III European Conference on Computational Mechanics: Solids, Structures and Coupled Problems in Engineering: Book of Abstracts

III EUROPEAN CONFERENCE ON COMPUTATIONAL MECHANICS

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering: Book of Abstracts

Edited by

C. A. MOTA SOARES J. A. C. MARTINS H. C. RODRIGUES JORGE A. C. AMBRÓSIO C. A. B. PINA C. M. MOTA SOARES E. B. R. PEREIRA and

J. FOLGADO Instituto Superior Técnico, Lisbon, Portugal

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN-10 1-4020-4994-3 (HB) ISBN-13 978-1-4020-4994-1 (HB)

Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com

Printed on acid-free paper

All Rights Reserved © 2006 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed in the Netherlands.

Table of Contents Preface Plenary Lectures M. Bendsoe Computational Challenges for Multi-Physics Topology Optimization ...................1 J.A. Cottrell, A. Reali, Y. Bazilev, T.J.R. Hughes Computational Geometry and the Analysis of Solids and Structures......................2 E. Oñate, S.R. Idelsohn, M.A. Celigueta, R. Rossi Advances in the Particle Finite Element Method for Fluid-Structure Interaction Problems................................................................................................3 W. Schiehlen, R. Seifried Elastic and Plastic Impacts in Multibody Dynamics ...............................................4 A. Suleman, P. Moniz Active Aeroelastic Aircraft Structures.....................................................................5 Keynote Lectures O. Allix Multiscale Strategy for Solving Industrial Problem ................................................6 F. Armero, C. Zambrana Numerical Integration of the Nonlinear Dynamics of Elastoplastic Solids .............7 T. Belytschko, S. Wang Computational Methods for Dynamic Crack Propagation.......................................8 P. Bergan, K. Bakken, K.C. Thienel Analysis and Design of Sandwich Structures Made of Steel and Lightweight Concrete ..................................................................................................................9 R. Borja Multiscale Modeling of Pore Collapse Instability in High-Porosity Solids...........10 R. de Borst, M.A. Abellan, J. Réthoré Instabilities and Discontinuities in Two-Phase Media ..........................................11 C. Bottaso Towards Maneuvering Aeroelasticity - Progress in the Simulation of Large Fluid-Structure Interaction Problems......................................................12 v

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K. K. Choi, L. Du A Design Optimization Formulation for Problems with Random and Fuzzy Input Variables Using Performance Measure Approach ............................13 I. Doltsinis Strength of Porous Ceramics - Mechanical Testing and Numerical Modelling ....14 E. A. Dowel, K.C. Hall, J.P. Thomas, R.E. Kielb,M.A. Spiker, C.M. Denegri, Jr. Reduced Order Models in Unsteady Aerodynamic Models, Aeroelasticity and Molecular Dynamics.......................................................................................15 G. S. Dulikravich, H.R.B. Orlande, B.H. Dennis Inverse Engineering...............................................................................................16 J. Fish, W. Chen Multiscale Approaches for Bridging Discrete and Continuum Scales ..................17 P. L. George Adapative Mesh Generation in 3 Dimensions by Means of a Delaunay Based Method - Applications to Mechanical Problems.........................................18 C. Hellmich, K. Hofstetter, C. Kober Computational Micromechanics of Biological Materials: Bone and Wood ..........19 G. Holzapfel,C.T. Gasser, D. Kiousis Mechanobiology: Computation and Clinical Application .....................................20 B. Karihaloo, Q.Z. Xiao Recent Developments of Hybrid Crack Element: Determination of its Complete Displacement Field and Combination with XFEM...............................21 P. Ladevèze, P. Enjalbert, G. Puel, T. Romeuf Structural Model Validation and the Lack-of-Knowledge Theory........................22 J.V. Lemos Modeling of Historical Masonry with Discrete Elements .....................................23 G.-R. Liu, B.B.T. Kee A Regularized Strong-form Meshfree Method for Adaptive Analysis..................24 W. K. Liu Multiresolution Analysis for Material Design.......................................................25 R. Lackner, R. Blab, J. Eberhardsteiner, H. A. Mang Characterization and Multiscale Modeling of Asphalt – Recent Developments in Upscaling of Viscous and Strength Properties ..........................26 J.F. Dëu, W. Larbi, R. Ohayon Dissipative Interface Modeling for Vibroacoustic Problems – A New Symmetric Formulation.........................................................................................27

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M. Papadrakakis, N.D. Lagaros, M. Fragiadakis Seismic Design Procedures in the Framework of Evolutionary Based Structural Optimization .........................................................................................28 L.F. González, L.H. Damp, J. Periaux, K. Srinivas High-fidelity Multi-Criteria Aero-Structural Optimisation Using Hierarchical Parallel Evolutionary Algorithms .....................................................29 E. Ramm, C. Förster, M. Neumann, W.A. Wall Interaction of Shells and Membranes with Incompressible Flows ........................30 J. N. Reddy, R.A. Arciniega Nonlinear Analysis of Composite and FGM Shells Using Tensor-Based Shell Finite Elements.............................................................................................31 R. Rolfes, G. Ernst, D. Hartung, J. Teßmer Strength of Textile Composites – A Voxel Based Continuum Damage Mechanics Approach .............................................................................................32 B. Schrefler, F. Pesavento, D. Gawin, M. Wyrzykowski Concrete at Early Ages and Beyond: Numerical Model and Validation ...............33 G. Schueller Uncertainty & Reliability Analysis of Structural Dynamical Systems..................34 Z. Waszczyszyn, L. ZiemiaĔski Neural Networks: New Results and Prospects of Applications in Structural Engineering ...........................................................................................................35 T. Ekevid, P. Kettil, H. Lane, N. E. Wiberg Computational Railway Dynamics........................................................................36 P. Wriggers and M. Hain Micro-Meso-Macro Modelling of Composite Materials .......................................37 A Computational Methods Organizers: Pereira, E.

M. Baitsch, T. Sikiwat, D. Hartmann (ID-1667) An Object-Oriented Approach to High Order Finite Element Analysis of Three-Dimensional Continua ............................................................. 38 E. Boerner, D. Mueller-Hoeppe, S. Loehnert, P. Wriggers A Finite Element Formulation Based on the Theory of a Cosserat Point - Extension to Ogden Material ..................................................................... 39

(ID-1984)

J. Eom, B. Lee A Macro Tetrahedral Element with Vertex Rotational D.O.F.s ................ 40

(ID-2456)

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S. Fialko (ID-1659) Application of Aggregation Multilevel Iterative Solver to Problems of Structural Mechanics......................................................................... 41 A. Hansen, F. Rochinha Convergence Analysis of a Domain Decomposition Method with Augmented Lagrangian Formulation..................................................................... 42

(ID-2357)

Y. Kholodov, A. Kholodov, N. Kovshov, S. Simakov, D. Severov, A. Bordonos, A. Bapayev (ID-2279) Computational Models on Graphs for Nonlinear Hyperbolic and Parabolic System of Equations .............................................................................. 43 Ü. Lepik Haar Wavelet Method for Solving Integral Equations and Evolution Equations .............................................................................................. 44

(ID-1478)

H. Ostad-Hossein, S. Mohammadi A New Approach for Elimination of Dissipation and Dispersion Errors in Particle Methods..................................................................................... 45

(ID-2368)

Z. Pawlak, J. Rakowski Solution of Stability Problem of Infinite Plate Strips................................ 46

(ID-2155)

R. Robalo, M. Coimbra, A. Rodrigues Modeling Time-Dependent Partial Equations with Moving Boundaries by the Moving Finite Element Method............................................... 47

(ID-1838)

D. Rypl (ID-2072)

Discretization of Three-Dimensional Aggregate Particles ........................ 48

K. Sato Bending of an Elliptical Plate on Elastic Foundation and Under the Combined Action of Lateral Load and In-Plane Force.................................... 49

(ID-1733)

M. Silva, E. Pereira A Modal Analysis Approach Using an Hybrid-Mixed Formulation to Solve 2D Elastodynamic Problems ................................................................... 50

(ID-2560)

L. Veiga, J. Niiranen, R. Stenberg A New Finite Element Method for Kirchhoff Plates................................. 51

(ID-1970)

E. Zieniuk, A. Boltuc Non-Element Method for Solving 2D Boundary Problems Defined on Connected Polygonal Domains Described by Navier Equation.......... 52

(ID-2406)

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B Computational Solid Mechanics Organizers: Martins, J.

S. Aitalyev, R. Baimakhan, Z. Masanov, N. Kurmanbekkyzy, G. Ylyasova (ID-2497) Computational Methods of Anisotropic Massif Mechanics Under Different Types of External Actions ..................................................................... 53 N. Alexandrova, P. Real Effect of Plastic Anisotropy on the Size of Elastic-Plastic Boundary in a Rotating Disk Problem................................................................... 54

(ID-1019)

E. Artioli, F. Auricchio, L. Veiga Numerical Testing on Return Map Algorithms for Von-Mises Plasticity with Nonlinear Hardening ..................................................................... 55

(ID-2068)

M. Asik (ID-2250)

A Model for the Analysis of Plates on a Layered Elastic Medium ........... 56

S. Benke Modeling of Solid State Transformations Using a Phase Field Model with Transformation Plasticity ................................................................... 57

(ID-1817)

S. Bosiakov, M. Zhuravkov Computer Modeling of Three-Dimensional Wave Movements in Anisotropic Elastic Environments ......................................................................... 58

(ID-1574)

R. Cardoso, V. Silva, H. Varum Visco-Elastic Regularization and Strain Softening ................................... 59

(ID-2411)

J. Cela, A. Piriz, M. Serna Numerical Simulations of the Rayleigh-Taylor Instability in Accelerated Solids ................................................................................................. 60

(ID-1108)

B. Diouf, F. Houdaigui, S. Poortmans, B. Verlinden, A. Habraken A Phenomenological Model to Simulate Mechanical Tests on Ultrafine-Grained Aluminum Produced by ECAE................................................ 61

(ID-2439)

E. Emmrich, O. Weckner The Peridynamic Equation of Motion in Non-Local Elasticity Theory ................................................................................................................... 62

(ID-1797)

A. Eraslan, E. Arslan Numerical Solution of Partially Plastic Curved Beam Problem................ 63

(ID-1232)

J. Fernandes, A. Alvim, R. Rocha A Proposal of Strain-Gage Rosette for Measurement Residual Stress Around a Circular Hole in a Plate with Circular Hole ................................ 64

(ID-1610)

J. Gómez, J. Royo, F. Martínez, E. Liarte, M. Jiménez Prediction of Dynamic Stiffness of Filled Rubber Mounts ....................... 65

(ID-1871)

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P. Grammenoudis, C. Tsakmakis (ID-1546) Size Effects in Finite Deformation Micropolar Plasticity ......................... 66 M. Grekov An Application of a Boundary Perturbation Method to Some Problems of Elasticity............................................................................................ 67

(ID-1888)

E. Grosu, I. Harari Spatial Stabilization of Semidiscrete Elastodynamics............................... 68

(ID-1005)

M. Grymer, M. Ekh, K. Runesson, T. Svedberg Modeling the Grain Size Effect Using Gradient Hardening and Damage in Crystal (Visco) Plasticity .................................................................... 69

(ID-1931)

T. Hata Stress-Focusing Effect Following Dynamically Transforming Strains in a Spherical Zirconia Inclusion............................................................... 70

(ID-1208)

E. Heikkola, S. Mönkölä, A. Pennanen, T. Rossi Controllability Method for the Solution of Linear Elastic Wave Equations............................................................................................................... 71

(ID-1469)

C. Husson, J. Richeton, S. Ahzi Development of a Flow Stress Model From Metals Using the Strain Rate/ Temperature Superposition Principle ................................................ 72

(ID-1590)

T. Kato, M. Kawahara Analysis of Elastic Body Using Kalman Filter Finite Element Method .................................................................................................................. 73

(ID-1827)

A. Lew, A. Eick Discontinuous Galerkin Methods for Nonlinear Elasticity ....................... 74

(ID-2100)

S. Marguet, P. Rozycki, L. Gornet A Rate Dependent Constitutive Model for Carbon-Fibre / EpoxyMatrix Woven Fabrics Submitted to Dynamic Loadings ...................................... 75

(ID-1870)

V. Matveyenko Solution of Viscoelasticity Problems Using Special Forms of Elastic Solutions .................................................................................................... 76

(ID-1798)

M. Mazdziarz Finite Elements Method Analysis of Influence of Contact Phenomena on Structure-Subsoil Interaction ........................................................ 77

(ID-1832)

M. Mizuno, Y. Sanomura Simulation of Inelastic Deformation of Polyethylene in Multiaxial State of Stress by Viscoelastic Constitutive Equtaion ........................................... 78

(ID-1286)

I. Pavlov Modeling of Granular Media by the 2D Discrete Lattice.......................... 79

(ID-1982)

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M. Rabahallah, B. Bacroix, S. Bouvier, T. Balan (ID-2510) Crystal Plasticity Based Identification of Anisotropic Strain Rate Potentials for Sheet Metal Forming Simulation .................................................... 80 G. Romano, M. Diaco, R. Barretta A Geometric Approach to the Algorithmic Tangent Stiffness .................. 81

(ID-2044)

V. Sadovskii, O. Sadovskaya Parallel Computation of 3D Problems of the Dynamics of ElasticPlastic Granular Material Under Small Strains ..................................................... 82

(ID-1277)

Z. Uthman, H. Askes A Hyperelastodynamic Ale Formulation Based on Spatial and Material Forces...................................................................................................... 83

(ID-1328)

Y. Zhuk Monoharmonic Approach to Investigation of Heat Generation in the Viscoplastic Solids Under Harmonic Loading ................................................ 84

(ID-1073)

C Coupled Problems Organizers: Rodrigues, H.

S. Bargmann, P. Steinmann (ID-1021) A Continuous Galerkin Finite Element Method for Thermoelasticity Without Energy Dissipation ...................................................... 85 J. Blaszczuk, Z. Domanski The Model Coupling Liquid Bridge Between Ellipsoidal Grains ............. 86

(ID-2171)

M. Brehm, C. Bucher Reliability of Wavelet Packet System Identification................................. 87

(ID-1652)

L. Ecsi, P. Elesztos An Attempt to Simulate More Precisely the Behavior of a Solid Body Using New Energy Conservation Equation ................................................. 88

(ID-1112)

J. Gawinecki Mathematical Aspects of the Initial-Boundary Value Problems in Nonlinear Thermoelasticity of Simple and Non-Simple Materials ....................... 89

(ID-1361)

M. Guerich, S. Chaabane Effect of Parameter Uncertainties on a Vibro-Acoustic Design................ 90

(ID-2610)

G. Khodabakhshi, V. Nassehi, L. Shojai, R. Wakeman A Numerical Method for Solid-Liquid Interaction ................................... 91

(ID-2533)

A. Kiselev, O. Nekhaeva, A. Privalsky Computational Simulation of Irreversible Deforming and Fracture of Damageable Solids and Structures .................................................................... 92

(ID-1121)

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M. Kropac, J. Murin (ID-2469) Solution of the Coupled Light-Mechanical Problems ............................... 93 I. Kurzhöfer, J. Schröder, H. Romanowski Simulation of the Ferroelectric Hysteresis Using a Hybrid Finite Element Formulation ............................................................................................. 94

(ID-1322)

R. Lanrivain, L. Silva, T. Coupez A Two-Phase Numerical Modelling of the Liquid Solid Transition in Polymer Processing ........................................................................................... 95

(ID-1338)

C. Leppert, D. Dinkler A Two-Phase Model for Granular Flows Applied to Avalanches............. 96

(ID-2596)

L. Margetts, I. Smith, J. Leng Parallel 3D Finite Element Analysis of Coupled Problems....................... 97

(ID-2218)

P. Porta, C. Vega Numerical Simulation of Rubber Curing Process with Application to Bladders Manufacture ....................................................................................... 98

(ID-1360)

E. Stupak, R. Kacianauskas, A. Dementjev, A. Jovaisa Coupled Finite Element Analysis of Composite Laser Rods Thermal Characteristics Under Longitudinal Diode Pumping .............................. 99

(ID-1475)

D Computational Structural Mechanics Organizers: Ambrósio, J.

T. Akis, T. Tokdemir, C. Yilmaz (ID-2480) On the Modeling of Nonplanar Shear Walls in Shear Wall-Frame Building Structures.............................................................................................. 100 F. Alamo, H. Weber, H. Espinoza Directional Drillstring Dynamics ............................................................ 101

(ID-2624)

M. Alinia, A. Rahai, S. Kazemi Mvm Energy Method for Buckling Analysis of Tapered Plates ............. 102

(ID-2651)

A. Alvarenga, R. Silveira Considerations on Advanced Analysis of Steel Portal Frames ............... 103

(ID-2119)

N. Azevedo, J. Lemos, J. Almeida A Discrete Element Model for the Fracture Analysis of Reinforced Concrete............................................................................................ 104

(ID-2569)

A. Baptista Analytical Criteria for the Evaluation of the Internal Forces at the Elastic and Plastic Limit States of Lozenge and Triangular Cross-Sections ....... 105

(ID-2700)

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M. Benachour, A. Hadjoui, M. Benguediab, N. Benachour, F. Hadjoui (ID-2252) Comparative Study of Aluminum Alloy Plate 2024/7050 Under the Effect of Internal Damping............................................................................ 106 A. Correia, F. Virtuoso Nonlinear Analysis of Space Frames ...................................................... 107

(ID-2648)

A. Davaran, A. Kashefi, S. Amiri Investigation of Shear Wall Behavior with Composite Boundary Elements .............................................................................................................. 108

(ID-1504)

M. Fard, S. Amiri, A. Kashefi An Investigation on Dynamic Behavior of Shear Walls on Flexible Foundation............................................................................................. 109

(ID-1096)

T. Fiedler, A. Öchsner, J. Grácio Influence of the Morphology of Adhesive Joining on the Mechanical Properties of Periodic Metal Hollow-Sphere-Structures.................. 110

(ID-1084)

H. Figueiredo, R. Calçada, R. Delgado Dynamic Behaviour of a Composite Twin Girder Bridge in a High Speed Interoperable Line..................................................................................... 111

(ID-2620)

A. Foces, J. Garrido, A. Moreno A Finite Element Model for Beam to Column Bolted Connections with Semi Rigid Behaviour ................................................................................. 112

(ID-1702)

G. Garcea, A. Madeo Rational Strain Measures - the Implicit Corotational Method................. 113

(ID-2353)

V. Kulbach, J. Idnurm Discrete and Continuous Analysis of Different Cable Structures ........... 114

(ID-1662)

Q. Li, V. Iu Three-Dimensional Vibration Analysis of Crystal Plates Via Ritz Method ................................................................................................................ 115

(ID-1072)

G. Lykidis, K. Spiliopoulos A 3D Solid Finite Element for Reinforced Concrete Analysis Allowing Slippage of Reinforcement .................................................................. 116

(ID-2233)

M. Lyly, J. Niiranen, R. Stenberg Some New Results on Mitc Plate Elements ............................................ 117

(ID-2000)

J. Neto, A. Assan Nonlinear Analysis of Reinforced Concrete Beams Considering the Slip Between Steel and Concrete................................................................... 118

(ID-1341)

J. Rijn Novel Semi-Analytical Methodology to Determine Model Parameters for a Simple Finite Element Bolt Model........................................... 119

(ID-2276)

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W. Rust, J. Overberg (ID-1803) Accounting for Fuselage Instabilities in the Coarse Model of an Aircraft Fuselage by Means of a Material Law ................................................... 120 J. Sosa, D. Owen, E. Neto, N. Petrinic Modelling of Reinforced Materials by a Subcycling Algorithm ............. 121

(ID-1530)

K. Spiliopoulos, T. Patsios Limit Analysis of Cable-Tied Structures................................................. 122

(ID-2329)

R. Steenbergen, J. Blaauwendraad Smart Super Elements in Slender Structures Subjected to Wind ............ 123

(ID-1516)

B. Trogrlic, A. Mihanovic, Z. Nikolic Nonlinear Analysis of Space R/C Frames with Non-Uniform Torsion ................................................................................................................ 124

(ID-2121)

G. Turvey, P. Wang An Fe Analysis of the Stresses in Pultruded GRP Single-Bolt Tension Joints and Their Implications for Joint Design ...................................... 125

(ID-1921)

J. Veiga, A. Henriques, J. Delgado An Efficient Evaluation of Structural Safety Applying Perturbation Techniques...................................................................................... 126

(ID-2688)

G. Yaoqing, L. Ke Semi-Analytical Analysis of Super Tall Building Bundled-Tube Structures............................................................................................................. 127

(ID-1214)

N. Zivaljic, A. Mihanovic, B. Trogrlic Large Displacements in Nonlinear Numerical Analyses for Cable Structures............................................................................................................. 128

(ID-1892)

E Industrial Applications Organizers: Pina, C.

V. Abadjiev, D. Petrova, E. Abadjieva (ID-1325) Mathematical Modeling for Synthesis and Design of NonOrthogonal Worm Gears with a Straight-Line Tooth Contact............................. 129 N. Ahmed, A. Mitrofanov, V. Silberschmidt, V. Babitsky Computational Modeling of Ultrasonically Assisted Turning................. 130

(ID-1675)

G. Anjos, R. Cunha, P. Kuklik, B. Miroslav Numerical Evaluation of Bored Piles in Tropical Soils by Means of the Geotechnical Engineering “Geo4” Fine Software..................................... 131

(ID-1694)

B. Atahanovich, B. Gayratovich Kinematics and Force Interaction of Screw Shaft with Variable Screw Course....................................................................................................... 132

(ID-1176)

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J. Barglik, B. Ulrych (ID-2343) Optimal Construction of the Thermo-Elastic Actuator ........................... 133 M. Barros, C. Oliveira Determination of Moment-Curvature Diagrams and MomentDeflection Curves in Reinforced Concrete Beams .............................................. 134

(ID-2490)

P. Berke, T. Massart Numerical Simulation of the Nanoindentation Experiment: Sensitivity Analysis of the Experimental Parameters.......................................... 135

(ID-2274)

M. Chuda-Kowalska, A. Garstecki, Z. Pozorski Numerical Evaluation of Wrinkling Stress in Sandwich Panels ............. 136

(ID-1599)

I. Conde, M. Jiménez, J. Bielsa, E. Liarte, M. Laspalas Application of Fea As a Predictive Tool in the Corrugated Paperboard Industry ............................................................................................ 137

(ID-1881)

J. Dib, F. Bilteryst, J. Batoz, I. Lewon Implementation of 3D Homogenization Techniques for the Thermo-Elastic FEM Analysis of Brazed Plate-Fin Heat Exchangers ................ 138

(ID-2631)

R. Hoffmann Hierarchical Treecode for Optimized Collision Checking in Dem Simulations – Application on Electrophotographic Toner Simulations .............. 139

(ID-1003)

J. Kazanecki, Z. Pater, J. Bartnicki 3D FEM Analysis of Basic Process Parameters in Rotary Piercing Mill ...................................................................................................................... 140

(ID-2559)

M. Khoshravan, J. Shahimehr Numerical Evaluation of the Influence of Stiffener Rings on the Critical Buckling Pressure of the Vessels............................................................ 141

(ID-2239)

G. Kiziltas Topology Optimization and Fabrication of Multi-Material Dielectrics for Antenna Performance Improvements .......................................... 142

(ID-1754)

P. Klinge Reduction Method Independent Substructure Synthesis ......................... 143

(ID-1882)

M. Lancini, A. Magalini, D. Vetturi Discrete Models for the Simulation of Rubber Components Dynamics............................................................................................................. 144

(ID-1321)

A. Liao, H. Zhang, C. Wu Contact Analysis of Impeller-Shaft Assembly and Reasonably Designing the Amount Interference of Turbocompressors.................................. 145

(ID-1102)

Y. Sheng, C. Lawrence, B. Briscoe Study of the Stress Induced Granular Consolidation Process by 3D DEM Simulation............................................................................................ 146

(ID-1483)

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J. Shiau, C. Smith (ID-2577) Numerical Analysis of Passive Earth Pressures with Interfaces ............. 147 W. Wang, J. Wang, W. Shen, Z. Xu, W. Fan Three-Dimensional Finite Element Analysis of a Multi-Propped Deep Excavation in Shanghai Soft Deposit......................................................... 148

(ID-1273)

M. Zhuravkov, S. Bosiakov, S. Pronckevich The Computer Analysis of the Temperature Fields Arising in Bearing Node at Rotation of a Rotor ................................................................... 149

(ID-1579)

MS.01 Acoustics Structural Interactions Organizers:Tadeu, A.

V. Decouvreur, P. Ladevèze, P. Bouillard (ID-2023) Updating 3D Acoustic Models with the Constitutive Law Error Method. a Two Step Approach for Absorbing Material Characterization........... 150 J. García-Andujar, L. Fritz, J. López-Díez Analysis of Fluid-Structure Coupling by Statistical Energy Analysis............................................................................................................... 151

(ID-2109)

K. Ito, J. Toivanen Efficient Iterative Solution of Time-Harmonic Scattering by Objects in Layered Fluid ..................................................................................... 152

(ID-1608)

B. Neuhierl, E. Rank Computational Aeroacoustics by Coupling the Finite-Element and the Lattice-Boltzmann-Method............................................................................ 153

(ID-1186)

A. Panteghini, F. Genna, E. Piana Analysis of a Perforated Panel for the Correction of Low Frequency Resonances in Domestic Rooms........................................................ 154

(ID-1132)

A. Pereira, A. Tadeu Sound Insulation Provided by a Multi-Layer System Containing a Heterogeneity: a BEM Approach ........................................................................ 155

(ID-1252)

B. Pluymers, W. Desmet, D. Vandepitte, P. Sas An Efficient Wave Based Method for Steady-State VibroAcoustic Transmission Calculations ................................................................... 156

(ID-1864)

P. Santos, A. Tadeu Modeling Sound Radiation by Structures Caused by a Ground Impact Load: a BEM Approach........................................................................... 157

(ID-2270)

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MS.02 Smart Structures and Materials Organizers: Suleman, A., Benjeddou, A.

I. Arias, S. Serebrinsky, M. Ortiz (ID-1763) Cohesive Model of Electromechanical Fatigue for Ferroelectric Materials and Structures ...................................................................................... 158 H. Atzrodt, S. Herold, D. Mayer Simulation of Active Systems in a NVH Full Car Model ....................... 159

(ID-1500)

F. Auricchio, L. Petrini, A. Reali Toward an Exhaustive Modeling of the Macroscopic Behaviour of Shape Memory Alloys......................................................................................... 160

(ID-1911)

A. Benjeddou, J. Ranger Vibration Damping Using Resonant Shunted Shear-Mode Piezoceramics...................................................................................................... 161

(ID-1207)

I. Figueiredo, G. Stadler Optimal Control of Piezoelectric Anisotropic Plates .............................. 162

(ID-2619)

W. Gambin, A. Zarzycki Residual Internal Forces in Stiffened Thermal-Bimorph Actuator After Forming Process......................................................................................... 163

(ID-1559)

A. Gomes, A. Suleman Spectral Level Set Methodology in the Optimal Design of Adaptive Aeroelastic Structures .......................................................................... 164

(ID-1523)

D. Marinova, D. Lukarski, G. Stavroulakis, E. Zacharenakis Nondestructive Identification of Defects for Smart Plates in Bending Using Genetic Algorithms .................................................................... 165

(ID-2195)

J. Pereira, M. Pacheco, P. Pacheco, R. Aguiar, M. Savi Modeling Shape Memory Alloy Plane Truss Structures Using the Finite Element Method ........................................................................................ 166

(ID-2097)

G. Pirge, N. Kiliç, O. Uçan, S. Altintas Evaluation of Nimnga Magnetic Shape Memory Alloys Using Cellular Neural Networks.................................................................................... 167

(ID-2575)

C. Ramos, R. Oliveira, R. Campilho, A. Marques Modelling of Fibre Bragg Grating Sensor Plates .................................... 168

(ID-2645)

M. Trindade, A. Benjeddou Refined Finite Element Model for Vibration Analysis of Sandwich Beams with Shear Piezoelectric Actuators and Sensors ..................... 169

(ID-1309)

S. Ueda, H. Kondo Thermoelectromechanical Response of a Parallel Crack in a Functionally Graded Piezoelectric Strip.............................................................. 170

(ID-1724)

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Y. Uetsuji, E. Nakamachi (ID-1833) Multi-Scale Finite Element Modeling of Piezoelectric Materials by a Crystallographic Homogenization Method.................................................. 171 H. Uyanik, Z. Mecitoglu Vibration Control of a Laminated Composite Plate Subjected to Blast Loading ...................................................................................................... 172

(ID-1490)

C. Vasques, R. Moreira, J. Rodrigues Experimental Identification of GHM and ADF Parameters for Viscoelastic Damping Modeling ......................................................................... 173

(ID-1960)

L. Wang, R. Melnik Numerical Aspects of Modelling Thermo-Mechanical Wave Propagation with Phase Transformations ............................................................ 174

(ID-1438)

A. Zabihollah, R. Ganesan, R. Sedaghati Analysis and Design Optimization of Smart Laminated Composite Beams Using Layerwise Theory.......................................................................... 175

(ID-2191)

MS.03 Asphalt Mechanics and Pavement Engineering Organizers: Lackner, R., Blab, R.

H. Benedetto, B. Delaporte, C. Sauzéat, M. Neifar (ID-2475) A Thermo-Viscoplastic Model for Bituminous Materials....................... 176 J. Croll From Asphalt to the Arctis: New Insights Into ThermoMechanical Ratchetting Processes....................................................................... 177

(ID-1149)

A. Holanda, E. Junior, T. Araujo, L. Melo, F. Junior, J. Soares An Object-Oriented System for Finite Element Analysis of Pavements............................................................................................................ 178

(ID-2557)

B. Lenhof, P. Kettil, K. Runesson, N. Wiberg On the Treatment of Convective Terms in Coupled HydroMechanics for Porous Media Subjected to Dynamic Loading ............................ 179

(ID-1804)

A. Molenaar Asphalt Mechanics, a Key Tool for Improved Pavement Performance Predictions...................................................................................... 180

(ID-1187)

MS.04 Biomechanical Simulations Organizers: Eriksson, A.

P. Blanco, I. Larrabide, S. Urquiza, R. Feijóo (ID-1858) Sensitivity of Blood Flow Patterns to the Constitutive Law of the Fluid .................................................................................................................... 181

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xix

A. Completo, F. Fonseca, J. Simoes (ID-2035) The Influence of Stem Design on Strains and Micromotion in Revision Total Knee Arthroplasty: Finite Element Analysis .............................. 182 A. Eriksson Optimization of Targeted Movements .................................................... 183

(ID-1192)

B. Heidari, D. Fitzpatrick, D. Mccormack, K. Synnott Persistence of Axial Rotation in Idiopathic Scoliosis Due to the Structural Changes of the Intervertebral Disc ..................................................... 184

(ID-1697)

H. Iwase, R. Himeno Numerical Simulation of Hemodynamcs in a Cerebral Artery ............... 185

(ID-1571)

A. John, M. Mazdziarz, J. Rojek, J. Telega, P. Maldyk Analysis of Some Contact Problems in Human Joints After Arthroplasty......................................................................................................... 186

(ID-1851)

P. Khayyer, A. Zolghadrasli, F. Daneshmand, A. Najafi Cardiovascular Disease Diagnosis Before Birth by Means of Chaotic Analysis on the Heart Rate Signal.......................................................... 187

(ID-1343)

I. Larrabide, R. Feijóo, E. Taroco, A. Novotny Configurational Derivative As a Tool for Image Segmentation.............. 188

(ID-1845)

G. Link, M. Kaltenbacher, R. Lerch Numerical Simulations to Analyze and Optimize the Human Substitute Voice .................................................................................................. 189

(ID-1670)

C. Nabais, R. Guedes, J. Simoes The Thaw Time of Frozen Cancellous Bone for Mechanical Testing................................................................................................................. 190

(ID-1843)

T. Olsson, J. Martins Modeling of Passive Behavior of Soft Tissues Including Viscosity and Damage......................................................................................................... 191

(ID-2414)

M. Racila, J. Crolet Nano and Macro Structure of Cortical Bone: Numerical Investigations....................................................................................................... 192

(ID-2288)

S. Rues, H. Schindler, K. Schweizerhof, J. Lenz Calculation of Muscle and Joint Forces in the Masticatory System........ 193

(ID-2403)

V. Vondrák, J. Rasmussen, M. Damsgaard, Z. Dostál The Algorithms of Mathematical Programming in Muscle Recruitment and Muscle Wrapping Problems ..................................................... 194

(ID-2177)

C. Yu, Y. Wang, Y. Liu, X. Sun Three-Dimensional Numerical Simulation of Airflow and Vibration Analysis for Upper Airway of Humans............................................... 195

(ID-2011)

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MS.05 Computational Biomechanics Organizers: Rodrigues, H.

D. Balzani, J. Schröder, D. Gross (ID-1317) Computer Simulation of Anisotropic Damage and Residual Stresses in Atherosclerotic Arteries..................................................................... 196 J. Folgado, R. Andrade, P. Fernandes Computational Study on Stability and Bone Remodeling for a Hip Replacement Using a "Minimal Invasive" Femoral Stem ................................... 197

(ID-2493)

A. Fritsch, L. Dormieux, C. Hellmich Porous Polycrystals Built Up by Uniformly and Axisymmetrically Oriented Needles: Homogenization of Elastic Properties ................................... 198

(ID-1549)

U. Görke, H. Günther, M. Wimmer A Poroviscoelastic Overlay Model for Finite Element Analyses of Articular Cartilage at Large Strains..................................................................... 199

(ID-1924)

R. Guimaraes, S. Taylor, G. Blunn A Finite Element Study of Strain Distribution in an Instrumented Knee Prosthesis for Full Force Measurement in Vivo......................................... 200

(ID-2341)

I. Larrabide, R. Feijóo, E. Taroco, A. Novotny Topological Derivative Applied to Image Enhancement ........................ 201

(ID-1850)

J. Mcgarry, A. Pathak, L. Valdevit, A. Evans, P. Mchugh, R. Mcmeeking Determination of Contractile Forces Generated by Actin Fibre Networks ............................................................................................................. 202

(ID-2219)

R. Ruben, P. Fernandes, J. Folgado, H. Rodrigues Hip Prosthesis Design Using a Multi-Criteria Formulation .................... 203

(ID-2476)

S. Simakov, A. Kholodov, Y. Kholodov, A. Nadolskiy, A. Shushlebin Global Dynamical Model of the Cardiovascular System ........................ 204

(ID-1464)

S. Simakov, A. Kholodov, Y. Kholodov, A. Nadolskiy, A. Shushlebin Computational Study of the Vibrating Disturbances to the Lung Function............................................................................................................... 205

(ID-1467)

B. Simon, P. Rigby, T. Newberg, R. Park, S. Williams Abaqus-Based, Coupled Porohyperelastic Transport Finite Element Models for Soft Hydrated Biological Structures ................................... 206

(ID-1736)

J. Soares, J. Moore, K. Rajagopal Theoretical Modeling of Cyclically Loaded, Biodegradable Cylinders ............................................................................................................. 207

(ID-1979)

K. Subramani, M. Oliveira, J. Simoes Validation of a Non-Linear Wear Model for UHMWPE ........................ 208

(ID-1699)

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xxi

L. Zach, S. Konvickova, P. Ruzicka (ID-1782) FEA of Human Knee Joint Replacement Using Real Bone Models ....... 209 MS.06 Boundary Elements Organizers: Leitão, V.

G. Dziatkiewicz, P. Fedelinski (ID-1318) Subregion Boundary Element Method for Piezoelectric Structures........ 210 M. Guminiak, R. Sygulski Vibrations of System of Plates Immersed in Fluid by Bem .................... 211

(ID-1260)

S. Gupta, G. Degrande, H. Chebli, D. Clouteau, M. Hussein, H. Hunt A Coupled Periodic FE-BE Model for Ground-Borne Vibrations From Underground Railways .............................................................................. 212

(ID-2365)

A. Iban, J. Garcia-Teran, I. Rico On the Application of the Bem to Rubber-Elastic Materials................... 213

(ID-2163)

J. Rungamornrat Modeling of Darcy's Flow in Generally Anisotropic Porous Media Containing Discontinuity Surface by SGBEM-FEM Coupling........................... 214

(ID-1568)

E. Sapountzakis, V. Protonotariou A Displacement Solution to Transverse Shear Loading of Beams by BEM ............................................................................................................... 215

(ID-1002)

M. Schanz, T. Rüberg Non-Conforming Coupled Time Domain Boundary Element Analysis............................................................................................................... 216

(ID-2103)

MS.08 Advanced Composites Organizers: Reddy, J. N., Mota Soares, C.M., Benjeddou, A.

F. Ashida, S. Sakata, K. Matsumoto (ID-1749) Control of Thermal Stress in a Piezoelectric Composite Disk by a Stepwise Applied Electric Potential Distribution ................................................ 217 J. Baucom, M. Qidwai, J. Thomas Mitigation of Free-Edge Effects by Meso-Scale Structuring .................. 218

(ID-2372)

J. Bekuit, D. Oguamanam, O. Damisa A Quasi-2D Finite Element Formulation for Static and Dynamic Analysis of Sandwich Beams .............................................................................. 219

(ID-2151)

J. Belinha, L. Dinis A Numerical Comparison of Distinct Meshless Methods for the Analysis of Composite Laminates....................................................................... 220

(ID-1521)

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Table of Contents

A. Benjeddou, S. Vijayakumar, I. Tawfiq (ID-1178) A New Damage Identification and Quantification Indicator for Piezoelectric Advanced Composites ................................................................... 221 A. Blom, S. Setoodeh, J. Hol, Z. Gürdal Design of Variable-Stiffness Conical Shells for Maximum Fundamental Frequency ...................................................................................... 222

(ID-1941)

P. Camanho, J. Mayugo, P. Maimí, C. Dávila A Micromechanics Based Damage Model for the Strength Prediction of Composite Laminates .................................................................... 223

(ID-1661)

J. Cardoso, N. Benedito, A. Valido Finite Element Analysis of Geometrically Nonlinear Thin-Walled Composite Laminated Beams.............................................................................. 224

(ID-2555)

J. Cardoso, A. Valido Design Sensitivity Analysis of Composite Thin-Walled Profiles Including Torsion and Shear Warping................................................................. 225

(ID-2626)

A. Carpentier, J. Barrau, L. Michel, S. Grihon Buckling Optimisation of Composite Panels Via Lay-Up Tables........... 226

(ID-2530)

J. Cognard, R. Créac'Hcadec Numerical Approach for the Design of Adhesively-Bonded Assemblies .......................................................................................................... 227

(ID-1335)

G. Dvorak, Y. Bahei-El-Din Enhancement of Blast Resistance of Sandwich Plates ............................ 228

(ID-2689)

C. Friebel, I. Doghri, V. Legat Mechanics and Acoustics of Viscoelastic Composites by a MicroMacro Mean-Field Approach .............................................................................. 229

(ID-1337)

P. Fuschi, A. Pisano Numerical Evaluation of Upper and Lower Bounds to the Collapse Limit Load for Composite Laminates................................................... 230

(ID-2346)

K. Kalnins, J. Auzins, R. Rikards Material Degradation Assessment for Stiffened Composite Shells Using Metamodelling Approach ......................................................................... 231

(ID-2049)

M. Karama, K. Afaq, S. Mistou A New Model for the Behaviour of the Multi-Layer Material Interfaces ............................................................................................................. 232

(ID-1981)

L. Kärger, J. Baaran, J. Teßmer Fe-Tool Codac for an Efficient Simulation of Low-Velocity Impacts on Composite Sandwich Structures ....................................................... 233

(ID-1101)

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xxiii

J. Kato, A. Lipka, E. Ramm (ID-2603) Preliminary Investigation for Optimization of Fiber Reinforced Cementitious Composite Structures .................................................................... 234 P. Kere, M. Lyly On Post-Buckling Analysis and Experimental Correlation of Cylindrical Composite Shells .............................................................................. 235

(ID-1801)

M. Laspalas, S. Maynar, C. Crespo, M. Jiménez, B. García Application of Failure Criteria to Short Fiber Reinforced Composites and Experimental Validation ........................................................... 236

(ID-1891)

C. Lopes, Z. Gürdal, P. Camanho Tow-Placed, Variable-Stiffness Composite Panels: Damage Tolerance Improvements Over Traditional Straight-Fibre Laminates................. 237

(ID-2153)

F. Melo, R. Moreira, J. Rodrigues A Mixed-Formulation Four-Node Rectangular Element in the Modeling of Laminate Composite Beam Stuctures............................................. 238

(ID-2052)

T. Messager, P. Chauchot, B. Bigourdan Optimal Design of Stiffened Composite Underwater Hulls .................... 239

(ID-1333)

J. Moita, C. Soares, C. Soares Higher Order Model for Analysis of Magneto-Electro-Elastic Plates ................................................................................................................... 240

(ID-1115)

F. Moleiro, C. Soares, C. Soares, J. Reddy Mixed Least-Squares Finite Element Model for the Static Analysis of Laminated Composite Plates ............................................................ 241

(ID-1508)

M. Nader, H. Garssen, H. Irschik Applications of Distributed Piezoelectric Electrode Patches for Active Noise and Vibration Control.................................................................... 242

(ID-1289)

E. Ng, A. Suleman Computational Elastoplatic Modeling of Multi-Phase FiberReinforced Composites ....................................................................................... 243

(ID-1554)

K. Niessen, J. Moreno, R. Gadow Evolution and Analysis of Stresses in Thixoforged Metal Matrix Composites .......................................................................................................... 244

(ID-2236)

M. Pietrzakowski Piezoelectric Control of Composite Plate Vibration: Effect of Electric Field Distribution ................................................................................... 245

(ID-1222)

H. Santos, C. Soares, C. Soares, J. Reddy A Finite Element Model for the Analysis of 3D Axisymmetric Laminated Shells with Embedded Piezoelectric Sensors and Actuators ............. 246

(ID-1082)

xxiv Table of Contents

J. Santos, H. Lopes, M. Vaz, C. Soares, C. Soares, M. Freitas (ID-1239) Damage Localization in Laminated Composite Plates Using Double Pulse-Electronic Holographic Interferometry ......................................... 247 E. Sapountzakis Shear Deformation Effect in Nonlinear Analysis of Spatial Composite Beams in Variable Axial Loading by BEM ...................................... 248

(ID-1139)

C. Schuecker, H. Pettermann Constitutive Ply Damage Modeling, FEM Implementation, and Analyses of Laminated Structures....................................................................... 249

(ID-1759)

W. Wagner, C. Balzani Simulation of Delamination in Stringer Stiffened FiberReinforced Composite Shells .............................................................................. 250

(ID-1204)

MS.10 Computational Fracture Mechanics Organizers: Leung, A.

A. Andreev (ID-1632) Development of Methods of Numerical Solution of Singular Integro-Differential Equations for Solid Mechanics Problems ........................... 251 I. Arias, J. Knap, V. Chalivendra, S. Hong, M. Ortiz, A. Rosakis Validation of Large Scale Simulations of Dynamic Fracture.................. 252

(ID-2247)

E. Giner, A. Vercher, O. González, J. Tarancón, F. Fuenmayor Crack Growth in Fretting-Fatigue Problems Using the Extended Finite Element Method ........................................................................................ 253

(ID-2196)

S. Jox, P. Dumstorff, G. Meschke Aspects of Crack Propagation and Hygro-Mechanical Coupling Using X-FEM ...................................................................................................... 254

(ID-2536)

S. Klishin Destruction of Rocks by Directional Hydraulic Fracturing on the Basis of Models of Plasticity with Internal Variables ......................................... 255

(ID-1287)

N. Krukova, I. Lavit The Finite-Element Method in Linear Fracture Mechanics Problems.............................................................................................................. 256

(ID-1638)

R. Larsson, M. Fagerström Finite Deformation Fracture Modelling of a Thermo-Mechanical Cohesive Zone..................................................................................................... 257

(ID-1471)

N. Myagkov, T. Shumikhin Critical Behavior and Energy Dependence of Mass Distribution in High-Velocity Impact Fragmentation.................................................................. 258

(ID-1145)

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xxv

S. Reese, P. Wriggers (ID-2419) One 3D Adaptive Fragmentation Procedure for the Explicit Simulation of Brittle Material Cracking .............................................................. 259 G. Tsamasphyros, T. Papathanassiou Finite Element Analysis of Cracked Plates with Circular Stress Raisers Used for S.I.F. Reduction ....................................................................... 260

(ID-2441)

N. Vlasov, I. Fedik Simulation of Materials Damage in the Field of Internal Stresses .......... 261

(ID-2230)

MS.11 Computational Mathematics Organizers: Stenberg, R., Figueiredo, I.

F. Bourquin (ID-2063) Computational Methods for the Fast Boundary Stabilization of Flexible Plates ..................................................................................................... 262 L. Costa, I. Figueiredo, P. Oliveira Sensor and Actuator Capabilities of a Laminated Piezoelectric Plate Model ......................................................................................................... 263

(ID-1835)

A. Loula, M. Correa Numerical Analysis of Stabilized Finite Element Methods for Darcy Flow .......................................................................................................... 264

(ID-2187)

C. Lovadina, R. Stenberg An Error Estimator for the Reissner-Mindlin Plate Problem .................. 265

(ID-1223)

N. Troyani, E. Gutiérrez A Convergence Study of the Numerical Solution of Two BiDirectionally Coupled Partial Differential Equations in Thermoelectricity ........ 266

(ID-1069)

MS.12 Computational Methods for Anisotropic Material Behaviour at Large Strains Organizers: Sansour, C.

M. Böl, S. Reese (ID-1440) Numerical Simulations of Rubber-Like Materials Under Changing Directions ............................................................................................................ 267 A. Bucher, U. Görke, R. Kreißig About an Efficient and Consistent Numerical Strategy for the Solution of the Initial-Boundary Value Problem................................................. 268

(ID-1920)

H. Hein Vibrations of Composite Beams with Multiple Delaminations............... 269

(ID-1482)

xxvi Table of Contents

I. Karsaj, C. Sansour, J. Soric (ID-2382) Computational Aspects of Anisotropic Finite Strain Plasticity Based on the Multiplicative Decomposition........................................................ 270 A. Menzel Adaptation of Biological Tissues - a Fibre Reorientation Model for Orthotropic Multiplicative Growth ................................................................ 271

(ID-1883)

J. Schröder, P. Neff Polyconvex Anisotropic Hyperelastic Energies ...................................... 272

(ID-1589)

J. Wang, V. Levkovitch, B. Svendsen Micromechanically Motivated Phenomenological Modeling of Induced Flow Anisotropy and Its Application to Sheet Forming Processes........ 273

(ID-1486)

MS.13 Computational Modelling of Masonry Structures Organizers: Lourenço, P.

A. Barbieri, A. Cecchi, A. Tommaso (ID-1199) 3D Homogenization Procedure for Load Bearing Masonry Columns .............................................................................................................. 274 C. Calderini, S. Lagomarsino Non Linear Modelling of Masonry Structures Under Cyclic Loads ....... 275

(ID-1809)

S. Casolo, C. Sanjust Macroscale Modelling of Structured Materials with Damage by a Specific Rigid Element Model ............................................................................ 276

(ID-1862)

A. Cecchi Plate Micromechanical Models for 3D Periodic Brickworks.................. 277

(ID-1027)

A. Cecchi, G. Milani, A. Tralli Limit Analysis of Out-Of-Plane Loaded Running Bond Masonry Walls Under Mindlin-Reissner Plate Hypotheses ............................................... 278

(ID-1140)

M. Malena, A. Bilotta, A. Lanzo Nonlinear Analysis of Brittle Materials .................................................. 279

(ID-1762)

T. Massart, R. Peerlings, M. Geers Damage Localisation in Computational Homogenisation of Masonry and Its Incorporation in a Two-Scale Computational Framework........ 280

(ID-2179)

G. Milani, P. Lourenço, A. Tralli 3D Homogenized Limit Analysis of Masonry Buildings Subjected to Horizontal Loads ............................................................................................. 281

(ID-1131)

F. Peña, P. Lourenço, J. Lemos Modelling the Dynamic Behaviour of Masonry Walls As Rigid Blocks.................................................................................................................. 282

(ID-1256)

Table of Contents xxvii

G. Zingone, G. Canio, L. Cavaleri (ID-2514) On the Improvement of Monumental Structure Safety: a Case Study.................................................................................................................... 283 A. Zucchini, P. Lourenço Homogenization of Masonry Using a Micro-Mechanical Model: Compressive Behaviour....................................................................................... 284

(ID-1056)

MS.14 Computational Stochastic Failure Mechanics Organizers: Gutierrez, M. A.

R. Jimenez-Rodriguez, L. Lacoma (ID-1756) Uncertainty Characterization and Settlement Analyses: the Importance of Distribution Types........................................................................ 285 M. Vorechovsky, R. Chudoba, J. Jerábek Adaptive Probabilistic Modeling of Localization, Failure and Size Effect of Quasi-Brittle Materials ......................................................................... 286

(ID-1885)

MS.15 Computational Stochastic Structural and Uncertainty Analysis Organizers: Schueller, G.

D. Alvarez (ID-1625) On the Use of Infinite Random Sets for Bounding the Probability of Failure in the Case of Parameter Uncertainty.................................................. 287 J. Colliat, M. Krosche, M. Krosche, M. Hautefeuille, A. Ibrahimbegovic, R. Niekamp, H. Matthies (ID-2470) Stochastic Analysis of Coupled Nonlinear Thermo-Mechanical Problems: SFEM Model ...................................................................................... 288 J. Crempien-Laborie Response of a Single Degree of Freedom Elastic Perfectly Plastic System Under Non-Stationary Gaussian Seismic Excitation .............................. 289

(ID-1064)

D. Degrauwe, G. Roeck, G. Lombaert Fuzzy Frequency Response Function of a Composite Floor Subject to Uncertainty by Application of the GĮd Algorithm............................. 290

(ID-1999)

J. Fernandes, M. Pinho, A. Alvim, A. Pithon Comparative Numerical Evaluation of Angra I Auxiliary Feedwater System Reliability by the Method of Suplementary Variables .......... 291

(ID-1611)

M. Galffy, M. Baitsch, A. Wellmann-Jelic, D. Hartmann Lifetime Estimation of Vertical Bridge Tie Rods Exposed to Wind-Induced Vibrations .................................................................................... 292

(ID-1088)

xxviii Table of Contents

G. Giunta, E. Carrera (ID-1544) Stochastic Static Analyses of FE Models by Means of Newton's Series Expansions................................................................................................ 293 M. Grigoriu A Galerkin Solution for Stochastic Algebraic Equations ........................ 294

(ID-1262)

R. Iwankiewicz, M. Vasta Approximate Method for Probability Density of the Response of a Linear Oscillator to a Non-Poisson Impulse Process........................................ 295

(ID-1906)

H. Kang, Y. Lee, J. Huh, B. Kwak Comparative Study of RBDO Algorithms Based on Form and FAMM................................................................................................................. 296

(ID-2625)

M. Mailhé, S. Chaabane, F. Léné, G. Duvaut, S. Grihon Probabilistic Analysis and Optimization of a Fully Composite Cylinder............................................................................................................... 297

(ID-2589)

A. Martowicz, L. Pieczonka, T. Uhl Assessment of Dynamic Behaviour of Spot Welds with Uncertain Parameters Using Genetic Algorithms Application............................................. 298

(ID-1316)

M. Munck, D. Moens, W. Desmet, D. Vandepitte Optimisation Algorithms for Non-Deterministic Dynamic Finite Element Analysis of Imprecisely Defined Structures.......................................... 299

(ID-1926)

M. Pellissetti, H. Pradlwarter, G. Schuëller Relative Importance of Uncertain Parameters in Aerospace Applications ........................................................................................................ 300

(ID-2699)

V. Potapov Stability of Elastic and Viscoelastic Systems Under Stochastic Non-Gaussian Excitation..................................................................................... 301

(ID-1175)

J. Santos, B. Mace Modelling Uncertainty in Mechanical Joint Parameters Using Component Modal and Fuzzy Approaches.......................................................... 302

(ID-2348)

E. Sibilio, J. Beck, M. Muto, M. Ciampoli Bayesian Model Updating Approach for Ground-Motion Attenuation Relations .......................................................................................... 303

(ID-2650)

C. Verhoosel, M. Gutiérrez, S. Hulshoff Iterative Solution of the Random Eigenvalue Problem ........................... 304

(ID-1205)

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xxix

MS.16 Contact Mechanics Organizers: Martins, J.

