation of a semi-implicit finite element method for viscous incompressible flows. Comp. Fluids, 22:725-747, 1993. 19. C.B. Yiang and M. Kawahara. A three step finite element method for unsteady incompressible flows. Comput. Mech., 11:355-370, 1993. 20. G. Ren and T. Utnes. A finite element solution of the time dependent incompressible NavierStokes equations using a modified velocity correction method. International Journal for Numerical Methods in Fluids, 17:349-364, 1993. 21. B.V.K.S. Sai, K.N. Seetharamu and P.A.A. Narayana. Solution of transient laminar natural convection in a square cavity by an explicit finite element scheme. Num. Heat Transfer, Part A, 25:593-609, 1994. 22. M. Srinivas, M.S. Ravisanker, K.N. Seetharamu and P.A. Aswathanarayana. Finite element analysis of internal flows with heat transfer. Sadhana - Academy Proc. Eng., 19:785-816, 1994.
References 107 23. EM. Gresho, S.T. Chan, M.A. Christon and A.C. Hindmarsh. A little more on stabilized q(1)q(1) for transient viscous incompressible flow. International Journal for Numerical Methods in Fluids, 21:837-856, 1995. 24. Y.T.K. Gowda, P.A.A. Narayana and K.N. Seetharamu. Mixed convection heat transfer past in-line cylinders in a vertical duct. Num. Heat Transfer, Part A, 31:551-562, 1996. 25. A.R. Chaudhuri, K.N. Seetharamu and T. Sundararajan. Modelling of steam surface condenser using finite element methods. Communications in Applied Numerical Methods, 13:909-921, 1997. 26. P. Nithiarasu, T. Sundararajan and K.N. Seetharamu. Finite element analysis of transient natural convection in an odd-shaped enclosure. International Journal of Numerical Methods for Heat and Fluid Flow, 8:199-220, 1998. 27. P.K. Maji and G. Biswas. Three-dimensional analysis of flow in the spiral casing of a reaction turbine using a differently weighted Petrov Galerkin method. Computer Methods in Applied Mechanics and Engineering, 167:167-190, 1998. 28. P.D. Minev and P.M. Gresho. A remark on pressure correction schemes for transient viscous incompressible flow. Communications in Applied Numerical Methods, 14:335-346, 1998. 29. J. Blasco, R. Codina and A. Huerta. A fractional-step method for the incompressible NavierStokes equations related to a predictor-multicorrector algorithm. International Journal for Numerical Methods in Fluids, 28:1391-1419, 1998. 30. B.S.V.P. Patnaik, P.A.A. Narayana and K.N. Seetharamu. Numerical simulation of vortex shedding past a cylinder under the influence of buoyancy. Int. J. Heat Mass Transfer, 42:3495-3507, 1999. 31. O.C. Zienkiewicz and R. Codina. Search for a general fluid mechanics algorithm. In D.A. Caughey and M.M. Hafez, editors, Frontiers of Computational Fluid Dynamics, pages 101-113. John Wiley & Sons, New York, 1995. 32. O.C. Zienkiewicz and R. Codina. A general algorithm for compressible and incompressible flow - Part I: The split, characteristic-based scheme. International Journal for Numerical Methods in Fluids, 20:869-885, 1995. 33. O.C. Zienkiewicz, B.V.K.S. Sai, K. Morgan, R. Codina and M. Vfizquez. A general algorithm for compressible and incompressible flow- Part II. Tests on the explicit form. International Journal for Numerical Methods in Fluids, 20:887-913, 1995. 34. O.C. Zienkiewicz and P. Ortiz. A split characteristic based finite element model for shallow water equations. International Journal for Numerical Methods in Fluids, 20:1061-1080, 1995. 35. P. Ortiz and O.C. Zienkiewicz. Modelizacin por elementos finitos en hidrulica e hidrodinmica costera. Technical report, Centro de Estudios y Experimentation de Obras Publicas, CEDEX, Madrid, 1995. 36. O.C. Zienkiewicz. A new algorithm for fluid mechanics. Compressible and incompressible behaviour. In Proc. 9th Int. Conf. Finite Elements Fluids- New Trends and Applications, pages 49-55, Venezia, October 1995. 37. R. Codina, M. Vfizquez and O.C. Zienkiewicz. A fractional step method for compressible flows: boundary conditions and incompressible limit. In Proc. 9th Int. Conf. Finite Elements FluidsNew Trends and Applications, pages 409-418, Venezia, October 1995. 38. P. Ortiz and O.C. Zienkiewicz. Tide and bore propagation in the Severn Estuary by a new algorithm. In Proc. 9th Int. Conf. Finite Elements Fluids - New Trends and Applications, pages 1543-1552, Venezia, October 1995. 39. O.C. Zienkiewicz, B.V.K.S. Sai, K. Morgan and R. Codina. Split characteristic based semiimplicit algorithm for laminar/turbulent incompressible flows. International Journal for Numerical Methods in Fluids, 23:1-23, 1996. 40. P. Ortiz and O.C. Zienkiewicz. An improved finite element model for shallow water problems. In G.E Carey, editor, Finite Element Modelling of Environmental Problems, pages 61-84. John Wiley & Sons, New York, 1996.
108 CBS algorithm: compressibleand incompressibleflow 41. R. Codina, M. V~izquez and O.C. Zienkiewicz. General algorithm for compressible and incompressible flows, Part III - A semi-implicit form. International Journal for Numerical Methods in Fluids, 27:13-32, 1998. 42. B.V.K.S. Sai, O.C. Zienkiewicz, M.T. Manzari, P.R.M. Lyra and K. Morgan. General purpose vs special algorithms for high speed flows with shocks. International Journal for Numerical Methods in Fluids, 27:57-80, 1998. 43. R. Codina, M. V~izquez and O.C. Zienkiewicz. A fractional step method for the solution of compressible Navier-Stokes equations. In M. Hafez and K. Oshima, editors, Computational Fluid Dynamics Review 1998, volume 1, pages 331-347. Word Scientific Publishing, 1998. 44. N. Massarotti, P. Nithiarasu and O.C. Zienkiewicz. Characteristic-based-split (CBS) algorithm for incompressible flow problems with heat transfer. International Journal for Numerical Methods in Fluids, 8:969-990, 1998. 45. O.C. Zienkiewicz, P. Nithiarasu, R. Codina, M. V~izquez and P. Ortiz. Characteristic-based-split algorithm, Part I: The theory and general discussion. In K.D. Papailou et al., editors, Proc. of ECCOMAS CVD 1998, volume 2. Athens, Greece, 1998. 46. N. Massarotti, P. Nithiarasu and O.C. Zienkiewicz. Characteristic-based-split algorithm, Part II: Incompressible flow problems with heat transfer. In K.D. Papailiou et al., editors, Proc. of ECCOMAS CFD 1998, volume 2, pages 17-21. Athens, Greece, 1998. 47. O.C. Zienkiewicz and P. Ortiz. The CBS (characteristic-based-split) algorithm in hydraulic and shallow water flow. In Second Int. Sym. River Sedimentation and Env. Hydraulics. University of Hong Kong, 3-12 December 1998. 48. O.C. Zienkiewicz, J. Rojek, R.L. Taylor and M. Pastor. Triangles and tetrahedra in explicit dynamic codes for solids. International Journal for Numerical Methods in Engineering, 43:565583, 1999. 49. O.C. Zienkiewicz, P. Nithiarasu, R. Codina, M. V~izquez and P. Ortiz. The characteristic based split procedure: an efficient and accurate algorithm for fluid problems. International Journal for Numerical Methods in Fluids, 31:359-392, 1999. 50. P. Nithiarasu and O.C. Zienkiewicz. On stabilization of the CBS algorithm. Internal and external time steps. International Journal for Numerical Methods in Engineering, 48:875-880, 2000. 51. P. Nithiarasu. Boundary conditions for the characteristic based split (CBS) algorithm for fluid dynamics. International Journal for Numerical Methods in Engineering, 54:523-536, 2002. 52. R. Codina and O.C. Zienkiewicz. CBS versus GLS stabilization of the incompressible NavierStokes equations and the role of the time step as stabilization parameter. Communications in Numerical Methods in Engineering, 18:99-112, 2002. 53. P. Nithiarasu. An efficient artificial compressibility (AC) scheme based on the characteristic based split (CBS) method for incompressible flows. International Journal for Numerical Methods in Engineering, 56:1815-1845, 2003. 54. N. Massarotti, P. Nithiarasu and A. Carotenuto. Microscopic and macroscopic approach for natural convection in enclosures filled with fluid saturated porous medium. International Journal of Numerical Methods for Heat and Fluid Flow, 13:862-886, 2003. 55. P. Nithiarasu, J.S. Mathur, N.P. Weatherill and K. Morgan. Three-dimensional incompressible flow calculations using the characteristic based split (CBS) scheme. International Journal for Numerical Methods in Fluids, 44:1207-1229, 2004. 56. P. Nithiarasu and R. Codina, Characteristic Based Split (CBS) Scheme, Mini Symposium, 4th European Congress on Computational Methods in Applied Sciences and Engineering, 24-28 July 2004, Jyv~iskyl~i,Finland. 57. O.C. Zienkiewicz, J. Szmelter and J. Peraire. Compressible and incompressible flow: an algorithm for all seasons. Computer Methods in Applied Mechanics and Engineering, 78:105-121, 1990.
References 109 58. O.C. Zienkiewicz. Explicit or semi-explicit general algorithm for compressible and incompressible flows with equal finite element interpolation. Technical Report 90/5, Chalmers Technical University, Gothenburg, 1990. 59. O.C. Zienkiewicz. Finite elements and computational fluid mechanics. In SEMNI Congress, Metodos Numericos en Ingenieria, pages 51-61, Gran Canaria, 1991. 60. O.C. Zienkiewicz and J. Wu. Incompressibility without tears! How to avoid restrictions of mixed formulations. International Journal for Numerical Methods in Engineering, 32:11841203, 1991. 61. O.C. Zienkiewicz and J. Wu. A general explicit or semi-explicit algorithm for compressible and incompressible flows. International Journalfor Numerical Methods in Engineering, 35:457--479, 1992. 62. P. Nithiarasu and O.C. Zienkiewicz. Analysis of an explicit fractional step method, Computer Methods in Applied Mechanics and Engineering (submitted, 2005) 63. C. Thomas and P. Nithiarasu. Influences of element size and variable smoothing on inviscid compressible flow solution. International Journal of Numerical Methods in Heat and Fluid Flow, 15:420--428, 2005. 64. O.C. Zienkiewicz, S. Qu, R.L. Taylor and S. Nakazawa. The patch test for mixed formulations. International Journal for Numerical Methods in Engineering, 23:1873-1883, 1986. 65. O.C. Zienkiewicz and R.L. Taylor. The finite element patch test revisited: a computer test for convergence, validation and error estimates. Computer Methods in Applied Mechanics and Engineering, 149:523-544, 1997. 66. A. Jameson and D.J. Mavripolis. Finite volume solution of the two dimensional Euler equations on a regular triangular mesh. InAIAA 23th Aerospace Sciences Meeting, Reno, NV, 1985. Paper AIAA-85-0435. 67. F. Brezzi and J. Pitk~iranta. On the stabilization of finite element approximations of the Stokes problem. In W. Hackbusch, editor, Efficient Solution of Elliptic Problems, Notes on Numerical Fluid Mechanics, volume 10. Vieweg, Wiesbaden, 1984. 68. T.C. Papanastasiou, N. Malamataris and K. Ellwood. A new outflow boundary condition. International Journal for Numerical Methods in Fluids, 14:587-608, 1992.
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The problems of incompressible flows dominate a large part of the fluid mechanics scene. For this reason, they are given special attention in this book and we devote three chapters to this subject. In the present chapter we deal with various steady-state and transient Newtonian situations in which the flow is forced by appropriate pressure gradients and boundary forces. In the next chapter we shall consider non-Newtonian flows including metal forming and viscoelastic flows. Free surface flows in which gravity establishes appropriate free surface patterns as well as the so-called buoyancy force in which the only driving forces are density changes caused by temperature variations are discussed in Chapter 6. It is mentioned in reference 1 that certain difficulties are encountered with incompressibility when this is present in the equations of solid mechanics. We shall find that exactly the same problems arise again in fluids especially with very slow flows where the acceleration can be neglected and viscosity is dominant (so-called Stokes flow). Complete identity with linear elasticity is found here. The essential difference in the governing equations for incompressible flows from those of compressible flows is that the coupling between the equations of energy and the other equations is very weak and thus frequently the energy equations can be considered either completely independently or as an iterative step in solving the incompressible flow equations. To proceed further we return to the original equations of fluid dynamics which have been given in Chapters 1 and 3; we repeat these below for problems of small compressibility.
Conservation o f mass Op 10p - ~ -- c 2 0 t
=
OUi Ox i
(4.1)
and C2 .._ K/p where K is the bulk modulus. Here in the incompressible limit, the density p is assumed to be constant and in this situation the term on the left-hand side is simply zero.
Introduction and the basic equations 111
Conservation of momentum
OUi Ot
0
0~ij
~(ujU~) -40xj Oxj
Op
Ox~ t- Pgi
(4.2)
In the above we define the mass flow fluxes as
(4.3)
Ui -- pui
Conservation of energy is now uncoupled and can be solved independently:
O(pE) 0 0 (k OT ) 0 0 Ot = -Ox---~(ujpE)+-~xi \ ~ --~xj (Ujp)+-~xi ('rijuj)+pgiui+qH
(4.4)
In the above Ui are the velocity components, E is the specific energy (cvT), p is the pressure, T is the absolute temperature, Pgi represents the body force and other source terms, qH represents heat generation and 7"ij are the deviatoric stress components given by Eq. (1.1 lb). Tij "-- ~
( Ou~ Ouj 2~ Ouk) ~
-~ C~Xi
~ ij ~
(4.5)
With the substitution made for density changes we note that the essential variables in the first two equations become those of pressure and velocity. In exactly the same way as these, we can specify the variables linking displacements and pressure in the case of incompressible solids. It is thus possible to solve these equations in one of many ways described in Chapter 12 of reference 1 though, of course, the use of the CBS algorithm is obvious. Unless the viscosity and in fact the bulk modulus have a strong dependence on temperature the problem is very weakly linked with the energy equation which can be solved independently. The energy equation for incompressible materials is best written in terms of the absolute temperature T avoiding the specific energy. The equation now becomes simply (neglecting source terms) cop
~
"3I- Uj
"-- ~
~ OXi
"3!- -~xi (TijUj) - -~xj (UjN)
(4.6)
and we note that this is now a scalar convection--diffusion equation of the type we have already encountered in Chapter 2, written in terms of the variable temperature as the unknown. In the above equation, the last two work dissipation terms are often neglected for fully incompressible flows. Note that the above equation is derived assuming the density and co (specific heat at constant volume) to be constants. In this chapter we shall in general deal with problems for which the coupling is weak and the temperature equations do not present any difficulties. However, in Chapter 6 we shall encounter buoyancy effects caused by atmospheric or general circulation induced by small density changes due to temperature differences. If viscosity is a function of temperature, it is very often best to proceed simply by iterating over a cycle in which the velocity and pressure are solved with the assumption of known viscosity and that is followed by the solution of temperature. Many
112 IncompressibleNewtonian laminar flows practical problems have been so treated very satisfactorily. We shall show some of these applications in the field of metal forming in the next chapter. In the main part of this chapter we shall consider the solution of viscous, Newtonian fluids and we shall generally use the CBS algorithm described in Chapter 3, though on occasion we shall depart from this due to the similarity with the equations of solid mechanics and use a more direct approach either by satisfying the B B stability conditions of Chapter 11 in reference 1 for the velocity and pressure variables, or by using reduced integration in the context of a pure velocity formulation with a penalty parameter (see Chapters 10 and 11 of reference 1).
4.2.1 The fully explicit artificial compressibility form The fully explicit scheme for incompressible flow approximation is obtained by substituting 0.5 < 01 < 1.0 and 02 = 0 into Eqs (3.55) and (3.58). It is obvious that c --+ 0 in incompressible flow approximations and therefore needs to be replaced with an artificial parameter/3 as discussed in Sec. 3.5.1.2' 3 With such an approximation, reaching the steady-state solutions will be straightforward. However, for transient unsteady state the dual time stepping procedure described in Sec. 3.5.2 needs to be followed. We consider both steady- and unsteady-state solutions here. The artificial compressibility based CBS scheme is generally referred to as the CBS-AC scheme in the present text.
4.2.2 The semi-implicit form For problems of incompressibility with K being equal to infinity or indeed when K is very large, we have a choice of using the fully explicit procedure with artificial compressibility as discussed in the above section or using the CBS algorithm in its semi-implicit form with 0.5 < 01 < 1.0 and 0.5 < 02 < 1 (Chapter 3, Sec. 3.4.2). 4 This of course will use an explicit solution for the momentum equation followed by an implicit solution of the pressure Laplacian form (the Poisson equation). The solution which has to be obtained implicitly involves only the pressure variable and we will further notice that, from the contents of Chapter 3, at each step the basic equation remains unchanged and therefore the solution can be repeated simply with different right-hand side vectors. The convergence rate to steady state of course depends on the time step used and here we have the time step limitation given by the Courant number h Atl < A t c r i t - - ~
lul
(4.7)
for inviscid problems and for viscous problems h2 At2 _< A/crit = 2---~
(4.8)
Use of the CBS algorithm for incompressible flows 113 is an additional limitation. Here we note immediately that the viscosity lowers the limit quite substantially and therefore convergence to steady state may not be exceedingly rapid. The examples which we shall show nevertheless indicate its good performance.
Example 4.1
Incompressible flow in a lid-driven cavity
The classical problem on which we shall judge the performance is that of the closed cavity driven by the motion of a lid. 5-9 There are various ways of assuming the boundary conditions but the most common is one in which the velocity along the top surface increases from the comer node to the driven value in the length of one element (socalled ramp conditions).* The solution was obtained for different values of Reynolds number thus testing the performance of the viscous formulation. The problem has been studied by many investigators but probably the most detailed investigation was that of Ghia et al. 5 in which they quote many solutions and data for different Reynolds numbers. We shall use those results for comparison. In the first figure, Fig. 4.1, we show the geometry, boundary conditions and finite element meshes. The problem definition is shown in Fig. 4.1(a). The top lid of the cavity is assumed to move with a prescribed velocity in one direction and all other walls are stationary. No-slip conditions for velocity are applied on all solid walls. Pressure is prescribed at one point as shown Fig. 4.1(a). Several meshes have been tested in the past 2 but only three meshes are shown in Fig. 4.1. The first mesh is a non-uniform structured mesh with smaller elements close to the walls. The second mesh is a uniform structured mesh of 100 x 100 size and the last mesh given in Fig. 4.1 is a non-uniform unstructured mesh with higher resolution close to the cavity walls. Figure 4.2 shows the stream traces and pressure contours for R e = 100 and 400 generated from the unstructured mesh shown in Fig. 4.1(d). As seen the results are smooth and free of oscillations. Expected secondary vortices on both bottom comers are predicted excellently by the scheme. In Fig. 4.3 the results produced by the unstructured mesh [Fig. 4.1(d)] and uniform structured mesh [Fig. 4.1 (c)] are compared for a Reynolds number of 5000. As noticed both the results are smooth and in good agreement with the other reported results. The important aspect of this figure is that a small comer vortex at the bottom fight comer, which normally requires a very high mesh resolution, has been predicted by the CBS scheme. The horizontal and vertical velocity component distributions are shown in Figs 4.4 and 4.5. All the results in these figures are produced using the unstructured mesh [Fig. 4.1(d)]. As noted the results are in excellent agreement with the benchmark fine mesh (121 x 121) solutions reported by Ghia et al. 5 A three-dimensional lid-driven cavity solution is shown in Fig. 4.6 at R e = 400. As seen the solutions obtained are smooth and the velocity distribution is in good agreement with the 2D results. However, it should be noted that at higher Reynolds numbers (> 1000) no steady state exists and the results show a fully three-dimensional solution.3, 8, 9
*Some investigators use the leaking lid formulation in which the velocity along the top surface is constant and varies to zero within an element in the sides. It is preferable, however, to use the formulation where velocity is zero on all nodes of the vertical sides.
114 IncompressibleNewtonian laminar flows
u~ = 1, u2=0
o
o
II II
II II
p=0
U 1 = /./2=0
(a) Geometry and boundary conditions.
(b) Non-uniform structured mesh, elements: 2888, nodes: 1521.
i i ?i i
(c) Uniform structured mesh, elements: 20000, nodes: 10201.
i i!
(d) Non-uniform unstructured mesh, elements: 10596, nodes: 5515.
Fig. 4.1 Incompressibleflow in a lid-driven cavity. Geometryand meshes.
Figure 4.7 gives the steady-state convergence histories of the lid-driven cavity problem for semi-implicit and fully explicit schemes. The L2 norm of velocity residual is calculated as
"~No.Nodesv/( i=1 lul n+l
_ luln) ~
~--.~No.Nodes V/( luln+') 2 i=,
(4.9)
Use of the CBS algorithm for incompressible flows 115
(a) Re= 100, stream traces.
(b) Re= 100, pressure contours. v
(c) Re = 400, stream traces. Fig. 4.2
(d) Re = 400, pressure contours.
Incompressible flow in a lid-driven cavity. Stream traces and pressure contours at Re = 100 and 400.
The steady state was assumed when the above L2 norm reached a value below 10 -5. It is clear from Fig. 4.7 that both the fully explicit and semi-implicit schemes converge almost at the same rate at Re -- 5000. However, at Re = 400, the convergence rate of the semi-implicit form is slightly better than that of the fully explicit scheme. Both the results were produced using local time stepping.
Example 4.2 Steady flow past a backward facing step
In this example, another widely used benchmark problem of flow past a backward facing step is considered. ~~ The problem definition is shown in Fig. 4.8. The inlet is
116
Incompressible Newtonian laminar flows
(a) Re = 5000, stream traces on the unstructured mesh.
(b) Re = 5000, stream traces on the uniform structured mesh.
/-
(c) Re = 5000, pressure contours on the unstructured mesh. Fig. 4.3
(d) Re = 5000, pressure contours on the uniform structured mesh.
Incompressible flow in a lid-driven cavity. Stream traces and pressure contours at Re = 5000.
situated at a distance of four times the step height from the step. The inlet height is twice the height of the step itself. The total length is 40 times the height of the step. The inlet Reynolds number based on the step height and the average inlet velocity is 229. At inlet a nearly parabolic horizontal velocity profile (experimental) is assumed and the vertical velocity component is assumed to be equal to zero. On the walls no-slip boundary conditions apply and at exit constant (zero) pressure conditions are prescribed. The pressure boundary condition can be relaxed if the fully explicit scheme is employed. 2
Use of the CBS algorithm for incompressible flows
~
0.8 rO
t~ 0.6 .1_, .~_ "10 m t~
.o_ 0.4 12
i
i
8 =5
i
> 0.2
CBS
0 012 014 Horizontal velocity
o16 o18
I
04
02
0
(a) R e = O. i
i
i
i
-
:
0.8
8
0.6
._o 0.4
-~ 0.4
>I1)
>
1=
i
-0.4
i
-0.2
6
i
i
i
0.2 0.4 0.6 Horizontal velocity
i
i
9 i
0.8
i
1
i
CBS Ghia et aL
0.2
Ghia et al. ~
08
0.6
=5
" ~ ~
02 04 06 Horizontal velocity
0.8
"0
0.2
i
(b) R e = 400. i
O r
.__.
i
0.6
"~ 0.4
0.2
i
i
0.8
r
>
--0.4-0.2
i
0
-0.4
(c) R e = 1000.
,
-0.2
0
0.2 0.4 0.6 Horizontal velocity
* I
0.8
(d) R e = 5000.
Fig. 4.4 Incompressible flow in a lid-driven cavity. Horizontal velocity distribution at different Reynolds numbers along the mid-vertical line.
In Fig. 4.9, the unstructured mesh used and the contours of horizontal velocity component and pressure are given. The mesh is finer near the walls and coarser away from the walls. The pressure and velocity distributions shown in Figs 4.9(b) and (c) are in good agreement with the available data. In Fig. 4.10, the numerical data are compared against the experimental data in the recirculation zone. As seen the agreement between the numerical and experimental datal~ is excellent.
Example 4.3 Steady flow past a sphere
The next problem considered is fully three dimensional and shows flow past a sphere. The computational domain is a rectangular imaginary box of length 25 D, where D is the diameter of the sphere, with the downstream boundary located at 20D from the centre of the sphere. The four side walls are located at a distance of 5 D from the centre of the sphere. All four confinement walls are assumed to be slip walls with normal velocity equal to zero. The inlet velocity is assumed to be uniform and the no-slip condition prevails on the sphere surface. This problem is solved using the fully explicit form of the CBS scheme.
117
118
Incompressible Newtonian laminar flows 0.6,
i
i
,
0.6
,
0.4
cBs t
0.4 l
-~-~8020
CBS
-~~o0.2
>
0
0
>a~ -0.2
>
-0.4
-0.4
-0 6 "0
, 0.2
, , 0.4 0.6 Horizontal distance
0.8
-0.6
1
0'.2
(a) Re = 0.
0.6
.
.
.
0'.6
0'.8
(b) Re = 4 0 0 .
.
0.6
0.4
0.4 cBs
@9 0.2
O mO
>
0~.4
Horizontal distance
GhiaetaL
*
o
02
cBs
~
GhiaetaL
*
0.
. uO
>
-0.2
_
-0.4 -0"60
-0.4 i 0.2
0.4 0.6 Horizontal distance (c) Re = 1000.
0.8
1
-0 6 "0
, 0.2
, , 0.4 0.6 Horizontal distance
0.8
1
(d) Re = 5 0 0 0 .
Fig. 4.5 Incompressible flow in a lid-driven cavity. Vertical velocity distribution at different Reynolds numbers along the mid-horizontal line.
For this problem, an unstructured grid containing 953 025 tetrahedral elements has been used. This mesh is generated using the PSUE mesh generator. TM12 Figure 4.11 (a) shows a portion of the surface mesh and Fig. 4.11 (b) shows a sectional view. The mesh is refined close to the sphere surface and in the rear where recirculation is expected. Figure 4.12 shows the contours of the Ul component of the velocity and the pressures computed at Reynolds numbers of 100 and 200. The coefficient of pressure, Cp, values on the surface along the flow axis are shown in Fig. 4.13. The non-dimensional Cp is calculated as Cp = 2 ( p - Pref) (4.10) where eref is a reference pressure (free stream value). Note that the results used for comparison were generated using very fine structured meshes. TM14 It should also be noted that all the results differ from each other close to the separation zone.
Example 4.4
Transient flow past a circular cylinder
This is a popular test case for validating the transient part of numerical schemes. Many other problems of interest can also be solved for transient accuracy but in this section only flow over a single circular cylinder is considered.
Use of the CBS algorithm for incompressible flows 119
(a) Unstructured mesh.
(c) u3 contours.
(b) Ul contours.
(d) Pressure contours.
Fig. 4.6 Incompressibleflow in a 3D lid-driven cavity. Mesh and contours at Re = 400.
Problem definition is standard. The inlet flow is uniform and the cylinder is placed at the centreline between two slip walls. The distances from the inlet and slip walls to the centre of the cylinder are 4D, where D is the diameter of the cylinder. Total length of the domain is 16D. A no-slip condition is applied on the cylinder surface. The initial values of horizontal velocity were assumed to be unity and the vertical component of velocity was assumed to be zero all over the domain.
120 Incompressible Newtonian laminar flows Re = 400
1
0.1
'Explicit-Semi-impli.---
0.1
,
Re = 5000
,
' Explicit'--Semi-impli. --..
0.01
0.01
0.001 E
0.001 0.0001
0.0001 1e-05
1e-05
I e-06
1e-06 I e-O;
,
0
1000
2000
3000
4000
5000
6000
1e-07
0
5(~00
No. time steps
10C)00 15()00 NO. time steps
20()00
25 ()00
(b) Re = 5000.
(a) Re = 400.
Fig. 4.7 Lid-driven cavity. Steady-state convergence histories for Re = 400 and 5000. Comparison between fully explicit (artificial compressibility) and semi-implicit schemes.
Experimental Ul and u2 = 0
\ UlU
o
p=O
Fig. 4.8 Incompressible flow past a backward facing step. Geometry and boundary conditions.
(a) Unstructured mesh.
(b) Ul velocity contours.
(c) Pressure contours. Fig. 4.9 Incompressible flow past a backward facing step. Mesh and contours of ul velocity and pressure,
Re = 229.
Use of the CBS algorithm for incompressible flows !
[]
I.
\
Exp. [] ~ CBS-AC ........
2.5 (=
n =.
i
I
i
n
0
II
~
0 r
=-
-o 1.5 m
b
=. "
=.-
nl
,i
,4,..o r .m
i .i
.=
r .B
i
1=
>
" | i
1 .i
.
0.5
0
2
.. i
4
.,
~
"
.a
[] !
|. ._
=.-
g
/
|
_
|
10 6 8 Horizontal velocity
|
12
=,.
14
16
Fig. 4.10 Incompressible flow past a backward facing step. Comparison between experimental 1~ and numerical data, Re = 229.
(a) Unstructured surface mesh. Fig. 4.11
(b) Unstructured mesh, cross-section.
Incompressible flow past a sphere. Unstructured mesh.
121
122
Incompressible N e w t o n i a n laminar flows
: :
: iiiiii!!i~ ii!!::i!ili !::!ii i:.!ii::i:i::ili i ::: ::i~ iiii~!!i ii i :: ~' :~::~i::ii!!ii~ii~i~i~:~!i~ii!~:!iiii::i:~i!ii~i~i~!!i~ii!i~ii!~i::i~i~iii!ii!iii!~i~i~!~ii~i~ii~iiii::~
................................ .................. ~:~,:~:.............. ~ :::::~~i~::~i~i~:~:~i~!i~`i!ii~i!~!ii~i~ii~~
~:~i~~'ii:
~i.
.
: : ~ii~i~. ii::::~. :: i i ~,.": .~i~!~'~:~'.....
i~
................ ~:~:~:~:~:!~!~:~:~~i!!i~:~:~:~:~:~~i :~:~~i ~i ~i ~i :~:~:~ ~i
:::!i ':!i!i !i i i i!i i i i i i!i i i i !i i i i i i i i~i i :'.i!.i!!i~ii .!:~i..... ::: ~::i~i::i~:i:.i~i:!ii~:i~!~i~i~iii:,i:i:ii~iii~iii:!ii:i:i:::iii:~. ,~ii~i::i if! :i~:.i:;.i:.~::~iii'~',i:,':t~;ii:...... ,iii'~ii!i:,ii
9
iii!
..........
,,
;i........
~!;iii;,,i:,iiiii~:iiiii~i~iiiii~iiii,i:!iiiii!ii!ii,i!i!!i
.....
i:i::,i!iii!,,i,~,iiiiiii:,!~,!i!%i!i~i~,iiiii~iii,,i .::: 9 :~ iiii:.::iiiiiiiii!iiiii'i!!iiiiiii!i!i!i!i':il
,:.: ~.;~~:; i:,:::,:::',il ............................................................. iiiiiiiiiiiiii!~iiii~:ii~i iiii!!iliiiii~
:
(b) Pressure contours, Re = 100.
(a) u~ contours, Re= 100.
........ ~........................................... :~',~ :!:"~:: ~ !:: ~::i!':'~!!~i ' i!! i i~!!i!!!i!i',i !~!~i!:i~i i!i !i ~!~iiii!i~i!i!i~i! i~! ......'~:,::::":::J:';::......'~"~i ~!:::::::::::::::::: i~!:,!i',~i!~'i,i:.::~: .::.::::::~::..-.:: i:!i~i~!~i!~i~:.::..:..:..:. i~:~:~i!~i~!~i~i!~i~!i!~ ................. ..:~..:..:.: 9 ............
....... ,.-~.
.....
!i'~i: ! !i~i~i'!i!i!
.::.:~~.:..::i~:.:
.i~!!:.~!~!i!i!,!. ii: ::ii :~i:::;i.....l
(d) Pressure contours, Re = 200.
(c) Ul contours, Re = 200. Fig. 4.12
Incompressible flow past a sphere. Contours of ul velocity and pressure.
I
~
0.6
i
i
G u ,cat PrdsAs?t:~;97~
•
/
I
0.6
0.2
ro~.0"2
-0.2
-0.2
-0.6
-0.6
-1
0
i
90
i
i
180 270 Theta (degrees) (a) Re= 100.
360
"'~
-1
I
0
i
i
Present scheme G. ulcat and Asian (1997)
90
I
•
/ ~/
I
180 270 Theta (degrees) (b) Re = 200.
360
Fig. 4.13 Incompressible flow past a sphere. Coefficient of pressure distribution on the surface along the flow direction.
Adaptive mesh refinement 123 Figure 4.14 shows the mesh used in two dimensions and the solution obtained. As seen the mesh close to the cylinder is very fine in order to capture the boundary layer and separation. Figures 4.14(b) and (c) show the time history of vertical velocity component at the mid-exit point and drag coefficient. Both the histories are in good agreement with many reported results.
4.2.3 Quasi-implicit solution We have already remarked in Chapter 3 that the reduction of the explicit time step due to viscosity can be very inconvenient and may require a larger number of time steps. The example of the cavity is precisely in that category and at higher Reynolds numbers the reader will certainly note a very large number of time steps are required before results become reasonably steady. In quasi-implicit form, the viscous terms of the momentum equations are treated implicitly. Here the time step is governed only by the relation given in Eq. 4.7. We have rerun the problems with a Reynolds number of 5000 using the quasi-implicit solution15 which is explicit as far as the convective terms are concerned. The solution obtained is shown in Fig. 4.15 for Re = 5000.
We have discussed the matter of adaptive refinement in Chapters 13 and 14 of reference 1 in some detail. In that volume generally an attempt is made to keep the energy norm error constant within all elements. The same procedures concerning the energy norm error can be extended of course to viscous flow especially when this is relatively slow and the problem is nearly elliptic. However, the energy norm has little significance at high speeds and here we revert to other considerations which simply give an error indicator rather than an error estimator. Two procedures are available and will be used in this chapter as well as later when dealing with compressible flows. References 1674 list some of the many contributions to the field of adaptive procedures and mesh generation in fluid dynamics.
4.3.1 Second gradient- (curvature) based refinement Here the meaning of error analysis is somewhat different from that of the energy norm and we follow an approach where the error value is constant in each element. In what follows we shall consider first-order (linear) elements and the so-called h refinement process in which increased accuracy is achieved by variation of element size. The p refinement in which the order of the element polynomial expression is changed is of course possible. Many studies are available on hp refinements where both h and p refinements are carried out simultaneously. This has been widely studied by Oden et al.27, 28, 44, 45 but we believe that such refinements impose many limitations on mesh generation and solution procedures and as most fluid mechanics problems involve an explicit time marching algorithm, the higher-order elements are not popular.
124 IncompressibleNewtonian laminar flows
(a) Unstructured mesh. 0.4o" ~ ,-, E o o
0.30.2-
o
0
o
i ~
t
i
_~
I
0.1
>
"~ -0.1 ._o 1:
>
-0.2 -0.3 -0.4 0
1.55
1.5
50
100 150 200 Non-dimensional real time (b) Vertical velocity fluctuation at the exit mid-point.
II
1
1.45
1.4
1.35
1.3
I
0
I
50
I
I
100 150 Non-dimensional real time (c) Drag history.
Fig. 4.14 Transient flow past a circular cylinder, Re = 100.
200
Adaptive mesh refinement 125 Pressure contours
Streamlines
ul velocity distribution along mid-vertical line i
i
-0.4
-0.2
l
i
0.2 Ul
0.4
'Ghia' m 'Present'---
0.8
0.6
# 0.4
0.2
0
Fig. 4.15
-0.6
0
I
I
0.6
0.8
1
Lid-driven cavity. Quasi-implicit solution for a Reynolds number of 5000.
The determination of error indicators in linear elements is achieved by consideration of the so-called interpolation error. Thus if we take a one-dimensional element of length h and a scalar function q~, it is clear that the error in q5is of order O (h 2) and that it can be written as (see reference 21 for details) e
--
d2~ d2~ h #p_ ~h = ch2_d~xZ ~ ch 2 dx z
where ~bh is the finite element solution and c is a constant.
(4.11)
126
Incompressible Newtonian laminar flows x2
Exact
Linear
A v
lw
nl
n2
A v
n I , n 2, nodes
h, element size > Xl
Fig. 4.16
Interpolation error in a one-dimensional problem with linear shape functions.
If, for instance, we further assume that q5 = q5h at the nodes, i.e. that the nodal error is zero, then e represents the values on a parabola with a curvature of d 2q5h/dx 2. This allows c, the unknown constant, to be determined, giving for instance the maximum interpolation error as (see Fig. 4.16) emax
-- l h 2 d2~bh 8 dx 2
(4.12)
or an RMS departure error as 1
h2
eRMS --
d 2~bh dx 2
(4.13)
In deducing the expressions (4.12) and (4.13), we have assumed that the nodal values of the function ~b are exact. As is shown in reference 1 this is true only for some types of interpolating functions and equations. However, the nodal values are always more accurate than those noted elsewhere and it would be sensible even in one-dimensional problems to strive for equal distribution of such errors. This would mean that we would now seek an element subdivision in which h 2d2~h dx 2
=
C
(4.14)
To appreciate the value of the arbitrary constant C occurring in expression (4.14) we can interpret this as giving a permissible value of the limiting interpolation error and simply insisting that h2d2~ h dx 2 where
e p --- C
< ep
-
is the user-specified error limit.
(4.15)
Adaptive mesh refinement 127 If we consider the shape functions of ~bto be linear then of course second derivatives are difficult quantities to determine. These are clearly zero inside every element and infinity at the element nodes in the one-dimensional case or element interfaces in two or three dimensions. Some averaging process has therefore to be used to determine the curvatures from nodally computed ~bvalues. However, before discussing the procedures used for this, we must note the situation which will occur in two or three dimensions. The extension to two or three dimensions is of course necessary for practical engineering problems. In two and three dimensions the second derivatives (or curvatures) are tensor valued and given as
OXi OXj
(4.16)
and such definitions require the determination of the principal values and directions. These principal directions are necessary for element elongation which is explained in the following section. The determination of the curvatures (or second derivatives) of ~bh needs of course some elaboration. With linear elements (e.g. simple triangles or tetrahedra) the curvatures of ~bh which are interpolated as q5h =Nq~
(4.17)
are zero within the elements and become infinity at element boundaries. There are two convenient methods available for the determination of curvatures of the approximate solution which are accurate and effective. Both of these follow some of the matter discussed in Chapter 13 of reference 1 and are concerned with recovery. We shall describe them separately.
4.3.2 Local patch interpolation In the first method we simply assume that the values of the function such as pressure or velocity converge at a rate which is one order higher at nodes than that achieved at other points of the element. If indeed such values are more accurate it is natural that they should be used for interpreting the curvatures and the gradients. Here the simplest way is to assume that a second-order polynomial is used to interpolate the nodal values in an element patch which uses linear elements. Such a polynomial can be applied in a least square manner to fit the values at all nodal points occurring within a patch which assembles the approximation at a particular node. For triangles this rule requires at least five elements that are assembled in a patch and this is a matter easily achieved. The procedure of determining such least squares is given fully in Chapter 13 of reference 1 and will not be discussed here. However, once a polynomial distribution of, say, q~ is available then immediately the second derivatives of that function can be calculated at any point, the most convenient one being of course the point referring to the node which we require. On occasion, as we shall see in other processes of refinement, it is not the curvature which is required but the gradient of the function. Again the maximum value of the gradient, for instance of ~b, can easily be determined at any point of the patch and in particular at the nodes.
128 IncompressibleNewtonian laminar flows
4.3.3 Estimation of second derivatives at nodes In this method we assume that the second derivative is interpolated in exactly the same way as the main function and write the approximation as (02q5)
OXiOXj
h
(
02q~ ) *
(4.18)
= N OxiOxj
This approximation is made to be a least square approximation to the actual distilbution of curvatures, i.e. OXi OXj
d ~2 = 0
(4.19)
and integrating by parts to give
(OXiOXj
OXiOXj) d~'2-- -M- (
0NT 0N d~.~) ~
Oxi Oxj
(4.20)
where M is the mass matrix given by M - ~ NTN dr2
(4.21)
which of course can be 'lumped'.
4.3.4 Element elongation Elongated elements are frequently introduced to deal with 'one-dimensional' phenomena such as shocks, boundary layers, etc. The first paper dealing with such elongation was presented as early as 1987 by Peraire et al. 21 But the possible elongation was limited by practical considerations if a general mesh of triangles was to be used. An alternative to this is to introduce a locally structured mesh in shocks and boundary layers which connects to the completely unstructured triangles. This idea has been extensively used by Hassan et al., 46' 49 Zienkiewicz and W u 43 and Marchant et al. 58 in the compressible flow context. In both procedures it is necessary to establish the desired elongation of elements. Obviously in completely parallel flow phenomena no limit on elongation exists but in a general field the elongation ratio defining the maximum to minimum size of the element can be derived by considering curvatures of the contours. Thus the local error is proportional to the curvature and making h 2 times the curvature equal to a constant, we immediately derive the ratio hmax/hmin. In Fig. 4.17, X~ and X2 are the directions of the minimum and maximum principal values of the curvatures. Thus for an equal distribution of the interpolation error we can write for each node* h2nl~22 1-
I-c
(4.22)
*Principal curvatures and directions can be found in a manner analogous to that of the determination of principal stresses and their directions. Procedures are described in standard engineering texts.
Adaptive mesh refinement 129 X1 X2
~
S hmin = hmax
Fig. 4.17 Elementeloncjation~ and minimum and maximumelementsizes.
which gives us the stretching ratio s as
s =
hmax = min
O2~b
(4.23)
With the relations given above, we can formulate the following steps to adaptively refine a mesh: 1. Find the solution using an initial coarse mesh. 2. Select a suitable representative scalar variable and calculate the local maximum and minimum curvatures and directions of these at all nodes. 3. Calculate the new element sizes at all nodes from the maximum and minimum curvatures using the relation in Eq. (4.22). 4. Calculate the stretching ratio from the ratio of the calculated maximum to minimum element sizes [Eq. (4.23)]. If this is very high, limit it by a maximum allowable value. 5. Remesh the whole domain based on the new element size, stretching ratios and the direction of stretching. To use the above procedure, an efficient unstructured mesh generator is essential. We normally use the advancing front technique operating on the background mesh principle 21 in most of the examples presented here.* The information from the previous solution in the form of local mesh sizes, stretching ratio and stretching direction are stored in the previous mesh and this mesh is used as a background mesh for the new mesh. In the above steps of anisotropic mesh generation, to avoid very small and large elements (especially in compressible flows), the minimum and maximum allowable sizes of the elements are given as inputs. The maximum allowable stretching ratio is also supplied to the code to avoid bad elements in the vicinity of discontinuities. It is generally useful to know the minimum size of element used in a mesh as many flow solvers are conditionally stable. In such solvers the time step limitation depends very much on the element size. *Another successful unstructured mesh generator is based on Delaunay triangulation. The reader can obtain more information by consulting references 53-55, 64, 66, 69-73.
130
IncompressibleNewtonian laminar flows The procedure just described for an elongated element can of course be applied for the generation of isotropic meshes simply by taking the maximum curvature at every point. The matter to which we have not yet referred is that of suitably choosing the variable q~to which we will wish to assign the error. We shall come back to this matter later but it is clear that this has to be a well-representative quantity available from the choice of velocities, pressures, temperature, etc.
4.3.5 First derivative- (gradient) based refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
: .......
The nature of the fluid flow problems is elliptic in the vicinity of the boundaries often forming so-called viscous boundary layers. However, at some distance from the boundaries the equations become almost hyperbolic. For such hyperbolic problems it is possible to express the propagation type error in terms of the gradient of the solution in the domain. In such cases the error can be considered as
o4,
h--~ - C
On
(4.24)
where n is the direction of maximum gradient and h is the element size (minimum size) in the same direction. The above expression can be used to determine the minimum element size at all nodes or other points of consideration in exactly the same manner as was done when using the curvature. However, the question of stretching is less clear. At every point a maximum element size should be determined. One way of doing this is of course to return to the curvatures and find the curvature ratios. Another procedure to determine the maximum size of element is described by Zienkiewicz and W u . 43 In this the curvature of the streamlines is considered and hmax is calculated as
hmax _<_~R
(4.25)
where R is the radius of curvature of the streamline and 3 is a constant that varies between 0 and 1. Immediately the ratio between the maximum and minimum element size gives the stretching ratio.
4.3.6 Choice of variables In both methods of mesh refinement, i.e. those following curvatures and those following gradients, a particular scalar variable needs to be chosen to define the mesh. The simplest procedure is to consider only one of the many variables and here the one which is efficient is simply the absolute value of the velocity vector, i.e. lul. Such a velocity is convenient both for problems of incompressible flow and, as we shall see later, for problems of compressible flow where local refinement is even more important than here. (Very often in compressible flows the Mach number, which in a sense measures the same quantity, has been used.) Of course other variables can be chosen or any combination of variables such as velocities, pressures, temperatures, etc., can be used. Certainly in this chapter the
Slow flows - mixed and penalty formulations
absolute velocity is the most reasonable criterion. Some authors have considered using each of the problem variables to generate a n e w m e s h . 47' 52, 54, 55, 62 However, this is rather expensive and we believe velocity alone can give accurate results in most cases.
4.3.7 An example - ........................................
~
.
:
...........
: .......
. ...........................................................................................................................................................................................................................
Here we show some examples of incompressible flow problems solved using the abovementioned adaptive mesh generation procedures. In the first problem driven flow in a cavity which we have previously examined is again used. We use an initial uniform mesh with 481 nodes and 880 elements. Final meshes and solutions obtained by both curvature- and gradient-based procedures are shown in Fig. 4.18. In general the curvature-based procedure gives a wide band of refined elements along the circulation path [Fig. 4.18(a)]. However, the number of refined elements along the circulation path is smaller when the gradient-based refinement is used [Fig. 4.18(b)]. Both the meshes give excellent agreement with the benchmark solution of Ghia et al. 5 [Fig. 4.18(c)]. In the adaptive solutions shown here we have not used any absolute value of the desired error norm as the definition of a suitable norm presents certain difficulties, though of course the use of energy norm in the manner suggested in reference 1 could be adopted. We shall use such an error requirement in some later problems.
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In the preceding sections we have indicated various adaptive methods using complete mesh regeneration with error indicators of the interpolation kind. Obviously other methods of mesh refinement can be used (mesh enrichment or r refinement) and other procedures of error estimation can be employed if the problem is nearly elliptic. One such study in which the energy norm is quite effectively used is reported by Wu et al. 33 In that study the full transient behaviour of the v o n K a r m a n vortex street behind a cylinder was considered and the results are presented in Fig. 4.19. In this problem, the mesh is regenerated at fixed time intervals using the energy norm error and the methodologies largely described in Chapter 14 of reference 1. Similar procedures have been used by others and the reader can refer to these works.34, 35 iiiliiiiii lii!iiHi!~ii~iiiii ili ii liiili~~~~~i!iiii!il ili ~~~~~ii i ili!liiilii~~~~~ii~'! ~iiliiiiiiiiillii!iii!iiiil!iililiilii!iililii ! ii~~'~ii!~~"iiliii!iili!i!ilii!i!i!iiii!iiiii!iiiiii! i!iii!iiiilili !iiililii!~iiiiiiiii!i!H~i~~ilHi~~iiii~~ !i!ii!ii!iii!iii!i!i!iilii!i!~iiii!i!i!iiil!ii!iiiliil!iiil!!i!i~'~~'~!i!!!ii i!i!i !i i!iii!ililil!i!iii!iiii~iiiiiiilili! i!i!i!ilili!i!iiliiiiiiiili!i!i!ii!ii !iii!iii!i!i!iil!iilii:~'~'~i!i~............... ~~i!i!i!:i'ii li!i ~'i !~i~ii iiii li!!iii!i!ii!iiiiiili!i!iiiii!ii!iiiiii!iiiiiiiiili!i i li!i ii!iiiiili~~'!i~!iii!ii!ii i!~i~i!i~ ~!~ ~!iiliii!iililililii!i!i!i!i!iiiii!i!iiiiililiiliii!i!iii!iiiliiiiiii!i!iiiil !iii!ii!iilililii l!ii !i!i iiii!ilili!iiiiiililiiHi!i !iii!i!i iili!i iiilililiii!i!iiiiiiii
4.5.1 Analogy with incompressible elasticity Slow, viscous incompressible flow represents the extreme situation at the other end of the scale from the inviscid problem of Sec. 1.3. Here all dynamic (acceleration) forces are, a p r i o r i , neglected and Eqs (4.1) and (4.2) reduce, in indicial form, to
Oui
OXi
= ~ - 0
(4.26)
131
132
Incompressible Newtonian laminar flows Adapted mesh
Streamlines
Pressure contours
(a) Curvature based procedure, Nodes: 2389, Elements: 4599
(b) Gradient based procedure, Nodes: 1034, Elements: 1962 'Ghia' 'curvature' ..... 'gradient .......
,/
0.8
-
/r
/
/7
I2
0.6 -
0.4 -
0.2
0 -0.6
-0.4
-0.2
0
0.2
0.4
!
I
0.6
0.8
1
U1
(c) Comparison of velocity at mid-vertical plane Fig. 4.18 Lid-driven cavity, Re = 5000. Adapted meshes using curvature- and gradient-based refinements and solutions.
c~
c~ ,I.,-
~
.Q v
oO_
~.om 0 c-
in Q~
c3.J
E
v~
.~
c~
oo_ c~ v~j
o~_ E c~
t'-
O cO u
v~
CIJ
v 'J
CL
E cIJ
L_ CIJ
Q_ c~
.>_
c~ Lr~
~P
U
.~
- 0c -
cB O_ L C~
~o Lsl Ltl L_ Q_ O
E ~m
c~
.cl~
9
,~ Q~ •
Lll
II
~.>
~m
~
134
IncompressibleNewtonian laminar flows and
OXj
OXi
(4.27)
Pgi -- 0
The above are completed of course by the constitutive relation ~ij
#
( ~Xj 0ui -4r~x/0UJ--(~ij-~XkJ 32 0uk ~
(4.28)
which is identical to the problem of incompressible elasticity in which we replace: 1. the displacements by velocities, 2. the shear modulus G by the viscosity #, and 3. the mean stress by negative pressure.
4.5.2 Mixed and penalty discretization ....................................................................................................................
: .........................................................................
~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
- ........................
---: ....................
--: .....................
-----:~-
.................
- -----:
.........................................
- .......
: ...............
---: ..........
The discretization of u and p is given by the independent approximations u -- Nufi
p = Npp
(4.29)
and may be used to construct a mixed form. The problem may be given as a penalty form by augmenting Eq. (4.26) by P / 7 where "7is a large parameter, thus m r,Su + p = 0 "7
(4.30)
When suitable discontinuous Np are used, penalty terms computed using reduced integration are equivalent to the mixed form 75 (see Chapter 1t of reference 1 for further details). The use of penalty forms in fluid mechanics was introduced in the early 1970's 76-78 and is fully discussed elsewhere. 79-81. As computationally it is advantageous to use the mixed form and introduce the penalty parameter only to eliminate the ~ values at the element level, we shall presume such penalization is done after the discretization of the mixed problem. The discretized equations of the mixed problem will always be of the form (4.31) in which K
-
f~ #BTl0Bdf2
where B - ,-qNu
G - f BTmNpdf2;
V - f NpVNpdS2
f = f NTpgdf2 + f r NTtdF t
(4.32)
Slow flows - mixed and penalty formulations
O (h)
(T3BI/3C)
O (h 2)
(T6/3C)
O (h 2)
(Q9/4C)
O (h 2)
(T6BI/3D)*
O(h)
(Q4/1D)*
O (h 2)
(Q9/4D)*
O (h 2)
(Q9/3D)
0 (h)
(T6/1 D)
(a) Continuous p interpolation
v
v
v
(b) Discontinuous p interpolation
o Velocity node v Pressure node * Denotes elements failing Babu,~ka-Brezzi test but still performing reasonably
Fig. 4.20 Someuseful velocity-pressureinterpolations and their asymptotic, energynorm convergencerates.
135
136 IncompressibleNewtonian laminar flows and the penalty number, q,, is introduced purely as a numerical convenience. This is taken generally as 79' 81 7 = (107-108)# There is little more to be said about the solution procedures for creeping incompressible flow with constant viscosity. The range of applicability is of course limited to low velocities of flow or high viscosity fluids such as oil, blood in biomechanics applications, etc. It is, however, important to recall here that the mixed form allows only certain combinations of Nu and N p interpolations to be used without violating the convergence conditions. This is discussed in detail in Chapter 11 of reference 1, but for completeness Fig. 4.20 lists some of the available elements together with their asymptotic convergence rates. 82 Many other elements useful in fluid mechanics are documented elsewhere, 83-85 but those of proven performance are given in the table. It is of general interest to note that frequently elements with Co continuous pressure interpolations are used in fluid mechanics and indeed that their performance is generally superior to those with discontinuous pressure interpolation on a given mesh, even though the cost of solution is marginally greater. It is important to note that the recommendations concerning the element types for the Stokes problem carry over unchanged to situations in which dynamic terms are of importance. The fairly obvious extension of the use of incompressible elastic codes to Stokes flow is undoubtedly the reason why the first finite element solutions of fluid mechanics were applied in this area.
Incompressible Newtonian fluid dynamics, including vortex shedding, has been discussed in this chapter. Several more two- and three-dimensional problems can be found in recent publications on the CBS scheme. In addition to coveting several benchmark problems of incompressible flows, we have also discussed several adaptive procedures for fluid dynamics problems. The chapter concludes by briefly describing the penalty methods.
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1. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method: Its Basis and Fundamentals. Elsevier, 6th edition, 2005. 2. P. Nithiarasu. An efficient artificial compressibility (AC) scheme based on the characteristic based split (CBS) method for incompressible flows. International Journalfor Numerical Methods in Engineering, 56:1815-1845, 2003. 3. P. Nithiarasu, J.S. Mathur, N.P. Weatherill and K. Morgan. Three-dimensional incompressible flow calculations using the characteristic based split (CBS) scheme. International Journal for Numerical Methods in Fluids, 44:1207-1229, 2004. 4. R. Codina, M. Vfizquez and O.C. Zienkiewicz. General algorithm for compressible and incompressible flows, Part III - A semi-implicit form. International Journal for Numerical Methods in Fluids, 27:13-32, 1998. 5. U. Ghia, K.N. Ghia and C.T. Shin. High-resolution for incompressible flow using the NavierStokes equations and multigrid method. J. Comp. Phys., 48:387--411, 1982.
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138 IncompressibleNewtonian laminar flows 25. B. Palmerio. A two-dimensional FEM adaptive moving-node method for steady Euler flow simulation. Computer Methods in Applied Mechanics in Engineering, 71:315-340, 1988. 26. R. L6hner. Adaptive remeshing for transient problems. Computer Methods in Applied Mechanics and Engineering, 75:195-214, 1989. 27. L. Demkowicz, J.T. Oden, W. Rachowicz and O. Hardy. Toward a universal h-p adaptive finite element strategy, Part 1. Constrained approximation and data structure. Computer Methods in Applied Mechanics and Engineering, 77:79-112, 1989. 28. J.T. Oden, L. Demkowicz, W. Rachowicz and T.A. Westermann. Toward a universal h-p adaptive finite element strategy, Part 2. A posteriori error estimation. Computer Methods in Applied Mechanics and Engineering, 77:113-180, 1989. 29. O.C. Zienkiewicz. Adaptivity-fluids-localization - the new challenge to computational mechanics. Transactions of Canadian Society of Mechanical Engineers, 15:137-145, 1991. 30. E.J. Probert, O. Hassan, K. Morgan and J. Peraire. An adaptive finite element method for transient compressible flows with moving boundaries. International Journal for Numerical Methods in Engineering, 32:751-765, 1991. 31. E.J. Probert, O. Hassan, J. Peraire and K. Morgan. An adaptive finite element method for transient compressible flows. International Journal for Numerical Methods in Engineering, 32:1145-1159, 1991. 32. A. Evans, M.J. Marchant, J. Szmelter and N.P. Weatherill. Adaptivity for compressible flow computations using point embedding on 2-d structured multiblock meshes. International Journal for Numerical Methods in Engineering, 32:895-919, 1991. 33. J. Wu, J.Z. Zhu, J. Szmelter and O.C. Zienkiewicz. Error estimation and adaptivity in NavierStokes incompressible flows. Computational Mechanics, 61:30--39, 1991. 34. J.F. Hetu and D.H. Pelletier. Adaptive remeshing for viscous incompressible flows. AIAA Journal, 30:1986-1992, 1992. 35. J.E Hetu and D.H. Pelletier. Fast, adaptive finite element scheme for viscous incompressible flows. AIAA Journal, 30:2677-2682, 1992. 36. J. Peraire, J. Peiro and K. Morgan. Adaptive remeshing for 3-dimensional compressible flow computations. J. Comp. Phys., 103:269-285, 1992. 37. E.J. Probert, O. Hassan, K. Morgan and J. Peraire. Adaptive explicit and implicit finite element methods for transient thermal analysis. International Journal for Numerical Methods in Engineering, 35:655-670, 1992. 38. J. Szmelter, M.J. Marchant, A. Evans and N.P. Weatherill. 2-dimensional Navier-Stokes equations with adaptivity on structured meshes. Computer Methods in Applied Mechanics and Engineering, 101:355-368, 1992. 39. R. L6hner and J.D. Baum. Adaptive h-refinement on 3D unstructured grid for transient problems. International Journal for Numerical Methods in Fluids, 14:1407-1419, 1992. 40. M.J. Marchant and N.P. Weatherill. Adaptivity techniques for compressible inviscid flows. Computer Methods in Applied Mechanics and Engineering, 106:83-106, 1993. 41. W. Rachowicz. An anisotropic h-type mesh refinement strategy. Computer Methods in Applied Mechanics and Engineering, 109:169-181, 1993. 42. P.A.B. de Sampaio, P.R.M. Lyra, K. Morgan and N.P. Weatherill. Petrov-Galerkin solutions of the incompressible Navier-Stokes equations in primitive variables with adaptive remeshing. Computer Methods in Applied Mechanics and Engineering, 106:143-178, 1993. 43. O.C. Zienkiewicz and J. Wu. Automatic directional refinement in adaptive analysis of compressible flows. International Journal for Numerical Methods in Engineering, 37:2189-2210, 1994. 44. J.T. Oden, W. Wu and M. Ainsworth. An a posteriori error estimate for finite element approximations of the Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering, 111:185-202, 1994.
References 139 45. J.T. Oden, W. Wu and V. Legat. An hp adaptive strategy for finite element approximations of the Navier-Stokes equations. International Journal for Numerical Methods in Fluids, 20:831-851, 1995. 46. O. Hassan, E.J. Probert, K. Morgan and J. Peraire. Mesh generation and adaptivity for the solution of compressible viscous high speed flows. International Journal for Numerical Methods in Engineering, 38:1123-1148, 1995. 47. M.J. Castro-Diaz, H. Borouchaki, P.L. George, F. Hecht and B. Mohammadi. Error interpolation minimization and anisotropic mesh generation. Proc. Int. Conf. Finite Elements Fluids - New Trends and Applications, 1139-1148, Venezia, 15-21 October 1995. 48. R. Lt~hner. Mesh adaptation in fluid mechanics. Eng. Frac. Mech., 50:819-847, 1995. 49. O. Hassan, K. Morgan, E.J. Probert and J. Peraire. Unstructured tetrahedral mesh generation for three dimensional viscous flows. International Journal for Numerical Methods in Engineering, 39:549-567, 1996. 50. D.D.D. Yahia, W.G. Habashi, A. Tam, M.G. Vallet and M. Fortin. A directionally adaptive methodology using an edge based error estimate on quadrilateral grids. International Journal for Numerical Methods in Fluids, 23:673-690, 1996. 51. M. Fortin, M.G. Vallet, J. Dompierre, Y. Bourgault and W.G. Habashi. Anisotropic mesh adaption: theory, validation and applications. Computational Fluid Dynamics '96, 174-180, 1996. 52. M.J. Castro-Diaz, H. Borouchaki, P.L. George, F. Hecht and B. Mohammadi. Anisotropic adaptive mesh generation in two dimensions for CFD. Computational Fluid Dynamics '96, 181-186, 1996. 53. K.S.V. Kumar, A.V.R. Babu, K.N. Seetharamu, T. Sundararajan and P.A.A. Narayana. A generalized Delaunay triangulation algorithm with adaptive grid size control. Comm. Num. Meth. Eng., 13:941-948, 1997. 54. H. Borouchaki, P.L. George, F. Hecht, P. Laug and E. Saltel. Delaunay mesh generation governed by metric specifications, Part I. Algorithms. Finite Elem. Anal. Des., 25:61-83, 1997. 55. H. Borouchaki, P.L. George and B. Mohammadi. Delaunay mesh generation governed by metric specifications, Part II. Applications. Finite Elem. Anal. Des., 25:85-109, 1997. 56. W. Rachowicz. An anisotropic h-adaptive finite element method for compressible Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering, 146:231-252, 1997. 57. J. Peraire and K. Morgan. Unstructured mesh generation including directional refinement for aerodynamic flow simulation. Finite Elem. Anal. Des., 25:343-356, 1997. 58. M.J. Marchant, N.P. Weatherill and O. Hassan. The adaptation of unstructured grids for transonic viscous flow simulation. Finite Elem. Anal. Des., 25:199-218, 1997. 59. H. Borouchaki, F. Hecht and P.J. Frey. Mesh gradation control. International Journal for Numerical Methods in Fluids, 43:1143-1165, 1998. 60. H. Borouchaki and P.J. Frey. Adaptive triangular-quadrilateral mesh generation. International Journal for Numerical Methods in Engineering, 41:915-934, 1998. 61. H. Jin and S. Prudhomme. A posteriori error estimation of steady-state finite element solution of the Navier-Stokes equations by a subdomain residual method. Computer Methods in Applied Mechanics and Engineering, 159:19-48, 1998. 62. P. Nithiarasu and O.C. Zienkiewicz. Adaptive mesh generation procedure for fluid mechanics problems. International Journal for Numerical Methods in Engineering, 47:629-662, 2000. 63. P.J. Green and R. Sibson. Computing Dirichlet tessellations in the plane. Comp. J., 21:168-173, 1978. 64. D.F. Watson. Computing the N-dimensional Delaunay tessellation with application to Voronoi polytopes. Comp. J., 24:167-172, 1981. 65. J.C. Cavendish, D.A. Field and W.H. Frey. An approach to automatic three dimensional finite element mesh generation. International Journal for Numerical Methods in Engineering, 21: 329-347, 1985.
140 Incompressible Newtonian laminar flows 66. W.J. Schroeder and M.S. Shephard. A combined octree Delaunay method for fully automatic 3-D mesh generation. International Journal for Numerical Methods in Engineering, 29:37-55, 1990. 67. N.P. Weatherill. Mesh generation for aerospace applications. Sadhana- Acc. Proc. Eng. Sci., 16:1-45, 1991. 68. N.P. Weatherill, P.R. Eiseman, J. Hause and J.E Thompson. Numerical Grid Generation in Computational Fluid Dynamics and Related Fields. Pineridge Press, Swansea, 1994. 69. N.P. Weatherill and O. Hassan. Efficient 3-dimensional Delaunay triangulation with automatic point generation and imposed boundary constraints. International Journal for Numerical Methods in Engineering, 37:2005-2039, 1994. 70. G. Subramanian, V.V.S. Raveendra and M.G. Kamath. Robust boundary triangulation and Delaunay triangulation of arbitrary planar domains. International Journal for Numerical Methods in Engineering, 37:1779-1789, 1994. 71. N.P. Weatherill, O. Hassan and D.L. Marcum. Compressible flow field solutions with unstructured grids generated by Delaunay triangulation. AIAA J., 33:1196-1204, 1995. 72. R.W. Lewis, Y. Zhang and A.S. Usmani. Aspects of adaptive mesh generation based on domain decomposition and Delaunay triangulation. Finite Elem. Anal. Des., 20:47-70, 1995. 73. H. Borouchaki and P.L. George. Aspects of 2-d Delaunay mesh generation. International Journal for Numerical Methods in Engineering, 40:1957-1975, 1997. 74. J.E Thompson, B.K. Soni and N.P. Weatherill, editors. Handbook of Grid Generation. CRC Press, 1999. 75. D.S. Malkus and T.J.R. Hughes. Mixed finite element method reduced and selective integration techniques: a unification of concepts. Computer Methods in Applied Mechanics and Engineering, 15:63-81, 1978. 76. O.C. Zienkiewicz and P.N. Godbole. Viscous, incompressible flow with special reference to non-Newtonian (plastic) fluids. In J.T. Oden et al., editors, Finite Elements in Fluids, volume 2, pp. 25-55. Wiley, Chichester, 1975. 77. T.J.R. Hughes, R.L. Taylor and J.F. Levy. A finite element method for incompressible flows. In Proc. 2nd Int. Symp. on Finite Elements in Fluid Problems ICCAD. Sta Margharita Ligure, Italy, pp. 1-6, 1976. 78. O.C. Zienkiewicz and P.N. Godbole. A penalty function approach to problems of plastic flow of metals with large surface deformation. J. Strain Anal., 10:180-183, 1975. 79. O.C. Zienkiewicz and S. Nakazawa. The penalty function method and its applications to the numerical solution of boundary value problems. ASME, AMD, 51:157-179, 1982. 80. J.T. Oden. R.I.P. methods for Stokesian flow. In R.H. Gallagher, D.N. Norrie, J.T. Oden and O.C. Zienkiewicz, editors, Finite Elements in Fluids, volume 4, Chapter 15, pp. 305-318. Wiley, Chichester, 1982. 81. O.C. Zienkiewicz, J.P. Vilotte, S. Toyoshima and S. Nakazawa. Iterative method for constrained and mixed approximation. An inexpensive improvement of FEM performance. Computer Methods in Applied Mechanics and Engineering, 51:3-29, 1985. 82. O.C. Zienkiewicz, Y.C. Liu and G.C. Huang. Error estimates and convergence rates for various incompressible elements. International Journal for Numerical Methods in Engineering, 28:2191-2202, 1989. 83. M. Fortin. Old and new finite elements for incompressible flow. International Journal for Numerical Methods in Fluids, 1:347-364, 1981. 84. M. Fortin and N. Fortin. Newer and newer elements for incompressible flow. In R.H. Gallagher, G.F. Carey, J.T. Oden and O.C. Zienkiewicz, editors, Finite Elements in Fluids, volume 6, Chapter 7, pp. 171-188. Wiley, Chichester, 1985. 85. M.S. Engleman, R.L. Sani, P.M. Gresho and H. Bercovier. Consistent v. reduced integration penalty methods for incompressible media using several old and new elements. International Journal for Numerical Methods in Fluids, 4:25-42, 1982.
i!iiiiii iiiiiiiii i i i i i iii iiliiUiDi! Incompressible non-Newtonian flows
!ii~i!ii~i~ii~iii!i!iii~i!i!ii!iii!i~ii~~i":ii~ii:!'~ii~!ii~iil!ii~i!~!iii@ iiiiiiiiil !iiii~qi~iHi!iiHiH!ii~i~l~i~!iiiii~i!iiiiHiii !i!i!ii~Hiii!i•i~i••ii!iiii!i!iiii~i!i~iiiii!iiiilii•iiH!•i•!iiiiii~iiilii!i!iiHiiiiiili !ii••iii~ii!i~!iii!iii•ii!}iiiiiiii i{iii~i!i~i•i!iiiiiii•ii!iiiiiiiiiii!iiiiiiiiiiiii!iiiiiiiiiii In this chapter we discuss the non-Newtonian effect in flow problems. In non-Newtonian flow cases, the viscosity of the material is dependent on non-linear relations of the stress and/or strain rate. Several models are available for non-Newtonian fluids, the simplest and widely used among these is the one based on a power law. We start with such a formulation for metal and polymer forming in Sec. 5.2. Fluids that partially return to their original form when applied stress is released are called viscoelastic. The models for viscoelastic fluids can be quite complicated. In Sec. 5.3 we present some simplified model forms and discuss their numerical solution using the CBS procedure described in Chapter 3. In Sec. 5.4 we briefly discuss a direct displacement method based on the CBS approach to solve impact problems. ~ii i il@' i ~!~''~~'i!~~~ ii!~'i~'i~l~'~~i~ii!ii!iil ~l ~~'~ii~'~'~iiili @i~~~iili!~~~~~ iiii~'~'~~i~'~'i i i@i@i ~'~i i~@ i!i!i ii~~ i~i i i!iliili i i @~~ iii~'~:i ~~:i':~!~ iliiiili ii i!iili i@ i ii i~~ ii@@ i ~~~ii !i ~ii~iiii~~~@iiii~:':~i' i~ @i!@~~~~ i!@ii"~:~iiiiiiiii!iiiii! l ~~!~~~i~ ~iil iiilii~"ii!ii i i i i i@iililiiiiili @~": i !~~~!!~ '@ii iii@i@il~i~~~ i ii@iii! i i i!ii i !iii i ii!i!
5.2.1 Non-Newtonian flows including viscoplasticity and plasticity In many fluids the viscosity, though isotropic, may be dependent on the rate of strain ~ij a s well as on the state variables such as temperature or total deformation. Typical here is, for instance, the behaviour of many polymers, hot metals, etc., where a power law of the type # = #0 e (5.1)
-(m-l)
with #0 = #0( T, ~) governs the viscosity-strain rate dependence where m is a physical constant. In the above ~ is the second invariant of the deviatoric strain rate tensor defined from Eq. (3.35), T is the (absolute) temperature and ~ is a total strain invariant. This secant viscosity can of course be obtained by plotting the relation between the deviatoric stresses and deviatoric strains or their invariants, as Eq. (3.34) simply defines the viscosity by the appropriate ratio of the stress to strain rate. Such plots are shown in Fig. 5.1 where O denotes the second deviatoric stress invariant. The above power
142 Incompressible non-Newtonian flows
m
(a) Linear,newtonian,fluid Iml < 1
G
(b) Non-newtonianpolymers
(Bingham)
o ,r
7=0 I
(Ideal plasticity) ~y
(c) Viscoplastic-plasticmetals Fig. 5.1 Stress ~, viscosity/1, and strain rate ~ relationships for various materials.
law relation of Eq. (5.1) is known as the Oswald de Wahle law (or power law) and is illustrated in Fig. 5.1(b). In a similar manner viscosity laws can be found for viscoplastic and indeed purely plastic behaviour of an incompressible kind. For instance, in Fig. 5.1(c) we show a viscoplastic Bingham fluid in which a threshold or yield value of the second stress invariant has to be exceeded before any strain rate is observed. Thus for the viscoplastic fluid illustrated it is evident that a highly non-linear viscosity relation is obtained. This can be written as -
where
O'y is the value of the second deviator invariant at yield.
(5.2)
Non-Newtonian flows - metal and polymer forming 143 The special case of pure plasticity follows of course as a limiting case when the fluidity parameter -y = 0, and now we have simply O'y
# = --=_
(5.3)
Of course, once again O'y c a n be dependent on the state of the fluid, i.e. O'y
---
fly(T, ?:)
(5.4)
The solutions (at a given state of the fluid) can be obtained by various iterative procedures, noting that Eq. (4.31) continues to be valid but now with the matrix K being dependent on viscosity, i.e. K = K(#) = K(~) = K(u)
(5.5)
thus being dependent on the solution. A total iteration process can be used simply here (see reference 1). Thus rewriting Eq. (4.31) as A{p}
= {~}
(5.6)
and noting that A = A(fi, [~) we can write -- A n l
0
A~ = A(fi, ~)n
(5.7)
Starting with an arbitrary value of # we repeat the solution until convergence is obtained. Such an iterative process converges rapidly (even when, as in pure plasticity, # can vary from zero to infinity), providing that the forcing f is due to prescribed boundary velocities and thus immediately confines the variation of all velocities in a narrow range. In such cases, five to seven iterations are generally required to bring the difference of the nth and (n + 1)th solutions to within the 1 per cent (euclidian) norm. The first non-Newtonian flow solutions were applied to polymers and to hot metals in the early 1970s. 2-4 Application of the same procedures to the forming of metals was introduced at the same time and has subsequently been widely developed. 5-3~ It is perhaps difficult to visualize steel or aluminium behaving as a fluid, being conditioned to use these materials as structural members. If, however, we note that during the forming process the elastic strains are of the order of 10 -6 while the plastic strain can reach or exceed a value of unity, neglect of the former (which is implied in the viscosity definition) seems justifiable. This is indeed borne out by comparison of computations based on what we now call flow formulation with elastoplastic computation or experiment. The process has alternatively been introduced as a 'rigid-plastic' f o r m , l l, 12 though such modelling is more complex and less descriptive.
144
Incompressible non-Newtonian flows
Today the methodology is widely accepted for the solution of metal and polymer forming processes, and only a limited selection of references of application can be cited. The reader would do well to consult references 21, 34, 35 and 36 for a recent survey of the field.
5.2.2 Steady-state problems of forming Two categories of problems arise in forming situations. Steady-state flow is the first of these. In this, a real, continuing, flow is modelled, as shown in Fig. 5.2(a) and here velocity and other properties can be assumed to be fixed in a particular point of space. In Fig. 5.2(b) the more usual transient processes of forming are illustrated and we shall deal with these later. In a typical steady-state problem if the state parameters T and ~ defining the temperature and total strain invariant are known in the whole field, the solution can be carried out in the manner previously described. We could, for instance, assume that the 'viscous' flow of the problem of Fig. 5.3 is that of an ideally plastic material under isothermal conditions modelling an extrusion process and obtain the solution shown in Table 5.1. For such a material exact extrusion forces can be calculated 37 and the table shows the errors obtained with the flow formulation using Prescribed
Prescribed
velocity
Extrusion
i _I___ I .... -I
-I--'r-L
I "-'I-'I -- --I -- -_I_ I---"I"17 I I ... _1 . . . I__ I "I' ~ - I "I" -- L" "I" I ~
Rolling (a) Steady rate Fig. 5.2 Formingprocessestypically used in manufacture.
. I. I
/
m I
_
Non-Newtonian flows - metal and polymer forming
Moving mesh _
-
_
_
.~.,. J J J J I ,I ~ ~j~
.
.
.
.
.
.
.
.
~--------
Extrusion
Rolling
!11
IIII i11| Forging
!
IIII IIII IIII
IIII IIII Cutting
Sheet forming (deep drawing) (b) Transient Fig. 5.2 Continued.
I
145
146 Incompressiblenon-Newtonian flows Slip boundary
)oundary
v=O u=l
specified
Fig. 5.3 Plane strain extrusion (extrusion ratio 2:1) with ideal plasticity assumed.
different triangular elements of Fig. 5.3 and two meshes. 38 The fine mesh here was arrived at by using error estimates and a single adaptive remeshing. In general the problem of steady-state flow is accompanied by the evolution of temperature (and other state parameters such as the total strain invariant ~) and here it is necessary to couple the solution with the heat balance and possibly other evolution equations. The evolution of heat has already been discussed and the appropriate conservation equations such as Eq. (4.6) can be used. It is convenient now to rewrite this equation in a modified form. Table 5.1 Comparisons of performance of several triangular mixed elements of Fig. 5.3 in a plane extrusion problem (ideal plasticity assumed) 38 Mesh 1 (coarse)
Mesh 2 (fine)
Element type
Ext. force
Force error
CPU(s)
Ext. force
Force error
CPU(s)
T6/1D T6B 1/3D T6B 1/3D* T6/3C
28 901.0 31043.0 29 031.0 27 902.5
12.02 20.32 12.52 8.15
67.81 75.76 73.08 87.62
25 990.0 26 258.0 26 229.0 25 975.0
0.73 1.78 1.66 0.67
579.71 780.13 613.92 855.38
Exact
25 800.0
0.00
25 800.0
0.00
-
-
Non-Newtonian flows - metal and polymer forming
First, we note that the kinetic energy is generally negligible in the problems considered and that with a constant specific heat d per unit volume we can write
pE ~ pe = OT
(5.8a)
Second, we observe that the internal work dissipation can be rewritten by the identity
0 0 OX---~(pui ) - -'~Xj (7"ji Ui ) ~ --O'ji ~ji
(5.8b)
where, by Eq. (1.8a), O'ji "-- 7"ji --
t~jip
(5.8C)
and, by Eq. (1.2),
l(Ouj
9
OUi)
(5.8d)
We note in passing that in general the effect of the pressure term in Eq. (5.8b) is often negligible and can be omitted if desired. Using the above and inserting the incompressibility relation we can write the energy conservation as [for an alternative form see Eq. (4.6)] C ---~- .2f_ Olg i _~x i
__ ~
k _~ixi
_ ( o.i j ~ ij + p g i u i ) - 0
(5.9)
where ~ - pcv. The solution of the coupled problem can be carried out iteratively. Here the term in the last bracket can be evaluated repeatedly from the known velocities and stresses from the flow solution. We note that the first bracketed term represents a total derivative of the convective kind which, even in the steady state, requires the use of the special weighting procedures discussed in Chapter 2. Such coupled solutions were carried out for the first time as early as 1973 and later in 1978, 8, 9 but are today practised routinely.
Example 5.1 Steady state rolling problem
Figure 5.4 shows a typical thermally coupled solution for a steady-state rolling problem from reference 9. It is of interest to note that in this problem boundary friction plays an important role and that this is modelled by using thin elements near the boundary, making the viscosity coefficient in that layer pressure dependent. 22 This procedure is very simple and although not exact gives results of sufficient practical accuracy.
5.2.3 Transient problems with changing boundaries These represent the second, probably larger, category of forming problems. Typical examples here are those of forging, indentation, etc., and again thermal coupling can be included if necessary. Figures 5.5 and 5.6 illustrate typical applications. The solution for velocities and internal stresses can be readily accomplished at a given configuration providing the temperatures and other state variables are known at
147
0
I'0 !
-IT
w
Jl
!
0 0
J_l_
--A
E
0
~ _
II I'.--
~1001~ = / .
0 it)
E
o u3 II X
E
II
U3 II
E
(1)
~" (~
~
~-
v II
e-
o ~
L_ 0
C
=
-(3
.~_ 0
o
E
.~
E
~ OOL =.L
o u3
E
u3 ~--
II X
E o
I'~ u3 II X
E o ~
II X
slu!0d leP0N
~" 0 0
./ Q..
o
o.._ ""
ou=
"--
E ~
m
QOh r
0
|
+..,
,==. LL
Non-Newtonian flows - metal and polymer forming
I
uAt
!
t.l, ~.
"-"
,"
/
-'-,~-"f"w--r
. . . . . . . . .
10% u~ ~ m /
I
/
(b) t = 15At
(a) t = 0
l
w~~~mm
f'-'T~ . . . . . . . . . . . . .
/ /----~ \ \
__i i -~ |
(c) t= 30At
(d) t = 45At
Fig. 5.5 Punch indentation problem (penalty function approach). 5 Updated mesh and surface profile with 24 isoparametric elements. Ideally plastic material; (a), (b), (c) and (d) show various depths of indentation (reduced integration is used here).
that instant. This allows the new configuration to be obtained both for the boundaries and for the mesh by writing explicitly
AX i -- uiAt
(5.10)
as the incremental relation. If thermal coupling is important increments of temperature need also to be evaluated. However, we note that for convected coordinates Eq. (5.9) is simplified as the convected terms disappear. We can now write ~ -c3T ~
OXiCg(OT) k ~ x i - (crij~ij + pgiui +
gh) -- 0
(5.11)
where the last term is the heat input known at the start of the interval and computation of temperature increments is made using either explicit or implicit procedures discussed in Chapter 3. Indeed, both the coordinate and thermal updating can make use iteratively of the solution on the updated mesh to increase accuracy. However, it must be noted that any
149
OO II LL
0 o 0
t--
0
E
II
t~
"0 "0 O.
o~ II
O
LL
0 o rE
O. "0
o~ II
,I--
ii
O
El 00 O 0D rO)
E O)
O)
O..
._> "O
or)
o (...
E
(u
.__.
o'1 m._ oJ (-..
