Recent Developments and Innovative Applications in Computational Mechanics

Recent Developments and Innovative Applications in Computational Mechanics

Dana Mueller-Hoeppe, Stefan Loehnert, and Stefanie Reese (Eds.)

Recent Developments and Innovative Applications in Computational Mechanics

ABC

Dana Mueller-Hoeppe Leibniz Universität Hannover Institute of Continuum Mechanics (IKM) Appelstr. 11 30167 Hannover Germany E-mail: [email protected]

Stefanie Reese RWTH Aachen University Institute of Applied Mechanics Mies-van-der-Rohe-Str. 1 52074 Aachen Germany E-mail: [email protected]

Stefan Loehnert Leibniz Universität Hannover Institute of Continuum Mechanics Appelstr. 11 30167 Hannover Germany E-mail: [email protected]

ISBN 978-3-642-17483-4 DOI 10.1007/978-3-642-17484-1 c 2011 Springer-Verlag Berlin Heidelberg  This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Data supplied by the authors Cover Design: WMX Design, Heidelberg Printed on acid-free paper 987654321 springer.com

Prof. Dr.-Ing. habil. Peter Wriggers

Preface

This Festschrift is dedicated to Professor Dr.-Ing. habil. Peter Wriggers on the occasion of his 60th birthday. Peter Wriggers’ main academic stations have been Hannover and Darmstadt. In Hannover, he achieved the degree of a Dipl.-Ing. as well as his PhD and habilitation. After having been in charge of the Institute of Mechanics and Computational Mechanics (IBNM) for almost ten years he became head of the Institute of Continuum Mechanics (IKM) in 2008. In Darmstadt, he was appointed to a full professorship at the Department of Mechanics in 1990. Other important places in Peter Wriggers’ professional life are the University of California at Berkeley (USA) as well as the University of Newcastle in Australia, to both of which he returns on a regular basis. This large number of international connections already indicate that Peter Wriggers is a scientist of extraordinary international reputation. This is underlined by the fact that the present Festschrift comprises contributions from almost every continent and from a very diverse group of people: friends, collaborators, former and current PhD students as well as his own mentor. A wide range of topics is covered in this book, from contact mechanics to finite element technology and micromechanics, among others. These contributions either represent Peter Wriggers’ own research activities or topics he takes interest in. In addition, the dedications of the contributing authors show that Peter Wriggers has represented more than just a scientist to a great number of people, to whom he also serves as friend, supporter and source of inspiration. We are glad to be in a position to edit this Festchrift and would like to thank Springer Verlag for the collaboration regarding this project, the authors for their contribution to the success of this book and Martin Lippmann for his patient support of joining the individual documents.

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In the name of all authors, we congratulate Peter Wriggers to his 60th birthday and wish him happiness, health, success and continued creativity for the years to come. Hannover, October 2010

Dana Mueller-Hoeppe Stefan Loehnert Stefanie Reese

Contents

New Applications of Mortar Methodology to Extended and Embedded Finite Element Formulations . . . . . . . . . . . . . . . . . . . . . Tod A. Laursen, Jessica D. Sanders 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Stability Issues Associated with Contact on Enriched Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Adaptation to the Embedded Interface Case . . . . . . . . . . . . . . 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermo-Mechanical Coupling in Beam-to-Beam Contact . . . . Daniela P. Boso, Przemyslaw Litewka, Bernhard A. Schrefler 1 Thermo-Mechanical Beam Finite Element . . . . . . . . . . . . . . . . 2 Weak Form for Thermo-Mechanical Contact . . . . . . . . . . . . . . 3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On Regularization of the Convergence Path for the Implicit Solution of Contact Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Giorgio Zavarise, Laura De Lorenzis, Robert L. Taylor 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Structure of the Consistent Tangent Stiffness . . . . . . . . . . . . . . 3 Large Penetration Basic Algorithm . . . . . . . . . . . . . . . . . . . . . . 3.1 Strategy Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Modified Stiffness and Residual during Phase One . . . 3.3 Limitations of the Strategy . . . . . . . . . . . . . . . . . . . . . . . 4 Large Penetration Enhanced Algorithm . . . . . . . . . . . . . . . . . . 4.1 Solution of the Problem for r < 1 . . . . . . . . . . . . . . . . . .

1 1 2 4 7 8 9 9 11 13 15 15

17 17 19 20 20 21 22 23 23

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5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On Different Variational Formulations of the Nitsche Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ridvan Izi, Alexander Konyukhov, Karl Schweizerhof 1 Nitsche Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Choice of the Lagrange Multiplier Set μ . . . . . . . . . . . . 1.2 Physical Meaning of the Non-penetration Terms . . . . . 2 Types of the Nitsche Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 3 FE Implementation of the Nitsche Approaches . . . . . . . . . . . . 3.1 Gauss Point-Wise Substituted Formulation . . . . . . . . . 3.2 Bubnov-Galerkin-Wise Partial Substituted Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Challenges in Computational Nanoscale Contact Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Roger A. Sauer 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Nanoscale Contact Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Nanoscale versus Macroscale Contact . . . . . . . . . . . . . . . . . . . . 4 Adhesion Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Multiscale Contact Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On the Four-node Quadrilateral Element . . . . . . . . . . . . . . . . . . . . Ulrich Hueck, Peter Wriggers 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Element Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stability of Mixed Finite Element Formulations – A New Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stefanie Reese, Vivian Tini, Yalin Kiliclar, Jan Frischkorn, Marco Schwarze 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Linear Elasticity - Mixed Variational Formulation . . . . . . . . . 3 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Compatible Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Enhanced Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Element Stiffness Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 27 28

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5 Eigenvalue Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Non-linear Finite Element Technology . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Finite Element Formulation Based on the Theory of a Cosserat Point – Modification of the Torsional Modes . . . . . . . Eiris F.I. Boerner, Dana Mueller-Hoeppe, Stefan Loehnert 1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 A Brief Introduction to the Cosserat Point Element . . . . . . . . 2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Brick Element for Finite Deformations with Inhomogeneous Mode Enhancement . . . . . . . . . . . . . . . . . . . . . . . . . Dana Mueller-Hoeppe, Stefan Loehnert 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Enhanced Strain Assumption . . . . . . . . . . . . . . . . . . . . . 2.2 Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Irregularly Meshed Beam . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Nearly Incompressible Block . . . . . . . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Automatic Differentiation Based Formulation of Computational Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Joˇze Korelc 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Automatic Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Automatic Differentiation in Computational Mechanics . . . . . 4 Automatic Differentiation Based Computational Models . . . . 4.1 ADB Form of Hyperelastic Models . . . . . . . . . . . . . . . . . 4.2 ADB Form of Elasto-plastic Models . . . . . . . . . . . . . . . . 4.3 Numerical Efficiency of ADB Form . . . . . . . . . . . . . . . . 4.4 ADB Form of Contact Formulations . . . . . . . . . . . . . . . 4.5 ADB Form in Stability Analysis . . . . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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61 61 62 62 63 64 65 68 68

69 69 70 71 72 72 73 73 74 76 76

79 79 80 81 82 82 82 83 84 85 85 86

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Nonlinear Finite Element Shell Formulation Accounting for Large Strain Material Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . Friedrich Gruttmann 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Variational Formulation of the Shell Equations . . . . . . . . . . . . 3 Mixed Hybrid Shell Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Numerical Example: Stretching of a Rubber Sheet . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hybrid and Mixed Variational Principles for the Geometrically Exact Analysis of Shells . . . . . . . . . . . . . . . . . . . . . . Paulo de Mattos Pimenta 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Geometrically-Exact First-Order-Shear Shell Model . . . . 3 Some Multi-field Variational Principles . . . . . . . . . . . . . . . . . . . 3.1 Principle of Total Potential Energy . . . . . . . . . . . . . . . . 3.2 Three-Field Principle of Veubeke-Hu-Washizu Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Two-Field Principle of Hellinger-Reissner Type . . . . . . 3.4 Two-Field Principle of Total Complementary Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Hybrid Principle of Hellinger-Reissner Type . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Shell Theory with Scale Effects, Higher Order Gradients, and Meshfree Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Carlo Sansour, Sebastian Skatulla 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Deformation and Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Generalized Shell Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An Electro-mechanically Coupled FE-Formulation for Piezoelectric Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W. Wagner, K. Schulz, and S. Klinkel 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Finite Element Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87 87 88 90 91 94 94

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Non-intrusive Coupling: An Attempt to Merge Industrial and Research Software Capabilities . . . . . . . . . . . . . . . . . . . . . . . . . . Olivier Allix, Lionel Gendre, Pierre Gosselet, Guillaume Guguin 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The General Principles of Non-intrusive Coupling . . . . . . . . . . 2.1 Piecewise Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Iterative Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Choice of the Interface Boundary Condition for the Local Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Examples Using Abaqus/Standard . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constitutive Models and Failure Prediction for Al-Alloys in Industrial Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Christian Leppin 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Factors Influencing Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Work-Hardening of Aluminum Alloys . . . . . . . . . . . . . . . . . . . . 4 Yield Locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Fracture Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Phenomenological Damage Model to Predict Material Failure in Crashworthiness Applications . . . . . . . . . . . . . . . . . . . . . Markus Feucht, Frieder Neukamm, and Andr´e Haufe 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Process Chain of Sheet Metal Part Manufacturing . . . . . 3 Failure Modelling in Forming and Crashworthiness Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 A Generalized Scalar Damage Model . . . . . . . . . . . . . . . 3.2 Failure Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Path-Dependent Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Stress and Strain Measures . . . . . . . . . . . . . . . . . . . . . . . 4.2 Nonlinear Accumulation of the Instability Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Post Critical Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Damage-Dependent Yield Stress . . . . . . . . . . . . . . . . . . . 5.2 Energy Dissipation and Fadeout . . . . . . . . . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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125 126 126 127 128 130 130 132 132

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A Computational Approach for Mixed-Lubrication Effects in Sealing Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Markus Andr´e 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Solid Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Fluid Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Coupled Fluid Film Computation . . . . . . . . . . . . . . . . . . . . . . . 4 Friction Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deformations of a Large Hall: Structural Design and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Klaus-Dieter Klee, Reinhard Kahn 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Steel Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Bearing Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Roof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Stiffening Components . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Support of Partial Halls . . . . . . . . . . . . . . . . . . . . . . . . . 3 Construction and Computation . . . . . . . . . . . . . . . . . . . . . . . . . 4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recovering Micropolar Continua from Particle Mechanics by Use of Homogenisation Strategies . . . . . . . . . . . . . . . . . . . . . . . . Wolfgang Ehlers 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Particle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Homogenisation Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modelling of Microstructured Materials with Micromorphic Continuum Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Britta Hirschberger, Paul Steinmann 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Micromorphic Continuum . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Micromorphic Continuum Framework . . . . . . . . . . . . . . 2.2 Hyperelastic Constitutive Framework . . . . . . . . . . . . . . 2.3 Numerical Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Application to Material Interfaces with Heterogeneous Micromorphic Mesostructure . . . . . . . . . . . . . . . . . . . . . . . . . . . .

155 155 156 156 157 159 160 162 162

163 163 164 165 166 168 168 171 177 177

179 179 180 183 186 188 189

191 191 192 192 193 193 195

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XV

3.1

Scale Transition between Interface and Micromorphic RVE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 A Computational Homogenization Approach for Micromorphic Meso-heterogeneous Material Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On Computational Homogenisation of Heterogeneous Media with Debonded Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Peri´c, D.D. Somer, E.A. de Souza Neto, W. Dettmer 1 Introduction and Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Multi-scale Constitutive Theory: Overview . . . . . . . . . . . . . . . . 2.1 RVE Kinematical Constraints . . . . . . . . . . . . . . . . . . . . . 2.2 Finite Element Approximation . . . . . . . . . . . . . . . . . . . . 2.3 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Frictional Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Assessment of Yield Surfaces of Heterogeneous Media with Debonded Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Computational Homogenisation Based Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Estimated Yield Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Assessment of Homogenization Errors in Transient Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. Runesson, F. Su, F. Larsson 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Transient Heat Flow – A Model Problem . . . . . . . . . . . . . . . . . 2.1 Space-Variational Format . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Explicit Homogenization Results . . . . . . . . . . . . . . . . . . 3 RVE-Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Dirichlet Boundary Conditions . . . . . . . . . . . . . . . . . . . . 3.2 Neumann Boundary Conditions . . . . . . . . . . . . . . . . . . . 4 Computational Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Problem Definition – Substructure Characteristics . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

195

196 196 197 198

199 199 200 201 201 202 202 202 202 203 204 205 206 206

207 207 208 208 209 210 210 211 212 212 214 214

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Multiscale Modeling of Metal Foams Using the XFEM . . . . . . Lovre Krstulovic-Opara, Stefan Loehnert, Dana Mueller-Hoeppe, Matej Vesenjak 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Modified XFEM for Heterogeneous Materials . . . . . . . . . . . . . 3 Incorporation of Finite Plasticity . . . . . . . . . . . . . . . . . . . . . . . . 4 Comparison of Metal Foams with and without Filler Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3D Multiscale Projection Method for Micro-/Macrocrack Interaction Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stefan Loehnert, Dana Mueller-Hoeppe 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Multiscale Technique in Three Dimensions . . . . . . . . . . . . 2.1 Stress Projection from the Fine Scale to the Coarse Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Projection of the Displacement Field from the Coarse Scale to the Fine Scale . . . . . . . . . . . . . . . . . . . . 3 Numerical Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Goal-Oriented Residual Error Estimates for XFEM Approximations in LEFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marcus R¨ uter, Erwin Stein 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 XFEM Approximations in LEFM . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Model Problem of LEFM . . . . . . . . . . . . . . . . . . . . . 2.2 XFEM Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . 3 A Posteriori Error Estimation in the Energy Norm . . . . . . . . 3.1 Error Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 An Implicit Residual Error Estimator . . . . . . . . . . . . . . 3.3 Equilibration of Tractions . . . . . . . . . . . . . . . . . . . . . . . . 4 Goal-Oriented Error Estimation in LEFM . . . . . . . . . . . . . . . . 4.1 Linearization of the J-Integral . . . . . . . . . . . . . . . . . . . . 4.2 Duality Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

215

216 216 218 218 221 221

223 223 224 224 227 228 230 230

231 231 232 232 233 234 234 234 235 236 236 236 237 238 238

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Multi-field Coupling Strategies for Large Scale Particle-Fluid Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.R.J. Owen, Y.T. Feng, K. Han, C.R. Leonardi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 LB Formulations for Turbulent Incompressible Fluid Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Standard LB Formulation . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Turbulence Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Hydrodynamic Forces for Fluid-Particle Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Fine Particle Modelling - Non-newtonian Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Thermal Lattice Boltzmann Method . . . . . . . . . . . . . . . . . 4 Numerical Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Particle Transportation in Turbulent Fluid Flows . . . . 4.2 Fine Particle Migration in a Block Cave . . . . . . . . . . . . 4.3 Modelling Heat Transfer in (Particle-)Fluid Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Simulation of Particle-Fluid Systems . . . . . . . . . . . . . Bircan Avci, Peter Wriggers 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Mathematical Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Equations for Fluid Motion . . . . . . . . . . . . . . . . . . . . . . . 2.2 Equations for Particle Motion . . . . . . . . . . . . . . . . . . . . . 3 The Discrete Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Collision Model for Normal Contact . . . . . . . . . . . . . . . 3.2 Frictional Tangential Contact Model . . . . . . . . . . . . . . . 4 Coupling of the Fluid and Particle Phase . . . . . . . . . . . . . . . . . 4.1 Evaluation of the Hydrodynamic Forces . . . . . . . . . . . . 4.2 Coupling Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Concurrent Multiscale Approach to Non-cohesive Granular Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Christian Wellmann, Peter Wriggers 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Discrete Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Homogenization and Elasto-plastic Parameters . . . . . . . . . . . . 4 Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

XVII

239 240 241 241 242 243 243 244 245 245 245 247 248 248 249 249 250 250 250 251 251 252 253 253 254 255 255 255

257 257 258 259 261

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5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 On Some Features of a Polygonal Discrete Element Model . . . Ekkehard Ramm, Manfred Bischoff, Benjamin Schneider 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Discrete Element Method with Polygonal Particles . . . . . . . . . 2.1 Models for Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Models for Cohesion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Model Material without Cohesion . . . . . . . . . . . . . . . . . 3.2 Model Material with Cohesion . . . . . . . . . . . . . . . . . . . . 3.3 Concrete with Microstructure . . . . . . . . . . . . . . . . . . . . . 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

265

Isogeometric Failure Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Clemens V. Verhoosel, Michael A. Scott, Michael J. Borden, Ren´e de Borst, Thomas J.R. Hughes 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Isogeometric Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Higher-Order Gradient Damage Formulation . . . . . . . . . . . . . . 3.1 Constitutive Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 L-Shaped Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Cohesive Zone Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Constitutive Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Single-Edge Notched Beam . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

275

A Method for Enforcement of Dirichlet Boundary Conditions in Isogeometric Analysis . . . . . . . . . . . . . . . . . . . . . . . . . Toby J. Mitchell, Sanjay Govindjee, Robert L. Taylor 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Dirichlet Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Examples from Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Infinite Half-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Infinite Plate with Circular Hole under Tension . . . . . 3.3 Infinite Plate with Elliptical Hole under Tension . . . . . 4 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

265 266 266 268 269 269 270 271 272 272

275 276 277 278 278 279 280 280 281 282

283 284 285 287 288 289 291 292 293

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Application of Isogeometric Analysis to Computational Contact Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ˙ Ilker Temizer 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Contact Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . . . 3 Isogeometric Treatment with NURBS . . . . . . . . . . . . . . . . . . . . 4 Knot-to-Surface Contact Algorithm . . . . . . . . . . . . . . . . . . . . . . 4.1 Contact of a Grosch Wheel . . . . . . . . . . . . . . . . . . . . . . . 4.2 Contact of Two Deformable Bodies . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stochastic Galerkin Method for the Elastoplasticity Problem with Uncertain Parameters . . . . . . . . . . . . . . . . . . . . . . . . . Bojana V. Rosic, Hermann G. Matthies 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Problem Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Numerical Analysis of the Problem . . . . . . . . . . . . . . . . . . . . . . 3.1 Discretisation of Input . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Stochastic Galerkin Method . . . . . . . . . . . . . . . . . . . . . . 4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Time-Discontinuous Galerkin Approach for the Numerical Solution of the Fokker-Planck Equation . . . . . . . . . . Udo Nackenhorst, Friederike Loerke 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 FPE Expression of Stochastic Dynamic Problems . . . . . . . . . . 3 Numerical Solution of the Fokker-Planck Equation with TDG Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interface Modelling in Computational Limit Analysis . . . . . . . . A.V. Lyamin, K. Krabbenhøft, S.W. Sloan 1 Discrete Formulation of Bound Theorems . . . . . . . . . . . . . . . . . 2 Velocity Discontinuities as a Patch of Thin Elements . . . . . . . 3 Stress Discontinuities as a Patch of Thin Elements . . . . . . . . . 4 Interfaces between Material Domains . . . . . . . . . . . . . . . . . . . . 5 Interfaces at Segments Subject to Loading or Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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295 295 296 296 298 299 300 300 302

303 303 304 304 305 305 306 306 307 310 310

311 312 313 314 317 317 318 321 321 323 324 325 326

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6 Moment-Free Interfaces for Modelling of Joints . . . . . . . . . . . . 7 Interfaces for Overlapping Connections . . . . . . . . . . . . . . . . . . . 8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

328 329 330 330

On the Coexistence of Intermeshed Hostile Populations . . . . . Tarek I. Zohdi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Direct Interaction Models: Rules of Engagement . . . . . . . . . . . 3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Identification of System Parameters: Genetic Algorithms . . . 5 An Example of Parity Identification . . . . . . . . . . . . . . . . . . . . . 6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

331 332 333 333 334 335 338 339 340

Chapter 1

New Applications of Mortar Methodology to Extended and Embedded Finite Element Formulations Tod A. Laursen and Jessica D. Sanders Dedicated to Peter Wriggers on the occasion of his 60th birthday (T.A. Laursen).

Abstract. The past decade or so has shown the mortar method to be a helpful foundation for formulation of new methods for contact/impact analysis. As originally formulated, the advantage of the mortar method is that it preserves inf-sup conditions associated with interfacial constraints, such that stability and convergence are more or less guaranteed, at least when some regularity of the solution is expected. In more recent work, interest has been shown by a number of researchers in extending many of these ideas to treatment of interfaces in extended finite element frameworks, and also to embedded surface techniques. In these cases, use of the mortar method with na¨ıve multiplier space choices can readily lead to instability. This paper highlights some of these issues, and contemplates stabilization methods which seem to be effective in such settings.

1 Introduction The subject of this brief article is the extension of mortar methods for contact analysis to the case where the geometric description of the interface does not coincide with the underlying discretization on one or both sides of the interface. Two instances are explicitly considered: contact occurring over enriched interfaces in the context of the extended finite element method (XFEM); and the embedding of an Tod A. Laursen Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708, USA e-mail: [email protected] Jessica D. Sanders Department of Civil and Environmental Engineering, Duke University, Durham, NC 27708, USA e-mail: [email protected]

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T.A. Laursen and J.D. Sanders

FEM mesh in another such that the interface “cuts” volumetrically through elements of the background mesh. The latter case is important, for example, in fluid-structure interaction (FSI) problems where the solid or structure might be represented by a Lagrangian type of description, overlaying an underlying Eulerian fluid mesh.

2 Stability Issues Associated with Contact on Enriched Interfaces Imposition of contact constraints within enriched approaches to elasticity was considered intensively in [4], considering in particular mesh tying constraints. In an extended finite element representation of an interfacial situation, the solution and weighting spaces S and V are allowed to be discontinuous over an interface Γ∗ as follows: S = {ui (x)|ui (x) ∈ H 1 , ui (x) = gi on Γd , ui (x) discontinuous on Γ∗ } V = {wi (x)|wi (x) ∈ H 1 , wi (x) = 0 on Γd , wi (x) discontinuous on Γ∗ }. In such methods, the displacement over an element cut by a single interface would take the form ui (x) = uˆi (x) + H(x)u˜i(x) , (1) where H(x) is an enrichment function representing the character of the solution in the near field of the discontinuity Γ∗ , satisfying also the partition of unity in the domain. If one were to take H(x) as a Heaviside function (a usual choice), the two sides of each interface would be completely free to drift apart (or, indeed, to interpenetrate). A schematic of this idea, in the instance of a polygranular structure whose boundaries are described by enriched boundaries, is given in Fig. 1. The strategy to be followed here involves use of a space of Lagrange multipliers, M h , to weakly enforce whatever constraints or constitution might be desirable for the interface to describe the physics at hand. In a standard contact problem, where the interface is defined by two domains surfaces coming into mechanical interaction with each other, the standard mortar choice would be to select one of the two surfaces as the domain for the mortar multiplier (the so-called non-mortar surface), and then to pick the trial functions over that domain to be the same order as the underlying discretization of the primary variable. Here, by contrast, we must construct a set of basis functions over a domain which does not conveniently coincide with the original discretization of the kinematic fields. In [4], we made perhaps the simplest choice, which was to divide the interface into a set of segments, Γe , having nodal coordinates corresponding to the intersection of Γ∗ with element edges from the background grid. Piecewise constant multiplier interpolation was then used over this “intersection grid”. This choice almost uniformly fails to satisfy the stability conditions of the inf-sup test, and frequently

New Applications of Mortar Methodology

3

(a)

(b)

Fig. 1 Schematic of enrichment to introduce interfaces in a polygranular grain structure, before contact multipliers are introduced: a) schematic showing contrived loading to pull grains part without adhesion; and b) resulting deformation, displaying jumps in displacement across interfaces. Elements separated by the enrichment are apparent in b)

results in uncontrolled oscillations in solution of the dual variable. The basic problem can be seen in the discrete form of the mixed formulation, which is summarized as      F d K GT , (2) = 0 m G0 where d contains displacement variables (including enriched degrees of freedom), and m is the vector of unknowns associated with the Lagrange multiplier space. In this case, the matrix G is assembled from local contributions, 

ge,l =

Γe

ˆ l Ne ΔΓe N

,

(3)

ˆ l contains the Lagrange multiplier shape functions associated where the matrix N with interface segment l, while the matrix Ne contains the local element shape functions. The na¨ıve choice of interpolation mentioned above causes the rows of GT to be linearly dependent, so that the kernel of GT is not the null vector. This frequently gives rise to an oscillatory component of the traction field at the interface. One approach to this problem is to simply make more intelligent choices of the multiplier spaces. In the method of vital vertices proposed in [2], this is done by carefully selecting only some of the intersection points to become nodes for the support of the multiplier field. The choice is made in such a way that ker(GT ) contains only the null vector, ensuring the stability of the method. Although this approach is promising, fully general extensions of it (particularly in three dimensions) are still

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T.A. Laursen and J.D. Sanders

forthcoming. In our work, by contrast, we ask whether simple stabilization procedures might produce effective and general methods even when the multiplier space choice is unstable on its own. In [4] the piecewise constant discretization of the interface is used, but the background displacement field is enriched with additional degrees of freedom. This approach can also be shown to be essentially equivalent to a method of weak constraint enforcement originally proposed by Nitsche [3], which in the case of an elasticity problem amounts to finding a displacement field u ∈ S such that 

w(i, j) σi j d Ω −

Ω



(1)

[[wi ]]σi j n j ΔΓ −

Γ∗



(1)

[[ui ]]σ (w)i j n j + α

Γ∗

 Γ∗

[[ui ]][[wi ]]ΔΓ∗ = 0 (4)

for all w ∈ V where

(1)

ai j  =

(2)

(ai j + ai j )

(1)

(2)

, and [[ai ]] = (ai − ai ) . 2 The operator σ (w)i j is defined for linear elasticity as σ (w)i j = Ci jkl w j,k . Nitsche’s method passes a patch test even for a zero value of the stabilization parameter (α = 0), and has been shown to be stable and convergent for the enriched formulation of tied contact. Figure 3, adapted from reference [4], shows the spatial convergence observed using Lagrange multiplier, penalty, and stabilized approaches to the tying constraint in a simple beam bending configuration, for a simple structured mesh shown in Fig. 2. The lack of convergence in the surface tractions is noteworthy in the case of the Lagrange multiplier treatment (corresponding, effectively, to a na¨ıve mortar approach). Notably, L2 measures of error in the bulk tend to obscure these stability problems, but accuracy of the surface tractions is vitally important if one wishes to reliably describe sophisticated interface constitution within an enriched framework.

Fig. 2 A simple beam bending test problem, using a structured mesh. Interface cuts the eighth column of elements from the left

3 Adaptation to the Embedded Interface Case We now consider an embedded formulation of linear elasticity, where a domain is discretized with two meshes and one (the embedded mesh), overlaps the other. This

New Applications of Mortar Methodology

5

(a) Fig. 3 Spatial convergence in the L2 surface norm of traction errors for beam bending problem, structured mesh

may be convenient for inclusions of material of one constitution in a broader domain with a second constitution, or as a precursor to the fluid/structure interaction case where a solid having its own discretization is immersed in a fluid mesh. In this method, the solution spaces S and V over a domain Ω = Ω (1) ⊕ Ω (2) are continuous over the respective domains, but discontinuous at the boundary Γ∗ , due to the nonmatching discretizations. In a similar manner to the enriched case considered in Sect. 2, use of a standard mortar method to impose matching conditions can be problematic from a stability standpoint. If constraints are overly represented in the discrete sense, mesh locking can lead to non-optimality in the L2 and H 1 norms, while inadequate representation of constraints leads to inaccurate enforcement of the constraints and poor resolution of surface fluxes. Our experience shows that despite these possible outcomes, use of standard mortar representations can work reasonably well in a variety of cases. However, we have identified a rather pathological case (see [1]) where the potential for locking is rather graphically illustrated. This occurs in the case where a finely meshed inclusion of a stiff material overlaps a coarser mesh of a softer constitution in a rectangular domain and a pure bending moment is applied, in a problem very similar to that used for testing of the extended finite element approach in the last section. For the mesh shown in Fig. 4a, where the overlapping grid has a stiffness difference from the background grid by a factor of 100, mesh locking is clearly present when pure bending is applied. Nitsche’s method can be shown to be applicable to this problem, but we have found an adaptation of this method to be very useful (possibly even necessary) in eliminating the above pathology. In general, Nitsche’s method can be interpreted as a consistent penalization of mortar continuity constraints. Specifically, the

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T.A. Laursen and J.D. Sanders

(a)

(b)

Fig. 4 Embedded interface treatment: bending stresses in a rectangle under pure bending conditions with a) a standard mortar enforcement of continuity at Γ∗ and b) a weighted Nitsche’s method enforcement

Fig. 5 Embedded interface treatment: error convergence in the L2 norm for various methods of continuity enforcement

consistency error arising from the nonconforming and non-vanishing elements of the discrete trial space on Γ∗ is corrected in these methods through augmentation of the penalty term in Eq. (4) (i.e., that containing α ) by the additional second and third terms in that expression. Equation (4) contemplates using a simple average of the stresses in the two domains in formulation of the Nitsche terms. It can be shown,

New Applications of Mortar Methodology

7

Fig. 6 Embedded interface treatment: error convergence in the energy norm for various methods of continuity enforcement (1)

however, that an equally consistent reformulation of the method can use either σi j (2)

or σi j alone in the Nitsche terms, or even a weighted combination of both, instead of an unbiased average as appears in (4). To relieve the aforementioned pathology, we have found that weighting these operators by the relative stiffnesses on either side of the interface produces very good results, relieving mesh locking (Fig. 4) and restoring optimal convergence in the L2 and energy norms (Figs. 5 and 6). Note also from Fig. 5 that the displacement norm may not in all cases distinguish clearly between an adequate and inadequate method, while the energy norm (Fig. 6) clearly reveals the shortcomings of unstabilized methods and also shows the superior behavior of the weighted Nitsche approach in this case.

4 Conclusion This paper has briefly described the extension of mortar contact strategies to two cases of great current interest: extended finite element formulations on one hand, and embedded interface descriptions on the other. Although the technical details involved in these two cases are somewhat different, the commonality between them is that often some form of stabilization is required to produce reliable results for arbitrary combinations of mesh topology, material properties, and interface orientation. Ongoing work is examining the extension of these results (which are for completely tied response) to more general forms of interface constitution.

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T.A. Laursen and J.D. Sanders

References 1. Laursen, T.A., Puso, M.A., Sanders, J.D.: Mortar contact formulations for deformabledeformable contact: past contributions and new extensions for enriched and embedded interface formulations. Comput. Meth. Appl. M. (in press, 2010) 2. Mo¨es, N., B´echet, E., Tourbier, M.: Imposing Dirichlet boundary conditions in the extended finite element method. Int. J. Numer. Meth. Eng. 67, 1641–1669 (2006) ¨ 3. Nitsche, J.: Uber ein Variationsprinzip zur L¨osung von Dirichlet-Problemen bei Verwendung von Teilr¨aumen, die keinen Randbedingungen unterworfen sind. Comput. Method. Appl. M. 191, 1122–1145 (1971) 4. Sanders, J.D., Dolbow, J.E., Laursen, T.A.: On methods for stabilizing constraints over enriched interfaces in elasticity. Int. J. Numer. Meth. Eng. 78, 1009–1036 (2009)

Chapter 2

Thermo-Mechanical Coupling in Beam-to-Beam Contact Daniela P. Boso, Przemyslaw Litewka, and Bernhard A. Schrefler Dedicated to Peter Wriggers for his 60th birthday, in recognition of our fruitful and long lasting collaboration, which we hope will continue in the future (B.A. Schrefler).

Abstract. Contact including coupling between mechanical and thermal fields constitutes a complicated problem because the mutual influences between displacements or strains and temperature are manifested in many different ways. A detailed description of various issues related to the thermo-mechanical coupling in contact can be found in the monograph [6]. In this paper a formulation of a beam-to-beam contact element for the coupled thermo-mechanical field is presented. Contact surfaces are considered ideally smooth, so that the heat transfer due to radiation and convection in the micro cavities is neglected. In the physical model of the beam material, only the linear thermal expansion is included and all the physical parameters are treated as independent of time and temperature.

1 Thermo-Mechanical Beam Finite Element The co-rotational 3D beam finite element derived in [4] is used, suitably improved to take into consideration the thermo-mechanical coupling. The axial force S in a bar subjected to a temperature change can be obtained from the following relation S = EAαt τ (1) where αt is the (constant) coefficient of the linear thermal expansion, EA is the axial beam stiffness and τ is the mean relative temperature in the element related to the Daniela P. Boso · Bernhard A. Schrefler Dipartimento di Costruzioni e Trasporti. Universit`a di Padova, Via F. Marzolo 9, 35131 Padova, Italy e-mail: [email protected], [email protected] Przemyslaw Litewka Institute of Structural Engineering, Poznan University of Technology, ul. Piotrowo, 60-965 Pozna´n, Poland e-mail: [email protected]

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D.P. Boso, P. Litewka, and B.A. Schrefler

initial temperature, for instance to the assembly temperature. The expanded vector of the degrees of freedom with respect to the mentioned Crisfield’s element has 14 entries and can be put into the following form   q = uT , τ T , (2) where u is the vector of the generalised nodal displacements and τ is the vector of the relative nodal temperatures. Assuming a linear distribution of temperature along the element and considering relation (1), the addition to the residual vector in the expanded beam finite element can be written as   RT = RT1 , 0, 0, 0, −RT1 , 0, 0, 0, qM1 , qM2 . (3) The vector R1 groups the nodal forces due to the thermal expansion of the element RT1 = EAαt τ {c1 , c2 , c3 }

.

(4)

The temperature τ is obtained as the mean value of the relative nodal temperatures

τ=

τ1 + τ2 2

,

(5)

and the values ci (i = 1, 2 or 3) are the cosines of angles between the current axis of the beam and the axes xi from the global set of coordinates. The quantities denoted with qM1 and qM2 in vector (3) are the nodal heat fluxes for the considered finite element. The addition to the tangent stiffness matrix can be put in the form ⎡ ⎤ K1 0 K2 0 K3 ⎢ 0 0 0 0 0 ⎥ ⎢ ⎥ ⎥ K=⎢ (6) ⎢ −K1 0 −K2 0 −K3 ⎥ , ⎣ 0 0 0 0 0 ⎦ 0 0 0 0 KT where the component matrices K1 and K2 result from the dependence of the cosines on the position vectors. The matrix K3 presented in (6) groups the terms resulting from the dependence of the axial force (1) on the nodal temperatures. The matrix KT in (6) requires a separate comment. In order to derive its components, the one-dimensional steady-state heat flow equation q = −k

∂τ ∂x

(7)

is used together with the principle of virtual temperature [1]. In (7) k denotes the heat conduction coefficient for the material. One can write the following relation for a finite element

Thermo-Mechanical Coupling in Beam-to-Beam Contact

q1 δ τ1 + q2 δ τ2 = k

L 0

11

∂τ δ τ dx . ∂x

(8)

With this at hand and with the assumption of linear distribution of temperature along the element, the element matrix of thermal conduction can be found in the form   1 −1 KT = k . (9) −1 1

2 Weak Form for Thermo-Mechanical Contact In the present analysis the temperature, denoted by τ (ξ ), is assumed to be constant in the cross-section. Hence, the temperature difference at the contact between beams m and s can be determined using the temperature values τmn and τsn for the crosssections corresponding to the contact points Cmn and Cms gH = τmn − τsn

(10)

.

The heat flow in contact spots is a complicated phenomenon. It depends on the micro-scale structure of the surfaces, the normal force and the temperature gradient. There are many proposals to formulate physical laws for this phenomenon, e.g. see [6, 10, 7]. In the present analyses a simple model is assumed, in which only the heat flow through the contact area between ideally smooth surfaces is considered. In such a case, besides the constraints resulting from the purely mechanical aspects of contact, the thermal constraint is also imposed in the following form gH = 0

(11)

.

This constraint is enforced using the penalty method, which yields the following expression for the heat flow q = εH qH , (12) where the thermal penalty parameter εH is introduced. In this formulation the penalty parameter is constant, but it is worth to note that Eq. (12) could be replaced by a thermal constitutive law and εH may be a physical value, see [7, 10, 2]. In the weak form and its linearisation the variation, the linearisation and the linearisation of the variation of the temperature difference (10) calculated with respect to mechanical and thermal unknowns are necessary. In the following the subscripts u for the displacements and T for the temperatures are introduced in these operators for the sake of clarity. The variables related to the displacements can be expressed as

Δu gH = Δu (τmn − τsn ) = τmn,m Δu ξmn − τsn,s Δu ξsn Δu δT gH = δT (Δu gH ) = δt τmn,m Δu ξmn − δT τsn,s Δu ξsn

,

.

(13)

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D.P. Boso, P. Litewka, and B.A. Schrefler

In this derivation one must consider that the local coordinate depends on the nodal displacements. Hence, the variation of temperature in contact depending on the local coordinate cannot be neglected. The variables related to the temperature take the form

δt gH = δT τmn − δT τsn ΔT gH = ΔT τmn − ΔT τsn ΔT δT gH = 0 .

, , (14)

The linearisation of the variation of the temperature difference vanishes because in the adopted finite element model the approximation of temperature along the element is a linear function of the nodal temperatures. The general form of the thermo-mechanical contact formulation allows to introduce the mutual relations between displacement and temperature fields. However, within the present assumptions, the penetration function gN depends only on displacements, not on temperature. Hence, the variables calculated from this gap function with respect to temperature are zero

Δ T gN = 0 , ΔT δu gN = 0 .

(15)

In order to carry out the analysis of a system with the thermo-mechanical coupling one must consider, besides the principle of virtual work, also an appropriate equation for the heat flow. As in [3], the global set of equations for the thermo-mechanical contact can be written in the following form ⎧ ⎨δu ΠmM + δu ΠsM + ∑ δu ΠcM = 0 act (16) ⎩δT ΠmH + δT ΠsH + ∑ δT ΠcH = 0 . act

The additional subscripts M and H are introduced in (16) to distinguish between the components of the functional resulting from the mechanical and thermal phenomena occurring in both beams. The components δT ΠmH and δT ΠsH in (16)2 are related to the heat flow through the beams m and s. The component of (16)2 , resulting from the contact in the active points takes the form

δT ΠcH = qδT gH

.

(17)

The linearisation of (16), necessary for the Newton-Raphson solution method of the non-linear set of equations of contact with thermo-mechanical coupling must be carried out separately for both groups of unknowns: mechanical – displacements and thermal – temperatures. The following relations are obtained

Thermo-Mechanical Coupling in Beam-to-Beam Contact

Δu δu ΠcM ΔT δu ΠcM Δu δT ΠcH ΔT δT ΠcH

13

= Δ u FN δu gN + FN Δ u δu gN , = Δ T FN δu gN + FN Δ T δu gN , = Δ u qδT gH + qΔ u δT gH , = Δ T qδT gH + qΔT δT gH .

(18)

Equation (18)1 gives the purely mechanical part of the contact formulation, which was discussed in [8, 9, 5]. The relations (18)2 and (18)3 lead to the elements responsible for the thermo-mechanical coupling. However, due to the adopted assumption that the mechanical properties do not depend on temperature, the linearisations in (18)2 vanish. Finally, the last of the relations (18) gives the purely thermal part of the contact. Considering relation (12) the coupling term can be found as

Δu δT ΠcH = εH Δu gH δT gH + εH gH Δu δV gV

.

(19)

The component responsible for the purely thermal part of contact, taking into account (18)3 , can be written as

ΔT δT ΠcH = εH ΔT gH δT gH

.

(20)

3 Numerical Example In this example contact between two cantilever beams with circular cross-sections is analyzed. The axes of the beams in the initial configuration are shown in Fig. 1. At the free end of beam 1 displacement Δ = 0.5 is applied. Besides, there are imposed temperatures on the beams ends: 0 and 5 on beam 1 as well as 50 and 45 on beam 2. Displacements and temperature are applied in 20 increments. The following data are used in the calculations: beam 1 – E = 200 · 105 , ν = 0.3, k = 1, αt = 1.2 · 10−5 , radius of circular cross-section 0.1, length 6.0; beam 2 – E = 75 · 105 , ν = 0.2, k = 1, αt = 1.2 · 10−5 , radius of circular cross-section 0.1, length 6.0; initial gap between the beams 0.01; normal contact penalty parameter εN = 1000. In this example the influence of the thermal penalty parameter on the results is investigated. Three different values are taken: εH = 1, 10 or 100. Deformed configuration of the beams axes and graphs of the temperature distribution along the beams are presented in Fig. 2. In these graphs the influence of the thermal penalty parameter, which plays a role of the heat conduction coefficient in the contact, can be observed. Its increase leads to equalling of temperature in the beams cross-sections at the contact. The temperature difference decreases from 12.51 for εH = 1, through 1.72 for εH = 10 to 0.18 for εH = 100. The linear temperature distribution for the no-contact case is also presented in the graphs. The discrepancy between the no-contact case and the presented non-linear distributions for the contact case is an effect of the displacements influence and the resulting contact between the beams on the temperature field. The opposite influence does not exist directly in the considered contact formulation due to the adopted

14

D.P. Boso, P. Litewka, and B.A. Schrefler

Fig. 1 Example 1 – initial configuration of beams axes

Fig. 2 Example 1: a) deformed configuration of beams axes; b) temperature distribution

assumptions. Still, this influence can be observed in Fig. 3, where the displacements of points on beam 1 along its initial axis (X) for the case εH = 100 with and without the thermal coupling are presented. It can be seen that the temperature field, modified by the contact, feeds back to the system and modifies displacements. In this sense the thermo-mechanical coupling in the considered form is bilateral.

Thermo-Mechanical Coupling in Beam-to-Beam Contact

15

Fig. 3 Example 1 – temperature influence on beam 1 displacements

4 Conclusions In this paper a new contact beam-to-beam finite element is presented. It is used to solve problems in the coupled thermo-mechanical fields, when the contacting surfaces can be considered perfectly smooth. Consistent linearization of the set of equations is performed, and the tangent stiffness matrix and residual vector for the contact finite element are presented. They correspond to the vector of nodal displacements and temperatures. In this sense they are easy to be implemented in any finite element package. The governing equations are solved simultaneously with the monolithic scheme. A numerical example is considered and discussed to check the performance of the proposed element. Future development may include adding of friction, a non-linear constitutive law for the contact area and considering microscopically rough contacting surfaces. Acknowledgements. Support for this work was partially provided by Italian National Research Grant STPD08JA32 004 and Italian Research Project EX 60%-60A09-8711/09. This support is gratefully acknowledged.

References 1. Bathe, K.-J.: Finite Element Procedures. Prentice Hall, Upper Saddle River (1996) 2. Boso, D.P., Zavarise, G., Schrefler, B.A.: A formulation for electrostatic-mechanical contact and its numerical solution. Int. J. Numer. Meth. Eng. 64, 382–400 (2005) 3. Boso, D.P., Litewka, P., Schrefler, B.A., Wriggers, P.: A 3D beam-to-beam contact finite element for coupled electric-mechanical fields. Int. J. Numer. Meth. Eng. 64, 1800–1815 (2005) 4. Crisfield, M.A.: A consistent co-rotational formulation for non-linear, three-dimensional beam-elements. Comput. Meth. Appl. M. 81, 131–150 (1990) 5. Litewka, P., Wriggers, P.: Frictional contact between 3D beams. Comput. Mech. 28, 26–39 (2002)

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6. Wriggers, P.: Computational Contact Mechanics. John Wiley & Sons Ltd., Chichester (2002) 7. Wriggers, P., Zavarise, G.: Thermomechanical contact – a rigorous but simple numerical approach. Comput. Mech. 46, 47–53 (1993) 8. Wriggers, P., Zavarise, G.: On contact between three-dimensional beams undergoing large deflections. Commun. Numer. Meth. En. 13, 429–438 (1997) 9. Zavarise, G., Wriggers, P.: Contact with friction between beams in 3-D space. Int. J. Numer. Meth. Eng. 49, 977–1006 (2000) 10. Zavarise, G., Wriggers, P., Schrefler, B.A., Stein, E.: Real contact mechanisms and finite element formulation – a coupled thermomechanical approach. Int. J. Numer. Meth. Eng. 35, 767–786 (1992)

Chapter 3

On Regularization of the Convergence Path for the Implicit Solution of Contact Problems Giorgio Zavarise, Laura De Lorenzis, and Robert L. Taylor There is no good science without good friendships. To Peter, for his 60th birthday, and for the long years of friendship (G. Zavarise, L. De Lorenzis and R.L. Taylor).

Abstract. Following up to a previous investigation, this paper proposes a strategy to deal with frictionless contact problems involving large penetrations, in the context of the node-to-segment formulation and of the penalty method. The rationale is based on two main considerations: the first one is that, within an iteration scheme, the use of consistent linearization is only convenient when the field of the unknowns is sufficiently close to the solution point; the second one is that, if the order of magnitude of the maximum contact pressure can be estimated a priori, this information can be exploited to approach the solution in a faster and more reliable way. The proposed strategy is based on a check of the nodal contact pressure, to select the technique that has to be used to perform each iteration. If the contact pressure is smaller than a predefined limit, the problem is solved in the standard way, using consistent linearization and Newton’s method. When the contact pressure exceeds the limit, a modified method is used. This one is based on the enforcement of a contact pressure limit and on the use of a simplified secant stiffness, where the geometric stiffness term is disregarded. The strategy has to be integrated with a specific “safeguard algorithm” to guarantee convergence to the correct solution also in cases where the maximum contact pressure has been underestimated. Two alternative procedures for this purpose are proposed.

1 Introduction The benefits of consistent linearization for the solution of any type of non-linear computational problem are well known. However, consistent linearization Giorgio Zavarise · Laura De Lorenzis Universit`a del Salento, Lecce, Italy Robert L. Taylor University of California at Berkeley, USA

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G. Zavarise, L. De Lorenzis, and R.L. Taylor

guarantees a quadratic rate of asymptotic convergence, provided that the field of the unknowns is close to the solution value. It is also true that, the bigger the loading or time step used in the analysis, the greater is the distance of the starting point from the solution one. This aspect is usually disregarded in classical solution schemes, and standard Newton procedures are commonly advocated from the first iteration. Contact algorithms are usually activated a posteriori, i.e. they first let the bodies penetrate each other, then they detect the penetration as a violation of the impenetrability constraint conditions. Regardless of the method adopted for the enforcement of the contact constraint, the current state which violates the impenetrability condition is used to compute the contact contribution to the virtual work. In the implicit scheme this equation set is linearized to solve the non-linear problem with a Newton type method. The strategy is usually effective for small penetrations and smooth evolution of the contact forces. In this case, convergence is usually achieved. However, the first few iterations are needed simply to stabilize the solution before a quadratic rate of convergence takes place. More importantly, if large penetrations occur between the contacting bodies due to large loading or time steps, the direct application of Newton’s method usually produces significant difficulties during the first few iterations. In such cases, the contact pressures resulting from the penetrations can be very large. This affects both the tangent stiffness and the residual vector, and generally produces a large local distortion of the mesh. Quite often a Newton method cannot recover to a smooth deformation state, and catastrophic divergence ensues. Therefore, it is currently necessary to use smaller loading or time increments which limit the amount of penetration. Following up to a previous investigation [5], this paper proposes a strategy both to perform large steps in the presence of large penetrations and to increase the convergence rate in case of normal penetrations. The rationale of the strategy is based on two main considerations. The first one is that the use of consistent linearization is only convenient when the field of the unknowns is sufficiently close to the solution point. The second one is that, if the order of magnitude of the maximum contact pressure can be estimated a priori (as usually happens in engineering problems), then this information can be exploited to approach the solution in a faster and more reliable way. In the proposed strategy, the solution is split into two phases and a different method is used for each one. Phase one takes place during the first iterations of each time or loading step. Within this phase, it is straightforward to demonstrate that the full Newton strategy with consistent linearization is often useless, as the contact pressures can be many orders of magnitude larger than the real ones. Phase two takes place when, due to the iterations performed during phase one, the contact penetrations have been significantly reduced. Hence the problem has been driven close to the solution point, then a Newton strategy with consistent linearization guarantees the best possible convergence rate. A smooth transition occurs between the two solution phases. For more details about the proposed procedure, its background, and additional examples, see [4].

Regularization of the Convergence Path of Contact Problems

19

2 Structure of the Consistent Tangent Stiffness For a better understanding of what happens during the first iterations using consistent linearization, we start by analyzing the characteristics of the consistent tangent stiffness matrix and of the residual vector for a typical contact problem. The penalty method is considered herein, and the formulation is developed in the framework of the node-to-segment algorithm. However, the proposed approach could be easily extended to other algorithms, such as mortar type methods (see e.g. [1]). For simplicity, the methodology will be presented with reference to 2D problems. For more details, see also [4-6]. The matrix form for the residual and for the consistent tangent stiffness can be established as, respectively, R = −Ac ε gN NS (1) and

(b) KT = KM + K(a) G + KG

(2)

where  Ac ε g N  N0 TTS + TS NT0 l

Ac ε g2N N0 NT0 l2 (3) In the above equations, gN is the normal penetration, evaluated at the slave node, Ac is the area of competence of the slave node, ε is the penalty parameter, l is the length of the master segment, and the vectors are defined as follows ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ n 0 t NS = ⎣ − (1 − ξ )n ⎦ N0 = ⎣ −n ⎦ TS = ⎣ − (1 − ξ )t ⎦ (4) −ξ n n −ξ t KM = Ac ε NS NTS

K(a) G =−

K(b) G =−

where n and t are the unit normal and tangent vectors to the master segment, and ξ is the normalized projection of the slave node onto the master segment. Component KM stems from the contribution related to gN δ gN in the contact contribution to the virtual work, and can be considered the core part of the stiffness. This part is clearly independent of the amount of penetration. It is very important to note that all terms of this matrix have bounded values. Hence, no matter what is the current value of the penetration during the iteration process, the terms of KM are bounded to well-defined limits, and may only become large as a result of a large penalty parameter multiplying the matrix. (b) Components K(a) G and KG stem from the term related to δ gN in the contact contribution to the virtual work. It can be shown that KG can be interpreted as the geometric component of the contact stiffness. More in detail, it is related to the change of orientation of the master segment and it vanishes in case of a constant (fixed) normal vector. Once again, the terms in the matrices (see Eqs. (3)b and (3)c) have bounded values. However, the factors multiplying the matrices depend strongly on the amount of penetration. In particular, the K(b) G component is proportional to

20

G. Zavarise, L. De Lorenzis, and R.L. Taylor

the square of gN . In case of large penetrations it is evident that this contribution to the stiffness can be several orders of magnitude different from its value close to the solution point. With the reduction of the penetration, the first term of the stiffness becomes more and more important with respect to the second one. Despite the presence of the second term is necessary for the quadratic rate of convergence, the dominating stiffness term when convergence is achieved is the first one, at least for usual values of the penalty parameter. It is then clear that, in case of unrealistic contact forces due to large penetrations, the geometric stiffness term is both useless, because the geometry is simply too far from the final one, and dangerous, because it strongly affects the local properties of the stiffness matrix.

3 Large Penetration Basic Algorithm 3.1 Strategy Outline In the proposed strategy the solution is split into two phases, and a different method is used for each one. Phase one takes place during the first iterations of each time or loading step. Within this phase, the full Newton strategy with consistent linearization is often useless, as the problem is very far from the solution point and the contact pressures can be many orders of magnitude larger than the real ones. Moreover, due to a rotation of the master segment, the geometric term of the consistent tangent stiffness matrix may feature large negative diagonal terms, with consequent catastrophic effects on the convergence performance. Phase two takes place when, due to the iterations performed during phase one, the contact penetrations have been significantly reduced. The problem has thus been driven close to the solution point, hence a Newton strategy with consistent linearization guarantees the best possible convergence rate. A smooth transition occurs between the two solution phases. The objective of this work is thus to set up a strategy performing better than a consistent full Newton linearization for the aforementioned phase one. What is needed is a criterion to limit the contact pressures and to construct a modified contact contribution to the stiffness matrix and to the residual vector. With this method, local contact instabilities can be controlled, and a fast solution path to an almost converged point can be constructed. The basic idea of the proposed large penetration (LP) strategy consists in avoiding the introduction into the system of large, physically meaningless contact forces which originate from unconstrained large penetrations [5]. For this purpose, it is sufficient to estimate a priori the order of magnitude of the maximum contact pressure that the contacting bodies may experience. Such an estimate represents an information which is easily available to the engineers. It can be shown that estimates of the correct maximum contact pressure within one order of magnitude, or even more, are sufficient for the present purposes. Therefore the availability of the estimate is not a limiting assumption. The estimated value is then used to set a bound for the contact pressures. The employment of the upper bound, combined with some modifications

Regularization of the Convergence Path of Contact Problems

21

of the standard Newton procedure, permits the execution of steps of unusual size. When using the proposed strategy, the limit on the step size is generally due to the large distortion of the continuum elements, and not to the contact.

3.2 Modified Stiffness and Residual during Phase One To overcome the difficulties cited earlier, two modifications are made to the standard Newton procedure during phase one of the solution. The first modification consists in disregarding the geometric stiffness term during this phase. The resulting expression of the tangent stiffness matrix is then KT,mod = KM = Ac ε NS NTS

.

(5)

This term is the starting point to build the phase one stiffness matrix. In order to do this, we have to focus first on the second modification. This is related to the contact force which goes into the residual. For large penetrations, the contact forces computed as Ac ε gN are grossly in error. The redistribution of these forces to the nodes by the vector NS , as per Eq. (1), is the second element of instability of the solution. To limit this force or, equivalently, the corresponding pressure, we propose to modify the linear relationship between contact pressure and normal penetration by using a cut-off with a maximum value independent of the penetration, pME (see Fig. 1). In this way the introduction of unrealistic forces into the system is prevented. Note that pME is an estimated value of the maximum contact pressure arising between the contacting surfaces at convergence. Hence, regardless from the amount of penetration, during phase one the contact pressure computed as pN = ε gN is replaced with the cutoff pressure, pME . It has to be remarked that the cut-off alone is not sufficient to perform large steps. In fact, if consistent linearization is used we get a zero derivative when the cut-off limit is enforced. Hence no contact stiffness is associated to the residual, even if penetration persists. The contact forces are then applied without any contact resistance. Once again this will have dangerous effects, because the resulting displacements in most cases lead to release, and then a new instability often takes place with part of the contacting surfaces switching from an open to a closed status during one iteration and viceversa for the next one. A very good performance has been achieved for phase one by introducing a secant stiffness. In most cases the secant stiffness is able to keep the gap closed and rapidly relax the contact conditions to approach values of the penetration for which consistent linearization can then be employed. The secant stiffness is related to the amount of penetration and to the maximum contact force by a variable penalty, computed at each iteration as

εs =

pME gN

.

(6)

This is then used directly in Eq. (5) in place of ε , see also Fig. 1. It is worth noting that, due to the constant pressure limit, the secant stiffness depends only on the

22

G. Zavarise, L. De Lorenzis, and R.L. Taylor

Fig. 1 Use of the secant stiffness

penetration and increases with a reduction of the penetration, until it reaches the standard penalty value. Subsequently the contact solution procedure shifts smoothly from phase one to phase two, where standard consistent linearization is performed.

3.3 Limitations of the Strategy For the next discussion, it is useful to introduce the following definition r=

pME pMR

(7)

where pMR is the value of the maximum contact pressure at convergence. The parameter r is therefore the ratio of the estimated to the real maximum contact pressure. The r value is > 1 or < 1 if the maximum contact pressure is, respectively, overestimated or underestimated. If the correct maximum contact pressure is overestimated by a reasonable amount, the LP algorithm performs very efficiently. If the r ratio is excessively large, convergence may be no longer achieved. This clearly results from a too early shift into phase two, which implies the introduction into the system of still large, unrealistic contact forces like for the standard consistent linearization. If the r ratio is large enough, the contact pressure limit may even exceed the product of the maximum initial (unchecked) penetration by the penalty parameter. In such a case the LP algorithm is not even activated, hence standard consistent linearization is always performed.

Regularization of the Convergence Path of Contact Problems

23

For any value of r > 1, provided that convergence is achieved, the converged solution is the correct one. “Correct solution” is here intended as the solution whose degree of approximation is related to the chosen value of the penalty parameter, ε . Hence, there is no influence of the r ratio on the accuracy of the converged solution, which only depends on the value of ε . The correctness of the converged solution is proved by the absence of contact elements which are still in phase one at convergence. As r gets closer to 1, or even r = 1 (i.e. if the correct value of the maximum contact pressure is known a priori), the number of iterations to convergence increases. The reason is that, as the limit pressure decreases, phase two is reached at a very late stage, hence most iterations are performed with a sub-quadratic rate of convergence. For r < 1, i.e. if the maximum contact pressure is underestimated, it is obviously not possible to obtain the correct solution, as the correct distribution of contact pressures includes values which are larger than the imposed limit. In such cases, typically convergence is achieved, however there is a number of contact elements that have not reached phase two at convergence. In other words, the problem converges to an incorrect solution, i.e. to a solution whose degree of approximation is worse than that related to the chosen penalty parameter. Therefore, it is necessary to add one more feature to the LP strategy, to be used when r < 1.

4 Large Penetration Enhanced Algorithm 4.1 Solution of the Problem for r < 1 Two alternative procedures to solve the problem described in Section 3.3 have been tested in this study and are described herein. The need to activate one of these procedures for a given contact element is identified by monitoring the evolution of the contact status between subsequent iterations. An easy criterion is a check of the variation of the normal penetration, e.g. through the following quantity ⎧ (i) ⎪ gN (i) (i−1) ⎪ | if gN < gN ⎨ | 1 − (i−1) gN Rg = (8) (i−1) gN ⎪ (i) (i−1) ⎪ | 1 − | if g > g ⎩ (i) N N gN

(i−1)

(i)

where gN and gN are the values of the penetration at the iterations number (i − 1) and i, respectively. From the definition, it follows that 0 ≤ Rg ≤ 1. If Rg is close to the unity, the penetration is rapidly changing within the iterations and no special procedure is needed. Conversely, a small value of Rg (close to zero) indicates that the penetration is undergoing very little change between subsequent iterations. During phase one this occurs when, due to the underestimation of pMR , the contact pressures are forced to remain below the too strict limit imposed by the LP algorithm. In such a case, a special procedure needs to be activated to guarantee convergence to the correct solution. Two procedures for this purpose are illustrated as follows. The first proposed procedure consists in increasing the pressure limit within the iterations. The resulting algorithm is indicated as LP-IP. The evolution of the contact

24

G. Zavarise, L. De Lorenzis, and R.L. Taylor

status has to be monitored to decide whether and when the increment can take place. Whenever Rg is less than a specified threshold, the contact pressure limit for that contact element is updated. The update strategy consists in adding the initial value of the pressure limit to its current value, i.e. (u)

(u−1)

(0)

pME = pME + α pME

u≥1

(0)

(9) (u)

(u−1)

where pME is the initial value of the estimated limit pressure, pME and pME are the limit pressures after the uth and (u-1)th update, respectively, and α is a weighting factor, which based on our experience can be set to 1. As a result, the current pressure limit is equal to one, two, three, etc. times its initial value as Rg is subsequently attained. Obviously, the more largely the correct maximum pressure is underestimated, the more updates need to be performed for the pressure limit. This reflects on the number of iterations needed to converge to the correct solution. Different update strategies may also be used. In any case, the increment of the pressure limit should be done very carefully, because it presents similar aspects to the increase of contact stiffness within augmentations. Since the attainment of a normal pressure below the cut-off limit coincides with the introduction of the geometric term into the tangent stiffness, a too rapid increase of the cut-off may result in a too early shift into phase two. In the second procedure, whenever Rg reaches the specified threshold for a contact element, augmentation is performed for that element, i.e. the current contact force is introduced into the system as an “external force”. In symbols (a)

(a−1)

p N = pN (a)

(a−1)

+ β εs gN

a≥1

(10)

where pN and pN are the normal pressures after the ath and (a-1)th augmentation, respectively, and β is an acceleration factor. For contact problems, using a β factor larger than one has been proved unsuccessful, hence also in this case a good choice could be β = 1. Whenever Rg reaches again the threshold in subsequent iterations, the value of the augmented force is updated. The resulting algorithm is indicated as LP-AU in the following. With respect to the LP-IP method, the LP-AU one has the additional advantage to reduce the penetration error inherent to the penalty method, due to the introduction of an augmentation scheme. The more largely the correct maximum pressure is underestimated, the more updates are performed for the augmented forces, the smaller is the norm of the normal penetration at convergence. Therefore, this method improves the quality of the solution, in terms of enforcement of the impenetrability condition. Correspondingly, the number of iterations to convergence also slightly increases. In the implemented procedures, the computation of Rg according to Eq. (8) is made for each active contact element. However, as soon as Rg for a given contact element is less than the specified threshold, the pressure limit is increased for all active contact elements (LP-IP procedure) or augmentation is performed for all contact elements (LP-AU).

Regularization of the Convergence Path of Contact Problems

25

5 Example The example deals with a wedge indented between two deformable bodies, see Fig. 2a. The lower bodies, made of the softer material (E = 2500, ν = 0.25), are restrained on the bottom and on the external sides. The wedge, made of the stiffer material (E = 25000, ν = 0.25), is subjected to an imposed vertical displacement on its top surface. The lateral surfaces of the cantilever are the slave ones, and the inclined surfaces of the indentor are the master ones. The contact penalty parameter is ε = 105 . The continuum is discretized with 4-node large deformation elastic plane-strain elements. All computations have been performed with the finite element program FEAP [2, 3]. Using standard consistent linearization (CL), the maximum vertical displacement of the wedge for which convergence is achieved is 3.0. Conversely, with the LP algorithm, convergence is reached for imposed displacements up to 7.5. The final, converged geometry for the latter value of imposed displacement is shown in Fig. 2b. The maximum value of the contact pressure is equal to 390. Figure 3 illustrates the deformed shape after the first four iterations, when CL is used. It is shown that the geometry evolves soon towards a highly distorted configuration, until catastrophic divergence occurs. Conversely, Fig. 4 shows the analogous deformed shapes when the LP algorithm is used. It is evident that the LP strategy prevents unrealistic values of the contact pressure to be attained. As a result, the geometry rapidly approaches the deformed configuration at convergence. In Fig. 5, the residual norm and the number of contact elements where the limit on the contact pressure is enforced are plotted versus the iteration number. The limitation on the contact pressure takes place initially for the whole active contact area (30 elements). At each subsequent iteration, the contact pressure becomes less than the cut-off value for an increasing number of slave nodes. Hence, such nodes enter phase two and for them CL is activated. For the remaining contact elements, the pressure limit is enforced and the modified tangent stiffness and residual are

(a) Initial geometry

(b) Geometry at convergence Fig. 2 Example geometry

26

G. Zavarise, L. De Lorenzis, and R.L. Taylor

(a) First iteration

(c) Third iteration

(b) Second iteration

(d) Fourth iteration

Fig. 3 Example: deformed shape after the first four iterations - CL

(a) First iteration

(b) Second iteration

(c) Third iteration

(d) Fourth iteration

Fig. 4 Example: deformed shape after the first four iterations - LP with r = 2.5

constructed. At this stage, the residual norm decreases but the rate of convergence is less than quadratic. After the first 8 iterations, all contact elements have entered phase two. From this iteration, CL is used for all contact elements and a quadratic rate of convergence is achieved. With 3 more iterations, convergence takes place. The results shown in Figs. 4 and 5 have been obtained for r = 2.5, i.e. with the estimated maximum contact pressure equal to two and one half times the correct value. However, results are qualitatively similar if other estimates are made, provided that r > 1, i.e. that the correct maximum contact pressure is overestimated.

Regularization of the Convergence Path of Contact Problems

27

Fig. 5 Example: residual norm and number of contacts over limit - LP with r = 2.5

6 Conclusions As a follow-up to a previous investigation [5], this paper has proposed a strategy to deal with contact problems involving large penetrations, in the context of the NTS formulation and of the penalty method. The strategy has shown a very good capability to deal with contact problems where the standard application of Newton’s method does not achieve convergence. The employment of an upper bound for the maximum contact pressure, coupled with the use of a secant contact stiffness contribution, permits to enforce gradually the violated impenetrability condition within the iterations. The proposed strategy performs efficiently if the estimated maximum contact pressure is reasonably in excess of the correct value. To deal with cases where the correct maximum contact pressure is underestimated, two alternative procedures have been devised. These procedures integrate the basic LP strategy enabling an automatic increase of the contact pressure over the initial limit. While both procedures have been shown to perform satisfactorily, the LP-AU one has the additional advantage to improve the quality of the solution, in terms of the degree of approximation in the enforcement of the impenetrability condition with the penalty method. Finally, the proposed strategy cannot only be used to achieve convergence in cases where standard CL would produce catastrophic divergence, but also to accelerate convergence for moderate values of initial penetrations. In these cases, when using a standard CL approach the first few iterations are typically needed to stabilize the solution before a quadratic rate of convergence is obtained. Using the LP strategy usually shortens this stabilization phase, thereby reducing the number of iterations and ultimately enhancing the computational efficiency.

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References 1. Fischer, K.A., Wriggers, P.: Frictionless 2D contact formulations for finite deformations based on the mortar method. Comput. Mech. 36, 226–244 (2005) 2. Taylor, R.L.: FEAP – A Finite Element Analysis Program, user manual. University of California, Berkeley, http://www.ce.berkeley.edu/projects/feap 3. Taylor, R.L., Zavarise, G.: FEAP – A Finite Element Analysis Program, contact manual. University of California, Berkeley, http://www.ce.berkeley.edu/projects/feap 4. Zavarise, G., De Lorenzis, L., Taylor, R.L.: A non-consistent start-up procedure for contact problems with large load-steps (submitted) 5. Zavarise, G., Taylor, R.L.: A force control method for contact problems with large penetrations. In: Owen, D.R.J., O˜nate, E., Hinton, E. (eds.) Proceedings of the V International Conference on Computational Plasticity, pp. 280–285. Pineridge Press, Barcelona (1997) 6. Zavarise, G., Wriggers, P., Nackenhorst, U.: A Guide for Engineers to Computational Contact Mechanics. The TCN Series on Simulation Based Engineering and Sciences, Vol. 1. Consorzio TCN, Trento (2006)

Chapter 4

On Different Variational Formulations of the Nitsche Method Ridvan Izi, Alexander Konyukhov, and Karl Schweizerhof

Abstract. Various numerical approaches in the case of contact problems are mainly dealing with additional terms enforcing constraints. Within the Nitsche approach the inclusion of constraints for both the non-penetration condition and the equilibrium of stresses on the contacting surfaces is carried out in a fully variational sense. Taking into account a specific choice and the physical meaning of the encountered Lagrange multipliers two different schemes for the Nitsche formulation are obtained: a) with Gauss point-wise substitution of Lagrange multipliers, b) with Bubnov-Galerkin-wise partial substitution of Lagrange multipliers. Both types of the Nitsche approach are implemented for 3D situations and a verification with a numerical example involving large sliding is presented.

1 Nitsche Formulation Contact problems within numerical approaches like the finite element method are mainly dealing with well known contact formulations like the Penalty or Lagrange multiplier method. For some particular problems these methods are not efficient enough and, thus, other approaches like the so-called Nitsche approach have to be investigated. In the current contribution two different Nitsche schemes applicable to contact problems derived in the sense of optimization theory are presented. The common basis of both Nitsche schemes is that the Nitsche approach is taking two contact constraints into account. Besides the well known non-penetration constraint (ξ 3 > 0 ≡ no contact) there is also the equilibrium of stresses between both bodies considered. Starting with the minimization problem considering both contact constraints the following statement for the contact problem is obtained: Ridvan Izi · Alexander Konyukhov · Karl Schweizerhof KIT, Institut f¨ur Mechanik, Kaiserstr. 12, 76131 Karlsruhe e-mail: [email protected]

30

R. Izi, A. Konyukhov, and K. Schweizerhof

Minimize the potentials WM + WS → min

(1)

subjected to constraints on the contact boundary: non-penetration ξ 3 = (rS − ρ M ) · nM = 0 equilibrium of stresses σ M · nM + σ S · nS = 0 .

(2) (3)

Here WM and WS denote the work of the structural part of the problem for the master and slave body, respectively. ρ M and rS define the position vectors on the master and slave surface. Together with the normal vector nM on the master body these position vectors are enabling the computation of the penetration ξ 3 according to [5] (see Fig. 1) with the so-called closest-point procedure. Eq. (3) gives the equilibrium on the mutual contact area on both bodies expressed by the stress tensor σ and the normal vector n.

Fig. 1 Definition of a spatial coordinate system specific for closest-point procedure

The surface stress vector σ · n, thereby, can be presented in parts of normal and tangent components leading to the formulation

σ · n = [(σ · n) · n]n + T = σn n + T .

(4)

For the nonfrictional problem considered here this decomposition of the stress vector enables to set the component T representing sticking and sliding cases to zero. Furtheron, for contact the direction of the normals of both bodies is equal (nM = −nS ). Thus, the equilibrium of stresses in Eq. (3) can be expressed by [(σ M · nM ) · nM ]nM + [(σ S · nS ) · nS ]nS = (σnM − σnS )nM = 0

.

(5)

In order to enforce the considered constraints in Eq. (2) and (5) in a fully variational sense – in accordance with optimization theory – an additional set of Lagrange multipliers on the master (μ M ) and on the slave surface (μ S ) are defined. Finally, the Nitsche approach is gained as the minimization problem for the following functional

On Different Variational Formulations of the Nitsche Method

N = WM + WS + WD + 

 SM

μ M · (σnM nM ) dSM −

μ S · (σnS nM ) dSS  μ S if SS with a set of Lagrange multipliers μ = μ M if SM −

31

→ min

SS

(6)

.

Hereby, the constraint of the equilibrium of stresses (Eq. (5)) between the contacting surfaces of the master and slave body (SM and SS ) is expressed by the last two terms. The functional constraint either in Penalty   WD represents the non-penetration  1 3 2 3 SC 2 εN (ξ ) dSC or Lagrange SC λ ξ dSC formalities with SC denoting the contacting area. SC is equal to the contact area measured either on the master surface (SM ) or on the slave surface (SS ). The parameter εN within the Penalty expression gives the penalty factor. In the following sections some remarks have to be pointed out in order to achieve a convenient formulation out of Eq. (6). The first remark is regarding the choice of the second set of Lagrange multipliers μ . Furthermore, the physical meaning of the Lagrange multiplier λ and its counterpart the Penalty term εN ξ 3 for the standard enforcement of the non-penetration constraint have to be depicted.

1.1 Choice of the Lagrange Multiplier Set μ Obviously, the additional Lagrange multipliers μ are representing position values due to their introduction in Eq. (6) for terms concerning stress vectors. The choice of μ as a specific position vector follows by taking the variation in the sense of Gˆateaux for Eq. (6) as

δ WM + δ WS + δ WD +

 SC

{δ μ M · (σnM nM ) + μ M · (δ σnM nM ) −

−[δ μ S · (σnS nM ) + μ S · (δ σnS nM )]} dSC = 0 .

(7)

Transforming this functional then step by step into the strong formulation by excluding all terms regarding the strong form of equilibrium of the bodies and the non-penetration constraint (Eq. (2)) the following expression remains  SC

{[(δ ρ M − δ rS ) + (δ μ M − δ μ S )] · (σnM nM ) + (μ M − μ S ) · (δ σnM nM )} dSC = 0 .

(8)

The boundary terms arising within this transformation out of the work expressions WM and WS are, hereby, expressed in contact kinematic formalities using δ ρ M and δ r S .

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R. Izi, A. Konyukhov, and K. Schweizerhof

In order to fulfill the requirement – first part in square brackets to become zero – the additional set of Lagrange multipliers μ is chosen specifically as

μ M = −ρ M μ S = −rS

(9)

leading to  SC

[(rS − ρ M ) · (δ σnM nM )] dSC =

 SC

ξ 3 δ σnM dSC = 0

(10)

and resulting again in a strong formulation for the non-penetration constraint (ξ 3 = 0).

1.2 Physical Meaning of the Non-penetration Terms The second remark before dealing with the formulation given in Eq. (6) is the physical meaning of the non-penetration terms. Thus, here the mechanical interpretation of the non-penetration terms for both, the Penalty and the Lagrange multiplier method, are repeated. The interpretation is necessary due to the Nitsche method – as will be shown later on – being a Lagrange multiplier method with specially chosen Lagrange multipliers. 1.2.1

Lagrange Multiplier λ

Taking the standard Lagrangian contact formulation L = WM + WS +

 SC

λ ξ 3 dSC → min

(11)

in variational form with the Gˆateaux derivative as

δ WM + δ WS +

 SC

(ξ 3 δ λ + λ δ ξ 3 ) dSC = 0

(12)

into account and removing with the help of the Gauss theorem all terms regarding the strong form – equilibrium of the bodies and the non-penetration constraint – the functional (12) can be written as  SM

(σnM nM ) · δ ρ M dSM −

 SS

(σnS nM ) · δ rS dSS +

 SC

λ δ ξ 3 dSC = 0 . (13)

Considering the contact kinematics ξ 3 = (rS − ρ M ) · nM the following equation  SC

(−σnM + λ ) δ ξ 3 dSC = 0

(14)

On Different Variational Formulations of the Nitsche Method

33

remains. Thus, leading in the strong formulation to the requirement that the Lagrange multiplier defined on the contact surface is equal to the normal stress on the ! master surface (σnM = λ ). 1.2.2

Penalty Term εN ξ 3

In order to gain the mechanical interpretation of the Penalty term the following functional is minimized 

P = WM + WS +

SC

1 εN (ξ 3 )2 dSC → min 2

(15)

by taking its variation into account

δ WM + δ WS +

 SC

εN ξ 3 δ ξ 3 dSC = 0 .

(16)

Transforming the variation step by step into the strong formulation using the Gauss theorem results in  SC

[(σnM nM ) · δ ρ M − (σnS nM ) · δ rS + εN ξ 3 δ ξ 3 ] dSC = 0 .

(17)

!

Finally, the requirement σnM = εN ξ 3 is obtained for the Penalty formulation. Summarizing the considerations within the Lagrange multiplier and the Penalty method the Lagrange multiplier λ used for the non-penetration constraint can be expressed physically as a combination of a normal stress σn and a penalty term εN ξ 3 .

2 Types of the Nitsche Approach The necessity to obtain the minimum of the functional in Eq. (6) together with the specific choice of the Lagrange multiplier set μ defined in Sect. 1.1 leads now to the following variational equation

δ WM + δ WS + δ WD + 

+ SC

[(δ rS − δ ρ M ) · (nM σnM ) + (rS − ρ M ) · (nM δ σnM )] dSC = 0 .

(18)

Considering the contact kinematics ξ 3 = (rS − ρ M ) · nM and its variation δ ξ 3 = (δ rS − δ ρ M ) · nM the functional

δ WM + δ WS + δ WD +

 SC

(δ ξ 3 σnM + ξ 3 δ σnM ) dSC = 0

(19)

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R. Izi, A. Konyukhov, and K. Schweizerhof

is achieved. Furtheron, expressing the functional WD representing the nonpenetration constraint according to the Penalty formulation in Sect. 1.2.2 the Nitsche approach is gained as

δ WM + δ WS +

 SC

εN ξ 3 δ ξ 3 dSC +

 SC

(δ ξ 3 σnM + ξ 3 δ σnM ) dSC = 0 .

(20)

This formulation is described by Wriggers and Zavarise in [6] and is also obtainable by considering a standard Lagrangian contact formulation like stated in (12) by using the physical meaning of the Lagrange multiplier λ (see Sect. 1.2) and substituting it with σnM + 12 εN ξ 3 . This can be interpreted as a Gauss point-wise substitution of the Lagrange multiplier in Eq. (12) and therefore is furtheron denoted as the Nitsche approach with Gauss point-wise substituted Lagrange multipliers. The representation of the Lagrange multiplier as

λ = σnM + εN ξ 3

(21)

can also be substituted in Eq. (12) using the Bubnov-Galerkin procedure with δ λ as a weighting function. Thus, we obtain the following equation system

δ WM + δ WS +  SC

 SC

(ξ 3 δ λ + λ δ ξ 3) dSC = 0

δ λ (λ − σnM − εN ξ 3 )dSC = 0 .

(22)

This approach is denoted as the Nitsche approach with Bubnov-Galerkin-wise partial substituted Lagrange multipliers and is used by Heintz and Hansbo, see [2].

3 FE Implementation of the Nitsche Approaches The discretization is carried out for a geometrically nonlinear solid element with Neo-Hookean material law. Thereby, a suitable decoupled representation of the strain-energy function for compressible isotropic hyperelastic materials is used (see [3]). Surface-To-Analytical-Surface (STAS) kinematics is implemented where one of the contacting bodies is a rigid body with an analytically parameterized surface like a rigid plane or rigid cylinder, see [4]. The surface of the rigid body is denoted as the slave surface.

3.1 Gauss Point-Wise Substituted Formulation Starting with Eq. (20) and using the approximation matrix A for the discretization of the penetration ξ 3 within the contact kinematics the expression

ξ 3 = (rS − ρ M ) · nM = nTM A uB

(23)

On Different Variational Formulations of the Nitsche Method

35

is gained where uB is the nodal displacement vector of the contacting surface of the solid element. nM denotes the normal vector on the contacting master surface of the nonlinear solid element. Furtheron, for implementation purposes the vector n2M is defined by entries of the normal vector nM as   n2M = n1 n1 , n2 n2 , n3 n3 , 2n1 n2 , 2n2 n3 , 2n1 n3 (24) enabling to compute the normal projection of the stress vector out of the stress tensor T as σnM = n2M DBu. This leads to the discretization

δ uT

 V

(BT DB) dV u + εN δ uTB

− δ uTB



 SC

(AT n nT A) dSC uB

T

SC

(AT n n2M DB) dSC u − δ uT

 SC

(BT DT n2M nT A) dSC uB = 0

, (25)

finally, ending up with matrices of the same order as

δ uT (Ku + εN Kξu

3ξ 3

− Kξu

3σ

n

− Kσu n ξ ) u = 0 . 3

(26)

The assembly directly results in the element stiffness matrix and its additions in the case of contact.

3.2 Bubnov-Galerkin-Wise Partial Substituted Formulation Discretization of Eq. (22) leads to the following two equation sets

δ uT δλ

T

 V

(BT DB) dV u + δ uTB



SC

(AT nC) dSC λ = 0

(εN C n A) dSC uB − δ λ T

SC



T

T



T

SC

(CT n2M DB) dSC u −

δλ T

 SC

(CT C) dSC λ = 0

. (27)

C indicates, hereby, the matrix consisting of shape functions for the Lagrange multiplier λ defined on the contacting surface of the solid element. The assembly of Eq. (27) leads to ξ3

δ uT Kuu u + δ uT Kuλ λ = 0 δ λ T (εN Kξλ u − Kσλ nu ) u − δ λ T Kλ λ λ = 0    3

Kλ u

(28)

36

R. Izi, A. Konyukhov, and K. Schweizerhof

where the Lagrange multipliers can be preeliminated as they are only elementwisely defined by applying the Schur complement to the equation system of      ξ3 u 0 Kuu Kuλ . (29) = λ 0 Kλ u −Kλ λ Regarding the check of contact an active set strategy is applied. Within the Gauss point-wise substituted formulation an active set is defined as the set of Gauss points with negative penetration. For the Bubnov-Galerkin-wise partial substituted formulation a special searching algorithm with a set of Gauss points enabling the inversion of the matrix Kλ u is used.