J. Bielsa, R. Rodríguez, L. Vila, M. Jiménez (ID-2076) Parametrized Finite Element Analysis of Tribological Instabilities on Polymer-Metal Sliding Contacts .................................................................... 305 D. Boso, P. Litewka, B. Schrefler, P. Wriggers Thermo-Electro-Mechanical Coupling in Beam-to-Beam Contact ......... 306

(ID-2104)

M. Campo, J. Fernandez, K. Kuttler, M. Shillor Numerical Analysis of a Dynamic Frictional Viscoelastic Contact Problem with Damage ......................................................................................... 307

(ID-1133)

A. Chernov, M. Maischak, E. Stephan Hp-Mortar Boundary Element Method and FE/BE Coupling for Multibody Contact Problems with Friction ......................................................... 308

(ID-1991)

G. Chevallier, D. Nizerhy Friction Induced Vibrations in a Clutch System. Consequences on the Apparent Friction Torque. ............................................................................. 309

(ID-2328)

A. Chudzikiewicz, A. Myslinski Thermoelastic Wheel - Rail Contact Problem with Temperature Dependent Friction Coefficient ........................................................................... 310

(ID-1704)

M. Cocou, M. Raous, M. Schryve Analysis of a Dynamic Contact Problem with Adhesion and Friction in Viscoelasticity.................................................................................... 311

(ID-2078)

G. Drossopoulos, G. Stavroulakis, C. Massalas Influence of the FRP Strengthening, the Shape and the Movement of Abutments on the Collapse of Arch Stone Bridges......................................... 312

(ID-2194)

R. Dzonou, M. Marques, L. Paoli Sweeping Process for Vibro-Impact Problem with a General Inertia Operator ................................................................................................... 313

(ID-1020)

A. Eddhahak, L. Chevalier, S. Cloupet On a Simplified Method for Wear Simulation in Rolling Contact Problems.............................................................................................................. 314

(ID-1251)

G. Festa, J. Vilotte Spectral Element Simulations of Rupture Dynamics Along Planar and Kinked Frictional Faults .............................................................................. 315

(ID-1312)

M. Foerg, T. Geier, L. Neumann, H. Ulbrich R-Factor Strategies for the Augmented Lagrangian Approach in Multi-Body Contact Mechanics........................................................................... 316

(ID-1480)

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L. Fourment, S. Guerdoux (ID-1217) A Simple Smoothing Procedure of 3D Surfaces for Accurate Contact Analysis: Application to Metal Forming Problems................................ 317 D. Gabriel, J. Plesek, F. Vales, M. Okrouhlík Symmetry Preserving Algorithm for a Dynamic Contact-Impact Problem ............................................................................................................... 318

(ID-2524)

H. Georgiadis, D. Anagnostou Problems of Concentrated Loads in Microstructured Solids Characterized by Dipolar Gradient Elasticity...................................................... 319

(ID-1342)

F. Gutzeit, M. Wangenheim, M. Kröger An Experimentally Validated Model for Unsteady Rolling .................... 320

(ID-1925)

S. Hartmann, S. Brunssen, E. Ramm, B. Wohlmuth A Primal-Dual Active Set Strategy for Unilateral Non-Linear Dynamic Contact Problems of Thin-Walled Structures ...................................... 321

(ID-2576)

Y. Kanno, J. Martins Arc-Length Method for Frictional Contact with a Criterion of Maximum Dissipation of Energy ........................................................................ 322

(ID-1573)

H. Khenous, P. Laborde, Y. Renard A Energy Conserving Approximation for Elastodynamic Contact Problems.............................................................................................................. 323

(ID-2170)

A. Konyukhov, K. Schweizerhof, P. Vielsack On Models of Contact Surfaces Including Anisotropy for Friction and Adhesion and Their Experimental Validations............................................. 324

(ID-2320)

R. Krause Fast and Robust Solution Methods for Dynamic Contact Problems ....... 325

(ID-2147)

M. Marques, L. Paoli A Velocity-Based Time-Stepping Method for Frictional Dynamics....... 326

(ID-1865)

T. Meyer, A. Gal, M. Klüppel Mechanical Modeling of Friction and Adhesion of Elastomers at Rough Interfaces.................................................................................................. 327

(ID-1785)

C. Miranda, M. Neto, G. Fainer A 9M Drop Test Simulation of a Dual Purpose Cask for Nuclear Research Reactors Spent Fuel Elements.............................................................. 328

(ID-1347)

U. Nackenhorst, M. Ziefle A Discontinuous Galerkin Approach for the Numerical Treatment........ 329

(ID-1745)

P. Neittaanmäki, A. Kravchuk, I. Goryacheva An Iterative Method with BEM Discretization for the Friction Contact Problems ................................................................................................ 330

(ID-1297)

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xxxi

J. Nettingsmeier, P. Wriggers (ID-1299) Frictional Contact of Elastomer Materials on Rough Rigid Surfaces ............................................................................................................... 331 M. Oliveira, J. Alves, L. Menezes Optimizing the Description of Forming Tools with Bézier Surfaces in the Numerical Simulation of the Deep Drawing Process.................. 332

(ID-1856)

E. Pratt, A. Léger Exploring the Dynamics of a Simple System Involving Coulomb Friction ................................................................................................................ 333

(ID-2367)

F. Rauter, J. Pombo, J. Ambrósio, M. Pereira Multibody Modeling of Pantographs for Catenary-Pantograph Interaction............................................................................................................ 334

(ID-1653)

M. Renouf, V. Acary Comparison and Coupling of Algorithms for Collisions, Contact and Friction in Rigid Multibody Simulations...................................................... 335

(ID-2436)

M. Renouf, A. Saulot, Y. Berthier Third-Body Flow During Wheel-Rail Interaction .................................. 336

(ID-2440)

J. Sá, S. Grégoire, P. Moreau, D. Lochegnies Modelling Thermal Contact Resistance on Glass Forming Processes with Special Interface Finite Elements................................................ 337

(ID-2352)

J. Shiau Numerical Investigation of Shakedown Residual Stresses Under Moving Surface Loads ........................................................................................ 338

(ID-1734)

N. Strömberg Frictional Contact/Impact Between a Hyperelastic Body and Moving Rigid Obstacles ...................................................................................... 339

(ID-1578)

J. Svígler Incorrect Contact of Screw Surfaces and Its Consequences.................... 340

(ID-1485)

K. Uenishi Three-Dimensional Rupture Instability of a DisplacementSoftening Interface Under Nonuniform Loading................................................. 341

(ID-1104)

K. Willner, D. Görke Contact of Rough Surfaces - a Comparison of Numerical and Experimental Results........................................................................................... 342

(ID-1288)

xxxii Table of Contents

MS.17 Continnum Models for Nano-Structures Organizers: Kompis, V.

M. Arroyo, T. Belytschko (ID-1543) Continuum Mechanics Modelling and Simulation of Carbon Nanotubes............................................................................................................ 343 L. Bochkareva, M. Kireitseu, G. Tomlinson Comparison of Computational Efficiency of Modeling Approaches to Prediction of Damping Behavior................................................. 344

(ID-1144)

P. Dluzewski, M. Mazdziarz, G. Jurczak, P. Traczykowski, S. Nagao, R. Nowak, K. Kurzydlowski (ID-1849) A Hybrid Atomistic-–Continuum Finite Element Modelling of Nanoindentation Test on Copper......................................................................... 345 V. Kompis, M. Stiavnický, M. Kaukic Continuum Models for Composites Reinforced by Micro/Nano Fibers................................................................................................................... 346

(ID-1741)

A. Kushima, Y. Umeno, T. Kitamura First Principles Evaluation of Ideal Strength of Cu Nanowire ................ 347

(ID-2339)

S. Lisina, A. Potapov Variation Descriptions of Nano-Structured Media.................................. 348

(ID-1668)

T. Messager, P. Cartraud Homogenization of Single-Walled Carbon Nanotubes ........................... 349

(ID-1234)

M. Rabia Phonon Scattering by Perturbed Multichannel Waveguides ................... 350

(ID-1010)

H. Wu, X. Wang An Atomistic-Information-Based Continuum Inhomogeneous Material Model for Metal Nanorod ..................................................................... 351

(ID-1086)

MS.18 Coupling Problems Organizers: Schrefler, B.

F. Duda, L. Guimarães, A. Souza, J. Barbosa (ID-2349) On the Modeling of Deformation-Diffusion-Damage Coupling in Elastic Solids ....................................................................................................... 352 J. Gatica, V. Pita, N. Brum Frost Growth on Cold Flat Plate: a Correlation for the Diffusion Resistance Factors ............................................................................................... 353

(ID-1698)

Table of Contents xxxiii

D. Khalmanova, F. Costanzo (ID-1532) Discontinuous Space-Time Galerkin Finite Element Method in Linear Dynamic Fully Coupled Thermoelastic Problems with Strain and Heat Flux Discontinuities .................................................................................... 354 M. Landervik, R. Larsson Pore Gas Interaction in Polymeric Foams with Respect to Energy Absorption........................................................................................................... 355

(ID-1463)

F. Meftah, H. Sabeur A Thermo-Hydro–Damage Model for the Dehydration Creep of Concrete Subjected to High Temperature............................................................ 356

(ID-2290)

D. Néron, L. Pierre, B. Schrefler A Time-Space Framework Suitable for the LATIN Computational Strategy for Multiphysics Problems .................................................................... 357

(ID-2483)

W. Oliveira, M. Savi, P. Pacheco, L. Souza Finite Element Analysis of the Thermomechanical Coupling in Quenching of Steel Cylinders Using a Constitutive Model................................. 358

(ID-2176)

B. Pichler, C. Hellmich, H. Mang A Combined Fracture-Micromechanics Model for Tensile StrainSoftening in Brittle Materials .............................................................................. 359

(ID-1550)

Ö. Sen, D. Turhan Transient Dynamic Response of Thermoelastic Cylindrical Layered Media..................................................................................................... 360

(ID-1491)

MS.19 Damage Organizers: Alfaiate, J.

H. Askes, M. Gutiérrez, A. Rodriguez-Ferran (ID-2083) Novel Nonlocal Continuum Formulations. Part 1: Gradient Elasticity Based on Nonlocal Displacements and Nonlocal Strains .................... 361 T. Bennett, S. Kulasegaram On the Use of a Damage Model Based on Non-Local Displacements in the Element-Free Galerkin Method......................................... 362

(ID-1514)

T. Domingues, J. Alfaiate Modelling of Reinforced Concrete Beams Strengthened with PreStressed CFRP..................................................................................................... 363

(ID-2350)

T. Fiedler, L. Cunda, A. Öchsner, G. Creus, J. Grácio Numerical and Experimental Studies of Damage in Porous Materials.............................................................................................................. 364

(ID-1203)

xxxiv Table of Contents

R. Frizzell, C. Mccarthy, D. Cronin, M. Mccarthy, R. O'Higgins (ID-2369) The Development of a Continuum Damage Model for Fibre Metal Laminate Structures............................................................................................. 365 M. Hassanzadeh, G. Fagerlund Residual Strength of the Frost-Damaged Reinforced Concrete Beams .................................................................................................................. 366

(ID-1872)

M. Konrad, R. Chudoba, B. Kang Numerical and Experimental Evaluation of Damage Parameters for Textile Reinforced Concrete Under Cyclic Loading..................................... 367

(ID-2057)

P. Neto, J. Alfaiate, J. Vinagre Modeling the Behavior of Reinforced Concrete Beams Strengthened with FRP........................................................................................ 368

(ID-2380)

S. Oliveira, N. Gaspar, P. Dinis Cracking Analysis in Concrete Dams Using Isotropic Damage Models. Objectivity of Numerical Solutions ....................................................... 369

(ID-2649)

R. Pedersen, A. Simone, B. Sluys Continuous-Discontinuous Modelling of Dynamic Failure of Concrete Using a Viscoelastic Viscoplastic Damage Model............................... 370

(ID-1962)

J. Pituba On the Formulation of Damage Constitutive Models for Bimodular Anisotropic Media ............................................................................. 371

(ID-2089)

L. Rosa, E. Carvalho, B. Danziger Soil-Structure Interaction - Case History Analysis Involving Structural Damage............................................................................................... 372

(ID-2540)

Y. Sanomura, K. Saitoh Evolution Equation of Creep Damage Under Stress Variation ............... 373

(ID-1737)

A. Sichaib, G. Mounajed, C. Laborderie, H. Boussa, H. Quoc Concrete Damage Model Adaptation for Cyclic Loading....................... 374

(ID-1746)

C. Silva, L. Castro Hybrid and Mixed Finite Element Formulations for Softening Materials.............................................................................................................. 375

(ID-1107)

D. Sornin, K. Saanouni Theoretical and Computational Aspects of an Elastoplastic Damage Gradient Non Local Model ................................................................... 376

(ID-1326)

J. Wu, J. Li On a New Framework for Anisotropic Damage Model .......................... 377

(ID-1616)

Table of Contents xxxv

MS.20 Design Optimization Under Uncertainty Organizers: Choi, K. K. (USA), Kwak, B. M., Gorsich, D.

N. Banichuk, S. Ivanova, E. Makeev, A. Sinitsin (ID-2572) Shell Optimization Under Constraint on Damage Accumulation ........... 378 J. Casaca, A. Gomes Design of Acceptance-Sampling Plans Under Bayesian Risk................. 379

(ID-1249)

E. Cherkaev, A. Cherkaev Optimal Design for the Worst Case Scenario.......................................... 380

(ID-2192)

K. Choi, I. Lee, D. Gorsich Dimension Reduction Method for Reliability-Based Robust Design Optimization............................................................................................ 381

(ID-2522)

R. Dippolito, S. Donders, L. Hermans, M. Hack, J. Peer, N. Tzannetakis A Fatigue Life Reliability-Based Design Optimization of a Slat Track Using Mesh Morphing .............................................................................. 382

(ID-1231)

B. Kwak, J. Chang, J. Kim A New Approach of Robust Design Based on the Concept of Allowable Load Set ............................................................................................. 383

(ID-2571)

T. Lee, J. Jung, D. Jung A Sampling Technique Enhancing Accuracy and Efficiency of Metamodel-Based RBDO: Constraint Boundary Sampling ................................ 384

(ID-2593)

D. Moens, D. Vandepitte Interval Sensitivity Analysis of Dynamic Response Envelopes for Uncertain Mechanical Structures......................................................................... 385

(ID-2139)

C. Poloni, P. Geremia, A. Clarich Multi-Objective Robust Design Optimization of an Engine Crankshaft ........................................................................................................... 386

(ID-1904)

Y. Ryu Development and Application of a New Metropolis GA for the Structural Design Optimization ........................................................................... 387

(ID-2570)

B. Youn, Z. Xi, L. Wells, D. Lamb Stochastic Response Surface Using the Enhanced DimensionReduction (EDR) Method for Reliability-Based Robust Design......................... 388

(ID-2539)

MS.21 Differential Quadrature, Generalized Methods and Related Discrete Element Analysis Methods Organizers: Chen, C.

R. Balevicius, R. Kacianauskas (ID-1499) DEM Analysis of Granular Flow in Pyramidal Hoppers ........................ 389

xxxvi Table of Contents

C. Chen (ID-2085) DQEM and DQFDM for Computational Mechanics Problems .............. 390 C. Cinquini, M. Bruggi, P. Venini An Innovative Truly-Mixed Method for Cohesive-Crack Propagation Problems.......................................................................................... 391

(ID-1191)

D. Rosillo, F. Pérez Compactly Supported Fundamental Functions for Spline-Based Differential Quadrature ....................................................................................... 392

(ID-1876)

F. Tornabene, E. Viola Differential Quadrature Solution for Parabolic Structural Shell Elements .............................................................................................................. 393

(ID-2347)

MS.23 Enriched and Enhanced Finite Element Technology Organizers: Areias, P.

E. Budyn, L. Henry, T. Hoc (ID-2167) Multiple Crack Growth Failure in Cortical Bone Under Tension by the Extended Finite Element Method ............................................................. 394 F. Cirak Subdivision Shells................................................................................... 395

(ID-1710)

C. Foster, R. Borja Capturing Slip Weakening and Variable Frictional Response in Localizing Geomaterials Using an Enhanced Strain Finite Element................... 396

(ID-2040)

L. Hazard, P. Bouillard, J. Sener A Partition of Unity Finite Element Method Applied to the Study of Viscoelastic Sandwich Structures ................................................................... 397

(ID-1972)

A. Kölke, A. Legay An Enriched Space-Time Finite Element Method for FluidStructure Interaction - Part Ii: Thin Flexible Structures ...................................... 398

(ID-2318)

A. Legay, A. Kölke An Enriched Space-Time Finite Element Method for FluidStructure Interaction- Part I: Prescribed Structural Displacement....................... 399

(ID-2316)

I. Moldovan, J. Freitas Hybrid-Trefftz Finite Element Models for Bounded and Unbounded Elastodynamic Problems.................................................................. 400

(ID-2246)

P. Rozycki, E. Bechet, N. Möes Explicit Dynamic with X-FEM to Handle Complex Geometries............ 401

(ID-2010)

L. Stankovíc, J. Mosler Prediction of Macroscopic Material Failure Based on Microscopic Cohesive Laws .................................................................................................... 402

(ID-1255)

Table of Contents xxxvii

MS.24 Error Analysis and Adaptivity Organizers: Wiberg, N. E., Moitinho de Almeida, J., Diez, P.

L. Chamoin, P. Ladevèze (ID-2111) Strict, Sharp and Practical Bounds of Computed Outputs of Interest for Evolution Problems........................................................................... 403 É. Florentin, P. Ladevèze, J. Bellec Error Bounds on Outputs of Interest for Linear Stochastic Problems.............................................................................................................. 404

(ID-2299)

M. Jasinski, G. Zboinski An Hp-Adaptive Analysis of Some Linear Free Vibration Problems.............................................................................................................. 405

(ID-1313)

H. Lane, P. Kettil, N. Wiberg Moving Mesh Adaptivity Applied to Railway Dynamics ....................... 406

(ID-2373)

J. Mosler, M. Ortiz Finite Strain R-Adaption Based on a Fully Variational Framework ....... 407

(ID-1254)

E. Rank, V. Nübel, A. Düster Extension Processes, Adaptivity and Remeshing for Elasto-Plastic Problems.............................................................................................................. 408

(ID-2351)

J. Ródenas, J. Albelda, C. Corral, J. Mas Efficient Implementation of Domain Decomposition Methods Using a Hierarchical H-Adaptive Finite Element Program ................................. 409

(ID-2107)

W. Wall, T. Erhart, E. Ramm Adaptive Remeshing in Transient Impact Processes with Large Deformations and Nonlinear Material Behavior ................................................. 410

(ID-2615)

MS.25 Evolutionary Methods for Design Organizers: Periaux, J.

B. Bochenek, P. Forys (ID-1826) Particle Swarms in Engineering Design Problems .................................. 411 E. Campana, G. Fasano, D. Peri, A. Pinto Particle Swarm Optimization: Efficient Globally Convergent Modifications....................................................................................................... 412

(ID-2025)

G. Dulikravich, I. Egorov, N. Jelisavcic Evolutionary Optimization of Chemistry of Bulk Metallic Glasses........ 413

(ID-2228)

T. Naskar Introduction of Control Points in Splines for Synthesis of Optimized Cam Motion Programme ................................................................... 414

(ID-1576)

xxxviii Table of Contents

G. Oliveira, S. Saramago, P. Oliveira (ID-1837) On the Use of Differential Evolution in the Trajectory Modeling of Parallel Architecture Robot............................................................................. 415 M. Teixeira, M. Brandão Evolutionary Topologic Optimization Using the Finite Element Method ................................................................................................................ 416

(ID-2327)

MS.27 Fluid-Structure Interactions Organizers: Idelsohn S.

E. Aulisa, S. Manservisi, P. Seshaiyer (ID-1566) A Multilevel Domain Decomposition Methodology for Solving Coupled Problems in Fluid-Structure-Thermal Interaction ................................. 417 A. Boer, M. Schoot, H. Bijl Moving Mesh Algorithm for Unstructured Grids Based on Interpolation with Radial Basis Functions........................................................... 418

(ID-1781)

F. Daneshmand, S. Niroomandi Vibrational Analysis of Fluid-Structure Systems Using Natural Neighbour Galerkin Method................................................................................ 419

(ID-2310)

A. Kraker, D. Rixen, R. Ostayen Fluid-Structure Interaction in FEM Journal Bearing Simulations........... 420

(ID-1923)

A. Kupzok, R. Wüchner, K. Bletzinger Numerical Simulation of Wind-Structure Interaction for Thin Shells and Membranes......................................................................................... 421

(ID-1109)

J. Li, H. Chen, J. Chen Reliability Analysis of Prestressed Egg-Shaped Digester ....................... 422

(ID-1119)

P. Matyushin, V. Gushchin The Vortex Structures in the Sphere Wakes in the Wide Range of the Reynolds and Froude Numbers ..................................................................... 423

(ID-1303)

R. Niesner, M. Haupt, P. Horst Transient Analysis Methods for Hypersonic Applications with Thermo-Mechanical Fluid-Structure Interaction ................................................. 424

(ID-1965)

S. Sarkar, H. Bijl Stall Induced Vibration & Flutter in a Symmetric Airfoil....................... 425

(ID-1520)

R. Unger, M. Haupt, P. Horst Application of Lagrange Multipliers for Computational Aeroelasticity....................................................................................................... 426

(ID-1946)

Table of Contents xxxix

MS.28 Fracture and Fatigue Mechanics Organizers: Karihaloo, B.

J. Assis, V. Monine, S. Filippov, S. Iglesias (ID-1717) Computer Simulation of Diffraction Technique Applied for Measurements of Surface Stress Gradients ......................................................... 427 K. Enakoutsa, J. Leblond, G. Perrin Numerical Assessment of a Micromorphic Model of Ductile Rupture ................................................................................................................ 428

(ID-2408)

H. Hosseini-Toudeshky, B. Mohammadi, S. Bakhshandeh Fatigue Crack Trajectory Analysis of Single-Side Repaired Thin Aluminum Panels with Various Composite Patch Lay-Up Configurations ........ 429

(ID-1555)

S. Jog, R. Baddam Prediction of the Crack Initiation Life of Turbine Blade ........................ 430

(ID-1120)

M. Khoshravan, A. Hamidi Numerical Analysis of the Influence of Location of the Stopping Holes and Their Diameter in the Crack Growth of Ductile Metals ..................... 431

(ID-2240)

M. Kopecky Computed Analysis to Determine Service Life Criteria of Special Elements and Applications .................................................................................. 432

(ID-1110)

T. Luther, C. Könke Analysis of Crack Initiation and Propagation in Polycrystalline Meso and Microstructures of Metal Materials..................................................... 433

(ID-1796)

E. Szymczyk, A. Derewonko, J. Jachimowicz (ID-1669) Analysis of Displacement and Stress Distributions in Riveted Joints.................................................................................................................... 434 MS.29 Genetic Algorithms Organizers: António, C.

C. António (ID-2124) The Synergetic Effects of Hybrid Crossover Operators in Structural Optimisation ....................................................................................... 435 G. Bugeda, J. Ródenas, E. Pahl, E. Oñate An Adaptive Mesh Generation Strategy for the Solution of Structural Shape Optimization Problems Using Evolutionary Methods ............. 436

(ID-2428)

J. Dias, R. Corrêa Multiobjective Optimization of Multibody Systems with Genetic Algorithms........................................................................................................... 437

(ID-2472)

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Table of Contents

L. Iuspa, V. Minutolo, E. Ruocco (ID-2064) Buckling Optimization of Grid Structures Via Genetic Algorithms ....... 438 D. Kim, Y. Kim Multiscale Multiresolution Genetic Algorithm Using Diverse Population Groups............................................................................................... 439

(ID-1740)

P. Montrull, O. Querin, C. Gomez An Adaptive Correction Function for Structural Optimization with Genetic Algorithms ............................................................................................. 440

(ID-2450)

L. Sousa, C. Castro, C. António Optimization of the Topology of Masonry Units From the Thermal Point of View Using a Genetic Algorithm ............................................ 441

(ID-2292)

M. Victoria, P. Marti Topology Optimization of Bidimensional Continuum Structures by Genetic Algorithms and Stress Iso-Lines ....................................................... 442

(ID-2452)

MS.31 Impact and Control Organizers: Holnicki-Szulc, J.

I. Ario, P. Pawlowski, J. Holnicki-Szulc (ID-1476) Dynamic Analysis of Folding Patterns for Multi-Folding Structures............................................................................................................. 443 J. Cardoso, P. Moita, A. Valido Mechanical Systems Design and Control Optimization with Varying Time Domain......................................................................................... 444

(ID-2627)

S. Pashah, E. Jacquelin, J. Lainé, M. Massenzio Modelling for the Determination of the Interaction Force of Impacted Structures ............................................................................................. 445

(ID-1750)

MS.33 Intelligent Computing in Solid and Structural Mechanics Organizers: Burczynski, T.

W. Beluch (ID-1812) Evolutionary Identification and Optimization of Composite Structures............................................................................................................. 446 A. Dlugosz Parallel Evolutionary Optimization of Heat Radiators by Using MSC MARC/MENTAT Software....................................................................... 447

(ID-1853)

G. Dziatkiewicz, P. Fedelinski Evolutionary Algorithm and Boundary Element Method for Solving Inverse Problems of Piezoelectricity...................................................... 448

(ID-1930)

Table of Contents

xli

W. Kus Evolutionary Optimization of Preform and Die Shape in Forging Using Computational Grid .................................................................................. 449

(ID-1813)

W. Kus, T. Burczynski Optimization of Mechanical Structures Using Serial and Parallel Artificial Immune Systems.................................................................................. 450

(ID-1918)

P. Nazarko, L. Ziemianski Experiments of Damage Detection in Strips Based on Soft Computing Methods and Wave Propagation....................................................... 451

(ID-2581)

P. Orantek The Optimization and Identification Problems of Structures with Fuzzy Parameters ................................................................................................ 452

(ID-1811)

A. Poteralski Topology Optimization of the 3-D Structures for Various Criteria Using Evolutionary Algorithm ............................................................................ 453

(ID-1818)

A. Skrobol, T. Burczynski Computational Intelligence System in Non-Destructive Identification of Internal Defects......................................................................... 454

(ID-1816)

M. Szczepanik Optimization of Topology and Stiffeners Locations in 2-D Structures Using Evolutionary Methods.............................................................. 455

(ID-1825)

N. Wang, K. Tai A Structural Optimization Problem Formulation for Design of Compliant Gripper Using a Genetic Algorithm................................................... 456

(ID-2638)

MS.34 Intelligent Optimization Organizers: Sousa, J. C.

R. Almeida, S. Vieira, J. Sousa, U. Kaymak (ID-2061) The Prediction of Bankruptcy Using Weighted Fuzzy Classifiers.......... 457 P. Pinto, T. Runkler, J. Sousa Optimization of a Logistic Process by Ant Colonies, Wasp Swarms and Genetic Post-Optimization.............................................................. 458

(ID-1128)

S. Sakata, F. Ashida, M. Zako Kriging-Based Estimation with Noisy Data ............................................ 459

(ID-1735)

C. Silva, J. Faria, D. Naso Distributed Optimization Using ACO for Concrete Delivery ................. 460

(ID-1894)

F. Viana, G. Kotinda, V. Steffen Tuning a Vibrating Blade Dynamic Vibration Absorber by Using Ant Colony Optimization and Finite Element Modeling..................................... 461

(ID-1100)

xlii

Table of Contents

MS.35 Interrelation of Numerical and Asymptotical Approaches in Solid and Structural Mechanics Organizers: Manevitch, L., Lamarque, C. H.

C. Lamarque, E. Gourdon (ID-2579) Energy Pumping of Systems Connected to a Nonlinear Energy Sink Device ......................................................................................................... 462 V. Lyakh, V. Meleshko Stress Analysis of Curved Elastic Bar and Elastic Wedge Under Bending Load; Infinite Systems and Asymptotic ................................................ 463

(ID-1712)

A. Potapov, I. Miloserdova, I. Potapov Nonlinear Oscillations in Discretely Continual System .......................... 464

(ID-1672)

F. Romeo, G. Rega Propagation Properties of Bi-Coupled Nonlinear Oscillatory Chains.................................................................................................................. 465

(ID-1519)

MS.36 Inverse Engineering Organizers: Dulikravich, G. S., Orlande, H.

S. Abboudi, E. Artioukhine (ID-2202) Numerical Analysis of the Estimation of Three Boundary Conditions in Two Dimensional Inverse Heat Conduction Problem................... 466 J. Auzins, S. Ruchevskis, R. Rikards, A. Chate Metamodeling for the Identification of Composite Material Properties............................................................................................................. 467

(ID-2037)

T. Baranger, S. Andrieux An Energy Approach for a Cauchy Problem in Elasticity....................... 468

(ID-2062)

R. Fedele, G. Maier, L. Marazza Flat-Jack Tests and Parameter Identification for Diagnostic Analysis of Dams ................................................................................................ 469

(ID-1955)

H. Gualous, D. Zibret, E. Artioukhine Convective Boiling in Mini-Channels: Flow Visualization and Inverse Thermal Characterization ....................................................................... 470

(ID-2693)

E. Katamine, H. Azegami, M. Hirai Solution of Shape Identification Problem on Thermoelastic Solids........ 471

(ID-1716)

L. Lage, A. Cuco, F. Folly, F. Soeiro, A. Neto Stochastic and Hybrid Methods for the Solution of an Inverse Mass Transfer Problem........................................................................................ 472

(ID-2437)

P. Masson, S. Rouquette, T. Loulou, E. Artioukhine Inverse Heat Conduction Problem: Estimation of a Source Term for an Electron Beam Welding; Theoretical and Experimental........................... 473

(ID-1346)

Table of Contents

xliii

H. Nakamura, M. Kawahara (ID-1829) The Parameter Identification of Elastic Modulus at Futatsuishi Site....................................................................................................................... 474 K. Ogura, M. Kawahara Parameter Identification of the Attenuation Using First Order Adjoint Method ................................................................................................... 475

(ID-1819)

K. Sassi, S. Andrieux Parameters Identification of a Nonlinear Viscoelastic Model Via an Energy Error Functional ................................................................................. 476

(ID-1292)

F. Soeiro, A. Neto Inverse Radiative Transfer Problems in Two-Layer Participating Media................................................................................................................... 477

(ID-2435)

N. Thomson, H. Orlande Computation of Sensitivity Coefficients and Estimation of Thermophysical Properties with the Line Heat Source Method.......................... 478

(ID-1026)

H. Velho, S. Sambatti, L. Chiwiacowsky Combining a Parallel Genetic Algorithm with Variational Approach for Assessing Structural Damage........................................................ 479

(ID-1558)

MS.37 Large Scale Shape and Topology Optimization Organizers: Pedersen, P., Bendsoe, M., Sigmund O.

M. Abdalla, S. Setoodeh, Z. Gurdal (ID-1621) Design of Variable-Stiffness Composite Panels for Maximum Buckling Load ..................................................................................................... 480 G. Cheng, L. Liu, J. Yan Optimum Structure with Homogeneous Optimum Truss-Like Material ............................................................................................................... 481

(ID-1722)

P. Clausen, C. Pedersen Non-Parametric Large Scale Structural Optimization............................. 482

(ID-2583)

A. Diaz, R. Mukherjee Optimal Joint Placement and Modal Disparity in Control of Flexible Structures............................................................................................... 483

(ID-1708)

A. Gersborg-Hansen Topology Optimization of 3D Stokes Flow Problems ............................ 484

(ID-1157)

J. Jensen Efficient Optimization of Dynamic Systems Using Pade Approximants ...................................................................................................... 485

(ID-1800)

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M. Kim, G. Jang, Y. Kim (ID-1743) Ground Structure Based Joint Stiffness Controlling Method for Joint Compliant Mechanism Design.................................................................... 486 S. Lambert, E. Pagnacco, L. Khalij, A. Hami Topology Optimization of Structures Subject to Random Excitations with Fatigue Life Constraints ........................................................... 487

(ID-2099)

E. Lemaire, P. Duysinx, V. Rochus, J. Golinval Improvement of Pull-In Voltage of Electromechanical Microbeams Using Topology Optimization ........................................................ 488

(ID-2065)

E. Lund Large Scale Optimization of Compression Loaded Composite Structures............................................................................................................. 489

(ID-1879)

C. Narváez, A. Tovar, D. Garzon Topology Synthesis of Compliant Mechanisms Using the Hybrid Cellular Automaton Method with an Efficient Mass Control STR ..................... 490

(ID-2215)

N. Olhoff, J. Du Topological Design for Minimum Sound Radiation From Structures Subjected to Forced Vibration............................................................ 491

(ID-1792)

P. Pedersen Aspects of 3D Shape and Topology Optimization with Multiple Load Cases .......................................................................................................... 492

(ID-1250)

N. Pedersen On Shape, Material and Orientational Design of Plates in Relation to Dynamics......................................................................................................... 493

(ID-1910)

N. Perchikov, M. Fuchs Optimal Layouts of Stiffeners for Plates in Bending - Topology Optimization Approach ....................................................................................... 494

(ID-1678)

U. Schramm How Topology Optimization Changed the Design Process .................... 495

(ID-2156)

Y. Sia, O. Querin Structural Shape Optimisation by Using Multi-Direction Boundary Points Movement Method................................................................... 496

(ID-1141)

O. Sigmund On Topology Optimization with Manufacturing Constraints.................. 497

(ID-1895)

K. Svanberg, M. Werme Sequential Integer Programming Methods for Stress-Constrained Shape and Topology Optimization ...................................................................... 498

(ID-1769)

Table of Contents

xlv

A. Tovar, W. Quevedo, N. Patel, J. Renaud (ID-2185) Topology Optimization with Stress and Displacement Constraints Using the Hybrid Cellular Automaton Method ................................................... 499 G. Yoon, J. Jensen, O. Sigmund Topological Design of Acoustic-Structure Interaction Structures with the Mixed Finite Element Method............................................................... 500

(ID-2008)

MS.38 Composite Molding-Numerical Simulations and Applications Organizers: Dimitrovova, Z., Bassir, D.

N. Abdelkader, Y. Chevalier, S. Aguib, N. Chikh (ID-2643) Cinematique Influence on the Vibrations of Stratified Plates ................. 501 D. Bassir, W. Zhang, S. Guessasma Optimization of Resign Transfer Molding Process by a Virtual Manufacturing and a Genetic Algorithms ........................................................... 502

(ID-1604)

Z. Dimitrovova Post-Processing Techniques Suitability for Mesolevel Free Boundary Flows .................................................................................................. 503

(ID-2335)

M. Nossek, M. Sauer, K. Thoma Adaptive Simulation of Cohesive Interface Debonding for Crash and Impact Analyses............................................................................................ 504

(ID-1942)

S. Shigehisa, Y. Miyata, H. Ochiai Failure Analysis of Cement-Treated Soil by FEM Implemented with Particle Discretization ................................................................................. 505

(ID-2454)

A. Zoheir, T. Nabil, R. Ayad, A. Samir, B. Malek An Homogenisation Procedure for Cardboard and Stitched Sandwiches Using Respectively Analytical and Numerical Simulation ............. 506

(ID-2551)

MS.39 Material Models for Composites at Different Length Scales Organizers: Rolfes, R.

D. Ballhause, M. König, B. Kröplin (ID-1489) Modelling of Woven Fabrics with the Discrete Element Method........... 507 M. Böl, S. Reese A New Approach for the Simulation of Damage Effects in Rubber-Like Materials Using Chain Statistics .................................................... 508

(ID-1441)

L. Gornet, S. Marguet, G. Marckmann Numerical Modelling of Nomex® Honeycomb Cores : Failure and Effective Elastic Properties........................................................................... 509

(ID-2066)

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G. Haasemann, M. Kästner, V. Ulbricht (ID-2024) Multi-Scale Modelling and Simulation of Textile Reinforced Materials.............................................................................................................. 510 L. Kaczmarczyk, Z. Waszczyszyn Enforcing Boundary Conditions in Micro-Macro Transition for Second Order Continuum .................................................................................... 511

(ID-2562)

M. Klüppel, J. Meier, M. Ramspeck Micro-Structure Based Modeling of Elastomer Materials ...................... 512

(ID-1511)

F. Lipperman, M. Ryvkin, M. Fuchs Analysis and Effective Properties of Honeycombs with NonSymmetric Unit Cells .......................................................................................... 513

(ID-1802)

M. Luxner, J. Stampfl, A. Woesz, P. Fratzl, H. Pettermann Influence of Defects and Perturbations on the Performance of 3D Open Cell Structures ........................................................................................... 514

(ID-1752)

A. Muhr Mechanics of Elastometer-Shim Laminates............................................ 515

(ID-2012)

J. Skocek, J. Zeman, M. Šejnoha Application of the Mori-Tanaka Method to Analysis of Woven Composites with Imperfections ........................................................................... 516

(ID-1636)

S. Smaoui, B. Abelwahed, D. Irini, D. Hélène An Homogenization Iterative Process for Nonlinear Materials Applied to Compacted Clays............................................................................... 517

(ID-2241)

M. Timmel, M. Kaliske, S. Kolling, R. Mueller A Micromechanical Approach for the Simulation of Rubberlike Materials with Damage........................................................................................ 518

(ID-1938)

W. Wajda, H. Paul Influence of Grains Misorientation on Material Hardening on Example of Aluminum Bicrystals Deformed in Channel Die ............................. 519

(ID-2544)

MS.40 Meshless Methods Organizers: Alves, C.

C. Alves, P. Antunes (ID-2532) The Method of Fundamental Solutions Applied to the Calculation of Eigensolutions for Simply Connected Plates .................................................. 520 C. Alves, S. Valtchev Comparison Between Meshfree and Boundary Element Methods Applied to BVPS in Domains with Corners ........................................................ 521

(ID-2558)

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xlvii

C. Alves, N. Martins (ID-2618) The Method of Fundamental Solutions Applied to a Heat Conduction Inverse Problem ............................................................................... 522 M. Arroyo, M. Ortiz Local Maximum-Entropy Approximation Schemes................................ 523

(ID-1540)

J. Belinha, L. Dinis Analysis of 2D Problems Resorting to a New Meshless Method............ 524

(ID-1935)

M. Bitaraf, S. Mohammadi Solving the Chloride Diffusion Equation in Concrete Structures for Prediction of Initiation Time of Corrosion .................................................... 525

(ID-1887)

S. Chantasiriwan Solution of the Stationary Three-Dimensional Navier-Stokes Equations by Using Radial Basis Functions........................................................ 526

(ID-1732)

T. Most, C. Bucher An Enhanced Moving Least Squares Interpolation for the Element-Free Galerkin Method ........................................................................... 527

(ID-1233)

H. Netuzhylov Enforcement of Boundary Conditions in Meshfree Methods Using Interpolating Moving Least Squares.................................................................... 528

(ID-1969)

V. Rosca, V. Leitão A Simple and Less-Costly Integration of Meshless Galerkin Weak Form .................................................................................................................... 529

(ID-2574)

C. Tiago, P. Pimenta Geometrically Exact Analysis of Shells by a Meshless Approach.......... 530

(ID-2159)

O. Valencia, F. Gómez-Escalonilla, F. Urbinati, J. López-Díez Weight Functions Analysis in Elastostatic Problems for Meshless Element Free Galerkin Method ........................................................................... 531

(ID-2398)

MS.41 Metal Forming Organizers: Cesar Sá, J., Pietrzyk, M.

F. Abbassi, O. Pantale, A. Zghal, R. Rakotomalala (ID-1575) Prediction of Sheet Metal Formability (FLD) by Using Diverse Method ................................................................................................................ 532 J. Alves, M. Oliveira, L. Menezes Modeling Drawbeads in Deep Drawing Simulations .............................. 533

(ID-1848)

S. Benke, G. Laschet On a 2-Phase Finite Element Model for the Coherent Mushy Zone in Casting Applications ....................................................................................... 534

(ID-1501)

xlviii Table of Contents

H. Christophe, A. Said, D. Loic (ID-1588) Modeling of Ductile Behavior of Metals Under a Wide Range of Loading Rates: Semi-Empirical Approach.......................................................... 535 S. Ding, B. Daniel, P. Meehan A New Relaxation Method for Roll Forming Problems.......................... 536

(ID-1362)

M. Glowacki, M. Hojny Computer Modelling of Deformation of Steel Samples with Mushy Zone......................................................................................................... 537

(ID-1753)

B. Haddag, F. Abed-Meraim, T. Balan Prediction of Strain Localisation in Forming Process Using Advanced Elastic-Plastic Behaviour Models Coupled with Damage.................. 538

(ID-2055)

R. Jorge, A. Roque, M. Parente, A. Fernandes, R. Valente Symulation of Hydroforming on Tailor-Welded Tubular Blanks Using Solid-Shell Finite Elements ...................................................................... 539

(ID-1988)

T. Lelotte, L. Duchêne, A. Habraken Fast Method to Predict an Earing Profile Based on Lankford's Coefficients and Yield Locus .............................................................................. 540

(ID-2402)

L. Madej, A. Zmudzki, P. Hodgson, M. Pietrzyk Possibilities of Application of the Multi Scale Strain Localization Café ..................................................................................................................... 541

(ID-2231)

K. Nabil, K. Yahia Thermal Modeling of D. C. Continuous Casting Process of a AlMg Alloy ............................................................................................................. 542

(ID-1915)

J. Ponthot, R. Boman, L. Papeleux, Q. Bui A 3D Arbitrary Lagrangian Eulerian Formulation for the Numerical Simulation of Forming Processes. ..................................................... 543

(ID-2053)

M. Poursina, H. Ebrahimi, J. Parvizian A Study for the Constitutive Equations of 1.4021 and 1.4841 Stainless Steels in Hot Deformation .................................................................... 544

(ID-2503)

W. Rasp, A. Yusupov Modelling of Spread and Side-Form Function in Hot Rolling by Different Upper-Bound Approaches ................................................................... 545

(ID-2243)

J. Sá, C. Zheng Non Local Models and Length Scale Effects on Metal Forming Processes ............................................................................................................. 546

(ID-2336)

M. Salmanitehrani, M. Poursina Analysis of Thermal Cracking of an Industrial Duct Using Finite Element Simulation ............................................................................................. 547

(ID-2500)

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xlix

M. Schwarze, S. Reese (ID-2393) Analysis of Forming Processes with Efficient Finite Element Procedures ........................................................................................................... 548 P. Teixeira, F. Pires, A. Santos, J. Sá Numerical Investigation of Fracture Onset in Sheet Metal Forming ............................................................................................................... 549

(ID-2413)

MS.42 Modeling in Mechanobiology Organizers: Lekszycki, T.

A. Andreykiv, F. Keulen (ID-2446) Effect of Surface Geometry and Local Mechanical Environment on Periimplant Tissue Differentiation:A Finite Element Study........................... 550 C. Bonifasi-Lista, E. Cherkaev Identification of Bone Structure From Effective Measurements............. 551

(ID-2190)

M. Cioffi, F. Galbusera, M. Raimondi, F. Boschetti, G. Dubini Computational Modeling of Mechanical Environment Within Tissue Engineered Cartilage................................................................................ 552

(ID-1868)

H. Kim, P. Clement, J. Cunningham Stress-Based Optimum and Bone Architecture....................................... 553

(ID-2340)

P. Kowalczyk Parameterized Orthotropic Cellular Microstructures As Mechanical Models of Cancellous Bone ............................................................. 554

(ID-1151)

M. Nowak Modeling of Cancellous Bone Surface Adaptation Based on the 3Dimensional Trabeculae Topology Evolution.................................................... 555

(ID-1004)

J. Pierre, C. Oddou Theoretical Analysis of the Remodeling Processes in Bony Tissue Engineered Implants............................................................................................ 556

(ID-2294)

G. Sciarra, T. Lekszycki Bone Remodeling Description Based on Micro Mechanical/Biological Effects ............................................................................ 557

(ID-1768)

MS.43 Modelling of Functionally graded materials and structures Organizers: Batra, R., Ferreira, A.

L. Aebi, J. Vollmann, J. Dual (ID-1497) Two-Dimensional Elastodynamic Wave Propagation in Graded Structures............................................................................................................. 558

l

Table of Contents

H. Argeso, A. Eraslan (ID-1184) A Computational Study on Functionally Graded Rotating Solid Shafts: Analysis of Preliminary Results .............................................................. 559 R. Batra, J. Xiao, D. Gilhooley, M. Mccarthy, J. Gillespie Static Analysis of Thick Functionally Graded Plates by Using a Higher-Order Shear and Normal Deformable Plate Theory ................................ 560

(ID-2013)

M. Bocciarelli, G. Bolzon, G. Maier Three-Point-Bending and Indentation Tests for the Calibration of Functionally Graded Material Models by Inverse Analysis ................................ 561

(ID-1747)

A. Ferreira, G. Fasshauer, C. Roque, R. Jorge Static Deformations and Natural Frequencies of Functionally Graded Plates by a Hybrid Meshless Method...................................................... 562

(ID-1339)

S. Hamza-Cherif, A. Houmat, A. Hadjoui Graded Fourier P-Element Calculation of Steady State Heat Conduction in Functionally Graded Materials .................................................... 563

(ID-1111)

T. Nguyen, K. Sab, G. Bonnet A Reissner-Mindlin Plate Model for Functionally Graded Materials.............................................................................................................. 564

(ID-2254)

W. Szymezyk A Review of the Chosen Problems of FEM Modeling of Surface Coatings............................................................................................................... 565

(ID-1823)

MS.44 Multibody Dynamics Organizers: Ambrósio, J.

C. Bottasso, D. Dopico, L. Trainelli (ID-1764) On the Optimal Scaling of Index Three DAEs in Multibody Dynamics............................................................................................................. 566 S. Breun, R. Callies Redundant Optimal Control of Manipulators Along Specified Paths .................................................................................................................... 567

(ID-1874)

M. Ebbesen, M. Hansen, T. Andersen Optimal Tool Point Control of Hydraulically Actuated Flexible Multibody System with an Operator-In-The-Loop.............................................. 568

(ID-2433)

D. Franitza, T. Lichtneckert Slim Elastic Structures with Transversal Isotropic Material Properties Under Finite Deformations................................................................. 569

(ID-2281)

M. Hajzman, P. Polach Modelling of Flexible Rods Falling in Fluid with Possible Contacts............................................................................................................... 570

(ID-1912)

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li

H. Irschik, M. Nader, C. Zehetner (ID-1934) Tracking of Displacements in Smart Elastic Beams Subjected to Rigid Body Motions ............................................................................................ 571 E. Lens, A. Cardona Dynamic Analysis of Constrained Nonlinear Multibody Systems with Intermittent Contact..................................................................................... 572

(ID-2509)

D. Leshchenko, L. Akulenko, S. Suksova Evolution of Rotation of a Triaxial Satellite Under the Action of Gravitational and Light Pressure Torques ........................................................... 573

(ID-1085)

S. Leyendecker, P. Betsch, P. Steinmann Mechanical Integrators for Nonlinear Flexible Multibody Dynamics............................................................................................................. 574

(ID-2309)

K. Lipinski Some Really Simple But Useful Model of Substitutable Elasticity Modelled As Elasticity in Six Subsequent Joints ................................................ 575

(ID-1757)

K. Lipinski Optimal Dampers Localization for a Body Under Double Load and the Body Behaviour for Some Intermediate Loads....................................... 576

(ID-1758)

P. Polach, M. Hajzman Design of Characteristics of Air Pressure Controlled Hydraulic Shock Absorbers in an Intercity Bus ................................................................... 577

(ID-1860)

R. Santos, V. Steffen, S. Saramago Robot Path Planning in a Constrained Workspace by Using Optimal Control Techniques ............................................................................... 578

(ID-1628)

G. Schanzer, R. Callies Optimal Control of Multi-Link Manipulators with Rivalling Actuators ............................................................................................................. 579

(ID-1771)

L. Sousa, P. Verissimo, J. Ambrósio Development of Validated Generic Road Vehicles for Crashworthiness Through Optimization Procedures ........................................... 580

(ID-2363)

M. Szczotka, I. Adamiec-Wójcik, S. Wojciech Developing Mathematical and Computer Models for Car Dynamics Using Joint Co-Ordinates and Homogenous Transformations ........... 581

(ID-1577)

MS.45 Multiphysics Modelling in Geomechanics Organizers: Borja, R. I., Montáns, F.J., Tamagnini, C.

J. Andrade, R. Borja (ID-1563) Finite Element Simulation of Deformation Bands in Saturated Granular Media with Inhomogeneous Porosities at the Meso-Scale ................... 582

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M. Caminero, F. Montáns (ID-1700) Computational Framework for Multilayer Plasticity Based on Critical State Soil Mechanics .............................................................................. 583 L. Hu, T. Hueckel Creep of Geomaterials Due to Coupled Damage and Spontaneous Mineral Dissolution............................................................................................. 584

(ID-2222)

R. Kohler, G. Hofstetter Validation of an Extended Cap Model for Partially Saturated Soils ....... 585

(ID-1257)

A. Mesgouez, G. Lefeuve-Mesgouez, A. Chambarel Partially Saturated Porous Medium Vibration Induced by an Impulsional Load................................................................................................. 586

(ID-1964)

M. Murad, C. Moyne Macroscopic Behavior of Smectitic Clays Derived From Nanostructure ...................................................................................................... 587

(ID-1103)

P. Rehermann, R. Borja, D. Pollard Mechanical Modeling of Multi-Layer Sedimentary Rock Folding ......... 588

(ID-1567)

D. Souza, A. Coutinho, J. Alves Fast Numerical Simulation of Porous Media Flows................................ 589

(ID-1536)

MS.46 Multiscale Mechanics of Biological Materials and Other Natural Composites Organizers: Hellmich, C.

F. Genna (ID-1129) A Micromechanically-Based Interface Model for the Periodontal Ligament.............................................................................................................. 590 Q. Grimal, K. Raum, P. Laugier Computation of Cortical Bone Macroscopic Properties From Microscopic Elastic Data..................................................................................... 591

(ID-1300)

R. Grytz, G. Meschke Computational Homogenization in Multi-Scale Shell Analysis at Large Strains........................................................................................................ 592

(ID-1238)

D. Katti, P. Ghosh, K. Katti Mineral Proximity Influences Protein Unfolding: a Molecular Dynamics Study .................................................................................................. 593

(ID-1998)

K. Katti, R. Bhowmik, D. Katti, D. Verma Modeling the Role of Interfaces on Mechanical Response in Composite Bone Biomaterials ............................................................................. 594

(ID-2018)

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liii

P. Laugier (ID-2022) Finite-Difference Computations of Ultrasound Wave Propagation in Bone ................................................................................................................ 595 F. Moravec, M. Muller Microstructural Model of the Viscoelastic Behaviour of Biological Tissues ............................................................................................... 596

(ID-1815)

K. Okumura Scaling Views on Strength of Soft/Hard Composites ............................ 597

(ID-2501)

MS.47 Multiscale Method for Structural Non-Linear and Dynamic Problems Organizers: Allix, O., Rey, C.

A. Amini, D. Dureisseix, P. Cartraud, N. Buannic (ID-1310) A Micro-Macro Strategy for Ship Structural Analysis with FETIDP Method .......................................................................................................... 598 E. Baranger, O. Allix, L. Blanchard On the Use of Fourier Expansions for the Simulation of Elastic Composite Pipes with Defects............................................................................. 599

(ID-2550)

B. Bourel, A. Combescure, L. Valentin A Multigrid Approach for Non-Linear Structural Analysis in Explicit Dynamics ............................................................................................... 600

(ID-2647)

V. Carvelli, C. Corazza, C. Poggi Mechanical Behaviour of Textile Structures: Two-Scales Approach ............................................................................................................. 601

(ID-1336)

P. Cresta, O. Allix, C. Rey, S. Guinard On Multilevel Strategies for Nonlinear Computations with Domain Decomposition: Application to Post-Buckling ...................................... 602

(ID-2138)

M. Kuprys, R. Barauskas Textile Fabric Simulator: Collisions Handling at the Level of Yarns ................................................................................................................... 603

(ID-1839)

J. Pebrel, P. Gosselet, C. Rey A Computational Strategy for Contact Simulation.................................. 604

(ID-2345)

C. Rickelt, S. Reese A Simulation Strategy for Life Time Calculations of Large, Partially Damaged Structures .............................................................................. 605

(ID-2361)

Y. Shiihara, O. Kuwazuru, N. Yoshikawa Finite Element Method in First-Principles Calculation........................... 606

(ID-2355)

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M. Spiridonakos, S. Fassois (ID-1840) Modelling and Simulation of Earthquake Ground Motion Via Functional Series TARMA Models with Wavelet Basis Functions .................... 607 MS.48 Neural Networks and Soft Computing in Solid and Structural Mechanics Organizers: Waszczyszyn, Z.

A. Borowiec, L. Ziemianski (ID-2582) Identification of Damage in Multispan Beams Using ParameterDependent Frequency Changes and Neural Networks ........................................ 608 A. Kucerova, M. Leps, J. Zeman Microplane Model Parameters Estimation Using Neural Networks ....... 609

(ID-1992)

K. Kuzniar Anns and Linguistic Variables in the Analysis of Mine Induced Rockbursts Transmission to the High Building................................................... 610

(ID-2592)

W. Lu, L. Duan, F. Mora-Camino, R. Faye Differential Flatness of Aircraft Flight Dynamics and Neural Inversion.............................................................................................................. 611

(ID-1591)

MS.49 Nonlinear Dynamics of Moving Structures Organizers: Zahariev, E., Mayo Nunez, J.

A. Ayestarán, C. Graciano (ID-2635) Finite Element Analysis of an Energy Absorbing Crush Zone Using Expanded Metal ........................................................................................ 612 L. Desouza Design of Satellite Control System Using the Optimal Nonlinear Theory ................................................................................................................. 613

(ID-1017)

J. Dias, F. Antunes, M. Pereira Design for Crashworthiness of Train Structures with Simplified Multibody Models ............................................................................................... 614

(ID-2499)

J. Gerstmayr, M. Matikainen Analysis of Stress and Strain in the Absolute Nodal Coordinate Formulation with Nonlinear Material Behavior .................................................. 615

(ID-2612)

S. Hornstein, O. Gottlieb Nonlinear Multimode Dynamics of a Moving Microbeam for Noncontacting Atomic Force Microscopy........................................................... 616

(ID-1172)

Y. Lin, P. Nikravesh Model Reduction with Mean-Axes in Deformable Multibody Dynamics............................................................................................................. 617

(ID-2105)

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lv

J. Mayo (ID-1841) Impacts with Friction in Flexible Multibody Dynamics ......................... 618 J. Orden, R. Ortega A Conservative Augmented Lagrangian Algorithm for the Dynamics of Constrained Mechanical Systems................................................... 619

(ID-1968)

M. Piovan, R. Sampaio Non Linear Model for Coupled Axial/Torsional/Flexural Vibrations of Drill-Strings................................................................................... 620

(ID-1561)

A. Poulimenos, M. Spiridonakos, S. Fassois Identification of Time-Varying Structures Under Unobservable Excitation: an Overview and Experimental Comparison..................................... 621

(ID-1810)

I. Romero Variational Integrators for the Rigid Body Dynamics............................. 622

(ID-1961)

E. Zahariev Planning and Optimization of Maneuver Strategy of Large Flexible Space Structures .................................................................................... 623

(ID-1548)

R. Zander, H. Ulbrich Free Plain Motion of Flexible Beams in MBS - a Comparison of Models................................................................................................................. 624

(ID-1515)

MS.50 Non-Linear Vibration of Structures Organizers: Ribeiro, P.