II
('o
<~:
E
n~
o'1
(-n~
n~
a.)
"o (.('o
n~ "-o
"o
E
s ...C:: aJ
E
(u d.-, n~
o a~
E
. m
v to d-i ('o
aJ
n~ c-oJ
t',-
r'--
T "
o
II
v
u'l ,.,. LL
Non-Newtonian f l o w s - metal and polymer forming
2~~ . ~ ~ (0.8.0,8/~ J-.....~
_ .~.~..
~.
.
.
.
.
.
'Effective' strain (~:) t = 2.9 s
~624w
.
.
.
.
.
Temperature (7") t = 2.9 s (b) Contours of state parameters at t = 2.9 s
0.4
0.3
g ~0.2
3
0.1
I
2
I
4
I
6
I
8 Time (s)
I
10
I
12
I
14
I
16
(c) Load versus time Fig. 5.6 (Continued) A transient extrusion problem with temperature and strain-dependent yield.4~ Adaptive mesh refinement uses T6/1D elements of Fig. 4.18.
continuous mesh updating will soon lead to unacceptable meshes and some form of remeshing is necessary.
Example 5.2 Punchindentation problem
In the example of Fig. 5.5, 5 in which ideal plasticity was assumed together with isothermal behaviour, it is necessary only to keep track of boundary movements. As
151
152
Incompressible non-Newtonian flows temperature and other state variables do not enter the problem the remeshing can be done simply- in the case shown by keeping the same vertical lines for the mesh position.
Example 5.3 Transientextrusion problem
Figure 5.6 shows a more realistic problem. 39' 40 Here, when a new mesh is created an interpolation of all the state parameters from the old to the new mesh positions is necessary. In such problems it is worthwhile to strive to obtain discretization errors within specified bounds and to remesh adaptively when these errors are too large. In the present examples similar methods have been adopted with success 41' 42 and in Fig. 5.6 we show how remeshing proceeds during the forming process. It is of interest simply to observe that here the energy norm of the error is the measure used. The details of various applications can be found in the extensive literature on the subject. This also deals with various sophisticated mesh updating procedures. One particularly successful method is the so-called ALE (arbitrary Lagrangian-Eulerian) method. 43-47 Here the original mesh is given some prescribed velocity ~ in a manner fitting the moving boundaries, and the convective terms in the equations are retained with reference to this velocity. In Eq. (5.9), for instance, in place of OT CUi OXi
we write
OT C(Ui -- ~)i) OX---~
etc., and the solution can proceed in a manner similar to that of steady state (with convection disappearing of course when Vi - " b/i; i.e. in the pure updating process). For more details on ALE framework, readers are referred to Chapter 6. It is of interest to observe that the flow methods can equally well be applied to the forming of thin sections resembling shells. Here of course all the assumptions of shell theory and corresponding finite element technology are applicable. Because of this, incompressibility constraints are no longer a problem but other complications arise. The literature of such applications is large, but much of the relevant information can be found in references 48-61. Practical applications ranging from the forming of beer cans to those of car bodies abound. Figures 5.7 and 5.8 illustrate some typical problems.
5.2.4 Elastic springback and viscoelastic fluids In Sec. 5.2.1 we have argued that omission of elastic effects in problems of metal or plastic forming can be justified because of the small amount of elastic straining. This is undoubtedly true when we wish to consider the forces necessary to initiate large deformations and to follow these through. There are, however, a number of problems in which the inclusion of elasticity is important. One such problem is for instance that of 'springback' occurring particularly in metal forming of complex shapes. Here it is important to determine the amount of elastic recovery which may occur after removing the forming loads. Some possible suggestions for the treatment of such effects have been presented in reference 22 as early as 1984. However, since that time much attention has been focused on the flow of viscoelastic fluids which is relevant to the above problem as well as to the problem of transport of fluids such as synthetic rubbers, etc. The procedures used in the study of such problems are quite complex and
I 0 0 0 0 0 0 0
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154
Incompressible non-Newtonian flows
xI
v
Y
(a) Mesh of 856 elements for sheet idealization
(b) Mesh for establishing die geometry Fig. 5.8 Finite element simulation of the superplastic forming of a thin sheet component by air pressure application. This example considers the superplastic forming of a truncated ellipsoid with a spherical indent. The original flat blank was 150 x 100 mm. The truncated ellipsoid is 20 mm deep. The original thickness was 1 mm. Minimum final thickness was 0.53 mm; 69 time steps were used with a total of 285 Newton-Raphson iterations (complete equation solutions). 51
belong to the subject of numerical rheology. 62-7~ Obviously the subject is vast and here we shall give a brief summary of the topic.
~,iiiii{ii i
~,,,~i,~i~,:~
ii iii~ii~:~/,!i,:,,~/,,!!,,,
In this section we present an introduction to viscoelastic flow equations and their finite element solution using the CBS scheme. A good example of a viscoelastic fluid is a polymeric liquid. A viscoelastic fluid can be modelled by a number of springs and dashpots connected appropriately. In such a system, springs represent elastic effects and dashpots represent viscous behaviour. One such system in series is shown in Fig. 5.9. When such models have a free elastic component they exhibit a springback upon unloading.
Viscoelastic flows
(c) Deformed shapes of various times Fig. 5.8 Continued.
G
/l 1
Fig. 5.9 Physicalrepresentation of Maxwell model using spring and dashpot.
155
156 Incompressiblenon-Newtonian flows Assuming ~ to be the strain rate of the series, the stress r will satisfy the following relation Or
r+aN =~1~
(5.12)
where #1 is the viscosity and A = ~1/G is a relaxation time with G being the spring constant. The above equation is widely known as the Maxwell equation, however, for fluid flow it does not obey the 'objectivity rule'. The generalization of the Maxwell model to satisfy the objectivity rule is carried out by replacing the time derivative Or/0t with the upper convected time derivative, i.e. replacing Eq. (5.12) with T + ,~V_ ~1~
(5.13a)
where
v =g+uNOr Or (0u
0u)
(5.13b)
In multidimensions the constitutive equation for isotropic behaviour becomes
23 6ij~Oul~)
(5.14a)
V OqTij C~7"ij I O n i t~Uj~ 7"ij "-- Ot -~- Uk-~X k -- -~xkTkj -Jr-Tik Oxk ]
(5.14b)
TiJ "~- /~V-T'ji -- ~ I
On i C~ j -~Xj ~ OX i
where
and is known as the Maxwell-B model. In viscoelastic flows, rij calculated from the above constitutive equation replaces the deviatoric stress in the momentum equation. Another popular model is the Oldroyd-B model. Here, stresses generated by a Newtonian solvent and polymer solution are combined. In addition to the spring and dashpot shown in Fig. 5.9 the Oldroyd-B model adds a Newtonian dashpot to the system in parallel as shown in Fig. 5.10. If we assume r v is the stress developed by the polymeric solution (Maxwell-B model) and Y' the stress developed by the Newtonian solvent, the Oldroyd-B model may be written as _ #2~
I I
(5.15)
T
Fig. 5.10 Physicalrepresentation of Oldroyd-B model using spring and dashpots.
Viscoelastic flows
where ~2 is the viscosity of the Newtonian solvent. Rearranging and substituting # = #1 + #2 we get ~-+ Av + #
"~+ AR'~
(5.16)
where AR is a retardation time given as #2A/#. In the following subsection we present a generalized form of the Maxwell-B and Oldroyd-B models together with the equilibrium equations to solve viscoelastic flows. Further details on constitutive modelling of viscoelastic flows may be found in references 63 and 72.
5.3.1 Governing equations The governing equations of viscoelastic flows may be written as
Continuity Op ~and
Momentum
OUz
0
(Ui) =
0
0
(5.17)
Op ~- 07i~
+
= -Ox-S-
O,r~
(5.18)
+
Where U / = pui, p is the density, u/are the velocity components, p the pressure and x/are coordinate directions. The superscript n in the above momentum equation indicates the Newtonian stress and superscript v indicates the non-Newtonian viscoelastic part of the stress. The Newtonian part of the stress relation is written as
n (OUi OUj 2 Onk(~ij) Tij = #2 k OXj -~ OXi 30x,
(5.19)
Note that for an incompressible flow the last term in the above equation disappears. In the above equations, ~ij is the Kroneker delta and #2 is the Newtonian dynamic viscosity. The non-Newtonian extra stress tensor is expressed in conservation form as
v
(OTi k JOt _0
v)(OUj
OUi) ( OUi OUj 2 6ij C~k) &i
3
(5.20)
In the above equation, #1 is the viscosity of viscoelastic contribution and A is the relaxation time. It should be noted that the transient density term in the continuity equation can be replaced by the following pseudo pressure term for incompressible flows as discussed in Chapter 3.
Op
lop
- ~ - = fl2 0t
(5.21)
Where/3 is an artificial compressibility parameter (or artificial wave speed). The problem is completed by specifying appropriate initial and boundary conditions for ui,
157
158 Incompressiblenon-Newtonian flows p and extra stresses. The non-dimensional form of the governing equations can be obtained by employing the following scales * : Ui
t* =
Ui . U cxz
p*
-
P . Por
-
tu~ p, pL ; --~ ; L lZUo~
* Xi 2,
=
7"iJ
_
_
Xi.
L
'
(5.22)
"r/~L ~U o~
where uo~ is the free stream velocity, p~ is the free stream density and L is any characteristic length. The non-dimensional form of the equations are:
Continuity equation Op* OUT I = 0 ot, Ox;
(5.23)
Momentum equation
1 01)*
OUT 0 , o%-7 + Ox--j ( u j v ; ) =
1 f Or}~* O'ri~*)
Re Ox* t- -~e ~,ct-~x~ + Ox-----f
(5.24)
and the Constitutive equation
~_ioj.,+ De ( Ori~*
O
Or* -ll- OX k-"----~( U k * T/j* )
-- De
7i~*Oxk* + 7)~ ----~ Oxk
)
+ (1 - a)
OXj*
+
(~Xi*
(5.25)
In the above non-dimensional equations n,
TiJ =
IOn*
Ouj
Ox----f+ Ox*
20u~t~ij) 30x---~k
(5.26)
Re is the Reynolds number and De is the Deborah number defined as po~uo, L Au~ Re = ~ ; De = rI L
(5.27)
here a = #2/#- With a -- 0 we recover the Maxwell-B model and a 7~ 0 gives the Oldroyd-B model. For incompressible viscoelastic flows the density term in Eq. (5.24) disappears. The equations discussed above along with appropriate boundary conditions will form the governing equation system for solving viscoelastic flow problems. The solution procedure adopted here is the CBS scheme explained in Chapter 3. The basic three steps are the same as for Newtonian incompressible flows. However, an additional fourth step is necessary to get solutions to the constitutive equations. The simple explicit characteristic-Galerkin approach is adopted here to solve the constitutive equations. Additional dissipation is necessary at higher Deborah numbers to smooth overshoots in
Viscoelastic flows
Horizontal velocity profile I ~ 4R 1_.
I-
12R
16R
.._1_..
-I-
._I
--I
Fig. 5.11 Viscoelasticflow past a circular cylinder. Geometryand boundary conditions. the solution if the fully explicit scheme is used. The artificial dissipation methods are discussed in Chapter 7. The artificial dissipation method used in the example below is a second-order method with a pressure switch. 73
Example 5.4
Viscoelastic flow past a circular cylinder
The problem definition is shown in Fig. 5.1 l. A circular cylinder of radius R is placed between two solid walls of a channel. The distances to inlet and exit from the centre of the cylinder are 12R and 16R respectively. The top and bottom walls are assumed to be at a distance of 2R from the centre of the cylinder. The Deborah number De is defined based on the radius R of the cylinder as
De =
/~uoo R
(5.28)
The following non-dimensional form of boundary conditions is employed in the flow calculations. 74 At inlet and exit, ul = 1.5 ( 1 - - ~ ) a n d
9(De)x~ 7-~2 =
3 4x2
u2 = 0
(5.29)
(5.30) (5.31)
and 7-~2 = 0. On solid walls (both channel walls and cylinder surface), no-slip conditions (ul = 0 and u2 = 0) are assumed. The stress boundary conditions on the solid channel walls are given as
% = 2aDe \ Ox2 J
Oul
7-~2 -- a Ox2
(5.32) (5.33)
and 722 = 0. On the cylinder wall the stress boundary conditions are similar to the one above but in cylindrical coordinates. Thus the stress boundary conditions on the cylinder surface are
7"o0 = 2aDe
(5.34)
159
160 Incompressiblenon-Newtonian flows 0u0
TrO -- C~
(5.35) Or and 7 " r r " - - O. Note that a transformation of quantifies from cylindrical to cartesian coordinates is essential in order to apply boundary conditions on the cylinder surface. To compare the present results with those available in the literature, 74' 75 the Reynolds number is assumed to be equal to zero and c~ = 0.41 for this problem. 74-76 The axial drag force on the cylinder per unit length is calculated in non-dimensional form as Fx =
[ ( - p + T~I -'1"7"~1)C0S0"~-(T~2 + r~2)sin0 ] RdO
(5.36)
The above axial drag force will be used to compare the present results against the results reported in the literature. Four different unstructured meshes have been employed to make sure that the proposed scheme converges consistently with mesh refinement. The details of the meshes are given in Table 5.2 (for a full cylinder). The following minimum error criterion is imposed on each and every case of the flow studied to define a steady-state solution. /]~nodes(~n+l __ r
-i
-~-i
W/]~nodes (d)n+l)2
--i
.ri
< 10 -3
(5.37)
-
where 05 is the any variable except pressure, v6 From a mesh sensitive study it was clear that the Mesh4 gives a solution, which is accurate enough to carry out the calculations. Figure 5.12 shows the Mesh4, which is used in all the calculations. Note that due to symmetry, 74' 75 only one half of the domain is used in the calculations. The close-up of the mesh in the vicinity of the cylinder is shown in Fig. 5.12(b). As seen the mesh is very fine close to the cylinder surface. Figure 5.13 shows the contours of velocity and pressure for Newtonian flow at Re = 0. These patterns are in excellent agreement with the results reported in the literature, vv The drag force calculated for this Newtonian case on Mesh4 is compared with reported fine mesh results in Table 5.3. The results of the present calculations agree excellently with the majority of the reported results. Figure 5.14 gives the contours of non-dimensional velocity components, pressure and extra stress components for a Deborah number of 0.5. All variable distributions are generally smooth without exhibiting any appreciable oscillatory behaviour.
Table 5.2 Details of unstructured meshes employed
Mesh Mesh Mesh Mesh Mesh
1 2 3 4
Nodes
Elements
Typical element size on the cylinder surface
3151 5848 12 217 21 238
5980 11 272 23 832 40 768
0.1 0.051 0.022 0.016
Viscoelastic flows
(a) Mesh
(b) Mesh in the vicinity of the cylinder Fig. 5.12 Viscoelastic flow past a circular cylinder. Unstructured mesh. Nodes: 10 619; Elements: 20 384.
(a) u1 velocity contours,
(b)
U2
velocity contours,
Ulmin = O, Ulmax =
U2min =
-0.895,
2.94.
U2max = 0 . 8 9 3 .
(c) Pressure contours, Pmin = -29.28, Pma• ; 36.04.
Fig. 5.13 Stokes flow past a circular cylinder. Re = 0, De = 0.0. Contours of velocity components and pressure.
161
162 Incompressiblenon-Newtonian flows Table 5.3 Comparisonof drag force for the Newtoniancase [74]
[78]
[75]
[77]
[79]
CBS
131.74
132.29
132.28
132.34
132.34
132.39
All the extra stress contours presented in Fig. 5.14 are in excellent qualitative agreement with the reported numerical results. 74' 75 These figures show existence of strong stress gradient on the cylinder surface and on the solid side wall in the vicinity of the cylinder, which is an indication of the converging--diverging effect between the cylinder and the wall. Figure 5.15 shows the comparison of drag forces calculated by different authors and the present study. The difference in the drag force values of different methods are small up to a Deborah number value of 0.4. However, at higher D e differences do exist between the methods. Results of Sun e t al. 75 and Liu e t al. 79 are produced on structured
(a) u 1 velocity contours, Ulmin = O, Ulmax = 2.99.
(b) u2 velocity contours, U2min - -0.929, U2max - 0.884.
(c) Pressure contours, Pmin = - 2 8 . 3 5 , Pmax = 34.97.
(d) "c~lcontours, 1;~lmin = - 0 . 9 7 9 , ~lmax = 76.45.
(e) I;~2
contours,
1;~2mi n -
-18.59,
(f) z~ contours, 1;v22 m i n - - 0 . 4 6 3 ,
1;t~2 max =
22.92.
17V22max "-
16.82.
Fig. 5.14 Viscoelastic flow past a circular cylinder. Re = 0, De = 0.5. Contours of velocity, pressure and elastic stresses.
Direct d i s p l a c e m e n t 170
,
~9n ,vv -
Dou ~=n 9 Dou ,,.,v - Dou Dou ~An 9 Dou
,
and and and and and
, , Liu e t al. Sun e t al. Fan e t al. Phan-Thienl Phan-Thien2 Phan-Thien3 Phan-Thien4 Phan-Thien5
"-'"-
,
approach to transient metal forming
,
,
,
,
i ,9
x
[] o A
C B S
9
9
9
9
9
9
,
9
9 ,
,
9
110
0
~ 0.2
~ 0.4
~ 0.6
,~ 0.8
,I
1
,,,
-
t 1.2
xXl
~, 1.4
~ 1.6
1.8
De
Fig. 5.15 Viscoelastic flow past a circular cylinder. Comparison of drag force distribution with other available numerical data. Dou and Phan-Thienl = Plain Oldroyd-B formulation without stress splitting; Dou and PhanThien2 = EVSS; Dou and Phan-Thien3 = DEVSS-w; Dou and Phan-Thien4 = DAVSS-w; Dou and Phan-Thien5 = Extrapolated results for zero mesh size.
meshes. Understandably they are more accurate compared to other reported results. It is also noticed that the extrapolated drag force for a zero mesh size given by Dou and Phan-Thien TM matches excellently with the results of Liu et al. 79 and Sun et al. 75 and to some extent agrees with the results of CBS. It should be noted that the present CBS results are produced on a high resolution unstructured mesh with a typical minimum element size on the cylinder surface of around 0.01. However, the minimum element size used by Dou and Phan-Thien TM is 0.0402. It is obvious that the mesh resolution used by Dou and Phan-Thien TM is not fine enough to accurately predict the drag force. Present results agree closely with the extrapolated results and those results produced on structured meshes up to a Deborah number of unity. Beyond unity, the CBS overpredicts the drag force. The main reason for the deviation is the extra artificial dissipation added to smooth the extra stress solution when D e > 1. This additional diffusion is responsible for extra artificial drag force, which results in the overprediction. The results can be improved via better artificial dissipation schemes and further mesh refinement.
iili!ii!iiiS iii~~~i~@ii!iiil iiiiiE!iiiiii~i~~Hi i~iiiiii~ii!iiiiiiiiiii!@ i ~iii~iii~i~i~9iiMii!i~iiiiii !i~ iiiiiiiiiiii ~iiiiiliiii!iiii iiiiiii!iiiiii!iiiii~iiiiiiJiii~iiiiiiii~iiiiPiiii!iP ~iii!iii~i!iiiiiiii! ii iiiii!ii!iiiiii i~iiiiii~%iiiiiii~i~iiiii~iii!iiiiiiiii iiiii~~i:::~~iiiiiii iii~!~i!i!iiiiiiiiiiiiiii i iiiiiiiiiiiiiii ii iiiii!iiiiii!iiiiiiii! iii~iii~iiiiiii i!iliiiiiiii iiiiiiii iiiiliii!i@iiiiiiiii!iiii iiiiiii!iiiiiiiiii;iiiii iliiiiii!iiiiiiiiiiil!iliiiiiiili!i84
ii!!!!!!!i!!!!!!i!!!!~!!~!!~~i! ~!!ii!!ii!i!i!!D!!!iiii!!!i!i!!i!!i!!!!i!!i!i!i!!iiii!!!!i iiii!ii!iii!!i!!i!i!ii!iii!ii!!i!iil!!ii!!ii!ili il!i i!ii!i!.!i....i..ii.i.!.i.!... Explicit dynamic codes using quadrilateral or hexahedral elements have achieved considerable success in modelling short-duration impact phenomena with plastic deformation. The prototypes of finite element codes of this type are DYNA2d and DYNA3d developed at Lawrence Livermore National Laboratory. 8~ 81 For problems of relatively slow metal forming, such codes present some difficulties as in general the time step is governed by the elastic compressibility of the metal and a vast number of time steps would be necessary to cover a realistic metal forming problem. Nevertheless much use has been made of such codes in slow metal forming processes by the simple expedient
163
164
Incompressible
non-Newtonian
flows w
~
(a)
(b)
tl
~ i i ~ IIIIIIIIIIIIIIIIIIli I I I i I I i Illliillllllllllllllllllllli
(c)
L , ' q ~ i,,'I~A
/L,,"b/llq./ lib,"
A/VV_V_V V V V ~
_
N~ll
(dl
L I I I
i I I I
| 1 I I
I
I
I
I
I
I
i I I I I I
1 I I I I I
I.i I I I I I
I I I I I
i I I I i I
I I I I I I
I I I I
I
I
i I I I I I
~ ] ~ I ~ 1 1 1 I I I I I ! .
i
i
i l ltl i
i
I
I_L_L~L~\~I IT-V~L~ll
!!!H!!tt~L-~"~l
\~(lll ~I
I IIIIIIIIIItllllli
(e) Fig. 5.16 Axisymmetric solutions to the bar impact problem: (a)initial shape; (b)linear triangles - displacement algorithm; (c) bilinear quadrilaterals - displacement algorithm; (d)linear triangles - CBS algorithm; (e) bilinear quadrilaterals - CBS algorithm.
of increasing the density of the material by many orders of magnitude. This is one of the drawbacks of using such codes whose description tightly belongs to the matter discussed in reference 1. However, a further drawback is the lack of triangular or tetrahedral elements of a linear kind which could compete with linear quadrilaterals or hexahedra currently used and permit an easier introduction of adaptive refinement. It is well known that linear triangles or tetrahedra in a pure displacement (velocity) formulation will lock for incompressible or nearly incompressible materials. However, we have already found that the CBS algorithm will avoid such locking even if the same (linear) interpolation is used for both velocities and pressure. 82 It is therefore possible to proceed in each step by solving a simple Stokes problem to evaluate the Lagrangian velocity increment. We have described the use of such velocity formulation in the previous chapter. The update of the displacement allows
Concluding remarks 165
(a)
(b)
Fig. 5.17 Three-dimensionalsolution: (a) tetrahedral elements - standard displacement algorithm; (b) tetrahedral elements - CBS algorithm.
new stresses to be evaluated by an appropriate plasticity law and the method can be used without difficulty as shown by Zienkiewicz et al. 82
Example 5.5
Impact of a circular bar
In Fig. 5. ] 6, we show a comparison between various methods of solving the impact of a circular bar made of an elastop]astir metal using an a•162 formulation. In this
figure we show the results of a linear triangle displacement [Fig. 5.1 6(b)] form with a single integrating point for each element and a similar study again using displacement linear quadrilaterals [Fig. 5.16(c)] also with a single integration point. This figure also shows the same triangles and quadrilaterals solved using the CBS algorithm and now giving very accurate final results [Fig. 5.16(d) and (e)]. In Fig. 5.17 we show similar results obtained with a full three-dimensional analysis. Similar methods for this problem have been presented by Bonet and Burton. 83
:ii:i~iiiiii!i~:!:~iiiiiiii~~!i~i!~igi!ilil: ii~i~iii~iii~;~{~iii~:i~ii!;i~ii{ii~,!iiii!iiiiiiii~ii~iiiiii~iiiii ii!i~iiiii;i~i~lii!~iiiii i;~ iiiliiiiiiiii~~i!ii~i~iiiiiiiiiiiiiiiiiiiiiiiiiiiiiii=iiii iii~ii,iiiiiii~iiii!iiiili!iii~;~{iiii•ii•!•i•i!i!•i•i•ii•i•i"ii!i•i•i•:i•ii!...... i!ii•i•i•i....ii•i•i•i•ii:iiii!iiiiiiiiiiil i~•i!~i!•ii•ii~•i•i~•i•i!ii~:iiiiiii:iiji: ~i~i!i~iiiiii!:ii:iii!iiiiiii:::::ii iii~i~!~i~ii;ii!i!iii!ii~i!iii!:~iilii~:ii:~ii!il~:i!ii~i:i!i~:i.ii•i~•ii~•ii•!i~ii•i!~!ii~iiiii!i:iiiiii •ii~ii!~i!i!i~•i•li•!ii(i•i!i•!i:iiiil •ijii•i•iiiiiil iiiii•iii!iiiiii!il!i!l:i!ii!iiiiiii•iiii•i!i!•i!i!i!ii!i•i• The range of examples for which an incompressible formulation applies is very large as we have shown in this chapter. Indeed many other examples could have been included but for lack of space we proceed directly to Chapter 6 where the incompressible formulation is used for problems in which free surface or buoyancy occurs with gravity forces being the most important factor.
166 Incompressiblenon-Newtonian flows iiiiiiii~iiiiiiiiiiiiiiiiiiii~iiiiiiiiiiiiiiiii!iii ii!i!!!ii{il
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1. O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method for Solid and Structural Mechanics, 6th Edition, Elsevier, 2005. 2. K. Palit and R.T. Fenner. Finite element analysis of two dimensional slow non-Newtonian flows. AIChE J., 18:1163-1169, 1972. 3. K. Palit and R.T. Fenner. Finite element analysis of slow non-Newtonian channel flow. AIChE J., 18:628-633, 1972. 4. B. Atkinson, C.C.M. Card and B.M. Irons. Application of the finite element method to creeping flow problems. Trans. Inst. Chem. Eng., 48:276--284, 1970. 5. O.C. Zienkiewicz and P.N. Godbole. A penalty function approach to problems of plastic flow of metals with large surface deformation. J. Strain Anal., 10:180-183, 1975. 6. G.C. Cornfield and R.H. Johnson. Theoretical prediction of plastic flow in hot rolling including the effect of various temperature distributions. J. Iron and Steel Inst., 211:567-573, 1973. 7. C.H. Lee and S. Kobayashi. New solutions to rigid plastic deformation problems using a matrix method. Trans. ASME J. Eng. for Ind., 95:865-873, 1973. 8. J.T. Oden, D.R. Bhandari, G. Yagewa and T.J. Chung. A new approach to the finite element formulation and solution of a class of problems in coupled thermoelastoviscoplasticity of solids. Nucl. Eng. Des., 24:420, 1973. 9. O.C. Zienkiewicz, E. Onate and J.C. Heinrich. Plastic Flow in Metal Forming, pp. 107-120, ASME, San Francisco, December 1978. 10. O.C. Zienkiewicz, E. On~te and J.C. Heinrich. A general formulation for coupled thermal flow of metals using finite elements. Int. J. Num. Meth. Eng., 17:1497-1514, 1981. 11. N. Rebelo and S. Kobayashi. A coupled analysis of viscoplastic deformation and heat transfer: I. Theoretical consideration: II. Application. Int. J. Mech. Sci., 22:699-705, 1980 and 22:707718, 1980. 12. P.R. Dawson and E.G. Thompson. Finite element analysis of steady state elastoviscoplastic flow by the initial stress rate method. Int. J. Num. Meth. Eng., 12:47-57, 382-383, 1978. 13. Y. Shimizaki and E.G. Thompson. Elasto-visco-plastic flow with special attention to boundary conditions. Int. J. Num. Meth. Eng., 17:97-112, 1981. 14. O.C. Zienkiewicz, P.C. Jain and E. On~te. Flow of solids during forming and extrusion: some aspects of numerical solutions. Int. J. Solids Struct., 14:15-38, 1978. 15. S. Nakazawa, J.ET. Pittman and O.C. Zienkiewicz. Numerical solution of flow and heat transfer in polymer melts. In R.H. Gallagher et al., editors, Finite Elements in Fluids, volume 4, Chapter 13, pp. 251-83. Wiley, Chichester, 1982. 16. S. Nakazawa, J.ET. Pittman and O.C. Zienkiewicz. A penalty finite element method for thermally coupled non-Newtonian flow with particular reference to balancing dissipation and the treatment of history dependent flows. Int. Symp. of Refined Modelling of Flows, 7-10 September 1982, Paris. 17. R.E. Nickell, R.I. Tanner and B. Caswell. The solution of viscous incompressible jet and free surface flows using finite elements. J. Fluid Mech., 65:189-206, 1974. 18. R.I. Tanner, R.E. Nickell and R.W. Bilger. Finite element method for the solution of some incompressible non-Newtonian fluids mechanics problems with free surface. Comp. Meth. Appl. Mech. Eng., 6:155-174, 1975. 19. J.M. Alexander and J.W.H. Price. Finite element analysis of hot metal forming, 18th MTDR Conf., pp. 267-74, 1977. 20. S.I. Oh, G.D. Lahoti and A.T. Altan. Application of finite element method to industrial metal forming processes. Proc. Conf. on Industrial Forming Processes, pp. 146-53. Pineridge Press, Swansea, 1982. 21. J.ET. Pittman, O.C. Zienkiewicz, R.D. Wood and J.M. Alexander, editors. Numerical Analysis of Forming Processes. Wiley, Chichester, 1984.
References 167 22. O.C. Zienkiewicz. Flow formulation for numerical solutions of forming problems. In J.ET. Pittman et al., editors, Numerical Analysis of Forming Processes, Chapter 1, pp. 1-44. Wiley, Chichester, 1984. 23. S. Kobayashi. Thermoviscoplastic analysis of metal forming problems by the finite element method. In LET. Pittman et al., editors, Numerical Analysis of Forming Processes, Chapter 2, pp. 45-70. Wiley, Chichester, 1984. 24. J.L. Chenot, E Bay and L. Fourment. Finite element simulation of metal powder forming. Int. J. Num. Meth. Eng., 30:1649-1674, 1990. 25. EA. Balaji, T. Sundararajan and G.K. Lal. Viscoplastic deformation analysis and extrusion die design by FEM. J. Appl. Mech.- Trans. ASME, 58:644-650, 1991. 26. K.H. Raj, J.L. Chenot and L. Fourment. Finite element modelling of hot metal forming. Indian J. Eng. Mat. Sci., 3:234-238, 1996. 27. J.L. Chenot. Recent contributions to the finite element modelling of metal forming processes. J. Mat. Proc. Tech., 34:9-18, 1992. 28. J. Bonet, E Bhargava and R.D. Wood. The incremental flow formulation for the finite element analysis of 3-dimensional superplastic forming processes. J. Mat. Proc. Tech., 45:243-248, 1994. 29. R.D. Wood and J. Bonet. A review of the numerical analysis of super-plastic forming. J. Mat. Proc. Tech., 60:45-53, 1996. 30. J.L. Chenot and Y. Chastel. Mechanical, thermal and physical coupling methods in FE analysis of metal forming processes. J. Mat. Proc. Tech., 60:11-18, 1996. 31. J. Rojek, E. On~te and E. Postek. Application of explicit FE codes to simulation of sheet and bulk metal forming processes. J. Mat. Proc. Tech., 80-81:620-627, 1998. 32. J. Bonet. Recent developments in the incremental flow formulation for the numerical simulation of metal forming processes. Eng. Comp., 15:345-357, 1998. 33. J. Rojek, O.C. Zienkiewicz, E. On~te and E. Postek. Advances in FE explicit formulation for simulation of metal forming processes. Proc. International Conference on Advances in Materials and Processing Technologies, AMPT 1999 and 16th Annual Conference of the Irish Manufacturing Committee, IMC16, volume 1, M.S.J. Hashmi and L. Looney, editors, Aug. 1999, Dublin City University. 34. J.-L. Chenot, R.D. Wood and O.C. Zienkiewicz, editors. Numerical Methods in Industrial Forming Processes, NUMIFORM 92. Valbonne, France, 14-19 September 1992, A.A. Balkema, Rotterdam. 35. J. Hu6tink and EET. Baaijens, editors. Simulation of Materials Processing: Theory, Methods and Applications, NUMIFORM 98. Enschede, Netherlands, 22-25 June 1998, A.A. Balkema, Rotterdam. 36. S. Ghosh, J.M. Castro and J.K. Lee, editors. Materials Processing and Design: Modelling Simulation and Applications, NUMIFORM 2004, American Institute of Physics. 37. W. Johnson and EB. Mellor. Engineering Plasticity. Van Nostrand-Reinhold, London, 1973. 38. O.C. Zienkiewicz, Y.C. Liu and G.C. Huang. Error estimates and convergence rates for various incompressible elements. Int. J. Num. Meth. Eng., 28:2191-2202, 1989. 39. G.C. Huang. Error estimates and adaptive remeshing in finite element anlysis of forming processes. Ph.D. thesis, University of Wales, Swansea, 1989. 40. G.C. Huang, Y.C. Liu and O.C. Zienkiewicz. Error control, mesh updating schemes and automatic adaptive remeshing for finite element analysis of unsteady extrusion processes. In J.L. Chenot and E. On~te, editors, Modelling of Metal Forming Processes, pp. 75-83. Kluwer Academic, Dordrecht, 1988. 41. O.C. Zienkiewicz, Y.C. Liu and G.C. Huang. An error estimate and adaptive refinement method for extrusion and other forming problems. Int. J. Num. Meth. Eng., 25:23-42, 1988. 42. O.C. Zienkiewicz, Y.C. Liu, J.Z. Zhu and S. Toyoshima. Flow formulation for numerical solution of forming processes. II - Some new directions. In K. Mattiasson, A. Samuelson, R.D. Wood and
168 Incompressiblenon-Newtonian flows
43. 44. 45.
46.
47. 48. 49. 50. 51.
52.
53.
54.
55. 56. 57. 58. 59. 60. 61.
62.
O.C. Zienkiewicz, editors, Proc. 2nd Int. Conf. on Numerical Methods in Industrial Forming Processes, NUMIFORM 86. A.A. Balkema, Rotterdam, 1986. T. Belytchko, D.P. Flanagan and J.M. Kennedy. Finite element methods with user controlled mesh for fluid structure interaction. Comp. Meth. Appl. Mech. Eng., 33:669-688, 1982. J. Donea, S. Giuliani and J.I. Halleux. An arbitrary Lagrangian-Eulerian finite element for transient dynamic fluid-structure interaction. Comp. Meth. Appl. Mech. Eng., 75:195-214, 1989. P.J.G. Schreurs, F.E. Veldpaus and W.A.M. Brakalmans. A Eulerian and Lagrangian finite element model for the simulation of geometrical non-linear hyper elastic and elasto-plastic deformation processes. Proc. Conf. on Industrial Forming Processes, pp. 491-500. Pineridge Press, Swansea, 1983. J. Donea. Arbitrary Lagrangian-Eulerian finite element methods. In T. Belytchko and T.J.R. Hughes, editors, Computation Methods for Transient Analysis, Chapter 10, pp. 474-516. Elsevier, Amsterdam, 1983. J. van der Lugt and J. Huetnik, Thermo-mechanically coupled finite element analysis in metal forming processes. Comp. Meth. Appl. Meth. Eng., 54:145-160, 1986. E. On~te and O.C. Zienkiewicz. A viscous sheet formulation for the analysis of thin sheet metal forming. Int. J. Mech. Sci., 25:305-335, 1983. N.M. Wang and B. Budiansky. Analysis of sheet metal stamping by a finite element method. Trans. ASME, J. Appl. Mech., 45:73, 1976. A.S. Wifi. An incremented complete solution of the stretch forming and deep drawing of a circular blank using a hemispherical punch. Int. J. Mech. Sci., 18:23-31, 1976. J. Bonet, R.D. Wood and O.C. Zienkiewicz. Time stepping schemes for the numerical analysis of superplastic forming of thin sheets. In J.L. Chenot and E. On~te, editors, Modelling of Metal Forming Processes, pp. 179-86. Kluwer Academic, Dordrecht, 1988. E. Massoni, M. Bellet and J.L. Chenot. Thin sheet forming numerical analysis with membrane approach. In J.L. Chenot and E. On~te, editors, Modelling of Metal Forming Processes. Kluwer Academic, Dordrecht, 1988. R.D. Wood, J. Bonet and A.H.S. Wargedipura. Simulation of the superplastic forming of thin sheet components using the finite element method. NUMIFORM 89 Proc., pp. 85-94. Balkhema Press, 1989. J. Bonet, R.D. Wood and O.C. Zienkiewicz. Finite element modelling of the superplastic forming of a thin sheet. In C.H. Hamilton and N.E. Paton, editors, Proc. Conf. on Superquality and Superplastic Forming. The Minerals, Metals and Materials Society, USA, 1988. E. On~te and C.A. Desaracibar. Finite element analysis of sheet metal forming problems using a selective bending membrane formulation. Int. J. Num. Meth. Eng., 30:1577-1593, 1990. W. Sosnowski, E. On~te and C.A. Desaracibar. Comparative study of sheet metal forming processes by numerical modelling and experiment. J. Mat. Proc. Tech., 34:109-116, 1992. J. Bonet and R.D. Wood. Incremental flow procedures for the finite element analysis of thin sheet superplastic forming processes. J. Mat. Pro. Tech., 42:147-165, 1994. W. Sosnowski and M. Kleiber. A study on the influence of friction evolution on thickness changes in sheet metal forming. J. Mat. Proc. Tech., 60:469-474, 1996. J. Bonet, P. Bhargava and R.W. Wood. Finite element analysis of the superplastic forming of thick sheet using the incremental flow formulation. Int. J. Num. Meth. Eng., 40:3205-3228, 1997. M. Kawka, T. Kakita and A. Makinouchi. Simulation of multi-step sheet metal forming processes by a static explicit FEM code. J. Mat. Proc. Tech., 80:54-59, 1998. S.K. Esche, G.K. Kinzel and J.K. Lee. An axisymmetric membrane element with bending stiffness for static implicit sheet metal forming simulation. J. Appl. Mech.- Trans. ASME, 66:153-164, 1999. M.J. Crochet and R. Keunings. Finite element analysis of die swell of highly elastic fluid. J. Non-Newtonian Fluid Mech., 10:339-356, 1982.
References 169 63. M.J. Crochet, A.R. Davies and K. Waiters. Numerical Simulation of Non-Newtonian Flow. Rheology series: Volume 1. Elsevier, Amsterdam, 1984. 64. R. Keunings and M.J. Crochet. Numerical simulation of the flow of a viscoelastic fluid through an abrupt contraction. J. Non-Newtonian Fluid Mech., 14:279-299, 1984. 65. S. Dupont, J.M. Marchal and M.J. Crochet. Finite element simulation of viscoelastic fluids of the integral type. J. Non-Newtonian Fluid Mech., 17:157-183, 1985. 66. J.M. Marchal and M.J. Crochet. A new mixed finite element for calculating visco elastic flow. J. Non-Newtonian Fluid Mech., 26:77-114, 1987. 67. M.J. Crochet and V. Legat. The consistent streamline upwind Petrov-Galerkin method for viscoelastic flow revisited. J. Non-Newtonian Fluid Mech., 42:283-299, 1992. 68. H.R. Tamaddonjahromi, D. Ding, M.E Webster and P. Townsend. A Taylor-Galerkin finite element method for non-Newtonian flows. Int. J. Num. Meth. Eng., 34:741-757, 1992. 69. E.O.A. Carew, P. Townsend and M.E Webster. A Taylor-Galerkin algorithm for viscoelastic flow. J. Non-Newtonian Fluid Mech., 50:253-287, 1993. 70. D. Ding, P. Townsend and M.E Webster. Finite element simulation of an injection moulding process. Int. J. Num. Meth. Heat Fluid Flow, 7:751-766, 1997. 71. H. Matallah, P. Townsend and M.E Webster. Recovery and stress-splitting schemes for viscoelastic flows. J. Non-Newtonian Fluid Mech., 75:139-166, 1998. 72. N. Phan-Thien. Understanding Viscoelasticity. Basics of Rheology. Springer-Verlag, Berlin, 2002. 73. P. Nithiarasu. A fully explicit characteristic based split (CBS) scheme for viscoelastic flow calculations. Int. J. Num. Meth. Eng., 60:949-978, 2004. 74. H-S. Dou and N. Phan-Thien. The flow of an Oldroyd-B fluid past a cylinder in a channel: adaptive viscosity vorticity (DVSS-co) formulation. J. Non-Newtonian Fluid Mech., 87:47-73, 1999. 75. J. Sun, M.D. Smith, R.C. Armstrong and R.A. Brown. Finite element method for viscoelastic flows based on the discrete adaptive viscoelastic stress splitting and the discontinuous Galerkin method: DAVSS-G/DG. J. Non-Newtonian Fluid Mech., 86:281-307, 1999. 76. T.N. Philips and A.J. Williams. Viscoelastic flow through a planar contraction using a semiLagrangian finite volume method. J. Non-Newtonian Fluid Mech., 87:215-246, 1999. 77. E. Mitsoulis. Numerical simulation of confined flow of polyethylene melts around a cylinder in a planar channel. J. Non-Newtonian Fluid Mech., 7:327-350, 1998. 78. Y. Fan, R.I. Tanner and N. Phan-Thien. Galerkin/least-square finite element methods for steady viscoelastic flows. J. Non-Newtonian Fluid Mech., 84:233-256, 1999. 79. A.W. Liu, D.E. Bornside, R.C. Armstrong and R.A. Brown. Viscoelastic flow of polymer solutions around a periodic, linear array of cylinders: comparisons of predictions for microstructure and flow fields. J. Non-Newtonian Fluid Mech., 77:133-190, 1998. 80. G.L. Goudreau and J.O. Hallquist. Recent developments in large scale Lagrangian hydrocodes. Comp. Meth. Appl. Mech. Eng., 33:725-757, 1982. 81. J.O. Hallquist, G.L. Goudreau and D.J. Benson. Sliding interfaces with contact-impact in large scale lagrangian computations. Comp. Meth. Appl. Mech. Eng., 51:107-137, 1985. 82. O.C. Zienkiewicz, J. Rojek, R.L. Taylor and M. Pastor. Triangles and tetrahedra in explicit dynamic codes for solids. Int. J. Num. Meth. Eng., 43:565-583, 1998. 83. J. Bonet and A.J. Burton. A simple average nodal pressure tetrahedral element for incompressible and nearly incompressible dynamic explicit applications. Comm. Num. Meth. Eng., 14:437-449, 1998.
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Free surface and buoyancy driven flows ~;i~i;~i;i~i~i~i~i~i~i~i~!~i~iiii~i~i~ii~i~i~i~i~i~ii~i~i~i~iii~iiii~ iiiiiiiiiii!iii i i ill
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In Chapter 4 we have introduced the reader to the solution of incompressible flow problems and have illustrated these with many examples of Newtonian (Chapter 4) and non-Newtonian (Chapter 5) flows. In the present chapter, we shall address two separate topics of incompressible flow, which were not dealt with in the previous chapters. This chapter is thus divided into two parts. The common theme is that of the action of the body force due to gravity. We start with a section addressed to problems of free surfaces and continue with the second section, which deals with buoyancy effects caused by temperature differences in various parts of the domain. The first part of this chapter, Sec. 6.2, will deal with problems in which a free surface of flow occurs when gravity forces are acting throughout the domain. Typical examples here would be for instance given by the disturbance of the free surface of water and the creation of waves by moving ships or submarines. Of course other problems of similar kinds arise in practice. Indeed in Chapter 10, where we deal with shallow water flows, a free surface is an essential condition but other assumptions and simplification have to be introduced. In the present chapter, we shall deal with the full incompressible flow equations without any further physical assumptions. There are other topics of free surfaces which occur in practice. One of these for instance is that of mould filling, which is frequently encountered in manufacturing where a particular fluid or polymer is poured into a mould and solidifies. In Sec. 6.3, we invoke problems of buoyancy and here we can deal with natural convection when the only force causing the flow is that of the difference between uniform density and density which has been perturbed by a given temperature field. In such examples it is a fairly simple matter to modify the equations so as to deal only with the perturbation forces but on occasion forced convection is coupled with such naturally occurring convection.
6.2.1 General remarks In many problems of practical importance a free surface will occur in the fluid (always liquid). In general the position of such a free surface is not known and the main problem
Free surface flows
is that of determining it. In Fig. 6.1, we show a set of typical problems of free surfaces; these range from flow over and under water control structures, flow around ships, to industrial processes such as filling of moulds. All these situations deal with a fluid which is incompressible and in which the viscous effects either can be important or may be neglected. The only difference from solving the type of problem which we have discussed in the previous two chapters is the fact that the position of the free surface is not known a priori and has to be determined during the computation. There are several ways of dealing with such free surface flows. We broadly classify them into three categories. They are (1) Pure Lagrangian methods, (2) Eulerian methods and (3)Arbitrary-Lagrangian-Eulerian (ALE) methods.
Lagrangian methods
In this method we write the equations for the fluid particles whose position is changing continuously in time. Such Lagrangian methods almost always are used in the study of solid mechanics but are relatively seldom applied in fluid dynamics due to the fact
Water level
~
/Sluice gate
Overflow i
surface
Water
Water Floor (a)
Free surface waves
(b) Free surface
Ship
Sea
Sea floor
(c)
(d)
Feeder Free surface
Free surface .
Sea floor
.
.
.
.
Mould cavity
.
Hydrofoil
(e) Fig. 6.1 Typical problems with a free surface.
Metal inlet
Die
(f)
171
172
Free surface and buoyancydriven flows that here very often steady-state flow occurs. There is an immediate advantage of Lagrangian formulation in the fact that convective acceleration is non-existent and the problem is immediately self-adjoint. Further, for problems in which a free surface occurs it allows the free surface to be continuously updated and maintained during the fluid motion. 1-4
Eulerian methods
In Eulerian methods, for which we have established the equations in Chapter l, the boundaries of the fluid motion are fixed in position and so indeed are any computational meshes. For free surface problems an immediate difficulty arises as the position of the free surface is not known a priori. The numerical method will therefore have to include an additional algorithm to trace the free surface positions. 5-15
Arbitrary-Lagrangian-Eulerian (ALE) methods
With both Lagrangian and Eulerian methods certain difficulties and advantages occur and on occasion it is possible to provide an alternative which attempts to secure the best features of both Lagrangian and Eulerian description by combining these. Such methods are known as ALE methods. These methods are generally complex to implement and we shall give further description of these methods later. 16-30
6.2.2 Lagrangian method As mentioned before the mesh moves with the flow in the Lagrangian methods. Thus after completion of the CBS steps involved it is necessary to determine the new positions of the mesh. The fluid dynamics equations solved are the same as the ones presented in Chapter 4 except that the convective terms are absent. The governing equations for isothermal Lagrangian fluid dynamics can be written as
OtI~ O G i Ot + ~ +Q=O
(6.1)
with dPT = (p, pUl, pu2) = P, U1, U2
(6.2)
being the independent variable vector, (6.3)
(;iT - (o,-~-il,-~-i~)
defining the diffusion fluxes. The source term Q represents the body force due to gravity. In the above, deviatoric stress components Tij are related to velocity as
7"ij -- #
Obli OUj ~ "~ OXi
2 0uk ~ij) 30xk
(6.4)
In the above equations, r is the Kroneker delta, Ui are the velocity components, p is the density, p the pressure and # is the dynamic viscosity.
Free surface flows 173
The continuity and dynamic momentum equations are repeated in a detailed indicial form as Continuity 0
(Ui) = 0
(6.5)
019 -q- Orij -4- Pgi OXi ~Xj
(6.6)
Op ~-
and Momentum OUi _ Ot --
Where Ui = pui and gi is the gravity force. It should be noted that the transient density term in the continuity equation can be replaced by the following relation
Op
1@
/320t
(6.7)
Where/3 is an artificial wave speed (Chapter 3). The problem is completed by specifying appropriate initial conditions for ui and p together with boundary conditions. The solution procedure follows one of the time discretizations discussed in Chapter 3. For using semi-implicit schemes the transient pressure term is omitted in Eq. (6.7). For fully explicit schemes, this term is retained along with the artificial compressible wave speed. However, a dual time stepping scheme is necessary when the artificial compressibility method is employed. After every real time step the coordinates of the nodes are updated using 2 1 X in+l = X in + -~ A t ~. iZun+l+ U ni )
Example 6.1 Lagrangian free surface method for a model broken dam problem
(6.8)
An example problem of somewhat idealized dam failure as shown in Fig. 6.2 is solved using the semi-implicit form of CBS. Although not realistic, this problem is frequently used as a benchmark for validating Lagrangian algorithms. The experimental data are indeed available and will be used here for comparison. As seen from Fig. 6.2 the problem consists of two slip walls on which slip boundary conditions are applied (normal velocity zero or tangential traction zero). The initial fluid position is as shown in Fig. 6.2 (left) with velocities at all nodes equal to zero. The dimensions of the dam are: H = 7 and W = 3.5. The gravity was assumed to act with a magnitude equal to unity (non-dimensional). The viscosity was assumed to be 0.01 (non-dimensional quantity). At t = 0, the gate was opened and the fluid from the dam was assumed to flow freely. The quantity of interest is the extreme horizontal free surface position L as shown in Fig. 6.2 (fight). The unstructured mesh used consists of 339 nodes and 604 elements. Figs 6.3 and 6.4 give the distorted mesh and contours at various time levels. As seen the results are generally smooth over all the domain. The pressure contours are free of
174
Free surface and buoyancy driven flows Initial fluid level
/
H~
Gate
g
~/,
~
~
k
~
surface
Slip walls
_..
~"
I-"
W
"-
W
'-"
_1
'Yl
L
v,
Fig. 6.2 Brokendam problem. Problemdefinition and schematicof the free surface.
/
(a) Mesh
(c) u2 velocity contours Fig. 6.3 Brokendam problem. Mesh and contours after t = 2.0.
(b) ul velocity contours
(d) P r e s s u r e c o n t o u r s
Free surface flows
Fig. 6.4
(a) Mesh
(b) u 1 velocity contours
(c) u 2 velocity contours
(d) Pressure contours
Broken dam problem. Mesh and contours after t = 5.0.
oscillation, which shows the effect of the good pressure stabilization properties of the CBS scheme. Figure 6.5 shows the comparison of extreme horizontal position reached by the free surface with the experimental data. 31 As seen the numerical results are in good agreement with the reported experimental data. The non-dimensional time in the horizontal coordinate is calculated as t ~/2g/W. Figures 6.3 and 6.4 show how a reasonably regular mesh at t = 0 becomes distorted after a certain number of time steps. While the results show a good agreement with the experimental data the irregularity of the mesh can cause errors and indeed in complicated problems this irregularity may cause element overlapping and unphysical results. Here lies one of the serious disadvantages of Lagrangian updating and care has to be taken to avoid extreme errors. Indeed, from time to time it may be necessary to remesh the whole problem to avoid element inversion. An interesting procedure showing continuous remeshing with a rather novel approach is discussed by Idelshon et al. 32-34 We shall not give the details of the procedure here but interested readers should consult references 33 and 34.
I
4-
I
- CBS Exp.
I
I
I
I
+
:::!3 I I
0015
Fig. 6.5
1'
, , 1.5 2, 2.5 3, Non-dimensional time
3 .i5
4
Broken dam problem. Comparison of numerical results with experimental data 31
175
176
Free surface and buoyancydriven flows
6.2.3 Eulerian methods As mentioned before the flow domain in Eulerian methods is fixed and the fluid is allowed to pass through the domain. Thus an additional procedure is necessary to track the free surface when Eulerian methods are employed. In order to correctly track the free surface, we need both dynamic and kinematic conditions to be satisfied on the free surface. Thus, on the free surface we have at all times to ensure that (1) the pressure (which approximates the normal traction) is known (dynamic condition) and (2) that the material particles of the fluid belonging to the free surface remain on this at all times (kinematic condition). These conditions are expressed as p--p Un
--
m
nTu - 0
(6.9)
on the free surface Ff. In the above equation b/n is the velocity component in the normal direction to the free surface and ff is the known pressure. For a non-breaking free surface the kinematic condition (material surface) can be restated for x3 - ~(t, Xl, x2) as D~7 Dt
(6.10)
"--U3
or 0/7 ...~_.U l ~ x 1 "]- U2 Ox2 0/7 -&
u3 = 0
(6.11)
where U l, /'/2 and U 3 are the velocity components in X1, X2 and x 3 directions shown in Fig. 6.6. At this point it is important to remark that two independent Eulerian approaches in solving free surface flow problems exist. In the first method the free surface is determined by solving Eq. 6.10 along with the dynamic conditions. Once the free surface position ~7is determined remeshing follows either once at the end of the solution procedure or frequently within the time stepping iterations. This way of mesh updating is well suited for solving problems of steady-state nature. If this method is employed for the solution of transient problems, some details of Lagrangian approach need to be included close to the free surface. In the second Eulerian method the mesh is often fixed throughout the calculation but the free surface is tracked to satisfy the kinematic condition. Once the free surface is tracked the dynamic boundary conditions are applied to satisfy the pressure/traction conditions. The standard procedure used in tracking the free surface is the so-called Volume of Fluid (VOF) method or one of its variants. 5, 13, 53-56 The major drawback of this method is that a rough idea about the external free surface is necessary a priori in order to generate a mesh which contains the free surface all the time.
Mesh updating or regeneration methods
Figure 6.6 shows a typical problem of ship motion together with the boundaries limiting the domain of analysis. This is an example of external free surface flow problem. In the interior of the domain we can use either the full Navier-Stokes equations or, neglecting
Free surface flows
Datum
Free surface
Uo
Ship
x3
Fig. 6.6 A typical problem of ship motion.
viscosity effects, a pure potential or Euler approximation. Both assumptions have been discussed in the previous chapters but it is interesting to remark here that the resistance caused by the waves may be four or five times greater than that due to viscous drag. Clearly surface effects are of great importance for ship design. Historically many solutions that ignore viscosity totally have been used in the ship industry with good effect by involving so-called boundary solution procedures or panel methods. 35-45 Early finite element studies in the field of ship hydrodynamics have also used potential flow equations. 46 A full description of these is given in many papers. However, complete solutions with viscous effects and full non-linearity are difficult to deal with. In the procedures that we present in this section, the door is opened to obtain a full solution without any extraneous assumptions and indeed such solutions could include turbulence effects, etc. The non-dimensional form of the incompressible flow equations can be written as OUi
Oxi
-- 0
(6.12)
and On i O(UiUj) Ot } OXj "-"
013 1 02ui OX i ! Re Oxei
X3
Fr 2
(6.13)
The viscous terms in the above equation are simplified using the conservation of mass equation (6.12). In the above equations Re is the Reynolds number defined in
177
178
Freesurface and buoyancydriven flows Chapters 3 and 4 and Fr is the Froude number given as
Fr =
Ucx~
(6.14)
where u~ is a reference velocity and L is a reference length. Further details of the equations can be found in Sec. 4.1 of Chapter 4 and indeed the same CBS procedure can be used in the solution. However, considerable difficulties arise on the free surface, despite the fact that on such a surface normal traction is known. The difficulties are caused by the fact that at all times we need to ensure that this surface is a material one and contains the particles of the fluid. Equation (6.11) for free surface height ~7is a pure convection equation (see Chapter 2) in terms of the variables t, u ~, u2 and u3 in which u3 is a source term. At this stage it is worthwhile remarking that this surface equation has been known for a very long time and was dealt with previously by upwind differences, in particular those introduced on a regular grid by Dawson. 36 However in Chapter 2, we have already discussed other perfectly stable, finite element methods, any of which can be used for dealing with this equation. On occasion additional artificial dissipation is added to stabilize the convection equation (6.11) in its discrete form at inlet and exit of the domain. ~~ It is important to observe that when the steady state is reached we simply have Or/ -
+
(6.15)
o--
which ensures that the velocity vector is tangential to the free surface. The solution method for the whole problem can now be fully discussed. The first of these solutions is that involving mesh updating, where we proceed as follows. Assuming a known reference surface, say the original horizontal surface of the water, we specify that the pressure on this surface is zero and solve the resulting fluid mechanics problem by the methods of the previous chapter. We start with known values of the velocities and find the necessary increment obtaining u ~+~ and pn+l from initial values. Immediately following this step we compute the increment of ~7using the newly calculated values of the velocities. We note here that this last equation is solved only in two dimensions on a mesh corresponding to the projected coordinates Of Xl and x2. At this stage the surface can be immediately updated to a new position which now becomes the new reference surface and the procedure can then be repeated to reach steady state.
Hydrostatic adjustment
Obviously the method of repeated mesh updating can be extremely costly and in general we follow the process described as hydrostatic adjustment. In this process we note that once the incremental ~7has been established, we can adjust the surface pressure at the reference surface by Pref -- P +
AT]npg
(6.16)
After each time step the above pressure value is forced on the free surface without altering the mesh. Of course this introduces an approximation but this approximation can be quite happily used for starting the following step.
Free surface flows
If we proceed in this manner until the solution of the basic flow problem is well advanced and the steady state has nearly been reached we have a solution which is reasonably accurate for small waves but which can now be used as a starting point of the mesh adjustment if so desired. In all practical calculations it is recommended that many steps of the hydrostatic adjustment be used before repeating the mesh updating which is quite expensive. In many ship problems it has been shown that with a single mesh quite good results can be obtained without the necessity of proceeding with mesh adjustment. We shall refer to such examples later. The methodologies suggested here follow the work of Hino et al., Idelsohn et al., Lrhner et al. and On~te et al. 8' 10, 12, 15,47-49 The methods which we discussed in the context of ships here provide a basis on which other free surface problems can be started at all times and are obviously an improvement on a very primitive adjustment of surface by trial and error.
Numerical examples using mesh regeneration methods Example 6.2 A submerged hydrofoil
We start with the two-dimensional problem shown in Fig. 6.7, where an NACA0012 aerofoil profile is used in submerged form as a hydrofoil which could in the imagination of the reader be attached to a ship. This is a model problem, as many two-dimensional situations are not realistic. Here the angle of attack of the flow is 5 ~ and the Froude number is 0.5672. In Fig. 6.8 we show the pressure distribution throughout the domain and the comparison of the computed wave profiles with the experimental 5~ and other numerical solutions. 48 In Figs 6.7 and 6.8, the mesh is moved after a certain number of iterations using an advancing front technique. 12' 51
Fig. 6.7 A submerged hydrofoil. Mesh updating procedure. Eulerflow. Mesh after 1900 iterations.
179
180
Free surface and buoyancy driven flows
(a) Surface Pressure Contour 0.15
:i :',::: ......
0.10 0.05
Duncan5~ " "
-
/
A~
/
~
0
-0.05 -0.10 -0.15 -2
000
I -1
I 0
''~
! 1
I 2
X/L (b) Comparison of Wave Profiles
I 3
! 4
5
Fig. 6.8 A submergedhydrofoil. Meshupdating procedure. Eulerflow. (a) Pressuredistribution. (b) Comparison with experiment. Figure 6.9 shows the same hydrofoil problem solved now using hydrostatic adjustment without moving the mesh. For the same conditions, the wave profile is somewhat underpredicted by the hydrostatic adjustment [Fig. 6.9(b)] while the mesh movement overpredicts the peaks [Fig. 6.9(b)]. In Fig. 6.10, the results for the same hydrofoil in the presence of viscosity are presented for different Reynolds numbers. As expected the wake is now strong as seen from the velocity magnitude contours [Figs 6.10(a)-(d)]. Also at higher Reynolds numbers (5000 and above), the solution is not stable behind the aerofoil and here an unstable vortex street is predicted as shown in Figs 6.10(c) and 6.10(d). Figure 6.10(e) shows the comparison of wave profiles for different Reynolds numbers.
Example 6.3 Submarine
In Fig. 6.11, we show the mesh and wave pattern contours for a submerged DARPA submarine model. Here the Froude number is 0.25. The converged solution is obtained by about 1500 time steps using a parallel computing environment. The mesh consisted of approximately 321 000 tetrahedral elements.
Free surface flows
(a) Surface pressure contour 0.2 Experiment Idelsohn et all2
0.15 0.10 o tO
0.05 .
.
.
.
.
(D
> -0.05
-0.1o -0.15
-0.20
I -2
I 0
(b) Comparison of wave profiles
I 2 Distance
I 4
I 6
8
Fig. 6.9 A submerged hydrofoil. Hydrostatic adjustment. Euler flow. (a) Pressure contours and surface wave pattern. (b) Comparison with experiment,s~
Example 6.4 Sailing yacht 12
The last example presented here is that of a sailing yacht. In this case the yacht has a 25 ~ heel angle and a drift angle of 4 ~ Here it is essential to use either Euler or Navier-Stokes equations to satisfy the Kutta-Joukoski condition as the potential form has difficulty in satisfying these conditions on the trailing edge of the keel and rudder. Here we used the Euler equations to solve this problem. Figure 6.12(a) shows a surface mesh of hull, keel, bulb and rudder. A total of 104 577 linear tetrahedral
181
182
Free surface and buoyancy driven flows
(b) R e = 1000
(a) Re = 500
/
(d) Re = 10000
(c) Re = 5000 Re = 500 R e = 1000 . . . .
0.04
= 5000 . . . . . . . R~= 10000 .............. Re
~ t k t/'~
.3'
o
0
....
P
",;,k-,,._~\
t;l',. i i II
i
lJ a
7
: !
,
-0.02 -
-0.04
-0.06
(e)
-4
I -2
I 0
I 2 Distance
I
4
I
6
8
Fig. 6.10 A submerged hydrofoil. Hydrostatic adjustment? 2 Navier-Stokes flow. (a)-(d) Magnitude of total velocity contours for different Reynolds numbers. (e) Wave profiles for different Reynolds numbers.
Free surface flows
(a) l=
i
(b) Fig. 6.11 Submerged DARPAsubmarine model.8 (a) Surface mesh. (b) Wave pattern.
183
184 Freesurface and buoyancydriven flows elements was used in the computation. Figure 6.12(b) shows the wave profile contours corresponding to a sailing speed of 10 knots.
6.2.4 Arbitrary-Lagrangian-Eulerian (ALE) method ....................................................................................................
- ....................................................................................
---:
..........
-----:
....................
-
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
-
....
_. . . . . . . . . . .
-:-
....
. ......
-
. . . . . . . . . . . . . . . . . . . . . . .
As mentioned before ALE methods have some features of both Lagrangian and Eulerian methods. This procedure is more complex to implement but there are several algorithms available in the literature. First, let us deal with the basics of an ALE description of flow (for an alternative description see reference28). Assume a scalar convection-diffusion
~'~~ "-~5~ . . . . .
(a)
(b) Fig. 6.12 A sailing boat. (a) Surface mesh of hull, keel, bulb and rudder. (b) Wave profile.
~,
Free surface flows
equation of the form
--
Dt
Oxi
(6.17)
~xi
where D / D t is the total time derivative. Let q~be transported to a new position P f in a time increment of At with a velocity of u as shown in Fig. 6.13. 20 The ALE method allows independent movement of grid points, i.e. the grid point is moved to a new position Pr in the time increment At with a grid velocity of Ugi. The scalar variable ~bfor position P f at t + At may be expressed using Taylor series expansion (in both space and time) as
ot
(6.18a)
+"
substituting n
04'7,,
#ppz = ~
(6.18b)
+ uiAt--~x i + . . .
into the above equation and neglecting second and higher order terms, we get n
#PPz = ~n+l = #PP + At--~-- + u i A t - ~ x i + . . .
(6.18c)
where u i are the convective velocity components. In a similar fashion ~ at position Pr at time t may be expanded (only in space) as ~Pr "-"
(~n =
Cp + U g i A t - ~ x i + . . .
(6.19)
here Ug~ are the grid velocity components. The relative value of the scalar variable ~b between actual particle and grid motion may be written as A ~ = c~n+l -- ckn = At--~-- + (ui -- u g i ) A t Ox------[
(6.20)
The above equation may be rewritten as At --+ 0 and dropping super and subscripts, Dq9 Oq9 O~b D t -- --~ + (ui - Ugi) --
(6.21)
OX i
x2
I
mr
(u - % ) At
UgAt//~~
.j
•
Fig. 6.13 ALE description in Cartesian coordinates.
Pf =xl
185
186 Freesurface and buoyancy driven flows
Substituting Eq. (6.21) into Eq. (6.17) gives the following equation in an ALE framework
0r ~_(Ui __Ugi) O~ -~OX---~
0 (Ot~) OXi k~x i
=0
(6.22)
Similarly the incompressible fluid dynamics equations in terms of total derivatives can be written as (for simplicity it is written in terms of primitive variables) Continuity
Dp OUi Dt + P-~ixi -- 0
(6.23)
and Momentum _ 10p 10rij O t -- ---p Ox---~-]- -p ~xj -~- g i
Dui
(6.24)
The ALE settings for the incompressible flows can be written as Continuity
Onxi/ = 0 -Op ~ - ~- (Ui __ Ugi) ~Op x/+ P~
(6.25)
and Momentum
OUi OUi 1 0 p ~- 10wij --~ -]- ( Ui -- Ugi ) ox----f -- -- p COX---7- p
+ gi
(6.26)
For constant density flows the continuity equation becomes Oui Oxi
= 0
(6.27)
It is easy to observe that Eq. (6.26) becomes a Lagrangian equation if grid velocity ugi is equal to fluid velocity ui and it becomes a Eulerian equation if ugi -- O. Thus by using an ALE method it will be possible to shift between the Lagrangian and Eulerian frameworks if necessary.
ALE implementation
As mentioned before implementation of the ALE method is difficult and varies depending on the problem to be solved. However, many have used this method to solve free surface problems with relatively small free surface displacement. 16-3~The most difficult problem in an ALE algorithm is allocating an appropriate mesh velocity Ugi. There is no universal method of determining the mesh velocity. Most of the time mesh velocity is problem dependent. It is standard practice in ALE algorithms to split the scheme into three phases. They are (1) Lagrangian solution, (2) Mesh rezoning action, and (3) Eulerian calculation. However, seldom are these phases discriminated in a computer code. The ALE procedure starts with a Lagrangian step as discussed in Sec. 6.2.2. If the mesh is expected to undergo too much distortion then a mesh rezoning procedure
Free surface flows
should be employed. There are various methods available for rezoning that include remeshing. 17 Once the rezoning is carded out a mesh velocity can be calculated from the nodal displacements to use it in the calculation of the convection velocity appearing in Eq. (6.26). There are several mesh rezoning procedures available for triangular and tetrahedral elements. One of such procedures was introduced by Giuliani 57 in which a function is constructed from measures of distortion and squeeze and minimized. This procedure works well for domains with fixed boundaries. Several improvements have been later carried out by many authors. 58 A variable smoothing method based on a combination of Laplacian and Winslow's method was introduced by Hermansson and Hansbo 59 which preserves the element stretching. It is also possible to use simple smoothing procedures widely employed in mesh generation. For instance we have employed a Laplacian type smoothing procedure in which coordinates of a node are recalculated as an average of the coordinates of the surrounding nodes. Depending on requirement this smoothing procedure can be employed several times within a single time step. 3~
Example 6.5 Sofitary wave propagation
We now consider a simple example of a solitary wave propagation between two walls. Figure 6.14 shows the problem definition. It consists of a liquid with free surface constrained within three walls, two vertical walls and one bottom horizontal wall. The total horizontal length of the domain is 16 and d = 1. The gravity direction is downward vertical with g = 9.81. The viscosity of the fluid was assumed to be 0.01. The time step employed was 0.025. The walls are assumed to be slip walls and initial conditions are calculated based on the work presented by Laitone 6~ for an infinite domain. The relationships for total wave height, velocity components and pressure are given as h - d + Hsech 2
U 1 --
V/-~dSeCh e
~
(Xl - ct)
~-~(X
1 --
(6.28)
ct)
(6.29)
1
]
tan,
Bottom wall Fig. 6.14 Solitary wave propagation. Problem definition.
(6.30)
187
188
Free surface and buoyancy driven flows
and p -- p g ( h
-
x2)
(6.31)
In the above equation c is given as - 1 -~ 2 d
20
+ O
(6.32)
The initial solution and mesh are generated by substituting t = 0 into Eqs (6.28)(6.30). In Figs 6.15 and 6.16 we show the meshes and the velocity vectors at various time levels for H i d = 0.3. The total number of elements and nodes are unchanged during the calculation, they are 3838 and 2092 respectively. The semi-implicit form of the CBS scheme was used in the calculations. The small portion close to the bottom wall was assumed to be Eulerian at all the time. After every time step the Lagrangian portion of the mesh is moved with the fluid and followed by mesh smoothing. The convection term is a part of the ALE scheme. However, during the Lagrangian phase the convection velocity is automatically calculated as equal to zero and after the mesh smoothing phase the convection velocity may have a value due to the mesh movement. Note that after mesh smoothing (ui - Uig) will be non-zero on the nodes, which are moved relative to the fluid motion. Figure 6.17 shows the fluid height at fight and left walls with respect to time. As seen the first peak at the fight wall is reached around a time of 2.28 and at the left wall it is reached at 6.9. Figure 6.18 shows the comparison of maximum height reached D against the experimental data of Maxworthy. 61 As seen the agreement between the numerical and experimental data is excellent.
(a) t= 0.0.
(b) t= 2.28.
(c) t = 4.56.
(d) t= 6.84.
(e) t = 9.12. Fig. 6.15
Solitary wave propagation. Meshes at various time levels.
iiiii~i~iiiiliiiiiiiiii~iiiii~
Buoyancy driven flows
(a) t = 0.0.
(b) t= ::,.::,8.
(c) t = 4 . 5 6 .
~~)~;~:~:.i:i:i:i``~:~:.:~:~y..~:.:~:~:~:.`.`:`...~v..vv......`~``..``.`uu~.~............i
............--.--:..-v.v..:..'.;:7::::'-"-X;:::::s ~:i;.'.::::::%::::':":}}))))):-":::::--".:.s : ! ; / ! ',",'",'~2~,;:dt !i!:i;{;!~~
,,~-~,~ ! I ~i 4" !"!"" " " "~:':'X':':--'! ;--"-'--'' -"-"-".":' ! -:--:: ::i ::: ::::::::: ! i ::::::::'':: ::::::',::::I
(d) t= 6.84.
(e) t= 9.12. Fig. 6.16 Solitary wave propagation. Velocity vector distribution at various time levels.
1.8
.
1.6
~-
.
.
.
1.8 |
"
1.6
1.4
.
.
.
.
.
. -
~1.4 ~'-
1.2
-'-- 1.2
1 0"80
.
/
1 '
2
'
4
'
'
'
6 8 10 Time (a) Right wall.
'
12
'
14
0.8
0
'
2
'
4
'
'
'
6 8 10 Time (b) Left wall.
'
12
'
14
Fig. 6.17 Wave heights with respect to time on the right and left side walls.
In some problems of incompressible flow the heat transport equation and the equations of motion are weakly coupled. If the temperature distribution is known at any time, the density changes caused by this temperature variation can be evaluated. These may on occasion be the only driving force of the problem. In this situation it is convenient to note that the body force with constant density can be considered as balanced by an initial hydrostatic pressure and thus the driving force which causes the motion is in fact
189
190 Free surface and buoyancy driven flows I
1.4
Numerical Exp.
1.2
"c:} 0.8 0.6 0.4
~J
I
I
I
+
-
J
0.2 0.2
0.25
0.3
0.35
Hid
0.4
0.45
Fig. 6.18 Solitary wave propagation. Comparison of wave heights with experimental data 61
the body force caused by the difference of local density values. We can thus write the body force at any point in the equations of motion as
P
]
07"/j
OXi -+- ~
+ -~xj(UjUi) --
-- gi ( P - Por
(6.33)
where p is the actual density applicable locally and p~ is the undisturbed constant density. The actual density entirely depends on the coefficient of thermal expansion of the fluid as compressibility is by definition excluded. Denoting the coefficient of thermal expansion as "Tr, we can write %, = P
~-~
(6.34)
where T is the absolute temperature. The above equation can be approximated to 1 p-p~
%, ~ -
(6.35)
pT-T~
Replacing the body force term in the momentum equation by the above relation we can write
p
[ Oui
~ ] + -~xI(UjUi) -"
Op
+ ~
JI-~/Tgi(Zoo -- T)
(6.36)
For perfect gases, we have
P (6.37) RT and here R is the universal gas constant. Substitution of the above equation (assuming negligible pressure variation) into Eq. (6.34) leads to p =
1 7T = -~
(6.38)
The various governing non-dimensional numbers used in the buoyancy flow calculations are the Grashoff number (for a detailed non-dimensionalization procedure see references 62-64) g y T ( T ~ - T)L 3 Gr = (6.39) //Ol
Concluding remarks 191 and the Prandtl number y
Pr = --
(6.40)
o~
where L is a reference dimension, and u and ce are the kinematic viscosity and thermal diffusivity respectively and are defined as # u = -,
a =
p
k
(6.41)
pep
where # is the dynamic viscosity, k is the thermal conductivity and Cp the specific heat at constant pressure. In many calculations of buoyancy driven flows, it is convenient to use another non-dimensional number called the Rayleigh number ( R a ) which is the product of Gr and Pr. In many practical situations, both buoyancy and forced flows are equally strong and such cases are often called mixed convective flows. Here in addition to the abovementioned non-dimensional numbers, the Reynolds number also plays a role. The reader can refer to several available basic heat transfer books and other publications to get further details. 64-8~
Example 6.6
Buoyancy driven flow in an enclosure
Fundamental buoyancy flow analysis in closed cavities can be classified into two categories. The first one is flow in closed cavities heated from the vertical sides and the second is bottom-heated cavities (Rayleigh-Benard convection). In the former, the CBS algorithm can be applied directly. However, the latter needs some perturbation to start the convective flow as they represent essentially an unstable problem. Figure 6.19 shows the results obtained for a closed square cavity heated at a vertical side and cooled at the other. 62 Both of the horizontal sides are assumed to be adiabatic. At all surfaces both of the velocity components are zero (no-slip conditions). The mesh used here was a non-uniform structured mesh of size 51 x 51. As the reader can see, the essential features of a buoyancy driven flow are captured using the CBS algorithm. The quantitative results compare excellently with the available benchmark solutions as shown in Table 6.1. 62 The adapted meshes for two different Rayleigh numbers are shown in Fig. 6.20. The adaptive methods are discussed in Chapter 4 for incompressible flows.
ii!i!i~iiiiiiii~iiiiiiiiiiiii~iiiiiiii~ii~!i!!iiiiiii~iiiiiiiiiiiii~i~i!iiii~iiiiii~iiiiiiii~!i!iiiiiiiiiiiii~i~iiiiiii~!i~i~i~i! iii!!iiiiiiiiiiiiiiliiiiiii!iii!ii!i!i!i!i!i!i!iii ~i:i~iiii~iiiiiiiiiiili!i!iiiiiiiiiii iiiii~!!!!iii!i ili!!!!iiiiil
i
i
i
i
ii
ii
iiiiiii
We have summarized all major methods for dealing with free surface flow analysis in this chapter. The numerical formulation of the free surface problems were only briefly discussed. Although brief, we believe that we have provided the reader with some essential techniques to start research on free surface flows. The section on the buoyancy driven flows is kept purposely brief. Interested readers should consult the cited references for additional details.
192
Free surface and buoyancy driven flows
(a)
(b)
(c) Fig. 6.19 Naturalconvection in a square enclosure. Streamlinesand isothermsfor different Rayleighnumbers.
References 193 Table 6.1 Natural convection in a square enclosure. Comparison with available numerical solutions. 62 References are shown in square brackets Average Nusselt number
~3ma x
Vmax
Ra
[79]
[80]
CBS
[79]
[80]
CBS
[79]
[80]
CBS
103 104 105 106
1.116 2.243 4.517 8.797
1.118 2.245 4.522 8.825 16.52 23.78
1.117 2.243 4.521 8.806 16.40 23.64
1.174 5.081 9.121 16.41 -
1.175 5.074 9.619 16.81 30.17 -
1.167 5.075 9.153 16.49 30.33 43.12
3.696 19.64 68.68 221.3 -
3.697 19.63 68.64 220.6 699.3 -
3.692 19.63 68.85 221.6 702.3 1417
10 7
-
4 x 107
-
(a)
(b)
Fig. 6.20 Natural convection in a square enclosure. Adapted meshes for (a) Ra = 105 and (b) Ra = 106.
1. E Bach and O. Hassager. An algorithm for the use of the Lagrangian specification in Newtonian fluid mechanics and applications to free-surface flow. J. Fluid Mech., 152:173-190, 1985. 2. B. Ramaswamy and M. Kawahara. Lagrangian finite element analysis applied to viscous free surface fluid flow. Int. J. Num. Meth. Fluids, 87:953-984, 1987. 3. E Muttin, T. Coupez, M. Bellet and J.L. Chenot. Lagrangian finite element analysis of timedependent viscous free-surface flow using an automatic remeshing technique: application to metal casting flow. Int. J. Num. Meth. Eng., 36:2001-2015, 1993. 4. Y.T. Feng and D. Peric. A time adaptive space-time finite element method for incompressible flows with free surfaces: computational issues. Comp. Meth. Appl. Mech. Eng., 190:499-518, 2000. 5. C.W. Hirt and B.D. Nichols. Volume of fluid (VOF) method for the dynamics of free surface boundaries. J. Comp. Phy., 39:210-225, 1981. 6. J. Farmer, L. Martinelli and A. Jameson. Fast multigrid method for solving incompressible hydrodynamic problems with free surfaces. AIAA Journal, 32:1175-1182, 1994. 7. M. Beddhu, L.K. Taylor and D.L. Whitfield. A time accurate calculation procedure for flows with a free surface using a modified artificial compressibility formulation. Applied Mathematics and Computation, 65:33-48, 1994.
194 Freesurface and buoyancydriven flows 8. R. Lrhner, C. Yang, E. Ofiate and I.R. Idelsohn. An unstructured grid based, parallel free surface solver. AIAA 97-1830, 1997. 9. G.D. Tzabiras. A numerical investigation of 2D, steady free surface flows. Int. J. Num. Meth. Fluids, 25:567-598, 1997. 10. R. Lrhner, C. Yang and E. Ofiate. Free surface hydrodynamics using unstructured grids. 4th ECCOMAS CFD Conf., Athens, 7-11 September 1998. 11. M. Beddhu, M.-Y. Jiang, L.K. Taylor and D.L. Whitfield. Computation of steady and unsteady flows with a free surface around the wigley hull. Applied Mathematics and Computation, 89:67-84, 1998. 12. I.R. Idelsohn, E. Ofiate and C. Sacco. Finite element solution of free surface ship wave problems. Int. J. Num. Meth. Eng., 45:503-528, 1999. 13. S. Shin and W.I. Lee. Finite element analysis of incompressible viscous flow with moving free surface by selective volume of fluid method. Int. J. Heat Fluid Flow, 21:197-206, 2000. 14. E.H. van Brummelen, H.C. Raven and B. Koren. Efficient numerical solution of steady flee-surface Navier-Stokes flow. J. Comp. Phy., 174:120-137, 2001. 15. E. Ofiate, J. Garcfa and S.R. Idelsohn. Ship hydrodynamics. Encyclopedia of Computational Mechanics, Wiley, 2004. 16. C.W. Hirt, A.A. Amsden and J.L. Cook. An arbitrary Lagrangian-Eulerian computing method for all flow speeds. J. Comp. Phy., 14:227-253, 1974. 17. B. Ramaswamy and M. Kawahara. Arbitrary Lagrangian Eulerian finite element method for unsteady, convective incompressible viscous free surface fluid flow. Int. J. Num. Meth. Fluids, 7:1053-1075, 1987. 18. S.E. Navti, K. Ravindran, C. Taylor and R.W. Lewis. Finite element modelling of surface tension effects using a Lagrangian-Eulerian kinematic description. Comput. Methods Appl. Mech. Eng., 147:41-60, 1997. 19. S. Ushijima. Three dimensional Arbitrary Lagrangian Eulerian numerical prediction method for non-linear free surface oscillation. Int. J. Num. Meth. Fluids, 26:605-623, 1998. 20. J.G. Zhou and P.K. Stansby. An arbitrary Lagrangian-Eulerian o- (ALES) model with nonhydrostatic pressure for shallow water flows. Comput. Methods Appl. Mech. Eng., 178:199-214, 1999. 21. H. Braess and P. Wriggers. Arbitrary Lagrangian Eulerian finite element analysis of free surface flow. Comput. Methods Appl. Mech. Eng., 190:95-109, 2000. 22. M. Iida. Numerical analysis of self-induced free surface flow oscillation by fluid dynamics computer code splash-ale. Nuclear Eng. Design, 200:127-138, 2000. 23. J. Sung, H.G. Choi and J.Y. Yoo. Time accurate computation of unsteady free surface flows using an ALE- segregated equal order FEM. Comput. Methods Appl. Mech. Eng., 190:1425-1440, 2000. 24. M. Souli and J.P. Zolesio. Arbitrary Lagrangian-Eulerian and free surface methods in fluid mechanics. Comput. Methods Appl. Eng., 191:451-466, 2001. 25. M.-H. Hsu, C.-H. Chen and W.-H. Teng. An arbitrary Lagrangian-Eulerian finite difference method for computations of free surface flows. J. Hydraulic Res., 39:1-11, 2002. 26. S. Rabier and M. Medale. Computation of free surface flows with a projection FEM in a moving framework. Comput. Methods Appl. Mech. Eng., 192:4703-4721, 2003. 27. D.C. Lo and D.L. Young. Arbitrary Lagrangian-Eulerian finite element analysis of free surface flow using a velocity-vorticity formulation. J. Comp. Phy., 195:175-201, 2004. 28. J. Donea andA. Huerta. Finite Element Method for Fluid Flow Problems. Wiley, Chichester, 2003. 29. J. Donea, A. Huerta, J.-Ph. Ponthot and A. Rodriguez-Ferran. Arbitrary Lagrangian-Eulerian methods. In Encyclopedia of Computational Mechanics, Chapter 14. Wiley, Chichester, 2004. 30. P. Nithiarasu. An arbitrary lagrangian eulerian formulation for free surface flows using the Characteristic Based Split (CBS) scheme. Int. J. Num. Meth. Fluids, 48:1415-1428, 2005. 31. J.C. Martin and W.J. Moyce. An experimental study of the collapse of liquid columns on a rigid horizontal plane. Philos. Trans. Roy. Soc. London Ser. A, 244:312-324, 1952.