4 Numerical Example For verification purposes both Nitsche formulations are used for the simulation of the frictionless free bending of a metal sheet on two rigid clyinders which involves large sliding. The geometrical setup is presented in Fig. 2a and is following the setup given in [1].

u R=2 12 24

(a)

(b)

Fig. 2 Setup of metal sheet on two rigid cylinders: a) geometrical properties; b) deformed situation with predefined u

The Neo-Hookean material properties chosen are equaling E = 105 and ν = 0.3, the element size used is 1 × 0.25 × 0.5. For a displacement controlled loading the deformed state is given in Fig. 2b. As a reference formulation for the Nitsche approaches the standard Penalty scheme is used. Within the given experimental setup both Nitsche formulations perform similar to the Penalty scheme, but show higher sensitivity regarding the amount of Gauss points (see Fig. 3a) which is more obvious at the initiation of sliding (see Fig. 3b). These oscillations are due to the fact that rather non-smooth contact checking is performed, which can be improved using more contact points – either more Gauss points or higher mesh refinement (see [1]). The Nitsche approach with the BubnovGalerkin-wise partial substituted Lagrange multipliers is performing more robust, thus, allowing a larger range of values for Gauss points and Penalty parameters. On

On Different Variational Formulations of the Nitsche Method Penalty with εN= 104 Nitsche (Gauss Point − wise) with εN=3*105 Nitsche (Bubnov−Galerkin − wise) with εN= 105

12 10

E = 105, ν=0.3 8 Gauss Points

8 Reaction Force

37

6

PP

4

PP P

2

PP P

PP P

0 −2

0

2

4

6

8

PP P

10

Displacement

(b)

(a)

Fig. 3 Displacement-reaction force diagram with 8 Gauss points and best performing εN for each formulation: a) overall result; b) cutout at initiation of sliding

Penalty Nitsche (Gauss Point − wise) Nitsche (Bubnov−Galerkin − wise)

12 10

!! !! ! ! !! ! ! !! ! ! !! ! ! !! E = 105, ν=0.3 5 εN=10

Reaction Force

8 6 4 2 0 −2

0

2

6 4 Displacement

(a)

8

10

(b)

Fig. 4 Displacement-reaction force diagram with 10 Gauss points and εN = E for each formulation: a) overall result; b) cutout for performance of the Nitsche approach with Gauss point-wise substituted Lagrange multipliers

38

R. Izi, A. Konyukhov, and K. Schweizerhof

the other hand the Gauss point-wise substituted Lagrange multipliers formulation is highly sensitive, even for εN = E, see Fig. 4a. While for this specific choice of the Penalty parameter the approach with Bubnov-Galerkin-wise partial substituted Lagrange multipliers is performing again similar to the Penalty scheme, the cutout in Fig. 4b shows the interruption of the computation for the Gauss point-wise substituted formulation at an early stage. Simpler examples like closure examples found in literature, see [6], do not show such a high sensitivity. For other complex setups like this metal forming problem, however, such sensitivities are observable. The focus in further studies will be on circumventing the sensitivity and on an automatic determination of correct parameters.

References 1. Harnau, M., Konyukhov, A., Schweizerhof, K.: Algorithmic aspects in large deformation contact analysis using “solid-shell” elements. Comput. Struct. 83, 1804–1823 (2005) 2. Heintz, P., Hansbo, P.: Stabilized Lagrange multiplier methods for bilateral elastic contact with friction. Comput. Meth. Appl. M. 195, 4323–4333 (2006) 3. Holzapfel, G.A.: Nonlinear solid mechanics. Wiley, New York (2000) 4. Konyukhov, A.: Geometrically exact theory for contact interactions. Habilitationsschrift. KIT, Karlsruhe (2010) 5. Konyukhov, A., Schweizerhof, K.: Contact formulation via a velocity description allowing efficiency improvements in frictionless contact analysis. Comput. Mech. 33, 165–173 (2004) 6. Wriggers, P., Zavarise, G.: A formulation for frictionless contact problems using a weak form introduced by Nitsche. Comput. Mech. 41, 407–420 (2008)

Chapter 5

Challenges in Computational Nanoscale Contact Mechanics Roger A. Sauer I have known Peter Wriggers since summer 2005 when he came to visit Berkeley and we discussed my doctoral research on computational nanoscale contact mechanics [7]. After graduation I took the opportunity to work with him at the Leibniz University Hannover. There I had the chance to teach the graduate courses ‘continuum mechanics’ and ‘contact mechanics’, coordinate various research projects, and, perhaps most challenging, get familiar with the German academic system. Since January 2010 I work at the Graduate School AICES in Aachen. I wish Peter Wriggers all the best for the future (R.A. Sauer).

Abstract. This paper outlines the differences between nanoscale and macroscale contact descriptions and gives an overview of the challenges encountered at the nanoscale. The adhesive instability, common to nanoscale contact, is illustrated by a simple example. Further emphasis is placed on multiscale approaches for contact.

1 Introduction Nanoscale contact mechanisms are essential for many applications, like adhesives, small scale surface characterization and machining, MEMS and NEMS (Micro- and Nano-electro-mechanical systems), self-cleaning surfaces, gecko adhesion, cohesive fracture and peeling problems. At this scale it becomes necessary to integrate the fundamental physical phenomena [3, 5] into the approaches of computational contact mechanics [4, 15]. The challenges encountered in this are discussed in the following sections.

2 Nanoscale Contact Challenges At small length scales several physical and numerical challenges present themselves that need to be accounted for in a computational framework. These challenges are: Roger A. Sauer Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH Aachen University, Templergaben 55, 52056 Aachen, Germany e-mail: [email protected]

40

R.A. Sauer

1. Computational contact mechanics: numerical accuracy, efficiency and stability, closest point projection onto discrete surfaces, friction algorithms, wear and lubrication modeling 2. Bridging the scales between atomistic and continuum description (see Sect. 3) 3. Efficient and accurate algorithms for nanoscale contact (see Sect. 3) 4. Complex surface microstructure at different length scales (see Fig. 1) 5. Physical instabilities caused by strong adhesion (see Sect. 4) 6. Peeling computations: numerical instabilities due to discretization error 7. Multiscale modeling and homogenization approaches (see Sect. 5) 8. Nanoscale contact dynamics: efficient and accurate integration algorithms 9. Interaction between nanoscale friction and adhesion 10. Multifield contact problems, e.g. thermal equilibrium and chemical reactions at nanoscale interfaces 11. Nanoscale material models for specific applications: soft adhesives, liquids, granular media 12. Parameter identification and determination Substantial work has been done to address the first challenge [4, 15]. The challenges posed by complex microstructures are illustrated by the examples in Fig. 1. An efficient formulation for stable peeling computations is presented in [12]. Challenges 2, 3, 5 and 7 are addressed in the following sections. Challenges 8–12 are mostly open research topics that call for further theoretical, experimental and computational research. Contact models that successfully describe various contact aspects need to be integrated into holistic top-down and bottom-up approaches. Such approaches attempt to find a unified description of various phenomena across different length scales and thus try to link macroscopic and microscopic model parameters. A helpful modeling framework for this is the bottom-up contact model outlined in the following section.

3 Nanoscale versus Macroscale Contact In this section the different descriptions commonly used for nanoscale and macroscale contact are contrasted. Considering conservative systems in both cases, the total potential energy can be written as

Π = Πint + Πc − Πext

(1)

,

where the individual contributions Πint , Πext and Πc denote the internal, external and contact energy. For macroscopic scales, contact between continua B1 and B2 is characterized by the the impenetrability constraint g(x1 , x2 ) ≥ 0 ∀ x1 ∈ ∂ B2 , x1 ∈ ∂ B2

,

(2)

which states that the gap g between arbitrary surface points must remain positive. The impenetrability causes the tractions tc acting on the contact surface between the

Challenges in Computational Nanoscale Contact Mechanics

41

a)

b) Fig. 1 a) Surface microstructure of the self-cleaning lotus leaf, adapted with permission from Eye of Science; b) Microstructure of the gecko adhesion mechanism [1], adapted with permission from the Journal of Experimental Biology

bodies (see Fig. 2). Utilizing the gap vector g, the contact energy can be expressed as  Πc = tc · g dA . (3) ∂ Bc

On the other hand, at the nanometer scale and below, contact is resolved into the interactions of individual atomic particles (see Fig. 2). The interaction across the contact interface can be described by pair potentials like the Lennard-Jones potential

φ (r) := ε

 r 12 0

r

− 2ε

 r 6 0

r

,

(4)

which, for example, is suitable to describe van-der-Waals adhesion between the bodies. Here r is the distance between the two particles, and r0 and ε are material constants characterizing the interaction. Given the pair potential φ , the contact energy of the discrete particle system follows from the sum

42

R.A. Sauer

a)

b) Fig. 2 Nanoscale (a) versus macroscale (b) contact description

n1 n 2

Πc = ∑ ∑ φ (xi − x j ) , i

(5)

j

which is taken over all interacting particles of the two bodies. A seamless transition between both approaches can be generated if the discrete sum in Eq. (5) is replaced by the continuous integral [13]

Πc =





B1 B2

β1 β2 φ (x1 − x2 ) dv2 dv1

,

(6)

where β1 and β2 denote the molecular densities of the bodies. According to this formulation the contact forces between the bodies follow as gradients of potential φ . This approach yields accurate results down to length scales of a few nanometers [8, 10]. At large scales this formulation resembles phenomenological constitutive adhesion and cohesion models [6, 16] that are enforced computationally by barrier or cross-constrained methods [15, 17]. This transition, as well as further computational details and efficient contact algorithms, are discussed in [13, 8, 10, 12].

4 Adhesion Instability During strong adhesion of soft bodies an instability can occur: The adhesive forces can become so strong that they overpower the internal forces of the solids. This phenomenon can be illustrated by the simple 1D example shown in Fig. 3a [7]: Two particles are considered that interact with the Lennard-Jones potential (4). The internal deformation of the solids is modeled by a spring with constant stiffness k. The lower particle is considered fixed, while the upper particle is pushed downward by an imposed displacement u that requires the force P. For r = r0 and u = 0,

Challenges in Computational Nanoscale Contact Mechanics

a)

43

b) Fig. 3 a) Adhesive contact example; b) Load-displacement curve

the force in the system is P = 0. The total potential energy of this system is then given by 2 1  Π (r) = φ (r) + k u + (r − r0 ) . (7) 2  For a fixed displacement u, equilibrium follows from ∂∂Πr = 0u=fixed , from which we can find a relation between r and u, namely u(r) =

F(r) − r + r0 k

,

(8)

where F(r) := − ∂∂φr is the interaction force of the Lennard-Jones potential. From Eq. (8) the load-displacement curve follows as   P(u) = k u + r(u) − r0 . (9) It can be displayed as P(r) vs. u(r) as is shown in figure 3b. This graph shows that, as u becomes large, the repulsion between the particles is so strong that the deformation is determined purely by the deformation of the spring. For large negative u, on the other hand, the attraction between the particles is very weak so that P → 0 and u → r0 − r, since the spring is barely deforming. 2 The stability of the system can be investigated by examining ∂∂ rΠ2 |u=fixed . Setting this derivative equal to zero, we can identify the critical spring stiffness  4 4/3 ε kcr = 36 13 r02

.

(10)

For k > kcr the system will always be stable. However, if k < kcr the system develops an instability. The unstable section of the equilibrium path is shown as a dashed line in the figure above. In this case, as we push the two particles together, their mutual

44

R.A. Sauer

attraction will suddenly overpower the spring and the particles snap together into a new equilibrium position. Likewise, when pulling the particles apart, they will suddenly snap free. This behavior carries over to continuous systems [2, 14].

5 Multiscale Contact Modeling Even though computational power has increased immensely in the past, it remains impossible to resolve even micrometer-scale problems at full atomic resolution. Therefore multiscale methods are needed that combine different modeling levels into one holistic model. To find appropriate multiscale models one must decide which details and effects to include at the various levels. To a large degree this presumes the knowledge of the characteristics that are emphasized and the characteristics that are lost between the scales and it thus becomes necessary to validate and refine chosen models. The demand for multiscale modeling lies both in the

Fig. 4 Multiscale modeling hierarchy of the adhesion mechanism used by the gecko

Challenges in Computational Nanoscale Contact Mechanics

45

development of theoretical formulations, that unify different descriptions at various length scales, and in the development of efficient computational formulations, that achieve to span a large range of length scales. Helpful modeling components are coarse-graining techniques, reduced order modeling, adaptive model refinement and FE2 strategies. The selection of an appropriate multiscale approach for contact depends on the specific problem at hand. An example is the adhesion mechanism of the gecko shown in Fig. 4. To model the adhesion mechanism of the gecko toes, five modeling levels are considered: A directional lamella model, at the millimeter scale, a seta model at the 10 μ m scale, a spatula model at the 100 nm scale, an effective contact ˚ model at the nanometer scale, and a molecular interaction model at the Angstrom scale. Advances in this direction have appeared in [9, 11].

6 Conclusion This paper discusses some of the challenges encountered in nanoscale contact mechanics. Some of these have been addressed and partly resolved satisfactory by recent research activity. Other challenges are still open topics that call for further theoretical, experimental and computational research. Among those are multiscale methods, time integration algorithms, nanoscale friction modeling, multifield methods and nanoscale material modeling. Acknowledgements. The author is grateful to the German Research Foundation (DFG) for supporting this work under project SA1822/5-1 and grant GSC 111.

References 1. Autumn, K., Dittmore, A., Santos, D., Spenko, M., Cutkosky, M.: Frictional adhesion: a new angle on gecko attachment. J. Exp. Biol. 209, 3569–3579 (2006) 2. Crisfield, M.A., Alfano, G.: Adaptive hierarchical enrichment for delamination fracture using a decohesive zone model. Int. J. Numer. Meth. Eng. 54, 1369–1390 (2002) 3. Israelachvili, J.N.: Intermolecular and Surface Forces. Academic Press, London (1991) 4. Laursen, T.A.: Computational Contact and Impact Mechanics: Fundamentals of modeling interfacial phenomena in nonlinear finite element analysis. Springer, Heidelberg (2002) 5. Persson, B.N.J.: Sliding friction: Physical principles and application. Springer, Heidelberg (2002) 6. Raous, M., Cang´emi, L., Cocu, M.: A consistent model coupling adhesion, friction, and unilateral contact. Comput. Meth. Appl. M. 177, 383–399 (1999) 7. Sauer, R.A.: An atomic interaction based continuum model for computational multiscale contact mechanics. Ph.D. thesis. University of California, Berkeley (2006) 8. Sauer, R.A., Li, S.: An atomistically enriched continuum model for nanoscale contact mechanics and its application to contact scaling. J. Nanosci. Nanotech. 8, 3757–3773 (2008) 9. Sauer, R.A.: Multiscale modeling and simulation of the deformation and adhesion of a single gecko seta. Comput. Meth. Biomech. Biomed. Eng. 12, 627–640 (2009)

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10. Sauer, R.A., Wriggers, P.: Formulation and analysis of a 3D finite element implementation for adhesive contact at the nanoscale. Comput. Meth. Appl. M. 198, 3871–3883 (2009) 11. Sauer, R.A.: A computational model for nanoscale adhesion between deformable solids and its application to gecko adhesion. J. Adhes. Sci. Technol. 24, 1807–1818 (2010) 12. Sauer, R.A.: Enriched contact finite elements for stable peeling computations. Submitted to Int. J. Numer. Meth. Eng. (2010) 13. Sauer, R.A., Li, S.: A contact mechanics model for quasi-continua. Int. J. Numer. Meth. Eng. 71, 931–962 (2007) 14. Sauer, R.A., Li, S.: An atomic interaction-based continuum model for adhesive contact mechanics. Finite Elem. Anal. Des. 43, 384–396 (2007) 15. Wriggers, P.: Computational Contact Mechanics. Springer, Heidelberg (2006) 16. Xu, X.-P., Needleman, A.: Numerical simulations of fast crack growth in brittle solid. J. Mech. Phys. Solids 42, 1397–1434 (1994) 17. Zavarise, G., Wriggers, P., Schrefler, B.A.: A method for solving contact problems. Int. J. Numer. Meth. Eng. 42, 473–498 (1998)

Chapter 6

On the Four-node Quadrilateral Element Ulrich Hueck and Peter Wriggers From January 1992 until January 1994, I worked on my PhD with Professor Peter Wriggers in Darmstadt. During that time, he also enabled for me a fruitful visit of three months to the University of Cape Town in South Africa. The paper “On the Four-node Quadrilateral Element” summarizes the results of the PhD. Since 1994, I am working for Siemens in the power generation business. For some projects, the knowledge about numerical methods and programming was helpful. However, the key aspect of my PhD was learning to work in complete uncertainly – and finding it challenging rather than unsettling (U. Hueck).

1 Introduction A new formulation for the quadrilateral is presented. The standard bilinear element shape functions are expanded about the element center into a Taylor series in the physical co-ordinates. Then the complete first order terms insure convergence with mesh refinement. Incompatible modes are added to the remaining higher order term, all of these being expanded into a second order Taylor series. The minimization of potential energy yields a constraint equation to eliminate the additional incompatible degrees of freedom on the element level. With the resulting constant and linear gradient operators being uncoupled, the stiffness matrix is written in terms of underintegration and stabilization. Therefore, the new quadrilateral is labeled QS6. Based on a one field variational principle, the formulation exhibits all desirable element properties. Numerical solutions of high accuracy are obtained. The element is suitable for problems involving bending, incompressibility and high mesh distortion. Furthermore, the approach is applicable to nonlinear analysis. Ulrich Hueck Siemens AG e-mail: [email protected] Peter Wriggers Institute of Continuum Mechanics, Leibniz Universit¨at Hannover, Appelstr. 11, D-30167 Hannover, Germany e-mail: [email protected]

48

U. Hueck and P. Wriggers

2 Element Formulation From the coordinates of the element nodes, constant coefficients are obtained as ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ x y a 1 b1 −1 1 1 −1 ⎜ 1 1 ⎟ ⎝ a2 b2 ⎠ = 1 ⎝ 1 −1 1 −1 ⎠ ⎜ x2 y2 ⎟ (1) ⎝ x3 y3 ⎠ 4 a 3 b3 −1 −1 1 1 x4 y4 J0 = a1 b3 − a3 b1

J1 = a1 b2 − a2 b1

J2 = a2 b3 − a3 b2

.

(2)

The Taylor series expansion of the element basis functions leads to the operators ⎛ ⎞ ⎛ ⎞ l1 0 g1 g 2 0 0 L T = ⎝ 0 l2 ⎠ G T = ⎝ 0 0 g3 g 4 ⎠ (3) l2 l1 g3 g 4 g1 g 2 with the polynomials  & '  1 J2 J1 l1 = − b1 ξ + b1 − b3 ξ η − b3 η J0 J0 J0 

l2 =

&

'

1 J2 J1 a1 ξ + a1 − a3 ξ η − a3 η J0 J0 J0

  2 J2 g1 = − b3 ξ + ξ η J0 J0   2 J1 g2 = b 1 η + ξ η J0 J0

(4)



  2 J −2 g 3 = a3 ξ + ξη J0 J0 

g4 = −



(5)

2 J1 a1 η + ξ η . J0 J0

For the integrations, a modified change of variables is used where the determinant  1 1 of the Jacobian, J, is replaced by J0 : dV = −1 J −1 0 dξ dη . On the element level, a (4 × 2) matrix, Φ , is then obtained by solving the equilibrium constraint & ' & ' G E GT dV Φ + G E LT dV = 0 (6) where E denotes the matrix of material stiffnesses. Three stabilization coefficients and the γ -vector are calculated to generate the element stiffness matrix, K, in which K0 is obtained from a 1-point quadrature of the standard quadrilateral: & ' ε1 γγ T ε2 γγ T K = K0 + (7) ε2 γγ T ε3 γγ T

On the Four-node Quadrilateral Element

&

ε1 ε2 ε2 ε3

'

 

=

L + ΦT G

49



 E

 LT + GT Φ dV

⎞ ⎛ ⎞ ⎛ ⎞⎤ 1 −1 −1 ⎟ ⎜ ⎟ ⎜ ⎟⎥ 1 ⎢⎜ ⎜ −1 ⎟ − J2 ⎜ 1 ⎟ − J1 ⎜ −1 ⎟⎥ γ= ⎢ ⎣ ⎝ ⎠ ⎝ ⎠ ⎝ 1 1 1 ⎠⎦ 4 J0 J0 −1 −1 1

(8)

⎡⎛

.

(9)

3 Numerical Example A cantilever beam subject to a moment tip-load is calculated. The sensitivity to geometric distortion is compared with other existing elements:

Fig. 1 Error of the beam tip deflection w.r.t. the distortion

References 1. Hueck, U., Wriggers, P.: On the four-node quadrilateral element. In: The Third World Congress on Computational Mechanics, Extended abstracts, Chiba, Japan, vol. 2, pp. 1802–1803 (1994) 2. Hueck, U., Wriggers, P.: A formulation for the 4-node quadrilateral element. Int. J. Numer. Meth. Eng. 38, 3007–3037 (1995)

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3. Pian, T.H.H., Sumihara, K.: Rational approach for assumed stress finite elements. Int. J. Numer. Meth. Eng. 20, 1685–1695 (1984) 4. Simo, J.C., Rifai, M.S.: A class of mixed assumed strain methods and the method of incompatible modes. Int. J. Numer. Meth. Eng. 29, 1595–1638 (1990) 5. Taylor, R.L., Beresford, P.J., Wilson, E.L.: A non-conforming element for stress analysis. Int. J. Numer. Meth. Eng. 10, 1211–1219 (1976) 6. Wriggers, P., Hueck, U.: A formulation of the QS6 element for large elastic deformations. Int. J. Numer. Meth. Eng. 39, 1437–1454 (1996) 7. Yuan, K.-Y., Huang, Y., Pian, T.H.H.: New strategy for assumed stresses for 4-node hybrid stress membrane element. Int. J. Numer. Meth. Eng. 36, 1747–1763 (1993)

Chapter 7

Stability of Mixed Finite Element Formulations – A New Approach Stefanie Reese, Vivian Tini, Yalin Kiliclar, Jan Frischkorn, and Marco Schwarze I met Peter for the first time in 1984 as a first semester student of civil engineering in Hannover. After completing my Diploma degree in 1990, Peter offered me a Ph.D. position in his group in Darmstadt. Not only did I enjoy the highly interesting, challenging and always fruitful scientific discussions with him, but also many other very important events such as the ski seminars and the evenings in the Irish pubs, beer and wine gardens. Peter displayed in all regards a tremendous endurance which his students like me had a difficult time in keeping up with. He also supported my postdoc stay in Berkeley and served as my habilitation advisor back in Hannover. After becoming myself professor, I have come to appreciate more and more his tremendous efficiency and intellectual abilities. They are the basis of his great success in his scientific work on the one hand, and his remarkable strategic planning skills on the other. In particular, the latter have been employed most successfully to better the position of mechanics in the worlds of science and university politics. I wish him the best for the coming years (S. Reese).

Abstract. To guarantee stability of non-linear mixed finite element formulations is still an unsolved problem. In the present contribution firstly a unified finite element technology for linear-elastic problems is described where the effect of locking can be well explained and the issue of instability is not relevant. The extension to large deformation models reveals the difficulty of differentiating between physcially relevant and artificial bifurcations. Powerful finite element technologies should be able to exhibit the first kind but not show the second kind of bifurcations. In the paper a strategy is developed to detect and to avoid such non-physical instabilities.

1 Introduction Most of the commonly used technologically treated finite element formulations belong to either the enhanced strain, the B-Bar or the reduced integration methods. Typical representatives of the first group can be found in Simo and Rifai [17], Simo and Armero [18], Wriggers and Hueck [20] and Korelc and Wriggers [6]. Stefanie Reese · Vivian Tini · Yalin Kiliclar · Jan Frischkorn · Marco Schwarze Institute of Applied Mechanics, RWTH Aachen University e-mail: [email protected]

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Methods of the second type have been introduced by Nagtegaal et al. [11], Malkus and Hughes [9] and Simo et al. [19]. The third class of elements is discussed in e.g. Belytschko et al. [3], Doll et al. [4] and Reese [10]. Interesting alternatives which combine different earlier approaches are found in Loehnert et al. [8] and Mueller-Hoeppe et al. [10]. Certainly this overview of the literature is far from being complete. All three element families already include representatives designed for complex problems as e.g. forming processes, biomechanics or challenging applications in medical technology. Thus, in principal, the problem of embedding large deformations and inelastic material laws has been solved. Nevertheless the question of stability is still open and causes difficulties in even simple calculations as the homogeneous compression of a hyperelastic block (Wriggers and Reese [13], Reese and Wriggers [14]). See for further investigations of this topic e.g. Pantuso and Bathe [12], Armero [1] and Dvorkin and Assanelli [5]. All three finite element technlogies mentioned above can be cast into the mathematical framework of a mixed variational formulation. At first we will confine ourselves to linear elasticity before we extend the investigation to finite kinematics and non-linear material laws in Sect. 6. An unified finite element technology will be derived which reveals the differences between physical and non-physical instabilities. The method is validated by a numerical example.

2 Linear Elasticity - Mixed Variational Formulation Most enhanced strain methods are derived from a three-field functional of HuWashizu type where the independent stress tensor σ˜ , the total strain tensor ε and the displacement vector u serve as independent variables. In the pioneering work of Simo and Rifai [17] it is shown that by requiring the spaces of admissible variations of σ˜ and the so-called enhanced strain tensor ε e := ε − sym (grad u) to be L2 -orthogonal the three-field functional reduces to a two-field functional. One finally arrives at the two-field functional g1 (u, ε e ) =

 B

σ · δ ε c dV + gext = 0,

g2 (u, ε e ) =

 B

σ · δ ε e dV

(1)

where the stress tensor σ is computed by means of the linear elasticity law σ = C [ε ] with C representing the linear elasticity tensor. The independent stress field σ˜ has been eliminated. Note further that we use from now on the short hand notation ε c := sym (grad u) for the compatible part of the total strain tensor ε . The term gext refers to the virtual work of the external loading. For the following analysis it is important to realize that the B-Bar method in the original format of Nagtegaal et al. [5] (see also Simo et al. [19]) can be also derived from (1). This is done by assuming the enhanced strain tensor to be purely D D volumetric, i.e. the deviatoric part ε D e of ε e vanishes and the relation ε = ε c is obtained. The scalar product σ · δ ε e reduces to p tr ε e where the pressure p is defined by p := tr σ /3. We finally add the two equations and obtain

Stability of Mixed Finite Element Formulations – A New Approach

g (u, ε e ) =



σ · δ ε dV + D

B



D

B

p tr δ ε dV + gext = 0

53

.

(2)

In what follows we refer always to (1) which includes the format (2).

3 Interpolation 3.1 Compatible Strain The interpolation of the two-dimensional standard linear isoparametric displacement formulation is often represented in the form u(e) = ∑4I=1 NI uI where the shape functions (node index I = 1, ..., 4) NI = NI (ξ , η ) = 14 (1 + ξ ξI ) (1 + η ηI ) depend on the local coordinates ξ and η defined on a quadratic reference element [−1, 1] × [−1, 1] denoted from now on by ◦. Note that italic letters are used to denote matrices. In this way the 2 x 1 matrix u(e) includes the displacements ux and uy in x- and y-direction, respectively. We further introduce the shape function vector N = r + ξ gξ + η gη + ξ η h. The vectors r, gξ , gη and h are given by 4 rT T = {1, 1, 1, 1}, 4 gTξ = {−1, 1, 1, −1}, 4 gTη = {−1, −1, 1, 1} and 4 hT = {1, −1, 1, −1}, respectively. The nodal displacements are included in the vector Ue . We can finally write ux = N T (Ue )x and uy = N T (Ue )y where UeT = {(Ue )Tx , (Ue )Ty } holds. For the finite element technology to be introduced in the present contribution it is suitable to split the shape function vector N into linear and “hourglass” parts. This procedure has been described already several times in the literature (see for the original derivation Belytschko et al. [3] and for further applications e.g. Reese et al. [16]). The part of N which depends linearly on x is determined  by N lin = r + [bx by ] (x − x0 ) where the vectors bx and by are defined by [bx by ] = gξ gη J0−1 . The matrix J = ∂ x/∂ ξ denotes the Jacobi matrix. The index 0 indicates quantities evaluated in the centre of the element which coincides with the origin of the local coordinate system (ξ = {ξ , η }T = o2 ). The vector x = {x, y}T includes the cartesian coordinates of an arbitrary point within the element. The hourglass part Nhg = N − Nlin is given by Nhg = (I4 − [bx by ] xnode ) ξ η h = ξ η γ . The first row of the 2 x 4 matrix xnode contains the x-coordinates, the second row the y-coordinates of the element nodes I = 1, ..., 4. The last equation defines the so-called stabilization vector γ (Belytschko et al. [3]). This vector has an important meaning in finite element technology as will be shown in the following sections. Let us now use the shape function vector N to interpolate the compatible strain tensor ε c . For this purpose we represent the latter quantity in the Voigt notation εˆ Tc = {(ux ),x , (uy ),y , 0, (ux ),y + (uy ),x } with εˆ c = BUe . In the plane strain case the third component of εˆ c is usually not included in the Voigt notation since it is zero. Note that in the context of the B-Bar method the compatible strain vector εˆ c will be split into deviatoric and volumetric parts the zz-components of which are not zero. The B-operator is decomposed into constant and “hourglass” parts: B = B0 + Bhg. The second part includes for general element shapes rational functions of ξ and η which reduce in the case of a parallelogram-shaped element to a linear function of ξ and η .

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3.2 Enhanced Strain 



Using the L2 -orthogonality ◦ σ˜ · δ ε e det J dξ dη t = 0 ⇒ ◦ δ ε e det J dξ dη t = 0 of Simo and Rifai [17] it has been additionally assumed that σ˜ is constant within the element. This is the easiest way to fulfill the patch test. Note that in the con text of the B-Bar method the latter equation will boil down to ◦ tr δ ε e detJ dξ dη t. The letter t refers to the element thickness. Obviously the interpolation of εˆ e has to fulfill the orthogonality. Working again with the Voigt notation we choose the interpolation εˆ e = GU Ue + GW We where the vector We includes additional, so-called enhanced degrees-of-freedom. The latter notation for εˆ e is not common in the literature. It is chosen here because in this way still all three finite element technologies, (1) the standard displacement formulation, (2) the classical enhanced strain method according to Simo and Rifai [17] and (3) the so-called B-Bar method of e.g. Malkus and Hughes [9] (see also Simo et al. [19]), are still included. Interestingly all three methods can be cast into the same framework of a condensed displacement formulation.

4 Element Stiffness Matrix Inserting the interpolations derived in the previous section into the weak forms stated in Sect. 2 yields for the element stiffness matrix after several calculation steps −1 the result Ke = ◦ BT red Cˆ B red J dξ dη t with B red = B − GW KWW KWU where B is given by B = B + GU . Let us further introduce with Bred = Bred  − B0 the so-called reduced B-operator. Interesting is at this point the reduction to a method for reduced integration with hourglass stabilization which works for all three finite element technologies considered here. Working with the assumption J = J0 which is exactly true only for parallelogram-shaped elements the matrix Bred depends linearly on ξ and η . The element stiffness matrix then splits into two parts: 

Ke = 4 BT0 Cˆ B0 J0 t + 

 K0



BTred Cˆ Bred J0 dξ dη t    ◦

.

Kstab

In the case of the standard displacement formulation the reduced B-operation is equal to the hourglass part Bhg of B. For the classical enhanced strain method Bred −1 takes the more complex form Bred = Bhg − j0 Lenh KWW KWU . The matrices j, Lenh , KWW and KWU are e.g. defined in Reese [13]. Finally using the B-Bar method Bred represents the deviatoric part of Bhg . In all three cases Bred is a linear function of ξ and η . Obviously the first part of Ke is the same for all three finite element technologies.

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55

5 Eigenvalue Analysis Investigating the eigenvalues of the element stiffness matrix Ke it turns out that the eigenvalues of K0 are physically reasonable, which points to the fact that the constant part B0 of the B-operator is correct and does not contain any artefacts. The situation is different for Kstab , the so-called hourglass stabilization matrix. For the standard formulation its eigenvalues show the well-known effects of volumetric and shear locking, i.e. the eigenvalues grow to infinity in the limit of incompressibility and extreme aspect ratios of the elements. This is a non-physical situation which can be overcome by the enhanced strain method, thus by working with another Bred . In the latter case the eigenvalues of Kstab are bounded in all situations. On the other hand, the B-Bar method yields eigenvalues of Kstab which grow to infinity for very thin elements but remain bounded for very large bulk moduli. Obviously this method eliminates the locking only partially. In addition it can be shown easily that all eigenvalues remain always positive. The element is stable no matter which Bred is used. Obviously the situation becomes much more complicated in the non-linear modelling where the element stiffness matrix strongly depends on the deformation. This is discussed in the following.

6 Non-linear Finite Element Technology The additive split of ε is replaced by the additive split of the deformation gradient F = I + Hc + He where Hc denotes the displacement gradient Grad u and He the enhanced strain contribution (the incompatible part of F). Using the short hand notation δ¯ h = (δ Hc + δ He ) F−1 we can express the sum of the two equations of the two-field functional (see also Simo and Armero [18]) by means of  B

σ (F) · δ¯ h dv + gext = 0 .

(3)

As in the linear theory the interpolation of δ¯ h takes different forms depending on the finite element technology used. In the standard case δ¯ h = grad δ u holds. Thus δ¯ h is equal to the spatial gradient of δ u. The classical non-linear enhanced strain method means to choose the ξ ξ - and the ηη -components of He to be functions of ξ and η , respectively. These are the terms which are missing for the displacement interpolation to be complete up to quadratic order. The crucial idea of the B-Bar method is to choose the volumetric part of δ¯ h to be constant within the element. To achieve this we replace the determinant of F by the independent field Θ which represents the ratio of the current volume of the element divided by the original element volume. The variation δ¯ h finally reads δ¯ h = (1/3) (δΘ /Θ ) I + (grad δ u)D . Using the assumption J = J0 , as in the linear theory the element stiffness matrix Ke can be split into two parts:

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Ke = K0 +

BTred Cˆ Bred J0 dξ dη t    ◦

(4)

.

Kstab

This introduces an error into the formulation which, however, vanishes with increasing mesh density (see e.g. Arunakirinathar and Reddy [2], K¨ussner and Reddy [7]). It is crucial to understand that the matrix Cˆ depends in a strongly non-linear manner on the deformation as well as on the internal variables as for example the plastic strain. For this reason the status of the element changes continuously. Therefore one usually observes that in the undeformed state all eigenvalues are positive and take on physically reasonable values. However, during the deformation nonphysical situations might develop which are due to the construction of Kstab . The matrix K0 is equal for all three finite element technologies. Thus, in order to study the stability behaviour at the element the eigenvalues of Kstab have to be monitored. Certainly this investigation should be coupled with the global eigenvalue analysis which shows whether bifurcation points have been passed. The latter statement can be understood if one investigates the energy UeT (K0 + Khg )Ue ≈ UeT K0 Ue in the compression of a rubber block (see Fig. 1). Important are not the absolute numbers of these energy terms but the ratio between UeT K0 Ue and UeT Khg Ue . Obviously the above statement is even fulfilled for coarse meshes. The situation becomes even more clear if one looks at the same example computed by a fine mesh (Fig. 2). The result can be explained by the fact that with increasing number of elements the strain state in one element resembles more and more the situation in a material point. In other words, the strain state is approximately constant for fine meshes. This means on the other hand that the hourglass part in the energy UeT Ke Ue becomes negligible.

U_e^T K_0 U_e

U_e^T K_hg U_e

1.47E+00 2.00E-01

5.74E-02 1.00E-02

1.80E+03 3.60E+03

2.59E-01 5.08E-01

5.40E+03 7.20E+03

7.57E-01 1.01E+00

9.00E+03 1.08E+04

1.26E+00 1.50E+00

1.26E+04 1.44E+04

1.75E+00 2.00E+00

1.62E+04 1.80E+04

2.25E+00 2.50E+00

2.56E+05

4.92E+00

Time = 2.50E+01

Time = 2.50E+01

Fig. 1 a) energy UeT K0 Ue , range: 1.47 - 258000 Nm; b) energy UeT Khg Ue , range: 0.0674 4.92 Nm

Actually this statement does not only hold for the term UeT K Ue , but also for the corresponding expression written in terms of the element eigenvector contribution: ωelem = ϕ Te (K0 + Khg ) ϕ e ≈ ϕ Te K0 ϕ e . This gives us a clear criterion to differentiate between a physically reasonable and an artificial (hourglass) bifurcation.

Stability of Mixed Finite Element Formulations – A New Approach

U_e^T K_0 U_e

57

U_e^T K_hg U_e

7.71E-02 2.00E-01

3.74E-04 1.00E-02

1.80E+03 3.60E+03

2.59E-01 5.08E-01

5.40E+03 7.20E+03

7.57E-01 1.01E+00

9.00E+03 1.08E+04

1.26E+00 1.50E+00

1.26E+04 1.44E+04

1.75E+00 2.00E+00

1.62E+04 1.80E+04

2.25E+00 2.50E+00

8.39E+05

6.95E+00

Time = 2.50E+01

Time = 2.50E+01

Fig. 2 a) energy UeT K0 Ue , range: 0.013 - 1130000 Nm; b) energy UeT Khg Ue , range: 2.26·10−5 - 7.90 Nm

Increasing the number of elements the hourglass part of ωelem becomes negligible. Consequently, if a zero element eigenvalue is of physical nature it should be observed also for fine meshes, certainly corresponding to the same eigenmode the appearance of which should also have converged. This means that the physical stability behaviour is included in K0 and should be also exhibited by this part of the element stiffness matrix. Such a situation can be observed in uniaxial compression (see Fig. 3) where ϕ Te K0 ϕ e has positive and negative values which are distributed in such a way in the structure that the corresponding eigenmode can be clearly recognized. The sum over all elements yields ωglob = 0, i.e. a bifurcation point has been detected. The values of ϕ Te Khg ϕ e are much smaller, practically negligible. Thus, here the element stiffness part Khg does not influence the stability behaviour neither at element nor at global level.

_________________ S T R E S S 12

_________________ S T R E S S 13

-1.93E-02 -1.00E-04 -8.00E-05 -6.00E-05 -4.00E-05 -2.00E-05 5.55E-21

4.27E-14 1.00E-14 1.00E-05 2.00E-05 3.00E-05 4.00E-05 5.00E-05

2.00E-05 4.00E-05 6.00E-05 8.00E-05 1.00E-04

6.00E-05 7.00E-05 8.00E-05 9.00E-05 1.00E-04

3.10E-03

5.47E-04

Time = 9.40E+00

Time = 9.40E+00

Fig. 3 Rubber block under uniaxial compression: a) first relevant eigenmode; b) term ϕ Te K0 ϕ e ; c) term ϕ Te Khg ϕ e

The situation shown in Fig. 4 is different. It is clearly recognized that the physically unreasonable eigenmode (in uniaxial compression) is characterized by the fact that the term ϕ Te Khg ϕ e is negative in many elements. In contrast, the part ϕ Te K0 ϕ e is positive in the entire structure. Thus, this “physical” measure indicates that there is no physical bifurcation. The non-physical bifurcation is due to the fact that the absolute value of the negative ϕ Te K0 ϕ is large enough to lead to a vanishing zero global eigenvalue. This causes the artificial bifurcation point.

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_________________ S T R E S S 12

_________________ S T R E S S 13

2.55E-04

-5.21E-03

4.10E-04 5.65E-04

-1.00E-03 -8.00E-04

7.20E-04 8.75E-04 1.03E-03

-6.00E-04 -4.00E-04 -2.00E-04

1.19E-03 1.34E-03

5.55E-20 2.00E-04

1.50E-03

4.00E-04

1.65E-03 1.81E-03

6.00E-04 8.00E-04

1.96E-03 2.12E-03

1.00E-03 1.49E-03

Time = 7.10E+00

Time = 7.10E+00

Fig. 4 Rubber block under uniaxial compression: a) non-physical eigenmode; b) term ϕ Te K0 ϕ e ; c) term ϕ Te Khg ϕ e

To avoid such non-physical bifurcations the following requirements can be stated: • • • •

A negative element eigenvalue should not be caused by Khg Consequently, Khg has to be positive definite A sufficient condition for that is that Cˆ is positive definite This cannot be guaranteed because Cˆ includes both, material and geometrical non-linearity. Also in physical situations the matrix Cˆ loses its positive definiteness. • Thus, for the element technologies discussed in this paper, it cannot be guaranteed that non-physical bifurcation points do not occur. • It should be mentioned further that this observation does not have any relation to the LBB condition which is not fulfilled anyway.

The problem can be tackled in the following way. Due to the fact that in the present method of reduced integration plus hourglass stabilization the hourglass part is decoupled, we have the possibility to replace the matrix Cˆ which has the potential to lose its positive definiteness by a matrix Cˆmod whose positive definiteness is guaranteed. Additionally we take into account that the so-called hourglass strain (the difference between the strain within the element minus the strain in the centre of the element) is small and vanishes with increasing mesh density. Thus, we can say that for elastic problems Cˆmod represents the deviatoric part of the elasticity matrix. The volumetric part is omitted in order to eliminate volumetric locking. In the case of inelasticity one has to be aware of the fact that the hourglass strain is not decomposed into elastic and inelastic parts because the latter decomposition takes place only in the centre of the element. Thus we require that the product Cmod εhg (here represented in one dimension) yields the correct stress. The physical meaning of Cmod is indicated in Fig. 5 where the equivalent stress over the equivalent strain is plotted schematically. The results for a challenging example (extreme inhomogeneous compression of a rubber block) are shown in Fig. 6. The length of the rubber block is 4 mm, the height 1 mm. The reference loading is p0 = 0.2 N/mm2 . It acts on the middle 2 mm (thickness of the structure: 1 mm). The material is modelled by means of the Neo-Hooke model ( μ = 80.2 N/mm2 , Λ = 400889 N/mm2 ). In particular the coarse mesh is extremely distorted. In spite of that the computation does not break down. No artificial bifurcation points occur. The finite element technology developed in this work

Stability of Mixed Finite Element Formulations – A New Approach

59

Fig. 5 Schematic plot of the relation between equivalent stress and strain

proves to be very suitable for this example where many other finite element technologies fail.

Fig. 6 Inhomogeneous compression of a rubber block

References 1. Armero, F.: On the locking and stability of finite elements in finite deformation plane strain problems. Comput. Struct. 75, 261–290 (2000) 2. Arunakirinathar, K., Reddy, B.D.: Some geometrical results and estimates for quadrilateral finite elements. Comput. Method. Appl. M. 122, 307–314 (1995) 3. Belytschko, T., Ong, J.S., Liu, W.K., Kennedy, J.M.: Hourglass control in linear and non-linear problems. Comput. Method. Appl. M. 43, 251–276 (1984) 4. Doll, S., Schweizerhof, K., Hauptmann, R., Freischl¨ager, C.: On volumetric locking of low-order solid and solid-shell elements for finite elastoviscoplastic deformations and selective reduced integration. Eng. Computation: International Journal for ComputerAided Engineering 17, 874–902 (2000) 5. Dvorkin, E.N., Assanelli, A.P.: Implementation and stability analysis of the QMITCTLH elasto-plastic finite strain (2D) element formulation. Comput. Struct. 75, 305–312 (2000)

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6. Korelc, J., Wriggers, P.: Improved Enhanced Strain Four-Node Element with Taylor Expansion of the Shape Functions. Int. J. Numer. Meth. Eng. 40, 407–421 (1997) 7. K¨ussner, M., Reddy, B.D.: The equivalent parallelogram and parallelepiped, and their application to stabilized sinite elements in two and three dimenstions. Comput. Method. Appl. M. 190, 1967–1983 (2001) 8. Loehnert, S., Boerner, E.F.I., Rubin, M.B., Wriggers, P.: Response of a nonlinear elastic general Cosserat brick element in simulations typically exhibiting locking and hourglassing. Comput. Mech. 36, 255–265 (2005) 9. Malkus, D.S., Hughes, T.J.R.: Mixed finite element methods-reduced and selective integration techniques: a unification of concepts. Comput. Method. Appl. M. 15, 63–81 (1978) 10. Mueller-Hoeppe, D.S., Loehnert, S., Wriggers, P.: A finite deformation brick element with inhomogeneous mode enchancement. Int. J. Numer. Meth. Eng. 78, 1164–1187 (2008) 11. Nagtegaal, J.C., Parks, D.M., Rice, J.R.: On numerically accurate finite element solutions in the fully plastic range. Comput. Method Appl. M. 4, 153–178 (1974) 12. Pantuso, D., Bathe, K.-J.: On the stability of mixed finite elements in large strain analysis of incompressible solids. Finite Elem. Anal. Des. 28, 83–104 (1997) 13. Reese, S.: On a consistent hourglass stabilization technique to treat large inelastic deformations and thermomechanical coupling in plane strain problems. Int. J. Numer. Meth. Eng. 57, 1095–1127 (2003) 14. Reese, S.: On a physically stabilized one point finite element formulation for three- dimensional finite elasto-plasticity. Comput. Method. Appl. M. 194, 4685–4715 (2005) 15. Reese, S., Wriggers, P.: A stabilization technique to avoid hourglassing in finite elasticity. Int. J. Numer. Meth. Eng. 48, 79–109 (2000) 16. Reese, S., Wriggers, P., Reddy, B.D.: A new locking-free brick element technique for large deformation problems in finite elasticity. Comput. Struct. 75, 291–304 (2000) 17. Simo, J.C., Armero, F.: A class of mixed assumed strain methods and the method of incompatible modes. Int. J. Numer. Meth. Eng. 29, 1595–1638 (1990) 18. Simo, J.C., Armero, F.: Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes. Int. J. Numer. Meth. Eng. 33, 1413–1449 (1992) 19. Simo, J.C., Taylor, R.L., Pister, K.S.: Variational and projection methods for the volume constraint in finite deformation elasto-plasticity. Comput. Method. Appl. M. 51, 177–208 (1985) 20. Wriggers, P., Hueck, U.: A Formulation of the QS6 Element for Large Elastic Deformations. Int. J. Numer. Meth. Eng. 39, 1437–1454 (1996) 21. Wriggers, P., Reese, S.: A note on enhanced strain methods for large deformations. Comput. Method. Appl. M. 135, 201–209 (1996)

Chapter 8

A Finite Element Formulation based on the Theory of a Cosserat Point – Modification of the Torsional Modes Eiris F.I. Boerner, Dana Mueller-Hoeppe and Stefan Loehnert Peter Wriggers is an exceptional person to me in several ways: his broad interest in many different subjects, his ability to always come up with excellent ideas, his trust, his ability to teach and motivate and to give a huge amount of support and freedom to his students have always inspired me. He is a great mentor and I am glad and honoured to have been his PhD student. I wish him all the best (E.F.I. Boerner).

Abstract. The generalization of the Cosserat point element to include irregular initial element shapes for the three dimensional case is based on analytical solutions to inhomogeneous deformation modes. It is desired to continue to use the inhomogeneous deformation modes bending, torsion and higher order hourglassing that have been proposed within the context of the original Cosserat point element. However, it became clear that the torsional modes cannot be used as it was done for the original Cosserat point element posing the necessity to modify them. The reasons for the modification as well as the modified modes and results are presented in this paper.

1 Motivation The generalization of the original Cosserat point element proposed in [5] exhibits problems for the case of initially distorted element shapes that differ from rectangles in two or rectangular parallelepipeds in three dimensions. This has been shown in [4]. A remedy to this problem for the two dimensional plane strain case has been given in [2]. An approach similar to the solution for the two dimensional plane strain case can be used for the more general three dimensional case, however, problems Eiris F.I. Boerner · Dana Mueller-Hoeppe · Stefan Loehnert Institute of Continuum Mechanics, Leibniz Universit¨at Hannover, Appelstrasse 11, D-30167 Hannover, Germany e-mail: [email protected], [email protected],[email protected]

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occured with the torsional modes proposed for the original Cosserat point element. The difficulties will be explained and a remedy will be presented in the following.

2 A Brief Introduction to the Cosserat Point Element The general idea behind the Cosserat point element is the decoupling of the strain energy density function Ψ into a part related to homogeneous and another part related to inhomogeneous deformations. The decoupling is additive, it is given by

Ψ = Ψhom + Ψinh

(1)

.

2.1 Kinematics The kinematical approach chosen to achieve the decoupling is a mapping using director vectors according to 7

7

X = ∑ N i (θ )Di

x = ∑ N i (θ )di

,

i=0

i=0

7

,

u = ∑ N i (θ )δ i

(2)

,

i=0

where X and x denote the position vectors in the initial and current configuration, Di and di denote the director vectors in the initial and current configuration, u is the displacement vector and δ i denote the displacement director vectors. θ are the reference coordinates and N i finally denote the shape functions defined below. N0 = 1

,

N1 = θ 1 , N2 = θ 2 , N3 = θ 3 , N4 = θ 1θ 2 N 5 = θ 1 θ 3 , N 6 = θ 2θ 3 , N 7 = θ 1θ 2 θ 3

, (3)

In order to split the deformation into a part related to homogeneous and inhomogeneous deformations respectively, see Eq. (1), different deformation measures are introduced, starting with the so-called homogeneous deformation gradient Fhom and the homogeneous part of the right Cauchy-Green tensor Chom Fhom =

1 D1/2V



F dV

,

Chom = FThom · Fhom

(4)

,

P0

where D1/2V is the initial element volume. Additionally, the scalar valued inhomogeneous strain measures κij are defined by

κ11 = W β 1 · D1 κ12 = H β 1 · D2 κ13 = Lβ 1 · D3 κ21 = Lβ 2 · D1

κ22 = W β 2 · D2 κ23 = H β 2 · D3

κ31 = H β 3 · D1

κ32 = Lβ 3 · D2

(5)

κ33 = W β 3 · D3

κ41 = W Lβ 4 · D1 κ42 = HLβ 4 · D2 κ43 = HW β 4 · D3

,

CPE – Modification of the Torsional Modes

63

with Di , (i = 1, 2, 3) being the contravariant director vectors and with

β m = F−1 con · dm+3 − Dm+3

3

Fcon = ∑ di ⊗ Di

(m = 1, ..., 4) ,

(6)

.

i=1

The reference configuration is a rectangular parallelepiped with edge lengths H, W and L, the latter equal the approximated element dimensions as shown in Fig. 1. They are obtained according to 1 |X0 + X1 + X2 + X3 − X4 − X5 − X6 − X7 | 4 1 W = |X2 + X3 + X6 + X7 − X0 − X1 − X4 − X5 | 4 1 L = |X0 + X3 + X4 + X7 − X1 − X2 − X5 − X6 | 4 H=

(7) (8) (9)

,

where Xi denotes the nodal position of the ith node in the initial configuration, the respective node numbers are given in Fig. 1 below.

Fig. 1 Cosserat point reference element P2 in three dimensions

0

4 θ2

1

5 θ3

W

θ1

3

7 L

2 H

6

2.2 Equilibrium The weak form of equilibrium in the current configuration is used for the derivation of the discretized weak form of equilibrium for the CPE, here, it is given for one element. ⎛ ⎞   7 3  i ∂ N (10) ∑ η i · ⎝ ∑ ∂ θ j σ · g j dv − N i f dv − N i t da⎠ = 0 i=0 j=1 Pt Pt ∂P        t   ti

fi

mi

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Above, σ is the Cauchy stress tensor, g j denotes the contravariant director basis, is called director vector of internal forces, fi is called director vector of body forces and mi is called director vector of surface loads, such that the discretized weak form of equilibrium in the current configuration of one Cosserat point element reads ti

ti − fi − mi = 0

(i = 0, ..., 7) .

(11)

2.3 Constitutive Equations It can be shown that the definition of the homogeneous and the inhomogeneous part of the strain energy density function given below decouples the stresses and the resulting strain energies for the case of purely homogeneous and purely inhomogeneous deformations respectively.

Ψhom = Ψ (Chom ) ,

1 Ψinh = κ˜ T · K · κ˜ 2

(12)

Above, Ψ denotes any hyperelastic strain energy density function e.g. a Neo-Hooke or an Ogden material could be chosen. K is a symmetric 12 × 12 matrix containing the so-called inhomogeneous material constants and κ˜ T is defined by   κ˜ T = κ11 , κ12 , κ13 , κ21 , κ22 , κ23 , κ31 , κ32 , κ33 , κ41 , κ42 , κ43 . (13) This definition of Ψinh leads to a linear constitutive relation between the stresses and the deformations for the inhomogeneous part of the deformation. Since the inhomogeneous material constants in K have to reproduce the solution for small deformations, exact solutions from linear theory are used. It can be shown that twelve different linear independent inhomogeneous deformation modes have to be used for the determination of the inhomogeneous material constants. This corresponds to the number of eigenvalues of a three-dimensional 8-node brick element that are not related to rigid body motions or homogeneous deformations. In [5], analytical solutions to six bending modes, three torsion modes and three modes describing higher order hourglassing are used, one of each is examplarily depicted in Fig. 2.

Fig. 2 Examples for the different inhomogeneous deformation modes: bending, torsion and higher order hourglassing

CPE – Modification of the Torsional Modes

65

The analytical solutions for all these modes are given e.g. in [1] or [5]. They are linear boundary value problems. It can be proven that any linear combination of linear independent inhomogeneous deformation modes is suitable for the determination of the inhomogeneous material constants in K . The analytical solutions used in [5] are valid for a rectangular parallelepiped shaped geometry. For the determination of the inhomogeneous material constants the analytical solutions for the displacement field u∗ and for the stress field σ ∗ are given for each of the twelve inhomogeneous deformation modes. In the following, ∗ indicates analytical solutions. For each of the modes, the analytical solution for the displacements u∗ is used for the determination of the director vectors of internal forces ti , while the director related vectors of surface loads mi employ the analytical solutions for the stresses σ ∗ . The analytical solutions are valid for the case of linear elasticity, such that the displacement director vectors δ i based on u∗ as well as the analytical solutions for the stresses σ ∗ are inserted in the linear Cosserat strain and stress measures. In the following, linear measures are indicated by the subscript LIN . The inhomogeneous material constants are then solved from the equilibrium equation according to tiLIN (K , u∗ ) = miLIN (σ ∗ ) (i = 1, ..., 7) .

(14)

3 Torsion The central problem that results in using a different approach for torsion is related to the warping functions φi , (i = 1, 2, 3). These warping functions are part of the analytical solution for the three torsion modes. They appear in the analytical displacement as well as the analytical stress field and are given e.g. in [1], [5] or [6]. The warping functions become zero for cubic elements and therefore yield a singular equation system for the determination of the inhomogeneous material constants for cubic elements. Figure 3 depicts the corresponding deformed meshes. The respective undeformed mesh is a cube. It can be recognized from Figure 3 that the negative torsion mode 1 on the left is the sum of mode 2 and mode 3, and hence a linear combination of those modes. This singularity becomes clear, when a closer look is taken at the analytical solution for the inhomogeneous deformation

ω1 = −0.2

ω2 = 0.2 Fig. 3 Deformed meshes torsion mode, H = W = L

ω3 = 0.2

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measures κi,LIN for torsion for a cubic element, hence H = W = L = Hc and φi = 0, (i = 1, 2, 3). Using the relation between the displacement director vectors and the analytical displacements for cubic elements given e.g. in [1] or [5], one obtains j

κ˜ ω1 ,LIN = Hc (0 0

ω 1 0 −ω 1 0

κ˜ ω2 ,LIN = Hc (0 0 −ω2 0 κ˜ ω3 ,LIN = Hc (0 0

00

00

0 0 0 0 0 0)T

ω2 0 0 0 0 0)T

.

ω3 0 −ω3 0 0 0 0 0)

(15)

T

From here, it is easy to see that the three torsion modes are linear dependent: since a linear approach is used for the determination of the inhomogeneous material constants, the magnitude of the mode ωi as well as the dimension Hc cancel out. Evaluation of the corresponding director vectors of internal forces ti∗ LIN and surface loads mi∗ will yield the same linear dependence, thereby causing difficulties for an evalLIN uation of the inhomogeneous material constants due to a singular equation system. In addition to the singular equation system obtained for cubic elements, no standard integration scheme can be applied for the warping functions in general as they are constructed by an infinite series of trigonometric functions describing a highly non-smooth function for which Gauss point integration is no longer suitable. These fundamental problems associated with the inhomogeneous torsion modes call for substitution of the torsion modes by a set of other appropriate modes. The only requirement for these substitution modes is linear independence of the bending and higher order hourglassing modes. Motivated by the analytical solution for torsion, the following displacement field u∗ is chosen for the substitution of the original one. The corresponding analytical Cauchy stress tensor σ ∗ is obtained from u∗ using the linearized strain measure ε and the linear Cauchy stress tensor σ LIN given by ( & ' ) 1 ∂u ∂u T ε= + , σ LIN = 2 με + Λ tr(ε )1 , (16) 2 ∂X ∂X where μ and Λ are Lam´e constants, such that σ ∗ can be obtained from u∗ according to ⎛ ⎞ ⎛ ⎞ ω3 ηˆ ζˆ 0 μ (ω3 + ω2 )ζˆ μ (ω1 + ω3 )ηˆ ⎜ ⎟ ⎜ ⎟ u∗ = ⎝ ω2 ξˆ ζˆ ⎠ , σ ∗ = ⎝ μ (ω3 + ω2)ζˆ 0 μ (ω2 + ω1 )ξˆ ⎠ .

ω1 ξˆ ηˆ

μ (ω1 + ω3 )ηˆ μ (ω2 + ω1 )ξˆ

0

(17) It is straightforward to prove the linear independence of u∗ above from the other inhomogeneous deformation modes for bending and higher order hourglassing. A modified solution for the torsion modes has also been presented in [3]. This solution provides a good approximation of the deformation field in interior elements of a mesh modeling pure torsion of a right cylindrical bar with a rectangular cross-section, however, the approach in [3] still contains the warping functions φi , (i = 1, 2, 3). Therefore, this solution cannot be used to obtain the inhomogeneous material constants numerically which is necessary for the development of an

CPE – Modification of the Torsional Modes

67

improved Cosserat point element capable of handling initial element shapes differing from rectangular parallelepipeds. The difference of the inhomogeneous material constants K18 given in [5] and the modified torsion modes presented above can best be observed by means of an eigenvalue analysis of the inhomogeneous constant matrix K . Therefore, a cube using the inhomogeneous material constants K18 from [5] on one hand and the modified torsion modes given in Eq. (17) together with the original modes for bending and higher order hourglassing on the other hand is examined. Figure 4 shows the model and also gives the material properties used here.

w

h

Geometry

Material

l = 2 mm

K = 1 000 MPa

w = 2 mm

μ = 600 MPa μ Ψ = (IC˜ − 3) + K (J − 1 − ln J) 2

h = 2 mm l

Fig. 4 Model eigenvalue analysis of K using different approaches for torsion Table 1 Results eigenvalue analysis of K using different approaches for torsion Torsion original (K18 ) Torsion modified (K ) 1.6869E+02 1.6667E+02 1.6667E+02 1.6667E+02 1.0000E+02 1.0000E+02 1.0000E+02 4.2173E+01 4.2173E+01 1.7500E+01 1.7500E+01 1.7500E+01

2.0000E+02 1.6667E+02 1.6667E+02 1.6667E+02 1.0000E+02 1.0000E+02 1.0000E+02 5.0000E+01 5.0000E+01 1.7500E+01 1.7500E+01 1.7500E+01

Table 1 shows the results of the eigenvalue analysis of K for both approaches. As expected, three eigenvalues, corresponding to the number of modified torsion modes, differ in Table 1. In this specific example, the eigenvalues obtained from K18 are slightly lower than the ones obtained by use of the modified solution for torsion. Hence, the new torsion modes result in a stiffer inhomogeneous material constant matrix K . Although this might be undesirable in the context of locking, it

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is unfortunately by no means possible to neither numerically nor analytically determine the inhomogeneous material constants for torsion in K18 . This is the case due to the fact that the inhomogeneous material constants in [5] are simply defined rather than obtained by means of the corresponding analytical solution. Their definition is based on the so-called Bubnov - Galerkin inhomogeneous material constants for torsion, which describe a complete trilinear approximation and hence a Q1 - element formulated within the Cosserat point element approach. These Bubnov - Galerkin constants for torsion are multiplied with a correction factor b, which appears in the exact analytical solution for torsion but is evaluated for a cubic reference element in order to obtain K18 . The inhomogeneous material constants based on the modified torsion mode given in Eq. (17), however, are not only close to those given in [5] for K18 , but they are truly based on an analytical solution which is necessary for the generalization of the Cosserat point element and which is also more desirable in the context of a consistent formulation.

4 Conclusions A modified solution for torsion for the Cosserat point element replacing the analytical solution for torsion of a rectangular parallelepiped as e.g. given in [6] is presented. The reason is the singularity that occurs when the original analytical solution for torsion is used for the determination of the inhomogeneous material constants for a cubic Cosserat point element. This is highly undesirable in the context of generalizing the Cosserat point element to handle irregular shaped initial elements. The modified solution for torsion is linear independent from the other nine inhomogeneous deformation modes used for the determination of K , and, as intended, it can be used for the analytical and numerical determination of the inhomogeneous material constants for the generalized Cosserat point element.

References 1. Boerner, E.F.I.: A finite element formulation based on the theory of a Cosserat point. Dissertation, Institute of Continuum Mechanics, Leibniz University of Hannover, Germany (2008) 2. Boerner, E.F.I., Loehnert, S., Wriggers, P.: A new finite element based on the theory of a Cosserat point - extension to initially distorted elements for 2D plane strain. Int. J. Numer. Meth. Eng. 71, 454–472 (2007) 3. Jabareen, M., Rubin, M.B.: Modified torsion coefficients for a 3-D brick Cosserat point element. Comput. Mech. 41, 517–525 (2008) 4. Loehnert, S., Boerner, E.F.I., Rubin, M.B., Wriggers, P.: Response of a nonlinear elastic general Cosserat brick element in simulations typically exhibiting locking and hourglassing. Comput. Mech. 36, 266–288 (2005) 5. Nadler, B., Rubin, M.B.: A new 3-D finite element for nonlinear elasticity using the theory of a Cosserat point. Int. J. Solids Struct. 40, 4585–4614 (2003) 6. Sokolnikoff, I.S.: Mathematical Theory of Elasticity, 2nd edn. McGraw-Hill, New York (1956)

Chapter 9

A Brick Element for Finite Deformations with Inhomogeneous Mode Enhancement Dana Mueller-Hoeppe and Stefan Loehnert

Professor Wriggers has always inspired me and being his student was the main reason for me to turn towards computational mechanics. His continuous support of my goals and projects is matchless. I am glad and honored to be his PhD student now. Congratulations and many more successful, healthy and happy years to come! (D. Mueller-Hoeppe).

Abstract. The basic idea of this finite deformation enhanced assumed strain element is to split the element deformation into a homogeneous and an inhomogeneous part. The enhancement is only applied to the inhomogeneous part which is treated as linear elastic, while a compressible Neo-Hooke material is used for the homogeneous part. To this end, the deformation gradient is multiplicatively split into its homogeneous and inhomogeneous part. Thus, the approach ensures objectivity of the element formulation and can be interpreted as a linear inhomogeneous deformation superposed by a finite homogeneous deformation. Examples show that the element is locking and hourglassing free and performs well for initially distorted meshes.

1 Introduction In the last two decades, the development of enhanced assumed strain (EAS) elements has played a major role in the improvement of finite element formulations such that they do not show locking behavior in bending dominated problems or for incompressible materials [10, 12]. The element presented in [10] was extended for the finite deformation case in [8], while [9] proposed a modified finite deformation EAS element (QM1/E12) showing improved behavior for initially distorted elements. Many different enhancements for small and finite deformation problems were studied by [4, 13]. Dana Mueller-Hoeppe · Stefan Loehnert Institute of Continuum Mechanics, Leibniz Univesit¨at Hannover, Appelstr. 11, D-30167 Hannover, Germany e-mail: {mueller,loehnert}@ikm.uni-hannover.de

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Recently, [2] developed a set of mixed formulations including a parameter to modify them for linear as well as finite deformations. This finite element works well for bending-dominated problems or nearly incompressible materials depending on the choice of the parameter. Another recent approach proposed by [3] is to use the bending energy for the construction of the enhanced modes and thus achieving the exact strain energy for a rectangular parallelepiped element. This method is a rough approximation for initially distorted elements. Although most of these finite element formulations perform well for many applications, some of them show hourglassing under circumstances like e.g. large compression. An element that does not show this behavior is the Cosserat point element introduced by [7]. However, as shown in [5], this element does not perform well for initially distorted elements. A remedy to this problem is presented in [1] for 2D plane strain. In the following, we present an element for which the idea of splitting the strain energy density into its homogeneous and inhomogeneous part as in the Cosserat point element is used. The inhomogeneous part of the displacement gradient is enhanced. If three enhanced modes are used, the element is denoted by Q1/EI9, if four modes are used, it is denoted by Q1/EI12.

2 Theoretical Background A domain Ω and its boundary ∂ Ω as shown in figure 1 are considered, with the equilibrium equation for a body subjected to body forces ρ0 b given by Div(P) + ρ0b = 0 .

(1)

Here, P denotes the first Piola-Kirchhoff stress tensor. The displacement boundary conditions on ∂ Ωu and the traction boundary conditions on ∂ Ωt are prescribed by u = u∗ on ∂ Ω u P · N = t∗ on ∂ Ωt

(2) (3)

where N is the outward unit normal in the initial configuration.

Ωt N

∂Ω Ω

Fig. 1 Domain

Ωu

A Brick Element for Finite Deformations

71

2.1 Enhanced Strain Assumption As mentioned above, the basis of the presented element formulation is to pick up the idea of [7] and additively decompose the element deformation into its homogeneous and inhomogeneous part. Enhancement is only applied to the inhomogeneous part, as it is responsible for locking behavior. To this end, the total deformation gradient F is multiplicatively split into its homogeneous and inhomogeneous part * + F = F¯ · F

(4)

with the volume average of the deformation gradient given by 1 F¯ = V

 Ω

F(x) dΩ

(5)

,

where V is the element volume in the initial configuration and x = X + u is the material point in the deformed configuration. In contrast to an additive decomposition, this multiplicative split guarantees objectivity of the element formulation. Analoguously to [7], the strain energy density function is additively split into a homogeneous and an inhomogeneous part, * + ¯ +Winh(F) W (F) = Whom (F) 2.1.1

(6)

.

Homogeneous Part of the Element Strain Energy

A compressible Neo-Hooke material model in terms of shear modulus μ , bulk modulus K and a parameter β accounting for different types of volumetric strain energy density functions is used for the homogeneous part of the deformation,     ¯ − 3 + K J¯−β − 1 + β ln(J) ¯ = μ J¯−2/3 tr(C) ¯ Whom (F) 2 β2

.

(7)

Here, the overall deformation measures are replaced by their homogeneous part ¯ J¯ = det(F) T ¯ = F¯ F¯ . C 2.1.2

(8) (9)

Inhomogeneous Part of the Element Strain Energy

The strain energy density function for the inhomogeneous part of the element deformation is given by the linear relation & ' & ' & ' 1 * * * + + + Winh F = F−1 : C : F−1 (10) 2

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with a constant elasticity tensor & ' 2 Ciklm = K − μ δik δlm + μ (δil δkm + δim δkl ) 3

.

(11)

The inhomogeneous part of the deformation gradient is enhanced such that * +=F + + H, * F

+ = F¯ −1 · F(x) F

(12)

.

* is given in A more detailed discussion of the enhanced displacement gradient H Sect. 3.

2.2 Variational Formulation As a variational principle, the Hu-Washizu principle in terms of the material point * and the first in the deformed configuration x, the enhanced displacement gradient H Piola-Kirchhoff stress tensor P is used, & '      * * + * ¯ ¯ Π (x, H, P) = Whom F + Winh F − P : F · H dΩ − Pext , (13) Ω

where Pext are the applied loads. Assuming the element stress P to be constant, variation of Eq. (13) yields 

∂ Whom δ F¯ : dΩ + ∂ F¯ Ω

 Ω

∂ Winh dΩ − δ Pext = 0 * + ∂F  * : ∂ Winh dΩ = 0 . δH * Ω + ∂F

+: δF

(14) (15)

3 Discretization For the presented element formulation, an eight-node brick element as depicted in Fig. 2 with standard trilinear isoparametric shape functions NI is used, 1 NI (ξ , η , ζ ) = (1 + ξ ξI )(1 + ηηI )(1 + ζ ζI ) , 8

(16)

the coordinates of node I in the reference element being denoted by ξI , ηI and ζI . * are given by three The ansatz functions for the enhanced displacement gradient H quadratic functions M1 , M2 and M3 as introduced in [12] for the Q1/EI9 element. For the Q1/EI12 element, the volumetric function M4 proposed in [9] is used in addition. M1 = (1 − ξ 2 ),

M2 = (1 − η 2 ),

M3 = (1 − ζ 2),

M4 = ξ ηζ

(17)

A Brick Element for Finite Deformations

73

η X2

Fig. 2 Eight-node brick element

ζ

ξ

X1

X3

4 Numerical Examples In the following, the proposed element formulations are tested in terms of accuracy, mesh distortion sensitivity and locking free response to bending dominated and near incompressibility problems. Additionally, the elements should be hourglassing free. The Q1/EI9 and Q1/EI12 element are compared with the standard triquadratic element (Q2) as well as the QM1/E12 element [9] and the mixed Q1P0 element for finite deformations [11] which is especially suited for incompressible materials. It is shown that the Q1/EI9 and Q1/EI12 element fulfill the patch test. Also, the response is independent of superposed rigid body rotations, thus showing objectivity of the element formulation [6]. Of the numerical examples presented in [6], a selection corresponding most closely to the properties demanded above is introduced in the following.

4.1 Irregularly Meshed Beam To examine the performance of the elements for bending dominated problems as well as initially distorted meshes, a shear load is applied to the free end of a beam as shown in Fig. 3, where also the geometry and material data is given and the boundary conditions are indicated. The vertical deflection vP of point P is studied for the Q1/EI9, Q1/EI12, Q2 and QM1/E12 element. The deformed configuration also showing the initial element distortion is depicted in Fig. 4.

F

w

x3 x2 x1

P

h F

a l

Geometry l = 10 mm h = 2 mm w = 1 mm a = 3 mm

Material K = 1000 MPa μ = 600 MPa β = −2

Load F = 6N

Fig. 3 Irregularly meshed beam: system, load and material data

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Fig. 4 Irregularly meshed beam: deformed configuration using the Q1/EI9 element

vP [mm]

1.03

Fig. 5 Irregularly meshed beam: displacement vP for the Q1/EI9, Q1/EI12, Q2 and QM1/E12 element

1.02 Q1/EI9 Q1/EI12 Q2 QM1/E12

1.01 1000

10000

100000

dof

The deflection vP w.r.t. the degrees of freedom is plotted in Fig. 5. All enhanced elements converge faster than the Q2 element. Not surprisingly, the Q1/EI9 and Q1/EI12 element yield almost identical results, as the additonal volumetric mode of the Q1/EI12 element does not have a strong influence in this bending dominated test.

4.2 Nearly Incompressible Block A nearly incompressible block consisting of a rubberlike material is loaded with an equally distributed surface load at its center as shown in Fig. 6, where also the geometry and material data is given and the boundary conditions are indicated. Due to symmetry reasons, only a quarter of the block is discretized. The deformed configuration is shown in Fig. 7. In Fig. 8, the convergence of the vertical deflection wP of the center point P is studied for the Q1/EI9, Q1/EI12, Q2, Q1P0 and QM1/E12 element. All enhanced strain elements and the Q1P0 element show a softer response than the Q2 element.

A Brick Element for Finite Deformations

a

q

75

Geometry h = 50 mm w = 100 mm l = 100 mm a = 25 mm b = 25 mm

b P h

l

Material K = 501 MPa μ = 1.61148 MPa β = −2

Load q = 3 MPa

w x3 x2

x1

Fig. 6 Nearly incompressible block: system, load and material data

Fig. 7 Nearly incompressible block: deformed configuration using the Q1/EI9 element

wP [mm]

20

19.5 Q1/EI9 Q1/EI12 Q2 Q1P0 QM1/E12

19

Fig. 8 Nearly incompressible block: displacement wP for the Q1/EI9, Q1/EI12, Q2, Q1P0 and QM1/E12 element

18.5 1000

10000 dof

100000

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Here, the additional volumetric mode used for the Q1/EI12 leads to a significantly better convergence compared to the Q1/EI9 element. The QM1/E12 element shows unphysical hourglassing behavior for all but the two coarsest meshes und thus fails to converge. Robustness is tested by comparing the minimum number of equally sized load steps necessary for convergence. It can be shown that the Q1/EI9 and Q1/EI12 element are as robust as the Q2 element [6].

5 Conclusions A new enhanced assumed strain formulation with a split of the strain energy density function into its homogeneous and inhomogeneous part is presented. Enhancement is only applied to the inhomogeneous part of the deformation, for which linear material behavior is assumed. For the homogeneous part of the deformation, a compressible Neo-Hooke material is used. As shown by a selection of numerical tests from [6], this leads to a very robust finite element formulation not showing hourglassing under circumstances where many other EAS formulations show unphysical behavior. Also, the element yields accurate results for coarse and distorted meshes and for bending dominated as well as nearly incompressible problems. Due to the split of the strain energy density function, the extension of the element formulation to inelastic material models is not straightforward.

References 1. Boerner, E.F.I., Loehnert, S., Wriggers, P.: A new finite element based on the theory of a Cosserat point – Extension to initially distorted elements for 3d plane strain. Int. J. Numer. Meth. Eng. 71, 454–472 (2007) 2. Chavan, K.S., Lamichhane, B.P., Wohlmuth, B.I.: Locking-free finite element methods for linear and nonlinear elasticity in 2d and 3d. Comput. Method. Appl. M. 196, 4075–4086 (2007) 3. Fredriksson, M., Ottosen, N.S.: Accurate eight-node hexahedral element. Int. J. Numer. Meth. Eng. 72, 631–657 (2007) 4. Korelc, J., Wriggers, P.: Consistent gradient formulation for a stable enhanced strain method for large deformations. Eng. Computation 13, 103–123 (1996) 5. Loehnert, S., Boerner, E.F.I., Rubin, M.B., Wriggers, P.: Response of a nonlinear elastic general Cosserat brick element in simulations typically exhibiting locking and hourglassing. Comput. Mech. 36, 266–288 (2005) 6. Mueller-Hoeppe, D.S., Loehnert, S., Wriggers, P.: A finite deformation brick element with inhomogeneous mode enhancement. Int. J. Numer. Meth. Eng. 78, 1164–1187 (2009) 7. Nadler, B., Rubin, M.B.: A new 3-d finite element for nonlinear elasticity using the theory of a Cosserat point. Int. J. Solids Struct. 40, 4585–4614 (2003) 8. Simo, J.C., Armero, F.C.: Geometrically nonlinear enhanced strain mixed methods and the method of incompatible modes. Int. J. Numer. Meth. Eng. 33, 1413–1449 (1992)

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9. Simo, J.C., Armero, F.C., Taylor, R.L.: Improved versions of assumed enhanced strain trilinear elements for 3d finite deformation problems. Comput. Method. Appl. M. 110, 359–386 (1993) 10. Simo, J.C., Rifai, M.S.: A class of mixed assumed strain methods and the method of incompatible modes. Int. J. Numer. Meth. Eng. 29, 1595–1638 (1990) 11. Simo, J.C., Taylor, R.L., Pister, K.S.: Variational and projection methods for the volume constraint in finite deformation elasto-plasticity. Comput. Method. Appl. M. 51, 177–208 (1985) 12. Wilson, E.L., Taylor, R.L., Doherty, W.P., Ghaboussi, J.: Incompatible displacement models. In: Fenves, S.J., Perrone, N., Robinson, A.R., Schnobrich, W.C. (eds.) Numerical and Computer Models in Structural Mechanics. Academic Press, New York (1973) 13. Wriggers, P., Korelc, J.: On enhanced strain methods for small and finite deformations of solids. Comput. Mech. 18, 413–428 (1996)

Chapter 10

Automatic Differentiation Based Formulation of Computational Models Joˇze Korelc Dedicated to Professor Peter Wriggers honoring his contribution to innovative computational methods (J. Korelc).

Abstract. The approach to automation of computational modeling is presented. It is shown that the unification of the classical mathematical notation of computational models and the actual computer implementation can be achieved by means of extended automatic differentiation technique combined with automatic code generation. The paper presents automatic differentiation based form (ADB form) of a classical mathematical notation of solid and contact mechanics and stability analysis.

1 Introduction The use of advanced software technologies plays a central role in the process that leads to the ultimate goal, i.e. a complete automation of computational modeling. The problem of automation of computational methods has been explored by researches from the fields of mathematics, computer science and computational mechanics, resulting in a variety of approaches (e.g. a hybrid object-oriented approach [1] and a hybrid symbolic-numeric approach [4]) and available software tools (e.g. computer algebra systems, AD tools [2], problem solving environments and numerical libraries). Automation can address all steps of the finite element solution procedure from the strong form of the boundary-value problem to the presentation of results [8], but more often it is used only for the automation of selected steps of the whole procedure. Yet, the true advantages of automation become apparent only if the description of the problem, the notation and the mathematical apparatus used are changed as well. It is demonstrated in the paper that this can be achieved using the AD technique. Thus, the basis for the automation of computational modeling is Joˇze Korelc University of Ljubljana, Faculty of Civil and Geodetic Engineering, Jamova 2, Ljubljana, Slovenia e-mail: [email protected]

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an automatic differentiation based form or ADB form of basic equations used to describe the problem. The introduced notation does not only simplify the derivation of the corresponding equations, but also reflects much more closely the actual algorithmic implementation. In this way, the mathematical formulation and computer implementation become indistinguishable. The paper presents the ADB form of a classical mathematical notation of solid mechanics, contact mechanics and stability analysis. In the actual implementation of the described methodology a general-purpose automatic code generator AceGen [3] was used to derive and code characteristic finite elements quantities (e.g residual vector and stiffness matrix) at the level of individual finite elements and a general-purpose finite element environment to solve the global problem. The automatic differentiation is briefly described in Section 2, followed by a general discussion about the use of AD in mechanics of solids in Section 3. In Section 4 the ADB form of the classical computational models is presented.

2 Automatic Differentiation The automatic differentiation technique is based on the fact that every computer program executes a sequence of elementary operations with known derivatives, thus allowing the evaluation of exact derivatives via the chain rule for an arbitrary complex formulation [2]. The details of the approach can be found in [5], and only a brief overview is given here. Let a be a set of mutually independent variables and f an arbitrary function of a. The “computational derivative” is then defined with the following formalism δˆ f (a) . (1) δˆ a , The above formalism has to be taken in an algorithmic way. The operator δˆ f (a) δˆ a represents differentiation of function f with respect to variables a performed by the AD algorithm. Thus, in the context of the paper the operator has dual purpose, i.e. to indicate the mathematical operation of differentiation as well as to indicate the algorithm used to obtain the required quantity. It also indicates that, in the process of derivation, a software tool for AD has been called. However, as powerful as AD technique is, the result of AD procedure might not automatically correspond to the mathematical formalisms (total derivative, partial derivative, consistent derivative, directional derivatives, etc.) used to describe the problem. The unification can be achieved by introducing additional information to the process of AD that will define exceptions within the AD procedure and modify the standard chain rule in a way that the result corresponds to the required mathematical formalism. Corresponding extended functionality of the traditional AD procedure called “automatic differentiation exceptions” has been introduced in [5]. Let b be a set of mutually independent intermediate variables that are part of evaluation of function f , G a set of arbitrary functions of a such that b := G(a) , and M an arbitrary matrix. The definition of AD exception is indicated by the following formalism

Automatic Differentiation Based Formulation of Computational Models

 δˆ f (a, b(a))  ∇ fA :=   Db δˆ a

.