R. Arquier, B. Cochelin (ID-2051) Numerical Computation of Non Linear Modes of Elastic Structures............................................................................................................. 625 S. Bellizzi, R. Bouc Nonlinear Modes: Amplitude-Phase Formulation and Bifurcation Analysis............................................................................................................... 626

(ID-2455)

M. Haterbouch, P. Ribeiro, R. Benamar Multi-Modal Non-Linear Free Vibration of Thin Isotropic Circular Plates ..................................................................................................... 627

(ID-1603)

M. Kireitseu, G. Tomlinson Computational Approaches to Prediction of Damping Behavior of Nanoparticle-Reinforced Coatings and Foamy Structures .................................. 628

(ID-1143)

A. Kocsis, G. Károlyi Buckling Under Conservative and Nonconservative Load ..................... 629

(ID-1531)

N. Krancevic, M. Stegic, N. Vrankovic Bifurcation of Periodic Solutions in the Two-Degree-of-Freedom System with Clearances....................................................................................... 630

(ID-1218)

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R. Lewandowski (ID-1873) Perturbation Method for Strongly Non-Linear Free Vibrations of Beams .................................................................................................................. 631 P. Ribeiro, B. Cochelin, S. Bellizzi Vibrations of Shallow and Deep Shells by the P-Version Finite Element Method .................................................................................................. 632

(ID-2129)

N. Srinil, G. Rega Resonant Non-Linear Dynamic Responses of Horizontal Cables Via Kinematically Non-Condensed/Condensed Modeling.................................. 633

(ID-1627)

J. Thomsen Computing Effective Properties of Nonlinear Structures Exposed to Strong High-Frequency Loading at Multiple Frequencies .............................. 634

(ID-2552)

J. Zapomel Implementation of a Vapour Cavitation Into Computational Models of Rotors Supported by Long Journal Bearings...................................... 635

(ID-1105)

MS.51 Optimization and Robust Design for Industrial-sized Problems Organizers: Bletzinger, K. U., Duddeck, F., Meyer M.

F. Duddeck (ID-1470) Multi-Criteria Optimizations and Robustness Estimations for Crashworthiness, Structural Dynamics, and Acoustics ....................................... 636 K. Grossenbacher, F. Duddeck, M. Ganser, P. Hora Process Robustness in Sheet Metal Forming by an Integrated Engineering Strategy ........................................................................................... 637

(ID-2030)

M. Hansen, T. Andersen, J. Hansen, O. Mouritsen Topology Optimization of Robots Using Mapping Techniques.............. 638

(ID-2606)

H. Müllerschön, M. Hove, B. Mlekusch Optimization Strategies for Highly Non-Linear FE-Applications As Crashworthiness Applications........................................................................ 639

(ID-2388)

M. Ohsaki Local and Global Searches of Approximate Optimal Designs of Regular Frames.................................................................................................... 640

(ID-1174)

J. Reger, T. Schneider, C. Ehlert Optimisation of Car Body Parts Regarding Equivalent Radiation Power Using a Genetic Algorithm and Morphing ............................................... 641

(ID-1596)

K. Sedlaczek, P. Eberhard Grid-Based Topology Optimization of Rigid Body Mechanisms Using Different Problem Formulations ............................................................... 642

(ID-1498)

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lvii

H. Wenzel (ID-1903) Combining Optimization and Robust Engineering Methods in the Engineering Product Design Process................................................................... 643 MS.56 Shape and Topological Sensitivity Analysis: Theory and Applications Organizers: Feijóo, R.., Taroco, E.

G. Allaire, F. Jouve, F. Gournay, A. Toader (ID-2691) Combining Topological and Shape Derivatives in Structural Optimization........................................................................................................ 644 F. Barthold, K. Wiechmann A Comparison of Displacement and Mixed Finite Element Formulations for Variational Design Sensitivity Analysis .................................. 645

(ID-2003)

M. Bonnet Inverse Acoustic Scattering by Small-Obstacle Expansion of Misfit Function .................................................................................................... 646

(ID-2677)

J. Faria, R. Feijóo, A. Novotny, E. Taroco, C. Padra Second Order Topological Sensitivity Analysis...................................... 647

(ID-2029)

M. Langelaar, F. Keulen Sensitivity Analysis of Shape Memory Alloy Shells .............................. 648

(ID-1905)

L. Miegroet, T. Jacobs, E. Lemaire, P. Duysinx Stress Constrained Optimization Using X-FEM and Level Set Description .......................................................................................................... 649

(ID-2026)

A. Myslinski Level Set Method for Optimization of Contact Problems ....................... 650

(ID-2067)

H. Pham, C. Bucher An Iterative Procedure for Model Updating Based on Selective Sensitivity............................................................................................................ 651

(ID-1083)

MS.57 Shell and Spatial Structures Organizers: Ramm, E.

M. Birsan (ID-1462) Extension, Bending and Torsion of Cylindrical Cosserat Shells Made From a Porous Elastic Material ................................................................. 652 N. Dung, G. Wells A Study of Discontinuous Galerkin Methods for Thin Bending Problems.............................................................................................................. 653

(ID-2449)

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M. Frenzel, M. Bischoff, W. Wall (ID-2611) Discrete Strain Gap (Dsg) Solid Finite Elements at Large Deformations for Non-Linear Analysis of Shells and Solids .............................. 654 E. Gal, M. Zelkha, R. Levy A Simple Co-Rotational Geometrically Non-Linear Membrane Finite Element Wrinkling Analysis ..................................................................... 655

(ID-1629)

B. Izzuddin Co-Rotational System Definitions for Large Displacement Triangular and Quadrilateral Shell Elements ...................................................... 656

(ID-1609)

P. Khosravi, R. Ganesan, R. Sedaghati Non-Linear Analysis of Composite Plates and Shells Using a New Shell Element ...................................................................................................... 657

(ID-1614)

J. Mäkinen, H. Marjamäki Total and Updated Lagrangian Geometrically Exact Beam Elements .............................................................................................................. 658

(ID-1683)

E. Maunder, B. Izzuddin Large Displacement Analysis of Plates Using Hybrid Equilibrium Elements .............................................................................................................. 659

(ID-1541)

H. Noguchi, F. Fujii, Y. Ishihara Buckling Modes of Large-Scale Shell Structures Automatically Detected From Linearized Stiffness by Iterative Solvers .................................... 660

(ID-2271)

I. Oliveira, E. Campello, P. Pimenta Finite Element Analysis of the Wrinkling of Orthotropic Membranes .......................................................................................................... 661

(ID-2415)

R. Schlebusch, B. Zastrau Theory and Numerics of a Surface-Related Shell Formulation............... 662

(ID-2284)

M. Spalatelu-Lazar, F. Léné, N. Turbé Modelling and Optimization of Sails ...................................................... 663

(ID-1989)

M. Tanaka, H. Noguchi Instability Analysis of Thin-Walled Structures Using Incompressible Hyperelastic Shell Elements....................................................... 664

(ID-2253)

MS.58 Simulation of Non-Gaussian Stochastic Processes and Fields with Applications to Structural Engineering Problems Organizers: Papadrakakis, M., Stefanou, G.

J. Cruz, M. Gutiérrez, L. Koene (ID-1206) Stochastic Simulation of Pitting Corrosion ............................................. 665

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lix

V. Denoël, H. Degée (ID-1227) Non Gaussian Response of Bridges Subjected to Turbulent Wind Effect of the Non Linearity of Aerodynamic Coefficients .................................. 666 M. Grigoriu Simulation of Non-Gaussian Stochastic Processes and Fields with Applications to Structural Engineering Problems ............................................... 667

(ID-1261)

N. Lagaros, G. Stefanou, M. Papadrakakis A Novel Approach for the Efficient Simulation of Highly Skewed Non-Gaussian Stochastic Fields .......................................................................... 668

(ID-1932)

M. Schevenels, G. Lombaert, G. Degrande, D. Degrauwe, B. Schoors The Wave Propagation in a Vertically Inhomogeneous Soil with a Random Dynamic Shear Modulus....................................................................... 669

(ID-1854)

G. Stefanou, M. Papadrakakis On the Karhunen-Loeve Expansion and Spectral Representation Methods for the Simulation of Gaussian Stochastic Fields ................................. 670

(ID-1940)

MS.59 Soft Tissue Organizers: Jorge, R.

J. Barbosa, R. Jorge, M. Parente, A. Fernandes, T. Mascarenhas, B. Patricio (ID-1452) Study of Mechanical Properties of Human Skin ..................................... 671 R. Basto, C. Costa, W. Pereira, M. Krüger, H. Orlande, H. Fonseca Thermophysical Properties of Different Samples of TissueMimicking Materials for Ultrasound Hyperthermia Phantoms ........................... 672

(ID-2160)

A. Cavicchi, L. Gambarotta, R. Massabò Different Computational Approaches in the Modeling of Wrinkling of Biological Membranes................................................................... 673

(ID-2426)

F. Gentil, R. Jorge, A. Ferreira, M. Parente, M. Moreira, E. Almeida Dynamic Study of the Middle Ear........................................................... 674

(ID-1496)

S. Kuzukami, N. Yoshikawa, O. Kuwazuru Image-Base Inverse Problems to Identify Three-Dimensional Displacement Field.............................................................................................. 675

(ID-1425)

P. Martins, R. Jorge, A. Ferreira, F. Gentil Experimental Study of the Middle Ear Biological Support Structures............................................................................................................. 676

(ID-1527)

T. Miyashita, H. Yamauchi, M. Inui, H. Yamakawa A Real-Time FEM Simulation for Cutting Operation Using Haptic Device.................................................................................................................. 677

(ID-2338)

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Table of Contents

M. Parente, R. Jorge, A. Fernandes, T. Mascarenhas, J. Martins (ID-1473) The Biomechanical Behavior of the Pelvic Floor Muscles During a Vaginal Delivery............................................................................................... 678 MS.60 Stability and Non-Linear Behaviour of Thin-Walled Members and Structures Organizers: Camotim, D.

S. Ádány, N. Silvestre, B. Schafer, D. Camotim (ID-1443) Buckling Analysis of Unbranched Thin-Walled Members: Generalised Beam Theory and Constrained Finite Strip Method ........................ 679 A. Andrade, D. Camotim, P. Costa Lateral-Torsional Buckling Analysis of Singly Symmetric WebTapered I-Beams Using Finite Elements and Finite Differences ........................ 680

(ID-1974)

J. André, A. Baptista Stability of Telescopic Props for Temporary Structures ......................... 681

(ID-1545)

C. Basaglia, D. Camotim, N. Silvestre Formulation of a GBT-Based Finite Element to Analyse the Global Buckling Behaviour of Plane/Spatial Thin-Walled Frames..................... 682

(ID-1943)

R. Bebiano, N. Silvestre, D. Camotim GBT-Based Finite Element to Analyse the Buckling Behaviour of Thin-Walled Members Subjected to Non-Uniform Bending............................... 683

(ID-2039)

N. Boissonnade, H. Degée A Non-Linear 3-D Beam Finite Element for the Study of Steel Frames with Tapered Members. .......................................................................... 684

(ID-1248)

R. Castro, J. Silva, P. Vellasco, S. Andrade, L. Lima, L. Neves Non-Linear Dynamical Response of Steel Portal Frames with Semi-Rigid Connections...................................................................................... 685

(ID-1726)

N. Chen, C. Soares Buckling Analysis of Stiffened Composite Panels.................................. 686

(ID-2002)

H. Degée, N. Boissonnade Interactive Buckling of Thin-Walled Rectangular Hollow Sections - Comparison Between Modified Beam Models and .......................................... 687

(ID-1228)

R. Degenhardt, A. Kling, K. Rohwer Design and Analysis of Composite Panels.............................................. 688

(ID-1914)

P. Dinis, D. Camotim On the Use of Shell Fea to Assess the Local Buckling & PostBuckling Behaviour of Cold-Formed Steel Thin-Walled Members.................... 689

(ID-1828)

N. Fallah A Finite Volume Method for Plate Buckling Analysis ........................... 690

(ID-1353)

Table of Contents

lxi

A. Formisano, G. Matteis, M. Maruzzelli, F. Mazzolani (ID-2517) Design of Slender Steel Shear Panels: a Numerical Study...................... 691 R. Goncalves, P. Grognec, D. Camotim Plastic Bifurcation Fea of Thin-Walled Members: Thin Shell Elements Vs. GBT-Based Beam Elements.......................................................... 692

(ID-1953)

R. Gonçalves, M. Ritto-Corrêa, D. Camotim A Large Displacement and Finite Rotation Thin-Walled Beam Finite Element Formulation................................................................................. 693

(ID-2017)

R. Goñi, E. Bayo A New Method to Assess the Rotation Capacity of Structural Hollow Sections Based in Multibody Theory ..................................................... 694

(ID-1474)

M. Haßler, K. Schweizerhof On the Stability Analysis of Thin Walled Shell Structures Containing Gas Or Fluid ..................................................................................... 695

(ID-2376)

M. Khedmati, M. Zareei Sensitivity Analysis on Ultimate Strength of Stiffened Aluminum Plates Under Combined Inplane Compression and Lateral Presuure .................. 696

(ID-2326)

H. Ovesy, S. Ghannadpour Large Deflection Behavior of Functionally Graded Plates Under Pressure Loads, Using Finite Strip Method......................................................... 697

(ID-1243)

H. Ovesy, H. Hosseini-Toudeshky, M. Kharazi Buckling Analysis of Laminates with Multiple Through-TheWidth Delaminations by Using Spring Simulated Model ................................... 698

(ID-1744)

P. Real, N. Lopes, L. Silva, C. Rebelo Numerical Validation of the Eurocode 3 Design Rules for LateralTorsional Buckling of I-Beams ........................................................................... 699

(ID-2126)

C. Rebelo, L. Silva, P. Real, N. Lopes Statistical Evaluation of the Eurocode 3 Design Rules for LateralTorsional Buckling of I-Beams ........................................................................... 700

(ID-2416)

M. Ritto-Corrêa, D. Camotim On the Interpolation of Rotations and Rigid-Body Motions in Nonlinear Beam Finite Elements......................................................................... 701

(ID-2050)

K. Rzeszut, A. Garstecki Stability of Beams and Columns Made of Thin-Walled ColdFormed Sections Accounting for Imperfections.................................................. 702

(ID-1944)

W. Schneider, M. Gettel Equivalent Geometric Imperfections for Steel Shell Structures Subject to Combined Loading ............................................................................. 703

(ID-2004)

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Table of Contents

I. Shufrin, M. Eisenberger (ID-2404) Shear Buckling of Thin Plates with Constant In-Plane Stresses ............. 704 N. Silva, N. Silvestre, D. Camotim GBT Formulation to Analyse the Buckling Behaviour of Frp Composite Branched Thin-Walled Members ...................................................... 705

(ID-2282)

N. Silvestre, N. Silva On the Influence of Material Couplings on the Buckling Behaviour of FRP Thin-Walled Columns – a GBT-Based Approach ................. 706

(ID-2272)

C. Soares, R. Luís Ultimate Strength of Plate Assemblies with Localized Imperfection Subjected to Compressive Loads ................................................... 707

(ID-1948)

R. Vieira, F. Virtuoso, E. Pereira Higher Order Analysis of a Thin-Walled Beam...................................... 708

(ID-2370)

MS.61 Structural and Multidisciplinary Optimization Organizers: Herskovits, J.

M. Abolbashari, M. Majdi (ID-2387) Structural Optimization Using Optimizer Program................................. 709 D. Babkin, E. Kligman, V. Matveyenko, N. Yurlova Optimization of Dissipative Characteristics of Structures on the Basis of Problems on Natural Vibrations of Viscoelastic Solids......................... 710

(ID-1807)

K. Bhagate, P. Pawar, A. Singh, J. Ye Accuracy of Design Sensitivity Analysis for Optimization of Structures for Small Strain Theory by Finite Element Method ........................... 711

(ID-1477)

D. Bojczuk Method of Optimal Reinforcement of Structures Based on Topological Derivative........................................................................................ 712

(ID-1601)

R. Botez, A. Dinu, I. Cotoi Optimisation of Unsteady Aerodynamic Forces for Aircraft Aeroservoelastic Studies ..................................................................................... 713

(ID-1606)

R. Cadete, J. Dias, M. Pereira A Multidisciplinary Design Optimization Framework Applied to Mechanical Systems ............................................................................................ 714

(ID-2371)

A. Canelas, P. Mappa, J. Herskovits, J. Telles Shape Optimization Using the Boundary Elements and a Sand Interior Point Algorithm for Constrained Optimization ...................................... 715

(ID-2448)

P. Coelho, P. Fernandes, J. Cardoso, J. Guedes, H. Rodrigues A Three-Dimensional Hierarchical Model for Topology Optimization of Structures .................................................................................. 716

(ID-2481)

Table of Contents

lxiii

A. Csébfalvi, G. Csébfalvi (ID-2019) A New Hybrid Meta-Heuristic Method for Optimal Design of Space Trusses with Elastic-Plastic Collapse Constraints..................................... 717 B. Desmorat The Concept of Homogeneous Thermodynamical Potentials for Nonlinear Structural Rigidity Optimization ........................................................ 718

(ID-2042)

L. Desouza Satellite Attitude Control System Parameters Optimization ................... 719

(ID-1097)

V. Dubeux, J. Herskovits, S. Mazorche A Limited Memory Quasi-Newton Preconditioner for Large Scale Optimization........................................................................................................ 720

(ID-2600)

M. Ebbesen, M. Hansen, N. Pedersen Design Optimization of Conveyor Systems ............................................ 721

(ID-2001)

J. Herskovits, E. Goulart, M. Aroztegui Sparse Quasi-Newton Matrices for Large Size Optimization with FAIPA, the Feasible Arc Interior Point Algorithm.............................................. 722

(ID-1977)

S. Holopainen Sensitivity and Sizing of Nonlinear Structures Made of Anisotropic Rubber-Like Material ...................................................................... 723

(ID-1241)

L. Johansen, E. Lund Optimization of Laminated Composite Structures Using Delamination Criteria and Adaptive Models ....................................................... 724

(ID-1896)

J. Kock, C. Strauss, C. Pohl, P. Wyk, P. Botes Yeast Biomechanics ................................................................................ 725

(ID-1057)

J. Kruzelecki, M. Król Optimization of Postbuckling Path for Cylindrical Shells Under External Pressure................................................................................................. 726

(ID-2123)

R. López, A. Tovar, C. Narváez Structural Analysis in Continuum Media Using Cellular Automata ....... 727

(ID-1742)

K. Mikhail, I. Egorov, S. Yuri, F. Konstantin Multidisciplinary Optimization of Complex Technical Systems ............ 728

(ID-2211)

J. Roche Adaptive Shape Optimization Method .................................................... 729

(ID-1235)

M. Secanell, B. Carnes, A. Suleman, N. Djilali A Pem Fuel Cell Cathode Model for Gradient-Based Optimization ....... 730

(ID-1553)

I. Shevtsov, V. Markine, C. Esveld, M. Markina Optimisation of a Railway Wheel Profile ............................................... 731

(ID-1774)

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Table of Contents

J. Stegmann, M. Stolpe (ID-1861) Discrete Material Optimization of Laminated Composites - Simp Vs. Global Optimization...................................................................................... 732 K. Svanberg, M. Werme First and Second Order Sensitivities of Functions with Respect to Binary Variables and Their Application in Topology Optimization ................... 733

(ID-1867)

A. Vincenti, P. Vannucci Optimal Design of Smart Composite Laminates by the Polar Method and the Genetic Algorithm Bianca ......................................................... 734

(ID-2184)

C. Wang, B. Duan, F. Zheng, H. Cao On Optimization Platform for Coupled Structural-Electromagnetic Performances of Large Reflector Antennas......................................................... 735

(ID-1570)

H. Wang, J. Croll, N. Yamamoto, S. Yamada Optimising Buckling Capacities for Composite Shells ........................... 736

(ID-2511)

B. Wilczynski, Z. Mróz Multiaxial Plastic Hardening Models Used in Shape Optimization with Respect to Fatigue Life................................................................................ 737

(ID-2269)

A. Wit, A. Lipka, E. Ramm, F. Keulen Multi-Level Optimization of Material and Structural Layout ................. 738

(ID-2588)

MS.62 Structural Dynamics Organizers: Azevedo, J.

R. Almeida, J. Silva (ID-1727) A Stochastic Modelling of the Dynamical Response of Highway Bridge Decks Under Traffic Loads ..................................................................... 739 S. Amiri, M. Barghian, M. Fard, J. Safadoust Comparisation of Two Dimensional Nonlinear Analysis of Integral Abutment Bridge and Simply Supported Bridge ................................... 740

(ID-1095)

J. Chang, I. Huang, W. Hou, P. Chang Dynamic Stability Analysis of Truss Structures Under Nonconservative Constant and Pulsating Follower Forces ................................. 741

(ID-2597)

A. Davaran, M. Fard, S. Amiri, A. Kashefi Comparison of Concrete Tall Building Behavior Using an Intermittent Shear Walls Form in One Frame ..................................................... 742

(ID-1602)

S. François, G. Degrande An Iterative Coupled Boundary-Finite Element Method for the Dynamic Response of Structures......................................................................... 743

(ID-2666)

Table of Contents

lxv

S. Fransen, D. Rixen (ID-2090) Pendulum Mode Control in the Dynamic Analysis of Lift-Off of Launchers ............................................................................................................ 744 M. Gao, J. Pan, J. Xiong Dynamic Response of Long Span Cable-Stayed Bridge Subjected to Earthquake and Moving Train......................................................................... 745

(ID-2291)

M. Gutiérrez, H. Askes Parametrisation of the Newmark Time Integrator for Non-Linear Solid Dynamics ................................................................................................... 746

(ID-1201)

S. Krenk Global Formulation of Conservative Time Integration by the Increment of the Geometric Stiffness .................................................................. 747

(ID-2477)

D. Makovicka, D. Makovicka Response Analysis of Building Loaded by Groundborne Transient Vibration.............................................................................................................. 748

(ID-1220)

H. Marjamäki, J. Mäkinen Different Approaches in Modelling Boom Lifting Movement................ 749

(ID-1528)

B. Möller, W. Graf, A. Hoffmann, J. Sickert, F. Steinigen Textile Reinforced Concrete Structures Under Uncertain Dynamic Loading Processes ............................................................................................... 750

(ID-2232)

M. Ursu Shock Response Spectrum Analysis for Measured Earthquake Data ..................................................................................................................... 751

(ID-1973)

MS.64 System Identification and Finite Element Updating Organizers: Cunha, A.

A. Frigerio, E. Bom, G. Mazzà (ID-1444) Histride: an Integrated Software for Dynamic Structural Identification ....................................................................................................... 752 J. Lee, J. Kim, Y. Kim Damage Detection by the Topology Design Formulation Using Modal Parameters................................................................................................ 753

(ID-1739)

F. Magalhães, B. Costa, A. Cunha, E. Caetano Experimental Validation of the Finite Element Modelling of Pinhão Bridge ...................................................................................................... 754

(ID-2661)

S. Oliveira, P. Mendes Development of a Cabril Dam Finite Element Model for Dynamic Analysis Using Ambient Vibration Tests Results ............................................... 755

(ID-2077)

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Table of Contents

D. Rebolho, L. Souza, E. Belo, F. Marques (ID-2054) Application of Eera Method for Identification of Modal Parameters of a Simulated Aircraft ..................................................................... 756 E. Reynders, G. Roeck Reference-Based Combined Deterministic-Stochastic Subspace Identification for Experimental and Operational Model Analysis....................... 757

(ID-2508)

MS.65 Temperature and time dependent effects in steel and concrete structures Organizers: Silva, L. S., Júlio, E.

D. Costa, V. Silva, E. Júlio (ID-1253) Numerical Modelling of Time-Dependent Behaviour of High Strength Concrete Beams .................................................................................... 758 G. Ranzi, M. Bradford, P. Ansourian Behaviour of Composite Steel-Concrete Beams with Longitudinal and Transverse Partial Interaction in Fire............................................................ 759

(ID-1721)

G. Ranzi, P. Ansourian, L. Dezi, S. Zhang Partial Interaction Analysis of Multi-Layered Composite Beams Accounting for Time Effects ............................................................................... 760

(ID-1731)

A. Santiago, L. Silva, P. Real Numerical Behaviour of Steel Sub-Frame System in Fire ...................... 761

(ID-1814)

MS. 66 Vehicle Dynamics Organizers: Schiehlen, W.

J. Borges, M. Leal, R. Filho, J. Rezende (ID-2556) Structure Design and Dynamic Analysis of Vehicle Using Metamodeling and Optimization Techniques...................................................... 762 F. Braghin, E. Sabbioni, F. Cheli Race Driver Model: Identification of the Driver’s Inputs ....................... 763

(ID-1449)

F. Braghin, S. Melzi, F. Cheli Race Driver Model: Trajectory Planning ................................................ 764

(ID-1450)

F. Cheli, E. Giangiulio, E. Sabbioni, A. Concas A Simplified Abs Numerical Model for Actively Controlled Vehicle Dynamic Simulations: Validation with Experimental Data ................... 765

(ID-2096)

F. Cheli, G. Diana, F. Ripamonti, G. Tomasini, G. Zanetti Aerodynamic Sensitivity Analysis of the New Emuv250 Train to Cross Wind by Wind Tunnel Tests and CFD Analysis ....................................... 766

(ID-2391)

Table of Contents

lxvii

K. Dufva, K. Kerkkänen, L. Maqueda, A. Shabana (ID-1169) Three-Dimensional Large Deformation Finite Elment Analysis of Belt Drives........................................................................................................... 767 A. Freitas, R. Silva, J. Dias Multibody and Finite Element Models for the Design of Motorcyclist’s Roadside Protections ................................................................... 768

(ID-2397)

J. Landre, L. Saturnino, M. Becker, L. Patrício, C. Barcellos An Integrated Educational Tool for Vehicle Dynamical Response Studies ................................................................................................................. 769

(ID-1162)

J. Massat, A. Bobillot, J. Laine Robust Methods for Detecting Defects in Overhead Contact Line Based on Simulation Results ............................................................................... 770

(ID-1479)

B. Mavroudakis, P. Eberhard Mode Decoupling Vehicle Suspension System Applied to Race Car ....................................................................................................................... 771

(ID-1751)

J. Meijaard, A. Schwab Linearized Equations for an Extended Bicycle Model............................ 772

(ID-1715)

L. Patrício, M. Becker, J. Landre, C. Barcellos A New Vehicle 3D Model with 7 Degrees of Freedom for Vehicle Dynamical Response Studies .............................................................................. 773

(ID-1163)

J. Pombo, J. Ambrósio A Hertzian Contact Formulation for the Wheel-Rail Contact Problem in Railway Dynamics ............................................................................ 774

(ID-2639)

R. Portal, J. Dias Multibody Models for Vehicle Accident Reconstruction........................ 775

(ID-2396)

G. Rill (ID-2242)

First Order Tire Dynamics ...................................................................... 776

A. Schwab, J. Meijaard, J. Kooijman Experimental Validation of a Model of an Uncontrolled Bicycle ........... 777

(ID-1624)

G. Tomasini, A. Collina, E. Leo, F. Resta, F. Cheli Numerical-Experimental Methodology for Runnability Analysis and Wind-Bridge-Vehicle Interaction Study ....................................................... 778

(ID-2392)

P. Verissimo, J. Ambrósio Improved Bushing Models for Vehicle Dynamics .................................. 779

(ID-2362)

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MS.68 Behaviour of Structures Submitted to Extreme Event Organizers: Simões. L.C.

A. Dias Simulation of Large Deformations on Timber Joints Using 3D FEM Models........................................................................................................ 780

(ID-1967)

A. Freire, J. Negrão Nonlinear Dynamics of Flexible Partially Collapsed Structures............. 781

(ID-1787)

K. Jármai, J. Rodrigues Optimal Steel Frame Design for Fire Resistance .................................... 782

(ID-1116)

M. Larcher, L. Stempniewski Simulation of Shock Wave Loaded Concrete with Discrete Cracks ....... 783

(ID-1869)

A. Lopes, A. Cunha, L. Simões CFD Based Evaluation of the Lock-In Phenomenon of a Bridge Under Wind Load................................................................................................ 784

(ID-2443)

A. Lopes, D. Gomes, L. Simões CFD Based Aerodynamic Study to Discrete Optimization of Bridge Cross Sections.......................................................................................... 785

(ID-2444)

K. Menchel, P. Bouillard, T. Massart Progressive Collapse Simulation in RC Structures ................................. 786

(ID-1986)

V. Terzi, M. Alexoudi, K. Pitilakis, T. Hatzigogos Vulnerability Assessment for Pipelines Under Permanent Ground Deformatioon. Comparison Between Analytical and Empirical ......................... 787

(ID-1775)

R. Vicente, H. Rodrigues, H. Varum Seismic Performance and Strengthening of Traditional Masonry Buildings in the City Centre of Coimbra............................................................. 788

(ID-1359)

Preface This book contains the edited version of the Abstracts of Plenary and Keynote Lectures and Papers, and a companion CD-ROM with the full-length papers, presented at the III European Conference on Computational Mechanics: Solids, Structures and Coupled Problems in Engineering (ECCM-2006), held in the National Laboratory of Civil Engineering, Lisbon, Portugal 5th - 8th June 2006. The book reflects the state-of-art of Computation Mechanics in Solids, Structures and Coupled Problems in Engineering and it includes contributions by the world most active researchers in this field. ECCM-2006 is a continuation of the very successful Conferences held in Munich, Germany (1999) and Cracow, Poland (2001) and it is organized by the European Committee of Computational Solid and Structural Mechanics (ECCSM) of the European Community on Computational Methods in Applied Science (ECCOMAS) in collaboration with the Portuguese Association of Theoretical, Applied and Computational Mechanics (APMTAC), the Technical University of Lisbon and the National Laboratory of Civil Engineering. ECCM-2006 is attended by about 1000 participants from 70 countries. More than 1300 Abstracts were submitted to ECCM-2006. Altogether, 6 plenary lectures, 35 keynote lectures and 800 papers are presented in 58 organized symposia. A companion book, containing the edited version of the majority of Plenary and Keynote Lectures, entitled Computational Mechanics: Solids, Structures and Coupled Problems in Engineering is also published by Springer 2006. The Proceedings of the Conference could not be possible without the sponsorship and financial support of: European Community on Computational Methods in Applied Science (ECCOMAS); European Committee of Computational Solids and Structural Mechanics (ECCSM); International Association of Computational Mechanics (IACM); Portuguese Association of Theoretical, Applied and Computational Mechanics (APMTAC); Foundation of Science and Technology (Portugal); Calouste Gulbenkian Foundation (Portugal); Technical University of Lisbon (Portugal); National Laboratory of Civil Engineering (Portugal); Instituto Superior Técnico (Portugal). The Editors are grateful to all authors and to the reviewers that helped ensuring the scientific quality, allowing for this book to be published before ECCM2006. We acknowledge the support of Mr. Pedro Pinto of the Technical University of Lisbon, in the editing of the book. The Editors are also grateful to all Members of Executive, Organizing, Advisory and Scientific Committees and to the organizers of the Symposia, whose work made possible the success of ECCM-2006. We are grateful to Andrea Marques and Ana Catarina Amador for their effort and valuable assistance in the conference and preparation of this book. Technical University of Lisbon, Portugal, June 2006 Carlos A. Mota Soares, João A.C. Martins Helder C. Rodrigues, Jorge A.C. Ambrósio, Carlos A.B. Pina Cristóvão M. Mota Soares, Eduardo B.R. Pereira, João Folgado lxix

lxxi

Executive Committee E. Oñate (President of IACM), Polytechnic University of Catalunya, Spain A. Mang (President of ECCOMAS), Technical University of Vienna, Austria C. A. Mota Soares (Co-Chairperson), Technical University of Lisbon, Portugal M. Papadrakakis (Co-Chairperson), National Technical University of Athens, Greece B. Schrefler, University of Padua, Italy J. Teixeira de Freitas, Technical University of Lisbon, Portugal

Organizing Committee E. Arantes e Oliveira (Honorary Co-Chairperson), Technical Univ. of Lisbon, Portugal E. Stein (Honorary Co-Chairperson), University of Hannover, Germany J. Ambrósio, Technical University of Lisbon, Portugal M. Bernadou, University of Leonard de Vinci, France T. Burczynski, Silesian University of Technology, Poland P. Diez, Polytechnic University of Catalunya, Spain M. Kleiber, IPPT PAN, Poland O. Mahrenholz, Technische Universitat Hamburg, Germany J. Martins, Technical University of Lisbon, Portugal C.M. Mota Soares, Technical University of Lisbon, Portugal P. Neittaanmaki, University of Jyväskylä, Finland E. Pereira, Technical University of Lisbon, Portugal J. Periaux, Dassault Aviation, France C. Pina, National Laboratory of Civil Engineering, Portugal P.Steinmann, University of Kaiserslautern, Germany E. Ramm, University of Stuttgart, Germany H.C. Rodrigues, Technical University of Lisbon, Portugal N. E. Wiberg, Chalmers University of Techonology, Sweden O. C. Zienkiewicz, University of Wales, Swansea, United Kingdom

Management Committee Carlos Pina, National Laboratory of Civil Engineering, Portugal Helder Rodrigues, Technical University of Lisbon, Portugal João Martins, Technical University of Lisbon, Portugal

Organizing Institution Portuguese Association of Theoretical, Applied and Computational Mechanics LNEC, Av. do Brasil 101, 1700-066 Lisboa, Portugal

lxxii

Advisory Committee F. Auricchio, Associazione Italiana di Meccanica Teorica e Applicata, Italy N. Bicanic, Association for Computer Methods in Engineering (ACME), UK M.H. Boduroglu, Turkish Committee on Computational Mechanics, Turkey T. Burczynski, Polish Association for Computational Mechanics (PACM), Poland M. Casteleiro, Sociedad Española de Métodos Numéricos en Ingenieria (SEMNI), Spain M. Cerrolaza, Sociedad Venezolana de Métodos Numéricos en Ingenieria, Venezuela P. Chauchot, Computational Structural Mechanics Association (CSMA), France C. K. Choi, Korean Association on Computational Mechanics (KACM), Korea J. Crempien, Sociedade Chilena de Mecánica Computacional (SCMC), Chile R. Delgado, Portuguese Association of Theoretical, Applied and Computational Mechanics (APMTAC), Portugal A. Eriksson, The Nordic Association for Computational Mechanics (NOACM), Denmark, Estonia, Finland, Iceland, Latvia, Lithuania, Norway, Sweden J. Fish, U. S. Association for Computational Mechanics (USACM), USA I. Harari, The Israel Association of Computational Methods in Mechanics, Israel I. Herrera, Sociedad Mexicana de Métodos Numéricos en Ingenieria, México S.R. Idelsohn, Asociación Argentina de Mecánica Computational (AMCA), Argentina W. Kanok-Nukulchai, Thailand Society for Computational Mechanics (TSCM), Thailand T. Kant, India Association of Computation Mechanics, India M. Kleiber, The Central European Association for Computational Mechanics (CEACM), Austria, Croatia, Hungary, Poland, Slovenia, The Czech Republic V. Kompis, Slovakia Association for Computational Mechanics, Slovakia G. R. Liu, Singapore Association for Computational Mechanics (SACM), Singapore P.R. Lyra, Brazilian Association for Computational Mechanics (ABMEC), Brazil J. Miller, Irish Society for Scientifc and Engineering Computation (ISSEC), Ireland H. Ohtsubo, Japan Society of Computational Engineering Science (JSCES), Japan M. Papadrakakis, The Greek Association of Computational Mechanics (GRACM), Greece E. Ramm, German Association of Computational Mechanics (GACM), Germany B. D. Reddy, South Africa Association for Theorical and Applied Mechanics (SAAM), South Africa S. Valliapan, Australian Association of Computational Mechanics, Australia T. Yabe, Japan Association for Computational Mechanics (JACM), Japan M. W. Yuan, Chinese Association of Computational Mechanics, China

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Scientific Committee S. Adali (South Africa) J. Alfaiate (Portugal) O. Allix (France) C. Alves (Portugal) C. António (Portugal) F. Amero (USA) J. Azevedo (Portugal) N. Banichuk (Russia) A.L. Baptista (Portugal) K.J. Bathe (USA) J. L. Batoz (France) P. Beckers (Belgium) T. Belytschko (USA) M. Bendsoe (Denmark) A. Benjeddou (France) P. Bergan (Norway) R. Borja (USA) R. de Borst (Netherland) F. Branco (Portugal) C.L. Bottasso (USA) E. Carrera (Italy) D. Camotim (Portugal) M. Casteleiro (Spain) J. Cesar Sá (Portugal) C. Chen (Taiwan) K.K. Choi, (USA) C. Cinquini (Italy) S. Cowin (USA) A. Cunha (Portugal) M. Doblaré (Spain) E.H .Dowel (USA) G. Dulikravich (USA) I. Doltsinis (Germany) R. Feijoo (Brazil) I. Figueiredo (Portugal) J. Fish (USA) L. Gaul (Germany) P.L. George (France) M. Geradin (Italy) E. Van der Giessen (Netherlands) M. Goicolea (Spain) C. Hellmich (Austria) J. Herskovits (Brazil) J. Holnicki-Szulc (Poland) G. Holzapfel (Austria) T.J.R. Hughes (USA) S. R. Idelsohn (Argentina)

R. Jorge (Portugal) J.J. Judice (Portugal) B. Karihaloo (UK) N. Kikuchi (USA) M. Kojic (USA) P. Ladevèze (France) V. Leitão (Portugal) T. Lekszycki (Poland) J. Vieira de Lemos (Portugal) A. Leung (China) W. K. Liu (USA) P. Lourenço (Portugal) B. Mace (UK) P. Martins (Portugal) R. Melchers (Australia) J. Moitinho de Almeida (Portugal) C. Navarro (Spain) P. Nikravesh (USA) J.T. Oden (USA) R. Ohayon (France) N. Olhoff (Denmark) X. Oliver (Spain) H. Orlande (Brasil) P. Pedersen (Denmark) H. Pina (Portugal) P. Prendergast (Ireland) A. Preumont (Belgium) J. N. Reddy (USA) P. Ribeiro (Portugal) R. Rolfes (Germany) W. Schiehlen (Germany) G.I. Schueller (Austria) O. Sigmund (Denmark) L. Simões (Portugal) J.C. Sousa (Portugal) R. Stenberg (Finland) A. Suleman (Portugal) A. Tadeu (Portugal) A. Torres Marques (Portgual) Z. Waszcyszyn (Poland) P. Wriggers (Germany) M. Zmindak (Slovak Republic) T.I. Zohdi (USA) Liu Gui-Rong (Singapore) M. Pietrzyk (Poland)

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International Correspondents M. Bercovier, The Hebrew University, Jerusalem, Israel S. Botello, University of Guanajuato, Mexico M. Cerrolaza, Central University of Venezuela, Venezuela C. K. Choi, Korea Advanced Inst. of Sc. & Techn., Korea R. Correa, Universidad de Chile, Chile S. M. Desphande, Indian Inst. of Science, India P. R. B. Devloo, Universidade Estadual de Campinas, Brazil A. Gaona, Universidad Nat. de Asuncion, Paraguay I. Herrera, UNAM, Mexico J. Herskovits, Federal University of Rio de Janeiro, Brazil J. Hurtado, Universidad Manizales, Colombia W. Kanok-Nukulchai, Asian Institute of Technology, Thailand M. Kawahara, Chuo University, Japan T. Kawai, University of Science in Tokyo, Japan A. M. Kharitonov, Russian Academy of Sciences, Russia T. Kobayashi, Japan E. B. Las Casas, Brasilian Association for Comp. Mech., Brazil C. H. Lee, University of Aeronautics, China A. Leung, City University of Hong-Kong, China C. A. Lin, National Tsing Hua University, Taiwan L. Quiroz, Universidad de Concepcion, Chile C. V. Ramakrishnan, Indian Institute of Technology, India R. Sampaio, PUC Rio, Rio de Janeiro, Brazil L. Súarez, Universidad de Puerto Rico, Puerto Rico K.Y. Sze, The University of Hong Kong, China S. Valliappan, University of New South Wales, Australia K. William, University of Colorado USA J. Yagawa, University of Tokyo, Japan Y.B. Yang, National University of Taiwan, Taiwan M. Yuan, University of Beijing, China W.X. Zhong, Dalian University of Technology, China F. G. Zhuang, CNSA Chinese Aerodynamics, China M. Xiao, NUAA, China

Sponsors European Community on Computational Methods in Applied Science International Association of Computational Mechanics Portuguese Association of Theoretical, Applied and Computational Mechanics Foundation of Science and Technology (Portugal); Calouste Gulbenkian Foundation (Portugal); Technical University of Lisbon (Portugal); National Laboratory of Civil Engineering (Portugal); Instituto Superior Técnico (Portugal)

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Computational Challenges for Multi-Physics Topology Optimization Martin P. Bendsøe Department of Mathematics, Technical University of Denmark Matematiktorvet B303, DK-2800 Kgs. Lyngby, Denmark [email protected] ABSTRACT Topology design involves working with multiple fields. Of primary interest is the density distribution of material that makes a certain objective function minimal while other objectives are satisfied as constraints in a mathematical programming statement. The evaluation of the objective and constraint functions will involve state variables that are fields that relate the design variables to physical behavior. The state fields are scalar or vector fields and multiple fields representing various physical responses may be involved; these fields will typically be coupled in multi-physics applications. Computational procedures for topology design (and for design optimization as a whole) thus encompass discretization schemes for design and state fields together with algorithms for optimization and for analysis. The prevailing computational approach to structural design and topology design in particular is to view the optimization procedure as a problem in the design variables only. This means that analysis is treated as a function call that provides information on function values and derivatives as a function of design. In optimization terms this is a nested format. An alternative to the nested format is to treat design and state fields on equal terms and formulate one unified optimization problem that involves also state equations as constraints. This is the typical approach in the areas of Mathematical Programs with Equilibrium Constraints (MPECs) and PDE-constrained optimization. Finally, in some cases it turns out to be advantageous to treat the design variables as functions of the states, for example where an explicit calculation of the optimal design for a fixed state is possible. This then leads to a variational statement for the optimal state field in itself. Computational challenges are thus by the nature of the problem two-fold and successful implementations rely on both efficient analysis (and the associated sensitivity analysis) and on the efficiency of optimization algorithms. The discretization of the analysis and design fields play here a significant role for stability and for obtaining relevant results, and it is typical that the optimization will utilize a poor model to give results of little physical meaning. For the design field, the computational model will typically also include a relaxation of an integer valued field and suitable ways to handle this is a crucial issue especially when extending topology design from continuum structural applications to multi-physics settings. The different approaches and the associated computational issues involved in their resolution will be illustrated by considering some recent work on design of multi-physics devices and the design of articulated mechanisms.

References [1] M.P. Bendsøe & O. Sigmund, Topology Optimization - Theory, Methods, and Applications. Springer-Verlag, Berlin-Heidelberg, 2003 (revised printing, 2004). [2] K.K. Choi & N.-H. Kim, Structural Sensitivity Analysis and Optimization, Vol 1 & 2. SpringerVerlag, New York, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Computational Geometry and the Analysis of Solids and Structures J.A. Cottrell , A. Reali†, Y. Bazilevs , T.J.R. Hughes Institute for Computational Engineering and Sciences The University of Texas at Austin 201 East 24th Street 1 University Station C0200 Austin, TX 78712, USA {jac3, bazily, hughes}@ices.utexas.edu †European

School for Advanced Studies in Reduction of Seismic Risk [email protected]

ABSTRACT The concept of Isogeometric Analysis is described. Basis functions generated from NURBS (NonUniform Rational B-Splines) are employed to construct exact geometric models. For purposes of analysis, the basis is refined and/or its order elevated without changing the geometry or its parameterization. Analogues of finite-element, h-refinement and p- refinement schemes are presented, and a new, more efficient, higher-order concept, k-refinement, is introduced. Refinements are easily implemented and exact geometry is maintained at all levels without the necessity of subsequent communication with a CAD (Computer Aided Design) description. In the context of structural mechanics, it is established that the basis functions are complete with respect to affine transformations, meaning that all rigid body motions and constant strain states are exactly represented. Standard patch tests are likewise satisfied. Numerical examples exhibit optimal rates of convergence for linear elasticity problems and convergence to thin elastic shell solutions. The concept of k-refinement is explored and shown to produce more accurate and robust results than standard finite elements for problems of structural vibrations. Through the use of nonlinear parameterizations, optical branches of frequency spectra are eliminated for k-refined meshes. Optical branches have been identified as contributors to Gibbs phenomena in wave propagation problems and the cause of rapid degradation of higher modes in p-method finite elements. A geometrically exact model of the NASA Aluminum Testbed Cylinder is constructed and frequencies and mode shapes are computed and shown to compare favorably with experimental results. It is argued that Isogeometric Analysis is a powerful generalization of standard, polynomial-based, finite element analysis.

References [1] T.J.R. Hughes, J.A. Cottrell, Y. Bazilevs. Isogeometric Analysis: CAD, finite elements, NURBS, edxact geometry, and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 194:4135–4195, 2005. [2] J.A. Cottrell, A. Reali, Y. Bazilevs, T.J.R. Hughes. Isogeometric Analysis of structural vibration. Computer Methods in Applied Mechanics and Engineering, in press, 2005. Also as an ICES report http://www.ices.utexas.edu.research/reports/2005/0527.pdf.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Advances in the Particle Finite Element Method for Fluid-Structure Interaction Problems Eugenio Oñate, Sergio R. Idelsohn, Miguel A. Celigueta and Ricardo Rossi International Center for Numerical Methods in Engineering (CIMNE) Universidad Politécnica de Cataluña Campus Norte UPC, 08034 Barcelona, Spain [email protected]

ABSTRACT There is an increasing interest in the development of robust and efficient numerical methods for analysis of engineering problems involving the interaction of fluids and structures accounting for large motions of the fluid free surface and the existence of fully or partially submerged bodies. Examples of this kind are common in ship hydrodynamics, off-shore structures, spillways in dams, free surface channel flows, liquid containers, stirring reactors, mould filling processes, etc. We present a general formulation for analysis of fluid-structure interaction problems using the particle finite element method (PFEM). The key feature of the PFEM is the use of a Lagrangian description to model the motion of nodes (particles) in both the fluid and the structure domains. Nodes are thus viewed as particles which can freely move and even separate from the main analysis domain representing, for instance, the effect of water drops. A mesh connects the nodes defining the discretized domain where the governing equations, expressed in an integral from, are solved as in the standard FEM. The necessary stabilization for dealing with the incompressibility condition in the fluid is introduced via the finite calculus (FIC) method. A fractional step scheme for the transient coupled fluid-structure solution is described. Examples of application of the PFEM method to solve a number of fluid-structure interaction problems involving large motions of the free surface and splashing of waves are presented.

References [1] E. Oñate, S.R. Idelsohn, F. Del Pin, R. Aubry, The particle finite element method. An overview, Int. J. Comput. Meth., Vol. 1 (2), 267-307, 2004. [2] S.R. Idelsohn, E. Oñate, F. Del Pin, The particle finite element method: a powerful tool to solve incompressible flows with free-surfaces and breaking waves, Int. J. for Num. Meth. in Engrg. Vol. 61, 964-989, 2004. [3] www.cimne.com/pfem

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Elastic and Plastic Impacts in Multibody Dynamics Werner Schiehlen, Robert Seifried Institute of Engineering and Computational Mechanics Paffenwaldring 9, 70569 Stuttgart, Germany [email protected] [email protected]

ABSTRACT Many mechanical systems are subject to impacts modeled as unilateral constraints using the multibody system approach and the coefficient of restitution found from measurements. A multi-scale method is presented for the computation of the coefficient of restitution considering elastic wave propagation and plastic deformation which may occur simultaneously. Different models are presented for the impact period: a continuum model and a modal model with elastostatic Hertzian contact resulting in a boundary approach, a linear modal model with precomputed and concurrently computed finite elements in the contact region, and a completely nonlinear finite element model. In engineering, impacts are usually emerging from repeated processes with repeated collisions occurring on a previously deformed contact area. Therefore, the efficient modal models are extended for the evaluation of repeated impacts. The influence of the initial velocity prior to the impact as well as the shape and the yield stress of the bodies involved are investigated. For the experiments a special test bench is designed using Laser-DopplerVibrometers for velocity and displacement measurements on a fast time scale.

References [1] C. Glocker, On frictionless impact models in rigid-body systems. Philosophical Transactions of the Royal Society of London, A359, 2385–2404, 2001. [2] W. Goldsmith, Impact: The Theory and Physical Behaviour of Colliding Solids, Edward Arnold Ltd, London, 1960. [3] B. Hu, W. Schiehlen, Multi-time scale simulation for impact systems: from wave propagation to rigid body motion. Archive of Applied Mechanics, 72, 885–897, 2003. [4] W. Schiehlen, R. Seifried, Three approaches for elastodynamic contact in multibody systems. Multibody System Dynamics, 12, 1–16, 2004. [5] W. Schiehlen, R, Seifried, P. Eberhard, Elastoplastic phenomena in multibody impact dynamics. Computer Methods in Applied Mechanics and Engineering, in press, [doi:10.1016/j.cma.2005.08.011]. [6] R. Seifried, W. Schiehlen, P. Eberhard, Numerical and experimental evaluation of the coefficient of restitution for repeated impacts. International Journal of Impact Engineering, 32, 508–524, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Active Aeroelastic Aircraft Structures A. Suleman and P. A. Moniz Instituto de Engenharia Mecânica, Instituto Superior Técnico (IDMEC-IST), Lisbon, Portugal. Av. Rovisco Pais 1, 1049-001 Lisboa PORTUGAL [email protected]

ABSTRACT Aeroelasticity results in problems such as structural divergence, aileron reversal, and flutter stability due to insufficient torsional stiffness of the wings and “aeroelastic weight penalty” became a widely used expression by engineers in aircraft design. Aeroelastic solutions generally involve increasing the structure stiffness or mass balance (passive solutions), which typically involve increase of weight and cost while decreasing performance. In the seventies, composite materials with highly anisotropic directional stiffness properties enabled the introduction of aeroelastic tailoring methods where the composition of thickness and orientation of the individual material layers could be tailored to minimize the added structural weight necessary to minimize the detrimental effects due to aeroelastic behavior. This technology paved the way for looking at aeroelasticity from a different perspective. The new paradigm consists of looking at the structural deformations, caused by aerodynamic forces, to be used intentionally and in a beneficial way in order to improve aerodynamic performance and help to create the required control forces. In the last two decades, a new actuation concept for structural control has emerged. This concept uses the multifunctional materials properties to control the structural stiffness and shape of composite materials. Several studies have been performed to demonstrate applications of adaptive structures in aircraft, helicopters and submarines. This paper presents the research and development of novel active aeroelastic control strategies, aimed at improved aircraft performance (structural weight, better control effectiveness) by controlling structural deformations to modulate the desired aerodynamic deformations. The proposed research was carried out in the framework of the European research project 3AS (Active Aeroelastic Aircraft Structures). To this end, the following research issues to enable active aeroelastic aircraft structures have been addressed: • Demonstrate the application of piezoelectric actuators and sensors to dynamic aeroelastic control of a structure and design an experimental and computational setup that allows the quantification of the performance in flutter suppression, buffeting vibration reduction, and attenuation of other external mechanical vibrations. • Develop an airborne flight test platform (RPV - Remotely Piloted Vehicle) to demonstrate the proposed concepts both in the wind tunnel and in actual flight conditions; • Develop a methodology to design and integrate the proposed adaptive structures technology in real aircraft while reducing the weight of a wing for a given flight envelope.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Multiscale Strategy for Solving Industrial Problems O. Allix L.M.T., E.N.S. de Cachan/ C.N.R.S./Univ. P. et M. Curie 61, avenue du Président Wilson, 94235 Cachan, France [email protected]

ABSTRACT Multi-scale strategies are now mature for most of linear spatial problem. They can also be applied as solver in the case of non-linear problem but it is our opinion that more robust and efficient methods can be designed in that case. Two examples will serve as illustration. The case of crack propagation [1] and the case of post-buckling analyses of aeronautical structures [2]. In [3] Ladevèze proposed a multi-scale strategy for solving non-linear problem. This method offers several possibilities to adapt it in order that the main difficulties of the problem are specifically treated. One of its appealing features is to split the macro part from the micro one at interface level only. This allow to work on the most adapted splitting of the interfacial quantities. The classical choice which allows to incorporate homogenization at the macro-level automatically is the one where the interfacial macro extractor is associated with the linear part. In the case of crack propagation it appears more adapted to introduce a discontinuous scheme at the macro level, the use of the PUM method [4] for crack allows improving the micro description of the solution. In the case of the post-buckling analysis of large aeronautical, the non-linear scheme used, allows to iterate where it is needed only, that is in the sub-structures prone to the highest degree of nonlinearity. As non-linearities are often localized (i. e; corresponds to a local buckling giving rise to large displacement of the whole structure) the computational effort can be reduced drastically in comparison with a classical [2].

References [1] P.-A. Guidault, O. Allix, L. Champaney , J.-P. Navarro “A micro-macro approach for crack propagation with local enrichment”, to appear in CST [2] Ph. Cresta, O. Allix, C Rey, S Guinard "Nonlinear localization strategies for domain decomposition methods: application to post-buckling analyses" to apppear in CMAME [3] P. Ladevèze, O. Loiseau, D. Dureisseix, “A micro-macro and parallel computational strategy for highly heterogeneous structures”, IJNME, 52 (1–2) 121–138,2001. [4] J. Melenk, I. Babuska, “The partition of unity finite element method: Basic theory and applications”, Computer Methods in Applied Mechanics and Engineering 139, 289–314, 1996.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Numerical Integration of the Nonlinear Dynamics of Elastoplastic Solids Francisco Armero & Christian Zambrana Structural Engineering, Mechanics and Materials University of California at Berkeley Berkeley, CA 94720 [email protected]

ABSTRACT Classical numerical schemes for the numerical integration of the equations of solid and structural dynamics in time show serious limitations when applied to the finite deformation range. These includes the classical Newmark and HHT type schemes. The unconditional stability property of some of these schemes is lost in the nonlinear range, with the simulations showing an unbounded growth of energy. Furthermore, the law of conservation of angular momentum is, in general, not inherited by the schemes in this nonlinear range either. To handle these difficulties, the formulation of the so-called energymomentum schemes has received a great deal of attention lately. As illustrated in [1], these difficulties are also present in the physically dissipative problem of finite strain plasticity. The schemes do not lead to dissipative discrete dynamical systems. We present in this contribution a new class of integration schemes for multiplicative finite strain plasticity that exhibit the non-negative energy dissipation characteristic of these physical systems and that preserve the conservation laws of linear and angular momenta in time. The exact physical dissipation is obtained, hence recovering previously developed energy-momentum schemes with the exact energy conservation for elastic problems in elastic steps of the elastoplastic simulations. Furthermore, the schemes allow the consideration of extensions showing a controllable high-frequency (numerical) energy dissipation to handle the numerical schemes of typical problems of interest. The new schemes rely on an alternative integration of the plastic evolution equations, or return mapping algorithm, that define the discrete in time evolution of the plastic internal variables and the exact enforcement of the yield constraint on the stresses driving the motion. We built on the developments presented in [1] and incorporate an additional set of new considerations for the volumetric contributions, thus arriving to algorithms that also preserve exactly the isochoric character of the plastic flow in some common models of finite strain plasticity (e.g. J2 -flow theory based on von Mises yield criteria). Moreover, we discuss the implementation of the new time-stepping schemes in the context of new formulations of assumed strain and mixed finite elements for the handling of the well-known volumetric locking in these cases, while obtaining the aforementioned conservation and dissipation properties in time. Several numerical simulations are presented illustrating the performance of the new schemes.