References 195 32. S.R. Idelsohn, N. Calvo and E. Ofiate. Polyhedrization of an arbitrary 3D point set. Comput. Methods Appl. Mech. Eng., 192:2649-2667, 2003. 33. S.R. Idelsohn, N. Calvo, E. O~ate and ED. Pin. The meshless finite element method. International J. Num. Meth. Eng., 58:893-912, 2003. 34. S.R. Idelsohn, E. Ofiate and ED. Pin. A lagrangian meshless finite element method applied to fluid-structure interaction problems. Computers and Structures, 81:655-671, 2003. 35. J.V. Wehausen. The wave resistance of ships. Adv. Appl. Mech., 1970. 36. C.W. Dawson. A practical computer method for solving ship wave problems. Proc. of the 2nd Int. Conf. on Num. Ship Hydrodynamics, USA, 1977. 37. K.J. Bai and J.H. McCarthy, editors. Proc. of the second DTNSRDC Workshop on Ship Wave Resistance Computations. Bethesda, MD, USA, 1979. 38. F. Noblesse and J.H. McCarthy, editors. Proc. of the second DTNSRDC Workshop on Ship Wave Resistance Computations. Bethesda, MD, USA, 1983. 39. G. Jenson and H. Soding. Ship wave resistance computation. Finite Approximations in Fluid Mechanics II, 25, 1989. 40. Y.H. Kim and T. Lucas. Non-linear ship waves. Proc. ofthe 18th Symp. on Naval Hydrodynamics. MI, USA, 1990. 41. D.E. Nakos and Pd. Scavounos. On the steady and unsteady ship wave patterns. J. Fluid Mech., 215:256-288, 1990. 42. H. Raven. A practical non-linear method for calculating ship wave making and wave resistance. Proc. 19th Sym. on Ship Hydrodynamics. Seoul, Korea, 1992. 43. R.E Beck, Y. Cao and T.H. Lee. Fully non-linear water wave computations using the desingularized method. Proc. of the 6th Sym. on Num. Ship Hydrodynamics. Iowa City, Iowa, USA, 1993. 44. H. Soding. Advances in panel methods. Proc. 21st Sym. on Naval Hydrodynamics. Trondheim, Norway, 1996. 45. C. Janson and L. Larsson. A method for the optimization of ship hulls from a resistance point of view. Proc. 21st Sym. on Naval Hydrodynamics. Trondheim, Norway, 1996. 46. C.C. Mei and H.S. Chen. A hybrid element method for steady linearized free surface flows. Int. J. Num. Meth. Eng., 10:1153-1175, 1976. 47. T. Hino. Computation of free surface flow around an advancing ship by Navier-Stokes equations. Proc. of the 5th Int. Conf. Num. Ship Hydrodynamics, 103-117. Hiroshima, Japan, 1989. 48. T. Hino, L. Martinelli and A. Jameson. A finite volume method with unstructured grid for free surface flow. Proc. of the 6th Int. Conf. on Num. Ship Hydrodynamics. Iowa City, IA, 173-194, 1993. 49. T. Hino. An unstructured grid method for incompressible viscous flows with free surface. AIAA-97-0862, 1997. 50. J.H. Duncan. The breaking and non-breaking wave resistance of a two dimensional hydrofoil. J. Fluid Mech., 126:507-516, 1983. 51. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method. Its Basis and Fundamentals. Elsevier, 2005. 52. F.H. Harlow and J.E. Welch. Numerical calculations of time dependent viscous incompressible flow of a fluid with a free surface. Phys. Fluids, 8:2182, 1965. 53. W.E. Johnson. Development and application of computer programs related to hypervelocity impact. Systems Science and Software Report 3SR-353, 1970. 54. R. Codina, U. Sch~ier and E. Ofiate. Mould filling simulation using finite elements. Int. J. Num. Meth. Heat Fluid Flow, 4:291-310, 1994. 55. K. Ravindran and R.W. Lewis. Finite element modelling of solidification effects in mould filling. Finite Elements in Analysis and Design, 31:99-116, 1998. 56. R.W. Lewis and K. Ravindran. Finite element simulation of metal casting. Int. J. Num. Meth. Eng., 47, 2000. 57. S. Giuliani. An algorithm for continuous rezoning of the hydrodynamic grid in arbitrary Lagrangian-Eulerian computer codes. Nuc. Eng. Desn., 72:205-212, 1982.
196 Free surface and buoyancydriven flows 58. J. Sarrate and A. Huerta. An improved algorithm to smooth graded quadrilateral meshes preserving the prescribed element size. Comm. Numer. Meth. Eng., 17:89-99, 2001. 59. J. Hermansson and P. Hansbo. A variable diffusion method for mesh smoothing. Comm. Numer. Meth. Eng., 19:897-908, 2003. 60. E.V. Laitone. The second approximation to cnoidal and solitary waves. J. Fluid Mech., 9:430-444, 1960. 61. T. Maxworthy. Experiments on collisions between solitary waves. J. Fluid Mech., 76:177-186, 1976. 62. N. Massarotti, P. Nithiarasu and O.C. Zienkiewicz. Characteristic-based-split (CBS) algorithm for incompressible flow problems with heat transfer. Int. J. Num. Meth. Heat Fluid Flow, 8:969-990, 1998. 63. P. Nithiarasu, T. Sundararajan and K.N. Seetharamu. Finite element analysis of transient natural convection in an odd-shaped enclosure. Int. J. Num. Meth. Heat Fluid Flow, 8:199-220, 1998. 64. R.W. Lewis, P. Nithiarasu and K.N. Seetharamu. Fundamentals of the Finite Element Method for Heat and Fluid Flow. Wiley, Chichester, 2004. 65. EP. Incropera and D.P. Dewitt. Fundamentals of Heat and Mass Transfer, 3rd edition. Wiley, New York, 1990. 66. A. Bejan. Heat Transfer. Wiley, New York, 1993. 67. Y. Jaluria. Natural Convection Heat Transfer. Springer-Verlag, New York, 1980. 68. O.C. Zienkiewicz, R.H. Gallagher and P. Hood. Newtonian and non-Newtonian viscous incompressible flow. Temperature induced flows. Finite element solutions. In J. Whiteman, editor, Mathematics of Finite Elements and Applications, volume. II, Chapter. 20, pp. 235-267. Academic Press, London, 1976. 69. J.C. Heinrich, R.S. Marshall and O.C. Zienkiewicz. Penalty function solution of coupled convective and conductive heat transfer. In C. Taylor, K. Morgan and C.A. Brebbia, editors, Numerical Methods in Laminar and Turbulent Flows, pp. 435-447. Pentech Press, 1978. 70. M. Strada and J.C. Heinrich. Heat transfer rates in natural convection at high Rayleigh numbers in rectangular enclosures. Numerical Heat Transfer, 5:81-92, 1982. 71. J.C. Heinrich and C.C. Yu. Finite element simulation of buoyancy driven flow with emphasis on natural convection in horizontal circular cylinder. Comp. Meth. Appl. Mech. Eng., 69:1-27, 1988. 72. B. Ramaswamy. Finite element solution for advection and natural convection flows. Comp. Fluids, 16:349-388, 1988. 73. B. Ramaswamy, T.C. Jue and J.E. Akin. Semi-implicit and explicit finite element schemes for coupled fluid thermal problems. Int. J. Num. Meth. Eng., 34:675-696, 1992. 74. B.V.K. Sai, K.N. Seetharamu and P.A.A. Narayana. Finite element analysis of the effect of radius ratio on natural convection in an annular cavity. Int. J. Num. Meth. Heat Fluid Flow, 3:305-318, 1993. 75. C. Nonino and S. Delgiudice. Finite element analysis of laminar mixed convection in the entrance region of horizontal annular ducts. Numerical Heat Transfer, Part A, Applications, 29:313-330, 1996. 76. S.C. Lee and C.K. Chen. Finite element solutions of laminar and turbulent mixed convection in a driven cavity. Int. J. Num. Meth. Fluids, 23:47-64, 1996. 77. S.C. Lee, C.Y. Cheng and C.K. Chen. Finite element solutions of laminar and turbulent flows with forced and mixed convection in an air-cooled room. Numerical Heat Transfer, Part A, Applications, 31:529-550, 1997. 78. Y.T.K. Gowda, P.A.A. Narayana and K.N. Seetharamu. Finite element analysis of mixed convection over in-line tube bundles. Int. J. Heat Mass Transfer, 41:1613-1619, 1998. 79. G. De Vahl Davis. Natural convection of air in a square cavity: a benchmark numerical solution. Int. J. Num. Meth. Fluids, 3:249-264, 1983. 80. P. Le Quere and T.A. De Roquefort. Computation of natural convection in two dimensional cavity with Chebyshev polynomials. J. Comp. Phy., 57:210-228, 1985.
!i iiiliiiii iiiiii i !iliiliii !il !!ii Compressible high-speed gas flow
Problems posed by high-speed gas flow are of obvious practical importance. Applications range from the exterior flows associated with flight to interior flows typical of turbomachinery. As the cost of physical experiments is high, the possibilities of computations were explored early and the development concentrated on the use of finite difference methods. It was only in the 1980s that the potential offered by the finite element forms were realized. One of the main advantages in the use of the finite element approximation here is its capability of fitting complex forms and permitting local refinement where required. However, the improved accuracy attainable by finite element methods is also of substantial importance as practical problems will often involve three-dimensional discretization with a very large number of degrees of freedom much larger than those encountered in typical structural problems. For such large problems direct solution methods are obviously not practicable and iterative methods based generally on transient computation forms are invariably used. Here of course we follow and accept much that has been established by the finite difference applications but generally will lose some computational efficiency associated with structured meshes. However, the reduction of the problem size which, as we shall see, can be obtained by local refinement and adaptivity will more than compensate for this loss (though of course structured meshes are included in the finite element forms). In Chapters 1 and 3 we have introduced the basic equations governing the flow of compressible gases as well as of incompressible fluids. Indeed in the latter, as in Chapter 4, we can introduce small amounts of compressibility into the procedures developed there specifically for incompressible flow. In this chapter we shall deal with high-speed flows with Mach numbers generally in excess of 0.5. Such flows will usually involve the formation of shocks with characteristic discontinuities. For this reason we shall concentrate on the use of low-order elements and of explicit methods, such as those introduced in Chapters 2 and 3. Here the pioneering work of Morgan, L/3hner, Peraire, Hassan and Weatherill must be acknowledged. 1-41 It was this work that opened the doors to practical finite element analysis in the field of aeronautics. We shall refer to their work frequently. In the first practical applications the Taylor-Galerkin process outlined in Appendix E for vector-valued variables was used almost exclusively. In this book we recommend, however, the CBS algorithm discussed in Chapter 3 as it presents a better
198 Compressiblehigh-speed gas flow approximation and has the advantage of dealing directly with incompressibility, which invariably occurs in small parts of the domain, even at high Mach numbers (e.g. in stagnation regions).
The Navier-Stokes governing equations for compressible flow were derived in Chapter 1. We shall repeat only the simplified form of Eq. (1.25) and here again using indicial notation. We thus write, for i = 1, 2, 3, 0tI)
0F i
0Gi
--~-+~x/+-~-x/+Q=O
(7.1)
with ~ T __ [p, pUl, pU2, pU3, pE]
(7.2a)
F T -" [pui, pului "l- P~li, pu2ui -'1-P~2i, pu3ui-I" P~3i, pHui]
(7.2b) (7.2c)
and QT = [0, --Pfl, --P f2, --P f3, --pfiui
-- qn]
(7.2d)
In the above Ouj
20uk
7"ij = ~[(~j -JI--~Xi ) --t~iJ3 -~Xk]
(7.2e)
The above equations need to be 'closed' by addition of the constitutive law relating the pressure, density and energy (see Chapter 1). For many flows the ideal gas law 42 suffices and this is P
p=
(7.3)
RT
where R is the universal gas constant. In terms of specific heats R = ( C p - co) = ( 7 -
1)co
(7.4)
where ,y= Cp
Cv
is the ratio of the constant pressure and constant volume specific heats. The internal energy e and total specific energy E are given as
(1)p
e =coT= E - e+
7-1 1
--_UiU i
2
p
(7.5)
Boundary conditions - subsonic and supersonic flow
and hence pE =
(1) ,~._1
pH = pE + p -
UiUi p + p~ 2
"Y-1
199
(7.6a) UiUi P W P~
(7.6b)
The variables for which we shall solve are usually taken as the set of Eq. (7.2a), i.e. p, pui
and
pE
but of course other sets could be used, though then the conservative form of Eq. (7.1) could be lost. In many of the problems discussed in this chapter inviscid behaviour will be assumed, with Gi = 0 and we shall then deal with the Euler equations. In many problems the Euler solution will provide information about the main features of the flow and will suffice for many purposes, especially if augmented by separate boundary layer calculations (see Sec. 7.13). However, in principle it is possible to include the viscous effects without much apparent complication. Here in general steady-state conditions will rarely arise as the high speed of the flow will be associated with turbulence and this will usually be of a small scale only capable of resolution with very small sized elements. If a 'finite' size of element mesh is used then such turbulence will often be suppressed and steady-state answers will be obtained. We shall in some examples include such full Navier-Stokes solutions using a viscosity dependent on the temperature according to Sutherland's law. 42 In the SI system of units for air this gives a viscosity # =
1.45T3/2
• 10 -6 (7.7) T+ll0 where T is in kelvin. Further turbulence modelling can be done by using the Reynolds averaged viscosity and solving additional transport equations for some additional parameters in the manner discussed in Chapter 8. iiiiii!iiiii~' ~'~' i~il!~'ii~' ~' i!iic'i~i!~ i'l~' iiiiii~iiii!iiii!iiiii!~i!~'ii'!i!i~ii!ii!!i!iiiiiiii~i~!!iii~i!iiiiiiiii!i!i!ii!ii!~iiii~ii!ii!ii~iiiiii~iii~i~ii~!ii!~ii~iii!iiii~i!iiiiiiii~iiiiiiiiiii~!i!ii~iii~iiiiiiiiiiiiiiiii~i~iiiiiiiiiiii~iiii~ii~iiiiiiiiiiii!iiiiiiiiii!i~iiiiiiiiiiii~iii!iiii! iiii
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The question of boundary conditions, which can be prescribed for Euler and NavierStokes equations in compressible flow, is by no means trivial. Demkowicz et al., 43 have addressed the issue of boundary conditions in a general sense and determined their influence on the existence of uniqueness of solutions determining their influence on the existence and uniqueness of solutions. In the following we shall discuss the case of the inviscid Euler form and of the full Navier-Stokes problem. We have already discussed the general question of boundary conditions in Chapter 3 dealing with numerical approximations. Some of these matters have to be repeated in view of the special behaviour of supersonic flow problems.
200
Compressiblehigh-speed gas flow
7.3.1 Euler equation Here only first-order derivatives occur and the number of boundary conditions is less than that for the full Navier-Stokes problem. For a solid wall boundary, F,, only the normal component of velocity Un needs to be specified (zero if the wall is stationary). Further, with lack of conductivity the energy flux across the boundary is zero and hence pE (and p) remain unspecified. In general the analysis domain will be limited by some arbitrarily chosen external boundaries, Fs, for exterior or internal flows, as shown in Fig. 7.1 (see also Sec. 3.8, Chapter 3). Here, it will in general be necessary to perform a linearized Riemann analysis in the direction of the outward normal to the boundary n to determine the speeds of wave propagation of the equations. For this linearization of the Euler equations three values of propagation speeds (characteristics) can be found in one dimension. 42' 44 CO ~
Un
(7.8)
c + = Un + C C_ --Un
~C
where u, is the normal velocity component and c is the compressible wave celerity (speed of sound) given by C
=/7_~ I
(7.9)
r
As of course no disturbances can propagate at velocities greater than those of Eq. 7.8 and in the case of supersonic flow, i.e. when the local Mach number is M-
lUnl C
> 1
(7.10)
we shall have to distinguish two possibilities:
1. Supersonic inflow boundary where Un >C
and the analysis domain cannot influence the upstream position (all characteristics are directed into the domain at the inlet; see Fig. 7.2), for such boundaries all components of the vector 9 must be specified; and
F$ Fu Fu
External flow
Fs~l
~
~/F s
Internal flow
Fig. 7.1 Boundaries of a computation domain. F u, wall boundary; F s, fictitious boundary.
Boundary conditions - subsonic and supersonic flow
2. Supersonic outflow boundaries where Un >C
and here by the same reasoning no components of 9 are prescribed (all characteristics are directed out of the domain). For subsonic boundaries the situation is more complex and here the values of 9 that can be specified are the components of the incoming Riemann variables. However, this may frequently present difficulties as the incoming wave may not be known and the usual compromises may be necessary as in the treatment of elliptic problems possessing infinite boundaries (see Chapter 3, Sec. 3.8). It is often convenient to prescribe boundary conditions for a subsonic flow by once again considering the direction of characteristics from Eq. (7.8). Since U n < C in subsonic flows, one characteristic at the inlet will be directed outwards and two will be directed inwards. At the exit one characteristic is directed inward and two outwards as shown in Fig. 7.2. Thus, it is necessary to prescribe two boundary conditions at inlet and one at exit for the subsonic condition shown in Fig. 7.2. However, this procedure is not easy to follow in multidimensional problems.
7.3.2 Navier-Stokes equations In the Navier-Stokes equations Gi ~= 0 and due to the presence of second derivatives, additional boundary conditions are required. For the solid wall boundary, Fu, all the velocity components are prescribed assuming, as in the previous chapter for incompressible flow, that the fluid is attached to the wall. Thus for a stationary boundary we put Ui " - - O
Further, if conductivity is not negligible, boundary temperatures or heat fluxes will generally be given in the usual manner. CO
c+
Supersonic exit
ic inle
Co /
C_
c _ ~
Co
c+
c_ ubsonic exit
c inlet c+
CO
Fig. 7.2 Characteristic directions at inlet and exit for supersonic and subsonic flows.
201
202 Compressiblehigh-speed gas flow For exterior boundaries Fs of the supersonic inflow kind, the treatment is identical to that used for Euler equations. However, for outflow boundaries a further approximation must be made, either specifying tractions as zero or making their gradient zero in the manner described in Sec. 3.8, Chapter 3. ii!i!iiiiiiiE iJiiiiiiii'~iii ~'~'~''~':":'iiiiiiirl~::'~,::!i ~illii2',iii'iN ,!i~""~~iiii~'~i~"iiiiiiiiiiiiiiiiiiiiiii ~ ~ ~ ii',iiiiiii i iiiiiiilii~iiiiiiii~!iii ! ~ iliii',~:~:':iiiiiiiiiii!iiiiiiiiiiiiiiiiiiiiiiiii}iiiiiii!ii!! i~,iiiiiiii~iiiiiiiiiii!iiiiii' i ~ ~ i ~,:~,iii'i~i,Hil ii~i' ~ ,iiiiii!ilii':iiiii!iiiiiiH',I:i!i',iiiiiiiiiiii!i:~ ~iii'i'~,iiiiii' i :~,ii'iii!,iiii',ii'!i,i'ii,i::~i'~,~:':~i'::iiiiiii}iii~ i~::'~':~'i!i:':i'',~iiiii iii',i!~,iiiiiiiiiiiili ~ ii~i~:,~,~~'~,~',,'~,,~,'',::,~:iiiii~:: : ~::'::'~::ili '~i'ii~'i:,i'~,iii:,iiiii~ '~i'~':~'i:'~ii'i'~,ii,i','i','i,i'~i',i'i!,i',,iiiii',!'iii,ii!iilili:,iiii~ ~ :,',i:::'::~i'i'::,~',':"ii',~i'iii',i,iiiiii!'iiili~,,i:,i'i~ili'iil,iiiii i{iiiii i~':,i~i:,':,ii'i':,ii':,l,ii'',iii'',i'iiiiiii', ,i'iiii', :iii'i,i!il i iii::i'iliiii' :i :i
Various forms of finite element approximation and of solution have been used for compressible flow problems. The first successfully used algorithm here was, as we have already mentioned, the Taylor-Galerkin procedure either in its single-step or twostep form. We have outlined both of these algorithms in Appendix E. However, the most generally applicable and advantageous form is that of the CBS algorithm which we have presented in detail in Chapter 3. We recommend that this be universally used as not only does it possess an efficient manner of dealing with the convective terms of the equations but it also deals successfully with the incompressible part of the problem. In all compressible flows in certain parts of the domain where the velocities are small, the flow is nearly incompressible and without additional damping the direct use of the Taylor-Galerkin method may result in oscillations there. We have indeed mentioned an example of such oscillations in Chapter 3 where they are pronounced near the leading edge of an aerofoil even at quite high Mach numbers (Fig. 3.4). With the use of the CBS algorithm such oscillations disappear and the solution is perfectly stable and accurate. In the same example we have also discussed the single-step and two-step forms of the CBS algorithm. We recommend the two-step procedure, which is only slightly more expensive than the single-step version but more accurate. As we have already remarked if the algorithm is used for steady-state problems it is always convenient to use a localized time step rather than proceed with the same time step globally. The full description of the local time step procedure is given in Sec. 3.4.4 of Chapter 3 and this was invariably used in the examples of this chapter when only the steady state was considered. We have mentioned in the same section, Sec. 3.4.4, the fact that when local time stepping is used nearly optimal results are obtained as Atext and Atin t are the same or nearly the same. However, even in transient problems it is often advantageous to make use of a different At in the interior to achieve nearly optimal damping there. We now summarize the explicit solution procedure below do i = l,number of time steps stepl- calculation of intermediate momentum step2- calculation of density step3- correction of m o m e n t u m step4- energy equation step5- pressure calculation from energy and density enddo !i
For further details on the algorithm readers are referred to Chapter 3. One of the additional problems that we need to discuss further for compressible flows is that of the treatment of 'shocks'. In a discrete solution the presence of shocks leads to oscillations in the solution and it is necessary to introduce some form of smoothing
Shock capture 203 to obtain viable solutions. In the CBS scheme the treatment of shocks is integrated into the first step of the above solution process as described in the next section.
Clearly with the finite element approximation in which all the variables are interpolated using Co continuity the exact reproduction of shocks is not possible. In all finite element solutions we therefore represent the shocks simply as regions of very high gradient. The ideal situation will be obtained if the rapid variations of variables are confined to a few elements surrounding the shock. Unfortunately it will generally be found that such an approximation of a discontinuity introduces local oscillations and these may persist throughout quite a large area of the domain. For this reason, we shall usually introduce into the finite element analysis additional viscosities which will help us in damping out any oscillations caused by shocks and, yet, deriving as sharp a solution as possible. Such procedures using artificial viscosities are known as shock capture methods. It must be mentioned that some investigators have tried to allow the shock discontinuity to occur explicitly and thus allowed a discontinuous variation of an analytically defined kind. This presents very large computational difficulties and it can be said that to date such trials have only been limited to one-dimensional problems and have not been used to any extent in two or three dimensions. For this reason we shall not discuss such s h o c k fitting methods further. 45' 46 The concept of adding additional viscosity or diffusion to capture shocks was first suggested by von Neumann and Richtmyer 47 as early as 1950. They recommended that stabilization can be achieved by adding a suitable artificial dissipation term that mimics the action of viscosity in the neighbourhood of shocks. Significant developments in this area are those of Lapidus, 48 Steger, 49 MacCormack and Baldwin 5~ and Jameson and Schmidt. 51 At Swansea, a modified form of the method based on the second derivative of pressure has been developed by Peraire et al. 12 and Morgan et al. 52 for finite element computations. This modified form of viscosity with a pressure switch calculated from the nodal pressure values is used subsequently in compressible flow calculations. Recently an anisotropic viscosity for shock capturing 53 has been introduced to add diffusion in a more rational way. The implementation of artificial diffusion is very much simpler than shock fitting and we proceed as follows. In this we first calculate the approximate quantities of the solution vector by using the direct explicit method. Now we modify each scalar component of these quantities by adding a correction which smoothes the result. Thus for instance if we consider a typical scalar component quantity q5 and have determined the values of ~bn+l, we establish the new values as below:
q~sn+l ._ (~n+l +
At#a~x i
(7.11)
where #a is an appropriate artificial diffusion coefficient. It is important that whatever the method used, the calculation of #a shou.ld be limited to the domain which is close to the shock as we do not wish to distort the results throughout the problem. For this reason many procedures add a switch usually activated by such quantities as gradients
204
Compressible high-speed gas flow of pressure. In all of the procedures used we can write the quantity ~a as a function of one or more of the independent variables calculated at time n. Below we only quote two of the possibilities.
7.5.1 Second derivative-based methods In these it is generally assumed that the coefficient #a must be the same for each of the equations dealt with and only one of the dependent variables ~ is important. It has usually been assumed that the most typical variable here is the pressure and that we should write 5~
lZa = feh3lU] -t'-c I OZP [ p
Oxi Oxi e
(7.12)
where Ce is a non-dimensional coefficient, u is the velocity vector, c the speed of sound, D is the average pressure and the subscript e indicates an element. In the above equation, the second derivative of pressure over an element can be established either by averaging the smoothed nodal pressure gradients or by using any of the methods described in Chapter 4, Sec. 4.3. A particular variant of the above method evaluates approximately the value of the second derivative of any scalar variable ~b (e.g. p) as 52 hZO~b ~x2 ~ (M-
ML)~b
(7.13)
where M and ML are consistent and lumped mass matrices respectively and the tilde indicates a nodal value. Though the derivation of the above expression is not obvious, the reader can verify that in the one-dimensional finite difference approximation it gives the correct result. The heuristic extension to multidimensional problems therefore seems reasonable. Now #a for this approximate method can be rewritten in any space dimensions as [Eq. (7.12)]
~-La
"
-
-
Ceh
lul + c
P
(M - ML)~
(7.14)
Note now that ~]~a is a nodal quantity. However, a further approximation can give the following form of ~a over elements" #ae "--' Ceh
where as 52
Se is the element pressure
(lU] -t- c) Se
switch which is a mean of nodal switches
Sa
=
[~e(Pa -- Pb)] ~ e l P a -- Pb]
(7.15)
Sa calculated (7.16)
It can be verified that Sa = 1 when the pressure has a local extremum at node a and when the pressure at node b is the average of values for all nodes adjacent to
Sa -- 0
Variable smoothing 205 node a (e.g. if p varies linearly). The user-specified coefficient Ce normally varies between 0.0 and 2.0. The smoothed variables can now be rewritten with the Galerkin finite element approximations [from Eqs (7.11) and (7.15)] as
r
s
.._ r
+ AtM~ 1
Ceae
Ate
(M - ML)
~)n
(7.17)
h/Ate
Note that, in Eq. (7.15), (lul + c) is replaced by to obtain the above equation. This method has been widely used and is very efficient. The cut-off localizing the effect of added diffusion is quite sharp. A direct use of second derivatives can, however, be employed without the above-mentioned modifications. In such a procedure, we have the following form of smoothing [from Eqs (7.11) and (7.12)]
@:+l-'@"+l-AtM-ilCeh31Ul+Cl~I ( f ONTONclf2)dp n P
We"
OxiOxi
(7.18)
This method was successful in many viscous problems. Another alternative is to use residual-based methods.
7.5.2 Residual-based methods In these methods ~ a i --- #(Ri), where Ri is the residual of the ith equation. Such methods were first introduced in 1986 by Hughes and Malett 54 and later used by many others. 55-58 A variant of this was suggested by Codina. 53 We sometimes refer to this as anisotropic shock capturing. In this procedure the artificial viscosity coefficient is adjusted by subtracting the diffusion introduced by the characteristic-Galerkin method along the streamlines. We do not know whether there is any advantage gained in this but we have used the anisotropic shock capturing algorithm with considerable success. The full residual-based coefficient is given by #ai = Ce
IRil
[V(~)i-----~
(7.19)
We shall not discuss here a direct comparison between the results obtained by different shock capturing diffusivities, and the reader is referred to various papers already published.59, 60 Another smoothing procedure occasionally used in low Mach number flows is referred to as 'variable smoothing' and this method is discussed in the following section.
i"i"i"ii~i "ii":qi"iir~iir":'~iiii"i:qi~~~i}~iii~~~i~iiiiii :qri"~:~:r1i62i',i'~i~ii'!"iiiili r:162 ili~S',ii',i~iiiii'ri162 i@ii@ r~i162 i',~iiW ',i ~"iiiiiiiii' i',i',i'~,i',i'r,ii'i,i',i',i',iiiiiiii iiii!~iriiiiiii'iii,!'i,iii',i~ii,i!iiiiiii!i i:iiii'~ili!iiiiiiiiiiiiiiiii ii!i i iiiiiiiiiiiiiiiil i iiiiiiili'i,iiiiiiiiiii iiiiiiii i i i i iiiiiiPii,iiiiiiiiiiiiiiiii!iiiii '~iiiiiiiiiiii!iiiiiiii ii!iiiii!ii',iiiiiiii!ii ':iii!ii!iiiiiiiii ii'i~iii'',ii':l,i'!i,iiiiii ii'i,i'l'i,',i',ii!i At low Mach numbers, we found it often necessary to include a variable smoothing procedure if the coupling between the energy and other variables exists. Of course isothermal approximations are possible to get a smooth solution. However, to solve a compressible flow problem at low Mach numbers (<0.8) without removing energy
206
Compressiblehigh-speed gas flow
Fig. 7.3 Subsonicinviscid flow past an NACA0012 airfoil at a Mach number of 0.25 and zero angle of attack. Smoothed density contours. 61
coupling we recommend a variable smoothing approach. 61 In the proposed variable smoothing approach the conservation variable, ~, at a node is smoothed by applying the following redistribution
=
1 c~ ML 1(M - MD)O] 1 + 0.5ce ~ + 1 + 0.5ce
(7.20)
where c~ is a variable smoothing parameter which varies between 0 and 0.05, M is the consistent mass matrix, MD is the principal diagonal of the consistent mass matrix and ML is the lumped mass matrix. By increasing c~ the weighting on the node in question is decreased while the influence of the surrounding nodes is increased. In Fig. 7.3 we show the smoothed density contours of subsonic flow past an NACA0012 airfoil at a Mach number of 0.25. The results without smoothing gives a very oscillatory solution as shown in Fig. 7.4 in which the pressure coefficients from smoothed and unsmoothed solutions are compared. At transonic and supersonic speeds, an additional shock capturing dissipation (Sec. 7.5) is necessary to capture and to smooth local oscillations in the vicinity of shocks.
,~,~,~,~!~,,~,~,~~,,~,~,~,~, ,~,~,~~,~,~~~,~~,~,!~~!~~i~ ,~,~~~!~~ ~~~~~~~~~~~1~~~~!~,~,~i~~~,~~,'!~,~~,!,~,!,,~,~,~,~,~ ~,~~~!,~~,~~,,~,~,~,~,~,~,~,~,~,~,~,~,~,~,~,~, The computation procedures outlined above can be applied with success to many transient and steady-state problems. In this section we illustrate its performance on a few relatively simple examples.
Some preliminary examples for the Euler equation 0.6 0.4 0.2 0 -0.2 -0.4 ~"
I
I
I
I
/
-0.6 ,,, -0.8 -1 -1.2 -1.4 -1.6
I
0
I
I
I
I
...............
"i
Without smoothing With smoothing ...... _
9
I
I
I
I
I
I
I
I
I
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Horizontal coordinate
1
Fig. 7.4 Subsonic inviscid flow past an NACA0012 airfoil at Mach number of 0.25 and zero angle of attack. Comparison between smoothed and unsmoothed pressure coefficients. 61
Example 7.1 Riemann shock tube - a transient problem in one dimension
This is treated as a one-dimensional problem. Here an initial pressure difference between two sections of the tube is maintained by a diaphragm which is destroyed at t - 0. Figure 7.5 shows the density, velocity and energy contours at the seventieth time increment, and the effect of including consistent and lumped mass matrices is illustrated. The problem has an analytical, exact, solution presented by S o d 62 and the numerical solution is from reference 1.
Example 7.2 Isothermal flow through a nozzle in one dimension
Here a variant of the Euler equation is used in which isothermal conditions are assumed and in which the density is replaced by pa where a is the cross-sectional area ~ assumed to vary as 63 (x - 2.5) 2 a = 1.0 + for 0 < x < 5 (7.21) 12.5 -
The speed of sound is constant as the flow is isothermal and various conditions at inflow and outflow limits were imposed as shown in Fig. 7.6. In all problems steady state was reached after some 500 time steps. For the case with supersonic inflow and subsonic outflow, a shock forms and Lapidus-type artificial diffusion was used to deal with it, showing in Fig. 7.6(c) the increasing amount of 'smearing' as the coefficient CLap is increased.
Example 7.3 over a step
Two-dimensional transient supersonic flow
This example concerns the transient initiation of supersonic flow in a wind tunnel containing a step. The problem was first studied by Woodward and Colella 64 and the results of reference 4 presented here are essentially similar.
207
208
Compressible high-speed gas f l o w 1.000
t=O ..,_-. ~ t =
18.5
0.425
Lumped mass
/
./
2.5 2.0
/
/
t
/
I
/
/," . . . . . '~""-~" ' ~ L I.~ Consistent / I mass
i
!
!
I
i..
i
l,
Fig. 7.5 The Riemann shock tube problem. 1,62 The total length is divided into 100 elements. Profile illustrated corresponds to 70 time steps ( A t -- 0.25). Lapidus c o n s t a n t CLa p -'- 1.0.
In this problem a uniform mesh of linear triangles, shown in Fig. 7.7, was used and no difficulties of computation were encountered although a Lapidus constant CLap -- 2.0 had to be used due to the presence of shocks.
Example 7.4
Inviscid flow past an RAE2822 airfoil
Here one of the popular test cases of inviscid steady flow past an RAE2822 airfoil is studied at a subsonic free stream Mach number of 0.75 and an angle of attack of 3.0 ~ The total external circular domain of 25 chord length was found to be sufficient to carry out the calculation. It is often useful to include far field vortex correction methods into the boundary condition in order to reduce the error generated by insufficient distance to the external boundaries. 65 The mesh used was an unstructured mesh with more nodes close to the airfoil surface. The total number of elements and nodes used are 18 342 and 8451 respectively.
Some preliminary examples for the Euler equation 209 2.0
-
2.0
1.5
1.5
p 1.0
Ul.0
.5
0.5 .
0
.
.
1.0
.
.
.
2.0 x
.
.
0
.
3.0
4.0
1.0
2.0 X
3.0
4.0
(a) Subsonic inflow and outflow 2.0
2,0--
1.5 ~"~x..~~
J
1.5
p 1.0
ul.0 0.5
0.5--
o-
0
1.0
2.0 X
3.0
I~
0
4.0
0
1.0
2.0 x
3.0
4.0
(b) Supersonic inflow and outflow 2.0
2.0
1.5
1.5
p 1.0
u 1.0
~-
r
0.5 0
1.0
2.0 x
3.0
4.0
0
0
. . . . . . .
1.0
2.0 x
3.0
4.0
(c) Supersonic inflow-subsonic outflow with shock .... Exact . . . . . . ,.C~ap= 2.0 ........... CLap = 1.0
u2 C1
0.0
5 . ( ) - - C1
i........................................................... i Fig. 7.6 Isothermalflow through a nozzle.1 Fortyelements of equal size used. Figure 7.8 shows the pressure contours in the vicinity of the airfoil. As seen the upper surface has a shock between a distance of 70 and 80% from the upstream end. In Figure 7.9 the pressure coefficient distribution on both top and bottom surfaces is compared against the benchmark AGARD results. 65 As seen the results are quite close although the CBS solution uses an unstructured mesh which is coarser than the one used by the AGARD paper.
210 Compressiblehigh-speed gas flow
v
v
v
v
v
v
v
v
v
v
v
~
~
I n f l o w ~ ~
.
.
~
y ~
0.6
x
2016 elements
1089nodes
3.0
(a) Structured uniform mesh
t=0.5
t= 1.5
t=2.5
t-4.0 (b) Solution - contours of presure at various times
Fig. 7.7 Transientsupersonicflow over a step in a wind tunnel4 (problem of Woodward and Colella64). Inflow Mach 3 uniform flow.
Some preliminary examples for the Euler equation
i
l
i
i .
-...,
l
\
f
i //
i
:
~ ------ ..... ..
,!
'
:
/
i
/ / ...............~o,\,
/
"-
\
\i
\ t
tj
\iil
,,
/---u\il
i
! /
)
i
/
/
---kl',I i
,i:,,/ / ~i'
',,
j
1
!~
i
:
,,
i
\ \i//',/:-" /A~j .~ .i~ ,/ '/
-.
9
i/"
".\>
<
/ i /
\
............... '~)
..:.., -""
/
\ /:// \
'
i
, .....,,-:, 't: ......: ....,i.....,.\. \'~-.-"
t.=,.,
7: ~i , \
\~
II
----
/
. . . . -........ .....
:
''-.< '''ki
--. . . . . . . . . . . . . . . . . . . . .
Fig. 7.8 Transonic inviscid flow past RAE2822 aerofoil. Pressure contours. Mach number = 0.75, angle of attack = 3.0 ~
.
.
.
.
1.5 l
,,
,,,'
. ,,,
ix
.....X,,,,,X..,X..,'~,;x~
, t . . ...... . . . ...... . ...... . ..... .......
0.5 I
o
tF
x
...... x'~';~ x x ~ ~ ~ . ~ ~ ...~'xxX x
-0.5 ~ : I ~ " j ~ -1 ~ -1.5
0
~
K.x.~.Xx ~ "
~x"~"~....~.~ ~ ^x.% 9.~X~,;~,~<
.............CP for upper surface ....... CP for lower surface 0.2
0.4
x/c
0.6
0.8
1
Fig. 7.9 Transonic inviscid flow past RAE2822 aerofoil. Pressure coefficient. Mach number = 0.75, angle of attack = 3.0 ~
211
212 Compressiblehigh-speed gas flow
7.8.1 General The examples of the previous section have indicated the formation of shocks both in transient and steady-state problems of high-speed flow. Clearly the resolution of such discontinuities or near discontinuities requires a very fine mesh. Here the use of 'engineering judgement', which is often used in solid mechanics by designing a priori mesh refining near singularities posed by comers in the boundary, etc., can no longer be used. In problems of compressible flow the position of shocks, where the refinement is most needed, is not known in advance. For this and other reasons, the use of adaptive mesh refinement based on error indicators is essential for obtaining good accuracy and 'capturing' the location of shocks. It is therefore not surprising that the science of adaptive refinement has progressed rapidly in this area and indeed, as we shall see later, has been extended to deal with Navier-Stokes equations where a higher degree of refinement is also required in boundary layers. We have discussed the history of such adaptive development and procedures for its use in Sec. 4.3, Chapter 4.
7.8.2 The h-refinement process and mesh enrichment Once an approximate solution has been achieved on a given mesh, the local errors can be evaluated and new element sizes (and elongation directions if used) can be determined for each element. For some purposes it is again convenient to transfer such values to the nodes so that they can be interpolated continuously. The procedure here is of course identical to that of smoothing the derivatives discussed in Sec. 4.3, Chapter 4. To achieve the desired accuracy various procedures can be used. The most obvious is the process of mesh enrichment in which the existing mesh is locally subdivided into smaller elements still retaining the 'old' mesh in the configuration. Figure 7.10(a) shows how triangles can be readily subdivided in this way. With such enrichment an obvious connectivity difficulty appears. This concerns the manner in which the subdivided elements are connected to ones not so refined. A simple process is illustrated showing element halving in the manner of Fig. 7.10(b). Here of course it is fairly obvious that this process, first described in reference 9, can only be applied in a gradual manner to achieve the predicted subdivisions. However, element elongation is not possible with such mesh enrichment. Despite such drawbacks the procedure is very effective in localizing (or capturing) shocks, as we illustrate in Fig. 7.11. In Fig. 7.11, the theoretical solution is simply one of a line discontinuity shock in which a jump of all the components of ,I, occurs. The original analysis carded out on a fairly uniform mesh shows a very considerable 'blurring' of the shock. In Fig. 7.11 we also show the refinement being carried out at two stages and we see how the shock is progressively reduced in width. In the above example, the mesh enrichment preserved the original, nearly equilateral, element form with no elongation possible.
Adaptive refinement and shock capture in Euler problems 213
(a) Triangle subdivision
(b) Restoration of connectivity Fig. 7.10 Mesh enrichment. (a) Triangle subdivision. (b) Restoration of connectivity.
Whenever a sharp discontinuity is present, local refinement will proceed indefinitely as curvatures increase without limit. Precisely the same difficulty indeed arises in mesh refinement near singularities for elliptic problems 66 if local refinement is the only guide. In such problems, however, the limits are generally set by the overall energy norm error consideration and the refinement ceases automatically. In the present case, the limit of refinement needs to be set and we generally achieve this limit by specifying the smallest element size in the mesh. The h refinement of the type proposed can of course be applied in a similar manner to quadrilaterals. Here clever use of data storage allows the necessary refinement to be achieved in a few steps by ensuring proper transitions. 67
7.8.3 h-refinement and remeshing in steady-state two-dimensional problems Many difficulties mentioned above can be resolved by automatic generation of meshes of a specified density. Such automatic generation has been the subject of much research in many applications of finite element analysis. We have discussed this subject in Sec. 4.3, Chapter 4. The closest achievement of a prescribed element size and directionality can be obtained for triangles and tetrahedra. Here the procedures developed by Peraire et al. 1~' ~2 are most direct and efficient, allowing element stretching in prescribed directions (though of course the amount of such stretching is sometimes restricted by practical considerations). We refer the reader for details of such mesh generation to the original publications. In the examples that follow we shall exclusively use this type of mesh adaptivity.
214
Compressiblehigh-speed gas flow
Initial configuration
........~=~i~!~!~t~ii~:~',:
Density a
Density after 200 steps
~:~. "
After 201 steps
~,i
Exact solution
.':
Density after 250 steps
Fig. 7.11 Supersonic, Mach 3, flow past a wedge. Exact solution forms a stationary shock. Successive mesh enrichment and density contours.
Example 7.5 Inviscid flow with shock reflection from a solid wall
In Fig. 7.12 we show a simple example 11 of shock wave reflection from a solid wall. Here only a typical 'cut-out' is analysed with appropriate inlet and outlet conditions imposed. The elongation of the mesh along the discontinuity is clearly shown. The solution was remeshed after the iterations nearly reached a steady state.
Example 7.6
Hypersonic inviscid flow past a blunt body In Fig. 7.13 a somewhat more complex example of hypersonicflow around a blunt, two-dimensional obstacle is shown. Here it is of interest to note that: 1. A detached shock forms in front of the body. 2. A very coarse mesh suffices in front of such a shock where simple free stream flow continues and the mesh is made 'finite' by a maximum element size prescription. 3. For the same minimum element size a reduction of degrees of freedom is achieved by refinement which shows much improved accuracy.
E
O "O
e-
<\
/ /
/ J
J J
p~ / /
v
~
.!..o c--
E
m~
o t--
b4 "o
..Q v
r--
E
p~ (",4
o
c.-
E (1;
o
g~ m
g8
~
~
o~
O
c o
o r.--
._o E
o
216
Compressiblehigh-speed gas flow
t Analysis domain (a) Sequence of meshes employed
t
velocity
(b) The corresponding density
t
(c) The corresponding pressure contours Fig. 7.13 Hypersonicflow past a blunt body11 at Mach 25, 22~ angle of attack. Initial mesh, nodes: 547, elements: 978; first mesh, nodes: 383, elements: 696; final mesh, nodes: 821, elements: 1574.
For such hypersonic problems, it is often claimed that special methodologies of solution need to be used. References 68-70 present quite sophisticated methods for dealing with such high-speed flows.
Three-dimensional inviscid examples in steady state
Example 7.7 Supersonic inviscid flow past a full circular cylinder
In Figs 7.14 and 7.15, we show the results of supersonic Mach 3 flow past a full cylinder. 59 The mesh [Fig. 7.14(b)] is adapted along the shock front to get a good resolution of the shock. The mesh behind the cylinder is very fine to capture the complex motion. In Figs 7.14(c) and 7.14(d), the Mach contours are obtained using the CBS algorithm with second derivative-based shock capture and residual-based shock capture respectively. In Fig. 7.15, the coefficient of pressure values and Mach number distribution along the mid-height through the surface of the cylinder are presented. Here the results generated by the MUSCL 68 scheme are also plotted for the sake of comparison. As seen the comparison is excellent, especially for the anisotropic (residual-based) scheme.
Example 7.8 Inviscid shock interaction
Figure 7.16 shows a yet more sophisticated e• in which an impinging shock interacts with a bow shock. An e• fine mesh d~stnbufion was used here to compare results with those obtained experimentally, 17 which were reproduced with high precision.
i"ii!i!iiii~~i~ :!!iii"::~':~~::':~'::~:'~::ii'~i"~' i!~'i::'~::' i' i'~'~~'~:'i~h:'~'~I''~ii,iiiii~iiiiii~i',!i!ili!~~,',il~,','~iii~,i''~,iii~i~i~',i!'i,~li:~!',i'!:i'~:i':,i~!~iiiiiiiii' '~',',i~ iiii!iiii'~,~','~i,iiii'i,i',i',':',i',i',i'~i,!'':'ii'~,'~',iiii'ii'~',iii'iii~,i',i'~',~'i','~,',ii',i',:i~iiiiii~, ,i~,i',i'~'~iii!:il,iii!',":':il"~i !'~i,'~',i~i '~iii~,iiiiii',ii',ii!iii''~,i'i~~,iii'i,i~!~'i,i ,i'~iii',i'iili,,iiii'',~i~':ii!iii!i~',il'~i~iiiii'?':i'',iiii!i'~~~::~:i:::iiii~ ,',',ii',ili'i'~~',',:i'i'i','i',ii,i:,'~i~,ii',i,l~i'~',i',ili'ii:,i!',!iiii~,'i,!:,',iilii~il',ii!il',iii~ i ',i':':"'~',iii:,i~i~,',iiii',iil',ii!',~,i!i~i',!ii'~,~','i~ ~, iii~'i~i','~i,iiiiii~,iii',!',~~~d~i~ i!',,i',ii!!iii ',iii~~~','i,'i,i':,~'~i, ,iii~i',i'ii~iiiii!i!i~',!'ii,!i~',~',i~',iii'',',:i""i'',i',i,',iii~, ','i,ii~i~ii:iiii',~~ii'~:,~ii:'ii~,iiiiiiiiiil',i~,i!il',~:::i~' :iii!'~if'~,i,~',il,iiii~i'iiiii' ,ii ~i!iii!i!i!ili!iiiiiiiiiiiiiii!
Two-dimensional problems in fluid mechanics are much rarer than two-dimensional problems in solid mechanics and invariably they represent a very crude approximation to reality. Even the problem of an aerofoil cross-section, which we have discussed in Chapter 3, hardly exists as a two-dimensional problem as it applies only to infinitely long wings. For this reason attention has largely been focused on, and much creative research done in, developing three-dimensional codes for solving realistic problems. In this section we shall consider some examples derived by the use of such threedimensional codes and in all these the basic element used will be the tetrahedron which now replaces the triangle of two dimensions. Although the solution procedure and indeed the whole formulation in three dimensions is almost identical to that described for two dimensions, it is clear that the number of unknowns will increase very rapidly when realistic problems are dealt with. It is common when using linear order elements to encounter several million variables as unknowns and for this reason here, more than anywhere else, iterative solution processes are necessary. Indeed much effort has gone into the development of special procedures of solution which will accelerate the iterative convergence and which will reduce the total computational time. In this context we should mention three approaches which are of help.
The recasting of element formulation in an edge form
Here a considerable reduction of storage can be achieved by this procedure and some economies in computational time achieved. We have not discussed this matter in detail but refer the reader to reference 26 where the method is fully described and for completeness we summarize the essential features of edge formulation in Appendix F.
217
218
Compressible high-speed gas flow _4
5
M=3 r=l
1
U1 = 1
u2=O
10
(a)
(b)
(c)
(d)
Fig. 7.14 Supersonicflow past a full cylinder.60 M = 3, (a) geometry and boundary conditions, (b) adapted mesh, nodes: 12 651, elements: 24 979, (c) Mach contours using second derivative shock capture and (d) Mach contours using anisotropic shock capture.
Three-dimensional inviscid examples in steady state 1.80 1.60 1.40 1.20 1.00 ~
0.80 -
0.60 0.40 0.20 0-
(a)
-0.20 -3.00
I
-2.00
I
-1.00
I
I
I
0 X
1.00
2.00
3.00
' MUSCL ................ CBS: Anisotropic Shock Capture CBS: Second Derivative Shock Capture
/I
4.00
ij
3.00 -
..Q
E '- 2.00 cO
f
1.00 -
(b)
0 -3.00
I
-2.00
I
-1.00
0 X
1.00
2.00
3.00
Fig. 7.15 Supersonic flow past a full cylinder.6~ M = 3, comparison of (a) coefficient of pressure, (b) Mach number distribution along the mid-height and cylinder surface.
219
220
Compressible high-speed gas flow Analysis domain
Fig. 7.16 Interaction of an impinging and bow shock wave.17 Adapted mesh and pressure contours.
Multigrid approaches
In the standard iteration process we proceed in a time frame by calculating point by point the changes in various quantities and we do this on the finest mesh. As we have seen this may become very fine if adaptivity is used locally. In a multigrid solution, as initially introduced into the finite element field, the solution starts on a coarse mesh, the results of which are used subsequently for generating the first approximation to the fine mesh. Several iterative steps are then carried out on the fine mesh. In general a return to the coarse mesh is then made to calculate the changes of residuals there and the process is repeated on several meshes done subsequently. This procedure can be used on several meshes and the iterative process is much accelerated. We discuss this process in Appendix G in a little more detail. However, we quote here several 71 76 references - in which such multigrid procedures have been used and these are of considerable value. Multigrid methods are obviously designed for meshes which are 'nested', i.e. in which coarser and finer mesh nodes coincide. This need not be the case generally. In
Three-dimensional inviscid examples in steady state many applications completely different meshes of varying density are used at various stages.
Parallel computation
The third procedure of reducing the solution time is to use parallelization. We do not discuss it here in detail as the matter is potentially coupled with the computational aspects of the problem. Here the reader should consult the current literature on the subject. 28-38 In what follows we shall illustrate three-dimensional applications on a few inviscid examples as this section deals with Euler problems. However, in Sec. 7.12 we shall return to a fully three-dimensional formulation using viscous Navier-Stokes equations.
7.9.1 Solution of the flow pattern around a complete aircraft In the early days of numerical analysis applied to computational fluid dynamics which used finite differences, no complete aircraft was analysed as in general only structured meshes were admissible. The analysis thus had to be carried out on isolated components of the aircraft as shown in Fig. 7.17. Later construction of distorted and partly structured meshes increased the possibility of analysis. Nevertheless the first complete aircraft analyses were done only in the mid-1980s. In all of these, finite elements using unstructured meshes were used (though we include here the finite volume formulation which was almost identical to finite elements and was used by Jameson et a/.77). The very first aircraft was the one dealt with using potential theory in the Dassault establishment. The results were published later by Periaux and coworkers. 78 Very shortly after that a complete supersonic aircraft was analysed by Peraire et al. 16 in Swansea in 1987.
Example 7.9 Inviscidflow past full aircraft
Figure 7.18 shows the aircraft analysed in Swansea 16 which is a supersonic fighter of genetic type at Mach 2. The analysis was made slightly adaptive though adaptivity was not carried very far (due to cost). Nevertheless the refinement localized the shocks which formed. In the analysis some 125 000 elements were used with approximately 70 000 nodes and therefore some 350 000 variables. This of course is not a precise analysis and many more variables would be used currently to get a more accurate representation of flow and pressure variables. A more sophisticated analysis is shown in the plate at the front of the book. Here a civil aircraft in subsonic flow is modelled and this illustrates the use of multigrid methods. In this particular multigrid application three meshes of different refinement were used and the iteration is fully described in reference 26. In this example the total number of unknown quantities was 1 616 000 in the finest mesh and indeed the details of the subdivision are given in the legend of the plate.
Example7.10 Inviscidengineintake
There are many other three-dimensional examples which could at this stage be quoted but we only show here a three-dimensional analysis of an engine intake 16 at Mach 2. This is given in Fig. 7.19.
221
222 Compressiblehigh-speed gas flow
Fig. 7.17 Inviscid flow past an ONERA M6 wing. Density contours. Mach number = 0.78, angle of attack to horizontal = 2.8 ~
7.9.2 THRUST- the supersonic car 27, 28, 79 A very similar problem to that posed by the analysis of the whole aircraft was given much more recently by the team led by Professor Morgan. This was the analysis of a car which was attempting to create the world speed record by establishing this in the supersonic range. This attempt was indeed successfully made on 15 October 1997. Unlike in the problem of aircraft, the altemative of wind tunnel tests was not available. Whilst in aircraft design, wind tunnels which are supersonic and subsonic are well used in practice (though at a cost which is considerably more than that of a numerical analysis) the possibility of doing such a test on a motor car was virtually non-existent. The reason for this is the fact that the speed of the air flow past the body of the car and the speed of the ground relative to the car are identical. Any test would therefore require the bed of the wind tunnel to move at a speed in excess of 750+ miles an hour. For this reason calculations were therefore preferable. The moving ground will of course create a very important boundary layer such as that which we will discuss in later sections. However, the simple omission of viscosity permitted the inviscid solution by a standard Euler-type program to be used.
Three-dimensional inviscid examples in steady state
Fig. 7.18 Adaptive three-dimensional solution of compressible inviscid flow around a high speed (Mach 2) aircraft. 16 Nodes: 70 000, elements: 125 000.
It is well known that the Euler solution is perfectly capable of simulating all shocks very adequately and indeed results in very well-defined pressure distributions over the bodies whether it is over an aircraft or a car. The object of the analysis was indeed that of determining such pressure distributions and the lift caused by these pressures. It was essential that the car should remain on the ground, indeed this is one of the conditions of the ground speed record and any design which would result in substantial lift overcoming the gravity on the car would be disastrous for obvious reasons.
223
224
Compressible high-speed gas flow
(a) Mesh on analysis surface
(b) Mesh on analysis surface
(c) Pressure contours
Fig. 7.19 Three-dimensional analysis of an engine intake16 at Mach 2 (14 000 elements).
The complete design of the supersonic car was thus made with several alternative geometries until the computer results were satisfactory. Here it is interesting, however, to have some experimental data and the preliminary configuration was tested by a rocket driven sled. This was available for testing rocket projectiles at Pendine Sands, South Wales, UK. Here a 1:25 scale model of the car was attached to such a rocket and 13 supersonic and transonic runs were undertaken. In Fig. 7.20(a), we show a photograph 27 of the car concerned after winning the speed record in the Nevada desert. In Fig. 7.20(b) a surface mesh is presented from which the full three-dimensional mesh at the surrounding atmosphere was generated (surface mesh, nodes: 39528, elements: 79060; volume mesh, nodes: 134 272, elements:
Three-dimensional inviscid examples in steady state
(a)
(b)
Fig. 7.20 Supersonic car, THRUST SSC.27 (a) Car and (b) finite element surface mesh. (Image used in (a) courtesy of SSC Programme Ltd. Photographer Jeremy C.R. Davey.) 887 634). We do not show the complete mesh as of course in a three-dimensional problem this is counterproductive. In Fig. 7.21, pressure contours 27 on the surface of the car body are given and somewhat similar contours are shown on the covers of this book. In Fig. 7.22, a detailed comparison of CFD results 27 with experiments is shown. The results of this analysis show a remarkable correlation with experiments. The data points which do not appear close to the straight line are the result of the sampling point being close to, but the wrong side of, a shock wave. If conventional correlation techniques for inviscid flow (viscous correction) are applied, these data points also lie on the straight
225
226
Compressible high-speed gas flow
(a)
(b) Fig. 7.21
-"
Supersonic car, THRUST SSC27 pressure contours. (a) Full configuration, (b) front portion.
line. In total, nine pressure points were used situated on the upper and lower surfaces of the car. The plot shows the comparison of pressures at specific positions on the car for Mach numbers of 0.71, 0.96, 1.05 and 1.08.
In all of the previous problems the time stepping was used simply as an iterative device for reaching the steady-state solution. However, this can be used in real time and the transient situation can be studied effectively. Many such transient problems have been dealt with from time to time and here we illustrate the process on three examples.
Example 7.11 Exploding pressure vessel
The first one concerns an exploding pressure vessel 8~ as the two-dimensional model as shown in Fig. 7.23. Here of course adaptivity had to be used and the mesh is regenerated every few steps to reproduce the transient motion of the shock front.
Viscous problems in two dimensions 227 Key x M = 0.71 0 M= 1.08 + M=0.96
CFD results
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6 -
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Fig. 7.22 Supersoniccar, THRUSTSSC27 comparison of finite element and experimental results.
Example 7.12
Shuttle launch
A similar computation is shown in Fig. 7.24 where a diagrammatic form of a shuttle launch is modelled again as a two-dimensional problem. 8~ Of course this twodimensional model is purely imaginary but it is useful for showing the general configuration. In Fig. 7.25, however, we show a three-dimensional shuttle approximating closely to reality. 29 The picture shows the initial configuration and the separation from the rocket.
iiiii!i!iiii~ii!iii~!!~ilili!iii!!!iiii~~iUiSiiiiiilP~~!i~~!i!i!ililil ii! ~!iil!!~~ !iiiii!d~~i~ |i~i~ Siili!!ii!ii!i!iiiiiiilii!!iii!ii!!ii!i!iiiilii!!iiilil!i!i!ili ii!iiiiliiliiiiiiiii!iiiii iiiiiii!iiiiiiiii!!!iiiiii!ii!iii Clearly the same procedures which we have discussed previously could be used for the full Navier-Stokes equations by the introduction of viscous and other heat diffusion terms. Although this is possible we will note immediately that very rapid gradients of velocity will develop in the boundary layers (we have remarked on this already in Chapter 4) and thus special refinement will be needed there. In the first example we illustrate a viscous solution by using meshes designed a priori with fine subdivision near the boundary. However, in general the refinement must be done adaptively and here various methodologies of doing so exist. The simplest of course is the direct use of mesh refinement with elongated elements which we have also discussed in Chapter 4. This will be dealt with by a few examples in Sec. 7.11.1. However in Sec. 7.11.2 we shall address the question of much finer refinement with very elongated elements in the boundary layer. Generally we shall do such a refinement with a structured grid near the solid surfaces merging into the general unstructured meshing outside. In that section we shall introduce methods which can automatically separate structured and unstructured regions both in the boundary layer and in the shock regions. The methodology is of course particularly important in problems of three dimensions.
228
Compressible high-speed gas flow Confining walls
r~ f f J f L," r~
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i
Fig. 7.23 A transient problem with adaptive remeshing.8O Simulationof a suddenfailure of a pressurevessel. Progression of refinement and velocity patterns shown. Initial mesh 518 nodes.
The special refinement which we mentioned above is well illustrated in Fig. 7.26. In this we show the possibility of using a structured mesh with quadrilaterals in the boundary layer domain (for two-dimensional problems) and a three-dimensional equivalent of such a structured mesh using prismatic elements. Indeed such elements have been used as a general tool by some investigators. 8~-83
Viscous problems in two dimensions 229
NE = 7870 NP = 4130
NE = 7377 NP = 3867
i
NE = 6847 NP = 3580
NE = 8 4 5 9 NP = 4379
Fig. 7.24 A transient problem with adaptive remeshing. 8~ Model of the separation of shuttle and rocket. Mach 2, angle of attack - 4 ~ initial mesh 4130 nodes.
Example 7.13
Viscousflow past a plate
The example given here is that in which both shock and boundary layer development occur simultaneously in high-speed flow over a flat plate. 84 This problem was studied extensively by Carter. 85 His finite difference solution is often used for comparison purposes although some oscillations can be seen even there despite a very high refinement. A fixed mesh which is graded from a rather fine subdivision near the boundary to a coarser one elsewhere is shown in Fig. 7.27. We obtained the solution using as usual
230 Compressible high-speed gas flow
(a)
(b)
Fig. 7.25 Separation of a generic shuttle vehicle and rocket booster.29 (a)Initial surface mesh and surface pressure; (b) final surface mesh and surface pressure.
the CBS algorithm. In Fig. 7.28, comparisons with Carter's 85 solution are presented and it will be noted that the CBS solution appears to be more consistent, avoiding oscillations near the leading edge.
7.11.1 Adaptive refinement in both shock and boundary layer In this section we shall pursue mesh generation and adaptivity in precisely the same manner as we have done in Chapter 4 and previously in this chapter, i.e. using elongated finite elements in the zones where rapid variation of curvature occurs. An example of this application is given in Fig. 7.29. Here now a problem of the interaction of a boundary layer generated by a fiat plate and externally impinging shock is presented. 86 In this problem, some structured layers are used near the wall in addition to the direct approach of Chapter 4. The reader will note the progressive refinement in the critical area. In such a problem it would be simpler to refine near the boundary or indeed at the shock using structured meshes and the idea of introducing such refinement is explored in the next section.
7.11.2 Special adaptive refinement for boundary layers and shocks As with the direct iterative approach, it is difficult to arrive at large elongations during mesh generation, and the procedures just described tend to be inaccurate. For this reason
Viscous problems in two dimensions 231
Unstructured, adaptively refined triangles 10-15 'body' layers of quadrilateral elements with length/, corresponding adaptive layer thickness d and number of layers decided by user I
(a) A two-dimensional sublayer of structured quadrilaterals
I
d
'Body' layer subdivision in three dimensions joining a tetrahedral mesh (b) A three-dimensional sublayer of prismatic elements Fig. 7.26 Refinement in the boundary layer.
it is useful to introduce a structured layer within the vicinity of solid boundaries to model the boundary layers and indeed it is possible to do the same in the shocks once these are defined. Within the boundary layer this can be done readily as shown in Fig. 7.26 using a layer of structured triangles or indeed quadrilaterals. On many occasions triangles have been used here to avoid the use of two kinds of elements in the same code. However, if possible it is better to use directly quadrilaterals. The same problem can of course be done three dimensionally and we shall in Sec. 7.12 discuss application of such layers. Again in the structured layer we can use either prismatic elements or simply tetrahedra though if the latter are used many more elements are necessary for the same accuracy. It is clear that unless the structured meshes near the boundary are specified a priori, an adaptive procedure will be somewhat complicated and on several occasions fixed boundary meshes have been used. However alternatives exist and here two possibilities should be mentioned. The first possibility, and that which has not yet been fully exploited, is that of refinement in which structured meshes are used in both shocks and boundary layers and the width of the domains is determined after some iterations. The procedure is somewhat involved and has been used with success in many trial problems as shown by Zienkiewicz and Wu. 87 We shall not describe the method in detail here but essentially structured meshes again composed of triangles or at least quadrilaterals divided into two triangles were used near the boundary and in the shock regions. The subdivision and accuracy obtained were excellent. In the second method we could
232 Compressiblehigh-speed gas flow
(a)
(b)
(c) Fig. 7.27 Viscous flow past a flat plate (Carter problem). 84 Mach 3, Re = 1000. (a) Mesh, nodes: 6750, elements: 13, 172 contours of (b) pressure and (c) Mach number.
Viscous problems
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Fig. 7.28 Viscous flow past a flat plate (Carter problem).84 Mach 3, Re = 1000. (a) Pressure distribution along the plate surface, (b) exit velocity profile.
imagine that normals are created on the boundaries, and a boundary layer thickness is predicted using some form of boundary layer analytical computation. 3~ 88 Within this layer structured meshes are adopted using a geometrical progression of thickness. The structured boundary layer meshing can of course be terminated where its need is less apparent and unstructured meshes continued outside. Figure 7.30 illustrates supersonic flow around an NACA0012 aerofoil using the automatic generation of structured and unstructured domains taken from reference 87. The second method, in which normals are grown from the solid surface to create a structured layer, is illustrated in Fig. 7.31 on a two-component aerofoil.
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Fig. 7.29 Continued.
Example 7.14 Transonic viscous flow past an NA CAO012 aerofoil
The external flow past an NACA0012 aerofoil is one of the popular benchmark problems of compressible fluid dynamics. 89' 90, 91 The transonic viscous flow is especially difficult to handle with many numerical schemes. The explicit schemes are generally difficult to use without additional acceleration procedures such as multigrid methods. Here, however, we provide a solution using the explicit scheme without any additional acceleration technique. The problem consists of an NACA0012 aerofoil placed at the centre of a circular domain of diameter 20 times the chord length. The inlet Mach number was assumed to be 0.85 and the Reynolds number was 2000. On the solid wall the no-slip conditions were assumed. The angle of attack in this problem was assumed to be zero. All the inlet conditions were assumed to be known. In Fig. 7.32 we show the mesh used for the calculations. As shown the mesh close to the solid wall is generated by constructing structured layers. Away from the wall the
235
236
Compressible high-speed gas flow
f.,i"
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Initial mesh, 1804 points, 3487 elements
(a)
Adaptive mesh, 916 points, 1830 elements
/
Mach number
Mach number
(b)
/
/
\,
1162 points, 2258 elements
(c)
Mach number
Fig. 7.30 Hybrid mesh for supersonic viscous flow past an NACA0012 aerofoil, 87 Mach 2, and contours of Mach number, (a)initial mesh, (b) first adapted mesh, (c) final mesh, (d) mesh near stagnation point (shown opposite).
i-
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8
238 Compressible high-speed gas flow
9
,,.
,
x
lr
Fig. 7.31 Structuredgrid in boundarylayer for a two-component aerofoil.30Advancing boundarynormais.
(a) Finite element mesh
(b) Structured layers close to the wall
Fig. 7.32 Transonicviscousflow past an NACA0012aerofoil. Mach number0.85, Reynoldsnumber = 2000. Finite element mesh.
Viscous problems in two dimensions mesh is purely unstructured. In this way we will be able to capture the strong boundary layer effects close to the walls. A total of 16 496 elements and 8425 nodes were employed in the calculation. Figure 7.33 shows the Mach contours. The minor oscillations noted are due to the coarse unstructured mesh used away from the walls. In Fig. 7.34 we show the surface quantity distribution. The quantity distribution in general is in excellent agreement with the fully structured mesh solutions. 89' 90, 91
Fig. 7.33 Transonicviscous flow past an NACA0012 aerofoil. Mach number 0.85, Reynolds number = 2000. Mach contours. ,5
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Fig. 7.34 Transonicviscous flow past an NACA0012 aerofoil. Mach number 0.85, Reynolds number = 2000. Surface pressure and skin friction coefficients distribution.
239
240
Compressible
high-speed
gas flow
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(a) Adapted mesh
(b) Structured layers close to the wall
! (c) Close-up of structured layers Fig. 7.35 Hypersonic viscous flow past a double ellipsoid. Unstructured mesh with structured mesh layers close to the walls.
Boundary layer-inviscid Euler solution coupling 241
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It is well k n o w n that h i g h - s p e e d flows w h i c h exist without substantial flow separation develop a fairly thin b o u n d a r y layer to w h i c h all the viscous effects are confined. T h e flow outside this b o u n d a r y layer is purely inviscid. Such p r o b l e m s have for s o m e years been solved a p p r o x i m a t e l y by using pure Euler solutions f r o m w h i c h the pressure distribution is obtained. C o u p l i n g these solutions with a b o u n d a r y layer a p p r o x i m a t i o n written for a very small thickness near the solid body provides the c o m p l e t e solution. T h e theory by w h i c h the separation b e t w e e n inviscid and viscous d o m a i n s is predicted is that b a s e d on the w o r k of Prandtl and for w h i c h m u c h d e v e l o p m e n t has taken place since his original work. Clearly various m e t h o d s of solving b o u n d a r y layer p r o b l e m s can be used and m a n y different techniques of inviscid solution can be i m p l e m e n t e d . In the b o u n d a r y layer full N a v i e r - S t o k e s equations are used and generally these equations are specialized by introducing the a s s u m p t i o n s of a b o u n d a r y layer in w h i c h no pressure variation across the thickness occurs. A n altemative to solving the equations in the w h o l e b o u n d a r y layer is the integral a p p r o a c h in w h i c h the b o u n d a r y layer equations need to be solved only close to the solid surface. Here the 'transpiration velocity m o d e l ' for l a m i n a r flOWS93 and the 'lage n t r a i n m e n t ' m e t h o d 94 for turbulent flows are notable approaches. Further extensions of these p r o c e d u r e s can be found in m a n y available research articles. 95-99 M a n y studies illustrate further d e v e l o p m e n t s and i m p l e m e n t a t i o n p r o c e d u r e s of v i s c o u s - i n v i s c i d coupling. ~~176176 A l t h o u g h the use of such v i s c o u s - i n v i s c i d coupling is not directly applicable in p r o b l e m s w h e r e b o u n d a r y layer separation occurs, m a n y studies are available to deal with separated flows. 1~176 We do not give any further details of v i s c o u s - i n v i s c i d coupling here and the reader can refer to the quoted references and A p p e n d i x H.
242 Compressiblehigh-speed gas flow
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
::::::::::::::::::::::::::ii ::::~::~::::~i:::: ::::::::::::::::::::::::::::::
This chapter describes important and far-reaching applications of the finite element method for use in the design of aircraft and other high-speed vehicles. The solution techniques described and examples presented illustrate some of the possibilities to obtain realistic three-dimensional results, however, there are still many unsolved problems. Most of these relate either to techniques for solving the equations or to satisfactorily modelling viscous and transient effects. The paths taken for simplifying the formulation to obtain more efficient calculations include possibilities such as multigrid methods, edge formulation, and parallel computing. However, full modelling of transients and boundary layer effects is much more difficult, especially for high speed flows. Use of boundary layer theory as an interface to inviscid flow is of course only an approximation and much 'engineering art' is needed to achieve acceptable results. This knowledge is acquired from available experimental data and becomes necessary whether approximations of any type are used or the full equations are solved directly. In either case, a finite element approach provides the user with the freedom of choice to decide which modelling method suits each particular situation. As indicated above, the only way to eliminate simplifying assumptions is via Direct Numerical Simulation (DNS) using the Navier-Stokes equations. However, performing a full three-dimensional, transient analysis by this approach generally will involve billions of elements in a realistic simulation at high Reynolds number and this approach is currently not feasible. Thus, in the next chapter we discuss the topic of turbulent flow and the use of time averaged models to describe the turbulent behaviour. Improved turbulence modelling approaches have been the topic of research during the last thirty years and provide a viable means to compute reliable time averaged quantifies.
1. R. LShner, K. Morgan and O.C. Zienkiewicz. The solution of non-linear hyperbolic equation systems by the finite element method. Int. J. Num. Meth. Fluids, 4:1043-1063, 1984. 2. R. LShner, K. Morgan and O.C. Zienkiewicz. Domain splitting for an explicit hyperbolic solver. Comp. Meth. Appl. Mech. Eng., 45:313-329, 1984. 3. O.C. Zienkiewicz, R. LShner, K. Morgan and J. Peraire. High speed compressible flow and other advection dominated problems of fluid mechanics. In R.H. Gallagher, G.F. Carey, J.T. Oden and O.C. Zienkiewicz, editors, Finite Elements in Fluids, volume 6, Chapter 2, pp. 41-88. Wiley, Chichester, 1985. 4. R. LShner, K. Morgan and O.C. Zienkiewicz. An adaptive finite element procedure for compressible high speed flows. Comp. Meth. Appl. Mech. Eng., 51:441--465, 1985. 5. R. L/3hner, K. Morgan, J. Peraire, O.C. Zienkiewicz and L. Kong. Finite element methods for compressible flow. In K.W. Morton and M.J. Baines, editors, ICFD Conf. on Numerical Methods in Fluid Dynamics, volume II, pp. 27-53. Clarendon Press, Oxford, 1986. 6. R. LShner, K. Morgan, J. Peraire and M. Vahdati. Finite element, flux corrected transport (FEM-FCT) for the Euler and Navier-Stokes equations. Int. J. Num. Meth. Eng., 7:1093-1109, 1987. 7. R. LShner, K. Morgan and O.C. Zienkiewicz. Adaptive grid refinement for the Euler and compressible Navier-Stokes equation. In Proc. Int. Conf. on Accuracy Estimates and Adaptive Refinement in Finite Element Computations, Lisbon, 1984.
References 243 8. R. Lrhner, K. Morgan, J. Peraire and O.C. Zienkiewicz. Finite element methods for high speed flows. AIAA paper 85-1531-CP, 1985. 9. O.C. Zienkiewicz, K. Morgan, J. Peraire, M. Vahdati and R. Lrhner. Finite elements for compressible gas flow and similar systems. In 7th Int. Conf. in Computational Methods in Applied Sciences and Engineering, Versailles, December 1985. 10. R. Lrhner, K. Morgan and O.C. Zienkiewicz. Adaptive grid refinement for the Euler and compressible Navier-Stokes equations. In I. Babuska, O.C. Zienkiewicz, J. Gago and E.R. de A. Oliveira, editors, Accuracy Estimates and Adaptive Refinements in Finite Element Computations, Chapter 15, pp. 281-298. Wiley, Chichester, 1986. 11. J. Peraire, M. Vahdati, K. Morgan and O.C. Zienkiewicz. Adaptive remeshing for compressible flow computations. J. Comp. Phys., 72:449-466, 1987. 12. J. Peraire, K. Morgan, J. Peiro and O.C. Zienkiewicz. An adaptive finite element method for high speed flows. In AIAA 25th Aerospace Sciences Meeting, Reno, Nevada, AIAA paper 87-0558, 1987. 13. O.C. Zienkiewicz, J.Z. Zhu, Y.C. Liu, K. Morgan and J. Peraire. Error estimates and adaptivity: from elasticity to high speed compressible flow. In J.R. Whiteman, editor, The Mathematics of Finite Elements and Application (MAFELAP 87), pp. 483-512. Academic Press, London, 1988. 14. L. Formaggia, J. Peraire and K. Morgan. Simulation of state separation using the finite element method.Appl. Math. Modelling, 12:175-181, 1988. 15. O.C. Zienkiewicz, K. Morgan, J. Peraire, J. Peiro and L. Formaggia. Finite elements in fluid mechanics. Compressible flow, shallow water equations and transport. In ASME Conf. on Recent Developments in Fluid Dynamics, AMD 95, American Society of Mechanical Engineers, December 1988. 16. J. Peraire, J. Peiro, L. Formaggia, K. Morgan and O.C. Zienkiewicz. Finite element Euler computations in 3-dimensions. Int. J. Num. Meth. Eng., 26:2135-2159, 1989. (See also same title: AIAA 26th Aerospace Sciences Meeting, Reno, AIAA paper 87-0032, 1988.) 17. J.R. Stewart, R.R. Thareja, A.R. Wieting and K. Morgan. Application of finite elements and remeshing techniques to shock interference on a cylindrical leading edge. Reno, Nevada, AIAA paper 88-0368, 1988. 18. R.R. Thareja, J.R. Stewart, O. Hassan, K. Morgan and J. Peraire. A point implicit unstructured grid solver for the Euler and Navier-Stokes equation. Int. J. Num. Meth. Fluids, 9:405-425, 1989. 19. R. Lrhner. Adaptive remeshing for transient problems with moving bodies. In National Fluid Dynamics Congress, Ohio, AIAA paper 88-3736, 1988. 20. R. Lrhner. The efficient simulation of strongly unsteady flows by the finite element method. In 25th Aerospace Sci. Meeting, Reno, Nevada, AIAA paper 87-0555, 1987. 21. R. Lrhner. Adaptive remeshing for transient problems. Comp. Meth. Appl. Mech. Eng., 75:195214, 1989. 22. J. Peraire. A finite element method for convection dominated flows. Ph.D. thesis, University of Wales, Swansea, 1986. 23. O. Hassan, K. Morgan and J. Peraire. An implicit-explicit scheme for compressible viscous high speed flows. Comp. Meth. Appl. Mech. Eng., 76:245-258, 1989. 24. N.P. Weatherill, E.A. Turner-Smith, J. Jones, K. Morgan and O. Hassan. An integrated software environment for multi-disciplinary computational engineering. Eng. Comp., 16:913-933, 1999. 25. P.M.R. Lyra and K. Morgan. A review and comparative study of upwind biased schemes for compressible flow computation. Part I: 1-D first-order schemes. Arch. Comp. Meth. Eng., 7:1955, 2000. 26. K. Morgan and J. Peraire. Unstructured grid finite element methods for fluid mechanics. Rep. Prog. Phys., 61:569-638, 1998.