81

(2)

Da =M

Notation (2) indicates that during the AD procedure the total derivatives of intermediate variables b with respect to independent variables a are set to be equal to matrix M. Thus, the “true” way how variables b are evaluated is neglected for the evaluation of derivatives of b. A special case is when the intermediate variables b algorithmically depend on a, but for some reason they have to be kept constant. This can be achieved by setting matrix M in (2) to zero as follows  δˆ f (a, b(a))  ∇ fB := . (3)   Db δˆ a Da =0

In this case, the direct use of AD would not give the correct results without the user intervention.

3 Automatic Differentiation in Computational Mechanics The AD tools were primarily developed for the evaluation of the gradient of objective function used within the gradient-based optimization procedures. Large finite element environments usually employ a large variety of finite elements, solution procedures, and they commonly use commercial numerical libraries. It would be difficult in such a case to directly apply the AD tools to get e.g. the global stiffness matrix of a large-scale problem. However, the AD technology can still be used for the evaluation of specific quantities that appear as a part of FE simulation. For example, one can use AD at the individual element level to evaluate element specific quantities such as strain and stress tensors, nonlinear coordinate transformations, residual vector, a consistent stiffness matrix and sensitivity pseudo-load vectors. The formulation of a computational model where all the derivatives are replaced by the corresponding computational derivatives is called the ADB form of a computational model. There are two approaches for the implementation of the AD often called the forward and the backward mode of AD, see e.g. Griewank [2]. The numerical efficiency of the differentiation of N scalar-valued functions F = { fi }, i = 1, 2, ..., N with respect to M independent variables a = {a j }, j = 1, 2, ..., M can,be measured by the numerical work ratio defined as wratio(F(a)) = cost(F(a), ∂∂ Fa ) cost(F(a)). The ratio is in general proportional to the number of independent variables (wratio(F(a)) ∝ M) in case of the forward mode and proportional to the number of functions (wratio(F(a)) ∝ N) in case of the backward mode. Consequently, the automatically generated code is numerically efficient, if the number of functions to differentiate and the number of calls to the AD procedure are kept to a minimum. One of the consequences of this rule is that, in general, formulations where the element residual vector is derived as a gradient of a scalar function, e.g. the variational potential,

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lead to a more efficient numerical code than those based on the weak form of the equilibrium equations. Thus, the possibility of transforming the weak form into the pseudo-potential scalar function is worth exploring.

4 Automatic Differentiation Based Computational Models 4.1 ADB Form of Hyperelastic Models In case of hyperelastic material responses, a principle of stationary elastic potential can be formulated. Within this formulation, the functional of the strain energy density function W can be formulated as a function of Np generalized displacement parameters pe of the element. The Gauss point contribution of the individual element Rg to the global residual is then obtained by automatic differentiation of the scalar W with respect to the displacement variables pe and integrated over the element domain by a numerical integration rule as follows Rg :=

δˆ W (pe ) δˆ p

(4)

e

and the Gauss point contribution of the individual element Kg to the global tangent matrix is obtained by automatic differentiation of Rg with respect to the displacement variables pe and given by KT g :=

δˆ Rg (pe ) δˆ p

.

(5)

e

The cost of element residual evaluation is cost(Re ) ≈ Ng cost(W ) and the cost of tangent matrix evaluation is cost(Ke ) ≈ Ng Np cost(W ), where Ng is the number of integration points.

4.2 ADB Form of Elasto-plastic Models The elasto-plastic problem is defined by a hyperelastic strain energy density function W , a yield condition f and a set of algebraic constraints Qg,n+1 (hg,n+1 ) = 0 to be fulfilled at Gauss point level when the material point is in plastic state. The vector of local unknowns hg,n+1 is composed of an appropriate measure of plastic strains, hardening variables and the consistency parameter λ , while Qg,n+1 are composed of the corresponding set of discretized evolution equations that describe the evolution of plastic strains and hardening variables and the consistency condition f = 0. The ADB form of the contribution of the internal forces to the weak form of equilibrium equations is for the 1. Piola-Kirchhoff stress tensor P and the deformation gradient F given by δˆ F Rg := P · , (6) δˆ p e,n+1

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where the evaluation of the stress vector depends on the elastic or plastic state of material point as follows f (τ trial ) < 0 elastic

P :=

δˆ W (F,hgn+1 ,hgn ) δˆ F



f (τ trial ) ≥

0 plastic P :=

δˆ W (F,hgn+1 ,hgn )   Dhgn+1 δˆ F DF =0

.

(7)

The AD exception Dhgn+1 DF = 0 in (7) prevents an implicit algorithmic dependency of hg,n+1 on pe,n+1 introduced by the solution of Qg,n+1 (hg,n+1 ) = 0. The direct application of the automatic differentiation procedure to obtain P would consider this algorithmic dependency and the evaluated stress tensor would not be correct. The weak form of the equilibrium equation is less appropriate for evaluation using the AD, thus it should be transformed into the pseudo-potential. Since (7)a implies (7)b, the “basic equation of symbolic plasticity” follows from (6) and leads to  δˆ W  Rg := . (8)  δˆ pe,n+1  Dhg,n+1 =0 Dpe,n+1

The evaluation of the consistent tangent is then obtained by AD of residual Rg as follows  δˆ Rg  KT g := (9) ( )−1  δˆ Qgn+1 δˆ Qg,n+1 δˆ pe,n+1  Dhg,n+1 DF

where AD exception

Dhg,n+1 DF

& =−

δˆ Qgn+1 δˆ h gn+1

=−

'−1

δˆ hgn+1

δˆ Qg,n+1 δˆ F

δˆ F

defines an implicit depen-

dency of hg,n+1 on pe,n+1 . More details about the procedure can be found in [5].

4.3 Numerical Efficiency of ADB Form The standard 2D, bi-linear, quadrilateral, plane strain, isoparametric element (Q1) and its 3D equivalent (H1) are derived following the procedures described in previous sections and analyzed for four different cases: linear elasticity, hyperelasticity, small strain elasto-plasticity and finite strain elasto-plasticity. In Table 1 the size of the generated user-subroutine code and the time needed for the numerical evaluation of Re and KTe are compared. The presented comparison is done for the example where the rectangular bar is stretched, thus all Gauss points are either in elastic or in plastic state. The results are normalized with respect to the linear elastic Q1 element. The code size is in the range from 9 to 105 Kbytes and the generation of the H1 finite strain elasto-plastic element on a 2GHz PC takes approximately 200 seconds with AceGen code generator. Thus, both the code size and the derivation time are small enough to allow “real time” automatic derivation of complex nonlinear finite elements.

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Constitutive model

Q1 Q1 Q1 Q1 H1 H1 H1 H1

linear elastic hyperelastic small strain, elasto-plastic finite strain, elasto-plastic linear elastic hyperelastic small strain, elasto-plastic finite strain, elasto-plastic

Code size Evaluation time AceGen time (Kbytes) (normalized) (normalized) 9 1 1 9 1.6 1.3 24 3.0 7.4 48 9.5 25 18 6.1 4.2 21 9.4 4.5 46 13.3 23.2 105 41.8 69.0

4.4 ADB Form of Contact Formulations The automation of slave node - master segment penalty contact formulations within the context of finite element formulations follows from the contribution of the contact forces to the weak form of equilibrium equations 

(TN δ gN + TT · δ gT ) dS

(10)

S

where TN is the normal contact traction, gN the normal gap, TT the tangential traction and gT the tangential slip. The determination of the closest-point projection from slave node to master segment leads to an additional system of algebraic equations Qe (he ) = 0 that has to be solved for each contact element, where the unknowns he = {ξe1 , ξe2 , gN,e } are the coordinates of the projection point and the normal gap. The ADB form of (10) is obtained by replacing the variations with AD notation   δˆ gN,e  δˆ ξeα  Re := Ae (TN,e + TT α ,e ) (11)   δˆ pe  Dhe =−A−1 δˆ Qe δˆ pe  Dhe =−A−1 δˆ Qe Dpe

where the AD exception

Dhe Dpe

e

δˆ pe

Dpe

e

δˆ pe

δ Qe = −A−1 defines an implicit dependence of he on e ˆ ˆ

δ pe

pe introduced by the solution of Q(h) = 0 and Ae is a contact element tributary area. By introducing additional AD exceptions, pseudo-potential for the frictional contact formulation can also be formulated, leading to the identical but numerically more efficient ADB form of penalty treatment of contact constraints as follows   δˆ TN,e gN,e + TT α ,e δˆ ξeα   Re := Ae . (12)  δˆ pe  Dhe DTT α ,e −1 δˆ Qe DTN,e Dpe =−Ae δˆ pe , Dpe =0, Dpe =0

For more details see [7], where the ADB form of augmented Lagrangian treatments of contact constraints and several contact smoothing techniques are also introduced.

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4.5 ADB Form in Stability Analysis The automation of the formulation of stability analysis within the context of finite element formulations can be derived for both major approaches to stability analysis [9]: (1) computation of bifurcation points based on a linearized stability analysis (linear buckling analysis); (2) a direct quadratically convergent computation of limit and bifurcation points based on extended system formulation. The initial stability analysis can be reduced to the classical eigenvalue problem (KL + λ Kσ ) φ =0 where KL is linear the elastic tangent matrix, Kσ the geometric matrix and λ the load factor. For more details see [9]. Thus, the automation of initial stability analysis requires automation of derivation of KL and Kσ . The Gauss point contribution KLg to the linear elastic tangent matrix-can be derived as a Hessian of the standard Hooke’s strain energy potential W = Λ 2(trε )2 + μ tr(ε .ε ) and leads to ( ) δˆ δˆ W KLg := (13) δˆ p δˆ p e

e

where ε is a small strain tensor and Λ , μ are the Lam´e constants. While the automation of the derivation of KLg is straightforward, the automation of the derivation of geometric matrix requires introduction of a new pseudo-potential Wσ = tr(σ .E), , ˆ ˆ where E is the Green-Lagrange strain tensor and σ :=δ W δ ε is linear elastic stress tensor. The Gauss point contribution Kσ g to the geometric tangent matrix can then be derived as a Hessian of the pseudo-potential using the automatic differentiation procedure as follows ⎛ ⎞     δˆ ⎜ δˆ Wσ  ⎟ Kσ g := . (14)   ⎝ ⎠ δˆ pe δˆ pe  Dσ =0  Dσ Dp e

Dpe =0

The automation of procedures for direct determination of stability points requires automation of the linearization of the critical point test function, e.g. the determinant of the global tangent matrix or a minimum diagonal element of the LU decomposed matrix. In [6] an algorithm for the backward AD of the LU matrix decomposition is developed for the linearization of the eigenvector-free critical point test functions. Since the critical point test function is a scalar, the numerical cost for the linearization of the determinant by AD remains proportional to the cost of evaluating the determinant [6].

5 Conclusions The ADB form is presented for a general formulation of hyperelastic and elastoplastic finite elements, contact finite elements and stability analysis. The presented methodology enables derivation of the finite element subroutines for elements with

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arbitrary topology (two dimensional, three dimensional, shell, plate, axisymmetric, etc.) and based on arbitrary displacements or assumed strain formulations (B-bar, Fbar, under-integrated, enhanced strain, etc.). Numerical experiments show that the numerical efficiency of automatically derived program codes based on the presented ADB notation is comparable to that of the manually coded finite elements.

References 1. Eyheramendy, D., Zimmermann, T.: Object-oriented symbolic derivation and automatic programming of finite elements in mechanics. Eng. Comput. 15, 12–36 (2000) 2. Griewank, A.: Evaluating derivatives: principles and techniques of algorithmic Differentiation. SIAM, PA (2000) 3. Korelc, J.: AceGen and AceFEM user manual, www.fgg.uni-lj.si/symech/ 4. Korelc, J.: Multi-language and multi-environment generation of nonlinear finite element codes. Eng. Comput. 18, 312–327 (2002) 5. Korelc, J.: Automation of primal and sensitivity analysis of transient coupled problems. Comput. Mech. 44, 631–649 (2009) 6. Korelc, J.: Direct computation of critical points based on Crout’s elimination and diagonal subset test function. Comput. Struct. 88, 189–197 (2010) 7. Lengiewicz, J., Korelc, J., Stupkiewicz, S.: Automation of finite element formulations for large deformation contact problems. Accepted in Int. J. Numer. Meth. Eng. (2010) 8. Logg, A.: Automating the finite element method. Arch. Comput. Method. E. 14, 93–138 (2007) 9. Wriggers, P.: Nonlinear Finite Element Methods. Springer, Berlin (2008)

Chapter 11

Nonlinear Finite Element Shell Formulation Accounting for Large Strain Material Models Friedrich Gruttmann I met Peter Wriggers for the first time in 1982 when I was civil engineering student in Hannover. He lectured theory of elasticity and finite element methods. Between 1984 and 1989 we were colleges in the group of Professor Erwin Stein at the Institut f¨ur Baumechanik und Numerische Mechanik. It was a very fruitful time with intensive discussions on computational mechanics. He became a further advisor of my doctoral thesis. Several joint papers have been published, especially on finite shells elements. Peter encouraged me to spend a year as a postdoctoral researcher in Berkeley. After that we worked together in a research project on thin composite structures. We met again in Darmstadt for a short period before Peter returned to Hannover in 1998 (F. Gruttmann).

Abstract. Based on the kinematic assumptions of Mindlin–Reissner a three–field variational formulation with independent displacements, stress resultants and shell strains is presented. Within the finite element formulation the interpolation of the independent shell strains consists of two parts. The first part corresponds to the interpolation of the stress resultants. Within the second part thickness strains are incorporated, which allows direct implementation of nonlinear three-dimensional constitutive equations. A mixed hybrid quadrilateral shell element is formulated. The essential feature of the element is the remarkable robustness in nonlinear applications.

1 Introduction Various approaches on shell formulations considering finite strains have been published. As an example we mention Ref. [9], where the Mooney–Rivlin material involving large membrane strains has been implemented in a finite shell element. The zero normal stress condition in thickness direction is enforced with an update of the thickness change at the end of each equilibrium iteration. In [6] an algorithm to satisfy the stress condition was proposed, which requires storage of some history Friedrich Gruttmann Chair of Solid Mechanics, TU Darmstadt, Hochschulstraße 1, 64289 Darmstadt e-mail: [email protected]

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variables on the element level. An approach with a quadratically convergent iteration at each integration point has been developed in [7] and generalized in [11]. An axisymmetrical Quasi–Kirchhoff–type shell element accounting for large inelastic strains was developed in [18] and generalized for arbitrary geometry in [8]. The theory in [19] with resultants of the Biot stress tensor avoids neglect of higher order moments, which is often done along with integrals of the 2. Piola-Kirchhoff stress tensor. A further possibility to introduce 3d-material laws is to use a higher-order shell model with extensible director kinematic. There is general agreement that formulations accounting for the through-the-thickness strain should represent the normal strain in thickness direction at least linearly through the shell thickness, e.g. [4]. This can be achieved assuming a quadratic distribution of the displacements through the thickness, e.g. [15]. This approach leads to finite element formulations with seven parameters at the nodes.Another way is to enhance the thickness strains applying the enhanced assumed strain method (EAS–method), e.g. [5, 2, 3]. The associated finite element possesses six displacement–like parameters at the nodes and a strainlike through-the-thickness variable, which is condensed out at the element level. Comparisons of both approaches are given in [4]. A 5-parameter shell element was developed in [10] using the EAS-approach and a polynomial expansion of the thickness strains with constant and linear parts.

2 Variational Formulation of the Shell Equations Let B be the three–dimensional Euclidean space occupied by the shell in the reference configuration with thickness h. With ξ i we denote a convected coordinate system of the body. The thickness coordinate h− ≤ ξ 3 ≤ h+ defines the reference surface Ω at ξ 3 = 0. The position vectors of the initial reference surface and current surface are X(ξ 1 , ξ 2 ) and x(ξ 1 , ξ 2 ), respectively. Furthermore, a director vector D(ξ 1 , ξ 2 ) with |D(ξ 1 , ξ 2 )| = 1 is introduced as a vector perpendicular to Ω . Accordingly, the unit director d of the current configuration follows by orthogonal transformations. With d · x,α = 0 transverse shear strains are accounted for. Here, the comma denotes partial differentiation with respect to the coordinates ξ α . With the introduced kinematic assumptions of Mindlin and Reissner neglecting higher order curvatures one obtains membrane strains εαβ , curvatures καβ and shear strains γα as follows 1 εαβ = (x,α ·x,β −X,α ·X,β ) 2 1 (1) καβ = (x,α ·d,β +x,β ·d,α −X,α ·D,β −X,β ·D,α ) 2 γα = x,α ·d − X,α ·D . The strains depend on v = [u, ω ]T with displacements u = x − X and rotational parameters ω of the reference surface. The components are organized in a vector ε g (v) = [ε11 , ε22 , 2ε12 , κ11 , κ22 , 2κ12 , γ1 , γ2 ]T , where the subscript g indicates geometric strains which are related to the displacement field.

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89

We postulate the existence of a strain energy density Wˆ (C) as function of the right Cauchy-Green tensor C = FT F with the deformation gradient F. The covariant components of the Green-Lagrangean strain tensor E = 12 (C − 1) are written in vector form E = [E11 , E22 , E33 , 2E12 , 2E13 , 2E23 ]T . The relation of E to the shell strains ε is given by E = Aε ⎡

1 ⎢0 ⎢ ⎢0 A1 = ⎢ ⎢0 ⎢ ⎣0 0

A = [A1 , A2 ] 0 1 0 0 0 0

0 0 0 1 0 0

ξ3 0 0 0 0 0

0 ξ3 0 0 0 0

0 0 0 ξ3 0 0

0 0 0 0 1 0

⎤ 0 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎦ 1

⎡

0 ⎢0 ⎢ ⎢1 A2 = ⎢ ⎢0 ⎢ ⎣0 0

⎤ 0 0 ⎥ ⎥ ξ3 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎦ 0



ε= 

εz =

εp εz



0 ε33 1 ε33

.

(2)



Here, ε p contains the physical shells strains in the arrangement according to ε g . 0 and ξ 3 ε 1 denote the constant and linear part of the independent Furthermore, ε33 33 thickness strains, with the internal virtual work of the  respectively. We continue shell δ Wi = (Ω ) (h) δ Wˆ (C) dV = (Ω ) δ ε T ∂ ε W dA with dV = μ¯ d ξ 3 dA and the ˆ (C) = (Aδ ε ) : determinant of the shifter tensor μ¯ . Inserting δ Wˆ (C) = 2δ E : ∂CW ˆ 2∂CW (C) with δ E = A δ ε in δ Wi yields the vector of the stress resultants 

∂ εW =

∂ε p W ∂ε z W





:=

h+ AT 1 h−

AT2

 S μ¯ dξ 3

.

(3)

Here, S = 2 ∂CWˆ (C) denotes a vector with all components of the 2. Piola-Kirchhoff stress tensor. The shell is loaded statically by surface loads p¯ on Ω and by boundary forces ¯t on Γσ . Furthermore, displacements u¯ and rotations ω¯ are prescribed on Γu , where . Γ = Γσ Γu denotes the boundary of Ω . The static and geometric field equations, the constitutive equations and the equation for the resultants of the thickness stresses are summarized as follows ⎫ 1 α ¯ εg − ε p = 0 ⎪ ⎪ j ( j n ),α +p = 0 ⎬ 1 α α in Ω . (4) ( j m ), +x, ×n = 0 ∂ W − σ = 0 α α ε p j ⎪ ⎪ ⎭ ∂ε z W = 0 Here we denote by σ = [n11 , n22 , n12 , m11 , m22 , m12 , q1 , q2 ]T the vector of independent stress resultants with membrane forces nαβ = nβ α , bending moments mαβ = mβ α and shear forces qα . The static and geometric boundary conditions read j (nα να ) − ¯t = 0 , j (mα να ) = 0 on Γσ (5) v − v¯ = 0 on Γu

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F. Gruttmann

where να denote components of the normal vector on the shell boundary and v are prescribed boundary displacements and rotations. We introduce θ := [v, σ , ε ]T , admissible test functions δ θ := [δ v, δ σ , δ ε ]T and obtain with σ˜ = [σ , 0]T , integration by parts and consideration of the static boundary conditions yields the weak form of the boundary value problem 

g(θ , δ θ ) =

(Ω )

−

(Ω )

[δ ε T (∂ε W − σ˜ ) + δ σ T (ε g − ε p ) + δ ε Tg σ ] dA

δ uT p¯ dA −



δ uT ¯t ds = 0 .

(6)

(Γσ )

The geometric boundary conditions have to be fulfilled as constraints.

3 Mixed Hybrid Shell Element The present finite element formulation for quadrilaterals is based on the isoparametric concept. A map of the coordinates {ξ , η } ∈ [−1, 1] from the unit square to the reference surface of the initial and current configuration is applied. Nodal position vectors XI and cartesian basis systems [A1I , A2I , A3I ], I = 1, 2, 3, 4 are generated within the mesh input. Here, A3I is perpendicular to Ω and A1I , A2I are constructed in such a way that boundary conditions can be accommodated. The current nodal director vector dI = a3I is obtained by an orthogonal transformation akI = RI AkI , k = 1, 2, 3. The tensor RI is a function of rotational parameters ω I = [ω1I , ω2I , ω3I ]T , and is evaluated using the Euler-Rodrigues formula. Hence the position vectors and the director vectors of the reference surface and current surface are interpolated with bi-linear functions NI = 14 (1 + ξI ξ )(1 + ηI η ). The independent field of stress resultants σ is approximated introducing the shape function matrix Nσ σ h = Nσ σˆ . (7) The vector σˆ ∈ R14 contains 8 parameters for the constant part and 6 parameters for the varying part of the stress resultant field, respectively. The interpolation of membrane forces and bending moments corresponds to the approach in [16]. The approximation of the independent shell strains

ε h = Nε εˆ

(8)

with shape function matrix Nε and parameter vector εˆ consists of two parts. The first part with 14 parameters corresponds to the interpolation (7). The shape functions of the second part with a variable number of parameters are constructed orthogonal to the stress interpolation. The membrane and bending strains in the second part may be interpolated with linear or bi-linear functions considering 2 or 4 parameters. The thickness strains can be interpolated with 1, 2 or 8 parameters, see also H¨uttel and Matzenmiller [10]. The parameter vectors σˆ and εˆ are condensed out on the element level. The resulting mixed hybrid shell element with 5 or 6 displacement degrees of

Nonlinear Finite Element Shell Formulation Accounting

91

freedom at the 4 nodes has been implemented in an extended version of the general purpose finite element program FEAP, documented in Taylor [17]. All matrices are specified in detail in Klinkel et al. [13].

4 Numerical Example: Stretching of a Rubber Sheet This example has been analyzed in various publications, e.g. [14, 2, 1]. Here, we show by comparison with the solid shell formulation [12] that the present element approximates finite thickness strains correctly. A square sheet with a hole is subjected to a stretching load. The quasi incompressible material response is governed by a Mooney-Rivlin model, [13]. Fig. 1 shows the problem with geometrical and material data. With respect to symmetry only one quarter is discretized with finite elements. At the left and right edge the degrees of freedom are linked together, such that same horizontal but no vertical displacements occur. The sheet is stretched up to twice of the original length, see Fig. 2. Along the hole (point A) a stability problem occurs induced by compressive stresses. In order to trace the secondary equilibrium path a perturbation load 10−7 F perpendicular to the plane is applied at point A. The secondary path is characterized by an out-of-plane deflection w. In Fig. 2 the load F is depicted versus the horizontal displacement u. The present element is used with two parameter sets for membrane/bending/thickness within the second part of the strain interpolation. Results are presented for the sets 2/2/1 and 2/2/2 and compared with calculations employing the solid shell element [12] with only one element through the thickness and the shell element [11]. In [11] a zero stress condition is imposed within a local Newton iteration. For all element types

L

L

u

L 2F

R

Geometry: L = 10 R=3 h = 0.1

2F

A L

Mooney-Rivlin material: μ1 = 50 α1 = 2 μ2 = −14 α2 = −2 Λ = 1000

Fig. 1 Geometry and material data of the rubber sheet with hole

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90

shell [11] solid shell [12] present 2/2/1 present 2/2/2

80 70 load F

60 50 40 30 20 10 0 0

2

4

6 8 displacement u

10

12

Fig. 2 Load deflection curve for the horizontal displacement u and deformed rubber sheet with hole at F = 90

90 80 70 load F

60 50 40 30

shell [11] solid shell [12] present 2/2/1 present 2/2/2

20 10 0 0

0.1

0.2

0.3

0.4

0.5

0.6

displacement w Fig. 3 Load deflection curve for the out of plane displacement w and perspective view of the hole of the deformed rubber sheet at F = 90

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E_33 [%]

solid shell [12]

0.00 -3.78 -7.56 -11.33 -15.11 -18.89 -22.67 -26.44 -30.22 -34.00

present 2/2/2 Fig. 4 Thickness strain of the deformed rubber sheet at F = 90; only one quarter of the deformed configuration is shown

Table 1 Convergence of the residual within the equilibrium iterations load increment iteration 1 2 3 4 5 6 7 8 F =2→5 shell [11] 100 10−1 10−2 10−1 10−3 10−4 10−6 10−9 present 100 10−1 10−1 10−3 10−5 10−10 load increment iteration 1 2 3 4 5 6 7 8 F = 10 → 50 shell [11] no convergence present 101 101 100 10−1 10−2 10−4 10−8

a two point Gauss quadrature through the thickness is used. In Fig. 3 the load F is plotted versus the out of plane deflection w at point A. The present formulation 2/2/1 considers only a constant approximation of the thickness strains. This assumption leads to an overestimation of the buckling load. In contrast to that the parameter set 2/2/2 contains according to Eq. (2) a constant and a linear function for the thickness strains. It leads to results which coincide very well with the shell and the solid shell element, see the diagrams of Figs. 2 and 3. Fig. 4 shows a plot of the thickness strains. The strains are evaluated for both element types at the upper layer of the Gauss integration points. The example demonstrates that the present element is able to approximate problems with finite strains correctly. In Tabular 1 the convergence rates of the global equilibrium iteration for different load steps are listed.

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It is observed that the present formulation is very robust with respect to the load step size.

5 Conclusions Independent thickness strains are included in a three field variational formulation, which allows consideration of arbitrary nonlinear three-dimensional constitutive equations. The numerical tests show that two parameters are sufficient to obtain good results. Furthermore it is shown that especially for finite deformations the formulation allows very large load steps and requires essentially less equilibrium iterations in comparison to displacement based elements.

References 1. Basar, Y., Itskov, M.: Finite element formulation of the Ogden material model with application to rubber-like shells. Int. J. Numer. Meth. Eng. 42, 1279–1305 (1998) 2. Betsch, P., Gruttmann, F., Stein, E.: A 4-node finite shell element for the implementation of general hyperelastic 3d-elasticity at finite strains. Int. J. Numer. Meth. Eng. 37, 2551–2568 (1994) 3. Bischoff, M., Ramm, E.: On the physical significance of higher order kinematic and static variables in a three-dimensional shell formulation. Int. J. Solids Struct. 37, 6933–6960 (2000) 4. Brank, B., Korelc, J., Ibrahimbegovic, A.: Nonlinear shell models with seven kinematic parameters. Comput. Meth. Appl. M. 194, 2336–2362 (2002) 5. B¨uchter, N., Ramm, E., Roehl, D.: Three-dimensional extension of non–linear shell formulation based on the enhanced assumed strain concept. Int. J. Numer. Meth. Eng. 37, 2551–2568 (1994) 6. De Borst, R.: The zero-normal-stress condition in plane–stress and shell elastoplasticity. Commun. Appl. Numer. M. 7, 29–33 (1991) 7. Dvorkin, E., Pantuso, D., Repetto, E.: A formulation of the MITC4 shell element for finite strain elasto-plastic analysis. Comput. Meth. Appl. M. 125, 17–40 (1995) 8. Eberlein, R., Wriggers, P.: Finite element concepts for finite elastoplastic strains and isotropic stress response in shells: theoretical and computational analysis. Comput. Meth. Appl. M. 171, 243–279 (1999) 9. Hughes, T.J.R., Carnoy, E.: Nonlinear finite element shell formulation accounting for large membrane strains. Comput. Meth. Appl. M. 39, 69–82 (1983) 10. H¨uttel, C., Matzenmiller, A.: Consistent discretization of thickness strains in thin shells including 3d-material models. Commun. Appl. Numer. M. 15, 283–293 (1999) 11. Klinkel, S., Govindjee, S.: Using finite strain 3d-material models in beam and shell elements. Eng. Computation. 19, 902–921 (2002) 12. Klinkel, S., Gruttmann, F., Wagner, W.: A robust non-linear solid shell element based on a mixed variational formulation. Comput. Meth. Appl. M. 195, 179–201 (2006) 13. Klinkel, S., Gruttmann, F., Wagner, W.: A mixed shell formulation accounting for thickness strains and finite strain 3d material models. Int. J. Numer. Meth. Eng. 74, 945–970 (2008) 14. Parisch, H.: Efficient non-linear finite element shell formulation involving large strains. Eng. Computation 3, 121–126 (1986)

Nonlinear Finite Element Shell Formulation Accounting

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15. Sansour, C.: A theory and finite element formulation of shells at finite deformations involving thickness change: circumventing the use of a rotation tensor. Arch. Appl. Mech. 65, 194–216 (1995) 16. Simo, J.C., Fox, D.D., Rifai, M.S.: On a stress resultant geometrically exact shell model. Part ii: The linear theory; computational aspects. Comput. Meth. Appl. M. 73, 53–92 (1989) 17. Taylor, R.L.: Feap - manual, http://www.ce.berkeley˜rlt/feap/manual.pdf 18. Wriggers, P., Eberlein, R., Gruttmann, F.: An axisymmetrical Quasi–Kirchhoff–type shell element for large plastic deformations. Arch. Appl. Mech. 65, 465–477 (1995) 19. Wriggers, W., Gruttmann, F.: Thin shells with finite rotations formulated in Biot stresses: Theory and finite element formulation. Int. J. Numer. Meth. Eng. 36, 2049–2071 (1993)

Chapter 12

Hybrid and Mixed Variational Principles for the Geometrically Exact Analysis of Shells Paulo de Mattos Pimenta

Friends are so precious that they need a special safe: our hearts (P. Pimenta).

Abstract. This paper addresses the development of some alternative hybrid and mixed variational formulations for the geometrically-exact three-dimensional firstorder-shear shell boundary value problem. In the framework of the complementaryenergy-based formulations, a Legendre transformation is used to introduce the complementary energy density in the variational statements as a function of the cross-sectional resultants only. The corresponding variational principles are shown to feature stationarity within the framework of the boundary-value-problem. The main features of the principles are highlighted, giving special attention to their relationships from both theoretical and numerical point of view.

1 Introduction Variational principles constitute the core of the development of numerical methods in solid mechanics. The utility of such principles is two-fold: first, they provide a very convenient method for the derivation of the governing equations and natural boundary conditions of the boundary value problem and, second, they provide the mathematical foundation required to produce consistent numerical approximations. In this second role, the variational methods have been most useful in computational solid mechanics. Many different variational principles can be constructed depending on the equations enforced in the weak form. Perhaps the most remarkable variational principle is the principle of stationary total potential energy, which states that, among all kinematically admissible displacement fields, those that satisfy the equilibrium conditions in the domain and at the boundary, lead to a stationary value of the total potential energy functional. Throughout the text, italic Latin or Greek lowercase letters (a, b, ..., α , β , ...) denote scalar quantities, bold Latin or Greek Paulo de Mattos Pimenta Polytechnic School at the University of S˜ao Paulo, PO Box 61548, 05424-970 S˜ao Paulo, Brazil e-mail: [email protected]

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lowercase letters (a, b, ..., α , β , ...) denote vectors, while bold Latin or Greek capital letters (A, B, ...) denote second-order tensors. Summation convention over repeated indices is adopted, with Greek indices ranging from 1 to 2 and Latin indices from 1 to 3.

2 The Geometrically-Exact First-Order-Shear Shell Model In this work, we recall the geometrically-exact six-parameter shell formulation presented in Campello et al. [1] (which is one of the existing shell models undergoing large strains and finite rotations, see [1] for further references). Although it may be not necessary, our approach defines energetically conjugated cross sectional stresses and strains, based on the concept of shell director with a standard Reissner-Mindlin kinematical assumption. Due to the use of cross sectional quantities, the derivation of equilibrium equations in strong and weak forms is considerably simpler, and the linearization of the latter leads naturally to a symmetric bilinear form for hyperelastic materials and conservative loadings (even far from equilibrium states). The resulting expressions are much similar to those obtained for geometrically-exact spatial rods, rendering a very convenient pattern for the simultaneous coding of rod and shell finite elements. Finite rotations may be treated here by the Euler-Rodrigues formula in a total or updated Lagrangian way. Two different parameterizations may be considered: • the usual Euler rotation vector and • the Rodrigues parameters (see also [1]). The first is singularity-free for any rotation increment while the latter delivers computationally more efficient expressions. It is assumed that the middle surface of the shell is plane at the initial reference configuration. Initially curved shells can be regarded as a stress-free  deformed state from the plane initial position, as shown in [2]. Let er1 er2 er3 be an orthogonal system, with the vectors erα placed on the shell reference mid-plane and er3 normal to this plane, as shown in Fig. 1. The position of any shell material point in the reference configuration can be described by

ξ = ζ + ar

,

(1)

r where the vector ζ = ξα erα defines a point on the reference mid-surface  b and  a is r r t the shell director at this point, given by a = ζ e3 . Here ζ ∈ H = −h , h is the t thickness coordinate, with h = hb +  h being the shell thickness in the reference configuration (observe that ξα , ζ sets a three-dimensional Cartesian frame). In the current configuration the position x of any material point can be expressed by the vector field x = z+a , (2)

where z = zˆ (ξα ) describes the current position of a point in the middle surface and a is the current director at this point, obtained as a = Qar , with Q as the rotation

Hybrid and Mixed Variational Principles

99

Fig. 1 Shell description and basic kinematical quantities

tensor. Notice that no thickness change is assumed during the motion and that first order shear deformations are accounted for since a is not necessarily normal to the current mid-surface. Let now {e1 , e2 , e3 } be a local orthogonal system on the current configuration, with ei = Qeri , as depicted in Fig. 1. ˆ (θ ) may be expressed in terms of the Euler rotation The rotation tensor Q = Q( vector θ , by means of the well-known Euler-Rodrigues formula Q=I+

sin θ 1 Θ+ θ 2

&

sin(θ /2) θ /2

'2

Θ2

(3)

in which θ is the rotation angle given by θ = ||θ ||. Still in (3) Θ = Skew(θ ) is the skew-symmetric tensor whose axial vector is θ . Here, we will keep the presentation in terms of θ and for simplicity we will adopt the total Lagrangian description for the rotation. Having defined z and ζ , the displacements of any point of the reference middle plane can be computed by u = z−ζ . (4) The components of u and θ on a global Cartesian system constitute the 6 degreesof-freedom of this shell model. Two skew-symmetric tensors that describe the specific rotations of the director can be defined as Kα = Q,α QT , where we have introduced the notation (•),α = ∂ (•) /∂ ξα for derivatives. One can show that the corresponding axial vectors are κ α = axial(Kα ) = Γ θ ,α , with the tensor Γ given in [1]. From differentiation of (2) with respect to ξ one can evaluate the deformation gradient F. This leads to the definition of the following material cross-sectional strain vectors

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η rα = QT z,α − erα

and

κ rα = Γ T θ ,α

(5)

.

These two vectors may be regarded as cross-sectional generalized strains, with z,α = erα + u,α . One may understand that the components η rα · erβ of η rα operate as membrane strains, while η rα · er3 as transversal shear strains. Expressions (5) are the back-rotated counterparts of η α = z,α − eα and κ α , respectively. Let the 1. Piola-Kirchhoff stress tensor be written as P = τ i ⊗ eri . The quantities τ i are nominal stress vectors, and act on cross-sectional planes whose normal vectors on the reference configuration are eri . Integration of two first vectors stress vectors along the shell thickness allows the definition of generalized cross-sectional stresses, i.e. 

nα =

τ α dζ



and

H

mα =

a × τ α dζ

.

(6)

H

The third vector is assumed to perform no work in this shell model (see a discussion on the plane stress issue in [1]). ˜ be the applied external forces and moments reOn the other hand, let n˜ and m spectively, both per unit area of the middle surface in the reference configuration. The shell local equilibrium can be stated by statics in a standard way. The result in the shell domain Ω ⊂ R2 is nα ,α + n˜ = 0 and ˜ =0 . mα ,α + z,α × nα + m

(7)

On the other hand, the back-rotated cross-sectional resultants are defined by nrα = QT nα

and

mrα = QT mα

(8)

.

The boundary of the domain Ω is denoted by Γ = ΓD ∪ ΓN , subdivided, as usual, in Dirichlet and Neumann parts according to the following boundary conditions u = u¯ and θ = θ¯ on ΓD r ¯ on ΓN να nα = n¯ and ναr mα = m

,

(9)

respectively, where ναr = ν r · erα are the components of the unitary normal to the boundary. In (9), u¯ and θ¯ are respectively the prescribed displacements and rotations ¯ are respectively the applied external forces and moments on ΓD , as well as, n¯ and m per unit reference length on ΓN . For the sake of convenience, let us define the following assembled vectors        r   r  n˜ n¯ u ηα nα r r , εα = , q˜ = , q¯ = and d = , σα = mrα κ rα μ˜ μ¯ θ (10)

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101

˜ and μ¯ = Γ T m ¯ have been introduced (see the where the pseudo-moments μ˜ = Γ T m remark in [1] on this matter). Note that (5) can be recast as

ε rα = εˆ rα (d)

(11)

.

Let us also introduce the following operators     Q 0 Q 0 Λ= and Hα = 0 Q 0 Γ

(12)

.

Making use of this generalized vectors and operators, the equilibrium boundaryvalue problem, described by (7) and (9), is reduced to the following synthetic form Tαe (d)σ rα + q˜ = 0 in Ω ν rα Hσ rα − q¯ = 0 on ΓN

(13)

where (with Z,α = Skew(z,α ))  Tαe (d) = Ψ eα Δ eα Γ

(no sum),

with Ψ eα =

 I 0 0 and 0 Γ T Z,α Γ T ⎡ ∂ ⎤ I 0 ⎢ ∂ ξα ⎥ ⎢ ⎥ Δ eα = ⎢ I 0 ⎥ ⎣ ∂ ⎦ 0 I ∂ ξα

. (14)

Let us introduce the trial displacement space given by D = {d ∈ H 1 (Ω ) | d = d¯ on ΓD } ,

(15)

where H 1 is the Sobolev space of first order. D contains functions that satisfy the Dirichlet boundary conditions. Let us also introduce the corresponding the test function linear space given by

δ D = {δ d ∈ H 1 (Ω ) | δ d = d¯ on ΓD } .

(16)

The weak form of the equilibrium boundary-value problem (13) is 

[Tαe (d)σ rα + q˜ ] · δ d dΩ +



[q¯ − ναr Hα σ rα ] · δ d dΓ = 0,

∀δd ∈ δD

, (17)

ΓN

Ω

which, after the application of the divergence theorem, delivers  Ω

[−σ rα · Tαc (d)δ d + q˜ · δ d] dΩ +



ΓN

q¯ · δ d dΓ = 0,

∀δd ∈ δD

,

(18)

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where 

Tαc (d) = Λ T Ψ cα Δ cα

(no sum),

 I 0 Z,α Γ and 0 Γ Γ ,α ⎡ ∂ ⎤ I 0 ⎢ ∂ ξα ⎥ ⎢ ∂ ⎥ Δ cα = ⎢ ⎥ I ⎣0 ⎦ ∂ ξα 0 I

with Ψ cα =

. (19)

(18) is a necessary and sufficient condition for (13). The expression of Γ ,α can be found in [1]. For hyperelastic material behavior, a strain energy density per unit reference area Ψ = Ψˆ (ε r1 , ε r2 ) can be defined such that

σ rα = ∂α Ψ =

∂Ψ ∂ ε rα

(20)

.

Let us assume that Ψ is a differentiable and convex function with respect to its arguments. On the basis of these assumptions, Eq. (20) establishes a one-toone correspondence between the stress-resultant vectors σ rα and the generalized cross-sectional strains ε rβ . Legendre has shown that this type of equation can be transformed into a conjugate form by introducing a new function Ψˆ ∗ (σ r1 , σ r2 ) defined by Ψˆ ∗ (σ r1 , σ r2 ) = σ r · ε r − Ψˆ (ε r1 , ε r2 ) , (21) which is called complementary strain energy density. Differentiation of this function leads to ∂Ψ ∗ ε rα = ∂α Ψ ∗ = . (22) ∂ σ rα The strong form of the boundary value problem governing the response of our shell model with hyperelastic material behavior consists of the following three sets of differential equations to be solved in Ω Tαe (d)σ rα + q˜ = 0 σ rα − ∂αΨ = 0 ε rα − εˆ rα d = 0

Ω Ω in Ω in in

(23)

representing equilibrium, constitutive and compatibility conditions in the domain, respectively, and, additionally, a set of prescribed boundary conditions on the boundary Γ = ΓD ∪ ΓN , divided into Dirichlet (kinematical) and Neumann (statical) conditions, respectively, as follows d − d¯ = 0 on ΓD ναr Hσ rα − q¯ = 0 on ΓN

.

(24)

Hybrid and Mixed Variational Principles

103

3 Some Multi-field Variational Principles We present now some multi-field variational principles that correspond to problem (23) with boundary conditions (24). (see also [3]).

3.1 Principle of Total Potential Energy We start with the well-known principle of stationary total potential energy for didactic purposes. The total potential energy is defined by the one-field functional U : D → R given by 

U(d) =

Ψ ◦ εˆ rα (d) dΩ −

Ω



q˜ · d dΩ −



q¯ · d dΓ

(25)

.

ΓN

Ω

In order to (25) be stationary, one gets the necessary condition δ U = 0, ∀ δ d ∈ δ D where   δ U = [−∂α Ψ · Tαc (d)δ d + q˜ · δ d] dΩ + q¯ · δ d dΓ (26) ΓN

Ω

is the Gˆaeaux derivative of (25). Introducing the constitutive equation (23)2 in (26), one arrives at (18), which, after the application of the divergence theorem, delivers (17). Hence, it can be concluded that a geometrically exact first-order-shear shell model is in equilibrium if and only if its total potential energy has a stationary value at the solution. This result is known as the principle of stationary total potential energy.

3.2 Three-Field Principle of Veubeke-Hu-Washizu Type The principle of stationary total potential energy can be generalized through the well-known method of Lagrange multipliers, which allows to introduce the compatibility equation (23)3 and the kinematical boundary condition (24)1 , assumed now as subsidiary conditions, into the framework of the variational expression (25). The result is the three-field functional of Veubeke-Hu-Washizu type, ΠV HW : H 1 (Ω ) × H 0 (Ω ) × H 0 (Ω ) → R, given by

ΠV HW (d, σ rα , ε rα ) =



[−Ψ + σ rα · (ε rα − εˆ rα (d)) + q˜ · d] dΩ +



q¯ · d dΓ

ΓN

Ω

+



¯ dΓ ναr Hσ rα · (d − d)

.

(27)

ΓD

Equations (23) and both boundary conditions (24) render this functional stationary. The major advantage of (27) is the fact that it avoids the inversion of the constitutive

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equation (23)2 . On the other hand it is a mixed functional, with the well-known stability problems in the approximation of saddle-points.

3.3 Two-Field Principle of Hellinger-Reissner Type It is interesting to note that, the two-field Principle of Hellinger-Reissner owes its origin to the idea that, in geometrically non-linear problems, the difficulties with the Complementary Potential Energy could be circumvented by considering stresses and displacements as primal variables. The two-field Principle of Hellinger-Reissner Type in our framework can be obtained from (27), considering the inverse of the constitutive equation given by (21). ∗ : H 1 (Ω ) × H 0 (Ω ) → R given by The result is a two-field functional ΠHR ∗ ΠHR (d, σ r ) =



[Ψ ∗ − σ r · εˆ r (d) + q˜ · d] dΩ

Ω



+ ΓN

q¯ · d dΓ +



¯ dΓ ναr Hσ rα · (d − d)

(28)

.

ΓD

The corresponding Euler-Lagrange equations are Tαe (d)σ rα + q˜ = 0 in Ω ∂α Ψ ∗ − εˆ rα (d) = 0 in Ω

(29)

and the boundary conditions (24). It is a mixed functional, with the known stability problems in the approximation of saddle-points.

3.4 Two-Field Principle of Total Complementary Potential Energy Although in the framework of the two-field Principle of Hellinger-Reissner Type, the equilibrium equations (29)1 have been obtained as Euler-Lagrange equations, they can be instead considered as subsidiary conditions to be satisfied a priori. Accordingly, assuming that equilibrium holds, when subjecting (28) to the equilibrium equations by means of appropriate Lagrange multipliers, the following augmented Lagrangian is obtained ∗ L∗ (d, σ rα ) = ΠHR (d, σ rα ) −



˜ · d dΩ + [Tαe (d)σ rα + q]



¯ · d dΓ [ναr Hσ rα − q]

,

ΓN

Ω

(30) which, after the use of the divergence theorem, delivers the following functional U ∗ (d, σ rα ) =

 Ω

[Ψ ∗ + σ rα · (Tαc (d)d − εˆ rα (d))] dΩ −



ΓD

ναr Hσ rα · d¯ dΓ

.

(31)

Hybrid and Mixed Variational Principles

105

Now we introduce the function space  e  2  T (d)σ r + q˜ = 0 in Ω and α α  E (d) = σ rα ∈ H 1 (Ω )   ναr Hσ rα = q¯ on ΓN and the linear space 

δ E (d) =

 e 2  T (d)δ σ r = 0 in Ω and α  α r 1 δ σ α ∈ H (Ω )   ναr Hδ σ rα = 0 on ΓN

,

in order to write U ∗ : H 1 (Ω ) × E (Ω ) → R and to get the Gˆateaux derivative of (31), which is

δU ∗ =



[∂α Ψ ∗ − εˆ rα (d)] · δ σ rα dΩ −

 

 d − d¯ · ναr Hδ σ rα dΓ

.

(32)

ΓD

Ω

Hence, the corresponding Euler-Lagrange equations emanating from δ U ∗ = 0, ∀ δ d ∈ H 1 (Ω ), ∀ δ σ r ∈ δ E (Ω ), are

∂α Ψ ∗ − εˆ rα (d) = 0 in Ω d − d¯ = 0 on ΓD

(33)

which are clearly the compatibility equations in the domain and at the boundary.

3.5 Hybrid Principle of Hellinger-Reissner Type Relaxing the continuity requirements of the displacements along the interelement boundaries (here indicated by ΓIE ) by means of a Lagrange multiplier qΓ , a generalized hybrid form of the functional (28) can be built. It is a (2+2)-field hybrid-mixed 2 : H 1 (Ω ) × H 0 (Ω ) × functional of Hellinger-Reissner type indicated by ΠHR 0 0 H (ΓIE ) × H (ΓIE ) → R and defined as 2 ΠHR (d, σ r , dΓ , qΓ ) = ΠHR (d, σ r ) +



¯ dΓ + qΓ · (dΓ − d)

ΓD



qΓ · (d − dΓ ) dΓ

.

ΓIE

(34) For this functional, the Euler-Lagrange equations are (29) and q¯ − qΓ = 0 on dΓ − d¯ = 0 on Γ q − ναr Hσ r = 0 on d − dΓ = 0 on

ΓN ΓD ΓIE ΓIE

.

(35)

A hybrid form of the two-field principle of total complementary potential energy (31) is also possible. Due to the lack of space, it will not be presented. This

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functional will be used in the derivation of equilibrated co-rotational finite elements. This is our next goal.

References 1. Campello, E.M.B., Pimenta, P.M., Wriggers, P.: A triangular finite shell element based on a fully nonlinear shell formulation. Comput. Mech. 31, 505–518 (2003) 2. Pimenta, P.M., Campello, E.M.B.: Shell curvature as an initial deformation: geometrically exact finite element approach. Int. J. Numer. Meth. Eng. 78, 1094–1112 (2009) 3. Santos, H.A.F.A., Pimenta, P.M., Almeida, J.P.M.: Hybrid and multi-field variational principles for geometrically exact three-dimensional beams. Int. J. Nonlinear Mech. 45, 809–820 (2010)

Chapter 13

A Shell Theory with Scale Effects, Higher Order Gradients, and Meshfree Computations Carlo Sansour and Sebastian Skatulla

Dedicated to Peter Wriggers in acknowledgement of his contributions to computational mechanics and of his influence on shaping the first author’s career especially during the time in Darmstadt (C. Sansour).

1 Introduction In recent times, scale effects have been given great attention due to the fact that they characterise the material behaviour at lower scales. The later being considered an important area of research in material sciences and engineering. Many applications, however, are concerned with so-called thin domains (e.g. thin films, nano tubes) which makes it more effective to run the computations via a shell theory. Hence the motivation to develop a shell theory which includes in a natural way scale effects. This is done by extending and modifying previous 3-d generalized formulations of the authors [5] to accommodate for the shell. Applications are discussed for the buckling of thin shells at small scales. In recent years various experiments on reinforced thin-walled cylindrical shells under axial compression were undertaken. [2] studied the residual strength of unstiffened cylindrical shells made of carbon fibre reinforced plastics subjected to torsion and/or axial compression in the post-buckling regime depending on the fiber orientation. [3] tested thin-walled steel cylinders with small-sized stringers. The number of stiffeners and shell length was varied and the critical buckling load determined. [7] employed a strain gradient theory implemented in a meshfree code Carlo Sansour School of Civil Engineering, The University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom e-mail: [email protected] Sebastian Skatulla CERECAM, Department of Civil Engineering, The University of Cape Town, South Africa e-mail: [email protected]

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to study higher order effects modelling the buckling behaviour of nanotubes under torsion and axial compression, respectively. The plan of this paper is as follows: In Sect. 2 the theory of the generalized continuum is outlined followed by Sect. 3 which outlines the shell theory. The applicability of the generalized shell formulation in conjunction with MLS is illustrated in Sect. 4 by some applications utilizing linear and nonlinear hyperelastic material laws.

2 Deformation and Strain The strain gradient theory which will be outlined in the following is based on the theoretical framework for a generalized continuum proposed in [5]. This framework makes use of the mathematical concept of a fibre bundle, where, in the simplest case, the generalized space is constructed as the Cartesian product of a macro space B ⊂ E(3) and a micro space S which we write as G := B × S . This definition assumes an additive structure of G which implies that the integration over the macro˜ ∈ G is and the micro-continuum can be performed separately. Each material point X related to its spatial placement x˜ ∈ Gt at time t ∈ R by the mapping ϕ˜ (t) : G −→ Gt . The generalized space can be projected to the macro-space in its reference and its ˜ = X and πt (˜x) = x respectively, where π0 as well as current configuration by π0 (X) πt represent projection maps, and X ∈ B and x ∈ Bt . The tangent space T G in the reference configuration and T G t in the current configuration are given by the pairs ˜ i × Iα ) and G ˜ i and Iα , respectively, which are defined by (G ˜ ˜ i = ∂X G ∂ϑi

and Iα =

˜ ∂X ∂ζα

,

g˜ i =

∂ x˜ ∂ϑi

and iα =

∂ x˜ ∂ζα

.

(1)

˜ ∈ G ) is of an Now, we assume that the placement vector x˜ of a material point P (X additive nature and is the sum of its position in the macro-continuum x ∈ Bt and in the micro-continuum ξ ∈ St as follows       x˜ ϑ k , ζ β ,t = x ϑ k ,t + ξ ϑ k , ζ β ,t . (2) In order to formulate generalized strain measures we proceed in analogy to the definition of the classical right Cauchy-Green deformation tensor and define its generalized equivalent expression as ˜ = F˜ T F˜ C

.

(3)

Neglecting higher order terms in ζ α as well as for the sake of mathematical simplic˜ k ⊗ Iβ ity and computational performance discarding contributions with respect to G and Iα ⊗ Iβ we have    k ˜ = x,k · x,l + ζ α x,k · gα ,l + gα ,k · x,l G ˜ ⊗G ˜ l = C + ζ α Kα C (4)

A Shell Theory with Higher Order Gradients

109

˜ still includes the conventional as well as the higher which is reasonable, because C order strains. C represents the conventional right Cauchy-Green deformation tensor and Kα the higher order contributions of Eq. (4). The scalar products of vectors are denoted by a dot.

3 Generalized Shell Theory To derive a shell theory with scale effects the above framework is modified and extended by assuming the base continuum B to consist of the Cartesian product M × L , where M is a two-dimensional surface parameterized by the curvilinear coordinates ϑ i , L is a one-dimensional space parameterized by the curvilinear coordinate z. Hence we assume now that G is of the following form: G := {M × L } × S

(5)

.

Here, and in what follows, Latin indices take the values 1 or 2 and Greek indices 1, ... to n. The tangent space T G in the reference configuration is defined by the ˜ i ×N ˜ × I˜α ) given by triple (G ˜ ˜ i = ∂X G ∂ϑi

,

˜ ˜ = ∂X N ∂z

and

˜ ∂X I˜α = ∂ζα

,

(6)

˜ i, N ˜ and I˜α , where the corresponding dual contra-variant vectors are denoted by G respectively. The corresponding tangent space in the current configuration T G t is spanned by the triple (˜gi × n˜ × ˜iα ) given by g˜ i =

∂ x˜ ∂ϑi

,

n˜ =

∂ x˜ ∂z

and

˜iα = ∂ x˜ ∂ζα

.

(7)

˜ is the unit normal vector on M , vector n, ˜ in general, is neither Note that, while N normal to the deformed surface Mt nor a unit vector. The generalized space is to be projected to the macro-space consisting of the spaces M and L in their reference and current configurations. Two types of projection maps can be defined: ˜ =X π0M (X)

and

π M (˜x) = x ,

(8)

˜ = X+Z π0L (X)

and

π L (˜x) = x + z

(9)

as well as

respectively, where X ∈ M , Z ∈ L , x ∈ Mt and z ∈ Lt . The exact definition of these projections depends on the geometry of the shell, i.e. its curvature. Clearly, while π0M , π0L are to be given a priori, their time dependent counter part will depend on the deformation itself. Also, the choice Z = zN is a natural one. At the shell surface the triple Gi , N defines the natural covariant base of the macro-space (or

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base continuum) M × L . The unit normal is defined by N = εi j Gi × G j , where ε i j denotes the components of the two-dimensional Ricci tensor. Now, similar to Eq. (2) we choose the placement vector x˜ of a material point P ∈ G to be the sum of its position in the macro-continuum given by x ∈ Mt as well as z ∈ Lt and its position in the micro-continuum ξ ∈ St as follows       x˜ = x ϑ k ,t + z ϑ k , z,t + ξ ϑ k , z, ζ β ,t . (10) With respect to z it is meaningful to consider a quadratic displacement field which allows for the application of a classical three-dimensional constitutive law [4]. We arrive at          x˜ = x ϑ k ,t + z + z2 λ ϑ k ,t d ϑ k ,t + ζ α aα ϑ k , z,t . (11) Higher order gradients are introduced by choosing the basis vectors of the microspace in the current configuration Mt × Lt as the micro-directors aα aα =

∂ x˜ | α , with α = 1 or 2 , ∂ ϑ α ζ =0

a3 =

∂ x˜ | α , with α = 3 . (12a,b) ∂ z ζ =0

No other degrees of freedom besides the displacement ones are introduced to describe scale-related effects. Taking the spatial derivatives of the generalized position vector in the current configuration (Eq.( 11)) with respect to the macro-coordinates ϑ i and z as well as the micro-coordinates ζ α the generalized deformation gradient is written as follows     ˜i F˜ = x,i + z + z2 λ d,i + z2 λ,i d + ζ α aα ,i ⊗ G α

˜ + aα ⊗ I˜ + ((1 + 2z λ ) d + ζ α aα ,z ) ⊗ N

.

(13)

Similar to Eq. (3) a generalized Cauchy-Green deformation tensor is formulated as ˜ I = (x,i · x, j + z x,i · d, j + z d,i · x, j C ˜i⊗ G ˜j + ζ α (z d,i · aα , j + z aα ,i · d, j + x,i · aα , j + aα ,i · x, j )) G

(14)

˜ II = (x,i · d + 2 z λ x,i · d + z d,i · d C

ζ α (z d,i · aα ,z + 2z λ aα ,i · d + x,i · aα ,z + aα ,i · d))   ˜ i⊗N ˜i ˜ +N ˜ ⊗G G

˜ III = (d · d + 4z λ d · d + ζ α (2 d · aα ,z + 4z ζ α λ d · aα ,z )) N ˜ ⊗N ˜ C

(15) (16)

˜k⊗ where higher order terms in z and ζ α as well as contributions with respect to G I II III β ˜ =C ˜ +C ˜ +C ˜ is assumed ˜ ⊗ I˜ and Iα ⊗ Iβ are disregarded. The tensor C Iβ , N to be the strain measure of the shell. The constitutive law is numerically integrated

A Shell Theory with Higher Order Gradients

111

over B resulting in a corresponding stress measure. The resulting expressions (not reproduced here) will account for both shell effects as well as scale-related ones. In addition, kinetic energy is introduced along similar lines which allow for kinematic computations involving scale effects as well.

4 Numerical Example The following study aims to explore possible application of the proposed generalized shell theory to size-scale effects and oriented material behaviour in elastic buckling of shells. Inspired by experiments of [3] which investigated the influence of longitudinal stiffeners and fibre reinforcement, respectively, towards the buckling behaviour of cylindrical shells, we want to show the developed shell theory can predict similar buckling behaviour in terms of stiffness increase as well as transition between various buckling modes. In this sense no quantitative predictions are sought but the general role of scaling effects and material orientation on buckling of shells. Therefore, we confine ourselves to perfect shell geometry and material. However, as only one eighth of the cylinder shown in Fig. 1 is modelled applying adequate symmetry conditions which can be viewed as an imperfection, we expect a lower critical buckling load and constraints on the evolution of the buckling pattern. The shell subjected to axial compression and the classical hyperelastic Saint-VenantKirchhoff constitutive law is utilized involving as material parameters the Young’s modulus E and Poisson’s ratio ν . The strain gradient theory outlined in the preceding sections is implemented in a moving least square-based meshfree code [1, 6]. The the dynamical modelling resorts to an explicit time integration scheme based on the midpoint rule. The Dirichlet boundary conditions are enforced by the modified boundary collocation method [8]. The micro-continuum S attached to each macroscopic point X ∈ B is chosen to be one-dimensional and its only director as defined in Eq. (12) follows the circumferential tangent on the cylindrical shell. The internal length scale parameter associated with the micro-director is kept constant throughout the entire macro-space B. The numerical integration over the micro-space is carried out with the help of Gauss quadrature, the order of which is chosen to be second according to the basis polynomial used for the MLS-approximation scheme, which is here a quadratic one. The discretization consists of 30 equally spaced particles in longitudinal, 20 in circumferential and 3 in thickness direction. This layout is found to provide a converged solution. The simulation is driven by a constant axial compression velocity vx = 10.0 mm/s and the resultant compression force is computed by integrating internal traction vector over the cylinder’s upper rim. Four different simulations are undertaken, the resulting load-compression curves are illustrated in Fig. 2. The dotted curve indicates the solution obtained by a classical Green strain tensor-based formulation [6], and the dashed curves show solutions determined by the generalized shell theory. The classical continuum formulation predicts a buckling behaviour which undergoes three modes, each with lower numbers of buckles. At first, when a critical buckling load of 1.37 kN at an axial compression of ux = 0.38 mm is reached, a large number of equally spaced

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circumferential wriggles appears and the compression load drops. Then ellipsoidal buckles form embedded in the circumferential wriggles. While these ellipsoidal buckles are being established, the circumferential wriggles disappear. The load continues to fall fast, as some of the ellipsoidal buckles merge. At ux = 1.25mm the compression loads arrives at an intermediate minimum of 0.37 kN, when the merging process is finalized as shown in Fig. 3. After a short reprive the buckling pattern is changing to its final configuration, as some of the remaining buckles begin to merge and the load is falling except for a short interruption, when the final stable pattern is about to manifest itself at ux = 3.69 mm with a load of 0.24 kN. Once this final configuration illustrated in Fig. 4 has settled in, the load is continuously decreasing at a very small rate without further change in the buckling mode. Making use of the generalized shell theory proposed in the previous section an internal length scale parameter l = 0.3 results almost in the same buckling behaviour except that the load level is slightly increased. Choosing l = 1.0 the instability point is

4.5 E = 1.003 x 104 kN/mm 2 í = 0.3 3 ñ = 1.2 kg/mm -4 c = 1.0 x 10 kNs/mm H = 200 mm R = 100 mm t = 0.25 mm

4

classical generalized l = 0.3 generalized l = 1.0 generalized l = 3.0

3.5 compression force [kN]

v

3 2.5 2 1.5 1 0.5

v = 10 mm/s

0 0

1

2

3

4

5

compression [mm]

Fig. 1 Problem definition of cylindrical shell subjected to axial compression

Fig. 2 Compression force vs. compression displacement diagram with different internal length scale parameters l

Fig. 3 Deformed configuration at |ux | = 1.25 mm (classical formulation)

Fig. 4 Deformed configuration at |ux | = 5.64 mm (classical formulation)

A Shell Theory with Higher Order Gradients

Fig. 5 Deformed configuration at |ux | = 0.79 mm with l = 1.0

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Fig. 6 deformed configuration at |ux | = 4.41 mm with l = 1.0

reached at ux = 0.46 mm for a critical buckling load of 1.61 kN. However, the forming of ellipsoidal buckles is delayed until the compression reaches ux = 0.68 mm for a corresponding load of 1.93 kN which is the absolute peak load. It can be clearly seen that the initial buckles form at the symmetry boundaries. Thus, there is clear grounds for speculation that the critical buckling load would be higher, if the entire cylinder were modelled. Subsequently, the load rapidly drops and a pattern of small ellipsoidal buckles depicted in Fig. 5 evolves similar to the classical case but at a larger compression of ux = 0.79 mm and higher load of 1.55 kN. At a compression of ux = 1.38 mm these buckles start to merge while load is falling down to 0.54 kN at a compression of ux = 3.19 mm. There, a short plateau of constant load is reached, as a large portion of the buckles has vanished. This is followed by another major merging process and a final load drop to 0.42 kN at ux = 4.41 mm leading to the final buckling pattern. This consists of four large buckles in the middle, the top as well as the bottom as shown in Fig. 6. Finally, Choosing l = 3.0 the critical buckling load of 2.72 kN at 0.73 mm is considerably higher than in the cases before. Similar to l = 1.0 the load further increases up to 4.19 kN at a compression of 1.48 mm, albeit at a much higher niveau. The following subsequent decrease of compression load is accompanied by the formation of a mixture of ellipsoidal and circular buckles. Their number however, is less than for l = 0.3 and l = 1.0, not so clear in shape, i.e. rather shallow, and larger in size. A last load drop occurs at a compression of 2.07 mm for 2.57 kN which leads to the same final buckling pattern as observed for l = 1.0. Summarizing, the one-dimensional micro-continuum increases the structural stiffness depending on the value of the internal length scale parameter and leads to a higher critical buckling load as well as delays the actual instability outset. If the choice of l is high enough, the final buckling mode is of lower order than predicted with the classical model.

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References 1. Belytschko, T., Gu, L., Lu, Y.Y.: Fracture and crack growth by element-free Galerkin methods. Model. Simu. Mater. Sc. 2, 519–534 (1994) 2. Bisagni, C., Cordisco, P.: An experimental investigation into the buckling and postbuckling of cfrp shells under combined axial and torsion loading. Compos. Struct. 60, 391–402 (2003) 3. Krasovsky, V.L., Kostyrko, V.V.: Experimental studying of buckling of stringer cylindrical shells under axial compression. Thin. Wall. Struct. 45, 877–882 (2007) 4. Sansour, C.: A theory and finite element formulation of shells at finite deformations involving thickness change: circumventing the use of a rotation tensor. Arch. Appl. Mech. 65, 194–216 (1995) 5. Sansour, C.: A unified concept of elastic-viscoplastic cosserat and micromorphic continua. J. Phys. IV. Proceedings 8, 341–348 (1998) 6. Skatulla, S., Sansour, C.: Essential boundary conditions in meshfree methods via a modified variational principle. Applications to shell computations. Comput. Ass. Mech. Eng. Sci. 15, 123–142 (2008) 7. Sun, Y., Liew, K.M.: Mesh-free simulation of single-walled carbon nanutubes using higher order cauchy-born rule. Comp. Mater. Sci. 42, 445–452 (2008) 8. Wagner, G.J., Liu, W.K.: Application of essential boundary conditions in mesh-free methods: a corrected collocation method. Int. J. Numer. Meth. Eng. 47, 1367–1379 (2000)

Chapter 14

An Electro-mechanically Coupled FE-Formulation for Piezoelectric Shells W. Wagner, K. Schulz, and S. Klinkel

From the beginning of my studies at the University of Hannover Erwin Stein inspired me to work in the field of mechanics. Since 1975 I was student assistant at the IBNM which brings me into contact with Peter, who was just finishing his education in civil engineering. It follows a joint work in Hannover until 1990. Especially I remember the (at that time) adventuresome computer based contacts during his visit at Berkeley in 1983-1984 in the ’pre-internet/email-time’. Afterwards in 1985-1990 we had a very fruitful cooperation, working together in one office of the IBNM. This leads to a number of joint activities, publications and our Habilitations. Finally Peter gets in 1990 a chair of Mechanics at the Technische Universit¨at Darmstadt and I am changing in 1994 as head of the Institute for Structural Analysis to the Karlsruher Institute of Technology (KIT) which completes nearly two fascinating decades (W. Wagner).

Abstract. Based on a mixed multi field variational formulation, a finite element formulation for piezoelectric shell structures is developed. Including the independent fields of displacements, electric potential, strains, electric fields, stresses, and dielectric displacements, the classical shell assumptions are extended to the electromechanically coupled problem. An arbitrary reference surface of the shell can be modeled with a four node element. The nodal degrees of freedom are three displacements, three rotations, and the difference of the electric potential in thickness direction. The formulation incorporates nonlinear kinematic assumptions, thus geometrical nonlinearities can be analyzed. According to a Reissner-Mindlin theory, the shell element accounts for constant transversal shear strains. The formulation incorporates a three-dimensional transversal isotropic material law, thus the kinematic W. Wagner · K. Schulz Institut f¨ur Baustatik, Karlsruher Institut f¨ur Technologie (KIT), Kaiserstr. 12, 76131 Karlsruhe, Germany e-mail: [email protected],[email protected] S. Klinkel Statik u. Dynamik d. Tragwerke, TU Kaiserslautern, Paul-Ehrlichstr. 14, 67663 Kaiserslautern, Germany e-mail: [email protected]

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in thickness direction of the shell is considered. The normal zero stress condition and the normal zero dielectric displacement condition of shells are enforced for the stress resultants by the corresponding independent fields.

1 Introduction Due to the inherent coupling of mechanical and electrical properties, piezoelectric materials can be used for sensor and actuator systems. Their applications range from precise positioning systems to structural health monitoring and vibration control. In order to reduce the costs and accelerate the design process, simulation via finite elements can be beneficial particularly for shell structures. Here, the consideration of geometrically and materially nonlinear behavior is required to get accurate results. With respect to a three dimensional material law, the strain and the electric field in thickness direction can be incorporated, thus ferroelectric hysteresis phenomena due to switching processes of the material polarization can be comprised.

2 Kinematics In order to display the geometry of the shell in the three dimensional Euklidean space B, the structure is modeled by a reference surface Ω with the boundary Γ . We denote a convected coordinate system of the body ξ i and an origin O with the global cartesian coordinate system ei . The shell has the initial thickness h in the reference configuration, thus we define the arbitrary reference surface by the thickness coordinate ξ 3 = 0 with h− ≤ ξ 3 ≤ h+ . By means of the convective coordinates, the position vectors of the shell surface Ω are given as X(ξ 1 , ξ 2 ) and x(ξ 1 , ξ 2 ) for the reference and the current configuration, respectively. The covariant tangent vectors for the reference and the current configuration are Ai and ai , respectively. D(ξ 1 , ξ 2 ) characterizes the director vector, which is given perpendicular to Ω and holds |D(ξ 1 , ξ 2 )| = 1. In the reference configuration it is defined D = A3 . For the current configuration, we obtain the corresponding inextensible director vector d by the orthogonal transformation d = RD with the rotation tensor R. Including a Reissner-Mindlin kinematic, we consider transverse shear strains, thus it holds d · x,α = 0. A displacement u can be displayed by the difference of the current and the initial position vectors u = x − X. The electric field Ei is defined as the gradient field of the electric potential ϕ . Due to the shell geometry we assume that the electrodes are arranged at the lower and upper surface and the piezoelectric material is poled in thickness direction. We therefore only consider the difference of the electric potential in thickness direction of the shell Δ ϕ . In the following, we refer to the notation that Greek indices range from 1 to 2, whereas we use the summation convention for repeated indices. Commas denote a partial differentiation with respect to the coordinates ξ α . We write the membrane strains εαβ , the curvatures καβ , the  shear strains γα , and the geometric electric field ( E g )α of the shell as

An Electro-mechanically Coupled FE-Formulation for Piezoelectric Shells

1 (x,α ·x,β −X,α ·X,β ) 2 1 καβ = (x,α ·d,β +x,β ·d,α −X,α ·D,β −X,β ·D,α ) 2 γα = x,α ·d − X,α ·D

117

εαβ =



( E g )α = −(Δ ϕ ),α

(1)

.

We summarize the strains and the electric field of the shell in a generalized geometric strain vector ε g (v). Herein it holds v = [u, ω , Δ ϕ ]T with the displacements u, the rotational parameters ω and the difference of the electric potential Δ ϕ . We define the relation between the Green-Lagrangean continuum strains ε and the independent shell strains ε¯ including the electric field by

ε = A ε¯ .

(2)

Here, the transformation matrix A contains constant approaches for the in-plane electric field and linear approaches for the strain and the electric field in thickness direction.

3 Constitutive Equations We write the linear constitutive equations with the Green-Lagrangean strain E, the  Lagrangean electric field E, the second Piola-Kirchhoff stresses S, and the dielectric  displacements D as      S E C −eT  =  . (3) −e −p −D E        σ ε ¯ C For a generalized formulation, we summarize the strain and the electric field in the vector ε and the dependent variables of the stresses and the dielectric displacements in σ . The three dimensional elasticity matrix C, the permittivity matrix p and the ¯ In C we assume transversal piezoelectric coupling modulus e are arranged in C. isotropic material behavior, which can be specified by five independent parameters. The polarization in thickness direction of the structure leads to a piezoelectric modulus e with three independent coefficients and a permittivity matrix p with two ¯ including independent coefficients. One obtains the stress resultants of the shell S, the mechanical stress resultants and the dielectric displacements, from the thickness integration S¯ =

 h+ h−

A σ μ¯ dξ 3 = T

 h+



h−

¯ A μ¯ dξ 3 ε¯ AT C   D

.

(4)

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Applying the material law, we derive the material matrix D of the shell. For slightly curved shells the determinant of the Shifter tensor μ¯ can be assumed as μ¯ ≈ 1. Regarding layered structures, we calculate D as the sum of the material matrix of every layer k as n

D=

n

∑ Dk = ∑

 hk+

k=1 hk−

k=1

¯ k A μ¯ dξ 3 AT C

.

In order to fulfill the normal zero stress condition of shells, we assume the vector of the independent stress resultants and the independent dielectric displacements as  T Sˆ¯ = Sˆ 0

(5)

.

Here, we define the stress and the dielectric displacement in thickness direction as zero.

4 Finite Element Approximation We model the shell structure by means of a reference surface. For the finite element formulation, we choose a four node element with bilinear shape functions NI . With respect to the isoparametric concept, the geometry, the director vector, and the mechanical nodal degrees of freedom uh and the electric potential Δ ϕ h are approximated on element level by the same shape functions as Xh =

4

∑

I=1

NI XI

Dh =



4

∑

I=1

NI DI

 4 uh = ∑ NI vI = N ve h Δϕ I=1

(6)

with vTe = [vT1 vT2 vT3 vT4 ] and vI = [u1 u2 u3 ω1 ω2 ω3 Δ ϕ ]TI . The superscript h denotes the characteristic size of the element discretization and indicates the finite element approximation. The nodal position vector XI and the local cartesian coordinate system [A1I , A2I , A3I ] are generated from the mesh input. Here, DI = A3I is set perpendicular to the surface Ω and A1I , A2I are built fulfilling the boundary conditions. We remark that the orthogonality holds true only for the nodal points. We calculate a local cartesian basis system ti according to [5] for every element with t3 as the normal vector in the midpoint of the element. t1 and t2 span the tangent plane in the element midpoint. The reference surface in the current configuration is approximated by the position vector xh and the director vector dh , both of which are set up analogously to (6). xI = XI + uI characterizes the position vector in the current configuration for every node and we obtain dI = a3I by the orthogonal transformation akI = RI AkI with k = 1, 2, 3. The rotational parameters ωkI are arranged in the vector ω I = [ω1I , ω2I , ω3I ]T . The rotation tensor RI as a function of ω I is calculated via the Rodrigues formula. The only electrical degree of freedom at each element node is the difference of the electric potential in thickness direction of the shell Δ ϕ . Thus the nodal degrees

An Electro-mechanically Coupled FE-Formulation for Piezoelectric Shells

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of freedom are three local displacements, three local rotations and the difference of the electric potential. We approximate the generalized variational vector of the geometric strain and the geometric electric field δ ε g by means of the matrix B

δεg =

4

∑ BI δ vI = B δ ve

(7)

.

I=1

In order to avoid shear locking phenomena, the shear strains based on linear displacement interpolations in (1)3 have to be substituted by the strains defined in [1]. Thus, we formulate the approximation of the shell strains according to [2]. The independent fields of the strains and the electric field are interpolated by ¯ ε α¯ , εˆ¯ = N

ˆ ε α, εˆ = N

Sˆ = Nσ β

.

(8)

Here, εˆ¯ characterizes the complete vector of the assumed strains and the assumed electric fields, whereas εˆ specifies the reduced vector without the components in thickness direction. α¯ , α , and β contain independent variables that can be eliminated by static condensation on element level. Sˆ characterizes the vector of the independent stress resultants and the independent dielectric displacements without the components in thickness direction. The interpolation of the membrane and bendˆ¯ T and ing strains corresponds to the procedure introduced by [4]. With θ [v, εˆ¯ , S] the stored energy W , which is a quadratic function of the independent strains and the electric field, the variational formulation of the electro-mechanically coupled problem reads  3

G(θ , δ θ ) =



+

Ω

Ω

+

 4 δ ε Tg Sˆ − δ vT b¯ dA − δ vT ¯t ds

Ω

Γσ

T ˆ¯ dA δ εˆ¯ (∂εˆ¯ W − S)

δ Sˆ T (ε g − εˆ ) dA

=0 .

(9)

Incorporating the interpolations of the strains, the electric fields, the stress resultants and the dielectric displacements in Eq. (9), we formulate the approximation of the variational formulation on element level as G(θ , δ θ ) = δ vTe



+ δ α¯ Te +

δ β Te



Ωe





Ωe Ωe

 BT Nσ β − fa dA  

  T T ¯ α Nε ∂εˆ¯ W dA − δ e

NTσ

ˆ εα ε g − NTσ N



 Ωe

dA

 ˆ Tε Nσ β dA N (10)

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with fa as the vector of the external loads. The linearized variational formulation is given by L [G(θ , δ θ ), Δ θ ] := G(θ , δ θ ) + D G · Δ θ = 0  3 4 T ˆ¯ + δ Sˆ T (Δ ε − Δ εˆ ) + δ Δ ε T sˆ dA D G ·Δθ = δ ε Tg Δ Sˆ + δ εˆ¯ (D Δ εˆ¯ − Δ S) g g Ωe

(11) which can be solved by means of static condensation. Considering an actuator use of piezoelectric shell structures, we assume a linear distribution of the electric potential in thickness direction. Thus, a constant value is assigned to the nodal degree of freedom Δ ϕ . We define the corresponding electric field in thickness direction as the average value for every element by 

E e3 = −

1 4

Δ ϕI I=1 h 4

∑

(12)

.

According to the gradient relation, the electric potential is divided by the thickness.

5 Numerical Example In order to test the element formulation, we analyze the steering of a smart antenna. The system consists of a parabolic antenna shell with four isochronously arranged piezoelectric segments related to [3]. The system and the material data are shown in Fig. 1. We completed the missing material parameters, marked by ()∗ , with empirical data of similar material behavior. The geometry is depicted in Fig. 2, as top view and as cut through the symmetry plane at AC. The antenna has four piezoelectric actuator segments, holding an angle of 11.25◦ each and isochronously attached at the lower surface. The shell and the segments have a thickness of

Aluminum shell: E = 20.0 · 109 mN2 ν = 0.3 Piezo actuators: E = 66.0 · 109 mN2 ν = 0.178 e13 = −5.6 mC2 e33 = 17.3 mC2

e∗15 = 13.0 mC2 2 p∗11 = p∗33 = 1.53 · 10−8 NCm2 Fig. 1 System and material parameters of the antenna with piezoelectric segments

An Electro-mechanically Coupled FE-Formulation for Piezoelectric Shells

D

[mm]

121

120°

11.25° 11.25°

C

A

11.25° 11.25°

B

C

z 175

x

175

A

175

75

3°

Fig. 2 Geometry of the parabolic antenna

t = 0.203 mm. The geometry of the antenna shell is characterized by the equation 7.5 2 2 z = 17.5 2 (x + y ) − 7.5. We model the structure with 64 elements in circumferential direction and 16 elements from the inner hole to the top edge. According to the anchorage of the antenna by means of a screw in the middle of the antenna shell, the displacement degrees of freedom at the inner hole are fixed. Furthermore, we fix the rotation around the local z-axis at the hole and at the top edge. Holding a polarization in thickness direction, the piezoelectric material is attached to the structure in the way, that the polarization direction is set from outside to the inside of the shell. In order to change the range of the antenna, we impress a voltage on the piezoelectric segments. Here, we distinguish between the following loading cases. Loading Case 1 2 3

Loading Δ ϕ on piezoelectric actuator segments Segment A Segment B Segment C Segment D +300V +300V +300V +300V +300V -300V +300V -300V +300V +300V -300V -300V

Fig. 3 shows the original state of the antenna and the deformed structure along the axis AC and BD, respectively, for the different loading options. Here, we introduce appropriate scaling factors (see Fig. 3) for an easier evaluation of the deformation. For the loading case 1, the antenna completely expands, while the total displacements are small. Thus, the whole range of the antenna can be extended. The displacement in z-direction for the points A,B,C, and D is 0.715 mm. The loading case 2 enlarges the range in one axis and diminishes it perpendicular to this axis. Here, the displacement at the points A and C holds −4.86 mm and at the points B and D +4.86 mm. A rotation of the antenna to the left hand side can be reached by the loading of case 3. The displacement in z-direction of A and B counts −1.359 mm and for C and D +1.359 mm.

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Y

(a) Loading case 1 (A=B=C=D=300V), scaling factor 20; x = X + 20u.

(b) Loading case 2 (A=C=300V, B=D=-300V) , scaling factor 5; x = X + 5u.