References [1] F. Armero, Energy-Dissipative Momentum-Conserving Time-Stepping Algorithms for Finite Strain Multiplicative Plasticity, Computer Methods in Applied Mechanics and Engineering, in press.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Computational Methods for Dynamic Crack Propagation Ted Belytschko, Song, Wang Northwestern University, Department of Mechanical Engineering 2145 North Sheridan Road, Evanston, IL 60208-3111 [email protected]

ABSTRACT Several methods for dynamic computational fracture mechanics are reviewed, and their performance in some benchmark problems is reviewed. The methods considered include element deletion, interelement unzipping models and the extended finite element method. Element deletion consists of simply deleting the element when a material criterion is met. The inter-element crack models are of the type proposed by Xu and Needleman, and by Ortiz and Pandolfi. The extended finite element method permits arbitrary crack propagation within the mesh. Advancement of the crack by several criteria is studied. Among the results that are compared are the crack paths, the speed of the crack and the energy dissipated by the fracture process.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Analysis and Design of Sandwich Structures Made of Steel and Lightweight Concrete Pål G. Bergan*, Kåre Bakken*, Karl-Christian Thienel† *

Det Norske Veritas Research NO-1322 Høvik Norway [email protected] [email protected]

†

Universität der Bundeswehr DE-85577 Neubiberg Germany [email protected]

ABSTRACT Reinforced concrete has been used in ship building for more than 150 years; particularly during war periods with shortage of steel. Unfortunately, this inexpensive material has proven not to be commercially viable, primarily because of added weight. A new concept for design of ships and marine structures has been developed using sandwich plates composed of steel skins with lightweight concrete as core material. New types of light-weight concretes have been developed and tested for this purpose. Very large structures can be built without increasing the core thickness considerably; scalability is achieved by way of a cellular structural concept. A Panmax type bulk carrier ship has been designed and analyzed by way of linear finite elements. The study shows great promise for this new sandwich concept in that it appears that it saves 30 to 40 percent of the steel and comes out with about the same weight compared with a standard steel ship design. It also seems clear that the sandwich concept may offer important advantages when it comes to safety, design life, and operational performance. The current type of concretes is very light with wet, demoulding density of less than 1000 kg/m3. Its use in sandwich panels has been extensively tested in the laboratory. Some testing has focused on strength of the light-weight concrete in compression and shear, whereas other tests have dealt with bond and pull-out strength in the steel-concrete interaction. The laboratory program has included static and fatigue testing of series of sandwich beams; it has to a large extent focused on interaction effects and the failure of the core material, particularly in shear failure. The testing shows how cracks initiate and develop. The cracks in the test specimen have a highly jagged appearance, indicating extensive capability of shear force transfer across cracks (shear locking). The tests also show that delamination primarily occurs as a secondary phenomenon, normally a short distance away from the contact plane between steel and concrete. The most remarkable finding is that the beams exhibit extensive ductility and ability to absorb energy far beyond initial cracking. A key to this behaviour is that the surface skins are locked by end plates at the end of the beams. The study reveals that there is a need for developing good mathematical models for this unconventional concrete. There is also a need for further developing nonlinear finite elements models that can simulate the complex behaviour and interaction effects of steel-concrete sandwich plates during all stages of damage development. The practical potential of this technology is very extensive.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Multiscale Modeling of Pore Collapse Instability in High-Porosity Solids Ronaldo I. Borja* *

Department of Civil and Environmental Engineering Stanford University, Stanford, CA 94305-4020, USA [email protected]

ABSTRACT High-porosity solids include elastomeric foams and other cellular materials, Aeolian sands, poorly cemented coquina, diatomite, and chalk. A majority of these materials are known to exhibit several regions of behavior in simple uniaxial or conventional triaxial compression: a nearly linearly elastic behavior at small strain, plastic behavior at larger strain, a plateau region in which strain increases at nearly constant stress, and, finally, a densification region characterized by pore collapse. In this paper we address the problem of pore collapse instability as a local bifurcation from a homogeneous solution driven by a singular constitutive tangent operator. We identify different eigenmodes (emodes) of bifurcation and propose an approach for constitutive branching useful for multiscale modeling of the pore collapse/densification process.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Instabilities and Discontinuities in Two-Phase Media René de Borst 1, 2, Marie-Angèle Abellan 3, Julien Réthoré 1 1

2

Faculty of Aerospace Engineering, Delft University of Technology P.O. Box 5058, 2600 GB Delft, The Netherlands [email protected]

LaMCoS – UMR CNRS 5514, INSA de Lyon, 69621 Villeurbanne, France 3

LTDS-ENISE – UMR CNRS 5513, Saint-Etienne, France

ABSTRACT Within the framework of the generalised theory of heterogeneous media, the complete set of equations is derived for a three-dimensional fluid-saturated porous medium. Subsequently, dispersion analyses are carried out for an infinite one-dimensional continuum, that has been deforming homogeneously prior to the application of the perturbation. A dispersive wave is obtained, but the internal length scale associated with it vanishes in the short wave--length limit, at least for the assumptions made regarding the constitutive behaviour of the solid and of the fluid. This result leads to the conclusion that, upon the introduction of softening, localisation in a zero width will occur and no regularisation will be present. This conclusion is corroborated by the results of numerical analyses of wave propagation in a finite one-dimensional bar [1]. The result has severe implications for finite element analyses of damaging multiphase media, since they will be mesh-dependent. To avoid mesh dependence, either the constitutive model for the solid must be equipped with a nonvanishing internal length scale, or any damage that occurs must be modelled in a strictly discrete manner. The second part of the contribution therefore focuses on the proper modelling of discontinuities in fluid-saturated porous media. A two-field finite element formulation has been set up with the displacements and the fluid pressure as the fundamental unknowns [2]. At discontinuities in the body, e.g. cracks, the displacement and the fluid pressure fields are allowed to be discontinuous. Numerically, the discontinuities in the displacements and the fluid pressure are incorporated using the partition-of-unity property of finite element shape functions. The tractions at the interface are related to the displacement jumps using a cohesive zone model, where the behaviour in the direction normal to the interface can be different from that in the tangential direction. Regarding the pore fluid flow, it has been assumed that the normal flux to the interface is proportional to the jump in pore pressures at both sides of the discontinuity, which can be conceived as the discrete analogon of Darcy’s relation for fluid flow in porous media. The paper concludes with some examples of finite element analyses of fluid-saturated porous media with discontinuities.

References [1] M.-A. Abellan, R, de Borst, Wave propagation and localisation in a softening two-phase medium. Computer Methods in Applied Mechanics and Engineering, doi: 10.1016/j.cma.2005.05.056. [2] R. de Borst, J.J.C. Remmers, A. Needleman, M.-A. Abellan, Discrete vs smeared crack models for concrete fracture: Bridging the gap. International Journal for Numerical and Analytical Methods in Geomechanics, 28, 583-607, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Towards Maneuvering Aeroelasticity — Progress in the Simulation of Large Fluid-Structure Interaction Problems Carlo L. Bottasso Politecnico di Milano Via La Masa 34, 20156 Milano, Italy [email protected]

ABSTRACT Advanced multidisciplinary software tools enable the development of sophisticated models of fixed and rotary wing vehicles. Such comprehensive vehicle models account for the interactions between the structural and aerodynamic fields, often coupled with hydraulic and electromechanical subsystem models and with control laws (aero-servo-elasticity). With these models it is now possible to predict with a growing level of confidence a variety of phenomena that are of crucial importance in the design phase, including loads, performance, stability, vibratory response, handling qualities and flight mechanics characteristics of the vehicle. The quick pace of the evolution of comprehensive vehicle models must be accompanied by a similar growth in the range of problems that can be addressed by simulation. In particular, it is clear that quite often the limiting factors that constrain the design are found in the maneuvering regime at the boundaries of the flight envelope. For example, a design engineer might be interested in flying a minimum time turn with a virtual model of a helicopter while not exceeding a maximum allowable load factor, in order to assess the vibratory characteristics of the machine in this extreme turn. Current aeroelastic simulation tools are not directly equipped to solve this class of problems, since in fact they all compute the dynamic response of the model under the action of assigned control inputs. Unfortunately, control inputs that will fly a given maneuver are in general not available, and must be computed. In this paper we describe a methodology for maneuvering aeroelasticity that is applicable to arbitrarily complex vehicle models. The approach is based on model-based virtual pilots that, on the basis of a formal maneuver description, first plan the path of the vehicle throughout the maneuver and then track it by driving the vehicle model along it, as proposed in Reference [1]. Both planning and tracking pilots are based on adaptive reduced models of the system and have the ability to learn, and hence improve their driving performance, as they steer the vehicle. We illustrate the use of this emerging technology with the help of relevant examples in the area of rotorcraft technology.

References [1] C.L. Bottasso, C.-S. Chang, A. Croce, D. Leonello, L. Riviello, Adaptive planning and tracking of trajectories for the simulation of maneuvers with multibody models. Computer Methods in Applied Mechanics and Engineering, in press, available online 10 October 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A Design Optimization Formulation for Problems with Random and Fuzzy Input Variables Using Performance Measure Approach K.K. Choi and Liu Du Department of Mechanical and Industrial Engineering The University of Iowa, Iowa City, IA 52242, U.S.A. [email protected] [email protected]

ABSTRACT To obtain reliable designs, aleatory and epistemic uncertainties are considered recently in the structural analysis and design optimization. The reliability based de-sign optimization (RBDO) method [1] is used when the amount of input data is sufficient enough to create accurate statistical distribution. On the other hand, when the sufficient input data are not available due to limitations in time, human, and facility resources, the optimum design may not be reliable if RBDO method is used. To deal with the situation that input uncertainties have insufficient information, a possibility (or fuzzy set) method can be used for structural analysis and possibility based design optimization (PBDO) [2]. However, in many industry design problems, we may have to deal with design problems that involve with the mixed input statistical random and fuzzy variables simultaneously. For these problems, RBDO may yield unreliable optimum designs because of insufficient data. On the other hand, treating the random variables as fuzzy variables and invoking PBDO to solve the mixed design variable problem may yield too conservative designs with higher optimum costs. This paper proposes a new mixed variable design optimization (MVDO) problem based on the performance measure approach (PMA) [1]. To evaluate the possibilistic constraint in MVDO, a sub-optimization problem for inverse analysis is carried out using a hyper-cylinder domain. To solve this sub-problem efficiently and effectively, a new numerical algorithm, maximum failure search (MFS) method, is proposed in this paper by combining the enhanced hybrid mean value (HMV+) method [3] for the inverse reliability analysis in RBDO and the maximal possibility search (MPS) method [2] for the inverse possibility analysis in PBDO. Some mathematical examples are used to demonstrate the efficiency and effectiveness of the proposed numerical MFS method. Some physical design examples are used to compare the proposed MVDO results with RBDO and PBDO results.

References [1] Youn BD, Choi KK, Park YH (2003) Hybrid analysis method for reliability-based de-sign optimization. Journal of Mechanical Design, ASME 125(2): 221-232 [2] Du L, Choi, KK, Youn BD. An Inverse Possibility Analysis Method For Possibility-Based Design Optimization. AIAA Journal, to be appear. [3] Youn BD, Choi KK, Du L (2005) Enriched Performance Measure Approach for Reliability-Based Design Optimization. AIAA Journal 43(4): 874-884

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Strength of Porous Ceramics – Mechanical Testing and Numerical Modelling Ioannis Doltsinis Faculty of Aerospace Engineering and Geodesy, University of Stuttgart Pfaffenwaldring 27, D-70569 Stuttgart [email protected]

ABSTRACT The lecture addresses modelling of the damage of porous ceramics on the microstructural level, and the failure of structural components. The research has been occasioned by the industrial interest in the strength of ceramic filter supports used in nanofiltration, where internal pressure is of importance. Porous ceramics subjected to fluid pressure in the pores are prone to brittle microfracturing which may progress to brittle or quasi-brittle rupture of the component. Apart from physical and numerical modelling on the microscale, subtle laboratory testing of components has been of paramount importance; brittleness suggests statistical evaluation. The study focuses on mechanical issues, but conceptual thoughts on stress-enhanced corrosion are included [4]. The phenomenon is significant to aging which determines the life-time of parts exposed to chemically aggressive media. A brief discussion on the formalism for the fracturing continuum [1] is followed by the modelling of cracking by separation of grain boundaries. The associated algorithm simulates progressive damage on artificial microstructures generated in the computer for given material characteristics [2], and determines rupture statistics in dependence of various parameters. Laboratory measurements on the structural parts of interest (circular cylinders with longitudinal channels) refer to the diametral compression (Brazilian) test which adequately replaces the condition of channel pressure [3], as justified by finite element stress analysis. The impact of the two loading cases on Weibull statistics is formally explained, effects of basic material and doping on component rupture are investigated, the damage tolerance confirmed, corrosion discussed. Synthesis incorporates material strength statistics in the finite element model and estimates critical locations in the structural part under internal pressure. There is quantitative agreement with pressure levels actually registered at rupture. The account refers to cooperative research performed at the Universities of Stuttgart and Caen [4].

References [1] I. Doltsinis, Issues in modelling distributed fracturing in brittle solids with microstructure, in: S.R. Idelsohn et al. (Eds.), Computational Mechanics - New Trends and Applications. (CDROM) CIMNE, Barcelona, 1998. [2] I. Doltsinis and R. Dattke, Modelling the damage of ceramics under pore pressure, Comput. Meths. Appl. Mech. Engng., 191, 29-46, 2001. [3] F. Osterstock, I. Doltsinis, O. Vansse O, The Brazilian reliability test and micromechanical modelling for channelled cylinders of multiphase porous ceramics, in: High-Performance Ceramics II, Trans. Tech. Publications, 2004. [4] I. Doltsinis and F. Osterstock, Modelling and experimentation on the strength of porous ceramics, Arch. Comput. Meth. Engng., 12, 303-336, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Reduced Order Models in Unsteady Aerodynamic Models, Aeroelasticity and Molecular Dynamics Earl H. Dowell*, Kenneth C. Hall*, Jeffrey P. Thomas*, Robert E. Kielb*, Meredith A. Spiker*, Charles M. Denegri, Jr. † *

†

Duke University, Durham, North Carolina, United States [email protected]

U.S. Air Force SEEK EAGLE Office, Eglin Air Force Base, Florida, United States

ABSTRACT The state of reduced order modeling of unsteady aerodynamic flows for the efficient calculation of fluid-structure interaction (aeroelasticity) is discussed. Reduced order modeling is a set of conceptually novel and computationally efficient techniques for computing unsteady flow about airfoils, wings, and turbomachinery cascades. Starting with either a time domain or frequency domain computational fluid dynamics (CFD) analysis of unsteady aerodynamic flows, a large, sparse eigenvalue problem is solved. Then, using just a few of the resulting aerodynamic eigenmodes, a Reduced Order Model (ROM) of the unsteady flow is constructed. The aerodynamic ROM can then be combined with a similar ROM for the structure to provide a Reduced Order Aeroelastic Model that reduces computational model sized and cost by several orders of magnitude. Moreover, the aerodynamic and aeroelastic eigenvalue and eigenmode information provides important insights into the physics of unsteady flows and fluidstructure interaction. The method is particularly well suited for use in the active control of aeroelastic (fluid-structural) and unsteady aerodynamic phenomena as well as in standard aeroelastic analysis. As an alternative to the use of aerodynamic eigenmodes, Proper Orthogonal Decomposition (POD) has also been explored. POD is an attractive alternative because of the greater simplicity of calculating POD modes rather than fluid eigenmodes per se. Moreover once the POD modes have been used to construct a Reduced Order Model, this ROM may be used to find a good approximation to the dominant aerodynamic eigenmodes. After the Hopf Bifurcation (flutter) condition is determined for the fluid-structural system, a novel High Dimensional Harmonic Balance (HDHB) solution method for the fluid (and structural) model(s) proves to be a very efficient technique for determining limit cycle oscillations in fluid-structural systems. In this approach one exploits the knowledge of the aeroelastic eigenmode determined from the aeroelastic ROM. Several examples will be discussed including the limit cycle oscillations (LCO) of the F-16 aircraft and the limit cycle oscillations (LCO) of the Von Karman vortex street behind a cylinder in a cross-flow. The latter is a prototypical example of self-excited fluid oscillations that occur for bluff bodies including wings at high angles of attack. Correlation of theoretical calculations with experiment will also be shown.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Inverse Engineering George S. Dulikravich1, Helcio R. B. Orlande2 and Brian H. Dennis3 1

2

3

Florida International University, Department of Mechanical & Materials Eng. 10555 West Flagler Street, Room EC 3474, Miami, Florida 33174, U.S.A. [email protected]

Federal University of Rio de Janeiro, Department of Mechanical Eng., COPPE, Cid. Universitaria, Cx. Postal 68503, Rio de Janeiro, RJ, 21941-972, Brazil [email protected]

Department of Mechanical and Aerospace Eng., University of Texas at Arlington, Arlington, Texas 78712, U.S.A. [email protected]

ABSTRACT Inverse problems are rapidly becoming a multi-disciplinary field with many practical engineering applications. The objective of this lecture is to present several such multi-disciplinary concepts and applications. In some examples, sophisticated regularization formulations were used. In other examples, different optimization algorithms were used as tools to solve de facto inverse problems. Due to the mathematical complexity of these multi-disciplinary and often multi-scale inverse problems, the most widely acceptable formulations eventually result in a need for minimization of a certain norm or a simultaneous extremization of several such norms. These single-objective and multiobjective minimization problems are then solved using appropriate robust evolutionary optimization algorithms. Specifically, we focus here on inverse problems of determining spatial distribution of a heat source for specified thermal boundary conditions, finding simultaneously thermal and stress/deformation boundary conditions on inaccessible boundaries, and determining chemical compositions of steel alloys for specified multiple properties.

References [1] M.N. Özisik, HRB Orlande, Inverse Heat Transfer: Fundamentals and Applications, Taylor & Francis, New York, NY, USA, 2000. [2] P.M.P. Silva, H.R.B. Orlande, M.J. Colaco, P.S. Shiakolas, G.S. Dulikravich, Estimation of Spatially and Time Dependent Source Term in a Two-Region Problem. In Proceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practic (Lesnic, D. ed.) July 11-15, 2005, Cambridge, United Kingdom [3] B.H. Dennis, G.S. Dulikravich, Z.-X. Han, Determination of Temperatures and Heat Fluxes on Surfaces and Interfaces of Multi-domain Three-Dimensional Electronic Components. ASME Journal of Electronic Packaging, Vol. 126, No. 4, December 2004, pp. 457-464. [4] B.H. Dennis, G.S. Dulikravich, S. Yoshimura, A Finite Element Formulation for the Determination of Unknown Boundary Conditions for Three-Dimensional Steady Thermoelastic Problems. ASME J. of Heat Transfer, Vol. 126, February 2004, pp. 110-118. [5] I.N. Egorov-Yegorov, G.S. Dulikravich, Inverse Design of Alloys for Specified Stress, Temperature and Time-to-Rupture by Using Stochastic Optimization. International Symposium on Inverse Problems, Design and Optimization – IPDO (Eds: Colaco, M., Orlande, H., Dulikravich, G.), Rio de Janeiro, Brazil, March 17-19, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Multiscale Approaches for Bridging Discrete and Continuum Scales Jacob Fish and Wen Chen Rensselaer Polytechnic Institute [email protected], [email protected]

ABSTRACT In this talk, I will present information-passing and concurrent discrete-to-continuum scale bridging approaches. In the concurrent approach both, the discrete and continuum scales are simultaneously resolved, whereas in the information-passing schemes, the discrete scale is modeled and its gross response is infused into the continuum scale. For the information-passing multiscale methods to be valid both the temporal and spatial scales should be separable. Among the information-passing bridging techniques, I will present the Generalized Mathematical Homogenization (GMH) theory [1,2] and the Multiscale Enrichment based on the Partition of Unity (MEPU) method [3]. The GMH constructs an equivalent continuum description directly from molecular dynamics (MD) equations. The MEPU approach gives rise to the enriched quasicontinuum formulation, capable of dealing with heterogeneous inter-atomic potentials, nonperiodic fields and high velocity impact applications. The second part of the talk will focus on multiscale systems, whose response depend inherently on physics at multiple scales, such as turbulence, crack propagation, friction, and problems involving nano-like devices. For these types of problems, multiple scales have to be simultaneously resolved in different portions of the problem domain. Among the concurrent bridging techniques, attention will be restricted to multilevel-like methods [4,5]. A space-time multilevel method for bridging discrete scales with either coarse grained discrete or continuum scales will be presented. The method consists of the wave-form relaxation scheme aimed at capturing the high frequency response of the atomistic vibrations and the coarse scale solution (explicit or implicit) intended to resolve the coarse scale features (in both space and time domains) of the discrete medium

References [1] J. Fish and C. Schwob, “Towards Constitutive Model Based on Atomistics,” International Journal of Multiscale Computational Engineering, Vol. 1 pp. 43-56, (2003). [2] W. Chen and J. Fish, “A Generalized Space-Time Mathematical Homogenization Theory for Bridging Atomistic and Continuum Scales,” to appear in Int. J. Num. Meth. Eng., (2005). [3] J. Fish and Z. Yuan, “Multiscale Enrichment based on Partition of Unity,” International Journal for Numerical Methods in Engineering, (2004), in print. [4] J. Fish and W. Chen, “Discrete-to-Continuum Bridging Based on Multigrid Principles,” Comp. Meth. Appl. Mech. Engng., Vol. 193, pp. 1693-1711, (2004). [5] H. Waisman and J. Fish, “Space-Time multigrid method for bridging discrete scales,” submitted to International Journal for Numerical Methods in Engineering (2005)

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Adaptive Mesh Generation in 3 Dimensions by Means of a Delaunay Based Method. Applications to Mechanical Problems. Paul Louis George ∗ INRIA,

Domaine de Voluceau, F 78153 Le Chesnay Cedex [email protected]

ABSTRACT Mesh adaptation is recognized as a powerful tool to compute accurate solution while minimizing the required resources (CPU time and memory space), therefore avoiding using parallel computing. It is also a way to accurately capture the physical behavior of the PDE problem in hand and, in some cases, this is the only way to access to a reasonable solution. Mesh adaptation in two dimensions can be considered as mature and is used in various problems. Right now, in three dimensions, the question is much more tedious and only a limited number of works can be reported. Mesh adaptation can be considered in two different ways. The first makes use of local modification of the current mesh so as to adapt it. Modification tools include well known operators such as point relocation, collapse (mesh coarsening), edge flips, point addition (mesh enrichment). Relatively easy to implement, such methods proved to give nice results in a number of cases but are not so flexible in specific when anisotropic features are desired. The second is based on the full generation of a new (adapted) mesh based on the current one and metric data provided at the nodes of this mesh. The generation method is then a variant (widely different in various aspects) of the well know mesh generation method. The aim being not only to mesh at the best a given domain but to match the given metric, which is much more demanding. We are concerned with this second approach and we propose a Delaunay based mesh generation method capable to complete adapted meshes. The mesh generation aspect is driven by metric data (element size and directional specification) which are defined by means of error estimates. Isotropic and anisotropic meshes can be produced. Concrete application examples will demonstrate the flexibility of the proposed method, show the low cost of the approach which compares well with the first approach.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

19

Computational Micromechanics of Biological Materials: Bone and Wood Christian Hellmich∗ , Karin Hofstetter∗ , Cornelia Kober† ∗ Vienna

†

University of Technology (TU Wien), Institute for Mechanics of Materials and Structures Karlsplatz 13/202, A-1040 Wien (Vienna), Austria [email protected], [email protected]

Osnabr¨uck University of Applied Science, Faculty of Engineering and Computer Science Albrechtsstraße 30, D-49009 Osnabr¨uck, Germany [email protected] ABSTRACT

Despite complex hierarchical organization of bone and wood, it was recently possible to identify a few elementary components at the micro and nanolevel of these material classes for the explanation of the diversity of macroscopic (poro-)elastic properties of different bones and woods [2, 3]. The mechanical properties (i.e. elasticity) of these elementary components are (up to experimental scattering) the same across a variety of different bones and woods, respectively; they are ’universal’, i.e., independent of tissue-type, species, and anatomical location. The mechanical interaction between these elementary components (mechanical morphology) and the dosages of these components in different tissues determine the macroscopic material properties. Having in mind that, as regards bone, these dosages are dependent on complex biochemical control cycles (defining the metabolism of the organism), the purely mechanical theory can be linked to biology, biochemistry, and, on the applied side, to clinical practice. Drug-driven or genetically driven changes in metabolisms lead to changes in the dosages of elementary components. The effects of these metabolic changes on the mechanical behavior of skeletal (sub)systems under well-defined loading conditions (e.g., downfall of elderly persons with osteoporosis) can then be studied by feeding structural models (e.g., Finite Element models [4]) of whole bones with the aforementioned macroscopic material properties - the output of the micromechanical models. This is probably highly relevant for patient-specific non-invasive bone disease diagnosis and therapy. As regards wood, our nano-to-macro approach is expected to support optimization of technological processes, such as drying.

References [1] L. Dormieux and F.-J. Ulm, editors. CISM Vol.480 – Applied Micromechanics of Porous Media. Springer, Wien - New York, 2005. [2] Ch. Hellmich. Microelasticity of bone, pages 289 – 331. In Dormieux and Ulm [1], 2005. [3] K. Hofstetter, C. Hellmich, and J. Eberhardsteiner. Development and experimental validation of a continuum micromechanics model for the elasticity of wood. European Journal of Mechanics A Solids, 24:1030 – 1053, 2005. [4] C. Kober, B. Erdmann, Ch. Hellmich, R. Sader, and H.-F. Zeilhofer. Validation of interdependency between inner structure visualization and structural mechanics simulation. International Congress Series, 1281:1373, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Mechanobiology: Computation and Clinical Application Gerhard A. Holzapfel∗† , Christian T. Gasser∗ , Dimitris Kiousis∗ ∗ Royal

Institute of Technology (KTH), School of Engineering Sciences Osquars backe 1, 100 44 Stockholm, Sweden {gh,tg,dk}@hallf.kth.se

† Graz

University of Technology, Computational Biomechanics Schiesstattgasse 14-B, 8010 Graz, Austria [email protected] ABSTRACT

Some reasons for the present worldwide procession of biomechanics are the exciting new developments in biology and the rapidly expanding field of mechanobiology, which aims to understand how cells respond to changes in their mechanical environment. Mechanobiology also studies the mechanical factors that may be important in, e.g., triggering the onset of atherosclerosis or aneurysms. Because of the inherent geometric, structural and material complexities of biological tissues [1], and the spatially non-uniform and time-varying boundary conditions, these type of problems demand computational methods [2], sufficient computational resources and graphics capability to display three-dimensional results. Computational models offer the potential to simulate multifield coupled processes encountered in the micro-heterogeneous biological tissues, and to realistically predict physiological functional interactions. Computational mechanobiology of biological tissue is increasing our ability to address multidisciplinary problems of academic, industrial and clinical importance. This presentation deals with the 3D modeling of balloon angioplasty, in particular of the interaction of balloon, stent and an atherosclerotic carotid artery. The artery is modeled as a heterogenous structure composed of three layers and a plaque. Distinctive attention is paid to the 3D contact of the artery with a stent and a balloon catheter, inflated with the goal to enlarge the area of the vessel. We use a smooth contact representation which is shown to be superior with respect to facet elements discretizing the interacting bodies. In addition, the dissection of the plaque, typically occurring during balloon inflation, is modeled by means of strong discontinuities and the application of the theory of cohesive zones using the partition of unity finite element method [3]. Contact pressure between the balloon-stent structure and the arterial wall leads to non-physiological stress concentration that can trigger adverse biological responses of the cells culminating in in-stent restenosis. Indeed, it has been shown that the design of a stent is a major risk factor for restenosis. A strategy of designing novel stents is shown with the goal to minimize vascular injury and to optimize long-term success [1]. Acknowledgement. Financial support for this research was partly provided by the Austrian Science Foundation under START-Award Y74-TEC, and KTH.

References [1] G. A. Holzapfel and R. W. Ogden, editors. Mechanics of Biological Tissue. Springer, Heidelberg, 2005. [2] G. A. Holzapfel. Computational biomechanics of soft biological tissue. In E. Stein, R. de Borst, and T. J. R. Hughes, editors, Encyclopedia of Computational Mechanics. Volume 2 Solids and Structures, pages 605– 635, Chichester, 2004. John Wiley & Sons. [3] T. C. Gasser and G. A. Holzapfel. Physical and numerical modeling of dissection propagation in arteries caused by balloon angioplasty. In M. H. Hamza, editor, Proceedings of the 3rd IASTED International Conference on Biomechanics, pages 229–233. ACTA Press, Anaheim, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Recent Developments of Hybrid Crack Element: Determination of Its Complete Displacement Field and Combination with XFEM B.L. Karihaloo, Q.Z. Xiao School of Engineering, Cardiff University, Queen’s Buildings, The Parade, Newport Road, Cardiff CF24 3AA, UK [email protected] (BL Karihaloo), [email protected] (QZ Xiao)

ABSTRACT The hybrid crack element (HCE) [1] is one of the most accurate and convenient finite elements (FEs) for the direct calculation of the stress intensity factor (SIF) and coefficients of the higher order terms of the Williams expansion [2, 3]. It represents a crack by only one super-element which is connected compatibly with the surrounding elements. It is very efficient for analysing bodies with many cracks [4]. The HCE is formulated from a simplified variational functional using truncated asymptotic crack tip displacement and stress expansions and interelement boundary displacements compatible with the surrounding regular elements. In the implementation, a general FE mesh can be used by forming the HCE from elements surrounding the crack tip [5]. The HCE can thus be included in any commercial package as conveniently as normal hybrid stress elements. However, the exclusion of the rigid body modes in the truncated asymptotic displacements creates jumps between these displacements and element boundary displacements. In this study, the rigid body modes are recovered by minimising these jumps via a least squares method. If the HCE only is used, the part of the crack inside the HCE need not conform to the mesh. However, crack faces away from the crack tip (outside the HCE) need to conform to the mesh. This disadvantage can be avoided by combining the HCE with the extended FEM (XFEM) [6]. The XFEM enriches the standard local FE approximations with a displacement discontinuity across a crack, and the asymptotic solution at the crack tip, with the use of the partition of unity (PU). It avoids using meshes conforming with the discontinuity and also adaptive remeshing as the discontinuity grows as is the case with the FEM. XFEM offers great flexibility in the modelling of the fracture process. However, the accuracy of the displacements and/or stresses in a few layers of elements surrounding the crack tip is low. The combined method using both HCE and XFEM inherits the flexibility of the XFEM and the high accuracy of the HCE. Typical static and propagating crack problems will be presented to demonstrate the efficiency and accuracy of this method.

References [1] P. Tong, T.H.H. Pian, S.J. Lasry, A hybrid element approach to crack problems in plane elasticity. Int J Numer Meth Engng, 7, 297-308, 1973. [2] B.L. Karihaloo, Q.Z. Xiao, Accurate determination of the coefficients of elastic crack tip asymptotic field by a hybrid crack element with p-adaptivity. Engng Fract Mech, 68, 1609-30, 2001. [3] Q.Z. Xiao, B.L. Karihaloo, X.Y. Liu, Direct determination of SIF and higher order terms of mixed mode cracks by a hybrid crack element. Int J Fract, 125, 207-25, 2004. [4] D. Zeng, N. Katsube, J.M. Zhang, W. Soboyejo, Hybrid crack-tip element and its applications. Finite Elem Anal Des, 38, 319–35, 2002. [5] B.L. Karihaloo, Q.Z. Xiao, Implementation of HCE on a general FE mesh for interacting multiple cracks. Proc ECCOMAS 2004. Jyväskylä, Finland, 24 - 28 July, 2004. CD-ROM. [6] N. Moës, J. Dolbow, T. Belytschko, A finite element method for crack growth without remeshing. Int J Numer Meth Eng, 46, 131-150, 1999.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Structural Model Validation and the Lack-of-Knowledge Theory Pierre Ladevèze *,†, Paul Enjalbert *, Guilllaume Puel •, Thierry Romeuf ռ *

LMT-Cachan (ENS Cachan/CNRS/Paris 6 University) 61 avenue du Président Wilson, F-94235 Cachan Cedex, France {pierre.ladeveze,paul.enjalbert}@lmt.ens-cachan.fr †

EADS Foundation Chair Advanced Computational Structural Mechanics • LMMSMat (École Centrale Paris/CNRS) Grande Voie des Vignes, F-92295 Châtenay-Malabry Cedex, France [email protected] ռ EADS Space Transportation Route de Verneuil BP96, F-78133 Les Mureaux Cedex, France [email protected]

ABSTRACT Today, the validation of complex structural models - i.e. the assessment of their quality compared to an experimental reference - remains a major issue. Most advanced approaches rely on the updating of deterministic dynamic parameters (stiffness, mass, damping) based on free or forced vibration tests. Uncertainties and probabilistic models can also be taken into account. In these works, the model validation is performed in a restricted sense. The true validation problem should be addressed through the comparison between the model - whether deterministic or not - used classically and the complete reality: such an issue raises philosophical questions. Here, we introduce a tentative answer through the Lack-Of-Knowledge (LOK) Theory, whose aim is to ”model the unknown”. In a certain way, this can be interpreted as an extension of what design engineers do when they introduce safety factors. Of course, the theory takes into account all the sources of uncertainties, including modeling errors, through the concept of basic LOKs. So far, two types of basic LOKs have been introduced: stiffness and excitation. For example, the structure being considered as an assembly of substructures, the basic stiffness LOKs are defined on the substructure level: each LOK is a scalar internal variable which quantifies the substructure’s LOK state in terms of structural stiffness; in mathematical terms, this variable is bounded by two stochastic bounds which follow probabilistic laws. Finally, a set of basic LOKs is added to the classical model to constitute the true model. This leads to an envelope of the actual responses; in particular, we can derive for the whole structure the effective LOK of a quantity of interest D, resulting in an interval with stochastic bounds. Another major question is the reduction of the LOKs using additional experimental information; the starting point could be an overestimated initial LOK level coming from experience. The paper focuses on the basic ideas of the Lack-Of-Knowledge Theory and on its first applications. Academic examples as well as industrial cases will be presented.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Modeling of Historical Masonry with Discrete Elements J. V. Lemos Laboratório Nacional de Engenharia Civil (LNEC) Av. do Brasil, 101, 1700-066 Lisboa, Portugal [email protected]

ABSTRACT Masonry is an inherently discontinuous material, formed by various components (stones, bricks, mortar), and its mechanical behavior reflects this internal structure. Engineering modeling of masonry structures is often based on continuum representations, using appropriate constitutive models, which provide an adequate solution for many practical cases. However, discontinuum models, which attempt to represent more closely the masonry components, are today applied with increasing frequency. They are preferred in research studies intended at understanding the mechanical behavior of masonry structures, but they are now also applied in engineering analysis, as the progress in computational resources is making them more accessible. Discontinuum representations may be achieved with various formulations of finite element methods. The present paper focuses on discrete elements models, a designation that covers a variety of representations of a structure as a system of blocks (rigid or deformable) or particles. Simplified contact formulations are generally used, to allow the analysis of very large systems, and explicit large displacement algorithms are employed. The paper discusses, in particular, two types of discrete element model, in relation to the specific requirements of masonry analysis. The first type is block models (rigid or deformable), which have proved very effective in the seismic analysis of historical stone masonry structures. Several applications are reviewed and key issues arising in this field are addressed, namely the variability of response observed in rigid block dynamics and the simplifications required in the analysis of large or complex structures. The second type of discrete element models examined is circular particle models, which represent each masonry block as a set of disks linked by contacts with tensile and shear bonds. Particles may also be used to simulate mortar or infill, and contacts between the various components are assigned appropriate constitutive behavior. An example of analysis of a multi-leaf wall is presented. The potential uses of these models in the study of the fundamental mechanics of masonry, in particular irregular masonry constructions, are discussed.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A Regularized Strong-form Meshfree Method for Adaptive Analysis G.R. Liu, Bernard B.T. Kee Centre for ACES, Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, Singapore 117576 [email protected]; [email protected]

ABSTRACT This talk presents an adaptive meshfree method which is based on strongform formulation and does not use any mesh predefined through node connectivity. In this present formulation, a radial point collocation procedure is used to discretize the system governing equations. Techniques are presented to stabilize the solution to obtain stable and accurate results. Adaptive scheme adopted in this work uses an error indicator based on residuals. Simple and practical refinement procedures are also presented for additional node insertion at each adaptive step. Numerical examples are presented to demonstrate that the proposed adaptive meshfree method can obtain efficiently stable solutions of desired accuracy.

References [1] G. R. Liu and Y. T. Gu, An introduction to meshfree methods and their programming, Springer, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Multiresolution Analysis for Material Design Wing Kam Liu Department of Mechanical Engineering, Northwestern University, Evanston, IL, 60208 [email protected]

ABSTRACT The relationship between material microstructure and properties is the key to optimization and design of lightweight, strong, tough materials. Material properties are inherently a function of the microscale interactions at each distinct scale of deformation in a material. Currently, we rely on empirical data to define the structure-property link in the material design chain. A model is proposed here in which a material is physically and mathematically decomposed to each individual scale of interest. Material deformation can subsequently be resolved to each of these scales. Constitutive behavior at each scale can be determined by analytically or computationally examining the micromechanics at each scale. The proposed multiresolution technique is capable of linking overall material properties to the underlying microstructure via the micromechanics at each scale of interest. The small scale deformation phenomena which have a profound impact on macroscale properties are captured. The technique is general enough to be used in any material which exhibits different constitutive behavior at each scale. It can be implemented in a general finite element framework. This is illustrated for a polycrystalline material, a granular material, an alloy containing particles at two scales. A potential use for a bio-inspired self healing composite is also discussed. The theory can then be applied computationally in a finite element framework to determine the overall material properties in terms of the constitutive behavior at each scale, without resorting to empiricism.

References [1] Cahal McVeigh, Franck Vernerey, Wing Kam Liu and L. Cate Brinson, Multiresolution Analysis for Material Design, To appear in the special issue of Computer Methods in Applied Mechanics and Engineering, in memory of Professor J H Argyris. [2] S. Li, W.K. Liu, Meshfree Particle Methods, Springer (2004) [3] S. Hao, W.K. Liu, C.T. Chang, Computer Implementation of Damage Models by Finite Element and Meshfree Methods, Computational Methods for Applied Mechanics and Engineering 187, (2000) 401-440 [4] S. Hao, W.K. Liu, B. Moran, F. Vernerey, G.B. Olson, Multi-scale constitutive model and computational framework for the design of ultra- high strength, high toughness steels, Computer Methods in Applied Mechanics and Engineering 193 n17-20 (2004) 1865-1908 [5] F. Vernerey, W.K. Liu, B. Moran, Continuum theory for multi- scale micromorphic materials, Northwestern University Report, Illinois (2005) [6] H. Kadowaki, W.K. Liu, Bridging Multiscale method for localization problems, Computer methods in applied mechanics and engineering, 193 (2004) 3267-3302 [7] L.C. Brinson, One Dimensional Constitutive Behavior of Shape Memory Alloys: Thermomechanical derivation with non-constant material functions, Journal of Intelligent Material Systems and Structures 4(2) (1993) 229-242 [8] W.K. Liu, E.G. Karpov, S. Zhang, H.S. Park, An introduction to computational nanomechanics and materials, Computer Methods in Applied Mechanics and Engineering, 193 n17-20 May (2004) 1529-1578

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Characterization and Multiscale Modeling of Asphalt - Recent Developments in Upscaling of Viscous and Strength Properties Roman Lackner*, Ronald Blab†, Josef Eberhardsteiner*, and Herbert A. Mang* *

Institute for Mechanics of Materials and Structures, Vienna University of Technology (TU Wien) Karlsplatz 13/202, 1040 Vienna, Austria {Roman.Lackner,Josef.Eberhardsteiner,Herbert.Mang}@tuwien.ac.at †

Institute for Road Construction and Maintenance, Vienna University of Technology (TU Wien) Gußhausstraße 28/233, 1040 Vienna, Austria [email protected]

ABSTRACT The assessment and prediction of the performance of multi-composed materials, such as e.g. asphalt, requires suitable procedures for identification of their mechanical properties. In case of asphalt used for trafficked pavements, these properties vary with the underlying mix design (volume fractions and used constituents) and additives (e.g., polymers). In the past, the mix design and the allowance of additives were optimized, aiming at (a) a low viscosity at high temperatures (T > 135 °C) for the construction and compaction process of high-quality asphalt layers, (b) a significantly higher viscosity at medium temperature in order to minimize the development of permanent deformations (rutting), and (c) sufficient relaxation behavior at sub-zero temperatures, avoiding low-temperature cracking (see failure modes in Figure (left)). Motivated by the large variety of asphalt mixtures resulting from this optimization process and the necessity of predicting the future performance of pavements, a multiscale model for asphalt is currently developed at TU Wien. It relates macroscopic properties to finer-scale information (such as volume fractions, morphology, and the behavior of material phases) by introducing, in addition to the so-called macroscale (i.e., the scale at which prediction analyses are performed), four finer scales of observation, ranging down to the so-called bitumen-scale (see Figure (right)).

Figure: (left) failure modes in asphalt pavements and (right) multiscale model for asphalt In this lecture, recent results on upscaling information from finer observation scales towards the macroscale are presented. For this purpose, both analytical methods (continuum micromechanics) and numerical schemes (limit analysis) are employed. In addition to the theoretical work, the presented multiscale model for asphalt requires a significant amount of experimental work, covering both the identification of properties of material phases at the different observation scales and the validation of the employed upscaling schemes. Test results as well as advanced test methods employed at the Christian Doppler Laboratory for “Performance-based optimization of flexible pavements” at TU Wien for the characterization of asphalt at different scales are presented.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Dissipative Interface Modeling for Vibroacoustic Problems - A New Symmetric Formulation ¨ Walid Larbi, Roger Ohayon Jean-Franc¸ois Deu, Structural Mechanics and Coupled Systems Laboratory Conservatoire National des Arts et M´etiers Chair of Mechanics - case 353 292 rue Saint-Martin, 75141 Paris Cedex 03, France [email protected]

ABSTRACT This work concerns the variational formulation and the numerical computation of vibroacoustic interior problems with interface damping. The coupled system consists of an elastic structure (described by a displacement field) containing an inviscid, compressible and barotropic fluid (described by a pressure field), gravity effects being neglected. Within the context of noise reduction techniques, we propose to investigate the effect of introducing a thin layer of damping material at the fluid-structure interface. The originality of this work lies in the introduction of an additional unknown field at the fluid-structure interface, namely the normal fluid displacement field [1, 2]. With this new scalar unknown, various interface damping models can be introduced in the variational formulation. Moreover, the associated finite element matrix system can be solved in frequency and time domains. Here, a Kelvin-Voigt rheological model is used to take into account the interface damping. For a given material, the damping parameters can be found from the experimental acoustic impedance in a particular frequency range [3]. Following the procedure developed in [4], the proposed variational formulation is written in a symmetric form through the introduction of a displacement potential of the fluid. Numerical examples are presented in order to validate and analyze the new formulation.

References [1] J.-F. De¨u, W. Larbi and R. Ohayon, Structural-acoustic vibration and transient problems with interface damping. Third M.I.T. Conf. on Computational Fluid and Solid Mechanics, 14-17 June, 2005, Cambridge, USA. [2] W. Larbi, J.-F. De¨u and R. Ohayon, A new finite element formulation for internal acoustic problems with dissipative walls. International Journal for Numerical Methods in Engineering, accepted for publication, 2006. [3] V. Kehr-Candille and R. Ohayon, Elastoacoustic damped vibrations – Finite element and modal reduction methods. P. Ladev`eze, O.C. Zienkiewicz eds. New Advances in Computational Structural Mechanics, Elsevier, Amsterdam, 321–334, 1992. [4] H.J.-P. Morand and R. Ohayon, Fluid-Structure Interaction. Wiley, New York, 1995.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Seismic Design Procedures in the Framework of Evolutionary Based Structural Optimization Manolis Papadrakakis, Nikolaos D. Lagaros, Michalis Fragiadakis

Institute of Structural Analysis & Seismic Research, National Technical University of Athens 9 Iroon Polytechniou Str., Zografou Campus, 15780 Athens, Greece {mpapadra, nlagaros, mfrag}@central.ntua.gr

ABSTRACT Since the early seventies structural optimization has been the subject of intensive research and several different approaches have been advocated for the optimal design of structures in terms of optimization methods or problem formulation. Most of the attention of the engineering community has been directed towards the optimum design of structures under static loading conditions with the assumption of linear elastic structural behaviour. For a large number of real-life structural problems assuming linear response and ignoring the dynamic characteristics of the seismic action during the design phase may lead to structural configurations highly vulnerable to future earthquakes. Furthermore, seismic design codes suggest that under severe earthquake events the structures should be designed to deform inelastically due to the large intensity inertia loads imposed. The objective of this work is to evaluate various design procedures adopted by seismic codes and their influence on the performance of real-scale structures under an objective framework provided by structural optimization. Several studies have appeared in the literature where seismic design procedures based on non-linear response (e.g. [1,2]) are presented and compared. However, this task can be accomplished in a complete and elaborate manner only in the framework of structural optimization, where the designs obtained with different procedures can be directly evaluated by comparing the value of the objective function of the optimization problem and the seismic performance of the optimum solution achieved. In this work evolutionary methods are implemented [3-5] to address the optimization problem and replace the conventional trial and trial and adjustmentbased procedures.

References [1] Han SW, Wen YK, Method for reliability-based seismic design: Equivalent nonlinear systems, II: Calibration of code parameters. ASCE Journal of Structural Engineering, 123(3): 256-270, 1997. [2] Bazzuro P, Cornell CA, Shome N, Carballo JE. Three proposals for characterizing MDOF nonlinear response. ASCE Journal of Structural Engineering 1998; 124(11): 1281-1289. [3] Foley CM. Optimized performance-based design for buildings. In Burns S.A. (Ed.) Recent Advances in Optimal Structural Design, ASCE Publications, 2002: 169-240. [4] M. Fragiadakis, N.D. Lagaros and M. Papadrakakis, Performance based earthquake engineering using structural optimization tools, International Journal of Reliability and Safety, 2005. [5] M. Fragiadakis, N.D. Lagaros and M. Papadrakakis Performance-based optimum design of steel structures considering life cycle cost, Structural and Multidisciplinary Optimization, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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High-Fidelity Multi-Criteria Aero-Structural Optimisation using Hierarchical Parallel Evolutionary Algorithms L. F. González*, L. H. Damp*, J. Périaux** and K. Srinivas* *

**

The University of Sydney, Sydney, NSW 2006, Australia {gonzalez, lloyd.damp, ragh}@aeromech.usyd.edu.au

INRIA Sophia Antipolis, OPALE project associate and CIMNE/UPC Barcelona [email protected]

ABSTRACT An emerging technique for the solution of Multi-criteria and Multidisciplinary Optimisation problems are Evolutionary Algorithms (EAs). This paper describes a parallel multi-criteria (multi-objective) evolutionary algorithms (PEAs) for aero-structural problems. The foundations of the algorithm are based upon traditional evolution strategies and incorporate the concepts of a multi-objective optimisation, hierarchical topology, asynchronous evaluation and parallel computing. The algorithm works as a black-box optimiser and has been coupled to several aerodynamic and aircraft conceptual design solvers. The paper describes the features of the method and the application of the method for aero- structural design problems. The coupling of the algorithm with a higher order panel method, a commercial FEA solver and an automated aerodynamic and structural mesh generation program is described. In this automated process the optimiser defines design variables for the automatic mesh generation program which defines and produces the external geometry of the wing (Root chord, tip chord, sweep, etc) and the internal geometry (number of spars, number of ribs skin thickness, etc). This geometry is then analysed by the aerodynamic solver and maps the pressure forces into the structural model. Single or multiple objectives can be defined by the optimiser to find non-dominated –Pareto optimal solutions or the Nash equilibrium. The parallel computing capabilities in combination with hierarchical levels of aerodynamic and FEA solvers are exploited to reduce computational expense. Comparisons will be presented between a traditional analytical approach -weakly couple, and a high fidelity –strongly coupled- approach for the aero-structural analysis of a high aspect ratio aircraft/UAV wing.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Interaction of Shells and Membranes with Incompressible Flows Ekkehard Ramm∗, Christiane F¨orster∗, Malte Neumann∗, Wolfgang A. Wall† ∗

Institute of Structural Mechanics, University of Stuttgart Pfaffenwaldring7, 70569 Stuttgart, Germany {ramm,foerster,neumann}@statik.uni-stuttgart.de

†Chair of Computational Mechanics, Technical University of Munich

Boltzmannstraße 15, 85747 Garching, Germany [email protected]

ABSTRACT For the dynamic behavior of lightweight structures like thin shells and membranes exposed to fluid flow the interaction between the two fields is often essential. Computational fluid-structure interaction provides a tool to predict this interaction and complement or eventually replace expensive wind tunnel experiments. Partitioned analyses techniques enjoy great popularity for the numerical simulation of these interactions. This is due to their computational superiority over simultaneous, i.e. fully coupled monolithic approaches, as they allow the independent use of suitable discretization methods and modular analysis software. We use, for the fluid, GLS stabilized finite elements on a moving domain based on the incompressible instationary Navier-Stokes equations, where the formulation guarantees geometric conservation on the deforming domain. The structure is discretized by nonlinear, three-dimensional shell elements. Commonly used sequential staggered coupling schemes may exhibit instabilities due to the so-called artificial added mass effect. As best remedy to this problem subiterations should be invoked to guarantee kinematic and dynamic continuity across the fluid-structure interface. Since iterative coupling algorithms are computationally very costly, their convergence rate is very decisive for their usability. To ensure and accelerate the convergence of this iteration the updates of the interface position are relaxed. The time dependent, ’optimal’ relaxation parameter is determined automatically without any user-input via exploiting a gradient method or applying an Aitken iteration scheme. A variety of numerical examples will show the capabilities of the presented methods.

References [1] Ramm, E., Wall, W. A.: Shell Structures - A Sensitive Interrelation between Physics and Numerics. International Journal for Numerical Methods in Engineering 60, 381-427, 2004. [2] Neumann, M., Tiyyagura, S.R., Wall, W.A., Ramm, E.: Robustness and Efficiency Aspects for Computational Fluid Structure Interaction. Proc. of the Second Russian-German Advanced Research Workshop on Computational Science and High Performance Computing, Stuttgart, Germany, March 14-16, 2005. In: Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM), Springer, 2005. ¨ [3] Forster , Ch., Wall, W.A., Ramm, E.: On the Geometric Conservation Law in Transient Flow Calculations on Deforming Domains. Int. J. Num. Meth. Fluids, 2005 (accepted).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Nonlinear Analysis of Composite and FGM Shells using Tensor-Based Shell Finite Elements J. N. Reddy* and R. A. Arciniega Advanced Computational Mechanics Laboratory Texas A & M University, College Station, TX 77843-3123 *[email protected]

ABSTRACT In this paper, a finite element model for the nonlinear analysis of laminated shell structures and through-thickness functionally graded shells is presented. A tensor-based finite element formulation is presented to describe the deformation and constitutive laws of a shell in a natural and simple way by using curvilinear coordinates. In addition, a family of high-order elements with Lagrangian interpolations is used to avoid membrane and shear locking; no mixed interpolations are employed. A first-order shell theory with seven parameters is derived with exact nonlinear deformations and under the framework of the Lagrangian description. This approach takes into account thickness changes and, therefore, 3D constitutive equations are utilized. Numerical comparisons of the present results with those found in the literature for typical benchmark problems involving isotropic and laminated composite plates and shells as well as functionally graded plates and shells are found to be excellent and show the validity of the developed finite element model. Moreover, the simplicity of this approach makes it attractive for applications in contact mechanics and damage propagation in shells. Acknowledgement. The research results reported herein were obtained while the authors were supported by the Structural Dynamics Program of the Army Research Office (ARO) through Grant . 45508EG.