244 Compressiblehigh-speed gas flow 27. R. Ayers, O. Hassan, K. Morgan and N.E Weatherill. The role of computational fluid dynamics in the design of the Thrust supersonic car. Design Optim. Int. J. Prod. & Proc. Improvement, 1:79-99, 1999. 28. K. Morgan, O. Hassan and N.E Weatherill. Why didn't the supersonic car fly? Mathematics Today, Bulletin of the Institute of Mathematics and its Applications, 35:110-114, August 1999. 29. O. Hassan, L.B. Bayne, K. Morgan and N.E Weatherill. An adaptive unstructured mesh method for transient flows involving moving boundaries. ECCOMAS '98. Wiley, New York, 1998. 30. O. Hassan, E.J. Probert, N.E Weatherill, M.J. Marchant and K. Morgan. The numerical simulation of viscous transonic flow using unstructured grids, AIAA-94-2346, 20-23 June, Colorado Springs, USA, 1994. 31. O. Hassan, E.J. Probert, K. Morgan and J. Peraire. Mesh generation and adaptivity for the solution of compressible viscous high speed flows. Int. J. Num. Meth. Eng., 38:1123-1148,1995. 32. O. Hassan, K. Morgan, E.J. Probert and J. Peraire. Unstructured tetrahedral mesh generation for three dimensional viscous flows. Int. J. Num. Meth. Eng., 39:549-567, 1996. 33. O. Hassan, E.J. Probert and K. Morgan. Unstructured mesh procedures for the simulation of three dimensional transient compressible inviscid flows with moving boundary components. Int. J. Num. Meth. Fluids, 27:41-55, 1998. 34. M.T. Manzari, O. Hassan, K. Morgan and N.E Weatherill. Turbulent flow computations on 3D unstructured grids. Finite Elements in Analysis and Design, 30:353-363, 1998. 35. E.J. Probert, O. Hassan and K. Morgan. An adaptive finite element method for transient compressible flows with moving boundaries. Int. J. Num. Meth. Eng., 32:751-765, 1991. 36. K. Morgan, N.E Weatherill, O. Hassan, EJ. Brookes, R. Said and J. Jones. A parallel framework for multidisciplinary aerospace engineering simulations using unstructured meshes. Int. J. Num. Meth. Fluids, 31:159-173, 1999. 37. K. Morgan, EJ. Brookes, O. Hassan and N.E Weatherill. Parallel processing for the simulation of problems involving scattering of electromagnetic waves. Comp. Meth. Appl. Mech. Eng., 152:157-174, 1998. 38. R. Said, N.E Weatherill, K. Morgan and N.A. Verhoeven. Distributed parallel Delaunay mesh generation. Comp. Meth. AppL Mech. Eng., 177:109-125, 1999. 39. K.A. Sorensen, O. Hassan, K. Morgan and N.E Weatherill. Agglomerated multigrid on hybrid unstructured meshes for compressible flow. Int. J. Num. Meth. Fluids, 40:593-603, 2002. 40. K.A. Sorensen, O. Hassan, K. Morgan and N.E Weatherill. A multigrid accelerated timeaccurate inviscid compressible fluid flow solution algorithm employing mesh movement and local remeshing. Int. J. Num. Meth. Fluids, 43:517-536, 2003. 41. K.A. Sorensen, O. Hassan, K. Morgan and N.E Weatherill. A multigrid accelerated hybrid unstructured mesh method for 3D compressible turbulent flow. Computational Mechanics, 31:101-114, 2003. 42. C. Hirsch. Numerical Computation of Internal and External Flows. Volume I, Fundamentals of Numerical Discretization. Wiley, Chichester, 1988. 43. L. Demkowicz, J.T. Oden and W. Rachowicz. A new finite element method for solving compressible Navier-Stokes equations based on an operator splitting method and h-p adaptivity. Comp. Meth. Appl. Mech. Eng., 84:275-326, 1990. 44. C. Hirsch. Numerical Computation of Internal and External Flows. Volume 2, Computational Methods for Inviscid and Viscous Flows. Wiley, Chichester, 1988. 45. J. Vadyak, J.D. Hoffman and A.R. Bishop. Flow computations in inlets at incidence using a shock fitting Bicharacteristic method. AIAA Journal, 18:1495-1502, 1980. 46. K.W. Morton and M.E Paisley. A finite volume scheme with shock fitting for steady Euler equations. J. Comp. Phys., 80:168-203, 1989. 47. J. von Neumann and R.D. Richtmyer. A method for the numerical calculations of hydrodynamical shocks. J. Math. Phys., 21:232-237, 1950. 48. A. Lapidus. A detached shock calculation by second order finite differences. J. Comp. Phys., 2:154-177, 1967.
References 245 49. J.L. Steger. Implicit finite difference simulation of flow about two dimensional geometries. AIAA J., 16:679--686, 1978. 50. R.W. MacCormack and B.S. Baldwin. A numerical method for solving the Navier-Stokes equations with application to shock boundary layer interaction. AIAA paper 75-1, 1975. 51. A. Jameson and W. Schmidt. Some recent developments in numerical methods in transonic flows. Comp. Meth. Appl. Mech. Eng., 51:467-493, 1985. 52. K. Morgan, J. Peraire, J. Peiro and O.C. Zienkiewicz. Adaptive remeshing applied to the solution of a shock interaction problem on a cylindrical leading edge. In P. Stow, editor, Computational Methods in Aeronautical Fluid Dynamics, pp. 327-344. Clarenden Press, Oxford, 1990. 53. R. Codina. A discontinuity capturing crosswind dissipation for the finite element solution of the convection diffusion equation. Comp. Meth. Appl. Mech. Eng., 110:325-342, 1993. 54. T.J.R. Hughes and M. Malett. A new finite element formulation for fluid dynamics IV: a discontinuity capturing operator for multidimensional advective-diffusive problems. Comp. Meth. Appl. Mech. Eng., 58:329-336, 1986. 55. C. Johnson and A. Szepessy. On convergence of a finite element method for a nonlinear hyperbolic conservation law. Math. Comput., 49:427-444, 1987. 56. A.C. Gale~o and E.G. Dutra Do Carmo. A consistent approximate upwind Petrov-Galerkin method for convection dominated problems. Comp. Meth. Appl. Mech. Eng., 68:83-95, 1988. 57. P. Hansbo and C. Johnson. Adaptive streamline diffusion methods for compressible flow using conservation variables. Comp. Meth. Appl. Mech. Eng., 87:267-280, 1991. 58. E Shakib, T.J.R. Hughes and Z. Johan. A new finite element formulation for computational fluid dynamics X. The compressible Euler and Navier-Stokes equations. Comp. Mech. Appl. Mech. Eng., 89:141-219, 1991. 59. P. Nithiarasu, O.C. Zienkiewicz, B.V.K.S. Sai, K. Morgan, R. Codina and M. V~izquez. Shock capturing viscosities for the general algorithm. In M. Hafez and J.C. Heinrich, editors, lOth Int. Conf. on Finite Elements in Fluids, pp. 350-356, 5-8 January 1998, Tucson, Arizona, USA. 60. E Nithiarasu, O.C. Zienkiewicz, B.V.K.S. Sai, K. Morgan, R. Codina and M. V~izquez. Shock capturing viscosities for the general fluid mechanics algorithm. Int. J. Num. Meth. Fluids, 28:1325-1353, 1998. 61. C.G. Thomas and E Nithiarasu. Influences of element size and variable smoothing on inviscid compressible flow solution. International Journal for Numerical Methods in Heat and Fluid Flow, 15,420-428, 2005. 62. G. Sod. A survey of several finite difference methods for systems of non-linear hyperbolic conservation laws. J. Comp. Phys., 27:1-31, 1978. 63. T.E. Tezduyar and T.J.R. Hughes. Development of time accurate finite element techniques for first order hyperbolic systems with particular emphasis on Euler equation. Stanford University paper, 1983. 64. E Woodward and E Colella. The numerical simulation of two dimensional flow with strong shocks. J. Comp. Phys., 54:115-173, 1984. 65. T.H. Pulliam and J.T. Barton. Euler computations of AGARD working group 07 airfoil test cases, AIAA-85-O018, AIAA 23rd Aerospace Sciences Meeting, 14-17 January 1985, Reno, Nevada. 66. O.C. Zienkiewicz and J.Z. Zhu. A simple error estimator and adaptive procedure for practical engineering analysis. Int. J. Num. Meth. Eng., 24:337-357, 1987. 67. J.T. Oden and L. Demkowicz. Advance in adaptive improvements: a survey of adaptive methods in computational fluid mechanics. In A.K. Noor and J.T. Oden, editors, State of the Art Survey in Computational Fluid Mechanics. American Society of Mechanical Engineers, 1988. 68. M.T. Manzari, ER.M. Lyra, K. Morgan and J. Peraire. An unstructured grid FEM/MUSCL algorithm for the compressible Euler equations. Proc. VIII Int. Conf. on Finite Elements in Fluids: New Trends and Applications, pp. 379-388. Pineridge Press, Swansea, 1993. 69. ER.M. Lyra, K. Morgan, J. Peraire and J. Peiro. TVD algorithms for the solution of compressible Euler equations on unstructured meshes. Int. J. Num. Meth. Fluids, 19:827-847, 1994.
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Compressiblehigh-speed gas flow 70. ER.M. Lyra, M.T. Manzari, K. Morgan, O. Hassan and J. Peraire. Upwind side based unstructured grid algorithms for compressible viscous flow computations. Int. J. Eng. Anal. Des., 2:197-211, 1995. 71. R.A. Nicolaides. On finite element multigrid algorithms and their use. In J.R. Whiteman, editor, The Mathematics of Finite Elements and Applications III, MAFELAP 1978, pp. 459-466. Academic Press, London, 1979. 72. W. Hackbusch and U. Trottenberg, editors. Multigrid Methods. Lecture Notes in Mathematics 960, Springer-Verlag, Berlin, 1982. 73. R. Lrhner and K. Morgan. An unstructured multigrid method for elliptic problems. Int. J. Num. Meth. Eng., 24:101-115, 1987. 74. M.C. Rivara. Local modification of meshes for adaptive and or multigrid finite element methods. J. Comp. Appl. Math., 36:79-89, 1991. 75. S. Lopez and R. Casciaro. Algorithmic aspects of adaptive multigrid finite element analysis. Int. J. Num. Meth. Eng., 40:919-936, 1997. 76. P. Wesseling. An Introdution to Multigrid Methods. R.T. Edwards Inc., 2004. 77. A. Jameson, T.J. Baker and N.P. Weatherill. Calculation of inviscid transonic flow over a complete aircraft. AIAA 24th Aerospace Sci. Meeting. Reno, Nevada, AIAA paper 86-0103, 1986. 78. V. Billey, J. Periaux, P. Perrier and B. Stoufflet. 2D and 3D Euler computations with finite element methods in aerodynamics. Lecture Notes in Math., 1270:64-81, 1987. 79. R. Noble, A. Green, D. Tremayne and SSC Program Ltd. THRUST. Transworld, London, 1998. 80. E.J. Probert. Finite element method for convection dominated flows. Ph.D. thesis, University of Wales, Swansea, 1986. 81. Y. Kallinderis and S. Ward. Prismatic grid generation for 3-dimensional complex geometries. AIAA J., 31:1850-1856, 1993. 82. Y. Kallinderis. Adaptive hybrid prismatic tetrahedral grids. Int. J. Num. Meth. Fluids, 20:10231037, 1995. 83. A.J. Chen and Y. Kallinderis. Adaptive hybrid (prismatic-tetrahedral) grids for incompressible flows. Int. J. Num. Meth. Fluids, 26:1085-1105, 1998. 84. O.C. Zienkiewicz, P. Nithiarasu, R. Codina, M. V~izquez and P. Ortiz. The Characteristic-BasedSplit procedure: an efficient and accurate algorithm for fluid problems. Int. J. Num. Meth. Fluids, 31:359-392, 1999. 85. J.E. Carter. Numerical solutions of the Navier-Stokes equations for the supersonic laminar flow over a two-dimensional compression comer. NASA TR-R-385, 1972. 86. O. Hassan. Finite element computations of high speed viscous compressible flows. Ph.D. thesis, University of Wales, Swansea, 1990. 87. O.C. Zienkiewicz and J. Wu. Automatic directional refinement in adaptive analysis of compressible flows. Int. J. Num. Meth. Eng., 37:2189-2210, 1994. 88. M.J. Castro-Diaz, H. Borouchaki, P.L. George, E Hecht and B. Mohammadi. Anisotropic mesh adaptation: theory, validation and applications. In J.-A. Drsidrri et al., editors, Computational: Fluid Dynamics ' 9 6 - Proc. 3rd ECCOMAS Conf., pp. 181-186. Wiley, Chichester, 1996. 89. L. Cambier. Computation of viscous transonic flows using an unsteady type method and a zonal grid refinement technique. In M.O. Bristrau, R. Glowinski, J. Periaux and H. Viviand, editors, Numerical Simulation of Compressible Navier-Stokes Flows, volume 18 of notes of numerical fluid mechanics. Vieweg, Wiesbaden, 1987. 90. N. Satofuka, K. Morinishi and Y. Nishida. Numerical simulation of two-dimensional compressible Navier-Stokes equations using rational Runge-Kutta method. In M.O. Bristrau, R. Glowinski, J. Periaux and H. Viviand, editors, Numerical Simulation of Compressible NavierStokes Flows, volume 18 of notes of numerical fluid mechanics. Vieweg, Wiesbaden, 1987. 91. S. Mittal. Finite element computation of unsteady viscous compressible flows. Comput. Meth. Appl. Mech. Eng., 157:151-175, 1998.
References 247 92. V. Schmitt and E Charpin. Pressure distributions on the ONERA M6 wing at transonic Mach numbers. AGARD Report AR-138, Pards, 1979. 93. M.J. Lighthill. On displacement thickness. J. Fluid Mech., 4:383, 1958. 94. J.E. Green, D.J. Weeks and J.W.F. Brooman. Prediction of turbulent boundary layers and wakes in compressible flow by a lag-entrainment method. Aeronautical Research Council Repo. and Memo Rept. No. 3791, 1973. 95. P. Bradshaw. The analogy between streamline curvature and buoyancy in turbulent shear flow. J. Fluid Mech., 36:177-191, 1969. 96. J.E. Green. The prediction of turbulent boundary layer development in compressible flow. J. Fluid Mech., 31:753-778, 1969. 97. H.B. Squire and A.D. Young. The calculation of the profile drag of aerofoils. Aeronautical Research Council Repo. and Memo 1838, 1937. 98. R.E. Melnok, R.R. Chow and H.R. Mead. Theory of viscous transonic flow over airfoils at high Reynolds number. AIAA Paper 77-680, June 1977. 99. J.C. Le Balleur. Calcul par copulage fort des 6coluements visqueux transsoniques incluant sillages et d6collemants. Profils d'aile portant. La Recherche Aerospatiale, May-June 1981. 100. J. Szmelter and A. Pagano. Viscous flow modelling using unstructured meshes for aeronautical applications. In S.M. Deshpande et al., editors, Lecture Notes in Physics 453. Springer-Verlag, Berlin, 1994. 101. J. Szmelter. Viscous coupling techniques using unstructured and multiblock meshes. ICAS Paper ICAS-96-1.7.5, Sorrento, 1996. 102. J. Szmelter. Aerodynamic wing optimisation. AIAA Paper 99-0550, January 1999. 103. J.C. Le Balleur. Viscous-inviscid calculation of high lift separated compressible flows over airfoils and wings. Proceedings A GARD/FDP High Lift Aerodynamics, A GARD-CP515, Banff, Canada, October 1992. 104. J.C. Le Balleur. Calculation of fully three-dimensional separated flows with an unsteady viscous-inviscid interaction method. 5th Int. Symp. on Numerical and Physical Aspects of Aerodynamic Flows, Long Beach CA (USA), January 1992. 105. J.C. Le Balleur and P. Girodroux-Lavigne. Calculation of dynamic stall by viscous-inviscid interaction over airfoils and helicopter-blade sections. AHS 51st Annual Forum and Technology Display, Fort Worth, TX, USA, May 1995.
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Turbulent flows iililiiiil8~i~ilii~ii!ii~~i i~i~~i i~OdUCt| i~~iJi~ ~ ~ ON ~ ~ ~ ~ ~ ~ ~ ~ ~ ; ~ ~ ~ n ~ : In the majority of moderate and high speed flow problems some form of random variation of flow variables exists. The 'laminar' treatment discussed in the previous chapters is generally not applicable when such variations occur. Turbulent flow is defined as a flow with random variation of various flow quantities such as velocity, pressure and density. Turbulence is a property of a flow, not that of a fluid. Numerical solutions of the transient Navier-Stokes equations are sufficient to resolve the turbulent behaviour if an adequate fine mesh resolution and time increment are used. However, this requires extremely large computer resources and with present day computers a direct numerical simulation is possible only at relatively low Reynolds numbers. Despite significant progress in understanding turbulent behaviour during the last century, the modelling of turbulence is still an unresolved problem and will remain so for the foreseeable future. In this chapter we provide an overview for the numerical solution of turbulent fluid dynamics equations based on Reynolds decomposition of solution variables employed along with some existing turbulence models and the Characteristic Based Split (CBS) scheme. We also make reference to various other works on turbulent flows and numerical modelling. Before going into details on current modelling procedures we summarise some of the important fundamental properties of turbulence. At this point it is worth noting that a turbulent flow is an irregular dissipative three dimensional process that occurs at relatively high Reynolds numbers. A turbulent flow is marked with random variation of quantities about an average flow as shown in Fig. 8.1.
8.1.1 Turbulence scales In a real turbulent flow the kinetic energy is transferred from larger scales to smaller scales. At the smallest scale kinetic energy is transformed into intemal energy and this process is called dissipation. The process of energy transfer between the scales is called 'the cascade process'. The smallest turbulent length scale is defined by the molecular viscosity and dissipation rate. Such a length scale is often referred to as the Kolmogorov length scale, 1 71, and given as r/--
--
(8.1)
Introduction Ui I
Fig. 8.1 Random variation of velocity in a turbulent flow with respect to time.
where u is the kinematic viscosity of the fluid and e is the turbulent dissipation rate. Similarly the Kolmogorov velocity and time scales are denoted by
P = (/2(~)1/4 and ~- --
(8.2)
(/])1/2
(8.3) e The turbulent dissipation rate occurring at small scales can be linked to the energy of large eddies as U3 e= (8.4) 1 where U is the large eddy velocity scale and I is the large eddy length scale. The above relations are given here to make the reader aware that length scales and dissipation rates are closely related. Most turbulence modelling procedures are developed based on these relationships.
8.1.2 Time averaging As mentioned previously extremely high mesh resolution is required to obtain numerical solutions at the smallest turbulence scales. This is very expensive and presently not possible for high Reynolds number flows. It is, therefore, obvious that other alternatives are necessary to obtain a viable approximate solution. The current standard procedure is to employ time averaged Navier-Stokes equations along with a turbulence model to determine the essential time averaged quantities. This approach reduces to a reasonable level the excessive grid resolution otherwise needed. With reference to Fig. 8.1 any turbulence quantity of interest may be expressed as
r162162
(8.5)
w
and the time averaged quantity may be obtained from r
dt
(8.6)
To demonstrate time averaging we consider the one dimensional steady state incompressible momentum equation d
ldp
dx
p dx
dx \
dx,]
0
249
250 Turbulentflows
Substituting a variation of the form of Eq. (8.5) for velocity u and pressure p into Eq. (8.7) and time averaging, we have d
ld
+ u')
d--x [(g + u')(g + u')] + P ~xx(~ + p')
- 0
(8.8)
m
In the above equation the average of fluctuating components u' and p' are equal to zero. Hence, the following simplified form of the above equation may be written as dx
p dx + dx k dx J - dxx (u-7-u-Tu')
(8.9)
Rearranging and rewriting the above momentum equation in multi-dimensional form and adding the time rate term, we have
O(pUi) 0
T where
013 Or-l-ij -~ -~xj (pF'lj~li) - -Ox---~ -~- p (~xj
0 (pul
Ox---~
,)
Uj
2 0F.lk (~ij )
Oil i Ott j Tij -- V ~ + OXi
(8.10) (8.11)
30xk
is the time averaged deviatoric stress and u Iu~ is a new unknown referred to as Reynolds stress. A fundamental property of the Reynolds stress is the turbulent kinetic energy of a flow, ~, which is defined as I~ -- -UiU i (8.12) !
!
2
where u I is the fluctuating component of the velocity as shown in Fig. 8.1. The Boussinesq assumption expresses the Reynolds stress as
--puIu~ -- ~T
(Obl i Ollj ~Xj -~- OXi
or
~ij = --Uilgl -- L/T
(Oll_~~j
Oblj + OXi
20Uk)
-3-~Xk(~ij --
~
pl~(~ij
20lg k (~ij) 2 30X---~k -- -~l~(~ij
(8.13)
(8.14)
From Eq. (8.14) it is clear that the unknown quantity to be modelled is the turbulent kinematic viscosity Ur -- # r / P , the turbulent kinetic energy, ~ and the turbulent dissipation rate c. Often the turbulent kinetic energy term in the above equation is dropped for simplicity. It is important to remark here that non-linear forms of turbulence modelling procedures are becoming popular. 2-8 However, for simplicity the present discussion is limited to the above form of equations.
8.1.3
Relation b e t w e e n
e and
IJr
The turbulent kinematic viscosity (or turbulent eddy viscosity) ur has the same dimensions as the laminar viscosity u. Thus, we can express the turbulent eddy viscosity in terms of velocity and length scales of a large eddy, i.e. ur -- CUI
(8.15)
Treatment of incompressible turbulent flows
where C is a constant and U and I are described in Sec. 8.1.2. Based on dimensional considerations U in Eq. (8.15) may be replaced with ~/-~. With such a substitution the turbulent eddy viscosity may be determined by solving a scalar transport equation for t~ and assuming an appropriate turbulent length scale I. This approach yields one equation ~ - l models. However, a better expression for turbulent eddy viscosity may be obtained by substituting Equation 8.4 into Eq. 8.15 and writing /~2 /'/T - - C / z - -
(8.16)
where c# is a constant. To employ the above equation, we need to solve two transport equations, one for t~ and another for e. This approach leads to t~-e or two equation models. Details of some one and two equation models are provided in the following sections. In the sections that follow we treat the incompressible and compressible flows separately and identify the differences. In addition to discussions on the Reynolds Averaged Navier-Stokes (RANS) models, we also provide a brief summary of Large Eddy Simulation (LES) and currently popular topics such as Detached Eddy Simulation (DES) approach. Several examples are also provided to illustrate the turbulent flow modelling approaches presented. For further details on turbulence and turbulence modelling the reader is referred to standard textbooks on this topic. 9-11
8.2.1 Reynolds averaged Navier-Stokes equations For turbulent flow computations, Reynolds averaged Navier-Stokes equations of motion are written in conservation form as follows
Mean-continuity
] Op
~2 ot
I
Opbl i
Oxi
=0
(8.17)
Mean-momentum OQl.li CO 10p O?ij --~ -t- -~xj (~j5k) . . . . . p Oxi ~- ~ -+---Oxj
(8.18)
where/3 is an artificial compressibility parameter, Ui a r e the mean velocity components, p is the pressure, p is the density, ~-ij is the laminar shear stress tensor given by Eq. (8.11 ) and the Reynolds stress tensor is given in Eq. (8.14).
8.2.2 One-equation models Wolfstein
~ -
I
modeP 2
In this model the turbulent eddy viscosity is determined from a mixing length and turbulent kinetic energy as lJ T -- C1#/4l~1/21 m (8.19)
251
252
Turbulentflows
where c~ is a constant equal to 0.09, ~ is the turbulent kinetic energy and lm is a mixing length. The mixing length lm is related to the length scale of the turbulence L as
lm-
(,3) C#
1/4
L
(8.20)
t
where Co and cv are constants. The transport equation for turbulent kinetic energy ~ is expressed by
01"~ 0 0 {(liT) 0t~ I O~li --~ -[- -~iXi ( ~li N) -- ~X i Y -'[" --0"~ ~ + ~ --~Xj
e
(8.21)
where cr~ is the diffusion Prandtl number for turbulent kinetic energy. The dissipation, e, is modelled as /~3/2 e = Co~ (8.22) L Near solid walls, the Reynolds number tends to zero and the highest mean velocity gradient occurs at the solid boundary. Thus, the one-equation model has to be used in conjunction with empirical wall functions, i.e. ur is multiplied by a damping function f~, -- 1 - e - 0 " 1 6 0 R ~ and e is divided by fb - - 1 - e - 0 " 2 6 3 R ~ , where R~ -- ~ / - ~ y / u . Here y is the shortest distance to the nearest wall. The constants are usually taken at crk = C o = 1 and the length scale L may be assumed to be equal to the characteristic length of the problem.
Spalart-AIImaras (SA) modeP 3
The Spalart-Allmaras (SA) model was first introduced for aerospace applications and is currently being adopted for incompressible flow calculations. This is another oneequation model, which employs a single scalar equation and several constants to model turbulence. The scalar transport equation used by this model is
Ot ~ OXj
= Cbl S~"t---(7 ~
(l] + g,) ~
+ Cb2 ~
'] E1' --Owlfw
(8.23)
where -- S -~- (/.)/k2y 2) fv2
fo2 = 1 - X / ( 1
+
Xf~l)
(8.24) (8.25)
In Eq. (8.24), S is the magnitude of vorticity and y is the shortest distance from a node to the nearest solid wall. The eddy viscosity is calculated as /IT = I)L1
(8.26)
Treatment of incompressible turbulent flows 253 where
3 fvl -- X3/( X3 -]-Cvl) X-
(8.27)
fJ/u
(8.28)
The parameter fw is given as
6 1 + %3
fw
33 g6 _+.Cw
g
I 1/6
(8.29)
where g -- r
+ Cw2(r 6 -
r)
(8.30)
// ^
r --
The constants are given by Cbl (1 + Cb2)/Cr, Cw2 - - 0.3,
Cbl/k 2 +
-Cw3
(8.31)
~k2y 2
0.1355, cr -- 2/3, Cb2 -- 2 and Col = 7.1.
--
0.622, k -- 0.41,
Cwl =
8.2.3 Two-equation models The standard ~ - E model In this model, the transport equation for n is the same as that in the one-equation model of Sec. 8.2.2. The second transport equation for calculating the turbulence energy dissipation rate e is
OC + --~Xi 0 (1-1iE) -- ~0 I ( Y "~ l/~e) ~OC I -+-Cel--Tij ---C RO~ i -'~ I~ OXj
C , 2C - -2
I~
(8.32)
where C~I = 1 . 4 4 and C~2 = 1.92. In addition, the diffusion Prandtl number for isotropic turbulence energy dissipation rate, (7~, is set to 1.3 as proposed by Jones and Launder. 14 In addition, ur is evaluated by ur = % - -
(8.33)
E
For near-wall treatments, modifications to the source terms of the e equation are needed in the near-wall region. Multiplying the coefficients c~, C~I and C~2 by the turbulence damping functions f~, f~l and f~2 appropriate low Reynolds number behaviour near the walls is achieved. Numerous wall damping functions have been proposed. We employ the ones suggested by Lam and Bremhorst~5 for steady flows. fu - (1
-
e -0"0165R~)2
fcl-- l -t- - ~
(20.5) 1 -+- ~
(8.34)
(8.35)
254 Turbulentflows and fe2
1 - e -R'~
--
(8.36)
where R t -- N2/pc. In the examples presented in later sections we have also employed the damping functions of Fan e t al. 16 f~ -- 0.4 fw
fw (1--0.4~t)
Rr [1--exp(42.63)
(8.37)
1
where f~ -- 1 - exp {
2.30
~-
fe2 --
2.30
1
1 - exp
8.89
0.4 - -1.8 -exp
[- (~)2]
-
(8.38)
-~
(8.39)
} fw2
and f~l - 1. The constants are c, - 0.09, ~rk - 1.0, cr~ - 1.3, 1.8.
Cel
=
1.4 and
Ce2 =
8.2.4 Non-dimensional form of the governing equations The turbulent flow solution is obtained by solving Eqs (8.17) and (8.18) with appropriate boundary conditions along with one of the turbulence models. The following nondimensional scales may be used in the calculations bli
Xi
-, = ~. ui u~
xi
* = __. D
r~* = eD
/,/2
b/3 ,
c~
cx~
p p~u 2
Ur --
9
ur
t*
tu~
D
,
/:~
D
Ycx~
(8.40)
where D is a characteristic dimension and the subscript c~ indicates a reference value. Substituting the non-dimensional scales into Eqs (8.17) and (8.18) and dropping asterisks leads to
10(p) ~2 Obl i
--ffi- +
0l
+
0 j (a j i) -
O(pfii) OX i
Oj9 Ox i
and ~-ij + ~ - (1.0Re + Ur) ( 0-~XJ ~ X / 0-+t ~0fij i
=0
(8.41)
O(~-ij -+- ~ )
Ox :
(8.42)
Treatment of incompressible turbulent flows
The non-dimensional density in Eq. (8.43) is unity for incompressible flow problems. The Reynolds number, Re, in the above equation is defined as
Re =
(8.44)
Ycx~
The viscosity, u, is assumed to be constant and equal to u~ in the above equations. Note that the turbulent kinetic energy term may be dropped from Eq. (8.43) for simplicity. The non-dimensional forms of the turbulence transport equations are given below.
One-equation model
The non-dimensional form of the ~ equation is
0/~
0
1 0 {(YT)
01~ I OUi 1 + __o.t~ ~ .ql_~ -~Xj
--~ -~- ~Xi (~liK,) -- Re Oxi
e
(8.45)
The mixing length and the turbulence length scales are normalized using the characteristic dimension D. In this study we assume L = D. The non-dimensional form of R~ is ~/-~yRe.
5palart-AIImaras model
The non-dimensional form of the transport equation is I
-- cb, S~q-
IEO/
(1 q- D)-~-
+ Cbz
Ion/] c,ReEy] (8.46)
where 1
- s + ~ ( ~ / k Z Y 2) f~2 r =
(8.47) (8.48)
Re Sk2 y 2
The structures of all the remaining parameters are unchanged. -
~
model
The ~ equation is identical to that of the one-equation model and the dissipation equation is given as
Oe 0 10( IJT) OC C ROI-Ii -~ -Jr -~Xi (~lie) = Re OXi 1 -~" io.e ~ "71-Cel--i~Tij Oxj
C2
Ce2--/~
(8.49)
The parameter Rt in its non-dimensional form is n2Re/e. The form of the other turbulent parameters is unchanged in non-dimensional form.
255
256 Turbulentflows
8.2.5 Shortest distance to a solid wall In all the turbulence models described above, distance from a node to the nearest wall is essential. For meshes with a small number of nodes it is a matter of checking the number of solid wall nodes against the total number of nodes in a mesh. However, such a procedure will consume a significant part of the computation time if the number of nodes is very large. Nowadays, it is common to solve compressible flow equations over several million number of nodes. Hence, we need to have faster and reliable wall distance calculation procedures in place. The wall distance calculation may be accelerated by creating linelets emanating from solid wall nodes as discussed in references 17 and 18. This method works by allocating all non-wall nodes to a linelet and then finding a short distance to the wall. This method was proved to reduce computing time to calculate wall distance. In time dependent flow problems which involve moving solid walls, the standard procedures of calculating the wall distances will again be expensive. In recent years, the differential equation-based wall distance calculation procedures have been developed to reduce the cost of repeated computation. We refer the readers to relevant papers on this topic. 19'20
8.2.6 Solution procedure It is evident from the RANS equations discussed above that the governing turbulent flow equations take the form of standard fluid dynamics equations discussed in the previous chapters. However, we have additional scalar transport equation(s) that need to be solved in conjunction with the RANS equations. We therefore recommend the CBS scheme along with any valid procedure for solving convection--diffusion equations from Chapter 2 for the solution of scalar turbulence transport equations. The transient solutions may be obtained via an artificial compressibility scheme using a dual time stepping algorithm. 21 Alternatively a semi- or quasi-implicit form of the scheme may be employed to obtain unsteady-state solutions. Unsteady-state RANS solutions are often referred to as URANS solutions. 21-26
Example 8.1
Turbulent flow past a backward facing step
A standard test case commonly employed for testing turbulent incompressible flow models at a moderate Reynolds number is the recirculating flow past a backward facing step. Unlike in channel flow, here the model has to handle the recirculation region immediately downstream of the step. The definition of the problem is shown in Fig. 8.2. The characteristic dimension of the problem is the step height (L). All other dimensions are defined with respect to this. The inlet is located at a distance of four times the step height upstream of the step. The inlet channel height is 2L. The total length of the channel is 40L. The inlet velocity profile is obtained from measurements reported by Denham e t al. 27 No-slip conditions apply on the solid walls. For the one-equation and two-equation models the inlet n and r profile were obtained by solving a channel flow problem. For the SA model, a fixed value of 0.05 for the turbulent scalar variable at the inlet
Treatment of incompressible turbulent flows Experimental turbulent profile p=O
x",, u5 =~2=o I..
4L
..._1_..
i ~,,,,
36L}
,,,.-- -...,,,,
Fig. 8.2 Turbulent flow past a two-dimensional backward facing step. Problem definition.
was prescribed. On the walls ~ was assumed to be equal to zero. The wall condition for c is e-
Re
(8.50)
dy
as discussed in reference 28. The scalar variable of the SA model was also assumed to be zero on the walls. The convergence histories to steady state for all three turbulence models are shown in Fig. 8.3. As can be seen both the one-equation, two-equation model of Fan et al. and the SA model reach the prescribed residual tolerance faster than the two-equation model of Lam and Bremhorst. 15 It is also noticed that the convergence to steady state is not monotonic, which is expected from an explicit time discretization. Figure 8.4 shows the comparison of velocity profiles against the experimental data of Denham et al. 27 It is obvious that the one-equation model failed to predict the 0.01
i~,.
i
j~.
i
i
i
I
I
_
0.001
i
i
i
i
I
I
I
I
One-equation - - Two-equation (Lam-Bremhorst) .... Two-equation (Fan et aL) ..... SA model .........
,i
.E
O t" o4 ..4
0.0001
"..k, ~ . ~r"-~',~~... 1e-05
le-06
le-07
0
I
I
5000 10000 15000 20000 25000 30000 35000 40000 45000 No. time steps
Fig. 8.3 Turbulent flow past a two-dimensional backward facing step. Steady-state convergence histories at =3025.
Re
257
258 Turbulentflows 3
~-...~...~
Re = 3025
,
"3.
u
1.5F
i,
.l."
~, '.
=
O Exp. ne-equation model
B B B N B N M
.-"
M
!
R
0"5 1 o
0
,~*-*/
i
2
M
6
4
w
Horizontal velocity
n
W
10
12
'
-
14
(a) One equation model "
"
Re = 3025
~u
Exp. = SA model
2.5 ot - -
%
2
,m
-o 1.5 .m 1=
>
1 0.5 0
4
0
6 8 Horizontal velocity (b) SA model
I
10
12
14
Re = 3025
3-.~. R
2.5-
, ;* !,
2
!
* ;* ~,
Lam and Bremhorst Fan et aL . . . . . . .
1.5 1,rJ ,~ U ~
""
0.5 0
n
0
2
4
6 8 Horizontal velocity (c) Two equation model
10
12
14
Fig. 8.4 Incompressible turbulent flow past a backward facing step. Velocity profiles at various downstream sections at Re = 3025.
Treatment of incompressible turbulent flows
(a) Mesh
(b) u1 contours
(c) ; contours
(d) Pressure contours
Fig. 8.5 Incompressibleturbulent flow past a backward facing step. Structured mesh (elements: 8092, nodes:
4183), velocity contours, O contours and pressure contours at Re = 3015 using the SA model.
recirculation region accurately. The SA model and the two-equation models on the other hand predict the recirculation better than the one-equation model with the SA model able to predict the recirculation more accurately. Both structured and unstructured meshes were employed in the calculation. Figures 8.5 and 8.6 show the SA model solutions on structured and unstructured meshes respectively. The qualitative difference between the two results is almost nil. The quantitative difference between the two solutions was also found to be negligibly small.
Example 8.2
Unsteady turbulent flow past a circular cylinder
This is an example in which we demonstrate the application of dual time stepping to predict time dependent flows. The example considered is the standard test case of transient turbulent incompressible flow past a two-dimensional circular cylinder at a Reynolds number of 10 000. The domain consists of a circular cylinder placed at a distance of 4D from the inlet, where D is the diameter of the cylinder. The distance from the centre of the cylinder to the top and bottom sides is equal to 4D. The exit of the domain is placed at a distance of 12D from the centre of the cylinder (Fig. 8.7). The finite element mesh used is shown in Fig. 8.7. The mesh in the vicinity of the cylinder and along the wake region are refined to capture the transient feature of the problem.
259
260 Turbulent flows
(a) Mesh
(b) ul contours
(c) r~contours
t,i.)
f,I/,I////S//i
(d) Pressure contours
Fig. 8.6 Incompressible turbulent flow past a backward facing step. Unstructured mesh (elements: 47359, nodes: 24 336), velocity contours, g, contours and pressure contours at Re = 3025 using the 5A model.
Both the SA model and Fan et al. models were tested on this problem. Uniform velocity conditions were assumed at the inlet. The turbulent scalar variable was assumed to be equal to 0.05 at the inlet for the SA model. On the top and bottom sides slip conditions were assumed and no turbulence quantity was prescribed. On the cylinder walls no-slip conditions were assumed and the turbulent scalar variable of the SA model was assumed to be zero. For the two-equation model, ~ and c values at the inlet were assumed to be 0.0025 and on the solid wall ~ was set to zero. The condition for c was the same one used in the previous example [viz. Eq. (8.50)]. Dual time stepping was employed in this problem. A true transient term was added to both step 3 of the CBS scheme as discussed in Chapter 3 and the turbulence transport equations. The pseudo time step used within each real time step is local and varies between the nodes depending on the local flow field and mesh size. The L2 norm of velocity residual was reduced to less than 10 -6 within every real time step in order to make sure that local steady state is achieved within each real time step. Instantaneous patterns of the horizontal velocity component for the SA model are shown in Fig. 8.8 at different real times, tR, and show that the vortex shedding is present.
Treatment of incompressible turbulent flows
Fig. 8.7 Incompressible turbulent flow past a circular cylinder. Finite element mesh. (Elements: 46433, nodes: 23 452.)
At tn = 10, the initial velocity field immediately behind the cylinder is nearly symmetric, however, the velocity field at tn = 20 and beyond shows unsymmetric shedding behaviour. From Figs 8.8(b), (c) and (d) it is obvious that the origin of the vortex street shifts between the areas above and below the central axis. The behaviour qualitatively confirms the periodic vortex shedding phenomenon. Figures 8.9 and 8.10 show respectively temporal drag and lift coefficients for the SA model analysis. Periodic flow and vortex shedding are clearly evident from the
(a) tR= 10
:i!i i i!i~i!i i~i i i~i,i;:i!i~i::?!i
iiiiiiiiii
(b) tR = 20
:
i i i i iii!i i!iiiiiiiiiii ii!iiiiiiiiiiiii. . . (c) tR = 30
(d) t R = 40
Fig. 8.8 Incompressible turbulent flow past a circular cylinder. Velocity contours at different non-dimensional real times at Re = 10 000 using the SA model.
261
262
Turbulent flows Re = 10 000 I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
1.5
,Z1 0.5
0
10 20 30 40 50 60 70 80 90 100 Non-dimensional real time
Fig. 8.9 Incompressible turbulent flow past a circular cylinder. Drag coefficient distribution with respect to non-dimensional real time at Re = 10 000 using the SA model.
graphs. The average experimental value of drag coefficient is around 1.12 and the Strouhal number is around 0.2. 29 Present prediction shows that the average value of drag coefficient is around 1.375 and the Strouhal number* is 0.167. The large difference in predicted drag coefficient is not surprising as all the URANS models have accuracy limitations. 6 For instance the standard n - e-based URANS reported in reference 25 underpredicts the drag coefficient by almost 50%. Even the 2D LES model reported in reference 30 significantly overpredicts the average drag coefficient. It appears that some of the non-linear RANS models give results better than the standard RANS models. 3~ The lift coefficient distribution with respect to real time using the SA model is 9shown in Fig. 8.10. Once again the pattern is perfectly periodic and the magnitude of Re = 10 000
2
Spalart-AIImaras model
1
-1 -2 0
10
20
I
I
I
I
I
30 40 50 60 70 80 Non-dimensional real time
I
90 100
Fig. 8.10 Incompressible turbulent flow past a circular cylinder. Lift coefficient distribution with respect to non-dimensional real time at Re = 10 000 using the SA model.
*Strouhalnumberis defined as St = n D / u o o wheren is the frequencyof vortex shedding, whichmay be defined as the inverse of the time taken for one shedding cycle. In non-dimensionalform St is equal to the inverse of the non-dimensional time for one shedding cycle.
Treatment of incompressible turbulent flows 263 0.5j
,
.../
,
,
Re = 10000 ,
,
,
,
,
SAmodel
,
.
I
/
.~ 0.2
-0 4 -0"5
0
10 20
30 40 50 60 70 80 Non-dimensional real time
90 100
Fig. 8.11 Incompressibleturbulent flow past a circular cylinder. Vertical velocity distribution at an exit point at Re = 10 000 using the SA model.
the lift coefficient is in qualitative agreement with other reported results. 3~ However, it should be noted that the turbulent flow over a circular cylinder at a Reynolds number of Re = 10 000 is essentially three dimensional. 32 Figure 8.11 shows the variation of vertical velocity component at an exit point on the horizontal centreline of the domain and is consistent with the drag and lift data. Figures 8.12, 8.13 and 8.14 show the drag coefficient, lift coefficient and vertical velocity variation at an exit point using the two-equation turbulence model. Although the velocity distribution is similar between the two-equation and SA models, the drag and lift coefficient distributions are quite different. The average drag coefficient value obtained is around 0.845, which is much smaller than that of the one predicted by the SA model. However, the two-equation model results are very similar to the one reported in reference 25 for a higher Reynolds number. The average lift coefficient obtained by the two-equation model is zero, which is consistent with the results from reference 25. The Strouhal number predicted by the two-equation model is 0.155. 1.4
Re = 10 000 I
I
1.2
I
I
I
I
I
I
I
Two-equation model -..--
|
t 0.8
cZ
0.6 0.4 0.2 I 1'0 20
1 I I I 30 40 50I 60 70 80I Non-dimensional real time
I
90 100
Fig. 8.12 Incompressible turbulent flow past a circular cylinder. Drag coefficient distribution with respect to non-dimensional real time at Re = 10 000 using the two-equation model.
264
Turbulent
flows
Re = 10 000
0.4
i
~
;
;
'
;
;
.;
~
;
I
0.3 0.2 0.1 ~
0
-0.1 -0.3 -0.4
0
10 20
30 40 50 60 70 80 Non-dimensional real time
90 100
Fig. 8.13 Incompressible turbulent flow past a circular cylinder. Lift coefficient distribution with respect to non-dimensional real time at Re = 10 000 using the two-equation model. Re = 10000
0.4
~
I
i
i
I
I
-
I
,
I
I
I
I
0.30.2
~ 0.1 "~ -0.1
~ -0.2
> -0.3
-0.4
o
2'o 3'o 20
r
Non-dimensional real time
Fi 9. 8.14 Incompressible turbulent flow past a circular cylinder. Vertical velocity distribution at an exit point at Re = 10 000 using the two-equation model.
The most important point noted here is that the explicit scheme along with an implicit dual time stepping approach can satisfactorily model unsteady turbulent incompressible flows.
The conservation equation for compressible flows may be rewritten from Chapters 1 and3. Mass conservation Op 10p OUi Ot c2 0 t Oxi where c is the speed of sound and depends on E, p and p.
(8.51)
Treatment of compressible flows M o m e n t u m conservation
OUi
o
&ij
OXj ( u j U i )
Ot
~
op
Ox i ~- Pgi
Oxj
(8.52)
Energy conservation O(pE) C~
0 - ~ ( u i p E ) -Jr- ~ C~Xi
0(OT) k -ff~xi
-
0
c9
--~ixi ( li i p ) -k- -~ixi ( 7i j u j ) -nI- p g i u i
(8.53)
In all of the above Ui are the velocity components, p is the density, E is the specific energy, p is the pressure, T is the absolute temperature, Pgi represents body forces and other source terms, k is the thermal conductivity, and ~-ij are the deviatoric stress components given by [Eq. (1.12b)] 7"ij = ~
IOUi OU j ~ "~ OXi
2(~ijIouk ) 3
~ Xk
(8.54)
In general, #, the dynamic viscosity, in the above equation is a function of temperature, #(T), and appropriate relations will be used if necessary.
8.3.1 Mass-weighted (Favre) time averaging Conventional time averaging discussed in Sec. 8.1.1 introduces additional terms in compressible flows and needs additional equations to close the system. To avoid this it is often useful to introduce a mass-weighted averaging defined by pui
ui --- _
P
(8.55)
where the overline in the above equation indicates a standard time averaged quantity. Here, the time dependent velocity may be written as Ui -- Ui -~- U "i -
(8.56)
The above equation has the similar form as standard time averaging relation (8.5). However, in the above equation the mass-weighted averaging term Ui replaces the conventional time averaging term fii. Now employing the mass-weighted velocity and standard time averaging for p and p we get the conservation of mass and momentum equations as
0~
0
-~- -~- ~X/(~)Ui) -- 0
(8.57)
and
0 0 -~(f,r,i) + ~ ( f , r , ir, j) = tit
tz~ j
op 0 + - ~ - ( % - puyuj) + ~gi Oxz u.~ j
(8.58)
265
266
Turbulent flows
It should be noted here that in standard mass averaging --7 ui 0 and pu~ =7/=0 and in "--~ l/ mass-weighted averaging u i ~ 0 and pu i 0. In Eq. (8.58) pu if/ Ujf/ are the Reynolds stresses. In a similar fashion the mass-weighted averaging for energy gives :
-
O(-~E)
Ot
(9 _ O ( cgT -- (OXi" (-'f tl i E) .qt_ ~ k --~Xi
-
,,\
pE""u i )
0
__
0
_~Xi ( ~li -'ff) ._11__.~Xi ( 7-ij U j ) _Jr_D g i ~l i
(8.59)
The extra terms in Eqs (8.58) and (8.59) are modelled to solve the compressible turbulent flow equations. The mass-weighted averaged turbulence transport equations for compressible flows are discussed further in Appendix I.
Example 8.3
Turbulent flow past an ONERA-M6 wing
This is an example of turbulent flow calculation using the n - ~ model (see Appendix I). The area in the vicinity of the wing is covered with structured layers to resolve the turbulent boundary layer accurately. Away from the surface the mesh is purely unstructured as shown in Fig. 8.15(a).
(a) Fig. 8.15 Turbulent, viscous, compressible flow past an ONERA M6 wing. 34 (a) Surface mesh, elements: 48 056.
Large eddy simulation
(b) Fig. 8.15 Continued. (b) Pressure contours.
Figure 8.15(b) shows the surface pressure contours. As can be seen the contours are marked with shocks. Figure 8.15(c) compares the pressure coefficient value against the experimental data. 35 As seen the agreement is good.
The idea of LES is developed based on splitting large-scale motions from small scales using a filtering operation such as
(x) - f~ f (x')G(x, x') dx'
(8.60)
where G is a filter function. Examples of such filters include Gaussian and tophat functions.
267
268 Turbulentflows 1.5
1.5
1.0
1.0
0.5 I
0.5
0
I
0 -0.5 -1.0
.
!
I
i
I
I
I
I
I
I
0.5 1.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 X/C
1.5
1.5
1.0
1.0
0.5
0.5
r
I
-0.5 -1.0
-1.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 X/C
1.5
1.0
1.0
oo
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 X/C
~0~ 0.5
0
i
-0.5 -1.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 X/C
-0.5
1.5
0.5
~
0
0 -0.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 X/C
-1.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 X/C
(c) Fig. 8.15 Continued. (c) Coefficient of pressure distribution at 20%, 44%, 65%, 80%, 90% and 95% of wing span, line-numerical 34 and circle-experimental. 3s
If the variables of the the incompressible Newtonian equations are subjected to the above filtering operation, we get O~i =0 Oxi
(8.61)
Large eddy simulation 269 and Oqbii 0 Ot .ql_ -~xj (U---~) -
10-fi
Owij
P OXi .~_ ~
Owij SGS
_~_ OXj
(8.62)
where ~iijGS ~" UiU j -- UiUj
(8.63)
-rsos in the above equation is generally modelled using various subgrid scale (SGS) models. The standard SGS models, 36 dynamic models 37 and non-linear models are a few examples. It is a vast area of research and difficult to cover all the theory behind these models in a chapter. For the sake of completeness, we provide the standard SGS model below. The SGS stress of Eq. (8.63) is represented exactly as Eq. (8.14) without the last term. However, the eddy viscosity is modelled differently here.
Standard SGS model
The eddy viscosity here is defined as
//T "- (CA) 2(-~3
(8.64)
where C is a constant, A is the filter width and ~ is the magnitude of the resolved strain rate tensor. This most widely used eddy-viscosity model was proposed by the meteorologist Smagorinsky. 36 Smagorinsky was simulating a two-layer quasigeotrophic model in order to represent large- (synoptic) scale atmospheric motions. He introduced an eddy viscosity that was supposed to model three-dimensional turbulence in the subgrid scales. In Smagorinsky's model, a mixing-length assumption is made, in which the eddy viscosity is assumed to be proportional to the subgrid scale characteristic length A and to a characteristic turbulent velocity based on the second invariant of the filtered field deformation tensor (i.e. strain-rate tensor). In other words, the well-known Smagorinsky's model, where the SGS time scaling, ~, in Eq. (8.64) is set as the magnitude of the local resolved strain-rate tensor, namely = ISI
=
(2~ij~dij) 1/2
(8.65)
and C = C~
(8.66)
An approximate value of this constant is 1
Cs ,~ -
7r
( 3 C /~ ) -3/4 ~
(8.67)
For a Kolmogorov constant CK of 1.4, which is obtained by measurements in the atmosphere, this yields Cs ,~ 0.18. Most workers prefer Cs = 0.1 - a value for which Smagorinsky's model behaves reasonably well for free-shear flows and for channel flow. However, the Smagorinsky constant Cs is required to have a sensible value to avoid excessive damping of resolved structures and the grid size A, as an indication of characteristic length scale, separates large- and small-scale eddies from each other and
270
Turbulentflows
is considered to be an average cell size. It is calculated for three-dimensional elements as follows A -- f(AXl,
Ax2,
Ax3)
1/2
(8.68)
Note that the above definition of mesh size can be changed to the usual finite element mesh size as discussed in Chapter 4. Despite increasing interest in developing more advanced subgrid scale stress models, Smagorinsky's model is still successfully used. ~@~@~s~@~~s~~~s~~8i 5 ..:etached
Eddy
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The Large Eddy Simulation (LES), despite less use of empirical relations, needs a very high computational overhead compared to RANS models. This is due to the fact that LES needs a very fine grid in the flow direction to capture flow aligned streak like structures occurring in the boundary layers. Motivated by this disadvantage of LES, Spalart et al. 38 suggested an approach which attempts to combine the best features of RANS and LES. This approach is referred to as the Detached Eddy Simulation (DES). This hybrid method reduces to RANS near solid boundaries and LES away from the wall. A minor modification to the SA model presented previously does the trick. Such a model will take advantage of RANS in the thin shear layers close to the walls where RANS models are calibrated. Away from the wall in the separated regions large eddies are resolved. The model used will be the same as the SA model described in Sec. 8.2.2 except that the shortest distance to the wall, y, is modified in such a way that the model calculates a RANS eddy viscosity close to the walls and SGS eddy viscosity away from the wall. The modification is simple and given as -- min(y, CDES A)
(8.69)
A has the same meaning as discussed in the previous section. The constant CDES was calibrated for homogeneous turbulence as 0.65. It is now possible to see that close to the walls CDESA is larger than y and the model becomes a RANS model. However, away from the wall the model becomes one for calculating the SGS eddy viscosity. 26 Obviously the difficulty of where and how to fix the grid interface between the RANS and LES zones arises. Two possibilities exist to define this interface. The interface location for the differing models is either explicitly specified, or, based on length scale compatibility, allowed to naturally locate. With the latter approach the location is strongly grid controlled. When explicitly specified (based on turbulence physics grounds), to enhance results length scale smoothing is necessary. 39 Another approach currently considered is referred to as Monotonically Integrated LES (MILES). In the MILES approach numerical diffusion is used to drain turbulence. 41-42
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The direct numerical simulation method can be used to solve all turbulence length scales including the Kolmogorov length scale given by Eq. (8.1). The standard Navier-
References 271 Stokes equation without any modelling is adequate to compute all turbulence scales. The difficulty here is the prohibitively expensive computing cost required to carry out a calculation even at a very small Reynolds number. The enhanced computational overhead is mainly due to the extremely fine mesh necessary to carry out the calculation and the need for higher order accurate numerical schemes. The number of nodes necessary to resolve all scales in a three-dimensional flow problem may be written in terms of the Reynolds number as (derived based on the assumption that the Kolmogorov length scales are solved) No. Nodes
(8.70)
= R e 9/4
Similarly the time step in a calculation is limited by the Kolmogorov time scale in Eq. (8.3). The time step size calculated from the time scale is 43 At =
0.003H
(8.71)
U T ( R e T ) 1/2
where Ur is the shear velocity and g e T "~ UT H / 2 u is the turbulent Reynolds number and H is the channel height. 43 Although DNS can resolve all turbulence length scales, the currently available computing facilities allow only very small Reynolds number turbulent flow calculations. The turbulence modelling approaches discussed in the previous sections (predominantly RANS) will not be replaced for the foreseeable future.
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The topic of turbulence is currently one of the very active areas of computational fluid dynamics research. It is also one of the topics where finite element solutions require significant computing power. As mentioned in the concluding part of the previous chapter DNS calculation generally requires billions of elements which is not possible on any currently available computer. To some extent this also is true for LES calculations. Extending the DNS and LES calculations to unstructured meshes is again a research topic and much progress is needed before such calculations can be routinely carried out. This is one reason why we summarised the RANS models and discussed LES and DNS only in passing. Despite the mentioned difficulties work on many facets of LES and DNS is increasing. We are optimistic that future developments in better computing and numerical methods will help us resolve the solution of turbulence problems in full. However, we also believe that realistic three-dimensional computations should only be used in regions where resolving these effects is essential- leaving the rest to the simpler Eulerian flow modelling presented in the previous chapter.
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i!iii!ii•i•i!•iii!iiiiiiiii•iiiiiiiiiii!iii•i•i•ii••i•ii•i•iii••i••iiii•ii••i•i•i•i•i••i••i•i•••i••••ii•i••!ii•••••!••••i•i•••i••iii•i•••••i••i••••i•i•i•i•i•i•!!•!•ii•••i•!•i•
1. A.N. Kolmogorov. Local structure of turbulence in incompressible viscous fluid for very large Reynolds number. Doklady AN. SSR, 30:299-303, 1941. 2. ES. Lien and M.A. Leschziner. Assessment of turbulence transport models including non-linear RNG eddy viscosity formulation and 2ND moment closure for flow over a backward facing step. Computers and Fluids, 23:983-1004, 1994.
272 Turbulentflows 3. D. Apsley, W.L. Chen, M. Leschziner and ES. Lien. Non-linear eddy-viscosity modelling of separated flows. Journal of Hydraulic Research, 35:723-748, 1997. 4. B.E. Launder. CFD for aerodynamic turbulent flows: progress and problems. Aeronautical Journal, 104:337-345, 2000. 5. P.G. Tucker. Prediction of turbulent oscillatory flows in complex systems. International Journal for Numerical Methods in Fluids, 33:869-895, 2000. 6. P.G. Tucker. Computation of Unsteady Internal Flows. Kluwer Academic Publishers, Dordrecht, 2001. 7. K. Abe, Y.J. Jang and M.A. Leschziner. An investigation of wall-anisotropy expressions and length-scale equations for non-linear eddy-viscosity models. International Journal of Heat and Fluid Flow, 24:181-198, 2003. 8. C. Wang, Y.J. Jang and M.A. Leschziner. Modelling two- and three-dimensional separation from curved surfaces with anisotropy-resolving turbulence closures. International Journal of Heat and Fluid Flow, 25:499-512, 2004. 9. B.E. Launder and B. Spalding. Mathematical Models of Turbulence. Academic Press, New York, 1972. 10. T. Cebeci and A.M.O. Smith. Analysis of turbulent boundary layers. Academic Press, New York, 1974. 11. D.C. Wilcox. Turbulence Modelling for CFD. DCW Industries Inc., La Canada, CA, 1992. 12. M. Wolfstein. Some solutions of plane turbulent impinging jets. Transacations of the ASME Journal of Basic Engineering, 92:915-922, 1970. 13. P.R. Spalart and S.R. Allmaras. A one-equation turbulence model for aerodynamic flows. AIAA Paper 92-0439, AIAA 30th Aerospace Sciences Meeting, 1992. 14. W.P. Jones and B.E. Launder. The prediction of laminarization with a two-equation model of turbulence. International Journal of Heat and Mass Transfer, 15:301-314, 1972. 15. C.K.G. Lam and K. Bremhorst. A modified form of the ~ - c model for predicting wall turbulence. Transactions of the ASME Journal of Fluids Engineering, 103:456-460, 1981. 16. S. Fan, B. Lakshminarayana and M. Barnett. Low-Reynolds-number ~ - c model for unsteady turbulent boundary layer flows. AIAA Journal, 31:1777-1784, 1993. 17. O. Hassan, K. Morgan and J. Peraire. An implicit finite element method for high speed flows. International Journal for Numerical Methods in Engineering, 32:183-205, 1991. 18. K.A. Sorensen. A multigrid accelerated procedure for the solution of compressible fluid flows on unstructured hybrid meshes. Ph.D. thesis, School of Engineering, University of Wales, Swansea, UK, 2001. 19. E. Fares and W. Schr6der. A differential equation for approximate wall distance. International Journal for Numerical Methods in Fluids, 39:743-762, 2002. 20. P.G. Tucker. Differential equation based wall distance computation for DES and RANS. Journal of Computational Physics, 190:229-248, 2003. 21. P. Nithiarasu and C.-B. Liu. An explicit Characteristic Based Split (CBS) scheme for incompressible turbulent flows. Computer Methods in Applied Mechanics and Engineering (to appear 2005). 22. S.H. Johansson, L. Davidson and E. Olsson. Numerical simulation of vortex shedding past triangular cylinders at high Reynolds number using a ~ - c model. International Journal for Numerical Methods in Fluids, 16:859-878, 1993. 23. P.A. Durbin. Separated flow computations using the K - c - ~ model.AIAA Journal, 33:659-664, 1995. 24. G. Bosch and W. Rodi. Simulation of vortex shedding past a square cylinder. International Journal for Numerical Methods in Fluids, 28:601-616, 1998. 25. M. Tutar and A.E. Holdo. Computational modelling of flow around a circular cylinder in sub-critical flow regime with various turbulence models. International Journal for Numerical Methods in Fluids, 35:763-784, 2001.
References 273 26. G. Constantinescu, C. Matthieu and K. Squires. Turbulence modelling applied to flow over a sphere. AIAA Journal, 41:1733-1742, 2003. 27. M.K. Denham, P. Briard and M.A. Patrick. A directionally-sensitive laser anemometer for velocity measurements in highly turbulent flows. Journal of Physics E: Scientific Instruments, 8:681-683, 1975. 28. Z. Yang and T.H. Shih. New time scale based n - e model for near-wall turbulence. AIAA Journal, 31:1191-1198, 1993. 29. H. Schlichting. Boundary-Layer Theory. McGraw Hill Book Company, New York, 1968. 30. R.P. Selvam. Finite element modelling of flow around a circular cylinder using LES. Journal of Wind Engineering and Industrial Aerodynamics, 67-68:129-139, 1997. 31. M. Saghafian, P.K. Stansby, M.S. Saidi and D.D. Apsley. Simulation of turbulent flows around a circular cylinder using nonlinear eddy-viscosity modelling: steady and oscillatory ambient flows. Journal of Fluids and Structures, 17:1213-1236, 2003. 32. V. Kalro and T. Tezduyar. Parallel 3D computation of unsteady flows around a circular cylinder. Parallel Computing, 23:1235-1248, 1997. 33. D.C. Wilcox. Reassessment of the scale determining equation for advanced turbulence models. AIAA Journal, 26:1299-1310, 1988. 34. M.T. Manzari, O. Hassan, K. Morgan and N.P. Weatherill. Turbulent flow computations on 3D unstructured grids. Finite Elements in Analysis and Design, 30:353-363, 1998. 35. V. Schmitt and E Charpin. Pressure distributions on the ONERA M6 wing at transonic Mach numbers. A GARD Report AR-138, Paris, 1979. 36. J. Smagorinsky. General circulation experiments with the primitive equations, I Basic experiment. Monthly Weather Review, 91:99, 1963. 37. M. Germano. Turbulence: the filtering approach. Journal of Fluid Mechanics, 238:325, 1992. 38. P.R. Spalart, W.H. Jou, M. Strelets and S.R. Allmaras. Comments on the feasibility of LES for wings, and on a Hybrid RANS/LES approach. In C. Liu and Z. Liu, editors, Advances in DNS/LES: First AFOSR International Conference on DNS/LES. Greyden, Columbus, OH, 1997. 39. P.G. Tucker and L. Davidson. Zonal k-1 based large eddy simulations. Computers and Fluids, 33:267-287, 2004. 40. J.P. Boris, EE Grinstein, E.S. Oran and R.L. Kolbe. New insights into large eddy simulation. Fluid Dynamics Research, 10:199-228, 1992. 41. W.J. Rider and L. Margolin. From numerical analysis to implicit subgrid turbulence modelling. AIAA Paper, AIAA 2003-4101, 2003. 42. P.G. Tucker. Novel MILES computations for jet flows and noise. International Journal of Heat and Fluid Flow, 25:625-635, 2004. 43. J. Kim, P. Moin and R. Moser. Turbulence statistics in fully developed channel flow at low Reynolds numbers. Journal of Fluid Mechanics, 177:133-166, 1987.
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i!iiiiiii!i!i!iii!i!iiiiiii i!ii!!iiiiii!ii!i!iiiililili I Generalized flow through porous media ~ y~ i~~ ~ ~ i ~ : ~ : ~~ :i ~~ ~ ~
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Flow through saturated porous media has been recognized as a fluid dynamics topic which has applications in a variety of engineering fields including seepage through soil, concrete, insulating media and packed beds; flows in heat exchangers and alloy solidification; and cooling of electronic components. Several books on porous media flow and heat transfer have been published coveting both analytical and numerical solution methodologies. 1-4 It appears that the usage of porous media flow model is divided into two parts. In the first of these we generally consider materials of low porosity (Figure 9.1 (a)) and relate a priori by physical law the quantity of flow passing through all the pores in the appropriate coordinate directions. Here we find that at low velocities, as generally occur here, the relationship is linear and the quantity of the flow is related linearly to the pressure gradient. Thus, we generally write that (Darcy's law 5) nO ui = - - ~ ( p
Id OXi
+ pgkxk)
(9.1)
where ui are the average velocity components, # is the dynamic viscosity of the fluid and n is permeability expressed in m 2. The permeability may be directional and in such situations, n will be a tensor. We now concentrate on the balance of total quantities and consider a unit volume of porous medium to which we apply the mass conservation (incompressible flow) Oui/Oxi - O. Using directly the linear relationship (Eq. (9.1)) we immediately find that the following equation is obtained (ignoring the gravity effects). OX i
~ OX i
-- 0
(9.2)
The problem now becomes simply a solution to potential equation if either the pressure or its gradients are known at the boundaries and we have discussed such solutions using the finite element method in Reference 1 and also in Chapter 1 of this book. However, more recently an alternative application of porous media has been used at high porosities. We refer to this second category as high porosity model in which the porosity is large often approaching unity. Such a model is extremely useful
A generalized porous medium flow approach Fluid J
@
Solid
@
J
@ (a) Low porosity medium
@ @
@ @
@
Fluid J
@
Solid
J
@
(b) High porosity medium
Fig. 9.1 Typical examples of porous media. for some problems. For instance a network of conduits distributed throughout the fluid and kept rigidly in space by external means is an example (Figure 9.1 (b)). This collection of obstructions will exert a force on the fluid in a complex manner. Now the relation between the force and the velocity is no longer linear as in the first model but can be determined by suitable experiments. Such models are generally useful for examples such as cooling of electronic components, flow past heat exchanger pipes, etc.. At very high porosity we generally consider the equations to be almost those of Navier-Stokes to which an experimental addition of measured force values caused by the solid is added. 6-1~Various quadratic equations relating this force per unit volume to the velocity field and to the solid matrix geometry are available in the literature. Clearly such models are generally valid if the volume of the obstructions is fairly small compared to the total volume. However, in principal this model can be used at low porosities but its accuracy will be reduced and will be very expensive. Thus, the low and high porosity models can be considered as two models dealing with different phenomena. A number of examples with the first type of the model is included in Reference 1 and we shall not consider this further again. However, we shall show how the problems can be dealt with using the generalised high porosity models. In the following sections we derive the high porosity model and present some examples.
In this section, a generalized model for solving porous medium flows will be presented. Let us consider the balance of mass, momentum, and energy for two-dimensional flow in a fluid saturated porous medium of variable porosity. We shall assume the medium to be isotropic with constant physical properties, except for the medium porosity. Let af be the fraction of area available for flow per unit cross-sectional area (Fig. 9.2), at a location in a given direction. In fact, af is an averaged quantity, the average being taken over the length scale of the voids (or the length scale of the particles, if the porous bed is made up of particles), in the flow direction. For an isotropic porous bed, af will be identical in all directions and can also be equal to the local bed porosity, e. In spite of averaging over the void length scale, the fractional area af may vary from location
275
276 Generalizedflow through porous media
Fluid
Solid
@ @+ @@
z~r 2
Ir Fig. 9.2 Fluid saturated porous medium. Infinitesimal control volume.
to location on the macro-length scale 'L' of the physical problem, due to the variation of the bed porosity. The porosity, c, of the medium is defined as void ~ -- total
volume
a f Ax1Ax2
volume-
A x l AX2
(9.3)
-- a f
Now, the mass balance of an arbitrary control volume, as shown in Fig. 9.2, gives Op I
O(pyu~y)
tot
OXi
= 0
(9.4)
where the subscript ' f ' stands for fluid, p is the density and Ui are the velocity components in the xi directions respectively. The volume averaged velocity components may be defined as, 3 Ui -- 6-Uif (9.5) Equation (9.4) can be simplified for an incompressible flow (constant density) as follows OUi = 0 (9.6) OXi
Similarly, the equation for momentum balance can be derived as T
~
.qL ~
o luiu, ] -T
6- OXi ( p f e ) -'t e Ox2
Di + Pfgi
(9.7)
where #e is the equivalent viscosity, pf the fluid pressure, g the acceleration due to gravity and Dx, are the matrix drag per unit volume of the porous medium. Several experimental correlations are available in the literature for drag. One of them being the experimental data reported by Ergun. ~1 The Ergun correlation is written as Di = ~fUi Jr"
1.75 p f
lul
Ui
(9.8)
A generalized porous medium flow approach 277 where u is the velocity vector in the field. Similarly, other drag components can also be derived and the final form of the governing equations for incompressible flow through a porous medium in dimensional form can be given as Continuity
OUi
= 0
OXi
(9.9)
Momentum
v
1 ~fUi
1 0 ~e 02Ui s OXi ( p f e ) -4 e Ox 2
-
1.75 Pl lul -- 1 ~ ~ ~ s ui -- Pgi
(9.10)
If the porosity of the medium is small (c --+ 0, ~ --+ 0) the velocity values will be small and the non-linear term in Eq. (9.10) will also be small compared to the linear drag term. Thus, if the porosity approaches zero, the linear drag term (Darcy term) becomes prominent and all other terms will be negligibly small, leading to an approximation of Eq. (9.1). On the other hand if the permeability approaches infinity (c ~ 1), the incompressible flow equations are recovered. It is therefore possible to solve problems in which both single phase fluid and porous media are part of the same domain. 12 Thus applications such as alloy solidification, which involves solid, liquid and porous mushy regions, can be tackled using the proposed approach as explained by Heinrich et al. 13 The energy conservation equation is also derived in a similar manner. The final form of the energy equation is Energy
[~(pc,)~ + (1 - ~)(pc,)s] -~- + (pc,)~,,-~x ~ = k ~
(9.11)
In the above equation, Cp is the specific heat at constant pressure, T is the temperature and k is the equivalent thermal conductivity. The subscripts f and s stand for the fluid and solid phases respectively. It should be noted that the permeability and thermal conductivity values can be directional, in which case they are tensors.
9.2.1 Non-dimensional scales The following final form of the non-dimensional equations may be obtained by suitable scaling. The non-dimensional scales used here are taken based on the assumption that the energy and momentum equations are weakly coupled via local density variation (see Chapter 6). Continuity equation
Ou~ Oxp'
= 0
(9.12)
278 Generalizedflow through porous media M o m e n t u m equations lOu* + l u ,
O
(u*)
__
10
u*i ReDa
J (OZu~)
1.75 lu*l u~' 14i3-64- 3/ +
(9.13) GrT,
Energy equation
~r--~- + u* Ox* = RePr
(9.14)
OX.2
In the previous equations, the parameters goveming the flow and heat transfer are the Darcy number (Da), Reynolds number (Re), Prandtl number (Pr), Grashoff number (Gr), the ratio of heat capacities (or), porosity of the medium (e), conductivity ratio (k*) and viscosity ratio (J). The definitions for the scales and non-dimensional parameters are ,
Xi
Xi. -- -Z'
* - - Ui" ui
- - uoo
a=
t*--tu~.
L ' p*f
--
Pf
9
T*
p fuoo 2 '
T-
Too
~_(pCp)f -+- (1 -- ~)(pCp)s, k* k 9 = m ; Re= (pCp)f kf Pr
=
~v f:
.
#e
~Tw - To~ ' J
Da = -{-i 9 Gr --
#f
pfuo~L
;
(9.15)
#f
g/3ATL 3
where/3 is the coefficient of thermal expansion and superscript refers to a free stream value. The above scales are suitable for most forced and mixed convection problems. However, for buoyancy driven flows, it is convenient to handle the equations using the following definition of the Rayleigh number (Ra), i.e. Ra =
gflA T L 3
ua
(9.16)
where the following different scales need to be employed in solving natural convection problems 9 ui---~Z"t* tcef. p . pL 2 ui = a f ' = L 2' -- p f a } (9.17) The non-dimensional governing equations for natural convection are Continuity equation
= 0
(9.18)
Discretization procedure 279
Momentum equations l Ou*
l u, 0 (u_~*~)
1 0 Ox* (ep*f )
=
1.75
Pru* Da
lu*l u7
~
JPr(OZu~) Ox.~2
c3/2 f- ~ c
(9.19)
+ "yiRaPrT * Energy equation OT* OT* = k, (OET * ) cr---~- + u~ Ox----f Ox.2
(9.20)
Other alternative scales are possible and the appropriate references should be consuited to learn more about scaling. In the above formulation, the buoyancy effects are incorporated by invoking the Boussinesq approximation as discussed in Chapter 6. The kinematic viscosity u, used in the above scales, is defined as # u- P
(9.21)
and a is the thermal diffusivity, given as
kf o~f = (pCp)f
(9.22)
It may be observed that the scales and non-dimensional parameters are defined by using the fluid properties. Often, a quantity called the Darcy-Rayleigh number is used in the literature as a governing non-dimensional parameter for Darcy flow. This is the product of the Darcy (Da) and fluid Rayleigh (Ra) numbers as defined previously. For isothermal flow problems, the energy equation is not solved.
i~i,'!i,'i~ !ii~,i~/,i!ii!i'!i~!i,!'i!!i!ii,'i:~!i!!iiii,iilili!i,ii!ii!i!iiliii!!~!iii!,i'i!i!i,!iliiii,i!~,i~,i~!il,i!i!iii!!ii!iiii!i!i!i~,i~,iii:,:i!~i',i'!ii!~!iliiiii~iiiii,!:i'!~ii~ilil!il
,~' i!,i'iiii~,:i'li~!i~i~:~i~:~i~:~i~i!~~i~:i!~~:~i~i~i~i!~~i',i'~!i~i,i!~!',~ii,'~i',',i',',',~',','~:~':,i'~,~'i,~' ,~,~i~i,~:,,~:',9~,i',~
The CBS scheme will be employed to solve the porous medium flow equations. In this context the same four steps will be utilized with minor modifications, as previously discussed in Chapter 3. In the following subsections, the temporal and spatial discretization schemes are given, which will then be employed to solve the porous medium equations. Use will be made only of simple, linear triangular elements to study porous medium flow problems.
280
Generalizedflow through porous media
The momentum equation is subjected to the characteristic-Galerkin procedure, as discussed in Chapter 3, viz.,
un+l __ uin
_
10(pe) n+O__ [Uj 0
eat
--
e Oxi +
~
[
1 02ui eRe Ox 2
ln+~ [ -
( )__U 1i e
n+O'
lul Ui
Ui
ReDa + C ~
e3/2
1+o3
(9.23)
+ CG terms
The body force terms are neglected in the above equation in order to simplify the presentation. In the above equation 'CG terms' stands for higher order terms introduced by the characteristic Galerkin time discretization (viz. Chapter 3). In Eq. (9.23), the parameter 'C' is a constant equal to 1.75/~/150 [see Eq. (9.10)]. The parameters 0, 01, 02 and 03 all vary between zero and unity and with appropriate values, different schemes of interest can be established. The superscript 0 should be interpreted as fn+O __ o fn+l + (1 - O) f n (9.24) where the superscript n indicates the nth time iteration. In the CBS scheme the velocities are calculated by splitting Eq. (9.23) into two parts as below. In order to simplify the presentation, 01, 02 and 03 are assumed to be equal to zero. It is important to note, however, that such an assumption severely restricts the time step which can be employed in the calculations. The semi- and quasi-implicit schemes, as discussed in Sec. 9.3.1, are the schemes widely employed for porous media flow calculations. In Step 1, the pressure term is completely removed from Eq. (9.23) and the intermediate velocity components fii are calculated (similar to Step 1 of the standard CBS scheme discussed in Chapter 3), as
Atli
IUjcO
Ui -- Ui n
m
eat
~xj
eat
( Uwi ) ] n .qt_ [ 1 02nil n e
E 1 Ui "-[-C ReDa
eRe Ox 2
lul e3/2 b/i ] n -'[- CG
(9.25) terms
The velocities can be corrected using the following equation, which has been derived by subtracting Eq. (9.25) from Eq. (9.23), i.e. Aui cat
--_
u n+l "
"
Afii
-- ui ---eat At
10(pe) "+~
e Ox i
(9.26)
However, the value of the pressure in the above equation is not known. In order to establish the pressure field, a pressure Poisson equation can be derived from the above equation and may be written as 1 02
0u 7
c Ox 2 (Pe)"+~ -" Oxi
(9.27)
Discretization procedure The above simplified equation has been derived by substituting the equation of continuity. We have a total of three steps to obtain a solution for the momentum and continuity equations. As discussed in Chapter 3, Eq. (9.25) is solved at the first step followed by Eq. (9.27) in the second step and Eq. (9.26) in the third step. Additional steps, such as temperature or concentration calculations, can be added to the above three steps. In problems where non-isothermal and mass transfer effects are involved, then after velocity correction, additional equations will be solved. If no coupling exists between the velocities and the other variables, such as temperature and concentration and only the steady-state solution is of interest, then the steady velocity and pressure fields can be established first and the rest of the variables can be calculated using the steady-state velocity and pressure values.
9.3.1 Semi- and quasi-implicit forms ................................................................................................................................................................
-..................................................................
---:-: ................................
--: . . . . . . . - - = - - - - - :
....... ----: ......................................................................................................
-=-~ .............
Single phase incompressible fluid flow problems can be solved in a fully explicit form, which is quite popular in fluid dynamics calculations. 14 However, a solution for the generalized porous medium equations using a fully explicit form has been less successful. This is mainly due to the large values of the solid matrix drag terms, especially at small porosity. In order to eliminate some of the time step restrictions imposed by these terms, schemes other than the fully explicit forms are discussed below. In the semi-implicit (SI) form, 15 the porous medium source terms and pressure equation are treated implicitly. In other words, 0 = 03 = 1 and 01 = 02 = 0. Although this scheme has good convergence characteristics to steady state, further complications are introduced due to implicit time integration of drag terms, i.e.
U~ i -- Uni -~- 1 ~li + C ~ [ul EAt ReDa ~
Ui
-- -
e 3/2
[~
--
(Ui)] n -31- [ 1
0
V
~
~2ui]n
ERe Ox 2
(9.28)
+ CG terms
or
1
~'li - ' ~
1
lul
1 )
+ ReDa + C ~
un EAt
=
e 3/2
[uj ~xj (ui)]
+
I leRe~2Ui] (9.29)
+ CG terms
The Step 2 pressure calculation becomes 1i92
e Ox2 (Pe)n+O --
(
~
1
Step 3 is also different and is given as 1
1
1
-~- R e D a
C
lul)
- ~ e + R e D a ~ % / ~ e3/2
( -
1
1 + ReVa +
C -~ ~
lu[)o~i e 3/2 OXi
(9.30)
n+l
Ui
C lul)l(Ope) Oxi
n+~
(9.31)
281
282 Generalizedflow through porous media Although extra complications were introduced in the semi-implicit form at Step 1 for steady-state solutions, we can avoid simultaneous solution of the algebraic equations by dividing Eq. (9.29) and Eq. (9.31) by the coefficient
A -
1
1
- ~ e + R e D a -~ ~
C lul) s
(9.32)
All the standard steps which we have used in Chapter 3 can be repeated here. The quasi-implicit (QI) 16 form is very similar to that of the above scheme but now the viscous, second-order terms are also treated implicitly (02 = 1).9 The important difference, however, is that the quasi-implicit scheme does not benefit from mass lumping when solving for the intermediate velocity values. A simultaneous solution of the LHS matrices is essential here. It has been proved that both the QI and SI schemes generally perform well as shown elsewhere. ~7
i',i i 'i,i i',i !'~':i!~:i':i@,!~ii~i',i!i]!i~i~,ii~i:~:~i~i~i!i~i~ii~ii~!~i'i,~!~i~,i~'~!~i'~i ,i':~,',ili!i! ii~~,!i'~,!~i',i~i',i!'i~i'il~!,'i'~,i~if,~!i~i~',i!il!i!ii~i'~i',i',i~i',i~,i'!,~i~,ii;~ili~i!~!~,i',i~i!i,!i'~,',:i~i,i~i',~':i,~!i'i,l;];~,~',',i',i',i!i!~i'i,'!!ii~i~!J!!!!' !',i ,i i !!!ii!i 'i,~iil:ii!i!i::i!i '~i !i !i i!i i i'~!ii!i iii i i !i!i!il',i i i i i ',i i i!',i i!ili i !i!i!i iiiiiiiii ii i i i i i i i i i i i i i !i~i,ii!i i ii i i!i i i i iii i i i i i i!i i ii iiiiii i]i i i i i i i!ii i i i i!i i i i i iiii!i i':i i i i i i ili
Several examples of porous medium flow problems are non-isothermal in nature. The main focus in such a case will be to demonstrate non-isothermal flow through a porous medium. As mentioned previously, an energy equation needs to be solved in addition to the momentum and pressure equations if the flow is non-isothermal. For steadystate problems, if no coupling exists between the momentum and energy equation, the temperature field can be established after calculation of the velocity fields. The temporal discretization of the energy equation can be written in a similar form to the momentum equation and is given as
cr
where
01
At
and
02
=
--
Ui
+ RePr
+ C G terms
(9.33)
have the same meaning as previously discussed in Sec. 9.3.1.
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Flow through packed beds is important in many chemical engineering applications. Generally, the grain size in the packed beds will vary depending on the application. As the particle size increases, the packing close to the walls will become non-uniform thereby creating a channelling effect close to the solid walls. In such cases, the porosity value can be close to unity near the walls but will decrease to a free stream value away from the walls. In such situations, the ability to vary the porosity within the domain itself is essential in order to obtain a correct solution. Although the theoretical determination of the near wall porosity variation is difficult, there are some experimental correlations available
Forced convection 283
p=O
=u2= T=I
60
'
Parabolic inlet profile for u 1 and u2 = 0
r= 0
Fig. 9.3 Forced convection in a channel filled with a variable porosity medium. Geometry and boundary conditions. to tackle this issue. One such widely employed correlation, given by Benenati and Brosilow,18 will be used, i.e. s
= s
E
1 + exp
-
(9.34)
where s is the free stream bed porosity taken to be equal to 0.39, and c is an empirical constant (c = 2 for dp = 5 mm). In general, the problem in this case is formulated based on the particle size dp, i.e. the Reynolds number is based on the particle size.
Example 9.1 Forced convection heat transfer in a packed-bed channel
Figure 9.3 shows the problem definition of forced flow through a packed bed. The inlet channel width is 10 times the size of the grain. The length of the channel is 10 times that of the inlet width. Zero pressure conditions are assumed at the exit. The inlet velocity profile is parabolic and no-slip boundary conditions apply on the solid side walls. Both the walls are assumed to be at a higher, uniform temperature than that of the inlet fluid temperature. The analysis is carried out for different particle Reynolds numbers ranging from 150 to 350. The quasi-implicit (QI) scheme with 0 = 1, 01 = 0 and 02 = 03 = 1 has been employed to solve this problem. A non-uniform mesh with triangular elements was also used in the analysis. The mesh is fine close to the walls and coarse towards the centre. The total number of nodes and elements used in the calculation are 3003 and 5776 respectively. Figure 9.4 shows a comparison of the calculated steady-state average Nusselt number distribution on a hot wall with the available experimental and numerical data. The Nusselt number is calculated using a non-dimensional temperature distribution as
hL ~o"LOT Nu - --if- ~xl
dXl
(9.35)
284
Generalized flow through porous media 245
,.. 2 2 0
(D JE)
E 195
ti
n
(~
170
(~
z
145 120
100
,T,,i
125
....
w....