(c) Loading case 3 (A=B=300V, C=D=-300V) , scaling factor 10; x = X + 10u. Fig. 3 Deformation of the parabolic antenna for the loading cases 1, 2 and 3 along the axis AC and BD ( ——– original state, – – – deformed state)

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References 1. Dvorkin, E., Bathe, K.-J.: A continuum mechanics based four node shell element for general nonlinear analysis. Eng. Computation 1, 77–88 (1984) 2. Gruttmann, F., Wagner, W.: Structural analysis of composite laminates using a mixed hybrid shell element. Comput. Mech. 37, 479–497 (2006) 3. Gupta, V.K., Seshu, P., Kurien Issac, K., Shevgaonkar, R.K.: Optimal steering of paraboloid antenna using piezoelectric actuators. Smart. Mater. Struct. 16, 67–75 (2007) 4. Simo, J.C., Fox, D.D., Rifai, M.S.: On a stress resultant geometrically exact shell model. Part II: the linear theory. Comput. Method. Appl. M. 73, 53–92 (1989) 5. Taylor, R.L.: Finite element analysis of linear shell problems. In: Whiteman, J.R. (ed.) The Mathematics of Finite Elements and Applications VI. Academic Press, London (1987)

Chapter 15

Non-intrusive Coupling: An Attempt to Merge Industrial and Research Software Capabilities Olivier Allix, Lionel Gendre, Pierre Gosselet, and Guillaume Guguin I have had the great pleasure to cooperate for several years now closely with Peter Wriggers who has been several times invited in LMT-Cachan. I have always been impressed by the profound understanding and practical knowledge of Peter regarding computational and material mechanics. I know that Peter has a deep concern regarding the application of fundamental researches which is the motivation of this paper. We hope to connect it more closely in the future to the seminal work of Peter, especially regarding contact and multiscale stochastic modeling of heterogeneous materials and damage. The recently accepted International Research Training Group Virtual Material and Structures and their Validation is therefore an exceptional possibility for us to continue and reinforce a close relationship with Peter, a great scientist and a great friend (O. Allix).

Abstract. In computational mechanics, it is often difficult to test research innovations on industrial problems because of software limitations: many of the commercial finite element packages commonly used in the industry lack flexibility and openness, whereas in-house research developments are usually very specific and may lack features required for “real-life” industrial simulations. Non-intrusive coupling is a tentative answer to this problem. It consists in introducing local enhancements and refinements into an existing industrial problem through a separate nonlinear local model that comes with its own solver; the two models are coupled by the means of an iterative exchange algorithm inspired from domain decomposition methods and multiphysics solution techniques, using both models and solvers without any modification. So far, the method has been implemented around the finite element package Abaqus and has been used to introduce local plasticity and geometric details into a linear elastic global problem. While current developments include the simulation of localized damage in slender composite structures, we think that the method could be adapted to a wide class of problems including hybrid experimentalsimulation approaches. Olivier Allix · Lionel Gendre · Pierre Gosselet · Guillaume Guguin LMT-Cachan, 61 avenue du Pr´esident Wilson, F-94235 Cachan Cedex, France e-mail: [email protected],[email protected], [email protected],[email protected]

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1 Introduction In the last decade, many innovative modelling or solution techniques have been introduced in the field of computational mechanics. These techniques, such as enriched finite elements or multiscale models, enable performing complex simulations that are out of reach of conventional finite element analysis (FEA) tools, in terms of computational or human costs. However, although these techniques have proved their performance by extensive testing on academic applications, they are scarcely applied on actual industrial problems because they cannot be conveniently implemented into commercial FEA software packages, which are the basis of most industrial computational environments. Non-intrusive coupling [12, 11] is a tentative answer to these limitations. It takes advantage of the fact that, in many industrial simulations, sources of difficulties (which are usually nonlinear phenomena, sometimes occurring at fine scales) are localized in small regions, and that the innovative techniques mentioned above were specifically designed to overcome such difficulties efficiently. Thus, the essential idea of non-intrusive coupling is to enhance an existing industrial simulation, that involves a complex model data set and a commercial FEA solver, by the means of a separate local model that is analyzed with its own dedicated solver. This way, the local model may contain innovative features that cannot be implemented conveniently into the commercial solver. The term “non-intrusive” means that in the process, neither the models nor the solvers need to be modified; they are used as “black boxes”, and a script is used to run the analyses and exchange displacements and forces between them. Of course, this non-intrusive framework also has a significant drawback: one has to do with the limitations of the commercial FEA solver that is used. This can impact performance, particularly when using software that are not optimized for implicit solver coupling schemes. However, we believe that this possibly non-optimal performance is a fair price to pay for the convenience of such a black-box tool; in addition, depending on the solver, several adjustments can be performed to reduce computational costs dramatically. At the moment, the computational efficiency of such a non-intrusive strategy is still an open question. What is certain is that this framework provides a way of performing enriched simulations for which no “monolithic” software is available at the present time. The rest of this chapter is organized as follows. Sect. 2 reviews the essential ideas of non-intrusive coupling and the different ways it can be made more efficient. Sect. 3 presents an application of this strategy to localized plasticity and first results on damage problems.

2 The General Principles of Non-intrusive Coupling The proposed analysis strategy starts from an existing “industrial” model, analyzed using commercial FEA software; its behaviour is supposed to be completely linear

Non-intrusive Enhancement of Industrial Models

127

(linear elasticity under small perturbations without contacts) and static. This model is called the global model and is schematized on Fig. 1a. Let us assume that in reality, nonlinear phenomena may occur in a small region of the structure Ω , denoted ΩI and called the area of interest; in the remaining region ΩC = Ω \ ΩI , called the complement area, the global model is assumed to be valid. In order to take those nonlinear phenomena into account, we would like to use innovative techniques or models that cannot be implemented conveniently into the global solver, as explained in the introduction. In addition, we suppose those phenomena may interact with small local details (such as holes or cracks) that were omitted in the global model. Therefore, we suggest defining a separate local model, limited to ΩI alone, that contains all the desired enhancements and is analyzed using its own dedicated nonlinear solver.This model is schematized on Fig. 1b. In this article, it is supposed to possess a standard finite element formulation and to be geometrically and kinematically compatible with the global model on the interface Γ = ΩI ∩ ΩC . However, this is a simplification rather than a fundamental limitation of the method, and noncompatible discretizations could very well be used as long as appropriate transfer operators are defined (for example, using mortar techniques [4]).


























(a) Global

(b) Local

(c) Reference

Fig. 1 Global, local and reference problems

Starting from these models, a coupling method is then used. The following sections present the two main ideas of this method.

2.1 Piecewise Substitution Though the two models are overlapping, contrary to classical “patch” methods [2, 3, 8, 10] which define the solution as a combination of a global and a local term, we prefer to refer to non-overlapping formulations (as used in fluid-structure interaction [22], multiscale simulations [25] and nonlinear domain decomposition methods [7, 21]) which limit the exchanges to surface data and seem more prone to the separate handling of nonlinearity. Therefore, we wish to eliminate the overlap from the formulation. For that purpose, the reference problem that we wish to solve is defined by piecewise substitution of the local model into the global model,

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as shown on Fig. 1c. Likewise, the solution to this problem is sought by piecewise substitution of the local solution into the global solution, that is: s|Ω = sL I

and

s|Ω = sG |Ω C

(1)

C

where s is the set of the solution fields (i.e. the displacement field and the Cauchy stress field) and superscripts G and L respectively denote quantities from the global and the local problems. In other terms, the global solution that lies “under” the local model is not retained: the overlap is eliminated. This way, the solution to the reference problem can be obtained from the two models and solvers by finding two solutions sL and sG such that (see left column): sL verifies every equation written in ΩI and its boundary. the restriction of sG to ΩC verifies every 2. equation written in ΩC and its boundary 1.

3. sL and sG match on Γ

hL (wL ) = λ L SCG wG − bCG = λCG wG = wL λCG + λ L = 0

The right column corresponds to the condensed version of the equation, where w is the interface nodal displacement and λ the interface nodal reaction force. The linear operator SCG is the Schur complement of the complement area (from the global model) and bCG is the associated condensed right-hand side vector. The nonlinear operator hL formally represents the local problem’s reaction to prescribed interface displacements. This set of three conditions is called the global-local formulation and its solution is trivially equal to the reference solution (assuming it is unique). It defines a surface coupling between the two models. The corresponding equations can be found in [12] in a continuous form, and in [11] in a discrete, condensed form as above. From a computational point of view, a crucial advantage of this formulation is the ability to use a nonlinear local solver to handle local nonlinearity, instead of relying on global iterations only; this principle, as used in [7, 21] in presence of localized but pronounced nonlinearity, can lead to huge savings in computational costs.

2.2 Iterative Coupling To enforce this coupling and solve the global/local formulation, several approaches can be imagined. In order to be both exact and non-intrusive at a reasonable cost, using an iterative algorithm is the most relevant choice, as suggested in early literature on global/local analysis [18, 23]. In accordance to the non-intrusive framework, the algorithm only consists in computing the responses to prescribed loads on the different models; no direct matrix manipulations are used. To meet those requirements, we have chosen a simple modified Newton method on interface

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quantities. This method starts from the initial elastic solution, then each iteration goes as follows. 1. Local analysis: the local nonlinear problem is solved, with part of the current global solution prescribed as a boundary condition on Γ . This condition can be prescribed displacements, prescribed nodal forces or a mixed condition, as it will be detailed below. For example, when using the displacement condition, we compute the reaction λ L as:

λ L = hL (wL ) .

(2)

2. Residual computation: an interface load vector called the residual is computed. It measures the non-verification of interface coupling equations. Convergence is tested here: if the residual’s norm is small enough, iterations are stopped. For example, if prescribed displacements were used, then the residual is the sum of nodal reaction forces between the two models (which should be zero if they were balanced): r = λ L + λCG (3) where λCG is extracted from the current global solution, either by classical postprocessing using elementary integration of stresses or by using an additional linear model to get reaction forces to the current global displacements. 3. Global correction: otherwise, the residual is injected into the global problem as an additional interface load. This is done by first solving a corrective global problem loaded only with the residual (all other loads and boundary conditions are set to zero) : Δ wG = (SCG + SIG)−1 r . (4) This computation is analogous to the linear step of a modified Newton iteration. Therefore, the effectiveness of the correction step can be improved at no extra cost by using a quasi-Newton update formula such as SR1 or BFGS [1, 19], written in a non-intrusive form as shown in [12]. Finally, the corresponding corrective solution Δ sG is added to the current global solution before going back to step 1. This class of algorithms has two important properties. First, if the algorithm converges, it is easy to prove that its limit is the fully coupled solution – that is, the reference solution [12, 11]. Therefore, the coupling method is reliable: the error can be estimated through the norm of the residuals and reduced as much as needed. Second, the method is indeed non-intrusive since the solvers operate as black boxes, and the model data sets are never modified. It only requires sending boundary conditions to the local solver or additional loads to the global solver, and reading interface nodal displacements and forces which are routine FEA operations. As a consequence, the algorithm’s implementation should be light and can use a high-level scripting language as provided by many FEA packages.

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2.3 Choice of the Interface Boundary Condition for the Local Step The simplest boundary condition that can be used on the local model is prescribed displacements, the value of which is extracted from the current global solution. This choice enables to work with kinematically admissible displacement fields. For that reason, it is popular in global/local [6, 18, 23] or multiscale analysis [9, 20], and submodelling with displacement conditions is natively available in many commercial FEA packages. Algorithms using this condition were studied in [12] for local plasticity. However, prescribing a displacement field is arguably not the most realistic way of emulating the influence of the rest of the structure on the local model. Robin conditions are known to be more efficient. They have been experimented in the field of global/local analysis [13, 14] and widely studied in domain decomposition methods [15, 17]. They consist in prescribing a linear combination of displacement and efforts; the factor that appears in this linear combination is an interface stiffness matrix. The choice of this matrix has a huge impact on performance. Intuitively, it emulates the “mechanical impedance” of the rest of the structure, and it is wellknown that it should give a correct approximation of the Schur complement of the linear region’s contribution. In [11] we proposed a two-scale approximation where the macro part is evaluated by taking the global model’s response to selected loads that represent “long-range” effects, whereas the micro part is given by the Schur complement of a narrow strip of elements adjacent to the interface.

3 Examples Using Abaqus/Standard We consider the 2D toy problem represented on Figs. 1a and 1b. The local problem’s constitutive model is elastic-plastic with linear isotropic hardening; the load is applied in one single increment because of software constraints, and its intensity is such that the elastic limit is slightly exceeded in the area of interest. To assess the coupling scheme’s performance, we solved the reference problem and computed, at each iteration, the relative error on the maximum cumulative plastic strain (with respect to its reference value). This particular quantity was chosen because it is a common goal of the analysis on many elastic-plastic applications (such as estimating the lifespan of ductile structures submitted to cyclic loadings) and because it is highly sensitive to the local model’s boundary condition. The evolution of this relative error during the iterations is shown on Fig. 2, for four different variants of the algorithm (with prescribed displacements or Robin conditions, with or without quasi-Newton acceleration). It appears that except for the simplest version (prescribed displacements without acceleration), all variants converge very quickly: an error of 10−3 on the maximum plastic strain (which is a quite strict threshold, by engineering standards) is reached after 2 or 3 iterations. The corresponding cumulative plastic strain maps are shown in Fig. 3, at three instants of the strategy.

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Fig. 2 Convergence of maximum plastic strains on the 2D example with localized plasticity

(a) Initial, displacements

(b) Initial, mixed

(c) Converged (reference) Fig. 3 Cumulative plastic strain maps at the first iteration and at convergence

As a first study on damage problems, we also considered the same geometry with an isotropic damage law (with bounded rate of damage). For moderate damage, the method behaves the same way (slow convergence for basic displacement approach, effective acceleration with SR1 and better performance with mixed conditions). The softening associated to damage might be the source of many difficulties that we have to deal with, like localization, instabilities (which require a local control), and dependency on the interface’s location.

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4 Conclusion We proposed a coupling approach to enhance a global linear model by a local model with refined geometry and non-linear constitutive equation. The approach is nonintrusive so that it can link industrial FEA software to in-house research code. Performance can be improved by using relevant boundary conditions on the local model and using acceleration techniques, leading to an efficient method which was validated on large industrial 3D localized plasticity problems. Prospects concern coupling between plates and 3D models, local treatment of many loading steps, application to damage problems like delamination in composites and to problems involving contact [24] in both local and global models.

References 1. Akg¨un, M.A., Garcelon, J.H., Haftka, R.T.: Fast exact linear and nonlinear structural reanalysis and the Sherman-Morrison-Woodbury formulas. Int. J. Numer. Meth. Eng. 50, 1587–1606 (2001) 2. Babuska, I., Andersson, B.: The splitting method as a tool for multiple damage analysis. SIAM J. Sci. Comput. 26(4), 1114–1145 (2005) 3. Ben Dhia, H.: Multiscale mechanical problems: the Arlequin method. Comptes-rendus de l’Acad´emie des Sciences IIb 326, 899–904 (1998) 4. Bernardi, C., Maday, Y., Patera, A.T.: A new nonconforming approach to domain decomposition: the mortar element method. In: Br´ezis, H., Lions, J.L. (eds.) Nonlinear partial differential equations and their applications, Coll`ege de France seminar, Pitman, London (1991) 5. Boucard, P. A., Ladev`eze, P., Poss, M., Roug´ee, P.: A nonincremental approach for large displacement problems. Comput. Struct. 64(1–4), 499–508 (1997) 6. Cormier, N.G., Smallwood, B.S., Sinclair, G.B., Meda, G.: Aggressive submodelling of stress concentrations. Int. J. Numer. Meth. Eng. 46, 889–909 (1999) 7. Cresta, P., Allix, O., Rey, C., Guinard, S.: Nonlinear localization strategies for domain decomposition methods: application to post-buckling analyses. Comput. Meth. Appl. M. 196(8), 1436–1446 (2007) 8. D¨uster, A., Rank, E., Steinl, G., Wunderlich, W.: A combination of an h- and a p-version of the finite element method for elastic-plastic problems. In: Wunderlich, W. (ed.) CDROM proceedings of the European Conference on Computational Mechanics, ECCM 1999, Munich (1999) 9. Feyel, F., Chaboche, J.L.: FE2 multiscale approach for modelling the elastoviscoplastic behavior of long fiber SiC/Ti composite materials. Comput. Meth. Appl. M. 126, 17–38 (2000) 10. Fish, J.: The s-version of the finite element method. Comput. Struct. 43(3), 539–547 (1992) 11. Gendre, L., Allix, O., Gosselet, P.: A two-scale approximation of the Schur complement and its use for non-intrusive coupling. Submitted to: Int. J. Numer. Meth. Eng. (2010) 12. Gendre, L., Allix, O., Gosselet, P., Comte, F.: Non-intrusive and exact global/local techniques for structural problems with local plasticity. Comput. Mech. 44, 233–245 (2009) 13. Hirai, I., Wang, B.P., Pilkey, W.D.: An efficient zooming method for finite element analysis. Int. J. Numer. Meth. Eng. 20, 1671–1683 (1984)

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14. Jara-Almonte, C.C., Knight, C.E.: The specified boundary stiffness and force (SBSF) method for finite element subregion analysis. Int. J. Numer. Meth. Eng. 26, 1567–1578 (1988) 15. Ladev`eze, P., Dureisseix, D.: A micro/macro approach for parallel computing of heterogeneous structures. Int. J. Comput. Civ. Struct. Eng. 1, 18–28 (2000) 16. Ladev`eze, P., Simmonds, J.: New concepts for linear beam theory with arbitrary geometry and loading. Eur. J. Mech. A/Solid 17(3), 377–402 (1998) 17. Magoul`es, F., Roux, F.X., Series, L.: Algebraic approximations of Dirichlet-to-Neumann maps for the equations of linear elasticity. Comput. Meth. Appl. M. 195, 3742–3759 (2006) 18. Mao, K.M., Sun, C.T.: A refined global-local finite element analysis method. Int. J. Numer. Meth. Eng. 32, 29–43 (1991) 19. Matthies, H., Strang, G.: The solution of nonlinear finite element equation. Int. J. Numer. Meth. Eng. 14(11), 1613–1626 (1979) 20. Oden, J.T., Zohdi, T.I.: Analysis and adaptive modeling of highly heterogeneous structures. Comput. Meth. Appl. M. 148, 367–392 (1997) 21. Pebrel, J., Rey, C., Gosselet, P.: A nonlinear dual domain decomposition method: application to structural problems with damage. Int. J. Multiscale. Comp. Eng. 6, 251–262 (2008) 22. Piperno, S., Farhat, C., Larrouturou, B.: Partitioned procedures for the transient solution of coupled aeroelastic problems. Comput. Meth. Appl. M. 20, 1638–1685 (1995) 23. Whitcomb, J.D.: Iterative global-local finite element analysis. Comput. Struct. 40(4), 1027–1031 (1991) 24. Wriggers, P.: Computational Contact Mechanics. Springer, Heidelberg (2006) 25. Zohdi, T.I., Wriggers, P., Huet, C.: A method of substructuring large-scale computationalmicromechanical problems. Comput. Meth. Appl. M. 190(4344), 5639–5656 (2001)

Chapter 16

Constitutive Models and Failure Prediction for Al-Alloys in Industrial Applications Christian Leppin

In 1993, when Peter Wriggers took me on as a doctoral student in his group in Darmstadt, my knowledge in mechanics and finite element methods was rather limited in retrospect. In this environment of scientific ambition and creativity which he provided it was then a very steep learning curve for me. Besides his superior expertise, I very much appreciated his patience and the freedom he would give his students. In my current work at an engineering department in industrial R&D in the area of material science I still benefit very much from all the things I learned under his tutelage (C. Leppin).

Abstract. There are some specific aspects of aluminum alloys which are to be considered in forming and crashworthiness simulations of aluminum components used in industrial applications. Firstly, some details about a material need to been known, since not just the alloy composition, but also the production process of the semifinished product can significantly influence the deformation properties. Due to its FCC lattice structure, the elasto-plastic deformation of aluminum shows some features different from standard metals. For good results in FEA of forming processes, especially when fracture is to be predicted, the specific characteristics of the yield locus and the work-hardening behavior need to be reflected appropriately in the numerical model.

1 Introduction Most prominently the automotive industry has adopted numerical modeling in its product development process to save time and costs. Other industries such as Christian Leppin 3A Technology & Management AG, Badische Bahnhofstrasse 16, 8212 Neuhausen am Rheinfall, Switzerland e-mail: [email protected]

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packaging have started to do the same. Regardless of the application, appropriate models for the prediction of the deformation and fracture behavior are crucial. The following paper gives a brief overview on topics and features specific to aluminum alloys, with stamping, bending and hydroforming of the semi-finished categories sheet and foil material, as well as extrusions in mind.

2 Factors Influencing Properties There is a large choice of aluminum alloys featuring different properties adapted to various specific applications. A key factor defining specific properties of an alloy certainly is the chemical composition; however the production process of an aluminum product can be just as important. As an example, Fig. 1 shows that for the same extrusion alloy mechanical properties such as strength, hardening and ductility can be customized to diverging needs of different automotive components by thermal treatment. Similarly, different process routes in a rolling mill can induce very different properties to the same alloy. It is important to keep this in mind for numerical modeling of forming processes with aluminum alloys, since a data set for a specific alloy does not necessarily apply to every situation.

Fig. 1 Example for customizing properties by thermal treatment – tensile test results of extruded AlMgSi0.5V (AA6014) for different temper conditions

3 Work-Hardening of Aluminum Alloys Fig. 2 shows examples of tensile test results for several aluminum alloys and temper conditions, giving an impression of their work-hardening behavior. Obviously the characteristics can be quite dissimilar. Noteworthy is the alloy AA5754-O (AlMg3) with its distinctive yield point and the pronounced serrated yielding due to the Portevin - Le Chatelier effect. Although the underlying micro-structural processes can have quite an effect on the meso-scale, both effects generally are ignored in numerical modeling with good results.

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Fig. 2 Examples of tensile test data for different aluminum alloys

Various empirical models for approximations of the work-hardening behavior exist, e.g.: Hollomon: Ludwik: Swift: Voce: Hocket-Sherby:

σ (ε ) = K ε n σ (ε ) = σ0 + K ε n σ (ε ) = K(ε0 + ε )n

(1) (2) (3)

σ (ε ) = σ∞ − (σ∞ − σ0 )e

−nε

σ (ε ) = σ∞ − (σ∞ − σ0 )e

−nε m

(4) .

(5)

All approximations include some form of exponential function. Unfortunately, for many aluminum alloys the exponent n is in fact not constant irrespective which model is applied. Therefore, great care has to be taken during parameter identification. Using the extrusion alloy 6008 T4 as an example, Fig. 3 shows the results of parameter identifications for the first four approximation models listed above, based on least square fits of the tensile test data. Furthermore, the uniform elongation of the tensile test is indicated. In addition, the uniform elongation for each curve fit is shown, based on the Consid´ere criterion for the onset of diffuse necking in a uni-axial tensile test:

∂σ =σ ∂φ

,

(6)

i.e. the intersection of each stress curve with its derivative. In the actual case the Voce approximation comes fairly close to the uniform elongation originally measured in the test, while the other approximations significantly overestimate uniform elongation, thereby potentially overestimating the general formability of the material in predictive forming simulations. Plotting the stress derivative against the stress itself is a good method to verify how physically meaningful an approximation of given test data is, as shown in Fig. 4.

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Fig. 3 Parameter identification for the work-hardening model – least square fits of the test data for different approximations, plus resulting uniform elongation for 6008 T4

Fig. 4 Parameter identification for the work-hardening model – verification of least square fit results for different approximations for 6008 T4

4 Yield Locus Besides the work-hardening model, the yield locus is another key feature of elastoplastic material models commonly used in forming simulation, since it defines the multi-axial response of a material to a given loading. Fig. 5 shows a typical yield locus of an aluminum alloy computed by a crystal plasticity model based on the analysis of the micro-structural texture in comparison with a von Mises approximation. Typical for metals with a FCC lattice structure, the shape of the yield locus is not elliptical; instead it resembles a Tresca criterion, featuring pronounced corners with rather straight, low curvature sections in between. Furthermore, aluminum alloys can show so-called ”anomalous” behavior, meaning a combination of r-ratios, respectively Lankford coefficients less than 1 with an equi-biaxial stress point at a higher level than the uni-axial stresses. Moreover, aluminum alloys generally show anisotropic behavior, typically with r-ratios in rolling direction (or extrusion direction) being lower than r-ratios in transverse direction, while at the same time strength in rolling direction (or extrusion direction) often tends to be higher than in transverse direction, see also summary of typical aluminum alloy phenomena related to the yield locus in Fig. 6. Finally, some aluminum alloys show 6 to 8 ears after

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Fig. 5 Typical yield locus of an aluminum alloy (based on texture analysis) and comparison with a von Mises approximation

Fig. 6 Summary of typical aluminum alloy phenomena related to the yield locus

drawing of a simple circular cup. In recent years increasingly complex yield locus model formulations, in 2D as well as in 3D, have been developed to account for the mentioned phenomena: see Barlat and Lian [5], Hill [7], Vegter [13], Barlat [4], Banabic et al. [1], Bron [6] and Barlat [2]. Depending on the number of model parameters, the experimental effort to identify these parameters can be considerable. Typically this involves the following mechanical testing program: Tensile tests in 0◦ , 45◦ and 90◦ with respect the rolling direction including measurement of the r-ratios. Furthermore, for identification of the equi-biaxial stress point, as well as to improve the representation of the hardening beyond uniform elongation, a hydraulic bulge test or alternatively a uni-axial layered compression test is recommended. Finally, a shear test is advisable to identify the situation in the shear region. Such an elaborate mechanical testing program to characterize a material is rather expensive and still gives rather limited insight into the overall behavior of a material. As a promising alternative, the yield locus can also be identified by analysis of the micro-structural texture via x-ray diffraction measurements and a crystal plasticity model to transfer

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the crystal orientation distribution function into potential responses of the material to external loading. Fairly good results have been achieved with a comparatively simple full constraint Taylor model, however it has also failed for some alloys and there is ongoing research for a more generally applicable approach.

5 Fracture Prediction Generally, the following different fracture mechanisms can be observed in metals: material failure by shear fracture and/or normal fracture, observed e.g. during trimming and hemming; furthermore, with thin-walled materials, localized necking due to membrane instability with subsequent material failure, as it typically is observed in stamping processes. Consequently, fracture criteria in FEA are chosen accordingly. Hereby, strain based formulations have the advantage of higher sensitivity over stress based formulations, due to the typically weak hardening of a formable material at the point of failure. Fig. 7 shows the concept of a simple model for material failure due to shear fracture or normal fracture. In this model, for shear and normal fracture respectively, the equivalent plastic strain at fracture is given as a function of the ratio of the mean stress and the von Mises equivalent stress (i.e. stress“triaxialit”), and as a function of the strain-rate. Note, that unlike other metals, the fracture strain of aluminum alloys typically increases with strain-rate. This fracture criterion concept can be further refined to account for strain-path dependency and anisotropic effects, as has been demonstrated by [8]. For fracture criteria to function properly, good predictions of stresses and strains are crucial. As a consequence, the quality of the model for the plastic deformation of the material (see Sects. 3 and 4) is just as important as the fracture criterion itself. This is especially true for failure by localized necking due to lack of hardening of the material, leading to membrane instability, see e.g. the investigation by [9]. Here, physically, the plastic deformation behavior itself defines when failure starts. This is not reflected in the approach of classical forming limit curves (FLC) in a major and minor strain representation:

ε1 = f(ε2 ) .

Fig. 7 Criteria for shear fracture and normal fracture

(7)

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Fig. 8 Classical and alternative FLC representations for necking failure prediction

Points of fracture are experimentally identified for specific strain paths in a very specific orientation with respect to the orthotropic axis of the material, and do not necessarily transfer to differing strain paths and/or differing orientations of deformation, which can easily lead to erroneous predictions for complex forming processes, see left-hand side examples in Fig. 7 with different two step strain paths where fracture occurs significantly below the original FLC. In fact, due to slight differences of the strain paths, even the FLC considerably depends on the test procedure which is used, as discussed in [10]. Following the findings of M¨uschenborn and Sonne [12], the problems inherent to classical FLC can be avoided by choosing a different representation based on the equivalent plastic strain:

εeq = f(α )

with

α=

ε˙2 ε˙1

.

(8)

If the intrinsic basic assumptions of classical FLCs hold (isotropy, von Mises yield locus, no kinematic hardening) this alternative FLC representation is load path independent, see example on right-hand side in Fig. 8, where instead of the ratio α the angle β is used to avoid the singularity when the rate of the first principal strain is zero. β represents the angle of the strain rate vector (ε˙11 , ε˙22 ) with respect to the 1-direction. Marciniak and Kuczynski [11] published a perturbation model where necking failure is approached by its original physical mechanism of a membrane instability problem. Further refinements of this original approach have been developed since then and are now available in commercial FEA codes for industrial applications, see e.g. [8]. Overall this approach can be considered very powerful, provided that the plastic deformation model used to predict the instability point actually is a good representation of the true behavior of the material.

6 Conclusions Major improvements have lately been achieved in the area of numerical modeling of forming processes with aluminum alloys for industrial applications. This especially concerns the representation of the yield locus and models for fracture prediction with arbitrary load paths. However, existing models still are insufficient to fully

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understand quite a number of processes. Such an example is the cold forming of a beverage can body, where equivalent plastic strains typically reach roughly 250% at the end of the process, although seemingly very brittle materials are used (e.g. AA3104: yield stress 290 MPa, ultimate tensile stress 330 MPa and uniform elongation 6%). Therefore, further improvements are needed. Considering the fact that Coulomb friction is currently the only option available in commercial software to model the often complex tribological conditions in a contact zone, the focus of near future research activity and model implementations should be rather in this area than in constitutive modeling of materials.

References 1. Banabic, B., Aretz, H., Comsa, D.S., Paraianu, L.: An improved analytical description of orthotropy in metallic sheets. Int. J. Plasticity 21, 493–512 (2005) 2. Barlat, F., Aretz, H., Yoon, J.W., Karabin, M.E., Brem, J.C., Dick, R.E.: Linear transformation-based anisotropic yield functions. Int. J. Plasticity 21, 1009–1039 (2005) 3. Barlat, F., Brem, J.C., Yoon, J.W., Chung, K., Dick, R.E., Lege, D.J., Pourgoghrat, F., Choi, S.H., Chu, E.: Plane stress yield function for aluminum alloy sheets. Part 1: theory. Int. J. Plasticity 19, 1297–1319 (2003) 4. Barlat, F., Lege, D.J., Brem, J.C.: A six-component yield function for anisotropic materials. Int. J. Plasticity 7, 693–712 (1991) 5. Barlat, F., Lian, J.: Plastic behavior and stretchability of sheet metals. Part 1: a yield function for orthotropic sheets under plane stress conditions. Int. J. Plasticity 5, 51–66 (1989) 6. Bron, F., Besson, J.: A yield function for anisotropic materials. Application to aluminum alloys. Int. J. Plasticity 20, 937–963 (2004) 7. Hill, R.: Constitutive modelling of orthotropic plasticity in sheet metals. J. Mech. Phys. Solids 38, 405–417 (1990) 8. Hooputra, H., Gese, H., Dell, H., Werner, H.: A comprehensive failure model for crashworthiness simulation of aluminium extrusions. Int. J. Crashworthiness 9, 449–463 (2004) 9. Lange, C., Bron, F., H¨anggi, P., M¨oller, T., Friebe, H., Gese, H., Daniel, D., Leppin, C.: Forming simulation of aluminum car body sheet with different yield models and comparison with experiment. In: Proceedings IDDRG 2010 (2010) 10. Leppin, C., Li, J., Daniel, D.: Application of a method to correct the effect of nonproportional strain paths on Nakajima test based forming limit curves. Proceedings Numisheet 4 (2008) 11. Marciniak, Z., Kuczynski, K.: Limit strains in the processes of stretch-forming sheet metal. Int. J. Mech. Sci. 9, 609–620 (1967) 12. M¨uschenborn, W., Sonne, H.-M.: Einfluss des Form¨anderungsweges auf Grenzform¨anderungen des Feinblechs. Arch. Eisenh¨uttenwes 9, 597–602 (1975) 13. Vegter, H., An, Y., Pijlman, H.H., Carleer, B.D., Huetink, J.: Advanced material models in simulation of sheet forming processes and prediction of forming limits. Proceedings Esaform 1, 499–514 (1998)

Chapter 17

A Phenomenological Damage Model to Predict Material Failure in Crashworthiness Applications Markus Feucht, Frieder Neukamm, and Andr´e Haufe I met Professor Wriggers for the first time in 1993 as a PhD student in Darmstadt, while attending his nonlinear FEM lecture. He brought the fascinating world of FEM simulation to me and largely influenced the topic of my PhD thesis. I always enjoyed the highly interesting and fruitful scientific discussions with him, especially when they took place past midnight in an Irish pub or in the Hobbit, where often new theories have been developed on a napkin. But I also have to mention other great events like mentally exhausting skiing seminars and one unique sailing trip in the Baltic Sea, where the mysteries of a sand bank remain unsolved until today (M. Feucht).

Abstract. The present contribution will focus on one of the most urging challenges in crashworthiness simulations, namely alternative or enhanced constitutive formulations to predict failure and cracking of structural parts made from high strength steel sheets. In order to achieve a more accurate prediction, the consideration of pre-damage resulting from a foregoing deep-drawing process seems to be an important extension of the proven processes. This leads to the necessity of considering damage in Finite Element Simulations of forming processes, allowing for the results to be used further on in what has to become the simulation process chain of sheet metal part manufacturing. In a broader sense, this simulation process chain may be termed as “producibility-to-serviceability”. Up to now, the driving force behind forming simulations usually is the question if a certain part may be produced on certain press equipment with a defined number of forming stages, starting from specific sheet material of given initial thickness. Carrying over the forming results to other simulation disciplines, like crashworthiness or NVH, where the serviceability Markus Feucht Daimler AG, EP/SPB, W059/HPC X271, D-71059 Sindelfingen, Germany e-mail: [email protected] Dipl.-Ing. Frieder Neukamm Daimler AG, EP/SPB, W059/HPC X271, D-71059 Sindelfingen, Germany e-mail: [email protected] Dr.-Ing. Andr´e Haufe DYNAmore GmbH, Industriestrasse 2, D-70565 Stuttgart, Germany e-mail: [email protected]

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of the designed structure is investigated further, will eventually give more insight into the effects of pre-straining and possible pre-damaging emerging from the production process to the target discipline. Nowadays, the crashworthiness of bodies in white is assessed to a major extent by finite element simulations without taking the production history into account. Therefore, making use of pre-damage data offers promising opportunity to the simulation engineer to enhance simulation accuracy. Regarding this, experience shows that high strength steel qualities are expected to be more problematic. The present contribution discusses engineering driven approaches to improve the predictiveness of crashworthiness simulations, and to close the constitutive gap between the forming and crashworthiness world.

1 Introduction The production history of sheet metal may have an enormous effect on part performance in crashworthiness applications. Similar effects may be investigated for other materials and other disciplines, i.e. polymers and plastics in occupant safety simulation. It is clear though, that the process chain of sheet metal part manufacturing not only starts at sheet metal forming but instead the blank has already some history of production before it is actually formed into some automotive part. Great effort is being put into the theory and application of numerical models that are able to predict constitutive properties of every single stage during sheet metal production. However, these effects will be neglected here. The blank will be treated as initially orthotropic plastic material with no initial damage.

2 The Process Chain of Sheet Metal Part Manufacturing The individual process steps of sheet metal production are illustrated in Fig. 1 together with the principal properties the respective model should take into account in order to enable predictive numerical studies. Within a research project supported by the German Federal Ministry of Education and Research (BMBF), grant #03X0501E, most of the simulation problems along the process chain were adressed. As will be described below, an emphasis in the author’s work was laid on the prediction of crashworthiness properties depending on the forming history of a part.

3 Failure Modelling in Forming and Crashworthiness Simulations On behalf of improvements for crashworthiness simulations, great effort has been done throughout the past years regarding the treatment of crack formation and propagation. Current state of the art here is the use of failure models that accumulate damage on an incremental basis. Most models are based on the observations of Bridgman [3], who found that failure strain in metallic materials depends on the

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Fig. 1 Process chain modeling: principle work- and data-flow

hydrostatic pressure. Examples of models in use are the Gurson model with extensions by Tvergaard and Needleman [6], and the failure model of Johnson and Cook [5]. As a shortcoming, the mechanical properties of sheet metal parts for crashworthiness calculations are usually assumed to be as in a maiden-like material delivery state. This disregards the changes in constitutive properties resulting from previous treatment in the process chain of sheet metal part manufacturing, including i.e. deep-drawing. In the easiest case, a local increase of the yield stress due to work hardening can be expected which may play an important role for low-speed impact cases. Since plastic pre-straining also results in a reduction of the remaining strain up to failure, the effect of pre-damaging should be phenomenologically taken into account in crashworthiness simulations. This in turn leads to the fact that not only plastic strains but also the damage state evolved during forming simulations should be modelled. For crashworthiness computations, the constitutive models used are usually isotropic and based on the von Mises flow rule or the Gurson, Tvergaard & Needleman approach (see Fig. 2b). For forming simulations, a more sophisticated and anisotropic description of yield loci – often based on the Hill-criterion – is considered important (see Fig. 2a), which makes it necessary to use different constitutive models for both parts of the process chain. A damage model suitable to be used for both disciplines therefore has to be able to correctly predict damage regardless of the details of the constitutive model formulation. In order to ensure this, a modular concept was implemented which allows for nearly arbitrary combinations of

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Fig. 2 Anisotropic Hill model vs. GTN model

Fig. 3 Modular concept: transfer of history variables along the process chain

constitutive models to be combined with the damage model. Due to this, a consistent damage prediction is possible by using the same damage model for both sides of the process chain.

3.1 A Generalized Scalar Damage Model In the following, the damage model GISSMO (Generalized Incremental Stress-State dependent damage Model), which is currently under development at Daimler and DYNAmore will be presented. The model is a combination of proven features of failure description provided by damage models for crashworthiness calculations, together with an incremental formulation for the description of material instability and localization. A user-friendly and simple input of material parameters is intended, which is being achieved by a phenomenological formulation of ductile damage. Special attention is paid to consider the point of instability or localization, as this is a central issue in forming simulations. In general, it can be expected that stress states will usually not be the same in a forming process compared to a following crash loading scenario. The model therefore includes not only the description of failure, but also functionality to provide an incremental and therefore path-dependent treatment of instability. This is needed

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to avoid a limitation of the traditional forming limit curve (FLC), which considers only the final state of deformation at the end of a forming process, and therefore does not take into account possible changes in strain path. Therefore the conventional FLC can not be used for multistage deformation processes, as which the two steps – forming and crash – of the sheet metal process chain can be considered. In order to allow for the treatment of arbitrary strain paths in the prediction of localization and failure, incremental formulations were chosen for both. The concept is to independently accumulate a measure for forming intensity F, and a measure for damage D, respectively.

3.2 Failure Prediction In order to accumulate damage increments following an arbitrary path of plastic deformation, a formulation driven by increments of equivalent plastic strain was chosen. This equation represents a generalization of the well-known linear accumulation rule for damage as proposed by Johnson and Cook [5].

ΔD =

n (1− 1 ) D n Δ εv εf

.

(1)

In this equation, the exponent n allows for a nonlinear accumulation of damage until failure. This introduces a possibility to model multistage material tests, as several publications indicate that damage – as an effect on the material microscale – usually will not follow linear evolutions, see Weck et al. [11]. The actual equivalent plastic strain increment is denominated as Δ εv . The quantity ε f represents the triaxialitydependent failure strain, which is used as a weighting function in this relation. The input of this failure strain is realized as a load curve of failure strain values vs. triaxiality, which allows for an arbitrary definition of triaxiality-dependent failure strains. This is needed to ensure flexibility when used for a wide range of different metallic materials. As the most simple option available, failure of an element can be manifested by abruptly deleting it as the damage measure D reaches unity. Yet, in order to be able to model the complex failure modes of structures made of high strength materials, more sophisticated techniques were implemented.

4 Path-Dependent Localization In the following, methods of treating material instability or localized deformation as applied in the GISSMO model will be described. The basic idea is to determine the strains at the onset of localization from tests under constant stress state (proportional loading). For example, tensile tests with various notch radii, shear tests and biaxial tests can be used. The resulting forming limit curve is used as an input for the constitutive model. Furthermore the curve is used as weighting function for the path-dependent accumulation of necking intensity up to the expected point of instability.

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In general, the localization behaviour of materials in numerical simulations depends on yield locus and evolution of the yield stress. As a direct determination of yield curves from specimen tests is not possible for the post-critical range of deformation, stress extrapolation based on engineering assumptions (or models) is used. Due to this, and as a cause of the inherent mesh-dependency of results in the postcritical range, the used parameters of an extrapolation would determine the material properties in the post-critical range, and lead to mesh-dependent results. Therefore, a damage-based regularization for the post-critical range is proposed in the present contribution. A more comprehensive description of localization issues can be found in De Borst et al. [4]. A motivation for the treatment of instability is to determine the beginning of material softening, which is used as a damage threshold for the coupling of damage to the yield stress in crashworthiness applications.

4.1 Stress and Strain Measures The traditional way of treating possible instabilities in sheet metal forming processes is the comparison of resulting strains in the final stage with a fixed curve of principal strain values (Forming Limit Curve - FLC). It is well known that the forming limit curve does not take into account any changes in strain path as it considers only the final stage of deformation. A practical approach for a strain-path dependent forming limit determination was made by M¨uschenborn and Sonne [9]. They proposed a transformation of the FLC from principal strain (ε1 , ε2 ) - space to a notation using the equivalent plastic strain εv . The idea in treating non-proportional strain paths was to consider the FLC curve as the locus of equivalent strain to necking, depending on the respective strain state. The usual notation for crashworthiness purposes is a characterization of load state using the invariants of the stress tensor. This is sufficient for isotropic material models, since the invariant notation is independent of the respective material direction considered. For the plane stress case, which is a common assumption for sheet metal problems, in-plane strain increments can be directly related to stress values. Therefore, the strain-based notation of the FLC can be transformed to a notation in invariants of the stress tensor. In crashworthiness computations, a notation using the stress triaxiality η is common practice:

η=

σm σv

(2)

with σm (mean stress) being the first invariant of stress. σv is the equivalent or von Mises stress. Using these quantities, the FLC can be directly transformed to this notation. It will be used in the following since the GISSMO model has been developed with respect to these quantities. Both strain- and stress-based notations are equivalent for the isotropic and plane stress case and proportional loading, therefore a

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determination of the necking locus can as well be formulated in strain-based notation. Similar transformations to a number of different notations can also be found in Bai and Wierzbicki [1]. Fig. 4 depicts a FLC in principal strain coordinates transformed to the corresponding equivalent plastic strain/triaxiality coordinates. The

Fig. 4 FLC in principal strain coordinates and in principal strain/triaxiality coordinates

usual way would be to compare the actual value of accumulated equivalent plastic strain to the limit value for a respective triaxiality. This corresponds to using the principal strain notation and would inherently result in the same limitations as there is no consideration of strain path changes.

4.2 Nonlinear Accumulation of the Instability Criterion Recent publications indicate a possible nonlinearity in the relation of damage and equivalent plastic strain, even for proportional strain paths. Weck et al. [11] performed measurements on a model material, that showed a rather exponential relation between strain and damage manifested by the nucleation and growth of voids. It seems a reasonable assumption that the development of plastic strain up to necking also obeys a nonlinear relation, yet no method that would allow for a direct measurement of this quantity is known to the authors. Despite this, a nonlinear means of accumulation is introduced to the GISSMO model, using the same relation as for the accumulation of ductile damage to failure. An identification of parameters for this relation will hardly be possible from direct tests, rather by means of reverse engineering simulations of multi-stage forming processes. The introduction of an additional parameter n therefore allows to fit the model to existing test data. In the same manner as the damage parameter, the “forming intensity” parameter F follows a nonlinear incremental evolution similar to the damage parameter itself (Eq. (1)). 1 n ΔF = F (1− n ) Δ εv (3) εv,loc introducing the accumulation exponent n ≥ 1. For n = 1, Eq. (3) reduces to a linear form. For proportional loading, or – in general – constant values of εv,loc , Eq. (3) can

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be integrated to yield a relation between the “forming intensity” F and the equivalent plastic strain: & ' εv n F= for εv,loc = const. (4) εv,loc For n = 1, Eq. (3) is a linear relation of current equivalent plastic strain and equivalent plastic strain to failure as depicted in Fig. 5. Using these relations, the forming

Fig. 5 Nonlinear accumulation εv,loc = 0, 68

intensity parameter F is accumulated the same way as the damage parameter D. The difference is limited to the use of a different weighting function, which is defined as a curve of limit strain depending on triaxiality for F, whereas for the failure parameter D the fracture strain as a function of triaxiality is input.

5 Post Critical Behaviour The post-critical range of deformation usually is not of interest for forming simulations, since the occurrence of instability or necking phenomena are already considered as failure due to the fact that a part showing these effects will not pass quality assurance for production. However, for crashworthiness purposes it is important to capture the post-critical behaviour of a material, since a maximum in energy absorption can be achieved only through a complete use of material ductility. The modelling of the post-critical behaviour of metals using the finite element method always introduces an undesired meshsize dependency on results. As soon as the instability develops, deformation reduces to a localized area and is no longer uniform. From this point on, no mesh convergence can be achieved. Through discretization,

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an artificial length scale is introduced to the model, which will lead to unphysical results if no countermeasures are taken. For the correct description of post-critical behaviour, different flow curves for each mesh size considered would have to be used since the amount of energy that has to be dissipated in post-critical regime strongly depends on the mesh size. Instead of using this rather impractical approach, the meshsize regularization is realized through the damage formulation. Energy regularization is done through the definition of a meshsize dependent failure strain and the coupling of damage to the stress tensor in post-critical deformation. The GISSMO model uses the effective stress concept which was proposed by Lemaitre [8].

5.1 Damage-Dependent Yield Stress As was proposed by Lemaitre [8] the damage and stress tensor are related according to the effective stress concept:

σ ∗ = σ (1 − D)

.

(5)

In combination with the treatment of material instability as described above, a damage threshold can be defined. As the damage parameter D reaches the damage threshold, damage and flow stress will be coupled. The current implementation allows for to either entering a damage threshold as a fixed input parameter, or for using the damage value corresponding to the instability point detected as described above. Either way, as soon as the post-critical range of deformation is reached a value of critical damage Dcrit is determined and used for the calculation of the effective stress tensor: & ' D − Dcrit σ∗ = σ 1 − for D ≥ Dcrit . (6) 1 − Dcrit Here a fading exponent m is introduced which will be further described below.

5.2 Energy Dissipation and Fadeout In order to model the physical phenomena of failure, which include the formation of voids and micro-cracks, formation of a macroscopic crack, and crack propagation up to complete failure, it is necessary to take into account the amount of energy that is dissipated throughout the process. Also, for numerical reasons, it is not of help for model stability to simply delete elements which are still holding considerable amounts of stress. The strategy followed in the GISSMO model is the definition of an element-size dependent fading exponent m, see eqn. (6). Using this coefficient, one can directly influence the amount of energy that is dissipated during element fade-out. In Fig. 5a, the effect of different values for m shows through the differences in area below the true stress-true strain curve.

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Fig. 6 a) Influence of the fading rxponent m; b) Simulation results of tensile tests and experimental data; c) Tensile test specimen modeled with different element sizes

This allows regularization not only of fracture strains, but also of the energy con-sumed during post-critical deformation. Using this approach, one can achieve a reasonably good regularization of the resulting engineering stress – engineering strain curves in tensile tests with different mesh sizes, see Figs. 6b and c.

6 Conclusions In the present contribution the latest work on closing the constitutive gap between sheet metal forming simulations and crashworthiness has been shown. Clearly, there is much more work on the way in academia, at research institutes and other privately owned companies than shown in this paper. However, the main challenge of any model that may be applied successfully will be the ease of use and the compatibility with existing models. For many years companies spent a lot of money to calibrate material parameters of certain plasticity models by an enormous amount of test data – this holds for the forming as well as for the crashworthiness world. Any new model must integrate this past efforts into its approach. Otherwise it will not be used simply for economical reasons. In the present work a newly developed damage model has been introduced. The present state of the GISSMO damage model as described above shows some promising potential when used for the simulation of tensile, shear and biaxial test specimen as can be seen in Neukamm et al. [10]. Though phenomenologically based it introduces a number of features that might be suited to describe the physics of ductile damage and failure in a variety of stress states and for different materials. Yet, limitations in predictive performance result not only from deficiencies in material modelling, but also from coarse discretization especially in crashworthiness simulations. Further research has to be done to take modelling problems resulting from limited mesh sizes into account. Depending on the materials, a greater number of different specimen tests will be needed to identify the parameters for the damage model.

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References 1. Bai, Y., Wierzbicki, T.: Forming severity concept for predicting sheet necking under complex loading histories. Int. J. Mech. Sci. 50, 1012–1022 (2008) 2. Barlat, F., Lege, D.J., Brem, J.C.: A six-component yield function for anisotropic materials. Int. J. Plasticity 7, 693–712 (1991) 3. Bridgman, P.W.: Studies in large plastic flow and fracture, with special emphasis on the effects of hydrostatic pressure. McGraw Hill Inc., New York (1952) 4. De Borst, R., Sluys, L.J., M¨uhlhaus, H.-B., Pamin, J.: Fundamental issues in finite element analyses of localization of deformation. Eng. Computation. 10, 99–121 (1993) 5. Johnson, G.R., Cook, W.H.: Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures. Eng. Fract. Mech. 21, 31–48 (1985) 6. Haufe, A., Neukamm, F., Feucht, M.: Numerische Modellierung von Dualphasenst¨ahlen bei der Herstellung, Verarbeitung und im Bauteil: Teil 2: Ber¨ucksichtigung der Geschichtsvariablen aus der Umformsimulation in der Crash-Berechnung, 30. In: EFB Kolloquium Blechverarbeitung, Bad Boll, M¨arz 2-3 (2010) 7. Haufe, A., Neukamm, F., Feucht, M., Borvall, T.: A comparison of recent damage and failure models for steel materials in crashworthiness application in LS-DYNA. In: 11th International LS-DYNA Users Conference 2010, Detroit, MI, USA, June 6-8 (2010) 8. Lemaitre, J.: A continuous damage mechanics model for ductile fracture. J. Eng. Mater. 107, 83–89 (1985) 9. M¨uschenborn, W., Sonne, H.-M.: Influence of the strain path on the forming limits of sheet metal. Arch. Eisenh¨uttenwesen 46, 597–602 (1975) 10. Neukamm, F., Feucht, M., Haufe, A.: Considering damage history in crashworthiness simulations. In: 7th European LS-DYNA Conference, Salzburg, May 14 –15 (2009) 11. Weck, A., Wilkinson, D.S., Toda, H., Maire, E.: 2D and 3D visualization of ductile fracture. Adv. Eng. Mater. 8, 469–472 (2006)

Chapter 18

A Computational Approach for Mixed-Lubrication Effects in Sealing Applications Markus Andr´e

Professor Wriggers introduced me to the secrets of solid and computational mechanics. I enjoyed to work together with him at the Universities of Darmstadt and Hanover and I still profit from the achieved knowledge. I thank him for the continuous support and wish him a happy birthday! (M. Andr´e).

Abstract. Friction and wear of dynamic seals in hydraulic systems are strongly influenced by the lubrication state between the surfaces of the sealing structure and the moving piston. The contact and lubrication state is a result of the coupling between the deformable seal structure and hydrodynamic effects in the lubricating fluid film. This paper presents an efficient computational method for the determination of mixed-lubrication states in such contact regions. A standard finite element model is used in order to compute the deformation of the seal structure, while simultaneously a user-defined interface element solves the Reynolds equation that describes the fluid film. In this way an implicit and numerically stable coupling between the solid and the fluid region is achieved.

1 Introduction In hydraulic cylinder systems the rod seals have to provide a reliable sealing function at minimal friction and wear even at high fluid pressure and high rod velocities. These requirements are usually achieved through a sufficient hydrodynamic lubrication within the sealing contact. A suitable design of the sealing shape is necessary to generate such a fluid film. The industrial standard is to estimate the hydrodynamic fluid film from the inverse hydrodynamic lubrication (IHL) theory [2], [6], where the fluid film thickness is computed from the inverse solution of the steadystate Reynolds equation. In such approaches the contact pressure is considered to Markus Andr´e Robert Bosch GmbH, Waiblingen e-mail: [email protected]

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be known, e.g. from a solid mechanics computation, and a full lubrication state is assumed. Later on, the inverse methods have been extended to elasto-hydrodynamic lubrication theories [5], [3], in which the elastic deformation of the solid structure is considered and is coupled to the fluid film thickness. Nevertheless, almost all approaches assume linear-elastic behaviour. Recently some works have been published on the computational treatment of fluid-structure-coupling within hydrodynamically lubricated contacts [9], [8]. In the present paper an efficient computational method is presented, which allows the prediction of the fluid film thickness in the lubricated sealing contact within the framework of a two-dimensional finite element model. It is able to handle mixedlubrication states as well as arbitrary non-linear and even inelastic material behaviour of the seal structure. Since the approach is fully embedded into the finite element computation and is implicitly considered within the equilibrium iteration procedure, it does not require a coupling to external software tools. Consequently, it is highly efficient and very stable. The basic idea of the presented approach is to solve the Reynolds equation, which describes the fluid pressure in thin fluid films. This task is handled within a user-defined element, whose nodes are located on the edge of the sealing structure.

2 Basic Equations The computation of hydrodynamically lubricated contacts requires an adequate coupling of the seal structure to the hydraulic fluid region. In this work the solid part will be computed with a standard finite element approach, whose basic equations are introduced below in Sect. 2.1. Since the solid mechanics part is completely handled by a standard finite element software, only the details required for the coupling between solid and fluid region will be given here. The lubricating fluid region is assumed to behave according to the well-known Reynolds equation as introduced in Sect. 2.2.

2.1 Solid Mechanics The presented computational approach is based on the finite element method. Since it is currently restricted to planar or axis-symmetric structures, the equations given below will also assume this restriction. The seal’s geometry is meshed by a suitable number of finite elements, which are defined through a number of nodes with coordinates Xi , where i is the node number. Then, the deformation of the structure is determined through the displacements ui of these nodes. Thus, the nodal coordinates xi of the deformed structure are given by x = X+u , with the vector definition in two spatial dimensions

(1)

A Computational Approach for Mixed-Lubrication Effects in Sealing Applications

& ' & ' & ' x X u x= , X= , u= y Y v

.

157

(2)

We assume that the x-axis is directed parallel to the piston surface and the piston movement. Thus, the y-axis is directed perpendicular to the piston surface. The computation of strains and stresses is completely covered by the standard finite element environment, so that any material model which is available in the chosen software may be used in conjunction with the presented method. The task of the finite element software is to determine the nodal displacements ui in such a way, that equilibrium of momentum is fullfilled. This yields a system of equations G (a) = 0 ,

(3)

where the solution vector a is assembled from all nodal displacements ui of the meshed solid structure a = (u1 , u2 , ...)T . (4) Here, the iterative Newton-Raphson-Method is used in order to determine the solution vector a. It requires the computation of the residual force vector r = G (a)

(5)

and the tangent iteration matrix K=

∂ G (a) ∂a

(6)

.

Within the iteration process the solution vector a is corrected by

Δ a = −K−1 r

(7)

until the residual vector r is sufficiently small and, thus, the equilibrium condition is fulfilled. Details on the finite element approach and the iteration procedure are available in standard literature, e.g. in [11].

2.2 Fluid Mechanics The computation of the fluid behaviour within the sealing contact area is based on the integrated Reynolds equation according to [4]

∂ ph (x) h(x) − hˆ = 6U η 3 ∂x h (x)

.

(8)

Here h(x) denotes the height of the fluid film, i.e. the distance between the two surfaces in the lubricated contact area. U denotes the velocity of the piston surface and η is the dynamic viscosity of the fluid. Furthermore, ph denotes the fluid pressure within the fluid film and hˆ is an integration constant that needs to be determined

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from the boundary conditions. Equation (8) is restricted to planar flow within thin films, so it can be assumed that ph does not depend on the y-coordinate in thickness direction of the fluid film. Integration of Equation (8) with respect to x yields the fluid pressure ξ h

h

p (x) = p (xa ) +

6U η

xa

h(ξ ) − hˆ dξ h3 (ξ )

(9)

.

The solution is completed by the boundary conditions ph (xa ) = pha

ph (xb ) = phb

and

(10)

.

Furthermore, the viscous shear stress acting on the surface of the seal is given by

τ h (x) =

Uη h(x) ∂ ph (x) − h(x) 2 ∂x

(11)

.

b)

a)

v Seal

u

y

y Piston

U

x

Node i

h

Node i+1 x

Fig. 1 a) Sealing system with seal and piston; b) Interface nodes defining the fluid film

In order to compute the fluid domain solution, Eq. (9) needs to be expressed in terms of the nodal coordinates Xi and the displacements ui of the interface nodes defining the fluid film. This situation is depicted in Fig. 1. A numeric integration of Eq. (9) yields ( ) i hk − hˆ h h pi = pa + ∑ 6U η Δ xk . (12) (hk )3 k=2 The fluid film thickness hk at node k is determined from the nodal coordinate and displacement as hk = Yk + vk − r p where r p is the radius of the cylindrical piston.

,

(13)

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The integration distance in x-direction, parallel to the piston surface, is given by

Δ xk = xk−1 − xk

with

xk = Xk + uk

(14)

and the integration constant hˆ needs to be determined iteratively from the pressure boundary condition Eq. (10). Finally, the forces on the interface nodes are determined from the fluid pressure and fluid shear stresses according to Eq. (12) and Eq. (11) as fkh = phk · Ak

and

tkh = τkh · Ak

(15)

.

Here, Ak denotes the length that is associated to node k, determined by ⎧1 (x − xk+1 ) if k = 2, ..., n − 1 ⎪ ⎨ 2 k−1 Ak = 12 (xk − xk+1 ) if k = 1 ⎪ ⎩1 if k = n 2 (xk−1 − xk )

.

(16)

3 Coupled Fluid Film Computation Since the interface nodes shown in Fig. 1 are belonging to the solid domain as well as to the fluid domain, the forces caused by the internal stresses within the seal structure must be in equilibrium with the pressure forces of the hydrodynamic fluid film. The first contribution is as usual achieved from the standard finite element environment. In order to achieve a coupled solution, the fluid forces just need to be added to the global residual force vector of Eq. (5) r = G (a) + fh

,

(17)

where fh is a force vector containing the contributions of all interface nodes as given by Eq. (5). In this way the Newton-Raphson method can be used to achieve the solution of the coupled problem. Nevertheless, the integration constant hˆ of Eq. (12) has not been determined yet. For this purpose the set of equations is extended with the boundary condition (10). Together with Eq. (12) this yields ( ) n hk − hˆ h h pb = pa + ∑ 6U η Δ xk . (18) (hk )3 k=2 Consequently, the system of equations and the residual force vector r is extended with ( ) n ˆ h − h k r = phb − pha − ∑ 6U η Δ xk (19) (hk )3 k=2

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ˆ Fiand the global displacement vector a gets one additional degree of freedom h. nally, the Newton-Raphson-scheme requires the iteration matrix K, which is defined as K=

∂r ∂a

(20)

.

4 Friction Approach Section 3 sufficiently describes the interaction of the solid structure part and the lubricating fluid film in the fully lubricated state, where the fluid film thickness is sufficient to completely separate the two sealing surfaces and no solid contact exists. Nevertheless, in many applications a mixed lubrication state will appear, where the fluid film pressure is not sufficient to fully separate the surfaces. Then, roughness peaks in the structure of these surfaces will be in contact and an average solid contact pressure pc will be acting within the interface.

a)

b) h h0

R1 h R2

p p0

Fig. 2 a) Rough surfaces; b) Soft contact model

When rough surfaces are approaching each other, the contact pressure pc will be zero as long as the distance exceeds the sum of the maximum surfaces roughnesses R1 + R2 (see Fig. 2a). When further approaching, more peaks will come into contact and the contact pressure will increase. This behaviour is described by a soft contact model, where the contact pressure pc (x) is a function of the contact gap h(x). In this work a relationship as available in standard finite elemente software packages [1] according to & '& & ' ' p0 h(x) h(x) c p (x) = − +1 exp − +1 −1 (21) exp(1) − 1 h0 h0 is used. The parameters h0 and p0 depend on the surface properties and usually need to be determined experimentally. The general slope of this relationship is depicted in Fig. 2b). The solid contact model is completed by a Coulomb friction approach. Also here the friction parameter μ needs to be determined from suitable experiments.

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Fig. 3 Sealing system with user-defined interface element for fluid film computation

Since the above contact models are common standard within nonlinear finite element software packages, they can easily be combined with the fluid film computation as described in Sect. 3 and there is no need for additional implementation. The software will automatically combine the solid contact model and the fluid film model. Consequently, the total contact pressure ptot that is acting on the surface of the seal is a superposition of the fluid pressure and the solid contact pressure ptot (x) = ph (x) + pc (x)

a)

b)

c)

d)

.

(22)

Fig. 4 Results of fluid film computation for piston velocity of 100 mm/s (solid line) and 800 mm/s (dashed line)

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Besides, the parameter p0 must be chosen in such a way, that h(x) > 0, because negative values for the contact gap h(x) are not allowed and will abort the computation.

5 Example In the given example an O-ring sealing system has been computed at two different piston velocities of 100 mm/s and 800 mm/s. The computed fluid film thickness, the hydrodynamic fluid pressure, the fluid shear stress and the solid contact pressure in the contact region are depicted in Fig. 4. While at a velocity of 100 mm/s the system is still in the mixed-lubrication state, at 800 mm/s the hydrodynamic fluid film is sufficient to fully separate the two surfaces, what is obvious from the zero solid contact pressure depicted in Fig. 4b). Further results of this approach have been published in [7] and [10].

References 1. Dassault Systemes Simulia Corp.: Abaqus Online Documentation, Version 6.8. Providence (2008) 2. Blok, H.: Inverse problems in hydrodynamic lubrication and design directives for lubricated, flexible, surfaces. In: Muster, D., et al. (eds.) Proc. International Symposium on Lubrication and Wear, Houston, pp. 9–151 (1963) 3. Field, G.J., Nau, B.S.A.: Theoretical study of the elastohydrodynamic lubrication of reciprocating rubber seals. ASLE Trans. 18, 48–54 (1974) 4. Hamrock, B.J.: Fundamentals of fluid film lubrication. McGraw-Hill, New York (1994) 5. Herrebrugh, K.: Solving the incompressible isothermal problem in EHL through an integral equation. Trans. ASME J. Lubr. Tech. 90, 262–270 (1968) 6. Kanters, A.F.C., Verest, J.F., Visscher, M.: On reciprocating elastomeric seals: calculation of film thicknesses using the inverse hydrodynamic lubrication theory. Tribol. Trans. 33, 301–306 (1990) 7. Oenguen, Y., Andr´e, M., Bartel, D., Deters, L.: An axisymmetric hydrodynamic interface element for finite-element computations of mixed lubrication in rubber seals. In: Proc. IMechE Part J: J. Eng. Trib., vol. 222, pp. 471–481 (2008) 8. Salant, R.F., Maser, N., Yang, B.: Numerical model of a reciprocating rod seal. ASME J. Tribol. 129, 91–97 (2007) 9. Salant, R.F., Shen, D.: Hydrodynamic effects of shaft surface finish on lip seal operation. STLE J. Trib Trans. 45, 404–410 (2002) 10. Schmidt, T., Andr´e, M., Poll, G.: A transient 2D-finite-element approach for the simulation of mixed lubrication effects of reciprocating hydraulic rod seals. Tribology International (December 5, 2009) (in Press), http://www.sciencedirect.com/science/article/ B6V57-4XVRYT7-1/2/487e33b5e23d5a5ef9a5d29be09cf6d0, doi:10.1016/j.triboint.2009.11.012 11. Wriggers, P.: Nichtlineare Finite-Element-Methoden. Springer, Berlin (2001)

Chapter 19

Deformations of a Large Hall: Structural Design and Analysis Klaus-Dieter Klee and Reinhard Kahn For his birthday, we wish our dear friend Peter Wriggers all the best, happiness, creativeness and health. We look back gratefully to the long time we spent together at the Institute of Mechanics and Computational Mechanics, which was shaped by amicable and cooperative teamwork. This was also cultivated when we transferred theoretical findings to challenging building projects. For the future, we wish Peter to always have a handwidth of water under the keel, well-being in the circle of his family and ever a “Dulle” in his hand when playing “Doppelkopf” (K.-D. Klee and R. Kahn).

Abstract. The new hall 27 of the Hannover fair was completed in the year 2002. Due to the large size of the hall (201, 9 ×181, 9 m) extensive investigations were necessary to determine the structural systems with regard to constraints resulting from the architectural design. One of the goals was the minimization of the displacement of the entire stucture with respect to the construction of the facade. In the following different problems with respect to the construction (e.g. realization of necessary gaps), structural analysis and assembly of the structure will be discussed in detail.

1 Introduction During the trade fair boom after the second world war, the focus of hall construction on the Hannover fair grounds was on maximizing exhibition space. This led to large, dark and often multi-story halls. Natural lightening was not desired and minizing buildup time before an exhibition was not an issue. However, nowadays economic efficiency in terms of buildup and dismounting time play a major role. Additionally, the area required for booths has increased up to 10.000 m2 in the last couple of years. Consequently, instead of multi-story halls, modern trade fair halls should be • distinctive contructions with an interesting room layout • large structures serving an agreeable climate inside the hall and yielding optimal daylight illumination • grand, possibly column free spaces which still facilitate economic use of building materials. Klaus-Dieter Klee · Reinhard Kahn Ingenieurgemeinschaft Klee & Wriggers, M¨unchener Str. 18, 30880 Laatzen, Germany e-mail: [email protected],[email protected]

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Fig. 1 South and west view of hall 27

The challenge constructing these halls is to bring in line the structural concept and the architectural requirements. To do justice to the increasing demand of exhibition space, the Deutsche Messe AG decided to proclaim a contest for the new development of halls 27 and 28 including the entrance and cafeteria area in the southwest region of the fairgrounds in the year 2000. The idea of W + P Architekten to combine these seperate buildings under one roof led, among other things, to winning the competition [8]. The problems of constructing, computing and assembling such a large hall, especially considering deformation, are discussed in the following with the example of the newly constructed hall 27 with dimensions of 180 × 200 m. Eight fairfaced concrete cores along the facades give distinction to the almost quadrilateral hall. These 12, 7 m high buildings are not only part of the horizontal bearing of the steel roof contruction, but also accommodate extensive technics and service areas. These bridging buildings are supposed to be extended to a convention center. The 18 m high, in vertical direction self-supporting glass facade is interrupted by the concrete cores and additional smaller shear wall segments made of reinforced concrete. The core and shear wall segments penetrate the facade and are covered with light colored natural stone. A circumferential overhead light band with a height of 5 m above the concrete surface gives a floating character to the overhanging roof. The roof is intersected by two 20 m wide overhead light bands running almost along its entire length and width and six single overhead lights with dimensions of 20 × 20 m. This leads to a light-flooded hall with an overhead light area of 7600 m2. Translucent glass is used to avoid direct solar radiation.

2 Steel Construction In the following sections, construction issues regarding the entire bearing structure, roof, facade and columns, stiffening components and support of partial halls are discussed.

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2.1 Bearing Structure Due to the uncommon size of the hall with a total length of 201, 85 m, width of 181, 85 m and attic height of 18, 90 m an extensive pilot survey to determine the bearing structure was unavoidable. The goal of the structural specification was to minimize the deformation of the overall system, especially at the roof edge and the expansion gaps, due to the decisive load cases temperature and wind. The circumferential vertical, in its plane self-supporting transparent facade is connected to the edge of the roof and the columns, where a maximal horizontal displacement of 60 mm must not be exceeded. The constructional solutions to this demand are: • The vertically self-supporting facade is constructed as a chain of joints allowing for a displacement of 60 mm perpendicular to its plane at the upper ending. • Displacements of the steel structure in the facade plane are enabled by horizontal sliding connections. Thus, the steel structure can move relatively to the fixed facade, see Fig. 2. Due to the hall dimensions a restraint-free, statically determinate support of the entire roof was not possible, as the displacements would have been too large (±100 mm). The design of several fixed points yields very high bearing forces and restraint internal forces in the roof structure, making an economic construction

Fig. 2 a) Connection of the concrete plates to the facade support (horizontal sliding bearing); b) Connection of glass facade with rod thread

Fig. 3 Cross section of hall with expansion joint

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impossible. To develop a structure in line with the above stated demands, the hall was divided into four bodies, each of which is free-standing. To this end, • three traversing expansion gaps were introduced in the roof to enable independent parts of the structure • a “swimming” support of the self-supporting overhead lights was drawn on • the displacement was restricted with aid of a singly statically indeterminate support of two hall segments, using the “shear softness” of the wide-spanning roof • a statically determinate support was used for the other two hall segments • cup and ball bearings, anchor-joint bearings and special bridge bearings were used to facilitate a preferably restraint free movement of the steel structure towards the chosen fixed points. The division of the hall leads to a three-nave hall with expanse widths of 80− −20 − −80 m in transverse direction. The load resulting from the 20 m wide overhead lights and roof segments between axes 3 and 5 in transverse direction and J and L in longitudinal direction are gathered proportinally by the four bodies. The symmetrically arranged expansion gaps in axes J and L yield to symmetrical parts of the structure 5-21/A-J (mirror-inverted to hall part 5-21/L-T) and 1-3/A-J (mirror-inverted to hall part 1-3/L-T). The three expanses in transverse direction consist of hinged facade and binder bearings in the outer axes A and T als well as clamped center columns in the axes J and L. The center 20 m expanse is bridged by clamped slender stringers in the region of the overhead lights, in all others areas haunched plate girders and rolledsteel sections, respectively, are used. The 80 m wide side naves are spanned by truss girders (diagonal trusses with intermediate pillars for supporting the stringers). The top and bottom booms consist of horizontally located HD-profiles (fine grained steel “Histar 355”, HD 400 x 314/509), the diagonals are made of HEB 400 and HEA 400 profiles, where their flanges are completely screwed to the flanges of the horizontally located profiles by means of gusset plates. The top booms of the truss girders are at the same time the booms of the main bracing, see Fig. 4. The 80 m long trusses distribute the vertical load to the approximately 16, 5 m long strained main columns in the facade in axes A and T (tubes 457 x 20, St. 52) and to the strained interior columns (pre-strained anchors) in axes J and L (tubes 508 x 25), respectively. As the trusses are attached to the columns by means of their top booms, deflection forces result in the last node of the bottom booms, which necessitate lateral support of the nodes. This is done by means of bending resistant connection of the pillars and diagonals to the top boom, which in turn is given lateral support by the stringers.

2.2 Roof The roof covering is facilitated by trapezoidal sheets which span 5 m and dispense their load to 20 m spanning stringers. The paned overhead lights elevating from the roof plane are formed grid like by IPE 180 profiles with intermediate set T-welded profiles, U 240 profiles and edge profiles made of welded brackets. These grids are in turn supported by the girders. The girders, which are self-supporting over a span

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Fig. 4 Support of roof plate, main and secondary bracing members

of 20 m, need to absorb the vertical and wind loads at the boundary of the roof as well as • transmit the wind and stabilization forces due to inclination of the facade columns into the bracing • realize lateral stabilization and rotation bedding of the top booms • serve as booms of the main bracing in the roof plane • serve as booms of the so-called auxiliary bracings (wind and stabilization bracings, see Figs. 4 and 8. As a consequence of these different functions, six strained types of stringers were developed, leading to different top booms made out of haunched welded and rolledsteel profiles. All top booms adjust to the respective slope of the roof, the strained trusses constructed with round steel profiles and HEA pillars. The lateral stabilization of the stringers’ top booms is achieved by rotation bedding of the trapezoidal sheets as well as tie bolts applied at the quarter points and in level with the neutral axis, which transmit the stabilization forces into the auxiliary bracing by means of tension. The roof’s circumferential boundary bolts (HEB 320/550 at gables, welded profiles in axes A and J) dispense their vertical reaction forces to the gable and facade columns. These bearings are designed such that the failure of three adjoining columns (due to fire or impact) can be coped with.

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2.3 Stiffening Components In the roof plane, two different types of bracing exist (see Fig. 4): • the main bracing (expanse of 20 x 20 m) transferring wind and stabilization loads to the bearing points • the auxiliary bracings (expanses of 5 x 5 m, see Fig. 4) absorbing the wind loads at the gables and the stabilization loads of the stringers and the boundary bearings. The stabilization bracings are constructed and arranged in a way which enables their use as assemling bracings. Stabilization of stringers and boundary bearings is effected by a structured load bearing system constituted by the four parallel auxiliary bracings in combination with the load dispensing tie bolts. These auxiliary bearings are placed in the inclined neutral axis of the stringers. Thus, they form a fork bearing at the quarter points together with the incumbent trapeziodal sheets. In each of the four partial halls, structured main bracings which dispense the wind and stabilization loads to the main bearing points (expanse 20 × 20 m) are placed. The booms are formed by the top booms of the truss girders as well as the stringers and boundary bearings, respectively.

2.4 Support of Partial Halls The support of partial halls 1 and 2 (5-21/A-J and L-T) is effected by • a hinged fixed point in axis A/7 • a hinged loose bearing in A/19, undisplaceable in transverse direction as shown in Fig. 6 • a hinged loose bearing in J/21, undisplaceable in longitudinal direction as shown in Fig. 7 For the support in A/7, a special construction was developed such that this point is fixed in longitudinal direction of the hall by placing a rigid in compression diagonal at the truss top boom (Fig. 5), while the support in longitudinal direction of the girders (transverse direction of the hall) is formed by a Gumba bridge bearing at the bottom boom node (Fig. 6a). Thus, the rotation point of this bearing is outside of the girder (Fig. 9). The strained center column in axis J forms a quasi horizontally relocatable bearing for the truss girder due to its bending softness. Due to this exceptional bearing kinematics, the column head in axis J moves outward with a maximal amount of approximately 74 mm when the girder is bending. This desired displacement compensates the concurrent compression of the girder’s top boom of approximately 50 mm due to the very large compression forces. This way, the displacements due to vertical loading of the girder in the expansion gaps J and L are minimized. The chosen Gumba loose bearings are a monovalent bearing. They allow for bending of the girder in this place, at the same time, movement perpendicular to the girder plane is possible (displacement towards the fixed point for the temperature

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Fig. 5 Simply statically undetermined supported part of the hall 80 × 160 m (span of girder 80 m)

Fig. 6 a) Assembly of support ledger(lower chord of girder in axis A’/19; b) Final state of support; c) Bolt joint support of main column [5]; d) Calotte support (column A/1)

Fig. 7 80 m-truss girder, supported at upper chord with trussed roof beam. Support in J/21

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Fig. 8 State of assembly with girder and trussed roof beams

Fig. 9 a) Connection of the concrete plates of the facade to the facade support (horizontal slidng bearing) [8]; b) Connection of glass facade with rod thread

Fig. 10 Gumba support which is vertically and perpendiculary displaceable to the plane

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load case). The monovalent bearing in J/21 yields a reduction of the roof edge (and thus the facade) displacement. The restraining forces resulting from the statically undeterminate support are gathered by the temperature load cases of Δ T = +30o C and Δ T = −30o C, respectively (in contrast to the assembling temperature of 15o C. The rigid in compression diagonal receives a slantedly attached hinged joint a its base point facilitating a displacement of the top point in transverse direction of the hall, see Fig. 7. The main bracing of partial halls 3 and 4 (1-3/A-J and 1-3/L-T) are formed by statically determinate single-span beams spanning 80 m (diagonal expanse 20 × 20 m). In A/3 and J/3, a hinged fixed bearing and a hinged loose bearing, respectively, are designed as Gumba bridge bearings (Fig. 10).

3 Construction and Computation For the computation of the entire roof structure, two systems were studied: A mapping of the entire roof plate to a plane system with a preceding determination of the spring stiffnesses of all six stringer types and a conversion to statically equivalent ersatz bearings was accomplished. To this end, the computation was done completely geometrically non-linear with the program package ANSYS [2], where the diagonals of the main bracing were assumed to be compression slacking and therefore modeled as rope elements. Overall, 15 basic load cases and 40 load case combinations were considered. Based on these computations, diagrams of transposition showing the maximal and minimal displacements resulting from all load cases for the facade connection points were established. Due to the separation into four bodies, e.g. a wind load case where one or two longitudinal surfaces are loaded by wind while two other sides are not loaded is decisive for partial hall 1 (see Fig. 12 where the unsymmetrical wind loading is shown). Secondly, a control computation of the entire structure was accomplished with the program RSTAB [7] assuming idealized stiffnesses, see Fig. 11b. The strained stringers were computed using bending torsion theory of second order accounting for • rotation elastic bedding by the trapezoidal sheets • lateral clamping by the tie bolts at the quarter points • shear stiffness of the roof. To this end, the straining form and the six different cross sections of the top booms were optimized. The neutral axis of the straining has to connect excentrically to the bearing below the calculational bearing point to ensure stable load bearing behavior. This excentricity was optimized in a way that the unintentional inclination of the HEA 120 pillars may add up to 7, 5 cm at a maximum pillar length of 250 cm and a simultaneous deflection of the girder’s top boom of 6 cm. Excentric normal forces resulting from bracing properties were accounted for. For one of the strained bearings, an additional finite element computation using shell theory was carried out [2].

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a) Main load cases • Dead load - steel, roofing, false ceiling, overhead lights, installations • Additional installations • Snow • Temperature - +/- 30 K • Wind - load levels alluding to professional opinion of hall 13 - loading of facade in +/- direction - suction load at the roof - loading of overhead lights • Internal high and low pressure, respectively • Imperfections b) Concept of numerical analysis • Structural pre-analysis regarding entire system • Numerical problems - extreme compression of girder‘s top booms causes loos of stability of parallel spanning parts - consequent failure of stringers spanning perpendicular to girders due to lack of lateral clamping - capturing of main bracing as mere tension trusses • Solution by division into several load frames, stiffening by structured systems - strained stringers with stabilization by auxiliary bracing - truss girders - auxiliary bracing - pillars and bearings • Control computation on entire system - without auxiliary bracing - with girders as pendular trusses with equivalent longitudinal stiffness

Fig. 11 a) Main load cases; b) Concept of numerical analysis [5]

Fig. 12 Part of the hall (160 × 80 m), loading due to wind

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• Optimization of bearing excentricity of the straining to achieve stable bearing characteristics • Imperfections - lateral parabolic deflection of the top boom in combination with incliation of the pillars dv = 7,5 cm at h = 2,50 m in the opposite direction • Computations - analysis w.r.t. second order bending torsion theory mit continuous rotation springs, lateral clamping in height of the neutral fiber of the top boom at quarter points - additional geometrically nonlinear stability analysis with a 3D shell model

Fig. 13 Analysis of trussed roof beams

Fig. 14 Method for line of weld (detail)

Due to the vertical bedding of the truss girders at the top boom, the excentric horizontal support at the bottom boom and the thus resulting deflection forces, a spacial computation using second order theory with bending stiff nodes was applied for the trusses. Additionally, a stability analysis according to DIN 18800, part 2 for single rods was performed. As displacements play a major role in the design of this hall, influence of screw slippage (diagonals and vertical rods are screwed to top and bottom booms) was also considered.