References [1] R.A. Arciniega, On a tensor-based finite element model for the analysis of shell structures, PhD dissertation, Dept. of Mechanical Engineering, Texas A&M University, December 2005. [2] J.N. Reddy, An Introduction to Nonlinear Finite Element Analysis, Oxford University Press, Cambridge, UK, 2004. [3] J.N. Reddy, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, 2nd edition, CRC Press, Boca Raton, Florida, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Strength of Textile Composites – A Voxel Based Continuum Damage Mechanics Approach Raimund Rolfes*, Gerald Ernst*, Daniel Hartung†, Jan Teßmer† *

†

Institute for Structural Analysis, University of Hannover Appelstraße 9a, 30167 Hannover, Germany {r.rolfes, g.ernst}@isd.uni-hannover.de

Institute of Composite Structures and Adaptive Systems, DLR Braunschweig Lilienthalplatz 7, 38108 Braunschweig, Germany {jan.tessmer, daniel.hartung}@dlr.de

ABSTRACT Industrialised infusion processes enable a cost-effective possibility to produce textile composite structures compared to pre-impregnated composite systems (Prepregs). Particularly with regards to high performance structures one has to be familiar with the material behaviour and the failure characteristic to apply fibre reinforced composites profitable. In this publication a brief overview of the current research activities to characterise the mechanical behaviour, failure and strength of textile composites compared with prepreg systems is presented. Unlike pre-impregnated and filament winding composites, textiles are different in their mechanical behaviour due to various fibre architectures of the preforms (braided, woven, stitched, tufted, etc.). Therefore, a finite element analysis of a representative volume element (unit cell) on a micromechanical level is a promising possibility to analyse the mechanical behaviour and to predict the material failure. Currently different approaches are used to account for many material characteristics. The first results of an approach, which considers a continuum damage model to predict the first micromechanical material failure, will be presented. A lot of standards have been established to determine the material properties of Prepregs over the last years. Particularly textile composites require an adapted test setup to account for the characteristic material behaviour and to validate different failure criteria. The standards used to qualify a preimpregnated material and a short description of the requirements for textile composites are presented. While many failure theories were developed during the last years, there are some weaknesses even by the most popular failure theories. The results of the World Wide Failure Exercise have shown that there is a demand for an accurate failure prediction also for prepreg composites. Especially the comparatively complex failure of textile composites requires an advanced failure theory. The fracture plane concept originally proposed by Hashin is a promising method to describe the failure behaviour of prepreg composites. A three dimensional failure model was developed based on a fracture plane by Juhasz, who considers the characteristic material behaviour of orthogonally reinforced composites. This criterion was implemented in a three dimensional finite element to account for the three dimensional stress state in each layer of a lamina by Kuhlmann and Rolfes [1].

References [1] G. Kuhlmann and R. Rolfes, A hierarchic 3d finite element for laminated composites. International Journal for Numerical Methods in Engineering, 61, 96–116, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Concrete at Early Ages and Beyond: Numerical Model and Validation Bernhard A. Schrefler*, Francesco Pesavento*, Dariusz Gawin†, Mateusz Wyrzykowski† *

Dept. of Structural and Transportation Engineering, University of Padova, Italy Via F. Marzolo 9, 34131 Padova - Italy [email protected], [email protected]

†

Dept. of Building Physics and Building Materials, Technical University of Lodz, Poland Al. Politechniki 6, 93-590 Lódz, Poland [email protected]

ABSTRACT This work deals with a new mathematical/numerical model for the analysis of the behaviour of concrete considered as multiphase viscous porous material from early ages to long term periods. This is a solidification-type model where all changes of material properties are expressed as functions of hydration degree, and not maturity nor equivalent hydration period as in maturity-type models. A mechanistic approach has been used to obtain the governing equations, starting from micro-scale, by means of modified averaging theory, also called hybrid mixture theory, [1-2]. Constitutive laws are directly introduced at macroscopic level. An evolution equation for the internal variable, hydration degree, describes hydration rate as a function of chemical affinity, considering additionally to the existing models, an effect of the relative humidity on the process. The model takes into account full coupling between hygral, thermal and chemical phenomena, as well as changes of concrete properties caused by hydration process, i.e. porosity, density, permeability, and strength properties. Phase changes and chemical phenomena, as well as the related heat and mass sources are considered. Some examples showing possibilities of the model for analysis of autogenous self-heating and selfdesiccation phenomena, as well as autogenous shrinkage are presented and discussed. Creep processes are modelled considering concrete as viscous-elastic material with aging caused by solidification of non-aging constituent, i.e. solidification theory for the so-called basic creep [1-2]. A Kelvin-type chain has been chosen for the definition of the compliance function, which corresponds to an expansion of that function in a Dirichlet’s series. Shrinkage is defined using the effective stress principle, as usual in the mechanics of porous materials, and it is coupled to the creep model. In such a way it is possible to have creep strains even if the concrete structure is not externally loaded. Capillary shrinkage is, in fact, characterized from capillary tensions which can be seen as a sort of internal load for the microstructure of the material. A series of numerical computations compared to the experimental results are presented as validation of the model described above.

References [1] D. Gawin, F. Pesavento, B.A. Schrefler: Hygro-thermo-chemo-mechanical modelling of concrete at early ages and beyond. Part I: Hydration and hygro-thermal phenomena. Int. J. Num. Meth. Engng, in print. [2] D. Gawin, F. Pesavento, B.A. Schrefler: Hygro-thermo-chemo-mechanical modelling of concrete at early ages and beyond. Part II: Shrinkage and creep of concrete. Int. J. Num. Meth. Engng, in print.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Uncertainty & Reliability Analysis of Structural Dynamical Systems Gerhart I. Schuëller Chair of Engineering Mechanics, Leopold-Franzens University, Innsbruck, Austria, EU. [email protected]

ABSTRACT In this paper various methods to analyze structural systems under stochastic dynamic loading are qualitatively compared. While the Karhunen-Loève expansion proved to be advantageous when uncertainty estimation is required, (advanced) Monte Carlo simulation procedures are recommended for reliability estimates. The computational efficiency of the methods play an important role w.r.t. practical applications. A further quantitative benchmark study is recommend.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Neural Networks: New Results and Prospects of Applications in Structural Engineering Zenon Waszczyszyn*†, Leonard Ziemianski† *

Cracow University of Technology, Institute of Computer Methods in Civil Engineering Warszawska 24, 31-155 Krakow, Poland [email protected] †

Rzeszów University of Technology, Chair of Structural Mechanics W. Pola 2, 35-959 Rzeszów, Poland [email protected]

ABSTRACT NN is a new computational tool for data processing and this tool can be characterized as a “data dependent and model free” approach. Other features of NNs correspond to their applicability in the analysis of nonlinear direct and inverse problems. NNs can also be used in hybrid systems as a complementary part to conventional computational methods, especially to FEM. The applications of the feed-forward, multilayer, error back-propagation NN, called for short BPNN (back-propagation NN), are discussed focusing on two fields: 1) BPNN as a new independent computational tool, 2) hybrid FEM/BPNN systems. All the considered applications are based on data taken from tests on laboratory models or measurements performed on natural scale buildings. BPNN applications are illustrated on four selected problems. The first one is related to soil-structure interaction caused by paraseismic excitations. The mappings of displacement response spectra DRSg o DRSb were performed, where: DRSg  spectrum computed on the ground level at a monitored building, DRSb  neurally predicted spectrum inside the building on the basement level. It was proved that the application of Kalman filtering for the training of a BPNN leads to much more exact approximation than the application of Rprop learning method. The second problem is related to the identification of placement of an additional mass fastened to a steel plate. The dynamic response corresponding to natural eigenfrequencies of the plate with the mass was used as the BPNN input. Satisfactory results for the parametric identification of the mass location were obtained due to addition of the Gaussian white noise to perturb a small number of measured dynamic responses and due to the application of cascade architecture of BPNNs. The third problem deals with the application of hybrid FEM/BPNN Monte Carlo method in the reliability analysis of steel cylindrical panels. The results of laboratory tests were explored to update the FE model which was then used to compute the patterns for the BPNN training and testing. The trained network was explored to rapid computation of Monte Carlo simulations of the panel ultimate load. It was shown that the hybrid approach enables us to predict the probability of reliable curve very efficiently. The fourth problem is related to the hybrid approach of the FE model updating. It is discussed on the example of a simple plane frame tested on laboratory models. In the formulated hybrid system a FE program was used for generating the training and testing sets of patterns. The trained BPNN was then explored in the inverse analysis for calibrating of control parameters values which well corresponded to the measured eigenfrequencies of the frame laboratory models. Using BPNNs in the hybrid updating of FE models makes it possible to eliminate the optimization procedure in the corresponding inverse analysis.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Computational Railway Dynamics Torbjörn Ekevid∗ , Per Kettil† , Håkan Lane† , Nils-Erik Wiberg† ∗ School

of Technology and Design Växjö University 351 95 Växjö, Sweden [email protected]

† Department of Applied Mechanics Chalmers University of Technology 412 96 Göteborg, Sweden {per.kettil; hakan.lane; nils-erik.wiberg}@sem.chalmers.se

ABSTRACT High-quality and efficient means of transport is of high priority in the modern society. Railway traffic is environment friendly and economically very competitive for both freight and personal transports at mid-range distances. Although the railway technology has been improved substantially during the last decades, generated vibrations still impose annoyances to the surroundings environment and leads to deterioration of the track structure. To understand the physical phenomena and propose countermeasures/improvements, simulation tools to perform computations of the entire dynamical system including subground, track structure and the railway vehicle has been developed. In particular, large effort has been devoted to the special wave propagation problem related to high-speed trains running at soft ground materials. As the speed of the train approaches and exceeds the natural (Rayleigh) wave propagation velocity of the ground material, shock waves similar to sonic boom originates from the onrushing train. The problem area contains several computational challenges since it impose techniques to handle non-reflecting boundaries[1], time integration of large-scale problems, non-linear material response etc. Efficient solvers based on a combination of multigrid[2], error estimations and adaptive refinement has been developed to reach acceptable execution times. The solution time is substantially reduced compared to conventional implicit solvers based on factorization. Moreover, indefinite system from the Lagrange multiplier[3] approach to handle constraint equations imposes additional preconditioning to guarantee convergence and reducing the number of iterations. In the paper a number of numerical examples from railway applications are presented. Results from computation where the train is represented as a collection of moving loads as well as a multi body system with complete train-track interaction are demonstrated.

References [1] T. Ekevid, N.-E. Wiberg, Wave propagation related to high-speed train - a scaled boundary FEapproach for unbounded domains, Comput. Methods Appl. Mech. Engrg. 191, 3947–3964, 2002. [2] U. Trottenberg, C. Oosterlee, A. Schüller, Multigrid, Academic Press, London, 2001. [3] Z.-H. Zhong, Finite element procedures for contact-impact problems, Oxford Univ. Press, Oxford, 1993.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Micro-Meso-Macro Modelling of Composite Materials P. Wriggers, M. Hain Institute of Mechanics and Computational Mechanics University of Hannover Appelstr. 9a, D-30167 Hannover, Germany [email protected]

ABSTRACT Multi-scale models can be extremely helpful in the understanding of complex materials used in engineering practice. In the presentation the basic theoretical strategy is developed. Possible finite element methods to solve such problems are explained in detail and discussed. These are based on homogenization techniques but also on true multi-scale solutions. The developed methodology is then applied to a specific engineering material which is concrete. This construction material has to be investigated on three different scales, the hardened cement paste, the mortar and finally the concrete. Here a successive two-stage approach is followed in which first the multi-scale model of the cement paste and mortar is applied. The resulting homogenization can then used in a multi-scale mortar-concrete model. The model for the hardened cement paste is based on a three–dimensional computer–tomography at the micrometer length scale. For this a finite element model is developed with different constitutive equations for the three parts unhydrated residual clinker, pores and hydrated products. The volume fraction of the hydrated products is approximately 84 Vol.%. For this part, a visco–plastic material model of Perzyna-type including damage is applied. The other two parts are described with a linear– elastic material model. The constitutive equations at the micro–scale contains inelastic parameters, which cannot be obtained through experimental testings. Therefore, one has to solve an inverse problem which yields the identification of these properties. For computational efficiency and robustness, a combination of the stochastic genetic algorithm and the deterministic Levenberg-Marquardt method is used. In order to speed-up the computation time significantly, a client-server based system is used. Hence, all calculations are distributed automatically within a network environment. The resulting constitutive parameters on the micro-scale are then used in the homogenized constitutive model for the mortar. But also in the multi-scale model for the mortar. Both results are compared with each other but also with experimental data. A further interesting application occurs when the micro-structure of the cement paste is filled partly with water and a freezing process takes place. Due to frost, the moisture inside the microstructure freezes. A constitutive model for ice is applied to the water filled parts of the microstructure is then developed. The expansion of the ice leads to damage in the micro–structure which yields an inelastic material behavior on the macro–scale. If such a calculation is performed for different moistures and temperatures, a correlation between moisture, temperature and the inelastic material behavior is obtained. Numerical examples show, that the developed approach reproduces the material behavior realistically.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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An Object-Oriented Approach to High Order Finite Element Analysis of Three-Dimensional Continua M. Baitsch∗, T. Sikiwat, D. Hartmann∗ ∗ Department of Civil Engineering

Ruhr-University of Bochum, Germany [email protected] [email protected] ABSTRACT It has been shown recently that the p-version of the finite element method is well suited for the analysis of thin walled three-dimensional continua [2]. However, designing and implementing software for the p-version of FEM is a challenge because of the increased complexity compared to the h-version. In this paper, we present an object-oriented finite element system implemented in Java. It is pointed out how the object-oriented paradigm, suitable design patterns and thorough unit testing can help to develop and maintain a complex engineering application. The basic idea of the software design is to separate generally applicable mathematical concepts, like basis functions and geometrical mappings from concrete element formulations that contain the physics of the actual problem. In the mathematical package, there are interfaces representing the concepts of basis functions (forming an Ansatz space) and functions defined on R1 , R2 and R3 . Several concrete classes implement these basic interfaces and realize for instance Lagrangian basis functions, hierarchical basis functions, functions constructed by a linear combination of basis functions or functions constructed by the blending function method. The use of NURBS curves hereby allows for the representation of complex geometries like that of the shell structure shown below. The power of this approach lies in the fact that on the element level only the interface types are used. Thus, an element formulation solely contains the physics and is not limited to a certain type of Ansatz space. Also, new Ansatz spaces can be easily incorporated lateron. Generally Java is considered as being slow for numerically intense applications. On the other side there are many advantages that make Java attractive also for simulation software [1]. In this paper, a hybrid approach is presented where the overall program is implemented in Java but numerically intense linear algebra operations are delegated to native code. In the full paper, it is shown that the software can be easily applied to problems involving 75 000 DOFs and more.

Analysis of a Shell Structure: System and Deformation

References [1] R.F. Boisvert, J. Moreira, M. Philippsen, and R. Pozo. Numerical computing in Java. Computing in Science and Engineering, 3(2):18–24, 2001. [2] A. D¨uster, H. Br¨oker, and E. Rank. The p-version of the finite element method for three-dimensional curved thin walled structures. International Journal for Numerical Methods in Engineering, 52:673–703, 2001.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A finite element formulation based on the theory of a Cosserat point – Extension to Ogden material Eiris F.I. Boerner , Dana S. Mueller-Hoeppe , Stefan Loehnert , Peter Wriggers Institute for Mechanics and Computational Mechanics University of Hannover, Appelstrasse 9a, 30167 Hannover, Germany [email protected] Department of Mechanical Engineering Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA ABSTRACT The theory of Cosserat points is the basis of a finite element formulation for a solid three-dimensional continuum, which was presented by [1]. Previous investigations [2] have revealed, that this formulation is free of showing undesired locking or hourglassing-phenomena. It additionally shows excellent behaviour for large deformations in any type of incompressible material and for sensitive structures such as plates or shells. Within the theory of Cosserat points, the position vectors X and x of an 8-node-brick element are described through director vectors Di and di .

X=

7
i=0

N i (θ 1 , θ 2 , θ 3 )Di

,

x=

7


N i (θ1 , θ 2 , θ3 )di

i=0 3 4

N 0 = 1, N 1 = θ 1 , N 2 = θ 2 , N 3 = θ , N = θ1 θ2 , N 5 = θ1 θ3 , N 6 = θ2θ3 , N 7 = θ1 θ2 θ3 The special choice of shape functions N i allows to split the deformation as well as resulting stresses into homogeneous and inhomogeneous parts respectively. The stresses due to the inhomogeneous part of the deformation are obtained by incorporating analytical solutions to the deformation modes bending, torsion and higher-order hourglassing for a rectangular parallelepiped shaped reference element, see [1] and [2]. This work shows approaches on how to overcome the difficulty of initially distorted element geometries that differ strongly from the shape of a rectangular parallelepiped. The formulation initially was restricted to a Neo-Hookean material. This work will present the extension to a general elastic Ogden material as well as to metal plasticity for large deformations with isotropic hardening. It will also give insight to the properties of the Cosserat point element and its behaviour for rubber-like materials.

References [1] Nadler, B.; Rubin, M.B. (2003), A new 3-D finite element for nonlinear elasticity using the theory of a Cosserat point, Solids & Structures, 40, 4585-4614. [2] Loehnert, S.; Boerner, E.F.I.; Rubin, M.B.; Wriggers, P. (2005), Response of a nonlinear elastic general Cosserat brick element in simulations typically exhibiting locking and hourglassing, Computational Mechanics, Vol. 36:266-288.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A macro tetrahedral element with vertex rotational D.O.F.s Jaesung Eom*, Byungchai Lee† *

Department of mechanical engineering, Korea advanced institute of science and technology, 373-1 Guseong-dong, Yuseong-gu, Daejon, 305-701, Republic of Korea [email protected]

†

Department of mechanical engineering, Korea advanced institute of science and technology, 373-1 Guseong-dong, Yuseong-gu, Daejon, 305-701, Republic of Korea [email protected]

ABSTRACT In this study, a macro tetrahedral element with vertex rotations is presented with the systematic macro element fabrication and the inclusion of proper higher-order deformation modes. The individual element test (IET) proved sub-tetrahedral elements are basically combined for the macro element. During the construction of the macro element, a higher-order stiffness is emerged by moving the influence of removed virtual node stiffness into the deformation modes which are related to the vertex rotations. The fundamental decomposition into a basic stiffness and a higher-order stiffness allows scaling the performance of the macro element. To fulfill the accuracy and stability requirement of the element, the transformation between hierarchical rotations and nodal freedoms is employed to separate rotational freedoms into the constant strain related rotation and the higher-order behavior related one. Each parameter in the element stiffness is tuned through beam-type higher order patch test. The numerical performance of the proposed element is compared with other solid elements with vertex rotation in the benchmark test problem.

References [1] K.Y. Sze and Y. S. Pan, Hybrid stress tetrahedral elements with Allman’s rotational D.O.F.s, International journal for numerical methods in engineering, 48, 1055-1070, 2000. [2] Carlos A. Felippa, A study of optimal membrane triangles with drilling freedoms. Computer methods applied mechanicultibody System Dynamics, 1, 149-188, 1997. [3] P. G. Bergan and L. Hanssen, A new approach for deriving ‘good’ finite elements, Proc. MAFELAP II Conf., Academic Press, Brunel University, 483-497, 1976. [4] Jaesung Eom and Byungchai Lee, A study of a macro membrane triangle with drilling freedoms, Proc. WCCM VI Conf., Tsinghua University Press & Springer-Verlag, 23, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Application of aggregation multilevel iterative solver to problems of structural mechanics Sergiy Yu. Fialko* *

National University of Architecture and Constructions in Kiev, Ukraine & Software Company SCAD Soft 4, I. Klimenko str., Office 20 Kiev, Ukraine, 03037 [email protected]

ABSTRACT The application of aggregation multilevel iterative solver AMIS [2, 3] to the structural analysis of large-scale finite element problems is discussed. It is well-known that such problems usually are poorly conditioned and this leads to slow convergence of iterative methods [1]. The preconditioned conjugate gradient method with aggregation multilevel preconditioning is applied to solution the both: linear static problems and natural vibration ones. The effectiveness of several solution stages is considered: the creation of aggregation model, the application of sparse matrix technique to solution of the coarsest level problem, the smoothing algorithms and so on. The set of local rigid links are imposed to the given finite element model to decrease the number of degrees of freedom. The nodeby-node or element-by-element approaches are applied to obtain the reduced model on coarsest level. We try to keep as large number of equation on coarsest level as the computer resources permits it to do. The block sparse multifrontal solver [4] is applied to solve the coarsest level problem. The proper reordering method among multilevel reordering and multiple minimum degrees ones is chosen to reduce fill-inns. The several modifications of incomplete Cholesky factorization approaches are applied to smooth rapidly oscillating residuals after prolongation. The numerous numerical examples, taken from computational practice of SCAD Soft, illustrate the robustness and efficiency of proposed method.

References [1] A. Perelmuter, S. Fialko. Problems of computational mechanics relate to finite-element analysis of structural constructions. International Journal for Computational Civil and Structural Engineering, 1(2) 72-86, 2005. [2] S. Fialko. Application of iterative solvers in finite element analysis of structural mechanics. Linear statics and natural vibrations. Proceedings of 8-th international conference "Modern building materials, structures and techniques". May 19–22, 2004, Vilnius, Lithuania. P. 721 – 725. [3] S. Fialko. Aggregation multilevel iterative sSolver for analysis of large-scale finite element problems of structural mechanics: linear statics and natural vibrations. R.Wyrzykowski et al. (Eds.): PPAM 2001, LNCS 2328, pp. 663 –670, 2002. Springer-Verlag Berlin Heidelberg 2002.

[4] S. Fialko. A block sparse direct multifrontal solver in SCAD software. Proceedings of the CMM-2005 – Computer Methods in Mechanics. June 21-24, 2005, Czestochowa, Poland. 73 – 74.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Convergence Analysis of a Domain Decomposition Method with Augmented Lagrangian Formulation Alexandre Santos Hansen*, Fernando Alves Rochinha† * Petrobras/ENGENHARIA/IEEPT/EEPTM/EDI Rua Almirante Barroso, 81, 11º.andar, Rio de Janeiro, RJ, 21031-004, Brasil [email protected] †

Dept. de Engenharia Mecânica – COPPE - UFRJ Cx. 68503, 21945-970, Rio de Janeiro, RJ, Brasil [email protected]

ABSTRACT Through the combination of an augmented Lagrangian formulation with a preconditioned inexact Uzawa algorithm, we construct a domain decomposition based method for finite element approximation of linear second-order elliptic partial differential equations. With this approach, the proposed method shares the main features of Lagrange multipliers based domain decomposition methods, i.e. number of iterations bounded by the local element size (H/h) using a simple coarse space and direct application to decompositions with irregular sub domain geometry, with the advantage that inexact solvers at sub domain level are allowed at sub domain level. An analysis of the method applied to the Poisson equation is presented to justify the preconditioners choices and to derive bounds for the convergence factor in function of the local element size.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Computational Models on Graphs for Nonlinear Hyperbolic and Parabolic System of Equations Yaroslav A. Kholodov1, Alexander S. Kholodov2, Nikolai V. Kovshov3, Sergey S. Simakov4, Dmitri S. Severov5, Alexey K. Bordonos6, Azilkhan Bapayev7 1-4,6,7 Moscow Institute of Physics and Technology 9, Institutski Line, Dolgoprudny City, Moscow Region, 141700, Russia 1 [email protected], 2 [email protected], 3 [email protected], 4 [email protected], 6 [email protected], 7 [email protected] 5 Open Technologies company Bld. 1, 30, Obrucheva str., Moscow, 117997, Russia 5 [email protected]

ABSTRACT For G GgraphG edge G each G with length X k we consider 1D nonlinear hyperbolic system of equations vGt  Fxk (v )G f , v {v1 ,..., v I } , t t 0 , 0 d xk d X k , k 1,..., K (1) with initial conditions v (0, xk ) v 0 ( xkG) , k 1,..., K and the next boundary conditions: for graph enters (l 0 1,... L0 , xk 0G) M li0 (t , v (t , 0)) 0 , i 1,..., rk0 d I (2) , for graph exits (l * 1,...L* , xk X k ) Mli* (t , v G(t , X i 1,..., rk* d I (3) and for graph branchpoints l 1,..., L G k )) G0 , \ lm (t , wl , vl1 ,..., vlM l ) 0 m 1,..., M l (4). Here K is the number of graph edges, L0 - enters, L* th exits, G GL - branchpoints, M l - incoming and outgoing graph edges forG the l branchpoint, vl1 ,..., vlM l - required vectors in theG ends of edges adjoining to branchpoin l , wl - required vector for G the branchpoint l . The matrix wF / wv A {aij } i, j 1,..., I is Jacobi matrix and we can apply 1 the identity A : /: , where / {Oi } is the diagonal matrix of the matrix A eigenvalues, : is the nonsingular matrix whose rows are linearly independent left-hand eigenvectors of the matrix A ( Det: z 0 ) and : 1 is the matrix inverse to : . G ToG enclose Gboundary Gconditions (2)-(4) we can use compatibility conditions Zi dv dti 0 , ( dv / dti wv / wt  Oi wv / wxk ), i 1,..., I along the characteristics of the system (1) dx Oi dt directed inside integration domain. These compatibility conditions can be used to analyze correctness of the problem definition for system (1) through the all graph segments. The main idea of this approach is that the solution of the global 3D problem for the whole graph can be split out on the set of the 1D problem for the single graph elements (edges). Then on the each step of numerical integration we join these 1D problems by the additional equation systems in the graph nodes to get the common solution for the original problem. By this way we guarantee the global coupling for the all original problem variables. The same computational model can be applied for nonlinear parabolic equations. The numerical results are presented for the problem solution on the different graph systems: The global numerical models of blood circulation in the human body. The model of global regional electrical power systems. The model of bar structures and frames behavior under the different impacts. The model of the intensive information flows in the computer networks. The model of heavy traffic in the big cities. The model of flood water and pollution propagation in the large river systems.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

44

Haar wavelet method for solving integral equations and evolution equations Ülo Lepik Institute of Applied Mathematics, University of Tartu Liivi 2, 50409 Tartu, Estonia [email protected]

ABSTRACT An efficient numerical method for solving nonlinear integral equations and evolution equations based on the Haar wavelets approach is proposed. The method is tested for Fredholm and Volterra integral equations and for the Burgers and sine-Gordon equations. Results obtained by computer simulation are compared with other available solutions. These calculations show that the accuracy of the Haar wavelet approach is quite high even in the case of a small number of grid points.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

45

A new approach for elimination of dissipation and dispersion errors in particle methods Hassan Ostad-Hossein*, and Soheil Mohammadi† *School of Civil Engineering, University of Tehran,Tehran, Iran [email protected] † School Civil Engineering, University of Tehran,Tehran, Iran [email protected]

ABSTRACT Numerical errors may be introduced in some numerical methods of solving differential equations because of their nature. Discretizing a continuum medium would result in changing the wave velocity and inducing numerical errors into the solution. Some methods using strong formulations are based on the Taylor expansion. Therefore, using only a finite number of Taylor series terms for particle simulations introduces truncation errors. Truncation of the Taylor expansion is also the reason for developing two other types of error. The first, called dispersion error appears in the form of extra vibration in high frequency modes that can result in solution instability in some problems. Another type of error is dissipation and may cause decrease in wave amplitude. Particle methods such as SPH [1] and CSPM [2], are also involved with truncation errors. A number of methods have already been proposed for removing dispersion from particle methods such as adding artificial stress. However these methods become energy dissipative resulting in wave amplitude decays after several time steps. In this paper further investigation is performed to study the roots of dispersion and dissipation errors in particle methods. A new procedure is proposed for eliminating dispersion and stabilizing the solution, based on the CSPM particle method and the Newmark time integration scheme. The results are compared with other existing methods.

REFERENCES [1] Chen JK, Beraun JE, Jih CJ, An improvement for tensile instability in smoothed particle hydrodynamics, Comput. Mech. 23 (1999a) 279-287 [2] Chen JK, Beraun JE, Jih CJ, Completeness of corrective smoothed particle method for linear elastodynamics, Comput. Mech. 24 (1999b) 273-285 [3]Chen JK, Beraun JE, Jih CJ, A corrective smoothed particle method for transient elastoplastic dynamics, Comput. Mech. 27 (2001) 177-187 [4] M.B. Liu, G.R. Liu, Z. Zong, Constructing smoothing functions in smoothed particle hydrodynamics with applications, Journal of Computational and applied Mathematics, 155 (2003) 263-284 [5] Gray J.P., Monaghan J.J., Swift R.P., SPH elastic dynamics, Comput. Methods Appl. Mech. Engrg 190 (2001) 6641-6662

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

46

Solution of stability problem of infinite plate strips Zdzisław Pawlak*, Jerzy Rakowski† *

Institute of Structural Engineering, Poznan University of Technology ul. Piotrowo 5, 60-965 Poznań, Poland [email protected]

†

Institute of Structural Engineering, Poznan University of Technology ul. Piotrowo 5, 60-965 Poznań, Poland jerzy.rakowski@ put.poznan.pl

ABSTRACT The aim of the paper is to solve a stability problem of infinite plate strips using the finite strip method (FSM). Contrary to well-known solutions for 2D continuous systems the authors present an idea for solving a stability problem of infinite discrete plates. A continuous plate strip simply supported on its opposite edges is divided into a regular mesh of identical finite strips. According to the finite strip procedure the field of loading and displacement functions are expressed in the form of harmonic series [1]. Stiffness and geometrical matrices for a four-degree-of-freedom finite strip are determined. The unknowns are deflections and transverse slope amplitudes along the nodal lines. Equilibrium conditions are derived from the FSM formulation. An infinite set of linear equations being the equilibrium conditions for each nodal line is expressed in the form of two second-order difference equations [2]. For a regular discrete system these equations written in the recurrent form are equivalent to the FSM matrix formulation. The exact solution of these difference equations is found in an analytical form. The displacement function fulfils the boundary conditions of the analysed plate strip and is given as a discrete expression for an arbitrary nodal load [3]. The solution of eigenvalue problem of the difference equations enables one to determine the critical forces of the structure. The main advantage of the presented approach is the analytical form of the solution obtained in the discrete domain. This enables a detailed parametrical analysis and investigations of the influence of geometrical and physical properties on the critical force value.

References [1] Y. C. Loo, A. R. Cusens, The Finite Strip Method in Bridge Engineering, New York, Viewpoint Publications,1978. [2] J. Rakowski, A critical analysis of quadratic beam finite elements. International Journal for Numerical Methods in Engineering, 1991, vol. 31, pp. 949-966 [3] Z. Pawlak, J. Rakowski, Static problem of plate strip by finite strip method using difference equation method. In: Proceedings of the 15th International Conference on Computer Methods in Mechanics CMM-2003, Gliwice/Wisła, 3-6 June 2003, pp. 279-280.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

47

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

48

Discretization of Three-Dimensional Aggregate Particles Daniel Rypl Department of Mechanics Faculty of Civil Engineering, Czech Technical University in Prague Th´akurova 7, 166 29, Prague, Czech Republic [email protected] ABSTRACT The design of concrete with specified properties became of increasing importance with the wide use of high-performance concretes (HPCs), such as pumpable concrete or self compacting concrete (SCC). Many concrete properties, starting from the mechanical properties as the compressive strength and modulus of elasticity, over the rheological properties influencing the workability of fresh concrete, up to physical properties as diffusivity and thermal and electric conductivity, for example, can be assessed by appropriate computational model representing concrete as a multiscale random composite material with realistically described aggregates. However, incorporation of three dimensional aggregate particles into computational code requires their proper discretization. This is not straightforward due to rather difficult mathematical characterization of aggregate particles of random shape. Modern technologies as computer tomography (CT) or magnetic resonance tomography (MRT) offer a powerful nondestructive technique for digital representation of opaque solid objects. This voxel based representation can be then discretized using for example marching cubes algorithm [1]. The resolution of the resulting triangulation, however, is strongly dependent on the resolution of the digital representation which might be either too coarse (without important features being captured) or too fine (with unimportant features captured by excessive number of elements). In the present work, the digital representation is first used to derive a smooth representation of aggregate particle using the expansion into spherical harmonic functions [2]. Although this representation is not universal it is suitable for almost all aggregates used in structural concrete. The significant advantage of this approach is that resolution of the smooth representation can be flexibly controlled by the number of terms in the expansion. In the next phase, the surface of aggregate particle is subjected to discretization using the advancing front technique. Although the representation of the surface is parameterized (by two spherical angles), the actual triangulation is performed directly on the surface in the real space [3] and not in 2D parametric space with subsequent mapping to the real space. The advantage of this procedure consists in the fact that the anisotropic meshing of the parametric space as well as the demanding calculations related to the reparameterization or the inverse mapping are avoided.

References [1] W.E. Lorensen, H.E. Cline, Marching cubes: A high resolution 3D surface construction algorithm. Computer Graphics, 21, 163-169, 1987. [2] E.J. Garboczi, Three-dimensional mathematical analysis of particle shape using x-ray tomography and spherical harmonics: Application to aggregate used in concrete. Cement and Concrete Research, 32, 1621-1638, 2002. [3] D. Rypl, Sequential and parallel generation of unstructured 3D meshes. PhD. Thesis, CTU Reports, 2 (3), 1998.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

49

Bending of an Elliptical Plate on Elastic Foundation and under the Combined Action of Lateral Load and In-Plane Force Kenzo Sato Akita University Department of Engineerings and Information Sciences, Faculty of Education and Human Studies, 010-8502 Akita, Japan [email protected] ABSTRACT The plane plates resting on elastic foundations are of practical importance in many engineering fields as seen in plate-subgrade structure, floating plate structure, composite material and so on. In some cases, the plates are subjected to large temperature differences from which considerable in-plane forces in the plates result. As structural elements in the wide fields of engineering, various types of elliptical plates may be used in order to avoid the high stress concentration and improve the usability of the instrument and the beauty of the architecture. In recent years, the author has been studied the vibration, buckling and bending problems of elliptical plates subjected to the combined action of lateral load and in-plane force[1]-[5]. In the reference [2] has been discussed also the influence of elastic foundation on the vibration and buckling of a clamped elliptical plate. From the viewpoint of the usefulness of an elliptical plate as structural element and the importance of the analytical solution in the mathematical theory of elasticity, it is the aim of this report to develop the exact theoretical analysis on the bending of an elliptical plate resting on a Winkler-type elastic foundation and subjected to the combined action of uniform lateral load and in-plane force. Here is considered the case that the plate-edge conditions are clamped and simply supported. Based on the classical small-deflection theory, the theoretical analysis is rigorously made in the elliptical coordinate system and the deflection surface due to bending of the plate is obtained in the form of an infinite Mathieu function series. The influences of elastic foundation and in-plane force on the bending of the elliptical plate are calculated by digital computer, and the new results obtained here will be presented in tables and figures.

References [1] K. Sato, Free Flexural Vibrations of a Simply Supported Elliptical Plate Subjected to an In-Plane Force. Theoretical and Applied Mechanics, 50, 165–181, 2001. [2] K. Sato, Vibration and Buckling of a Clamped Elliptical Plate on Elastic Foundation and under Uniform In-Plane Force. Theoretical and Applied Mechanics, 51, 49–62, 2002. [3] K. Sato, Bending of a Clamped Elliptical Plate under the Combined Action of Uniform Lateral Load and In-Plane Force. Theoretical and Applied Mechanics, 53, 37–47, 2004. [4] K. Sato, Bending of a Simply Supported Elliptical Plate under the Combined Action of Uniform Lateral Load and In-Plane Force. Theoretical and Applied Mechanics, 54, 31–44, 2005. [5] K. Sato, Bending of an Elastically Restrained Elliptical Plate under the Combined Action of Lateral Load and In-Plane Force. JSME International Journal, Ser.A, 49, 130–137, 2006.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

50

A modal analysis approach using an Hybrid-Mixed formulation to solve 2D elastodynamic problems M. Vicente da Silva∗ , Eduardo Pereira† ∗ Universidade

Nova de Lisboa - FCT/UNL, Civil Engineering Department Quinta da Torre, 2829-516 Monte da Caparica [email protected]

† Instituto

Superior T´ecnico, Civil Engineering and Architecture Department Av. Rovisco Pais no 1, 1049-001 Lisboa [email protected] ABSTRACT

The purpose of this work is to present a Hybrid-Mixed finite element formulation to solve elastodynamic plate problems in frequency domain. This Hybrid-Mixed model is derived [2] establishing, non-locally, the dynamic equilibrium, compatibility and constitutive relations based on the Galerking weighted residual method. Two different approximations fields are used in the element domain, namely the stresses and the displacements fields. A third approximation independent from the previous ones is also required for the displacement field on the static boundary of the elements. Once obtained the governing system, the stresses and the boundary displacements degrees of freedom are eliminated, thus resulting a new condensed system, were only the domain displacements degrees of freedom (qV ) intervine: (K − ω 2 M )qV = QV (1) The condensed system assume a form analogous to the one obtained with the conventional FEM[1], however with a different and richer physical meaning. In a first stage natural frequencies and shape modes are identified. Then, modal analysis technique is performed to uncouple the governing system equations, and to assess the relevance of each mode to a specific action. Less relevant modes are eliminated, thus reducing substantially the computational costs without significant lost of accuracy. The Hybrid-Mixed model is implemented using 4-node serendipian standard master elements to define the finite element shape, and non-nodal, Legendre polynomials as approximation functions. This functions are suited for this purpose because they allow to introduce in the finite element code closed form solutions to compute the coefficients of the structural matrices, avoiding time consuming numerical integration. Good results can be reach using macro finite elements since h-refinements are easy to performe simply by increasing the maximum degree of the polynomials in the approximation functions. The efficiency and performance of the presented method is validated with the aid of numerical examples. Free vibrations natural frequencies are determined and transient response in undamped and viscous damped structures are studied.

References [1] R. W. Clough and J. Penzien. Dynamics of Structures. McGraw-Hill, New York, 2nd edition, copyright 1993. [2] E. M. B. R. Pereira and J. A. T. Freitas. A mixed-hybrid finite element model based on orthogonal functions. International Journal for Numerical Methods in Engineering, 39:1295–1312, 1996.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A new finite element method for Kirchhoff plates Lourenc¸o Beir˜ao da Veiga ∗, Jarkko Niiranen†, and Rolf Stenberg† ∗ Dipartimento di Matematica ”Federigo Enriques”

via Saldini 50, 20133 Milano, Italy [email protected] †Institute of Mathematics, Helsinki University of Technology

P.O. Box 1100, 02015 TKK, Finland jarkko.niiranen@tkk.fi, rolf.stenberg@tkk.fi

ABSTRACT Based on the ideas from [1] and [2] we present a new finite element method for the Kirchhoff plate bending model [3]. The method uses C 0 basis functions for the deflection and the rotation, i.e. the same approach as used for the Reissner–Mindlin model. To account for the effective shear force at the free boundary a stabilization term is added. We prove optimal a-priori and a-posteriori error estimates. The corresponding results from benchmark problems are also reported.

References [1] P. Destuynder and T. Nevers, Une modification du mod`ele de Mindlin pour les plaques minces en flexion pr´esentant un bord libre. RAIRO Mod´el. Math. Anal. Num´er., 22, 217–242, 1988. [2] R. Stenberg, A new finite element formulation for the plate bending problem. P. G. Ciarlet, L. Trabucho, and M. Via˜no eds. Asymptotic Methods for Elastic Structures, Proceedings of the International Conference, Lisbon, October 4–8, 1993, Walter de Gruyter & Co., Berlin – NewYork, 209–221, 1995. [3] L. Beir˜ao da Veiga and J. Niiranen and R. Stenberg, A family of C 0 finite elements for Kirchhoff plates. Helsinki University of Technology, Institute of Mathematics, Research Reports, A 483, January, 2006 (http://www.math.tkk.fi/reports).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

52


                  
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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

53

Computational Methods of Anisotropic Massif Mechanics Under Different Types of External Actions Sh. Aitalyev, R. Baimakhan, Zh. Masanov, N. Kurmanbekkyzy, G. Ylyasova Institute of Mechanics and Engineering Science Almaty, Kazakhstan, Aksay-5, house 7, ap.23 [email protected]

ABSTRACT Mechanics of mining rocks is developed adequately in Kazakhstan. It is connected with the fact that there are many deposits of solid minerals, for developing of which is necessary to build the mines and pits. For oil and gas production the wells are holed, now and then deep and super-deep. The underground railway is being built in the former capital of Kazakhstan – Almaty, which is situated in the high seismic zone. Mechanic engineers of Kazakhstan have created different calculating models of rocks massif: homogeneous isotropic and anisotropic, small-laminated and large-laminated, inhomogeneous with breaks, bloc layer, etc. In the paper the methods of calculating of the intensedeforming state of anisotropic massif round the spread underground constructions are proposed. Anisotropic is accepted as transversal anisotropic. The plane of isotropic reflects presence of stratified rocks in city. If to consider the spread underground constructions, differently oriented relative to the isotropic plane and horizontal plane, the calculating model is characterized by the eight parameters: five mechanical constants and three angles. ci, j = ci, j (E1 , E2 ,G2 ,ν 1 ,ν 2 ,ϕ ,φ , χ ) [1], [2], where

E1 , E 2 , γ 1 , γ 2 ,G - elastic constants, coefficients of Poisson, shear density modulus; these angles are the incline of isotropic plane to the horizon, deviation of the underground construction axis from the line of stretching, deviation of the isotropic plane from the horizon. Own weight of the massif, horizontal tectonic forces, seismic loadings are considered as external actions. In case of presence of all these forces, when the centers of earthquakes are far and only long-waved actions come, numerical results can be obtained easily enough by the method of conformal representations. Here are used three functions instead of the classic two analytical functions, conformable to particular space case under the conditions of generalized plain deformation. Under certain conditions the method of boundary integral equations – the method of boundary elements – is convenient. But the finite-element method is more efficient, especially when seismic loadings are given by the way of synthesized accelerograms of real earthquakes and problems of seismic waves non-stationary diffraction at the underground constructions into anisotropic massif are solved. In the paper, basically, results for rocks anisotropic massif, obtained by the finite-element method, are proposed.

References [1] Zh. Erzhanov, Sh. Aitaliev, Zh. Masanov, Seismic-stressed condition of underground structures in a stratified anisotropic massif. Nauka, Almaty, 1980. [2] R. Baimakhan, Calculation of seismic-stressed condition of underground structure in a heterogeneous stratum by finite element method. Dauir, Almaty, 2002.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

Effect of Plastic Anisotropy on the Size of Elastic-Plastic Boundary in a Rotating Disk Problem *

Nelli N. Alexandrova , Paulo M.M. Vila Real† *

Civil Engineering Department University of Aveiro, 3810-193 Aveiro, Portugal [email protected] † Civil Engineering Department University of Aveiro, 3810-193 Aveiro, Portugal [email protected]

ABSTRACT The problem of dependence of stress distribution and size of elastic-plastic boundary on the angular velocity, material anisotropy, boundary conditions and geometric parameters in rotating disks is of great importance due to a large number of applications. In particular, all of these parameters can have a significant effect on the development of plastic zones in such disks. The elastic stress distribution in rotating anisotropic disks with constant and variable thickness is well known. However, there are only a few studies incorporating plastic stress analysis for anisotropic materials. Considering the case of a single elastic perfectly-plastic annular rotating disk, subjected to typical stress boundary conditions, the influence of the leading design parameters - material anisotropy coefficients - on the size of elastic-plastic boundary arising due to the action of centrifugal forces is investigated. An axisymmetric problem is formulated assuming that the principal axes of anisotropy coincide with the radial and tangential directions in the plane of the disk. The Hill’s quadratic yield criterion is adopted in the plastic zone, and material properties in the elastic zone are considered to be isotropic obeying the general Hooke’s law. All stresses are assumed to be continuous across the elastic-plastic boundary. Three different characteristic values of rotational speed are chosen in numerical calculations. It is demonstrated that the size of the plastic zone is very sensitive both to the change in material anisotropy coefficients and the magnitude of angular velocity.

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Numerical Testing on Return Map Algorithms for von-Mises Plasticity with Nonlinear Hardening based on a Generalized Midpoint Integration Scheme Edoardo Artioli1, Ferdinando Auricchio2,3, Lourenço Beirão da Veiga4 §

DISTART Department, University of Bologna Viale del Risorgimento, 2 – 40136, Bologna, Italy [email protected] 3

2 Department of Structural Mechanics, University of Pavia Institute of Applied Mathematics and Information Technology, CNR Via Ferrata, 1 – 27100, Pavia, Italy [email protected] 4

Department of Mathematics, University of Milan Via Saldini, 50 – 20133, Milan, Italy [email protected]

ABSTRACT We consider an associative von-Mises elastoplastic constitutive model in the realm of small deformations [1]. The model takes into account both linear isotropic hardening and linear/nonlinear kinematic hardening. The aim of the work is to test integration algorithms based on a return mapping concept and adopting a generalized midpoint integration rule. The method under consideration was originally proposed by Ortiz and Popov [2] and further studied in the simpler case of nonhardening materials by Simo [3]. The tested method guarantees yield consistency at the end of the time step and results linearly or quadratically accurate depending on the choice of the integration parameter. The numerical algorithm adopts a return map update based on a projection along the midpoint normal-to-yield-surface direction onto the endpoint limit surface. A testing on the method accuracy and precision is carried out by comparison with a new exponentialbased integration algorithm [4]. The comparison is carried out solving zero-dimenisonal mixed prescribed stress-strain loading histories. Accuracy and precision are determined by plotting the instantaneous error graphs on stress and strain as well as iso-error maps on stress.

References [1] J. Lubliner, Platicity Theory. Macmillan Publishing Company, 1990. [2] M. Ortiz, W. Popov, Accuracy and stability of integration algorithms for elastoplastic constitutive relations. International Journal for Numerical Methods in Engineering, 21, 15611577, 1985. [3] J.C. Simo, Numerical analysis and simulation of plasticity. in Handbook of numerical Analysis, P.G. Ciarlet and J.L. Liond Eds., Elsevier, 1998. [4] A. Artioli, F. Auricchio, L. Beirão da Veiga, Second-order accurate integration algorithms for von-Mises plasticity with a nonlinear kinematic hardening mechanism. Computer Methods in Applied Mechanics and Engineering, submitted, 2006.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A Model for the Analysis of Plates on a Layered Elastic Medium Mehmet Zülfü Aúık* *

Middle East Technical University Dept. of Engineering Sciences [email protected]

ABSTRACT In the present study, a gross model is developed to represent the behavior of a plate on a layered medium. The model assumes that the displacements take place only in vertical direction, and the plate is only subjected to the vertical forces. Total depth of foundation is finite. The model developed in this study predicts the static response of a plate on a layered medium. The present model is a simple and a semi-analytical one. The method is based on the minimum potential energy and variational principles. Shear modulus of the foundation is assumed to be different and constant at each layer. The coupled equations derived for each layer by using minimum potential energy and variational principles, are solved by employing finite difference method and iterative procedure. Results are compared with the finite element model for the verification of the model. The computer code developed for the numerical model can be run even on small computers.

References [1] [2] [3] [4] [5] [6]

[7]

[8] [9]

T. Nogami, Two-parameter Layer Model for Analysis of Slab on Elastic Foundation , J. Engrg. Mech. ASCE, 113,1279-1291, 1987. R. Jones and J. Xenophontos, The Vlasov Foundation Model , Int. J. Mech. Sci., 19, 317-323, 1977. C. V. G. Vallabhan and Y. C. Das, Modified Vlasov Model for Beams on Elastic Foundations , J. Geotech. Engng., ASCE, 117:6, 956-966,1991. C. V. G. Vallabhan and Y. C. Das, The Analysis of Circular Tank Foundations , J. Engng. Mech., ASCE, 117:4, 789-797, 1991. C. V. G. Vallabhan and Y. C. Das, A Refined Model for Beams on Elastic Foundations , Int. J. Solids Structures, ASCE, 27:5, 629-637, 1991. M. Z. Asik, C. V. G. Vallabhan and Y. C. Das, Vertical Vibration Analysis of Rigid Circular Footings on a Soil Layer with a Rigid Base , Developments in Theoretical and Applied Mechanics, SECTAM,Volume XVIII, Alabama, 1996. M. Z. Asik, 3-D Model for the Analysis of the Rectangular Machine Foundations on a Soil Layer , 6th International Symposium on Numerical Models in Geomechanics, NUMOG VI, Montreal, Canada, 1997. M. Z. Asik, Dynamic Response Analysis of the Machine Foundations on a Nonhomogeneous Soil Layer , Computers and Geotechnics, 24:2, 141-153, 1999. M. Z. Asik, Analysis of Strip Footings on a Layered Medium , International Conference on Enhancement and Promotion of Computational Methods in Engineering Science, 2-5 Agust, 1999, Macao, China, 1999.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Modeling of solid state transformations using a phase field model with transformation plasticity S. Benke ACCESS e. V. Materials and Processes, RWTH Aachen, D-52072 Aachen, Germany Address [email protected]

ABSTRACT The evolution of the morphology of a second phase during solid state transformations is influenced by various factors like the degree of the misfit strain, distribution of the precipitates, anisotropy and inhomogeneity of the stiffness, interfacial energy, diffusion of solute atoms and others. From experimental observations it is well known that the evolution of a solid-state phase transformation in a stressed metal induces plastic deformation, even for very small applied stresses. Two transformation mechanisms can be distinguished: the transformation dilatation leading to a volume change (Greenwood-Johnson mechanism) and the martensitic transformation which results additional in shape changes (Magee effect). In the present contribution we focus our attention to volumetric phase transformations in low carbon steels caused by a diffusion controlled growth of ferrite in an austenitic matrix. The microstructural evolution is simulated by the use of a multi-phase field model which is based on the diffuse interface theory [1]. The model combines the diffusion of the species which defines the time scale, the curvature of the interfaces which defines the length scale and phase diagram information as well as the mechanical strain energy which both represent the energy scale. For the solution a combination of finite difference and finite element methods is used. The evolution of the microstructure is calculated with the software MICRESS [2] which solves the system of equations consisting of phase field equations and the diffusion equation using a control volume approach. MICRESS is coupled to a nonlinear finite element code which solves the mechanical subproblem and passes back the mechanical part of the driving force of the phase transformation in a staggered way within the solution procedure. From a mechanical point of view both phases are assumed to behave elasto-visco-plastically and the material properties in turn vary with the phase field parameters which describe the evolution of the microstructure. Using this model the influence of the strain energy on the γ − α transformation in a low alloyed carbon steel is studied. During the phase transformation the evolution of the inelastic strain due to creep and relaxation effects is tracked and the effect of different external loadings on the developed morphology is investigated.

References [1] I. Steinbach, F. Pezzolla, B. Nestler, M. Seeelberg, R. Prieler, G. J. Schmitz, J. L. L. Rezende, A phase field concept for multiphase systems. Physica D 94, 135–147, 1996. [2] www.micress.de

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Computer modeling of three-dimensional wave movements in anisotropic elastic environments S. Bosiakov, M. Zhuravkov Belarusian State University [email protected] [email protected]

ABSTRACT The theory of elastic waves in anisotropic bodies concerns to well enough investigated section of mechanics of continuous environments and the basic results in this direction belong to F. I. Fedorov, M. J. Musgrave, R. G. Payton, E. Dieulesaint, D. Royer [1, 2, 3], etc. However, three-dimensional representations of wave movements (a surface of reverse velocities, wave surfaces) are absent in full, that it is possible to explain complexity and bulkiness of the corresponding characteristic equations, and also absence of their exact analytical decision (exception transversal-isotropic environments make). Researches of features of propagation of flat elastic waves and classification of anisotropic environments are lead for special directions and crystallographic planes of an anisotropic body, which not always yields true and consistent results. The offered approach to visualization and the quantitative description of three-dimensional wave movements bases on sharing of system of computer mathematics Mathematica in aggregate with methods of characteristics of the theory of the differential equations with partial derivatives of hyperbolic type. Thus the basic stages of modeling of wave processes are the finding of the characteristic equation, its analytical decision, definition of velocities of propagation of elastic waves and coordinates of points of environment which energy of wave indignation has reached. The specified stages are put in a basis of the package of expansion of system Mathematica developed by authors which functionalities allow to execute construction of surfaces of velocities, reverse velocities and surfaces of ray velocities (three-dimensional wave fronts), and also to construct sections of these surfaces any plane which is passing through the beginning of coordinates for anisotropic environments of any class of symmetry. Surfaces of reverse velocities and wave surfaces enable to carry out the strict classifications of anisotropic environments basing the simultaneous account of features two quasitransversal waves. The obtained results also can be used both for correct statement of experiments, and for correct interpretation of experimental data as allow to generate evident representations about features of propagation of waves in anisotropic environments.

References [1] ɗ. Ⱦɶɟɥɟɫɚɧ, Ⱦ. Ɋɭɚɣɟ, ɍɩɪɭɝɢɟ ɜɨɥɧɵ ɜ ɬɜɟɪɞɵɯ ɬɟɥɚɯ. ɉɪɢɦɟɧɟɧɢɟ ɞɥɹ ɨɛɪɚɛɨɬɤɢ ɫɢɝɧɚɥɨɜ. Ɇɨɫɤɜɚ, ɇɚɭɤɚ, 1982. [2] Ɏ. ɂ. Ɏɟɞɨɪɨɜ, Ɍɟɨɪɢɹ ɭɩɪɭɝɢɯ ɜɨɥɧ ɜ ɤɪɢɫɬɚɥɥɚɯ. Ɇɨɫɤɜɚ, Ɇɢɪ, 1965. [3] M. J. Musgrave, R. G. Payton, Criteria for elastic waves in anisotropic media. Jornal of Elasticity, 14, 269-285, 1984.

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Visco-elastic regularization and strain softening Rui S. Cardoso *,1, Vitor D. Silva †, Humberto Varum *,2 * University of Aveiro Department of Civil Engineering Campus Universitário de Santiago, P-3810-193 Aveiro, Portugal 1 [email protected]

2

[email protected]

† University of Coimbra Department of Civil Engineering Polo II da universidade-Pinhal de Marrocos, P-3030-290 Coimbra, Portugal

[email protected] ABSTRACT In this paper it is intended to verify the capacity of regularization of the numerical solution of an elasto-plastic problem with linear strain softening. The finite element method with a displacement approach is used. Drucker-Prager yield criteria is considered. The radial return method is used for the integration of the elasto-plastic constitutive relations. An elasto-visco-plastic scheme is used to regularize the numerical solution. Two constitutive laws have been developed and implemented in a FE-program, the first represent the radial return method applied to Drucker-Prager yield criteria and the second is a time integration procedure for the Maxwell visco-elastic model. Attention is paid to finite deformations. An associative plastic flow is considered in the Drucker-Prager elasto-plastic model. The algorithms are tested in two problems with softening. Figures showing the capability of the algorithms to regularize the solution are presented.