150
i ....
175
I ....
200
I ....
225
Reynolds
i ....
i ....
250
275
i ....
300
i ....
325
350
number
Fig. 9.4 Forced convection in a channel. Comparison of Nusselt number with experimental data for different particle Reynolds numbers. Points - experimental [19]; dashed line - numerical [19]; solid - CBS.
2.5 ,
(D 0
:\\
w
"~ >
,
2
Forcheimmer
";..~..
tl:i t'-
....
Generalized .9
.....
Brinkman
.........
._o (D 1 . 5 (-.
E
"? eo Z
1 'd 0.5
I
0
0.5
I
I
I
1
1.5
2
Non-dimensional
I
2.5
3
length
Fig. 9.5 Forced convection in a channel. Comparison between the generalized model, Forcheimmer and Brinkman extensions to Darcy's law.
Figure 9.5 shows the difference between the generalized model and the Brinkman 2~ and Forcheimmer 21 extensions for the velocity profiles close to the wall in a variable porosity medium at steady state. As may be seen the Forcheimmer and Brinkman extensions fail to predict the channelling effect close to the wall. Whilst the Brinkman extension is insensitive to porosity values, the Forcheimmer model does not predict the viscous effect close to the channel walls.
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The fluid flow in a variable porosity medium within an enclosed cavity under the influence of buoyancy is another interesting and difficult problem to analyse. In order to study such a problem, an enclosure packed with a fluid saturated porous medium is considered in the following example.
Natural convection 285
Example 9.2 enclosure
Buoyancy driven convection in a particle packed
The aspect ratio of the enclosure is 10 (ratio between height and width). All the enclosure walls are subjected to 'no-slip' boundary conditions. The left vertical wall is assumed to be at a higher, uniform temperature than that of the fight side wall. Both the horizontal walls are assumed to be insulated (Fig. 9.6). The properties of the saturating fluid are assumed to be constant other than that of the density. The density variation is invoked by the Boussinesq approximation. Table 9.1 shows the steady-state quantitative results and a comparison with the available numerical and experimental data. These data were obtained on a non-uniform structured 61 x 61 mesh. The accuracy of the prediction can be improved by further refinement of the mesh. An extremely fine mesh is essential near the cavity walls in order to predict the channelling effect in this region. In Table 9.1, experimental data is obtained from reference 22 and the numerical data for comparison is obtained from reference 23. The following Nusselt number relation was used for this problem
1 foLOT Nu - -L ~ dx
(9.36)
Here T is the non-dimensional temperature and L is the non-dimensional length.
9.6.1 Constant porosity medium Problems where the variation in porosity is of less significance normally occur in porous media, which have small solid particle sizes. For instance, thermal insulation is one such example where the variation in porosity near the solid walls is not important but the uniform free stream porosity value can be very high. In order to investigate such
ul =u2=O
ul = u2 = 0
u 1 = u2 = 0
T=I
T=O
10
ul=u2=O
Fig. 9.6 Natural convection in a fluid saturated variable porosity medium. Problem boundary conditions.
286 Generalizedflow through porous media Table 9.1 Averagehot wall Nusseltnumberdistributionfor natural convectionin a variableporositymedium,aspect ratio = 10 Fluid dp (-e Pr k* Ra Experimental Numerical CBS
Water
5.7
0.39
7.1
Ethyl alcohol
5.7
0.39
2.335
1.929 15.4
1.830x 3.519 x 2.270 x 3.121 x
107 107 108 108
2.595 3.707 12.56 15.13
2.405 3.496 13.08 15.57
2.684 3.892 12.17 14.28
media, a benchmark problem involving buoyancy driven convection in a square cavity has been solved.
Example 9.3 Buoyancy driven flow in a cavity filled with fluid saturated constant porosity medium
The problem definition is similar to the one shown in Fig. 9.6, the difference being that the aspect ratio is unity here. The square enclosure is filled with a fluid saturated porous medium with constant and uniform properties except for the density, which is again incorporated via the Boussinessq approximation. A 51 • 51 non-uniform mesh (Fig. 9.7) is employed for this problem. The Darcy and non-Darcy flow regime classifications and the Darcy number limits have been discussed by many researchers. One important suggestion was given in the paper by Tong and Subramanian. 24 In Fig. 9.8, we show the velocity and temperature distribution at different Darcy and Rayleigh numbers. In this case the product of the Darcy and Rayleigh numbers is kept at a constant value in order to amplify the nonDarcy effects. It is clearly obvious that the maximum velocity in the Darcy flow regime,
Fig. 9.7 Buoyancydriven flow in a fluid saturated porous medium. Finite element mesh. Nodes" 2601, elements: 5000.
Natural convection
ttttitt, , , . . . . . . . . --
llltiit~
.
.
.
.
.
.
tllil~ IIIi:"' ...........
.
.
.
.
.
.
o
.
: : ::::::',iii',;~
.
#
: :
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tt~
.
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i
~
~~_~i --_=-_--i =_= =__=_z_~_----;iii;iiiiiiiill~""'ii| --:-_~i i _~ ~ ~ ~ ~ i'llii.-'iiiiii (a) Vector plot
-
I
-
(b) Temperature
R a = 10 8, D a = 10 -6
.,,,,:: .................. 9iii;::::iiii i i i i i i i iiii:',:iiiiii~ ,,,, tttttt,,
.
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(c) Vector plot
(d) Temperature R a = 10 6, D a = 10 -~
)
,,~'..w..-'..-'--'--'.'.- "--'.- ; ~ ; '=" t ,.. ,._ ~ ,._ ! ,=. !!!,=_'.=.,V=~!~ ~
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fpplpl,,~
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il
9 9149
iiiiiiiiiiiiiiiii;i i i i i i i i iiiiiiiiiiiiiiiiii (f) Temperature
(e) Vector plot R a = 10 4, D a = 10 -2
Fig. 9.8 Natural convection in a fluid saturated porous, square enclosure. Vector plots and temperature contours for different Rayleigh and Darcy numbers, Pr = 0.71.
287
288 Generalizedflow through porous media 9.2 Average Nusselt number comparison with analytical and numerical results.
Table
Nu Ra* = R a D a
Analytical
Numericall
10 50 100 500 1000
-
1.07
1.98 3.09 8.40 12.49
3.09 13.41
Numerical2 -
2.02 3.27 18.38
CBS 1.08 1.96 3.02 8.38 12.52
at a Darcy number of 10 -6, is located very close to the solid walls. The non-Darcy velocity profile, at a Darcy number of 10 -2, on the other hand looks very similar to that of a single phase fluid and the maximum velocity is located away from the solid walls. At a Darcy number of 10 -4 the flow undergoes a transition from a Darcy flow regime to a non-Darcy flow regime. The temperature contours also undergo noticeable changes as the Darcy number increases from 10 -6 to 10 -2. Both the scheme and the model implementation have been designed in such a way that as the Darcy number increases, the flow approaches a single phase fluid flow, which is evident from Fig. 9.8. In Table 9.2, the quantitative results obtained from the above analysis (only for the Darcy flow regime, Da < 10 -5) are compared with other available analytical and numerical results. As seen, the results are in excellent agreement with the reported results. In Table 9.2, analytical solution has been obtained from reference 25, 'Numericall' and 'Numerical2' have been obtained from references 26 and 27 respectively. It should be noted that results by Walker and Homsy 25 are analytical. The numerical results presented by Trevisan and Bejan 27 overpredict the results, which may be due to the coarse mesh employed.
Example 9.4 Buoyancy driven convection in an axisymmetric enclosure filled with fluid saturated constant porosity medium
In order to compare the present numerical results with experimental data, an axisymmetric model was developed and a buoyancy driven flow problem was studied. The boundary and initial conditions are the same as for the previous problem. The main difference being in the definition of the geometry. In this case, the geometry is an annulus with a radius ratio (ratio between outer and inner radii) of 5.338 (see Fig. 9.9). The fluid used to saturate the medium is water with a Prandtl number of 5. The results are generated for different Grashof numbers (Ra/Pr) and compared with the experimental Nusselt number predictions as shown in Fig. 9.10. In general the agreement is excellent for the range of Grashof numbers considered. iiiiiiiiiii9iiiiiiiiiiii~ iiiii:ii iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii ii ~ Oii~iiiim l iiiiiii Uiiiill diiiiiiiiiili!iiiiil i hiiiiiiiii g iiiiiiiiiiiiiiii!i!iiii~ iiiiiiliiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii r~m~Fk~ ~ ~ I ~ I ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ !
This chapter presents a brief summary of formulations to compute flow through porous media with high porosities. It is important to fully understand the concepts given
References
Insulated
"-0
1 Fig. 9.9 Natural convection in a fluid saturated constant porosity medium. Problem definition.
i,_
~D
E tin
el0t~
Z ll) r t__
.
_
i I
"00
0 0
Present, Pr = 5.0 (x:x3cx3 Exp., 3 mm Exp., 6 mm
> <
1 10
i
i
i
I
i
i
i I
i
i
i
i
i
Darcy-Grashof number
i
o'oo i
Fig. 9.10 Natural convection in a fluid saturated constant porosity medium within an annular enclosure. Comparison of hot wall steady-state Nusselt number with the experimental and numerical data. 28
in Chapter 3 before carrying out porous media flow calculations using this model. Though brief we have covered all essential features of finite element modelling of flow through fluid saturated porous media in this chapter.
iiiiiiiiii!iiiiiiiiiiii~iiiiiii~iiii!~iiii~!!~!~!~!~ii!iiiiiiiiiiiiiiii~ii~iiiiii~iiiiiiiiiiiiiiiiii!iiii~iiiiiiiiiiiii!ii!!!~i~i~i~i~iiiiii~i~i~i~i~i~i~i~iiiiii!iiiiiiiiiiii!i~iiiii!i!~iiiiiiii~iiii~iiiiii~i~iiii!iiii!i
1. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method. Its Basis and Fundamentals. Elsevier, 2005. 2. R.W. Lewis and B.A. Schrefler. The Finite Element Method in the Deformation and Consolidation of Porous Media. John Wiley & Sons, Chichester, 1998.
289
290
Generalized flow through porous media 3. 4. 5. 6. 7. 8. 9.
10.
11. 12.
13. 14.
15.
16.
17.
18. 19. 20. 21. 22.
23. 24.
25.
D.A. Nield and A. Bejan. Convection in Porous Media. Springer-Verlag, New York, 1992. M. Kaviany. Principles of Heat Transfer in Porous Media. Springer-Vedag, New York, 1991. H. Darcy. Les Fontaines Publiques de la Ville de Dijon. Dalmont, Paris, 1856. S. Whitaker. Diffusion and dispersion in porous media. American Institute of Chemical Engineering Journal, 13:420-427, 1961. K. Vafai and C.L. Tien. Boundary and inertia effects on flow and heat transfer in porous media. International Journal of Heat Mass Transfer, 24:195-203, 1981. C.T. Hsu and P. Cheng. Thermal dispersion in a porous medium. International Journal of Heat and Mass Transfer, 33:1587-1597, 1990. P. Nithiarasu, K.N. Seetharamu and T. Sundararajan. Natural convective heat transfer in an enclosure filled with fluid saturated variable porosity medium. International Journal of Heat and Mass Transfer, 40:3955-3967, 1997. P. Nithiarasu, K.N. Seetharamu and T. Sundararajan. Finite element modelling of flow, heat and mass transfer in fluid saturated porous media. Archives of Computational Methods in Engineering, State of the Art Reviews, 9:3-42, 2002. S. Ergun. Fluid flow through packed column. Chemical Engineering Progress, 48:89-94, 1952. N. Massarotti, P. Nithiarasu and O.C. Zienkiewicz. Porous medium- fluid interface problems. The finite element analysis by using the CBS procedure. International Journal for Numerical Methods in Heat Fluid Flow, 11:473-490, 2001. J.C. Heinrich, S. Fellicelli and D.R. Poirier. Vertical solidification of dendritic binary alloys. Computer Methods in Applied Mechanics and Engineering, 89:435-461, 1991. P. Nithiarasu. An efficient artificial compressibility (AC) scheme based on the Characteristic Based Split (CBS) method for incompressible flows. International Journal for Numerical Methods in Engineering, 56:1815-1845, 2003. P. Nithiarasu and K. Ravindran. A new semi-implicit time stepping procedure for buoyancy driven flow in a fluid saturated porous medium. Computer Methods in Applied Mechanics and Engineering, 165:147-154, 1998. P. Nithiarasu, K.N. Seetharamu and T. Sundararajan. Double-diffusive natural convection in an enclosure filled with fluid-saturated porous medium: a generalized non-Darcy approach. Numerical Heat Transfer, Part A - Applications, 30:413-426, 1996. P. Nithiarasu. A comparative study on the performance of two time stepping schemes for convection in a fluid saturated porous medium. International Journal of Numerical Methods for Heat and Fluid Flow, 11:308-328, 2001. R.E Berenati and C.B. Brosilow. Void fraction distribution in packed beds. AIChE Journal, 8:359-361, 1962. K. Vafai, R.L. Alkire and C.L. Tien. An experimental investigation of heat transfer in variable porosity media. ASME J. Heat Transfer, 107:642-647, 1984. H.C. Brinkman. A calculation of viscous force exerted by a flowing fluid on a dense swarm of particles. Applied Science Research, 1:27-34, 1947. P. Forcheimmer. Wasserbewegung durch bodem. Z. Ver. Deutsch. Ing., 45:1782, 1901. H. Inaba and N. Seki. An experimental study of transient heat transfer characteristics in a porous layer enclosed between two opposing vertical surfaces with different temperatures. International Journal of Heat and Mass Transfer, 24:1854-1857, 1981. E. David, G. Lauriat and P. Cheng. A numerical solution of variable porosity effects on natural convection in a packed sphere cavity. ASME J. Heat Transfer, 113:391-399, 1991. T.W. Tong and E. Subramanian. A boundary layer analysis for natural convection in vertical porous enclosures- use of Brinkmann- extended Darcy model. International Journal of Heat and Mass Transfer, 28:563-571, 1985. K.L. Walker and G.M. Homsy. Convection in a porous cavity. Journal of Fluid Mechanics, 87:449-474, 1978.
References 291 26. G. Lauriat and V. Prasad. Non-Darcian effects on natural convection in a vertical porous enclosure. International Journal of Heat Mass Transfer, 32:2135-2148, 1989. 27. O.V. Trevisan and A. Bejan. Natural convection with combined heat and mass transfer buoyancy effects in porous medium. International Journal of Heat and Mass Transfer, 28:1597-1611, 1985. 28. V. Prasad, F.A. Kulacki and M. Keyhani. Natural convection in porous media. Journal of Fluid Mechanics, 150:80, 1985.
i i i i i i i i i i i i i i !i i !i i i i!i i i i i i i i i i
Shallow water problems*
The flow of water in shallow layers such as occur in coastal estuaries, oceans, rivers, etc., is of obvious practical importance. The prediction of tidal currents and elevations is vital for navigation and for the determination of pollutant dispersal which, unfortunately, is still frequently deposited there. The transport of sediments associated with such flows is yet another field of interest. If the free surface flow is confined to relatively thin layers the horizontal velocities are of primary importance and the problem can be reasonably approximated in two dimensions. Here we find that the resulting equations, which include in addition to the horizontal velocities the free surface elevation, can once again be written in the same conservation form as the Euler equations studied in previous chapters [viz. Eq. (7.1)]. Indeed, the detailed form of these equations bears a striking similarity to those of compressible gas flow- despite the fact that now a purely incompressible fluid (water) is considered. It follows therefore that: 1. The methods developed in the previous chapters are in general applicable. 2. The type of phenomena (e.g. shocks, etc.) which we have encountered in compressible gas flows can occur again. It will of course be found that practical interest focuses on different aspects. The objective of this chapter is therefore to introduce the basis of the derivation of the equation and to illustrate the numerical approximation techniques by a series of examples. The approximations made in the formulation of the flow in shallow water bodies are similar in essence to those describing the flow of air in the earth's environment and hence are widely used in meteorology. Here the vital subject of weather prediction involves their daily solution and a very large amount of computation. The interested reader will find much of the background in standard texts dealing with the subject, e.g. references 1 and 2. A particular area of interest occurs in the linearized version of the shallow water equations which, in periodic response, are similar to those describing acoustic phenomena. In the next chapter we shall therefore discuss some of these periodic phenomena involved in the action and forces due to waves. 3
*Contributed mainly by E Ortiz, Associate Professor, University of Granada, Spain.
The basis of the shallow water equations
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In previous chapters we have introduced the essential Navier-Stokes equations and presented their incompressible, isothermal form, which we repeat below assuming full incompressibility. We now have the equations of mass conservation:
Oui OXi --0
(lO.la)
and momentum conservation:
0
lop
Olgj ~- (UiUj)~ -"~- ~ D Oxj
10
p Oxi
Tij -- gi -- 0
(10.1b)
with i, j being 1, 2, 3. In the case of shallow water flow which we illustrate in Fig. 10.1 and where the direction x3 is vertical, the vertical velocity u3 is small and the corresponding accelerations are negligible. The momentum equation in the vertical direction can therefore be reduced to
10p
- --
p Ox3
+ g = 0
(10.2)
where g3 = - g is the gravity acceleration. After integration this yields
p = p g ( ~ - x3) + Pa
(10.3)
as, when x3 = r/, the pressure is atmospheric (Pa) (which may on occasion not be constant over the body of the water and can thus influence its motion). On the free surface the vertical velocity u3 can of course be related to the total time derivative of the surface elevation as discussed in Chapter 6. With reference to Fig. 10.1 we can directly derive the depth averaged conservation of mass equation by considering an infinitesimal control volume in the horizontal plane as
Oh
O(h~ti)
Ot
OXi
=0
(10.4)
where Ui is the depth averaged velocity field. We can immediately see the similarity with the conservation of mass equation for compressible gas flows with h replacing density p here. It should be noted that Eq. (10.4) can also be derived by integrating Eq. (10.1a) between - H and ~7(see Fig. 10.1). 4 Now we shall perform depth integration on the momentum equations in the horizontal directions, which results in depth averaged momentum equations. Integrating Eq. (10.1 b) between the top and bottom surfaces, i.e.
f r lO[ u --~ iOH lOplOrij] -'[- ~Xj (uiuj) "-["p OXi P OXi
fi
dx3 - - 0
(10.5)
with i = 1, 2 and j~ in the above equations are Coriolis accelerations. In order to integrate the above equation, we need to use the following integration formula
fb J,
o
o fb Ob Oa (r, s) dr - -~s F (r, s) dr - F (b, s ) - ~ + F (a, s)--~ us
us
(10.6)
293
294 Shallow water problems k
x3
x s wind dra la, g
x2
Free surface P - Pa -
A
~ ~
~~ ~
(atmosphere)
Mean water level
(datum)~ "
171h3,bed 'friction' (a) Coordinates
A~, u~
I
./I
Average velocity (b) Velocity distribution
Fig. 10.1 The shallow water problem. Notation.
A simple example of using the above formula is (p constant)
H pOx----~i
~Xi
]
n P
,7 ~Xi
+
(o.) -H
---~Xi
(10.7)
After integration the depth averaged momentum equation (10.5) becomes
Oh u i Oh ~i u j - gh Orl 1 Ot ~ Oxj -~xi + _( ~i _ ~ai) + h f i p
h Opa pOxi
1 0
~-p-~Xj
f H
7"ij dx3
(10.8) Where superscripts s and b respectively indicate top and bottom (seafloor) surfaces. In Eq. 10.8 the shear stresses on the surface can be prescribed externally, given, say, the wind drag. The bottom shear is frequently expressed by suitable hydraulic resistance formulae, e.g. the Ch6zy expression, giving 7~3i _ p g ]El ]fi i Ch 2
(10.9)
where I~1 - ~ u i ;
i -- 1, 2
and C is the Ch6zy coefficient. The Coriolis accelerations, fi, in Eq. (10.8) are important in large-scale problems and defined as fl -- J~2
f 2 - -j~u,
(10.10)
The basis of the shallow water equations where j~ is the Coriolis parameter. At this stage we omit the stresses acting within the fluid and simply consider the surface and base drag which can be evaluated independently. The addition of shear stresses on vertical faces of the slice can be included but are neglected in this book. Thus, the simplified momentum equation without the fluid stress terms is
Ohii 0t
+
Ohuiuj
Oxj
h Opa
017 1 -- -gh-"~'m "21--(7"s3i - 7~3i) .ql_hfi uxi p
p OXi
(10.11)
If we compare the above equation with the Euler equations for compressible gas flows we find the equivalent pressure term in Eq. (10.11) appears in its non-conservation form. Also, an additional variable r/is introduced. In order to write the above equation in conservation form and to eliminate the additional variable ~7, let us consider the following alternative form of the equivalent pressure term with ~7= h - H.
_ghO0
O(h2-H g 2
OX---~--"
OXi
2)
OH
+ g(h - H) Ox---~
(10.12)
Substituting the above alternative form into Eq. (10.11) gives
O(hii) --y-+
0
(hCtii ) J
] 1
OXi -~g(h 2 - n 2) -+- ~(T~i- 7-b)
-
-
Opa
OH
h
OXi
P OXi
+hfi + g(h - H ) - -
(10.13)
Now the above equation is in a form identical to that ofinviscid compressible momentum equations with h replacing p. The first three terms in the above equations represent transient, convection and equivalent pressure terms. All the remaining four terms can be assumed to be source terms. Equations (10.4) and (10.13) form the shallow water problem. These equations may be rewritten in a compact form as
0(I)
OFi
-~- + ~x~ + Q -- 0 where
--
{h)
(10.14)
(10.15a)
hill hi2
hlli
Fi --
2 - H 2) hfi 1ii 71 (~li ~g(h 1 2 - H 2) hi2ii -t-"(~2i-~g(h 1
(10.15b)
295
296
Shallow water problems and
OH h - h ) ~ , ~ - g(h - I - I ) - ~ +
Opa
p OXl
O ~.
OH h Opa h ~ r , 1 - g ~h - I-l ) ~ . -~
P OX2
ox2
1 p
1 P
"~1 +
~-~ +
g~llfil Ch 2
(10.15c)
gfi21fil Ch 2
with i = 1, 2. The above conservative form of shallow water equations was first presented in references 4 and 5 and is generally applicable. However, many variants of the general shallow water equations exist in the literature, introducing various approximations. In the following sections of this chapter we shall discuss time stepping solutions of the full set of the above equations in transient situations and in corresponding steadystate applications. If we deal with the linearized form of Eqs (10.4) and (10.13), we see immediately that on omission of all non-linear terms, bottom drag, etc., and by letting hui ~ H u i etc., we can write these equations as
Oh O( H fi i )
0t
0 oxi
-I" -z--(HFti) = 0 oq
+ gH-z--(h - H) = 0 oxi
Noting that
rl=h-H
Oh
and
(lO.16a) (10.16b)
O,7
- ~ = 0t
the above becomes
0
077- i" ~ ~ O(Hui)
0t
(HFti) -- 0 Or]
+gH-z--- = 0 oxi
(10.17a) (10.17b)
Elimination of HFti in Eq. (10.17a) by substituting Eq. (10.17b) immediately yields Ot 2
OXi gH~xi
-- 0
(10.18)
or the standard Helmholtz wave equation. For this case, many special solutions are analysed in the next chapter. The shallow water equations derived in this section consider only the depth averaged flows and hence cannot reproduce certain phenomena that occur in nature and in which some velocity variation with depth has to be allowed for. In many such problems the basic assumption of a vertically hydrostatic pressure distribution is still valid and a form of shallow water behaviour can be assumed.
Numerical approximation 297 The extension of the formulation can be achieved by an a priori division of the flow into strata in each of which different velocities occur. The final set of discretized equations consists then of several, coupled, two-dimensional approximations. Alternatively, the same effect can be introduced by using several different velocity 'trial functions' for the vertical distribution, as was suggested by Zienkiewicz and Heinrich. 6 Such generalizations are useful but outside the scope of the present chapter.
iiHii i i3iiiHiHi
i
iiiii!iiiiHiiiii!iiiii!iiii iil !iiiiili iiiiiiiiiHiiiHiHi iiHi!!iiiiiii!Hiil iii
Both finite difference and finite element procedures have for many years been used widely in solving the shallow water equations. The latter approximation has been applied in the 1980s and Kawahara7 and Navon 8 survey the early applications to coastal and oceanographic engineering. In most of these the standard procedures of spatial discretization followed by suitable time stepping schemes are adopted. 9-16 In meteorology the first application of the finite element method dates back to 1972, as reported in the survey given in reference 17 and the range of applications has been increasing steadily.4, 5,18-51 At this stage the reader may well observe that with the exception of source terms, the isothermal compressible flow equations can be transformed into the depth integrated shallow water equations with the variables being changed as follows: p (density) --+ h (depth)
ui (velocity)--+ ui (mean velocity) 1
p (pressure) --+ -~g(h 2 - H 2) These similarities suggest that the Characteristic-Based-Split (CBS) algorithm adopted in the previous chapters for compressible flows be used for the shallow water equations. 52 By using the Cartesian system and notation of Fig. 10.1 the shallow water equations can now be rewritten in a convenient form for the general CBS formulation developed in Chapter 3 as:
Oh OUi --~ + Oxi
oui
=
0
o(-aj ui ) + ~0 [1-~g(h 2 - H 2)]
OXj
-k-Qi = 0
where Ui = hEli. The rest of the variables are the same as described before. The extension of effective finite element solutions of high-speed flows to shallow water problems has already been successful in the case of the Taylor-Galerkin method, a, 5, 48 However, the semi-implicit form of the general CBS formulation provides a critical time step dependent only on the current velocity of the flow U (for pure convection): d At < (10.19)
-IUI
298 Shallow water problems
where d is the element size, instead of a critical time step in terms of the wave celerity
C---~"
At < -
(10.20) c -4-IUI
which places a severe constraint on fully explicit methods such as the Taylor-Galerkin approximation and others, 4' 5, 32 particularly for the analysis of long-wave propagation in shallow waters and in general for low Froude number problems. Important savings in computation can be reached in these situations obtaining for some practical cases up to 20 times the critical (explicit) time step, without seriously affecting the accuracy of the results. When nearly critical to supercritical flows must be studied, the fully explicit form is recovered, and the results observed for these cases are also excellent. 53' 54 The three essential steps of the CBS scheme may be written in its semi-discrete form as
Step 1" AU[ - A t
[ 0
---~xj(~ljUi)-
At
0 ( ~ x j ) l
Qi + -~-u~-~x k
(bljUi) + Qi
n
(10.21)
Step 2: Ah - - A t
~xi
+ O10x------~t
OxiOx------~i-~- 02 OxiOx i
Step 3: mUi --
U?+1- U? -- AU/*- A
Op n+02 t ~
Oxi
At 2 c~2pn ~ f i k ~
2
OxkOxi
(10.23)
with p = 1/2g(h 2 H2). In the above equations, 0.5 _< 01 ~< 1.0 and 02 - - 0 for the explicit scheme and 0.5 _< 01 < 1.0 and 0.5 _< 02 _< 1.0 for the semi-implicit scheme. For further details on the time and spatial discretizations, the reader is referred to Chapter 3. In the examples that follow we shall illustrate several problems solved by the CBS procedure, and also with the Taylor-Galerkin method described in Appendix E. -
10.4.1 Transient one-dimensional problems - a performance assessment In this section we present some relatively simple examples in one space dimension to illustrate the applicability of the algorithms.
Examples of application
Wave P.fopagation / ~
- j
L
40
!
'i'"i"/-~~l
rl = a sech 2 1/2 (3a) 1~2(x---~-1)
I'
_[..o., ~/'~//.-, x=0
Initial
u = - ( I + ~/ea)rll(ax + q), a = 0.1, g = 1.0, a = 1/30~-----e~176
(a) Problem statement 40 elements
80 elements
160 elements
(b) Solution for 40, 80 and 160 elements at various times Fig. 10.2 Shoalingof a wave.
Example 10.1 Sofitary wave
The first, illustrated in Fig. 10.2, shows the progress of a solitary wave 55 onto a shelving b e a c h . T h i s f r e q u e n t l y s t u d i e d s i t u a t i o n 56' 57 s h o w s w e l l the p r o g r e s s i v e s t e e p e n i n g o f the w a v e o f t e n o b s c u r e d b y s c h e m e s that are v e r y dissipative.
ExamplelO.2
Dambreak
The second example of Fig. 10.3 illustrates the so-called 'dam break' problem diagrammatically. Here a dam separating two stationary water levels is suddenly removed
299
300
Shallow w a t e r problems
. . . . . . . . . . . .
40 elements in L
"',,\ .... \ \ 9 ''".
rli~,
h
\ 9
t=25
-....x.......,,~.l..-.,,,,
I
| c>
~ .......
"
H=l
t=7.s~ -
"
'-
, \
"~//////////// I
,,.,,:', ...'-,...J
'
"
t=5.0
~--t=7.5
f
'
~
,
.
9
9
**
.-
-
#
/
.=H=l
\ 7'
t-O
9
$ 9 IL
_
~, "1
. |
-
Fig. 10.3 /
Propagation of waves due to dam break (CLap = 0). Forty elements in analysis domain. C = ~ / q h - - 1, At=0.25. u
and the almost vertical waves progress into the two domains. This problem, somewhat similar to those of a shock tube in compressible flow, has been solved quite successfully even without artificial diffusivity.
ExamplelO.3
Bore
The final example of this section, Fig. 10.4, shows the formation of an idealized 'bore' or a steep wave progressing into a channel carrying water at a uniform speed caused by a gradual increase of the downstream water level. Despite the fact that the flow speed is 'subcritical' (i.e. velocity < ~/-~), a progressively steepening, travelling shock clearly develops. 10.4.2
Two-dimensional
periodic tidal motions
The extension of the computation into two space dimensions follows the same pattern as that described in compressible formulations. Again linear triangles are used to interpolate the values of h, U1 and/-/2. The main difference in the solutions is that of emphasis.
Example 10.4 Periodic wave
The first example of Fig. 10.5 is presented merely as a test problem with periodic surface elevation at the inlet. Here the frictional resistance is linearized and an exact solution
Examples of application 40 elements 2 11 Prescribed
water level
-1o
history
1.0
uI
0.5-
v
-0.5 Distance Fig. 10.4 A 'bore' created in a stream due to water level rise downstream (A). Level at A, r/= 1 - cos 7r t130 (0 < t < 30), 2 (30 < t). Levels and velocities at intervals of five time units, A t -- 0.5.
known for a periodic response 58 is used for comparison. This periodic response is obtained numerically by performing some five cycles with the input boundary conditions. Although the problem is essentially one dimensional, a two-dimensional uniform mesh was used and the agreement with analytical results is found to be quite remarkable.
Example 10.5 Bristol channel
In the second example we enter the domain of more realistic applications. 4, 5,52-54,59 Here the 'test bed' is provided by the Bristol Channel and the Severn Estuary, known for some of the highest tidal motions in the world. Figure 10.6 shows the location and the scale of the problem. The objective is here to determine tidal elevations and currents currently existing (as a possible preliminary to a subsequent study of the influence of a barrage which some day may be built to harness the tidal energy). Before commencement of the analysis the extent of the analysis domain must be determined by an arbitrary, seaward, boundary. On this the measured tidal heights will be imposed. This height-prescribed boundary condition is not globally conservative and can also produce undesired reflections. These effects sometimes lead to considerable errors in the calculations, particularly if long-term computations are to be carded out (like, for instance, in some pollutant dispersion analysis). For these cases, more general open
301
302
Shallow w a t e r problems
I'(.
AIOI
.................................
!
E ,2,o
~-
~
=
A
t=TI2
~>5
v
Analytical y = 0, :12 Ay y=O,•
TI40
Computed At =
8 0
I
I
I
~
I
I ~
x
L
I0~'---9 Analytical
,-.
A v
t=3TI4
5
y = 0, .+.2 Ay } y = 0, +_Ay
Computed At= TI40
Z
O v
Z"
~
~0
-5
i
i
I
I
:~
I
I
i
~
I
I
~-'~
I
x= L
R
/////
///
//.
/ ///
//
L = 22~Lx
Fig. 10.5 Steady-state oscillation in a rectangular channel due to periodic forcing of surface elevation at an inlet. Linear frictional dissipation? 2
Examples of application .i
E
Stockpole Quay
om m
t-l,_
Beachley w
Swansea
Newp~, j ~
Port Talbot
Worms Head
Porthcawl
E
Ilfracombe
E
/1" Clevedon
Barry Lynmouth
,,.,= ...,,,
Cardiff
,'
Weston-Super-Mare
Minehead Watchet Hinkley Pt w
Q)
I
0
I
I
I
I
I
50 km
I
Fig. 10.6 Locationmap. BristolChanneland SevernEstuary.
boundary conditions can be applied, as, for example, those described in references 35 and 36. The analysis was carried out on four meshes of linear triangles shown in Fig. 10.7. These meshes encompass two positions of the external boundary and it was found that the differences in the results obtained by four separate analyses were insignificant. The mesh sizes ranged from 2 to 5 km in minimum size for the fine and coarse subdivisions. The average depth is approximately 50 m but of course full bathygraphy information was used with depths assigned to each nodal point. The numerical study of the Bristol Channel was completed by a comparison of performance between the explicit and semi-explicit algorithms. 53 The results for the coarse mesh were compared with measurements obtained by the Institute of Oceanographic Science (lOS) for the M2 tide, 59 with time steps corresponding to the critical (explicit) time step (50 s), four times (200 s) and eight times (400 s) the critical time step. A constant real friction coefficient (Manning) of 0.038 was adopted for all of the estuary. Coriolis forces were included. The analysis proved that the Coriolis effect was very important in terms of phase errors. Table 10.1 represents a comparison between observations and computations in terms of amplitudes and phases for seven different points which are represented in the location map (Fig. 10.6), for the three different time steps described above. The maximum error in amplitude only increases by 1.4% when the time step of 400 s is used with respect to the time step of 50 s, while the absolute error in phases ( - 13 o) is two degrees more than the case of 50 s ( - 11 o). These bounds show a remarkable accuracy for the semi-explicit model. In Fig. 10.8 the distribution of velocities at different times of the tide is illustrated (explicit model).
303
e-o
9
E _m o o
"o o c(:o b,. to
2.i u_
o
o
E
o -(3 o c,~. co ,,-
.LJ 0
to
t~
o
,).-
co
ffl co
E
o o
(o (,o
o
o cOd O0 CO u..
r-. (D
(1)
E
8 c-
nO
.C o
o =~
00
0 0~ 9 o.
~ ~ o ~- ~~oE ~o~ >o
cD
D,.
~-- c~ c'o ,~- LO CO D,. O 0 0 ~
~'-~o o Or)
o
co
cO
rL.. O.J (I;
c-c'-
-0 c-cO
co r--
o
(I.;
c-
E
+.~ rE
.E
p,,,
LL.
OL
Examples of application 305 Table 10.1 BristolChannel and SevernEstuaryobserved results and FEM computation(FL mesh) of tidal half-amplitude (m x 102) Location Tenby Swansea Cardiff Porthcawl Barry Port Talbot Newport Ilfracombe Minehead
Observed
FEM
262 315 409 317 382 316 413 308 358
260 (- 1%) 305 (-3%) 411 (0%) 327 (+3%) 394 (+3%) 316 (- 1%) 420 (+2%) 288 (-6%) 362 (+ 1%)
Example 10.6 River Severn bore
In the analysis presented we have omitted details of the River Severn upstream of the eastern limit [see Figs 10.6 and 10.9(a)], where a 'bore' moving up the fiver can be observed. An approach to this phenomenon is made by a simplified straight extension of the mesh used previously, preserving an approximate variation of the bottom and width until the point G (Gloucester) (77.5 km from Avonmouth), but obviously neglecting the
".':i::: i
~e=
- I mls
Fig.10.8 Velocityvectorplots(FLmesh).
3 hours
306
Shallow water problems
Finite element mesh including river
(a)
B (b)
B
(c)
Fig. 10.9 Severnbore.
dissipation and inertia effects of the bends. Measurement points are located at B and E, and the results (elevations) are presented in Fig. 10.9(d) for the points A, B, E in time, along with a steady fiver flow. A typical shape for a tidal bore can be observed for the point E, with fast flooding and a smooth ebbing of water. (The flooding from the minimum to maximum level is in less than 25 minutes.)
Examples of application
Severn Bore 15 10 , [ -
0
_~
"'
0
,~,...~
\
9
"
.../.
.../.
...
.:
\
\
\.I
-5
-10
,/'N,.~
,;,>. '..., ,
"------- A ------- g E
0
I
.,,
l
5
, "-...1..."
l
10
/
l
15 t(h)
I
20
25
30
Water elevations for points A, B, E
8 ~6 _
/
A
~----
Computed Measured
10 ~-
c 4 _ .o 2
LU 0
8
MWL
;
-2 12
.... Computed
~'9
v
>
/'k
14
16
18 20 t(h)
22
24
26
61""5 12
'l 14
16
Point B
J.
i
18 20 t(h)
I 22
r-24
26
Point E Computed and measured elevations (d)
Fig. 10.9 Continued.
10.4.3 Tsunami waves A problem of some considerable interest in earthquake zones is that of so-called tidal waves or tsunamis. These are caused by sudden movements in the earth's crust and can on occasion be extremely destructive. The analysis of such waves presents no difficulties in the general procedure demonstrated and indeed is computationally cheaper as only relatively short periods of time need be considered. To illustrate a typical possible tsunami wave we have created one in the Severn Estuary just analysed (to save problems of mesh generation, etc., for another more likely configuration).
Example 10.7
Tsunami wave in Severn
Estuary
Here the tsunami is forced by an instantaneous raising of an element situated near the centre of the estuary by some 6 m and the previously designed mesh was used
307
308 Shallowwater problems (Fig. 10.7, FL). The progress of the wave is illustrated in Fig. 10.10. The tsunami wave was superimposed on the tide at its highest level - though of course the tidal motion was allowed for. This example was included in reference 5. One particular point only needs to be mentioned in this calculation. This is the boundary condition on the seaward, arbitrary, limit. Here the Riemann decomposition of the type discussed earlier has to be made if tidal motion is to be incorporated and note taken of the fact that the tsunami forms only an outgoing wave. This, in the absence of tides, results simply in application of the free boundary condition there. The clean way in which the tsunami is seen to leave the domain in Fig. 10.10 testifies to the effectiveness of this process.
10.4.4 Steady-state solutions On occasion steady-state currents such as may be caused by persistent wind motion or other influences have to be considered. Here once again the transient form of explicit computation proves very effective and convergence is generally more rapid than in compressible flow as the bed friction plays a greater role. The interested reader will find many such steady-state solutions in the literature.
ExamplelO.8 Steadystatesolution
In Fig. 10.11 we show a typical example. Here the currents are induced by the breaking of waves which occurs when these reach small depths creating so-called radiation stresses.6, 30,60 Obviously as a preliminary the wave patterns have to be computed using procedures to be given later. The 'forces' due to breaking are the cause of longshore currents and rip currents in general. The figure illustrates this effect on a harbour. It is of interest to remark that in the problem discussed, the side boundaries have been 'repeated' to model an infinite harbour series. 6~
Example lO.9 Supercriticalflow
Another type of interesting steady-state (and also transient) problem concerns supercritical flows over hydraulic structures, with shock formation similar to those present in high-speed compressible flows. To illustrate this range of flows, the problem of a symmetric channel of variable width with a supercritical inflow is shown here. For a supercritical flow in a rectangular channel with a symmetric transition on both sides, a combination of a 'positive' jump and 'negative' waves, causing a decrease in depth, appears. The profile of the negative wave is gradual and an approximate solution can be obtained by assuming no energy losses and that the flow near the wall turns without separation. The constriction and enlargement analysed here was 15 ~ and the final mesh used was of only 6979 nodes, after two remeshings. The supercritical flow had an inflow Froude number of 2.5 and the boundary conditions were as follows: heights and velocities prescribed in inflow (left boundary of Fig. 10.12), slip boundary on walls (upper and lower boundaries in Fig. 10.12) and free variables on the outflow boundary (fight side of Fig. 10.12). The explicit version with local time step was adopted. Figure 10.12 represents contours of heights, where 'cross'-waves and 'negative' waves are contained. One can observe the 'gradual' change in the behaviour of the negative wave created at the origin of the wall enlargement.
Examples of application
Time = 0
Fig. 10.10 Severntsunami. Generation during high tide. Water height contours (times after generation).
309
310
Shallow water problems
%
"
. . . .
-
---.. " - - " . "I..:.
'
'
,
-
,
,=,
_
,,
'
,Q ,
Fig. 10.11 Wave-inducedsteady-stateflow past a harbour.3~
Fig. 10.12 Supercritical flow and formation of shock waves in symmetric channel of variable width contours of h. Inflow Froude number = 2.5. Constriction" 15~
A special problem encountered in transient, tidal, computations is that of boundary change due to changes of water elevation. This has been ignored in the calculation
Shallow water transport
/•Boundary
at time tn
~
'
-
Boundaryat time t n +
Atn
!1
Fig. 10.13 Adjustmentof boundarydue to tidal variation. presented for the Bristol Channel-Severn Estuary as the movements of the boundary are reasonably small in the scale analysed. However, in that example these may be of the order of 1 km and in tidal motions near Mont St Michel, France, can reach 12 km. Clearly on some occasions such movements need to be considered in the analysis and many different procedures for dealing with the problem have been suggested. In Fig. 10.13 we show the simplest of these which is effective if the total movement can be confined to one element size. Here the boundary nodes are repositioned along the normal direction as required by elevation changes Aq. If the variations are larger than those that can be absorbed in a single element some alternatives can be adopted, such as partial remeshing over layers surrounding the distorted elements or a general smooth displacement of the mesh.
~i~ii~i~i!i~i~i!~i~i~i~i~~~i~i~~i~i~i~i~i~i~i~i!i~~ih~~i~i~~~!~~~~~~~~~!~ ~~~~~~~i~i~i~!~i~i~i~!~i~!~i~i~i~i~!~i~i~i~i~~~~~~ i~ Shallow water currents are frequently the carrier for some quantities which may disperse or decay in the process. Typical here is the transport of hot water when discharged from power stations, or of the sediment load or pollutants. The mechanism of sediment transport is quite complex 61 but in principle follows similar rules to that of the other equations. In all cases it is possible to write depth-averaged transport equations in which the average velocities Ui have been determined independently. A typical averaged equation can be written - using for a scalar variable [e.g. temperature (T) ] as the transported quantity - as
O(hT) C~
Jr-
O(hEtiT) OX i
0 (OT) hk ~xi
OX i
+R- 0
for i -- 1, 2
(10.24)
where h and ~i are the previously defined and computed quantifies, k is an appropriate diffusion coefficient and R is a source term. A quasi-implicit form of the characteristic-Galerkin algorithm can be obtained when diffusion terms are included. In this situation practical horizontal viscosity ranges (and
311
312
Shallowwater problems diffusivity in the case of transport equations) can produce limiting time steps much lower than the convection limit. To circumvent time step restriction imposed by the diffusion term, a quasi-implicit computation, requiring an implicit computation of the viscous terms, is recommended. The application of the characteristic-Galerkin method for any scalar transport equation is straightforward, because of the absence of the pressure gradient term. The computation of the scalar h T is analogous to the intermediate momentum computation, but now a new time integration parameter 03 is introduced for the viscous term such that 0 < 0 3 __<1. The application of the characteristic-Galerkin procedure gives the following final matrix form (neglecting terms higher than second order): At 2 (M + 03AtD) AT -- - A t [ C T n + MR n] - ----~-[Ku Tn + f R ] - At03 DTn + fb Z
(10.25)
where now T is the vector of nodal h T values" M - f a NTN dr2 C -Ku
--
fa
ON NTfiJ~xj dr2 ~ x (NT~k)~Xj (N~j)
D--
~ O N T ON --~x/k ~x/ dr2
fR-
2 ~0x (NTRk)N dr2
and fb -- A t
da
OT ~ Nk--~x i " ni dF
As an illustration of a real implementation, the parameters involved in the study of transport of salinity in an industrial application for a fiver area are considered here. The region studied was approximately 55 kilometres long and the mean value of the eddy diffusivity was of k = 40m s -1. The limiting time step for convection (considering eight components of tides) was 3.9 s. This limit was severely reduced to 0.1 s if the diffusion term was active and solved explicitly. The convective limit was recovered assuming an implicit solution with 03 = 1.0. The comparisons of diffusion error between computations with 0.1 s and 3.9 s had a maximum diffusion error of 3.2% for the 3.9 s calculation, showing enough accuracy for engineering purposes, taking into account that the time stepping was increased 40 times, reducing dramatically the cost of computation. This reduction is fundamental when, in practical applications, the behaviour of the transported quantity must be computed for longterm periods, as was this problem, where the evolution of the salinity needed to be calculated for more than 60 periods of equivalent Me tides and for very different initial conditions. The boundary conditions are the same as for tidal problems:
Concluding remarks 313
Time = 9 hours
Time = 36 hours
Fig. 10.14
= 18 hours
Time = 54 hours
Heat convection and diffusion in tidal currents. Temperature contours at several times after
discharge of hot fluid.
a partially closed region with an open boundary condition where the tide components are imposed. For this case a radiation condition based on Riemmann invariants has been used. For details, the reader can find a detailed description in reference 35. For the salinity transport equation, the open boundary condition also can be derived on the basis of Riemmann invariants in a similar way as for the rest of the equations. In Fig. 10.14 we show by way of an example the dispersion of a continuous hot water discharge in an area of the Severn Estuary. Here we note not only the convection movement but also the diffusion of the temperature contours.
In this chapter we summarized another important form of incompressible fluid dynamics equations to solve shallow water flows. The important features of various numerical solution procedures are also mentioned here. Turbulent fluid dynamics and accurate estimation of free surface in estuaries are growing areas of research but much more complicated approaches are necessary to tackle such problems. However, the procedure explained in this chapter, though it has limitations, is easy to follow and very effective in getting a quick solution to shallow water problems.
314 Shallowwater problems ii iiiliiiiiiiiiii',ii'~i','~iiiiii', ~,',ii:i',',',',',',','~',~,:',ii',':',',iiii~'~ ',', ~~,', ,ii',i',i:~,i'~',:',!',i'~i', ~, ,'~ii illi~iii',',:~,,!'~~,'~'~'~, ',',i'',,i~,i~,i','i~'~'~',i'~','i,',~,',!',','i,ii',i~','i,'i,'~i',~i',i'~',',i'~i',',~i:'~,,~i~iii' ,',~,i:iii'i',~,iii iii!iiii!'~ ',i ii',ii',iiii ~i~ ',i~ ' ,'i,'i,'i~',~,:~ ii',' ,iiiiiiii~,'~i',:','~',:ii','~,,~'~i'~i'~ !i~,i~'~~, ',iii',:,i'~','~',':i!::iii',iiii',iiiiii','~',i'~i ',i i','~'~',',',i',',i i i!',','~'~i'~i',',iii~,i ',', '~'~','i',,'~i!iii~, ', ~,~,~,',~,i',~',',~'i',,iiiii'~ii~,:,~~,i'~'~i'~ii',i'~,i,'~,'~ '~ ',i':~','~,~ ii',' '~'~',i'~i', ii '~',i~,',',~'i~i'~': ',~,',:',',',',',i','~',',',!':',~;,:,
1. M.B. Abbott. Computational Hydraulics. Elements of the Theory of Free Surface Flows. Pitman, London, 1979. 2. G.J. Haltiner and R.T. Williams. Numerical Prediction and Dynamic Meteorology. Wiley, New York, 1980. 3. O.C. Zienkiewicz, R.W. Lewis and K.G. Stagg, editors. Numerical Methods in Offshore Engineering. Wiley, Chichester, 1978. 4. J. Peraire. A finite element method for convection dominated flows. Ph.D. thesis, University of Wales, Swansea, 1986. 5. J. Peraire, O.C. Zienkiewicz and K. Morgan. Shallow water problems: a general explicit formulation. Int. J. Num. Meth. Eng., 22:547-574, 1986. 6. O.C. Zienkiewicz and J.C. Heinrich. A unified treatment of steady state shallow water and two dimensional Navier-Stokes equations. Finite element penalty function approach. Comp. Meth. Appl. Mech. Eng., 17/18:673--689, 1979. 7. M. Kawahara. On finite-element methods in shallow-water long-wave flow analysis. In J.T. Oden, editor, Computational Methods in Nonlinear Mechanics, pp. 261-287. North-Holland, Amsterdam, 1980. 8. I.M. Navon. A review of finite element methods for solving the shallow water equations. In Computer Modelling in Ocean Engineering, pp. 273-278. Balkema, Rotterdam, 1988. 9. J.J. Connor and C.A. Brebbia. Finite-Element Techniques for Fluid Flow. Newnes-Butterworth, London and Boston, 1976. 10. J.J. O'Brien and H.E. Hulburt. A numerical model of coastal upwelling. J. Phys. Oceanogr., 2:14-26, 1972. 11. M. Crepon, M.C. Richez and M. Chartier. Effects of coastline geometry on upwellings. J. Phys. Oceanogr., 14:365-382, 1984. 12. M.G.G. Foreman. An analysis of two-step time-discretisations in the solution of the linearized shallow-water equations. J. Comp. Phys., 51:454-483, 1983. 13. W.R. Gray and D.R. Lynch. Finite-element simulation of shallow-water equations with moving boundaries. In C.A. Brebbia et al., editors, Proc. 2nd Conf. on Finite-Elements in Water Resources, pp. 2.23-2.42, 1978. 14. T.D. Malone and J.T. Kuo. Semi-implicit finite-element methods applied to the solution of the shallow-water equations. J. Geophys. Res., 86:4029-4040, 1981. 15. G.J. Fix. Finite-element models for ocean-circulation problems. SlAM J. Appl. Math., 29: 371-387, 1975. 16. C. Taylor and J. Davis. Tidal and long-wave propagation, a finite-element approach. Computers and Fluids, 3:125-148, 1975. 17. M.J.P. Cullen. A simple finite element method for meteorological problems. J. Inst. Math. Appl., 11:15-31, 1973. 18. H.H. Wang, P. Halpem, J. Douglas, Jr and I. Dupont. Numerical solutions of the one-dimensional primitive equations using Galerkin approximations with localised basis functions. Mon. Weekly Rev., 100:738-746, 1972. 19. I.M. Navon. Finite-element simulation of the shallow-water equations model on a limited area domain. Appl. Math. Modelling, 3:337-348, 1979. 20. M.J.P. Cullen. The finite element method. In Numerical Methods Used in Atmosphere Models, volume 2, Chapter 5, pp. 330-338. WMO/GARP Publication Series 17, World Meteorological Organisation, Geneva, Switzerland, 1979. 21. M.J.P. Cullen and C.D. Hall. Forecasting and general circulation results from finite-element models. Q. J. Roy. Met. Soc., 102:571-592, 1979. 22. D.E. Hinsman, R.T. Williams and E. Woodward. Recent advances in the Galerkin finiteelement method as applied to the meteorological equations on variable resolution grids.
References 315
23.
24.
25.
26. 27. 28. 29. 30.
31. 32. 33. 34. 35. 36. 37. 38.
39.
40.
41. 42. 43.
In T. Kawai, editor, Finite-Element Flow-Analysis. University of Tokyo Press, Tokyo, 1982. I.M. Navon. A Numerov-Galerkin technique applied to a finite-element shallow-water equations model with enforced conservation of integral invariants and selective lumping. J. Comp. Phys., 52:313-339, 1983. I.M. Navon and R. de Villiers. GUSTAF, a quasi-Newton nonlinear ADI FORTRAN IV program for solving the shallow-water equations with augmented Lagrangians. Computers and Geosci., 12:151-173, 1986. A.N. Staniforth. A review of the application of the finite-element method to meteorological flows. In T. Kawai, editor, Finite-Element Flow-Analysis, pp. 835-842. University of Tokyo Press, Tokyo, 1982. A.N. Staniforth. The application of the finite element methods to meteorological simulations a review. Int. J. Num. Meth. Fluids, 4:1-22, 1984. R.T. Williams and O.C. Zienkiewicz. Improved finite element forms for shallow-water wave equations. Int. J. Num. Meth. Fluids, 1:91-97, 1981. M.G.G. Foreman. A two-dimensional dispersion analysis of selected methods for solving the linearised shallow-water equations. J. Comp. Phys., 56:287-323, 1984. I.M. Navon. FEUDX: a two-stage, high-accuracy, finite-element FORTRAN program for solving shallow-water equations. Computers and Geosci., 13:225-285, 1987. P. Bettess, C.A. Fleming, J.C. Heinrich, O.C. Zienkiewicz and D.I. Austin. A numerical model of longshore patterns due to a surf zone barrier. In 16th Coastal Engineering Conf., Hamburg, West Germany, October, 1978. M. Kawahara, N. Takeuchi and T. Yoshida. Two step explicit finite element method for tsunami wave propagation analysis. Int. J. Num. Meth. Eng., 12:331-351, 1978. J.H.W. Lee, J. Peraire and O.C. Zienkiewicz. The characteristic Galerkin method for advection dominated problems- an assessment. Comp. Meth. Appl. Mech. Eng., 61:359-369, 1987. D.R. Lynch andW. Gray. A wave equation model for finite element tidal computations. Computers and Fluids, 7:207-228, 1979. O. Daubert, J. Hervouet and A. Jami. Description of some numerical tools for solving incompressible turbulent and free surface flows. Int. J. Num. Meth. Eng., 27:3-20, 1989. G. Labadie, S. Dalsecco and B. Latteaux. Resolution des 6quations de Saint-Venant par une m6thode d'616ments finis. Electricit6 de France, Report HE/41, 1982. T. Kodama, T. Kawasaki and M. Kawahara. Finite element method for shallow water equation including open boundary condition. Int. J. Num. Meth. Fluids, 13:939-953, 1991. S. Bova and G. Carey. An entropy variable formulation and applications for the two dimensional shallow water equations. Int. J. Num. Meth. Fluids, 23:29-46, 1996. K. Kashiyama, H. Ito, M. Behr and T. Tezduyar. Three-step explicit finite element computation of shallow water flows on a massively parallel computer. Int. J. Num. Meth. Fluids, 21:885-900, 1995. S. Chippada, C. Dawson, M. Martinez and M.F. Wheeler. Finite element approximations to the system of shallow water equations. Part I. Continuous time a priori error estimates. TICAM Report, Univ. of Texas at Austin, 1995. S. Chippada, C. Dawson, M. Martinez and M.F. Wheeler. Finite element approximations to the system of shallow water equations. Part II. Discrete time a priori error estimates. TICAM Report, Univ. of Texas at Austin, 1996. O.C. Zienkiewicz, J. Wu and J. Peraire. A new semi-implicit or explicit algorithm for shallow water equations. Math. Mod. Sci. Comput., 1:31-49, 1993. M.M. Cecchi, A. Pica and E. Secco. A projection method for shallow water equations. Int. J. Num. Meth. Fluids, 27:81, 1998. M.M. Cecchi and Salasnich. Shallow water theory and its application to the Venice lagoon. Comput. Methods Appl. Mech. Eng., 151:63-74, 1998.
316 Shallowwater problems 44. EX. Giraldo. The Lagrange-Galerkin method for the two-dimensional shallow water equations on adaptive grids. Int. J. Num. Meth. Fluids, 33:789-832, 2000. 45. C.N. Dawson and M.L. Martinez-Canales. A characteristic-Galerkin approximation to a system of shallow water equations. Numerische Mathematik, 86:239-256, 2000. 46. V. Aizinger and C. Dawson. A discontinuous Galerkin method for two-dimensional flow and transport in shallow water. Advances in Water Resources, 25:67-84, 2002. 47. C. Dawson and J. Proft. Discontinuous and coupled continuous/discontinuous Galerkin methods for the shallow water equations. Comp. Meth. Appl. Mech. Eng., 191:4721-4746, 2002. 48. M. Quecedo and M. Pastor. A reappraisal of Taylor-Galerkin algorithm for drying-wetting areas in shallow water computations. Int. Num. Meth. Fluids, 38:515-531, 2002. 49. EX. Giraldo. Strong and weak Lagrange-Galerkin spectral methods for the shallow water equations. Comput. & Meth. Appl., 45:97-121, 2003. 50. C. Dawson and J. Proft. Discontinuous/continuous Galerkin methods for coupling the promitive and wave continuity equations of shallow water. Comp. Meth. Appl. Mech. Eng., 192:5123-5145, 2003. 51. C. Dawson and J. Proft. Coupled discontinuous and continuous Galerkin finite element methods for the depth integrated shallow water equations. Comp. Meth. Appl. Mech. Eng., 193:289-318, 2004. 52. O.C. Zienkiewicz and P. Ortiz. A split-characteristic based finite element model for the shallow water equations. Int. J. Num. Meth. Fluids, 20:1061-1080, 1995. 53. O.C. Zienkiewicz and P. Ortiz. The characteristic based split algorithm in hydraulic and shallowwater flows. Keynote lecture. In 2nd Int. Symposium on Environmental Hydraulics, Hong Kong, 1998. 54. P. Ortiz and O.C. Zienkiewicz. Tide and bore propagation by a new fluid algorithm. In Finite Elements in Fluids. 9th Int. Conference, 1995. 55. R. Lrhner, K. Morgan and O.C. Zienkiewicz. The solution of non-linear hyperbolic equation systems by the finite element method. Int. J. Num. Meth. Fluids, 4:1043-1063, 1984. 56. S. Nakazawa, D.W. Kelly, O.C. Zienkiewicz, I. Christie and M. Kawahara. An analysis of explicit finite element approximations for the shallow water wave equations. In Proc. 3rd Int. Conf. on Finite Elements in Flow Problems, volume 2, pp. 1-7, Banff, 1980. 57. M. Kawahara, H. Hirano, K. Tsubota and K. Inagaki. Selective lumping finite element method for shallow water flow. Int. J. Num. Meth. Fluids, 2:89-112, 1982. 58. D.R. Lynch and W.G. Gray. Analytic solutions for computer flow model testing. Trans. ASCE, J. Hydr. Div., 104(10):1409-1428, 1978. 59. Hydraulic Research Station. Severn tidal power. Report EX985, 1981. 60. D.I. Austin and P. Bettess. Longshore boundary conditions for numerical wave model. Int. J. Num. Mech. Fluids, 2:263-276, 1982. 61. C.K. Ziegler and W. Lick. The transport of fine grained sediments in shallow water. Environ. Geol. Water Sci., 11:123-132, 1988.
Long and medium waves*
The main developments in this chapter relate to linearized surface waves in water, but acoustic and electromagnetic waves will also be mentioned. We start from the wave equation (10.18), which was developed from the equations of momentum balance and mass conservation in shallow water. The wave elevation, ~7, is small in comparison with the water depth, H. If the problem is periodic, we can write the wave elevation, r~, quite generally as ~7(x, y, t) = fT(x, y)exp(iwt) (11.1) where w is the angular frequency and f7may be complex. Equation (10.18) now becomes W2 V T (HVfT) + --f7 -- 0 g
0
or
OXi
(O/v]) H ~
W2 + --f7 = 0 g
(11.2)
or, for constant depth, H, V2~ + k2~ = 0
or
~
02~
OXiOXi
+ k2f7 -- 0
(11.3)
where the wavenumber, k = co/~/g H, is related to the wave length, )~, by k = 2rr/)~. The wave speed is c = w~ k. Equation (11.3) is the Helmholtz equation [which was also derived in Chapter 10, in a slightly different form, as Eq. (10.18)] which models very many wave problems. This is only one form of the equation of surface waves, for which there is a very extensive literature. 1-4 From now on all problems will be taken to be periodic, and the overbar on ~7will be dropped. The Helmholtz equation (11.3) also describes periodic acoustic waves. The wavenumber k is now given by w/c, whereas in surface waves w is the angular frequency and c is the wave speed. This is given by c = ~/K/p, where p is the density of the fluid and K is the bulk modulus. Boundary conditions need to be applied to deal with radiation and absorption of acoustic waves. The first application of finite elements to acoustics was by Gladwell. 5 This was followed in 1969 by the solution of acoustic equations by Zienkiewicz and Newton, 6 and further finite element models by Craggs. 7 A more comprehensive survey of the development of the method is given by Astley. 8 Provided that the dielectric constant, e, and the permeability, #, are constant, then Maxwell's equations for electromagnetics can be *Contributed fully by Peter Bettess, School of Engineering, Universityof Durham, UK.
318
Longand medium waves
reduced to the form
V 2q5
e# 02q5 = c 2 oqt2
4rrp s
and
V2A
e# 02A __ c 2 oqt2
4rt#J c
(11.4)
where p is the charge density, J is the current, and 0 and A are scalar and vector potentials, respectively. When p and J are zero, which is a frequent case, and the time dependence is harmonic, Eqs (11.4) reduce to the Helmholtz equations. More details are given by Morse and Feshbach. 9 For surface waves on water when the wavelength, A = 27r/k, is small relative to the depth, H, the velocities and the velocity potential vary vertically as coshkz. 1,2, 10, 11 The full equation can now be written as
r V T(cCg 777) + --77 g = 0
or
(9(
Oq/'])
ccg ~
Oxi
(.02
+ - -g~ 7 - - 0
(11.5)
where the group velocity, Cg = nc, n = (1 + (2kH/sinh 2 k H ) ) / 2 and the dispersion relation w 2 = gk tanh kH
(11.6)
link the angular frequency, w, and the water depth, H, to the wavenumber, k.
~!i~i~iiiiii!i~ii~ii~i~i~ii~~' !i~i~i~iii~i"~'iiiii~ii~ii{i~!ii!~i!iii~iii~!i!ii~!i~i~~' i~i~i{ii~{ii{iiiiii!iiiiiiiiiiii' !ii!iiiiii}!iiii~ii:!~:~~i~~' iii~:~ii~i~ii!iiiiiiiiiiiiiiiiiiiiii{{iiiiii iiii{iiiiiiii{ii!ii{i~iiiiiiiiii i!i~ii~{i~'~ ii~i!i!~il~iiii!~i~ii~ii'i:~'i:~i~iiiiii!{i!!iiii!i~i~~iil ~ii~iiiii':i':i':i!iiiii iiiiiiiiiiiiiii i!~}~{~}~{~{~~{~{~{~~~~ i! ii!ili!i!iiiiilili ~iii~iiiii~i~i~!~}~iiii~iii!iii!iiiiiiiiii~i~i~ii}i[iiiii~iii~i!ii~i~i~iiiiiiiiii~i~iii!!iii~iiiii~i~ii~iiii!ii~!~i~i~i~ii!~i~iiiii~!iiiiiiii~iiiii~i!iii!ii~!iiiiiiii~iiiii~i~ii~iiiiiii~iiii~i!i!~i~!~ii~iiiiii~ii We now consider a closed domain of any shape. For waves on water this could be a closed basin, for acoustic or electromagnetic waves it could be a resonant cavity. In the case of surface waves we consider a two-dimensional basin, with varying depth. In plan it can be divided into two-dimensional elements, of any of the types discussed in reference 12. The wave elevation, r/, at any point (~, r/) within the element can be expressed in terms of nodal values, using the element shape function N, thus r / ~ f / = N~?
(11.7)
Next Eq. (11.2) is weighted with the shape function, and integrated by parts in the usual way, to give H~-Tx/
NT ) N
df2~ - 0
(11.8)
The integral is taken over all the elements of the domain, and ~7represents all the nodal values of r/. The natural boundary condition which arises is &7/On = 0, where n is the normal to the boundary, corresponding to zero flow normal to the boundary. Physically this corresponds to a vertical, perfectly reflecting wall. Equation (11.8) can be recast in the familiar form (K - w2M) ~7- 0
(11.9)
Waves in closed domains - finite element models
where M = f~ N T-glNdf2
K = J~ BTDB dr2
(11.10)
and the D matrix is constructed from the depths, H. H can vary with position.
~ H0H~ It is thus an eigenvalue problem as discussed in Chapter 16 of reference 12. The K and M matrices are analogous to structure stiffness and mass matrices. The eigenvalues will give the natural frequencies of oscillation of the water in the basin and the eigenvectors give the mode shapes of the water surface. Such an analysis was first carried out using finite elements by Taylor et al. 13 and the results are shown as Fig. 16.5 of reference 12. There are analytical solutions for harbours of regular shape and constant depth. 1' 3 The reader should find it easy to modify the standard element routine given in reference 12, Chapter 19, to generate the wave equation 'stiffness' and 'mass' matrices. In the corresponding acoustic problems, the eigenvalues give the natural resonant frequencies and the eigenvectors give the modes of vibration. The model described above will give good results for harbour and basin resonance problems, and other problems governed by the Helmholtz equation. In modelling the Helmholtz equation, it is necessary to retain a mesh which is sufficiently fine to ensure an accurate solution. A 'rule of thumb', which has been used for some time, is that there should be 10 nodes per wavelength. This has been accepted as giving results of acceptable engineering accuracy for many wave problems. However, recently more accurate error analysis of the Helmholtz equation has been carried out. 14' 15 In wave problems it is not sufficient to use a fine mesh only in zones of interest. The entire domain must be discretized to a suitable element density. There are essentially two types of error: 1. The wave shape may not be a good representation of the true wave, that is the local elevations or pressures may be wrong. 2. The wavelength may be in error. This second case causes a poor representation of the wave in one part of the problem to cause errors in another part of the problem. This effect, where errors build up across the model, is called a pollution error. It has been implicitly understood since the early days of modelling of the Helmholtz equation, as can be seen from the uniform size of finite element used in meshes. Babu~ka et al. 14, 15 show some results for various finite element models, using different element types, and the error as a function of element size, h, and wave number, k. The sharper error results show that the simple rule of thumb given above is not always adequate. Since the wave number, k, and the wavelength, A, are related by k = 27r/A, the condition of 10 nodes per wavelength can be written as kh ~ 0.6. But keeping to this limit is not sufficient. The pollution error grows as k3h 2. Babu~ka et al. propose a posteriori error indicators to assess the pollution error. See the cited references and Chapter 13, reference 12, for further discussion of these matters.
319
320
Long and medium waves
The main defects of the simple surface-wave model described above are the following: 1. Inaccuracy when the wave height becomes large. The equations are no longer valid when ~7becomes large, and for very large r/, the waves will break, which introduces energy loss. 2. Lack of modelling of bed friction. This will be discussed below. 3. Lack of modelling of separation at re-entrant comers. At re-entrant comers there is a singularity in the velocity of the form 1/~/7, where r is the distance from the comer. The velocities become large, and physically the viscous effects, neglected above, become important. They cause retardation, flow separation and eddies. This effect can only be modelled in an approximate way. Now the response can be determined for a given excitation frequency, as discussed in Chapter 16 of reference 12.
The engineering approach to the energy loss in the boundary layer close to the sea bed is to introduce a friction force, proportional to the water velocity. This is called the Ch6zy bed friction. Since the force is non-linear in the velocity if it is included in its original form it makes the equations difficult to solve. The usual procedure is to assume that its main effect is to damp the system, by absorbing energy, and to introduce a linear term, which in one period absorbs the same amount of energy as the Ch6zy term. The linearized bed friction version of Eq. (11.2) is
V T(HW7)
(.,02
+ - - ~ 7 - iwMrl = 0
g
or
Oq (07]) OXi
H
~
(02
+ - - r / - icoMr/= 0
g
(11.11) where M is a linearized bed friction coefficient, which can be written as M = 8Umax/371" C 2H, C is the Ch6zy constant and Umaxis the maximum velocity at the bed at that point. In general the results for ~7will now be complex, and iteration has to be used, since M depends upon the unknown Umax. From the finite element point of view, there is no longer any need to separate the 'stiffness' and 'mass' matrices. Instead, Eq. (11.11) is weighted using the element shape function and the entire complex element matrix is formed. The matrix fight-hand side arises from whatever exciting forces are present. The re-entrant comer effect and wave-absorbing walls and permeable breakwaters can also be modelled in a similar way, as both of these introduce a damping effect, due to viscous dissipation. The method is explained in reference 16, where an example showing flow through a perforated wall in an offshore structure is solved.
Short-w~ve diffraction problems are those in which the w~vdength is much smaller than any of the dimensions of the problem. Such problems ~Lrisein surface waves on water,
Waves in unbounded domains (exterior surface wave problems) acoustics and pressure waves, electromagnetic waves and elastic waves. The methods described in this chapter will solve the problems, but the requirement of 10 nodes or thereabouts per wavelength makes the necessary finite element meshes prohibitively fine. To take one example, radar waves of wavelength 1 mm might impinge on an aircraft of 10 m wing span. It is easy to see that the computing requirements are truly astronomical. This topic is considered in more detail in Chapter 12.
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Problems in this category include the diffraction and refraction of waves close to fixed and floating structures, the determination of wave forces and wave response for offshore structures and vessels, and the determination of wave patterns adjacent to coastlines, open harbours and breakwaters. In electromagnetics there are scattering problems of the type already described, and in acoustics we have various noise problems. In the interior or finite part of the domain, finite elements, exactly as described in Sec. 11.2, can be used, but special procedures must be adopted for the part of the domain extending to infinity. The main difficulty is that the problem has no outer boundary. This necessitates the use of a radiation condition. Such a condition was introduced in Chapter 18 of reference 12 as Eq. (18.18), for the case of a one-dimensional wave, or a normally incident plane wave in two or more dimensions. Work by Bayliss et al. 19,20 has developed a suitable radiation condition, in the form of an infinite series of operators. The starting point is the representation of the outgoing wave in the form of an infinite series. Each term in the series is then annihilated by using a boundary operator. The sequence of boundary operators thus constitutes the radiation condition. In addition there is a classical form of the boundary condition for periodic problems, given by Sommerfeld.17, 18 A summary of some of the available radiation conditions is given in Table 11.1.
11.6.1 Background to wave problems The simplest type of exterior, or unbounded wave problem is that of some exciting device which sends out waves which do not return. This is termed the radiation problem. The next type of exterior wave problem is where we have a known incoming wave which encounters an object, is modified and then again radiates away to infinity. This case is known as the scattering problem, and is more complicated, in as much as we have to deal with both incident and radiated waves. Even when both waves are linear, this can lead to complications. Both the above cases can be complicated by wave refraction, where the wave speeds change, because of changes in the medium, for example changes in water depth. Usually this phenomenon leads to changes in the wave direction. Waves can also reflect from boundaries, both physical and computational.
11.6.2 Wave diffraction We now consider the problem of an incident wave diffracted by an object. The problem consists of an object in some medium, which diffracts the incident waves. We divide
321
322
Long and medium waves
Table 11.1 Radiationconditionsfor exteriorwaveproblems Dimensions 1
2
3
Generalboundaryconditions Transient Ox +c-& -=~
B m ~ -- O, m --+ r
Bmf~ -- O, m ~
m(~.~O2j--~3/2)) nm'-jl~l "Jt"'-~t"~-~ "=
nm--j=lf i ( ~
Periodic
0r Ox
+ikr
lim(X)~/-7(~--~r+ ik~b)=0 r----~ or m Q0 --~rr +
0 2j--l) "~--"~t"lf
lim(X)r(-~rr r---~
+ik~b) =0 or
Bm r -- O, m "+ (x) Bm = ]I~l
oo
Bm r = O, m -~ oo
2j-/3/2))
ik + ~
Bm =
jill(0 "= ~r +ik+ 2 j r- l )
Symmetricboundaryconditions Transient Or +7-OT =~
0r +ikr Or
Or
+ - - + - c-~t =0 Axisymmetric Periodic
Or + - + - c -0-t-t= 0 Sphericallysymmetric
o+(, )
o+(, )
-~r + ~r+ik ~ = 0 Axisymmetric
-~-+ -+ikr ~b=0 Sphericallysymmetric
the medium as shown in Fig. 11.1 into two regions, with boundaries FA, FB, FC and FD. These boundaries have the following meanings. FA is the boundary of the body which is diffracting the waves. FB is the boundary between the two computational domains, that in which the total wave elevation (or other field variable) is used, and FD s
i
~
Objects in water
]-'A Boundary of objects Fig. 11.1 Generalwave domains.
Waves in unbounded domains (exterior surface wave problems)
that in which the elevation of the radiated wave is used. Fc is the outer boundary of the computational model, and FD is the boundary at infinity. Some of these boundaries may be merged. A variational treatment will now be used, as described in reference 12, Chapter 3. A weighted residual treatment is also possible. The elevation of the total wave, ~Tr, is split into those for incident and radiated waves, 7~i and ~TR. Hence r/r = ~71 + r/R- The incident wave elevation, ~Tz,is assumed to be known. For the surface wave problem, the functional for the exterior can be written as I"l =
. -~ C C g ( V T I ) T v ~ -
O')2Cg 7~21 d x d y
C
(11.12)
where making FI stationary with respect to variations in ~7corresponds to satisfying the shallow water wave equation (11.5), with natural boundary condition O0/0n = 0, or zero velocity normal to the boundary. The functional is rewritten in terms of the incident and radiated elevations, and then Green's theorem in the plane (reference 12, Appendix G) is applied on the domain exterior to FB. But the radiation condition discussed above should be included. In order to do this the variational statement must be changed so that variations in r/yield the correct boundary condition. Details are given by many authors, see, for example, Bettess and Bettess. 21 After some manipulation the final functional for the exterior is -~1 [ CCg(~7?]s)T~rT] s -- w ~ (ZTc]csg) ] 2 dx dy
II -- / f
b Oil i s d y + Jl CCg I-~xrl b
C
O1]i ~7s dx 1 + -~ If -~y
ikccgOTS)2 dF
(11.13)
d
The influence of the incident wave is thus to generate a 'forcing term' on the boundary 1-'n. For two of the most popular methods for dealing with exterior problems, linking to boundary integrals and infinite elements, the 'damping' term in Eq. (11.13), corresponding to the radiation condition, is actually irrelevant, because both methods use functions which automatically satisfy the radiation condition at infinity.
11.6.3 Incident waves, domain integrals and nodal values It is possible to choose any known solution of the wave equation as the incident wave. Usually this is a plane monochromatic wave, for which the elevation is given by ~Tt = a0 exp[ikr cos(0 - 7)], where 7 is the angle that the incident wave makes to the positive x axis, r and 0 are the polar coordinates and a0 is the incident wave amplitude. On the boundary FB, we have two types of variables, the total elevation, ~Tr, on the interior, and r/R, the radiation elevation, in the exterior. Clearly the nodal values of r/in the finite element model must be unique, and on this boundary, as well as the line integral, of Eq. (11.13), we must transform the nodal values, either to r/r or to r/R. This can be done simply by enforcing the change of variable, which leads to a contribution to the 'fight-hand side' or 'forcing' term. 21
323
324
Long and medium waves
There are several methods of dealing with exterior problems using finite elements in combination with other methods. Some of these methods are also applicable to finite differences. The literature in this field has grown enormously in the past few years, and this section cannot begin to pretend to be comprehensive. A book could be written on each of the sub-headings below. The monograph by Givoli 22 is devoted exclusively to this field and gives much more detail on the competing algorithms. It is a very useful source and gives many more algorithms than can be covered here. The book edited by Geers, 23 from an IUTAM symposium, gives a very useful and up-to-date overview of the field. Methods for exterior Helmholtz problems are also discussed by Ihlenburg in his monograph. 24 Four of the main methods are listed below. The first three are usually local in their effect. The last is always global, linking together all the nodes on the exterior of the finite element mesh. 1. 2. 3. 4.
Local Non-Reflecting Boundary Conditions (NRBCs). Sponge layers, Perfectly Matched Layers (PMLs). Infinite elements. Linking to exterior solutions, both series and boundary integral [also called Dirichlet to Neumann mapping (DtN)].
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The term comes from the mathematical literature. These conditions are also called boundary dampers by engineers because of the obvious physical analogy. As was seen in Chapter 18 of reference 12, we can simply apply the plane damper at the boundary of the mesh. This was first done in fluid problems by Zienkiewicz and Newton. 6 However, the low-order versions of the more sophisticated NRBCs or dampers proposed by Bayliss et al. 19' 20 can be used at little extra computational cost and a big increase in accuracy. The NRBCs are developed from the series given in Table 11.1. Details for low-order cases are given in reference 25. For the case of two-dimensional waves the line integral which should be applied on the circular boundary of radius r is a
=
2 g]2"~ 2
~S
ds
(11.14)
where ds is an element of distance along the boundary and c~ =
3/4r 2 - 2k 2 + 3ik/r 2 / r + 2ik
and
/3 -
1 2 / r + 2ik
(11.15)
For the plane damper, /3 - 0 and c~ - ik. For the cylindrical damper/3 - 0 and c~ - i k - 1/2r. The corresponding expressions for three-dimensional waves are different. Non-circular boundaries can be handled but the expressions become much more complicated. This is because as the higher-order terms are included in the
Local Non-Reflecting Boundary
Cylinder
10-
Conditions (NRBCs) 325
Incident waves -r
8642-
g
l-r 4%..
o
L._
e
l "li/
~ -2>
N -4--6--8-10
-16~ -34 ]~ -35
"/ i / !J
\
~
/ Outer radius dfofirnnit:element
Plane damper .......... Cylindricaldamper -----"-- Second-orderdamper
.,,IL
f
Fig. 11.2 Dampersolutionsfor wavesdiffractedby circularcylinder.Comparisonof relativeerrorsfor various outer radii, (ka = 1). Relativeerror= (abs (Tin)- abs (rla))labs (T/a). boundary condition, then higher-order derivatives in the normal direction are required. These can be transformed into derivatives in the circumferential direction by using the governing wave equation. However, even derivatives in the circumferential direction pose difficulties. So NRBCs based on the Bayliss expressions have, in practice, been limited to the lowest few terms. Some results are given by Bando et al. 25 Figure 1 1.2 shows the waves diffracted by a cylinder problem for which there is a solution, due to Havelock. 26 In this case the second-order damper is the highest-order modelled. The higher-order dampers are clearly a big improvement over the plane and cylindrical dampers, for little or no extra computational cost. Other researchers, of whom we mention only Engquist and Majda, 27' 28 Higdon29, 30 and Hagstrom et al. 31' 32 have also proposed NRBCs. Although the derivations vary considerably the effect is similar, in that a hierarchy of boundary operators is defined, but the resulting terms are different to those of Bayliss et al. For the Helmholtz equation the general form of these NRBCs can be written as
(11.16)
326
Long and medium waves
where q5 is the field variable, n is the normal to the boundary and Cj is the j term in the boundary condition. The expressions for C j vary, depending upon whose theory is used. One approach is simply to use a range of waves with different angles of incidence. (Obviously the plane damper case arises when N = 1 and C1 is the wave speed.) All these sets of NRBCs have in common the problem of the escalating order of derivatives, arising from expansion of the products of the operator terms in Eq. (11.1 6). This has been a problem but Givoli 33' 34 has recently demonstrated how the high-order derivative difficulty can be side-stepped through the use of auxiliary variables. The most straightforward way of using auxiliary variables leads to an unsymmetric matrix for the boundary condition terms. But Givoli 34 proves that this unsymmetric matrix can always be transformed into a symmetric form and gives the necessary construction. The method results in the addition of auxiliary variables on the boundary of the mesh, but no other difficulties. Givoli expands Eq. (11.16), using auxiliary variables so that it becomes
(o o) +c
C-u,
(11.17)
-]- Cj-..~
Ul -- u2
(11.18)
+ Cj
uN-1 - uN
(11.19)
where uj are the auxiliary variables. The resulting matrix (which in general is unsymmetric) representing the effect of these auxiliary variables on the boundary is then transformed according to Givoli's procedure into a symmetrical matrix. The procedure works for both transient and harmonic problems. The literature on these boundary conditions has grown remarkably in recent years and this section has only been able to give an outline of the possibilities available. For further information the reader is referred to the book by Givoli 22 and the volume edited by Geers, 23 which gives access to recent developments. The papers in the Geers volume by Bielak, Givoli, Hagstrom, Hariharan, Higdon, Pinsky and Kallivokas should be consulted.