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Additional studies were performed to determine which idealized area of the top booms of the girders in the plane, geometrically non-linear computation of the roof plate needs to be considered for the temperature load case. The truss booms were welded on site where a polygonal superelevation (pass of 250 mm) was achieved by the welding joint. Special care was taken about the construction of the butt joint, the shape of which enables a flawless examination of the weld quality by aid of ultrasound (Fig. 14). The static system of the overhead lights is designed as a grid of welded and rolled-steel profiles, which is located on the overhead light stringers. Additionally, these systems were calculated for loads in plane of the bearing structure. Respective loads result e.g. from unequal friction forces at the supports, which are constructed as sliding bearings. Possible forming of rhombi is prevented by choosing a matching stiffness for the grid. In addition, the grids are constructed such that sufficient lateral support and rotation bedding of the strained stringers is given. The support of the grid onto the stringers was realized in a way that displacement adjustment due to temperature is possible for a total length of the overhead light of up to 120 mm.

Fig. 15 Support of skylight with possibility to expand [5]

Fig. 16 Detail of sliding support edge beam, fork bearing, front and back view

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Special focus was directed to the design of the sliding bearings, as this kind of bearings frequently should only support restricted degrees of freedom. As an example, consider bearing L/21. Here, displacement in longitudinal direction of the boundary bearing should only be possible to a resticted extent (Fig. 16). This is facilitated by a tappet at the end of the sliding plates: Vertical lateral sheets form a fork bearing while an additional lateral sheet with an upper horizontal bracket serves as a protection against lift-off. The construction of the sliding bearing of the glass boundary bearing in axis J/19 is displayed in Fig. 16. At points J/5 and L/5 intersection extension gaps had to be constructed. The development, construction and computation of the sliding bearings and expansion gaps was a substantial part of the support structure planning for hall 27. Based on the displacement computation, pole plans were established to determine optimal arrangement of bearing joints, centering bars as well as cup and ball bearings. Some of the support points were examined by additional finite element computations to calculate e.g. strain paths of stud-bolts.

Fig. 17 Sliding support of edge beam J-L/19 (designed as fork bearing)

Fig. 18 Hall 13 of Deutsche Messe AG Hannover [3], internal view

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Fig. 19 Hall 9 of Deutsche Messe AG Hannover [6]

Hall 13 (1997)

Hall 9 (1999)

Hall 27 (2002)

Area (m2)

27.000

32.312

36.611

Total tonnage (t) thereof steel cast ropes

2.930 2.330 600 -

2.650 1.910 230 510

3.300 3.300 -

Weight of roof (t) column weight

2.420 510

2.300 350

2.867 433

Roof, steel (kg/m2) columns (kg/m2)

89,6 18,9

71,2 10,8

78,3 11,8

total steel (kg/m2)

108,5

82,0

90,1

load bearing roof (kg/m ) 51,0 total (kg/m2) 159,5

77,0 159

10,0 100,1

maximum column liberty 90 x 220 19.800 (m2)

105 x 203 21.315

80 x 200 16.000

steel construction cost (Mio DM)

17,0

21,5

15,0

DM/t

5.802

8.113

4.545

DM/m2

630

665

410

2

Fig. 20 Tonnage of steel and costs of halls 9, 13 and 27 of Deutsche Messe AG Hannover

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4 Summary Due to the exceptional size of the building and the respective displacements of the structure, an uncommon solution in terms of dividing the hall into four partial halls connected by expansion gaps was chosen. Special attention was given to the bearing and displacement kinematics with corresponding challenging design of the highly stresses sliding bearings in combination with swimmingly supported overhead lights. With an under roof area of approximately 36.611 m2, 3.300 tons of steel were used. The weight of the load-carrying roof construction including columns and bearings is 90, 1 kg/m2 with a maximum liberty of columns of 16.000 m2 . If the load bearing roof covering (trapezoidal sheets) is included, a weight of 100, 1 kg/m2 results. The slimness and lightness of the roof structure was achieved by consistent optimization and adjustment of all load bearing construction elements. For further reading as well as comparisons to the other new halls 9 (Fig. 19) and 13 (Fig. 18) the reader is referred to Fig. 20 as well as [1, 3, 4, 5, 6].

References 1. Ameling, F.-J. (c/o Horstmann & Partner): Tragwerksplanung Massivbau. Hannover (2000) 2. ANSYS: User Manual, rev. 5.3, vol. I-IV 3. Kahn, R., Klee, K.-D., Siebold, H., Stein, E.: Montage und Berechnung der Stahlkonstruktion der Messehalle 13. In: Finite Elemente in der Baupraxis., Verlag Ernst & Sohn (1998) 4. Klee, K.-D.: Messehalle 27 - Form¨anderungen einer sehr groen Halle und ihre konstruktive Bew¨altigung. Stahlbau 11, 767–777 (2003) 5. Klee & Wriggers + Tokarz Frerichs Leipold: Leitdetailpl¨ane ARGE Stahlbau. Hannover (2001) 6. Plieninger, S., Kahn, R., Klee, K.-D.: Berechnung, Fertigung und Montage der Stahlkonstruktion der Messehalle 8/9 in Hannover. Deutscher Stahlbauverband. 19. Steinfurter Stahlbauseminar (2000) 7. Dlubal GmbH: Programmsystem RSTAB. Tiefenbach 8. W + P Architekten: Entwurfsplanung zur Halle 27. Hannover (2000)

Chapter 20

Recovering Micropolar Continua from Particle Mechanics by Use of Homogenisation Strategies Wolfgang Ehlers Dear Peter, on the occasion of your sixtieth birthday, I wish you all the best, especially health, but of course also the necessary success to continue to be as successful as before. You have almost achieved everything you wanted in your position as professor and academic teacher. Use the time - it is precious (W. Ehlers).

Abstract. The present article aims at linking micropolar continua and particle mechanics by use of homogenisation techniques. In both cases, the continuous and the discontinuous formulation, micropolar properties arise as a result of local effects stemming from the microstructure of the material. By use of homogenisation strategies, it can be shown that particle dynamics corresponds to the micropolar Cosserat continuum rather than to the standard Cauchy one.

1 Introduction The mechanical behaviour of natural systems like soil as well as general granular aggregates can be observed on different scales. While the macroscopic, continuummechanical view on natural systems with elastic, elasto-plastic or elasto-viscoplastic material properties is generally very convenient for the description of large-scale problems, there are some features like localisation, fracturing and granular flow that cannot be described by simple continuum models. These effects stemming from the microstructure of the material can be integrated in the overall description on the basis of extended continuum models like those given by micropolar theories, as for example, the Cosserat theory [1]1 . On the other hand, a purely microscopic view on soil or granular matter considering these materials as an ensemble of rigid Wolfgang Ehlers University of Stuttgart, Institute of Applied Mechanics (CE), Pfaffenwaldring 7, 70771 Stuttgart, Germany e-mail: [email protected] 1

The reader, who is interested in a closer view on micropolar continua, is referred to the early work by G¨unther [7] and Schaefer [9] and to the later work by the author [3]. Extended work on microcontinua, in general, has been provided by Eringen, cf. e. g. [6]. Micropolar theories, in general, split into approaches including constrained and unconstrained microrotations. The approaches by G¨unther and Schaefer proceed from unconstrained rotations as well as the present article.

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particles interacting through a set of contact laws leads to huge numerical costs and is, therefore, not generally desirable. The present contribution exhibits the description of particle ensembles by means of Newton’s equations of momentum and angular momentum in combination with contact laws and neighbouring lists. As a result, one obtains a tool for the computation of both dynamical problems like the outflow of a hopper and quasi-static problems like the biaxial test of soil mechanics including the development of shear bands. Proceeding from volume averaging techniques, one does not only succeed in transferring contact forces towards stresses but also in transforming contact moments towards couple stresses. These results clearly demonstrate that a macroscopic description of granular material should include microscopic information through the consideration of micropolar media. The numerical examples address these effects by exhibiting the computation of dynamical and quasi-static problems including the description of shear bands.

2 The Particle Model Based on Newton’s equations and convenient neighbouring lists, a set of rigid spherical particles can be described and numerically treated on the basis of the following equations: • balance of linear momentum: N

(i) (i)

m p x¨ p =

∑ fc

(i)

(i)

+ mp g ,

(1)

c=1

• balance of angular momentum: (i) . (Θ (i) p ωp ) =

N

∑ lc

(i)

(i)

× fc

(2)

.

c=1

(i) ¨ Therein, m p and Θ (i) p p are the mass and the tensor of inertia of a particle P , x (i)

(i)

is the acceleration of its particle centre and ω p its rotational velocity vector, while ( · ) . characterises the material time derivative. Furthermore, g is the gravitational (i) force per unit mass, and fc is an external contact force acting at one of the N (i) contact points of the particle surface with the corresponding branch vector lc , cf. Fig. 1. (i) (i) (i) Splitting the contact forces fc into normal and tangential parts, fcn and fct , yields (i)

(i)

(i)

(i)

fc = fcn + fct

with

(i) (i j) (i j) fcn = [ En Δ (i j) + Dn Δ˙ (i j) ] n(i j) , (i)

(i j)

fct = Et

(i j)

[ u¯ t

(i j)

(i j)

− u¯ t p ] + Dt

(i j)

[ u˙ t

(3) (i j)

− u˙ t p ] .

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181

(i)

fc

(i)

lc P (i)

(i)

rp (i)

e2

xp (i) mp

O

e1

(i)

Θp

Fig. 1 Particle P (i) in contact

Here, it has been assumed that the normal contact force follows a viscoelastic contact law, while the tangential one behaves viscoelastic-plastically. With respect to (i j) (i j) (i j) (i j) the viscoelastic contacts, En and Dn as well as Et and Dt are the normal and tangential elastic and viscous moduli. Furthermore, ( j)

(i)

( j)

(i)

Δ (i j) = x p − x p  − r p − r p 

(4)

is the so-called overlap, and n(i j) is the outward-oriented unit surface normal directed from particle P (i) towards P ( j) . Regarding the relative motion between the (i j) particles in contact, u˙ t is the relative tangential velocity at the contact point, and (i j) (i j) u˙ t p is its frictional part. Finally, u¯ t defines the relative tangential displacement (i j)

between P (i) and P ( j) obtained from integration over the contact time, while u¯ t p is obtained as its frictional part. To explain these terms in detail, consider the equations of rigid body motion of particles P (i) , where vc = x˙ p + ω p × lc (i)

(i)

(i)

(i)

(5)

defines the velocity at a contact point of the particle’s boundary and (i)

(i)

vct = t(i j) · vc

(6)

its tangential projection with t(i j) as the tangential unit vector pointing in the direc(i) tion of fct . Following this yields

(i j)

u˙ t

= t

(i j)

(i) ( j) (vct − vct )

and

(i j)

t

u¯ t (t) = t(i j) (t) τc

(i)

( j)

[ vct (τ ) − vct (τ ) ] dτ

, (7)

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W. Ehlers

where the time span between τc and t defines the contact time. Since t(i j) is varying (i j) during the contact time, u¯ t , as a part of the constitutive setting (3), is not obtained (i j) as the time integral of u˙ t . (i j) (i j) (i j) (i j) Splitting u¯ t in viscoelastic and plastic parts, u¯ t − u¯ t p and u¯ t p , requires the following analogs of plasticity formulae, where μ (i j) is the friction coefficient [10]: 5 (i) (i) (i) • yield condition: F := fct · fct − μ (i j) fcn  ≤ 0, (i j) u˙ t p = Λ t(i j)

• plastic flow rule:

• Kuhn-Tucker conditions: F ≤ 0 ,

−→

Λ ≥ 0,

(i j) u¯ t p = t(i j) (t)

ΛF =0 .

t

(i j)

t(i j) (τ ) · u˙ t p (τ ) dτ ,

τc

(8) Once the above set of equations is given, the numerical scheme for the computation of initial-boundary-value problems within the Discrete Element Method (DEM) proceeds from convenient neighbouring lists and the Verlet algorithm obtained from (1) and (2) for an explicit time integration, viz.: • translational and rotational velocities at t + Δ t: (i9

x˙ p (t + Δ t) =

ϕ˙ (i) p (t + Δ t) =

f(i) (t) (i) mp

N

(i)

Δ t + x˙ p (t),

(i)

−1 (i) ¯ (t) Δ t + ϕ˙ (i) (Θ (i) p ) m p (t),

c=1 N

(i)

(i)

f(i) (t)

(i)

(i)

¯ (i) (t) = ∑ lc × fc (t) . where m c=1

• translational and rotational displacements at t + Δ t: x p (t + Δ t) =

(i)

where f(i) (t) = ∑ fc (t) + m p g ,

(9)

(i)

Δ t 2 + 2 x p (t) − x p (t − Δ t) , (i) mp (i) −1 (i) (i) ϕ (i) ¯ (t) Δ t 2 + 2 ϕ (i) p (t + Δ t) = (Θ p ) m p (t) − ϕ p (t − Δ t) .

(10)

To exhibit the possibilities of the above scheme, Fig. 2 displays the numerical computation of an outflow of spherical particles out of a hopper. In this particular computation, use was made of an initial setting of monodisperse particles in a triangular lattice. As a result, the particle ensemble forms shear bands under 60◦ while particles are falling away. It is furthermore seen from the pointers with a horizontal initial position that the particles rotate at the shear band and while falling. As a result of this finding, it is expected that a homogenisation over a convenient representative elementary volume (REV) will rather exhibit a Cosserat continuum than a Cauchy one. It is furthermore expected that the rotational degrees of freedom will not be activated in the overall domain but in the shear band zone.

Micropolar Continua and Particles

183

Fig. 2 Granular flow – outflow of a hopper

3 Homogenisation Technique Based on the Hashin’s MMM principle [8], any homogenisation process proceeds from the assumption that the REV (volume R, surface ∂ R) as the homogenisation volume is small compared to the overall domain. As a result, volume integrals are negligible in comparison to surface integrals: 

( · ) dv 



( · ) da

(11)

.

∂R

R

Corresponding to (1) and (2), the respective continuum-mechanical balances read [3]: • balance of linear momentum: 



ρ x¨ dv =

t da + ∂V

V



ρ g dv ,

(12)

V

• balance of angular momentum (Cauchy continuum): 



x × ρ x¨ dv =

V

∂V

x × t da +



x × ρ g dv ,

(13)

V

• balance of angular momentum (Cosserat continuum):  V

[ x × ρ x¨ + ρ ( Θ¯ ω¯ ) . ] dv =

 ∂V

¯ da + [ x × t + m]

 V

[ x × ρ g + ρ c¯ ] dv

.

(14)

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W. Ehlers

∂R

¯ (i) m f(i) (i)

c

lc

(i)

fc

P (i)

(i) xp

∂ P (i)

REV

Fig. 3 REV bounded by the curve connecting the boundary particle centres

In (12)-(14), V and ∂ V are the volume and the surface of the considered continuum, ρ is the mass density, t is the surface traction, Θ¯ is the mass-specific tensor of ¯ is the Cosserat surface couple, and microinertia, ω¯ is the local rotational velocity, m c¯ is the volume couple. Combining particle mechanics and continuum mechanics on the basis of an REV, cf. Fig. 3, the MMM principle (11 yields the following statement with respect to the momentum balances (1) and (12):  ∂R

B

t da = ∑ f(i) = 0 ,

where

f(i) =

i=1

N0

∑ fc

(i)

(15)

c=1

is the resultant contact force acting at the centre of P (i) . Furthermore, B is the number of boundary particles, while N0 is the number of external contacts at P (i) . Once the contact force resultant is placed at the particle centre, it is necessary to introduce ¯ (i) summing up the offsets of the individual contact forces, the additional moment m viz.: ¯ (i) = m

N0

∑ lc

(i)

(i)

× fc

.

(16)

c=1

Comparing the balance equations of angular momentum (2), (13) and (14), it is obvious in analogy to the procedure to obtain (15) that one concludes to  ∂R

B

¯ da = ∑ [ xp × f(i) + m ¯ (i) ] = 0 . [ x × t + m] (i)

(17)

i=1

Thus, an ensemble of particles obviously always behaves like a Cosserat contin(i) uum. On the other hand, shrinking the particle radii r p towards zero decreases the

Micropolar Continua and Particles

185

¯ and m ¯ (i) are vanishing as a result of vanishing branch vectors Cosserat effect until m (i) lc . This incorporates the standard continuum approach (Cauchy continuum) into a micropolar theory (Cosserat continuum). The question whether or not Cosserat effects become apparent in a numerical setting depends on two conditions: (1) the (i) granular microstructure must exhibit non-vanishing r p such that couple stresses are possible; (2) the problem under consideration must exhibit tangential contact forces which are usually activated by local rotations as they occur, for example, during shear banding of granular material. As a result, one observes couple stresses. Proceeding from standard arguments of continuum mechanics together with Cauchy’s theorem yields the Cauchy stress tensor T and the corresponding tensorial stress moment M as t = Tn

x × t = (x × T) n =: M n =: m .

and

(18)

¯ reads In addition, the couple stress tensor M ¯ n ¯ =M m

(19)

.

In case that a continuous REV is considered, the homogenisation of stresses, stress moments and couple stresses yields the average A =

1 VR



A dv ,

(20)

R

¯ It is furthermore concluded from [4] that where A is the substitute for T, M and M. the following identities hold: AT = I AT = (gradx) AT

,

div (x ⊗ A) = (gradx) AT + x ⊗ divA

.

(21)

Thus, it is easily seen that AT = div(x ⊗ A) − x ⊗ divA 1 AT = VR



,

[ div (x ⊗ A) − x ⊗ divA ] dv ,

(22)

R

where AT = AT  has been used. With the aid of the Gaussian integral theorem and Hashin’s MMM principle, (22)2 finally results in AT =

1 VR

 ∂R

(x ⊗ A) n da .

(23)

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W. Ehlers

¯ read: From (18), (19) and (23), the volume averages of T, M and M 1 T = VR M= ¯ M=

1 VR 1 VR



t ⊗ x da ,

∂R ∂R

m ⊗ x da ,

(24)

¯ ⊗ x da . m

∂R

The final step in the present homogenisation procedure stems from the comparison of (24) representing the homogenisation over continuous REV and the respective terms governing particle ensembles. Thus, (24)1 and (15) combine to the homogenised Cauchy stress T=

1 VR

B

∑ f(i) ⊗ x p

(i)

(25)

i=1

obtained from contact forces at the boundary particles of the REV and the location vectors of the corresponding particle centres. Analogously, (24)2 , (17) and (18) combine to M=

1 VR

B

∑ m(i) ⊗ x p =

i=1

(i)

1 VR

B

∑ (x p × f(i)) ⊗ x p = (i)

(i)

i=1

1 VR

B

∑ xp

(i)

(i)

× (f(i) ⊗ x p ) ,

i=1

(26) while the homogenised couple stress yields ¯ M=

1 VR

B

∑ m¯ (i) ⊗ x p

(i)

,

(27)

i=1

where (24)3 together with (16) and (19) has been used. From (25)-(27), it is seen that the REV of a particle ensemble basically behaves ¯ = 0 and, as a result, T = TT , see like a Cosserat continuum. This includes M also [3, 4, 2]. Numerical computations carried out on the basis of the DEM will clearly demonstrate this effect after homogenisation.

4 Numerical Example The following example concerns a biaxial test carried out on a dry Hostun sand specimen at the Laboratoire 3S at Grenoble. In the experiment, a rectangular sand specimen is pressed between top and bottom loading plates, while the sides are stabilised by a lateral hydraulic pressure. During the experiment, the specimen is covered by a latex membrane such that the lateral load can be applied by a fluid pressure.

Micropolar Continua and Particles

187

Fig. 4 Biaxial test – initial configuration (left) and final configuration (right) exhibiting a shear band after loading

By use of the DEM, the present example is computed with the equations described above, where, in addition, the latex membrane is modelled by elastic springs between the centres of the boundary particles. The present computation is based on an initial configuration of mono-disperse circular particles in their densest packing. After a certain amount of loading, a shear band occurs and the right upper triangle starts to slide over the left lower one, thereby forming the typical “noses” at both ends of the shear band. It can furthermore be observed that the shear band width is growing during the sliding process until it reaches its final value which, in the present example, reaches approximately five particle diameters. Proceeding from the homogenisation technique presented in (25) and (27), the present computation makes use of a particle-centre-based procedure such that each particle can be understood as the centre of an REV, cf. [4, 2]. Following this yields the results exhibited in Figure 6. To obtain these results, small REV of only a few

Fig. 5 Biaxial test – DEM computation exhibiting a definite primary shear band and two further secondary ones

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W. Ehlers

Fig. 6 Homogenised skew-symmetric stresses skw T (left) and homogenised couple stresses ¯ (right) displayed at the reference geometry M

particles around the centre particles have been used. Larger REV would include, also across the shear band domain, areas without micropolar rotations and, therefore, would not be able to precisely predict the shear band activity. This is also seen from Figure 6, where the couple stresses only occur in the shear zone, while the remainder ¯ = 0, the stresses T of the specimen does not show any Cosserat effect. Once M are no longer symmetric. This means that the skew-symmetric part skwT is non¯ Obviously, this domain is the shear zone, where zero in the same domain as M. the particles exhibit distinct rotations. A continuum-mechanical investigation of both Cauchy and Cosserat continua and various numerical applications of non-polar and micropolar materials can furthermore be found in [3]. The examples presented there show the same result, namely, that micropolar degrees of freedom and with them couple stresses and skew-symmetric stress parts only occur in zones with distinct micro-information (micro-rotation). Such zones obviously include shear bands as they have been presented here. Finally, it should be noted once again that a micropolar continuum always includes the possibility of exhibiting micropolar rotations and, as a result, the possibility of exhibiting couple stresses. However, micro-rotations are not always active, since the appearance of micro-rotations depends on the considered initial-boundaryvalue problem.

5 Conclusion In the present article, it has been shown that the discrete element method is capable to describe particle-dynamical problems such as the outflow of a hopper as well as geomechanical problems such as the quasi-static biaxial test. Local information of stress and couple stress tensors can be obtained by use of homogenisation

Micropolar Continua and Particles

189

techniques combined with a particle-centre-based strategy. The homogenised values of stresses and couple stresses directly correspond to respective results obtained by use of the micropolar Cosserat continuum approach. Concerning the determination of material parameters of micropolar elasto-plastic material, the interested reader is finally referred to [5].

References 1. Cosserat, E., Cosserat, F.: Th´eorie des corps d´eformables. A. Hermann et fils, Paris (1909); Theory of Deformable Bodies, NASA TT F-11 561 (1968) 2. D’Addetta, G.A., Ramm, E., Diebels, S., Ehlers, W.: A particle center based homogenization strategy for particle assemblies. Eng. Computation. 21, 360–383 (2004) 3. Ehlers, W.: Foundations of multiphasic and porous materials. In: Ehlers, W., Bluhm, J. (eds.) Porous Media: theory, experiments and numerical applications, pp. 3–86. Springer, Berlin (2002) 4. Ehlers, W., Ramm, E., Diebels, S., D’Addetta, G.A.: From particle ensembles to Cosserat continua: Homogenization of contact forces towards stresses and couple stresses. Int. J. Solids Struct. 40, 6681–6702 (2003) 5. Ehlers, W., Scholz, B.: An inverse algorithm for the identification and the sensitivity analysis of the parameters governing micropolar elasto-plastic granular material. Arch. Appl. Mech. 77, 911–931 (2007) 6. Eringen, A.C.: Microcontinuum field theories I: Foundation and solids. Springer, New York (1999) 7. G¨unther, W.: Zur Statik und Kinematik des Cosseratschen Kontinuums. Abhandlungen der Braunschweigischen Wissenschaftlichen Gesellschaft 10, 195–213 (1958) 8. Hashin, Z.: Analysis of composite materials – A survey. ASME J. Appl. Mech. 50, 481–505 (1983) 9. Schaefer, H.: Das Cosserat-Kontinuum. Zeitschrift f¨ur Angewandte Mathematik und Mechanik (ZAMM) 47, 485–498 (1967) 10. Scholz, B.: Application of a micropolar model to the localization phenomena in granular materials. Dissertation thesis, Report No. II-15 of the Institute of Applied Mechanics (CE), University of Stuttgart (2007)

Chapter 21

Modelling of Microstructured Materials with Micromorphic Continuum Approaches C. Britta Hirschberger and Paul Steinmann This contribution is dedicated to Peter Wriggers, with whom I have got the opportunity to work with since October 2009 in a both scientifically and personally pleasant and inspiring environment. I wish you all the best for your outstanding scientific career and most importantly health and happiness (C.B. Hirschberger).

Abstract. Micromorphic continuum theories provide the benefit of modelling size effects that arise in specimens with a relatively large microstructure. For the micromorphic continuum, we present the governing equations, which allow for a finite-element approximation to predict the size effects numerically. One application presented here is the computational multiscale framework for material layers with a heterogeneous micromorphic mesostructure.

1 Introduction Due to the ineptitude of classical local or rather Cauchy continua to account for size effects stemming from a relatively large intrinsic microstructure, generalized continua have been developed starting from the early 20th Century. The micromorphic continuum as a higher-order continuum goes back to Eringen [3] and has recently been rediscovered [7, 8, 11, 12]. It is characterised by each material point being endowed with a microcontinuum that may undergo affine, yet kinematically independent deformations. The micromorphic continuum comprises other prominent theories as special cases, such as the micropolar or the microstretch continuum. These Dr.-Ing. C. Britta Hirschberger Leibniz Universit¨at Hannover, Institute of Continuum Mechanics, Appelstr. 11, D-30167 Hannover, Germany e-mail: [email protected] Prof. Dr.-Ing. habil. Paul Steinmann Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg, Lehrstuhl f¨ur Technische Mechanik, Egerlandstraße 5, D-91058 Erlangen, Germany e-mail: [email protected]

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Fig. 1 Micropolar rotatable triad

ϕ X

x

F

B0

Bt ¯ G

Fig. 2 Microstretch extendable rotatable triad ¯ X B¯ 0 Fig. 3 Micromorphic deformable triad

F¯

x¯ B¯ t

Fig. 4 Micromorphic continuum and microcontinua with the macro and micro deformation maps

obey stricter assumptions on their micro deformation, as sketched with the aid of a vector triad in Figs. 1–3 and are advocated e. g. by [1, 2, 5, 6, 13, 16]. Compared to higher-gradient continua, the essential benefit of micromorphic continua lies in the straightforward numerical treatment with additional degrees of freedom rather than tedious higher continuity requirements.

2 The Micromorphic Continuum In a micromorphic continuum body B, each physical point P with material placement X is endowed with a microcontinuum B¯ 0 that may undergo affine, otherwise arbitrary deformations independently of that of the macro material point. The micro¯ placement within B¯ 0 is denoted by X.

2.1 Micromorphic Continuum Framework As depicted in Fig. 4, the kinematics of the macro continuum obeys the standard macro deformation map and its gradient. Moreover, a micro deformation map F¯ (being a tensor of second order) and its gradient with respect to macro placement are introduced:

Modelling of Microstructured Materials

x = ϕ (X) ¯ ¯, x¯ = F(X) ·X

193

F(X) := ∇X ϕ (X) ¯ ¯ G(X) := ∇X F(X)

(1) (2)

as illustrated in Fig. 4. With the power-conjugate Piola-type stress measures, i. e. , ¯ the local balances of the macro stress P, the micro stress P¯ and the double stress Q, momentum and higher-order momentum read DivP = 0 in B0 ,

¯ − P¯ = 0 in B0 DivQ

(3)

,

omitting body forces. For the present quasistatic case, the micromorphic strong form is closed by the Dirichlet and Neumann boundary conditions

ϕ = ϕ pre P·N =

pre t0

ϕ

on ∂ B0 , on

∂ B0P ,

pre F¯ = F¯

¯ · N = ¯tpre Q 0

¯

on ∂ B0F

,

(4)

¯ ∂ B0Q

.

(5)

on

2.2 Hyperelastic Constitutive Framework For the internally stored-energy density W0 we suggest a hyperelastic constitutive ansatz, ¯ G ¯ + 1 p [F¯ − F] : [F¯ − F] ¯ = W0NH + 1 μ l 2 G:· W0 (ϕ , F) (6) 2 2 that consists of a Neo-Hooke-type term W0NH on the macro scale, a straightforward quadratic formulation on the micro scale, W0mic , and an additional scale-transition term W0scale to couple both scales. Besides the Lam´e constants λ and μ as well as the spatial dimension ndim , the formulation incorporates an internal length parameter l, which accounts for the size of the microcontinua, and a scale-transition parameter p that controls the interaction between the macro and the micro deformation.

2.3 Numerical Aspects Numerically, the micro deformation is introduced as extra degrees of freedom in a micromorphic finite-element formulation. From the balance equations (3) a nonlinear coupled system of equations is obtained, as elaborated in [8]. With e. g. a Newton-Raphson algorithm the solution of quasistatic micromorphic boundaryvalue problems is rather straightforward. 2.3.1

Influence of the Micromorphic Material Parameters

By varying the internal length parameter in (6), we observe a size effect in the predicted behaviour, as displayed in Fig. 5. For larger internal length, which represents a smaller specimen size in the same microstructured material, the micromorphic theory predicts a stiffer response with a more even spatial distribution of stress.

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C.B. Hirschberger and P. Steinmann

l =0

l = L0 /20

l = L0 /5

l = L0 /2

Fig. 5 Specimen with static lateral crack under tension: Cauchy-type macro stress σ22 in longitudinal direction

Note that the scale transition parameter p controls the coupling between macro deformation and micro deformation. The larger this is, the stronger the micromorphic model resembles the second gradient continuum, see reference [12]. 2.3.2

Material Forces in a Micromorphic Continuum

To evaluate the sensitivity of a micromorphic body to defects, in reference [8] also the material force method [17] was applied to the micromorphic continuum. In a finite-element postprocessing, the macro material force at node L is computed as 

FL :=

Σ · ∇X NLΦ dV

.

(7)

B0

It is energetically conjugate to variations in the deformation map X = Φ (x), which indicates changes in the material configuration B . The tensor Σ is the Eshelby-type macro stress, which is obtained from the Piola-type stresses using the pull-back operation ¯ 1,2 ¯ . Σ = U0 I − Ft · P − G : Q

(8)

U0 is the total energy of the material configuration. Moreover, NLΦ are the nodal shape functions to approximate the inverse macro deformation. Fig. 6 shows the material forces for different internal lengths l. Particularly the force at the crack tip displays the size effect such that a defect in a material with a larger intrinsic microstructure is less eager to propagate.

Modelling of Microstructured Materials

l=0

l = L0 /20

195

l = L0 /5

l = L0 /2

Fig. 6 Specimen with static lateral crack under tension: Micromorphic material forces FL for a variation of internal length

3 Application to Material Interfaces with Heterogeneous Micromorphic Mesostructure The micromorphic continuum was employed in the modelling of thin material layers, whose underlying heterogeneous, micromorphic mesostructure gives rise to size effects, as for the material illustrated in Fig. 7. To evaluate the macroscopic response of such a layer, a multiscale framework was proposed [9, 10] which models the layer as a cohesive interface on the macro level, while for the representative volume elements on the meso and micro scale a micromorphic continuum was chosen, as depicted in Fig. 8.

ˆ N Γˆ 0

ˆ M

¯ X

h0

X

B¯ 0

B0 w0

Fig. 7 A microstructured material layer within a bulk material

Fig. 8 Continuum multiscale framework with standard cohesive in* on the macro level, and micromorphic continuum B on the terface Γ meso scale, which intrinsically incorporate the microstructure in the microcontinua B¯

3.1 Scale Transition between Interface and Micromorphic RVE The scale transition between the cohesive interface and the micromorphic RVE is achieved by mechanical equivalence assumptions on the one hand and corresponding boundary conditions on the other hand. The decisive mechanical quantities in the cohesive interface Γˆ 0 , i. e. , deformation jump ϕˆ , traction ˆt0 , and virtual work per interface surface, are postulated to be equivalent to the volume averages of deformation, stress and virtual work in the micromorphic RVE:

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C.B. Hirschberger and P. Steinmann

  ˆ /h0 ≡ F , I + ϕˆ  ⊗ N ˆ , ˆt0 ≡ P · N ¯ δ G ¯ ˆt0 : δ ϕˆ  ≡ h0 P : δ F + P¯ : δ F¯ + Q:·

(9) (10) .

(11)

Herein ◦ denotes a volume average over the entire RVE B0 . The last equation is a Hill–Mandel-type condition, which the following boundary conditions have to fulfil: Matching the interfacial geometry, the deformation jump is applied as a boundary condition on top and bottom of the RVE (adjacent to the bulk), while we allow for periodic meso and micro deformation along the interface plane .

3.2 A Computational Homogenization Approach for Micromorphic Meso-heterogeneous Material Layers Based on this continuum mechanics multiscale framework, a nested computational homogenization scheme in the tradition of FE2 [4, 14, 15] is set up, which allows to solve nonlinear interface–micromorphic RVE multiscale problems numerically. We straightforwardly use cohesive interface elements at the macro scale, for which the constitutive response at each integration point is evaluated in the RVE using a finite element discretization obeying the boundary conditions. The macro traction vector is evaluated from the nodal reaction forces fI at all prescribed nodes I in the finite-element discretized RVE B0   1 ˆ ˆt0 = fI ⊗ XI · N I ∈ ∂ B0pre . (12) V0 ∑ I Morever, the macroscopic constitutive tangent operator Dϕˆ  ˆt0 is extracted, which projects the increment Δ ϕˆ  on Δ ˆt0 for the Newton–Raphson algorithm, see [10].

3.3 Numerical Examples Two numerical examples are selected to demonstrate the size effect, which in the one case is extracted from the RVE only without macro iterations, and in the second case is observed moreover in the global macroscopic response. Figures 9–10 display the first benchmark example, in which the macroscopic deformation is fully prescribed. The choice of the internal length l reflects the size of the microstructure in the heterogeneous layer. The larger l is, the stiffer the RVE behaves, which leads to a stiffer macro traction–separation response displayed in Fig. 10. In Fig. 11, another example is presented with a macro specimen initially consisting of a square shape enclosing a circular centred hole with a material layer positioned at both lateral sides of the hole. While it is subjected to tensile loading, in the layer, a sample micromorphic mesostructure is used to determine the constitutive response of the layer. Fig. 12 reveals the influence of the mesostructure on the macro response.

Modelling of Microstructured Materials

197 500 400

tˆ0M*

300

l/h0 l/h0 l/h0 l/h0

=0 = 0.1 = 0.2 = 0.4

200 100 0

−100 0

0.1

0.2

0.3

0.4

ϕˆ M* 

0.5

Fig. 9 Benchmark shear problem: Deformed macro Fig. 10 Shear traction vs. separation for different internal-length parameters l mesh and RVE mesh

11 00

* Ξ 11 00 1100 00 11

11 00 0 1

Fig. 11 Multiscale problem under tension: spatial macro mesh with RVEs along the interface

4

2400

l/h0=0 l/h0=0.05 l/h0=0.1 l/h0=0.2

0.3

2

1.5 2000

*f * N

0.25

1800 l/h0=0 l/h0=0.05 l/h0=0.1 l/h0=0.2

1600

0.2 1400

0.5

0.6

0.7

* Ξ

0.8

0.9

x 10

2200

* tN*

ϕ*N* 

0.35

1

1200

0.2

0.25

ϕ*N* 

0.3

0.35

l/h0=0 l/h0=0.05 l/h0=0.1 l/h0=0.2

1

0.5

0 0

0.1

0.2

0.3

u*N*

0.4

0.5

Fig. 12 Resulting macroscopic quantities in the cohesive interfaces evaluated from the micromorphic RVE

4 Conclusion The micromorphic continuum represents a versatile framework for the modelling of boundary-value problems in which size effects play a role. Due to its numerical structure, which forgoes higher continuities requirements, it is particularly straightforward to apply the finite element method. Thus in computational mechanics,

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it is relevant for a broad range of applications and in particular the multiscale modelling of miniaturized structures. Extensions to inelastic micromorphic behaviour, also involving crystal kinematics, futher increase the range of applications.

References 1. Dietsche, A., Steinmann, P., Willam, K.: Micropolar elastoplasticity and its role in localization analysis. Int. J. Plasticity 9, 813–831 (1993) 2. Ehlers, W., Volk, W.: On theoretical and numerical methods in the theory of porous media based on polar and non-polar elasto-plastic solid materials. Int. J. Solids Struct. 35, 4597–4617 (1998) 3. Eringen, A.C.: Mechanics of micromorphic materials. In: G¨ortler, H. (ed.) Proc. 11th Int. Congress of Appl. Mech., pp. 131–138. Springer, New York (1964) 4. Feyel, F., Chaboche, J.L.: FE2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composites materials. Comput. Meth. Appl. M. 183, 309–330 (2000) 5. Forest, S., Sievert, R.: Nonlinear microstrain theories. Int. J. Solids Struct. 43, 7224–7245 (2006) 6. Grammenoudis, P., Tsakmakis, C.: Hardening rules for finite deformation micropolar plasticity: restrictions imposed by the second law of thermodynamics and the postulate of Il’Iushin. Continuum Mech. Therm. 13, 325–363 (2001) 7. Grammenoudis, P., Tsakmakis, C.: Micromorphic continuum. Part I: strain and stress tensors and their associated rates. Int. J. Nonlinear Mech. 44, 943–956 (2009) 8. Hirschberger, C.B., Kuhl, E., Steinmann, P.: On deformational and configurational mechanics of micromorphic hyperelasticity – theory and computation. Comput. Meth. Appl. M. 196, 4027–4044 (2007) 9. Hirschberger, C.B., Ricker, S., Steinmann, P., Sukumar, N.: Computational multiscale modelling of heterogeneous material layers. Eng. Fract. Mech. 76, 793–812 (2009) 10. Hirschberger, C.B., Sukumar, N., Steinmann, P.: Computational homogenization of material layers with micromorphic mesostructure. Philos. Mag. A 88, 3603–3631 (2008) 11. J¨anicke, R., Diebels, S., Sehlhorst, H.G., D¨uster, A.: Two-scale modelling of micromorphic continua. Continuum Mech. Therm. 21, 297–315 (2009) 12. Kirchner, N., Steinmann, P.: A unifying treatise on variational principles for gradient and micro-morphic continua. Philos. Mag. A 85, 3875–3895 (2005) 13. Kirchner, N., Steinmann, P.: Mechanics of extended continua: modeling and simulation of elastic microstretch materials. Comput. Mech. 40, 651–666 (2007) 14. Kouznetsova, V.G., Geers, M.G.D., Brekelmans, W.A.M.: Multi-scale second-order computational homogenization of multi-phase materials: a nested finite element solution strategy. Comput. Meth. Appl. M. 193, 5525–5550 (2004) 15. Miehe, C., Schr¨oder, J., Schotte, J.: Computational homogenization analysis in finite plasticity. Simulation of texture development in polycrystalline materials. Comput. Meth. Appl. M. 171, 387–418 (1999) 16. Steinmann, P.: A micropolar theory of finite deformation and finite rotation multiplicative elastoplasticity. Int. J. Solids Struct. 31, 1063–1084 (1994) 17. Steinmann, P.: Application of material forces to hyperelastostatic fracture mechanics. I. Continuum mechanical setting. Int. J. Solids Struct. 37, 7371–7391 (2000)

Chapter 22

On Computational Homogenisation of Heterogeneous Media with Debonded Inclusions D. Peri´c, D.D. Somer, E.A. de Souza Neto, and W. Dettmer This paper is dedicated to Professor Peter Wriggers as a recognition of his immense scientific achievements and as an appreciation of our long standing friendship. Peter has undoubtedly been one of the leading figures in the field of nonlinear computational mechanics, and with his seminal contributions to the finite element analysis of nonlinear solid mechanics problems, he has been instrumental in leading the field from uncertain beginnings to its relative maturity. He has always been an inspiration for younger generations and, when needed, he has generously provided his invaluable help and advice. It has been a privilege and pleasure to know Peter, and we wish he continues his outstanding contributions for many years to come (D. Peri´c).

Abstract. Modelling of weak interfaces is an important area of micro-mechanics, as many macroscopic phenomena are linked to the behaviour at the interfaces at different scales. In this work a computational homogenization procedure is used in constitutive description of heterogeneous media with debonded inclusions. More specifically, the objective is to determine the yield surface of an RVE that contains a fully debonded inclusion embedded within ideally plastic matrix, whereby the interface is modelled by a Coulomb type friction law. The macroscopic behaviour of the RVE is shown to coincide, in the limit cases, with the behaviour of material with voids for tensile loading conditions, whereas for compressive loading conditions, it is shown to approach the behaviour of material with fully bonded inclusions.

1 Introduction and Background Due to their suitability for implementation within non-linear finite element environments, multi-scale methods are particularly attractive for the description of complex macroscopic behaviour of heterogeneous composite materials. This is commonly D. Peri´c · D.D. Somer · E.A. de Souza Neto · W.G. Dettmer Civil and Computational Engineering Research Centre, Swansea University, Swansea SA28PP, UK e-mail: [email protected],[email protected], [email protected],[email protected]

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achieved by considering relatively simple RVEs that take into account the geometry of the micro structure and whose response is modelled by conventional continuum non-linear constitutive laws. One major use of such methods is the determination of the parameters of a canonical macroscopic constitutive model by fitting the data produced by finite element solutions of a single RVE under prescribed macroscopic strain histories (see, for instance, [2, 4, 7, 8, 9, 11]). Macroscopic behaviour of composites with weak interfaces is defined not only by the mechanical properties and volume ratios of the ingredients, but also by the geometry and the strength of the imperfect bond between them. Inter-phase cracking, debonding and sliding can cause local degradation, that continuum models would fail to capture, and computational homogenization has a substantial modelling potential in this field. Obtaining macroscopic yield surfaces of heterogeneous media, in which the heterogeneity is manifested by the presence of voids or inclusions, deserves special attention. Gurson [3] derived upper bound yield loci estimates for porous ductile materials by means of a semi-analytical method, which was extended by G˘ar˘ajeu and Suquet [1] to porous ideally plastic or viscoplastic materials containing rigid particles. Giusti et al. [2] used computational homogenization to predict macroscopic yield surfaces for RVEs as functions of the void ratio of the porous metal, and compared the results with the Gurson yield surface [3]. In this work, computational homogenization is employed to assess the yield surface of an RVE containing a nearly rigid debonded elastic inclusion. For the numerical treatment of the matrix-inclusion interface within the RVE, the standard geometrically non-linear frictional contact procedure is adopted (see Wriggers [12] for details). The yield surface is constructed by observing the deviatoric/pressure stresses which occur during the plastic collapse of the RVE under a specified sequence of loadings.

2 Multi-scale Constitutive Theory: Overview The class of homogenisation-based multi-scale constitutive models employed in the present study is characterised by the assumption that the strain and stress tensors at a point of the macro-continuum are volume averages of their respective microscopic counterpart fields over a pre-specified Representative Volume Element (RVE). These fields are obtained by solving the RVE equilibrium problem, characterised by the standard virtual work equilibrium equation (see e.g. [2, 8, 6, 10])  Ωμ

σ μ : ∇η dv = 0

∀ η ∈ Vμ

,

(1)

where Vμ is the (as yet not defined) space of virtual kinematically admissible displacements of the RVE.

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2.1 RVE Kinematical Constraints The space Vμ of virtual kinematically admissible displacements defines the kinematical constraints enforced upon the RVE. Periodic RVE boundary displacement fluctuations is the assumption usually employed in the modelling of media with periodic micro-structure [5]. The periodic kinematical constraint is defined by the choice   Vμ = per Vμ ≡ v, suff. reg. | v(x+ ) = v(v− ), ∀ pairs {x+ , x− } ∈ ∂ Ω μ , (2) where {x+ , x− } are pairs of points, defined by a one-to-one correspondence, lying on opposing sides of the RVE boundary.

2.2 Finite Element Approximation The corresponding fully discrete incremental version of (1) consists in finding a vector v˜ n+1 ∈ h Vμ of global nodal displacement fluctuations such that μ g≡

 hΩ

μ

4 3 n n+1 h * n+1 η T GT σ + G v˜ n+1 μ (ε¯ μ , ξ ) dv = 0 , ∀ η ∈ V μ

,

(3)

where G denotes the global discrete gradient matrix containing the appropriate n+1 shape function derivatives, σ* μ is the incremental constitutive functional at the RVE level that delivers the array of stress components, ε¯ n+1 is the array of macroscopic strain components. Problem (3) may differ from finite element versions of conventional solid mechanics problems only in the construction of the relevant finite-dimensional space h Vμ . To achieve this, we conveniently split the degrees of freedom of the finite element mesh into three sets so that for an arbitrary vector v ∈ h V μ of kinematically admissible nodal displacements fluctuations of the RVE, we have  T v = vi v f vd v p , (4) where the subscripts i, f , d and p denote, respectively, the degrees of freedom of the interior of the RVE, the free, dependent and prescribed degrees of freedom of the RVE boundary. The set p contains the minimum prescription needed to eliminate rigid body motion and the set d contains the degrees of freedom that depend on the free ones through the kinematical constraints embedded in the definition of per V μ (2). This (linear) dependence can be generally expressed in matrix form as vd = α v f

,

(5)

where α is the identity matrix, and vd and v f contain the degrees of freedom of pairing boundary nodes located at the points with coordinates x+ and x− , respectively, lying on opposing sides of h ∂ Ω μ .

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2.3 Solution Procedure The aim is to find the unknown set of interior (vi ) and free boundary displacements (v f ), as a response to the set of prescribed displacements (v∗ ), that satisfy the equilibrium and the kinematical constraint. The final reduced set of non-linear algebraic finite element equations to be solved is obtained by introducing representation (4,5) for η and v˜ n+1 in (3). After straightforward matrix manipulations, this gives μ 6

gi g f + α T gd

7

6 7 0 = 0

(6)

,

which is solved by using the Newton-Raphson procedure. Once a solution is found, the macroscopic stress is computed by using nodal reactions.

3 Frictional Contact Assume that bodies Ω μ1 and Ω μ2 come into contact during their deformation histories corresponding to the deformation mappings ϕ 1 and ϕ 2 . By assuming a uniform 1 2 contact boundary that permits existence of a point X¯ on body Ω μ1 for every point X¯ 1 2 on Ω μ2 , where the distance between X¯ and X¯ is minimized, a penetration function gN can be defined as: 2

1

gN = (X¯ − X¯ ) · n¯ ≥ 0

(7)

.

3.1 Boundary Value Problem The standard solid mechanics boundary value problem is extended in standard fashion to involve two bodies, as well as the contact contributions C con : '    2 & α α α α α α σ μ : ∇η dv − f · η dv − t · η da ∑ α =1

Ω μα con

+C

Ω μα

= 0 , ∀η α ∈ V μ

∂ Ω μαt

(8)

.

3.2 Constitutive Relations A number of different approaches exist to include the contact contributions C con into the boundary value problem, such as, the Lagrange multipliers approach, penalty approach and direct elimination. In this work we adopt a penalty approach, which is characterised by adopting a constitutive equation at the interface, defined by: C con =

 ∂ Ω μcon

t · δ g da =

 ∂ Ω μcon

(tN δ gN + tT · δ gT ) da .

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The constraint of impenetrability given by (7) is relaxed by postulating an elastic constitutive relationship for normal contact pressure as: tN = cN gN

(9)

,

where cN is the penalty parameter. Tangential contact is concerned with the relative movement between two bodies and has two basic ingredients, as follows: If the point of contact does not change, the stick condition sets a constraint on the tangential velocity/displacement as: g˙ T = 0 ⇔ tT = 0

.

(10)

By relaxing the constraint given by (10), we establish an elastic constitutive relationship for the stick as: tT = cT gT stick = cT (gT − gT slip ) ,

(11)

where cT is a constant. The constitutive model for frictional tangential slip is commonly assumed to be analogous to elastoplastic constitutive law, with the flow rule given as: ∂Φ g˙ T slip = λ˙ , (12) ∂ tT where, Φ is the bounding (yield) function used as a flow potential and λ˙ is the plastic parameter. A convenient choice for the bounding (yield) function is based on the Coulomb’s law:

Φ = ||gT || − μ tN

.

(13)

The flow rule is completed with the loading/unloading conditions given by:

Φ ≤ 0 ; λ˙ ≥ 0 ; λ˙ Φ = 0 .

(14)

4 Assessment of Yield Surfaces of Heterogeneous Media with Debonded Inclusions In his landmark work, Gurson [3] derived upper bound yield loci estimates for porous ductile materials by collecting a sufficient number of points on the macroscopic yield surface. Each point is defined by the deviatoric/hydrostatic components of the plastic collapse stress for a prescribed load, and the yield surface is obtained by curve-fitting these points in stress space. Further work covering materials containing rigid particles [1], or comparing yield surfaces obtained by computational homogenisation [2] is also worth noting. In what follows, we extend the investigation to the plastic collapse of an RVE containing a debonded nearly rigid elastic inclusion by using computational homogenisation.

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4.1 Computational Homogenisation Based Methodology A square RVE with a rigid inclusion at its centre, 10% by volume, is considered under conditions where the matrix and the inclusion are either bonded or fully debonded. For the debonded configuration, contact with (μ =0.3) and without (μ =0) friction is considered. The ideally plastic matrix material of the plane strain RVE is modelled by means of a standard von Mises elasto-plastic model. The loading programme consists of prescribing a macroscopic strain path ε (γ ) = γ ε¯ imposed upon the RVE, parametrized by a time factor γ , and a unit strain tensor satisfying ε¯ =1 with ε¯ defined as:  1    8 √ √1 0 0 ε¯ = α 2 √1 + 1 − α 2 √1 2 . (15) 0 2 0 2 Finite element simulations are performed by varying the load factor α between (-1,1) and the macroscopic von Mises stress and hydrostatic pressure are extracted from the homogenized stress. Figure 1 shows the variation of the macroscopic von Mises stress and the pressure for bonded/debonded/no inclusion configurations for load factor α = ±0.5. As expected, the debonded inclusion is not loaded at all for

0.2

0.32

0.15

0.24

0.1

0.16

0.05

0.08

0

0 0

0.5

1

0

0.2

0.5

1

0 -0.5

0.15 -1 0.1

-1.5 -2

0.05 -2.5 0

-3 0

0.5

(a)

1

0

(b)

0.5

(c)

1

(d)

Fig. 1 Plots for a) von Mises Stress (top) and Pressure (bottom) vs. pseudo time, and b) equivalent plastic strain for (top) fully bonded inclusion, fully debonded inclusion without friction (μ =0), fully debonded inclusion with friction (μ =0.3) and (bottom) no inclusion for α = 0.5; c) and d) for α = −0.5

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tensile loading conditions, thus all graphs coincide with an RVE that has a void, whereas for the compressive loading conditions it can be observed that the RVE with a fully debonded inclusion approaches the behaviour of an RVE with a fully bonded inclusion as the pressure is increased.

4.2 Estimated Yield Surfaces Plastic collapse is assumed to occur when the macroscopic stress reaches the limit value with increasing load. For tensile loading conditions this is manifested by the flattening of the von Mises stress curve. For compressive loads, for which the RVE stiffens, plastic collapse does not happen for the volume fraction considered. For visualisation, we have picked and interpolated points that intersect a 2% offset from the elastic part of the von Mises stress curve. Non-dimensional pressure and the von Mises stress p¯ q¯ of the plastic collapse stress are calculated by normalising with the yield stress. Each p¯ q¯ couple constitutes a yield surface point, which is plotted in Fig. 2. Note that for an RVE with a void, the yield surface obtained Giusti et al. [2] is recovered. Again, for tensile loading conditions, the RVE with a fully debonded inclusion approaches the behaviour of an RVE with a void, whereas for compressive loads, the RVE with a fully debonded inclusion approaches the behaviour of an RVE with a fully bonded inclusion.

1.2 1 Debonded Inclusion, μ =0 Debonded Inclusion, μ =0.3 Bonded Inclusion Hole Homogeneous Matrix

q

0.8 0.6 0.4 0.2 0 -5

-4

-3

-2

-1

0

1

2

3

4

5

p

Fig. 2 Estimated yield surfaces for inclusion that is (i) fully bonded (ii) fully debonded without friction (μ =0) (iii) fully debonded with friction (μ =0.3) and (iv) no inclusion

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5 Conclusion and Remarks Modelling of weak interfaces is an important area of micro-mechanics, as many macroscopic phenomena are linked to interfaces at different scales. An RVE with a fully debonded inclusion has been studied, in which the interface between the inclusion and matrix has been modelled using the Coulomb friction law. It has been demonstrated that, under tensile loading conditions, the RVE approaches the behaviour of material with voids, whereas for compressive loads, it approaches the behaviour of material with a fully bonded inclusion.

References 1. G˘ar˘ajeu, M., Suquet, P.: Effective properties of porous ideally plastic or viscoplastic materials containing rigid particles. J. Mech. Phys. Solids 45, 873–902 (1997) 2. Giusti, S.M., Blanco, P.J., de Souza Neto, E.A., Feij´oo, R.A.: An assessment of the Gurson yield criterion by a computational multi-scale approach. Eng. Computation 26, 281–301 (2009) 3. Gurson, A.L.: Continuum theory of ductile rupture by void nucleation and growth: part I - yield criteria and flow rules for porous ductile media. J. Eng. Mater. T. ASME. 99, 2–15 (1977) 4. Hain, M., Wriggers, P.: Numerical homogenization of hardened cement paste. Comput. Mech. 42, 197–212 (2008) 5. Michel, J.C., Moulinec, H., Suquet, P.: Effective properties of composite materials with periodic microstructure: a computational approach. Comput. Meth. Appl. M. 172, 109–143 (1999) 6. Miehe, C., Shotte, J., Schr¨oder, J.: Computational micro-macro transitions and overall moduli in the analysis of polycrystals at large strains. Comp. Mater. Sci. 16, 372–382 (1999) 7. Pellegrino, C., Galvanetto, U., Schrefler, B.A.: Computational techniques for periodic composite materials with non-linear material components. Int. J. Numer. Meth. Eng. 46, 1609–1637 (1999) 8. Speirs, D.C.D., de Souza Neto, E.A., Peri´c, D.: An approach to the mechanical constitutive modelling of arterial wall tissue based on homogenization and optimization. J. Biomech. 41, 2673–2680 (2008) 9. Temizer, I., Wriggers, P.: An adaptive method for homogenization in orthotropic nonlinear elasticity. Comput. Meth. Appl. M. 196, 3409–3423 (2007) 10. Terada, K., Saiki, I., Matsui, K., Yamakawa, Y.: Two-scale kinematics and linearization for simultaneous two-scale analysis of periodic heterogeneous solids at finite strain. Comput. Meth. Appl. M. 192, 3531–3563 (2003) 11. Watanabe, I., Terada, K., de Souza Neto, E.A., Peri´c, D.: Characterization of macroscopic tensile strength of polycrystalline metals with two-scale finite element analysis. J. Mech. Phys. Solids 56, 1105–1125 (2008) 12. Wriggers, P.: Computational Contact Mechanics. Wiley, New York (2002)

Chapter 23

Assessment of Homogenization Errors in Transient Problems K. Runesson, F. Su, and F. Larsson Dedicated to Professor Peter Wriggers on the occasion of his 60th birthday (K. Runesson, F. Su and F. Larsson).

Abstract. Model-based 1st order homogenization for stationary problems was recently extended to transient problems. Along with the classical averages, a higher order conservation quantity in the macroscale problem is then obtained. This effect depends on the size of the “subscale computational cell” (denoted RVE) that is subjected to different prolongation conditions (Dirichlet, Neumann). The issue addressed in this paper is how to choose the optimal size of the RVE in order to obtain the best possible fit to the single-scale solution. It turns out that there is a trade-off between the RVE-size and the macroscale mesh diameter.

1 Introduction Homogenization in space is a well-established technique for dealing with the effect of microstructural heterogeneity that can be considered as statistically homogeneous in the standard sense. In the case of “first order” homogenization of stationary problems, such as quasistatic elasticity and stationary diffusion, the concept of a Representative Volume Element (RVE) is well-defined. The convergence of a typical macroscale variable (such as the effective strain energy) for increasing RVE-size is smooth and (in general, although not always) monotonic. Moreover, the sensitivity of this convergence for the macroscale FE-mesh density is small. Clearly, the subscale modeling, e.g. boundary conditions and finite element discretization of the RVE, represents model errors in the corresponding macroscale “constitutive relations”. The picture is considerably less clear in the presence of transients, such as transient diffusion, deformation coupled to pore pressure development, etc., even when the classical first order homogenization assumption is adopted and the K. Runesson · F. Su · F. Larsson Chalmers University of Technology, SE-412 96 G¨oteborg, Sweden e-mail: [email protected],[email protected], [email protected]

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time-variation is assumed smooth. Such spatial homogenization problems have quite ¨ recently been treated in the literature; e. g. Ozdemir et al. [4], Temizer and Wriggers [5, 6], Larsson et al. [2]. Although based on asymptotic expansion, homogenization in both space and time was discussed in e.g. Fish et al. [1]. A most striking feature is the size-effect of the “RVE”1 that is introduced by the transient character. In fact, the corresponding “internal length” plays a role that is similar to that of socalled ”higher order” spatial homogenization models for stationary problems, cf. Kouznetsova et al. [3], and influences the “effective” response that is obtained (in practice) for sufficiently dense macroscale FE-mesh. Another way of expressing this fact is that the transient character introduces scale-mixing. The aim of this paper is to assess the influence of the transient and how the model error depends (in qualitative terms) on the macroscale FE-mesh as well as on the RVE-size. Particular focus is placed on the effect of ignoring the subscale transient (still preserving the macroscale scale-effect).

2 Transient Heat Flow – A Model Problem 2.1 Space-Variational Format As a model problem, consider the (uncoupled) energy balance equation dt Φ + q · ∇ = f

in Ω × [0, T ],

(1)

where Φ = Φ (u) is the stored volume-specific internal energy (which is a conservation quantity and taken as a function of the absolute temperature, u, in the simplest modeling approach), q = q(g) is the heat flux (which is taken as a funcdef

tion of the temperature gradient, g =∇u, in the simplest modeling approach), and where ∇ is the spatial gradient with respect to coordinates x in Ω . Moreover, f is a (bulk)volume-specific heat source (supply of energy) within Ω . Henceforth, we choose the simplest possible constitutive equations for Φ and q of the form Φ (u) = c u and q(g) = −k g, where we introduced the constant, but strongly microheterogenous, coefficients of thermal capacity, denoted c, and thermal conductivity, denoted k, respectively. The resulting model problem is, thus, completely linear. As to the boundary conditions, we shall henceforth adopt the standard subdivision of ∂ Ω into Dirichlet and Neumann parts with prescribed temperature (u = up ) and def

prescribed heat flux (z =q · n = zp ), respectively. To avoid technical difficulties in homogenization on the boundary, we henceforth assume that zp = 0. The classical approach to introduce “model-based homogenization” is via volume-averaging2: y(¯x,t) → y2 (¯x,t), whereby it is assumed that the RVE 1 2

Although not valid in the classical sense the notion of RVE is used subsequently for simplicity instead of the formally more adequate “subscale computational cell”. The subscale volume average on Ω2 of a function y is denoted def

y2 (¯x) = |Ω 1(x)| 2



Ω2 ydV,

x¯ ∈ Ω .

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occupies the subscale region Ω2 (with boundary Γ2 ). The RVE is centered at the macroscale position x¯ , i.e. x − x¯ 2 = 0 for any given x¯ ∈ Ω . Upon employing the dG(0)-method for integrating the (fine)scale problem in def time, we introduce the solution space (in standard fashion) U = n U associated with the time interval In = (tn−1 ,tn ) whose length is Δ t = tn − tn−1 and the relevant variational forms: n

def

A( u; δ u) = n

n

def

L(δ u) =

 Ω

[−Δ t n q · ∇[δ u]2 + n Φ δ u2 ] dV

(2)

  Ω

 Δ t n fˆ δ u2 + n−1Φ δ u2 dV

(3)



def where we introduced the time-averaged source n fˆ = I1n In f dt. The resulting “homogenized” problem then reads: For n = 1, 2, ..., N, find n u ∈ U that solves n

def

R(n u; δ u) = n L(δ u) − n A(n u; δ u) = 0

∀ δ u ∈ U0

.

(4)

To simplify notation, superscript n will be omitted henceforth.

2.2 Explicit Homogenization Results Inside each RVE, the total solution is split into a smooth part, uM , and a fluctuating part, us ; hence, u(¯x; x) = uM (¯x; x) + us (¯x; x). The scales are linked by setting ¯ is the smooth (macroscale) solution uM (¯x; x¯ ) = u(¯ ¯ x) inside each RVE, where u¯ ∈ U def

defined on Ω . The steps of connecting uM and us =us {uM}3 for given uM define the prolongation and allow for computing homogenized quantities. The homogenization properties are defined explicitly via a suitable assumption about the smoothness of the macroscale solution uM . First order homogenization (linear variation of uM within the RVE) will be assumed here in standard fashion uM (¯x; x) = u(¯ ¯ x) + g¯ (¯x) · [x − x¯ ] for x ∈ Ω2

,

(5)

def ¯ where g¯ = ∇ u¯ is the macroscale temperature gradient.

Remark: Before specifying boundary conditions on the RVE-problem, we note that u¯ = u2 and g¯ = ∇u2 = g2 since us 2 = 0 and ∇us 2 = 0 in general. Upon inserting the expression (5) into (2,3) and using the appropriate HillMandel macro-homogeneity condition, we may obtain the macroscale balance equa¯ that solves tion: Find u¯ ∈ U def ¯ ¯ u; ¯ u; R{ ¯ δ u} ¯ = L{ δ u} ¯ − A{ ¯ δ u} ¯ = 0,

¯0 ∀δ u¯ ∈ U

with the homogenized forms A¯ and L¯ defined explicitly as 3

Curly brackets {(•)} indicate implicit functional dependence on (•).

(6)

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def ¯ u; ¯ = A(u; δ uM ) = A{ ¯ δ u} def ¯ δ u} L{ ¯ = L(δ uM ) =

 3

 3Ω Ω

4 ¯ · δ g¯ dV¯ ¯ u¯ + Φ −Δ t q¯ · δ g¯ + Φδ

(7) 4

¯ u¯ + n−1Φ¯ · δ g¯ dV¯ Δ t f¯δ u¯ + Δ t ¯f · δ g¯ + n−1 Φδ

.(8)

The pertinent homogenized quantities of 1st and 2nd order are given as the (implicit) relations in terms of the macroscale variables u¯ and g¯ : 9 : q¯ {u, ¯ g¯ } = q2 , Φ¯ {u, ¯ g¯ } = Φ 2 , f¯ = fˆ 2 (9) def Φ¯ {u, ¯ g¯ } = Φ [x − x¯ ]2 ,

9 : ¯f def = fˆ[x − x¯ ] 2

.

(10)

¯ are vector-valued quanIt is noted that f¯ and Φ¯ are scalar-valued, whereas ¯f and Φ tities. In the present case of a linear problem, it is clear that the homogenized quantities can be expressed in terms of constant (within a given time increment) “effective” properties as q¯ = q¯ 0 + Y¯ u¯ − K¯ · g¯ ,

Φ¯ = Φ¯ 0 + C¯ u¯ + B¯ · g¯ ,

Φ¯ = Φ¯ 0 + C¯ u¯ + B¯ · g¯

(11)

¯ and C¯ are vectors, whereas K¯ and where it is noted that C¯ is a scalar quantity, Y¯ , B, ¯B are 2nd order tensors. For the present class of problems, the linear approximation does indeed introduce a size effect in analogy to the case of 2nd order homogenization, cf. Kouznetsova et al. [3]. This fact becomes evident in the appearance of the “2nd order” term Φ¯ , which will vanish when the size of the RVE becomes infinitely small. In order to compute the “effective” tensors, it is necessary to solve the pertinent sensitivity problem for the chosen prolongation condition on the RVE-problem. This task is considered next in terms of Dirichlet and Neumann boundary conditions.

3 RVE-Problem 3.1 Dirichlet Boundary Conditions The most straightforward choice of prolongation condition is defined by the con(D) dition us = 0 on Γ2 associated with the pertinent subscale solution space U2 = {u suff. regular : u = 0 on Γ2 }. We note the identity g¯ = ∇u2 = g2 while still u¯ = u2 . The space-variational RVE-problem associated with the time interval In (D) becomes: For given u, ¯ g¯ , find us ∈ U2 that solves a2 (uM (u, ¯ g¯ ) + us , δ us ) = l2 (δ us ), where we introduced the space-variational forms

(D)

∀δ u s ∈ U 2

(12)

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def

a2 (u, δ u) = δ u Φ (u)2 − Δ t∇[δ u] · q(g)2 = cδ u u2 + Δ tk ∇[δ u] · ∇u2 def

l2 (δ u) = Δ t fˆ δ u2 + n−1 Φδ u2 = Δ t fˆ δ u2 + c n−1 u δ u2

.

(13) (14) (15) (16)

Remark: (i) The transient of the RVE-problem becomes very rapid with reduced RVE-size, i.e. a stationary problem is retrieved in the limit when L2 becomes vanishingly small. (ii) To ignore the subscale transiet for a finite-sized RVE represents a model error on the macroscale. In order to establish the effective operators, one needs to compute “unit fluctuation fields” corresponding to a unit value of u¯ and g¯ , cf. Larsson et al. [2].

3.2 Neumann Boundary Conditions The other extreme prolongation condition is defined by the assumption that the RVE-boundary flux z is generated from an a priori unknown constant flux vector qˆ ˆ which corresponds to a boundary condition of the type z = qˆ · n + Qˆ and heat sink Q, on Γ2 . We may then introduce the pertinent space for the (local) temperature solu(N) tion as follows on any given RVE: U2 = {u suff. regular}. The space-variational RVE-problem associated with the time interval In becomes: For given u, ¯ g¯ , find (N) u ∈ U2 , qˆ ∈ R3 and Qˆ ∈ R that solve (N) ˆ = l2 (δ u), a2 (u, δ u) + c2,1 (δ u, qˆ ) + c2,2(δ u, Q) ∀δ u ∈ U2 c2,1 (u, δ qˆ ) = Δ t g¯ · δ qˆ , ∀δ qˆ ∈ R3 | Γ | ˆ = Δ t u¯ 2 δ Q, ˆ ∀δ Qˆ ∈ R c2,2 (u, δ Q) |Ω 2 |

(17)

where we introduced the space-variational forms def

a2 (u, δ u) = δ u Φ (u)2 − Δ t∇[δ u] · q(g)2 = cδ u u2 + Δ tk ∇[δ u] · ∇u2    Δt def c2,1 (u, qˆ ) = Δ t∇u2 · qˆ = n u dS · qˆ |Ω2 | Γ2    Δt ˆ def c2,2 (u, Q) = u dS Qˆ |Ω2 | Γ2 def l2 (δ u) = Δ t fˆ δ u2 + n−1Φδ u2 = Δ t fˆ δ u2 + c n−1 u δ u2

(18) (19) (20) (21) (22)

Remark: The condition (17)2 obviously ensures that ∇u2 = g¯ . However, (17)3 does not ensure that u2 = u; ¯ hence, u2 = u¯ in general.

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4 Computational Results 4.1 Problem Definition – Substructure Characteristics A very simple 1D-problem in Ω = (0, 1) is used to illustrate how the homogenization error depends on the RVE-size, L2 . In particular, it is of interest to assess the error of ignoring the subscale transient for given value of L2 . Simple linear FEapproximation of u¯ is adopted with 3 Gauss-points per element. The same type of element is adopted for the subscale FE-discretization. The transient problem is “driven” by the boundary conditions u(0,t) ¯ = t, g(1,t) ¯ = 0, and we consider only a single timestep, Δ t = 1. The initial temperature is homogeneous (both on the subscale and macroscale), which gives Φ0 = 0 and, as a consequence, Φ¯ 0 = 0 and ¯ = 0. We remark that, in the present simple 1D-situation, identical RVE-solutions Φ 0 are obtained for the Dirichlet and Neumann conditions. The fine-scale, or “exact”, solution (denoted ufine (x,t)) is obtained for the choice πx  πx  k = kmin + [kmax − kmin ] sin2 , c = cmin + [cmax − cmin ] sin2 (23) l l where l is the “wavelength” of the spatial heterogeneity. Here, we choose kmax = cmax = 1 and kmin = 0.1, cmin = 0.5. In terms of homogenization, it is assumed that (23) holds within each RVE with the ”global” coordinate x replaced by the RVEcoordinate x − x; ¯ thus representing a typical realization of a statistically homogeneous microstructure. The effect of the RVE-size is shown in Figs. 1 to 2 in terms of how the homogenization error varies with the mesh-density. Here, the relative error in “average response” is depicted. It is defined as E=|

Qhom (u) − 1| Q(ufine )

(24)

where  L

Q(u) = 0

 L

NG

u dx ≈ ∑ u(x¯i )Wi , i=1

Qhom (u) =

0

NG

u2 dx ≈ ∑ u2,iWi

(25)

i=1

where NG is the total number of Gauss integration points in the macroscale mesh defined by the typical mesh diameter H. Note that u2,i = u( ¯ x¯i ) in general. Subscale transient solutions are labelled (t), while subscale stationary solutions (which are approximate) are labelled (s). The figures show quite consistently that the difference between the (t) and (s) solutions increases with increasing RVE-size and, moreover, that this difference is more pronounced for a larger microstructure, l, at the same absolute RVE-size, L2 . A finite value of L2 induces a model error in comparison with the single-fine scale problem. However, it is of considerable interest to note that the optimal choice of L2 , that gives the minimal error E, depends strongly on the macroscale mesh size H. For example, Fig. 1 shows that minimum error is obtained for L2 = 9l = 0.36 for

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4.5

L2 /l = 1 (t) L2 /l = 1 (s) L2 /l = 5 (t) L2 /l = 5 (s) L2 /l = 9 (t) L2 /l = 9 (s)

4

Relative error, E, [%]

3.5 3 2.5 2 1.5 1 0.5 0

2

4

6

8

10

12

14

16

Number of macroscale elements Fig. 1 Convergence of the macroscale FE-solution for different relative RVE-size, L2 /l. Microheterogeneity is defined by the “wavelength” l = 0.04

3.5

L2 /l = 1 (t) L2 /l = 1 (s) L2 /l = 5 (t) L2 /l = 5 (s) L2 /l = 9 (t) L2 /l = 9 (s)

Relative error, E, [%]

3

2.5

2

1.5

1

0.5

0 2

4

6

8

10

12

14

16

Number of macroscale elements Fig. 2 Convergence of the macroscale FE-solution for different relative RVE-size, L2 /l. Microheterogeneity is defined by the “wavelength” l = 0.01

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as little as 4 macro-elements, whereas the error vanishes for L2 = 5l = 0.20 when 10 elements are used. Clearly there is no objective rule, and the optimal choice of L2 depends pathologically on H for each given heterogeneity measure l. For large l (such as those shown in Fig. 1), it appears that ignoring the subscale transient introduces a significant model error. On the other hand, for small l (as shown in Fig. 2), then it is clear that ignoring the subscale transient introduces less model error. Finally, the remaining model error introduced by homogenization is retrieved by considering the convergence with H.

5 Conclusions Variationally consistent 1st order homogenization of transient problems introduces a higher order conservation quantity in the macroscale problem, and its effect is enhanced by the RVE-size. For a given macroscale mesh diameter, there exists an optimal size of the RVE in order to obtain the best possible fit to the single-scale (fully resolved) solution. Moreover, to ignore the subscale transient introduces a model error for a finite-sized RVE.

References 1. Fish, J., Chen, W., Nagai, G.: Non-local dispersive model for wave propagation in heterogeneous media: one-dimensional case. Int. J. Numer. Meth. Eng. 54, 331–346 (2002) 2. Larsson, F., Runesson, K., Su, F.: Variationally consistent computational homogenization of transient heat flow. Int. J. Numer. Meth. Eng. 81, 1659–1686 (2010) 3. Kouznetsova, V., Geers, M.G.D., Brekelmans, W.A.M.: Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme. Int. J. Numer. Meth. Eng. 54, 1235–1260 (2002) ¨ 4. Ozdemir, I., Brekelmans, W.A.M., Geers, M.: Computational homogenization for heat conduction in heterogeneous solids. Int. J. Numer. Meth. Eng. 73, 185–204 (2008) 5. Temizer, I., Wriggers, P.: Thermal contact conductance characterization via computational contact homogenization: a finite deformation theory framework. Int. J. Numer. Meth. Eng. 83, 27–58 (2010) 6. Temizer, I., Wriggers, P.: Homogenization in finte thermoelasticity. J. Mech. Phys. Solids (2010), doi:10.1016/j.jmps.2010.10.004

Chapter 24

Multiscale Modeling of Metal Foams Using the XFEM Lovre Krstulovic-Opara, Stefan Loehnert, Dana Mueller-Hoeppe, and Matej Vesenjak I joined the group of Professor Wriggers in September 1997 when he was a head of the Institut f¨ur Mechanik IV at the University of Darmstadt. I was employed as a researcher on the Brite-Euram project. Together with Professor Wriggers and few other co-workers I moved to Institut f¨ur Baumechanik und Numerische Mechanik, University of Hannover in September 1998, where I continued to work on the same project. After my promotion in December 2000, I continued to work with the IBNM group till September 2001 when I moved to Croatia and later on got a position on the University of Split. I continued my collaboration with Prof. Wriggers trough two bilateral DAAD projects. Beside the great time spend together with the group of Professor Wriggers, his family and himself, our joined work resulted in seven journal articles and twelve conference contributions. We are still keeping on working together on several topics such as multi-scale modeling (L. Krstulovic-Opara).

Abstract. Irregular open cell cellular structures such as metal foams play an important role in light weight structures and shock absorption components. They are tested experimentally in quasi-static and dynamic compression tests, and they are modeled numerically with finite elements and geometrical information from CT scans. Filler material within the metal foams enhances the energy dissipation properties which are important for shock absorbers. For comparison reasons the same geometrical structure of the foam needs to be modeled with and without filler material. One way Lovre Krstulovic-Opara University of Split, FESB e-mail: [email protected] Stefan Loehnert · Dana Mueller-Hoeppe Institute of Continuum Mechanics, Leibniz Univesit¨at Hannover, Appelstr. 11, D-30167 Hannover, Germany e-mail: [email protected], [email protected] Matej Vesenjak University of Maribor, Faculty of Mechanical Engineering e-mail: [email protected]

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to do that is the application of level sets and the XFEM to account for the jump in the strain field within an element. Since only small parts of the foam structure can be modeled, a representative volume element of the structure is used for the calculations. A comparison of the experiments as well as the numerical results of the model with and without filler material is shown.

1 Introduction Cellular structures are common structural forms characterizing numerous bio-materials such as cork, bone, coral, wood, and engineering materials such as metal foams, insulation materials, etc. For the case of bio-materials, the cellular structures are often filled with second phase material, e.g. gelatinous or liquid substance. Our recent research [10, 5, 8, 4] was focused on filling regular cellular metal cubes with silicone filler material, where the goal was to increase the absorption capabilities by introducing the silicone filler. Our current research is experimental evaluation and numerical modeling of irregular metal foams, where a CT scan of the metal foam is the base for the numerical model presented herein. In reference [3], the quasistatic and dynamic modeling of irregular open-cell cellular structure without filler material is presented. The quasi-static results from [3] are addressed herein when making comparison to the metal foam without the filler included.

Fig. 1 The metal foam with and without filler

2 Modified XFEM for Heterogeneous Materials The discretization of complex three dimensional geometries with standard finite elements can be cumbersome. In general, meshes consist mainly of tetrahedral elements, and very fine meshes are required to obtain qualitatively good results. This is computationally expensive.

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An alternative is to use the eXtended Finite Element Method (XFEM) [1], where the material interface can be represented independently of the mesh [9], thus facilitating the use of a regular structured brick mesh. The idea of the XFEM is to incorporate the geometry of the interface by a level set function φ , where the signed distance to the interface is stored for each node of elements belonging to both phases and interpolated by trilinear shape functions within the element. The discontinuity in the strain and stress field is accounted for by enrichment functions, which include a priori known solution properties in the approximation space. The approximation of the displacement field uh is then given by uh (x) = ∑ Ni (x)ui + ∑ Ni (x) · ψi (x)ai

(1)

i∈I ∗

i∈I

where I is the set of all nodes in the domain and I ∗ is the set of enriched nodes, see Fig. 2 for illustration. The standard nodal displacement degrees of freedom are given by ui and the extra nodal degrees of freedom associated with the enrichment function ψ are given by ai , where Ni are the standard trilinear shape functions. For the enrichment function for heterogeneous materials the ramp function introduced by [2] for the 2d case and extended by [7] to the 3d case is used, 3 4 ψi (x) = |φ h (x)| − |φ h (xi )| · R(x) . (2) The ramp function R(x) =

∑ Ni (x)

(3)

i∈I ∗

ensures that in the blending elements depicted in Fig. 2, where only part of the nodes are enriched in the standard XFEM, the partition of unity is fulfilled by “fading out” the enrichment function. The originally intended enrichment used in the standard XFEM is maintained only in those elements where all nodes are enriched. In the enrichment function given in Eq. (2) the actual absolute value function is shifted by Fig. 2 Blending elements and enriched nodes of the original (left) and the corrected (right) XFEM

enriched node material interface element blending element

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the nodal value of the absolute value function. Only by using this shift in combination with the ramp function the discontinuity in the strain field is reflected accurately and does not introduce spurious displacements in the blending elements.

3 Incorporation of Finite Plasticity The base material of the investigated metal foam is aluminum alloy. It reacts elastically only for small strains. For larger strains the considered aluminum alloy shows extensive plasticity with a rather small hardening effect. The elastic strains remain small even when the plastic strains are large. An appropriate elasto-plastic material model for such a behavior is given by Hencky plasticity. The strain measure used is the logarithmic stretch εi = log(λi ) with λi being the principal stretches. This ensures validity for large plastic strains. The used strain energy density function for the stored elastic energy as a function of the elastic part of the logarithmic stretches is   1 K Ψ (ε1 , ε2 , ε3 ) = μ ε1e 2 + ε2e 2 + ε3e 2 − μ (log(J))2 + (log(J))2 3 2

.

(4)

Here, K is the bulk modulus, μ is the shear modulus and J = det(F) is the determinant of the deformation gradient. The corresponding Kirchhoff stress tensor is 2 τi = 2 μεie − μ log(J) + K log(J) (5) 3 where the multiplicative split of the deformation gradient into its elastic and plastic part is used. F = Fe · F p (6) The applied plasticity model is a simple von Mises model with linear isotropic hardening and associated flow rule. The yield function is ; 3 dev dev f (τ , ξ ) = (τ : τ ) − τy , τy = τy0 + H ξ (7) 2 where H is the hardening modulus and ξ is the hardening variable. The integration of the flow rule is numerically done by a radial return mapping algorithm.