References [1] V. D Silva, A simple model for viscous regularization of elasto-plastic constitutive laws with softening. Communications in Numerical Methods in Engineering, John Wiley & Sons, Ltd., Vol. 20, pp. 547-568, 2004. [2] R. Cardoso, Regularização visco-elástica de problemas elasto-plásticos com amaciamento, MS.c Thesis, University of Coimbra, 2001 (in Portuguese). [3] H. Varum, R. Cardoso, A geometrical non-linear model for cable systems analysis, Second International Conference on Textile Composites and Inflatable Structures, Eds. E. Onãte and B. Kröplin, Stuttgart 2-5 October 2005, pp. 234-242. [4] V. D. Silva, Mechanics and Strength of Materials, ISBN 3-540-25131-6, 529 p., 402 illus., Springer, 2006. [5] J. Argyris, I. St. Doltsinis, V. D. Silva, Constitutive modelling and computation of non-linear viscoelastic solids – Part I: Rheological Models and Numerical Integration Techniques. Computer Methods in Applied Mechanics and Engineering, Vol. 88 No.2, pp. 135-163, 1991.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Numerical simulations of the Rayleigh-Taylor instability in accelerated solids Juan J. López Cela1*, Antonio R. Piriz1 and María C. Serna1 1 Escuela Técnica Superior de Ingenieros Industriales Universidad de Castilla La Mancha 13071 Ciudad Real, Spain [email protected]

ABSTRACT The classical Rayleigh-Taylor (RT) instability is developed in the interface that separates a heavy fluid from a lighter one in the presence of a gravitational field. If the interface is not perfectly planar, its small perturbations will grow without bound. When a solid is submitted to a very high acceleration, for example by the application of an external pressure on one of its faces, this interface is also RT unstable. This kind of situations are found in several technological applications like explosive forming process, laser implosion of fusion targets, electromagnetic implosion of metal liners or experiments to achieve hydrogen metallization [1]. If the RT instability appears the experiment could fail. So it is important to study which are the mechanisms that produce stabilizing effects to alleviate the growth of the instability. When we are dealing with fluids the stabilizing mechanisms are, for example, gradient effects, viscosity and superficial tension. Dealing with solids, their intrinsic properties can produce such a stabilizing effect. In a very simple interpretation, the elastic forces that the solid develop can compensate the buoyancy force that leads the instability, but when the material plastifies it can again be unstable. Analytical solutions are only available for elastic solids. The exact analytical solution describing the entire process exists only for solid-vacuum interfaces [2]. For solid/solid and viscous fluid/solid interfaces there is an exact analytical model that describes only the asymptotic phase [3] and an approximate analytical model that we have presented elsewhere [4] that describes not only the asymptotic behaviour but also the initial transient phase. The initial transient phase is of great importance because the instability could enter in the non-linear regime or in the plastic flow before the asymptotic regime is reached. In this work we present some numerical simulations based on the Finite Element Method for a very thick solid layer. In particular we have verified that the stability criteria defined by previous models are consistent with the simulations and also we have studied the influence of the elastic material parameters.

References [1] N. A. Tahir et al, Metallization of hydrogen using heavy-ion-beam implosion of multilayered cylindrical targets, Physical Review E, 63, pp. 016402-016410, 2001. [2] B.J. Plohr and D.H. Sharp, Instability of accelerated elastic metal plates, Z. Angew. Math. Phys. 49, 786-804, 1998. [3] G. Terrones, Fastest growing linear Rayleigh-Taylor modes at solid/fluid and solid/solid interfaces, Physical Review E 71, pp. 036306-036316, 2005. [4] A. R. Piriz, J. J. López Cela, O.D. Cortázar, N.A. Tahir and D.H.H. Hoffmann, Rayleigh-Taylor instability in elastic solids, Physical Review E 72, pp. 056313, 2005.

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A Phenomenological Model to Simulate Mechanical Tests on Ultrafinegrained Aluminum Produced by ECAE B. Diouf*, F. El Houdaigui*, S. Poortmans†, B. Verlinden†, A. M. Habraken*. *

Department of Mechanics of Materials and Structures University of Liège, chemin des Chevreuils 1, 4000 Liège, Belgium {bdiouf, felhoudaigui, Anne.Habraken}@ulg.ac.be †

Department of Materials Engineering Katholieke Universiteit Leuven, Kasteelpark Arenberg 44, 3001 Leuven, Belgium {Stijn.Poortmans, Bert.Verlinden}@mtm.kuleuven.be

ABSTRACT Results of FEM simulations predicting the mechanical behaviour at room temperature of test specimens of ultrafine-grained aluminum produced by ECAE following route BC [1] for 8 passes are presented. The constitutive law is either based on a Hill model or on the Minty micro-macro [2] model and coupled with an isotropic hardening law and/or kinematic hardening law. The yield locus shape, its size and its position during tension, compression and torsion tests have been studied. Initial texture measurements allow identification of a constitutive law based on a set of representative crystals and crystal plasticity approach using a Full-Constraint Taylor model. Finite element simulations using the previous constitutive laws are compared with experimental investigations. The results show that applying an initial back stress identified by tensile and compression tests to the yield locus predicts the initial flow stress in torsion test. The Minty micromacro model coupled with a Voce type hardening model gives a good agreement with experimental results for the prediction of the shape at different stages of deformation of a compressed test specimen. The simulation of tensile tests underline the need of inverse modelling as, due to the test specimen shape, the test is far from being homogeneous. The link between test specimen length and the necking appearance is studied.

References [1] V.M. Segal, Materials processing by simple shear. Materials Science and Engineering: A, 197, 157-164, 1995. [2] A.M. Habraken, L. Duchêne, Anisotropic elasto-plastic finite element analysis using stress-strain interpolation method based on a polycrystalline model. International Journal of Plasticity, 20(89), 1525-1560, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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The peridynamic equation of motion in non-local elasticity theory Etienne Emmrich∗ , Olaf Weckner† ∗ Technische

Universit¨at Berlin, Institut f¨ur Mathematik Straße des 17. Juni 136, 10623 Berlin, Germany [email protected]

† Massachusetts

Institute of Technology, Department of Mechanical Engineering Cambridge, MA 02139, USA [email protected]

ABSTRACT During the last few years, non-local theories in solid mechanics that account for effects of long-range interactions have become topical again. One of these theories is the so-called peridynamic modelling, introduced by Silling [1]. The governing equation of motion is the partial integro-differential equation
f (x, x ˆ, u(x, t), u(ˆ x , t), t) dˆ x + b(x, t) , x ∈ V , t > 0 , ρ(x)∂t2 u(x, t) = H(x)

for the displacement field u = u(x, t) of a body that occupies the reference volume V, supplemented by initial conditions for u(·, 0) and ∂t u(·, 0). Here, ρ denotes the mass density, f the so-called pairwise force field that describes the interaction of material particles, and b collects outer forces. Moreover, H(x) = {ˆ x ∈ V : ˆ x − x ≤ δ} is the so-called peridynamic horizon for prescribed δ > 0. An essential feature is that f is independent of any spatial derivative. It is, therefore, a promising approach for the simulation of problems in which discontinuities emerge such as fracture or cracking. The peridynamic modelling has recently been successfully applied in numerical simulations of problems in solid mechanics such as the fracture of a plate with notches, the undirected growth of cracks, the wrinkling and tearing of membranes, the deformation of composite materials etc. In this talk, we give an overview of the peridynamic modelling as well as the numerical and theoretical results obtained so far. Concentrating on the description of a linear microelastic material, we present new results concerning the well-posedness of the problem. We then suggest a quadrature formula methods for the spatial approximation of the governing equation and show some numerical simulations. Moreover, the question of energy conservation and the comparison of elastic energy in both the peridynamic and the classical theory are discussed.

References [1] S. Silling, Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids, 48, 175–209, 2000.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Numerical Solution of Partially Plastic Curved Beam Problem Ahmet N. Eraslan, Eray Arslan Department of Engineering Sciences Middle East Technical University, Ankara 06531, Turkey {aeraslan, erarslan}@metu.edu.tr

ABSTRACT Numerical solutions are obtained for elastic and elastic-plastic bending of a curved beam via couples at its end sections, under plane stress presupposition. The model proposed is based on the von Mises' yield criterion, Henky's deformation theory, and nonlinear strain hardening material behavior. Using formal nondimensional variables, and an appropriate stress function, a single second order nonlinear differential equation describing the deformation behavior of the curved beam is obtained. A shooting technique using Newton iterations with numerically approximated tangents is used for the numerical integration of the governing equation. The computational model is verified both in elastic and partially plastic cases making use of published analytical and numerical FEM solutions. 1.2

stresses and displacement

0.8

I

I

VT

0.4

Vr 0.0

VT

u u10

-0.4

-0.8

rEP

1.081

rNA 1.239

rEP

1.434

-1.2 1.0

1.1

1.2

1.3

1.4

1.5

radial coordinate (r/a)

Figure 1: Stresses and displacement in a partially plastic curved beam.

The distributions of stress and displacement in a partially plastic nonlinearly hardening curved beam at T 0q plane is displayed in Fig. 1. Inner radius to outer radius ratio, a / b , has been taken as 1.5 . The stress variable I in this figure is computed from von Mises' yield equation so that I t 1 in a plastically deformed region. The two vertical lines labeled as rEP , which correspond to I

1,

indicate elastic-plastic borders. Other vertical line rNA points the neutral axis at which V T rNA 0 . The curved beam is composed of an inner plastic region in 1 d r d 1.081 , an elastic region in 1.081 d r d 1.434 , and an outer plastic region in 1.434 d r d 1.5 .

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A Proposal of Strain-Gage Rosette for Measurement Residual Stress Around a Circular Hole in a Plate With Circular Hole José Luiz Fernandes*, Antonio Carlos Marques Alvim†, Renato Seixas da Rocha† *

Federal Center of Technological Education – Mechanical Engineering Department Av. Maracanã, 229, BlocoE, Sala505.7 [email protected] †

Nuclear Engineering Programme – COPPE / UFRJ – Brazil Av. Maracanã, 229, BlocoE, Sala505.7 [email protected] †

R&D Center of Petrobras – CENPES – Brazil

Cidade Universitária – Avenida Jequitibá – Quadra 7 – Ilha do Fundão ZIP code 21949-900 – Rio de Janeiro – RJ – Brasil [email protected]

ABSTRACT Hole-drilling strain-gage rosette method is an experimental technique used to measure residual stresses near the surface of a mechanical component. Various types of rosettes are available with different geometric configurations and dimensions. The strain field high gradients observed near the hole, where the strain-gage rosette is bonded, makes the problem of choosing the optimal rosette position a critical one. In this work a methodology of residual stress measurement for the technique Hole-Drilling was developed to simulate the effect of a through circular hole in a thin plate submitted under nominal uniaxial and biaxial loads up to the yield strength (Se). In the biaxial case was considered relations between the nominal components load of Vnx/Vny = 0,25, 0,50, 0,75 and 1,00. It was simulated by finite elements the measurement residual stress methodology named “HoleDrilling" in plate submitted nominal biaxial big loading until the yielding stress for the API 5L X60. In this study was projected and a strain gage rosette containing 5 strain gages positioned 0, 90, 150, 225 and 300 degrees. The relative error was evaluated between the numeric solution and the theoretical solution proposed by ASTM E837 (elastic) and by Savin (elastoplastic). It was verified that the relative errors for the main strain H1 and H2, obtained by the strain gages redundant to decrease. It is worth to emphasize that the strain gages rosette proposed with five strain-gages didn't eliminate the problem of the eccentricity, but it quantified with larger precision this effect.

References [1] Beghini, M.; Bertini, L.; Raffaelli, P. “Numerical analysis of plasticity effects in the hole-drilling residual stress measurement”, Journal of Testing and Evaluation, JTEVA, Vol. 22, No 6, November, pp. 522  529, 1994. [2] Fernandes, J.L., A Methodology for the Analysis and Modeling of Residual Stresses, (in Portuguese), Doctor Degree Thesis, Department of Mechanical Engineering, PUC/RJ, 340p.,2002. [3] Fernandes, J.L., Castro, J.T.P., 2003, “Analysis Elastoplastic of Stress around a Circular Hole in a Plate Submitted Biaxial and Uniaxial Loading”, 58nd International Congress of Brazilian Metals Association, Rio de Janeiro, 2003. [4] Kabiri, M., 1984, “Non-uniform residual stress measurement by hole-drilling method”, Experimental Mechanics, pp. 328 – 336, 1984.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

65

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

66

Size effects in finite deformation micropolar plasticity P. Grammenoudis∗ , Ch. Tsakmakis∗ ∗ Darmstadt

University of Technology Department of Mechanics AG 1 Continuum Mechanics Hochschulstraße 1 D-64289 Darmstadt Germany [email protected] [email protected] ABSTRACT Micropolar theories offer a possibility to model size effects in the constitutive behavior of materials. Typical feature of such models is that they deal with a microrotation, which is supposed to represent an independent state variable, and its space gradient. As a consequence, the stress tensor is no longer symmetric and couple stresses enter the theory. Accordingly, a micropolar plasticity law exhibiting kinematic hardening effects should account for both, a back-stress tensor and a back-couple stress tensor. This has been considered in the micropolar plasticity model developed by Grammenoudis and Tsakmakis [1], [2]. To be more specific, we assume the multiplicative decomposition to apply for both the deformation gradient tensor of the macroscopic continuum and the micropolar rotation tensor of the assigned microstructure. From these, we obtain additive decompositions for the strain and curvature tensors as well as their associated rates. A yield function is assumed which reflects isotropic and kinematic hardening effects. The stress tensor and the back-stress tensor in the yield function are supposed to obey the structure of the so-called Mandel stress tensor within the framework of classical (non-polar) plasticity. All constitutive equations are defined in such a manner that the second law of thermodynamics is fullfilled in every admissible process. The capabilities of the constitutive theory are demonstrated by means of finite element calculations. To this end, the micropolar plasticity model is implemented into finite element code ABAQUS. Details of the implementation are given in Grammenoudis and Tsakmakis [2]. Here, a discussion of the influence of hardening on size effects is given, with reference to torsion deformation (see Grammenoudis and Tsakmakis [3]).

References [1] P. Grammenoudis and Ch. Tsakmakis, Hardening rules for finite deformation micropolar plasticity: Restrictions imposed by the second law of thermodynamics and the postulate of Il’iushin. Continuum Mechanics and Thermodynamics, 13, 325–363, 2001. [2] P. Grammenoudis and Ch. Tsakmakis, Finite element implementation of large deformation micropolar plasticity exhibiting isotropic and kinematic hardening effects. International Journal for Numerical Methods in Engineering, 62, 1691–1720, 2005. [3] P. Grammenoudis and Ch. Tsakmakis, Predictions of microtorsional experiments by micropolar plasticity. Proceedings of the Royal Society, 461, 189–205, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

67

An Application of a Boundary Perturbation Method to Some Problems of Elasticity M. Grekov St. Petersburg State University Universitetski Pr., 35, St. Petersburg, 198504, Russia [email protected]

ABSTRACT There are a lot of problems of great importance in continuum mechanics, which can be solved by means of the perturbation method. In many cases, this method relates to a perturbation of a reference boundary for which a similar boundary problem has a relatively simple solution or can be easy solved in a closed form. The weakness of all cited works is that they are confined to constructing the first order perturbation solutions and the perturbation technique is carried out in different manners even for the 2-D problems. At the same time, using Kolosov's complex potentials, a unified perturbation method for the 2-D problem of elasticity has been developed. Based on this method and algorithm constructed, one can easy compute any-order accurate solution of a wide range of boundary problems. The aim of the present paper is to illustrate the general approach to constructing an algorithm of the unified boundary perturbation method. A brief review of the works devoted to the application of the perturbation technique to the crack and interface problems is presented. A unified boundary perturbation method based on Goursat-Kolosov's complex potentials and Muskhelishvili's complex variable representations is expounded for the cases of a slightly curved interface and an interfacial crack slightly deviating from a straight one. An algorithm of deriving the complex potentials of anyorder accurate perturbation solution in a closed form has been developed for the problems under consideration. Explicit results are given for the first-order solutions when local deviations of the boundary from the straight one are described by power functions. Characteristics of a stress field are analyzed for locally curved interfaces and curvilinear interfacial cracks. These studies have been partly described in works [1–3].

References [1] M.A. Grekov. A perturbation method in a problem on deformation of composite with slightly curved interface. Vestnik of St. Petersburg University, Ser. 1, 1, 81-88, 2004 (in Russian). [2] M.A. Grekov, S.N. Makarov. Stress concentration at a slightly curved section of a surface of an elastic body. Izvestiya RAN. Mekhanika tverdogo tela, 6, 53-61, 2004 (in Russian, translated in Mechanics of Solids). [3] M.A. Grekov, J.V. Malkova. Application of the perturbation method to the problem on a curvilinear interfacial crack. Int. Conference on Stability and Control Processes, St. Petersburg, Russia, 3, 1655-1656, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Spatial Stabilization of Semidiscrete Elastodynamics Eran Grosu, Isaac Harari ∗ Department of Solid Mechanics, Materials, and Systems, Tel Aviv University 69978 Ramat Aviv, Israel ∗ [email protected] ABSTRACT Solutions of direct time integration schemes that converge in time to conventional semidiscrete formulations may be polluted at small time steps by noncausal oscillations. These pathologies are the deleterious effects of higher modes of spatially discrete formulations, which are approximated poorly. Algorithmic damping reduces the effect of these higher modes, but they are inevitably admitted into the computation as the time step is reduced on a fixed mesh, with convergence to the solution of the semidiscrete formulation. This degradation is not an artifact of the time-marching scheme, but rather a property of the solution of the semidiscrete formulation itself. The Rothe method (or horizontal method of lines) of first discretizing in time and then in space on each discrete time level is employed in order to characterize the onset of such pathologies and eliminate them. Standard finite elements are optimal in the energy norm for elastostatics, and thus need not be stabilized. In the transient case, the Rothe method reveals that the time-discrete equation that is solved at each time level of standard implicit schemes is, in fact, a Galerkin approximation of a steady elasticity equation with an additional reaction term. An analogy to such singularly perturbed elliptic problems provides a threshold for the onset of small time step oscillations. Numerical computations at smaller time steps might violate the principle of causality. A simple procedure of spatial stabilization removes this pathology from implicit time-integration schemes, without effecting unconditional temporal stability, leading to implicit time-integration that is free of noncausal oscillations. Computation of discontinuous phenomena with the trapezoidal rule (the implicit, unconditionally stable, second-order accurate algorithm of the Newmark family of methods) is accompanied by spurious oscillations, due to its lack of algorithmic damping. At time steps above the threshold for spatial stability these oscillations follow the wave front, but below this critical time step the oscillations precede the wave front, violating the principle of causality. In the spatially stabilized version of the trapezoidal rule the oscillations persist, but always behind the wave front, regardless of the time step. The HHT-α [1] and Generalized-α [2] algorithms are implicit, unconditionally stable, second-order accurate schemes that possess adjustable algorithmic damping, significantly reducing the spurious oscillations observed in the trapezoidal rule. Nonetheless, below the critical time step, spurious oscillations precede the wave front. Spatial stabilization of these algorithms eliminates the spurious precursors to the wave front at all time steps.

References [1] H. M. Hilber, T. J. R. Hughes, and R. L. Taylor, Improved numerical dissipation for time integration algorithms in structural dynamics. Earthquake Eng. Struct. Dyn., 5, 282–292, 1977. [2] J. Chung and G. M. Hulbert, A time integration algorithm for structural dynamics with improved numerical dissipation: The generalized-α method. Trans. ASME J. Appl. Mech., 60, 371–375, 1993.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Modeling the Grain Size Effect using Gradient Hardening and Damage in Crystal (Visco) Plasticity Mikkel Grymer, Magnus Ekh, Kenneth Runesson, Thomas Svedberg Chalmers University of Technology, Dept. of Applied Mechanics 412 96 G¨oteborg, Sweden [email protected]

ABSTRACT The macroscopic behavior of a polycrystalline material (metal) depends on the characteristics of the grain structure. Among the important properties are the size and morphology of the grains, volume fraction of different phases, and the subgrain material modeling. In this contribution we put emphasis on the modeling and numerical simulation of the grain size dependence on the macroscopic response (in terms of onset of yielding, hardening/softening characteristics, etc.). Within the framework of continuum thermodynamics and finite strains, we formulate a subgrain material model that comprises crystal (visco)plasticity, crystal damage and gradient hardening. The crystal damage is based on the concept of a fictitious (undamaged) configuration, and it is assumed to be driven by inelastic slip in each slip system. Furthermore, the gradient hardening gives a contribution from each slip system which is added to the well established local hardening. The grain interaction in a Representative Volume Element is resolved using finite elements. In order to solve the arising coupled field equations (for the displacements and the gradient hardening in the slip systems) a so-called dual mixed FE algorithm is adopted. Linear displacements and gradients are assumed in a basic set-up. As an alternative, quadratic displacements are introduced, while the linear gradient approximation is retained. Dirichlet boundary conditions on the RVE (corresponding to a given macro-scale deformation gradient) are adopted, and various prolongation conditions inside the RVE are investigated: The classical Taylor assumption, Generalized Taylor assumption (to grain boundaries only) and a fully unconstrained local displacement field. In particular, the two first approaches may be used to provide a good start solution for the fully unconstrained (most general) approach. All computations are restricted to 2D.

References [1] T. Svedberg and K. Runesson, Algorithm for gradient-regularized plasticity coupled to damage based on a dual mixed FE-formulation. Computer Methods in Applied Mechanics and Engineering, 161(1-2), 49–65, 1998. [2] M. Ekh R. Lillbacka and K. Runesson, A model framework for anisotropic damage coupled to crystal (visco)plasticity. International Journal of Plasticity, 20(12), 2143–2159, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Stress-Focusing Effect Following Dynamically Transforming Strains in a Spherical Zirconia Inclusion Toshiaki Hata Faculty of Education, Shizuoka University 836 Oya Aoiku, Shizuoka City, 422-8529, Japan [email protected] ABSTRACT Some composite materials, such as Zirconia toughened ceramics, are remarkable material, which has a high strength, a high elastic modulus, and an improved toughness, etc. Most of the good qualities are common in many ceramic composite materials. These good qualities are made possible through the phase transformation of composite particles. The mechanism in the toughening of ceramics is the stressinduced phase transformation of a Zirconia particle as shown in the work by Garvie et al. [1], which is accompanied by a volumetric expansion. Due to this expansion, the composite material consisting of Zirconia particles within a brittle matrix becomes more resistant to the thermal fracture. While in the dynamic state the mechanism in the toughening of ceramics is not clarified. The elastodynamic response of the transformation-toughened ceramics under an instantaneous phase transformation has been investigated in Ref.[2]. The transformation toughening utilizes the stress-induced phase transformation of particles, which is accompanied by a dynamic volumetric expansion. In this paper a phenomenological model is proposed to describe the situation, which involves a dynamic phase transformation in a spherical particle of Zirconia embedded in the infinite elastic matrix. When an infinite elastic medium with a spherical inclusion of Zirconia is suddenly subjected to an instantaneous transversely anisotropic phase transformation, stress waves occur at the surface of spherical inclusion the moment instantaneous transformation strains are applied. The stress wave in an inclusion proceeds radially inward to the center of the inclusion. The wave may accumulate at the center and show the stress-focusing effects, even though the initial stress should be relatively small. Stress waves, which develop following rapid transformation strains, display a stress-focusing effect as they proceed radially towards the center in this geometry. After the numerical calculations we conclude that the stress-focusing effects in a spherical inclusion play an important role in the fracture of matrix. In the steady state the fracture in the matrix with a spherical inclusion occurs at the interface of an inclusion, whereas in the dynamic case the fracture in the matrix with a spherical inclusion occurs at the center of an inclusion because of the stress-focusing effects. Therefore it should be noted that the stress-induced mechanism of phase transformation in the toughening of the Aluminum Oxide ceramics with a Zirconia inclusion in the steady state does not hold in the dynamic state.

References [1] R.C. Garvie, R.H. Hannink, and R.T. Pascoe, Ceramic Steel, Nature, 258, 703–704, 1975. [2] T.Hata, Stress-Focusing Effect in a Spherical Inclusion Embedded in an Infinite Medium Caused by Instantaneous Phase Transformation, Dynamics of Advanced Materials and Smart Structures, Kluwer Academic, 95–104, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Controllability Method for the Solution of Linear Elastic Wave Equation Erkki Heikkola1 , Sanna M¨onk¨ol¨a∗,2 , Anssi Pennanen2 , Tuomo Rossi2 1 Numerola

Oy, P.O. Box 126, FI-40101 Jyv¨askyl¨a, Finland of Mathematical Information Technology, P.O. Box 35 (Agora) FI-40014 University of Jyv¨askyl¨a, Jyv¨askyl¨a, Finland ∗[email protected].fi

2 Department

ABSTRACT We consider the use of controllability techniques for the numerical solution of time-harmonic elastic wave equations. Instead of solving directly the time-harmonic equation, we return to the corresponding time-dependent equation and look for time-periodic solution. The basic approach is to time-integrate the wave equation from initial conditions until the time-periodic solution is reached. Unfortunately, the convergence of such an approach is usually slow. We accelerate the convergence with a control technique by representing the original time-harmonic equation as an exact controllability problem for the time-dependent wave equation. This involves finding such initial conditions that after one timeperiod the solution and its time derivative would coincide with the initial conditions. Spatial discretization is done with spectral element method. It allows convenient treatment of complex geometries and varying material properties. The basis functions are higher order Lagrange interpolation polynomials, and the nodes of these functions are placed at Gauss-Lobatto collocation points. The integrals in the weak form of the equation are evaluated with the corresponding Gauss-Lobatto quadrature formulas. As a consequence of the choice, spectral element discretization leads to diagonal mass matrices which significantly improves the computational efficiency of the explicit time-integration used. Moreover, when using higher order elements, same accuracy is reached with less degrees of freedom than when using lower order finite elements. After discretization, exact controllability problem is reformulated as a least-squares optimization problem, which is solved with a preconditioned conjugate gradient algorithm. Each conjugate gradient iteration requires computation of the gradient of the least-squares functional, which involves the solution of the state equation and the corresponding adjoint equation, solution of a linear system with the preconditioner, and some matrix-vector operations. Computation of the gradient of the functional is an essential point of the method, and we have done it with the adjoint state technique directly for the discretized problem. Algebraic multigrid method is used for preconditioning the conjugate gradient algorithm.

References [1] M. O. Bristeau, R. Glowinski, and J. P´eriaux. Controllability methods for the computation of timeperiodic solutions; application to scattering. Journal of Computational Physics, 147(2), 265-292, 1998. [2] G. C. Cohen, Higher-Order Numerical Methods for Transient Wave Equations, Springer, 2002. [3] F. Kickinger, Algebraic multi-grid for discrete elliptic second-order problems. Multigrid methods V (Stuttgart, 1996), 157–172, Springer, 1998. [4] J.J. Heys, T.A. Manteuffel, S.F. McCormick, and L.N. Olson, Algebraic multigrid for higher-order finite elements, Journal of Computational Physics, 204(2), 520–532, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Development of a flow stress model for metals using the strain rate / temperature superposition principle Ch. Husson*, J. Richeton†, S. Ahzi† *

Institut Supérieur d’Ingénierie de la Conception - Equipe ERMeP 27 rue d’Hellieule, F-88100 Saint-Dié-des-Vosges [email protected] †

Université Louis Pasteur, IMFS-UMR 7507 2 rue Boussingault, F-67000 Strasbourg [email protected] - [email protected]

ABSTRACT The stress-strain response of metallic materials and alloys is influenced by the temperature at which deformation occurs and by the loading rate. To model this behavior under a wide range of strain rates and temperatures we propose to use the strain rate / temperature superposition principle: an increase in temperature will have the same effect on the yield stress as a decrease in strain rate. This type of approach has been recently proposed to model the yield behavior of solid polymers. Here we show that this approach can be extended to model the yielding and flow stress evolution in ductile metals. For a wide range of temperatures and strain rates, the superposition principle is applied for different metals such as copper and high strength steel. This comparison showed a good agreement with the experimental data found in the literature.

References [1] J. Richeton, S. Ahzi, L. Daridon, Y. Rémond, A formulation of the cooperative model for the yield stress amorphous polymers for a wide range of strain rates and temperatures. Polymer, 46, 6035-6043, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Analysis of Elastic Body using Kalman Filter Finite Element Method Taku Kato*, Mutsuto Kawahara† *

Department of Civil Engineering, Chuo University Kasuga 1-13-27, Bunkyo-ku, Tokyo 112-8551, Japan [email protected] † Department of Civil Engineering, Chuo University Kasuga 1-13-27, Bunkyo-ku, Tokyo 112-8551, Japan [email protected]

ABSTRACT This research is the estimation of elastic body using the Kalman Filter Finite Element Method. The ground is always continuing a minute vibration. And it is very difficult to identify data about production of any actions. The Kalman Filter is the method of estimating unknown parameter using observation data distorted by noise. The Kalman Filter is the filtering algorithm presented by Kalman and Bucy in 1960’s, which is based on the stochastic process theory of the state space model and the orthogonal projection for the linear system. In the Kalman Filter, the system and observation noise are included. The system noise is error that arises in approximating basic equation. The state space model is consisted by system model equation and the observation equation. The system model equation can be expressed state of phenomena. And the observation equation is denoting relation between the actual observation data and the state value. The Kalman Filter is a calculation algorithm is estimation problem it is divided in three workspace shown as followed, (1) Prediction is the problem calculated optimal estimated values in the future, (2) Filtering is calculated it at the present, (3) Smoothing is calculated it in the past. The Kalman Filter can estimate the state value in time direction. However, the Kalman Filter cannot estimate the state value in space direction. If combine the advantage of the Kalman Filter and Finite Element Method, it can estimate not only in time but also in space direction. And it is the Kalman Filter Finite Element Method. For the temporal discretaization, the Newmark β method is used. And For the spatial discretaization, the Galerkin method is applied. As the numerical analysis, it is presented that the estimation of the quarry. The quarry is Futatsuisi-mountain in Miyagi Japan. The actual data is used from September 20th in 2005.

References [1] Hikawa, A., Kawahara, M. and Kaneko, N., Parameter Identification of Ground Elastic Modulus at Excavation Site of Tunnel. Vol.1, Research Report of Professor M. Kawahara Lab, pp72-83, 2004. [2] Kato, Y. and Kawahara, M., Kalman Filter Finite Element Method Applied to Elastic Body Using Tunnel Model, Research Report of Professor M. Kawahara Lab. [3] Komine, H., Kawahara, M. and Koizumi N., Application of First Order Adjoint Method to Parameter Identification, Research Report of Professor M. Kawahara Lab.

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Discontinuous Galerkin methods for nonlinear elasticity Adrian Lew∗, Alex Ten Eick† ∗Stanford University Durand 275, Stanford, CA, 94305-4040, USA [email protected] † Stanford University Durand 212, Stanford, CA, 94305-4040, USA [email protected]

ABSTRACT In this talk we present a general formulation for discontinuous Galerkin methods for nonlinear elasticity. We discuss the efficiency, implementation, and stabilization of the method, and illustrate the results with numerical examples, including convergence rates, behavior under nearly incompressible conditions and under severely large deformations. We illustrate the performance of the method with applications to biomechanics problems.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A Rate Dependent Constitutive Model for Carbon-Fibre/Epoxy-Matrix Woven Fabrics Submitted to Dynamic Loadings S. Marguet∗, P. Rozycki†, L. Gornet† ∗Ecole Centrale de Nantes, GeM, UMR CNRS 6183 1 rue de la No BP92101, 44321 Nantes Cedex 3, France [email protected] †Ecole Centrale de Nantes, GeM, UMR CNRS 6183 1 rue de la No BP92101, 44321 Nantes Cedex 3, France [email protected], [email protected]

ABSTRACT This paper deals with the modelling until rupture of composite structures made of carbon/epoxy woven fabrics and submitted to dynamic loadings. The mechanical behaviour of the woven ply appears to be orthotropic, strongly non linear and rate dependent. In the warp and fill directions the ply behaves in a way slightly non linear, elastic and brittle whereas in shear, it can be observed a progressive loss of rigidity of the elastic modulus combined with inelastic strains. In this work, and in order to take into account the previous physical phenomena, we have adapted and strongly coupled a viscoplastic model with a delayed damage mesomodel [1]. In each direction of fibres, a damage variable drives the failure. In shear, the association of a damage variable with a plastic strain allows the model to represent the non linear irreversible effects described above. Special attention is paid to ensure a good representativeness of the model compared with the experiments on a large range of strain rates. This results in the introduction of viscous elastic moduli. The constitutive equations are solved implicitly throughout a backward Euler scheme which is implemented by the mean of a Closest Point Projector algorithm. To identifiy the parameters, an optimization procedure based on the direct search method is carried out. It leads to an accurate behaviour of the model in both tensile and shear directions for strain rates evolving from at least 1.10−3 s −1 to 1.103 s −1 . Furthermore, to check the ability of the model to avoid strain localization phenomenon or mesh dependency, a campaign of simulations is performed on the classical example of a bar in tension. Finally, and in order to test the model on structural applications, both the impact on composite plate and the dynamic crushing of thin-walled tube are simulated and compared with some experimental results from [2] and [3]. The model developed in this study appears to be able to predict quite precisely the collapse of the structures from the initiation of a macroscopic cracks towards the rupture.

References ` O. Allix, J-F. Deu, ¨ D. Lev ´ eque, ˆ [1] P. Ladeveze, A mesomodel for localization and damage computation in laminates, Computer Methods in Applied Mechanics Engeneering, Vol 183, pp 105-122, 2000. [2] A.F. Johnson, A.K. Pickett and P. Rozycki, Computational methods for predicting damage in composite structures, Composites Science and Technology, Vol 61, pp 2183-2192, 2001. [3] A.G. Mamalis, D.E. Manolakos, M.B. Ioannidis, D.P. Papapostolou, On the response of thinwalled CFRP composite tubular components subjected to static and dynamic axial compressive loadings: experimental, Composite Structures, Vol 69, pp 407-420, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Solution of viscoelasticity problems using special forms of elastic solutions Valery P. Matveyenko* *

Institute of Continuous Media Mechanics of Ural Branch of RAS Academician Korolev str., 1 Perm, 614013, Russia [email protected]

ABSTRACT The Volterra method is widely used for solving linear quasistatic viscoelasticity problems. According to this method, the construction of the viscoelastic solution reduces to the replacement of elastic constants by Volterra operators and to further interpretation of operator relations. The transition from the elastic solution to the viscoelastic one is usually attended with great calculation difficulties encountered in interpretation of operator relations. One way to meet this challenge is to represent the solution of the elastic problem in the form suitable for further interpretation. The first sample of such representation belongs to Volterra. One of most effective methods to overcome these difficulties was suggested by A.A.Ilyushin. This work presents an approach that enters into this group. The variants of this approach are suggested to develop elastic solutions for uniform solid as a series in integer powers of an elastic constant (Ilyushin’s parameter) and as a series in integer powers of two elastic constants (Ilyushin’s parameter and the volume modulus). The variant of construction of elastic solutions for piecewiseuniform solids as a series in integer powers of elastic constants (Ilyushin’s parameters) of each component is developed. The obtained forms of elastic solutions are used to find solutions to the following viscoelastic problems: for uniform or piecewise-uniform (made of different viscoelastic materials) solid made of materials with hereditary–elastic properties under shear deformations and elastic properties under volume deformations. The mathematical convergence of the considered calculation procedures is proved. For numerical realization of the problems, the standard procedures of the finite-element method are used. The obtained numerical results demonstrate the efficiency of the applied calculation procedure. The results obtained from the solution of viscoelasticity problems are provided. The proposed method made it possible to discover a new qualitative effect associated with the time behavior of stresses in piecewise-uniform solids. The essence of the effect is that at constant or monotonic time variable external load the change in stresses may be of non-monotonic character.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Finite elements method analysis of influence of contact phenomena on structure-subsoil interaction ´ Marcin Mazdziarz Institute of Fundamental Technological Research Warsaw,Poland [email protected] ABSTRACT The contact between structure and subsoil is formulated by using 3D elasto-viscoplastic relationships incorporating pore pressure. Finite element implementation is then performed for the geotechnical structure, subsoil and contact relationships. Contacts elements with zero thickness are carefully developed, allowing for slip and stick. Solid isoparamtric elements are also developed. The novelty of the contribution consists of using elasto-viscoplastic model incorporating pore pressure for 3D contact. Using implemented spatial elements in the program HYDRO-GEO elasto-viscoplastic analysis of interaction between structure and subsoil was carried out. Earth dam interacting with concrete weir and excursion trough in one of Polish earth dams (Dobczyce) was analyzed. Simulation of deformation of structure and slide of soil on surface of the retaining wall was studied. The goal of this study was to examine both from theoretical and numerical point of view the influence of contact phenomena on structure-subsoil interaction. The numerical analysis was carried out by using finite element methods including spatial contact elements with zero thickness. 3D formulation of the interacting structure with subsoil was developed. The program HYDRO-GEO enables to perform numerical calculations of the considered complex contact problem in the 3D case. The paper can be divided into two essential parts. The first part is concerned with discretized 3D description of basic relationships by using the finite element method. To do this 3D solid isoparametric elements were introduced. Pore pressure was taken into account. Spatial contact finite elements were developed and incorporated into the program. These elements cover elasto-viscoplastic contact behavior and take into account pore pressure, slip and stick range. In numerical part of the paper the developed discrete model was implemented into the computer program HYDRO-GEO. Then the elasto-viscoplastic interaction between structure and subsoil can be analyzed. A complex system consisting of the earth dam and concrete weir was studied (one of Polish earth dams in Dobczyce).

References ´ [1] Marcin Mazdziarz, Influence of contact phenomena on structure-subsoil interaction: finite elements method analysis, PhD thesis, Politechnika Warszawska, Warszawa (in Polish), 2003. [2] Zienkiewicz O. C., Taylor R. L. , The Finite Element Method. Vol 1-3. Oxford: ButterworthHeinemann , 2000.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Simulation of Inelastic Deformation of Polyethylene in Multiaxial State of Stress by Viscoelastic Constitutive Equtaion Mamoru Mizuno* and Yukio Sanomura† *

Dept. Machine Intelligence and Systems Engineering Akita Prefectural University Tsuchiya, Yuri-Honjo, Akita 015-0055, Japan [email protected] †

Dept. Mechanical Engineering Tamagawa University Tamagawa-Gakuen, Machida, Tokyo 194-8610, Japan [email protected]

ABSTRACT Polymers show significant strain recovery after reversal of the loading direction, and conventional constitutive equations can not describe inelastic deformations including the strain recovery properly. Then the present authors have proposed a viscoelastic constitutive equation for polyethylene in order to describe the inelastic deformations. In the formulation, a total strain was assumed to be the sum of an elastic strain and a viscous strain. The elastic strain was subjected to the Hooke’s law, and the viscous strain was derived from the kinematic hardening creep theory of Malinin and Khadjinsky, which was combined with the nonlinear kinematic hardening rule of Armstrong and Frederick. In order to describe the strain recovery, a loading surface was defined in a viscoelastic strain space, and a new parameter was defined by using the loading surface. Then the nonlinear kinematic hardening rule was modified by using the parameter. Inelastic deformations in a uniaxial state of stress were simulated by using the constitutive equation and the validity of the formulation and the modification was verified by comparing the simulations with experimental results of polyethylene. Then inelastic deformations under typical cyclic loadings in the uniaxial state of stress were predicted, and features of the deformations were discussed. In general, the viscoelastic constitutive equation will be employed for structural analyses such as a FEM. Thus the applicability and the capability of the constitutive equation to predict deformations in a multiaxial state of stress are important. In the present paper, inelastic deformations of polyethylene in a two-axial state of stress by the combination of a normal stress and a shear stress are simulated by using the viscoelastic constitutive equation. As a loading condition, proportional and non-proportional strain paths under the total strain control at constant strain rate are considered. Inelastic deformations under cyclic loadings are focused on in particular. Results of the simulations are compared with experimental results of polyethylene and the validity of the constitutive equation is verified in the multiaxial state. And features of the inelastic deformations of polyethylene are inspected by simulations under various loading conditions.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Modeling of granular media by the 2D discrete lattice Igor S. Pavlov* *

Mechanical Engineering Research Institute of the Russian Academy of Sciences 85, Belinsky street, 603024, Nizhny Novgorod, RUSSIA [email protected]

ABSTRACT A two-dimensional model of a granular medium is considered that represents a square lattice consisting of uniform round particles. The particles have both translational and rotational degrees of freedom. Each particle (granule) is supposed to interact directly with eight nearest neighbours in the lattice [1]. The nonlinear governing equations describing propagation and interaction of waves of various types in such a medium have been derived for a cubic potential of elastic interactions between the particles in the discrete and continuum approximations. The discrete equations are valuable, particularly, for numerical simulation of nonlinear wave processes in granular media. The continuum approximations of the model at issue are convenient for its comparison with known theories of solids. The continuum equations do not coincide with the classical theory of elasticity due to additional equation for the rotational wave. Such a wave exists when the frequency is larger than a threshold value. Its dispersion properties are similar to dispersion properties of the spin wave in a magnetelastic medium. From the numerical estimations of the rotational wave velocity in some cubic crystals follows that, as a rule, it is less than the translational wave velocities. When microturns of the particles are absent, the linear parts of the governing equations of the first continuum approximation degenerate into Lame equations for anisotropic medium with the cubic symmetry. The governing equations are structurally similar to equations of the anisotropic Cosserat continuum with centrally symmetric particles. However, the longitudinal wave velocity does not depend on the medium structure in the Cosserat continuum, while such dependence presents in the considered model. The last fact enables to explain, particularly, experimentally observing variations of this wave velocity when size of granules grows. The governing equations of the second (quasicontinuum) approximation contain summands with higher-order derivatives and give explanation to appearance of the longitudinal wave dispersion. The Cosserat theory is unable to explain this effect.

References [1] I.S. Pavlov, A.I. Potapov, and G.A. Maugin, A 2D Granular Medium With Rotating Particles. Int. J. of Solids and Structures, 2005 (in print).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

81

A geometric approach to the algorithmic tangent stiffness G. Romano∗, M. Diaco†, R. Barretta† ∗ Dipartimento di Scienza delle Costruzioni (DiSCo)

Universit`a di Napoli Federico II, via Claudio 21- 80125 Napoli, Italy [email protected] †Dipartimento di Scienza delle Costruzioni (DiSCo) Universit`a di Napoli Federico II, via Claudio 21- 80125 Napoli, Italy [email protected] - [email protected]

ABSTRACT The elastoplastic tangent stiffness is the linear operator which provides the stress rate corresponding to a prescribed strain rate. As such, it plays a central role in the computational aspects of elastoplastic problems. According to the usual approach to the nonlinear evolutive analysis of elastoplastic models, a finite time-step is considered and the evolution law describing the constitutive behavior is reformulated as a finite step flow rule. The algorithmic tangent stiffness was first introduced by Simo and Taylor in [1]. They showed that the adoption of the algorithmic tangent stiffness leads to a significant improvement of the asympthotic convergence rate. The expression of the algorithmic tangent stiffness provided in [1], and in all the subsequent references to their contribution, was based on an explicit formulation of the elastoplastic constitutive law in terms of a plastic scalar multiplier. The geometrical analysis developed in the present paper is based on a formulation of the constitutive problem in terms of the nonlinear projector, in complementary elastic energy, on the convex elastic domain. The algorithmic tangent stiffness is evaluated as the composition between the derivative of the nonlinear projector and the elastic stiffness. The key point consists in the evaluation of the derivative of the nonlinear projector. A direct geometric argument, based on hypersurface theory, shows that the derivative can be expressed as the difference between the linear projector on the hyperplane tangent at the trial stress point and the shape operator of the parallel hypersurface passing thru the trial stress-state, multiplied by the distance between the trial stress and the projected stress, evaluated in the complementary elastic norm. The composition of the linear projector with the elastic stiffness is in fact the rate elastoplastic tangent stiffness. Since the analytic expression of the parallel hypersurface thru the trial stress point is available only in special cases, an effective procedure consists in substituting it with the level set of the yield function passing thru the trial stress point. In this way, the exact expression of the algorithmic stiffness is got when the level sets of the yield function are homothetic hypersurfaces, as in von Mises plasticity criterion, and a simple useful approximation is obtained in the general case. Indeed the proposed procedure greatly simplifies the computations while preserving the benefit of an improved convergence rate, since it takes effectively into account the curvature of the yield hypersurface, thus leading to a reduced tangent stiffness, in comparison with the rate tangent stiffness.

References [1] J.C. Simo, R.L. Taylor, Consistent tangent operators for rate-independent elastoplasticity, Comp. Meth. Appl. Mech. Engrg. , 48, 101–118, 1985.

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Parallel Computation of 3D Problems of the Dynamics of Elastic-Plastic Granular Material under Small Strains Vladimir M. Sadovskii, Oxana V. Sadovskaya∗ Institute of Computational Modelling SB RAS Akademgorodok, Krasnoyarsk, Russia, 660036 ∗[email protected] ABSTRACT A process of waves propagation in elastic-plastic granular material under small strains is described on the basis of a new rheological model taking into account different resistance of the material with respect to tension and compression [1]. This model leads to the system of nonlinear partial differential equations of non-classical type which has an evident structure for numerical realization. Parallel decomposition of the used shock-capturing numerical method [2] is founded on the space-variable decomposition procedure. The proposed algorithm is accomplished as a program system for supercomputers with parallel architecture by means of SPMD (Single Program – Multiple Data) technology in Fortran-90 with the use of MPI (Massage Passing Interface) library. Various variants of 1D, 2D and 3D division of the spatial computational domain between the computational nodes are used. The program system allows to simulate a propagation of elastic-plastic waves generated by mechanical impacts in a body, aggregated of an arbitrary number of heterogeneous blocks with curvilinear boundaries. The exact one-dimensional solutions with plane shock waves are used for testing. Some problems of the waves refraction on the interior surfaces between blocks of granular materials with different mechanical properties are solved.

Stress field in different time moments

By means of the numerical experiments it is shown that the plane fronts of two waves, bending due to inhomogeneous loosening, can be reflected with the formation of transverse cumulative splash. The curved fronts of shock waves for different time moments and the cumulative splash (a typical zone of the compressive stresses, moving bottom-up) are represented in the Figure.

This work was supported by the grant of the Russian Foundation for Basic Research no. 04-01-00267, the Complex Program of the Presidium of RAS no. 14 “Fundamental Problems of Informatics and Informational Technologies” and the Russian Science Support Foundation.

References [1] V.M. Sadovskii, O.V. Sadovskaya, Parallel Computation of Elastic-Plastic Waves Propagation in Granular Material. Proc. of the 7th Intern. Conf. on Mathematical and Numerical Aspects of Wave Propagation. Brown University, Providence, 223–225, 2005. [2] O.V. Sadovskaya, Shock-Capturing Method as Applied to the Analysis of Elastoplastic Waves in a Granular Material. Comput. Math. & Mathematical Phys. 44, 1818–1828, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A hyperelastodynamics ALE formulation based on spatial and material forces. Z. Uthman, H. Askes University of Sheffield, Department of Civil and Structural Engineering. Mappin Street, Sheffield S1 3JD, United Kingdom [email protected] [email protected]

ABSTRACT This contribution aims at providing the formulation and implementation details of arbitrary Lagrangian Eulerian hyperelastodynamic problem classes. This ALE formulation is based on the dual balance of momentum in terms of spatial forces (the well-known Newtonian forces) as well as material forces (also known as configurational forces). The balance of spatial momentum results in the usual equation of motion, whereas the balance of the material momentum indicates deficiencies in the nodal positions, hence providing an objective criterion to optimize the finite element mesh. The main difference with traditional ALE approaches is that the combination of the Lagrangian and Eulerian description is no longer arbitrary, in other words the mesh motion is no longer user defined but completely embedded within the formulation. The present work aims at developing spatial and material variational equations based on the Hamiltonian principle. These equations will be discretised to obtain the weak form of the momentum and continuity equations. The discretized ALE Hamiltonian equations of the spatial motion problem introduces the balance of the discretised spatial momentum and the discretised spatial continuity equation while the corresponding material motion problem defines the balance of the discretised material (or configurational) momentum and the discretised material continuity equation. We will deal with two systems of partial differential equations: the scalar continuity equation and the vector momentum equation. The momentum and continuity equations will then be linearised. The time integration of both the spatial and the material equations is performed with the Newmark scheme. A monolithic solution strategy solving the spatial and the material momentum equations simultaneously has been carried out while updating of the spatial and the material densities was attained through solving the spatial and material continuity equations (mass conservation). The solution defines the optimal spatial and material configuration in the context of energy minimization.

References [1] T. Belytschko, W.K. Liu and B. Moran. Nonlinear Finite Elements for Continue and Structures. John Wiley & Sons Ltd, 2001. [2] E. Kuhl and P. Steinmann, A hyperelastodynamic ALE formulation based on referential, spatial and material settings of continuum mechanics: Acta Mechanica, 174, 201–222, 2005. [3] E. Kuhl, H. Askes and P. Steinmann, An ALE formulation based on spatial and material settings of continuum mechanics. Part 1: Generic hyperelastic formulation. Comput. Methods Appl. Mech. Engrg, 193, 4207-4222, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Monoharmonic Approach to Investigation of Heat Generation in The Viscoplastic Solids under Harmonic Loading Yaroslav A. Zhuk Timoshenko Institute of Mechanics 3, Nesterov str, Kiev 03057, Ukraine [email protected]

ABSTRACT Intensive forced vibration of structural elements can be accompanied with significant temperature rise (heating) caused by internal dissipation of mechanical energy. Therefore, the thermomechanical coupling can cause the degradation in material properties, high strains and stresses and results in fracture of various engineering structures. In studying the nonstationary, in particular resonant, processes, it is important to take into account inelastic deformation and heat emission [1]. Indeed, the thermoviscoplastic behavior of the material and the coupling of the mechanical and thermal fields may affect substantially the characteristics of vibration dampers for engineering structures. An approach to the solution of coupled problems of vibrations and dissipative heating of viscoplastic bodies is developed. Two problem statements are elaborated: “exact” and “ approximate” ones. The former statement consists of universal balance equations of thermodynamics and constitutive equations derived from the general thermodynamic theory of viscoplastic bodies with internal state variables. For this aim the theory version that corresponds to the Bodner-Partom generalized flow theory is used. The approximate or monoharmonic problem statement for the case of steady-state vibrations under the harmonic loading is obtained by the application of modified harmonic linearizing technique to the initial system of equations. Thereby the subsystem of mechanical equations is formulated in the complex-value form and material properties are described by means of amplitudedependent complex-value moduli. Numerical technique for the solution of the problems of vibration and dissipative heating of spatial and thin-wall elements of structures are developed. They are based on the iterative linearization procedures in association with FEM. The capabilities of the monoharmonic approach are studied by the example of the resonant and quasistatic vibrations and dissipative heating of a viscoplastic disk excited by a harmonic load [2].Additional attention is paid to the case of inhomogeneous (layered) solids. Physical and mechanical behavior of such systems is very complex because of the heterogeneity of the stress-strain state. Modeling such a behavior is additionally complicated by the necessity of allowing for the coupling of the mechanical and thermal fields when cyclic loading is intensive and for dynamic effects, in particular, when intensive vibrations occur in the quasistatic or resonant domain. Comparing the solutions obtained in the frame of the exact and approximate statements demonstrates high accuracy of the monoharmonic model for the problems investigated (axisymmetric vibrations and heating of layered or homogeneous viscoplastic disk and rectangle).

References [1] I. Senchenkov, Y Zhuk and V. Karnaukhov, Modeling the thermomechanical behavior of physically nonlinear materials under monoharmonic loading. International Applied Mechanics, 40, 943-969, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A continuous Galerkin finite element method for thermoelasticity without energy dissipation Swantje Bargmann∗ , Paul Steinmann∗ ∗ Chair

of Applied Mechanics, University of Kaiserslautern P.O. Box 3049, 67653 Kaiserslautern, Germany {bargmann,ps}@rhrk.uni-kl.de ABSTRACT

The classical thermal theory based on Fourier’s law leads to a diffusive regime. Contrary to that Green and Naghdi [3] developed a theory of thermoelasticity without energy dissipation whose temperature evolution equation is hyperbolic. Among others, the introduction of a new internal variable, i.e. the thermal displacement α with α˙ = T , leads to the theory without energy dissipation, or also called theory of type II, which does not involve energy dissipation. It incorporates thermal wave propagation in a very consistent way and is capable of modeling the second sound phenomenon. The governing equations of the dynamic, linear theory of isotropic and homogeneous thermoelasticity without energy dissipation are the temperature equation a (1) ρcbT˙ = ρr + ρ κ∆α − ρbT0 3wKI : ε˙ b and the balance of linear momentum ˙ = divσ + b, [ρv] (2) where the entropy flux p = − ρκ b ∇α is determined by the same potential which determines the mechanical stresses σ. This contribution concentrates on numerical aspects of the Green-Naghdi theory of type II. In order to perpetuate the consistency of their theory to the numerical setting we resort to a Galerkin finite element method in space and in time. As the theory itself does not admit energy dissipation, conserving time integration schemes that inherit the underlying conservation principles are of great interest. Customary implicit time-stepping schemes fail to conserve major invariants, for example the total energy. The coupled dynamic system of equations is discretized in time within the framework of finite element methods using a continuous Galerkin (cG) method. In general, the cG-method has proven to qualify well for hyperbolic problems. This fact also holds true for hyperbolic heat conduction and linear thermoelastostatics [1, 2]. The cG(k)- method approximates trial functions piecewise and continuously with polynomials of degree k and test functions piecewise and discontinuously across the element boundaries with polynomials of degree k − 1. The coupled system is solved monolithically. A numerical example is investigated in order to evaluate the performance of the proposed method.

References [1] S. Bargmann and P. Steinmann, Finite element approaches to non-classical heat conduction in solids. Comp. Model. Eng. Sci. 9(2), 133–150, 2005. [2] S. Bargmann and P. Steinmann Theoretical and computational aspects of non-classical thermoelasticity. submitted [3] A.E. Green and P.M. Naghdi A re-examination of the basic postulates of thermomechanics. Proc. R. Soc. Lond. 432, 171–194, 1991.