11.8.1
Sponge layers, Perfectly Matched Layers or PMLs
As was seen earlier, non-reflecting boundary conditions (NRBCs) attempt to absorb the outgoing wave on the boundary of the computational domain. An obvious extension of this idea is to absorb the wave over an artificial domain, external to the domain of interest. It is intuitively obvious that if the domain is made large enough and the correct damping is inserted, then the wave energy reflected back into the domain of interest must become very small. In surface waves the physical analogy is to the energy absorbers, often made of horse hair, at the ends of wave tanks. The theoretical
Mapped periodic (unconjugated) infinite elements 327 developments in this field go under two names, sponge layer which tends to be used in hydraulic computations and meteorology, and perfectly matched layers or PMLs, in electomagnetics, or Maxwell's equations, and related fields such as acoustics, quantum mechanics and elastodynamics. Some authors do use both terms. The first paper on the method appears to be that of Larsen and Dancy 35 in 1983. The PML was first applied to the Maxwell equations by Brrenger 36' 37 in 1994. In this method a damping factor is introduced in each equation in those places where a normal spatial derivative appears. That is the governing equations are modified so that in the sponge layer or PML region a damping term is introduced. The damping factor is selected in a semi-empirical way and typically varies in space. More details are given in references 38 and 39 and in the papers by Monk and Collino, Hayder and Driscoll in reference 23.
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iiiiiHi
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Infinite elements are described in the book by Bettess, 4~ which although out of date can be used as an introduction to the topic. More recent reviews are by Astley 41 and Gerdes. 42 The methods described in reference 12, Chapter 5, can be developed to include periodic effects. This was first done by Bettess and Zienkiewicz, using so-called 'decay function' procedures and they were very effective. 11,43 Comparison results with Chen and Mei 44' 45 for the artificial island problem are shown in Fig. 16.6 of reference 12. Later 'mapped' infinite elements were developed for wave problems, and as these are more accurate than those using exponentials, they will be described here.
The theory developed in reference 12, Chapter 5, for static infinite elements will not be repeated here. Details are given in references 11, 21, 40--43, 46-50. Finite element polynomials of the form P = ao + OLI~ "~ 0~2~ 2 -'~ 9 9 9 become
P = /~0 + -7 /~1 + ~5 /~2 + " "
(11.20)
in which/3 i can be determined from the c~s and a. If the polynomial is zero at infinity then/30 = 0. Many exterior wave problems have solutions in which the wave amplitude decays radially like 1/r (and higher-order terms) and an advantage of this mapping is that such a decay can be represented exactly. In some cases, however, the amplitude decays approximately as 1/V/-7, and this case needs a slightly different treatment. Accuracy can be increased by adding extra terms to the series (11.20).
11.10.1 Introducing the wave component In two-dimensional exterior domains the solution to the Helmholtz equation can be described by a series of combined Hankel and trigonometric functions, the simplest
328
Long and medium waves
solution to the Helmholtz equation being Ho(kr). For large r the zeroth-order Hankel function oscillates roughly like cos(kr) + i sin(kr), while decaying in magnitude as r -1/2. A series of terms 1/r, 1/r 2, etc., generated by the mapping, multiplied by r 1/2 and the periodic component exp(ikr) will be used to model the r -1/2 decay. The shape function is thus N (~, 7/) - M(~, ~])r 1/2 exp(ikr)
(11.21)
where r = A / (1 - ~). The shape function in Eq. (11.21) will now be, for compatibility with the finite elements,
(2)
1/2(
A ) 1/2 exp ( i --~-) k A exp ( i k A 1-
N (~, ~7) -- M(~, rl)
1
(11.22)
In the improved version of this element, 21 the constant, A, varies within the element. A is now determined on each radial line from the positions of the nodes. It is interpolated between these values. The original mapped infinite element did not include the possibility of varying the mapping, so that the infinite elements had to be placed exterior to a cylinder or sphere. There was also an uncertainty about the integrations in the infinite radial direction, which was resolved by Astley et al. 5~This arose because the boundary terms at infinity were incompletely stated, although the element, as presented in references 48 and 49, is correct. Following the introduction of a shape function of the form given in Eq. (11.22), the standard finite element methodology is used. That is to say the weighted residual or variational expression is formed and integrated over the infinite domain. The only novelties are the oscillatory nature of the shape function, and the infinite extent of the domain. Mapped wave envelope infinite elements were later developed, using the same methodology, but with a complex conjugate weighting (see Sec. 11.11). Later still Astley et al., 51 Cremers and Fyfe and Coyette 52' 53 generalized the mapping of these wave envelope infinite elements, so that it was no longer necessary to place them exterior to a sphere or cylinder. After this work Bettess and Bettess 21 generalized the original mapped wave infinite elements. Figure 11.3 shows some results from the diffraction of waves by an ellipse, for which there is an analytical solution. i',',',i':,'i',i'',~,',:i~i~iiii~, ,iiiiii'~i ::::'~~i~i'~,~','i::i,:::,~,~',~i,i,i,i~,,~',',i~,,i,:~,'~,7~,i',~::'i':,'i~,~ii~,',i i',i{i~,~,~~,~,i',li',''i,~iliif', ,i:::::' i!:::::i~' i i il;,~ili',i~:i~i'~:'~,'i~il'~i,~,ii!~' i,ii'i,il!',ii~,',~ii i'~iiiii'i,i::::::ii~,iliii'~i'~i',ili'~i~!i',:i:'~,',:i~~:i~:~i: '{ii~i~,i~i~i~iii'~il,!'i~i,!',iiii, i~,!:,iiil{i}~,i'i,ii~ !i',i',{iii'~,i'il,'i,'!~{iii!l',!i'~,i"::: :iii~i i '~i~,ii'~{'i'i',~'i~',ii'~:i:~~iiii'"'~,',iil!i',!ii','i,!iii',i~'~ilf~ii ',i',iiiiiif::::iiiiii~i~,ili!ilili{i~~i~,;~i'i~li'i~~ilii''~'i,ii'i~~,i',iili~,!i'i~iiiii~~ii~!iiiiii~,',::'!i~~'~ii',i'i',,iiiilil'~'~i'i~'~iii'iiiii',i',i~,iiiii'',?!i!i i i{i''~,','.~,.i... i~~'~,'~i'iilii, ii',i'~,'~iiii~, ',ili!ii::iiill iiiiiii~iiiiliiiiiiii:~ii::~~:iiilii~':i':!iil!ii'~ii~,i iii'iiii',i'~,ii{li'i~li~,iii~,i'i~i,',i'ii!~ili' , :i'~iiii'i!i''~,i!iiii!iii!ii',i~,ii:,ii',i:,i!i~,'~ii'ii',i,'i!i',i'~;',i',ii
iiiiiiiiiiiii~iiiiiiii!iiii!iiiiiii!ii iii ii!iiiiiiiiiiiiiiii~i~::iiiiiiii~i~i~iiiiiiiiliiiii
i iii!!ililiilil
iii
i
il
ii iiii
iii i i ,i iiiii::iiiiiii i iiiiii!iiii!ii i!!i!iiiiii!i!i!iiii!!iiii!!ii!ii !iliiiiii!iiiiiiiiiiiii !iiiiiiiiiiiiiiiiiiiiiiiiiiiliiiiiiiiiiiiiiiiiii{iiii' iiiii'~ii ~i'~iii
Bumett with Holford 54-56 proposed a completely new type of infinite element for exterior acoustics problems. This uses prolate or oblate spheroidal coordinates, and separates the radial and angular coordinates. Bumett also further clarified the variational statement of the problem and explained in more detail the terms at the infinite boundary. It is known that a scattered wave exterior to a sphere can be written in spherical polar coordinates as
P--
e-kr ~ Gn(O,q~,k) r
n---0
rn
(11.23)
Ellipsoidal type infinite elements of Burnett and Holford 329
Fig. 11.3 Real part of elevations of plane wave diffracted by an ellipse, of aspect ratio 2, Bettess.21
This proof was generalized to the case when the coordinates 0, r r are not simply spherical, but prolate or oblate spheroidal or ellipsoidal. There are several benefits to using such coordinate systems: 1. The volume integrals separate into radial and angular parts which can be carried out independently. This leads to economies in computation. 2. The radial integration is identical for every such infinite element, so that the only integration which needs to be carried out for every infinite element is along the finite element interface. 3. The radial integration is the only part containing the wave number. 4. The ellipsoidal coordinates can be used to enclose a large variety of different geometries in the finite element interior, while still retaining a guarantee of convergence in 3D. The angular shape functions are written in the conventional polynomial form. The radial shape functions take the form Ni =
e -ikr
m j=l
hij
(kr)J
(11.24)
The coefficients hij are given from the condition of circumferential compatibility between adjacent infinite elements. There is effectively no difference between this radial behaviour and that originally proposed in the mapped infinite wave elements by Bettess et al. 48' 49 The difference in the infinite element methodology lies in the fact that the radial variable, r, is now in ellipsoidal coordinates. Burnett and Holford 54-56 give
330
Long and medium waves 2
ka = 100 Shell quadratic through thickness
1
~o n" -1
-2
1
I
2
150
180
ka = 100 Shell quadratic through thickness
a~ 1
K
~o
e.-
_E
-1
0
30
60
90 120 Polar angle, 0
150
180
Fig. 11.4 Waves scattered by an elastic sphere for ka = 100, Burnett and Holford. s6
the necessary detailed information for the element integrations and the programming of these infinite elements, together with some results. The analytical expressions are too long to include here. The elements have been used on submarine fluid-structure interaction problems, and substantial efficiencies over the use of boundary integral models for the scattered waves have been claimed. In one case Burnett states that the finite and infinite element model ran for 7 hours on a workstation. His projected time for the corresponding boundary element model was about 3000 hours, the infinite elements giving a dramatic improvement! The Burnett elements have been tested up to very short-wave cases, up to k a = 1 O0 for an elastic sphere diffraction problem, which is shown in Fig. 11.4.
~ ~!!i~2 `~v~, ~~~~n ~~`~!~p~ ~ ` ~~ ` c~!Ug~t~d), ~ ~ ~ ~ ~ inf,n,t~~!~~.ts Astley introduced a new type of finite element, in which the weighting function is the complex conjugate of the shape function. 57' 58 The great simplification which this
Wave envelope (or conjugated) infinite elements 331 _
Incident mode number = 1 Reduced frequency ka = 11 Spiral mode number m~ = 8
,
2
"
" "'
'.-.. ~." 1
"
0
............ Conventional FEM Wave envelope FEM
s ,,
-"~ "',
..',
'~
.\-
A
',
" " , , "" " ' \ ' , , ,' ~'7,?,', , . : . . . . . ~,
-' ,
2
Plane wavelength
4 z/a
6
8
Fig. 11.5 Computed acoustical pressure contours for a hyperbolic duct (/9o = 70 ~ ka = 11, m 4, = 8). Conventional and wave envelope element solutions, Astley. 58
introduces is that the oscillatory function exp(ikr) cancels after being multiplied by e x p ( - i k r ) , and the remaining terms are all polynomials, which can be integrated using standard techniques, like Gauss-Legendre integration (see reference 12, Chapter 5). This type of element was originally large (i.e. many wavelengths in extent), but not infinite. Figure 11.5 shows an example from acoustics, that of acoustical pressure in a hyperbolic duct. Good results were obtained despite using a relatively coarse mesh. Astley's shape function was of the form
Ni (r, O) ri e_ik(r_ri ) r
(11.25)
where Ni is the standard shape function. The weighting function is thus
Ni (r, O) ri eik(r_ri ) r
(11.26)
Bettess 59 showed that for a one-dimensional synthetic wave type equation the infinite wave envelope element recovers the exact solution. The element matrix is now hermitian rather than symmetric (though still complex), which necessitates a small alteration to the equation solver. (There are not usually any problems in changing standard profile or front solvers to deal with complex systems of equations.) Unfortunately the problem tackled by Bettess did not include the essential feature of physical waves, in two and three dimensions. Later workers applied the wave envelope concept to true wave problems. In this case it can be shown that if the weighting function is simply the complex conjugate of the shape function, terms arise on the boundary at infinity.* This is discussed by Bettess. 4~ The terms can be evaluated, but they are not symmetrical (or hermitian), and therefore impose a change of solution technique. An alternative, which *Some writers, particularly mathematicians, prefer to call the usual wave infinite elements, unconjugated infinite elements, and the Astley type wave envelope infinite elements, conjugated infinite elements.
332
Long and medium waves
eliminates the terms at infinity, was proposed by Astley et al. 51 In this a 'geometrical factor' is included in the weighting function, which then takes the form
Ni (r, o) ( ri ) 3 eil~(r-ri)
(11.27) r It has been shown that this form of weighting functions gives very good results. Such wave envelope infinite elements have been further developed by Coyette, Cremers and Fyfe.52, 53 These elements have incorporated a more general mapping than that in the original Zienkiewicz et al. mapped infinite wave element. Cremers and Fyfe allow the mapping to vary in the local ( and r/directions.
The use of a complex conjugate weighting in the wave envelope infinite elements means that the original variational statement, Eq. (11.22), must be changed to allow the use of the different weighting function. This gives rise to a number of issues relating to the nature of the weighted residual statement and the existence of various terms. These issues were touched on by Bettess, 4~ but have been subsequently subjected to more detailed study. Gerdes and Demkowitz 6~ 61 analysed the wave envelope elements, and subsequently the wave infinite elements. 62 Some of this work is restricted to spherical scatterers. Other analysis is carried out by Shirron and Babu~ka, 63, 64 who reveal a somewhat paradoxical result. The original (unconjugated) infinite elements give better results in the finite element mesh, but worse results in the infinite elements themselves. But the wave envelope (conjugated) elements give worse results in the finite elements, and better results in the far field. This result, which is ascribed to ill-conditioning, does seem to be counterintuitive. Astley 41 and Gerdes 42 have also surveyed current formulations and accuracies. Infinite elements have traditionally been used with relatively small numbers of radial terms. Recent investigations have addressed the problems of ill-conditioning and accuracy of infinite elements when the number of terms in the radial direction is increased. 65-68 In general it is found that the more radial terms that are retained, the worse the conditioning of the infinite element. This is similar to the effect noted in plane wave basis finite elements, which are discussed in Chapter 12. It has been suggested by Dreyer and von Estorff 68 that the ill-conditioning effect can be reduced by the use of Jacobi polynomials in the radial direction.
11.13.1 Other applications Infinite elements have been applied to a large range of applications, and it is impossible to survey the field in the scope of this chapter. The reader is directed to references 40, 41 and 42. An interesting application of infinite elements to Maxwell's equations by Demkowitz and Pal 69 is worthy of mention.
~i~i~iiii!i~i!i~'i!~~ii~'!ii'ii~iiiiii!i~ii!ii~'ii~:~i~~i i~i:i~!ii'~i:i~iiii!ii~iii!i~iii~i~iii~ii!ii~ii~ii~i:i~ii~i~ii~i~ii~i~fii~i~i!i~ii~i~ii!i~iii~~i~i!~ii~i~i~!i~i~i!~iii!i~ii~i!~:i:~~iiiii~iiii~i~ii~iiii~~ii~i~~i ~ii!i~i!iii!ii!ii!iiii!i~i~i!iii~ii~ii!~ii~i~iiiiiii~iii~i~i~i~ii~i~iiii Harari, with various co-workers, 7~ has developed infinite elements for the Helmholtz equation in two and three dimensions. The elements are sectors, bounded by radial
Convection and wave refraction
lines defined by Os and Os+l, and the arc r = R. Then AOs = Os+l - Os. Hankel functions are used to describe the radial behaviour and the circumferential behaviour is modelled using linear polynomials in 0. For a two-noded element the shape functions can be written N1 --
Ho(kr)
Os+l - 0
no(kR)
AOs
and
Ho(kr) 0 - Os no(kR) AO~
N1 =
(11.28)
The advantage of this formulation is that because the Hankel functions are solutions of the Helmholtz equation the integrations over the infinite elements can be eliminated. Continuity between the finite element domain and the infinite element domain is enforced weakly, as is the continuity between adjacent infinite elements. Harari shows that the matrices which arise for the case of linear infinite elements, with an orientation as specified, are
kRH~(kR) AOs [ 2 Ho(kR) 6 1
]]
2
(11.29)
for the finite/infinite element interface and
k R -~s
-1
(11.30)
1
where
C~oo=
fo ~ Ho(kr)Ho(kr) dr Ho(kR)Ho(kR) r
(11.31)
for the boundaries between the infinite element radial boundaries. He also g o e s o n to develop higher-order infinite elements of similar type. He gives results for some academic type problems.
Conventional wave finite elements will deal satisfactorily with wave refraction caused by local changes in wave speed. This is seen in the example of Fig. 11.6, where changes in water depth lead to local changes in wave speed. In exterior domains infinite elements have difficulties if the problem has a wave speed which is a function of position. There are also difficulties with linking to exterior solutions, or DtN methods, if an analytical solution or Green's function is used, since these almost invariably assume constant wave speed, in which case the Helmholtz equation is homogeneous. However, acoustic waves in inhomogeneous media have been solved using finite elements in conjunction with large (but finite) wave envelope elements by Astley and Eversman. TM In this case the governing equation is the inhomogeneous wave equation = 0 where p is the density, c is the wave speed and p* is the acoustical pressure.
(11.32)
333
o.
o ~
-~:
\
,r_ L.__~
o
/
o
o.
1/
c
:
cJ
c-
.E
._o
0 4..-,
q,D
Transient problems 335 Wave refraction can also occur because the waves are superposed upon a underlying flow field. Important applications are surface waves on water currents and sound waves on air flows, for example in aircraft noise. This case is more difficult and the general equations become complicated. See, for example, Lighthill. 4 For the case of sound waves superposed on a flow field Astley 75 has applied wave envelope finite elements to problems in which a substantial amount of convection is present. In this case the governing equation is no longer the Helmholtz equation. A detailed discussion is beyond the scope of this chapter.
iiiiiiiii iiii i !iiii
iiiiiiiii i il iii ii iiiiiii i i iiiiiiiiiiiiiiiiiiiiiiii i i! iiiiiiiiii i i i
i!!iiiiiiiiiiiiiii!iii~iiii~i~ii~iEii!!i!~i~iiiii~!~i!ii~ii!~i~iiiiiii~ii~ii~!i!~iii~J~!ii!ii!!~i!!~!!!~i~ii~i~!i~iiiii~iiiiiiiiiiii~!~i~!~ii~i!iii~i!!ii~i~i~iiiii~i~i!iiiii!~!i~iii~iii~i~i!~iii!~i~iiii~!!~i~!~ii~!iiiii
Recently Astley 76-78 has extended his wave envelope infinite elements, using the prolate and oblate spheroidal coordinates adopted by Burnett and Holford, 54-56 and has shown that they give accurate solutions to a range of transient wave problems. With the geometric factor of Astley, which reduces the weighting function and eliminates the surface integrals at infinity, the stiffness, K, damping, C, and mass M matrices of the wave envelope infinite element become well defined and frequency independent, although unsymmetric. This makes it possible to apply such elements to unbounded transient wave problems. Figure 11.7 shows the transient response of a dipole. More results from the application of infinite elements to transient problems are given by Cipolla and Butler, 79 who created a transient version of the Burnett infinite element. There appear to be more difficulties with such elements than with the wave envelope elements, and a consensus that the latter are better for transient problems seems to be emerging. Dampers and boundary integrals can also be used for transient problems. Space is not available to survey these fields, but the reader is directed, again, to Givoli 22 and Geers. 23 One set of interesting results was obtained using transient dampers by Thompson and Pinsky. 8~ 1.5 1.0 ~ P ( C )
~0.5 o
8~
0
.~_ -0.5
I
i~ -1.0 -1.5
0
I 0.5
I I I 1.0 1.5 2.0 Dimensionless time T (= tD/c)
Fig. 11.7 Transient response of a dipole, Astley.6s
I 2.5
3.0
336
Longand medium waves
A general methodology for linking finite elements to exterior solutions was proposed by Zienkiewicz et al., 81'82 following various a d h o c developments, and this is also discussed in reference 12, particularly in Chapter 10. The linking of interior and exterior solutions is also sometimes called D i r i c h l e t to N e u m a n n or DtN mapping. Under this title it is discussed by many authors, including Givoli. 22 The exterior solution can take any form, and those chiefly used are (1) exterior series solutions and (2) exterior boundary integrals, although others are possible. The two main innovators in these cases were Berkhoff, l~ 83 for coupling to boundary integrals, and Chen and Mei 44' 45 for coupling to exterior series solutions. (Although there are earlier papers on the linking of finite elements and exterior solutions.) Astley 84 demonstrates that the Chen and Mei methods are effectively the same as what has been more recently termed the DtN mapping, at any rate for the wave equation. Although the methods which have been proposed for linking finite elements to exterior series solutions and boundary integrals are quite different in detail, it is useful to cast them in the same general form. More details of this procedure are given in reference 81. Basically the energy functional given in Eq. (11.23) is again used. If the functions used in the exterior automatically satisfy the wave equation, then the contribution on the boundary reduces to a line integral of the form
n = ~
(11.33)
O~dr
It can be shown 16' 81,82 that if the free parameters in the interior and exterior are b and a respectively, the coupled equations can be written IK
~T]{i}+{I}={I}
(11.34)
where K j i = -~
[(PNj)Ni+ Nj(PNi)]dF
and
f
dF
Kji -- ./r[(PNj)(Ni)]
(11.35) In the above P is an operator giving the normal derivative, i.e. P = 0/0n, l~l is the finite element shape function, N is the exterior shape function, and K corresponds to the normal finite element matrix. The approach described above can be used with any suitable form of exterior solution, as we will see. All the nodes on the boundary become coupled.
11.17.1 Linking to boundary integrals BerkhofflO, 83 adopted the simple expedient of identifying the nodal values of velocity potential obtained using the boundary integral, with the finite element nodal values.
Linking to exterior solutions (or DtN mapping) This leads to a rather clumsy set of equations, part symmetrical, real and banded, and part unsymmetrical, complex and dense. The direct boundary integral method for the Helmholtz equation in the exterior leads to a matrix set of equations O~ A~7 = B---:' On
(11.36)
(The indirect boundary integral method can also be used.) The values of r/and o'hT/0n on the boundary are next expressed in terms of shape functions, so that r / ~ f / = N~7
and
m0n~,~~0n0fl= M ( ~ - }
(11.37)
N and M are equivalent to N in the previous section. Using this relation, the integral for the outer domain can be written as lq = ~
MTN~7dr
(11.38)
where F is the boundary between the finite elements and the boundary integrals. The normal derivatives can now be eliminated, using the relation (11.36), and ~7 can be identified with the finite element nodal values, ~7, to give
rI=~1 bT(B-1A)T f r MTN dFb
(11.39)
Variations of this functional with respect to b can be set to zero, to give 0b = 2
(B -~A)
MTN dF +
(B -~A)
MTN dF
b = I(b
(11.40)
where I( is a 'stiffness' matrix for the exterior region. It is symmetric and can be created and assembled like any other element matrix. The integrations involved must be carried out with care, as they involve singularities. Results obtained for the problem of waves refracted by a parabolic shoal are shown in Fig. 5.6 of reference 82.
11.17.2 Linking to series solutions Chen and Mei 44,45 took the series solution for waves in the exterior, and worked out explicit expressions for the exterior and coupling matrices, K and K, for piecewise linear shape functions, N, in the finite elements. The series used in the exterior consists of Hankel and trigonometric functions which automatically satisfy the Helmholtz equation and the radiation condition: --
^
m
~7-- ~ j=O
Hj(kr)(aj cos jO -4- ~j sin jO)
(11.41)
337
338
Longand medium waves
The method described above leads to the following matrices.
"2H~ ... KT
__
-knLc
2H~ .-. 2H~ ... "
H~(cosnOp + cosrtO1) H n'(cos nO1 + cos nO2) H n'(cos nO2 + cos nO3)
o . .
...
... ...
"
o
.
.
H'(cosn01,_l + cosn0v)
H'(sinnOp_~ + sinn0v)
9
2H~ ...
H~(sinnOp + sinnO~) H'n(sin nO1 + sin nO2) H'(sin nO2 + sin nO3)
. . 9
...
(11.42)
g-Trrkh{diag[2HoH~
H;H1
H;H1
...
H'sH~ H~Hs]}
(11.43)
where m is the number of terms in the Hankel function series, r is the radius of the boundary, Lc is the distance between the equidistant nodes on Fc, p is the number of nodes, and Hn and H n' are Hankel functions and derivatives evaluated with argument
(kr).
Other authors have worked out the explicit forms of the above matrices for linear shape functions, and also it is possible to work them out for any type of shape function, using, if necessary, numerical integration. It will be noticed that the matrix I~ is diagonal. This is because the boundary l~s is circular and the Hankel functions are orthogonal. If a non-circular domain is used, I( will become dense. Chen and Mei 44 applied the method very successfully to a range of problems, most notably that of resonance effects in an artificial offshore harbour, the results for which are shown in reference 12, Chapter 16, Fig. 16.6. The method was also utilized by Houston, 8~ who applied it to a number of real problems, including resonance in Long Beach Harbour, shown in Fig. 1 1.8.
As has already been described, when the water is deep in comparison with the wavelength, the shallow water theory is no longer adequate. For constant or slowly varying depth, Berkhoff's theory is applicable. Also the geometry of the problem may necessitate another approach. The flow in the body of water is completely determined by the conservation of mass, which in the case of incompressible flow reduces to Laplace's equation. The free surface condition is zero pressure. On using Bernoulli's equation and the kinematic condition, the free surface condition can be expressed, in terms of the velocity potential, r as
02r
0r
&---T+ g-~- + 2 ( v r
[V
0r
1
(-0-7)] + 2 ( V r 1 6 2 1 6 2
=0
(11.44)
where the velocities are lg i - - O r i . This condition is applied on the free surface, whose position is unknown a priori. If only linear terms are retained, Eq. (11.44) becomes, for transient and periodic problems,
Oq2~
~-
Ot 2
O~
-=--
~- g o z
-
0
or
0(]5 ~Z
C02
= --q5 g
(11.45)
Three-dimensional effects in surface waves
/
Note: Number of node points = 1701 Number of elements = 2853
3000
Scale 0 3000 6000
feet
(a) Finite element grid, grid 3 Fig. 11.8 Finite element mesh and wave height magnification for Long Beach Harbour, Houston.8S
which is known as the Cauchy-Poisson free surface condition. It was derived in terms of pressure in reference 12, Chapter 18 as Eq. (18.13). Three-dimensional finite elements can be used to solve such problems. The actual three-dimensional element is very simple, being a potential element of the type described in reference 12, Chapter 7. The natural boundary condition is 0qS/0n = 0, where n is the outward normal, so to apply the free surface condition it is only necessary to add a surface integral to generate the •2 / g term from the Cauchy-Poisson condition [see Eq. (18.13) of reference 12]. Twodimensional elements in the far field can be linked to three-dimensional elements in the near field around the object of interest. Such models will predict velocity potentials, pressures throughout the fluid, and wave elevations. They can also be used to predict fluid-structure interaction. All the necessary equations are given in reference 12, Chapter 18. More details of fluid-structure interactions of this type are given by Zienkiewicz and Bettess. 86 Essentially the fluid equations must be solved for incident waves, and for motion of the floating body in each of its degrees of freedom (usually six). The resulting fluid forces, masses, stiffnesses and damping are used in the equations
339
340
Longand medium waves
/
~o ,~'c,
6
...,2~2
2
/
5/
(b) Contours of wave height amplification, grid 3. 232-s wave period
Fig. 11.8 Continued.
of motion of the structure to determine its response. Figure 11.9 shows some results obtained by Hara e t al. 87 for a floating breakwater. They obtained good agreement between the infinite elements and the methods of Sec. 11.17.
11.18.1 Large-amplitude water waves There is no complete wave theory which deals with the case when q is not small in comparison with the other dimensions of the problem. Various special theories are invoked for different circumstances. We consider two of these, namely, large wave elevations in shallow water and large wave elevations in intermediate to deep water. We have discussed a similar problem in Chapter 10.
11.18.2 Cnoidal and solitary waves The equations modelled in Chapter 10 can deal with large-amplitude waves in shallow water. These are called cnoidal waves when periodic and solitary waves when the
Three-dimensional effects in surface waves
c(9
10
-
0
O
oo
~~Ei~nfunction
(9
c-
I o
._o 0.5
.
E ffl
~
1
method
Present method (twodimensional model)
c-
(9
L/V 1 2
o
I
I
I
I I t~ I I 3 4 5 6 7 Wave period T (s) I
I
I
I
I 8 I
I 9 I
10 I
01 05 10 20 30 40 50 60 70 80 Wavelength ;L/B
Incident wave
Infinite elements Three-dim ~ ~ ~ ~ finite elements ,r ~,~,..=-'] Bed
~" surface Free ~8 m
/~~nit:
m
14
Incident wave
~=349m
q1=Im Fig. 1 1.9 Element mesh, contours of wave elevation and wave transmission coefficients for floating breakwater, Hara.87
341
342 Longand medium w a v e s period is infinite. For more details see references 1-4. The finite element methodology of Chapter 10 can be used to model the propagation of such waves. It is also possible to reduce the equations of momentum balance and mass conservation to corresponding wave equations in one variable, of which there are several different forms. One famous equation is the Korteweg-de Vries equation, which in physical variables is k X//-~
( 1+
+
h:X ~ ~ x 3
= 0
(11.46)
This equation has been given a great deal of attention by mathematicians. It can be solved directly using finite element methods, and a general introduction to this field is given by Mitchell and Schoombie. 88
11.18.3 Stokes waves When the water is deep, a different asymptotic expansion can be used in which the velocity potential, qS, and the surface elevation, ~7, are expanded in terms of a small parameter, e, which can be identified with the slope of the water surface. When these expressions are substituted into the free surface condition, and terms with the same order in e are collected, a series of free surface conditions is obtained. The equations were solved by Stokes initially, and then by other workers, to very high orders, to give solutions for large-amplitude progressive waves in deep water. There is an extensive literature on these solutions, and they are used in the offshore industry for calculating loads on offshore structures. In recent years, attempts have been made to model the second-order wave diffraction problem, using finite elements, and similar techniques. The first-order diffraction problem is as described in Sec. 11.8.1. In the second-order problem, the free surface condition now involves the first-order potential.
First order
Second order
005 (1)
(..02
0z
g
Oq~(2)
(.02
0z
g
~(1) ~. 0
(11.47)
q5(2) -- O~(D 2)
(11.48)
and
c ~ ) m (YDI _ (2) -~- OgDD (2)
c~(2) _ - i co qS~> ( 0 205~1) DI
2g
-0--~
-[-"12WV ~1) V~(D1) g
044'~ Oz
-i
4~'~
r 2 u= m g
(11.49)
u& (11.50)
Three-dimensional effects in surface waves
_(2)
-i
Cr'DD
2gV'O
Og2
u
+ i ~ (VqS~))2 g
(11.51)
The second-order boundary condition can be thought of as identical to the first-order problem, but with a specified pressure applied over the entire free surface, of value a. Now there is no a priori reason why such a pressure distribution should give rise to outgoing waves as in the first-order problem, and so the usual radiation condition is not applicable. The conventional procedure is to split the second-order wave into two parts, one the 'locked' wave, in phase with the first-order wave, and the other the 'free' wave, which is like the first-order wave but at twice the frequency, and with an appropriate wavenumber obtained from the dispersion relation. For further details of the theory, see Clark et al. 89 Figure 11.10 shows results for the second-order wave elevation around a circular cylinder, obtained by Clark et al. Although not shown, good agreement has been obtained with predictions made by boundary integrals. Preliminary results, for wave forces only, have also been produced by Lau et al. 9~ A much finer finite element mesh is needed to resolve the details of the waves at second order. The second-order wave forces can be very significant for realistic values of the wave parameters (those encountered in the North Sea, for example). The first-order problem is solved first and the first-order potential is used to generate the forcing terms in Eqs (11.50) and (11.51). These values have to be very accurate. In principle the method could be extended to third and higher orders, but in practice the difficulties multiply, and in particular the dispersion relation changes and the waves become unstable. 4
Real, {~2~/(H2/4a) }
0.5
0.0 "; ....-0.5
9 "" .... 9
0.0
-1.%
o.o
-0.5
-1.5~
-1.o:;;ii
1 .o
:~.o.o
Incident wave ~
4-1.5
~
-1 .o
0.5
I
1.5,.
v..,o.o
2.5 .~
1.o"~ 1.5."~ ~
0.5
9
0.0"::~ -0.5 "-., -1.0"
9,
".%
0.5
Fig. 11.10
~ 0.0
"y~
0.0
..... '0.5 Imaginary {~leb(H214a)}
Second-order wave elevations around a cylinder - real and imaginary parts (Clark et a/.89).
343
344
Long and medium waves i ',i!i!i'~i',',i',',i'~,~',', i ',!ii',iig',',i',',',i','~, 0 ~,C! U~i!~ M~!i~i~,~ ~ , ~ k S
i ~i
~i'!'~~,',',i',~~,',~i',~,i',i~',i'-~, ,~,',~,',i~'~,'~i','~i,'i,i',i'~',',~i',i',',i',i',~~i,'~,'~ii','~,',',i','~'i,i'~',',i'~,','~','i~',',','~,i',','~,','',~,',',',',i':i',i',',',~i','~',',i',',',i',',',',',,'',','i,~,'~,, ~,~,i',',',',~,i~,~',',~i',i~' ,!'~,'~'~i~'~:'~i'~i',i~.ii~i'~',',i',!i
For waves which are linear and of medium or large length, compared with the size of the problem, in closed and unbounded domains, finite elements are in a fairly mature state. Improvements in element efficiency, such as spectral methods, which are discussed in Chapter 12, are not so critically important if there are few wavelengths in the problem. Incremental improvements in methods for dealing with the exterior domain continue to be made. As well as higher accuracy from the exterior models, a better mathematical understanding is being obtained. The position is not so well developed in the case of elastic waves, which have not been covered in this chapter, for lack of space. The multiple wave speeds in elasticity make some of the exterior domain methods (discussed in Secs 11.6 to 11.17) either impossible, or highly complicated in practice. There are some results on error indicators for wave problems, which there has not been space to cover. In general, error indicators do not give such a great pay-off as they do in static and potential problems, because usually the waves extend throughout the problem and making the mesh coarse anywhere may have the result of polluting the solution in other regions. See the earlier discussion and that in Chapter 12. Error bounds for the exterior methods are thinner on the ground. Non-linear waves, such as surface waves of large elevation which have been discussed in Sec. 11.18, remain a significant challenge. Anyone who has seen a wave break on a shoreline must be aware of how difficult it is to model that process on a computer. But apart from surface waves there are not really that many non-linear waves of great importance. Effects like shocks arising from bores on water, or shock waves and sonic booms in compressible flow have been dealt with in other chapters.
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H. Lamb. Hydrodynamics. Cambridge University Press, 6th edition, 1932. G.B. Whitham. Linear and Nonlinear Waves. John Wiley, New York, 1974. C.C. Mei. The Applied Dynamics of Ocean Surface Waves. Wiley, New York, 1983. M.J. Lighthill. Waves in Fluids. Cambridge University Press, 1978. G.M.L. Gladwell. A variational model of damped acousto-structural vibration. Journal of Sound and Vibration, 4:172-186, 1965. O.C. Zienkiewicz and R.E. Newton. Coupled vibrations of a structure submerged in a compressible fluid. Proc. International Symposium on Finite Element Techniques, Stuttgart, pp. 360-78, 1-15 May 1969. A. Craggs. The transient response of a coupled plate acoustic system using plate and acoustic finite elements. J. Sound Vibration, 15:509-528, 1971. R.J. Astley. Finite elements in acoustics. Sound and Silence: Setting the balance, Proc. Internoise 98, Christchurch, New Zealand, volume 1, pp. 3-17, 16-18 November 1998. P.M. Morse and H. Feshbach. Methods of Theoretical Physics, volumes 1 and 2. McGraw-Hill, New York, 1953. J.C.W. Berkhoff. Linear wave propagation problems and the finite element method. In R.H. Gallagher et al., editors, Finite Elements in Fluids, 1, pp. 251-280 Wiley, Chichester, 1975. O.C. Zienkiewicz and P. Bettess. Infinite elements in the study of fluid structure interaction problems. Proc. 2nd Int. Symp. on Comp. Methods Appl. Sci., Versailles, 1975, also published in J. Ehlers et al., editors, Lecture Notes in Physics, volume 58, Springer-Verlag, Berlin, 1976.
References 345 12. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method: The Basis and Fundamentals. Elsevier, 6th edition, 2005. 13. C. Taylor, B.S. Patil and O.C. Zienkiewicz. Harbour oscillation: a numerical treatment for undamped natural modes. Proc. Inst. Civ. Eng., 43:141-156, 1969. 14. I. Babu~ka, E Ihlenburg, T. Stroubolis and S.K. Gangaraj. A posteriori error estimation for finite element solutions of Helmholtz' equations. Part I: the quality of local indicators and estimators. Int. J. Num. Meth. Eng., 40(18):3443-3462, 1997. 15. I. Babu~ka, E Ihlenburg, T. Stroubolis and S.K. Gangaraj. A posteriori error estimation for finite element solutions of Helmholtz equations. Part II: estimation of the pollution error. Int. J. Num. Meth. Eng., 40(21):3883-3900, 1997. 16. O.C. Zienkiewicz, P. Bettess and D.W. Kelly. The finite element method for determining fluid loadings on rigid structures: two- and three-dimensional formulations. In O.C. Zienkiewicz, R.W. Lewis and K.G. Stagg, editors, Numerical Methods in Offshore Engineering, Chapter 4. John Wiley, 1978. 17. A. Sommerfeld. Th6orie math6matique de la diffraction. Math. Ann., 47:317-374, 1896. 18. A. Sommerfeld. Partial Differential Equations in Physics. Academic Press, New York, 1949. 19. A. Bayliss, M. Gunzberger and E. Turkel. Boundary conditions for the numerical solution of elliptic equations in exterior regions. SlAM Journal Appl. Math., 42:430-451, 1982. 20. A. Bayliss, M. Gunzberger and E. Turkel. Boundary conditions for the numerical solution of elliptic equations in exterior regions. ICASE Report No. 80-1, 1980. 21. J.A. Bettess and P. Bettess. A new mapped infinite wave element for general wave diffraction problems and its validation on the ellipse diffraction problem. Comp. Meth. Appl. Mech. Eng., 164:17-48, 1998. 22. D. Givoli. Numerical Methods for Problems in Infinite Domains. Elsevier, Amsterdam, 1992. 23. T.L. Geers, editor. Computational Methods for Unbounded Domains. Kluwer Academic Publishers, Dordrecht, 1998. 24. E Ihlenburg. Finite Element Analysis of Acoustic Scattering. Springer, New York, 1998. 25. K. Bando, P. Bettess and C. Emson. The effectiveness of dampers for the analysis of exterior wave diffraction by cylinders and ellipsoids. Int. J. Num. Meth. Fluids, 4:599-617, 1984. 26. T.H. Havelock. The pressure of water waves on a fixed obstacle. Proceedings of the Royal Society of London, A, 175:409-421, 1950. 27. B. Engquist andA. Majda. Absorbing boundary conditions for the numerical simulation of waves. Math. Comp., 31:629-652, 1977. 28. B. Engquist and A. Majda. Radiation boundary conditions for acoustic and elastic wave calculations. Comm. Pure Appl. Mathematics, 32:313-357, 1979. 29. R.L. Higdon. Absorbing boundary conditions for difference approximations to the multidimensional wave equation. Math. Comput., 47(176):437-459, 1986. 30. R.L. Higdon. Radiation boundary conditions for dispersive waves. SIAM J. Numer. Anal., 31:64100, 1994. 31. T. Hagstrom and T. Warburton. High-order radiation boundary conditions for time-domain electromagnetics using an unstructured discontinuous Galerkin method. In K. Bathe, editor, Computational Fluid and Solid Mechanics, pp. 1358-1363, Elsevier, 2003. 32. T. Hagstrom and S.I. Hariharan. A formulation of asymptotic and exact boundary conditions using local operators. Appl. Numer. Math., 27:403, 1998. 33. D. Givoli. High-order nonreflecting boundary conditions without high-order derivatives. J. Comp. Phys., 170:849, 2001. 34. D. Givoli and B. Neta. High-order non-reflecting boundary scheme for time-dependent waves. J. Comp. Phys., 186:24-46, 2003. 35. J. Larsen and H. Dancy. Open boundaries in short wave simulations - a new approach. Coastal Engineering, 7:285-297, 1983. 36. J.P. B6renger. A perfectly matched layer for the absorption of electromagnetic waves. J. Comp. Phys., 114:185-200, 1994.
346
Long and medium waves 37. J.E Brrenger. Perfectly matched layer for the FDTD solution of wave-structure interaction problems. IEEE Trans. Antennas Propagat, 44:110-117, 1996. 38. T.G. Shepherd, K. Semeniuk and J.N. Koshyk. Sponge layer feedbacks in middle-atmosphere models. J. Geophys. Res., 101:23447-23464, 1996. 39. S. Abarbanel, D. Gottlieb and J.S. Hesthaven. Well-posed perfectly matched layers for advective acoustics. J. of Comput. Phys., 154:266-283, 1999. 40. P. Bettess. Infinite Elements. Penshaw Press, Sunderland, 1992. 41. R.J. Astley. Infinite elements for wave problems: a review of current formulations and an assessment of accuracy. Int. J. Num. Meth. Eng., 49:951-976, 2000. 42. K. Gerdes. Infinite elements for wave problems. J. Comput. Acoustics, 8:43-62, 2000. 43. P. Bettess and O.C. Zienkiewicz. Diffraction and refraction of surface waves using finite and infinite elements. Int. J. Num. Meth. Eng., 11:1271-1290, 1977. 44. H.S. Chen and C.C. Mei. Oscillations and wave forces in an offshore harbor. Parsons Laboratory, Massachusetts Institute of Technology, Report 190, 1974. 45. H.S. Chen and C.C. Mei. Oscillations and wave forces in a man-made harbor in the open sea. Proc. lOth. Symp. Naval Hydrodynamics, Office of Naval Research, pp. 573-594, 1974. 46. O.C. Zienkiewicz, C. Emson and P. Bettess. A novel boundary infinite element. Int. J. Num. Meth. Eng., 19:393-404, 1983. 47. O.C. Zienkiewicz, P. Bettess, T.C. Chiam and C. Emson. Numerical methods for unbounded field problems and a new infinite element formulation. ASME, AMD, 46:115-148, New York, 1981. 48. P. Bettess, C. Emson and T.C. Chiam. A new mapped infinite element for exterior wave problems. In R.W. Lewis, P. Bettess and E. Hinton, editors, Numerical Methods in Coupled Systems. John Wiley, Chichester, 1984. 49. O.C. Zienkiewicz, K. Bando, P. Bettess, C. Emson and T.C. Chiam. Mapped infinite elements for exterior wave problems. Int. J. Num. Meth. Eng., 21:1229-1251, 1985. 50. R.J. Astley, P. Bettess and P.J. Clark. Letter to the editor concerning Ref. 128. Int. J. Num. Meth. Eng., 32(1):207-209, 1991. 51. R.J. Astley, G.J. Macaulay and J.P. Coyette. Mapped wave envelope elements for acoustical radiation and scattering. J. Sound Vibration, 170(1):97-118, 1994. 52. L. Cremers, K.R. Fyfe and J.P. Coyette. A variable order infinite acoustic wave envelope element. J. Sound Vibration, 171(4):483-508, 1994. 53. L. Cremers and K.R. Fyfe. On the use of variable order infinite wave envelope elements for acoustic radiation and scattering. J. Acoust. Soc. Amer., 97(4):2028-2040, 1995. 54. D.S. Burnett. A three-dimensional acoustic infinite element based on a prolate spheroidal multipole expansion. J. Acoust. Soc. Amer., 95(5):2798-2816, 1994. 55. D.S. Burnett and R.L. Holford. Prolate and oblate spheroidal acoustic infinite elements. Comp. Meth. Appl. Mech. Eng., 158:117-141, 1998. 56. D.S. Burnett and R.L. Holford. An ellipsoidal acoustic infinite element. Comp. Meth. Appl. Mech. Eng., 164:49-76, 1998. 57. R.J. Astley and W. Eversman. A note on the utility of a wave envelope approach in finite element duct transmission studies. J. Sound Vibration, 76:595-601, 1981. 58. R.J. Astley. Wave envelope and infinite elements for acoustical radiation. Int. J. Num. Meth. Fluids, 3:507-526, 1983. 59. P. Bettess. A simple wave envelope element example. Comms. Applied Num. Meth., 3:77-80, 1987. 60. K. Gerdes and L. Demkowitz. Solution of 3D Laplace and Helmholtz equations in exterior domains of arbitrary shape, using HP-finite-infinite elements. Comp. Meth. Appl. Mech. Eng., 137:239-273, 1996. 61. L. Demkowitz and K. Gerdes. Convergence of the infinite element methods for the Helmholtz equation in separable domains. Numerische Mathematik, 79:11-42, 1998.
References 347 62. L. Demkowitz and K. Gerdes. The conjugated versus the unconjugated infinite element method for the Helmholtz equation in exterior domains. Comp. Meth. Appl. Mech. Eng., 152:125-145, 1998. 63. J.J. Shirron. Solution of exterior Helmholtz problems using finite and infinite elements. Ph.D. thesis, University of Maryland, 1995. 64. J.J. Shirron and I. Babu~ka. A comparison of approximate boundary conditions and infinite element methods for exterior Helmholtz problems. Comp. Meth. Appl. Mech. Eng., 164:121139, 1998. 65. R.J. Astley. Mapped spheroidal wave-envelope elements for unbounded wave problems. Int. J. Num. Meth. Eng., 41:1235-1254, 1998. 66. R.J. Astley. Infinite elements for wave problems: a review of current formulations and an assesment of accuracy. Int. J. Num. Meth. Eng., 49:951-976, 2000. 67. R.J. Astley and J.-P. Coyette. Conditioning of infinite element schemes for wave problems. Int. J. Num. Meth. Eng., 17:31-41, 2001. 68. D. Dreyer and O. van Estorff. Improved conditioning of infinite elements for exterior acoustics. Int. J. Num. Meth. Eng., 58:933-953, 2003. 69. L. Demkowitz and M. Pal. An infinite element for Maxwell's equations. Comput. Methods Appl. Mech. Eng., 164:77-94, 1998. 70. I. Harari, P. Barai and P.E. Barbone. Higher-order boundary infinite elements. Comput. Methods Appl. Mech. Eng., 164:107-119, 1998. 71. I. Harari. A unified variational approach to domain based computation of exterior problems of time-harmonic acoustics. Applied Numerical Mathematics, 27:417-441, 1998. 72. I. Harari, P. Barai and P.E. Barbone. Numerical and spectral investigations of Trefftz infinite elements. Int. J. Num. Meth. Eng., 46:553-577, 1999. 73. I. Harari, P. Barai, P.E. Barbone and M. Slavutin. Three-dimensional infinite elements based on a Trefftz formulation. J. Comput. Acoustics, 9(3):381-394, 2001. 74. R.J. Astley and W. Eversman. Wave envelope elements for acoustical radiation in inhomogeneous media. Computers and Structures, 30(4):951-976, 1988. 75. R.J. Astley. A finite element, wave envelope formulation for acoustical radiation in moving flows. J. Sound Vibration, 103(4):471-485, 1985. 76. R.J. Astley. Transient wave envelope elements for wave problems. J. Sound Vibration, 192(1 ):245-261, 1996. 77. R.J. Astley, G.J. Macaulay, J.P. Coyette and L. Cremers. Three dimensional wave-envelope elements of variable order for acoustic radiation and scattering. Part 1. Formulation in the frequency domain. J. Acoust. Soc. Amer., 103(1):49-63, 1998. 78. R.J. Astley, G.J. Macaulay, J.P. Coyette and L. Cremers. Three dimensional wave-envelope elements of variable order for acoustic radiation and scattering. Part 2. Formulation in the time domain. J. Acoust. Soc. Amer., 103(1):64-72, 1998. 79. J.L. Cipolla and M.J. Buffer. Infinite elements in the time domain using a prolate spheroidal multipole expansion. Int. J. Num. Meth. Eng., 43(5):889-908, 1998. 80. L.L. Thompson and P.M. Pinsky. A space-time finite element method for the exterior structural acoustics problem: time dependent radiation boundary conditions in two space dimensions. Int. J. Num. Meth. Eng., 39:1635-1657, 1996. 81. O.C. Zienkiewicz, D.W. Kelly and P. Bettess. The coupling of the finite element method and boundary solution procedures. Int. J. Num. Meth. Eng., 11(1):355-375, 1977. 82. O.C. Zienkiewicz, D.W. Kelly and P. Bettess. Marriage h la mode- the best of both worlds (finite elements and boundary integrals). In Energy Methods in Finite Element Analysis, Chapter 5. Wiley, 1978. 83. J.C.W. Berkhoff. Computation of combined refraction-diffraction. Proc. 13th Int. Conf. on Coastal Engineering, Vancouver, 10-14 July 1972.
348
Long and medium waves 84. R.J. Astley. FE mode matching schemes for the exterior Helmholtz problems and their relationship to the FE-DtN approach. Comm. Num. Meth. Eng., 12:257-267, 1996. 85. J.R. Houston. Long Beach Harbor: numerical analysis of harbor oscillations. US Army Engineering Waterways Experimental Station, Vicksburg, MS. Report 1, Misc. Paper H-76-20, 1976. 86. O.C. Zienkiewicz and P. Bettess. Fluid-structure interaction and wave forces. An introduction to numerical treatment. Int. J. Num. Meth. Eng., 13:1-16, 1978. 87. H. Hara, K. Kanehiro, H. Ashida, T. Sugawara and T. Yoshimura. Numerical simulation system for wave diffraction and response of offshore structures. Mitsui Engineering and Shipbuilding Co., Technical Bulletin, TB 83-07, October 1983. 88. A.R. Mitchell and S.W. Schoombie. Finite element studies of solitons. In R.W. Lewis, P. Bettess and E. Hinton, editors, Numerical Methods in Coupled Systems, Chapter 16, pp. 465-488. John Wiley, Chichester, 1984. 89. P.J. Clark, P. Bettess, M.J. Downie and G.E. Hearn. Second order wave diffraction and wave forces on offshore structures, using finite elements. Int. J. Num. Meth. Fluids, 12:343-367, 1991. 90. L. Lau, K.K. Wong and Z. Tam. Nonlinear wave loads on large body by time-space finite element method. Computers in Engineering, Proc. of the Int. Conf. on Computers in Engineering 3, pp. 331-337. ASME, New York, 1987.
iiiiii
Short waves*
The problem of modelling short waves is dealt with in this chapter. Short waves are taken to be waves in which the wavelength is much smaller than any other parameters in the problem. The literature on this topic has mushroomed in recent years, and this chapter can only give a brief outline of ongoing research activity.
As was seen in the previous chapter, classical finite element methods can deal relatively easily with linear waves in two and three dimensions, when there are few wavelengths in the problem domain. But many problems of practical importance contain many wavelengths. The computational demands of conventional numerical methods for an accurate solution cannot be met. So in recent years, attention has turned to new types of finite elements for such problems. Similar special variations have also been developed for other numerical methods, such as boundary elements. Surveys of these methods are available. 1'2 A variety of new finite elements have been developed, and they will be surveyed briefly in this chapter. A common feature is that most incorporate more of our knowledge of waves and wave behaviour into the algorithm. The problem of short waves in a large domain is part of the general class of multi-scale problems. 3' 4 That is the problem of large scale, but the overall behaviour is significantly influenced by small-scale behaviour. Other important examples are the behaviour of structures composed of materials with microstructure, for example composites or concrete, and turbulent flows. In the latter the details of the flow, with very small length scales strongly influence the behaviour of the entire system.
A prime example of a short wave problem is that of radar waves incident upon an aircraft. The wavelength of the radio wave may be of the order of several mm, whereas the size of the aircraft is of the order of tens of m. Since about 10 degrees of freedom *Contributed fully by Peter Bettess, School of Engineering, University of Durham, UK.
350
Short waves
are needed per wavelength in conventional numerical models this gives a scale factor of over 1000. Classical finite element methods generate large systems of equations with literally astronomical numbers of unknowns for the solution of such problems. This makes them virtually intractable. If we imagine a computational domain around an aircraft of a cube of 20 metres per side and a wavelength of the incident wave, of 2 cm, it is possible to compute the size of the computational problem. The above model requires n - 1012 nodal variables, all complex. The semi-bandwidth of a finite element model would be b = 108 degrees of freedom, leading to a storage requirement for a classical direct solver of nb= 1020 complex words, or 1.6 x 1021 bytes. The number of complex arithmetic operations would be nb2/2 = 0.5 x 1027. Twenty metres is not large for a model size and 2 cm is not small for a radar wavelength. It corresponds to a frequency of about 15 GHz, and radar can operate at higher frequencies. So it can be seen that conventional methods, even on supercomputers, struggle to make an impact on such problems. But the solutions are of great practical interest. For most short wave problems the frequency domain solutions are of the most interest. Even in radar applications, where the radar beam is transient, it still endures for many wave cycles so the problem is really in the frequency domain. There are important short wave transient problems, but these are in the minority. There are essentially two approaches to frequency domain problems. The first is to iterate a general transient solution to a steady state. The second is to factor the frequency out of the problem and to operate on the resulting complex equations. 1. Time domain solutions. These methods imply some kind of stepping forward in time, usually based on a finite difference in time scheme. Many different schemes have been proposed. The advantage of this approach is that any transient problem can be solved, including the steady-state problem. If an explicit scheme is used, together with a diagonal mass matrix, then the computation involved in a time step is very small, and the storage requirements are very small. Such methods are explained in detail in reference 5 and in Chapter 3 of the present book. 2. Frequency domain solutions. These methods solve the response for a given frequency. All terms in the problem are factored by exp(iwt), as explained above. The advantage is that a single solution, of a complex system matrix, gives a complete solution at all times. The disadvantage is that they cannot directly solve transient problems, although Fourier transforms can be used to move from a set of frequency domain solutions to a transient solution, for linear problems. The complex system matrix to be solved, though sparse for finite elements, is very large and direct solutions are very computationally expensive. Such problems have been described in reference 5. In recent years the problem described above, that of the limitation in the short waves, has been subjected to attack from a number of different directions, which have shown considerable promise. It is fair to say that in the last ten years, the wavelengths in the scattering problems which can be solved have decreased dramatically. As a consequence so the frequencies have increased.
Recent developments in short wave modelling
351
The historical trend is towards tailoring the numerical algorithms to the known nature of the wave problem, and to use the best possible algorithms, for example in integrating the various matrices which arise.
It was earlier stated that about 10 degrees of freedom are needed to model each wavelength. However, that is an oversimplification. There are two sorts of error which arise from wave modelling using polynomials. These are local errors and pollution errors. Error estimates for these errors have been given by several authors. 6-9 Ihlenburg and Babu~ka 6 give expressions for the error, e, in the H l-seminorm, in a domain of unit length, as e < - CI(p)
(hk) m ~pp
+ C2(p)k
(h~p) mq-I
(12.1)
where p is the order of the finite element polynomial, h is the element size and m is the minimum of 1 and p, where l + 1 is the regularity of the exact solution. C1 and Cz are constants depending only upon p. The first term in Eq. (12.1) is the local error and the second term is the pollution error, which increases with k, even for (hk) fixed. If the domain is large the second term will come to dominate. It is evident from Eq. (12.1) that, other things being equal, it is desirable to have the highest possible order of polynomial in the finite element, if we seek to minimize the error for a given discretization. The pollution error can be thought of as an error in the numerical value for the wavelength. Obviously, the larger the number of wavelengths in a problem, the more the errors from this source will accumulate.
i!iiiii~i!~'i'~i:i':i~i!iii~i~i i~i~i i~"~'i~~i~:i'!l~il~ii~iii~lili~li!iii!i~i~~~' !~ii~i'~!:i~i~i~iii~i~i~ii~i~!~!~ii!i!~i~!i~i!i~i~ii~i~~Ii~~iii~ii~i~i~:~i:i~i!~i~i~ii~i~i~!~i i~!!i~ii!~i!i~i~i~i~i~~ii~iii~i!i~i~i:i~::i~i~i!i~ii!~i{iii~i~~:ii!~i~i~ii~i{!~ii~!i~i~i!i~i!~i~!i~i~i~!i~i~i~i~i!~i!i!i~i~ii!~i~ii~!i~i~~i~i~ii~i~~~ii~i~i~i~ii~i~i~!i~i~i~i~i~ii~i~i~ ~i~i~ii~iiii~!iiiiiiiiiii~iiiiiiiiiiiiiii~ii~iiiiiiii~i~iiiiiiiiiiiiiii!~i~iiiiiiiiiiiiii!ii!iiiiiiiiiiiiiiiii~iiiiiiii~ii~!~iiiiiii~ii~iiiiiiiiiiiiiii~iiiiiii!~i~i!iiiiiii!~i~iii~i~iii!i!i~iii~i~i!iiii~i~i~i~iiii There have been several important developments in recent years, which have enabled the finite element method to be applied to wave scattering problems with much smaller wavelengths, or at much higher frequencies. The approaches can be outlined as follows: 1. Conventional finite element polynomials, with a transient solution scheme, possibly with characteristic based algorithms. 2. Finite elements which incorporate the wave shape: (a) Shape functions using products of waves and polynomials. (b) Shape functions using sums of waves and polynomials. (c) Ultra weak formulations. (d) Trefftz type finite elements which incorporate wave shapes. 3. Spectral schemes. 4. Discontinuous Galerkin methods in transient schemes. There have been similar developments within the boundary element field, which are too extensive to discuss here. Other surveys are available, l' 2
352
Short waves
The penalty in using a fine mesh of conventional finite elements in solving wave problems, referred to above, is the storage and solution of the system matrix. The approach of Morgan et al. 10, 11 iS not to assemble and solve the system matrix, and to treat the problem as transient. The Maxwell equations are
0E
e0-~- = curlH
and
OH
#0--~- = - c u r i e
(12.2)
where E and H are the electric and magnetic field intensity vectors respectively. The equations are combined and expressed in the conservation form
O U 3=O yjFl~J o tOXj = 0
where
U -
[E]H
(12.3)
"._.
and the flux vectors, F are derived from the curl operators. That is F 1 = [0,/-/3,-H2, 0 , - E 3 , E2] T F 2 = [-/-/3, 0, H1, E3, 0 , - E l ] r F3
(12.4)
-- [H2,-H1, 0 , - E 2 , El, 0] T
The algorithm used is the characteristic-Galerkin (or Lax-Wendroff) method as described in Chapter 2. Details of the algorithm as applied to the electromagnetic problem are given by Morgan et al. Improved CPU efficiency and reduced storage requirements are obtained by the use of a representation in which each edge of the tetrahedral mesh is numbered and the data structure employed provides the numbers of the two nodes which are associated with each edge. Because of the massive computations needed for problems of scattering by short waves, parallel processing has also been used. The problem of radar scattering by an aircraft is shown in Fig. 12.1(a) and Fig. 12.1 (b) shows the radar cross-section obtained for the aircraft. Morgan et al. present results for electromagnetic waves scattered by an aircraft of length 18 m, for a wavelength of 1 m. The model was run on a CRAY parallel computer and had about 1 500 000 elements.
!iiiiiiiiiiiii!i!!i!iiiiiiii~ii!i~iiiiiii!iiiiiiiiFiii~ii~~iiiiii~i!~i~i~~~~iiiii!~~ii~~~~!~~iii~~~~iii~i~~i~i~~iiiiiiiiiii!iiiiiiiiiiiiiiiiiiiii!!iiiiiiiiii~iiiiiiiiiiiiiiiiiii!i!iiiiiiiii In these elements the usual polynomial shape functions are extended with functions which enhance the solution space used in the element. The first attempt to do this was the infinite elements of Bettess and Zienkiewicz. 12' 13 The first attempt on finite elements was that of Astley, 14, 15 using his wave envelope, or complex conjugate weighting method. For more details see Chapter 11, Sec. 11.12. Following Astley's wave envelope technique, Chadwick, Bettess and Laghrouche 16 attempted to develop wave envelope finite elements in which the wave direction was unknown, a priori, and to iterate for the correct wave direction, using some type of residual. In
Finite elements incorporating wave shapes 353
(a)
rr"
0
(b)
I 45
I 90
I 135
I 180 Degrees
I 225
I 270
I 315
360
Fig. 12.1 Scattering of a plane wavelength 2 m by a perfectly conducting aircraft of length 18 m: (a) waves impacting aircraft, (b) computed distribution of RCS, Morgan. i~
the above methods single waves were included in the elements either in a radial or some predefined direction. Chadwick's approach had some success but has difficulties. The next step was the method proposed by Melenk and Babu~ka 17' 18 in which multiple plane waves are used. This is categorized as a form of the partition of a unity finite element method (see
354
Short waves
Chapter 16 of reference 5). Melenk and Babu~ka demonstrated that if such shape functions are used the method works for a plane wave propagated through a square mesh of square finite elements, even when the direction of the wave was not included in the nodal directions. The set of plane waves is complete, 19 so the choice is rational. Other wave functions have been suggested such as cylindrical waves, or a series of Hankel functions. Melenk and Babu~ka 17' 18 and Huttunen et al. 20'21 consider various choices for the wave functions. Mayer and Mendel 22 have also used shape functions which incorporate the wave shape. Similar ideas were also developed in the boundary integral method by de La Bourdonnaye, under the title of microlocal discretization. 23'24 All the methods mentioned above use trial functions, or shape functions, formed from the product of the usual polynomial shape functions and plane (or almost plane) waves. One of the problems with this approach is that the integrations over the element domain may become computationally intensive. It is also possible to add the wave solutions to the polynomial element shape functions or to use different methods which sidestep the integration difficulties (see Sec. 12.6.3).
12.6.1 Shape functions using products of polynomials and waves ~ _ ~ . , .
.
.
.
.
,,. .
.
.
.
,
, .
.
.
.
.
.
.
.
.
.
.
.
.
,,,
.
.
.
.
.
.
,,..
.
,,,,,
Laghrouche and Bettess applied the method originated by Melenk and Babu~ka 17' 18 to a range of wave problems, and enjoyed some success. 25' 26 The starting point is the standard Galerkin weighted residual form of the Helmholtz equation, which leads to L
(--'~TTw(~7'q~"~-k2W~b))dr2+
f r W(V~b)Tn d l - ' - - 0
The element approximation is now taken as ~ m ~ = ~--~ Na~31Ala a=l /=1
(12.5)
(12.6)
where Nj are the normal polynomial element shape functions,
~l -- eik(xcosOl+ysinOl)
(12.7)
and n is the number of nodes in the element, and m is the number of directions considered at each node. The shape function consists of a set of plane waves travelling in different directions, the nodal degrees of freedom corresponding to the amplitudes of the different waves and the normal polynomial element shape functions allowing a variation in the amplitude of each wave component within the finite element. The derivatives of the shape and weighting functions can be obtained in the normal way, but these now also include derivatives of the wave shapes. The new shape function, Pi, gives
OP(a- 1)m+/ Ox OP(a-1)m+l
0Na
Oy
bTy
ONa
+ ikNa
{coso) sin 01
r
(12.8)
Finite elements incorporating wave shapes 355 The global derivatives are obtained in the usual way from the local derivatives, using the inverse of the Jacobian matrix. The element stiffness and mass matrices are
Krs -- f
(VWr)Tl~TPsdf2
Mrs-- J2 WrPsd~
(12.9)
where r and s are integers which vary over the range of 1, 2 . . . . . (n x m). When calculating the element matrices, the integrals encountered are of the form (for a quadrilateral element)
Ial --
/1/1 1
f (~, 7]) e ik(xc~
e ik(xc~
d~ dr/
(12.10)
1
This integral has often been performed numerically. But when the waves are short, many Gauss-Legendre integration points are needed. Typically about 10 integration points per wavelength are needed. Laghrouche and Bettess solve a range of wave diffraction problems, including that of plane waves diffracted by a cylinder. The mesh and the results are shown in Fig. 12.2. As can be seen the results are in good agreement with the analytical series solution. In this problem = 87r, )~ = 0.25a, radius of cylinder, a = 1, and the mesh extends to r -- 7a. For a conventional radial finite element mesh, the requirement of 10 nodes per wavelength would lead to a mesh with 424 160 degrees of freedom. But in the results shown, with 36 directions per node and 252 nodes, there are only 9072 degrees of freedom. The dramatic reduction in the number of variables has prompted investigation and development of the method. The method still has a number of uncertainties regarding the conditioning of the system matrix and the stability of the technique. Various approaches have been suggested to reduce the cost of integrating the element matrix. For the choice of a plane wave basis any term in the element matrix can be written as the product of two plane waves, and a polynomial term. The general integral which arises, in two dimensions, is of the form
ka
s f (~,~/)e ig(('rl) dr2
(12.11)
where ~ and ~7are the local coordinates, f (~, r/) is a rational function which reduces to a polynomial if the Jacobian is constant, g(~, ?7) is some function related to the element mapping and f2 is the element domain. If the Jacobian is constant, then g((, ?7) = c~(+/~7 (c~ and/3 being constants), which greatly simplifies the integrations. Such integrals are highly oscillatory and difficult to evaluate. The first analyses using such elements used large numbers of Gauss-Legendre integration points. 17' 18,25,26 The approach of Ortiz and Sanchez 27 is to combine the two plane waves travelling in directions 01 and 02 to the x axis, to give a single wave travelling in the mean direction, 0 = (01 + 02)/2 with a local wavenumber, k' given by k' = 2k cos (01 - 02)/2. The integral then takes the simpler form
f f(~, f])e i'7~ dr2
(12.12)
where in Eq. (12.12), ~ is oriented in the mean wave direction, ?7is perpendicular and -y depends on the local wavenumber and element geometry. Ortiz and Sanchez used this
356
Short waves exact solution 9 numerical solution
2
=~0 5 4-"
13::
-2
i
i
i
i
i
i
i
100 150 200 250 300 350 400 Angle [degrees]
(a) Cylinder mesh.
(b) Real potential. exact solution numerical solution
1.0.ma:i (1)
0.5-
.i-, O
(v
..c: 0
C}. tl:i
O.O-
-0.5
-1.0 -
t,~
m
E
-1.5
-
-2.0 0
I
I
I
I
I
I
I
I
50 100 150 200 250 300 350 400 Angle [degrees] (c) Imaginary potential.
Fig. 12.2 Short waves diffracted by a cylinder, modelled using special finite elements, Laghrouche and Bettess.26 Reprinted from reference 26, with permissionfrom Journal of ComputationalAcoustics.
result to develop a special integration scheme in the direction of the wave resultant, ~, which contains all the oscillatory effects, in combination with classical integration in the orthogonal ~7 direction, where the variation is of a polynomial form. This has the advantage that the integration of the oscillatory function has been reduced by one dimension. The price that has to be paid is that a new local coordinate system must be set up for each pair of wave directions, and the limits of integration are no longer simple. To date their approach has only been applied to linear triangular elements. Whether the limits of integration will be easy to manipulate with higher-order elements, and in three dimensions, has not yet been investigated. Ortiz has extended his approach to refraction problems and this is described below in Sec. 12.7.
Finite elements incorporating wave shapes 357
Another approach to the integral of Eq. (12.10) was suggested by Bettess et al. 3~ In Bettess et al.'s method the constant Jacobian for simpler finite elements is exploited to transform the integrations into the local coordinates. This integral takes the following specific form for the linear triangular finite element.
~01~01-~
f (~, rl)eia~e i~o dr/d~
(12.13)
In the above ~ and r/are the local coordinates, and f (~, ~7) is a polynomial. Unfortunately the resulting expressions, though straightforward, become complicated and it is almost essential to use computer algebra. 3~ Nevertheless an integration scheme can be developed. Both the Ortiz and Bettess et al. approaches have been demonstrated to give large reductions in computer time, when the waves become short. Bettess et al. have also demonstrated that the integrations over rectangular and cuboid finite elements can be greatly simplified. Their method has also been applied to tetrahedral element integrations (unpublished work). The other feature of these approaches is that although the integration of each finite element matrix is computationally intensive, the calculations are highly parallelizable. There is no difficulty in obtaining an n times speed-up on a computer with n processors. Any repetition in the element geometries can also be exploited. A feature of these Partition of Unity Finite Elements (PUFEs) is that the system matrix is often ill-conditioned. This has concerned some authors. 18'22 Strouboulis et al. 4 suggest that in this general class of methods the equations may be ill-conditioned, but that this is not necessarily a bad thing. The final results may be acceptable, even if the coefficients of the wave components are not unique. Astley 29 has analysed the conditioning of the element matrices, for some simple cases, and has evolved a semiempirical rule relating condition number, number of wave directions and wavenumber. Babugka et al. 17' 18 also consider a posteriori error estimation for such elements. Mayer and Mande122 also discuss strategies for selecting wave directions. The method can be extended to three dimensions 29' 31 and can be linked to conventional infinite elements. 32
12.6.2 Shape functions using sums of polynomials and waves The complications arising from the integration of products of shape functions and wave functions can be eliminated by introducing the waves in a different form, for example by adding the set of waves to the conventional shape function polynomials. Two of these approaches will now be considered. These are 1. The discontinuous enrichment method of F a r h a t . 33-35 2. The ultra weak variational formulation of Despr6s and Cessenat, 36' 37 and Huttunen and Monk. 2~ 21
12.6.3 The discontinuous enrichment method The Farhat et a/. 33-35 approach uses the original polynomial shape function, and adds to it the set of plane waves. (This is in contrast to the Babugka and Melenk approach,
358
Short waves
described above in Sec. 12.3, in which the plane waves are multiplied by the polynomials.) Continuity with adjacent elements is enforced weakly by using Lagrange multipliers. The integrations relating to the plane waves, which Farhat terms enrichmentfunctions, can be reduced to integrations along element boundaries. Moreover these terms can be eliminated from the element matrix by the use of static condensation. There are three sets of variables in the finite elements: 1. ~bP, the usual nodal variables for the finite element polynomial which is used. 2. ~be, the variables relating to the solutions of the Helmholtz equation used for 'enrichment'. These may be coefficients of plane waves, Hankel sources or other wave solutions. 3. p, the Lagrange variables which are defined along the edges of the element. The boundary value problem is put in variational form. The Lagrange multipliers which enforce compatibility are defined as the trace, or outward normal, of vectors p. The contribution to the functional for an element is then
1L(VT~p.V~p--k2~pTq~)d~ fe .n~bdl-"
1-I(u, p) - ~
e
--
P
(12.14)
The potential q5 is spilit into the polynomial and enrichment functions as described above. That is ~b - ~be + ~be. The resulting element matrix is similar but not identical to those shown in Chapter 11 of volume 1. It takes the form
NEe K cP
Nee K ce
KeC 0
Ce pC
(12.15)
where the various matrices K ij arise from the variational statement of the Helmholtz equation. Because the 'enrichment' variables Ce are only defined internally to each element, they can be eliminated at element level using static condensation. This leads to a final element matrix with only polynomial ce and Lagrange multipliers p as variables. The element matrix then takes the form
~e ._ I ~PP ~PC ] fccP fccc
(12.16)
where (12.17)
~CP __~CP__~CE(~EE)-I~Ep, ~CC____~CE(~PP)-I~Ec
(12.18)
The vectors p used to generate the Lagrange multiplier must be chosen correctly. For a triangle element Farhat gives p(x y) -- s '
{ cl c x) c2 + c 3 y
(12.19)
where ci are constants. This gives a constant Lagrange multiplier along the element edge. Farhat points out that because the enrichment functions satisfy the homogeneous
Finite elements incorporating wave shapes 359 Helmholtz equation, the integrals over the domain can be reduced to integrals over the boundaries of the elements. The Lagrange multiplier terms can also be scaled to improve matrix conditioning. These are all very attractive features of the method. Farhat also reports better conditioning of the system matrix with this approach. The results given in his report are all quite academic, being mostly concerned with the propagation of plane waves through square meshes of square elements. Some studies of errors are given. Errors are very low when the plane wave is travelling in one of the directions used for the plane wave basis, or a direction close to it. For other directions, although the errors are larger, they are much smaller than in conventional Galerkin finite elements.
12.6.4 Ultra weak formulation Another promising approach is termed the ultra weak formulation. Among the leading innovators here have been Desprrs '36 Cessenat and Desprrs 37 and Huttunen et a/. 2~ 21 To quote Huttunen: 'In this approach integration by parts is used to derive a variational formulation that weakly enforces appropriate continuity conditions between elements via impedance boundary conditions.' The significant advantage of this approach is that the integrations over each element can be obtained in closed form. This is because the variational principle used relies on a test function which satisfies the Helmholtz equation. As a result all the integrals over the element domain vanish, leaving only integrals along element boundaries, which enforce compatibility between elements. This contrasts with the integration of classical plane wave basis (or other wave basis) element matrices which can be very computer intensive. As in the PUFE method described above, ill-conditioning of the system matrix may be a problem. Cessenat and Desprrs 37 developed the mathematics of the procedure and applied it to the scattering of electromagnetic waves by an NACA 0012 aerofoil, at 1500 MHz. Huttunen et al. 2~ considered an inhomogeneous version of the Helmholtz equation, which looks similar to the shallow water wave equation (11.2).
v. (iv)u P
p a-Ou
itTu
-- Q
+ - - u -- 0 P p a-Ou
iou
in ~
+ g
(12.20)
on F
(12.21)
where f2 is the domain and F is the smooth boundary. Both D(X1, X2) , and k(Xl, X2) are functions of position. Q is a parameter defined on the boundary, 1-', Q < 1. Although this formulation allows for spatial variation in p and k, in practice the treatment in the paper takes them to be piecewise constant in each element. The method is applied to problems in which different zones have different wave speeds. Richardson and stabilized bi-conjugate gradient iterative methods were used for solving the system matrix. Huttunen et al. 21 have also applied the method to elastic wave problems. Cessenat and Desprrs 37 present results for the scattering of an electromagnetic wave of frequency 1500 MHz by an NACA 0012 aerofoil. The mesh was laid down in three layers around the aerofoil. A total of 2976 elements with 1615 nodes were used. Figure 12.3 shows the radar cross-section of the aerofoil.
360
Short waves NACA 0012 wave on trailing edge Antenna Diagram - RCS dB.m LQ ~ .~.
........... ~ ........
',
~_
~
_ ~..'
. . ~ I- ~ ( " ~ ' ~ V ~' ~ /
i
;o ......... ;Lo ......... ;~o ......... ~o ....... o ~ ~ i i ~ ; ~ o : ,
.....-.:'.,-" t
= s
9
.t e
:
..
.o'"
9
,,el
., s
I
, I
"
j .~
t,
j"-12
UWVF 1500 MHz TM - - ' - - I IE 1500 MHz TM -" -- I UWVF 1500 MHz TE " " ' I
"L...
- ......... ;Lo ....
~" ~,.O.o
~: 2:,. ".,.'.'.:..:. ~.~ : ~ ' , . ' ~ : "~ ""
V
k ,
;
Fig. 12.3 Radar cross-section for NACA 0012 on trailing edge at 1500 MHz, Cessenat and Despr~s.37 Comparisons of UWVF and boundary integral results for frequency 1500 MHz, and TM and TE polarizations. Figure reprinted from reference 37, with permission from SIAM.
Huttunen et al. solved a number of example problems 2~ using the ultra weak variational method, including that shown in Figs 12.4 to 12.6. This is an inner domain of radius 5 cm, with a wave speed of 3000 ms -~ and a computational domain of radius hmax = 2 13 cm 10
E
v0
-5 -10
0
-5
0 x(cm)
5
10
Fig. 12.4 Ultra weak variational element mesh, Huttunen, Monk and Kaipio.2~ Reprinted from reference 20, with permission from Elsevier.
Finite elements incorporating wave shapes
:.
0.05
0.05
0.04
0.04
0.03
0.03
0.02
0.02
0.01
~:
:~
~:~
-~::
0.01
.~
0 9
-0.01
-0.01
-0.02
-0.02
-0.03
-0.03
-0.04
-0.04
-0.05
(a) 0.05 0.04 0.03 0.02
..
,.. i
0.01
.: :
--0.01 -0.02 -0.03 --0.04
(b) Fig. 12.5 Cylindricalwave scattered by a circularregion of different wave speed, Huttunen, Monk and Kaipio.2~ Top shows the analytical solution of the problem with f = 250 kHz. Bottom shows the UWVF solution of the problem using a uniform basis of 21 wave directions per node. Reprinted from reference 20, with permission from Elsevier.
10 cm, with a wave speed of 1500 ms -~. A cylindrical wave source, of the form ui n
i
.. (1)
-- ~/-/6
(k2r)
(12.22)
has an origin, for the radius, r, located at x = - 1 1 cm, y = 0 cm. The maximum element size is hmax - 2.13 cm. Figure 12.5 shows the analytical solution compared with the solution obtained using the Ultra Weak Variational Formulation (UWVF), for f = 250 kHz. Huttunen et al. report good results for frequencies up to 450 kHz, corresponding to about six wavelengths per element. Two strategies were used, one of
361
362
Short waves 100 % 1
,
I
I
~
I
Uniform basis Non-uniform basis
~ 10 . 2 .__. ~ r*r 1 0 - 4
hmax - 2.13 cm
i 10 000
i 20 000
. ._....._
~.
~
~,~_
hm~ = 0 . ~ . . . . ~
hmax = 1.21 cm
10-6
~
I
I
I
I
I
30000
40000
50000
60000
70000
80000
Number of degrees of freedom
(a)
1015
-
I
I
J
.,.-..
~1010 f
tO
IV I
X
~
05
100
I 10 000
I 20 000
I I I 30 000 40 000 50 000 Number of degrees of freedom
f
-(9- Uniform basis Non-uniform basis I
I
60 000
70 000
80 000
(b)
Fig. 12.6 Relative discrete/-2 errors and condition numbers for three different frequencies against the number of degrees of freedom, Huttunen, Monk and Kaipio.2~ Top figure shows the relative discrete L2 error for three different frequencies against the number of degrees of freedom. Bottom figure shows the condition number. Reprinted from reference 20, with permission from Elsevier.
uniform number of bases for each element and the other in which the number of bases was dictated by the condition number. The details of the strategy are too complicated to report here. However, better results were obtained by using the varying number of basis functions. The relative errors at different frequencies and the condition number are shown in Fig. 12.6.
12.6.5 Trefftz type finite elements for waves This method is Similar to that described in Sec. 13.5 of reference 5. There is now a considerable literature on the Trefftz type finite (or T-element) methods. Two of the key workers in this field are Jirousek and Herrera. Survey papers are available. 38, 39 It is assumed that the scattering form of the Helmholtz equation is to be solved. The theory has been developed by Jirousek and Stojek. 4~ 41 Background theory on suitable functions for the Trefftz method and T-completeness is given by Herrera. 42 The method has
Finite elements incorporating wave shapes 363 also been applied by Cheung et al. 43 to wave scattering and wave force problems. The theory here follows Stojek. 41 The problem domain f2 is divided into n computational subdomains, ~2k, each with their own local coordinate system, (rk, Ok). In each subdomain, f2~, the total wave potential is written as ~bk. It is assumed to be an approximate solution which satisfies the governing differential equation exactly. The total potential of the wave, ~b, is expressed as the sum of the incident potential ~i and the scattered potential ~bs. m
~k -- r .~_ Csk __ ~i .3t_ ~
(12.23)
NkjCkj -" r .~_ Nkek
j=l or m
cb~ -- ~
Nkjc~j = NkCk
(12.24)
j=l
where Nk is a truncated complete set of local solutions to the Helmholtz equation, and ek is a set of unknown complex valued coefficients, so that VZNk q- k2Nk = 0
in f2k
(12.25)
The corresponding outward normal velocity Vk = V~bk 9n on Of2k is given by Vk -- 1)i -JI- TkCk
or
vk
(12.26)
- - TkCk
The functional to be minimized enforces in a least-squares sense the Neumann boundary conditions on FB and the continuity in potential, ~b and normal derivative v on all the subdomain interfaces, Ft. It is given by
l (~) = I (c) - w2 fr l v 12dr + f B
l r -r
[2 dF + w2 fr l v+ + v I .2dF
I
I
(12.27) where the subscripts +, - in integrals along Fi indicate solutions from the two respective neighbouring Trefftz fields and w is some positive weight. It is assumed that for unbounded domains the selected functions automatically satisfy the Sommerfeld radiation condition. Setting the first variation of the functional I (~b) with respect to ~b equal to zero gives a set of linear equations to solve. Stojek 44 has applied this method to wave scattering problems involving numbers of circular and rectangular cylinders.