4 Comparison of Metal Foams with and without Filler Material The open-cell cellular specimens tested and modeled herein is aluminum alloy foam (Fig. 1) scanned by a CT scanner. The applied material parameters for the finite plasN , K = 42604 N , τ 0 = 91 N and H = 500 N . ticity model are μ = 19663 mm 2 mm2 y mm2 mm2 The filler material is a vacuum treated two-component molding silicone 2K-Z010. The molding silicone insures easy pealing off, preventing the silicone creating tension forces between foam and the filler material. More details about the silicone

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material characteristics are presented in [10]. The applied load is a displacement controlled uniaxial compression at speed of 0.2 mm s . The diagrams in Fig. 3 depict the force-displacement comparison of the foam with and without the filler included. As foam specimens are in a first step cut into cubes, then poured with silicone, and again sliced into cubes, the loading of filled specimens is characterized by the soft response at the beginning. This soft behavior caused by the specimen cutting imperfection, corresponding to the first 3 millimeters of the displacement in the loading curve (Fig. 3), represents the part where the metal foam lattice at the boundary is getting into contact with the compression plates. Thus the soft part at the beginning of the loading can be neglected. When comparing the response of the irregular open-cell cellular material with filler to the regular ones with filler presented in [10], the filler influences the results more in case of the irregular open-cell cellular specimens. This is caused by the higher portion of the filler material for the regular case. The goal of the presented research was to increase the energy dissipation during the compression without increasing the force for the yielding stress plateau. The dissipated energy is the surface under the load-deflection curve. The energy dissipation increase is clearly distinguishable for this irregular open-cell cellular material with filler included (i.e. the metal foam). That was not the case for the regular open-cell cellular material presented in [10]. The densification in the load-deflection curve occurs earlier in the diagram for the filler case. Although an increasing yielding stress can be observed, its inclination is not too large. Therefore there is a significant increase of the dissipated energy with a slight growth of the yield stress as a drawback in the case when material is used for energy dissipation members. However, the densification strain is not significantly decreased. The numerical simulations of the irregular open cell structure without filler material are performed by using a quadratic tetrahedral mesh generated from the CT scan

Fig. 3 Quasi-static loading of irregular open-cell foam with and without silicone filler material included

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Fig. 4 Von Mises stress within the deformed foam structure without filler material calculated with standard quadratic tetrahedral finite elements

Fig. 5 Von Mises stress within the deformed foam structure including filler material calculated with the XFEM

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geometry. From Fig. 4 it can be seen that in quite a few spots the material plasticizes at an early stage of the deformation already. Plastic deformations occur a long time before local buckling appears within the structure. The small hardening modulus of aluminum leads to the fact that plasticity abets buckling. For comparability reasons for the calculations including filler material it is necessary to use the same geometry of the aluminum foam matrix. Since it appears difficult to create a mesh for the filler material connected to the mesh of the aluminum matrix, level set values are computed at the nodes of a regular trilinear brick mesh in order to represent the geometry of the matrix and filler material implicitly. Using these level sets and the XFEM, computations including the filler material can be performed. In Fig. 5 the deformed configuration as well as the von Mises stress of the foam with filler is shown.

5 Conclusions The introduction of the molding silicone filler material results in significant energy dissipation increase. When comparing to the case of regular open-cell cellular material presented in [10] significant differences in the behavior are noticed. As the filler influence depends upon the foam-to-filler ratio as well as on the foam lattice structure, any variation in material structure should be tested, scanned by the CT, and numerically modeled. The general conclusion should not be carried out without testing and modeling of the material, which points out the importance of the modeling approach presented herein. The numerical computations of the RVE are meaningful only up to the first buckling point which does occur quite early within the structure. Beyond that point due to localization effects, the RVE concept fails. Multiscale techniques that are capable of handling instabilities and localization effects such as the multiscale projection method [6] need to be applied in future to handle the post-buckling domain of the open-cell cellular structure with or without filler material included. Acknowledgements. The financial support from the DAAD German-Croatian project “Dynamic multi-scale analysis of irregular open cell cellular structures” is gratefully acknowledged.

References 1. Belytschko, T., Black, T.: Elastic crack growth in finite elements with minimal remeshing. Int. J. Numer. Meth. Eng. 45, 601–620 (1999) 2. Fries, T.-P.: A corrected XFEM approximation without problems in blending elements. Int. J. Numer. Meth. Eng. 75, 503–532 (2008) 3. Krstulovic-Opara, L., Loehnert, S., Mueller-Hoeppe, D.S., Vesenjak, M.: Multi-scale modeling of regular open-cell cellular structures. In: Proceedings of ECCM 2010 Paris, IV European Conference on Computational Mechanics (2010)

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4. Krstulovic-Opara, L., Loehnert, S., Vesenjak, M., Mueller-Hoeppe, D.: Multi-scale modeling of regular open-cell cellular structures based on the homogenization principles. In: Papadrakakis, M., Kojic, M., Papadopoulos, V. (eds.) 2nd South-East European Conference on Computational Mechanics, SEECCM 2009 (2009) 5. Krstulovic-Opara, L., Loehnert, S., Vesenjak, M., Mueller-Hoeppe, D., Wriggers, P., Marendic, P.: Multiscale numerical modeling of regular open-cell cellular structures wiht elastic filler material. In: Schrefler, B.A., Perego, U. (eds.) Proceedings of the World Congress on Computational Mechanics (2008) 6. Loehnert, S., Belytschko, T.: A multiscale projection method for macro/microcrack simulations. Int. J. Numer. Meth. Eng. 71, 1466–1482 (2007) 7. Loehnert, S., Mueller-Hoeppe, D.S., Wriggers, P.: 3D corrected XFEM approach and extension to finite deformation theory. Accepted in Int. J. Numer. Met. Eng. (2010) 8. Loehnert, S., Krstulovic-Opara, L., Vesenjak, M., Mueller-Hoeppe, D., Wriggers, P.: Homogenization of regular open cell cellular structures accounting filler material. In: Smojver, I., Soric, J. (eds.) Proceedings of 6th International Congress of Croatian Society of Mechanics (2009) 9. Sukumar, N., Chopp, D.L., Mo¨es, N., Belytschko, T.: Modeling holes and inclusions by level sets in the extended finite-element method. Comput. Meth. Appl. M. 190, 6183–6200 (2001) ¨ 10. Vesenjak, M., Krstulovic-Opara, L., Ren, Z., Ochsner, A., Domazet, Z.: Experimental study of open-cell cellular structures with elastic filler material. Exp. Mech. 49, 501–509 (2009)

Chapter 25

3D Multiscale Projection Method for Micro-/Macrocrack Interaction Simulations Stefan Loehnert and Dana Mueller-Hoeppe

In the beginning of my studies Peter Wriggers inspired me with his great lectures to set a focus on mechanics and computational mechanics. He encouraged me to study abroad and eventually he became my mentor and supervisor for my PhD program. Thanks to him and his continuous support I had the opportunity to do research and to develop my fields of interest within numerical methods. It is a great pleasure to work together with him as a PhD student, post-doc and researcher (S. Loehnert).

Abstract. The presented 3D multiscale projection technique is an extension of the 2D version proposed in [2]. Especially in three dimensions it is essential to have efficient numerical methods at hand that are capable of dealing with complex mechanical problems such as localization phenomena on multiple scales. These phenomena are important to consider accurately since microstructural features can have a significant influence on the propagation behavior of a macrocrack. In the presented method, we employ the corrected version of the XFEM to account for the discontinuities in the displacement field due to cracks on all scales. Further improvements of the multiscale technique that lead to a reduction in computational time are shown.

1 Introduction Especially for the simulation of three dimensional problems that are strongly influenced by different characteristics on multiple scales like e.g. the interaction of a macrocrack with multiple microcracks in the vicinity of the macrocrack front, modern multiscale and multiphysics methods have gained enormous attention. A strong interaction of microcracks and a propagating macrocrack plays an important role e.g. in the failure analysis of ceramic materials as they are used for human bone or Stefan Loehnert · Dana Mueller-Hoeppe Institute of Continuum Mechanics, Leibniz Univesit¨at Hannover, Appelstr. 11, D-30167 Hannover, Germany e-mail: [email protected], [email protected]

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dental implants. In many cases due to the limitation of computational resources, a singlescale simulation of such naturally multiscale behavior is extremely difficult. In contrast, for such applications modern and efficient multiscale techniques can lead to very accurate results taking into account all the important effects on all the considered scales explicitly without unnecessarily rough assumptions. In many cases, multiscale techniques based on homogenization principles can be applied. For cracks however, these techniques are only applicable as long as the cracks do not propagate or develop since otherwise localization phenomena occur that contradict the assumptions made for a representative volume element which is the basis for standard homogenization techniques. Thus, alternative methods capable of handling localization phenomena as well need to be applied to crack development problems on multiple scales. One of these techniques was developed for the two dimensional case in [2]. It utilizes the extended finite element method (XFEM) [4] in combination with level set techniques on all scales to reflect the crack geometry and the displacement discontinuity within the finite element mesh without the need to change the discretization during a crack propagation process.

2 The Multiscale Technique in Three Dimensions The presented multiscale strategy is based on a projection of the stresses from the finer scale to the next coarser scale. For convenience, here we restrict ourselves to a two scale approach, even though the method is generally applicable to an arbitrary number of scales. On the fine scale, macrocracks as well as microcracks that cannot be directly considered on the coarse scale due to an insufficiently small coarse scale element size, are modelled explicitly. On the coarse scale however, only macrocracks are modelled explicitly. Every crack that is larger than a certain size depending on the typical element size on the coarse scale is considered to be a macrocrack. On the coarse scale, the influence of microcracks is considered only implicitly by a projection of the stresses from the fine scale to the coarse scale.

2.1 Stress Projection from the Fine Scale to the Coarse Scale We subdivide the displacement field u into a coarse scale displacement field u0 and fluctuations u¯ 1 due to fine scale features. u = u0 + u¯ 1

(1)

.

Both fields may contain discontinuities. The weak form of equilibrium for the coarse scale problem needs to consider the entire displacement field. It reads  Ω0

σ (u0 + u¯ 1 ) : gradsym (η 0 ) dΩ =

 Ω0

f · η 0 dΩ +



∂Ω0

t · η 0 d∂ Ω

.

(2)

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Here σ is the Cauchy stress tensor depending on the total deformation, f are body forces and t are tractions applied on the coarse scale boundary. Note that in (2) the coarse scale test functions η 0 defined in the entire coarse scale domain Ω 0 are used. On the fine scale domain Ω 1 ⊂ Ω 0 we define the fine scale test functions η 1 to obtain the fine scale weak form of equilibrium 

σ (u0 + u¯ 1 ) : gradsym (η 1 ) dΩ =

Ω1



f · η 1 dΩ

.

(3)

Ω1

Since on the entire fine scale domain boundary we employ displacement boundary conditions obtained from the coarse scale solution, surface tractions do not occur in the weak form for the fine scale computation. On both scales we employ the corrected version of the XFEM [1] to account for the discontinuities in the displacement fields as well as the stress singularities at the crack tips as they occur in linear elastic fracture mechanics. In contrast to the standard XFEM, in the corrected XFEM the partition of unity is also fulfilled in the so-called blending elements describing the transition zone between elements of which all nodes are enriched with the front enrichment functions and elements without enriched nodes. Even though the multiscale technique is capable of handling arbitrary material models as well as finite deformation theory on all scales, here we restrict ourselves to the case of linear elastic fracture mechanics. In the XFEM the displacement field is approximated by ( ) nα nα nenr n n α α α α ˆ αI · uˆ αI uh = ∑ NI uI + ∑ q( f j (x) − f j (xI ))a jI = ∑ N (4) I=1

j=1

I=1

where α = 0 indicates the coarse scale and α = 1 the fine scale, NIα are the nodal shape functions, uαI are the standard nodal degrees of freedom, f j are the enrichment functions and aαjI are the corresponding enriched degrees of freedom on the respective scale. In (4) we employ the shift of the actual enrichment functions such that at all the nodes the enriched degrees of freedom have no influence. The notaˆ α and the overall vector of nodal degrees tion using the enriched shape functions N I α of freedom uˆ I is used for simplicity only. Since they span the analytical solution of arbitrary mode 1, mode 2 and mode 3 combinations of a planar crack with a straight crack front in linear elasticity, for all nodes of elements that are cut by the crack front and for the nodes of all the surrounding elements we use the four enrichment functions     √ √ f1 (r, ϕ ) = r sin ϕ2 , f2 (r, ϕ ) = r cos ϕ2 , (5)     √ √ f3 (r, ϕ ) = r sin ϕ2 sin(ϕ ), f4 (r, ϕ ) = r cos ϕ2 sin(ϕ ) . Here r and ϕ are the smallest distance of the considered point to the crack front and the angle of the point to the tangent to the crack surface at the crack front. In (4) q is a trilinear ramp function used to fade out the effect of the front enrichment functions in the blending elements.

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q = ∑ NIα qαI

qαI =

with

6

I

1 : I node of a crack front element 0 : otherwise

(6)

When the ramp function is used, for each crack in 3D a sixfold linear dependence occurs in the global stiffness matrix. We avoid this linear dependence by dropping the last enrichment function. See [1] and [3] for more details. The geometry of the crack is reflected by two level set functions φ and ψ . The first level set function φ is a signed distance function describing the smallest distance to the crack surface, the second level set function ψ is a signed distance function determining the distance to the plane perpendicular to the crack surface cutting the crack front. Using those two level set functions it is easy to evaluate the geometrical data r and ϕ required for the enrichment functions. & ' 8 φ r = φ 2 + ψ 2 , ϕ = arctan (7) ψ For all nodes belonging to elements that are completely cut by the crack we use the jump enrichment function. 6 +1, φ ≥ 0 f1 = (8) −1, φ < 0 Similar to the displacement approximation the discretized form of the test functions can be setup for all scales. ( ) nα nα nenr n n α α α α ˆ αI · ηˆ αI η h = ∑ NI η I + ∑ q( f j (x) − f j (xI ))b jI = ∑ N (9) I=1

j=1

I=1

Defining the matrix BαI to compute the symmetric gradient of the test functions on all scales sym

grad

(η αh )

nα n

=

∑ BαI · ηˆ αI

(10)

I=1

the discretized form of the weak form of equilibrium on the coarse scale becomes n0n

∑

T

ηˆ 0I ·

I=1 n0n

∑



T

B0I : σ (u0 + u¯ 1 ) dΩ =

Ω0 T

ηˆ 0I ·

I=1



n0n

ˆ 0I T · f dΩ + ∑ ηˆ 0I T · N I=1

Ω0



ˆ 0I T · t d∂ Ω N

(11)

.

∂Ω0

Similarly the weak form of equilibrium on the fine scale reads n1n

∑

I=1

T ηˆ 1I

·



Ω1

T B1I

: σ (u + u¯ ) dΩ = 0

1

n1n

∑ ηˆ 1I

I=1

T

·



Ω1

ˆ 1I T · f dΩ N

.

(12)

3D Multiscale Projection Method for Micro-/Macrocrack Interaction Simulations

227

The quadrature in (11) and (12) is done using a Gauss point strategy and subdivisions of the brick elements to avoid quadrature across discontinuities as well as insufficiently accurate quadrature. See [3] for more details. However, applying this quadrature method within the projection technique leads to a very large number of Gauss points within coarse scale elements onto which the stresses from a fine scale solution are projected. A possible remedy is the aggregation of information from all Gauss points in each fine scale element, such that only the stresses at one point P within each fine scale element are projected onto the coarse scale mesh. This can be interpreted as an averaging technique.

σ PVM1

n1GP

=

∑ σ 1l Jl wl

(13)

l=1

Here, σ P is the projected stress tensor, VM1 is the volume of the fine scale element M the point P is located in, σ 1l is the stress tensor at each Gauss point of the fine scale element, Jl is the Jacobian and wl the weight of each Gauss point. Point P is assumed to be located at the reference coordinates (ξ , η , ζ ) = (0, 0, 0) in the fine scale element M. This approximation does not reduce the accuracy of the coarse scale simulation significantly while it drastically reduces the amount of computational time.

2.2 Projection of the Displacement Field from the Coarse Scale to the Fine Scale For the fine scale computation we apply pure displacement boundary conditions on the entire surface of the fine scale domain excluding crack surfaces which are kept traction free. To guarantee that the fluctuations of the displacements on the boundary of the fine scale domain vanish (u¯ 1 = 0) and that the continuity of the displacement field is ensured we apply a projection of the displacements from the coarse to the fine scale. In general a macrocrack cuts the surface of the fine scale domain. As a result nodes of the fine scale mesh located on the boundary of the fine scale domain will have different enrichments than nodes of the coarse scale mesh on the same boundary. Thus it is not straight forward to calculate all nodal degrees of freedom of all fine scale nodes on the boundary. The projection we use is a least squares fit of the type ( ) ( )  1 1 1 0 0 1 (14) ∑ Nˆ I · uˆ I − ∑ Nˆ J · uˆ J · ∑ Nˆ K · ηˆ K dΩ = 0 . Ω1

I

J

K

Even though only the displacements on the fine scale boundary need to be projected, the integral in (14) needs to be carried out over the entire fine scale domain. If it was evaluated only along the surface, linear dependencies would occur between the standard and enriched degrees of freedom. This is illustrated in Fig. 1. Here, the standard degrees of freedom and the jump enriched degrees of freedom of nodes A

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∂Ω1

macrocrack B C A

node enriched with the jump function Ω1

Fig. 1 Enrichments on the boundary of the fine scale domain

and B would depend on each other. This effect can only be eliminated by evaluating the integral in (14) in the whole element C. To avoid any loss of robustness of the least squares projection, we carry out the integral in the entire fine scale domain leading to some extra computational cost. However, the degrees of freedom of nonenriched nodes can be determined directly by evaluating the displacements at the same location in the coarse scale mesh. Thus in the least squares projection only all the nodal degrees of freedom of enriched nodes need to be determined. This significantly reduces the numerical effort.

3 Numerical Investigations The multiscale technique has been applied to a number of simple test examples showing some basic properties regarding its applicability and accuracy. In two dimensions the properties of the method are investigated in detail in [2]. In three dimensions, until now the fine scale domain is always chosen to be a tube (Fig. 2) with a circular cross section around the crack front of a macrocrack, since it can be assumed that in the vicinity of a macrocrack front the stresses and stress gradients are large such that microstructural features can have a significant influence on the overall behavior of the considered structure. The example shown here is a planar macrocrack with a curved crack front in a cube shaped macrostructure. Above and below the crack front of the macrocrack there are two elliptically shaped planar microcracks that are only considered explicitly in the fine scale computation. Due to the position and orientation of the two microcracks a crack shielding effect reduces the stresses at the macrocrack front in the vicinity of the microcracks. The deformed coarse scale geometry and a cut through the fine scale domain is shown in Fig. 3. To test the effect of the fine scale domain size on the solution, three different radii of the tube around the crack front are chosen. In Fig. 4 the stresses in vertical direction are shown. It can clearly be seen that the fluctuations of the stresses on the boundary of the respective fine scale domain become smaller the larger the fine scale domain is chosen. However, since the computational effort to compute the fine scale problem increases significantly with increasing fine scale domain size, it is

3D Multiscale Projection Method for Micro-/Macrocrack Interaction Simulations

macrocrack surface

229

macrocrack front tube defining the fine scale domain

Fig. 2 Macrocrack and fine scale domain around the crack front

Fig. 3 Deformed coarse scale geometry and cut through the fine scale domain of a macrocracked cube with two microcracks in the vicinity of the macrocrack front

Fig. 4 Stress distribution in a small, medium and large fine scale domain

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important to choose a fine scale domain size that is just large enough to ensure the desired accuracy. An indicator for the accuracy are the fluctuations of the stresses near the boundary. In the smallest fine scale domain in Fig. 4 the fluctuations along the boundary still seem to be rather large whereas in the medium and large fine scale domain the fluctuations of the stresses appear to be negligibly small.

4 Conclusion and Outlook The example computations presented in the last sections would not have been possible in a reasonable time on a modern computer system in a single scale analysis with a meshsize comparable to the fine scale mesh presented. Additionally it can be shown that the influence of fine scale features on the crack propagation behavior of a macrocrack can be significant. This indicates that multiscale methods capable of treating localization phenomena such as crack propagation are necessary. The presented multiscale technique including the XFEM is an efficient way to calculate the influence of microcracks on macrocracks. In future it will be necessary to introduce an error controlled mesh and model adaptivity to adjust the fine scale domain geometry as well as the coarse and fine scale mesh resolution to achieve a desired accuracy.

References 1. Fries, T.-P.: A corrected XFEM approximation without problems in blending elements. Int. J. Numer. Meth. Eng. 75, 503–532 (2008) 2. Loehnert, S., Belytschko, T.: A multiscale projection method for macro/microcrack simulations. Int. J. Numer. Meth. Eng. 71, 1466–1482 (2007) 3. Loehnert, S., Mueller-Hoeppe, D.S., Wriggers, P.: 3D corrected XFEM approach and extension to finite deformation theory. Int. J. Numer. Meth. Eng. (2010), doi: 10.1002/nme.3045 4. Mo¨es, N., Dolbow, J., Belytschko, T.: A finite element method for crack growth without remeshing. Int. J. Numer. Meth. Eng. 46, 131–150 (1999)

Chapter 26

Goal-Oriented Residual Error Estimates for XFEM Approximations in LEFM Marcus R¨uter and Erwin Stein The ”Hanseat” Peter Wriggers with his astute intellect and eminent scientific achievements is one of the best students from our Hannover school of Computational Mechanics. Considering additionally his extraordinary capabilities and successes in the organization of collaborative national and international research and teaching projects, he is the undisputed number one amongst my highly gifted students who became Professors at Universities. Professor Wriggers has a high ranked personality and deduces his integrity, confidence and consistency of thinking, willing and doing from his noble character. With hanseatic restraint and a certain distance as well as with his pleasant coolness, he surely will lead the Institute of Continuum Mechanics to further scientific summits. Many thanks, Peter, for many common goals and successes and for your fairness! (E. Stein).

Abstract. The objective is to derive and apply residual-type goal-oriented a posteriori error estimators for the discretization error obtained while approximately evaluating the nonlinear J-integral as a fracture criterion in linear elastic fracture mechanics (LEFM) using the extended finite element method (XFEM). The upperbound error estimator proposed is based on the solutions of Neumann problems for both the primal and an auxiliary dual problem on the element level for which equilibrated tractions are computed in terms of ansatz functions that are L2 -orthogonal to the XFEM ansatz functions.

1 Introduction In the classical theory of LEFM, a pre-existing macroscopic crack is modeled by traction-free Neumann boundary conditions. In this fashion, the finite element Marcus R¨uter Institute of Continuum Mechanics, Leibniz Universit¨at Hannover, Appelstr. 11, D-30167 Hannover, Germany e-mail: [email protected] Erwin Stein Institute of Mechanics and Computational Mechanics, Leibniz Universit¨at Hannover, Appelstr. 9a, D-30167 Hannover, Germany e-mail: [email protected]

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method (FEM) can be applied straightforwardly to compute a numerical solution of the crack problem. As a consequence, the elements align with the crack surface. In particular for crack propagation this turns out to be a drawback, since the direction of crack propagation is generally not known a priori and thus a remeshing strategy is often required. Furthermore, the stress singularity at the crack tip is not well captured using standard finite elements. To cope with both problems, XFEM was introduced in [3] and further extended in [8], [5] and [7]. Briefly speaking, XFEM allows to let the crack run through the elements. Moreover, enriched basis functions are used at the crack tip that are capable of capturing the stress singularity. As a numerical method, however, it is surprising that not much attention has been drawn so far to the analysis and adaptivity of the resulting XFEM discretization error, which is of particular importance for the approximation of the fracture criterion, here the J-integral. This type of error analysis leads to the approach of goal-oriented error estimation, as introduced in [4]. In recent years, goal-oriented error estimates have also been developed and successfully applied to finite element methods in LEFM, see e.g. [9], where an implicit residual-type error estimator based on the one introduced in [2] and [1] is presented. In this paper, we extend the approach presented in [9] to XFEM approximations in LEFM to obtain both optimal convergence and upper bounds on the error of the J-integral. The paper is organized as follows: In Sect. 2, the model problem of LEFM and the associated XFEM are introduced. Subsequently, in Sect. 3 we focus on an implicit residual-type error estimator for the energy norm which is then extended in Sect. 4 to the case of goal-oriented error control in LEFM. Finally, in Sect. 5 an illustrative and comparative numerical example is presented.

2 XFEM Approximations in LEFM 2.1 The Model Problem of LEFM Let the isotropic linear elastic body be given by the closure of a bounded open set Ω ⊂ R3 (the closure is denoted by a bar, i.e. Ω¯ ) with a piecewise smooth and polyhedral boundary Γ = ∂ Ω such that Γ = Γ¯D ∪ Γ¯N ∪ Γ¯c and ΓD ∩ ΓN ∩ Γc = 0, / where ΓD and ΓN are the portions of Γ where Dirichlet and Neumann boundary conditions are imposed, respectively. Furthermore, Γc is a traction-free boundary that models the pre-existing crack surface, see Fig. 1 as an example. The variational setting of the elliptic and self-adjoint model problem is given as follows: find the displacement field u ∈ V that satisfies a(u, v) = F(v)

∀v ∈ V

.

(1)

Here, a : V × V → R and F : V → R are bilinear and linear forms defined as 

a(u, v) =

Ω

ε (u) : C : ε (v) dV

(2a)

Goal-Oriented Residual Error Estimates for XFEM Approximations in LEFM 

and F(v) =

Ω

f · v dV +

 ΓN

¯t · v dA ,

233

(2b)

respectively. In the above, ε denotes the linear strain tensor, C is the symmetric elasticity tensor, f ∈ [L2 (Ω )]3 are body forces, and ¯t ∈ [L2 (ΓN )]3 are prescribed tractions on ΓN . The test and solution space V is defined as V = {v ∈ [H 1 (Ω \ Γc )]3 ; v|ΓD = 0} in order to account for the discontinuity of the solution u along Γc which appears since the traction-free boundary condition on Γc is not directly imposed into (1). The pre-existing crack Γc then starts to grow in the direction of e|| if the value of the J-integral exceeds the material-dependent threshold Jc . The domain expression of the J-integral, see [10], is defined as J(u) = −

 ΩJ

H(qe|| ) : Σ˜ (u) dV

.

(3)

Here, q is a weighting function with q = 1 at the crack tip and q = 0 on ΓJ , cf. Fig. 1. Moreover, H is the displacement gradient and Σ˜ denotes the Newton-Eshelby stress tensor Σ˜ = Ws I − HT · σ with the specific strain-energy function Ws , identity tensor I and Cauchy stress tensor σ = C : ε .

2.2 XFEM Approximations In order to establish the (corrected) XFEM discretization, see [5], associated with the continuous problem (1), we first introduce the approximation ( ) 3

uh =

∑ Ni uˆ i + ∑ Ni H bˆ i + ∑ Ni R ∑ f j cˆ i, j

i∈K

i∈L

i∈M

(4)

j=1

with nodal finite element ansatz functions Ni , Heaviside function H, ramp function R, vectors of nodal values u, ˆ bˆ and cˆ as well as the reduced (from 4 to 3) enrichment functions, see [7], 6 7 1 θ 1 θ 1 θ 2 2 2 { f j (r, θ )} = r sin , r cos , r sin sin θ . (5) 2 2 2 Furthermore, K is the set of all nodes, L is the set of nodes of elements that are cut by the crack except for the nodes of the crack tip element, and M is the set of nodes of the crack tip element and its surrounding elements. The XFEM discretization of the continuous problem (1) then reads: find an approximate solution uh ∈ Vh ⊂ V such that a(uh , vh ) = F(vh )

∀vh ∈ Vh

.

(6)

The XFEM approximation of the J-integral (3) is then obtained by computing the value for J(uh ).

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3 A Posteriori Error Estimation in the Energy Norm 3.1 Error Representation Substituting the XFEM approximation uh into the continuous problem (1) we obtain the weak form of the residual R : V → R defined by R(v) = F(v) − a(uh , v) .

(7)

In particular, inserting (6) into (7) yields the Galerkin orthogonality R(vh ) = 0 for all vh ∈ Vh which can also be expressed as a(e, vh ) = 0 for all vh ∈ Vh with the XFEM discretization error e = u − uh . Recalling the variational form (1), it is obvious that the error e ∈ V is the solution to the variational problem a(e, v) = R(v)

∀v ∈ V

(8)

.

The associated local problem of the error representation (8) consists in seeking a solution e|Ω¯ e ∈ Ve = {v|Ω¯ e ; v ∈ V } that satisfies ae (e|Ω¯ e , ve ) = Re (ve ) ∀ve ∈ Ve

.

(9)

Here, ae : Ve × Ve → R and Re : Ve → R denote the restrictions of a and R to an element Ω¯ e , respectively, i.e. Re is defined as 

Re (ve ) =

Ωe

f|Ω¯ e · ve dV +

 ∂ Ωe

te · ve dA −

 Ωe

σ (uh |Ω¯ e ) : ε (ve ) dV

(10)

with the exact traction field te ∈ [L2 (∂ Ωe )]3 on the element boundary ∂ Ωe .

3.2 An Implicit Residual Error Estimator The general idea for deriving an a posteriori error estimator is to substitute the exact tractions te in (10), which are apparently in equilibrium, with an approximation ˜te which then defines the equilibrated residual R˜ e : Ve → R. Thus, we are led to the problem of seeking an error approximation ψ e ∈ Ve such that ae (ψ e , ve ) = R˜ e (ve ) ∀ve ∈ Ve

.

(11)

Since (11) is a Neumann problem, the approximate tractions ˜te have to be in equilibrium as well to fulfill the well-posedness of the variational problem. Evidently, upon summing up the equilibrated residuals R˜ e over all ne elements Ω¯ e , its sum has to equal R and thus it follows that

Goal-Oriented Residual Error Estimates for XFEM Approximations in LEFM  ΓN

¯t · v dA = ∑ ne

 ∂ Ωe

˜te · v|Ω¯ dA ∀v ∈ V e

.

235

(12)

Hence, the equilibrated tractions ˜te have to coincide with the tractions ¯t on the Neumann boundary ∂ Ωe ∩ ΓN . Furthermore, they have to fulfill Cauchy’s lemma, i.e. ˜te = −˜t f on Ω¯ e ∩ Ω¯ f for all elements Ω¯ f = Ω¯ e . Bearing in mind the above observations concerning the weak form of the residual R and the exact error representation (8), we thus arrive at the important relation a(e, v) = ∑ ae (ψ e , v|Ω¯ e )

∀v ∈ V

.

(13)

ne

Following [1], it can be shown that upon applying the Cauchy-Schwarz inequality twice on (13) we obtain the following estimator for the XFEM discretization error measured in the energy norm ( |||e||| ≤

)1

∑ ae ( ψ e , ψ e )

2

(14)

ne

which has the virtue that it has no interpolation constant and therefore provides a guaranteed upper error bound. Strictly speaking, however, this only holds if the local Neumann problems (11) are solved exactly. Thus, higher-order ansatz functions are implemented in the XFEM code to retain the upper bound property of the estimator. Furthermore, elements that are cut by a crack have to be enriched with appropriate ansatz functions to model the discontinuity in the solution.

3.3 Equilibration of Tractions The quality of the residual error estimator (14) clearly depends on the choice of the equilibrated tractions. The approach presented to compute ˜te closely follows the ones originally proposed in [6], [11] and [12]. For the equilibrated tractions we introduce the following ansatz ˜te = th + β e f

on Ω¯ e ∩ Ω¯ f

(15)

with Ω¯ f = Ω¯ e and XFEM tractions th . The vector-valued interface function β e f : Ω¯ e ∩ Ω¯ f → R3 is determined by the equilibration condition R˜ e (ve ) = 0 for all ve ∈ Ze , i.e. the space of rigid body modes. In the spirit of the L2 -orthogonality of finite element ansatz functions, see [11], the key idea is to make an ansatz for β e f that is L2 -orthogonal to the XFEM ansatz functions (4) which results in a decoupled linear system of equations.

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4 Goal-Oriented Error Estimation in LEFM 4.1 Linearization of the J-Integral In computational LEFM, energy norm error control is not efficient and thus not recommendable. Therefore, the error of the nonlinear J-integral J(u) − J(uh ) = JT (e)

(16)

is controlled directly, see [9], in terms of the (linear) tangent form JT : V → R, defined as JT (v) = −



ΩJ

H(qe|| ) : CΣ (uh ) : H(v) dV

(17)

with the elasticity tensor CΣ = I ⊗ σ − I⊗ σ − HT · C.

4.2 Duality Techniques The error estimation of the J-integral requires an auxiliary dual problem based on ∗ ∗ the dual bilinear form a : V × V → R, see [4]. Since a is symmetric, cf. (2a), a ∗ coincides with a and the dual problem reads: find a solution u ∈ V that satisfies ∗

a(u, v) = JT (v) ∀v ∈ V

(18)

. ∗

As in the case of the primal problem (1), we can, at best, approximate u by the ∗ XFEM solution uh ∈ Vh of the associated discretized dual problem. Recalling (16) and the Galerkin orthogonality, setting v = e in (18) and introduc∗ ∗ ∗ ing the dual XFEM discretization error e = u − uh , the error of the J-integral can be exactly represented by J(u) − J(uh ) = a(e, e) = ∑ ae (e|Ω¯ e , e|Ω¯ e ) ∗

∗

(19)

ne

and thus, with the Cauchy-Schwarz inequality, estimated from above by |J(u) − J(uh )| ≤ ∑ |||e|Ω¯ e |||Ω |||e|Ω¯ e |||Ω ∗

e

ne

∗

≤ |||e||||||e|||

.

e

(20a) (20b)

With the error estimator (14) we obtain from (20a) a non-guaranteed upper bound, which still holds in most computations and usually is very efficient, whereas from (20b) we obtain a guaranteed upper bound which is not necessarily efficient. In the following, the estimators resulting from (20a) and (20b) are designated as “residual 1” and “residual 2”, respectively.

Goal-Oriented Residual Error Estimates for XFEM Approximations in LEFM

237

5 Numerical Example We consider a 2D K1 plane-stress crack propagation problem. The system is a borosilicate glass square plate as depicted in Fig. 1, where also the material data are given. The XFEM approximation (4) is based on Q1 shape functions. The solutions of the discrete primal and dual problem are plotted in Figs. 2 and 3, respectively. Here, the influence of the enrichment functions is visible which yield a curved crack line even on a coarse mesh. For the sake of comparison, both FEM and XFEM results concerning the error convergence based on uniform mesh refinements and the associated effectivity indices are visualized in Figs. 4 and 5, respectively. Obviously, FEM and XFEM yield the same order of convergence. However, the interpolation constant for XFEM is

t ΓN 500mm

ΓJ Γc

ΓD

500mm

ΩJ E = 64000 N/mm² ν = 0.2 |t| = 0.33 N/mm²

Ω t

Fig. 1 Structural system and loading for the primal problem

Fig. 2 Primal solution, magnified 104 times

Fig. 4 Estimated error |J(u) − J(uh )|

Fig. 3 Dual solution, magnified 106 times

Fig. 5 Effectivity indices

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much smaller such that the computational costs are reduced to about 10% (in average) of the corresponding FEM computation.

6 Conclusions Residual-type a posteriori error estimators for the XFEM discretization error of the J-integral were presented that are based on local Neumann problems. The required equilibrated tractions rely on L2 -orthogonal functions with respect to the XFEM ansatz functions. The numerical example showed that XFEM computations yield much more accurate results (about 90 %) than corresponding FEM computations in terms of considerably smaller errors, whereas the order of error convergence remains the same. Future research concerns adaptive XFEM, which should increase the order of convergence.

References 1. Ainsworth, M., Oden, J.T.: A posteriori error estimation in finite element analysis. John Wiley & Sons, New York (2000) 2. Bank, R.E., Weiser, A.: Some a posteriori error estimators for elliptic partial differential equations. Math. Comp. 44, 283–301 (1985) 3. Belytschko, T., Black, T.: Elastic crack growth in finite elements with minimal remeshing. Int. J. Numer. Meth. Eng. 45, 601–620 (1999) 4. Eriksson, K., Estep, D., Hansbo, P., Johnson, C.: Introduction to adaptive methods for differential equations. Acta Numer. 4, 106–158 (1995) 5. Fries, T.-P.: A corrected XFEM approximation without problems in blending elements. Int. J. Numer. Meth. Eng. 75, 503–532 (2008) 6. Ladev`eze, P., Leguillon, D.: Error estimate procedure in the finite element method and applications. SIAM J. Numer. Anal. 20, 485–509 (1983) 7. Loehnert, S., Mueller-Hoeppe, D.S., Wriggers, P.: 3D corrected XFEM approach and extension to finite deformation theory. Accepted in Int. J. Numer. Meth. Eng. (2010) 8. Mo¨es, N., Dolbow, J., Belytschko, T.: A finite element method for crack growth without remeshing. Int. J. Numer. Meth. Eng. 46, 131–150 (1999) 9. R¨uter, M., Stein, E.: Goal-oriented a posteriori error estimates in linear elastic fracture mechanics. Comput. Meth. Appl. M. 195, 251–278 (2006) 10. Shih, C.F., Moran, B., Nakamura, T.: Energy release rate along a three-dimensional crack front in a thermally stressed body. Int. J. Fract. 30, 79–102 (1986) 11. Stein, E., Ohnimus, S.: Coupled model- and solution-adaptivity in the finite-element method. Comput. Meth. Appl. M. 150, 327–350 (1997) 12. Stein, E., R¨uter, M.: Finite element methods for elasticity with error-controlled discretization and model adaptivity. In: Stein, E., de Borst, R., Hughes, T.J.R. (eds.) Encyclopedia of Computational Mechanics, 2nd edn. John Wiley & Sons, Chichester (2007)

Chapter 27

Multi-field Coupling Strategies for Large Scale Particle-Fluid Problems D.R.J. Owen, Y.T. Feng, K. Han, and C.R. Leonardi This paper is dedicated to Prof. Peter Wriggers in acknowledgement of his immense contributions to the field of computational mechanics. Peter is the premier figure in the numerical modelling of contact problems and given his seminal contributions to the topic over three decades it is hard to believe that this is the celebration of only his 60th birthday. Of course, he is an established authority in other areas and is recognised as a leading international researcher in non-linear computational modelling, where his most recent book is already a reference text. Peter has a long standing connection with the Swansea modelling group, where he was held in high esteem by the late Prof. Olek Zienkiewicz. The authors dedicate this paper to Peter and wish him continued health and a long-lived research career (D.R.J. Owen).

Abstract. This paper outlines a number of numerical techniques within a general combined Lattice Boltzmann-Discrete Element framework for the modeling of particle transport in incompressible fluid flows, with or without the presence of a thermal phase. The standard Lattice Boltzmann (LB) formulation for the simulation of incompressible fluid flows is reviewed, together with incorporation of the Smagorinsky turbulence model into the formulation. The hydrodynamic interactions between fluid and particles are realised through an immersed boundary condition. The extension of the linear LB formulation to non-Newtonian fluid regimes enables the modelling of fine particle migration problems in mass mining operations, while the double population based thermal LB approach permits the modelling of heat transfer phenomena in fluid-particle systems. The combination of some or all of these techniques offers an advanced and powerful numerical tool for the modelling of many important engineering problems. D.R.J. Owen · Y.T. Feng · K. Han · C.R. Leonardi Civil and Computational Engineering Centre, School of Engineering, Swansea University, SA2 8PP, UK e-mail: [email protected]

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1 Introduction The transport of solid particles within a fluid flow has a wide range of scientific and engineering applications. Understanding of the underlying particle-fluid dynamics in the problem has been somewhat limited by the lack of powerful analysis tools. In a particle-fluid system, the fluid dynamics of the flow is influenced by the presence of solid particles, and the motion of the solid particles is, in turn, driven by fluidinduced forces. In several applications of industrial relevance involving particulate media, the system response is also governed by the need to consider other physical phenomena, such as thermal effects. The current paper considers the essential issues necessary for an effective computational treatment of such coupled systems and specific problems that are addressed include (i) particle transport problems in which the particles being transported through the fluid are large and extend over several fluid grid cells and (ii) fines migration problems in which fine particles that are several orders of magnitude smaller than the main rock fragments flow through the moving particle system. A further problem addressed is heat transfer between a moving particle system at elevated temperatures. The development of effective numerical modeling frameworks for these types of problem is however very challenging due to the inherent complexity of the phenomena involved. The main objective of the current work is to develop a unified computational framework to model the problems concerned. In recent years, the lattice Boltzmann method (LBM) [2] has emerged as an alternative fluid solver, and offers various advantages, including high space-time resolution and full scalability on parallel computers. Since Ladd’s early work [11], the LBM has been widely employed to model fluid-particle interactions. Furthermore, employing the discrete element method (DEM) to account for particle-particle interactions gives rise to a combined LBM-DEM solution procedure [3]. In our previous work [7, 5], the coupled LBM-DEM strategy has been extended to simulate regular and irregular shaped particle transport problems with turbulent fluid flows. Many engineering problems involve both coarse and fine particles with or without the presence of a second fluid/gas phase. The modelling of this type of problem imposes an even greater challenge as the above coupled LB-DE methodology is no longer adequate. For cases with a binary mixture of both large particles and fine powder type particles, we propose to model the motion of the fine particles as a non-Newtonian fluid flow and thus the standard LB formulation for Newtonian fluids must be extended to capture the constitutive behaviour of granular media or dry bulk material. Another attractive feature of the LB formulation is that it can be extended to model heat transfer phenomena including pure heat conduction and forced or natural convection in (particle-)fluid flows. The formulation can also naturally account for heat conduction between solid particles. This paper outlines the essential numerical techniques discussed above in a general combined LB-DE solution strategy for the numerical analysis of various particle-fluid systems exhibiting a large number of particles and high Reynolds numbers, combined flow of particles interspersed with fine grained material, and/or heat

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241

transfer in the system. The discrete element method is not discussed as the details can be found elsewhere.

2 LB Formulations for Turbulent Incompressible Fluid Flows The lattice Boltzmann method describes the fluid in terms of fluid particle density functions at discrete lattice nodes and discrete times. It simulates fluid flows by tracking the evolution of fluid particle distributions instead of tracking single fluid particles. In the LBM, the problem domain is divided into regular lattice nodes. The fluid is modeled as a group of fluid particles that are allowed to move between lattice nodes or stay at rest. During each discrete time step of the simulation, fluid particles move to the nearest lattice node along their directions of motion, where they “collide” with other fluid particles that arrive at the same node [2].

2.1 Standard LB Formulation In the widely used D2Q9 model [13], the fluid particles at each node are allowed to move to their eight immediate neighbours with eight different velocities ei , (i = 1, · · · , 8). The primary variables in the LB formulation are the fluid density distribution functions, fi . The evolution of the density distribution functions at each time step is given by fi (x + ei Δ t,t + Δ t) = fi (x,t) −

 Δt  eq fi (x,t) − fi (x,t) τ

(i = 0, · · · , 8)

(1)

where τ is a non-dimensional parameter termed the relaxation time; and fieq are the equilibrium distribution functions defined as: & ' & ' 3 3 9 3 f0eq = w0 ρ 1 − 2 v · v ; fieq = wi ρ 1 + 2 ei · v + 4 (ei · v)2 − 2 v · v 2c c 2c 2c (2) (i = 1, · · · , 8), in which c is the lattice speed given by c = h/Δ t (Δ t is the discrete time step); wi are the weighting factors: w0 = 4/9; w1,2,3,4 = 1/9;w5,6,7,8 = 1/36. The LB discrete time evolution equation (1) comprises two computational phases: collision and streaming. The collision operation computes the right-hand side of Eq. (1) that only involves the variables associated with each node x, and therefore is a local operation. The streaming phase then explicitly propagates the updated distribution functions at each node to its neighbours x + ei Δ t, where no computations are required and only data exchange between neighbouring nodes is necessary. These features, together with the explicit time-stepping nature and the use of a regular grid, make LB computationally efficient, simple to implement and natural to parallelise. The macroscopic fluid variables, density ρ and velocity v, can be recovered from the distribution functions as

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ρ = ∑ fi , i=0

8

ρ v = ∑ f i ei

(3)

i=1

while the fluid√ pressure field p is determined by the equation of state p = c2s ρ , where cs = c/ 3 is the fluid speed of sound. The kinematic viscosity, ν , of the fluid is implicitly determined by the model parameters, h, Δ t and τ as & ' & ' 1 1 h2 1 1 ν= τ− = τ− ch . (4) 3 2 Δt 3 2

2.2 Turbulence Modelling Though the LB method is well established for a variety of fluid flows, turbulence modeling within the LB framework remains a challenge and only very limited work has been reported. A simple route to the incorporation of turbulence modeling to the LB formulation is to directly apply the concept of Large eddy simulation (LES) in conjunction with the incorporation of the widely used one-parameter Smagorinsky subgrid model [14]. Following this approach, the filtered form of the LB equation is expressed as  1 ˜ eq f˜i (x + ei Δ t,t + Δ t) = f˜i (x,t) − fi (x,t) − f˜i (x,t) τ∗

(i = 0, · · · , 8)

(5)

where f˜i and f˜ieq represent respectively the distribution function and the equilibrium distribution function at the resolved scale. The effect of the unresolved scale motion is modeled through an effective collision relaxation time scale τt . Thus in Eq. (5) the total relaxation time should be

τ∗ = τ + τt where τ and τt are respectively the relaxation times corresponding to the true fluid (molecular) viscosity ν and the turbulence viscosity νt defined by a subgrid turbulence model. Accordingly ν∗ is given by 1 1 1 1 ν∗ = ν + νt = (τ∗ − )c2 Δ t = (τ + τt − )C2 Δ t; 3 2 3 2

1 νt = τt c2 Δ t 3

.

By employing the Smagorinsky model, the turbulence viscosity νt is explicitly calculated from the filtered strain rate tensor S˜i j = (∂ j u˜i + ∂i u˜ j )/2 and a filter length scale (which is equal to the lattice spacing h) as

νt = (Sc h)2 Sˆ

(6)

where Sc is the Smagorinsky constant; and Sˆ the characteristic value of the filtered strain rate tensor S˜ which can be obtained directly from the second-order moments of the non-equilibrium distribution function.

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2.3 Hydrodynamic Forces for Fluid-Particle Interactions For the particle transport problems considered, modeling of the interaction between fluid and solid particles requires a physically correct “no-slip” velocity condition imposed on their interface. In the current work, an immersed moving boundary (IMB) method proposed by Noble and Torczynski [12] is employed where a control volume is introduced for each lattice node. Meanwhile, a local fluid to solid ratio γ is defined, which is the volume fraction of the nodal cell covered by the solid particle. The LB equations for those lattice nodes (fully or partially) covered by a solid particle is modified to enforce the “no-slip” velocity condition as  1 eq  fi (x + ei Δ t,t + Δ t) = fi (x,t) − (1 − β ) fi (x,t) − fi + β fim τ

(7)

where β is a weighting function depending on the local fluid/solid ratio γ ; and fim is an additional term that accounts for the bounce back of the non-equilibrium part of the distribution function. The formulations to calculate β and fim can be found in [12]. The total hydrodynamic forces and torque exerted on a solid particle over n particle-covered nodes are summed as  ( )  ( ) Ff = Ch

∑ n

βn ∑ fim ei i

;

Tf = Ch

∑(xn − xc ) × n

βn ∑ fim ei

. (8)

i

2.4 Fine Particle Modelling - Non-newtonian Fluid Flow When modelling the combined flow of particles interspersed with fine grained material, the combined LB-DE approach is no longer a viable computational option. The fine particles cannot be represented as individual rigid objects due to their extremely small size and the very large number involved. The central idea is to treat the fine particles as a continuous non-Newtonian fluid medium within the LB framework, while the large particles are still modelled by DEM. This approach requires the extension of the linear LB formulation to handle non-Newtonian behavior of fluid flows. The implementation of power law fluids within the LB formulation has been undertaken [1] to investigate both shear thinning and shear thickening behaviour. In a power law fluid the viscosity is defined as a continuous function of the strain rate, μ = μ0 sn−1 , where μ0 is a consistency constant and n is the power law index. The strain rate can be obtained from the symmetric strain rate tensor which can be conveniently obtained from the momentum flux tensor. In this way the local fluid viscosity at each LB grid point depends on the local strain rate at the same location. Now the shear-dependent viscosity change is enforced by varying the relaxation parameter 1  1 τ = + τ0 − sn−1 (9) 2 2

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in which τ0 is the Newtonian relaxation parameter corresponding to a viscosity of μ0 . It is clear the above strategy for modelling non-Newtonian fluid flows is very similar to that of turbulence phenomena.

3 The Thermal Lattice Boltzmann Method The thermal lattice Boltzmann method (TLBM) has been developed to solve heat transfer problems. Particularly, He et al. [10] introduce a double population approach, using a density distribution function to simulate hydrodynamics for fluid flows and an internal energy distribution function to simulate thermodynamics for heat transfer. In this approach the flow and the temperature fields are solved by the following two evolution equations f¯i (x + ei Δ t,t + Δ t) − f¯i (x,t) = − g¯i (x + ei Δ t,t + Δ t) − g¯i(x,t) = −

 1 ¯ fi (x,t) − fieq (x,t) τ + 0.5

(10)

  τg 1 eq g¯i (x,t) − gi (x,t) − fi Zi (11) τg + 0.5 τg + 0.5

where 0.5 f¯i = fi + ( fi − fieq ); τf

g¯i = gi +

0.5 Δt (gi − geq fi Zi i )+ τg 2

(12)

in which gi are the internal energy distribution functions with discrete velocity ei along the i-th direction; geq i are the equilibrium distribution functions; and τg is the non-dimensional internal energy relaxation time which controls the rate of change to equilibrium. The left-hand sides of Eqs. (10) and (11) denote the streaming process while the right-hand sides model the collisions through relaxation. The term Zi = (ei − u)·[∂ u/∂ t + (ei ·∇)u] represents the effect of viscous heating and is expressed as (ei − u) · [u(x + ei Δ t,t + Δ t) − u(x,t)] . (13) Δt √ In the D2Q9 model, for gas flows, c can be defined as c = 3RTm , where R is the gas constant and Tm the average temperature. The equilibrium distribution functions geq i are defined in the D2Q9 model as ⎧ 3 3(u · u) 4 ⎪ eq ⎪ g = w ρε − ⎪ 0 0 ⎪ ⎪ 2c2 ⎨ 3 3 3(e 9(ei · u)2 3(u · u) 4 i · u) eq (14) gi = wi ρε + + − (i = 1, 2, 3, 4) 2 ⎪ 2 2c 2c4 2c2 4 ⎪ 3 ⎪ 2 ⎪ 6(ei · u) 9(ei · u) 3(u · u) ⎪ ⎩ geq + − (i = 5, 6, 7, 8) i = wi ρε 3 + 2 4 c 2c 2c2 Zi =

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in which ρε is the internal energy which can be replaced by temperature T to simplify the calculation of gi if the flow is incompressible or the compressibility can be ignored. The macroscopic variables, internal energy per unit mass ε and heat flux q, can be calculated from the zeroth and first order moments of the distribution functions as Δt &ρε = ∑ g¯i − 2 ∑ fi Zi ' τg Δt q = ∑ ei g¯i − ρε u − e f Z i i i ∑ 2 τg + 0.5

.

(15)

4 Numerical Illustrations 4.1 Particle Transportation in Turbulent Fluid Flows The performance of the coupled LB-DE approach is demonstrated with the simulation of particle transport against gravity from the bottom of a fluid domain along a pipe under the suction action resulting from the negative pressure difference applied. The problem is simplified to a fluid-particle interaction simulation. The details are illustrated in Fig. 1a. The fluid domain is divided into a 800 × 800 square lattice with spacing h = 2.5 mm. The material properties of the fluid are chosen as: density ρ = 1000 kg/m3 and kinematic viscosity ν = 5 × 10−5 m2 /s. A constant pressure boundary condition with ρin = ρ is imposed at the two (inlet) boundaries as shown in the figure. A smaller pressure with ρout = 0.97ρ is applied to the outlet of the pipe. The remaining boundaries are assumed stationary walls. A total of 70 circular particles with different sizes uniformly distributed from 30mm to 80mm are randomly positioned at the bottom of the domain. Full gravity (g = 9.81 m/s2 ) is considered. The flow field in terms of the total velocity contour and the evolution of the particles at three time instances are depicted in Figs. 1b-d, from which the complex fluid flow patterns due to fluid-particle interactions, particle/particle and particle/wall collisions are clearly observed. Further analyses employing non-circular (polygon and super-quadrics) particles are reported in [7], while a full 3D validation against an experiment is available in [6].

4.2 Fine Particle Migration in a Block Cave Block caving is an underground mining technique in which the rock supporting the orebody is gradually undercut to induce gravitational stresses which act to fracture the orebody rock and subsequently promote caving. Fines migration in a block cave can be characterised by the faster movement of fine material towards the draw point in comparison to larger, blocky material. Use of the discrete element method to include the migration of fines would require the simultaneous solution of elements greater than 2 m (blocks) and smaller than 50 mm (fines). In an industrial sized 3D model this method would require in the order of 108 particles and subsequently be intractable. Consequently, the

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(a)

(c)

(b)

(d)

Fig. 1 a): Problem description; b)-d) Total velocity contours of the fluid flow at three time instances

current approach is to simulate the fines phase as a lattice Boltzmann continuum that is coupled to the discrete element method for the simulation of large blocks. A numerical example of the coupled LBM-DEM framework applied to fines migration is illustrated in Fig. 2. A simplified 2D cave geometry featuring two draw points is employed, and the LB domain is meshed with an optimised lattice with maximum dimensions of 722 × 596. Circular discrete elements with randomly distributed radii were used to fill the cave with “blocks”. The LBM fines are modelled with a Newtonian fluid and a gravitational body force is applied to the discrete elements and the fines. The draw of blocks and fines is handled by the intermittent deactivation of discrete elements from the right draw point only. Figure 2a shows a contour plot of the vertical displacement of the discrete element blocks at the conclusion of the simulation showing the formation of a draw cone. Figure 2b shows the progressive

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displacement of selected points in the LBM fines. Although not immediately evident from the plots of Figs. 2a and 2b, the analysis shows that the fines material migrates faster to the draw point than the discrete element blocks in the regions corresponding to the boundaries of the draw cone.

4.3 Modelling Heat Transfer in (Particle-)Fluid Flows This example simulates heat transfer in a system comprising 30 circular particles that are in contact with neighboring particles. The particles have different initial temperatures ranging from 0 to 1, and the four walls are adiabatic. The domain is divided into a 200 × 200 lattice. The model parameters for the fluid are chosen as Ra = 105 , Pr = 0.71, τ f = 0.05 and τg = 0.005. Fig. 3 depicts the temperature contours and total fluid velocity vectors of the system at two time instances, from which the complex patterns of thermal and fluid flows as well as a nearly steadystate thermal equilibrium can be observed.

(a)

(b)

Fig. 2 a) The vertical displacement (m) of the discrete element blocks; b) Progressive displacement of selected points in the LBM fines

(a) time step=0

(b) time step=40000

Fig. 3 Heat convection and conduction in a fluid-particle system

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5 Conclusions This paper presents a number of numerical techniques within a general combined LB-DE methodology for the modeling of particle transport in incompressible turbulent fluid flows. The extension of the linear LB formulation to non-Newtonian fluid regimes enables the modelling of fine particle migration problems in mining operations. The double population based thermal LB approach permits the modelling of heat transfer phenomena in fluid-particle systems. The combination of some or all of these techniques will offer an advanced and powerful numerical tool for the modelling of many important engineering problems. In addition, an effective method has been developed for particle-particle heat conduction in [4]. The extension of the methodology to magneto-rheological fluids is reported in [8, 9].

References 1. Aharonov, E., Rothman, D.: Non-Newtonian flow (through porous media): a latticeBoltzmann method. Geophys. Res. Lett. 20, 679–682 (1993) 2. Chen, S., Doolen, G.: Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 30, 329–364 (1998) 3. Cook, B.K., Noble, D.R., Williams, J.R.: A direct simulation method for particle-fluid systems. Eng. Computation 21(2-4), 151–168 (2004) 4. Feng, Y.T., Han, K., Li, C.F., Owen, D.R.J.: Discrete thermal element modelling of heat conduction in particle systems: basic formulations. J. Comput. Phys. 227, 5072–5089 (2008) 5. Feng, Y.-T., Han, K., Owen, D.R.J.: Coupled lattice Boltzmann method and discrete element modelling of particle transport in turbulent fluid flows: Computational issues. Int. J. Numer. Meth. Eng. 72(9), 1111–1134 (2007) 6. Feng, Y.T., Han, K., Owen, D.R.J.: Combined three-dimensional lattice Boltzmann method and discrete element method for modelling fluid-particle interactions with experimental validation. Int. J. Numer. Meth. Eng. 81(2), 229–245 (2010) 7. Han, K., Feng, Y.T., Owen, D.R.J.: Numerical simulations of irregular particle transport in turbulent flows using coupled LBM-DEM. Comput. Model. Eng. Sci. 18(2), 87–100 (2007) 8. Han, K., Feng, Y.T., Owen, D.R.J.: Modelling of magnetorheological fluids with combined lattice Boltzmann and discrete element approach. Commun. Comput. Phys. 7(5), 1095–1117 (2010) 9. Han, K., Feng, Y.T., Owen, D.R.J.: Three dimensional modelling and simulation of magnetorheological fluids. Int. J. Numer. Meth. Eng. (Published Online: May 2, 2010) 10. He, X., Chen, S., Doolen, G.R.: A novel thermal model for the lattice Boltzmann method in incompressible limit. J. Comput. Phys. 146, 282–300 (1998) 11. Ladd, A.: Numerical simulations of fluid particulate suspensions via a discretized Boltzmann equation (Parts I & II). J. Fluid Mech. 271, 285–339 (1994) 12. Noble, D., Torczynski, J.: A lattice Boltzmann method for partially saturated cells. Int. J. Mod. Phys. A 9, 1189–1201 (1998) 13. Qian, Y., d’Humieres, D., Lallemand, P.: Lattice BGK models for Navier-Stokes equation. Europhys. Lett. 17, 479–484 (1992) 14. Smagorinsky, J.: General circulation experiments with the primitive equations: I. The basic equations. Mon. Weather Rev. 91, 99–164 (1963)

Chapter 28

Numerical Simulation of Particle-Fluid Systems Bircan Avci and Peter Wriggers

This contribution is dedicated to my PhD advisor and teacher Professor Peter Wriggers. I would like to thank you for your continuous support, scientific and conceptual guidance during all the projects we run together. Your confidence in my work and the provided academic freedom makes this contribution possible. It is a pleasure for me to learn from you and work with you in such a friendly atmosphere at the institute (B. Avci).

Abstract. In this work a fictitious domain method is presented for the direct numerical modeling of 3D particulate flows. The flow field is described by the nonstationary incompressible Navier-Stokes equations and the motion of the particles is modeled by the Newton-Euler equations. For the computation of the two-phase flow the multigrid Finite Element Method (FEM) is coupled with the Discrete Element Method (DEM). The phase coupling is performed in an explicit manner by applying rigid body motion constraints. The combination of fast FEM solvers with efficient search algorithms for the DEM allows 3D simulations with a large number of particles.

1 Introduction Particulate flows can be found in many applications, such as sedimentation, fluidized beds or fluvial erosion. Thus, the description of particle-fluid systems is of great importance both from practical and fundamental points of view. In this area the works of Hu et al. [5], Johnson & Tezduyar [6], Glowinski et al. [3] and Peskin [14] have been pioneering. Here, the approaches in [5, 6] are based on the ALE-method implying a moving boundary fitted grid whereas the works [3, 14] apply the fictitious domain approach wherein a fixed background mesh is used. Various methods have been developed for the fictitious domain approach in order to couple the Eulerian Bircan Avci · Peter Wriggers Institute for Continuum Mechanics, Leibniz Univesit¨at Hannover, Appelstrasse 11, D-30167 Hannover e-mail: [email protected], [email protected]

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and Lagrangian descriptions of the flow field and the particle phase, respectively. For our numerical studies we perform an explicit phase coupling by applying a rigid body constraint to the Navier-Stokes equations. Here, the application of efficient search algorithms for the DEM in order to model the particles motion in combination with a fast multigrid FEM flow solver for the fluid phase allows computations of large 3D particulate systems.

2 Mathematical Description We consider in this presented method a multiphase domain Ω ∈ R3 which consists of the flow field Ω f (t) and N particles where each particle Pi occupies the domain Ω i (t). Hence, the composition of the two-phase field reads Ω = Ω f (t) ∪ {Ωi (t)}i=1,N .

2.1 Equations for Fluid Motion The motion of the flow field is modeled by the nonstationary incompressible NavierStokes equations which can be expressed by

ρf

∂u

∂t

f

 + u f · ∇u f − ∇ · σ = 0 and ∇ · u f = 0, ∀ x ∈ Ω f

.

(1)

Herein, u f is the fluid velocity, ρ f the fluid density and σ describes the Cauchy stress tensor. In our numerical studies we apply the Newtonian material law as the constitutive equation. It states a linear relation between shear stresses and strains. For a Newtonian flow it reads  1 σ = −pI + 2 μεε with ε = ∇u f + (∇u f )T , (2) 2 where p is the pressure, I the identity tensor and μ states the dynamic viscosity. The strain rate tensor is denoted by ε .

2.2 Equations for Particle Motion The motion of each particle Pi is governed by the Newton-Euler equations. Thus, its translational and angular velocities, Ui and ω i , have to satisfy the following equations dUi Mi = (ρi − ρ f )Vi g + Fi + F f ,i (3) dt dω i + ω i × (Θ i ω i ) = Ti + T f ,i . (4) dt Therein, Mi is the mass, ρi the mass density, g the gravity vector and Vi denotes the volume of Pi . In the Euler equations the tensor of inertia is represented by Θ i . The

Θi

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contact forces resulting from collision with neighbor particles and wall elements are stated as Fi whereby the hydrodynamic forces which act upon the surface ∂ Ωi of Pi from the flow field are defined as F f ,i . The torques that are caused by the external forces Fi and F f ,i with respect to the center of mass Mi are associated to the quantities Ti and T f ,i , respectively. They are given for the hydrodynamic part via 

F f ,i =



∂ Ωi

t dA and T f ,i =

∂ Ωi

r × t dA ,

(5)

where the traction vector t on ∂ Ωi is defined by t = σ n f . Therein, n f is the unit outward normal vector and r gives the position vector with respect to Mi .

3 The Discrete Element Model The DEM was initially introduced in the frame of rock mechanics through the pioneering work of Cundall & Strack [2]. The behavior of each single grain of the granular media is governed by the use of the theory of rigid body dynamics. In our studies the grains are modeled by spherical quasi rigid particles. In order to describe the contact forces between colliding particles, constitutive contact models must be introduced for the normal and tangential contact approach.

3.1 Collision Model for Normal Contact The collision between particles is treated by a penalty-force approach. In the case of overlapping surfaces, the particles are connected by a Kelvin-Voigt spring-dashpot element. Here, the resulting collision forces are constitutively governed by a nonlinear viscoelastic model. Many of the viscous contact models are a generalization of the Hertzian elastic contact law (Hertz [4]), cf., e.g, Kruggel-Emden [8]. Based on the Hertz model, Brilliantov et al. [1] proposed a constitutive viscoelastic collision model which can be given for the particles Pi and P j of different material properties and radii as described in P¨oschel & Schwager [15] through cn =

kh =

3 R R 41/2 4 i j 3(ki + k j ) Ri + R j

1 − νh2 Eh

for

h = i, j

F n = −(cn (gn )3/2 + d ng˙n ) .

(6)

(7) (8) dn

Herein, Eh is the Young’s modulus, Ri and R j the radii of the particles and = cn (gn )1/2 (Ai + A j )/2 the viscous damper parameter including the dissipative constants Ai and A j . The Poisson ratio is associated to the particles by νh and the interpenetration or the elastic deformation of the particles is given through

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gn = (ri + r j ) − l ≥ 0

(9)

for contact. Therein, l = ||l|| is the length of the distance vector l between Mi and M j , where l is defined by l = x j − xi with the vectors xi and x j representing the position vectors of Mi and M j , respectively. The penetration velocity vn is governed by the projection of the relative velocity onto the direction of the unit normal vector n = l/l of the contact area A via vn = g˙n = (Ui − U j ) · n .

(10)

Thus, the normal contact force results in Fn = F n n and contributes to Fi in (3).

3.2 Frictional Tangential Contact Model Frictional forces play mostly an important role in particle systems, e.g., in describing heap formation or pipe plugging. Cundall & Strack [2] suggest in their popular model the initiation of a tangential spring in the contact area of colliding particles. Basing on this model, a computational approach was proposed by Luding [12] for treating the tangential contact in the frame of the DEM considering static as well as dynamic friction. Previously, a similar computational approach was introduced by Wriggers [18, 19] for handling frictional contact problems in the frame of the FEM. The slip/stick behavior of the contact area is determined by a frictional law. The constitutive relation of Coulomb’s law couples the tangential force F t via the coefficient of friction to the normal force. Thus, one yields for sliding F t = μd F n and for sticking F t ≤ μs F n . Therein, the dynamic and static coefficient of friction are denoted by μd and μs , respectively. In general the following relationship holds: μd ≤ μs . The relative velocity at the contact point C of Pi and P j is obtained by vr = vCi − vCj

,

(11)

where vCi = Ui + ω i × ri and vCj = U j + ω j × r j are the surface velocities. The vectors pointing both from Mi and M j to C are associated by ri = Ri n and r j = R j (−n), respectively. By projection of (11) upon n, the tangential relative velocity at C is governed via (12) vt = vr − (vr · n) n . The computation of the tangential contact force is performed by employing a return mapping scheme. At this two-step scheme, we assume stick in the first step, followed by a slip check in the second step. A penalty formulation for the tangential trail force Fot = ||Fto || can be given such that it reads Fto = −(ct gt + d t vt ) .

(13)

Therein, gt is the elongation of the tangential spring, ct and d t are the tangential spring stiffness and the tangential dissipation parameter, respectively. By applying Coulomb’s law the following relationship is obtained

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6 f trail =: ||Fto || − μs ||Fn || ⇒

≤ 0 : Stick > 0 : Slip .

(14)

Subsequently, the tangential spring is incremented at the next time step in the stick case by the relation gt = gt + vt Δ tDEM where Δ tDEM denotes the time step of the DEM. In the slip case, the tangential spring is aligned according to Luding [12] by 1 gt = − (F t t + d t vt ) , ct

(15)

where t = Fto /||Fto || describes the unit tangential vector at C . The contact area may be rotated slightly between two successive time steps. Therefore, the tangential spring is projected into the rotated contact area A rot at the beginning of each new time step via gt = gt − (gt · n)n. Thus, the tangential contact force reads for stick Ft = ||Fto || and for slip Ft = μd Fn . As a result, the governed frictional contact forces Ft = F t t and Tt = ri × Ft contributes to Fi in (3) and Ti in (4), respectively.

4 Coupling of the Fluid and Particle Phase In the fixed grid approach the mesh of the flow field does not coincide with the particles’ surface. Therefore, one of the key points of the fictitious domain methods is the transfer of information between the Eulerian and the Lagrangian locations. Here, the goal is to model accurately complex moving boundaries on a fixed background mesh.

4.1 Evaluation of the Hydrodynamic Forces Lebedev and his coworkers introduced in a series of articles [11, 9, 10] quadrature rules of different algebraic order of accuracy for the integration of functions over a spherical surface. The number of Lebedev integration points NLeb is related to the order of the underlying grid L via NLeb = 2 + (L + 1)2 /3. The distribution of the integration points for NLeb = 302 is shown in Fig. 1a. Thus, the respective integrals

0

i −i

Fig. 1 a) Distribution of the Lebedev quadrature points for NLeb = 302; b) Classification of the elements: 0 =: fluid element, i =: element of Pi , −i =: boundary element of Pi ; c) Nonlinear weighted velocity update (◦ := element center point, × := fluid velocity node)

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in (5) can be evaluated by the use of the Lebedev quadrature scheme. The numerical integration takes the form 

F=

∂Ω

t dA =

NLeb

NLeb

k=1

k=1

∑ (dF)k = J ∑ wk tk

,

(16)

where tk = (σ n)k is the traction vector at the Lebedev point k, wk the corresponding integration weight and σ  specifies the averaged fluid stress tensor of the element in which the point k is located. The mapping from the unit sphere into the spatial sphere for Pi is performed via the Jacobian J = 4π R2i .

4.2 Coupling Constraints In order to distinguish the domains of the individual phases as well as to describe the phase interface a scalar level set function can be introduced for the coupling procedure, cf. e.g. Sussmann [16]. As a result, the elements can be kept track with respect to their phase affiliation. They are classified such that elements belonging to Ωi are labeled by its particle number Pi (interior element:= i, boundary element:= −i ) and by “0” for a flow field element, see Fig. 1b. Therein, the classification criteria for the phase affiliation is the location of the center point of the element. In order to model the interaction between the phases, a rigid body motion constraint is explicitly applied to the Navier-Stokes equations in the solid part of the domain for the velocity update of Ω f , cf. Takiguchi et al. [17] and Kajishima et al. [7]. For the interior elements, i.e. elements denoted by i, see Fig. 1b, the constraint for the velocity nodes V ∈ Ω i reads u f = Ui + ω i × r p

(17)

,

where r p describes the vector from Mi to the respective velocity node. The velocity update for nodes belonging to a boundary element, i.e. elements depicted by −i, is distinguished in nodes for which counts V ∈ Ω i and V ∈ / Ωi . In the first case, the constraint to the Navier-Stokes equations can be given by u f = (1 − φA)u f + φA (Ui + ω i × r p )

,

(18)

where φA is the projected area fraction of the intersected control area of the element which lay within Ωi . In the second case, the velocity is updated by a nonlinearweighted strategy, see Fig. 1c, as proposed by Luo et al. [13] via the constraint u f = (1 − φR)u f + φR (Ui + ω i × r p )

.

(19)

Herein, the interpolation factor φR = e−Rer α is a function of the relative Reynolds number Rer and the relative distance α , cf. Luo et al. [13].

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5 Numerical Example The presented numerical example shows a separation simulation of a particulate flow field in which 5184 particles are embedded having two different densities for each half. The density ratios of the phases ρ f /ρs are 1.1 and 0.9 such that half of the particles are heavier than the fluid and the other half are lighter, respectively, whereby the densities are randomly distributed. The particles are initially at rest and they are accelerating due to the action of gravity and buoyancy. As it can be seen in Fig. 2, the particles separate totally into the lower and upper domain of the flow field such that towards the end of the simulation distinct domains of pure fluid and the respective domains containing the particles can be observed.

Fig. 2 Snapshots of the separation process of 5184 particles at different times. The heavy and light particles separate from each other into the lower and upper part of the flow field, respectively

6 Conclusion The presented fictitious domain method is a relatively easy and efficient way to model particulate flows. The coupling is performed by only applying an explicit constraint to the Navier-Stokes equations. Therefore, the use of present fast FEM flow solvers in combination with efficient DEM routines and search algorithms allows powerful computations of particulate flows with large numbers of particles. The numerical tests showed that the method is suitable for obtaining good results in modeling particle-fluid interactions.

References 1. Brilliantov, N.V., Spahn, F., Hertzsch, J.M., P¨oschel, T.: Model for collisions in granular gases. Phys. Rev. E 53, 5382–5392 (1996) 2. Cundall, P.A., Strack, O.D.L.: Discrete numerical model for granular assemblies. Geotechnique 29, 47–65 (1979)

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3. Glowinski, R., Pan, T.W., Hesla, T.I., Joseph, D.D., Periaux, J.: A distributed Lagrange multiplier/fictitious domain method for flows around moving rigid bodies: application to particulate flow. Int. J. Numer. Meth. Fl. 30, 1043–1066 (1999) ¨ 4. Hertz, H.: Uber die Ber¨uhrung fester elastischer K¨orper. J. Reine. Angew. Math. 92, 156–171 (1882) 5. Hu, H.H., Joseph, D.D., Crochet, M.J.: Direct simulation of fluid particle motions. Theor. Comp. Fluid Dyn. 3, 285–306 (1992) 6. Johnson, A.A., Tezduyar, T.E.: Simulation of multiple spheres falling in a liquid-filled tube. Comput. Meth. Appl. M. 134, 351–373 (1996) 7. Kajishima, T., Takiguchi, S., Hamasaki, H., Miyake, Y.: Turbulence structure of particleladen flow in a vertical plane channel due to vortex shedding. JSME Int. J. B 44, 526–535 (2001) 8. Kruggel-Emden, H., Wirtz, S., Scherer, V.: Applicable contact force models for the discrete element method: the single particle perspective. J. Press. Vess.-T ASME 131 (2009); Pressure Vessels and Piping Conference of the American-Societyof-Mechanical-Engineers, Chicago, IL, July 27-31 (2008) 9. Lebedev, V.I.: The 59th order of algebraic accuracy quadrature formula for sphere. Dokl. Akad. Naukt. 338, 454–456 (1994) 10. Lebedev, V.I., Laikov, D.N.: Quadrature formula for the sphere of 131-th algebraic order of accuracy. Dokl. Akad. Naukt. 366, 741–745 (1999) 11. Lebedev, V.I., Skorokhodov, A.L.: Quadrature-rules for a sphere of 41-order, 47-order and 53-order of accuracy. Dokl. Akad. Naukt. 324, 519–524 (1992) 12. Luding, S.: Micro-macro transition for anisotropic, frictional granular packings. Int. J. Solids Struct. 41, 5821–5836 (2004) 13. Luo, K., Wang, Z., Fan, J.: A modified immersed boundary method for simulations of fluid-particle interactions. Comput. Meth. Appl. M. 197, 36–46 (2007) 14. Peskin, C.S.: Flow patterns around heart valves – numerical method. J. Comput. Phys. 10, 252–271 (1972) 15. P¨oschel, T., Schwager, T.: Computational Granular Dynamics. Springer, Heidelberg (2005) 16. Sussmann, M., Smereka, P., Osher, S.: A level set approach for computing solutions to incompressible 2-phase flow. J. Comput. Phys. 114, 146–159 (1994) 17. Takiguchi, S., Kajishima, T., Miyake, Y.: Numerical scheme to resolve the interaction between solid particles and fluid turbulence. JSME Int. J. B 42, 411–418 (1999) 18. Wriggers, P.: On consistent tangent matrices for frictional contact problems. In: Pande, G., Middleton, J. (eds.) Proceedings of NUMETA 1987 (1987) 19. Wriggers, P.: Computational Contact Mechanics. Springer, Heidelberg (2006)

Chapter 29

A Concurrent Multiscale Approach to Non-cohesive Granular Materials Christian Wellmann and Peter Wriggers This contribution is dedicated to Professor Peter Wriggers. As my Ph.D. advisor he guided me in the development of what is summarized on the next few pages. I really appreciated his sound advice as well as the fruitful but relaxed working atmosphere at the institute through the recent years. For this I am grateful and I wish him all the best for his 60th birthday. Congratulations! (C. Wellmann).

Abstract. A concurrent two-scale approach for frictional non-cohesive granular materials is presented. In domains of large deformation the material is modeled on the grain scale by a 3D discrete element method. Elsewhere the material is considered continuous and modeled by the finite element method using a non-associative MohrCoulomb model whose parameters are fit to the particle model via a homogenization scheme. The discrete and finite element model are coupled by the Arlequin method. Therefore an overlapping domain is introduced in which the virtual work is interpolated between both models and compatibility is assured by kinematic constraints. For this purpose the discrete particle displacements are split into a fine and coarse scale part and equality of the coarse scale part and the continuum solution is enforced through the penalty method.

1 Introduction Classically granular materials are modeled via a continuum approach using the finite element method (FEM) to solve engineering scale problems. A drawback of this approach is that due to the complex material behavior advanced constitutive equations are based on a huge number of parameters whose determination is an intricate process. Furthermore, granular materials are prone to localization of deformations in small regimes like shear bands where standard continuum assumptions are no longer valid. An alternative not prone to these problems is the discrete element method Christian Wellmann Institute of Continuum Mechanics, Leibniz Universit¨at Hannover, Appelstr. 11, D-30167 Hannover, Germany e-mail: [email protected]

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(DEM) modeling the material on the grain scale. However, it is cumbered with high computational costs limiting feasible problems in space and time. Here these two approaches are combined in a concurrent two-scale approach where domains of large eventually discontinuous deformations are modeled by the DEM while the remaining domain is modeled by the computationally efficient continuum approach. Similar approaches have been developed for the coupling of molecular dynamics (MD) and FEM. However, due to the differences between MD and DEM regarding the particle interaction, shape, and arrangement MD-FEM coupling schemes cannot be simply transferred to the DEM-FEM case. A general concept for the coupling of different models is the Arlequin method [1]. It is based on the introduction of a coupling domain Ω C in which the models are superposed. Within Ω C the energy is defined as interpolation of the individual energies and compatibility is ensured through kinematic constraints. Schemes of this kind have been applied to DEM-FEM couplings in [3] and [5] for cohesive granular materials like e.g. concrete. For these materials major particle rearrangements are only possible in case of inter-particle bond breakage. Hence, the deformation mechanisms are quiet distinct from those of non-cohesive granular materials. Therefore, the direct constraint between particle displacements and interpolated FE displacements in Ω C , which might be appropriate for cohesive materials, would result in unnatural particle displacements for non-cohesive materials. Consequently, in this approach the particle displacements are split into a coarse and a fine scale part and the coarse scale part is constrained to the FE displacements. In this way particle rearrangements are not impeded within Ω C and the typical force chain microstructure is not disturbed. The DEM is described in Sect. 2. In Sect. 3 the bulk behavior of the particle model is determined via a homogenization scheme and the resulting stress-strain curves are used to fit the parameters of the continuum model. The coupling scheme is described in Sect. 4 and applied to some numerical examples in Sect. 5.

2 Discrete Element Method The DEM uses superquadric particle geometries defined by (  2   2 ) εε1   2  x1  ε1  x2  ε1 2  x3  ε2 F (x) =   +   +   = 1 . r1 r2 r3

(1)

Herein the radius parameters ri control the elongation and the exponents εi control the angularity, cf. Fig. 1. Compared to standard spherical particles this variability enables a better representation of real grain shapes which has been shown to have an impact on the bulk behavior of particle packings, cf. [7]. Contact forces between adjacent particles are determined from small overlaps. Herein the particles are assumed elastic and the Hertzian contact theory is applied. Accordingly, the overlap is considered as elastic deformation at the contact point and the elastic repulsive force is given by

A Concurrent Multiscale Approach to Non-cohesive Granular Materials Fig. 1 Superquadric particles with radii r1 = r2 = r3 /2 and varying angularity parameters εi . Note that εi ∈ [0, 2] yields a convex geometry and especially εi = 0 a hexahedron, εi = 1 an ellipsoid, and εi = 2 an octahedron

f N = γ E ∗ δ 3/2

1 = 0.5, 2 = 0.5

with

259

1 = 0.5, 2 = 1

1 1 − ν12 1 − ν22 = + E∗ E1 E2

,

1 = 1.0, 2 = 1.5

(2)

where δ is the overlap distance, γ is a function of the principal radii of curvature of the contacting surfaces, and νi and Ei are Poisson’s ratio and Young’s modulus of the particles. For the tangential contact the solution of Mindlin is used in combination with the Coulomb friction law. The determination of the contact geometry is a relatively complex problem which is formulated as an unconstrained two-dimensional optimization problem in terms of the contact normal [9]. Finally, the particle equations of motion are integrated in time using an explicit central difference method for the translational part and a momentum conserving fourth order Runge-Kutta method for the rotational part.