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The Model Coupling Liquid Bridge Between Ellipsoidal Grains Jolanta Báaszczuk*, Zbigniew DomaĔski† Institute of Mathematics and Computer Science, Czestochowa University of Technology Dabrowskiego 73, 42-200 Czestochowa, Poland * [email protected] † [email protected]

ABSTRACT Granular materials are the subject of scientific studies due to their unusual physical properties which differ significantly from solid and liquid states of matter. In this paper we are interested in the role played by humidity on static and dynamic properties of systems consisting of non-spherical grains. The addition of a liquid to the material adds an attractive force to the system and then, increased its stability. Quantitative description of wetting thermodynamics is sensitive not only to the contact angle between solid and the liquid but also to the to the shape of grains and thus we analyze ellipsoidal grains and we assume that liquid spreads uniformly over the whole grain’s surface. We consider a model grain’s surface consisting of asperities of equal size uniformly distributed over the grain’s surface. We also suppose that each asperity may be either totally filled with liquid or stay empty. Thus, in our approach we consider two regimes of the inter-grain adhesive force versus volume of the wetting layer. For very small amount of liquid, the capillary force comes from the fluid accumulated around a small number of asperities at which two neighbouring grains are in contact. If the fluid wets the surface of the grains then all asperities are filled and inter-grain adhesive energy is determined mainly by the macroscopic curvature of the grain, and the surface roughness does not play a crucial role. In this case, the distribution of values of inter-grain energies is determined only by macroscopic quantities, i.e. the geometry of grains and material characteristics. Using toroidal approximation for the shape of liquid bridges and some simple probabilistic arguments we analyze the influence of amount of liquid on mutual grain – grain orientation. We found two energetically favorable orientations: one with mutually parallel and second with perpendicular axes of contacted grains.

References [1] J. Báaszczuk, Z. DomaĔski, Liquid – induced glassy behaviour of dense non-spherical grain ensembles, 16th International Conference on Computer Methods in Mechanics, CMM 2005, Publishing House of Czestochowa University of Technology, 347-348, 2005. [2] J. Báaszczuk, Z. DomaĔski, Toroidal approximation for capillary bridges between ellipsoidal grains, Scientific Research of the Institute of Mathematics and Computer Science 1(4) 2005, Publishing House of Czestochowa University of Technology, 13-17, 2005. [3] J. Báaszczuk, Dynamics and statics of packing granular materials. PhD thesis, Czestochowa University of Technology, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Reliability of Wavelet Packet System Identification Maik Brehm ∗, Christian Bucher† ∗Institute of Structural Mechanics, Bauhaus-University Weimar

Marienstraße 15, 99423 Weimar, Germany [email protected] † Institute of Structural Mechanics, Bauhaus-University Weimar

Marienstraße 15, 99423 Weimar, Germany [email protected] ABSTRACT In recent years various methods have been developed to identify structural system parameters. One approach for dynamical system identification is the application of wavelets. [1] uses Daubechies wavelets within a wavelet transformation algorithm and [2] investigates the biorthogonal wavelet transformation. The basic challenge of such wavelet-based methods is the determination of an optimal set of approximation and detail coefficients to set up the linear equation system for the identification of the unknown parameters. A major improvement compared to the simple wavelet transformation is achieved by using wavelet packets. [3] presents an automatic best basis search to determine a suitable set of coefficients. However, this algorithm is not unique and identifies only one of the best sets of coefficients. Due to the frequency decomposition of the wavelet analysis all these methods can handle a high noise level. Of course, the quality of the solution still depends on the noise level. The current practice of verifying the results are based on the known parameters or those obtained by other methods. Both ways are suboptimal to identify parameters of a genuine structure. This study investigates the possibilities to increase the reliability of the identification of the system parameters by means of wavelet transformation and the wavelet packet algorithms without the knowledge of the correct solution. Condition numbers and optimization methods are used to detect the best set of coefficients to set up an optimal equation system. Furthermore, the presented methods are discussed regarding their challenges and feasibilities for practical system identification. The developed algorithms have been implemented in the SLang Software package, which is available at the Bauhaus-University Weimar for research activities.

References [1] V. Zabel, Applications of Wavelet Analysis in System Identification. PhD Thesis, BauhausUniversity Weimar, 2003. [2] M. Brehm et al. , Applications of Biorthogonal Wavelets in System Identification. P. Neittaanm¨aki, T. Rossi, K. Majava, and O. Pironneau (eds.) Proceedings of the 4th European Congress on Computational Methods in Applied Sciences and Engineering, Jyv¨askyl¨a, 24-28 July, 2004. [3] M. Brehm et al. , Applications of Wavelet Packets in System Identification . 76. Annual Conference of the International Association of Applied Mathematics and Mechanics (GAMM), University of Luxembourg, March 28 - April 1, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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An attempt to simulate more precisely the behavior of a solid body using new energy conservation equation for fully coupled thermal structural analysis L. Écsi*, P. ÉlesztĘs† †*

Dept. of Strength of Mater, Faculty of Mech. Eng, Slovak University of Technology in Bratislava Námestie slobody 17, 812 31 Bratislava, Slovak Republic * [email protected], † [email protected]

ABSTRACT In this paper a numerical study of solid bar behavior under various mechanical loads is presented using fully coupled thermal structural analysis with large strain / large deformation formulation. The analysis is based on a new energy conservation equation [2], which the authors believe represents the most complete formulation of the first principle of thermodynamics in contemporary computational mechanics. In the numerical analysis was used the finite element method (FEM), utilizing the updated Lagrange method and the extended NoIHKH material model [1],[3], modified for large strain / large deformation cyclic plasticity of metals. In the stress update calculation [4] was used the Jaumann objective rate in the form of the Green-Naghdi objective rate, formulated with the aid of the rotated Cauchy stress tensor. The rotation tensor was expressed with the Rodriguez formula. The case was implemented into a finite element code using “proper” linearization, which means that no simplifications were used in a gradient or an element volume expression in the current configuration. The presented results represent some of the first outcomes of the fully coupled thermal structural analysis utilizing the new energy conservation equation with large strain/ large deformation formulation, which the authors consider to be positive. The new energy conservation equation can still be improved by introducing a heat source in it to take into account the amount of energy dissipated into heat during plastic deformation, but to propose a mathematical formula for the heat source is rather an experimental problem than a mathematical one. If the new energy conservation equation proves to be experimentally correct, in the future more complex problems will be able to be solved, mainly in the area of fast/ultra fast thermoelasticity or thermoplasticity.

References [1] L. Écsi, Extended NoIHKH model usage for cyclic plasticity of metals, In proceedings of the 7th. International Conference on Applied Mechanics (CD), Hrotovice, Czech Republic, 29.March1.April, 2005. [2] L. Écsi, Numerical behavior of a solid body under various mechanical loads using finite element method with new energy balance equation for fully coupled thermal structural analysis, In proceedings of the 6th. International Congress on Thermal Stresses, Vienna, Austria, 26-29. May, 2005, Vol. II, pp. 543-546. [3] J. Lemaitre, Handbook of Materials Behavior Models 1, Academic Press, NY, 2000 [4] E.A. Souza, D. Peric, D.R.J. Owen, Computational Plasticity, Small and Large Strain Finite Element Analysis of Elastic and Inelastic Solids, Swansea, to be published

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Mathematical aspects of the initial-boundary value problems in nonlinear thermoelasticity of simple and non-simple materials J.A. Gawinecki Institute of Mathematics and Cryptology, Faculty of Cybernetics Military University of Technology Str. Kaliskiego 2, 00-908 Warsaw, Poland [email protected] ABSTRACT In this lecture we report on diffrent coupled thermoelastic systems. They are models for the description of elastic heat conductive media. We will consider a coupled second-order hyperbolic system describing the thermoelasticity of simple materials and a coupled parabolichyperbolic systems of thermoelasticity of non-simple materials. It is known that the classical thermoelasticity theory (hyperbolic-parabolic) leads to a parabolic diffrential equation for the temperature distribution in rigid heat conductors. This implies that thermal perturbations are felt instantaneously in every part of the body. Although, at first sight, this outcome seems to contradict the physical intuition, it can be justified by resorting to the fact that molecular motion, which places a crucial part in transport phenomena, is very rapid except at extremely low temperatures. Hencen finite velocity of propagation for thermal perturbations is usually non-observable unless experiments are performed in some neighborhood of absolute zero as in the case of liquid helium. In fact, thermal waves, commonly known as second sound, are detected in some metals cooled approximately down to 20◦ K. In our lecture we consider the theory of thermoelasticity by considering the temperature-ratedependence and assigning an appropriate constitutive function to the entropy flux. Such a theory leads to a hyperbolic differential equation for thermal perturbations different from the equation describing propagation of thermal perturbation in classical thermoelasticity which is parabolic one. We consider also the nonlinear hyperbolic-parabolic system of coupled partial differential equation of fourth order describing the thermoelasticity of non-simple materials. We are interested in both the lineralized and the non-linear systems looking for the description of the asymptotic behavior of smooth solutions, for smoothing effects of the systems and specifically for the non-linear systems for the global existence in time of solutions. For hyperbolic systems it is well known that in many cases locally existing smooth solutions tend to develop singularities in finite time. The basic problem for the system in question here is whatever and in which way the added damping by heat conduction or viscosity will assure the global existence of solutions. From this point of view we investigate the global existence in time for large data in nonlinear thermoviscoelasticity.

References 1. 2.

J. A. Gawinecki, Global solutions to initial value problems in nonlinear hyperbolic thermoelasticity. Dissertationes Math. 344, 1–51, Warsaw Poland. J. A. Gawinecki. Global existence of solution for non-small data to non-linear spherically symmetric thermoviscoelasticity, Math. Met. Appl. Sci. 26, 970–936, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Effect of Parameter Uncertainties on a Vibro-Acoustic Design M. Guerich, S. Ben Chaabane Pôle Universitaire Léonard de Vinci ESILV – DER MS 92916 Paris la Défense Cedex {Francemohamed.guerich, samir.benchaabane}@devinci.fr

ABSTRACT The prediction of vibro-acoustic behaviour of structures is of the greatest interest for the design of less noisy machines. In the low frequency (LF) range, using finite element method (FEM) and/or boundary element method (BEM) in general insure such predictions. In the case where the FEM is used, in general this corresponds to an internal vibro-acoustic coupling problem, the vibro-acoustic model is constituted of structural and acoustical models. The structural model is performed using finite element of plates and/or shells and the acoustical model is constituted of volumic finite elements. When an internal vibro-acoustic coupling problem is studied, the modal method is in general used. It consists of the computation of structural and acoustical modes, which will be used as a basis of the displacement for the structure and of the pressure for the fluid. These modes depend on mechanical properties of the structure and the cavity and on geometric dimensions. Moreover, to predict the vibro-acoustic behaviour of the coupled system, appropriate caseload and cinematic boundary conditions must be considered. Thus, the vibro-acoustic model involves in general, a big number of parameters. These parameters can be given with some uncertainties.The purpose of this paper is to measure the effect of these uncertainties on the vibro-acoustic behaviour of the whole system. With such study the reliability of the vibro-acoustic system can be evaluated. To illustrate these effects, we have considered a simple case of vibro-acoustic response of a coupled plate-cavity to a harmonic mechanical force in a LF range.Some of the design parameters are considered as uncertain. These uncertainties are represented by Gaussian distributions of the parameters.A large sample of design points is defined and the responses are evaluated in these design points. The distribution of the response is then extracted and its coefficient of variation (COV) is compared to those of the parameters. Failure criteria (pressure level on the cavity and/or cinematic energy of the structure) can be defined to estimate the failure probability of the design.

References [1] Morand J.P., Ohayon R., Fluid structure interaction, Masson, Paris, 1992. [2] Lesueur C., Rayonnement acoustique des structures (in French), Eyrolles, Paris, 1988. [3] Guerich M., Hamdi M.A, A Numerical Method for Vibro-Acoustic Problems with Incompatible Finite Element Meshes Using B-Spline Functions, JASA, 105 (3), 1999, pp 1682-1694. [4] Melchers R. E., Structural Reliability Analysis And Prediction, Jhon Wiley & Sons Chichester, 1999 [5] ", S. Bouabdallah, S. Ben Chaabane, S. Missoum, Analyse fiabiliste du procédé de mise en forme des tôles minces, 7ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES, Giens 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A numerical method for solid-liquid interaction Goodarz Khodabakhshi, Vahid Nassehi, Leila Shojai, Richard J.Wakeman Chemical Engineering Department Loughborough University, Loughborough Leicestershire, LE11 3TU, United Kingdom {g.khodabakhshi, v.nassehi, l.shojai, r.j.wakeman}@lboro.ac.uk

ABSTRACT This paper deals with the mathematical modelling of coupled fluid flow and solid deformation problems. A novel mathematical technique for linking of the two sets of governing equations in a single model has been proposed. Results obtained by this technique using a range of power-law index for fluid flow simulation and elasticity modulus for the solid displacement are presented and discussed . Changing the rheological behaviour of the fluid has a significant effect on the deformation of the solid. These results are found to be self consistent and as expected from a theoretical point of view.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Computational Simulation of Irreversible Deforming and Fracture of Damageable Solids and Structures Alexey B. Kiselev, Olga V. Nekhaeva, Anton V. Privalsky Mechanics and Mathematics Faculty of Moscow M.V. Lomonosov State University Leninskie Gory, MSU, Main Bldg, Moscow 119992, Russia [email protected]

ABSTRACT Thermomechanical processes, which proceed in deformable solids under intensive dynamic loading, consist of mechanical, thermal and structural ones, which correlate themselves. The structural processes involve the formation, motion and interaction of defects in metallic crystals, phase transitions, the breaking of bonds between molecules in polymers, the accumulation of microstructural damages (pores, cracks), etc. Irreversible deformations, zones of adiabatic shear and microfractures are caused by these processes. Dynamic fracture is a complicated multistage process including an appearance, evolution and confluence of microdefects and a formation of embryonic microcracks, pores, their grow up to the break-up of a bodies with division into separate parts. The present paper include new results in the next scopes: 1) development the thermodynamically correct mathematical models of damageable thermoelastoviscoplastic medium (microfracture); 2) development the methods for determination of “nonstandart” constants of medium models, connected with microfracture of material; 3) numerical simulation of destruction (fragmentation) of constructions (macrofracture); 4) numerical investigation of some problems for damageable solids and structures (dynamical deforming and fracture of thick-walled cylindrical and spherical shells under explosion; dynamical deforming and fracture of thick-walled two-layer shell, filled with liquid, under impact and high velocity penetration; the problems of dynamic deforming and destruction of an oil-holding layer in gydraulic fracturing). Some of previously obtained results in consideration domains are published in the papers [1-5]. Russian Foundation for Basic Research (grants No. 06-01-00185 and No. 05-08-01435) and ISTC (grant No. 2992) are acknowledged for financial support.

References [1] A.B. Kiselev, Mathematical modeling of dynamical deformation and combined microfracture of a thermoelastoviscoplastic medium, Moscow Univ. Mech. Bull., 53, No. 6, 32-40, 1998. [2] A.B. Kiselev, A.A. Lukyanov and M. Thiercelin, Numerical simulation of dynamic propagation of curvilinear cracks of hydraulic fracturing, Moscow Univ. Mech. Bull., 59, No. 1, 36-41, 2004. [3] A.B. Kiselev and O.V. Nekhaeva, Numerial modeling of dynamical deformation and fracture of a thick-walled spherical shell, Moscow Univ. Mech. Bull., 59, No. 5, 53-58, 2004. [4] A.B. Kiselev and O.V. Nekhaeva, Numerial simulation of dynamical deformation and fracture of a thick-walled cylindrical shell, Moscow Univ. Mech. Bull., 60, No. 2, 33-37, 2005. [5] A.B. Kiselev and O.V. Nekhaeva, Mathematical modelling of dynamic processes of irreversible deforming, micro- and macrofracture of solids and structures, 11th Int. Conference on Fracture (Turin, Italy, March 20-25, 2005), Abstract Book, Turin, CCI, 228 (Proc. on CD-ROM, 6 p), 2005.

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Solution of the coupled light-mechanical problems Miroslav Kropáþ*, Justín Murín† * Visteon-Autopal, s.r.o. Nový Jiþín Lužická 14, 74101 Nový Jiþín, Czech Republik [email protected] †

Department of Mechanics, Faculty of Electrical Engineering and Information Technology, Slovak University of Technology Ilkoviþova 3, 812 19 Bratislava, Slovak Republik [email protected]

ABSTRACT

Dynamic optical systems (e.g. automotive headlights) provide the changes of their optical parameters, on the one hand by the motion of the whole lighting unite or on the other hand by the rigid motion of the individual elements of the optical system [1]. Our principle of reflector area controlled deformation is a new point of view in a required light beam modification of headlights during the car drive. It is necessary to solve coupled light-mechanical task to examine the influence of reflector area deformation on the quality of beam pattern by numerical methods. Light-mechanical task of weak coupled field is solved by the sequential method. In the first process step the deformation analyses of the reflector is made by the FEM method [2], following by the light analyses. In the second step the beam pattern of the optical system is under evaluation. Searching algorithm has been developed for the lighting field solving on the base of the backward ray tracing algorithm, which analyzes optical system and seeks out optical system elements orienting light rays into the examining parts of the illuminated half-space. Light performance of optical system is qualified as the beam pattern on the plane in the specific distance out of optical midpoint that represents detector. For the coupling between active optical system elements and detector nodes, the finite elements of light beam have been developed as well as the alternative direct computation. For the documentation of above-mentioned process we solved a bilinear optical system consist of a light source, a reflector and a detector. The response of the optical system has been sought (the curve of the illuminance spreading on the detector) on the controlled deformation of the reflector curve. The numerical simulation outcomes has been verified with the experiment measurements of the optical parameters of recall optical system.

References [1] M. Kropáþ, J. Murín, Design and analysis of the automotive headlights. ýasopis pre elektrotechniku a energetiku. 11, 89-92, 2005. [2] ANSYS 8.1 - Theory manual, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Simulation of the ferroelectric hysteresis using a hybrid finite element formulation I. Kurzh¨ofer∗ , J. Schr¨oder∗ , H. Romanowski∗ ∗

Institute of Mechanics, Department of Civil Engineering, University of Duisburg-Essen Universit¨atsstr. 15, 45117 Essen, Germany [email protected] ABSTRACT

Ferroelectric materials can be found in a wide range of applications in smart materials, e.g. vibration reducing sensors or fuel injection systems. A special characteristic feature of these materials is the appearance of a spontaneous polarization in a certain temperature range. This polarization can be reversed by an applied electric field of sufficient magnitude. The resulting nonlinear material behavior is expressed by characteristic dielectric and butterfly hysteresis loops. These effects are correlated to the structure of the crystal and especially to the axis of the spontaneous polarization. In the present work we present an electric hybrid element formulation where stresses and electric field are derived from constitutive relations. Therefore, we consider the electric displacement as additional degree of freedom, as presented in [1]. Furthermore, the finite element problem is condensed in a manner that the formulation is suitable for conventional boundary problems where the electric field is computed by the negative derivative of the basic field variable φ, used in the finite element approximation. The anisotropic material behavior is modeled within a coordinate-invariant formulation for an assumed transversely isotropic material, see [2] and [3]. The anisotropic response of the material is governed by isotropic tensor functions which depend on a finite set of invariants. The nonlinearities of ferroelectric materials are considered by the regard of internal variables which are governed by evolution equations. The key assumption of this procedure is the split of the strains and the electric displacement into reversible and remanent part. A switching surface is defined and a general mapping algorithm is applied to evaluate the remanent values at the actual timestep. The resulting hysteresis loops for a ferroelectric ceramic which are governed by one part of the Helmholtz free energy are discussed by considering a simple numerical example.

References [1] K. Ghandi & N.W. Hagood, A hybrid finite element model for phase transitions in nonlinear electro-mechanically coupled material. Smart Structures and Materials, Proceedings of SPIE, 3039, 97–112, 1997. [2] J. Schr¨oder & D. Gross, Invariant formulation of the electromechanical enthalpy function of transversely isotropic piezoelectric materials. Archive of Applied Mechanics, 73, 533–552, 2004. [3] H. Romanowski & J. Schr¨oder, Coordinate invariant modeling of the ferroelectric hysteresis within a thermodynamically consistent framework. A mesoscopic approach. Trends in Applications of Mathematics to Mechanics, Wang Y. & Hutter K. (eds.), Shaker Verlag, 419–428, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A Two-Phase Numerical Modelling of the Liquid Solid Transition in Polymer Processing R. Lanrivain†, L. Silva, T. Coupez CEMEF, Ecole des Mines de Paris 1 rue Claude Daunesse, 06904 Sophia Antipolis, FRANCE †[email protected] ABSTRACT In this paper, we present a two-phase model. It allows the numerical computation, during processing, of a polymer’s mechanical behavior at the liquid state, during the transition, and at the solid state. This work’s context is an industrial project aiming the accurate modelling of the injection molding process [1], [2]. It focuses on the 3D numerical simulation of the different stages encountered in this process : filling, packing, cooling and ejection. During processing, anisotropy of the stress state build-up affects its mechanical, optical or dimensional properties, and induces warpage once the part is ejected. In order to predict residual deformations in the mould, and after ejection, the whole process is simulated with a two-phase thermo-mechanical model, which includes dynamics of phase change. Thus, interactions between polymer in flow and solidified one are implicitly taken into account in the coupled mechanical formulation. Based on the two-phase flow theory [3], a diffuse interface model is derived. A phase field quantifies the presence of each phase at each point of the computational domain. This parameter is obtained by solving an evolution equation of the hyperbolic type, that can be function of temperature or strain. Different behavior laws can be associated to the liquid (like viscous or viscoplastic) and to the solid (like elastic or hyperelastic). So each phase is described by its specifical kinematic variables (velocity for the fluid, displacement for the solidified part). Furthermore, the system of conservation equations (mass and momentum of each phase) is closed by momentum and mass coupling relations (for example, friction type). The kinematics variables for both liquid and solid phase (velocity, liquid pressure, displacement, and solid pressure) are calculated from a single strongly coupled system (monolythic approach) by using the mixed finite elements method within an eulerian framework. Due to the large number of unknowns, parallel computation is obviously required. Validation tests of the model’s implementation are shown, using literature benchmark examples. Results obtained in 3D complex industrial parts underline the robustness and the efficiency of our model.

References [1] T. Coupez,C. Gruau, C. Pequet and J. Bruchon, Metric Map and Anisotropic Mesh Adaptation for Static and Moving Surfaces. The Sixth WCCM, Beijing, China, 2004. [2] L. Silva, R. Valette, and T. Coupez. Viscoelastic compressible modelling of 3d filling and postfilling of complex industrial parts. In V. Brucato, editor, The 6th Int. ESAFORM Conf. Mat. Form., 19-22. University of Salerno, 2003. [3] D.A. Drew, Mathematical modelling of two phase flow. Annu. Rev. Fluid Mech. 15, 261–291, 1983.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A two-phase model for granular flows applied to avalanches Caroline Leppert, Dieter Dinkler Institut f¨ur Statik Technische Univerist¨at Braunschweig, Germany [email protected] ABSTRACT Understanding of granular materials under rapid motion is of importance to many phenomena in nature and industrial applications. During landslides and avalanches as well as in mixing and storing processes as in silos granular materials undergo phase transitions from solid to fluid state of matter. At rest, the material is characterised by dilatant elasto-plastic behaviour that is rate independent. For low densities and high shear-rates fluid properties dominate. A continuum mechanical approach for dense granular flow including both aspects is presented. Therefore, frictional stresses, representing the rateindependent aspects, extend the viscous stresses using the generalised visco-plastic constitutive model with Coulomb friction as proposed by Chen and Ling [1]. The differentiation between solid and fluid state of matter results from a comparison between current shear stress and plastic yield criterion. The model is able to represent many experimental results for moderately fast granular flows introducing an artificial viscosity that includes internal friction and depends on the current strain rate. To simulate the flow of granular material that travels large distances the material is described by the Navier-Stokes-equations. This requires the formulation of all balance equations for momentum and mass and the constitutive equations using velocity variables. The space-time finite element method is applied to discretise the weak form of balance and constitutive equations. This allows the monolithic coupling between the flowing material and any possible surrounding structure [3]. The method is extended by the level set technique [4] to model the motion of the free surface between granular material and air in order to represent landslides and the complete discharge of silos

References [1] C. Chen and C. Ling, Granular-flow rheology: role of shear-rate number in transition regime, J. Engng. Mech., Vol. 122, 469–480, 1996 [2] S.B. Savage, Analyses of slow high-concentration flows of granular materials. J. Fluid Mech., Vol. 377, 1–26, 1998 [3] B. Hubner ¨ , E. Walhorn and D. Dinkler, A monolithic approach for fluid-structure interaction with space-time finite elements Comp. Meth. Appl. Mech. Eng., Vol. 193/23, 2069–2086, 2004 [4] S. Osher and R. Fedkiw, Level set methods: An overview and some recent results. J. Comp. Phys., Vol. 79, 463–502, 2001

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Parallel 3D Finite Element Analysis of Coupled Problems Lee Margetts *, Ian M. Smith† and Joanna M. Leng* * Manchester Computing, University of Manchester Kilburn Building, Oxford Road, Manchester M13 9PL [email protected] [email protected] †

School of Mechanical, Aeronautical and Civil Engineering, University of Manchester, P.O. Box 88, Manchester M60 1QD [email protected]

ABSTRACT Steady advances in computer power have enabled researchers to consider tackling increasingly complex problems. In the academic community, current focus is on multiscale modelling and multiphysics. The aim is for simulation to be more realistically representative of real world processes. This paper considers the simulation of coupled problems involving more than one physical process, multiphysics. In particular, the authors present some ideas and experiences regarding the use of the finite element method and parallel computers to solve 3D coupled problems. In the literature, two main approaches have been used to solve coupled problems. These are sometimes referred to as (i) fully coupled modelling and (ii) un-coupled multi-physics. Both methods have their advantages and disadvantages. In the paper, the authors discuss some of the issues that should be considered when selecting a particular strategy, to ensure computational efficiency. Particular attention is given to an example from the field of magnetohydrodynamics: three dimensional steady state flow in a perfectly insulated rectangular duct. The magnetohydrodynamics example involves solving a system in which both magnetic and hydrodynamic forces influence the behaviour of the fluid. Visualisation of the problem, using streamlines to represent fluid flow (Figure 1) shows that a three dimensional representation is essential to capture the full complexity of the flow. A fully coupled solution strategy is presented in which the full system is represented by a single “stiffness” matrix and solved by a single computer program. A parallel implementation of an element-by-element variant of BiCGStab(l) is used to solve the equations, demonstrating the efficient use of up to 128 processors.

slow

fast

Figure 1 Velocity Streamlines from Three Different Viewpoints Along a Rectangular Duct

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Numerical Simulation of Rubber Curing Process with Application to Bladders Manufacture Paulo Porta † ∗, Carlos Vega† † DARMEX S.A.C.I.F.I. Luis Mar´ıa Drago 1555 – B1852LGS Burzaco – Argentina [email protected] [email protected] ∗ Corresponding author

ABSTRACT A large number of polymer products are formed into their final shape by polymerization in situ. In particular, mold curing process is the final step in many rubber products manufacturing and determines both the quality of the resulting product as well as productions costs. During this process, important changes in the mechanical properties –e.g. viscosity and modulus– take place, changes which are generally hard to be experimentally characterised. In view of this, a mathematical model is proposed for rubber vulcanisation molding, its strategical value being two fold: from production standpoint, its ability to predict optimal production parameters –the optimal curing time being the most important– and from quality assessment perspective, its capacity of predicting molded part properties. Following the literature –see, e.g. [1]– the mathematical model is built from general mass–energy conservation principles. A series of plausible hypothesis are made in order to simplify the model, which results in a combination of i) the unsteady Fourier’s heat conduction equation –(1)– with a distributed internal heat source, resulting from the reaction –(2), ii) the reaction rate equation –(3)– and iii) closure (γ) constitutive equations, for the kinetic constants of the process, namely KC and tinc : ∂Θ ∂t

= div (k · gradΘ) +

q  = Hr dC dt

q ρcp

in Ω for t > 0

dC dt

(γ)

= KC (1 − C)γ

(1) (2)

for t > tinc

(3)

An algorithm is proposed to solve this coupled system. A coupled ODE–implicit in time finite element approximation is proposed and implemented under ALBERTA ([2]). After calibration of the computational model, the complete temperature and cross-links concentration is obtained for the typical bladder geometry. Performance results as well as a short discussion on error estimator behaviour are also presented.

References [1] In-Su Han and Chang-Bock Chung and Hyoeng-Gwan Jeong and Sung-Ju Kang and Seung-Jai Kim, Optimal cure steps for product quality in a tire curing process. Journal of Applied Polymer Science, 74, 2063–2071, 1999. [2] Schmidt, A. and Siebert, K., Design of adaptive finite element software, Springer, Berlin, 2005.

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Coupled Finite Element Analysis of Composite Laser Rods Thermal Characteristics under Longitudinal Diode Pumping E. Stupak*, R. Kaþianauskas*, A. S. Dementjev†, A. Jovaiša† *Department of Strength of Materials, Vilnius Gediminas Technical University, Saulơtekio av. 11, LT10223, Vilnius-40, Lithuania [email protected], [email protected] †

Nonlinear Optics and Spectroscopy Laboratory, Institute of Physics, Savanoriǐ av. 231, LT02300, Vilnius, Lithuania [email protected], [email protected]

ABSTRACT Diode laser pumped solid-state lasers are covering a wide range of applications. The longitudinal pumping geometry of laser rod is particularly favorable as it provides a high degree of spatial overlap between pump and lasing modes. However, high pump-power densities are required to achieve sufficient inversion in the laser material. This produces high thermal loading in laser crystals, which, in turn, leads to undesirable thermal effects, such as temperature-dependent index change, temperature-dependent stresses, and end-surface deformation. All those effects are qualified as thermal lensing in laser material. Yttrium aluminum garnet (YAG) crystals doped with Nd, Er, Yb and other ions are currently the most popular laser crystals for diode laser pumped solid-state lasers, especially in end-pumped configurations, that is why laser rods with YAG host material are chosen as the objects of modelling. A multiphysics approach and finite-element method are used to the numerical simulation of thermal lensing in such rods. Thermo-mechanical behavior is considered by applying the ANSYS software, while particular pre-processor for generation of the heat source as well as postprocessor for evaluation of the optical path difference (OPD) is developed. The difference between standard crystal blocks and composite structures with undoped end caps is investigated. The results show that OPD for one pass through the crystal in all cases is practically the same. Thus, the composite rod geometry does not reduce significantly thermal lensing, however temperature gradients are considerable smaller. The obtained results also show that the use of the composite AE practically does not decrease the thermally induced aberrations of spherical type. Therefore, the quality of the probe or laser beam will change mainly in the same way in both the conventional and the composite AE pumped with Gaussian beams.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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On the Modeling of Nonplanar Shear Walls in Shear Wall - Frame Building Structures Tolga Akis1, Turgut Tokdemir2, Cetin Yilmaz3 1 Department of Civil Engineering Atılım University, Ankara, øncek 06836, Turkey [email protected] 2 Department of Engineering Sciences Middle East Technical University, Ankara 06531, Turkey [email protected] 3

Department of Civil Engineering Middle East Technical University, Ankara 06531, Turkey [email protected]

ABSTRACT The objective of this study is to model the non-planar shear walls of asymmetric shear wall-frame building structures in elastic range. The modeling work is based on open and closed sections shear wall assemblies for which two different three-dimensional models are developed and verified in comparison to common shear wall modeling techniques. Two-dimensional modeling of symmetric building structures having planar shear walls may be a practical method. However, especially for an asymmetric shear wall-frame building system that contains non-planar shear wall assemblies, the structural system should be modeled in three dimensions. In addition, the three dimensional behavior of the shear wall assemblies should also be taken into consideration. The proposed models are based on conventional wide column analogy, in which a planar shear wall is replaced by an idealized frame structure consisting of a column and rigid beams located at floor levels. For open section shear walls, the connections of the rigid beams are released against torsion in the model. For modeling closed section shear walls, in addition to this the torsional stiffness of the wide columns are adjusted by using a series of equations. In the modeling studies, the rigid diaphragm floor assumption is also taken into consideration. As an example, plan view of an open section U shaped shear wall assembly and the corresponding model is given in Fig. 1. The main goals of the models that are given in this study are to reduce the required time and capacity for the threedimensional analysis of shear wall-frame building systems. In the verification studies, two single shear wall assemblies (an open and a closed section) and a six storey shear wall-frame building system are considered. Static lateral load analysis and dynamic analysis are performed on these structures where the proposed models are used. The results of these analyses are compared with the results obtained by using common shear wall modeling techniques and it is observed that both results agree well with each other.

Figure 1. U-Shaped Shear Wall Section and Proposed Model

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Directional Drillstring Dynamics Fredy Coral Alamo∗ , Hans Ingo Weber∗ and Harry Saavedra Espinoza† ∗ Pontif´ıcia Universidade Cat´olica do Rio de Janeiro Rua Marquˆes de S˜ao Vicente 225, 22453-900, Rio de Janeiro - RJ - Brazil {fjcoral,hans}@mec.puc-rio.br † Universidad Nacional de Ingenieria

Av. Tupac Amaru S/N, Lima - Peru [email protected]

ABSTRACT A rotating rod under load may execute vibration in different ways: transversal (bending), longitudinal (axial), torsional, or a combination of any of those. In this article, the dynamics of a rotating directional drilling system, constrained to rotate in a borehole, is investigated using the finite element formulation. To study the behavior of the system, a rotating 3D Cosserat rod element is used, this is a newly non linear element developed in this work. The equations of coupled bending, axial and torsional motion of the rotating elastic rod element is derived using the Cosserat rod theory. In general, for slender structures, the shear deformation can be neglected, consequently, to model the drillstring the Bernoulli hypothesis is considered and the shear deformations are neglected. The finite rod element developed has 12 degrees of freedom and takes into account all the geometric nonlinearities. Explicit expressions for the element mass, gyroscopic, stiffness, and non linear terms are derived using Hamiltons principle. Moreover, the finite element discretization is employed and numerical solutions are obtained for the nonlinear drillstring dynamics. Overall, the Cosserat model provides an accurate way of modelling long slender rods.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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MVM Energy Method for Buckling Analysis of Tapered Plates M. M. Alinia1, A. R. Rahai2, and S. Kazemi1 1,2

Amirkabir University of Technology 424 Hafez Ave. Tehran 15875-4413, Iran {m.alinia, a_rahai}@aut.ac.ir

ABSTRACT Tapered plates are being increasingly used in modern engineering structures. The increasing use is due to the distributed flexural stiffness that helps reduce the weight of structural elements and improve the utilization of the material. The flexural stiffness, vibrational, and buckling capacities of these plates may be significantly increased by appropriate tapering. On the other hand, provision of openings in perforated plates which provide access for inspection, services, and maintenance, can greatly enhance the applicability of these members in many structures such as platforms, naval and aeronautical structures. Beside these advantages, openings result in a reduction of the buckling capacity, which should be taken into account. n this paper a new exact solution procedure using energy method based on modified vibrational mode shapes is formulated for the buckling analysis of simply supported rectangular plates of abruptly varying stiffness subjected to uniform edge stresses. It is shown that the vibrational mode shapes of a tapered plate is in fact a linear combination of various mode shapes of intact plates. This phenomenon is used to estimate the vibrational mode shapes of taper plates, and is then incorporated to evaluate the buckling loads.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Considerations on Advanced Analysis of Steel Portal Frames Arthur R. Alvarenga and Ricardo A. M. Silveira Department. of Civil Engineering of School of Mines – Federal University of Ouro Preto – UFOP Campus Universitário, Morro do Cruzeiro, s/n, 35400-000 Ouro Preto – MG – Brazil [email protected], [email protected]

ABSTRACT This paper presents a little study about the necessary steps to qualify a second- order inelastic analysis as advanced one. Plastic-zone approach is applied to steel plane frames (portals) and the numerical formulation is based on finite element model of a Bernoulli-Euler beam-column member using the called “slice technique”. This element is set on a Lagrangian updated co-rotational system. The nonlinear problem is solved using Newton-Raphson iterative strategy and a new axial force iterative integration is shown. This process was implemented on a computer program PPLANAV* and the minimum requirements of advanced analysis (initial geometrical imperfections and residual stress) are automatically generated. Two examples show good agreement with other researcher’s answers, but there’s a great elapsed computing time.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A Discrete Element Model For The Fracture Analysis of Reinforced Concrete N. Monteiro Azevedo*, J. V. de Lemos† ,J. R. de Almeida+ * UNL-FCT 2829-516 Monte de Caparica [email protected] † LNEC 1700-066 Lisboa [email protected] + UNL-FCT 2829-516 Monte de Caparica [email protected]

ABSTRACT The Discrete Element Method was initially applied to the analysis of discontinuous media, e.g. in rock mechanics and soil mechanics. Recently the DEM has been used in fracture studies of non-homogeneous continuous media such as concrete and rock. A 2D circular rigid discrete element formulation based on the DEM that has been further developed to model concrete is adopted [1]. To simulate the concrete at the meso-level, random assemblies of particles based on a given sieve analysis have to be generated. The DEM model micro-properties also have to be previously calibrated through uniaxial tension and compression tests. The formulation of a 1D rigid discrete element that interacts with the discrete rigid particles through contact interfaces is presented. The DEM enhanced model with reinforcement capabilities is evaluated in a three point bending [2] and in a four point bending [3] tests experiments of reinforced concrete beams without stirrups. Under flexure loading conditions the model is shown to predict the expected final crack pattern, the crack propagation and the load displacement diagram. Under shear loading conditions the model is shown to predict the size effect behaviour that occurs on beams without stirrups failing under diagonal shear and also the expected crack propagation and final crack patterns.

References [1] N. Monteiro Azevedo, A rigid particle discrete element model for the fracture analysis of plain and reinforced concrete, PhD thesis, Heriot-Watt University, Scotland, 2003. [2] C. Bosco, A. Carpinteri and P. Debenardi, Minimum reinforcement in high strength concrete, Journal of Structural Engineering, 116, 427-437, 1990. [3] Z.P. Bazant and M. T. Kazemi, Size effect on Diagonal shear failure of Beams Without Stirrups, ACI-Structural Journal, 88, 268-276, 1991.

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Analytical Criteria for the Evaluation of the Internal Forces at the Elastic and Plastic Limit States of Lozenge and Triangular Cross-Sections António M. Baptista Laboratório Nacional de Engenharia Civil (LNEC) Av. Brasil 101 - 1700-066 Lisboa - Portugal [email protected]

ABSTRACT The elastic-plastic methods for the design of steel structures have been introduced in some national codes, in Europe and America, for almost half a century. This innovation has resulted from the recognition of these design methods for a better estimation of the ultimate resistance of some types of steel structures. These methods are often based on some hypotheses, such as the formation of plastic hinges in the most stressed cross-sections. The development of these plastic hinges is affected by the interaction between the axial force and the bending moment acting on the cross-section. The interaction criteria between these internal forces depend on the cross-section shape. Therefore, specific analytical criteria are required for each type of cross-sections, and for each combination of bending moments over the two main axis of inertia of the cross section (in the case of bi-axial bending). However, these criteria are not usually available in the design codes or text books for most of the cross-section shapes, especially for those that are less common in steel construction. Even in the case of the most usual cross-sections, the analytical criteria are often defined by means of simplified expressions. In the meanwhile, the use of new shapes is becoming more frequent, due to bold innovative structural solutions conceived by modern architects. This paper presents a set of analytical criteria for the evaluation of the internal forces (axial force and bending moment) in lozenge and triangular steel cross-sections, at their elastic and plastic limit states. They are written in a non-dimensional form, which makes them independent of the cross-sections dimensions and of their width-to-height ratio. These criteria are based on the hypothesis of a full yielding of the cross-sections at their plastic limit state. The material is supposed to present elastic-plastic behaviour without hardening. The effects of eventual shear or torsion deformations on the cross section are supposed to be negligible. These analytical criteria put in evidence the different behaviour of symmetrical and non-symmetrical cross-sections, regarding their axis of bending, at their elastic and plastic limit states. They constitute a basis for the elaboration of more complex analytical criteria, for hollow lozenge and triangular cross-sections, or for rectangular full or hollow cross-sections submitted to axial forces and bi-axial bending. A worked example shows an application of these criteria to the evaluation of the elastic and plastic limit states in a square section submitted to an axial force and bi-axial bending. Another worked example presents the expressions for the evaluation of the maximum eccentricities of an axial force in a triangular section, at its elastic and plastic limit states.

References [1] Ch. Massonnet, M.Save, Calcul Plastique des Constructions. Vol 1 – Structures dépendant d’un Paramètre. Ed. B. Nelissen, 3rd Edition, 1976.

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Comparative Study of Aluminum Alloy Plate 2024/7050 Under the Effect of Internal Damping M. Benachour *, A. Hadjoui*, M. Benguediab**, N. Benachour***, F. Hadjoui* *Automatic Laboratory of Tlemcen / Faculty of Engineering Sciences / UABB - Tlemcen BP 230 – Tlemcen 13000 – Algeria {mbenachour_99, hadjoui_ab, hadjoui_fethi}@yahoo.fr ** University of Sidi Bel Abbes / Faculty of Engineering Sciences [email protected] *** Faculty of Sciences / Physical Department / UABB - Tlemcen [email protected]

ABSTRACT In the present work we determine the influence of structural damping, of an aluminum alloy on the dynamic behavior of a plate with various boundary conditions. Structural damping (loss factor energy) is a very significant parameter by its influence on the dynamic behavior of the mechanical structures. The material of the studied plate is the aluminum alloy 2024 T3 and 7050 T7351. The behavior vibratory of the plate is studied by the finite element method and a comparative study is presented for theses materials. We present also the influence of the loss factor energy on the variation of the inertance relating to each node and each degree of freedom. In parallel, we study the influence of the geometrical parameters of the structure in the frequency domain. In addition, we identify the excited mode of vibration where a comparative example for steel is presented in order to show the excited mode of vibration. The effect of the internal damping is significant in resonance peak, where the reduction of the vibratory amplitude (inertance) is significant. The increase of the geometrical parameters (thickness, ratio length/width), decreases the vibratory amplitude, and on the other hand shift the peaks of resonance towards the high frequencies. The boundary conditions have a great influence on the rotational inertances in a resonance peak.

References [1] M. I. Friswell, The direct updating of damping and stiffness matrices. AIAA Journal, 36, n° 3, 491-493, March 1998. [2] J. C SNOWDON, Vibration and shock in damped mechanical. John Wiley and Sons Inc, 1968. [3] R. D. Mindlin, H. Deresiewics and Schacknow, Flexural vibrations of rectangular plates. Int. J. Eng. Sc., 07, 99-113, 1969. [4] C.F BEARDS, Structural vibration analysis: modeling, analysis and damping for vibrating structures. Ellis Horwood limited, England 1983. [5] S. SRINIVAD, C. V. RAO and A. K. RAO, An exact analysis for vibration of simply supported homogenous and laminated thick rectangular plates. J. of Sound and Vibration, 12, 187-199, 1970. [6] A. HADJOUI, M. BENACHOUR, Application of the matrix connection to the study of the thick plates. Fourth Mechanical Congress, 13 – 16 April 1999, Mohammadia, Morocco. [7] R. MALY Joseph, A. BENDER Kirsten and C. PENDLETON Scott, Complex stiffness measurement of vibration damped structural elements. International Modal Analysis Conference, IMAC-XVIII, San Antonio, Texas, February 2000.

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Nonlinear Analysis of Space Frames António A. Correia*, Francisco B. E. Virtuoso† *

ICIST and Instituto Superior Técnico, DECivil Av. Rovisco Pais, 1049-001 Lisboa, Portugal [email protected]

†

ICIST and Instituto Superior Técnico, DECivil Av. Rovisco Pais, 1049-001 Lisboa, Portugal [email protected]

ABSTRACT This work presents a method developed and implemented to consider material and geometrical nonlinearities in the behaviour of space frames. A geometrically nonlinear formulation is developed in which the compatibility and equilibrium relations are established in the structure’s deformed configuration using a co-rotational description of the movement. The use of total lagrangian, updated lagrangian and co-rotational descriptions of the movement is discussed. An exact method is presented to consider three-dimensional finite rotations, which cannot be added like vectors as assumed in geometrically linear analysis. The finite rotations are considered by using Euler’s finite rotation formula. That formulation allows obtaining the correct relationships between the rotational degrees of freedom considering large displacements and rotations. The material nonlinear behaviour considered in this work is due to the use of general nonlinear axial stress-axial strain relationships. Based on these assumptions for the material behaviour, and by using an approximation for the internal forces field throughout the frame element, a material nonlinear formulation for three-dimensional structures is presented. The use of an approximation for the internal forces field instead of the usual approximation for the internal displacements field is discussed. An unusual method of integration over the cross-section is presented, where the integrations are carried out over its perimeter, thus allowing any geometrical polygonal shape for it. In this work, despite large displacements and rotations being allowed, only small deformations are considered. This allows the material and geometrical nonlinearities to be treated in a completely independent way. The incremental and iterative strategies used are discussed. The capabilities of the formulations presented here are exemplified by the analysis of a few benchmark problems.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Investigation of shear wall behavior with composite boundary elements Ali Davaran1, Armin Kashefi and Shahin Nayeri Amiri Department of Civil Engineering, Tabriz University Tabriz, Iran [email protected] {davaran, sh.nayeri}@tabrizu.ac.ir

ABSTRACT Composite construction in steel and concrete offers significant advantages for use as the primary lateral resistance system in building structures subjected to seismic loading. While composite beam and joist floor system have been common for over a half a century, through the use of over the past decade a substantial amount of researches have been conducted world wide on a wide range of composite lateral resistance systems. The appropriate behavior of composite structures and the economical considerations are resulted in offering a new detail of composite shear wall and evaluation of the behavior of this system. The non-linear static analysis has been used in this research. In the detail which is offered in this research steel plates are used to reinforce the boundary elements of concrete shear wall and these steel plates are connected with shear studs to concrete. The research results show that the behavior of this detail with lateral loading is appropriate and ductility of this system is favorable, but the economical features of this detail must be verified when it is necessary to reduce the size of the boundary elements and it is suggested for further researches

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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An investigation on dynamic behavior of shear walls on flexible foundation Dr. Mikail Yousefzadeh Fard1, Shahin Nayeri Amiri2and Armin Kashefi Department of Civil Engineering, University of Islamic Azad Tabriz, Iran {mikail,sh.nayeri}@tabrizu.ac.ir Department of Civil Engineering, Tabriz University Tabriz Iran [email protected]

ABSTRACT

Nowadays, shear walls are used as efficient structural systems for resisting against external lateral loads such as wind and earthquake loads. Regarding to this fact that the pervious studies on the behavior of shear wall systems were performed without considering the effect of foundation and interaction between soil and structure, it seems necessary to conduct some research in the evaluation of shear walls dynamic behavior rested on flexible foundation and with considering the effect of soilstructure interaction. The linear modeling of a shear wall rested on flexible foundation is conducted which is accompanied with the modeling of soil by Winkler `s springs under foundation. The dynamic behavior of shear wall system is assessed. The results obtained show that depending on types of soils under foundation, the values of displacements are more than displacements when the flexibility of foundation is ignored. Also the internal forces at the base of shear walls and the periods of the first three modes of the structure have significant differences between two cases. In one case the interaction of soil-structure is ignored and on the other one the interaction is considered.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Influence of the morphology of adhesive joining on the mechanical properties of periodic metal hollow-sphere-structures ∗,†, J. Gr´acio∗, † ¨ T. Fiedler∗,†, A. Ochsner ∗ Centre for Mechanical Technology and Automation, University of Aveiro, Aveiro, Portugal † Department of Mechanical Engineering, University of Aveiro, Portugal

tfi[email protected], [email protected], [email protected]

ABSTRACT Hollow-sphere-structures (HSS) originate a new group of cellular materials characterised by a high reproducibility of geometry and therefore mechanical properties. Well-known advantages of cellular materials are a high ability of energy adsorption, good damping, excellent heat insulation and a high specific stiffness. Combination of these properties opens a wide field of potential applications, e.g. automotive, aviation or space-industry. Essential Limiting factors for the utilisation of cellular materials are inconstant material parameters and relatively high production costs. Both factors can be reduced by the application of hollow-sphere-structures. A new powder metallurgy based manufacturing route enables the economic production of metallic hollow spheres of defined geometry. Different joining technologies such as sintering, soldering and adhering can be applied to assemble hollow spheres to interdependent structures. Adhering provides the most economic way of joining and allows for further cost reduction and therefore the expansion of the field of potential applications. Another important advantage it the possible utilisation of the mechanical behaviour and morphology of the adhesive layer as a further design parameter for the optimisation of the structure’s mechanical properties for specific applications. The influence of the morphology and mechanical properties of the adhesive layer is discussed in the scope of this article. Finite element (FE) analysis is performed for periodic structures and the results are compared for different geometries. Two different approaches for adhering are considered: hollow spheres completely embedded in a polymer matrix (syntactic foam) and alternatively, concentration of the adhesive layer in the contact points of the spheres (Partial-HSS). In contrast to earlier approaches (e.g. [1, 2]), the geometry is discretised based on regular hexahedron elements. This approach is much more time-consuming, but important in order to achieve more accurate simulation of nonlinear-behaviour (e.g. plasticity).

References [1] W.S. Sanders and L.J. Gibson, Mechanics of BCC and FCC hollow-sphere foams, Materials Science and Engineering A, 352, 150–161, 2003. [2] S. Gasser, F. Paun, A. Cayzeele and Y. Br´echet, Uniaxial tensile elastic properties of a regular stacking of brazed hollow spheres, Scripta Materialia, 48, 1617–1623, 2003.

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Dynamic behaviour of a composite twin girder bridge in a high speed interoperable line Helder Figueiredo*, Rui Calçada†, Raimundo Delgado† *

Faculdade de Engenharia da Universidade do Porto Rua Dr. Roberto Frias, s/n [email protected]

†

Faculdade de Engenharia da Universidade do Porto Rua Dr. Roberto Frias, s/n [email protected], [email protected]

ABSTRACT A high-speed railway system is presently under implementation in Portugal, which will allow the connection of the country with a large European network. At the moment, many countries have already developed their own structural solutions for dealing with the effects of high-speed trains in bridges and a great number of structures have been in operation for several years. These solutions were initially designed for specific train types, thus being checked for the dynamic effects of only a small part of the actual European high speed rail traffic. Recent advances in the understanding of the behaviour of high speed railway bridges have been introduced in the EN 1990-Annex A2 and EN 1991-2, reflecting the work undertaken by the ERRI committee D214. In the case of interoperable lines where the high speed TSI is applicable, this being the case of the future Portuguese high speed network, additional checks for dynamic analysis using High Speed Load Model should be performed. One of the solutions that has proven to be very competitive in France is the composite twin girder bridge. This type of deck is used in continuous schemes, with spans lengths ranging from 40m up to 65m. In this paper the dynamic behaviour of this type of bridge is assessed using as reference a 333m long composite twin girder deck located on the French TGV Nord line. The bridge is continuous over its entire length, comprising 7 intermediate spans of 40m and 2 end spans of 28m and 25m. Dynamic analyses of the bridge were performed for both the European high speed trains and the HSLM load schemes, using various types of FE models of increasing level of detail. The response of the bridge is checked in terms of structural safety (amplification levels and fatigue requirements), deck acceleration and passenger comfort.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A Finite Element Model for Beam to Column Bolted Connections with Semi Rigid Behaviour Foces A.*, Garrido, JA.†, Moreno A.†† *

ETS de Ingenieros Industriales, Universidad de Valladolid Paseo del Cauce, s/n, 47011 Valladolid [email protected]

†

ETS de Ingenieros Industriales, Universidad de Valladolid Paseo del Cauce, s/n, 47011 Valladolid [email protected] ††

Escuela Politécnica Superior. Universidade da Coruña. Campus de Esteiro, C/ Mendizábal, s/n, 15403 Ferrol [email protected]

ABSTRACT Nowadays it is recognized that connections and members of steel frameworks have a semi rigid and nonlinear behaviour. One of the main concerns is how to incorporate the connection characteristics into an analysis. In the present study, beam-to-column bolted end plate connections, widely used because of the economy and simplicity of fabrication and assembly, are investigated for predicting their rotational behaviour (moment-rotation curve) that can be used in the frame analysis. In order to predict the rotational behaviour of this type of connection, a three-dimensional finite element model has been developed by the COSMOS/M® code. The proposed model takes into consideration the interaction between the various components that are comprised in the connection. Thus, the modelling domain includes the beam, end plate, bolts and nuts and the column. The main novelty now presented consists of accounting for the flexibility of the column components in the analysis. The column is modeled using solid elements. Other authors employ shell elements or use a rigid surface representing the column flange. Besides, the analysis incorporates the effects of material nonlinearity, for the plates and bolts, using the elastic-perfecty plastic stress-strain relationship. The results obtained from the finite element analysis are evaluated and verified by comparing the numerically predicted results with those of the corresponding tests carried out. The numerical results are also compared with a simplified theoretical model based on yield line analysis and the stub-tee analogy. The 3D finite element model presented in this study can be used to generalize the rotational behaviour of the end plate connection through more extensive parametric studies in order to take into consideration the connection flexibility and its effect on the performance of steel frames. So, extension of the methodology now presented may involve an improvement in analysis and design procedures of the proposed type of connection according to modern design codes.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Rational strain measures - The implicit corotational method G. Garcea, A. Madeo Dipartimento di Strutture, Universit´a della Calabria, 87030 Arcavacata di Rende (Cosenza), Italy [email protected], [email protected] ABSTRACT The nonlinear analysis of slender structures undergoing large displacements–small strains requires a proper description of the kinematic strain– displacement relationship to define a rational strain measure. In particular the strain measure must be unaffected by finite rigid body motions, that is it must be objective. Rational strain measures for structural models are not usually available or are too complex to be used in a FEM context, while the technical models which are usually adopted do not satisfy the requirements for objectivity. A simple method, called implicit corotational, to obtain rational strain measures is proposed. The starting point is the corotational idea [1] which is now, following Biot [2], applied at the continuum level. The body is thought of as subdivided in finite parts, each one with a corotational frame that follows the rigid motion of the part. In the corotational frame a linearized kinematic and a linear strain measure may be used in this way. The strain measures become then more accurate with a reduction in the part size. Furthermore it becomes exact by an appropriate limit process. The final results is a rational strain measure. As the displacement field in the corotational reference is infinitesimal, linearized kinematical models for beams, plates and shells can usefully be employed. Since linearized models are always available even for complex structural models (plates or shells) it is easy to obtain the corresponding rational strain measures using this approach. The correctness of the strain measures obtained for 3D beams and thin plates is clear when they are compared with those available in literature [3, 4]. Numerical analysis are performed using Koiter’s asymptotic approach [5]. In this context, which is particularly sensitive to the exactness of the strain measures , the accuracy of the results can be seen.