T-complete systems The functions used in the various domains of the problem must be complete. They are chosen as follows: 1. Bounded subdomain, Bessel and trigonometrical functions,
{Jo(kr), Jn(kr) cosn0, Jn(kr) sin n0}
(12.28)
2. Unbounded subdomain, Hankel and trigonometrical functions, { H0~1'(kr), Hn~1)(kr)cos nO, H~nl' (kr)sin nO}
(12.29)
364
Short waves
3. Special purpose functions for a bounded subdomain with a circular hole, which are linear combinations of the solutions above, satisfying the boundary conditions on the hole of radius b,
{{ (Jo(kr)- H~(kb----~J~ H~
(Jn(kr)- H,n(kb-----~J~(Hn(kr)) kb) cosnO,
(Jn(kr) - H'nJ~(kb)(kb------~ Hn(kr)) sinn0} /
(12.30)
4. Special purpose functions for the bounded angular comer subdomain,
{Jo(kr),Jn/~(kr)cos[~(O-O~)]l n
(12.31)
where 01 is the angle between the straight boundary and the x axis, and a is the total angle subtended by the angular comer.
As was discussed in Chapter 11, waves can be refracted in two ways: 1. By changes in the local wave speed. 2. By movements of the medium through which the waves pass. Examples of the former include transmission of acoustic waves through material with changing density and transmission of surface waves on water of changing depth. Examples of the latter include acoustic waves transmitted through a flow field, like the noise from jet engines and surface waves on an ocean, or nearshore currents. The wave equations take a different form in these two cases. In general the former problem is much simpler than the latter. Conventional finite elements can deal in a straightforward way with refraction caused by changes in the local wave speed, and no special procedures are needed. The goveming equation is simply the inhomogeneous Helmholtz equation and finite elements can solve this just as easily as the homogeneous Helmholtz equation. But for waves refracted by flows, the equation is much more complicated and depends upon the particular application.
12.7.1 Wave speed refraction The ultra weak variational formulation of Sec. 12.6.4 above has been applied by Huttunen a/. 2~ to cases where the wave speed changes between regions of the problem. This was shown in Figs 12.4 to 12.6. O r t i z 46 has extended his method to deal with refracted waves on the surface of water. Ortiz considers an inhomogeneous form of the Helmholtz equation which was developed by Berkhoff 45 for waves on water of gradually changing depth.
et
17(A(x, y)V~) + B(x, y)q5 = 0
(12.32)
Refraction 365 where
6021 A (x, y) -- CCg = k2 2
1+
2kh sinh 2kh
)
B(x, Y)
__ CO2 __Cg ~ . CO2 _1
c
2
(1\ +
sinh 2kh
)
(12.33) In this case, strictly speaking, the plane wave basis is no longer valid, since the plane waves are not solutions to the inhomogeneous Helmholtz equation. Ortiz assumes that in each element the plane wave functions still form a valid solution space, since the wave speed does not vary greatly over any element. He modifies the earlier integration scheme, 27 to deal with the Berkhoff mild slope wave equation. Again for each pair of wave directions in the element he generates a local coordinate system based on the averaged wave direction. He then integrates analytically in that direction, which contains the most oscillatory variations, and uses conventional integration in the orthogonal direction. His elements are all linear triangles. He uses a Higdon boundary condition to absorb the outgoing waves (see Chapter 11). He presents numerical results for waves scattered by a parabolic shoal and compares them with those of other researchers. 46 The above wave finite elements are strictly only effective for the Helmholtz equation, or other equations with a constant wave speed. The Ortiz formulation is a kind of halfway house, in which the depth is allowed to vary in the governing partial differential equation, but a set of plane waves is still used in the solution space. B e t t e s s 47 has suggested methods for dealing with refraction, but has produced no results to date. In his proposed method, the wave speed is supposed to vary linearly within triangular elements. It is well known that for a linear variation in the wave speed, waves which are initially plane will follow circular arcs. This effect is used in ray tracing techniques. He therefore proposes to replace the usual set of plane waves with waves following circular arcs. The arc trajectories are relatively straightforward to calculate. Apart from this innovation, the plane wave basis element theory goes over unchanged. Since no numerical results have yet been obtained it is an open question as to whether the method would work or not. Because of this uncertainty, no theory is given here as the method is still highly speculative. Another possibility which can occur is to have a step change in wave speed. This occurs, for example, in shallow water waves where the waves pass over a step, and in elastic waves when there is a discontinuity in the density or stiffness of the elastic material. This is the case modelled by Huttunen et al. 2~ 21 The plane wave basis finite elements have recently been extended to deal with this case by Laghrouche e t al. 48 Lagrange multipliers were used to link together the regions with the different wave speeds. The method is as explained by Zienkiewicz et al. 49 Consider a problem in which for simplicity there are only two domains with different wavenumbers, kl and k2. The starting point is the Helmholtz equation (V 2 + k2)@l = 0
in f21
(12.34)
We now consider a problem with subdomains with a different constant wave speed in each subdomain. It is sufficient to consider the two subdomain problem. We begin by considering the solution of the Helmholtz equation expressed in terms of the scalar potential ~bl in the subdomain f2~ bounded by F1 U 1-'int, where V 2 denotes the Laplacian
366
Short waves
operator and kl is the wavenumber in the subdomain f21. Robin boundary conditions are specified on the boundary F1. These are Vr
9n l d- i k l r
-- gl
on
(12.35)
F1
where gl is the boundary condition, V is the gradient vector operator, and nl is the outward normal to the line boundary I'~ U ['int.
Weighted residual scheme The differential equation (12.34) is multiplied by an arbitrary weight function W~ and then integrated by parts to give the weak form B(W1, r
(12.36)
= L(W1)
where B stands for the bilinear form
B(W1, r
-
L
(VW1. V r 1 - k2Wlr l
+ i k l f r WIr d F - J i
WlVr
.nldF
(12.37)
int
1
and f L(W1) = ]
dF 1
W l g l dF
(12.38)
Following the same procedure, we obtain for the subdomain f22 bounded by F2 U 1-'int the weak form B(W2, ~2) = L(W2) (12.39) where all functions and parameters are defined in a similar way as for the weak form (12.36) replacing the subscript 1 by 2.
Plane wave basis finite elements The two subdomains are divided into n-noded finite elements. In each finite element, the potential is first written as a polynomial interpolation of the nodal values of the potential. Then each nodal potential is approximated by a discrete sum of plane waves propagating in different directions in the plane. In our case, a number m j of plane waves are used in the approximating system at the node j which is similar to that of Eqs (12.5) and (12.6). Equation (12.5) is retained, but Eq. (12.6) now has different wavenumbers in the two adjacent elements. r
-" eik~ (x cos Ot+y sin 0t)
r
--" eik2 (x cos 0t+y sin Or)
(12.40)
The continuity of potential and the gradient condition are written as follows: ~1 ~
~2
(12.41)
Refraction 367
and 1 1 k-~V~bl . h i + k-~V~b2 . n : = 0 J
l
(12.42)
.
Lagrange multipliers are then introduced to enforce this condition between the two domains, so that
iv
A = k2
1
4)1 . n l - -~22Vth2 .n2
(12.43)
where k - max(kl, k2) and Ajt are the Lagrange multipliers at node j with respect to direction ~/. We put Qjl = Nj exp (ik[x cos Ol+ y sin Ol]) and then A = QA. A Galerkin scheme is used so that the weighting functions are chosen to be the same as the interpolating oscillatory functions. The Galerkin equations for the two domains may be written in a matricial form H1 A 1 - C1 ,~-- f l H2 A2 -+-C2 ,r "-- f2
(12.44)
where
H1-- j~ (VP~VP~ - k21P~P~)dfl + ikl fr P~P~ dF 1
(12.45)
1
C 1 ---
k2 fr P~Q
(12.46)
dF
int
and f
f ~ = ] pr g~ dF
(12.47) JF l For the second set of equations in the system (12.44), the matrices are obtained in the same way by replacing subscript 1 by 2. At this stage, there are more unknowns than equations. Therefore, we add the continuity condition as
f
QT[~2 -- (I)1] dF
0
(12.48)
int
Substituting the approximations for the fields, then writing compactly the above steps, gives the following system to solve
[.10 i]{al} {!:} 0
-cT
/-/2
C2
Aa
o
x
-
(12.49)
Example problems A study of the accuracy of the elements was carried out. When waves pass from a region of slow wave speed to a region of high wave speed, with a large angle between the normal to the interface and the direction of wave propagation, edge waves, or
368
Short waves
evanescent waves can appear. It is perhaps not obvious that a solution space of plane waves in the plane wave basis elements should be able to accurately represent such evanescent waves, although the proof of completeness would indicate that they should work. Figure 12.7 shows some results where the angle of incidence was chosen to be such that only evanescent waves would arise in the second medium. The element mesh which is not shown is rectangular and appropriate Robin boundary conditions were applied. The plane wave direction set included the incident and reflected wave directions. The results demonstrate that the evanescent effects are properly captured by these elements.
Plane scattered by stepped cylinder This example deals with the diffraction of a plane wave by a rigid circular cylinder, of radius rl, which is assumed to be vertical and the plane wave is incident horizontally
0.4
l
o
Exact solution
Numerical solution
0.2 0.0
~ -0.2 rr
-0.4
-s
"4
-~
"2
-'1
6 x (a)
1.0
t
~
~
~.
g
Exact solution 9
O.5
=o "6
0.0
~9 -o.5 t-
E -1.0 X
(b) Fig. 12.7 Evanescent modes, two subdomains, varying wave speed example. Laghrouche et al.48 Rectangular problem - 5 _< x _< 5, - 5 _< y _< 5. k I, - 5 _< x _< O; k 2, 0 _< x _< 5 Robin boundary conditions, angle of incidence 45 ~ critical angle 30 ~ k I and k 2 chosen to give evanescent transmitted wave, potential plotted on y = O. Reproduced by permission of Elsevier.
Refraction
~!ii
A
~i~;~:#~, i;:~
.::~!i:i.!i!~!i!i,~i............. :~;i :,~::
2.5
............................0.5
1
....~
~
~ ;~iili
-2.5
Fig. 12.8 Waves scattered by stepped cylinder, two subdomain, varying wave speed example. Laghrouche
et al.48 cylindrical problem inner domain (1) 1 > r > 2, k 1 = 107r Robin boundary conditions, plane wave
incident along x axis outer domain (2) 2 > r > 3, k 2 = 67r upper half plane wave basis finite elements, lower half plane analytical solution 3.4 dof per wavelength, L 2 error 0.4%, left real, right imaginary. Reproduced by permission of Elsevier.
on the surface of the water. Around the cylinder, the water is of depth h l up to a circular region of radius r2. Then for r > r2, the depth is h2. In Fig. 12.8, the outer region is deeper than the one around the cylinder (hi < h2). However, the theory remains the same if h~ > h2. The physical domain of this problem is infinite in extent. This means that it must be truncated at a finite distance from the scatterer to enable a numerical simulation. An analytical solution for the problem in terms of Bessel and Hankel functions was developed in reference 48. The problem was analysed for a range of different k values. The errors are summarized in Table 12.1. E"2 is the error in the L2 norm. The parameter 7-is the number of degrees of freedom per wavelength. Figure 12.8 shows contours of the real and imaginary components of the wave potential around the cylinder for kl -- 107r and k2 -- 67r. The results are similar to those of Huttunen et al. obtained using the ultra weak variational formulation.
12.7.2 Refraction caused by flows .................
.-..::::- .......................................................................
The problem of waves refracted by flows is more difficult than the case of refraction due to changing wave speed. The topic is dealt with briefly in Chapter 11. Recently Astley and Gamallo 5~ 51 have applied plane wave basis type methods to wave refraction in the presence of a known flow field in one and two dimensions. In the one-dimensional case the flow is along a duct with changing cross-sectional area. So the flow speed varies with position along the duct. x is the distance along the duct and A(x) is the corresponding cross-sectional area of the duct. The sound speed, Co(X), the density, Table 12.1
Plane wave scattering by a rigid circular cylinder, k l -- 2k2
kz
271"
471"
67/"
87r
107r
127r
1471"
167r
187r
207]"
7"
25.9
12.9
8.6
6.5
5.2
4.3
3.7
3.2
2.8
2.6
E'21%]
0.002
0.007
0.02
0.1
0.5
2.5
0.9
0.4
0.6
1.1
.--
,
369
370 Shortwaves
p(x), and the velocity, u0(x), along the duct have been previously found using the one-dimensional nozzle equation. The velocity is given as u(x) = dr where r is the acoustic velocity potential. The governing equation for small acoustic perturbations, derived from the linearized momentum and continuity equations, is then the convected wave equation
apodx
P~
-t- c 2 r
c~xx
_
co
~xx
c~
(12.50)
In the two-dimensional scheme the wavenumber varies in accordance with the prevailing flow. In this case uniform flow in the x direction, with Mach number M was considered in a duct, extending from x = 0 to x = L and of width a. The resulting convected wave equation in non-dimensionalized form is
o~y +
(1 - M 2 ) 02r ~ x
- 2ikM~x +
k2r - 0
(12.51)
The specific boundary conditions considered are as follows (1 - M 2 ) 0r ~ x- _ i k M r - -cos(mTry)
or
(1 - M 2 ) ~x - i k M r
0r Oy
= -ikr
=0
on
on on
x - 0
x = L/a
y--0,1
(12.52) (12.53) (12.54)
A polar plot of the wavenumber as a function of direction at any point takes the form of an ellipse. This is illustrated in Fig. 12.9. The local plane wave basis is also
ky/k
1/(1-M)
1/(1+M)
i
o
kx/k
(a) The continuous set of wavenumber vectors.
270 (b) A finite set of wavenumber vectors uniformly distributed.
Fig. 12.9 Wavenumber ellipse for the convective wave equation, Gamallo and Astley.sl Reproduced by permission from reference 51, copyright John Wiley & Sons Ltd.
Refraction
Trial basis function
l
Vlj(_~) = Nt(x,Y)exp(-ik(O/).x)
,, L
,
Local approximation function exp(-ik_(ej).x_)
"
"
'
I' I
Nodal interpolation function NI~
!
"-.. ,~. .............
". . . . . .
.~::::............
1 ~:,. I
, ~
"'-....~.::.~. . . . . . . . . . . . . .
Finite element mesh
~.. ........
Node I (a) Construction of the PUFEM trial basis Mean flow streamline FE mesh /~A
node
/
f~-
Local wavenumber ellipse aligned with the flow
(b) The local wavenumber ellipse Fig. 12.10 PUFEM basis, Gamallo and Astley. sl Reproduced by permission from reference 51, copyright John Wiley & Sons Ltd.
shown in Fig. 12.10. Astley and Gamallo report on the accuracy of the partition of unity elements. In summary they state that 'Clearly all of the PUM models offer a huge improvement over conventional low order FEM.' Astley and Gamallo report the same general experiences as in the zero flow case for such types of element, namely reduction in number of degrees of freedom compared with conventional elements and large condition numbers. The problem of completely general flow fields is more difficult. The plane waves are no longer solutions to the wave pattern in the presence of a
371
372
Short waves
0.8
0.6L~
Duct
"-- 0.4 ~ ~ , ~ , ~
0"2t t pinner'~ -o
t 1 i~ ', ---"~'---~--.-~ o skr,.... 'a9 / .~or, . " !
0.4 p ~
i
15
o.=~ o., o . ~
~
0,21
~.~ ~. ~.~.
~
x
:
......
S p i n ~
t
(a) Computational domain
/
.~,;Y" k,
i
,, 'o
.......d:6"'~
~.;\~ i. . . . . . .
o's . . . .
.....
,
\,
' .
",
: .o
lls ....... x
2
2s
(b) Mean flow Mach numbers
Fig. 12.11 Non-uniformaxisymmetricduct, Gamallo and Astley.51 Reproducedby permission from reference 51, copyright John Wiley & Sons Ltd.
general flow field. However, they are still a legitimate solution space, though with no guarantee of completeness. The refracted plane wave solutions can still be used on the assumption that they are a good approximation if the flow is not varying rapidly, and is fairly constant locally. Gamallo and Astley 5~ have applied the same elements to a general flow field through a non-uniform axisymmetric duct. Figure 12.11 shows the geometry, and Fig. 12.12 shows results obtained using PUFEM and Q-FEM (quadratic finite elements). They report that accurate solutions are obtained using 12 000 degrees of freedom with PUFEM, whereas 100000 degrees of freedom are needed for the quadratic conventional finite elements. If the conventional finite element mesh was made any coarser, pollution errors were encountered.
Spectral finite elements are elements which exploit choices in the selection of the finite element node locations and integration schemes. Conventional consistent mass matrices lead to a large though sparse system mass matrix. This is expensive in time stepping schemes, since even if the time stepping scheme is explicit the mass matrix has to be factorized and a back substitution carried out at each time step. If the mass matrix is lumped, the formulation is no longer strictly consistent and the results may be unreliable. If it were possible to integrate exactly the mass matrix by sampling only at the nodes within an element, then the mass matrix though theoretically consistent would be diagonal and explicit time stepping schemes would be much more economical. If
1.
Il
l !ii ill i ii ii :i !i~ I'
N i: i::~ li .~
o.5i
00 . . . .
~
0.5 . . . .
,
1
1.5
i)
~::~I~
2
2.5
X
(a) PUFEM, 12000 degrees of freedom (1500 points and 8 directions)
_15
-l:~llg
-
iti~:
I .
::!ii!i o.5' 0
0
"
.i,
~
-
! ii ,I
.
:..
i'i
ii~ ~!! 0.5
1
1.5
2
2.5
X
(b) Q-FEM, 100000 degrees of freedom
Fig. 12.12 Non-uniform axisymmetric duct. Acoustic pressure field (real part) for c~ = 25, Gamallo and Astley.51 Reproducedby permission from reference 51, copyright John Wiley & Sons Ltd.
Spectral finite elements for waves
a linear finite element is considered, then the general terms in the mass matrix are quadratic. If a Newton-Cotes formula is used to integrate the mass matrix exactly, then three sampling points are needed in one dimension. As there are only two nodes in a linear element, the number of nodes is inconsistent with an exact integration using a Newton-Cotes integration scheme. If Gauss-Legendre integration is used, then two sampling points are needed to exactly integrate the mass matrix. But these are not at the ends of the element and so the element cannot enforce continuity. However, the N point Gauss-Lobatto scheme integrates over the range - 1 to + 1, and samples at the two end points of the range and N - 2 internal points. The scheme integrates exactly powers of x up to 1 + 2(N - 2) - 1 = 2N - 3, for N sampling points. The scheme comes close to integrating the mass matrix exactly. For example, in a one-dimensional quintic finite element with six nodes, the highest power of x in the mass matrix would be 10. The corresponding Gauss-Lobatto scheme would integrate exactly powers of x up to 9. Such integration formulas appear to have been first adopted by Fried and Malkus. 52 They achieved orders of integration, using Gauss-Lobatto schemes which were sufficiently accurate to integrate the stiffness matrix exactly, though not the mass matrix. The Gauss-Chebyshev scheme is also close to integrating the mass matrix exactly. In both these types of element the internal nodes of an element are not equally spaced, but are located at the positions dictated by the integration formulas. The Gauss-LegendreLobatto scheme is given by
f
+l
1
2
f ( x ) d x ~,
n(n-
n-1
1)
+ y ~ wjf(xj)
[/(1) + / ( - 1 ) ]
j=2
(12.55)
where xj is the (j - 1)th zero of Pn'-I (X), where P(x) is a Legendre polynomial and the weights, w j, are given by Wj =
r
(12.56)
~
n ( n - 1)LPn_,(xj)J
2
The corresponding Gauss-Chebyshev-Lobatto points are given by 52-54 7ri -cos ~ (12.57) N where N is the number of integration points. The Gauss-Legendre-Lobatto elements have been investigated by Mulder and others 55-57 and the Gauss-Chebyshev-Lobatto elements by Dauksher, Gottlieb, Hesthaven and others. 53, 54,58-64 There are numerous references to these methods and the above citations are only given as examples of recent work. They are not meant to be exhaustive. In general the numerical investigations demonstrate that spectral elements outperform conventional finite elements in transient wave problems. They have lower dispersion and better phase properties. The Gauss-Chebyshev formulation has also been used for direct collocation approaches to wave problems. This will not be discussed here but references are available. 58-64 In their basic form the spectral elements are not applicable to problems meshed with triangles or tetrahedra. In order to do this, it is necessary to generalize the GaussLobatto and Gauss-Chebyshev-Lobatto schemes to triangles. It is notoriously difficult
373
374
Short waves
Fig. 12.13 Electromagnetic waves scattered by fighter aircraft. Projection of finite element mesh onto surface of aircraft. Results due to Hesthaven and Warburton and published in reference 63. Results reproduced with permission from Elsevier.
to generalize integration schemes from straight line segments, squares and cubes, to triangles and tetrahedra. Although there are proofs that efficient symmetrical integration schemes for triangles and tetrahedral exist there are no general (open ended) formulas for them, but only special results for certain numbers of points. 65' 66 Considerable effort has gone into generalizing the Gauss-Lobatto schemes for triangles and tetrahedra. The corresponding points are called Fekete points. The Fekete points are defined as the points which maximize the determinant of the Vandermonde matrix relating the coordinates of the sampling points to the polynomial coefficients. For large numbers of points the interpolating polynomials may become oscillatory and the Vandermonde matrix can become ill-conditioned. The Fekete points are closely related to the Lebesgue points, but are easier to determine. 67 Kamiadakis and Sherwin 68 give a very comprehensive treatment of spectral finite elements applied to CFD.
This method is frequently used in the time domain, although it can be applied to virtually any finite element problem and not just the Helmholtz wave equation. In the
Discontinuous Galerkin finite elements (DGFE) 375
discontinuous Galerkin method, elements are linked together by constraints, which approximately satisfy continuity of various quantities between elements. The jumps between elements are of the same magnitude as the truncation error. The most natural way of applying the constraints is probably the method of Lagrange multipliers. However, this has two disadvantages: the introduction of additional equations to solve and the indefiniteness of the resulting matrix. In practice the constraints are applied using the element variables themselves. This can be done using some kind of penalty formulation. The mathematical details of the method are explained in reference 5. A comprehensive set of papers on the method edited by Cockburn e t al. 69 are available. The paper by Zienkiewicz e t al. 49 gives an approachable introduction to the method. The method has been applied to a large range of problems, including waves. 7~Important applications to electromagnetic wave scattering problems are given by Hesthaven and Warburton. 71' 72 They explain in considerable detail the techniques needed to extract the full potential of the method. Some of the points are explained below. A significant property of the discontinuous Galerkin finite element method is that the associated mass matrix is local to each element. Furthermore, the affine nature of straight sided triangles and tetrahedra implies that their mass matrices differ only by a multiplicative constant. In practical computations, the relatively small reference triangle (or tetrahedron) mass matrix can be inverted in preprocessing, leading to an extremely efficient method. In the conventional finite element method the assembled stiffness and mass matrices are large sparse matrices, linking together all adjacent degrees of freedom. In the DGFE method the system stiffness matrix need not be formed, and only its product with the vector of field variables from the previous iteration need be retained. This greatly reduces the storage requirement of the method, which becomes O ( N ) , where N is the number of variables in the problem. The other features of the method, as developed by Hesthaven and Warburton and others, are that high-order finite elements are used. They show a 56 node tetrahedron, for example, and use tetrahedra with up to 286 nodes. The node locations are selected using special procedures, and the shape functions are formed using Lagrange polynomial interpolation. Error bound results, Eq. (12.1), show that for a given number of degrees of freedom in a wave problem, it is better to use higher-order polynomials. The use of higher-order elements involves the use of dense, local, reference element operator matrices. However, in the case of straight sided triangle meshes only one set of reference operator matrices is required. For a given operation, say differentiation, the reference element matrix is only loaded into the cache once and the field data in all elements can be differentiated with this one matrix and then physical derivatives are computed using the chain rule. Coupling this approach with standard, optimized, linear algebra mathematics libraries, leads to extremely efficient codes. Exact analytic integration is used where necessary. The authors analyse their scheme for consistency and accuracy. They present a number of results including the scattering of a plane wave by a sphere of radius a at k a - 10. Warburton 7~ also gives results for the scattering of electromagnetic waves by an F15 fighter aircraft. Results are shown in Figs 12.13 and 12.14. Eskilsson and Sherwin 7~ discuss the modelling of the shallow water equations using the discontinuous Galerkin method. They consider as an example the modelling of the
376 Shortwaves
Fig. 12.14 Electromagnetic waves scattered by fighter aircraft. Contours of electrical field on surface of aircraft. Results due to Hesthaven and Warburton and published in reference 63. Results reproduced with permission from Elsevier.
Port of Visby on the Baltic Sea. Figure 12.15 shows the mesh of elements which was used and the water depths, and Fig. 12.16 shows a snapshot of the surface elevations after 500 seconds.
1500 3
3
1000 ~'
i,j1 500
Berth no. 5 1. Generating boundary 2. Open boundary 3. Wall boundary
- direction /
0-
~
I
0
~
,
,
5
....
1obo ....
5 bo'
(a) Mesh and boundary conditions. Fig. 12.15 Harbour layout, Eskilsson and Sherwin.7~ Reproduced by permission from reference 70, copyright John Wiley & Sons Ltd.
Discontinuous G a l e r k i n f i n i t e e l e m e n t s (DGFE) 1500 d i i i i i i i i i i i!i!i-i l- 2
~i~ii - 3 -4 -5 -6 -7 -8 -9 -10
1000
500 -
0
Fig. 12.15
i
I 0
i
i
i
i
I 500
i
i
i
x (b) Depth.
i I 1000
i
i
i
i
I 1500
Continued.
Fig. 12.16 Snapshot of surface elevation after 500 seconds, Eskilsson and Sherwin. 7~ Reproduced by permission from reference 70, copyright John Wiley & Sons Ltd.
377
378
Short waves
,i'i!ii,'li!i!ii!i!ili~ilii~!:i'~!i,~' i~i~'i!i~il'iii,l'~,iiiliiiiii,i'i~i~ili'~ i'ii~,i'~i,~iii,~ii,i,~i~,ii,~' i!i~i,'~ilii!iii~i~iii,~,~,~::~ii~,'i~~ii,!~!i~i!~i~i~/~i!~~i~i~i~i~i~i~i~i~~ii~~ii~!i~ii!~i i~iii!iiii!!i!ii!ii~!!i!iiii!~ii~ii!~iii!i!i!i!ii~!~ii~iii~ii~ii~i~i~ii~!iii!i~ii~ii!ii~iiiiii!~iiiiiii~i~i,'!i~i~i~'ii~'ii~~i'~i',il,'i,'i!!~i i,'ili~:'i,~~i'i,!~'i,~' i,~i,!~i,~' i,~i,~i'',i~'i~i,~i,~'i,~ii,~:' The field of short wave modelling is currently the focus of intense research activity, not only using finite element, but also boundary integral and other domain- and boundarybased methods. At the m o m e n t there are a n u m b e r of promising algorithms, some of which have been described above. The most powerful finite element-based m e t h o d appears to be the discontinuous Galerkin method. Certainly this has achieved the solution o f problems containing the largest n u m b e r o f wavelengths. The other finite element-based algorithms do not seem to be quite so powerful. However, given the high level of research activity, this m a y change. The chief competing boundary-based m e t h o d seems to be the fast multipole method, based on a more efficient formulation of the boundary integral method.
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1. P. Bettess, O. Laghrouche and E. Perrey-Debain, editors. Short-wave scattering. Theme Issue, Philosophical Transactions Royal Society, Series A, volume 362, number 1816, 15 March 2004. 2. M. Ainsworth, P. Davies, D. Duncan, P. Martin and B. Rynne, editors. Topics in Computational Wave Propagation. Lecture Notes in Computational Science and Engineering, volume 51. Springer, Berlin, 2003. 3. T.J.R. Hughes. Multiscale phenomena: Green's functions, the Dirichlet to Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Comput. Meth. Appl. Mech. Eng., 127:387-401, 1995. 4. T. Stroubolis, I. Babu~ka and K. Copps. The design and analysis of the Generalised Finite Element Method. Comput. Meth. Appl. Mech. Eng., 181:43-69, 2000. 5. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method: The Basis and Fundamentals. Elsevier, Amsterdam, 6th edition, 2005. 6. F. Ihlenburg and I. Babu~ka. Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation. Int. J. Num. Meth. Eng., 38:3745-3774, 1995. 7. I. Babu~ka, E Ihlenburg, T. Stroubolis and S.K. Gangaraj. A posteriori error estimation for finite element solutions of Helmholtz' equations. Part I: the quality of local indicators and estimators. Int. J. Num. Meth. Eng., 40(18):3443-3462, 1997. 8. I. Babu~ka, E Ihlenburg, T. Stroubolis and S.K. Gangaraj. A posteriori error estimation for finite element solutions of Helmholtz equations. Part II: estimation of the pollution error. Int. J. Num. Meth. Eng., 40(21):3883-3900, 1997. 9. M. Ainsworth. Discrete dispersion relation for hp-finite element approximation at high wave number. SlAM J. Numer. Analysis, 42(2):553-575, 2004. 10. K. Morgan, O. Hassan and J. Peraire. A time domain unstructured grid approach to the simulation of electromagnetic scattering in piecewise homogeneous media. Comp. Meth. Appl. Mech. Eng., 152:157-174, 1996. 11. K. Morgan, P.J. Brookes, O. Hassan and N.P. Weatherill. Parallel processing for the simulation of problems involving scattering of electromagnetic waves. Comp. Meth. Appl. Mech. Eng., 152:157-174, 1998. 12. O.C. Zienkiewicz and P. Bettess. Infinite elements in the study of fluid structure interaction problems. In J. Ehlers et al. editors, Lecture Notes in Physics, 58. Springer-Verlag, Berlin, 1976. 13. P. Bettess and O.C. Zienkiewicz. Diffraction and refraction of surface waves using finite and infinite elements. Int. J. Num. Meth. Eng., 11:1271-1290, 1977.
References 379 14. R.J. Astley and W. Eversman. A note on the utility of a wave envelope approach in finite element duct transmission studies. J. Sound Vibration, 76:595-601, 1981. 15. R.J. Astley. Wave envelope and infinite elements for acoustical radiation. Int. J. Num. Meth. Fluids, 3:507-526, 1983. 16. E. Chadwick, P. Bettess and O. Laghrouche. Diffraction of short waves modelled using new mapped wave envelope finite and infinite elements. Int. J. Num. Meth. Eng., 45:335-354, 1999. 17. J.M. Melenk and I. Babu~ka. The partition of unity finite element method. Basic theory and applications. Comp. Meth. Appl. Mech. Eng., 139:289-314, 1996. 18. J.M. Melenk and I. Babu~ka. The partition of unity finite element method. Int. J. Num. Meth. Eng., 40:727-758, 1997. 19. P.M. Morse and H. Feshbach. Methods of Theoretical Physics, volumes 1 and 2. McGraw-Hill, New York, 1953. 20. T. Huttunen, P. Monk and J.P. Kaipio. Computational aspects of the ultra weak variational formulation. J. Comput. Phys., 182:27-46, 2002. 21. T. Huttunen, P. Monk, F. Collino and J. P. Kaipio. Computation aspects of the ultra weak variational formulation. SIAM J. Scientific Computing, 25(5): 1717-1742, 2004. 22. P. Mayer and J. Mandel. The finite ray element method for the Helmholtz equation of scattering: first numerical experiments. Report Number CU-CAS-00-20, College of Engineering, University of Colorado, Boulder, Colorado, USA. UCD/CCM Report 111, URL: http:// www-math.cudenver.edu/ccrn/reports.html, 1997. 23. A. de La Bourdonnaye. High frequency approximation of integral equations modelizing scattering phenomena. Mod. Math. Anal. Numer., 28(2):223-241, 1994. 24. A. de La Bourdonnaye. Convergence of the approximation of wave functions by oscillatory functions in the high frequency limit. C. R. Acad. Paris Sdr. L 318:765-768, 1994. 25. O. Laghrouche and P. Bettess. Short wave modelling using special finite elements - towards an adaptive approach. In J.R. Whiteman, editor, The Mathematics of Finite Elements and Applications X, pp. 181-194. Elsevier, 2000. 26. O. Laghrouche and P. Bettess. Short wave modelling using special finite elements. J. Comput. Acoustics, 8(1): 189-210, 2000. 27. P. Ortiz and E. Sanchez. An improved partition of unity finite element model for diffraction problems. Int. J. Num. Meth. Eng., 50:2727-2740, 2001. 28. O. Laghrouche, P. Bettess and R.J. Astley. Modelling of short wave diffraction problems using approximating systems of plane waves. Int. J. Num. Meth. Eng., 54:1501-1533, 2002. 29. O. Laghrouche, P. Bettess, E. Perrey-Debain and J. Trevelyan. Plane wave basis for wave scattering in three dimensions. Comm. Num. Meth. Eng., 19:715-723, 2003. 30. P. Bettess, J. Shirron, O. Laghrouche, B. Peseux, R. Sugimoto and J. Trevelyan. A numerical integration scheme for special finite elements for Helmholtz equation. Int. J. Num. Meth. Eng., 56:531-552, 2002. 31. E. Perrey-Debain, O. Laghrouche, P. Bettess and J. Trevelyan. Plane wave basis finite elements and boundary elements for three dimensions. Phil. Trans. Roy. Soc. - Theme Issue, 362(1816):561-578, 2004. 32. R. Sugimoto and P. Bettess. Coupling of mapped wave infinite elements and plane wave basis finite elements for the Helmholtz equation in exterior domains. Comm. Num. Meth. Eng., 19: 761-777, 2003. 33. C. Farhat, I. Harari and L. Franca. The discontinuous enrichment method. Report CU-CAS-00-20, Center for Aerospace Structures, University of Colorado, 2000. 34. C. Farhat, I. Harari and L. Franca. A discontinuous finite element method for the Helmholtz equation. Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2000, Barcelona, Spain, 11-14 September 2000, 1-15, 2000. 35. C. Farhat, I. Harari and L. Franca. The discontinuous enrichment method. Comput. Meth. Appl. Mech. Eng., 190:645-679, 2001.
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36. B. Despr6s. Sur une formulation variationelle de type ultra-faible. Comptes Rendus de l'Acad~mie des Sciences- Series L 318:939-944, 1994. 37. O. Cessenat and B. Despr6s. Application of an ultra weak variational formulation of elliptic PDEs to the two dimensional Helmholtz problem. SIAM J. Num. Anal., 35(1):255-299, 1998. 38. J. Jirousek and A. Wr6blewski. T-elements: state of the art and future trends. Arch. Comp. Meth. Eng. State Art Rev. 3, 323-434, 1996. 39. I. Herrera. Trefftz method: a general theory. Numerical Methods for Partial Differential Equations, 16(6):561-580, 2000. 40. J. Jirousek. Basis for development of large finite elements locally satisfying all field equations. Comput. Meth. Appl. Mech. Eng., 14:65-92, 1978. 41. M. Stojek. Least-squares Trefftz-type elements for the Helmholtz equation. Int. J. Num. Meth. Eng., 41:831-849, 1998. 42. I. Herrera. Boundary Methods: anAlgebraic Theory. Pitman, London, 1984, ISBN 0 273 08635 9. 43. Y.K. Cheung, W.G. Jin and O.C. Zienkiewicz. Solution of Helmholtz equation by the Trefftz method. Int. J. Num. Meth. Eng., 32:63-78, 1991. 44. M. Stojek. Finite T-elements for the Poisson and Helmholtz equations. Ph.D. thesis number 1491, Ecole Polytechnique Federale de Lausanne, 1996. 45. J.C.W. Berkhoff. Refraction and diffraction of waver waves: derivation and method of solution of the two-dimensional refraction-diffraction equation. Delft Hydraulics Laboratory, Report and Mathematical Investigation, W154, 1973. 46. P. Ortiz. Finite elements using plane wave basis. Phil. Trans. Roy. Soc., 362(1816):525-540, 2004. 47. P. Bettess. Special wave basis finite elements for very short wave refraction and scattering problems. Comm. Num. Meth. Eng., 20:291-298, 2004. 48. O. Laghrouche, P. Bettess, E. Perrey-Debain and J. Trevelyan. Wave interpolation finite elements for Helmholtz problems with jumps in the wave speed. Comput. Meth. Appl. Mech. Eng., 194:367-381, 2004. 49. O.C. Zienkiewicz, R.L. Taylor, S.J. Sherwin and J. Peir6. On discontinuous Galerkin methods. Int. J. Num. Meth. Eng., 58:1119-1148, 2003. 50. R.J. Astley and P. Gamallo. Special short wave elements for flow acoustics. Comput. Meth. Appl. Mech. Eng., to appear. 51. P. Gamallo and R.J. Astley. The partition of unity finite element method for short wave acoustic propagation on nonuniform potential flows. Int. J. Num. Meth. Eng., to appear. 52. I. Fried and D.S. Malkus. Finite element mass matrix lumping by numerical integration without convergence rate loss. Int. J. Solids Struct., 11:461-466, 1976. 53. W. Dauksher and A.E Emery. Accuracy in modeling the acoustic wave equation with Chebyshev spectral finite elements. Finite Elements in Analysis and Design, 26:115-128, 1997. 54. W. Dauksher and A.E Emery. An evaluation of the cost effectiveness of Chebyshev spectral and p-finite element solutions to the scalar wave equation. Int. J. Num. Meth. Eng., 45(8):1099-1114, 1999. 55. W.A. Mulder. Spurious modes in finite-element discretisations of the wave equation may not be all that bad. Appl. Num. Math., 30:425-445, 1999. 56. W.A. Mulder. Higher-order mass-lumped finite elements for the wave equation. J. Comput. Acoustics, 9(2):671-680, 2001. 57. M.J.S. Chin-Joe-Kong, W.A. Mulder and M. Van Veldhuizen. Higher-order triangular and tetrahedral finite elements with mass lumping for solving the wave equation. J. Eng. Math., 35:405-426, 1999. 58. J.S. Hesthaven, P.G. Dinesen and J.P. Lynov. Spectral collocation time-domain modeling of diffractive optical elements. J. Comput. Phys., 155:287-306, 1999. 59. J.S. Hesthaven. Spectral penalty methods. Appl. Num. Math., 33:23-4 1, 2000. 60. J.S. Hesthaven and C.H. Teng. Stable spectral methods on tetrahedral elements. SIAM J. Sci. Comput., 21 (6):2352-2380, 2000.
References 381 61. B. Yang, D. Gottlieb and J.S. Hesthaven. Spectral simulations of electromagnetic waves scattering, J. Comput. Phys., 134:216-230, 1997. 62. J.S. Hesthaven and D. Gottlieb. Stable spectral methods for conservation laws on triangles with unstructured grids. Comput. Methods Appl. Mech. Eng., 175:361-381, 1999. 63. D. Gottlieb and J.S. Hesthaven. Spectral methods for hyperbolic problems. J. Comput. Appl. Math. 128:83-131, 2001. 64. J.S. Hesthaven. From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex. SIAM. J. Numer. Anal., 35(2):655-676, 1998. 65. R. Cools and P. Rabinowitz. Monomial cubature rules since 'Stroud': a compilation. J. Comput. and Appl. Math., 48:309-326, 1993. 66. R. Cools. Monomial cubature rules since 'Stroud': a compilation- Part 2. J. Comput. and Appl. Math., 112:21-27, 1999. 67. M.A. Taylor, B.A. Wingate and R.E. Vincent. An algorithm for computing Fekete points in the triangle. SlAM J. Numer. Anal., 38(5): 1707-1720, 2000. 68. G.E.M. Karniadakis and S.J. Sherwin. Spectral/hp Element Methods for CFD. Oxford University Press, Oxford, 1999. 69. B. Cockburn, G.E. Karniadakis and C.-W. Shu. Discontinuous Galerkin Methods. Springer, Berlin, 2000. 70. C. Eskilsson and S.J. Sherwin. A triangular spectral/hp discontinuous Galerkin method for modelling 2D shallow water equations. Int. J. Num. Meth. Fluids, 45(6):605-624, 2004. 71. T. Warburton. Application of the discontinuous Galerkin method to Maxwell's Equations using unstructured polymorphic hp-finite elements. In B. Cockburn et al., editors, Discontinuous Galerkin Methods, pp. 451-458. Springer, Berlin, 2000. 72. J.S. Hesthaven and T. Warburton. Nodal high-order methods on unstructured grids - I time domain solution of Maxwell's equations. J. Comp. Phys., 181:186-221, 2002.
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Computer implementation of the CBS algorithm
In this chapter we shall consider some essential steps in the computer implementation of the CBS algorithm on structured or unstructured finite element grids. Only linear triangular elements will be used and the notes given here are intended for a two-dimensional version of the program. The sample program listing and user manual along with several solved problems are available to download from the website http://www.nithiarasu.co.uk or the publisher's site, http://books.elsevier.com, free of charge. The program discussed can be used to solve the following different categories of fluid mechanics problems: 1. Compressible viscous and inviscid flow problems. 2. Incompressible viscous flows. 3. Incompressible flows with heat transfer. With further simple modifications, many other problems such as turbulent flows, solidification, mass transfer, free surfaces, etc. can be solved. The procedures presented here are largely based on the theory presented in Chapter 3. The language employed is FORTRAN. It is assumed that the reader is familiar with FORTRAN 1'2 and finite element procedures discussed in this volume. We call the present program CBSflow since it is based on the CBS algorithm discussed in Chapter 3 of this volume. We prefer to keep the compressible and incompressible flow codes separate to avoid any confusion. However, an experienced programmer can incorporate both parts into a single code without much memory loss. Each program listing is accompanied by some model problems which helps the reader to validate the codes. In addition to the model inputs to programs, a complete user manual is available to users. Any error reported by readers will be corrected and the program will be continuously updated by the authors. The modules are (1) the data input module with preprocessing and continuing with (2) the solution module and (3) the output module. The program CBSflow contains the listing for solving transient Navier-Stokes (or Euler-Stokes) equations iteratively. Here there are many possibilities such as fully explicit forms, semi-implicit forms and quasi-implicit forms and fully implicit forms as discussed in Chapter 3. We concentrate
The data input module 383 mainly on the first two forms which require small memory and simple solution procedures compared to other forms. In both the compressible and incompressible flow codes, only non-dimensional equations are used. The reader is referred to the appropriate chapters (Chapters 3, 4 and 5) for different non-dimensional parameters. In Sec. 13.2 we shall describe the essential features of data input to the program. Here either structured or unstructured meshes can be used to divide the problem domain into finite elements. Section 13.3 explains how the steps of the CBS algorithm are implemented. In that section, we briefly remark on the options available for shock capturing, various methods of time stepping and different procedures for equation solving. In Sec. 13.4, the output generated by the program and postprocessing procedures are considered.
iiiiil~ilili3~ iiiiil~iiiiThe lililiIil~ili~iii~d~a iliiiifli!li!liiii!liRpUt i!ii~!i!il:i~iliil~liliili!imlilililiOdi i!iilii~Uii i!l~ii~i~~~~~~~.iii ~ii~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ i iiiii!iifiiilii i iiiilili ilil This part of the program is the starting point of the calculation where the input data for the solution module are prepared. Here an appropriate input file is opened and the data are read from it. The mesh generators are provided separately. By suitable coupling, the reader can implement various adaptive procedures as discussed in Chapters 4 and 6. Either structured or unstructured mesh data can be given as input to the program. Readers are referred to the web page for further details.
13.2.1 Mesh data - nodal coordinates and connectivity Once the nodal coordinates and connectivity of a finite element mesh are available from a mesh generator, they are allotted to appropriate arrays. The coordinates are allotted to coord(i, a) with i defining the appropriate Cartesian coordinates xl(i = 1) and x2 (i = 2) and a defining the global node number. Similarly the connectivity is allotted to an array intma(b, l). Here b is the local node number and I is the global element number. It should be noted that the material code normally used in heat conduction and stress analysis is not used but can be introduced if necessary.
13.2.2 Boundary data In CBSflow we mostly use the edges to store the information on boundary conditions. Some situations require boundary nodes (e.g. pressure specified at a single node) and in such cases corresponding node numbers are supplied to the solution module.
13.2.3 Other necessary data and flags In addition to the mesh data and boundary information, the user needs to input a few more parameters used in flow calculations. For example, compressible flow computations need the values of non-dimensional parameters such as the Mach number,
384 Computer implementation of the CBS algorithm Reynolds number, Prandtl number, etc. Here the reader may consult the non-dimensional equations and parameters discussed in Sec. 3.1, Chapter 3, and in Chapter 4. Several flags for boundary conditions, shock capture, etc. also need to be given as inputs. For a complete list of such data and flags, the reader is referred to the user manual and program listing at the author's web page.
13.2.4 Preliminary subroutines and checks A few preliminary subroutines are called before the start of the time iteration loop. Establishing the surface normals, element area (for direct integration), mass matrix calculation and lumping and some data initialization subroutines are necessary before starting the time loop.
The solution module primarily contains the following steps (explicit time stepping) pre-processing do iter = i, n u m b e r of time steps call a l o t i m ! allot a p p r o p r i a t e time step v a l u e call shock ! calculate shock capturing viscosity call stepl ! intermediate momentum call step2 ! calculate density/pressure call step3 ! correct momentum call e n e r g y ! e n e r g y e q u a t i o n call p r e s s ! r e l a t e d e n s i t y and p r e s s u r e u s i n g energy cal I b o u n d ! apply boundary conditions cal i c h e c k ! c h e c k s t e a d y state c r i t e r i o n e n d d o !iter post-processing
The time iteration is carried out over the steps of the CBS algorithm and over many other subroutines such as the local time step and shock capture calculations. As mentioned, the energy can be calculated after the velocity correction. However, for a fully explicit form of solution, the energy equation can be solved in step 1 along with the intermediate momentum variable if preferred. Most of the routines within the time loop are further subdivided into several other subroutines. For instance convection and diffusion are treated using separate routines within the first step.
13.3.1 Time step In general, three different ways of establishing the time steps are possible. In problems where only the steady state is of importance, so-called 'local time stepping' is used
Solution module (Chapter 3). Here a local time step at each and every nodal point is calculated and used in the computation. When we seek accurate transient solution of any problem, the so-called 'minimum step' value is used. Here the minimum of all local time step values is calculated and used in the computation. Another and less frequently used option is that of giving a 'fixed' user-prescribed time step value. Selection of such a quantity needs considerable experience from solving several flow problems. The time loop starts with a subroutine where the above-mentioned time step options are available ( a l o t i m ) . If the last option of the user-specified fixed time step is used, the local time steps are not calculated.
13.3.2 Shock capture The CBS algorithm introduces naturally some terms to stabilize the oscillations generated by the convective acceleration. However, for compressible high-speed flows, these terms are not sufficient to suppress the oscillations in the vicinity of shocks and some additional artificial viscosity terms need to be added (Chapter 7). We provide two different forms of artificial viscosities based on the second derivative of pressure in the program. Another possibility is to use anisotropic shock capturing based on the residual of individual equations solved. However, we have not included the second alternative in the program as the second derivative-based procedures give quite satisfactory results for all high-speed flow problems. In the first method implemented, we need to calculate a pressure switch (Chapter 7) from the nodal pressure values. We calculate the switch for internal node as (Fig. 13.1)
S1 =
14pl -- P2 -- P3 -- P4 -- Psi IPl - P21 + IPl - P31 + IPl
-
Pal +
(a) Fig. 13.1 Typical element patches: (a) interior node; (b) boundary node.
IPl -
(b)
Psi
(13.1)
385
386 Computerimplementation of the CBS algorithm and for the boundary node we calculate (Fig. 13. l b) 15pl S1 --
-- 2p2
--
P3
-
-
2p41
21pl - P21 + IPl - P31 + 21pl - P41
(~3.2)
The nodal quantifies calculated in the manner explained above are averaged over individual elements. In the next option available in the code, the second derivative of pressure is calculated from the smoothed nodal pressure gradients (Chapter 4) by averaging. Other approximations to the second derivative of pressure are described in Chapter 4. The user can employ those methods to approximate the second derivative of pressure if desired.
13.3.3 CBS algorithm. Steps In the subroutine s t ep 1 we calculate the temperature-dependent viscosity at the beginning according to Sutherland's relation (see Chapter 7). The averaged viscosity values over each element are used in the diffusion terms of the momentum equation and dissipation terms of the energy equation. The diffusion, convective and stabilization terms are integrated over elements and assembled appropriately into the RHS vector. The integration is carried out directly. Finally the RHS vector is divided by the lumped mass matrices and the values of intermediate momentum variables are established. In step two, in explicit form, the density/pressure values are calculated. The subroutine s t e p 2 is used for this purpose. Here the option of using different values of 01 and 02 is available. In explicit form 02 is identically equal to zero and 01 varies between 0.5 and 1.0. For compressible flow computations, the semi-implicit form with 02 greater than zero has little advantage over the fully explicit form. For this reason we have not included the semi-implicit form for compressible flow problems in the program. For incompressible flow problems, the semi-implicit form is given. In this 01, as before, varies between 0.5 and 1 and 02 is also in the same range. Now it is essential to solve the pressure equation implicitly in s t e p 2 of the algorithm. Here in general we use a conjugate gradient solver as the coefficient matrix is not necessarily banded. The third step is the one where the intermediate momentum variables are corrected to get the real values of the intermediate momentum. In all three steps, mass matrices are lumped if the fully explicit form of the algorithm is used. As mentioned in earlier chapters, this is the best way to accelerate the steady-state solution along with local time stepping. However, in problems where transient solutions are of importance, either a mass matrix correction as given in Chapter 2 or a simultaneous solution using a consistent mass matrix may be necessary.
13.3.4 Boundary conditions As explained before, the boundary edges are stored along with the elements to which they belong. Also in the same array i s 2 de ( 2, 3 ) the flags necessary to inform the solution module which type of boundary conditions are stored. In this array i = 1, 2 correspond to the node numbers of any boundary side of an element, i = 3 indicates
References 387
the element to which the particular edge belongs and i - 4 is the flag which indicates the type of boundary condition (a complete list is given in the user manual available at the authors' and publisher's web page). Here j is the boundary edge number.
13.3.5 Solution of simultaneous equations - semi-implicit form .................................................................................................................................................................
........ ----. ............
- - . . . . . . . . . . -_-:~. . . . . . . . . . . . . . . . . . . . . . . . . . . .
~ ..........
---~ . . . . . . . . . . .
---~
The simultaneous equations need to be solved for the semi-implicit form of the CBS algorithm. Two types of solvers are provided. The first one is a banded solver which is effective when structured meshes are used. For this the half-bandwidth is necessary in order to proceed further. The second solver is a conjugate gradient solver. The latter can be used to solve both structured and unstructured meshes. The details of procedures for solving simultaneous equations can be found in reference 3.
13.3.6 Different forms of energy equation In compressible flow computations only the fully conservative form of all equations ensures correct position of shocks. Thus in the compressible flow code, the energy equation is solved in its conservative form with the variable being the energy. However, for incompressible flow computations, the energy equation can be written in terms of the temperature variable and the dissipation terms can be neglected. In general for compressible flows, Eq. (3.13) is used, and Eq. (4.6) is used for incompressible flow problems.
13.3.7 Convergence to steady state The residuals (difference between the current and previous time step values of parameters) of all equations are checked at every few user-prescribed number of iterations. If the required convergence (steady state) is achieved, the program stops automatically. The aimed residual value is prescribed by the user. The program calculates L2 norm of residual of each variable over the domain. The user can use them to fix the required accuracy.
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If the imposed convergence criteria are satisfied then the output is written into a separate file. The user can modify the output according to the requirements of postprocessor employed. Here we recommend the education software developed by CIMNE (GiD) for post- and preprocessing of data. 4 The facilities in GiD include two- and threedimensional mesh generation and visualization.
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1. I. Smith and D.V. Griffiths. Programming the Finite Element Method. Wiley, 3rd Edition, Chichester, 1998. 2. D.R. Willr. Advanced Scientific Fortran. Wiley, Chichester, 1995.
388 Computer implementation of the CBS algorithm 3. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method: Its Basis and Fundamentals. Elsevier, 6th edition, 2005. 4. GiD. International Center for Numerical Methods in Engineering, Universidad Polit6cnica de CatalunL 08034, Barcelona, Spain.
Non-conservative form of Navier-Stokes equations To derive the Navier-Stokes equations in their non-conservative form, we start with the conservative form omitting the body forces. Conservation of mass:
Op { Ot
O(pui) Op OUi Op --" ~ -~"t- Ui --0 OXi Ot P~X i
(A.1)
Conservation of momentum:
O(ujpui)
O(pui) cot
Oxj
07"ij
Op
-- Ox----j-t- ~x/ = 0
(A.2)
O(OT) O(ujp) O('rijuj) Oxl k-~x~ + Oxj - Oxj = 0
(1.3)
Conservation of energy:
O(pE) O(ujpE) ot + Oxj
Rewriting the momentum equation with terms differentiated as
Op OTij OXj F ~x/-- 0
Oili Op Ouj Op On i P--'~ -JI-Ui(-'~ ~- P~xj "JI-UJ-~xj) '~- pUJ oxj
(A.4)
and substituting the equation of mass conservation [Eq. (A.1)] into the above equation gives the reduced momentum equation
On i
1 070 10p p Oxj F P Oxi = 0
On i
--~ -F uj Oxj
(A.5)
Similarly as above, the energy equation [Eq. (A.3)] can be written with differentiated terms as
Op I
Ouj O(uip) Oxi
Op t " x ,,,
~
OE x
Oxi
)- 0
OE
+ U ox
0 (kCgT oxi
~) (A.6)
390 AppendixA Again substituting the continuity equation into the above equation, we have the reduced form of the energy equation
OE OE --~ + uj Oxj
1 O ( O__ff_T'~ l O(uip) p Oxi \k Oxi) -! P Oxi
10('rijuj) p Ox~
(A.7)
Some authors use Eqs (A.1), (A.5) and (A.7) to study compressible flow problems. However, these non-conservative equations can result in multiple or incorrect solutions in certain cases. This is true especially for high-speed compressible flow problems with shocks. The reader should note that such non-conservative equations are not suitable for simulation of compressible flow problems.
i i ,i'~..,'~' '~'~~''~'~i'~'i~'i~' ~ii'~i~'i'~i']i~'i~i~'i'~i '~~i'~'~'~i'~i'~i'~ii~'il'i~'i~ii'i!ii Self-adjoint differential equations Let us consider the following system of linear partial differential equation to demonstrate the property of self-adjointness A(u) = Lu + b = 0
(B.1)
where L is a linear differential operator. For the above equation to be self-adjoint the operator L requires L ~ r (L')') df2 = L "yr (L~) dr2 + b.t.
(B.2)
for any two functions ~ and 7. In the above equation b.t. stands for boundary integral terms.
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The drag force is the resistance offered by a body which is equal to the force exerted by the flow on the body at equilibrium conditions. The drag force arises from two different sources. One is from the pressure p acting in the flow direction on the surface of the body (form drag) and the second is due to the force caused by viscosity effects in the flow direction. In general the drag force is characterized by a drag coefficient, defined as
Cd =
D 1
(C.1)
2
A f ~PooUoo
where D is the drag force, A f is the frontal area in the flow direction and the subscript c~ indicates the free stream value. The drag force D contains the contributions from both the influence of pressure and friction, i.e. (C.2)
D = Dp -k- D f
where Dp is the pressure drag force and Df is the friction drag force in the flow direction. The pressure drag, or form drag, is calculated from the nodal pressure values. For a two-dimensional problem, the solid wall may be a curve or a line and the boundary elements on the solid wall are one dimensional with two nodes if linear elements are used. The pressure may be averaged over each one-dimensional element to calculate the average pressure over the boundary element. If this average pressure is multiplied by the length of the element, then the normal pressure acting on the boundary element is obtained. If the pressure force is multiplied by the direction cosine in the flow direction, we obtain the local pressure drag force in the flow direction. Integration of these forces over the solid boundary gives the drag force due to pressure D p. The viscous drag force D f is calculated by integrating the viscous traction in the flow direction over the surface area. The relation for the total drag force in the Xl direction may be written for a two-dimensional case as Dx~
- fA [(--P +
Tll)nl
-}- T12n2]
dAs
s
where n l and n2 are components of the surface normal n as shown in Fig. C. 1.
(c.3)
Appendix C 393
u~
v
Fig. C.1 Normalgradient of velocity close to the wall.
iiiiHiii~i~ iii!iiHiiii"~"' ii~'i~ii'i!"il'~i~ii~iiiii~'i'i':iiii~ii' !iiii!ieiii~~ i!i'~!ii!ilii~i!'iii~i~iiilililil!iiiiii!ii!i!il!i!~iiil!il~ili!i~!i lili!iii~ii!~~i~ "i~'ii~~i i~~ ~ ~ iiiH~ ~ ~ ~ }~ ~ iiiiiiiiiiii } ilii ~iiiiHiiiiiiii ~ iiiiii ~iiiiiiiiiiiii ~ iiiiiiiiiiiiiill iiiiiii i i The local coefficient of friction on a solid surface is defined as
"Fw
Cf
--
1 ~p~u2~
(C.4)
where "rw is the local shear stress and subscript c~ indicates a reference quantity. The wall shear stress at the centre of a boundary surface element (one dimensional in 2D and two dimensional in 3D problems) is calculated as 7-w = ~.t _ (-r.n)n where n is the surface normal and
(C.5)
./.t is the total viscous traction given as "r t = r n
(C.6)
where 7-are the deviatoric stresses. The pressure coefficients are normally calculated
as
Cp = 2 ( p i - Prey)
(C.7)
where the subscript r e f i s a reference value (often the value at inlet or free stream). iiiiiiiiiliiRiiiiiii ~i~i~i~iiil}ii~i~i~!~iiiiiiiiiiiii!iii }i! ii~i~i~i~iiiiiiiiiiiiiiiiii!iiiiiiiiii!iiiiil iiiii!ii}iiiiiii}}i~iiiiiiiiiiiii iiiiiiiiiiiiiiii i~iiiiiiiiii iiliiiiiiiiiiiil ii}ii[ii!iiiiii}[ii}!i[ii!i}iiiii}iiiiiiii[H!iiiiiiiii!iiiiiiiiiiii!iiii[!iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii iiiiiiiiiii!iiil~iiii!!iliiiiii[i[iiiiiiiiii!iiiiiiiiiiiiiiiiiiiiii ilil iiiiiii[}ilil}iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiilliiiiiiiii
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In most fluid dynamics and convection heat transfer problems, it is often easier to understand the flow results if the streamlines are plotted. In order to plot these streamlines, or flow pattern, it is first necessary to calculate the stream function values at the nodes. The stream function is defined by the following relationships,
Ul ---
or OX2
0r U2 -'- C~X1
(C.8)
394 AppendixC where ~ is the stream function. If we differentiate the first relation with respect to x2 and the second with respect to Xl and then sum, we get the differential equation for the stream function as (C.9) A solution to the above equation is straightforward for any numerical procedure. This equation is similar to Step 2 of the CBS scheme and an implicit procedure immediately gives the solution. Unlike the pressure equation of Step 2, the stream function of a solution needs to be calculated once only.
Integration formulae
Let a, b and c be the nodes of a triangular element. Integrating over the triangular area gives 1
A-
Xla
X2a
f dxldX2 "- 1 1 Xlb
XZb
1
Xlc
(D.1)
X2c
where A is the area of the triangle. For a linear triangular element (shape functions are the same as local coordinates), the integration of the shape functions can be written as
J~ d e
d !e !f !2A N a N b N f dr2 = (d + e + f + 2)!
(D.2)
On the boundaries
f r NdN[~ d r -- (d +d!e!l e + 1)!
(D.3)
Note that a - b is assumed to be the boundary side. The above equation is identical to the integration formula of a one-dimensional linear element. In the above equation I is the length of a boundary side.
Let a, b, c and d be the nodes of a linear tetrahedron element. Integrating over the volume gives
V --
dx 1 dx2 dx3
1 -
-
--'a
a
1 1
x1 Xlb
X2
X~
x~
x~
1
C
xt
C
X2
X3
1
xd
x2d
x3d
C
(D.4)
396
AppendixD where V is the volume of a tetrahedron. For linear shape functions, the integration formula can be written as
f ueuIUcUd"=(e+ e!f f!g!h!6V +g+h+3)!
(D.5)
On the boundaries
fF NaN e bf N cdF g -
e! f!g!2A (e+f+g+2)!
(D.6)
Note that the above formula is identical to the integration formula of triangular elements within the domain. In the above equation A is the area of a triangular face.
iii Convection-diffusion equations: vector-valued variables i!!ii~i~i ili}JJJ~:::~::!~J!J:!:::i~ili :iJ!{i!J:::: d{iiJii~i!iii~j~!!d{ii~~::/~i~~!!i::,:~~i~!iJii!iiJ{J::s~!~i{~i~i:~::::::::::::::::::::::: :: i~~!~!:~i:i:iJJ}JJJJJi iJlJiJi i::!:Ji!~i~si~ii::::~:i:!~!:J~!::~::::Ji:: :!:i~:~d{:::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::: ~i~i::iiiis::::~::i{7;::!::!::ili ::~:;:~:~iiJl:,ii~:~:~::sif:i~!i~::i::;::::i!~J~JiJiJ~:iJ~:i,:i:::::::::::::::::::::::::::::::::::::: ::~sJi: ;!:#41i~i!{~J{i }:i::Jii i~ >,~d:::~Jiii~iii~i iiili::::: i~i!i:: sljl::::::::::::::::::::::::::::::: :)~iu:s:i::ili::~::::l{il{!Ji{i:::::: :::::::::::::::::::::::::::::::::::: ~:~:: :~i!!!:~i:i:{i~i:,:i!,:{!iiiii :~i:iiijli :~iiiii!ijJi!i{i~S~i iliii'!i~i~
The only method which adapts itself easily to the treatment of vector variables is that of the Taylor-Galerkin procedure. Here we can repeat the steps of Sec. 2.7 but now addressed to the vector-valued equation with which we started Chapter 2 [Eq. (2.1)]. Noting that now 9 has multiple components, expanding ,I~ by a Taylor series in time we have r )n+l
=
r ~n -~-
At--~--
.
At2 02@ I
+ 2
(E.1)
Ot2 n+O
where 0 is a number such that 0 < 0 < 1. From Eq. (2.1), n
= -- rL~X/
"~" -~-X/
(E.2a)
"~-
and differentiating ('~ OF i
Ot [
Q]
~G i
+~
+
n+O
(E.2b)
In the above we can write 0
OF i
0
OF i 0r
-O tggxi( ) ~~ ( o ~) where Ai
-----
~
-
0[A/(
OX i
-'~Xj -~- --~Xj ~t-
(E.2c)
OFi/Ot~ and if Q - Q ( ~ , x) and 0 Q / 0 ~ = S,
OQ Ot
OQ o ~ O~ Ot
(OFi
~i
Q)
=-svaz/+ ~7x/+
(E.2d)
398 AppendixE
We can therefore approximate Eq. (E. 1) as
A~I~n ~ r ~n+l __r ~n
[OFi
0 (O~i)
+b7
At2 ~X/ 0 [Ai ~Xj OFj ~ J Q] -~"T n { ( + 0Fj oqGj Q)} +s( + +
OGi
+
Q)] (E.3)
Omitting the second derivatives of Gi and interpolating the n + 0 between n and n + 1 values we have
AgI) ~ (I~n+l __ It)n
[Ore -- -At/~X/"+-
i O'qI- ~ Q ] (n[--] [ C At~ J i ] ) ~ 0 ~n+l
"q- ---2---[~X/{Ai(
+
n(1 --0)
\~Xj "]"
..~..T[_~xi{Ai(_~xj_.[ -
n-t-
1
(E.4)
(OFj -q- Q)] n(1 - 0)
At this stage a standard Galerkin approximation is applied which will result in a discrete, semi-implicit, time-stepping scheme. As the explicit form is of particular interest we shall only give the details of the discretization process for 0 = 0. Writing as usual
q,~N@ we have
(J~NTN dr2) A~
_ _At [ ~ Nx ( OFi\_~x~ -b -~xi AtfNT
2
0
OFj "~-Q ) ) d f 2
~X/(Ai(~
At~NTs(OFJ 2
Q) d~
\~xj +
Q) d~ 1
(E.5)
,
This can be written in a compact matrix form similar to Eq. (2.107) as M A ~ = - A t [ ( C + Ku + K ) ~ + f]" in which, with
0,~
Gi - -kij Oxj
(E.6a)
Appendix E 399 we have (on omitting the third derivative terms and the effect of S) matrices of the form of Eq. (2.108), i.e. C-
L
ON N TAi ~ dr2
K u - L 0NT (AiAj ~_~t)ON
K-
L GgNT
ON
--~xi kiJ~xj d~2
(E.6b)
f = L ( N T + ~AtA i cgNT'~ OX i ,] Q dr2 + boundary terms
M - L NTN d~ With 0 = 1/3 it can be shown that the order of approximation increases and for this scheme a simple iterative solution is possible. We note that with the consistent mass matrix M the stability limit for 0 = 1/3 is increased to C = 1. Use of 0 -- 1/3 apparently requires an implicit solution. However, similar iteration to that used in Eq. (2.117) is rapidly convergent and the scheme can be used quite economically. Jiii~ii~iiiiiiiiiiiiiiii~E~i~iiiiiiiii~ii~ii~iiiiii~iiiiiii~i~r~iii~iiiiiiiiiiii~iiii~iiiiiiiiiiii~iiiii~iiiiiii~iii~ii~ii~iii~iiiiii~i~iiiiiii~iiiii~i~iiiiiii~iiiiiiiii~i~i~iii~ii~ii~i~ii~iiiiiii~i~i~iiiiiii~iiiiii~iii~
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There are of course various alternative procedures for improving the temporal approximation other than the Taylor expansion used in the previous section. Such procedures will be particularly useful if the evaluation of the derivative matrix A can be avoided. In this section we shall consider two predictor-corrector schemes (of Runge-Kutta type) that avoid the evaluation of this matrix and are explicit. The first starts with a standard Galerkin space approximation being applied to the basic equation (2.1). This results in the form Md,~, - M ,I~9 = Pc + P o + f = ~ dt
(E.7)
where again M is the standard mass matrix, f are the prescribed 'forces' and ec(~I')
- L
NT ~x/dr2 cgFi
(E.8a)
NT~Gi ~x/dff2
(E.8b)
represents the convective 'forces', while PD(@) are the diffusive ones.
-- L
400 AppendixE
If an explicit time integration scheme is used, i.e. MAcI, _= M(~, "+~
- ~n) =
At~3n(~n)
(E.9)
the evaluation of the fight-hand side does not require the matrix product representation and Ai does not have to be computed. Of course the scheme presented is not accurate for the various reasons previously discussed, and indeed becomes unconditionally unstable in the absence of diffusion and external force vectors. The reader can easily verify that in the case of the linear one-dimensional problem "' "n 1 the fight-hand side is equivalent to a central difference scheme with ~I'nl and 'I'i+ only being used to find the value of ,1,7+1, as shown in Fig. E.l(a). The scheme can, however, be recast as a two-step, predictor-corrector operation and conditional stability is regained. Now we proceed as follows: Step 1. Compute cI:)n+l/2 using an explicit approximation of Eq. (E.9), i.e. ~n+l/2
__ t~n _~_
At M _ I ~ n 2
(E.10)
and Step 2. Compute t~ n+l inserting the improved value of of Eq. (E.9), giving t ~ n + l __ ~ n Jl-
t~ n+l/2
in the fight-hand side
AtM-I~ n+1/2
(E.11)
This is precisely equivalent to the second-order Runge-Kutta scheme being applied to the ordinary system of differential equations (E.7). Figure E.l(b) shows in the one-dimensional example how the information 'spreads', i.e. that now ~n+l will be dependent on values at nodes i - 2, . . . , i + 2. It is found that the scheme, though stable, is overdiffusive and numerical results are poor. An alternative is possible, however, using a two-step Taylor-Galerkin operation. Here we return to the original equation (2.1) and proceed as follows: Step 1. Find an improved value of cI,n+ 1/2 using only the convective and source parts. Thus ,~,n+l/2 _ ,~n _~At2 ( ~0Fin + )Qn (E.12a) which of course allows the evaluation of F n+l/2. We note, however, that we can also write an approximate expansion as
F7+1/2
-Fn+
A t OF n
20t At
n
A l A n Or
-Fi2 0Fj 0Gj
'
Q)n
(E.12b)
Appendix E 401
t~
(a) Single-step explicit
,i,n+,,
,n
i-1
~
i
,
i+1
......
x
....
i-1 (b) Standard predictor--corrector
i
+ 1/2At r
i+1
tTtn+z~tt l~O~I~0~~I
....
n
i-1 (c) Local prediction-corrector (two-step Taylor-Galerkin)
tn
i
tn +
x
1/2At
x
i+1
Fig. E.I Progressionof information in explicit one- and two-step schemes.
This gives An (OFj
aGj
Q)n
=
2
At
(Fn+l/2 ',--i
--
FT)
(E.12c)
Step 2. Substituting the above into the Taylor-Galerkin approximation of Eq. (E.5) we have
Mmt~ = _ AtILNT(OFi + ~oqGi 0 (Fn+l/2 x / + O )n dr2 + LNT ~X/,,__ i -- F n) dr2 +
L NTS(Fn+l/2 i - Fn i ) dr2 ] (E.lZd)
and after integration by parts of the terms with respect to the Xi derivatives we obtain simply
n d ~ + L N T[Q + S (Fn+l/2 - Fn)] dS2 M A ~ = - A t { L ~0NT (FT/+1/2+ G/)
+ fF NT---i (Fn+I/2 + Gni )n/dF }
(E.13)
We note immediately that: 1. The above expression is identical to using a standard Galerkin approximation on Eq. (2.1) and an explicit step with Fi values updated by the simple equation (E. 12a).
402 AppendixE 2. The final form of Eq. (E.13) does not require the evaluation of the matrices Ai resulting in substantial computation savings as well as yielding essentially the same results. Indeed, some omissions made in deriving Eq. (E.6a) did not occur now and presumably the accuracy is improved. A further practical point must be noted: 3. In non-linear problems it is convenient to interpolate Fi directly in the finite element manner as Fi -- N~i
rather than to compute it as Fi (~}). Thus the evaluation of F n+l/2 need only be made at the quadrature (integration) = n+l/2
points within the element, and the evaluation of 9 by Eq. (E.12a) is only done on such points. For a linear triangular element this reduces to a single evaluation of ,~n+l/2 and F n+l/2 for each element at its centre, taking of course ~n+l/2 and F n+l/2 as the appropriate interpolation average there. In the simple one-dimensional linear example the information progresses in the manner shown in Fig. E.l(c). The scheme, which originated at Swansea, can be appropriately called the Swansea two step, and has found much use in the direct solution of compressible high-speed gas flow equations. We have shown some of the results obtained by this procedure in Chapter 7. However in Chapter 3 we have discussed an alternative which is more general and has better performance. It is of interest to remark that the Taylor-Galerkin procedure can be used in contexts other than direct fluid mechanics. The procedure has been used efficiently by Morgan et al. in solving electromagnetic wave problems.
E.2.1 Multiple wave speeds When ~bis a scalar variable, a single wave speed will arise in the manner in which we have already shown at the beginning of Chapter 2. When a vector variable is considered, the situation is very different and in general the number of wave speeds will correspond to the number of variables. If we return to the general equation (2.1), we can write this in the form C~
0~I)
C~ i
+ A i ~ x/ + ~
+ Q = 0
(E.14)
where Ai is a matrix of the size corresponding to the variables in the vector ~. This is equivalent to the single convective velocity component A - U in a scalar problem and is given as 0Fi Ai = 0 ~
(E.15)
This in general may still be a function of cI,, thus destroying the linearity of the problem.
Appendix E 403 Before proceeding further, it is of interest to discuss the general behaviour of Eq. (2.1) in the absence of source and diffusion terms. We note that the matrices Ai can be represented as (E. 16)
Ai "- Xi Ai X~- 1
by a standard eigenvalue analysis in which Ai is a diagonal matrix. If the matrices Xi are such that Xi = X
(E.17)
which is always the case in a single dimension, then Eq. (E. 14) can be written (in the absence of diffusion or source terms) as O~ -
Ot -
-]-- X A i
X _ 1 0r ~ .. = 0
OXi
(E. 18)
Premultiplying by X -~ and introducing new variables (called Riemann invariants) such that t~ -- x - l ~ I ~
(E.19)
we can write the above as a set of decoupled equations in components 4) of q~ and corresponding A of A: -O~ & - ]- Ai ~O~ x/= 0
(E.20)
each of which represents a wave type equation of the form that we have previously discussed. A typical example of the above results from a one-dimensional elastic dynamics problem describing stress waves in a bar in terms of stresses (a) and velocities (v) as
&r
Ov
Ov Ot
1 0o pox
ot
=o =0
This can be written in the standard form of Eq. (2.1) with
crlp} The two variables of Eq. (E.19) become ~1 = O " - CV
where c =
~/E/p
and
~)2 = O" "+" CI)
and the equations corresponding to (E.20) are 0~1
0~)1
ot + c 0~2
Ot
=~ 0~2
c-- x = 0
representing respectively two waves moving with velocities +c.
404 Appendix E
Unfortunately the condition of Eq. (E.17) seldom pertains and hence the determination of general characteristics and therefore decoupling is not usually possible for more than one space dimension. This is the main reason why the extension of the simple, direct procedures is not generally possible for vector variables. Because of this we shall in Chapter 3 only use the upwinding characteristic-based procedures on scalar systems for which a single wave speed exists and this retains justification of any method proposed.
ii!i
i
Edge-based finite element formulation The edge-based data structure has been used in many recent finite element formulations for flow problems. As mentioned in Sec. 7, Chapter 7, this formulation has many advantages such as smaller storage, etc. To explain the formulation we shall consider the Euler equations and a few assembled linear triangular elements on a two-dimensional finite element mesh as shown in Fig. E 1. From Eq. (1.25) we rewrite the following Euler equations 0tI)
0F i
Ot + ~xi
--
(F.1)
0
where 9 are the conservative variables. If the element-based formulation for the above equation omits the stabilization terms, the weak form can be written as
f N k AAt~ d ~ 2 = - f ( N k ) TOFi ~x~ d r
(E2)
In a fully explicit form of solution procedure, the left-hand side becomes M(A cI,/At) and here M is the consistent mass matrix (see Chapter 3). We can write the RHS of the above equation for an interior node I [Fig. C. 1(a)] by interpolating Fi in each element and after applying Green's theorem as
~ Eel
(NkF/k)
=
E
Eel
3
OX i
E
(F~ + F/J + F/r)
(F.3)
where A E is the area and I, J and K are the three nodes of the element (triangle) E. This is an acceptable added approximation which is frequently used in the TaylorGalerkin method (see Chapter 2). In another form, the above RHS can be written as [Fig. F. 1(a)]
A1 ON1 ~(F/+ 3 OX i
F1 + F2) -t
A2 ON1
3
~(F/+
F2 + F3) -i
OX i
A3 ON1
~(F/+
3 Oxi
F3 + F1) (E4)
where A1, A2 and A3 are the areas of elements 1, 2 and 3 respectively. For integration over the boundary on the RHS, we can write the following in the element formulation N I (NkF/k) dFn8 B~I
B
B6I
406 AppendixF 2
2
3
1
(a)
(b)
Fig. F.I Typical patch of linear triangular elements: (a)inside node; (b) boundary node. where n is the boundary normal. The above equation can be rewritten for the node I in Fig. F. 1(b) as F B1 (2F[ + F~)na + FB2(2F/ + V~)n2 (F.6) 6 --6 where FB1 and FB2 are appropriate edge lengths and subscripts 1 and 2 indicate the edges in Fig. F. 1(b). The above Eqs (F.3) and (E5) can be reformulated for an edge-based data structure. In such a procedure, Eq. (E3) can be rewritten as (for node I)
L
l EON' ( F / + F [ ' ) I ~ON'(NkF/k) d a - - ~ m s ~ ( [Ae30xi (F.7) E6I E S--1 E~_IIs where m~ is the number of edges in the mesh which are directly connected to the node I and the summation ~-'~EEIIs extends over those elements that contain the edges I I~.
The coefficient in Eq. (E7) is
c:'s= It can be easily verified that
Ae
tONI]
-T L x/J E=Ils E ms
S=I
C/I" - 0
(F.8)
(F.9)
for all i. The user can now readily verify that the above equation is identically equal to the standard element formulation of Eq. (F.4) if we consider the node I in Fig. F. 1(a). For the boundary nodes, however, Eq. (F.9) is not satisfied and thus the element formulation is not reproduced. For the boundary edges, in addition to Eqs. (F.6) and (F.7) the following addition is necessary
-
[1-'--un, B1
1-'B2 ]
+ --unz r[
(F.10)
Alternatively the above contribution may be added to Eq. (E6) to complete the edge formulation.
"'iiiiHiiii'i'ii''ii i'i'iiiiiiiiii i'''ii iii,,i,ii,ii,,i i'i'iii' iiiiiii'iiiii'iiii iiiil iiiiiiiiiiiiiiii'''i''i'!iiiiii' ii'iiiiiii Multigrid method It is intuitively obvious that whenever iterative techniques are used to solve a finite element or finite difference problem it is useful to start from a coarse mesh solution and then to use this coarse mesh solution as a starting point for iteration in a finer mesh. This process repeated on many meshes has been used frequently and obviously accelerates the total convergence rate. This acceleration is particularly important when a hierarchical formulation of the problem is used. We have indeed discussed such hierarchical formulations in Chapter 4 of The Finite Element Method: Its Basis and Fundamentals (Zienkiewicz, Taylor and Zhu) and the advantages are pointed out there. The simple process which we have just described involves going from coarser meshes to finer ones. However, it is not useful if no return to the coarser mesh is done. In hierarchical solutions such returning is possible as the coarser mesh matrix is embedded in the finer one with the same variables and indeed the iteration process can be described entirely in terms of the fine mesh solution. The same idea is applied to the multigrid form of iteration in which the coarse and fine mesh solution are suitably linked and use is made of the fact that the fine mesh iteration converges very rapidly in eliminating the higher frequencies of error while the coarse mesh solution is important in eliminating the low frequencies. To describe the process let us consider the problem of
LO=f
in
f2
(G.1)
which we discretize incorporating the boundary conditions suitably. On a coarse mesh the discretization results in KCq~c = r c
(G.2)
which can be solved directly or iteratively and generally will converge quite rapidly if q~c is not a big vector. The fine mesh discretization is written in the form
=f:
(G.3)
and we shall start the iteration after the solution has been obtained on the coarse mesh. Here we generally use aprolongation operator which is generally an interpolation from which the fine mesh values at all nodal points are described in terms of the coarse mesh values. Thus qS[ = Pq~C_1 -k- AqS[
(G.4)
408 AppendixG where Ath f is the increment obtained in direct iteration. If the meshes are nesting then of course the matter of obtaining P is fairly simple but this can be done quite generally by interpolating from a coarser to a finer mesh even if the points are not coincident. Obviously the values of the matrices P will be close to unity whenever the fine mesh points lie close to the coarse mesh ones. This leads to an almost hierarchical form. Once the prolongation to (~f has been established at a particular iteration i the fine mesh solutions can be attempted by solving K f A~:
- ff - R {
(G.5)
where the residual R is easily evaluated from the actual equations. We note that the solution need not be complete and can well proceed for a limited number of cycles after which a return to the coarse mesh is again made to cancel out major low frequency errors. At this stage it is necessary to introduce a matrix Q which transforms values from the fine mesh to the coarse mesh. We now write for instance q~c = Qq~/f
(G.6)
where one choice for Q is, of course, pT. In a similar way we can also write R; = QR /
(G.7)
where Ri are residuals. The above interpolation of residuals is by no means obvious but is intuitively at least correct and the process is self-checking as now we shall start a coarse mesh solution written as "C "" KC(t~i+l - ~b c) = R iC
(G.8)
At this stage we solve for t~icq_1 using the values of previous iterations of q~c and putting the collected residuals on the fight-hand side. This way of transferring residuals is by no means unique but has established itself well and the process is rapidly convergent. In general more than two mesh subdivisions will be used and suitable operators P and Q have to be established for transition between each of the stages. The total process of solution is vastly accelerated and proceeds well as shown by the many papers cited in Chapter 7.