3 Homogenization and Elasto-plastic Parameters Within the two-scale approach the discrete particle and the continuum model represent the same material. Consequently, the bulk behavior of both models should coincide. Therefore, the DEM parameters are adapted to a granular material and the DE model is used as reference to fit the continuum parameters. For this purpose the bulk behavior of the DE model is determined by a homogenization scheme. As granular material Leighton Buzzard Sand fraction B is chosen which is a silica sand with rounded grains ranging between 0.6 mm and 1.18 mm in size. To adapt the DEM geometry parameters a detailed analysis of the grain shape performed in [2] was used. The elastic parameters of silica were chosen as E = 50 GPa and ν = 0.2 and the density is ρ = 2.65 g/cm3 . Finally, the friction between dry and wet particles has been analyzed in [6] and [4] yielding an approximate value of μ = 0.24 as inter-particle friction coefficient for dry silica grains. The bulk behavior is analyzed by tests on random periodic representative volume elements (RVEs). The samples have the shape of a rectangular box with side length Li , cf. Fig. 2a. Particles at one boundary face are in contact with those at the opposite face so that only inter-particle contacts occur. Within this setting strains can be easily applied by varying the dimensions Li . On the other hand, the application of stress boundary conditions is realized by an adaptive dimension control scheme [10] enabling the simulation of quasi-static triaxial tests: A cubical sample is isotropically

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compressed to a reference state from where the lateral stresses σ1 = σ2 are kept constant while the sample is compressed in x3 direction using a constant strain rate ε˙3 . The resulting compressive stress and volumetric strain for a lateral pressure of 100 kPa are shown in Fig. 2b. They are in good qualitative agreement with experimental results of medium dense Leighton Buzzard Sand fraction B. The bulk behavior should now be represented by a non-associative Mohr-Coulomb model whose yield function is given in terms of the principal stresses by Φ = σ1 − σ3 + (σ1 + σ3 ) sin φ + 2 c cos φ , (3) where φ is the friction angle and c the cohesion parameter. The plastic potential is of the same form but with the friction angle replaced by the so-called dilation angle ψ . Taking into account that the particle model is non-cohesive we set c = 0 leaving four material parameters to fit, i.e. two elastic constants and the friction and dilation angle. The elastic constants are fit to the initial phase of the triaxial curves yielding a Young’s modulus E = 60 MPa and a Poisson’s ratio ν = 0.145. The friction angle is fitted to the Mohr circles at the state of maximum compressive stress resulting in φ = 23.15◦ , cf. Fig. 3. Finally the constant dilation shown by the particle model is represented by ψ = 5.17◦ . a

b 250

3 2.5

200

L3

1.5 1

100

εV, %

-σ3, kPa

2 150

0.5 50

x3 x2 x1

L2

0 0

L1

0

σ3 εV 5

10 -ε3, %

-0.5 20

15

Fig. 2 a) Cubical periodic RVE with random particle package; b) Triaxial test results for a lateral pressure of 100 kPa

τ, kPa

ϕ = 23.15°, c = 15 Pa

Fig. 3 Fitting of the friction angle φ to the Mohr-Circles at the state of maximum compressive stress Results of four triaxial tests performed at different lateral pressures are used

140 120 100 80 60 40 20 0 -500

-450

-400

-350

-300

-250 σ, kPa

-200

-150

-100

-50

0

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4 Coupling The coupling is realized by the Arlequin method [1] which is based on the introduction of a coupling domain Ω C in which both models are superposed, cf. Fig. 4. Within Ω C the overall virtual work is interpolated between the individual model’s virtual works using a weighting function w(x)

δ W = δ W FE + δ W DE δW



FE

=

δ W DE =

Ω FE np

with

(1 − w) [σ : δ ε + ρ (¨x − b) · δ u] dΩ −  np

∑ δ Wα = ∑

α =1

α =1

Ωα

(4)



(1 − w) t · δ u dΓ , (5) 

Γ FE nα

w ρ (¨x − b) · δ uα dΩ −

∑ wαβ fαβ · δ uα

. (6)

β =1

Herein nα is the number of contacts of particle α , wαβ = w(xαβ ) is the weight factor at the point of contact of particles α and β , and fαβ is the corresponding contact force acting on particle α . The FE part (5) is evaluated using standard integration techniques. For the evaluation of the DE part (6) w is assumed to be a continuous, smooth, and monotonic function and replaced by its linearization about the particle center c within Ωα . Under the condition Ω α  Ω C this yields the weighted equations of motion nα

wc m c¨ = wc m b +

∑ wαβ fαβ

with

wc = w(c) ,

(7)

rαβ = xαβ − c ,

(8)

β =1

wc I · ω˙ + ω × wc I · ω =

nα

∑ wαβ rαβ × fαβ

with

β =1

where m is the mass and I is the inertia tensor with respect to c. The coupling is completed through the imposition of kinematic constraints within Ω C , which is not straightforward in the case of a particle and a continuum model. The particles are equipped with translational and rotational degrees of freedom (DOF). On the other ΩFE

Fig. 4 To couple the DE and FE method they are superposed within the coupling domain Ω C . Here the overall virtual work is interpolated between both models using the weight function w(x)

w=0

ΩC

w ∈ [0, 1]

ΩDE

w=1

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hand, a standard continuum approach without rotational DOF is used in Ω FE because this is sufficient to model the material behavior when no localizations occur. Therefore, constraints will only be formulated in terms of the translational DOF. Within the DE-FE coupling schemes for cohesive granular materials presented in [3] and [5] the particle displacements are directly constrained to the continuum displacements at the particle center. Therefore, these schemes enforce the particles residing in one element to translate according to the element ansatz. While this might be appropriate for cohesive materials it represents an unnatural constraint for noncohesive materials where fluctuations within the grain displacements arise due to the non-uniform particle shapes and an irregular particle arrangement. To overcome this problem an operator is defined which projects the particle displacements onto the FE ansatz space. This yields a decomposition of the particle displacements into a coarse and a fine scale part like it is used in the Bridging Scale Method [8] to couple atomic and continuum simulations. Within Ω C a continuous displacement field is introduced using the FE ansatz uDE (x) =

∑

NI (x) uDE I

(9)

,

I∈N C

where N C is the set of nodes belonging to Ω C . The unknown nodal values are determined by a least square fit min

∑

uDE I α ∈P C

Vα uα − uDE (cα )2

,

(10)

with the particle volume Vα and displacement uα . The solution of (10) yields a linear operator projecting the discrete particle onto the nodal displacements. Now the constraint term is formulated using the continuous field and the penalty method C=

ε 2

 ΩC

uDE − uFE 2 dΩ

,

(11)

where ε is the penalty parameter. Finally the variational formulation reads

δW + δC = 0 ,

(12)

where δ C gives the nodal and particle coupling forces.

5 Numerical Examples The cubical system shown in Fig. 5a is analyzed via a two-phase triaxial test consisting of a hydrostatic compression and a shear phase. The resulting principal stresses for a full DE, full FE, and the coupled DE-FE system are plotted in Fig. 5b. The DE and FE results reveal that the Mohr-Coulomb model is not able to model the pressure dependent stiffness of the particle model. Therefore, the FE principal stresses deviate from the DE ones in the high pressure regime. As to be expected the

A Concurrent Multiscale Approach to Non-cohesive Granular Materials a

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b 0 -100

σ, kPa

-200 -300 -400

σ1, full DE σ2, full DE σ3, full DE σ1, full FE σ2, full FE σ3, full FE σ1, DE-FE σ2, DE-FE σ3, DE-FE

-500 -600 -700 0

0.01

0.02

0.03

0.04

0.05

0.06

time, s

Fig. 5 a) Coupled cubical system with one element layer as coupling domain Ω C ; b) Principal stresses for two-phase triaxial test

a

b 16.1 mm 16.1 mm

v = 1 m/s

rigid

178.2 mm

8.05 mm Ω

DE

ΩC

745 mm

80.5 mm Ω z

y x

FE

1m

Fig. 6 a) Dimensions of plane strain pile installation example; b) Sample deformation during pile installation with color according to initial height

principal stresses of the coupled system lie in between the mono-model results. Fig. 6a shows the dimensions of a plane strain pile installation example. Large deformations are expected in the pile vicinity which is modeled by the DEM requiring about 540 000 particles. The rest is discretized by trilinear bricks and a linear variation of the weight function is used within Ω C . The friction between the particles and the pile is chosen as μ = 0.1. Fig. 6b reveals the deformation of the system after 0.1 s corresponding to an insertion distance of 100 mm. In the zone up to a distance of about three particle diameters from the pile the material is dragged with the pile for a huge distance. Up to a distance of about 15 particle diameters there is a steep gradient from material being dragged down to material being pushed up. Further away there is a monotonic decay of the distance the material is pushed up. Herein the contour lines run smoothly across the interface indicating a smooth transition between the models.

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6 Conclusion A concurrent two-scale two-method approach for non-cohesive granular materials is presented which uses a 3D DEM to model domains of large deformations and a continuum method elsewhere. The coupling yields a smooth transition between the two material models and enables the solution of problems not feasible by mono-method approaches. Future development steps are the enhancement of the DE model to get a quantitative match of the bulk behavior of granular materials, the implementation of an advanced continuum model able to represent the DE behavior more closely, and finally the development of criteria for an adaptive control of the DE domain.

References 1. Ben Dhia, H., Rateau, G.: The Arlequin method as a flexible engineering design tool. Int. J. Numer. Meth. Eng. 62, 1442–1462 (2005) 2. Clayton, C.R.I., Abbireddy, C.O.R., Schiebel, R.: A method of estimating the form of coarse particulates. Geotechnique 59, 493–501 (2009) 3. Frangin, E., Marin, P., Daudeville, L.: On the use of combined finite/discrete element method for impacted concrete structures. J. Phys. IV 134, 461–466 (2006) 4. Ishibashi, I., Perry, C., Agarwal, T.K.: Experimental determinations of contact friction for spherical glass particles. Soils Found. 34, 79–84 (1994) 5. Rojek, J., Onate, E.: Multiscale analysis using a coupled discrete/finite element model. Interact. Mult. Mech. 1, 1–31 (2007) 6. Rowe, P.W.: The stress-dilatancy relation for static equilibrium of an assembly of particles in contact. Proc. R. Soc. London A 269, 500–527 (1962) 7. Salot, C., Gotteland, P., Villard, P.: Influence of relative density on granular materials behavior: DEM simulations of triaxial tests. Granul. Matter 11, 221–236 (2009) 8. Wagner, G.J., Liu, W.K.: Coupling of atomistic and continuum simulations using a bridging scale decomposition. J. Comput. Phys. 190, 249–274 (2003) 9. Wellmann, C., Lillie, C., Wriggers, P.: A contact detection algorithm for superellipsoids based on the common-normal concept. Eng. Computation 25, 432–442 (2008) 10. Wellmann, C., Wriggers, P.: Homogenization of Granular Material modeled by a 3D DEM. In: O˜nate, E., Owen, D.R.J. (eds.) Particles-Based Methods. Fundamentals and Applications 2009, pp. 211–231. Springer, Heidelberg (in press, 2010)

Chapter 30

On Some Features of a Polygonal Discrete Element Model Ekkehard Ramm, Manfred Bischoff and Benjamin Schneider Dedicated to Professor Peter Wriggers for his great achievements in computational mechanics (E. Ramm).

Abstract. The contribution describes a two-dimensional discrete element method with polygonal particles in a two-dimensional setting allowing the simulation of granular as well as quasi-brittle material. Different models for soft contact as well as cohesion between the particles are presented. Emphasis is put on the specific features of the polygonal particles. Simulations are compared to results of small scale experiments with regular particles of steel nuts. In addition the capabilities of the method are demonstrated simulating complex concrete specimens with a distinct heterogeneous microstructure.

1 Introduction A discrete element model is introduced describing the mechanical behavior of materials consisting of separable solid particles on a mesoscopic scale. The materials may be granular like sand or powder. However the particles can also be initially glued together like in concrete or ceramics and may disintegrate during loading. These materials are usually called cohesive-frictional or quasi-brittle. In case the cohesion either initially is not present or vanishes during loading the particles interact only by contact. Special focus is put on modeling the failure of ensembles with polygonal particles. The method with its underlying equations is presented with emphasis on models for particle interaction by contact and cohesion. Thereafter, simulations are shown and compared to experimental results. Ekkehard Ramm · Manfred Bischoff · Benjamin Schneider Institute of Structural Mechanics, Universit¨at Stuttgart, Pfaffenwaldring 7, 70550 Stuttgart, Germany e-mail: [email protected], [email protected], [email protected]

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2 Discrete Element Method with Polygonal Particles The two-dimensional discrete element method of [8] has been adopted as basic model and extended, see [2] and [7]. It models materials as agglomerations of separable rigid bodies, the discrete elements. They are convex unbreakable polygons which interact through contact as well as cohesive forces, as sketched in Fig. 1. Each particle has three degrees of freedom, the translations in e1 - and e2 -direction as well as the rotation. The balance of linear and angular momentum yield the equation of motion for each discrete element g

g

g

Mg x¨ g = fct + fch + fext

(1)

.

Mg is the general mass matrix containing mass and mass moment of inertia and x¨ g is the array with the particle acceleration in e1 - and e2 -direction as well as the angular g acceleration. The right hand side comprises the forces and torques from contact fct g g and from cohesion fch as well as from the external loads fext . For all particles of a sample this results in a coupled system of ordinary differential equations of second order. The system is integrated in time by an explicit predictor–corrector method according to [5].

bo Ao fct,n

kt

fct,t

kn

e2 e1 a

b

c

Fig. 1 Interaction of particles: a) contact, b) cohesion by beams and c) cohesion by interfaces

2.1 Models for Contact Computational contact mechanics has been intensively investigated in the last years, see for example [13]. In particle dynamics the contact search is in general the most time consuming part [12]; for a contact detection algorithm for superellipsoids confer [11]. We adopt often used simplified contact models determining repulsing forces of two colliding particles. A small overlap Ao of the rigid particles is allowed resulting in contact forces as shown in Fig. 1a. The contact force is split into a normal and a tangential part: fct = fct,n + fct,t . They are applied in normal and tangential direction

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267

at the midpoint of the contact line bo , called the contact point, giving also rise to a torque around the center of mass. For the normal force we use two different models. The first one consists of an elastic and a viscous part and was introduced in [8] as ( ) (1) En A o fct,n = − + meff γn vn n . (2) dc (1)

En is the elastic contact stiffness, γn the damping coefficient and vn the relative velocity in normal direction. For the contacting particle pair dc denotes the characteristic length, meff is the effective mass and n the outward unit normal on the contact line. The second model also combines an elastic part as proposed in [4] and a viscous part (2)

fct,n = −(En Ao bo + meff γn vn ) n .

(3)

Its elastic part can be derived from a potential. For the tangential force again two different models have been used. The first one takes the minimum of a frictional and a viscous force according to [8] fct,t = −sgn(vt )min(μt fct,n , meff γt |vt |) t .

(4)

μt is the friction coefficient and t the unit tangent on the contact line. For the second one an elasto-plastic model with a Coulomb frictional yield limit similar to [1] is chosen fct,t = −Et uel t t ;

Ft = fct,t  − μtfct,n  ≤ 0

.

(5)

Herein uel t describes the relative elastic displacement in tangential direction and Ft is the yield function. The elasto-plastic model is able to reproduce sticking as well as sliding friction, whereas the first one can only model sliding friction. When particles move on a background plate, the force between each particle and background medium has to be modeled in addition to the usual contact. Similar to the tangential contact model (5) an elasto-plastic model with a Coulomb frictional yield limit is applied at each point on the particle surface contacting the background plate

τ ct,b = −Eb uel b

;

Fb = τ ct,b  − μb|σb | ≤ 0 .

(6)

Herein σb is the normal stress between the particle surface and the background plate. The shear stress τ ct,b is integrated numerically yielding the force and the torque acting on the particle.

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2.2 Models for Cohesion In addition to the contact force, cohesive or adhesive forces are introduced if the particles are initially glued together. The cohesive force between two particles is realized by three different models with increasing complexity. The first one inserts a small strain shear beam (Timoshenko beam) between connected particles as sketched in Fig. 1b. The beam is clamped to the centers of the discrete elements and does not possess any mass. It deforms with the degrees of freedom of the particles leading to additional forces and torques on both particles. The beam behaves linearly elastic until maximum axial deformation εmax and rotational deformation θmax are reached [2] and [6]   & ' εax 2 max |θi |, |θj | + =1 . (7) εmax θmax In this failure criterion εax indicates the axial strain and θ the relative rotation of the beam at its ends i and j. It is applied only under tensile axial strains. Fulfilling this criterion, the beam immediately fails in a brittle manner. The brittle beam model is the simplest and numerically cheapest one. In order to also reproduce less brittle and more ductile failure behavior, a second model is introduced. It is again a shear beam but accounts for the successive degradation of the cohesive connection. For this purpose, the elastic beam element is enhanced by a damage model with linear softening instead of using the brittle failure criterion. The degradation of the beam’s Young’s modulus E is described by the usual factor (1 − d) where d is the isotropic damage parameter. The damage is zero until a certain strain level ε0 is achieved. It increases to one until the maximum strain level εm is reached, where the beam is completely damaged and fails. The strain level ε˜ is determined by the equivalent strain at the middle cross section of the beam h ε˜ = < εax + |κ | > +|γ | . (8) 2 The measure combines the strains from axial elongation εax , curvature κ and shear deformation γ ; h is the height of the beam. The beam with damage is able to reproduce a more ductile failure behavior of materials still being numerically inexpensive. The third and most complex model inserts a continuous connection similar to a zero-thickness interface element at the surfaces as sketched in a simplistic way by springs in Fig. 1c. Each point of the interface exhibits a non-associated softening plasticity coupled in normal and tangential direction by a two-surface MohrCoulomb yield function [10]. The stress distribution is integrated numerically at a finite number of points along the interface to compute the resulting force and torque on the particle [2]. The softening interface is the numerically most expensive but physically most detailed model, since it represents the cohesive forces where they appear in reality, namely at the connected surfaces of the particles.

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3 Examples 3.1 Model Material without Cohesion In order to test the capabilities of the model for granular materials, uniaxial compression experiments with simplistic particle geometries are performed, see Fig. 2. Standard hexagonal steel nuts are displayed on a steel table in an ordered way; the upper load platen is moved downwards with slow, constant velocity vˆ2 = −6.23 · 10−4 m/s. In the numerical simulation contact force models (3) and (5) as well as background force model (6) are used. The geometry of each particle is simplified as a sharp-edged solid hexagon instead of the real steel nut geometry with a cylindrical hole and rounded edges. In Fig. 2 the initial stage of the sample and a stage when a pattern of diagonal shear bands has appeared are plotted. The shear bands are triggered by the missing particle and the inclination angles are prescribed by the particle geometry. In the simulation we see block by block movement of parts of the sample reproducing very well the primary failure phenomena. Detailed traces of the individual particles, especially in the later failure regimes, are difficult to reproduce. On the one hand, this is physically caused by imperfections of the geometry and material parameters which are not modeled in the present study. On the other hand, the numerical model exhibits high parameter sensitivity because of the non-smooth character of the contact force models.

Fig. 2 Top view of the uniaxial compression test on a granular model material: experiment (top) and simulation (bottom) at undeformed (left) and deformed stage (right)

  v2 ·10−3 ms

e2 e1

−1

0

1

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3.2 Model Material with Cohesion We use the same hexagonal steel nuts as before however bond them together by standard glue. In this case the sample with 22 particles stands between two glass panes in the testing device, see Fig. 3a. The friction between particles and panes is so low that it could be neglected. Again a prescribed velocity vˆ2 = −3.3 · 10−5 m/s is applied to the upper row of particles. For the present study the cohesion between particles (adhesion through glue) is modeled by the beam with damage. The data are ρ = 3.3 g/cm3 , E = 28 N/cm2 , ε0 = 1.1 · 10−3 and εm = 2.3 · 10−1. Figure 3b compares the particle samples for experiment and simulation; the figure also shows the detached glue connections in deformed stage (right) where the respected beams of the overlay lattice have already reached the descending softening branch and are damaged. The horizontal beams are mainly elongated whereas the diagonal ones are essentially sheared. The pattern of localization is identical to that of the simulation.

a

b

Fig. 3 a) Experiment with glued steel nuts and b) comparison of experiment and simulation at undeformed (left) and deformed stage (right)

0.00

Load (kN)

−0.25

Fig. 4 Load–displacement diagram: three experiments with glued particles (thin lines) and simulation (thick line)

Sim.

−0.50 Exp. −0.75 −0.15

−0.10 −0.05 Displacement (cm)

0.00

On Some Features of a Polygonal Discrete Element Model

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For three experiments the load–displacement diagrams are displayed in Fig. 4 and supplemented by the smooth curve from simulation. Although geometry and material of the nuts do not differ very much from each other the scatter of the results for all three experiments is clear; the deviation is caused by the rather primitive gluing process. On the other hand the simulation represents a kind of overall mean response having a lower failure load though. The rather smooth transfer into the post-critical regime in the experiments is due to the more ductile failure of the glue under shear compared to the material model applied in the analysis.

3.3 Concrete with Microstructure For a detailed investigation of heterogeneous quasi-brittle materials like concrete, their microstructure is explicitly modeled [3]. Since the microstructure of the experimental tests in [9] is unknown, an artificial microstructure is created. Therefore, the particles are partitioned into groups representing individual grains and others representing matrix of concrete as shown in Fig. 5b. The cohesive connections between grain particles are stiffer and have higher strength than the ones between matrix particles. The connections at the bond layer between a grain particle and a matrix particle are even less stiff and fail at lower loading. This accounts for the physical properties of standard concrete where matrix material is softer than the grains and failure mainly occurs along the grain–matrix bond. Using this microstructure enhanced discrete element model, uniaxial displacement driven compression tests are simulated. The models (2) and (4) for the contact and the softening interface model for cohesion are used. The results are compared to experiments documented in [9] using different loading platens, see Fig. 5a. 0

teflon [9]

σ¯22 (kN/cm2 )

-1

brushes [9]

-2 -3

sim∗i 

-4

sim∗1

1

-5 -6 a

dry platen [9] -7 -6

-5

-4 -3 -2 -1 ε22 (·10−3 )

0

b

Fig. 5 Compression test on concrete: a) vertical nominal stress σ¯ 22 versus nominal strain ε22 and b) sim∗1 at 1, matrix/grain particles in light grey/dark grey, eliminated interfaces as black lines

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In Fig. 5b the eliminated cohesive bonds are marked as black lines and tensile splitting type failure is observed. In Fig. 5a the nominal stress of one specific simulation sim∗1 and of the average over seven simulations with different statistically varied samples sim∗i  are plotted versus the nominal strain. The linear as well as the softening branch and the peak load of the experiments are well reproduced by the simulations.

4 Conclusions The polygonal discrete element method applying simplified contact models is able to account for dominant failure phenomena of granular materials. This is shown also for several other loading scenarios in [2]. For quasi-brittle materials the cohesive models of increasing complexity, from the brittle beam to the beam with damage and the softening interface, enable more and more reliable results. The introduction of an artificial microstructure allows detailed modeling of failure and qualitatively as well as quantitatively good correspondence with experiments. The beam with damage is a promising compromise in view of accuracy and numerical expense. Acknowledgements. The authors are indebted for the financial support of the German Research Foundation (DFG) within the research project “Fragmentierung koh¨asiver Reibungsmaterialien mit diskretem Partikelmodell” under grant no. RA 218/22-1 and the support within the Cluster of Excellence in Simulation Technology (EXC 310/1) at the University of Stuttgart.

References 1. Cundall, P.A., Strack, O.D.L.: A discrete numerical model for granular assembilies. G´eotechnique 29, 47–65 (1979) 2. D’Addetta, G.A.: Discrete models for cohesive frictional materials. Ph.D. thesis, Bericht Nr. 42, Institut f¨ur Baustatik, Universit¨at Stuttgart, Germany (2004) 3. D’Addetta, G.A., Ramm, E.: A microstructure-based simulation environment on the basis of an interface enhanced particle model. Granul. Matter 8, 159–174 (2006) 4. Feng, Y.T., Owen, D.R.J.: A 2D polygon/polygon contact model: algorithmic aspects. Eng. Comput. 21, 265–277 (2004) 5. Gear, C.W.: Numerical initial value problems in ordinary differential equations. PrenticeHall, Englewood Cliffs (1971) 6. Herrmann, H.J., Hansen, A., Roux, S.: Fracture of disordered, elastic lattices in two dimensions. Phys. Rev. B 39, 637–648 (1989) 7. Schneider, B., D’Addetta, G.A., Ramm, E.: Application of the dicrete element method to quasibrittle materials. In: O˜nate, E., Owen, D.R.J. (eds.) Proceedings of Particles 2009: Particle-based methods: Fundamentals and applications, pp. 97–100. CIMNE, Barcelona (2009) 8. Tillemans, H.-J., Herrmann, H.J.: Simulating granular solids under shear. Phys. A 217, 261–288 (1995)

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9. Vonk, R.: Influence of boundary conditions on softening of concrete loaded in compression. Report TUE/BKO 89.14, Faculteit Bouwkunde, Technische Universiteit Eindhoven, The Netherlands (1989) 10. Vonk, R.: Softening of concrete loaded in compression. Ph.D. thesis, Technische Universiteit Eindhoven, The Netherlands (1992) 11. Wellmann, C., Lillie, C., Wriggers, P.: A contact detection algorithm for superellipsoids based on the common-normal concept. Eng. Comput. 25, 432–442 (2008) 12. Williams, J.R., O’Connor, R.: Discrete element simulation and the contact problem. Arch. Comput. Methods. Eng. 6, 279–304 (1999) 13. Wriggers, P.: Computational contact mechanics, 2nd edn. Springer, Berlin (2006)

Chapter 31

Isogeometric Failure Analysis Clemens V. Verhoosel, Michael A. Scott, Michael J. Borden, Ren´e de Borst, and Thomas J.R. Hughes I have known Peter Wriggers since he was a visiting scholar hosted by the late Juan Simo in the Division of Mechanics and Computation at Stanford University. Peter had general interest in computational solid mechanics but focused especially on contact problems, and soon after became the preeminent expert on the subject. I have met Peter over the years at numerous international conferences, and just last year I visited his institute at Leibniz University in Hannover for the first time. I had a very enjoyable time socially with Peter and his wife Claudia. Peter, his assistant Ilker Temizer, and I began a fruitful research collaboration combining the themes of isogeometric analysis and contact problems. I hope that this begins an important new research direction in the solution of challenging contact problems. I wish Peter a very happy 60th birthday and the best professional successs and personal happiness for many years to come (T.J.R. Hughes).

Abstract. Isogeometric analysis is a versatile tool for failure analysis. On the one hand, the excellent control over the inter-element continuity conditions enables a natural incorporation of continuum constitutive relations that incorporate higherorder strain gradients, as in gradient plasticity or damage. On the other hand, the possibility of enhancing a basis with discontinuities by means of knot insertion makes isogeometric finite elements a suitable candidate for modeling discrete cracks. Both possibilities are described and will be illustrated by examples.

1 Introduction Understanding and predicting failure is of crucial importance for improving the design of engineering structures. Two distinct approaches have found their way into the literature: Smeared and discrete models. In smeared approaches, failure is Clemens V. Verhoosel · Ren´e de Borst Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands e-mail: [email protected],[email protected] Michael A. Scott · Michael J. Borden · Thomas J. R. Hughes University of Texas at Austin, 78712 Austin, USA e-mail: [email protected],[email protected], [email protected]

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described by the gradual degradation of the bulk material. In discrete failure models, cracks are described by discontinuities in the displacement field. We introduce isogeometric finite elements as a versatile tool for failure analysis. The isogeometric analysis concept was introduced by Hughes et al. [11] and has been applied successfully to a wide variety of problems in solids, fluids and fluidstructure interactions [7]. Use of smooth spline bases in isogeometric analysis has computational advantages over standard finite elements. In isogeometric analysis the geometry and solution space are fully coupled. This makes it possible to construct bases for complex geometries, which can be obtained directly from a computer aided design (CAD) tool [3]. From an analysis point of view isogeometric analysis can be considered as an element-based discretization technique [5]. This compatibility with traditional finite elements facilitates the application to industrial problems. The ability of isogeometric finite elements to control inter-element continuity conditions provides a unified framework for smeared and discrete failure models. In this contribution we use isogeometric finite elements to analyze gradient-enhanced continuum damage models, and to simulate discrete cracks that employ a cohesive zone concept.

2 Isogeometric Finite Elements The fundamental building block of isogeometric analysis is the univariate B-spline, e.g. [14, 7]. A univariate B-spline is a piecewise polynomial of order p defined over  a non-decreasing knot vector Ξ = ξ1 , ξ2 , . . . , ξn+p+1 , with n the number of basis functions. We refer to the knot intervals of positive length as elements. A B-spline of order p is defined as a linear combination of n B-spline basis functions n

a(ξ ) = ∑ Ni,p (ξ )Ai

(1)

i=1

where Ni,p (ξ ) represents a B-spline basis function of order p and Ai is called a control point or variable. The B-spline basis is defined recursively, starting with the zeroth order (p = 0) functions  1 ξi ≤ ξ < ξi+1 Ni,0 (ξ ) = (2) 0 otherwise from which the higher-order (p = 1, 2, . . .) basis functions follow by the Cox-de Boor recursion formula [8, 4] Ni,p (ξ ) =

ξi+p+1 − ξ ξ − ξi Ni,p−1 (ξ ) + Ni+1,p−1 (ξ ) . ξi+p − ξi ξi+p+1 − ξi+1

(3)

An example is shown in Fig. 1. Non-uniform rational B-splines (NURBS) are introduced as a generalization of B-splines by

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277 n Ni,p (ξ )Wi Ai = ∑ Ri,p (ξ )Ai w(ξ ) i=1 i=1 n

a(ξ ) = ∑

(4)

where w(ξ ) = ∑ni=1 Ni,p (ξ )Wi is the weighting function. The ability of NURBS to describe many objects of engineering interest has made them the industry standard in design. NURBS are p − mi times continuously differentiable over a knot i, where mi is the multiplicity of that knot. Smooth bases are therefore obtained naturally by increasing the order of the basis functions. Knot insertion can be used to enhance a basis with regions of reduced continuity. This concept is demonstrated in Fig. 1. Ξ = {0, 0, 0, 1, 1, 1}

Ni,2 (ξ)

0.8

N1,2

0.6

N2,2

1

N3,2

Ni,2 (ξ)

1

  Ξ = 0, 0, 0, 13 , 13 , 13 , 1, 1, 1

0.4

0.8

N3,2 N1,2

0.6

N2,2

N4,2

N6,2 N5,2

0.4 0.2

0.2

0

0 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

ξ

0.6

0.8

1

ξ

Fig. 1 Quadratic B-spline basis functions without (left) and with (right) a discontinuity

Since problems of engineering interest are almost exclusively two- or threedimensional, generally use is made of multivariate splines. Analysis suitable multivariate splines can be created by means of tensor product NURBS (e.g. [7]) or T-splines (e.g. [16]). From an analysis point of view, NURBS and T-splines can be treated in a unified way using B´ezier extraction [5].

3 Higher-Order Gradient Damage Formulation Continuum damage models [12] are commonly used for the simulation of diffuse fracture processes. Several modifications of the original theory have been proposed to overcome the mesh dependency problems associated with the absence of an internal length scale [6]. In the implicit gradient damage formulation [13] a nonlocal equivalent strain field is approximated by a Taylor expansion. The formulation based on a second-order Taylor expansion has found widespread use due to its compatibility with C0 -continuous finite elements. It has, however, been demonstrated that the accuracy of the second-order approximation can be limited [1]. For that reason it is important to study the effect of the higher-order terms in the Taylor approximation. The natural ability of isogeometric finite elements to construct smooth bases makes them an ideal candidate for the discretization of these higher-order formulations.

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3.1 Constitutive Behavior In isotropic continuum damage models, the Cauchy stress is related to the infinitesimal strain tensor by σi j = (1 − ω )Hi jkl εkl , (5) where ω ∈ [0, 1] is a scalar damage parameter and H is the Hookean elasticity tensor for undamage material (i.e. with ω = 0). When damage has fully developed (ω = 1) a material has lost all stiffness. Note that we adopt index notation with summation over repeated italic indices. The damage parameter is related to a history parameter κ by a damage law ω = ω (κ ). The history parameter evolves according to the Kuhn-Tucker conditions for the loading function f = η¯ − κ , where η¯ is a nonlocal strain measure. This nonlocal equivalent strain avoids the spurious mesh dependencies observed for local damage models [17]. For the implicit gradient formulation [13] the nonlocal equivalent strain η¯ (x) is related to the local equivalent strain η (x) by 1 ∂ 2 η¯ 1 ∂ 4 η¯ 1 ∂ 6 η¯ η¯ (x) − lc2 2 (x) + lc4 2 2 (x) − lc6 2 2 2 (x) + . . . = η (x) . (6) 2 ∂ xi 8 ∂ xi ∂ x j 48 ∂ xi ∂ x j ∂ xk This gradient approximation has proved to yield a computationally efficient approximation of the nonlocal formulation ( )  x − y2 η¯ (x) = η (y)g(x, y) dy with g(x, y) = exp − , (7) 2lc2 Ω from which it is observed that the width of the damage zone is controlled by the model parameter lc .

3.2 L-Shaped Specimen We consider an L-shaped specimen (Fig. 2). The diagonal failure zone resulting from the set-up requires mesh refinements in that direction, which is achieved using consistent T-splines. In the undamaged state a linear isotropic material is considered with modulus of elasticity 10 GPa and Poisson’s ratio 0.2. Plane stress conditions are assumed. A modified von Mises local equivalent strain is used [18], with the tensile strength being 10 times the compressive strength. The damage law [10]  0 κ ≤ κ0 ω (κ ) = (8) κ0 1 − κ {(1 − α ) + α exp[β (κ0 − κ )]} κ > κ0 −4 is used with parameters √ κ0 = 4 · 10 , α = 0.98 and β = 80. The nonlocal length scale is taken as lc = 5 2 ≈ 7.07 mm. Force-displacement curves are obtained using the cubic B´ezier mesh shown in Fig. 2. The mesh consists of 1686 B´ezier elements, over which 1543 basis functions

Isogeometric Failure Analysis 250 mm

279

250 mm

250 mm

F, u 250 mm

F, u

Fig. 2 Schematic representation and B´ezier mesh for the L-shaped specimen. The thickness of the specimen is 200 mm

20

Nonlocal d=2 d=4 d=6

Fig. 3 Force-displacement results for the L-shaped specimen using the nonlocal formulation and the second(d = 2), fourth- (d = 4) and sixth-order (d = 6) gradient formulations

F [kN]

16 12 8 4 0 0

0.5

1

1.5

2

2.5

u [mm]

are defined. Comparison with results obtained on a refined mesh has demonstrated the sufficient accuracy of the results for comparing the various formulations. In Fig. 3 the results of the gradient formulations are compared with the nonlocal formulation. Upon increasing the order of the formulation the approximation of the nonlocal result is improved. For the considered problem, the sixth-order formulation is observed to be very efficient, since it accurately approximates the nonlocal result, whereas the involved computational effort is negligible compared to the nonlocal formulation.

4 Cohesive Zone Formulation For many materials, failure is characterized by the appearance of discrete cracks. In contrast to purely brittle fracture, the failure process in most materials takes place in a zone that is larger than its atomistic microstructure. Discrete fracture models that incorporate a process zone, referred to as cohesive zone models, were introduced

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by Dugdale [9] and Barenblatt [2]. From the perspective of element technology, the challenge lies in flexibly capturing the internal traction boundaries, by which cracks are modeled. This is particularly so when propagating discontinuities are to be simulated. The excellent control over inter-element continuity conditions makes isogeometric analysis an excellent candidate for the discretization of cohesive zone formulations.

4.1 Constitutive Behavior In cohesive zone formulations a crack is described by an internal discontinuity boundary. A cohesive law describes the relation between the traction t acting on this internal boundary and the jump in the displacement field u over it t = t(u)

(9)

.

Generally a distinction is made between initially rigid and initially elastic tractionopening relations. In many cases one is interested in studying the evolution of cracks, and their effect on the load bearing capacity of a structure. In those cases, the internal discontinuity boundary gradually extends through the domain. The evolution of the discontinuity is governed by a fracture criterion, which requires the stress state at the crack tip to be equal to the fracture strength when the crack is propagating. The direction of propagation is usually taken perpendicular to that of the maximum principal stress.

4.2 Single-Edge Notched Beam We consider a single-edge notched (SEN) beam (Fig. 4). The bulk material is modeled as a linear elastic isotropic material with modulus of elasticity of 35 GPa and Poisson’s ratio 0.15, and plane strain conditions are assumed. The fracture process is described by the cohesive law [19]

20

180

20 20

180

20 20

5 x2 x1

1 11 P

80

10 P 11

Fig. 4 Schematic representation of the single-edge notched (SEN) beam. All dimensions are in millimeters, and the depth of the beam is 100 mm.

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Fig. 5 Contour plot showing the σx1 x1 Cauchy stress. Displacements are 100 times amplified

& ' tult tn = tult exp − κ , Gc

ts = dint exp(hs κ )us  .

(10)

The history parameter κ is determined by the loading function f = un  − κ and secant unloading is assumed. The fracture strength and fracture toughness are taken equal to tult = 2.8 MPa and Gc = 0.1 N/mm, respectively. The initial shear stiffness dint is taken equal to 1 MPa/mm and hs , which governs the degradation of the shear stiffness, is assumed to be zero. A cubic T-spline mesh consisting of 1204 elements and 1367 basis functions is used. A mesh convergence study was performed to confirm the adequacy of this mesh. In Fig. 5 a contour plot of the cracked SEN-beam is shown. Both the curved dominant crack and secondary crack emerging from the bottom of the specimen are in good agreement with experimental observations [15]. Note from the contour plot that both crack paths are smooth as a consequence of their higher-order parameterization.

5 Conclusions Isogeometric finite elements are shown to be a versatile tool for failure analysis. Isogeometric analysis allows for the construction of smooth basis functions on complex domains, providing an appropriate solution space for higher-order differential equations encountered in e.g. gradient damage formulations. The possibility of enhancing a spline basis with discontinuities through knot insertion makes isogeometric finite elements suitable for capturing discontinuities, in particular cracks. Acknowledgements. T. J. R. Hughes and M. A. Scott were partially supported by ONR Contract N00014-08-0992, T. J. R. Hughes was also partially supported by NSF Grant 0700204, and M. A. Scott was also partially supported by an ICES CAM Graduate Fellowship. M. J. Borden was supported by Sandia National Laboratories. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC0494AL85000.

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References 1. Askes, H., Pamin, J., de Borst, R.: Dispersion analysis and element-free Galerkin solutions of second- and fourth-order gradient-enhanced damage models. Int. J. Numer. Meth. Eng. 49, 811–832 (2000) 2. Barenblatt, G.I.: The mathematical theory of equilibrium cracks in brittle fracture. In: Advances in Applied Mechanics, pp. 55–129. Elsevier, Amsterdam (1962) 3. Benson, D.J., Bazilevs, Y., De Luycker, E., Hsu, M.C., Scott, M.A., Hughes, T.J.R., Belytschko, T.: A generalized finite element formulation for arbitrary basis functions: from isogeometric analysis to XFEM. Int. J. Numer. Meth. Eng. 83(6), 765–785 (2010), doi:10.1002/nme.2864 4. de Boor, C.: On calculating with B-splines. J. Approx. Theory 6, 50–62 (1972) 5. Borden, M.J., Scott, M.A., Evans, J.A., Hughes, T.J.R.: Isogeometric finite element data structures based on B´ezier extraction. Int. J. Numer. Meth. Eng. (2010), doi:10.1002/nme.2968 6. de Borst, R.: Damage, material instabilities, and failure. In: Stein, E., de Borst, R., Hughes, T.J.R. (eds.) Encyclopedia of Computational Mechanics, pp. 335–373. Wiley, Chichester (2004) 7. Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley, Chichester (2009) 8. Cox, M.G.: The numerical evaluation of B-splines. IMA J. Appl. Math. 10 (1972) 9. Dugdale, D.S.: Yielding of steel sheets containing slits. J. Mech. Phys. Solids 8, 100–104 (1960) 10. Geers, M.G.D., de Borst, R., Brekelmans, W.A.M., Peerlings, R.H.J.: Strain-based transient-gradient damage model for failure analyses. Comput. Meth. Appl. M. 160, 133–153 (1998) 11. Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Meth. Appl. M. 194, 4135–4195 (2005) 12. Lemaitre, J., Chaboche, J.L.: Mechanics of solid materials. Cambridge University Press, Cambridge (1990) 13. Peerlings, R.H.J., de Borst, R., Brekelmans, W.A.M., de Vree, J.H.P.: Gradient enhanced damage for quasi-brittle materials. Int. J. Numer. Meth. Eng. 39, 3391–3403 (1996) 14. Rogers, D.F.: An Introduction to NURBS. Academic Press, San Diego (2001) 15. Schlangen, E.: Experimental and Numerical Analysis of Fracture Processes in Concrete. Ph.D. thesis, Delft University of Technology (1993) 16. Sederberg, T.W., Zheng, J., Bakenov, A., Nasri, A.: T-splines and T-NURCCs. ACM T. Graphic. 22, 477–484 (2003) 17. Sluys, L.J., de Borst, R.: Dispersive properties of gradient-dependent and rate-dependent media. Mech. Mater. 18, 131–149 (1994) 18. de Vree, J.H.P., Brekelmans, W.A.M., van Gils, M.A.J.: Comparison of nonlocal approaches in continuum damage mechanics. Comput. Struct. 55, 581–588 (1995) 19. Wells, G.N., Sluys, L.J.: A new method for modelling cohesive cracks using finite elements. Int. J. Numer. Meth. Eng. 50, 2667–2682 (2001)

Chapter 32

A Method for Enforcement of Dirichlet Boundary Conditions in Isogeometric Analysis Toby J. Mitchell, Sanjay Govindjee, and Robert L. Taylor Peter Wriggers has a long association with the University of California, Berkeley. Following completion of his doctoral degree in the early 1980’s he spent a postdoctoral study period in the structural mechanics group of the Department of Civil & Environmental Engineering. During this period he implemented the first large deformation contact algorithms into our finite element system FEAP and also developed the now famous consistent tangent formulation for node-to-surface contact. In several subsequent visits he has collaborated with us on a variety of computational mechanics topics. We are pleased to contribute to the volume celebrating his 60th birthday and wish him many more years of continued success and good health (R.L. Taylor).

Abstract. Isogeometric finite element analysis is a technique that substitutes NURBS basis functions for the Lagrange polynomial basis functions used in standard finite element analysis. This allows finite element analysis to exactly replicate the CAD geometry on which it is based. However, the non-interpolatory nature of NURBS basis functions used in CAD means that imposition of Dirichlet boundary conditions can no longer be accomplished by collocation of exact values at the control points. A technique such as a global least-squares fit of the prescribed boundary data onto the span of the basis functions is required; however, this requires solution of potentially large sets of equations, leading to unacceptable computational costs, particularly in problems with time-varying Dirichlet boundary data. This Toby J. Mitchell Graduate Student University of California, Berkeley e-mail: [email protected] Sanjay Govindjee Chancellor’s Professor University of California, Berkeley e-mail: [email protected] Robert L. Taylor Professor of the Graduate School University of California, Berkeley e-mail: [email protected]

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paper presents a method to weakly impose Dirichlet boundary conditions in isogeometric finite element analysis that is shown (via numerical examples) to be significantly more efficient than a global least-squares fit while attaining nearly the same accuracy.

1 Introduction Today, problems in solid mechanics are routinely solved using the finite element method of analysis as described in common reference books (e.g., see [9] or [8]). To perform an analysis it is necessary to generate a mesh describing the specific geometry to be analyzed. For this task a CAD system is commonly employed to extract geometric data to describe the mesh. Each of these disciplines has evolved independently and recent research efforts have been directed at unifying the two areas with the combined approach designated as an isogeometric analysis (see, e.g. [5], [3]). In this approach the finite element model directly uses the CAD system geometry and interpolation scheme as a basic description of the problem. Commonly NURBS type interpolation is used to define the initial problem and it is necessary to refine the CAD geometry by interpolation elevation and knot insertion (called k-refinement in [3]). After refinement it is then necessary to describe the boundary conditions for the problem. A NURBS curve is defined in terms of control points that, in general, do not lie on the curve. Thus, determination of appropriate values to apply to the control points for Dirichlet type boundary conditions differs from the current practice for Lagrange-based interpolation where specified values are collocated at nodes. Only in the special case of straight domain boundaries, in which control points for NURBS lie on the boundary surface, can simple collocation be used. To date little attention has been devoted to this topic for problems in solid mechanics with curved boundaries where accurate enforcement is important. For fluid problems a weak method of enforcement has been presented by Bazilevs & Hughes [2, 1]. However, the approach requires additional terms in the weak form of the boundary value problem and may not provide the accuracy needed for inelastic problems or problems that involve contact. Beyond these efforts, Hughes et al. have noted that one can utilize a least-squares methodology to determine boundary control point values [5]. The present paper proposes an efficient least-squares based technique for the enforcement of Dirichlet boundary conditions in the general case of arbitrarily curved boundaries with arbitrary data. A similar type of least-squares approach is presented by Hinton & Campbell; however, they use the method to smooth discontinuous stresses between elements using reduced order shape functions [4]. We first note that, while a full least-squares solution will give an optimal L2 -norm fit of a NURBS curve to any prescribed boundary data, this requires solution of a potentially large set of matrix equations along the boundary, leading to computational costs that would be unacceptable in problems with time-varying Dirichlet boundary data (e.g. earthquake simulations). We propose a technique that satisfies the competing requirements of computational efficiency and numerical accuracy, and which

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is simple and straightforward to implement within a standard finite element analysis software architecture. A development of the method is presented along with several numerical examples to demonstrate its properties in comparison with the full least-squares solution.

2 Dirichlet Boundary Conditions Consider a body occupying the reference configuration Ω in a general n-dimensional space with Dirichlet boundary Γu and a function u(ξ ) that specifies the (exact) Dirichlet boundary data along Γu which is parameterized by the coordinates ξ . A least-squares fit of the finite element boundary data u( ¯ ξ ) to the boundary data u(ξ ) can be obtained by minimizing the least-squares functional

Π=

1 2

 Γu

[u( ¯ ξ ) − u(ξ )]2 dΓ =

1 2

 Γu

[N(ξ )b − u(ξ )]2 dΓ

,

(1)

where N is a row vector of global shape functions which are assumed to have local support (as is true for both Lagrange polynomial and NURBS basis functions) and b is a vector of (vector-valued) control variables that determine the shape of the boundary data curve u( ¯ ξ ). Thus u( ¯ ξ ) = N(ξ )b is simply the dot product of the global shape function vector with the control variables and is identical in every respect, save for the non-interpolatory nature of the control variables and NURBS basis functions, with Lagrange polynomial based finite element methods. Minimizing (1) with respect to the control variables b yields the system 

Mb = p where M =

Γu

NT N dΓ and p =

 Γu

NT u dΓ

.

(2)

Note that the least-squares fit matrix M is identical in form to a finite element surface mass matrix. The control variables b resulting from the solution of (2) thus provide the optimal L2 -norm fit of the NURBS boundary data curve to the prescribed data. Solution of this system requires the effective inversion of the (n · G) × (n · G) matrix M, where G is the total number of Dirichlet boundary control points and n is the number of nodal unknowns. For large systems, a significant portion of the total finite element solution time could be consumed by the solution of the least-squares fit of the boundary data. For problems with time-varying Dirichlet boundary conditions, this operation must be conducted within every time step (and within every iteration of every time step for nonlinear problems), likely leading to an unacceptable computational expense. Methods of accelerating the boundary solution are therefore needed to make Dirichlet boundary condition enforcement practical. Let C be the assembly operator, a rectangular Boolean matrix that maps local element degrees of freedom to global degrees of freedom, and let Me be an uncoupled block diagonal matrix containing all the local element matrices resulting from (2b). Similarly, let pe be a vector containing all local element load vectors resulting from (2c). It is easily verified that

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M = CMe CT and p = Cpe

(3)

and therefore the least-squares fit equation Mb = p can be rewritten as CMe CT b = Cpe

(4)

.

If we factor C out of both sides of the equation, we obtain Me CT b = pe + q ,

(5)

where q is an element of the null space of C. Let us ignore q for the time being and examine the following modified equation Me CT bˆ = pe

(6)

,

where bˆ is an altered solution vector that is not in general identical to the optimal least-squares solution b. We observe that due to the non-singular nature of the element matrices contained in Me , it is possible to invert Me and solve for a vector of local element solutions se : CT bˆ = (Me )−1 pe = se

(7)

.

ˆ but are now prevented from directly We wish to obtain the global solution vector b, doing so by CT , which is in general non-square and thus non-invertible. A straightforward solution is to premultiply both sides of the equation by C, leading to a least-squares type fit using the assembly operator C. This will yield an invertible square matrix CCT on the left-hand side. We can then obtain bˆ via bˆ = (CCT )−1 C(Me )−1 pe = Ase

,

(8)

where we have relabeled (CCT )−1 C as A. The local-to-global degree of freedom mapping of C is necessarily unique. By the uniqueness of this local-to-global map, there is only one unit-valued entry in each column of C; all other entries in that column must be zero. Therefore each row of C is orthogonal to every other row of C, and by this property CCT , must be a diagonal matrix. Thus the inversion of CCT is trivial. Furthermore, since each row of C contains a number of unity entries equal to the number of local degrees of freedom mapped to that row’s corresponding global degree of freedom, the diagonal entries of CCT are simply the number of elements connected to each node (or control point in the case of NURBS). It is therefore clear that the operator A = (CCT )−1 C simply calculates the uniformly weighted average of all entries of the local solution vector se belonging to any single node. At this point, we can write the resulting solution algorithm in a simple form: 1. Form element matrices Me and vectors pe . 2. Invert and solve one element at a time to obtain local solutions se = (Me )−1 pe . 3. Average local solutions at shared nodes to obtain global solution bˆ = Ase .

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The assembly and inversion of the global least-squares matrix M is thus entirely avoided in favor of the element-by-element inversion of each local matrix and the ˆ This subsequent averaging of the local solutions to obtain the global solution b. element-by-element solution (hereafter referred to as the averaged local solution) is clearly less computation- and memory-intensive than the full least-squares solution. The averaged local solution is not the same as the optimal least-squares solution (henceforth referred to as the “exact” solution, despite some degree of approximation being present). The error between the exact and averaged local solution is given as e = b − bˆ = A(Me )−1 q . (9) Rather than making a rigorous analysis of this error, we instead present several numerical examples that demonstrate its good properties.

3 Examples from Linear Elasticity Three two-dimensional linear elastic problems are selected to test the proposed method. The first is a point load acting vertically on an infinite elastic half-space (the Flamant problem), as described in [6]. The second is an infinite plate under tension weakened by a circular hole, as has been extensively used in the isogeometric analysis literature, e.g., see [5]. The last problem is the generalized version of the infinite plate under tension in which the hole can be elliptical and the tension applied at an angle relative to the axes of the ellipse. The analytical solution of the elliptical hole problem involves complex variable methods and can be found, for example, in [7]. All three problems involve infinite domains that must be truncated at some location for finite element analysis. The exact analytical solution is applied at these truncated boundaries, in our case as a Dirichlet boundary condition. In each case, the proposed method is used to apply the exact-solution boundary condition, and results are compared with the “exact” least-squares solution. In the case of the half-space, a comparison is also made with the results obtained by applying the exact-solution boundary condition as a Neumann boundary condition (with minimal Dirichlet boundary conditions applied to ensure uniqueness of the solution). The Neumann boundary condition solution can be used as a reference to determine the accuracy of both the least-squares fit and averaged local Dirichlet boundary condition solutions. For all cases, convergence rates of the isogeometric finite element solution at polynomial orders 2, 3, 4, and 5 are calculated and compared with the expected theoretical rates. The rate of convergence of the averaged local solution boundary condition values to the “exact” least-squares boundary condition values is also computed and compared to the rate of convergence of the isogeometric finite element solution. This is particularly important because, if the rate of convergence of the averaged local solution boundary data to the least-squares solution boundary data is greater than the rate of convergence of the finite element solution itself, then any penalty in accuracy incurred in substituting the averaged local method for the

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least-squares method will disappear as the mesh is refined. If this is the case, then there is fundamentally no drawback to using the averaged local solution.

3.1 Infinite Half-Space A point load aligned with the y-axis acts at the origin of a homogeneous elastic halfspace with its boundary aligned with the x-axis. The half-space is treated as a plane stress problem, with Young’s modulus E = 105 and Poisson ratio ν = 0.3. The half space is subjected to a load P = 105 . To model the solution the half space is truncated by a circular arc at a radius ro = 0.5. This particular choice leads to a truncatedboundary displacement field that is reasonably complicated. The analytical solution contains a stress singularity at the point of load application which could damage the convergence of the solution; the domain is therefore also truncated at an inner radius ri = 0.05 and the exact solution along this boundary is applied as a Neumann boundary condition, Fig. 1.

Fig. 1 Diagram of the half-space problem

Fig. 2 Initial single-element mesh. The thick line is the element boundary, thin lines mark equal distances in the parametric space of the element, and medium lines joined by circles indicate the control points of the mesh

The initial mesh is a single quadratic NURBS element (Fig. 2); this is sufficient to model the quarter-circle boundaries exactly. Higher levels of mesh refinement are obtained by k-refinement [3], i.e. by first elevating the polynomial order of the original single-element mesh, then inserting knots into the NURBS mesh to subdivide the mesh into multiple elements (Fig. 3). The elements are subdivided evenly along the circumferential direction and with a logarithmic scaling along the radial direction. This is found to produce a mesh that retained good element quality across the full range of element sizes, (see Fig. 4). The use of a single element in the original mesh ensures that the maximum possible degree of inter-element continuity, C(p−1) , is retained in each refined mesh.

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Fig. 3 Mesh at highest level of refinement: 32 Fig. 4 Close-up of most refined mesh near the elements in the circumferential direction and inner boundary. Logarithmic scaling of ele64 in the radial direction ment boundaries in the radial direction produced good element aspect ratios across the full range of element sizes

The RMS error of the finite element, σ f e , versus the exact stress, σex , is evaluated by Gauss quadrature from  E=

 Ω

σ f e − σex

2

dΩ

1/2 .

(10)

This is plotted versus the dimension of the maximum element diagonal for p = 2, 3, 4, 5, and for 6 levels of mesh subdivision (Fig. 5). The error of the averaged local boundary condition fit versus the least squares fit is calculated by simply summing the squares of the difference between the control variable values given by the two solutions and taking the square root of the resulting quantity, then plotting this error versus the same mesh size parameter (Fig. 6). Convergence rates agree with theoretical predictions and the Neumann and leastsquares Dirichlet boundary condition results are indistinguishable in Fig. 5. The rate of convergence of the averaged local fit to the least-squares solution is in fact equal to or greater than the rate of convergence of the finite element solution itself (Fig. 6). Thus the averaged local fit results, though initially deviating from the least-squares results at coarse levels of refinement, can be observed to merge with them at higher levels of refinement, Fig. 5, although the difference between the two solutions does become more pronounced as polynomial order is increased.

3.2 Infinite Plate with Circular Hole under Tension An infinite plate weakened by a circular hole with radius ri = 1 is subjected to tension along the x-axis. The plate is in a state of plane stress with elastic properties E = 105 and ν = 0.3. The stress at infinity is σx = 10. The x-axis and y-axis are fixed in the y- and x-direction, respectively. This problem is identical to that used

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Fig. 5 Convergence of the half-space problem. Solid lines are the least-squares fit, dashed lines are the averaged local fit, and dotted lines are the results using only Neumann boundary conditions. Neumann and least-squares Dirichlet results are not visibly distinct

Fig. 6 Boundary condition error for halfspace problem

in [5] except that we use an initial mesh composed of a single quadratic element truncated at a radius ro = 3 rather than two adjacent quadratic elements (Fig. 7). The exact displacements are applied as a Dirichlet condition at the outer boundary. The singularity at the sharp corner of the mesh used in [5] necessitates a special nonlinear knot-insertion algorithm for mesh refinement to avoid degradation in convergence rates; with our mesh, correct convergence is obtained using uniform knot spacings during mesh refinement, Fig. 8.

Fig. 7 The mesh used in [5]; note the singularity at the sharp corner

Fig. 8 The initial single quadratic element mesh with circular boundaries yields elements that retain good aspect ratios under mesh refinement

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Fig. 9 Convergence of the circular hole problem

291

Fig. 10 Boundary condition error for circular hole problem

Convergence rates agree with theoretical expectations (Fig. 9), and the error of the boundary condition control variable values from the averaged local solution relative to the least-squares solution decreases at a rate equal to or exceeding the finite element convergence rate (see Fig. 10). The averaged local solution reaches the machine precision limit for quintic polynomial order at the highest mesh refinement prior to degradation in the least-squares result; however, the averaged local solution performs well outside this limit.

3.3 Infinite Plate with Elliptical Hole under Tension As a final example, we generalize the plate with a circular hole problem to allow an elliptical hole of any aspect ratio and the application of tension at an arbitrary angle.

Fig. 11 Initial single-element Fig. 12 Mesh for highest level Fig. 13 Close-up of most reof mesh refinement: 64 ele- fined mesh near the inner mesh for elliptical hole ments in both the radial and boundary circumferential directions

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Fig. 14 Convergence of the elliptical hole Fig. 15 Boundary condition error for elliptical hole problem problem

Table 1 Convergence Rates of Solutions and Boundary Conditions

Order 2 3 4 5

Half-space Least Sq. 2.0513 3.0767 4.1263 5.1978

Average 2.0513 3.0823 4.2676 5.3520

BC Error 3.3048 3.4382 5.4074 5.0646

Circ. Hole Least Sq. 2.0276 3.0380 4.0469 5.0700

Average 2.0276 3.0402 4.0808 5.1876

BC Error 3.2936 3.4795 5.4438 4.8722

Ellip. Hole Least Sq. 2.0418 3.0855 4.1672 5.9128

Average 2.0419 3.0911 4.4055 6.5952

BC Error 3.4266 3.2681 5.2842 6.6217

As the aspect ratio of the ellipse is narrowed, the stress concentration approaches infinity; the mesh quality also declines as the aspect ratio is narrowed. Thus, a 2 : 1 ratio between major and minor axes of the ellipse is chosen to avoid problems with mesh quality and stress singularities. Parameters of the problem are identical with the previous example except for mesh dimensions. The initial mesh used is shown in Fig. 11. Uniform knot spacings are used to refine the mesh (Fig. 12); this is sufficient to ensure reasonable element aspect ratios, particularly in the critical area around the stress concentration (Fig. 13). The convergence rates agree with the theoretical expectations (Figs. 14, 15). For reference, the rates of convergence for the three problems given above are summarized in Table 1. Rates of convergence are computed as the slope of the line between the highest two levels of accurate refinement on a log-log plot.

4 Closure We have presented a method for the enforcement of Dirichlet boundary conditions for isogeometric finite element analysis that is more efficient than an L2 leastsquares fit and which converges to the exact solution with mesh refinement for three test problems. Although a proof of convergence is not presented, the numerical ex-

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amples shown suggest that the method performs nearly as well as a least-squares fit at significantly reduced computational cost. Given the demonstrated properties of the method, there appears to be no drawback to using it – at high mesh refinement, the results are essentially equivalent.

References 1. Bazilevs, Y., Michler, C., Calo, V.M., Hughes, T.J.R.: Weak Dirichlet boundary conditions for wall-bounded turbulent flows. Comput. Meth. Appl. M. 196, 4853–4862 (2007) 2. Bazilevs, Y., Hughes, T.J.R.: Weak imposition of Dirichlet boundary conditions in fluid mechanics. Comput. Fluids 36, 12–26 (2007) 3. Cottrell, J.A., Hughes, T.J., Bazilevs, Y.: Isogeometric analysis: toward integration of CAD and FEA. Wiley, Chichester (2009) 4. Hinton, E., Campbell, J.S.: Local and global smoothing of discontinuous finite element functions using a least squares method. Int. J. Numer. Meth. Eng. 8, 461–480 (1974) 5. Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Meth. Appl. M. 194, 4135–4195 (2005) 6. Sadd, M.H.: Elasticity: Theory, Applications, and Numerics. Elsevier/Academic Press (2009) 7. Sokolnikoff, I.S.: Mathematical Theory of Elasticity. McGraw-Hill, New York (1956) 8. Wriggers, P.: Nonlinear Finite Element Methods. Springer, Berlin (2008) 9. Zienkiewicz, O.C., Taylor, R.L., Zhu, J.Z.: The Finite Element Method: Its Basis and Fundamentals, 6th edn. Elsevier, Oxford (2005)

Chapter 33

Application of Isogeometric Analysis to Computational Contact Mechanics ˙Ilker Temizer When Tarek I. Zohdi suggested that it might be possible for me to visit Peter Wriggers’ institute after my Ph.D. studies, I welcomed such an opportunity with enthusiasm. Professor Zohdi inquired on my behalf and Professor Wriggers generously offered a position. It was a great pleasure to work with Professor Wriggers during the past four and a half years on a variety of exciting research topics. As I move on to the next stage of my career as an assistant professor at Bilkent University in Turkey, I look forward to establishing a productive cooperation between Hannover ˙ Temizer). and Ankara (I.

Abstract. An isogeometric analysis approach for computational contact mechanics problems is outlined. The key ingredient of the approach, in comparison to earlier surface smoothing techniques, is the use of NURBS surface discretizations that are directly inherited from NURBS volume discretizations. To treat the contact constraints, a knot-to-surface (KTS) algorithm is developed and discussed together with its extension towards a mortar-based formulation. Qualitative and quantitative studies deliver robust results for a variety of finite deformation contact problems.

1 Introduction Isogeometric analysis is a recent computational mechanics technique where the basis functions within the finite element analysis are inherited from the exact geometric description that is delivered by the computer aided design process. Numerous recent applications of isogeometric analysis to problems ranging from solid mechanics to fluid problems as well as fluid-structure interaction analysis have demonstrated the advantages of this novel technology [7, 1, 11, 4, 3, 14, 2]. The problem of non-smooth contact discretizations has attracted significant attention in the computational mechanics community. In order to improve the performance of contact algorithms, various surface smoothing algorithms have been ˙Ilker Temizer Institute of Continuum Mechanics, Leibniz Universit¨at Hannover, Appelstr. 11, D-30167 Hannover, Germany e-mail: [email protected]

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proposed for the master surface [5, 15, 12] as well as exact geometric descriptions for rigid obstacles [9, 8]. However, these approaches do not preserve a consistency between volume and surface discretizations. With the isogeometric analysis framework, smooth surface discretizations can be achieved by describing a surface geometry that is inherited automatically from the isogeometric description of the volume. In the following, aspects of such a description will be outlined in the context of NURBS-based isogeometric analysis.

2 Contact Boundary Value Problem In this work, finite deformation quasi-static contact problems will be considered in a purely mechanical setting. Denoting the reference and current configurations of a body B via R and R, related to each other by the motion x = χ (X), the strong form of the linear momentum balance is Div[P] = 0 in R

(1)

subject to appropriate boundary conditions. For the modeling of contact between two bodies B(1) and B (2) , the contact interface R c := R (1),c = R (2),c is pulled back (1),c (2),c to Roc := Ro = Ro . All integrals are subsequently evaluated on Roc . Friction effects are beyond the present scope of the investigations and hence are left out. Consequently, the interface Piola traction is p := p(1) = pN n(2) where n(2) is the outward unit normal to B (2) . Defining gN = −(x(1) − x(2) ) · n(2) to be the normal gap, the contact contribution to the weak form can be expressed as 

(δ x

(1)

− δ x ) · p dA = −



(2)

δ Roc

δ gN pN dA .

δ Roc

(2)

Limiting the studies to unilateral contact without adhesion, Karush-Kuhn-Tucker conditions for impenetrability on δ R c are gN ≥ 0

,

p N ≥ 0 , g N pN = 0 .

(3)

See [18] and [13] for further details. Within the numerical contact treatment, B(1) will be identified as a slave body whereas B(2) is the master. Within this convention, the master surface is parametrized via convective coordinates ξ α , α ∈ {1, 2}.

3 Isogeometric Treatment with NURBS Presently, the contact geometry will be a NURBS surface that is directly inherited from the volume NURBS discretization which is introduced next using standard NURBS terminology. The reader is referred to [16] and [4] for further details and extensive references.

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Let Ξ i be the open non-uniform knot vector associated with the i-th dimension of a patch:

Ξ i = { ξ0i , . . . , ξ pi i , ξ pi i +1 , . . . , ξni i , ξni i +1 , . . . , ξmi i } .       pi + 1 equal terms

(4)

pi + 1 equal terms

Here, mi = ni + pi + 1, pi is the polynomial order of the accompanying B-spline basis functions, ξ ji is the j-th knot and ni + 1 would be the number of accompanying control points in a one-dimensional setting. In a three-dimensional setting, a volume is parametrized by V(ξ 1 , ξ 2 , ξ 3 ) =

n1

n2

n3

∑ ∑ ∑

Rd1 d2 d3 (ξ 1 , ξ 2 , ξ 3 )Pd1 d2 d3

(5)

d1 =0 d2 =0 d3 =0

where Pd1 d2 d3 are the control points and Rd1 d2 d3 ≥ 0 are the rational B-spline (NURBS) basis functions. The latter are defined via a tensor product in a fourdimensional space based on homogeneous coordinates [16]. The projected form in the three-dimensional space is Rd1 d2 d3 (ξ 1 , ξ 2 , ξ 3 ) =

wd1 d2 d3 B1 (ξ 1 ) B2d2 (ξ 2 ) B3d3 (ξ 3 ) W (ξ 1 , ξ 2 , ξ 3 ) d1

(6)

with Bidi as a nonrational B-spline basis function. The normalizing weight W is given in terms of the weights wd1 d2 d3 > 0 and Bidi via W (ξ 1 , ξ 2 , ξ 3 ) =

n1

n2

n3

∑ ∑ ∑ wd1 d2 d3 B1d1 (ξ 1) B2d2 (ξ 2 ) B3d3 (ξ 3)

.

(7)

d1 =0 d2 =0 d3 =0

The knot vectors together with the associated control points and the accompanying weights constitute a patch. The continuity and order of Bidi depends on Ξ i only. If Ξ i has no repeated interior knot ξ ji , j ∈ [pi + 1, ni ], then the order-pi basis function Bidi has continuity C pi −1 . Every repetition of a knot decreases the continuity by one order at this knot. The order of NURBS parametrization will be denoted by N p in subsequent sections, while the order of Lagrange polynomials employed will be denoted by L p . The approximation spaces based on N 1 and L 1 are identical. In a finite element (FEM) setting, all degrees of freedom are discretized via the same NURBS basis functions used for the geometric description. The unique knot spans are conveniently chosen as the integration domains (elements). The counterparts of the h- and p-refinement procedures for FEM discretizations based on Lagrange polynomials are the knot insertion and order elevation procedures in the NURBS setting. While p-refinement preserves the number of nodes, order elevation leads to an increase in the number of control points. When the two must be conducted together, the k-refinement procedure will be employed where order elevation precedes knot refinement [4]. This has the advantage that a higher degree of

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smoothness can be achieved within the patch across non-repeated knot entries and the final number of control points is less compared to the alternative case. For the numerical evaluation of the weak forms emanating from Lagrange or NURBS based discretizations, 2p Gauss-Legendre quadrature points will be employed within each element for order-p approximations unless otherwise noted. This ensures a converged quadrature. See [10] for a recent discussion of efficient quadrature schemes appropriate for isogeometric analysis. NURBS surface parametrization is inherited from the volume parametrization in a straightforward fashion. For example, let ξ−1 := ξ01 . By construction [16] V(ξ−1 , ξ 2 , ξ 3 ) =

n2

n3

∑ ∑ R−d2d3 (ξ 2 , ξ 3)P−d2 d3

(8)

d2 =0 d3 =0

where P− d2 d3 := P0d2 d3 and, including the weighting factor, 2 3 R− d2 d3 ( ξ , ξ )

:=

w0d2 d3 B2d2 (ξ 2 ) B3d3 (ξ 3 )

∑d22 =0 ∑d33 =0 w0d2 d3 B2d2 (ξ 2 ) B3d3 (ξ 3 ) n

n

.

(9)

Hence, only the knowledge of the knot vectors Ξ 2 and Ξ 3 and a reduced set of control points together with the accompanying weights are sufficient to characterize the surface associated with ξ−1 . The same principle applies for ξ+1 := ξm11 and all other dimensions. Hence, in general, a surface patch (in particular a contact patch) is directly inherited from the volume patch and has the same parametrization as before but only with two dimensions α ∈ {1, 2} that correspond to any two of the three dimensions. The corresponding knot vectors are Ξ α with associated B-spline basis functions Bαdα and parametric space coordinates ξ α that are conveniently chosen as the convective coordinates for contact computations. The surface parametrization is therefore S(ξ 1 , ξ 2 ) =

n1

n2

∑ ∑ Rd1 d2 (ξ 1, ξ 2 )Pd1 d2

.

(10)

d1 =0 d2 =0

4 Knot-to-Surface Contact Algorithm The classical node-to-surface (NTS) algorithm of computational contact mechanics cannot be directly employed with NURBS because the control points are not interpolatory. The straightforward extension of NTS to the isogeometric setting corresponds to a knot-to-surface (KTS) algorithm. Herein, one enforces gN = 0 during contact directly at the contact element quadrature points. Each quadrature point corresponds to a unique value (ξ 1 , ξ 2 ) ∈ [0, 1] × [0, 1]. Following [6], the same procedure will be employed for Lagrange polynomial interpolations as well.

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In the following, some aspects of the KTS algorithm are investigated qualitatively. All the examples presented in this section employ the penalty method to regularize the contact constraints. Consequently, a penalty parameter εN is introduced such that pN = εN gN . (11) The penalty parameter is chosen sufficiently high to ensure that the contact constraints are accurately satisfied. The constitutive response will be limited to isotropic elasticity based on an Ogden material.

4.1 Contact of a Grosch Wheel In this section, the elastic contact of a Grosch wheel with a flat rigid surface is considered. During loading, the inner rim of the wheel is displaced rigidly towards the surface. The lowest order NURBS description for the wheel geometry is constructed by employing N 2 in the angular direction and N 1 in the radial and thickness directions. It is sufficient to employ a single patch with four elements in the angular direction and a single element in all other directions to exactly represent the geometry. The ends of the patch are connected in the angular direction. All directions are elevated to the same order. Here, the number of Gauss-Legendre quadrature points employed per direction within the contact elements is 12 for all examples, which provides a good resolution of the contact interface at coarse discretizations. A two-dimensional example is provided in Fig. 1. Here, the initial NURBS geometry description is shown together with intermediate and final compression stages. The KTS algorithm naturally enforces the contact constraints within a weak formulation. Clearly, there would not be any straightforward way of controlling the contact geometry directly via the control points. They lie significantly below the contact surface at all stages of contact.

(a) initial

(b) intermediate

(c) final

Fig. 1 Two-dimensional contact of a Grosch wheel with a rigid surface: N 6 case. The mesh consists of only four NURBS elements. The scale displays the magnitude of P

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(a) L 1 /N

1

(b) L 2

(c) N

2

Fig. 2 Two-dimensional contact of two deformable bodies

4.2 Contact of Two Deformable Bodies Figure 2 shows various discretizations for a deformable body compressed against another one in a purely mechanical setting at a coarse discretization. The upper (slave) body is ten times stiffer with respect to compression and shear compared to the lower (master) body. The L 1 /N 1 -discretization displays the typical problem in contact mechanics, namely that at coarse discretizations the interface resolution is unsatisfactory. The L 2 -discretization provides a much better resolution with the same number of degrees of freedom. However, the discontinuity in smoothness on the master surface is clearly visible at the contact interface. Such discontinuities lead to convergence difficulties in classical contact algorithms and various smoothing algorithms have been designed to alleviate such difficulties. The N 2 -discretization with the same number of elements does not involve any repeated interior knots. Consequently, a qualitatively well-resolved C 1 -continuous contact interface is visible. This is a potential advantage that would be beneficial particularly in frictional contact situations with large sliding as partially demonstrated next. The same example in a three-dimensional setting is provided in Fig. 3 employing a N 2 -discretization. Here, the slave body is compressed onto the master body (5 loading steps) and subsequently it is rotated through 45◦ (10 loading steps) and dragged along the diagonal of the master surface (30 loading steps). During the twist and drag stages, the slave surface traverses multiple element boundaries. Again, both the master and slave surfaces are C 1 -continuous. In all applications of the KTS algorithm, the active-set search is embedded within the Newton-Raphson iterations for the nonlinear system. The load steps are chosen such that convergence is achieved in at most 10 Newton-Raphson iterations. Within this setup and at the given resolution, one cannot achieve convergence with L 1 /N 1 - or L 2 -discretizations.

5 Conclusion In this work, contact constraints were treated via NURBS-based isogeometric analysis and compared with standard C 0 -continuous Lagrange finite elements. The basis for the discussion was built on an extension of the classical node-to-surface (NTS)

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(a) compression

(b) rotation

301

(c) drag

Fig. 3 Three-dimensional contact of two deformable bodies based on an N 2 -discretization

1.2

1.2

exact

1

N

exact

1

2

0.6

0.6

N

3

pX

0.8

pX

0.8

0.4

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0.2

0.2

0

0

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-0.2 0

0.2

0.4

0.6

X

0.8

1

1.2

0

0.2

0.4

0.6

0.8

1

1.2

X

Fig. 4 Accurate mortar-based isogeometric contact results in normalized coordinates for the Hertz problem on a coarse discretization. The contact interface is resolved through rrefinement

algorithm of contact mechanics for Lagrange discretizations to a knot-to-surface (KTS) algorithm that is also suitable for NURBS discretizations. In comparison with earlier surface smoothing techniques in the literature, the NURBS contact patches of the developed KTS algorithm are inherited directly from the NURBS volume parametrization in a straightforward fashion. Consequently, it is possible to achieve arbitrary smoothness (in particular C 1 -continuity) across contact element interfaces while preserving a consistency between surface and volume discretizations. The KTS algorithm delivered qualitatively satisfactory results for various twoand three-dimensional finite deformation contact problems. However, quantitative investigations on the classical Hertz contact problem highlight a need for the relaxation of the mechanical contact constraints. For this purpose, we are currently pursuing mortar KTS approaches [17] that deliver superior results in comparison with the standard KTS algorithm for NURBS discretizations as well as for its application to Lagrange discretizations (Fig. 4). An additional important next step is

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˙I. Temizer

the treatment of frictional contact between two finitely deformable bodies and the development of appropriate mortar-based constraint relaxation techniques. Acknowledgements. This contribution is a summary of a recent joint work [17] with Peter Wriggers and Thomas J.R. Hughes. The author gratefully acknowledges the stimulating discussions he had with the coauthors during the preparation stages of this work.

References 1. Bazilevs, Y., Calo, V.M., Hughes, T.J.R., Zhang, Y.: Isogeometric fluid-structure interaction: theory, algorithms and computations. Comput. Mech. 43, 3–37 (2008) 2. Bazilves, Y., Michler, C., Calo, V., Hughes, T.J.R.: Isogeometric variational multiscale modeling of wall-bounded turbulent flows with weakly enforced boundary conditions on unstretched meshes. Comput. Meth. Appl. M. 199, 780–790 (2010) 3. Cohen, E., Martin, T., Kirby, R., Lyche, T., Riesenfeld, R.: Analysis-aware modeling: understanding quality considerations in modeling for isogeometric analysis. Comput. Meth. Appl. M. 199, 334–356 (2010) 4. Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric Analysis. Wiley, Chichester (2009) 5. Eterovic, A.L., Bathe, K.-J.: An interface interpolation scheme for quadratic convergence in the finite element analysis of contact problems. In: Wriggers, P., Wagner, W. (eds.) Computational Methods in Nonlinear Mechanics. Springer, Berlin (1991) 6. Fischer, K.A., Wriggers, P.: Mortar based frictional contact formulation for higher order interpolations using the moving friction cone. Comput. Meth. Appl. M. 195, 5020–5036 (2006) 7. Gomez, H., Calo, V.M., Bazilevs, Y., Hughes, T.J.R.: Isogeometric analysis of the ChanHilliard phase-field model. Comput. Meth. Appl. M. 197, 4333–4352 (2008) 8. Hansson, E., Klarbring, A.: Rigid contact modelled by CAD surface. Eng. Computation. 7, 344–348 (1990) 9. Heege, A., Alart, P.: A frictional contact element for strongly curved contact problems. Int. J. Numer. Meth. Eng. 39, 165–184 (1996) 10. Hughes, T.J.R., Reali, A., Sangalli, G.: Efficient quadrature for NURBS-based isogeometric analysis. Comput. Meth. Appl. M. 199, 301–313 (2010) 11. Kiendl, J., Bletzinger, K.U., Linhard, J., W¨uchner, R.: Isogeometric shell analysis with Kirchhoff-Love elements. Comput. Meth. Appl. M. 198, 3902–3914 (2009) 12. Krstulovic-Opara, L., Wriggers, P., Korelc, J.: A C 1 –continuous formulation for 3D finite deformation frictional contact. Comput. Mech. 29, 27–42 (2002) 13. Laursen, T.A.: Computational Contact and Impact Mechanics. Springer, Berlin (2006) 14. Lipton, S., Evans, J.A., Bazilevs, Y., Elguedj, T., Hughes, T.J.R.: Robustness of isogeometric structural discretizations under severe mesh distortion. Comput. Meth. Appl. M. 199, 357–373 (2010) 15. Padmanabhan, V., Laursen, T.A.: A framework for development of surface smoothing procedures in large deformation frictional contact analysis. Finite Elem. Anal. Des. 37, 173–198 (2001) 16. Piegl, L., Tiller, W.: The NURBS Book, 2nd edn. Springer, Berlin (1996) 17. Temizer, ˙I., Wriggers, P., Hughes, T.J.R.: Contact treatment in isogeometric analysis with NURBS. Accepted for publication in Comput. Meth. Appl. M. (2010) 18. Wriggers, P.: Computational contact mechanics, 2nd edn. Springer, Berlin (2006)

Chapter 34

Stochastic Galerkin Method for the Elastoplasticity Problem with Uncertain Parameters Bojana V. Rosic and Hermann G. Matthies Dedicated to Peter Wriggers on the occasion of his 60th birthday (H.G. Matthies).

Abstract. The mathematical formulation and numerical simulation of an elasticplastic material with uncertain parameters in the small strain case is considered. Traditional computational approaches to this problem usually use some form of perturbation or Monte Carlo technique. This is contrasted here with more recent methods based on stochastic Galerkin approximations. In addition, we introduce the characterisation of the variational structure behind the discrete equations defining the closest-point projection approximation in stochastic elastoplasticity.

1 Introduction The uncertainties in inelastic systems arise from a variety of sources including the geometry of the problem, material properties, boundary conditions, initial conditions, or excitations imposed on the system. As a result, depending on the source of randomness, the behaviour of a system will have an uncertain character. In the deterministic sense the parameters describing elastic (reversible)/inelastic (irreversible) behaviour are determined by indentation techniques and then considered as constants. However, in a case of materials such as soil and bone the classical approach [5] will not properly describe the output due to large possible variations, which are uncertain. In order to give a more reliable description we introduce input parameters as random fields and processes with prescribed statistics. In other words, we model the governing equations as stochastic partial differential equations (SPDE) with random parameters. Bojana V. Rosic · Hermann G. Matthies TU Braunschweig, Institute of Scientific Computing, Hans-Sommer-Strasse 65, D-38106 Braunschweig, Germany e-mail: [email protected],[email protected]

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The most basic technique for solving SPDEs is the Monte Carlo statistical sampling approach, whose accuracy strongly depends on the number of realisations that one chooses and on the variance of the quantity being estimated [2]. However, this method is quite expensive. The perturbation methods expand random fields via Taylor series around their mean restricting the number of terms on second order, since the higher order approximation increases complexity of the system [1]. The moment equations method [6] tries to resolve this situation by direct computation of random solution moments. In this paper we will focus on the stochastic Galerkin method, which discretizes the stochastic part of the differential equation in such a way that one obtains a large single structured system. The paper is organised as follows: Sect. 2 gives the mathematical description of stochastic infinitesimal elastoplasticity focusing on the dual formulation of the initial-boundary value problem which governs the stochastic elastoplastic behaviour by extension of the deterministic theory as presented in [5]. Sect. 3 outlines the methods used to solve the obtained stochastic partial differential equations and the stochastic radial return map. Finally, we end with a discussion of the applicability of methods on some test examples.

2 Mathematical Formulation Inelastic theory describes the rate-independent inelastic behaviour of solids, meaning that the deformation of the material does not depend on the rate at which loads are applied. In the quasi-static state the inelastic output is determined by iterative solution of the convex minimisation problem, as will be further explained.

2.1 Problem Setting Let (Ω , B, P) be a probability space with Ω the set of elementary events ω , P the probability measure and B an σ -algebra. Let us introduce the random parameters K, G, σy and f as a random bulk, random shear modulus, yield stress and random force. The uncertainty in those parameters arises in a very natural way. In addition, let us consider the domain G with a piecewise smooth Lipshitz continuous boundary Γ on which are imposed Dirichlet (ΓD ) and Neumann (ΓN ) boundary conditions, such that ΓD ∩ ΓN = ∅ and Γ = Γ¯N ∪ Γ¯D . The time interval of interest is denoted with T ⊂ R+ . Let the space of displacements be represented by tensor space U := U ⊗ S = o

H 1 (G ) ⊗ L2 (Ω ), obtained as a tensorial product of the deterministic space U and the stochastic space S . The space of the right hand side is in duality with U and given as F := U∗ . In addition, let us introduce the generalised stress Σ = (σ , β ) ∈ Q, Q = B × B, B = L2 (G ) ⊗ L2 (Ω ), where σ represents the stress and β the internal force. Similarly, the generalised random plastic strain E p := (ε p , α ) is defined by a random plastic deformation ε p and random internal hardening variables α . Due to the assumption of small deformations a generalised random

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deformation E := (Du, 0) is additively decomposed to elastic Ee and plastic part E p almost surely for each ω ∈ Ω . The differential operator is defined in a weak sense such that for a single tensor product u1 (x)u2 (ω ) ∈ U ⊗ S we have D : u1 (x)u2 (ω ) → (Du1 (x))u2 (ω ). The relations between kinematic and dynamic variables are given by constitutive laws: Hooke’s law ε , Cε e  = ε , σ  and hardening law α , Hα  = α , β , where fourth-order tensor C(x, ω ) represents the constitutive tensor, H the hardening law and ·, · the stochastic duality operator ε , σ  := E ( G ε (x, ω )σ (x, ω ) dx) .

2.2 Variational Formulation In order to introduce a variational framework regarding the mathematical as well as numerical analysis of the problem, we start from the flow law in the form of [5]: E˙ p , T − Σ  ≤ 0,

∀T ∈ K

(1)

where E˙ p := (ε˙ p , α˙ ) represents the rate of change of generalised plastic strain. The unknown variables are considered to be the displacement u ∈ U and a generalised stress Σ ∈ Q. Due to the definition of the stress space we are able to introduce its convex subset K = {Σˆ ∈ Q : Σˆ ∈ K a.e. in G ⊗ Ω }, where K represents a closed, convex and nonempty set of admissible stresses defined with the help of the yield function φ , i.e. K = {Σ | φ (ω , Σ ) ≤ 0 a.e. ∀ω ∈ Ω }. In order to define the variational problem (also known as “dual”formulation [5]), one has to introduce a ˆ and symmetric, continuous bilinear forms b (Σ , Σˆ ) := σ , C−1 σˆ  + b, H−1 b a (v, Σ ) := ε (v), τ , (Σ = {τ , β }), as well as linear functional (t, v) := f, v. Problem. For a given with appropriate initial conditions find (u, Σ ) : T → U ⊗ K such that for almost ∀t ∈ T , ∀ω ∈ Ω the following equation and inequality are satisfied: a (v, Σ (t)) = (t, v),

∀v ∈ U

  b Σ˙ , Σˆ − Σ (t) + a(u(t), ˙ Σˆ − Σ (t)) ≥ 0,

(2)

, ∀Σˆ ∈ K

.

(3)

3 Numerical Analysis of the Problem The variational inequality Eq. (3) may be equivalently formulated as a convex minimisation problem [5], where one has to minimise the functional in one time step with respect to the displacement u and the stress field Σ . After discretisation in the space, the problem can be solved on the computer via a return mapping algorithm [9].

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3.1 Discretisation of Input The input random fields κ (x, ω ) (bulk and shear modulus, yield stress) are assumed to be exponential piecewise transformations of a Gaussian random field, i.e. the lognormal random fields. Their discretisation is done by a combination of truncated Karhunen-Lo`eve and polynomial chaos expansion (KLE/PCE) such that (γ ) one has κ (x, θ ) = ∑M l=0 ∑γ ∈J κl Hγ (θ )κl (x), where J = {γ | ∀ j > M : γ j = ∞ 0, |γ | := ∑ j=1 γ j ≤ p} represents the multi-index set, κl (x) the KL eigenfunctions, (γ )

κl the coefficients of polynomial chaos expansion of KL- random variables and Hγ (θ ) Hermite polynomials in Gaussian random variables θ [8, 7].

3.2 Stochastic Galerkin Method The generalised stress Σ (σ , β ) = Σ (C−1 (Du− ε p ), Hα ) = Σ (u, E p ) depends on the displacement u and generalised plastic strain E p = (ε p , α ). Following this, we may reformulate the bilinear term a in Eq. (2) to a form: a (v, Σ (u, E p )) = A(E p )[u]v = (t, v),

∀v ∈ U

(4)

which defines a hemicontinous operator A. In order to solve the previous equation we employ the time n (Euler backwards) and finite element spatial discretisation h [9, 5] such that one looks for the solution (uhn , Σnh ) which satisfies: a(vh , Σnh ) = (tn , vh ),

∀vh ∈ U h ⊗ S .

(5)

However, previous equation is semi-discretised and it cannot be solved without a stochastic discretisation [7]. The stochastic ansatz is chosen as a space of multivariate Hermite polynomials in mutually independent Gaussian random variables S I := span{Hγ | γ ∈ JM,p } ⊂ S . Inserting the stochastic ansatz to Eq. (5) and projecting the obtained residual in a standard Galerkin manner we obtain a system of equations:   (6) r(u) = [..., E Hβ (f(θ ) − A(θ , E p (θ ))[u(θ )]) , ...] = 0, which may be further solved by an iterative technique (Newton-Raphson, BFGS, etc.). 3.2.1

Stochastic Radial Return Map

Let us rewrite the functional Eq. (3) in a little bit different form. Starting from the definition of the stress rate σ˙ = Cε˙ e = C(ε˙ − ε˙ p ) we may formulate σ˙ = σ˙ e + σ˙ p , where σ˙ e = Cε˙ . Now taking that E(u) = (Du, 0) and Σ e = (Cε , 0) = GE (where G = diag(C, H)) one obtains:   b Σ˙ e − Σ˙ , Σˆ − Σ (t, ω ) ≤ 0, ∀Σˆ ∈ K . (7)

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The time discretisation of the previous form by implicit backward Euler’s scheme (the current step is n) lead us to a form:   b Σntr − Σn , Σˆ − Σn ≤ 0, ∀Σˆ ∈ K . (8) Here the term Σntr represents the trial stress obtained by a predictor step and  En = E(un ) − E(un−1 ) the increment of the generalised deformation. Let us now introduce the norm Σ b = b(Σ , Σ )1/2 such that the problem given by Eq. (8) collapses to the standard optimisation (i.e. minimisation) problem [5]: & ' 1 tr 2 Σn = argmin Σ − T b , ∀ω ∈ Ω (9) 2 n T ∈K The iterative algorithms for solving Eq. (6) and Eq. (8) consist of a predictor and corrector step in each iteration, also known as a stochastic radial return map [5, 8, 9]. In further text we will denote by n the current step and by k the current iteration. Predictor step. The predictor step calculates the polynomial chaos expansion of displacement ukn by solving the equilibrium equation Eq. (6). The displacement is  then used for the calculation of the strain increment Enk and the trial stress Σnk,tr = k + GE k assuming the step to be purely elastic. If the stress Σ k,tr lies outside of Σn−1 n n the admissible region K we proceed with the corrector step. Otherwise, Σnk = Σnk,tr represents the solution and we may move to the next step. Corrector step. The purpose of the corrector step is to project the stress outside the admissible region back onto a point in K . The problem of closest point projection becomes complicated since we deal with uncertain parameters, i.e. polynomial chaos variables (PCV), which require the introduction of new algebra called PC algebra [8, 4]. Namely, we try to find the solution Σn of the problem Eq. (9), which exists if there is a scalar λ such that Σ = Σ tr − λ Gφ (Σ ) where the Kuhn-Tucker conditions are satisfied, i.e. φ (Σ ) ≤ 0, λ ≥ 0, λ φ (Σ ) = 0, ∀ω ∈ Ω . This system further may be solved by the Newton method in general case .

4 Numerical Results Let us consider a 2D plate with a circular hole under a specified loading. The problem is defined by random shear modulus (mean 28000, deviation 10%) and yield stress (mean 243, deviation 10%) taken as the lognormal random fields. The reference solution is calculated as the result of Monte Carlo simulation with 100000 samples. Namely, we compared the probability distribution functions of stress σyy in the plasticity area (Fig. 1a), as well as the convergence of the mean value residuum for both approaches (Fig. 1b). From those figures we may conclude that the stochastic Galerkin method may be used for the simulation of stochastic elastoplastic problem

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a)

b)

Fig. 1 a) The comparison of probability density functions of the stress variable σyy in the plasticity area and b) the convergence of the stochastic Galerkin method for 20 and 10 terms in KL expansion applying the modified Newton Raphson method with line search compared to the Monte Carlo method. Monte Carlo results are calculated for 100000 samples

a)

b)

Fig. 2 a) One of realisations of the stress σxy (half of domain); b) values of the relative error of the mean displacement uyy given in percents and calculated for the reference solution obtained with Monte Carlo simulation

with the same accuracy as the Monte Carlo approach. In addition, we calculated the relative error of the y-displacement, which proves this statement (Fig. 2b). In contrast to the deterministic solution, the separate realisations of the stress and strain variables are not symmetric any more since the properties of the material are random. Due to this reason we have that the stress component σxy has nonsymmetric characteristic (Fig. 2a). In addition, the output solution depends on input statistics, as well as the covariance length. The covariance function here is taken in exponential form. Namely, we may conclude from Fig. 3 that the choice of the covariance length influences the stress output as it is expected. In the case of smaller length we have larger oscillations in the output and vice versa.

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Fig. 3 Influence of correlation lengths a on stress σyy

Fig. 4 10000 realisations of the yield stress σy in 2 different points of the domain (von Mises yield criteria)

When we deal with random initial yield stress, we have that its value changes from point to point of the considered domain inducing the change of the yield criteria as it shown on (Fig. 4), where we have plotted 10000 realisations of mentioned function. The dimensionality of the random space in the stochastic Galerkin method may be as large as possible, depending on the number of independent random variables involved in the parameterisation of the random shear modulus and yield stress. Hence, the computational cost may grow out of control. This problem is known as the “curse-of-dimensionality”. However, this problem is often reduced by application of adaptive algorithms and fast computers. On other side, the Monte Carlo simulation requires thousands of realisations in order to achieve the converged statistics, which may be very expensive in practice considering the dimensionality of the real elastoplastic problems.

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5 Conclusion The idea of random variables as functions in an infinite dimensional space approximated by elements of finite dimensional spaces has brought a new view to the field of stochastic elastoplasticity. In this paper, we have proposed an extension of the stochastic finite element method and related numerical procedures to the resolution of inelastic stochastic problems in the context of Galerkin methods. In some way this strategy may be understood in a sense of model reduction technique due to the applied Karhunen-Lo`eve and polynomial chaos expansion. A Galerkin projection minimises the error of the truncated expansion such that the resulting set of coupled equations gives the expansion coefficients. If the smoothness conditions are met, the polynomial chaos expansion converges exponentially with the order of polynomials. In contrast to the Monte Carlo, the Galerkin approach, when properly implemented, can achieve fast convergence and high accuracy and can be highly efficient in particular practical computations. Acknowledgements. The authors would like to acknowledge the financial support of Technical University Braunschweig and DAAD.

References 1. Anders, M., Hori, M.: Stochastic finite element methods for elasto-plastic body. Int. J. Numer. Meth. Eng. 46, 1897–1916 (1999) 2. Ballio, F., Guadagnini, A.: Convergence assessment of numerical Monte Carlo simulations in groundwater hydrology. Water Resour. Res. 40 (2004) doi:10.1029/2003 WR002876 3. Calfisch, R.E.: Monte Carlo and Quasi-Monte-Carlo methods. Acta Numer. 7, 1–49 (1998) 4. Debusschere, B.J., Najm, H.N., Pebay, P.P., Knio, O.M., Ghanem, R.G., Le Maˆıtre, O.P.: Numerical challenges in the use of polynomial chaos representations for stochastic processes. J. Sci. Computing 26, 698–719 (2005) 5. Han, W., Daya Reddy, B.: Plasticity: mathematical theory and numerical analysis. Springer, New York (1999) 6. Jeremi´c, B., Sett, K., Levent Kavvas, M.: Probabilistic elastoplasticity: formulation of evolution equation of probability density function. J. Eng. Mech. ASCE (2005) 7. Matthies, H.G., Keese, A.: Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comp. Meth. Appl. M. 194, 1295–1331 (2005) 8. Rosi´c, B., Matthies, H.G.: Computational approaches to inelastic media with uncertain parameters. Journal of the Serbian Society for Computational Mechanics 2, 28–43 (2008) 9. Simo, J.C., Hughes, T.J.R.: Computational inelasticity. Springer, New York (1998)

Chapter 35

A Time-Discontinuous Galerkin Approach for the Numerical Solution of the Fokker-Planck Equation Udo Nackenhorst and Friederike Loerke Professor Udo Nackenhorst is head of the Institute of Mechanics and Computational Mechanics at Leibniz Universit¨at Hannover since 2010. He joined the institute in 2000 as Associate Professor which has been headed by Professor Peter Wriggers at that time. There has been a quite fruitful cooperation with Peter Wriggers from the first day. We jointly established basic studies in Computational Engineering at Hannover, had very successful research cooperations, for example within a research unit on rubber friction and organized several international symposia at Hannover. The co-author, Friederike Loerke, had the opportunity enjoying Peter Wriggers as academic teacher, especially in finite element classes. To the occasion of his 60th birthday we wish the very best to him, especially health and creativeness. Looking forward for further decades of good collaboration (U. Nackenhorst).

Abstract. Structural dynamical systems under random white noise excitation can be described by time-dependent stochastic differential equations. Under white noise assumption the response process possesses Markov characteristics and the transient stochastic differential equations can be transformed into evolution equations for the probability density, the so called Fokker-Planck equations. The Fokker-Planck equation has its origin in the description of the motion of tiny particles in a fluid. Its mathematical structure is comparable with coupled advection and diffusion problems which represent a broad class of problems in engineering and natural sciences. However, the numerical treatment of advection-diffusion type partial differential equations remains a challenging task due to the non self-adjoint advection operator. Semi-discretization techniques are subject to strong limits for the relation of spatial and temporal discretization (often expressed by the Courant number) for prevention of artificially oscillatory or dispersive solutions. In this contribution a finite Udo Nackenhorst · Friederike Loerke Institute of Mechanics and Computational Mechanics, Leibniz Universit¨at Hannover, Appelstr. 9a, D-30167 Hannover, Germany e-mail: [email protected], [email protected]

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element approach in space and time, where the time space is discretized discontinuously is presented as superior method for the numerical treatment of hyperbolic advection-diffusion type partial differential equations. This method is referred to as Time-Discontinuous Galerkin method. It will be illustrated by the response of dynamical systems under uncertain excitation, which are described by the FokkerPlanck equation.

1 Introduction Structures in civil engineering are subjected to uncertain dynamic loads, e.g. wind and wave loading on offshore buildings or earth-quake loading on bridges and towers. In traditional engineering uncertainties in mechanical excitation are treated separately from the structural analysis, for buildings it is referred to EUROCODE EN 1990. Detailed computations on the deterministic response are performed, while uncertainties are tackled using heuristic security concepts. Furthermore, boundary conditions and system parameters should be regarded as random properties and their stochastic description should be incorporated in the system describing differential equations. The numerical treatment of stochastic differential equations (SDEs) occuring in continuum mechanical problems has recently gained increasing attention. One of the first state of the art reviews was published in [13]. Direct approaches to the solution of SDEs have been developed by [3], [4] and [16]. Those are based on a series expansion of the uncertain system parameters, usually combined with a polynomial chaos approximation of the structural response. A state of the art review on this type of approach is provided by [15]. Within the finite element concept on the computation of approximate solutions for partial differential equations this leads to high-dimensional sample spaces and enormous computational effort for engineering models. In addition, considerable effort is needed for the computation of statistic moments to be performed in a postprocessing procedure. Currently, model reduction techniques are under development to overcome these problems (see e.g. [9]). Another direct approach is the reformulation of the original SDEs, which under certain assumptions concerning the stochastic process leads to an evolution equation for the probability density of the motion of dynamical systems. Because of the analogy to the classical problem of Brownian motion of tiny particles in a fluid, these equations are referred to as Fokker-Planck equations (FPEs) [10]. They possess the mathematical structure of hyperbolic differential equations and are therefore analogous to advection-diffusion problems. Discussions on the numerical treatment of this type of differential equations have already lasted for a long time. In an early study [14], a Galerkin type approach, based on polynomial expansions, was used to solve high-dimensional stationary FPEs. This approach was further developed by [17] and others. The application of methods from computational fluid dynamics was promoted by [18], for example, who proposed a finite element approach and suggested domain decomposition for higher dimensions. Finite element and finite

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difference schemes were further applied to the FPE for various applications by [6], [11] and [7]. This contribution targets on stable and numerically efficient solution methods for FPEs. In comparison with more traditional semi-discretization techniques, using finite element approximation in space and finite difference discretization in time, the advantage of Time-Discontinuous-Galerkin methods will be shown.

2 FPE Expression of Stochastic Dynamic Problems Structural mechanical systems under random excitation are described by stochastic differential equations. The equation of motion of a linear oscillator, m¨x + c˙x + kx = f(t) ,

(1)

with the random force f(t) is used as an example. Fluctuations in boundary conditions can be described by Gaussian white noise for simplicity, which is characterized by the first and second moments of its components. The ensemble average is  f (t) = 0 and the process is delta-correlated, i.e.  f (t) f (t  ) = δ (t − t  ), with a Gaussian distribution. Due to the delta-correlation, the system response forms a Markov vector process, wherein the future state only depends on the known present state and is independent of previous states. To compute the system response, Kloeden and Platen [5] suggest Euler-Maruyama methods, based on space-time discretization or Taylor series expansions with time discretization only. A drawback of these methods is the computation of a large amount of superfluous information. Because of the Markov property of the stochastic process, the response statistics of the system can be computed solely from the transition probability density function. The evolution of the probability density function is described by the Fokker-Planck equation. The Fokker-Planck equation takes advantage of the advection-diffusion characteristics of the stochastic process. Denoting the probability density of the motion by n

p(x,t) = ∏ δ (xi − xi (t)), x = [x1 x˙1 . . . xn x˙n ]T

(2)

i=1

with the stochastic system response vector x(t) and the probability of a state in [xi , xi +dxi ] is p(x,t)dxi dt. An evolution equation for the probability density function is obtained from the time derivative of Eq. (2) n ∂ ∂ n ∂ xi p(x,t) = − ∑ ∏ δ (xi − xi (t))  ∂t ∂t i=1 ∂ xi i=1

which is transformed into

(3)

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∂ p(x,t) ∂ ∂2 =− (Di (x,t)p(x,t)) + (Di j (x,t)p(x,t)) , ∂t ∂x ∂x ∂x  i    i j   advection

(4)

diffusion

with the drift vector Di and the diffusion tensor Di j (see [10]). Numerical simulation techniques for the evolution of the probability density are of great interest.

3 Numerical Solution of the Fokker-Planck Equation with TDG Methods Due to the advection-diffusion characteristics of the FPE, solution schemes established in computational fluid dynamics can be applied. Traditionally finite element discretization in space combined with finite difference approximation in time is used, although it is known that this method only converges for Courant numbers Cr = ΔΔ xt Di = 1. In compact form the FPE (4) is rewritten as

∂p = −Di ∇p + ∇(Di j ∇p) ∂t

(5)

which is transformed into its weak form, for the purpose of finite element discretization in space:  Ω

HT H d Ω

∂ p˜ + ∂t

 Ω

di HT H,x dΩ p˜ −

 Ω

Di j HT,x H,x dΩ p˜ = 0 .

(6)

This results in a system of ordinary differential equations in time M 

∂ p˜ + Q p˜ − K p˜ = 0 , ∂t



(7)



with M = Ω HT H dΩ , Q = Ω di HT H,x dΩ and K = Ω Di j HT,x H,x dΩ . The application of a finite difference time-step algorithm in its general form leads to M

pnj − pn−1 j

Δt

+ (Q − K)θ

pnj+1 − pnj−1 2Δ x

+ (Q − K)(1 − θ )

n−1 pn−1 j+1 − p j−1

2Δ x

=0

. (8)

Depending on the parameter θ , this denotes an explicit Euler scheme for θ = 0, an implicit Euler scheme for θ = 1 and the Crank-Nicholson scheme for θ = 0.5. The standard Galerkin method leads to wiggles in the solution because of the nonself-adjoined character of the advection operator. Therefore, Brooks and Hughes [1] introduced the Streamline Upwind/Petrov Galerkin (SUPG) method, wherein advective information is obtained from upstream, using the weighting function η˜ = η + τ Di ∇η . This leads to a system of equations in time, as denoted in (7), and the time stepping scheme (8) is applied. An alternative approach has been

Time-Discontinuous Galerkin Approach for the Fokker-Planck Equation Fig. 1 Discontinuous ansatz functions in time

315

p(x,t) pn−

pn−1 − pn−1 + t n−1

Tn

pn+ tn

t

proposed by Cockburn and Shu [2] using a discontinuous finite element approximation in time, referred to as Time-Discontinuous Galerkin (TDG) method. A systematic comparison of traditional finite difference schemes and TDG-methods of different order, is for example given in [19]. In the context of the TDG method the weak form of the FPE is rewritten as '   & ∂p ∂ ∂2 η + (Di p) − (Di j p) dΩ dt = 0 . (9) ∂ t ∂ xi ∂ xi ∂ x j Tn Ω The probability density and the weighting function are approximated by finite element ansatz functions in space and time, while discontinuities are assumed at the boundaries of each time interval, pn+ = lim p(t n + s), pn− = lim p(t n − s) and [|pn |] = pn+ − pn− s→0

s→0

,

(10)

as depicted in Fig. 1. The time discrete form of Eq. (9) reads 

  N˜ T N˜ ,t dt + N˜ +T (t n−1 )N˜ + (t n−1 ) Ωe η˜ p˜ dΩ T  n    + 

Tn

 N˜ T N˜ Dt  

(11)

ϒa

  ∂ p˜ ∂ η˜ ∂ p˜ ∂ η˜ ∂ Di j ∂ Di ˜ ˜ − η p ˜ − D η + D + p ˜ dΩ i ∂ xi i j ∂ xi ∂ x j Ωe ∂ xi ∂ xi ∂ x j



ϒb

  = N˜ +T (t n−1 )N˜ − (t n−1 ) Ωe η˜ p˜n−1 dΩ   

(12)

, (13)

ϒc

with time matrices ϒi . With continuous ansatz functions in space a non-symmetric linear system of equations is obtained: ˆ = [ϒ c (t)Mpˆ n−1 ] , [ϒ a (t)M + ϒ b (t)Q][p]    Kˆ

with the mass matrix

(14)

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M=

Ωe

HT H d Ω

and the advection and diffusion matrix '  & ∂ Di j ∂ Di T T T T Q= − H H − Di H H,xi + Di j H,xi H,x j + H,xi H DΩ ∂ xi ∂xj Ωe       advection

(15)

.

(16)

diffusion

The mentioned solution schemes were thoroughly investigated. The TDG method emerged as being accurate even for Courant numbers Cr = 1. The order of the scheme can be easily increased by increasing the polynomial degree of the ansatz functions in time and oscillations as well as dispersions are much better suppressed than for any finite difference scheme, including the SUPG method (see Fig. 2). Furthermore, sound mathematical proofs can be derived for discontinuous Galerkin methods. Proofs on coercivity and boundedness for example have been presented by Sch¨otzau and Zhu [12] for the advection-diffusion equation.

p

x

p

x Fig. 2 Comparison of SUPG and TDG (linear/quadratic ansatz functions) for an advectiondiffusion problem (1D), Cr = 3

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4 Numerical Example As a first example the response of a non-linear Van-der-Pol oscillator under white noise excitation, x¨ − (α − β x2 )x˙ + x = ω (t) , (17) is investigated. The motion forms a stochastic process which possesses Markov characteristics. Therefore the probability density is easily computed and transformed into the corresponding FPE,

∂p ∂p ∂ ∂2p =− + [β x2 x˙ − α x˙ + x]p + D 2 ∂t ∂ x ∂ x˙ ∂ x˙

.

(18)

The non-linear oscillator is subjected to white noise at its initial state, thus the mean value of the motion is the position of rest. The probability density function at this state is a Gaussian distribution with the mean value μ = 0 and a constant standard deviation σ . The Van-der-Pol oscillator will swing into a limit cycle from any state apart from the state of rest. Therefore the probability density function is expected to possess maxima only on the limit cycle. With the Time-Discontinuous Galerkin method, this high-gradient solution can be computed exactly and efficiently (see Fig. 3).

5 Conclusions Instead of solving stochastic differential equations, the Fokker-Planck equation provides a direct representation of probability density functions. The time-discontinuous Galerkin approach has been proven to be a stable and efficient method for the solution of Fokker-Planck equations. There are no restrictions regarding the Courant

Fig. 3 Probability density function of the Van-derPol oscillator with maxima at the limit cycle in phase space

v

x

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criterion and the accuracy can be easily increased with the order of the ansatz functions. Regarding problems with more spatial degrees of freedom a drawback of the direct computation of the probability density emerges: the dimension of the system of equations rises to 2 × DOF stochastic dimensions plus time. This leads to large systems of equations with broad band structure. Direct solvers fill up the working memory already for very coarse discretizations. Iterative solvers, like stabilized biconjugate gradient methods, were applied successfully instead. To capture the high dimensionality, the application of sparse grid solvers is suggested to reduce the system size. It also seems appropriate to use multi-scale methods, like suggested in [8] or to apply a domain decomposition computational scheme.

References 1. Brooks, A.N., Hughes, T.J.R.: Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Method. Appl. M. 32, 199–259 (1982) 2. Cockburn, B., Shu, C.W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998) 3. Ghanem, R.G., Spanos, P.D.: Stochastic finite elements. Dover Publications, New York (2003) 4. Keese, A.: A review of recent developments in the numerical solution of stochastic partial differential equations (stochastic finite elements). Report No. Informatikbericht 2003, vol. 6 (2003) 5. Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1999) 6. Kumar, P., Narayanan, S.: Solution of Fokker-Planck equation by finite element and finite difference methods for nonlinear systems. Sadhana-Acad. P. Eng. S. 31, 445–461 (2006) 7. Lehtikangas, O., Tarvainen, T., Kolehmainen, V., Pulkkinen, A., Arridge, S.R., Kaipio, J.P.: Finite element approximation of the Fokker-Planck equation for diffuse optical tomography. J. Quant. Spectrosc. Ra. 111, 1406–1417 (2010) 8. Masud, A., Bergmann, L.A.: Application of multi-scale finite element methods to the solution of the Fokker-Planck equation. Comput. Method. Appl. M. 194, 1513–1526 (2005) 9. Nouy, A.: Recent developments in spectral stochastic methods for the numerical solution of stochastic partial differential equations. Arch. Comput. Method. E. 16, 251–285 (2009) 10. Risken, H.: The Fokker-Planck equation methods of solution and applications. Springer, Berlin (1996) 11. Schmidt, F., Lamarque, C.-H.: Computation of the solutions of the Fokker-Planck equation for one and two DOF systems. Commun. Nonlinear Sci. 14, 529–542 (2009) 12. Sch¨otzau, D., Zhu, L.: A robust a-posteriori error estimator for discontinuous Galerkin methods for convection-diffusion equations. Appl. Numer. Math. Eng. 59, 2236–2255 (2009) 13. Schueller, G.I., Bergman, L.A., Bucher, C.G., Dasgupta, G., Deotdatis, G., Ghanem, R.G., Grigoriu, M., Hoshiya, M., Johnson, E.A., Naess, N.A., et al.: A state-of-the-art report on computational stochastic mechanics. Probabilist. Eng. Mech. 12, 197–321 (1997) 14. Soize, C.: Steady-state solution of Fokker-Planck equation in higher dimension. Probabilist. Eng. Mech. 3, 196–206 (1988)

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15. Stefanou, G.: The stochastic finite element method: past, present and future. Comput. Method. Appl. M. 198, 1031–1051 (2009) 16. Sudret, B., Der Kiureghian, A.: Stochastic finite element methods and reliability. A stateof-the-art-report. Structural Engineering, Mechanics and Materials Program, University of California Berkeley, Report No. UCB/SEMM-2000/08 (2000) 17. von Wagner, U., Wedig, W.V.: On the calculation of stationary solutions of multidimensional Fokker-Planck equations by orthogonal functions. Nonlinear Dynam. 21, 289–306 (2000) 18. Yi, W., Spencer Jr, B.F., Bergman, L.A.: Solution of the Fokker-Planck equation in higher dimensions: application of the concurrent finite element method. In: Shiraishi, N., Shinozuka, M., Wen, Y.K. (eds.) 7th International Conference on Structural Safety and Reliability, ICOSSAR 1997 (1997) 19. Ziefle, M., Nackenhorst, U.: Numerical techniques for rolling rubber wheels: treatment of inelastic material properties and frictional contact. Comput. Mech. 42, 337–356 (2008)

Chapter 36

Interface Modelling in Computational Limit Analysis A.V. Lyamin, K. Krabbenhøft and S.W. Sloan To Peter in honour of his 60th birthday and for all his help in teaching us the subtleties of contact mechanics for geotechnical problems (A.V. Lyamin, K. Krabbenhøft and S.W. Sloan).

Abstract. In many geotechnical stability problems it is important to account for interface conditions between two or more adjoining bodies, e.g. retaining walls and footings with no-tension contact between soil and structure. These interfaces can be considered as discontinuities in stress and velocity fields developed in the system undergoing plastic collapse. Discontinuous variable fields are routinely employed in FE lower and upper bound limit analyses to improve the performance of lower order elements used to obtain rigorous bounds on the collapse factor. Traditionally, stress and velocity discontinuities have been implemented as a set of special equalities on the stress and velocity variables of adjacent nodes across inter-element boundaries. The major drawback of this approach is that the velocity discontinuities are restricted only to materials with Tresca or Mohr-Coulomb yield criteria. Recently, however, it was shown that velocity discontinuities can be represented by a patch of regular elements of zero thickness. This development opens the way for a discontinuous upper bound formulation to be used with general yield criteria in both two- and three-dimensions. By also treating stress discontinuities as a patch of zero thickness elements in a lower bound formulation, lower and upper bound FE methods can be used effectively to solve stability problems involving a wide variety of materials and interface conditions.

1 Discrete Formulation of Bound Theorems Consider a domain Ω with boundary Γt , as shown in Fig. 1. Let t and q denote, respectively, a set of fixed tractions acting on the part of the boundary Γ and a set of unknown tractions acting on the part of the boundary Γq . Similarly, let g and h be A.V. Lyamin · K. Krabbenhøft · S.W. Sloan Centre for Geotechnical and Materials Modelling, University of Newcastle, NSW, Australia e-mail: [email protected]

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Fig. 1 A domain subject to a system of surface and body forces

a system of fixed and unknown body forces which act, respectively, on the volume Ω . Under these conditions, the objective of a lower bound calculation is to find a stress distribution which satisfies equilibrium throughout Ω , balances the prescribed tractions t on Γt , nowhere violates the yield criterion, and maximises the integral 

Q=

q dΓ +

Γq



h dΩ

(1)

Ω

The objective of an upper bound calculation is to find a velocity distribution u which satisfies compatibility, the flow rule, the velocity boundary conditions w on the surface area Γu , and minimises the integral 

W internal =

Ω

σ ε˙ dΩ

(2)

An upper bound estimate on the true collapse load can be obtained by equating W internal to the power dissipated by the external loads 

W external = Γt

tT u dΓ +

 Γq

qT u dΓ +

 Ω

gT u dΩ +



hT u dΩ

.

(3)

Ω

To preserve the bounding properties of the numerical solutions, linear finite elements are used to discretise the continuum. In an effort to provide the best possible bounds, kinematically admissible velocity discontinuities and statically admissible stress discontinuities are permitted at all inter-element boundaries for, respectively, the upper and lower bound analyses [3, 4]. These discontinuities allow accurate estimates of the collapse load to be computed without using an excessive number of

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elements and can be efficiently implemented using the approach described in following sections.

2 Velocity Discontinuities as a Patch of Thin Elements A general approach for modelling a discontinuous velocity field in D dimensions, regardless of the yield criterion involved, can be derived by treating a discontinuity as a patch of D infinitely-thin elements [1]. The problem model is then simplified because the power dissipation given by (2) can be computed as the sum of contributions from all elements in the mesh. We will next show that using linear finite elements for the problem discretisation results in an a priori simplification for velocity discontinuities which are valid for general yield criteria.

Fig. 2 Discontinuity as a patch of interconnected thin elements – upper bound

For illustrative purposes let us consider a two-dimensional patch of interconnecting triangles shown in Fig. 2. Each triangle is a constant stress-linear velocity element with the velocity vector u, varying according to u = ∑ Ni (x, y)ui

,

Ni (x, y) =

i

ai + bi x + ci y 2Δ

(4)

where Δ is the area of the triangle and coefficients a, b, c are computed from nodal coordinates as follows ak = xl ym − xm yl

,

b k = yl − ym

,

ck = xm − xl

(5)

for node k and then with cyclic interchanges of indexes for nodes l and m. The compatibility matrix B is given by ⎡ ⎤ b 0 bl 0 bm 0 1 ⎣ k 1 ¯ ¯ ¯  1 ¯ 0 ck 0 cl 0 cm ⎦ = B = [Bk Bl Bm ] = Bk Bl Bm = B or B¯ = BΔ (6) 2Δ Δ Δ ck bk cl bl cm bm and the power dissipated in any triangular element, regardless of its area, is calculated from

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W= Δ

σ ε˙ dΔ =



¯ = σ ε¯˙ σ Bu dΔ = σ Bu

(7)

Δ

The flow rule can be presented in a similar way as ¯ = Δ × λ˙ ∇ f (σ ) = λ¯˙ ∇ f (σ ) Bu

(8)

Considering now the case of an infinitely thin element with side lm being collapsed, ¯ m , resulting in the compatibility matrix we find that B¯ k → 0 and B¯ l → −B  *= 0 B

¯ lm −B

B¯ lm



(9)

where B¯ l has been replaced by B¯ lm for notation convenience. It is readily seen that the strain rate in element k, l, m in this case can be expressed in terms of differences between velocities (velocity jumps) at nodes l and m leading to * = B¯ lm Δ ulm ε˙ = Bu

(10)

Krabbenhøft et al. [1] showed that expression (10), when employed for a MohrCoulomb criterion, leads to the conventional expressions for the flow rule and power dissipation in the discontinuities. But modelling the discontinuities as patches of infinitely thin elements avoids any need for special treatment. Indeed, the power dissipation is computed using (7) and the flow rule constraints are given by (8). These constraints are actually obtained automatically as part of the system of optimality conditions for the upper bound optimization problem.

3 Stress Discontinuities as a Patch of Thin Elements An efficient lower bound formulation requires statically admissible stress discontinuities between adjacent elements (Fig. 3). The constraints for these discontinuities are that only normal and shear stresses must be continuous across the inter-element boundary. We now show that this requirement is equivalent to the element equilibrium conditions written for the discontinuity elements in the patch. Using the notation introduced in the previous section, the equilibrium conditions for element k, l, m can be written as

Fig. 3 Discontinuity as a patch of interconnected thin elements – lower bound

Interface Modelling in Computational Limit Analysis

⎧ k⎫ ⎧ ⎫ ⎨σ ⎬ 1 3 4⎨ σ k ⎬  T T T ¯ B ¯ BT σ = BTk BTl BTm B¯ B σl = σ l = − (g + h) ⎩ m⎭ Δ k l m ⎩ m⎭ σ σ 

325

(11)

which is equivalent to ⎧ ⎫ 3 4⎨ σk ⎬ ¯ Tk B ¯ Tl B¯ Tm σ l = − (g + h) Δ B ⎩ m⎭ σ For a zero volume element with side lm being collapsed, Eq. (12) becomes ⎧ ⎫ 3 4⎨ σk ⎬ T T ¯ Tlm B¯ Tlm 0 −B σ l = 0 or B¯ lm σ l = B¯ lm σ m ⎩ m⎭ σ

(12)

(13)

The last of Eqs. (13) represents the equality of surface tractions between nodes l and m. After dividing the coefficients bl and cl of matrix B¯ lm by the length of the discontinuity L, we obtain the direction cosines βx x , βx y for the axis x . Thus (11) finally leads to  l2  m2 tx tx = (14) tym tyl Application of the conditions 14 is equivalent to setting the normal/shear stresses to be equal at nodes l and m, as these are linearly related to surface tractions by  l2    l2 tx σn βx x βx y = = (15) βy x βy y tyl στl Therefore, the approach of treating discontinuities as a patch of zero volume elements also suits the lower bound formulation, as no special “discontinuity constraints” need to be introduced. Indeed, simple application of the familiar equilibrium conditions is sufficient to ensure that the discontinuity is statically admissible.

4 Interfaces between Material Domains The arrangement of the elements at an interface and the locations of the stress and velocity nodes are presented in Fig. 4. Assuming an associated flow rule, the interface conditions are governed by the yield function in the zero thickness elements. For upper bound analysis, only one layer of discontinuity elements in the interface is needed as these elements have separate stress variables (Fig. 4a). For lower bound meshes this is not the case and two layers of zero thickness elements are essential to prescribe material properties to the stress points which are separate from the stress points of the domains adjoining the interface. To make the patch symmetric, two

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Fig. 4 Interface layout for upper (a), (b) and lower (c) bound formulations

layers of elements are used by default for both lower and upper bound limit analysis in proposed implementation. To demonstrate the feasibility of the approach the plane strain collapse of a surface footing on clay subjected to vertical eccentric loading (Fig. 5) is considered. Two kinds of interface conditions are modelled: full adhesion and tension cut off. The collapse mechanisms shown and corresponding bearing capacity values demonstrate the influence of the tension cut-off condition.

Fig. 5 Strip footing on clay subject to eccentric (e = 0.6) loading with full adhesion and tension cut-off interface conditions

5 Interfaces at Segments Subject to Loading or Boundary Conditions Modelling of surface effects under applied loading or boundary conditions can proceed in the same manner as it was done for implementing the interfaces between materials. Furthermore, only one layer of zero thickness elements is sufficient in this case as we already have stress nodes which are “outside” of the domain. Therefore, for these nodes any desired yield conditions can be applied without interference with the material assumed for the domain itself (Fig. 6). The simple example of vertical loading of a cohesive-frictional (c = 1, φ = 20◦ ) rectangular block laying on a flat surface is used (Fig. 7) to show a few possible scenarios of using the element patch interfaces between the applied loading or boundary

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Fig. 6 Surface interface layout for upper (a) and lower (b) bound formulations.

conditions and the problem domain. Three different cases of interface conditions are modeled with corresponding collapse mechanisms and limit loads shown in Fig. 7. The second example is that of equal-channel angular extrusion under plane strain conditions. The metal is modelled as a von Mises material while the Mohr-Coulomb friction law is used for the metal-wall interaction. The analysis considers a 90◦ channel and three different interface conditions. The corresponding velocity fields are shown in Fig. 8, together with the collapse pressures.

Fig. 7 Collapse of rectangular block (H/B = 0.8) subject to rigid vertical loading with: a) rough top/rough bottom; b) rough top/partially rough bottom; c) smooth top/rough bottom

Fig. 8 Equal-channel angular extrusion: FE mesh and velocity fields

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6 Moment-Free Interfaces for Modelling of Joints Usually structural elements have to be used to model joints with rotation. However, moment free connections can be implemented without any special elements by applying equality constraints on the stresses and velocities of the surface nodes of adjoining domains as shown in Fig. 9. These constraints ensure force and moment equilibrium across the joint for static formulations and rigid segment rotation for kinematic formulations, thus preserving the rigor of both lower and upper bound analyses. The case of dual leg footing failure is used here to check the implementation of joints for limit analysis applications (Fig. 10). Two extreme cases are considered: a) the foundation panel is fully attached to the legs; b) the panel is attached to the legs via moment free joints. The load applied is inclined at 30 degrees to the horizontal, thus inducing quite distinctive modes of collapse, namely sliding and rotational failure, as shown in Figs. 10a and 10b.

Fig. 9 Moment free joint constraints for upper a) and lower b) bound formulations

Fig. 10 Failure of panel resting on pair of footing legs and subject to inclined (30◦ ) loading: a) panel is fully attached to footings; b) panel is attached via moment free joints

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7 Interfaces for Overlapping Connections Quite often in geotechnical engineering a series of plane strain stability analyses is performed on critical sections of the original 3D problem to make the case computationally feasible. This practice is common for such problems as the bearing capacity of foundations and the stability of dams, slopes and retaining walls. For anchor supported retaining walls, a problem arises in the modelling of anchors/ties without simultaneously introducing an artificial reinforcement effect. One efficient solution is to take the connection between the wall and the anchor ”out” of the soil and make it overlap. This requires a special connection interface which preserves the wall interaction with the soil and at the same time connects it to the anchor tie. Such complex connections can be modelled efficiently by using multilayered zero thickness patches of elements, as shown in Fig. 11. The upper bound implementation of multilayered interfaces is straightforward, as the power dissipated at the interface is just a sum of powers dissipated in all interface elements. For a lower bound analysis, a small adjustment is needed to the single layer implementation. In this case, the shear and normal tractions for nodes on the unsplit side must be equal to the sum of the shear and normal tractions of each of the layers (Fig. 11b). Figure 12 shows an anchored sheet pile wall with the anchor tie implemented as a) interacting with the surrounding soil and b) overlapping the soil with a

Fig. 11 Dual connection interface for upper (a) and lower (b) bound formulations

Fig. 12 Building with anchored sheet pile wall support: a) anchor tie is in contact with soil; b) anchor tie overlaps the soil and is connected to the wall using dual layer interface

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double-layered interface connection to the wall. The dual layer connection interface employed for this problem also includes a moment free wall/tie connection and no-tension conditions between the wall and the adjacent soil. The difference in the collapse pattern and bound values underlines the importance of selecting the model appropriately.

8 Conclusions A new approach for modelling discontinuous velocity and stress fields in the framework of numerical limit analysis has been presented. The method is based on a patch of zero thickness “solid” elements of regular topology with the properties of the interface material governed by the assumed yield criterion enforced at the corresponding stress points. Since all the conditions of the limit theorems are satisfied, the resultant kinematic and static formulations furnish rigorous upper and lower bounds and can be used for two- and three-dimensional stability problems with various interface conditions that are governed by general types of yield criteria.

References 1. Krabbenhøft, K., Lyamin, A.V., Hijaj, M., Sloan, S.W.: A new discontinuous upper bound limit analysis formulation. Int. J. Numer. Meth. 63, 1069–1088 (2005) 2. Lyamin, A.V., Krabbenhøft, K., Abbo, A.J., Sloan, S.W.: General approach to modelling discontinuities in limit analysis. In: Proc. IACMAG, Turin, vol. 11 (2005) 3. Sloan, S.W., Kleeman, P.W.: Upper bound limit analysis using discontinuous velocity fields. Comput. Meth. Appl. M. 127, 293–314 (1995) 4. Sloan, S.W.: Lower bound limit analysis using finite elements and linear programming. Int. J. Numer. Anal. Met. 12, 61–77 (1988)

Chapter 37

On the Coexistence of Intermeshed Hostile Populations Tarek I. Zohdi Peter Wriggers is an exceptional scholar, who has made pioneering and fundamental contributions to computational mechanics. It is my pleasure to participate in this honorary volume. I had the great fortune of being a post-doctoral student of Peter and completing my Habilitation under his guidance. He is the ideal mentor and a remarkable human being. It is a genuine pleasure working with him! Dear Peter: “happy birthday and many more to come”! (T.I. Zohdi).

Abstract. From its inception, the field of computational mechanics has primarily been focussed on engineering applications involving solid and fluid mechanics. As we enter the 21st century, it is apparent that we enter a different period in human civilization where, perhaps, computational methods can also start to play a significant and constructive role. The recent dramatic increase in computational power available for mathematical modeling and simulation raises the possibility that numerical methods can play a significant predictive role in the analysis of populations. This fact has motivated the work that will be presented. As a model problem, the objective of this work is to begin the development of robust computational procedures to investigate the behavior of initially intermeshed hostile, potentially human, populations. The applications are numerous, stemming from human civil wars on the macroscale to cellular-level conflicts, for example, between cancer and healthy cells, on the microscale. Classical modeling approaches attempt to develop continuum type field equations, usually making somewhat unrealistic assumptions in order to obtain tractable partial differential equations. For example, when dealing with small populations, or populations which become quite small and heterogeneously dispersed during the time-history of interaction, the assumptions behind regularization techniques leading to continuum models, may not apply. With these issues in mind, and due to the dramatic increase in desktop computing, we develop a discrete interaction approach where one can easily modify “rules of engagement”, population sizes, Prof. Tarek I. Zohdi Department of Mechanical Engineering, 6195 Etcheverry Hall, University of California, Berkeley, CA, 94720-1740, USA e-mail: [email protected]

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reproductive rates, etc, to provide quantitative spatial and temporal information. A robust inverse algorithm is also developed to determine “parity points”, i.e. multiple parameter sets which yield population coexistence. For sufficiently complex models of population interaction, the determination of parity points falls within the category of non-convex, non-differential optimization, with constraints. Accordingly, a general approach, based on a stochastic genetic algorithm, is developed and applied to a sufficient complex model for two interacting populations in order to illustrate the overall method.

1 Introduction The simulation of population dynamics has been a topic of ongoing research for over 200 years, dating back, at least, to the work of Thomas Malthus in 1798. The celebrated Malthusian model relates the projected population, at some time t + Δ t, to the current population. Virtually all subsequent, more complex, models build upon the Malthusian approach. A typical model reads as P˙ = aP, where a is a growth rate parameter that may depend on many factors, such as birth and death rates, availability of resources, conflicts, etc, or even P itself. Extensions to competing populations are relatively straightforward. Notable are the well-known Lotka-Volterra predator-prey relations, developed in the 1920s, and further enhancements incorporating competition between populations. While such models have some qualitative predictive capability they fail to take into account spatial distributions of such populations. Furthermore, models which attempt to develop continuum type field equations, by passing to the limit as Δ t → 0 1 make somewhat unrealistic assumptions in order to obtain tractable partial differential equations.2 Historically, most approaches apply asymptotic analysis to the resulting equations in order to extract some qualitative estimates of the model behavior. Relatively recently, numerical discretization, both in time and space, has been used to perform more indepth analyses, in particular, focusing on transient model behavior. However, one must question the process of first “smearing out” (discrete) population behavior to develop continuum models resulting in partial differential equations and then discretizing them back again into nodal values. This process is not bijective, in other words, one does not recover the original discrete system (Fig. 1). Also, because of the simplifying assumptions that are typically made, in order to obtain tractable field equations, the resulting discrete equations are not as physically meaningful as the true discrete representation, which may be based on complex rule-driven processes of interaction, which are not amenable to smooth (tractable) continuum representations. In particular, when dealing with small populations, or populations which become quite small and heterogeneously dispersed during the time-history of interaction, the assumptions behind regularization techniques leading to continuum models, may not apply. 1 2

The same goes for spatial discretizations. Frequently, such models exhibit diffusive or wave-like behavior.

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333

HOMOGENIZATION

FUNDAMENTALLY DISCRETE CONTINUUM

NUMERICALLY

NUMERICAL DISCRETIZATION

DISCRETE

Fig. 1 Left: The usual process of developing a continuum model from an inherently discrete system, which is then re-discretized. Right: The interaction between two groups

1.1 Objectives Accordingly, the objective of this work is to develop a robust computational procedure to investigate the behavior of initially intermeshed hostile species populations by directly working at the rule-driven level of interaction. One would expect that, if the two hostile groups were initially uniformly dispersed over some area and were of initially equal number and had similar characteristics, that: 1) both groups would suffer massive fatalities, leaving only well separated, “enclaves” of homogeneous species, where one group had locally dominated over the other, 2) the enclaves would grow, unboundedly, until they would encounter another in “border conflicts” and 3) the growing, hostile, groups would develop well defined boundaries. The question addressed in this communication is: If the groups do not have the same characteristics, for example combat skills, reproductive rates, etc., can they coexist? Furthermore, what skill sets between groups allows for overall coexistence of both groups? In order to answer this question, in this work a computational framework is developed to determine “parity points”, i.e. multiple parameter sets which yield coexistence, between inter-meshed, hostile, populations. For sufficiently complex models of population interaction, the determination of parity points falls within the category of non-convex, non-differential optimization, with constraints, which we describe shortly.

2 Direct Interaction Models: Rules of Engagement We now construct a model problem based on discrete rule-driven interaction between members of two populations. Consider the following construction, for the

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“rules of engagement” for two hostile populations, which are either in close proximity to one another or intermeshed (Fig. 1): • If two members of the opposing populations, denoted (1) and (2), come within a (1) (2) (1−2) certain conflict distance, ||ri − r j || ≤ di j , then two are said to engage in a “local” conflict. • The members alert all “support members” of their respective populations, within (1) (1) (1−1) (2) (2) (2−2) a certain “support distance”, ||ri − r j || ≤ si j , and ||ri − r j || ≤ si j , in order to aid in the local conflict. • The “combat skills” of the two populations may be different. Consider a certain number of the members of population (1), p(1) , which are engaged in a local conflict and (2), p(2) , which are engaged in the same local conflict. The percentage of each group in the local conflict are φ1 = the following rules for victory:

p(1) p(1) +p(2)

and φ2 =

p(2) . p(1) +p(2)

Consider

1. If w(1) φ (1) = w(2) φ (2) then all members of both populations, that are involved in the local conflict, perish. 2. If w(1) φ (1) > w(2) φ (2) then all members of population (2) perish and min(p(1) , (w(1) φ (1) − w(2) φ (2) )(p(1) + p(2) )) of population (1) survive. 3. If w(1) p(1) < w(2) p(2) then all members of population (1) perish and min(p(2) , (w(2) φ (2) − w(1) φ (1) )(p(1) + p(2) )) of population (2) survive. • A member of either population cannot participate in two local conflicts simultaneously. • Once a member of either population perishes, it cannot participate in any further conflicts. • If a member of a population survives beyond a certain number of conflict periods then it produces two offspring, and then perishes. The offspring are placed randomly within an “offspring” radius, centered at the spatial location of the parent. The development of a continuum approximation and corresponding rediscretization, would be extremely tedious, if not impossible. The relative ease at which one can generate two populations, and step them through several conflict periods is rather obvious. This is easy to implement.

3 An Example As an example, consider two populations, each starting with 1000 members (Fig. 3), initially distributed randomly between −1 ≤ xi , yi ≤ 1, both centered at (x, y) = (0, 0), and which which grow and conflict according to the previously defined rules. Consider 75 conflict periods and the following parameters: • The initial number of members in population 1: P10 = 1000 • The initial number of members in population 1: P20 = 1000 (1−2) • The conflict distance: di j = 0.25

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The support distance for population 2: = 0.25 The local conflict manpower weight for population 1: w(1) = 0.5 The local conflict manpower weight for population 2: w(2) = 0.5 The offspring radius for population 1: o(1) = 0.25 The offspring radius for population 2: o(2) = 0.25 The number of conflict periods until viable offspring are produced for population 1: g(1) = 5. • The number of conflict periods until viable offspring are produced for population 2: g(2) = 5. Remarks: Figure 3 illustrates the overall growth of the populations, and the movement of their “mass (population) centers”. When the populations are strongly intermeshed, initially almost all perish. Afterwards, the populations grow, however, with distinct boundaries evolving between them. Typically, for such systems with a finite number of members, there will be slight variations in the behavior for different random starting configurations. This is discussed further later. However, if one wished to extract some overall statistical behavior, a number of different starting realizations must be tested and then the overall results averaged.

4 Identification of System Parameters: Genetic Algorithms Typically, for the class of problems considered in this work, the corresponding formulations depend in a nonconvex and nondifferentiable manner on the system parameters. Classical gradient-based deterministic optimization techniques are not robust, due to difficulties with objective function nonconvexity and nondifferentiability. Classical gradient-based algorithms are likely to converge only toward a local minimum of the objective functional if an accurate initial guess to the global minimum is not provided. Also, usually it is extremely difficult to construct an initial

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guess that lies within the (global) convergence radius of a gradient-based method. These difficulties can be circumvented by the use of a certain class of nonderivative search methods, usually termed “genetic” algorithms (GA), before applying gradient-based schemes. Genetic algorithms are search methods based on the principles of natural selection, employing concepts of species evolution, such as reproduction, mutation and crossover. Implementation typically involves a randomly generated population of fixed-length elemental strings, “genetic information”, each of which represents a specific choice of system parameters. The populations of individuals undergo “mating sequences” and other biologically-inspired events in order to find promising regions of the search space. There are a variety of such methods, which employ concepts of species evolution, such as reproduction, mutation and

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crossover. Such methods primarily stem from the work of John Holland [4]. For reviews of such methods, see, for example, Goldberg [2], Davis [1], Onwubiko [6], Kennedy and Eberhart [5] and Goldberg and Deb [3]. Implementation: Adopting the approaches found in Zohdi [7], a genetic algorithm has been developed to treat nonconvex inverse problems involving various aspects of multi-particle mechanics. The central idea is that the system parameters form a genetic string and a survival of the fittest algorithm is applied to a population of such strings. The overall process is a) A population (S members in total) of different parameter sets are generated at random within the parameter space, each represented by a (“genetic”) string of the system (N) parameters, b) The performance of each parameter set is tested, c) The parameter sets are ranked from top to bottom according to their performance, d) The best parameter sets (parents) are mated pairwise producing two offspring (children), i.e. each best pair exchanges information by taking random convex combinations of the parameter set components of the parents’ genetic strings and e) The worst performing genetic strings are eliminated, new replacement parameter sets (genetic strings) are introduced into the remaining population of best performing genetic strings and the process (a-e) is then repeated. The term “fitness” of a genetic string is used to indicate the value of the objective function. The most fit genetic string is the one with the smallest objective function. The retention of the top fit genetic strings from a previous optimization generation (parents) is critical, since if the objective functions are highly nonconvex (the present case), there exists a clear possibility that the inferior offspring will replace superior parents. When the top parents are retained, the minimization of the cost function is guaranteed to be monotone (guaranteed improvement) with increasing optimization generations. There is no guarantee of successive improvement if the top parents are not retained, even though nonretention of parents allows more new genetic strings to be evaluated in the next optimization generation. Numerical studies conducted by the author imply that, for sufficiently large populations, the benefits of parent retention outweigh this advantage and any disadvantages of “inbreeding”, i.e. a stagnant population. For more details on this “inheritance property” see Davis [1] or Kennedy and Eberhart [5]. In the upcoming algorithm, inbreeding is mitigated since, with each new optimization generation, new parameter sets, selected at random within the parameter space, are added to the population. Previous numerical studies of the author [7] have indicated that not retaining the parents is suboptimal due to the possibility that inferior offspring will replace superior parents. Additionally, parent retention is computationally less expensive, since these parameter sets do not have to be re-evaluated in the next optimization generation. An implementation of such ideas is as follows [7]: • STEP 1: Randomly generate a population of S starting genetic strings, Λ i , (i = 1, ..., S) : def Λ i ={Λ1i , Λ2i , Λ3i , Λ4i , ..., ...ΛNi } • STEP 2: Compute fitness of each string Π (Λ i ), (i=1, ..., S) • STEP 3: Rank genetic strings: Λ i , (i=1, ..., S)

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• STEP 4: Mate nearest pairs and produce two offspring, (i=1, ..., S) def def Λ i = Φ (I) Λ i + (1 − Φ (I))Λ i+1 , Λ i+1 = Φ (II) Λ i + (1 − Φ (II) )Λ i+1 • NOTE: Φ (I) and Φ (II) are random numbers, such that 0 ≤ Φ (I) , Φ (II) ≤ 1, which are different for each component of each genetic string • STEP 5: Kill off bottom M < S strings and keep top K < N parents and top K offspring (K offspring+K parents+M=S) • STEP 6: Repeat STEPS 1-6 with top gene pool (K offspring and K parents), plus M new, randomly generated, strings • OPTION: Rescale and restart search around best performing parameter set every few optimization generations • OPTION: We remark that gradient-based methods are sometimes useful for post-processing solutions found with a genetic algorithm, if the objective function is sufficiently smooth in that region of the parameter space.

5 An Example of Parity Identification Again, consider two populations, each starting with 1000 members, both centered at (0,0), and distributed randomly between −1 ≤ xi , yi ≤ 1, which grow and conflict according to the previously defined rules. Consider a desired parity condition that the populations have similar sizes after 20 conflict periods. Mathematically speakdef (1−2) (1−1) (2−2) ing, defining Λ ={di j , si j , si j , w(1) , w(2) , o(1−1), o(2−2) , g(1) , g(2) }, we write

1 −P2 | the following minΛ Π (Λ ) = |P |P1 +P2 | , where P1 and P2 are populations after 20 conflict periods for a given Λ . The goal is to find the sets of Λ that minimize Π , other than the trivial parity set, namely that both of the populations have identical characteristics. In particular, we are interested in determining what factors (parameters) can counterbalance one another for coexistence. The normalized parameters in the genetic algorithm were allowed to vary between:

• Initial number of members in population 1: P10 = 1000 • Initial number of members in population 1: P20 = 1000 (1−2) • Conflict distance: 0.1 ≤ di j ≤ 0.25 (1−1)

• Support distance for population 1: 0.1 ≤ si j • • • • • •

≤ 0.25

(2−2) Support distance for population 2: 0.1 ≤ si j ≤ 0.25 (1) Manpower weight for population 1: 0.15 ≤ w ≤ 1

Manpower weight for population 2: 0.15 ≤ w(2) ≤ 1 Offspring radius for population 1: 0 ≤ o(1) ≤ 0.25 Offspring radius for population 2: 0 ≤ o(2) ≤ 0.25 Number of conflict periods until viable offspring are produced for population 1: 1 ≤ g(1) ≤ 10. • Number of conflict periods until viable offspring are produced for population 2: 1 ≤ g(2) ≤ 10.

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Table 1 The top 6 performers after 35 optimization generations. The top two met the population parity condition, namely that populations were identical after 20 conflict periods

Rank 1 2 3 4 5 6

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di j 0.2199 0.1000 0.1000 0.1000 0.1000 0.1000

s1 0.1276 0.1031 0.1058 0.1050 0.1048 0.1047

s2 0.1238 0.1000 0.1000 0.1000 0.1000 0.1000

w1 0.5762 0.3271 0.6435 0.6459 0.6528 0.6676

w2 0.7445 0.5259 0.4823 0.4541 0.4371 0.3957

o1 0.1123 0.1142 0.2377 0.2388 0.2393 0.2394

o2 0.1027 0.1028 0.1700 0.1701 0.1700 0.1700

g1 /10 0.5716 0.9541 0.5217 0.5092 0.5094 0.5110

g2 /10 0.9499 0.9282 0.7851 0.7191 0.7303 0.7462

Π 0.0000 0.0000 0.1638 0.1653 0.1729 0.1781

Table 1 shows all possible combinations for population parity. A total of 706 genetic strings were tested, with 11.868 samples per string needed for proper statistical stabilization (a total of 8379 samples were tested). Table 1 illustrates parity points between population (1) and (2). For example, parity point RANK=1 indicates that if population (2) requires less “manpower” to win a local conflict, namely w2 = 0.7445 as opposed to w1 = 0.5762 for population (1) (more combatants for local victory), then population (1) must compensate by having a higher reproductive rate g1 = 5.716 as opposed to g2 = 9.499 (longer time to reproduce). Table 1 illustrates the top six parity points found by the algorithm. Figure 2 illustrates the convergence of the genetic search.

6 Concluding Remarks The utility of the presented computational approach is that one can trivially modify the “rules of engagement”, population sizes, reproduction rates, etc., and provide quantitative spatial and temporal information. In summary, the objective of this work was to develop a robust computational procedure to investigate the behavior of initially intermeshed hostile populations. Specifically, we studied a model problem consisting of two hostile groups, who, in total, comprised the entire population. Direct interaction models were developed, where one can easily modify “rules of engagement”, population sizes, reproductive rates, etc., and provide quantitative spatial and temporal information. A robust inverse algorithm was developed to determine “parity points”, i.e. multiple parameter sets which yield coexistence between inter-meshed, hostile populations, based on a stochastic genetic algorithm and it was applied to a sufficient complex model for two interacting populations in order to illustrate the technique. A further caveat, such a computational technique is easy to implement, and it is no extra effort to increase the number of population character parameters.

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References 1. Davis, L.: Handbook of genetic algorithms. Thompson Computer Press (1991) 2. Goldberg, D.E.: Genetic algorithms in search, optimization and machine learning. Addison-Wesley, Reading (1989) 3. Goldberg, D.E., Deb, K.: Special issue on genetic algorithms. Comput. Method. Appl. M. 186(2-4), 121–124 (2000) 4. Holland, J.H.: Adaption in natural and artifactial systems. Ann Arbor. Mich. University of Michigan Press (1975) 5. Kennedy, J., Eberhardt, R.: Swarm intelligence. Morgan Kaufmann Publishers, San Francisco (2001) 6. Onwubiko, C.: Introduction to engineering design optimization. Prentice-Hall, Englewood Cliffs (2000) 7. Zohdi, Z.I.: Computational design of swarms. Int. J. Numer. Meth. Eng. 57, 2205–2219 (2003)

Contents

New Applications of Mortar Methodology to Extended and Embedded Finite Element Formulations . . . . . . . . . . . . . . . . . . . . . Tod A. Laursen, Jessica D. Sanders 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Stability Issues Associated with Contact on Enriched Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Adaptation to the Embedded Interface Case . . . . . . . . . . . . . . 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermo-Mechanical Coupling in Beam-to-Beam Contact . . . . Daniela P. Boso, Przemyslaw Litewka, Bernhard A. Schrefler 1 Thermo-Mechanical Beam Finite Element . . . . . . . . . . . . . . . . 2 Weak Form for Thermo-Mechanical Contact . . . . . . . . . . . . . . 3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On Regularization of the Convergence Path for the Implicit Solution of Contact Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Giorgio Zavarise, Laura De Lorenzis, Robert L. Taylor 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Structure of the Consistent Tangent Stiffness . . . . . . . . . . . . . . 3 Large Penetration Basic Algorithm . . . . . . . . . . . . . . . . . . . . . . 3.1 Strategy Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Modified Stiffness and Residual during Phase One . . . 3.3 Limitations of the Strategy . . . . . . . . . . . . . . . . . . . . . . . 4 Large Penetration Enhanced Algorithm . . . . . . . . . . . . . . . . . . 4.1 Solution of the Problem for r < 1 . . . . . . . . . . . . . . . . . .

1 1 2 4 7 8 9 9 11 13 15 15

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5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On Different Variational Formulations of the Nitsche Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ridvan Izi, Alexander Konyukhov, Karl Schweizerhof 1 Nitsche Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Choice of the Lagrange Multiplier Set μ . . . . . . . . . . . . 1.2 Physical Meaning of the Non-penetration Terms . . . . . 2 Types of the Nitsche Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 3 FE Implementation of the Nitsche Approaches . . . . . . . . . . . . 3.1 Gauss Point-Wise Substituted Formulation . . . . . . . . . 3.2 Bubnov-Galerkin-Wise Partial Substituted Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Challenges in Computational Nanoscale Contact Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Roger A. Sauer 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Nanoscale Contact Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Nanoscale versus Macroscale Contact . . . . . . . . . . . . . . . . . . . . 4 Adhesion Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Multiscale Contact Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On the Four-node Quadrilateral Element . . . . . . . . . . . . . . . . . . . . Ulrich Hueck, Peter Wriggers 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Element Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stability of Mixed Finite Element Formulations – A New Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stefanie Reese, Vivian Tini, Yalin Kiliclar, Jan Frischkorn, Marco Schwarze 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Linear Elasticity - Mixed Variational Formulation . . . . . . . . . 3 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Compatible Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Enhanced Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Element Stiffness Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Eigenvalue Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Non-linear Finite Element Technology . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Finite Element Formulation Based on the Theory of a Cosserat Point – Modification of the Torsional Modes . . . . . . . Eiris F.I. Boerner, Dana Mueller-Hoeppe, Stefan Loehnert 1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 A Brief Introduction to the Cosserat Point Element . . . . . . . . 2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Brick Element for Finite Deformations with Inhomogeneous Mode Enhancement . . . . . . . . . . . . . . . . . . . . . . . . . Dana Mueller-Hoeppe, Stefan Loehnert 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Enhanced Strain Assumption . . . . . . . . . . . . . . . . . . . . . 2.2 Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Irregularly Meshed Beam . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Nearly Incompressible Block . . . . . . . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Automatic Differentiation Based Formulation of Computational Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Joˇze Korelc 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Automatic Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Automatic Differentiation in Computational Mechanics . . . . . 4 Automatic Differentiation Based Computational Models . . . . 4.1 ADB Form of Hyperelastic Models . . . . . . . . . . . . . . . . . 4.2 ADB Form of Elasto-plastic Models . . . . . . . . . . . . . . . . 4.3 Numerical Efficiency of ADB Form . . . . . . . . . . . . . . . . 4.4 ADB Form of Contact Formulations . . . . . . . . . . . . . . . 4.5 ADB Form in Stability Analysis . . . . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Nonlinear Finite Element Shell Formulation Accounting for Large Strain Material Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . Friedrich Gruttmann 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Variational Formulation of the Shell Equations . . . . . . . . . . . . 3 Mixed Hybrid Shell Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Numerical Example: Stretching of a Rubber Sheet . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hybrid and Mixed Variational Principles for the Geometrically Exact Analysis of Shells . . . . . . . . . . . . . . . . . . . . . . Paulo de Mattos Pimenta 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Geometrically-Exact First-Order-Shear Shell Model . . . . 3 Some Multi-field Variational Principles . . . . . . . . . . . . . . . . . . . 3.1 Principle of Total Potential Energy . . . . . . . . . . . . . . . . 3.2 Three-Field Principle of Veubeke-Hu-Washizu Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Two-Field Principle of Hellinger-Reissner Type . . . . . . 3.4 Two-Field Principle of Total Complementary Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Hybrid Principle of Hellinger-Reissner Type . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Shell Theory with Scale Effects, Higher Order Gradients, and Meshfree Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Carlo Sansour, Sebastian Skatulla 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Deformation and Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Generalized Shell Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An Electro-mechanically Coupled FE-Formulation for Piezoelectric Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W. Wagner, K. Schulz, and S. Klinkel 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Finite Element Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87 87 88 90 91 94 94

97 97 98 103 103 103 104 104 105 106

107 107 108 109 111 114

115 116 116 117 118 120 123

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Non-intrusive Coupling: An Attempt to Merge Industrial and Research Software Capabilities . . . . . . . . . . . . . . . . . . . . . . . . . . Olivier Allix, Lionel Gendre, Pierre Gosselet, Guillaume Guguin 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The General Principles of Non-intrusive Coupling . . . . . . . . . . 2.1 Piecewise Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Iterative Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Choice of the Interface Boundary Condition for the Local Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Examples Using Abaqus/Standard . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constitutive Models and Failure Prediction for Al-Alloys in Industrial Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Christian Leppin 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Factors Influencing Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Work-Hardening of Aluminum Alloys . . . . . . . . . . . . . . . . . . . . 4 Yield Locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Fracture Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Phenomenological Damage Model to Predict Material Failure in Crashworthiness Applications . . . . . . . . . . . . . . . . . . . . . Markus Feucht, Frieder Neukamm, and Andr´e Haufe 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Process Chain of Sheet Metal Part Manufacturing . . . . . 3 Failure Modelling in Forming and Crashworthiness Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 A Generalized Scalar Damage Model . . . . . . . . . . . . . . . 3.2 Failure Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Path-Dependent Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Stress and Strain Measures . . . . . . . . . . . . . . . . . . . . . . . 4.2 Nonlinear Accumulation of the Instability Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Post Critical Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Damage-Dependent Yield Stress . . . . . . . . . . . . . . . . . . . 5.2 Energy Dissipation and Fadeout . . . . . . . . . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

XIII

125 126 126 127 128 130 130 132 132

135 135 136 136 138 140 141 142

143 144 144 144 146 147 147 148 149 150 151 151 152 153

XIV

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A Computational Approach for Mixed-Lubrication Effects in Sealing Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Markus Andr´e 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Solid Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Fluid Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Coupled Fluid Film Computation . . . . . . . . . . . . . . . . . . . . . . . 4 Friction Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deformations of a Large Hall: Structural Design and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Klaus-Dieter Klee, Reinhard Kahn 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Steel Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Bearing Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Roof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Stiffening Components . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Support of Partial Halls . . . . . . . . . . . . . . . . . . . . . . . . . 3 Construction and Computation . . . . . . . . . . . . . . . . . . . . . . . . . 4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recovering Micropolar Continua from Particle Mechanics by Use of Homogenisation Strategies . . . . . . . . . . . . . . . . . . . . . . . . Wolfgang Ehlers 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Particle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Homogenisation Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modelling of Microstructured Materials with Micromorphic Continuum Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Britta Hirschberger, Paul Steinmann 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Micromorphic Continuum . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Micromorphic Continuum Framework . . . . . . . . . . . . . . 2.2 Hyperelastic Constitutive Framework . . . . . . . . . . . . . . 2.3 Numerical Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Application to Material Interfaces with Heterogeneous Micromorphic Mesostructure . . . . . . . . . . . . . . . . . . . . . . . . . . . .

155 155 156 156 157 159 160 162 162

163 163 164 165 166 168 168 171 177 177

179 179 180 183 186 188 189

191 191 192 192 193 193 195

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XV

3.1

Scale Transition between Interface and Micromorphic RVE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 A Computational Homogenization Approach for Micromorphic Meso-heterogeneous Material Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On Computational Homogenisation of Heterogeneous Media with Debonded Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Peri´c, D.D. Somer, E.A. de Souza Neto, W. Dettmer 1 Introduction and Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Multi-scale Constitutive Theory: Overview . . . . . . . . . . . . . . . . 2.1 RVE Kinematical Constraints . . . . . . . . . . . . . . . . . . . . . 2.2 Finite Element Approximation . . . . . . . . . . . . . . . . . . . . 2.3 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Frictional Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Assessment of Yield Surfaces of Heterogeneous Media with Debonded Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Computational Homogenisation Based Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Estimated Yield Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Assessment of Homogenization Errors in Transient Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. Runesson, F. Su, F. Larsson 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Transient Heat Flow – A Model Problem . . . . . . . . . . . . . . . . . 2.1 Space-Variational Format . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Explicit Homogenization Results . . . . . . . . . . . . . . . . . . 3 RVE-Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Dirichlet Boundary Conditions . . . . . . . . . . . . . . . . . . . . 3.2 Neumann Boundary Conditions . . . . . . . . . . . . . . . . . . . 4 Computational Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Problem Definition – Substructure Characteristics . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

195

196 196 197 198

199 199 200 201 201 202 202 202 202 203 204 205 206 206

207 207 208 208 209 210 210 211 212 212 214 214

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Contents

Multiscale Modeling of Metal Foams Using the XFEM . . . . . . Lovre Krstulovic-Opara, Stefan Loehnert, Dana Mueller-Hoeppe, Matej Vesenjak 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Modified XFEM for Heterogeneous Materials . . . . . . . . . . . . . 3 Incorporation of Finite Plasticity . . . . . . . . . . . . . . . . . . . . . . . . 4 Comparison of Metal Foams with and without Filler Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3D Multiscale Projection Method for Micro-/Macrocrack Interaction Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stefan Loehnert, Dana Mueller-Hoeppe 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Multiscale Technique in Three Dimensions . . . . . . . . . . . . 2.1 Stress Projection from the Fine Scale to the Coarse Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Projection of the Displacement Field from the Coarse Scale to the Fine Scale . . . . . . . . . . . . . . . . . . . . 3 Numerical Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Goal-Oriented Residual Error Estimates for XFEM Approximations in LEFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marcus R¨ uter, Erwin Stein 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 XFEM Approximations in LEFM . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Model Problem of LEFM . . . . . . . . . . . . . . . . . . . . . 2.2 XFEM Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . 3 A Posteriori Error Estimation in the Energy Norm . . . . . . . . 3.1 Error Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 An Implicit Residual Error Estimator . . . . . . . . . . . . . . 3.3 Equilibration of Tractions . . . . . . . . . . . . . . . . . . . . . . . . 4 Goal-Oriented Error Estimation in LEFM . . . . . . . . . . . . . . . . 4.1 Linearization of the J-Integral . . . . . . . . . . . . . . . . . . . . 4.2 Duality Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

215

216 216 218 218 221 221

223 223 224 224 227 228 230 230

231 231 232 232 233 234 234 234 235 236 236 236 237 238 238

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Multi-field Coupling Strategies for Large Scale Particle-Fluid Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.R.J. Owen, Y.T. Feng, K. Han, C.R. Leonardi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 LB Formulations for Turbulent Incompressible Fluid Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Standard LB Formulation . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Turbulence Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Hydrodynamic Forces for Fluid-Particle Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Fine Particle Modelling - Non-newtonian Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Thermal Lattice Boltzmann Method . . . . . . . . . . . . . . . . . 4 Numerical Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Particle Transportation in Turbulent Fluid Flows . . . . 4.2 Fine Particle Migration in a Block Cave . . . . . . . . . . . . 4.3 Modelling Heat Transfer in (Particle-)Fluid Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Simulation of Particle-Fluid Systems . . . . . . . . . . . . . Bircan Avci, Peter Wriggers 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Mathematical Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Equations for Fluid Motion . . . . . . . . . . . . . . . . . . . . . . . 2.2 Equations for Particle Motion . . . . . . . . . . . . . . . . . . . . . 3 The Discrete Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Collision Model for Normal Contact . . . . . . . . . . . . . . . 3.2 Frictional Tangential Contact Model . . . . . . . . . . . . . . . 4 Coupling of the Fluid and Particle Phase . . . . . . . . . . . . . . . . . 4.1 Evaluation of the Hydrodynamic Forces . . . . . . . . . . . . 4.2 Coupling Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Concurrent Multiscale Approach to Non-cohesive Granular Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Christian Wellmann, Peter Wriggers 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Discrete Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Homogenization and Elasto-plastic Parameters . . . . . . . . . . . . 4 Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

XVII

239 240 241 241 242 243 243 244 245 245 245 247 248 248 249 249 250 250 250 251 251 252 253 253 254 255 255 255

257 257 258 259 261

XVIII

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5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 On Some Features of a Polygonal Discrete Element Model . . . Ekkehard Ramm, Manfred Bischoff, Benjamin Schneider 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Discrete Element Method with Polygonal Particles . . . . . . . . . 2.1 Models for Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Models for Cohesion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Model Material without Cohesion . . . . . . . . . . . . . . . . . 3.2 Model Material with Cohesion . . . . . . . . . . . . . . . . . . . . 3.3 Concrete with Microstructure . . . . . . . . . . . . . . . . . . . . . 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

265

Isogeometric Failure Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Clemens V. Verhoosel, Michael A. Scott, Michael J. Borden, Ren´e de Borst, Thomas J.R. Hughes 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Isogeometric Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Higher-Order Gradient Damage Formulation . . . . . . . . . . . . . . 3.1 Constitutive Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 L-Shaped Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Cohesive Zone Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Constitutive Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Single-Edge Notched Beam . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

275

A Method for Enforcement of Dirichlet Boundary Conditions in Isogeometric Analysis . . . . . . . . . . . . . . . . . . . . . . . . . Toby J. Mitchell, Sanjay Govindjee, Robert L. Taylor 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Dirichlet Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Examples from Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Infinite Half-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Infinite Plate with Circular Hole under Tension . . . . . 3.3 Infinite Plate with Elliptical Hole under Tension . . . . . 4 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

265 266 266 268 269 269 270 271 272 272

275 276 277 278 278 279 280 280 281 282

283 284 285 287 288 289 291 292 293

Contents

Application of Isogeometric Analysis to Computational Contact Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ˙ Ilker Temizer 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Contact Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . . . 3 Isogeometric Treatment with NURBS . . . . . . . . . . . . . . . . . . . . 4 Knot-to-Surface Contact Algorithm . . . . . . . . . . . . . . . . . . . . . . 4.1 Contact of a Grosch Wheel . . . . . . . . . . . . . . . . . . . . . . . 4.2 Contact of Two Deformable Bodies . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stochastic Galerkin Method for the Elastoplasticity Problem with Uncertain Parameters . . . . . . . . . . . . . . . . . . . . . . . . . Bojana V. Rosic, Hermann G. Matthies 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Problem Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Numerical Analysis of the Problem . . . . . . . . . . . . . . . . . . . . . . 3.1 Discretisation of Input . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Stochastic Galerkin Method . . . . . . . . . . . . . . . . . . . . . . 4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Time-Discontinuous Galerkin Approach for the Numerical Solution of the Fokker-Planck Equation . . . . . . . . . . Udo Nackenhorst, Friederike Loerke 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 FPE Expression of Stochastic Dynamic Problems . . . . . . . . . . 3 Numerical Solution of the Fokker-Planck Equation with TDG Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interface Modelling in Computational Limit Analysis . . . . . . . . A.V. Lyamin, K. Krabbenhøft, S.W. Sloan 1 Discrete Formulation of Bound Theorems . . . . . . . . . . . . . . . . . 2 Velocity Discontinuities as a Patch of Thin Elements . . . . . . . 3 Stress Discontinuities as a Patch of Thin Elements . . . . . . . . . 4 Interfaces between Material Domains . . . . . . . . . . . . . . . . . . . . 5 Interfaces at Segments Subject to Loading or Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

XIX

295 295 296 296 298 299 300 300 302

303 303 304 304 305 305 306 306 307 310 310

311 312 313 314 317 317 318 321 321 323 324 325 326

XX

Contents

6 Moment-Free Interfaces for Modelling of Joints . . . . . . . . . . . . 7 Interfaces for Overlapping Connections . . . . . . . . . . . . . . . . . . . 8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

328 329 330 330

On the Coexistence of Intermeshed Hostile Populations . . . . . Tarek I. Zohdi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Direct Interaction Models: Rules of Engagement . . . . . . . . . . . 3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Identification of System Parameters: Genetic Algorithms . . . 5 An Example of Parity Identification . . . . . . . . . . . . . . . . . . . . . 6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

331 332 333 333 334 335 338 339 340