....
Boundary layer-inviscid flow coupling A few references on the topic of boundary layer-inviscid flow coupling are given in Chapter 6. In this appendix we shall briefly explain a simple procedure of this flow coupling procedure. To understand the process of coupling the Euler and integral boundary solutions we shall consider a typical flow pattern around a wing as shown in Fig. E. 1. Both turbulent and laminar regimes are shown in this figure. We summarize the procedure as follows: Step 1. Solve the Euler equations in the domain considered around the aerofoil. Here any mesh can be used independently of the mesh used for the boundary layer solution. The solution thus obtained will give a pressure distribution on the surface of the wing. Step 2. Solve the boundary layer using an integral approach over an independently generated surface mesh. If the surface nodes do not coincide with the Euler mesh, the pressure needs to be interpolated to couple the two solutions. The laminar portion near the boundary (Fig. E.1) is calculated by the 'Thwaites compressible' method and the turbulent region is predicted by the 'lag-entrainment' integral boundary layer model. Step 3. The Euler and integral solutions are coupled by transferring the outputs from one solution to the other. As indicated in Fig. H. 1, direct and semi-inverse couplings can be used for different regions. The semi-inverse coupling is introduced here mainly to stabilize the solution in the turbulent region close to separation. Figure H.2 shows the flow diagrams for the present boundary layer-inviscid coupling. Further details on the Thwaites compressible method and semi-inverse coupling can be found in the references discussed in Sec. 6.12, Chapter 7 (Le Balleur and coworkers). In Fig. H.2, Cp is the coefficient of pressure; s the coordinate along the surface; ~ the boundary layer thickness; 0 the momentum thickness; Cf the skin friction coefficient; H the velocity profile shape parameter; p the density; VN the transpiration velocity; K* is a factor developed from stability analysis; the subscript v marks the viscous boundary layer region; 6* the displacement thickness; the superscript i indicates inviscid region and the superscript m indicates the current iteration.
410 Appendix H Semi-inverse
~" , , 2 T ' , I I ~ F ~ ~9 '
Turbulent
II Transition Laminar ~
sss
... ~ o ~ ~,=, ~
" .
Turbulent i|
. ....
. . . . . . . . . . . . . . . )~',-~ . . . . . . . . . . . . Direct ',
- . . . . . . . . . . . . . . . . . . .
Semi-inverse
Semi-inVerse"~
Fig. H.1 Flow past an aerofoil. Typical problem for boundary layer-inviscid flow coupling.
The following are useful relations for some of the above quantities: H--if,
=
1
n, "
pvu~
K*=
/3= 4 1 - M
27r0'
2 (H.1)
where n is the normal direction from the wing surface. We have the following equations to be solved in the integral boundary layer lagentrainment model. Continuity
d----s- = d H
Ce - H~
~
- (H +
~
1)--uv ~
(H.2)
Momentum dO __ C f
ds
2
(H+2-M
(H.3)
~ 2) - 0- du, uo ds
Lag-entrainment
0
F 2.8 ( - F/ ds LH + H1
dCe
Ws)EQ
0.5
0 duo
0 duo (1 + 0.2M 2) ] uo ds (1 + 0.075M 2) (1 + O..i~-)
(H.4)
where F is a function of Ce and Cf and given as 0.02Ce + C 2 + F
(0.01 + Ce)
3 (H.5)
Appendix H 411
J
Unstructured grids or multiblock Euler inviscid method
Cp,8
Direct calculation d PVN=~(Pv UvS*)
T
i_
Lag-entrainment boundary layer viscous method
6, O, Cf, H
I-
Direct calculation
(a)
Unstructured grids or multiblock Euler inviscid method
Cp,S
Direct calculation
pvN
pv~n+1=pv~n+ K*[ e-du~
v
Uv ds
du i m Uv
Lag-entrainment boundary layer viscous method VN
---~6"'- I
Direct calculation
(b) Fig. H.2 Couplingtechniques: (a) direct; (b) semi-inverse.
In the above equations,/-/and
-
Hi are
the velocity profile shape parameters defined as
1 / ~ ( u1)- - -
H--O
Uv
n.,
H1 - -
0
(H.6)
412 AppendixH C e is the entrainment coefficient; uv the mean component of the streamwise velocity at the edge of the boundary layer; M the Mach number; CT the shear stress coefficient; A the scaling factor on the dissipation length; the subscripts EQ and EQo denote respectively the equilibrium conditions and equilibrium conditions in the absence of secondary influences on the turbulence structure. Once the above equations are solved, the transpiration velocity VN is calculated as shown in Fig. H.2 and is added to the standard Euler boundary conditions on the wall and plays the role of a surface source. The coupling continues until convergence. In practice, in one coupling cycle, several Euler iterations are carried out for each boundary layer solution.
Mass-weighted averaged turbulence transport equations iiiiii iiiiii~ii$~iiiiiiJ Jiiiii iiiiiiiiiiiiFiii iiiiiiiii iiiiiiii iiiiii
iiiliiiiiiiiiiii
In this subsection we provide two turbulence models commonly employed in the compressible flow calculations. Before discussing these models we write the Reynolds stress term and turbulent heat flux term in terms of turbulent eddy viscosity as
~ij -" IdT
( OLIi OUj ~Xj ~ OXj
2 0U k (~ij ) 2 30Xk --
5ptcrij
(I.1)
and
Cp OT q~ = -Izr Prr Oxj
(I.2)
One of the following turbulence models may be employed to calculate the turbulent viscosity.
5palart-AIImaras model
In this model the turbulent eddy viscosity is calculated as ~T
YT -- ~ ~- ~fol p
(1.3)
where fol
=
X3
x 3 + c31
(1.4)
with b'
x = -
//
(1.5)
414
Appendix I The viscosity variable ~ is calculated from Of,
Of,
leo {
Col[1 - ft2]~S~' + -O" ~X j
=
E
-- Cwl fw -- "-~ ft2
l[ l
(u-+- ~,) ~
+ Cb2 -~Xj
(I.6)
+ ftl A~2
The parameters used in the above equation are written as // --
K 2 y 2 fv2
CO +
X
fv2 "- 1--
f w -- g g
-
-
1 +Xfol
1 + cw3 g6 + cw6 3
r + cw2(r 6
r =min
-
-
r)
~K~d 2 , 1 0
ft; - ct3 e x p ( - c t 4 X 2) ft 1 --" Ctl gt exp
gt
--
CUt
-ct2 ~
2 2
[y2 + gt Yt ] (I.7)
0.1, ~3t Ax
min
where y is the distance from a given node to the nearest wall, ~ is the vorticity given as
"--
On3
OU2
OX2
OX3
+
OUl
OU3
OX3
OXl
"[-
OU2
OUl
OXl
OX2
(I.8)
Aft is the difference in velocity between the point and trip, yt is distance from a node to a trip point or curve, ~t is the vorticity magnitude at trip point or curve, Ax is the surface grid spacing at trip. Other constants used in the model are Cbl = 0.1355, Cb2 = 0.622, cr = 2/3, K = 0.41, cwl = Cbl/K 2 + (1 + Cb2)/cr, Cw2 = 0.3, Cw3 = 2, Col = 7.1, Ctl = 1, Ct2 = 2, Ct3 - " 1.1 and C t 4 = 2. The major difference between the model given here and the one used in Sec. 8.2 is that here we have a trip curve (3D) or trip point (2D) to trigger turbulence. The trip curve is often defined at a 3% distance from the leading edge of a solid surface. However, the model without a trip curve is widely employed as explained in Chapter 8. - w model
The basic idea of the n - a; model arises from the fact that vorticity is directly proportional to n2fl, i.e. /~2
- c~ 1
(I.9)
Appendix I 415 where c is a constant. The eddy viscosity may therefore be written as #r =/9-
(I.10)
W
The transport equations for ~ and w may be written as
~(p/~) -~- ~-X/(p/~Ui) =
+ ~X/(~/jUj) -- ~*p/'~W
(I.11)
and
~(9( ~ )
(9 ( ~ ) + ~x-Tx
- ~xiCg(#W~xi~ + o~WC9 ~-~/(~j) - ~:
(i.12)
where #~ = # + #r/o,~ and #~ = # + #r/tr~. The constants are c~ = 5 / 9 , / 3 = 3 / 4 0 , / 3 * = 9/100, cr~ = a~ = 2. The turbulence models discussed in this section can be non-dimensionalized as discussed in Sec. 8.2 if necessary.
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Author index
Page numbers in bold are for pages at the end of chapters with names of author references. Abarbanel, S., 327, 346 Abbott, M.B., 292, 314 Abe, K., 250, 271 Adey, R.A., 54, 56, 77 Ainsworth, M., 123, 138, 349, 351,378 Aizinger, V., 297, 316 Akin, J.E., 82, 93, 106, 191,196 Alexander, J.M., 143, 144, 166 Alkire, R.L., 284, 291 Allmaras, S.R., 252, 270, 272, 273 Almeida, R.C., 41, 75 Altan, A.T., 143, 166 Alvarez, G.B., 41, 75 Amsden, A.A., 172, 186, 194 Apsley, D.D., 250, 262, 271,273 Armstrong, R.C., 160, 162, 163, 169 Ashida, H., 340, 341,348 Asian, A.R., 118, 137 Astley, R.J., 317, 327, 328, 330, 331,332, 333, 334, 335,336, 344, 346, 347, 348, 352, 369, 370, 371, 372, 379, 380 Aswathanarayana, P.A., 82, 106 Atkinson, B., 143, 166 Atkinson, J.D., 41, 75 Austin, D.I., 297, 308, 315, 316 Ayers, R., 197, 222, 224, 225, 227, 228, 244 Baaijens, EET., 144, 167 Babau, A.V.R., 123, 129, 139 Babuska, I., 81, 97,105, 319, 332, 345, 347, 349, 35 l, 353, 354, 355,357, 378, 379 Bach, E, 172, 193 Bai, K.J., 177, 195 Baker, T.J., 221,246 Balaji, EA., 143, 167 Baldwin, B.S., 203, 204, 245 Bando, K., 324, 325, 327, 328, 329, 345, 346 Barai, E, 332, 347 Barbone, EE., 332, 347
Barnett, M., 254, 272 Barrett, K.E., 34, 74 Barton, J.T., 208, 209, 245 Batchelor, C.K., 4, 27 Baum, J.D., 123, 138 Baumann, C.E., 52, 76 Bay, F., 143, 167 Bayliss, A., 321,324, 345 Bayne, L.B., 197, 221,227, 230, 244 Beck, R.F., 177, 195 Beddhu, M., 172, 193, 194 Behr, M., 297, 315 Bejan, A., 191,196, 274, 275,276, 288, 290, 291 Bellet, M., 152, 168, 172, 193 Belytschko, T., 41, 75, 152, 168 Benque, J.P., 56, 77 Benson, D.J., 163,169 Bercovier, H., 136, 140 Bercovier, M., 56, 77 Berenati, R.E, 283, 290 B6renger, J.P., 327,345, 346 Berkhoff, J.C.W., 318, 336, 344, 347, 364, 380 Bermudez, A., 56, 77 Bettess, J.A., 21, 23, 27, 323, 327, 328, 345 Bettess, P., 21, 23, 27, 297, 308, 315, 316, 318, 320, 323,324, 325,327,328, 329, 331,332, 336, 339, 343,344, 345, 346, 347, 348,349, 351,352, 354, 355,356, 357, 365,368, 369, 378, 379, 380 Bhandari, D.R., 143, 147, 166 Bhargava, P., 143, 152, 167, 168 Bilger, R.W., 143, 166 Billey, V., 123, 137, 221,246 Bishop, A.R., 203, 244 Biswas, G., 52, 77, 82, 107 Blasco, J., 82, 107 Bonet, J., 143, 152, 154, 165, 167, 168, 169 Boris, J.P., 68, 78, 270, 273 Bornside, D.E., 162, 163, 169 Borouchaki, H., 123, 129, 131,139, 140, 233, 246
418 Author index Bosch, G., 256, 272 Bourgault, Y., 123, 139 Bova, S., 297, 315 Bradshaw, P., 241,247 Braess, H., 172, 186, 194 Brakalmans, W.A.M., 152, 168 Brebbia, C.A., 82, 106, 297, 314 Bremhorst, K., 253, 257, 272 Brezzi, E, 42, 75, 99, 109 Briard, P., 256, 257, 272 Brinkman, H.C., 284, 291 Brook, D.L., 68, 78 Brookes, P.J., 118, 137, 197, 221,244, 352, 378 Brooks, A.N., 39, 46, 47, 74, 76 Brooman, J.W.E, 241,247 Brosilow, C.B., 283,290 Brown, R.A., 160, 162, 163, 169 Budiansky, B., 152, 168 Burnett, D.S., 328, 329, 330, 335,346 Burton, A.J., 165, 169 Butler, M.J., 334, 347 Calvo, N., 175, 195 Cambier, L., 235, 239, 246 Cao, Y., 177, 195 Card, C.C.M., 143,166 Cardle, J.A., 52, 76 Carew, E.O.A., 154, 169 Carey, G., 297, 315 Carey, G.E, 41, 52, 75, 76 Carotenuto, A., 82, 108 Carter, J.E., 229, 230, 246 Casciaro, R., 220, 246 Castro, J.M., 144, 167 Castro-Diaz, M.J., 123, 131,139, 233,246 Caswell, B., 143, 166 Cavendish, J.C., 123, 139 Cebeci, T., 251,272 Cecchi, M.M., 297, 315 Cervera, M., 45, 76 Cessenat, O., 359, 360, 380 Chadwick, E., 352, 379 Chan, A.H.C., 274, 275, 290 Chan, S.T., 82, 107 Charpin, F., 247 Chartier, M., 297,314 Chastel, Y., 143, 167 Chaudhuri, A.R., 82, 107 Chen, A.J., 228,246 Chen, C.-H., 172, 186, 194 Chen, C.K., 191,196 Chen, C.Y., 191,196 Chen, H.S., 177, 195, 327, 336, 337, 338, 346 Chen, W.L., 250, 271 Cheng, P., 275,285, 290, 291 Cheng, S.I., 118, 137
Cheng, T.W., 113, 137 Chenot, J.L., 143, 144, 152, 167, 168, 172, 193 Cheung, Y.K., 363, 380 Chiam, T.C., 327, 328, 329, 346 Chin-Joe-Kong, M.J.S., 373,380 Chippada, S., 297, 315 Choi, H.G., 172, 186, 194 Chorin, A.J., 82, 83, 106 Chow, R.R., 24 l, 247 Christie, I., 34, 35, 37, 74, 299, 316 Christon, M.A., 82, 107 Chung, T.J., 143, 147, 166 Cipolla, J.L., 334, 347 Clark, EJ., 327, 328, 343,346, 348 Cockburn, B., 374, 381 Codina, R., 42, 43, 45, 47, 52, 57, 59, 75, 76, 78, 82, 91, 93, 97, 99, 102,107,108, 112, 123,136,137, 176,195, 203,205,218, 219, 229, 232, 245, 246 Colella, P., 207, 210, 245 Collino, E, 354, 357, 359, 364, 365,379 Comini, G., 82, 106 Connor, J.J., 297, 314 Constantineseu, G., 256, 270, 272 Cook, J.L., 172, 186, 194 Cools, R., 374, 381 Copps, K., 349, 357, 378 Cornfield, G.C., 143, 166 Coupez, T., 172, 193 Courant, R., 34, 74 Coyette, J.P., 328, 332, 334, 346, 347 Craggs, A., 317, 344 Cremers, L., 328, 332, 334, 346, 347 Crepon, M., 297, 314 Crochet, M.J., 154, 157, 168, 169 Cullen, M.J.P., 297,314 Currie, I.G., 4, 27 Dalsecco, S., 297, 303, 313, 315 Dancy, H., 327, 345 Darcy, H., 274, 290 Daubert, O., 297, 315 Dauksher, W., 373,380 David, E., 285,291 Davidson, L., 256, 270, 272, 273 Davies, A.R., 154, 169 Davis, J., 297, 314 Dawson, C., 297, 315, 316 Dawson, C.N., 297, 316 Dawson, C.W., 177, 195 Dawson, P.R., 143, 166 de La Bourdonnaye, A., 354, 379 De Roquefort, T.A., 191,196 de Sampaio, P.A.B., 52, 76, 123, 138 de Sampmaio, P.A.B., 47, 48, 76 De Vahl Davis, G., 191,196 de Villiers, R., 297, 315
Author index 419 Del Guidice, S., 82, 106 Delgiudice, S., 191,196 Demkowicz, L., 123, 138, 199, 213, 244, 245, 332, 346, 347 Denham, M.K., 115, 117, 137, 256, 257,272 Depr6s, B., 359, 360, 380 Derby, J.J., 113, 137 Derviaux, A., 123, 137 Desaracibar, C.A., 152, 168 Deshpande, M.D., 113, 137 Devloo, E, 123, 137 Dewitt, D.E, 191,196 Dinesen, EG., 373, 380 Ding, D., 154, 169 do Carmo, E.G.D., 41, 75 Dompierre, J., 123, 139 Donea, J., 37, 41, 42, 45, 69, 74, 75, 78, 82, 106, 152, 168, 172, 185, 186, 194 Dou, H-S., 159, 160, 162, 163, 169 Douglas, J. Jr, 56, 77, 297, 314 Downie, M.J., 343,348 Dreyer, D., 332, 347 Duncan, D., 349, 351,378 Duncan, J.H., 180, 181,195 Dupont, I., 297, 314 Dupont, S., 154, 169 Durany, J., 56, 77 Durbin, EA., 256, 272 Dutra Do Carmo, E.G., 205,245 Eiseman, ER., 123, 140 Ellwood, K., 101,109 Emery, A.E, 373,380 Emson, C., 324, 325,327, 328, 329, 345, 346 Engleman, M.S., 136, 140 Engquist, B., 325,345 Ergun, S., 276, 290 Esche, S.K., 152, 168 Eskilsson, C., 375,376, 377, 381 Evans, A., 123,138 Eversman, W., 330, 333,346, 347, 352, 379 Ewing, R.E., 56, 77 Fan, S., 254, 272 Fan, Y., 169 Fares, E., 256, 272 Farhat, C., 357, 379 Farmer, J., 172, 193 Feng, Y.T., 172, 193 Fenner, R.T., 143, 166 Feshbach, H., 318,344, 354, 379 Field, D.A., 123,139 Fix, G.J., 297, 314 Flanagan, D.E, 152, 168 Fleming, C.A., 297, 308,315 Fletcher, C.A., 40, 74
Forchheimer, E, 284, 291 Foreman, M.G.G., 297,314, 315 Formaggia, L., 197, 221,223, 243 Fortin, M., 123, 136, 139, 140 Fortin, N., 136, 140 Fourment, L., 143, 167 Franca, L., 357,379 Franca, L.E, 41, 75 Frey, EJ., 123, 139 Frey, W.H., 123, 139 Fried, I., 373,380 Fyfe, K.R., 328, 332, 346 Gale~o, A.C., 205,245 Gallagher, R.H., 34, 74, 191,196 Gamallo, E, 369, 370, 37 l, 372, 380 Gangaraj, S.K., 319, 345, 351,378 Garcfa, J., 172, 179, 194 Garg, V.K., 277, 290 Geers, T.L., 324, 326, 327, 335,345 George, EL., 123, 129, 13 l, 139, 140, 233,246 Gerdes, K., 327, 332, 346, 347 Germano, M., 269, 273 Ghia, K.N., 113, 131,136 Ghia, U., 113, 131,136 Ghosh, S., 144, 167 GiD, 387, 388 Giraldo, EX., 297, 316 Girodroux-Lavigne, E, 241,247 Giuliani, S., 82, 106, 152, 168, 187, 195 Givoli, D., 324, 326, 335,336, 345 Gladwell, G.M.L., 317,344 Godbole, EN., 134, 140, 143, 149, 151,166 Gottlieb, D., 327, 346, 373,374, 376, 381 Goudreau, G.L., 163, 169 Goussebaile, J., 56, 77 Gowda, Y.T.K., 82, 107, 191,196 Gray, W., 297, 315 Gray, W.G., 301,316 Gray, W.R., 297, 314 Green, A., 222, 246 Green, J.E., 241,247 Green, EJ., 123,139 Gregoire, J.E, 56, 77 Gresho, EM., 82, 107, 136, 140 Griffiths, D.E, 34, 35, 37, 74 Griffiths, D.V., 382, 387 Grinstein, EE, 270, 273 Gtil~at, 0., 118, 137 Gunzberger, M., 32 l, 324, 345 Guymon, G.L., 40, 75 Habashi, W.G., 123, 139 Hackbusch, W., 220, 246 Hagstrom, T., 325,345 Hall, C.D., 297,314
420 Authorindex Halleux, J.I., 152, 168 Hallquist, J.O., 163, 169 Halpern, E, 297, 314 Haltiner, G.L., 292, 314 Hansbo, E, 187, 196, 205,245 Hara, H., 340, 34 l, 348 Harari, I., 41, 75, 332, 347, 357, 379 Harbani, Y., 56, 77 Hardy, O., 123, 138 Hariharan, S.I., 325,345 Harlow, EH., 195 Hassager, O., 172, 193 Hassan, O., ll8, 123, 128, 129, 137, 138, 139, 140, 197, 216, 22 l, 222, 224, 225,227, 228,230, 233, 234, 238,240, 243, 244, 246, 256, 266, 268,272, 273, 352, 353,378 Hauguel, A., 56, 77 Hause, J., 123, 140 Havelock, T.H., 325,345 Hearn, G.E., 343,348 Hecht, E, 56, 77, 123, 128, 129, 131,139, 233,246 Heinrich, J.C., 34, 46, 52, 74, 76, 77, 143, 147, 166, 191,196, 297, 308, 314, 315 Hermansson, J., 187, 196 Herrera, I., 362, 380 Herrmann, L.R., 40, 75 Hervouet, J., 297, 315 Hesthaven, J.S., 327, 346, 373, 374, 375, 376, 380, 381 Hetu, J.E, 123, 13 l, 138 Higdon, R.L., 325,345 Hill, T.R., 7 l, 78 Hindmarsh, A.C., 82, 107 Hino, T., 179, 180, 195 Hinsman, D.E., 297, 314 Hirano, H., 299, 316 Hiriart, G., 21, 27 Hirsch, C., 4, 27, 198, 199, 200, 244 Hirt, C.W., 172, 176, 186, 193, 194 Hoffman, J.D., 203,244 Holdo, A.E., 256, 262, 263,272 Holford, R.L., 328, 329, 330, 335,346 Homsy, G.M., 288, 291 Hood, E, 34, 74, 19 l, 196 Houston, J.R., 338, 339, 348 Hsu, C.T., 275,290 Hsu, M.-H., 172, 186, 194 Huang, G.C., 136, 140, 146, 148, 151,152, 167 Huerta, A., 37, 42, 45, 74, 75, 82, 107, 172, 185, 186, 187, 194, 196 Hu6tink, J., 144, 167 Huetnik, J., 152, 168 Hughes, T.J.R., 39, 4 l, 42, 46, 47, 52, 74, 75, 76, 134, 140, 205,207,245, 349, 378 Hulbert, G.M., 4 l, 75 Hulbert, H.E., 297, 314
Huttunen, T., 354, 357, 359, 360, 361,362, 364, 365, 379 Huyakorn, P.S., 34, 46, 74 Hydraulic Research Station, 301,303,316 Idelsohn, I.R., 172, 179, 194 Idelsohn, S.R., 52, 76, 77, 172, 175, 179, 194, 195 Ihlenburg, E, 319, 324, 345, 351,378 Iida, M., 172, 186, 194 Inaba, H., 285,291 Inagaki,, 299, 316 Incropera, EP., 191,196 Irons, B.M., 143, 166 Isaacson, E., 34, 74 Ito, H., 297, 315 Jain, P.C., 143, 166 Jaluria, Y., 191,196 Jameson, A., 98, 109, 172, 179, 180, 193, 195, 203, 221,245, 246 Jami, A., 297, 315 Jang, Y.J., 250, 271,272 Janson, C., 177, 195 Jenson, G., 177, 195 Jiang, B.N., 52, 76 Jiang, M.Y., 172, 194 Jin, H., 123, 139 Jin, W.G., 363, 380 Jirousek, J., 362, 380 Johan, Z., 41, 47, 52, 75, 76, 205,245 Johansson, S.H., 256, 272 Johnson, C., 47, 52, 76, 205,245 Johnson, R.H., 143, 166 Johnson, W., 144, 167 Johnson, W.E., 176, 195 Jones, J., 197, 221,243, 244 Jones, J.W., 118, 137 Jones, W.P., 253,272 Jou, W.H., 270, 273 Jue, T.C., 82, 93, 106, 191,196 Kaipio, J.P., 354, 357, 359, 360, 361,362, 364, 365, 379 Kakita, T., 152, 168 Kallinderis, Y., 228, 246 Kalro, V., 262, 273 Kamath, M.G., 123, 129, 140 Kanehiro, K., 340, 341,348 Karniadakis, G.E.M., 374, 381 Kashiyama, K., 297, 315 Kawahara, M., 82, 106, 172, 173, 186, 193, 194, 297, 299, 303,314, 315, 316 Kawka, M., 152, 168 Kelly, D.W., 37, 46, 47, 74, 299, 316, 320, 336, 345, 347 Kennedy, J.M., 152, 168
Author index 421 Keunings, R., 154, 169 Keyhani, M., 289, 291 Kim, J., 113,137, 271,273 Kim, Y.H., 177, 195 Kinzel, G.K., 152, 168 Kleiber, M., 152, 168 Kobayashi, S., 143, 166, 167 Kodama, T., 297, 303,315 Kolbe, R.L., 270, 273 Kolmogorov,A.N., 248, 271 Kong, L., 197, 242 Koshyk, J.N., 327,346 Kulacki, F.A., 289, 291 Kumar, K.S.V., 123, 129, 139 Kumar, S.G.R., 82, 1116 Kuo, J.T., 297,314 Labadie, C., 56, 77 Labadie, G., 297, 303, 313,315 Laghrouche, O., 349, 351, 352, 354, 355, 356, 357, 365, 368, 369, 378, 379, 381) Lahoti, G.D., 143,166 Laitone, E.V., 187, 196 Lakshminarayana, B., 254, 272 Lal, G.K., 143, 167 Lam, C.K.G., 253,257, 272 Lamb, H., 4, 10, 27, 317, 318, 344 Landau, L.D., 4, 27 Lapidus, A., 67, 78, 203,245 Larsen, J., 327,345 Larsson, L., 177, 195 Larwood, B., 118, 137 Latteaux, B., 297, 303, 313,315 Lau, L., 343,348 Laug, P., 123, 129, 131,139 Launder, B.E., 250, 251,253, 271, 272 Lauriat, G., 285,288, 291 Laval, H., 82, 106 Lax, P.D., 61, 66, 78 Le Balleur,, 241,247 Le Quere, P., 191,196 Lee, C.H., 143, 166 Lee, J.H.W., 297, 298, 302, 315 Lee, J.K., 144, 152, 167, 168 Lee, R.L., 82, 106 Lee, S.C., 191,196 Lee, T.H., 177, 195 Lee, W.I., 172, 176, 194 Legat, V., 123, 139, 154, 169 Leonard, B.P., 37, 74 Lesaint, P., 71, 78 Leschziner, M.A., 250, 271,272 Levine, E., 56, 77 Lewis, R.W., 21, 27, 123, 129, 140, 172, 176, 186, 190, 191,194, 195, 196, 274, 290, 292, 314 Lick, W., 311,316
Lien, ES., 250, 271 Lifshitz, E.M., 4, 27 Lighthill, M.J., 241,247, 317, 334, 343, 344 Lin, P.X., 56, 77 Liou, J., 56, 77 Liu, A.W., 162, 163, 169 Liu, C-B., 256, 272 Liu, Y.C., 123, 136,137,14t), 146, 148, 151,152,167, 197, 243 Lo, D.C., 172, 186, 194 Ltihner, R., 56, 64, 65, 77, 78, 123,137,138,139, 172, 178, 179,194, 197,207,208,220, 228,242, 243, 246, 299, 316 Lopez, S., 220, 246 Lucas, T., 177, 195 Lynch, D.R., 297, 301,314, 315, 316 Lynov, J.P., 373,380 Lyra, P.M.R., 197,244 Lyra, P.R.M., 52, 76, 82,1t)8, 123,138,216,217,246 Macaulay, G.J., 328, 332, 334, 346, 347 McCarthy, J.H., 177, 195 MacCormack, R.W., 203, 204, 245 Majda,, 325,345 Maji, P.K., 52, 77, 82, 11)7 Makinouchi, A., 152, 168 Malamataris, N., 101,109 Malett, M., 205,245 Malkus, D.S., 134, 14t), 373,380 Malone, T.D., 297, 314 Mandel, J., 354, 357, 379 Manzari, M.T., 82,11)8, 197,216, 217, 221,244, 246, 266, 268,273 Marchal, J.M., 154, 169 Marchant, M.J., 123, 128, 138, 139, 197, 221,233, 238,244 Marcum, D.L., 123, 129, 140 Margolin, L., 270, 273 Marini, L.D., 42, 75 Marshall, R.S., 191,196 Martin, J.C., 175, 194 Martin, P., 349, 351,378 Martinelli, L., 172, 179, 180, 193, 195 Martinez, M., 297, 315 Martinez-Canales, M.L., 297, 316 Massarotti, N., 82, 11)8, 190, 191,196, 277, 290 Massoni, E., 152, 168 Matallah, H., 154, 169 Mathur, J.S., 82, 96, 108, 112, 136 Matthieu, C., 256, 270, 272 Mavripolis, D.J., 98, 11)9 Maxant, M., 56, 77 Maxworthy, T., 188, 190, 196 Mayer, P., 354, 357, 379 Mead, H.R., 241,247 Medale, M., 172, 186, 194
422
Authorindex Mei, C.C., 177, 195, 317, 327, 336, 337, 338, 344, 346 Melenk, J.M., 353, 354, 355,357, 379 Mellor, P.B., 144, 167 Melnok, R.E., 241,247 Minev, P.D., 82, 107 Mitchell, A.R., 34, 35, 37, 46, 74, 342, 348 Mitsoulis, E., 160, 169 Mittal, S., 235, 239, 247 Mohammadi, B., 123, 129, 131,139, 233,246 Moin, P., 113, 137, 271,273 Monk, P., 354, 357, 359, 360, 361,362, 364, 365,379 Morgan, K., 47, 52, 56, 64, 65, 76, 77, 78, 82, 91, 96, 99, 106, 107, 108, 112, 118, 123, 125, 128, 129,136,137,138,139, 197, 203,204,205,207, 208,213,214, 215,216, 217,218,219, 220, 221, 222, 223,224, 225,227,228, 230, 233,238,240, 242, 243, 244, 245, 246, 256, 266, 268,272, 273, 296, 297,298, 299, 301,308,314, 316, 352, 353, 378 Morinishi, K., 235,239, 247 Morse, P.M., 318, 344, 354, 379 Morton, K.W., 54, 56, 77, 203,245 Moser, R., 271,273 Mulder, W.A., 373,380 Muttin, F., 172, 193 Nakayama, T., 82, 106 Nakazawa, D.W., 37, 45, 46, 47, 74, 75 Nakazawa, S., 56, 77, 97, 109, 134, 136, 140, 143, 166, 299, 316 Nakos, D.E., 177, 195 Narayana, P.A.A., 82, 106, 107, 123, 129, 139, 191, 196 N~ivert, U., 47, 52, 76 Navon, I.M., 297, 314, 315 Navti, S.E., 172, 186, 194 Nesliturk, A., 41, 75 Neta, B., 326, 345 Newton, R.E., 317, 324, 344 Nguem, N., 52, 76 Nichols, B.D., 172, 176, 193 Nickell, R.E., 143, 166 Nicolaides, R.A., 220, 246 Nield, D.A., 274, 275, 276, 290 Nishida, Y., 235,239, 247 Nithiarasu, P., 21, 27, 47, 52, 57, 59, 76, 78, 82, 83, 93, 94, 95, 96,107,108,109, 112, 113, 116, 123, 131,136,139, 159,169, 172, 186, 190, 191,194, 196, 205,206, 207, 217, 218, 219, 229, 232,245, 246, 256, 272, 275, 277, 281,282, 290 Nitsche, J.A., 72, 78 Noble, R., 222, 246 Noblesse, E, 177, 195 Nonino, C., 191,196
O'Brien, J.J., 297, 314 O'Carroll, M.J., 21, 27 Oden, J.T., 123, 134, 137, 138, 139, 140, 143, 147, 166, 197, 199, 213,244, 245 Oh, S.I., 143,166 Ohmiya, K., 82, 106 Olsson, E., 256, 272 Ofiate, E., 44, 45, 49, 52, 75, 76, 77, 143, 147, 152, 153,166,167,168, 172, 175,176, 178, 179,194, 195 Oran, E.S., 270, 273 Ortiz, P., 57,59, 78, 82,107, 108, 229, 232, 246, 297, 298, 301,303,316, 355, 364, 365,379, 380 Pagano, A., 241,247 Paisley, M.E, 203,245 Pal, M., 332, 347 Palit, K., 143,166 Palmeiro, B., 123, 138 Palmeiro, E, 123, 137 Papanastasiou, T.C., 101,109 Pastor, M., 82,108, 164, 165,169, 274, 275,290, 297, 316 Patil, B.S., 319, 345 Patnaik, B.S.V.E, 82, 107 Patrick, M.A., 256, 257, 272 Patrik, M.A., 115, 117, 137 Peiro, J., 123, 137, 138, 197, 203,204, 213,216, 221, 223,243, 245, 246, 365,375,380 Pelletier, D.H., 123, 131,138 Peraire, J., 56, 64, 77, 78, 82, 101,108, 123, 125, 128, 129,137,138,139, 197,203,204, 207, 208,213, 214, 215, 216, 217, 221,223,233,240, 242, 243, 244, 245, 246, 256, 272, 293,296, 297, 298, 301, 302, 308, 314, 315, 352, 353,378 Periaux, J., 123, 137, 221,246 Peric, D., 172, 193 Perrey-Debain, E., 349, 351,357, 378, 379 Perrier, P., 221,246 Perry-Debain, E., 365,368,369, 380 Peseux, B., 357, 379 Phan-Thien, N., 157, 159, 160, 162, 163, 169 Philips, T.N., 160, 169 Pica, A., 297,315 Pin, ED., 175,195 Pinsky, P.M., 335,347 Pironneau, O., 56, 77 Pitk~anta, J., 47, 52, 76, 99, 109 Pittman, J.E, 45, 75, 143, 144, 166 Ponthot, J.-Ph., 172, 186, 194 Posse, M., 56, 77 Postek, E., 143, 167 Prasad, V., 289, 291 Price, J.W.H., 143, 166 Probert, E.J., 123, 128, 138, 139, 197, 221,227, 228, 229, 233, 238, 240, 244, 246
Author index 423 Proft, J., 297, 316 Prudhomme, S., 123, 139 Pulliam, T.H., 208, 209, 245 Qu, S., 97,109 Quartapelle, L., 82, 106 Quecedo, M., 297, 316 Rabier, S., 172, 186, 194 Rabinowitz, P., 374, 381 Rachowicz, W., 123, 138, 139 Raithby, G.D., 82, 106 Raj, K.H., 143, 167 Ramaswamy, B., 82, 93,106, 172, 173, 186, 191,193, 194, 196 Rannacher, R., 82, 106 Raveendra, V.V.S., 123, 129, 140 Raven, H., 177, 195 Raviart, P.-A., 71, 78 Ravindran, K., 172, 176, 186, 194, 195, 281,290 Ravisanker, M.S., 82, 106 Rebelo, N., 143,166 Reed, W.H., 71, 78 Rees, M., 34, 74 Reinhart, L., 56, 77 Ren, G., 82, 106 Reynen, J., 52, 76 Rice, J.G., 82, 106 Richez, M.C., 297,314 Richtmyer, R.D., 203,245 Rider, W.J., 270, 273 Rimon, Y., 118, 137 Rivara, M.C., 220, 246 Roach, P.J., 4, 27 Rodi, W., 256, 272 Rodriguez-Ferran, A., 172, 186, 194 Rojek, J., 82, 108, 143, 164, 165, 167, 169 Runchall, A.K., 34, 74 Russel, T.E, 56, 77 Russo, A., 42, 75 Rynne, B., 349, 351,378 Sacco, C., 172, 179, 194 Saghafian, M., 262, 273 Sai, B.V.K., 191,196 Sai, B.V.K.S., 47, 52, 76, 82, 91, 99, 106, 107, 108, 205, 217, 218, 219, 245 Said, R., 118, 137, 197, 221,244 Saidi, M.S., 262, 273 Sakhib, E, 41, 75 Salasnich, 297, 315 Saltel, E., 123, 129, 131,139 Sanchez, E., 355,365,379 Sani, R.L., 136, 140 Sarpkaya, T., 2 l, 27 Sarrate, J., 187, 196
Sastri, V., 56, 77 Satofuka, N., 235,239, 247 Scavounos, Pd., 177, 195 Sch~ier, U., 176, 195 Schlichting, H., 3, 27, 262, 272 Schmidt, W., 203, 245 Schmitt, V., 247, 267, 268, 273 Schneider, G.E., 82, 106 Schnipke, R.J., 82, 106 Schoombie, S.W., 342, 348 Schrefler, B.A., 274, 275,290 Schreurs, P.J.G., 152, 168 Schr6der, W., 256, 272 Schroeder, W.J., 123, 140 Scott, V.H., 40, 75 Secco, E., 297,315 Seetharamu, K.N., 21, 27, 82, 93, 106, 107, 123, 129, 139, 190, 191,196, 275,282, 290 Seki, N., 285, 291 Selvam, R.P., 262, 272 Semeniuk, K., 327,346 Shakib, E, 47, 52, 76, 205, 245 Shanker, P.N., 113, 137 Shephard, M.S., 123, 140 Shepherd, T.G., 327, 346 Sherwin, S.J., 365,374, 375, 376, 377, 380, 381 Shih, T.H., 257, 272 Shimizaki, Y., 143, 166 Shimura, M., 82, 106 Shin, C.T., 113, 131,136 Shin, S., 172, 176, 194 Shiomi, T., 274, 275,290 Shirron, J., 357, 379 Shirron, J.J., 332, 347 Shu, C.-W., 375,381 Sibson, R., 123,139 Silva, R.S., 41, 75 Sinha, S.K., 277, 290 Slavutin, M., 332, 347 Smagorinsky, J., 269, 273 Smith, A.M.O., 251,272 Smith, I., 382, 387 Smith, J.W., 113, 137 Smith, M.D., 160, 162, 163, 169 Smolinski, P., 41, 75 Sod, G., 207, 208, 245 Soding, H., 177, 195 Sommerfield, A., 321,345 Soni, B.K., 123, 140 Sorensen, K.A., 197, 221,244, 256, 272 Sosnowski, W., 152, 168 Souli, M., 172, 186, 194 Spalart, P.R., 252, 270, 272, 273 Spalding, B., 251,272 Spalding, D.B., 34, 74 Squire, H.B., 241,247
424 Author index Squires, K., 256, 270, 272 Srinivas, M., 82, 106 SSC Program Ltd., 222, 246 Stagg, K.G., 292, 314 Staniforth, A.N., 297, 315 Stansby, EK., 172, 185, 186, 194, 262, 273 Steger, J.L., 203,245 Stewart, J.R., 197, 217, 220, 243 Stojek, M., 362, 363, 380 Storti, M.A., 52, 76 Stoufflet, B., 221,246 Strada, M., 191,196 Strelets, M., 270, 273 Stroubolis, T., 319, 345, 351,378 Strouboulis, T., 123, 137, 349, 357, 378 Subramanian, E., 286, 291 Subramanian, G., 123, 129, 140 Sugawara, T., 340, 341,348 Sugimoto, R., 357, 379 Suli, E., 56, 77 Sun, J., 160, 162, 163, 169 Sundararajan, T., 82, 93,107, 123,129,139, 143,167, 190, 196, 275,277, 282, 290 Sung, J., 172, 186, 194 Szepessy, A., 47, 52, 76, 205,245 Szmelter, J., 82, 101,108, 123, 131,138, 241,247 Takeuchi, N., 297, 315 Tam, A., 123, 139 Tam, Z., 343,348 Tamaddonjahromi, H.R., 154, 169 Tang, L.Q., 113,137 Tanner, R.I., 143,166, 169 Taylor, C., 82, 106, 172, 186,194, 297, 314, 319, 345 Taylor, L.K., 172, 193, 194 Taylor, M.A., 374, 381 Taylor, R.L., 2, 4, 5, 14, 17, 19, 21, 23, 27, 30, 40, 52, 54, 74, 80, 81, 82, 92, 97, 105, 108, 109, 111, 112, 123, 127, 131,134, 136,136,140, 143, 152, 164, 165,166,169, 180,195,319,320,321,323, 327, 331,338, 339, 345, 350, 354, 362, 365,375, 378, 380, 387, 388 Temam, R., 4, 27 Teng, C.H., 373,380 Teng, W.-H., 172, 186, 194 Tezduyar, T., 297, 315 Tezduyar, T.E., 207,245 Tezduyar, T.T.I., 56, 77 Thareja, R.R., 197, 217, 220, 243 Thomas, C., 94, 109 Thomas, C.G., 206, 207, 245 Thompson, E.G., 143,166 Thompson, J.E, 123, 140 Thompson, L.L., 335,347 Tien, C.L., 275,284, 290, 291 Tong, P., 40, 74
Tong, T.W., 286, 291 Tottenberg, U., 220, 246 Townsend, E, 154, 169 Toyoshima, S., 134, 136, 140, 152, 167 Tremayne, D., 222, 246 Trevelyan, J., 357, 365,368, 369, 379, 380 Trevisan, O.V., 288, 291 Tsang, T.T.H., 113, 137 Tsubota, K., 299, 316 Tucker, EG., 250, 256, 262, 270, 271,272, 273 Turkel, E., 321,345 Turner-Smith, E.A., 197, 243 Tutar, M., 256, 262, 263,272 Tzabiras, G.D., 172, 194 Upson, C.D., 82, 106 Ushijima, S., 172, 186, 194 Usmani, A.S., 123, 129, 140 Utnes, T., 82, 106 Vadyak, J., 203,244 Vafai, K., 275, 284, 290, 291 Vahdati, M., 123, 125, 128, 129, 137, 197, 213, 214, 215, 216, 243 Vallet, M.G., 123, 139 van der Lugt, J., 152, 168 van Estorff, O., 332, 347 Van Veldhuizen, M., 373, 380 Vazquez, C., 56, 57, 59, 77, 78 V~quez, M., 47, 52, 76, 82, 93, 97, 99, 102,107,108, 112, 123,136,137,205,217,218,219,229,232, 245, 246 Veldpaus, EE., 152, 168 Verhoeven, N.A., 197, 221,244 Vilotte, J.P., 134, 136, 140 Vincent, R.E., 374, 381 von Neumann, J., 203,245 Walker, K.L., 288, 291 Waiters, K., 154, 169 Wang, C., 250, 272 Wang, H.H., 297, 314 Wang, N.M., 152, 168 Warburton, T., 325,345, 375,381 Ward, S., 228, 246 Wargedipura, A.H.S., 152, 168 Watson, D.E, 123, 129, 139 Weatherill, N.E, 52, 76, 82, 96, 108, 112, 118, 123, 129,136,137,138,139,140, 197, 221,222, 224, 225,227, 228,230, 233,238,243, 244, 246, 266, 268, 273, 352, 378 Webster, M.E, 154, 169 Weeks, D.J., 241,247 Wehausen, J.V., 177, 195 Welch, J.E., 195 Wendroff, B., 61, 66, 78
Author index 425 Wesswling, E, 220, 246 Westermann, T.A., 123, 138 Wheeler, M.E, 297, 315 Whitaker, S., 275,290 Whitfield, D.L., 172, 193, 194 Whitham, G.B., 317, 318, 344 Wieting, A.R., 197, 217, 220, 243 Wifi, A.S., 152, 168 Wilcox, D.C., 25 l, 272, 273 Will6, D.R., 382, 387 Williams, A.J., 160, 169 Williams, R.T., 292, 297, 314, 315 Wingate, B.A., 374, 381 Wolfstein, M., 34, 74, 25 l, 272 Wong, K.K., 343,348 Wood, R.D., 143, 144, 152, 154, 166, 167, 168 Wood, R.W., 152, 168 Woodward, E., 297, 314 Woodward, E, 207, 210, 245 Wriggers, E, 172, 186, 194 Wr6blewski, A., 362, 380 Wu, J., 82, 109, 123, 128, 130, 131, 138, 139, 231, 233, 236, 246, 297, 315 Yagewa, G., 143, 147, 166 Yahia, D.D.D., 123, 139 Yang, B., 373,381 Yang, C., 172, 178, 179, 194 Yang, Z., 257, 272 Yeckel, A, 113, 137 Yiang, C.B., 82, 106 Yoo, J.Y., 172, 186, 194
Yoshida, T., 297,315 Yoshimura, T., 340, 341,348 Young, A.D., 241,247 Young, D.L., 172, 186, 194 Yovanovich, M.M., 82, 106 Yu, C.C., 52, 76, 191,196 Zalesiak, S.T., 68, 78 Zhang, Y., 123, 129, 140 Zhou, J.G., 172, 185, 186, 194 Zhu, J.Z., 2, 4, 5, 14, 17, 19, 21, 23, 27, 80, 81, 92, 97, 105, 111,112, 123, 127, 131,134, 136, 136, 137, 152,167, 180,195, 197,213,243,245,319, 320, 321,323,327, 331,338,339, 345, 350, 354, 362, 375,378, 387,388 Ziegler, C.K., 311,316 Zienkiewicz, O.C., 2, 4, 5, 14, 17, 19, 21, 23, 27, 30, 34, 35, 37, 40, 45, 46, 47, 52, 54, 56, 57, 59, 64, 65, 74, 75, 76, 77, 78, 80, 81, 82, 91, 92, 93, 95, 97, 99, 101, 102, 105, 107, 108, 109, 111, 112, 123, 125,127, 128, 129, 130, 131,134, 136,136, 137,138,139,140, 143, 144, 146, 147, 148, 149, 151,152, 153, 154, 162, 164, 165,166,167,168, 169, 180, 190, 191, 195,196, 197,203,204,205, 207,208, 213, 214, 215,216, 217,218, 219,221, 223,229, 231,232, 233,236, 242, 243, 245, 246, 274,275,277,290, 292, 296, 297,298,299, 301, 302,303,308,314, 315, 316, 317, 318, 319,320, 321,323,324,327, 328, 329, 331,336, 338, 339, 344, 345, 346, 347, 348, 350, 352, 354, 362, 363, 365,375,378, 380, 387,388 Zolesio, J.E, 172, 186, 194
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Subject index Acoustic wave velocity, 14 Adaptive mesh generation, incompressible Newtonian laminar flows, for transient problems, 131,133 Adaptive mesh refinement, incompressible Newtonian laminar flows: about adaptive mesh refinement, 123 choice of variables, 130-1 element elongation, 128-30 estimation of second derivatives at nodes, 128 first derivative- (gradient) based refinement, 130 lid-driven cavity example, 131-2 local patch interpolation: superconvergent values, 127 second gradient- (curvature) based refinement, 123-7 interpolation errors, 125-6 principal values and directions, 127 see also Compressible high-speed gas flow, adaptive refinement and shock capture in Euler problems Aerofoil, potential flow solution example, 21-2 Anisotropic shock capturing, with high-speed gas flow, 205 Arbitrary-Lagrangian-Eulerian (ALE) methods, free surface flows, 172, 185-9 Artificial compressibility: and the character-based split (CBS) algorithm, 95-6 and dual time stepping, 96 Artificial diffusion concept, 40 Astley's shape function, with conjugated infinite elements, 331 Babu~ka-Brezzi restriction (BB), 81 circumvention of, 97-8 Balancing diffusion, 46 streamline balancing diffusion, 47 Bore, shallow water example, 300-1 Boundary conditions: with the CBS algorithm, 100-3
convection-diffusion-reaction equation, 72-3 Dirichlet type, 15 governing equations, 9 Neumann type, 15 radiation, 62-4 Boundary layer-inviscid flow coupling, 410-12 Boussinesq approximation, porous medium flow, 279, 285 Boussinesq assumption, with turbulent flows, 250 Brinkman extensions, porous medium flow, 284 Bristol channel, shallow water example, 301-5 Buoyancy driven incompressible flows: about buoyancy driven flows, 189-91 flow in an enclosure example, 191-3 Grashoff number, 190 Prandtl number, 191 Rayleigh number, 191 Burger equation, 68-70 Cauchy-Poisson free surface condition, waves, 339 Character-based split (CBS) algorithm: about the CBS algorithm and compressible and incompressible flow, 79-81,104-5 about the split, 82-3 artificial compressibility, 95-6 in transient problems (dual time stepping), 96 boundary conditions: application of real boundary conditions, 101-3 fictitious boundaries, 100-1 prescribed traction boundary conditions, 101 solid boundaries in inviscid flow (slip conditions), 101 solid boundaries with no slip, 101 Chorin split, 82-3 circumvention of the Babu~ka-Brezzi (BB) restrictions, 97-8 forms/schemes: about forms, 92 evaluation of time limits, 93-5 fully explicit form, 92 quasi- (nearly) implicit form, 93
428
Subjectindex Character-based split (CBS) algorithm- cont. semi-implicit form, 93 governing equations for, 79-82 with high-speed gas flow, 202-3 inviscid problem, performance of two- and single-step algorithms, 103-5 mass diagonalization (lumping), 91-2 with metal forming, transient, 164 with Porous medium flow, 280 with shallow water problems, 297-8 single step version, 98-9 spatial discretization and solution procedure, 86-91 split A, 86-91 split B, 91 temporal discretization, 83-5 split A, 84-5 split B, 85 with viscoelastic flows, 163 see a l s o Computer implementation of the CBS algorithm; Incompressible Newtonian laminar flow Chrzy bed friction, long and medium waves, 320 Chrzy coefficient, 294 Chorin split, 82-3 see also Character-based split (CBS) algorithm Cnoidal and solitary waves, 340-2 Coefficients of pressure and friction, postprocessing, 393 Compressible high-speed gas flow basics: about high-speed gas flow, 197-8 boundary conditions, subsonic and supersonic: Euler equation, 200-1 Navier-Stokes equations, 201-2 boundary layer-inviscid Euler solution coupling, 241 Euler equation examples: inviscid flow past an RAE2822 airfoil, 208-10, 211 isothermal flow through a nozzle in one dimension example, 207, 209 Riemann shock tube - transient problem in one dimension example, 207, 208 two-dimensional transient supersonic flow over a step example, 207-8, 210 governing equations: ideal gas law, 198 internal energy, 198 Navier-Stokes, 198 total specific energy, 198-9 numerical approximations and the CBS algorithm, 202-3 shock capture methods: about shock capture, 203-4 anisotropic viscosity shock capturing, 205 residual-based methods, 205
second derivative-based methods, 204-5 structured meshes, 197 variable smoothing, 205-6 subsonic inviscid flow past an NACA0012 airfoil, 206 see also Turbulent flows, compressible material Compressible high-speed gas flow, adaptive refinement and shock capture in Euler problems: about adaptive refinement, 212 h-refinement process and mesh enrichment, 212-13 h-refinement and remeshing in steady-state two-dimensional problems, 213-17 remeshing in steady-state two-dimensional problems examples: hypersonic inviscid flow past a blunt body example, 214-16 inviscid flow with shock reflection from a solid wall example, 214, 215 inviscid shock interaction example, 217, 220 supersonic inviscid flow past a full circular cylinder example, 217-19 Compressible high-speed gas flow, steady state three-dimensional inviscid examples: complete aircraft flow patterns: inviscid engine intake example, 221-2, 224 inviscid flow past full aircraft example, 221,223 multigrid approaches, 220-1 parallel computation, 221 recasting element formulations in an edge form, 217 THRUST- the supersonic car, 222-6 Compressible high-speed gas flow, transient twoand three-dimensional problems: exploding pressure vessel example, 226, 228 shuttle launch, 227, 229, 230 Compressible high-speed gas flow, viscous problems in three dimensions, 240 hypersonic viscous flow past a double ellipsoid, 240-1 Compressible high-speed gas flow, viscous problems in two dimensions: about viscous flow, 227-8, 231 adaptive refinement in both shock and boundary layer, 230, 234-5 special adaptive refinement for boundary layers and shocks, 230-3, 236--8 transonic viscous flow past an NACA0012 aerofoil, 235-9 viscous flow past a plate example, 229-30, 232 Compressible and incompressible flow see Character-based split (CBS) algorithm Computer implementation of the CBS algorithm: about computer implementation, 382-3 data input module, 383--4 boundary data, 383
Subject index 429 mesh data- nodal coordinates and connectivity, 383 necessary data and flags, 383-4 preliminary subroutines and checks, 384 output module, 387 solution module, 384-7 boundary conditions, 386-7 CBS algorithm; steps, 386 convergence to steady state, 387 different forms of energy equation, 387 shock capture, 385-6 solution of simultaneous equations semi-implicit form, 387 time step, 384-5 see also Character-based split (CBS) algorithm Conservation laws, 28-30 Conservation of mass: about conservation of mass, 1-2 governing equations, 6-7, 81 incompressible Newtonian laminar flow, 110 Conservation of momentum: for CBS algorithm, 80-1 dynamic equilibrium, 7 incompressible Newtonian laminar flow, 111 Constitutive equation, viscoelastic flows, 158 Continuity equation: with free surface flows, 173 viscoelastic flow, 157 Convection-diffusion equations: vector-valued variables: multiple wave speeds, 402-4 Swansea two step operation, 402 Taylor-Galerkin method, 397-9 two-step predictor-corrector methods, 399-400 two-step Taylor-Galerkin operation, 400-2 Convection-diffusion-reaction equation: about convection-diffusion equations, 28-30, 73 boundary conditions, 72-3 Galerkin process/method/procedure, 72-3 conservation laws, 28-30 convective flux quantities, 28 diffusive flux quantities, 28 Galerkin weighting, 30 pure convection treatment, 70-2 transport equation, 28 velocity field, 29 see also Steady state convection-diffusion equation in one dimension; Steady state convection-diffusion equation in two (or three) dimensions; Transient convection-diffusion equation; Waves and shocks, non-linear Convective acceleration effects, 1 Convective flux quantities, 28 Coriolis accelerations/parameters, with shallow water problems, 293-5,303
Dam break, shallow water example, 299-300 Darcy-Rayleigh number, porous medium flow, 279 Darcy's law/flow regime, porous medium flow, 274-5, 286-8 Deborah number, with viscoelastic flows, 158, 160, 162 Detached Eddy Simulation (DES), 270 Deviatoric stresses/deviatoric strain rates, 1, 5 Differential equations, self-adjoint, 391 Diffusion: artificial diffusion concept, 40 balancing diffusion, 46 cross-wind diffusion, 47 negative diffusion concept, 40 streamline balancing diffusion, 48-9 see also Convection-diffusion-reaction equation Diffusive flux quantities, 28 Direct Numerical Simulation (DNS), 270-1 Dirichlet type boundary conditions, 15 Discontinuous enrichment method, short waves, 357-9 Discontinuous Galerkin Finite Elements (DGFE), 374-7 Discretization procedure, porous medium flow, 279-82 Drag force calculation, postprocessing, 392-3 Dual time stepping, and artificial compressibility, 96 Dynamic viscosity, 1 Eddies, large eddy simulation, 267-9 eddy viscosity, 269 Kolmogorov cascade/constant, 269 Smagorinsky's model/constant, 269 standard SGS model, 269-70 Edge-based finite element formulation, 405--6 Eigenvalues, problems with waves in closed domains, 319 Elastic bulk modulus, 1, 14 Elastic springback, in non-Newtonian flows, 152-4 Electromagnetic scattering, 352-3 Energy conservation: for CBS algorithm, 80 and equation of state, 7-9 Enrichment functions, short waves, 358 Enthalphy, 8, 13 Equation of state and energy conservation, 7-9 Equations of fluid dynamics, 1-4 see also Governing equations of fluid dynamics; Inviscid, incompressible flow Ergun correlation, porous medium flow, 276-7 Euler equations, 9-11 compressible high speed gas flow examples, 206-11 with high-speed gas flow, 200-1
430 Subjectindex Euler problems s e e Compressible high-speed gas flow, adaptive refinement and shock capture in Euler problems Euler solutions, boundary layer-inviscid Euler solution coupling, 241 Eulerian form, 6 Eulerian methods, free surface flows, 176-84 Explicit characteristic-Galerkin procedures, 56-62 Extrusion: transient extrusion problem, 150-4 s e e a l s o Metal forming, transient, direct displacement approach; Non-Newtonian flows - metal and polymer forming; Viscoelastic flows Fekete points, 374 Finite element methods/approximation s e e Weighted residual and finite element methods Finite increment calculus (FIC): in multidimensional problems, 48-9 stabilization of the convection-diffusion equation, 43-4 Finite volume approximation/technique/methodology, 2, 23-4 Poisson equation in two dimensions example, 24-6 Forcheimmer extensions, porous medium flow, 284 Forming, steady-state problems, 144-7 Free surface incompressible flows: about free surface flows, 170-2 Arbitrary-Lagrangian-Eulerian (ALE) methods: about the ALE method, 172, 185-6 implementation of the ALE method, 186-7 solitary wave propagation example, 187-9 Eulerian methods: about Eulerian methods, 172, 176 hydrostatic adjustment, 178-9 mesh updating or regeneration methods, 176-84 sailing boat example, 184-5 ship motion problem, 177 submarine example, 181-3 submerged hydrofoil example, 179-82 Lagrangian methods, 171-5 continuity equation, 173 Kroneker delta, 172 model broken dam problem example, 173-5 momentum equation, 173 Frequency domain solutions, short waves, 350 Friction coefficients, postprocessing, 393
with transients, 51-3 Galerkin procedure: and boundary conditions for convection-diffusion, 72-3 simple explicit characteristic, 56-62 Galerkin scheme/equations, short waves, 367 Galerkin spatial approximation, 56, 59 and CBS algorithm, 80 Galerkin weighting, convection-diffusion-reaction equation, 30, 32 Gas flow s e e Compressible high-speed gas flow Gauss-Chebyshev-Lobatto scheme, 373-4 Gauss-Legendre integration, 355, 373 Gauss-Lobatto scheme, 373-4 Givoli's procedure, NRBCs with waves, 326 GLS s e e Galerkin least squares (GLS) approximation Governing equations of fluid dynamics: Babu~ka-Brezzi restriction, 81 balance of energy, 9 boundary conditions, 9 for character-based split (CBS) algorithm, 79-82 compressible flow, 11 conservation of energy and equation of state, 7-9, 80 non-dimensional form, 82 conservation of mass, 6-7 non-dimensional form, 81 conservation of momentum, 7, 80 non-dimensional form, 81 deviatoric stresses/deviatoric strain rates, 5 enthalphy, 8 Euler equations, 9-11 Eulerian form, 6 gradient operator, 6 indicial notation, 4 intrinsic energy, 8 inviscid flow, 11 Kroneker delta, 5, 80 Lam6 notation, 6 Navier-Stokes equations, 9-11 non-dimensional form (for CBS algorithm), 81-2 rates of strain, 5 stress-strain rate relations, 5 stresses in fluids, 4-6 turbulence/turbulent instability, 11 volumetric viscosity, 5-6 Gradient operator, 6 Grashoff number, with buoyancy driven flows, 190 Green's function, 43
Galerkin, finite element, method, 17-18 Galerkin formulation with two triangular elements example, 19-21 Galerkin least squares (GLS) approximation/method: in multidimensional problems, 48-9 in one dimension, 41-2
h-refinement: and mesh enrichment, 212-13 and remeshing in steady-state two-dimensional problems, 213-17 Hankel functions, 338 and infinite elements, 328
Subject index 431 with Trefftz type infinite elements, 333 Higdon boundary condition, short waves, 365 Hydrofoil, submerged, example, free surface flows, 179-82 Hydrostatic adjustment, free surface flows, 178-9 Ideal gas law, with high-speed gas flow, 198 Incompressibility constraint difficulties, 1-2 Incompressible flows, 13-14 about free surface and buoyancy driven flows, 170 about incompressible flows, 3 acoustic wave velocity, 14 elastic bulk modulus, 14 s e e a l s o Buoyancy driven incompressible flows; Character-based split (CBS) algorithm; Free surface incompressible flows Incompressible Newtonian laminar flow: about laminar flow, 110, 136 basic equations: conservation of energy, 111 conservation of mass, 110 conservation of momentum, 111 with CBS algorithm: fully explicit artificial compressibility form, 112 incompressible flow in a lid-driven cavity example, 113-20, 125 quasi-implicit form, 123 semi-implicit form, 112-23 steady flow past a backward facing step example, 115-21 steady flow past a sphere example, 117-22 transient flow past a circular cylinder example, 118-24 mixed and penalty discretization/formulations, 134-6 slow flows, analogy with incompressible elasticity, 131,134 s e e a l s o Adaptive mesh refinement, incompressible Newtonian laminar flows Incompressible non-Newtonian flows: about non-Newtonian effects, 141, 165 s e e a l s o Extrusion; Metal forming, transient, direct displacement approach; Non-Newtonian flows - metal and polymer forming; Viscoelastic flows Indicial notation, 4 Infinite elements: about infinite elements, 327 accuracy of, 332 Burnett and Holford ellipsoidal type infinite elements, 328-30 mapped periodic (unconjugated) infinite elements, 327-8 Hankel functions, 328 Trefftz type infinite elements, 332-3 Hankel functions, 333
wave envelope (conjugated) infinite elements, 330-2 Astley's shape function, 331 Integration formulae: linear tetrahedron, 395-6 linear triangles, 395 Interpolation errors, 125-6 Intrinsic energy, 8 Inviscid, incompressible flow, 11-13 irrotational flow, 12 stream function, 13 velocity potential solution, 11-13 Irrotational flow, 12 Kolmogorov cascade/constant, 269 Kolmogorov length scale, 248 Korteweg-de Vries wave equation, 342 Kroneker delta, 5 and CBS algorithm, 80 with free surface flows, 172 with viscoelastic flows, 157 Lagrangian methods, free surface flows, 171-5 Lam6 notation, 6 Laminar flow s e e Incompressible Newtonian laminar flow Lapidus type diffusivity, 67-8 Local Non-Reflecting Boundary Conditions (NRBCs), long and medium waves, 324-7 Long waves s e e Waves, long and medium Mass-weighted averaged turbulence transport equations, 413-15 Maxwell equation, with viscoelestic flows, 156 Medium waves s e e Waves, long and medium Mesh enrichment: and h-refinement process, 212-13 with high-speed gas flow problems, 212-13 Mesh refinement s e e Adaptive mesh refinement, incompressible Newtonian laminar flows Mesh updating, free surface flows, 176-8 Metal forming, transient, direct displacement approach, 163-5 and the CBS algorithm, 164, 165 impact of circular bar example, 165 Microlocal discretization, 354 Mixed and penalty discretization/formulations, 134-6 Modelling errors, short waves, 351 Mollifying/smoothing discontinuities, 39 Momentum equation: free surface flows, 173 viscoelastic flow, 157 Monotonically Integrated LES (MILES), 270 Multigrid method, 407-8
432
Subjectindex NACA0012 aerofoil: inviscid problems of subsonic and supersonic flow, 103-5, 206 transonic viscous flow past the aerofoil, 235-9 Navier-Stokes equations, 9-11, 26 with compressible high-speed gas flow, 198 derivation of non-conservative form, 389-90 with high-speed gas flow, 198, 201-2 with turbulent flows, 248, 249 Neumann type boundary conditions, 15 Newton-Cotes formula, 372-3 Newtonial dynamic viscosity, 157 Newtonion laminar flow s e e Adaptive mesh refinement, incompressible Newtonian laminar flows; Incompressible Newtonian laminar flow Non-Newtonian flows - metal and polymer forming: about viscosity, 141-2 elastic springback, 152-4 flow formulation, 143 Oswald de Wahle law, 142 prescribed boundary velocities, 143 steady-state problems of forming, 144-7 kinetic energy and work considerations, 147 steady state rolling example, 147-8 transient problems with changing boundaries, 147-52 punch indentation example, 149, 151-2 transient extrusion problem, 150--4 viscoelastic fluids, 152-4 viscoplastic fluids, 142 viscoplasticity and plasticity, 141-4 s e e a l s o Metal forming, transient, direct displacement approach; Viscoelastic flows Non-self-adjoint equations, 1 Oldroyd-B model, 156 ONERA-M6 wing, turbulent flow past example, 266-8 Ortiz formulation, short waves, 365 Oswald de Wahle law, 142 Partition of Unity Finite Elements (PUFEs), 357 Peclet number, 32, 33 Perfectly Matched Layers (PMLs), NRBCs with waves, 326-7 Petrov-Galerkin methods: with transients, 51-2, 60-1 for upwinding in one dimension, 34-9, 42 Poisson equation in two dimensions: finite volume formulation with triangular elements example, 24-6 Galerkin formulation with two triangular elements example, 19-21 Pollution error, waves in closed domains, 319 Polymeric liquids, 154
Porous medium flow: about flow through porous media, 274-5 Boussinesq approximation, 279 Brinkman extensions, 284 with CBS scheme, 280 continuity equation, 277, 278-9 Darcy-Rayleigh number, 279 Darcy's law, 274-5 discretization procedure, 279-82 energy equation, 277, 278, 279 Ergun correlation, 276-7 forced convection, 282-3 heat transfer in a packed channel example, 283-4 Forcheimmer extensions, 284 generalized approach, 275-9 momentum equation, 277, 278, 279 natural convection: about natural convection, 284 Boussinesq approximation, 285 buoyancy driven convection in an axisymmetric enclosure example, 288-9 buoyancy driven convection in a packed enclosure example, 285-6 buoyancy driven flow in a saturated cavity example, 286-8 constant porosity medium, 285-6 Darcy flow regime, 286-8 non-dimensional scales, 277-9 non-isothermal flows, 282 porosity definition, 276 semi- and quasi-implicit forms, 281-2 Postprocessing: coefficients of pressure and friction, 393 drag force calculation, 392-3 stream function, 393-4 Prandtl number, 82 with buoyancy driven flows, 191 Pressure coefficients, postprocessing, 393 PUFEs (Partition of Unity Finite Elements), 357 Radiation, boundary conditions, 62-4 RAE2822 airfoil, inviscid flow past example, 208-11 Rayleigh number, with buoyancy driven flows, 191 Refraction s e e Waves, short Regeneration methods, free surface flows, 176-8 Reynolds averaged Navier-Stokes equations, 251 Reynolds number: with turbulent flows, 248, 249 with viscoelastic flows, 158, 160 Reynolds stress, with turbulent flows, 250 Riemmann invariants, with shallow water transport, 313 Riemmann shock tube - high-speed gas flow example, 207 River Severn bore, shallow water example, 305-7 Robin boundary conditions, short waves, 366
Subject index 433 Sailing boat example, free surface flows, 184-5 Self-adjoint differential equations, 17, 391 SGS s e e Sub-Grid Scale (SGS) approximation/method Shallow water: about shallow water problems, 4, 292 drying areas, 310-11 equations for shallow water, basis of, 293-7 Ch6zy coefficient, 294 Coriolis accelerations/parameters, 293-5 Helrnholtz equation, 296 mass conservation with full incompressibility, 293 notation, 294 numerical approximation, 297-8 Characteristic-Based-Split (CBS) algorithm, 297-8 Taylor--Galerkin approximation, 298 Steady state solutions examples: steady state solution, 308, 310 supercritical flow, 308, 310 transient one-dimensional examples: bore, 300-1 dam break, 299-300 solitary wave, 299 transport of shallow water, 311-13 characteristic-Galerkin method/procedure, 311-12 depth-averaged transport equations, 311 Riemmann invariants, 313 tsunami wave in Severn Estuary example, 307-9 two dimensional periodic tidal motion examples: Bristol channel, 301-5 periodic wave, 300-2 River Severn bore, 305-7 Ship motion problem, free surface flow methods, 177 Shocks s e e Waves and shocks, non-linear Shuttle launch, high-speed gas problem flow example, 227, 228, 230 Slow flows, analogy with incompressible elasticity 131, 134 Smagorinsky's model, 269 Sommerfield radiation condition, 363 Spalart-Allmaras (SA) model, turbulence transport equations, 252-3, 413-14 Split, the s e e Character-based split (CBS) algorithm Sponge layers, NRBCs with waves, 326-7 Steady state convection-diffusion equation in one dimension: about the steady state problem, 31-2, 49-50 artificial diffusion concept, 40 balancing diffusion in one dimension, 39-40 continuity requirements for weighting functions, 37-9 convection diffusion example, 32-4 discretization, 31
finite increment calculus (FIC) stabilization, 43--4 Galerkin least squares approximation (GLS), 41-2 Galerkin weighting, 31-2 Green's function, 43 higher order approximations, 44-5 mollifying/smoothing discontinuities, 39 Peclet number, 32, 33 Petrov-Galerkin methods for upwinding, 34-9 sub-grid scale (SGS) approximation, 42-3 variational principle, 40-1 weight function for exact solution example, 34 Steady state convection-diffusion equation in two (or three) dimensions: about two or three dimensions, 45, 49-50 balancing diffusion, 46 finite increment calculus (FIC), 48-9 Galerkin least squares (GLS), 48-9 streamline (Upwind) Petrov-Galerkin (SUPG) weighting, 45-8 Stokes flow, 2-3 Stokes waves, 342-3 Stream function, 13 postprocessing, 393-4 Streamline balancing diffusion, 47-8 Streamline (Upwind) Petrov--Galerkin (SUPG) weighting, 45-8 Stresses in fluids, governing equations, 4-6 Strong and weak forms s e e Weighted residual and finite element methods Structures meshes, with compressible high-speed gas flow, 197 Sub-grid scale (SGS) approximation/method, 42-3 Submarine example, free surface flows, 181-3 Submerged hydrofoil example, free surface flows, 179-82 Supercritical flow, shallow water example, 308, 310 SUPG (Streamline (Upwind) Petrov-galerkin) weighting, 45-8 Swansea two step operation, convection-diffusion equations, 402 Taylor-Galerkin method/procedures, 52, 65-6 with shallow water problems, 298 used for vector-valued variables, 387-9 Tetrahedron, linear, integration formulae, 395-6 THRUST the supersonic car, Euler solution example, 222--6 Time domain solutions, short waves, 350 Transient convection-diffusion equation: about transients, 50--3 advection of a Gaussian cone in a rotating fluid, 62 boundary conditions - radiation, 62--4 characteristic directions, 51 characteristic-Galerkin method/procedures, 54-6, 61-3 discretization procedures, 51-3 -
434 Subjectindex Transient convection-diffusion equation - cont. explicit characteristic-Galerkin procedures, 56-62 Galerkin least squares (GLS) method, 51-3 mathematical background, 50-1 mesh updating and interpolation methods, 53-4 Petrov-Galerkin method, 51-2, 60-1 and the steady-state condition, 66 Taylor-Galerkin methods/procedures, 52, 65-6 see a l s o Waves and shocks, non-linear Transport equation, 28 Transport of shallow water, 311-13 Trefftz type finite elements for short waves, 362-4 Triangles, linear, integration formulae, 395 Tsunami wave in Severn Estuary example, 307-9 Turbulence transport equations, mass-weighted averaged: about turbulence models, 413 Spalart-Allmaras model, 413-44 Turbulent flows, basics: about turbulent flows, 248-9 Boussinesq assumption, 250 instability, 11 Kolmogorov length scale, 248 large eddy velocity scale, 249 Reynolds stress, 250 time averaging, 249-50 turbulent eddy/kinematic viscosity, 250-1 Turbulent flows, compressible material: detached Eddy Simulation (DES), 270 Direct Numerical Simulation (DNS), 270-1 energy conservation equation, 265 large eddy simulation, 267-9 Kolmogorov constant, 269 Smagorinsky's model, 269 standard SGS model, 269-70 mass conservation equation, 264 mass-weighted (Favre) time averaging, 265-6 momentum equation, 265 Monotonically Integrated LES (MILES), 270 turbulent flow past an ONERA-M6 wing example, 266-8 Turbulent flows, incompressible material: diffusion Prandtl number, 252, 253 governing equations, non-dimensional: i( - e model, 255 one-equation model, 255 Spalart-Allmaras model, 255 turbulent flow solution, 254-5 Reynolds averaged Navier-Stokes equations, 251 shortest distance to a solid wall, 256 solution procedure/examples: CBS scheme recommended, 256 turbulent flow past a backward facing step example, 256-9 unsteady turbulent flow past a circular cylinder example, 259-64
Spalart-Allmaras (SA) (one equation) model, 252-3 standard K - e (two equation) model, 253-4 Wolfstein i( - l (one equation) model, 251-2 Ultra weak formulation, short waves, 359-62 Ultra Weak Variational Formulation (UWVF), 361-2 Upwinding: optimal streamline upwinding, 59-60 using Petrov-Galerkin methods, 34-9 Vandermonde matrix, 374 Velocity field, 29 Viscoelastic flows: about viscoelastic flows, 154--7 and the CBS algorithm, 163 flow past a circular cylinder example, 159-63 governing equations: constitutive equation, 158 continuity, 157, 158 Deborah number, 158, 160, 162 Kroneker delta, 157 momentum, 157, 158 Newtonial dynamic viscosity, 157 Reynolds number, 158, 160 Maxwell equation, 156 Oldroyd-B model, 156 polymeric liquids, 154 Viscosity: polymers and hot metals, 141 secant velocity, 141 viscoelastic fluids, 152-4 viscoplastic fluids, 142 viscosity-strain rate dependence, 141-2 viscous flow problems, 3 volumetric viscosity, 5 see also Compressible high-speed gas flow, viscous problems .... Volumetric viscosity, 5-6 Wave envelope (conjugated) infinite elements, 330-2 Wave propagation, solitary wave example, free surface flows, 186-9 Waves, long and medium: about long and medium waves, 317-18, 344 bed friction, 320 Chrzy bed friction, 320 convection of waves, 333-5 linking finite elements to exterior solutions (DtN mapping): about linking, 336 Hankel functions, 338 to boundary integrals, 336-7 to series solutions, 337-8
Subject index local Non-Reflecting Boundary Conditions (NRBCs): about NRBCs, 324-6 Givoli's procedure, 326 Perfectly Matched Layers (PMLs), 326-7 sponge layers, 326-7 modelling difficulties, 320 refraction of waves, 333-5 three-dimensional effects in surface waves: about surface waves in deep water, 338-40 Cauchy-Poisson free surface condition, 339 cnoidal and solitary waves, 340-2 free surface condition, 338, 338-9 Korteweg--de Vries equation, 342 large amplitude waves, 340 Stokes waves, 342-3 transient problems, 335 unbounded problems, 324 waves in closed domains, finite element models, 318-19 eigenvalue problem, 319 pollution error, 319 waves in unbounded domains, 321-3 diffraction and refraction problems, 320-1 incident waves, domain integrals and nodal values, 323 radiation condition, 321-2 radiation problem, 321 scattering problem, 321 wave diffraction, 321-3 s e e a l s o Infinite elements; Shallow water Waves and shocks, non-linear, 66-70 Burger equation, 69-70 development of shock, 67-8 Lapidus type diffusivity, 67-8 propagation speeds, 67 steep wave modelling, 67-9 Waves, short: about short waves, 349-51 frequency domain solutions, 350 time domain solutions, 350 Discontinuous Galerkin finite elements (DGFE), 374-7 electromagnetic scattering, transient solution, 352-3 finite elements incorporating wave shapes: about finite elements with short waves, 352-4 discontinuous enrichment method, 357-9 enrichment functions, 358 Gauss-Legendre integration points, 355 microlocal discretization, 354 Partition of Unity Finite Elements (PUFEs), 357 shape functions using products of polynomials and waves, 354-7
shape functions using sums of polynomials and waves, 357 Sommerfield radiation condition, 363 Trefftz type finite elements for waves, 362--4 ultra weak formulation, 359--62 Ultra Weak Variational Formulation (UWVF), 361-2 modelling developments, 351 modelling errors, 351 refraction: about refraction, 364 acoustic velocity potential, 370 convected wave equation, 370 Galerkin scheme, 367 Higdon boundary condition, 365 Ortiz formulation, 365 plane scattered by stepped cylinder example, 368-9 plane wave basis finite elements, 366 refraction caused by flows, 369-72 Robin boundary conditions, 366 wave speed refraction, 364-9 weighted residual scheme, 366 spectral finite elements for waves, 372--4 Fekete points, 374 Gauss-Chebyshev-Lobatto scheme, 373-4 Gauss-Legendre integration, 373 Gauss-Lobatto scheme, 373-4 Newton-Cotes formula, 372-3 Vandermonde matrix, 374 T-complete systems, 363-4 Weak form of equations, 15 s e e a l s o Weighted residual and finite element methods Weighted residual and finite element methods: about strong and weak forms, 14-15 boundary conditions, Neumann type, 15 elements, 16 examples: free surface potential flow, 21-3 Poisson equation in two dimensions: Galerkin formulation with two triangular elements, 19-21 potential flow solution around an aerofoil, 21-2 shape functions for triangle with three nodes, 18-19 Galerkin, finite element, method, 17-18 nodal values, 16 self-adjoint differential equations, 17 test functions, 16 weak form of equations, 15 weighted residual approximation, 16-17 Wolfstein x-1 model, 251-2
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Plate 1 Analysis of subsonic flow around an aircraft (Dassault Falkon) Courtesy of Prof. Ken Morgan, School of Engineering, University ofWales Swansea. Source: J. Peraire, J. Peiro and K. Morgan, Multigrid solution of the 3-D compressible Euler equations on unstructured tetrahedral grids, Int. J. Num. Meth. Eng., 36, 1029-1044, 1993.
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Plate 2 Supersonic car, THRUST SSC, surface mesh and pressure contours Details near the nose. Part of the car is shown on the book cover. Courtesy of Prof. Ken Morgan, School of Engineering, University of Wales Swansea (Nodes: 39, 528 and Elements: 79060). Source: K. Morgan, 0. Hassan and N.P. Wetherill, Why didn't the supersonic car fly?, Mathernat~csToday, Bulletin of the Institute of Mathematics and its Applications, 35, 110-1 14, August 1999.
Plate 3 Wave elevation pattern behind a ship C60 hull for Froude number 0.238. Courtesy of Prof. E. Oiiate, CIMNE, Barcelona. Source: E. Oiiate, J. Garcia and S.R. Idelsohn, Ship hydrodynamics. In E. Stein, R. de Borst and 1I.R. Hughes, editors, Chapter 18, Encyclopediaof Computational Mechanics. John Wiley, 2004.
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Pressure contours. Courtesy of Prof. 0. Hassan, School of Engineerlng, University of Wales Swansea. Source: 0. Hassan, L.B. Bayne, K. Morgan and N.P. Wetherill, An adaptive unstructured mesh method for transient flows involving moving boundaries, ECCOMAS'98. John Wiley.
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Plate 5 3D dam-break. Wave interacting with two circular cylinders Courtesy of Prof. R. L~hner, George Mason University, USA. Reference: C. Yang, R. L~hner and S.C.Yim. Development of a CFD simulation method for extreme wave and structure interactions, 24th International Conferenceon Offshore Mechanics and Arctic Engineering, 12-17 June 2005, Halkidiki, Greece. Problem definition and free surface evaluation.
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/ (a) Simulation of waves hitting a breakwater in a harbour using the particle finite element method (PFEM)I, 2
(b) Simulation of the sinking of a tanker ship using the particle finite element (PFEM)I, 2
Plate 6 Particle finite element method Courtesy of Prof. E. Ohate, ClMNE, Barcelona. References: (1) E. OEate, S.R. Idelsohn, E Del Pin and R. Aubry. The particle finite element method. An overview, Int. J. Comp. Meth., 1:267-307, 2004. (2) S.R. Idelsohn, E. Ohate and F. Del Pin. The particle finite element method: a powerful tool to solve incompressible flows with free-surfaces and breaking waves. Int. J. Num. Meth. Eng., 61:964-989, 2004.