References [1] B. Nour-Omid, C.C.Rankin, Finite rotation analysis and consistent linearization using projectors. Computer Methods in Applied Mechanics and Engineering, 93, 353-384, 1991. [2] Maurice A.Biot, Mechanics of Incremental Deformations. J. Wiley & Sons, New-York, 1995. [3] Stuart S.Antman, Nonlinear Problems of Elasticity. Springer-Verlag, New-York, 1995. [4] J.C. Simo, L.Vu-Quoc, A three-dimensional finite-strain rod model. Part II: Computational aspects. Computer Methods in Applied Mechanics and Engineering, 58, 79-116, 1986. [5] W.T.Koiter, On the stability of elastic equilibrium. Thesis, Delft, 1945. English transl. NASA TTF10, 883 (1967) and AFFDL\TR70-25 (1970).

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Discrete and continuous analysis of different cable structures V. Kulbach, J. Idnurm Department of Structural Design Tallinn University of Technology, Ehitajate tee 5, Tallinn, Estonia [email protected] Department of Transportation Tallinn University of Technology, Ehitajate tee 5, Tallinn, Estonia [email protected]

ABSTRACT Continuous and discrete analysis of different cable structures is under investigation in our report. In cases of distributed loads the non-linear conditions of equilibrium and equations of deformation compatibility are taken as initial equations [1]. To eliminate the horizontal displacements, the equations of deformation compatibility were used in integrated form; corresponding integrals
(du/dx)dx were replaced by respective displacements of supporting structures under action of cable forces. Determination of deflections and inner forces for certain cable structures under action of distributed vertical loads may be carried out by means of exact analysis. A girder-stiffened structure has also an exact solution but in the form of complicated transcendental equations. A simpler, compact solution may be found with a proper approximation of the deflection function in the form of trigonometric dependences. Continuous analysis may be also applied to spatial cable structures in the form of hypar-networks with elliptical contour beam [2]. Suitable approximation of the deflection function with use of Galyorkin procedures brings us to values of the network’s deflection and cable forces very near to exact ones. In discrete analysis of girder-stiffened cable structures, the condition of equilibruim is to be composed for every node and the equation of deformations compatibility for every section of the cable [3]. For every joint on stiffening girder were used equation which consider girder’s node’s deformations, internal force in the hangers and load, which is balanced by the stiffening girder. The load may be by distributed load or concentrated force, and may be applied on any point of girder. Using this equations, and moment equilibrium conditions for girder’s supports, were calculated vertical displacements of cable nodes and displacements of girder nodes. Using iteration can be found such value of cable force, which gives actual displacement for every node. After found cable force, all required parameters can be calculated. Examples of analysis of hypar-network and the bridge for a 6100 m strait crossing are presented in the full text of our paper.

References [1] V. Kulbach, Investigation of prestressed cable structures at Tallinn Technical University, Proc. Estonian Acad. Sci., Eng., 8/2, 68 – 83, 2002. [2] I. Tärno, Effects of contour ellipticity upon structural behaviour of hyparform suspended roofs, Publ. of Royal Institute of Technology, Stockholm, 1998. [3] V. Kulbach, S. Idnurm, J. Idnurm, Discrete and continuous modeling of suspension bridges , Proc. Estonian Acad. Sci., Eng., 8/2, 121 – 133, 2002.

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Three-dimensional Vibration Analysis of Crystal Plates via Ritz Method Qian LI∗ and VaiPan IU† ∗ Department of Civil and Environmental Engineering, University of Macau

Macao SAR, PRC [email protected] †Department of Civil and Environmental Engineering, University of Macau

Macao SAR, PRC [email protected] ABSTRACT In this paper, three-dimensional vibration of rectangular Y-cut crystal plate has been investigated. The three displacement components of plate are expanded in series of Chebyshev polynomial multiplied by the boundary function R which makes expansions satisfy the essential boundary conditions along the edges. Chebyshev polynomial series are chosen as admissible functions for its two distinct advantages: One is that it is a set of complete and orthogonal series in the interval [−1, 1]; the other is that it includes constant and proportional terms. The constant term can easily express the whole rigid displacement of the body. The proportional term can easily reflect the shear force effect along the thickness of a finite plate. The maximum energy function of a plate is expressed in terms of Chebyshev polynomial series. The eigenvalue matrix for natural vibration frequencies is obtained by Ritz method and then solved by computer program. Example of an infinite plate excited by thickness-shear deformation parallel to one edge is solved and verified by exact solutions. Other examples of four clamped edges and four simply supported edges rectangular Y-cut crystal plates are carried out. The trial plates are of three different edge lengths and two different thicknesses. The first twenty frequencies of natural free vibration are compared with those from a finite element method. It also shows that the results from present method and finite element method have a good agreement. Besides, for the advantage of the constant and proportional terms in Chebyshev polynomial series, convergence study demonstrates the rapid rate and high efficiency. The frequencies monotonically decrease and approach certain values with the increase in the number of terms of admissible functions. Three terms are sufficient for the requirement of expansion in the thickness direction. Finally, the free vibration of clamped square Y-cut crystal is investigated. Due to the three-dimensional expansions, any relative displacements in any points of the plate body of modes can be determined very easily by back substitution of the eigenvalues. The deflected shapes of first eight modes show the flexural and thickness extensional modes explicitly.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A 3D Solid Finite Element for Reinforced Concrete Analysis Allowing Slippage of Reinforcement Georgios Ch. Lykidis, Konstantinos V. Spiliopoulos Institute of Structural Analysis and Aseismic Research Department of Civil Engineering, National Technical University of Athens, Zografou Campus, Athens, 157 73, Greece [email protected], [email protected]

ABSTRACT In order to evaluate the safety levels of the design of reinforced concrete structures it is essential to be able to predict their response under any type and level of loading. To this end the finite element method of analysis may be used. For such an analysis to be realistic, one must take into account all aspects of the nonlinear behaviour of reinforced concrete. A simple smeared crack material model for concrete behaviour [1], which is based on experiments and uses the uniaxial compressive concrete strength as the only prerequisite has been used recently by the authors to analyze structures loaded statically and dynamically ([2]). A numerical method that treats crack opening and closure in a unified way and presents no numerical instability has been presented. Steel bars are taken into account using an embedded reinforcement formulation [3] and assuming perfect bond with surrounding concrete. The latter assumption is avoided in the present paper with an additional degree of freedom. A realistic model [4] is used to describe the interface behaviour along a reinforcing bar. Comparative analyses of the model with and without bar slipping are performed for static loading cases. The analyses show that the whole procedure manages to give stable and realistic results. This enhanced, therefore, oneparameter concrete model may be used in the analysis of reinforced concrete structures more effectively.

References [1] M. D. Kotsovos, M. N. Pavlovic, Structural concrete, finite element analysis for limit state design, Thomas Telford, London, 1995. [2] K.V. Spiliopoulos, G. Ch. Lykidis, An efficient three dimensional solid finite element dynamic analysis of reinforced concrete structures, Earthquake Engineering & Structural Dynamics, 35, 137-157, 2006. [3] A. E. Elwi, T. M. Hrudey, “Finite Element Model for Curved Embedded Reinforcement”, ASCE, Journal of Engineering Mechanics, 115, 740-754, 1989. [4] R. Eligehausen, E. P. Popov, & V. V. Bertero, Local bond stress - slip relationships of deformed bars under generalized excitations, Rep. UCB/EERC 83-23, Earthquake Engng, Res. Ctr., University of California, Berkeley, 1983.

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Some new results on MITC plate elements Mikko Lyly∗, Jarkko Niiranen†, and Rolf Stenberg† ∗

CSC – Scientific Computing Ltd. P.O. Box 405, 02101 Espoo, Finland mikko.lyly@csc.fi †Institute of Mathematics, Helsinki University of Technology

P.O. Box 1100, 02015 TKK, Finland jarkko.niiranen@tkk.fi, rolf.stenberg@tkk.fi

ABSTRACT The approximation of the deflection for the MITC plate elements [1, 2] is shown to be superconvergent with respect to a special interpolation operator [3]. This property holds in the H 1 -norm and the interpolation operator is closely related to the reduction operator used in the MITC methods. A part of the superconvergence result is, roughly speaking, that the vertex values obtained with the MITC methods are superconvergent. This may be an explanation why these methods have become so popular. By utilizing the superconvergence property a postprocessing method has been introduced [3] in order to improve the accuracy of the approximation for the deflection. The new approximation is a piecewise polynomial of one degree higher than the original one and it is constructed element by element which implies low computational costs. We show various computational results illustrating the superconvergence properties of the original approximation and confirming the improved accuracy of the postprocessed approximation. In the numerical tests both uniform and non-uniform meshes are used and cases with different kinds of boundary conditions are studied.

References [1] K.-J. Bathe, F. Brezzi and M. Fortin, Mixed-interpolated elements for Reissner-Mindlin plates. Int. J. Num. Meths. Eng., 28, 1787–1801, 1989. [2] F. Brezzi, M. Fortin and R. Stenberg, Error analysis of mixed-interpolated elements for ReissnerMindlin plates. Math. Mod. Meth. Appl. Sci., 1, 125–151, 1991. [3] M. Lyly, J. Niiranen and R. Stenberg. Superconvergence and postprocessing of MITC plate elements. Helsinki University of Technology, Institute of Mathematics, Research Reports A 474, January, 2005 (http://math.tkk.fi/reports).

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Nonlinear Analysis of Reinforced Concrete Beams Considering the Slip Between Steel and Concrete. Joaquim Marins Neto*, Aloisio Ernesto Assan† *

Department of Structures, Faculty of Civil Engineering, State University of Campinas Av. Albert Einstein, 951, C.E.P.: 13084-971, P.O.Box 6021,Campinas, SP, Brazil [email protected]

†

Department of Structures, Faculty of Civil Engineering, State University of Campinas Av. Albert Einstein, 951, C.E.P.: 13084-971, P.O.Box 6021,Campinas, SP, Brazil [email protected]

ABSTRACT In this work aspects of interaction between steel and concrete, for reinforced concrete structures, with particular interest in the mechanism of slip that occurs in the steel-concrete interface are presented. A nonlinear computational model which considers the bond-slip behavior between reinforcing steel and concrete for the nonlinear analysis of beams subjected to bending is developed. The finite element method is used to predict the behavior of reinforced concrete structures based on the properties of the concrete, the reinforcing steel, and the relationship of the steel-concrete interface. The concept about equivalent uniaxial stress-strain model proposed by [1] is used to describe the nonlinear behavior of reinforced concrete which incorporates tensile cracking at a limiting stress and the strain-softening phenomenon beyond the maximum compressive strength from an incremental load procedure with an iterative approach to obtain an equilibrium position of the structure for each increment. The bond is modeled with interface element (bond-zone element) connecting the steel and concrete elements. The interface element presented by [2] has its stiffness based on the stages of relationship between the local bond stress and the relative slip of the bar, for incremental load process. Several numerical examples comparing results of bending beams are presented.

References [1] D. DARWIN; D. A. W. PECKNOLD, Inelastic model for cyclic biaxial loading of reinforced concrete. Civil Engineering Studies, University of Illinois, Illinois, USA, july, 1974. [2] A. K. DE GROOT; G. M. A. KUSTERS; T. MONNIER, Numerical modeling of bond-slip behavior. Heron, Conc. Mech., Vol. 26, 1981.

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Novel semi-analytical methodology to determine model parameters for simple finite element bolt model Joost C.F.N. van Rijn National Aerospace Laboratory NLR P.O. Box 153, 8300 AD Emmeloord, The Netherlands [email protected]

ABSTRACT Bolted joints are commonly used in aircraft structures to join the various parts. These joints are an important factor in the sizing of the construction, for composite to composite as well as for composite to metal joints, as a larger thickness is normally required in the joint area. Consequently joints have an important bearing on the weight of the structure and the material costs. Moreover the production costs of joining are significant. Aircraft structures are mostly modeled using a shell representation. Bolted joints are incorporated in these models using a simple model that represents the bolt by a beam element that connects to one node in each adherent finite element model. At these particular nodes a point force is exerted on the adherent models that will cause a certain, mesh dependent, deformation. Experimentally determined bolt stiffnesses are available from various sources. A finite element model of a bolted joint should provide the same bolt stiffness; otherwise the distribution of bolt loads will be erroneous especially in a multiple load path situation. The bolt stiffness can be changed by adaptation of the beam area or the moments of inertia. It is necessary to account for the deformation of the adherent mesh in order to obtain the right bolt stiffness. The novel semi-analytical methodology to determine the model parameters for a simple finite element bolt model, presented in this paper, provides for a better correspondence between computed and experimentally determined behavior of a bolted joint. It enhances the finite element representation of a bolt as it explicitly accounts for the local deformation in the finite element representation of the adherents. The influence of variation in mesh density and boundary conditions for the shell representation of the adherents is established through a simple finite element analysis. The influence of variations in material stiffness and adherent thickness for both metal and composite adherents is obtained with the presented methodology. A mathematical foundation is provided for the simple finite element bolt model that also enables a sensitivity analysis. The use of the methodology will be demonstrated as it was used to determined the bolt parameters for a total of 40 different configurations within a detailed model of a flap section The presented methodology was developed within the context of the Sixth Framework EU project Bond Assisted Single Step Assembly (BASSA). The financial contribution of the EU is gratefully acknowledged.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Accounting for Fuselage Instabilities in the Coarse Model of an Aircraft Fuselage by means of a Material Law Wilhelm Rust*, Josef Overberg† *

Fachhochschule Hannover – University of Applied Sciences Ricklinger Stadtweg 120, 30459 Hannover, Germany [email protected] † CAD-FEM GmbH Schmiedestr. 31, 31303 Burgdorf, Germany [email protected]

ABSTRACT In the typical dimensioning and evaluation process of an aircraft fuselage nominal stresses were obtained from a linear elastic fuselage or section model (barrel) discretized with finite elements in the coarsest possible way. This model does not account for nonlinearities. However, the stresses are compared with allowable ones obtained from either tests or finite element analyses of cutouts (panels) with fine meshes accounting for contact, material and geometric nonlinearities including stability problems. The results are force-displacement curves ending up in the ultimate load. Divided by related areas nominal allowable stresses are obtained. If in one coarse element the stress exceeds the allowable one the total fuselage section is considered as failing. This method does not account neither for changes of the force fluxes due to reaching the local ultimate load nor for the fact that the onset of buckling locally weakens the structure before failing. Therefore, a method is presented here which transfers the local nonlinear behavior analyzed for the panels with fine meshes to the coarse global model. For that purpose all nonlinear effects on the local force-displacement behavior are introduced to the global model by an elasto-plastic material law. It is shown how the stress-strain curves for shear and compression loading (of different shape) are calculated, how they interact in the yield condition, how the flow rule is chosen and how the hardening rule accounts for different ratios of shear and compression. It is shown that the assumptions work well for the described purpose and that the global system can reach a higher ultimate load compared with the standard method. Possible Extensions and Limits of the method are outlined.

References 1. P. Linde, J. Pleitner, W. Rust, Virtual Testing of Aircraft Fuselage Stiffened Panels. Proceedings of 24th Int. Congr. of the Aeronautical Sciences ICAS 2004 2. W. Rust, P. Linde, Ultimate Load Analyses of Aircraft Fuselage Structures within the Virtual Test Rig, Proceedings of IASS/IACM 2005, 5th International Conference on Computation of Shell & Spatial Structures, Salzburg 2005 3. P. Linde, A. Schulz, W. Rust, Influence of modelling and solution methods on the postbuckling behaviour of stiffened aircraft fuselage panels, to appear in Composite Structures

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Modelling of reinforced materials by a subcycling algorithm J.L. Curiel Sosa∗, D.R.J. Owen†, E.A. de Souza Neto†, N. Petrinic∗ ∗ Department of Engineering Science,University of Oxford

Parks Road, Oxford OX1 3PJ, UK [email protected] †Civil and Computational Engineering Centre, University of Wales Swansea

Singleton Park, Swansea SA2 8PP, UK [email protected]

ABSTRACT The high nonlinearity associated to the interface and constituents in reinforced materials -e.g. reinforced concrete- has motivated the development of this subcycling algorithm. The interface modelling and the complex material model used to represent the continuum implies a small critical time step when solving a spatial discretised finite element mesh with an explicit time integrator conditionally stable. Making two subcycles -one for the continuum and the other one for the reinforcement- the smallest critical time step does not rule the other sub cycle. The interface is modeled by transmission conditions including empirically-based bond stress-slip relationship. A set of pullout tests of reinforcing bar embedded in a surrounding continuum to demonstrate the efficiency of the scheme is presented and, then, validated against experimental results from the literature. The attractiveness of this scheme lies in the computational efficiency implied by running reinforcement and continuum at two different velocities of execution and solve the problem of nonlinearity created in the interface of very distinct materials.

References [1] J. L. Curiel Sosa, E. A. de Souza Neto, and D. R. J. Owen. A combined implicit-explicit algorithm in time for non-linear finite element analysis. Commun. Numer. Meth. Engng, 22:63–75, 2006. [2] W. J. T. Daniel. A study of the stability of subcycling algorithms in structural dynamics. Comput. Methods Appl. Mech. Engrg., 156:1–13, 1998. [3] I. W. Farmer. Stress distribution along a resin grouted rock anchor. Int. J. Rock Mech. Min. Sci. Geomech., 12:347–351, 1975. [4] M. O. Neal and T. Belytschko. Explicit-explicit subcycling with non-integer time step ratios for structural dynamic systems. Computer and Structures, 31(6):871–880, 1989.

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Limit Analysis of Cable-Tied Structures Konstantinos V. Spiliopoulos, and Theodoros N. Patsios Institute of Structural Analysis & Aseismic Research National Technical University of Athens 9 Iroon Politehniou, Zografou Campus, 157-73, Athens, Greece [email protected]

ABSTRACT Cables are frequently used to strengthen existing framed structures. They are also the main members that support bridge-decks in cable-stayed bridges. This type of structures may be idealised to consist of beam-type members under pure bending and cabled members under pure tension. In order to get an estimate of the strength of such structures a step by step elasto-plastic analysis must be used. This procedure, however, is time-consuming as it has to follow every single plasticization and any deplasticization that may occur up to collapse. Limit analyses, based on the upper or lower bound theorems of plasticity provide a better alternative. In the present work, a limit analysis procedure, based on the upper bound theorem and leading to a linear programming problem, is followed. Plastic rotation in the form of a plastic hinge at the end of a beam-type member marks the plasticization of the corresponding section when its ultimate moment is exceeded, whereas a plastic extension occurs at one point inside the cabled member when its ultimate axial force is exceeded. The whole approach is formulated within the mesh description of statics which is a generalisation of the force method. This method is known to be the computationally most effective one for linear programming structural problems. The process consists of three distinct parts. The first part deals exclusively with the cabled members of the structure. The influence of their force on the rest of the structure is taken into account by satisfying equilibrium along the shortest path between its two ends. In the second part the indeterminacy of the rest of the structure that now consists of beam-type members is catered for using an existing algorithm, also based on a shortest path technique between two points of a connected planar graph. Cantilevers that follow the shortest path of each load to the ground are used in the third part to satisfy equilibrium with the applied loads. The whole process renders a fully automatic and computationally efficient numerical method to find the limit load of the above-mentioned structures. Two examples of strengthened frames, as well as an example of a cable-stayed bridge are analysed.

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Smart Super Elements in Slender Structures Subjected to Wind Raphaël D.J.M. Steenbergen*, Johan Blaauwendraad† *

TU Delft, Faculty of Civil Engineering and Geosciences P.O. Box 5048, NL-2600 GA, Delft, The Netherlands. [email protected]

† Emeritus professor Structural Mechanics, TU Delft, Faculty of Civil Engineering and Geosciences [email protected]

ABSTRACT Structural analysis of tall buildings of asymmetric plan and irregular geometry subjected to wind load eventuates in complicated calculus. This is among others the case if parts of the building or stability elements stop at a lower height than the rest. FEM programs are at disposal; however the modeling takes a lot of time and a quick and deeper understanding of the force flow is not provided. In this paper this want is supplied by developing a closed-form super element method for two frequently occurring building types. For two types of a tall building of irregular geometry, an insight-providing closed-form analysis method of combining super elements is presented. The main-structure is subdivided in only two super elements. The super elements are based on closed-form solutions describing the force flow in the stability elements. Within an element no change of floor plan, wall and shaft geometry occurs. A node between elements is only chosen where the properties of the building change. The in-plane stiffnesses of the floors are included and act as distributed coupling springs between the stability elements. For each super element a set of simultaneous differential equations is derived and closed-form solutions are obtained; see [1]. For each super element the stiffness matrix is composed from the homogeneous solution and the load vector is composed from both the particular and the homogeneous solution. Foundation stiffness is accounted for. At each change of geometry (node) a marked disturbance in the moment and shear force diagram is found, attenuating along a number of storeys depending on the ratio of the characteristic length and the length of the building. Closed-form expressions for the influence lengths of these disturbances are obtained. Including the rotational stiffness of the foundation may result in substantial disturbances in the stress state at the base of the building. No disturbance occurs if the ratio of the rotational stiffnesses of wall and shaft equals the ratio of the base moments of wall and shaft for an ideal rigid foundation. Results have been presented in [1]. Because of the use of a very small number of super elements with closed-form solutions, the method contributes to the understanding of the behaviour of the considered tall buildings with a discrete change along the height. In a preliminary design stage a fast analysis can be made without spending much time in modelling. It is shown that the modelling and calculating time of the present method is reduced significantly in comparison with complete finite element analysis and accurate results are obtained.

References [1] R.D.J.M. Steenbergen, Static Analysis of Asymmetric Buildings. Delft University of Technology, Delft, 2005.

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Nonlinear analysis of space R/C frames with non-uniform torsion Boris Trogrlic, Ante Mihanovic and Zeljana Nikolic University of Split, Faculty of Civil Engineering and Architecture Matice hrvatske 15, 21000 Split, Croatia [email protected] [email protected] [email protected]

ABSTRACT This paper presents a numerical model of stability and load-bearing capacity of space reinforced concrete (R/C) frame structures taking into account the material and geometric nonlinearity. The developed model describes the behavior of space frames with composite cross sections under a monotonically increasing load, from zero up to the ultimate load, i.e. collapse of the structure. The collapse of the structure occurs due to exceeding the limit load and/or loss of stability of space beams or whole structure. The fibre decomposition procedure is developed to solve material and geometrical nonlinear behaviour of composite cross-section in three-dimensional frames. The filaments in the fibre decomposition model of the cross-section, which describe uniaxial behaviour of materials, are extended over corresponding finite element and create a separate prismatic body discretised by brick finite elements. After mapping of boundary forces on prismatic body, i.e. ‘comparative body’, the capture of non-uniform torsion is applied. The main attention in this approach is concentrated on the evaluation of the torsional stiffness, which are strongly nonlinear. Three integration levels exist: the first along the beam-column finite element, the second over the fibre decomposed cross-section and the third over a prismatic comparative body. Behaviour of the space frames shall be more realistically described in this way, especially flexural, lateral and torsional stability effects. The global procedure includes an incremental-direct iteration step approach. The incremental step model of gravitational load level is applied. Geometrical nonlinearity is assumed by Total Lagrange small displacement formulation. The perfect bond-slip effect between concrete and rebars as well as smeared crack model is assumed. Two examples are studied to verify the accuracy of the program and demonstrate its application in practical engineering.

References [1] B.A. Izzuddin, A.A.F.M. Siyam and D.L. Smith, An efficient beam–column formulation for 3D reinforced concrete frames, Computers and Structures, 80, Issues 7-8, 659-676, 2002. [2] H.-G. Kwak and S.-P. Kim, Nonlinear analysis of RC beams based on moment–curvature relation, Computers and Structures, 80, Issues 7-8, 615-628, 2002.

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An FE Analysis of the Stresses in Pultruded GRP Single-Bolt Tension Joints and Their Implications for Joint Design Geoffrey J. Turvey*, Pu Wang† *

Lancaster University Engineering Department, Bailrigg, Lancaster, LA1 4YR, UK [email protected] † Schlumberger Stonehouse Technology Centre, Stroudwater Industrial Estate, Stonehouse, Gloucestershire, GL10 3SX, UK [email protected]

ABSTRACT FE (Finite Element) analysis is used to determine stresses on critical planes and around the hole edge in a two-dimensional model of a single-bolt tension joint in pultruded GRP (Glass Reinforced Plastic) plate material. The analysis takes account of bolt – hole clearance and friction at the contact surfaces between the bolt shank and the hole. It is shown that even when the hole clearance is nominally zero (~0.2mm) critical stress distributions, normalized with respect to the far field stress are not invariant but change as the tension increases. Friction between the bolt shank and the hole and the small hole clearance are the principal factors which cause the zone of contact (defined by the angle it subtends at the centre of the bolt) to increase with increasing tension, and produce significant changes in the stress distributions at critical locations. These observations cast doubt on the validity of the simplified method of design, given in the EUROCOMP code [1], for bolted tension joints in pultruded GRP plate material, because the method relies on normalized critical stress distributions remaining unchanged as the tension load applied to the joint increases.

References [1] J.L. Clarke ed, Structural design of polymer composites – EUROCOMP design code and handbook, E. & F.N. Spon, London, 1996

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An Efficient Evaluation of Structural Safety applying Perturbation Techniques José M.G.C.Veiga*, António A. R. Henriques†, Jorge M. Delgado* *

†

Escola Superior de Tecnologia e Gestão - IPVC Viana do Castelo - Portugal [email protected] and [email protected]

Laboratório da Tecnologia do Betão e do Comportamento Estrutural Faculdade de Engenharia da Universidade do Porto - Portugal [email protected]

ABSTRACT The application of probabilistic techniques on structural safety evaluation has suffered a great expansion in the last years. However, one of the main problems in the introduction of these techniques is the long computational time consuming required, particularly when simulation methods as Monte-Carlo method are used, even when sampling reduction techniques are adopted [2]. In this paper is presented an efficient structural reliability method that couples perturbation techniques with the finite element method [1]. This method allows, in one only structural analysis, to evaluate the mean value and the standard deviation of the structural response, by defining previously the probability distribution of problem basic random variables. Consequently a much faster analysis is performed, when compared with the most frequent used methods based on reliability techniques. Considering a structural system, with n structural elements, submitted to a load defined by F·Φ = F·[Φ1, Φ2, …, Φn]; where F is the load intensity and [Φ1, Φ2, …, Φn] is the load distribution vector along the structure. According to the finite element method, the system equilibrium is defined by the following equation: K(u)·U = F·Φ ; where K(u) is the tangent stiffness matrix of the structure, defined as a function of the nodal displacements U and F·Φ is the nodal forces vector (it includes dead loads, live loads, wind, etc.). By applying perturbation techniques to this equation it is possible to quantify the mean structural response and its dispersion, in terms of displacements or forces. Finally comparative examples between the results obtained with this technique and other probabilistic methods are presented, allowing to appraise the potentialities of the proposed method.

References [1] J. Eibl and B. Schmidt-Hurtienne, General outline of a new safety format. New developments in non-linear analysis method, CEB Bulletin d’Information, 229, 33-48, 1991. [2] A. Haldar and S. Mahadevan, Probability, Reliability and Statistical Methods in Engineering Design. John Wiley & Sons, New York, 2000. [3] E. Altus, E. Totry and S. Givli, Optimized functional perturbation method and morphology based effective properties of randomly heterogeneous beams. Int. J. Solids Struct., 42, 2345-2359, 2005.

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Semi-Analytical Analysis of Super Tall Building Bundled-Tube Structures Gong Yaoqing *, Li Ke *

Civil Engineering School of Henan Polytechnic University, Henan Jiaozuo, 454000, China [email protected]

ABSTRACT A new semi-analytical method is developed for the analysis of interactions between the subgrade and the foundation and the superstructure of the super tall building bundled-tube structure by threedimensional model which is a combination of stiffened-thin-wall tubes on semi-infinite elastic body. The subgrade is idealized as a semi-infinite elastic body, and the rigidities of the elastic body pertinent to various deformations of the foundation have been expressed as analytical equations [1], with which the reactions between the foundation and subgrade can be quantified expediently. The foundation is taken as a part of the superstructure. In fact, the foundation is the extension of the superstructure toward the underground. The only difference is the size, since in most cases the foundation must be large enough to make the soil stable. The superstructure and its foundation of the super tall building bundled-tube structure are simplified equivalently and continuously to a combination of stiffened-thin-wall tubes on semiinfinite elastic subgrade. Then discretization is made by some nodal lines, the unknown functions defined on the lines are used as primary unknowns, and interpolating functions are implemented between the lines. So the displacement field of the computing model can be expressed by the unknown functions. After using the principle of minimum potential energy, the governing equations will then be obtained, which is actually a group of ordinary differential equations. Therefore, analysis of a tall building structure will be changed into the solution of the boundary problem of a group of ordinary differential equations that can be solved by the precise and powerful Ordinary Differential Equation Solver—COLSYS [2] , a kind of computational software. The interactions between the subgrade and the foundation and the superstructure of a super tall building bundled-tube structure due to static loadings are analyzed by the method based on the model. The numerical results show that the analytical model is reasonable and feasible. Therefore, a practicable method for the global analysis of the super tall building bundled-tube structure is obtained, and some valuable conclusions are acquired through analyzing the computing results as well.

References [1] Gong Yaoqing, Tall Building Structures on Elastic Subgrade and Research of Semi-Analytical Method [D]. Beijing: Tsinghua University, 1999 (in Chinese). [2] Yuan Si, Introduction of a common compute program for boundary problems of ordinary differential equation—COLSYS [J]. Computational Mechanics and Application, 2, 104-105, 1990 (in Chinese).

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Large Displacements in Nonlinear Numerical Analyses for Cable Structures Nikolina Zivaljic *, Ante Mihanovic †, Boris Trogrlic † *

Faculty of Civil Engineering and Architecture, University of Split Matice hrvatske 15, 21 000 Split, Croatia Nikolina.Zivaljic@ gradst.hr

†

Faculty of Civil Engineering and Architecture, University of Split Matice hrvatske 15, 21 000 Split, Croatia [email protected] Boris.Trogrlic@ gradst.hr

ABSTRACT Method for defining appropriate form of prestressed, tensile cable structures and for calculating stress and displacements for such structures is presented. The developed numerical model is taking into account the material and geometric nonlinearity. The described model represents a practical way of implementing the large displacements theory in the analysis of finding appropriate form of prestressed cable structures. The behavior of the structure under an increasing load, from zero up to final is described. The load usually applied in two phases. The first phase can be prestressing. In the second phase, the structure is computed taking into account the dead and the live gravity load. An approach to solving the problem of large displacements in the theory of structures is presented, based on an incremental approach of the Total Lagrange formulation with the small displacements. The model is based on the assumption that the FE are linear and small enough and thus tracking of large translational displacements can be approximated by a simple geometrical model. The resulting force, i.e. stress, inside FE are expressed within the large displacements, based on the successive approach of small displacements of each increment, using a singular quasi-tangent stiffness matrix. A solution of renewable of the internal forces and stress and their influence is presented. The renewal of the large translational displacements is based on their vector from increment to increment. The renewal of the geometry configuration is influenced to the renewal of basic stiffness, geometrical stiffness and large displacement stiffness. Spatial discretization of the system is on two-node line elements. The fiber discretization of the cross section is on triangular elements where mechanical properties of each fiber are presented by the V  H diagram. The numerical nonlinear material model is based upon nonlinear material properties defined in the form of a uniaxial V  H diagram. The developed model was tested on few practical examples. They are compared by the research computations of the other authors.

References [1] Dvornik J, Lazarevic D. Fractals and formfinding – magic with real numbers. International journal for Engineering Modelling 16, 1-11, 2003. [2] Tabarrok B, Qin Z. Nonlinear Analysis of tension structures, Computers and Structures; 45, 973984, 1992.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

129

Mathematical Modeling for Synthesis and Design of Non-orthogonal Worm Gears with a Straight-line Tooth Contact V. Abadjiev*, D. Petrova†, E. Abadjieva† *

Institute of Mechanics, Bulgarian Academy of Science Acad. G. Bonchev Str., block 4, Sofia 1113, Bulgaria [email protected]

ABSTRACT The development of the different kinds of transmissions in recent decades has pushed the processes related with the construction of electric transmissions when a preliminary given law of motions transformation is realized by applying a suitable electronic control. Created in this period field of mechatronics covers in particular the most of the mentioned mechanisms and its scientific achievements may be applied to them. Independently of the intensive development of this new part of the techniques, the practice shows that when constructing transmissions oriented to transform big powers, the power mechanical transmission have not found their alternatives. This circumstance motivates the researchers in their trials to create new types of mechanical gears and/or to improve the existing ones in order to find out their new exploitation qualities [1]. The presented research aims at defining of an adequate mathematical model and constructing of a computer program for insuring the process of constructive and technological synthesis and of the design of non-orthogonal worm gears of type Wildhaber. This class of hyperbolic gears is synthesized according to the second Olivier’s principle. The mentioned technological characteristic and the specific geometry of the active tooth surfaces of the gear pair allow this type of mechanical gear sets to be applied alike as power transmission as a kinematic one. The mathematical modeling for synthesis of worm gears of type Wildhaber is based on an approach upon “the region of mesh” [2]. Together with the gear pair design the worked out algorithm realizes a control of the quality of meshing in the whole mesh region or in its definite parts. This control consists of: defining of an optimal configuration of the region of mesh, and of suitable dimensions of the active tooth surfaces; a registration and a limitation of the singular points on the meshed tooth surfaces; a choice of a calculated variant of a gear pair with an optimum situation of the contact lines in the of mesh region from a point of view of hydrodynamic lubrication and related with it hydrodynamic loading capacity, etc. The computer program is directed to studying real models in the design process. Patterns having concrete practical applications are illustrated in the paper.

References [1] Litvin F. Gearing Geometry and Applied Theory. PTR Prentice Hall, A Paramount Communication Company, Englewood Eliffs, New Jarsy, 1994 [2] Abadjiev V. Mathematical Modelling for Synthesis of Spatial Gears. Journal of Process Mechanical Engineering. Proc Inst Mech Engrs, Vol. 216, Part E, 31-46, 2002

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Computational modeling of ultrasonically assisted turning Naseer Ahmed*, Alexander V. Mitrofanov†, Vadim V. Silberschmidt††, Vladimir I. Babitsky††† Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, UK *[email protected] † [email protected] †† [email protected] ††† [email protected]

ABSTRACT Ultrasonically assisted turning (UAT) is an advanced machining technique, where high frequency vibration (frequency f | 20 kHz, amplitude a | 15 Pm) is superimposed on the movement of the cutting tool. Compared to conventional turning (CT), this technique allows significant improvements in processing intractable materials, such as high-strength aerospace alloys, composites and ceramics. Superimposed ultrasonic vibration yields a noticeable decrease in cutting forces, as well as a superior surface finish [1]. The paper presents a three-dimensional thermomechanically-coupled finite element (FE) model of both UAT and CT that was recently developed as an extension of the initial 2D model [2]. The current model enables studies of various 3D effects in turning, such as oblique chip formation, as well as the influence of the tool geometry on process parameters, e.g. cutting forces and stresses generated in the workpiece material. The model allows transient, coupled thermomechanical simulations for elasto-plastic materials with strain-rate sensitivity. Chip shapes and forces acting on the cutting tool are analyzed. Stress, strain and temperature distributions in the cutting zone are studied. The effects of cutting parameters (such as the feed rate) and influence of friction on both UAT and CT are investigated. Numerical results are validated by the experimental tests performed at our in-house UAT prototype.

References [1] Babitsky, V.I., A. Kalashnikov, A. Meadows, and A. Wijesundara, Ultrasonically assisted turning of aviation materials. Journal of Materials Processing Technology, 132, 157-167, 2003. [2] Mitrofanov, A.V., V.I. Babitsky, and V.V. Silberschmidt, Thermomechanical finite element simulations of ultrasonically assisted turning. Computational Material Sciences, 32, 463-471, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Numerical evaluation of bored piles in tropical soils by means of the geotechnical engineering “GEO4” Fine Software Anjos, G.J.M.*, Cunha, R.P.†, Kuklik, P.‡, Miroslav, B.‡ *

Federal University of Pará, Belém, Brazil [email protected] †

University of Brasília, Brasília, Brazil [email protected]

‡

Czech Technical University, Prague, Czech Republic [email protected]; [email protected]

ABSTRACT This paper presents the back-analyses of field loading tests carried out with bored pile and drilled shaft founded in a tropical soil executed in the University of Brasília experimental research site. For this, a numerical simulation was carried via existing commercial application software denominated GEO4. This software computes the load-displacement curve of the pile’s head plus, distribution of normal and shear forces along the pile’s shaft. The Shear behavior of pile-soil interface is described using the elastic-plastic material model with Mohr-Coulomb yield condition. The complete response of any foundations is represented by determination of shaft and toe resistance plus settlement analyses. Hence, this paper focused in the determination of components of resistance (angle friction and cohesion) and settlement (Young Modulus) for this type of foundation, and explains and presents, in details, the software GEO4 from Fine Inc. Ltd. for foundation design. In relation to the cohesion, it was verified the important effect that this parameter have in the determination of shaft resistance. Moreover, few research topics nowadays deal with the determination of this particular parameter for bored piles. The assessment of geotechnical parameter is a vital component of geotechnical design and some formulation are also presented for this evaluation.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Kinematics and force interaction of screw shaft with variable screw course Bahadirov Gayrat Atahanovich*, Bahadirov Kudrat Gayratovich1 *700143,Uzbekistan, Tashkent, Akademgorodok, F.Khodjaev str. 31, Institute of mechanics and seismic stability of structures Uzbek Academy of sciences [email protected] 1

700100, Uzbekistan, Tashkent, Shahjahan str. 5, Institute of textile and light industry [email protected]

ABSTRACT The screw shafts with spiral blade are using in practice are not satisfied to modern demands in industry. For effective smoothing of sheet materials’ creases are necessary uniformly and simultaneously to smooth creases by screw blades from middle site toward ends of the shaft. For achieving of this goal the new construction of screw smoothing blade shaft are developed with so property like blade divergence from middle site toward ends of the shaft with variable (increasing) course. In this construction the next step are compiled by previous via multiplying it on the coefficient which taking into account a delay (sliding) of screw blade in direction of rotation axis of smoothing shaft. The laws of rotation movement of blades’ sides and contact surfaces of screw blade of the shaft with processed of sheet material are obtained. The equation of a curve line of a side of the screw blade is investigated and received too. The smoothing speed of creases and the blade’s speed of a rather sheet material is determined, also are determined it normal, tangent and complete acceleration. Radius of curvature of a side of the screw blade of shaft is determined. The mathematical interrelation of step changes of screw blade, velocity of contact site, and radius of curvature are constructed by depending on angle of shaft’s turn. At mechanical processing of sheet materials by taking into account technological demands directions and volume of acted forces are selected. The force interaction of the variable course blade and processed sheet material with taking into account resistance forces of processed material and variety of screw course, friction forces which arising in contact point of a blade’s surfaces and sides, axial forces of reaction of a support of the shaft, angle of ascent and friction of the screw blade are considered. The torque moment necessary for rotation of one blade and whole shaft is obtained in this work. The driving rate of dislocation of a contact point of a sheet material with the blade of the screw shaft are presented as relation of dislocation of a blade’s screw shaft around of an axis to the appropriate moving of a contact point of the blade on an rotation axis. The numerical solutions’ of equations are compiled by universal software MATHCAD for mathematical calculations with different variations of system’s parameters. Graphics allowing consider kinematical process occurring between blades and processed material are developed on the base of the obtained results. Obtained results used at designing of screw shaft with variable step.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Optimal construction of the thermo-elastic actuator J. Barglik1, B. Ulrych2 1 Silesian University of Technology Krasiēskiego 8, 40-019 Katowice, Poland [email protected] 2

University of West Bohemia Universitní 26, 306 14 Plzeė, Czech Republic [email protected]

ABSTRACT Electro-mechanical actuators seem to be one of devices which produce mechanical forces as a result of eddy-currents influence. The most optimal of them are electromagnetic actuators constructed as ferromagnetic devices and thermo-elastic actuators constructed as bimetallic or mono-metallic are shown in Fig. 1.

Figure 1: Arrangement of the thermo-elastic actuator and distribution of electromagnetic field: 1– dilatation element, 2 – field coil, 3 – sleeve, 4 – ring, 5 – frame, 6 – body The ferromagnetic cylinder 1,that plays the role of dilatation element, is clamped to non-ferromagnetic ring 4. The ring is pressed in a sleeve 3, which is fixed to frame 5 of the tool machine. A cylindrical coil 2 is replaced between the ring 4 and dilatation element 1. The coil supplied with harmonic current of effective value I ext and frequency f, generates time variable electromagnetic field. Due to consequent induction heating the temperature of dilatation element 1 rapidly rises. As a result of temperature rise element 1 is pushed to the body 6 with a contact force FC. This kind of the actuator is characterized by a relatively big dispersion of magnetic field. The paper deals with analysis of possibilities of construction optimization of the device in order to minimize losses of electromagnetic energy and consequently to increase total efficiency and to match electromagnetic compatibility requirements. The aim is to find the construction of the actuator making possible to use practically all the electromagnetic energy for induction heating of the dilatation element. The task formulated as a weakly- coupled electromagnetic – temperature - heat stress problem is solved by means of FEMbased numerical method. The results for illustrative example are presented and discussed.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

134

Determination of Moment-Curvature Diagrams and Moment-Deflection Curves in Reinforced Concrete Beams M. H. F. M. Barros*, C. Oliveira† * Civil Engineering Department, FCTUC Polo II Pinhal de Marrocos, 3030 Coimbra Portugal [email protected] † Civil Engineering Department, FCTUC Polo II Pinhal de Marrocos, 3030 Coimbra Portugal [email protected]

ABSTRACT The purpose of this paper is to establish an automated process in order to swiftly calculate momentcurvature and moment-deflection diagrams. Comparisons can, this way, be established between several compressed concrete behaviour theories. The model is implemented into a mathematical manipulation program. The consideration of concrete tensile stress-strain relations in structural analysis if often neglected, which leads to the consideration of a far different stiffness in analysis. Considering the tensile stress-strain model referred in Bazant et al[1], together with different compressive stress-strain models, a good comparison of the predicted theoretical behaviour of the several models is obtained

References [1] Deformation of Progressively Cracking Reinforced Beams – ZdenƟk P. Bazant and Byung H. OH

[2] – Eurocode 2: Design of Concrete Structures – CEN – European Committee for Standartization

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

135

Numerical simulation of the nanoindentation experiment: sensitivity analysis of the experimental parameters P. Berke∗ and T. J. Massart† Universit´e Libre de Bruxelles Structural and Material Computational Mechanics Department CP 194/5 avenue Roosevelt 50, B-1050 BRUSSELS (Belgium) ∗ [email protected] † [email protected]

ABSTRACT The use of certain metallic materials in micro-mechanical systems applications is promising for chirurgical applications because of their bio-compatibility and interesting mechanical and wear properties compared to the widely used silicon. The reliability of miniaturized components, the building blocks of such systems depends largely upon the reliability of the techniques applied to characterize the materials, in relation with numerical simulations. Nanoindentation is the method adapted to investigate the local mechanical properties of materials at the nanoscale. The inter-disciplinary nature of such an experiment makes the interpretation of the results difficult. The goal of the research is the use of a relatively simple but flexible computational tool for the simulation of the nanoindentation experiment in order to better understand the physics and the mechanics involved. A finite element code therefore has been developed and used to solve the simulation problem with all the non-linearities involved (finite deformations, plasticity, contact evolution), including isotropic plastic behavior with hardening and an accurate computational contact mechanics feature using the augmented Lagrangian scheme. These tools allow to investigate the most significant sources of dispersion in nanoindentation experiments and their influence. This analysis helps to fix the range of given experimental parameters for which the sensitivity of experimental results is important. The numerical simulation tool allows a parametric study to quantify the effects of some of these experimental conditions, such as the tip geometry and the surface roughness.

References [1] P.Wriggers, Computational Contact Mechanics. John Wiley and Sons Ltd, ISBN 0-471-49680-4, 2002 [2] T.Belytschko, W.Kam Liu, B.Moran, Nonlinear Finite Elements for Continua and Structures. John Wiley and Sons Ltd, ISBN 0-471-98773-5, 2000 [3] J.P.Ponthot, Unified stress update algorithms for the numerical simulation of large deformation elasto-plastic and elasto-viscoplastic processes, Int.J.Plas., 18, 91-126, 2002

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

136

Numerical evaluation of wrinkling stress in sandwich panels Monika Chuda-Kowalska, Andrzej Garstecki and Zbigniew Pozorski Poznan University of Technology ul. Piotrowo 5, 60-965 Poznan [email protected]

ABSTRACT Sandwich panels have been for years used as structural and cladding elements. Characteristic feature of these structures is that important role is played by temperature actions, creep of the core and local instability of thin faces of the panel [2]. Proper estimation of these phenomena has become a challenging issue because of strong tendency to optimize technical parameters and costs. The aim of the study is numerical analysis of bending of three-layered panels. In the classical approach linear constitutive equations for faces and core materials are assumed. Moreover, identical tension and compression elasticity modules of the core are introduced [1]. For the plate loaded by axial force P and uniform transversal load q, the following differential equilibrium equation is used: Bw IV + Pw′′ + cw = −q , where w denotes deformation form of compressed face and c is a stiffness coefficient of the core treated as a Winkler foundation. The term B represents the bending stiffness of the plate. In practice, wrinkling stress depends on many more factors, neglected in analytical solutions, though observed in experiments [3]. In this paper we use numerical methods and hence we can allow for the loss of face adhesion and anisotropy of the core. The analysis is carried out for various dimensions (span and depth of the panel and face thickness) and for various material parameters. By the way of the parametric analysis, the sensitivity of structural response to variations of dimensional and material parameters will be studied. The range of applicability of classical theoretical models will be discussed basing on numerical examples. The study presented in the paper was inspired by one of the biggest in the world producers of sandwich panels, with the aim to increase safety and economy.

References [1] K. Stamm, H. Witte, Sandwichkonstruktionen. Berechnung, Fertigung, Ausführung (in German). Springer-Verlag, Wien, Austria, 1974. [2] D. W. Sleight, J. T. Wang, Buckling analysis of debonded sandwich panel under compression. NASA Technical Memorandum 4701, Langley Research Center, Hampton, Virginia, 1995. [3] O. T. Thomsen, Y. Frostig, Localized bending effects in sandwich beams: photoelastic investigations versus high-order sandwich theory results. Composite Structures, 37, 97-108, 1997.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

137

Application of FEA as a Predictive Tool in the Corrugated Paperboard Industry 

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

138

Implementation of 3D homogenization techniques for the thermo-elastic FEM analysis of brazed plate-fin heat exchangers J. Dib†,*, F. Bilteryst* , J.L. Batoz* , I. Lewon† † Nordon-Cryogenie 25 bis, rue du Fort, BP 87 88194 Golbey, France [email protected]

* ERMeP Institut Supérieur d’Ingénierie de la Conception (GIP-InSIC) 27 rue d’Hellieule, 88100 Saint-Dié-des-Vosges, France [email protected]

ABSTRACT The present study results from a research collaboration between Laboratory ERMeP (GIP-InSIC) and the company Nordon-Cryogenie (Vosges, France), one of the major world manufacturers of heat exchangers for cryogenic processes. A general description of a multi-stream brazed aluminium plate-fin heat exchanger is presented in Fig 1. The problem for Nordon Cryogénie is to guarantee a thermal performance as well as the safety of the heat exchanger by ensuring the structural integrity of each stream subjected to pressure and temperature gradients. The current research program consists in the development of a dedicated FEM solver for the global thermo-elastic analysis of an exchanger. Since a complete FEM model would lead to several millions of structural elements (solid, shell) we propose a simplified 3D FEM model based on homogeneization techniques to obtain the equivalent (effective) stress-strain relations and equivalent thermal load vectors of the corrugated fins brazed with the plate layers (Fig. 1). Two approaches are considered (kinematical and mechanical models) while considering periodicity. The implementation of these two methods is performed using the free finite element software Code_Aster.

Dib J., Bilteryst F., Batoz J.-L., Lewon I. : Application des techniques d’homogénéisation pour l’analyse tridimensionnelle d’échangeurs de chaleur à plaques et ondes. Actes du 7ème Colloque National en Calcul des Structures, Giens 2005, Vol.1, 417-422, Lavoisier, France, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

139

Hierarchical treecode for optimized collision checking in DEM simulations – application on electrophotographic toner simulations Rainer Hoffmann* *

Océ Printing Systems Siemensallee 2, 85581 Poing, Germany [email protected]

ABSTRACT Discrete element modelling (DEM) according to Cundall and Strack [1] of a large set of particles with long-ranging forces like gravitation or electrostatics has the disadvantage of a computation time dependence on the number of particles of O(N²). This can be overcome by the usage of a hierarchical tree code [2] which groups particles which are far away to virtual pseudoparticles. This reduces the number of force calculations so that the computation time dependence can be reduced to O(N log N). The criterion which determines a possible grouping of particles is the theta parameter which is a measure for the reciprocal distance of the particle of interest to the neighboring particles. The paper here shows that the tree algorithm can be also used for an efficient collision checking routine since particles that the algorithm determines to be far enough from the particle of interest can certainly be excluded for a collision checking. It is shown, however, that for a particle set with a uniform radius distribution an upper limit for the theta parameter exists. When this upper limit is exceeded collisions will be suppressed or artificial collisions will occur so that the simulation result is severely falsified. This limit can be overcome by extending the theta parameter to include the radius of the particle of interest. This allows the theta parameter to be increased significantly over the previously found limit, thus reducing the computation time by a further 30 % without introducing much additional error. The prerequisite for such a high theta parameter is that the simulated particle set is rather densely packed (packing density > 10%) so that the total behaviour is dominated by collisions not by the long-ranging forces. The algorithm is applied to the simulation of electrostatically charged toner particles used for the electrophotographic print process. To provide a test case for the simulation a simple transfer experiment is chosen: A roller covered thickly with charged toner is positioned next to a second roller with a thin air gap between them. An external voltage is applied to the rollers causing the particles to jump to the second roller. The results of this experiment can be easily measured and the comparison with the simulation shows an error below 5%.

References [1] P. Cundall and O. Strack. A discrete numerical model for granular assemblies. Geotechnique, 29, 47-65, 1979 [2] Joshua Barnes and Piet Hut. A hierarchical o(n log n) force calculation algorithm. Nature, 324, 1986

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

140

3D FEM analysis of basic process parameters in rotary piercing mill *

†

Jan Kazanecki , Zbigniew Pater , Jarosław Bartnicki *

†

AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Cracow, Poland [email protected] †

Department of Computer Modelling and Technology of Metal Forming, The Lublin University of Technology, 36,Nadbystrzycka, 20-618 Lublin, Poland [email protected] [email protected]

ABSTRACT In this paper the 3D FEM analysis of basic process parameters in rotary piercing mill is presented. In this process the material is formed by means of two skew rolls, two Diescher’s discs and a plug. The material is dragged by the rolls, it moves axially forward and rotates. The FEM analyze of the rotary piercing process was made under the conditions of 3D state of strain with taking into consideration the thermal phenomena. The calculations were made by means of commercial software MSC.SuperFORM2004 with application of different rolls’ skew angles, different plug designs and working positions. All the mentioned above variants were calculated at a different area reduction ratio. In the result, the progression of shapes, temperature and distributions of stress and strain in various sections of the analyzed tubes were characterized. The numerical results of calculations were justified during the stand test with the use of 100Cr6 steel. The comparisons of the numerical and experimental tests confirm good agreement between the obtained results.

References [1] J. Kazanecki: The seamless tube manufacturing, AGH Kraków 2003, p. 1-622 [2] Z.Malinowski, J. Kazanecki, S. Urba ski: Thermal – mechanical model of tube elongation process in Diescher’s mill, Journal of Material Processing Technology 60, 1996, p. 513-516

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141

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

143

Reduction method independent substructure synthesis Paul J. Klinge VTT (Technical Research Centre of Finland), Industrial Systems P.O.Box 1000, FIN-02044 VTT, Finland [email protected]

ABSTRACT Today machine manufacturers are increasingly assembling their products from components manufactured by subcontractors. The component manufacturers often have made detailed FE models of their products. If the designer of the new built-up machine could use these models, he would save a lot of work. To utilise the models the companies must have FE programs than can transfer data. Even then there is a risk that all relevant data is not transfered properly. The FE programs may also differ so, that the results calculated with the transfered model are not the same as with the original program. And the combined FE model of the built-up product becomes large, often too large for practical design calculations. A natural solution is to use substructures. The subcontractors can make reduced sized substructures of their products and the designer uses them in his calculations. However, there are various substructure synthesis methods and file formats, so this works only if the companies use the same FE program. Here a new general method to connect substructures is presented. With this method submodels formed with almost any reduction/substructure method and program can be connected. Also a new kind of substructure is presented. It includes not only stiffness and mass but also damping which is not assumed to be proportional to mass and stiffness. Finally a standard file format for the new substucture is proposed.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

144

Discrete Models for the Simulation of Rubber Components Dynamics M. Lancini, A. Magalini, D. Vetturi Università degli Studi di Brescia – Dipartimento di Ingegneria Meccanica via Branze 38, 25123 Brescia, Italy [email protected]

ABSTRACT This contribution deals with improvements which have been undertaken on a modeling and simulation tool (developed by the authors [1]), aimed to numerically characterize the dynamic behavior of rubber to metal devices (such as: engine mountings, shock absorbers, vibration dampers, etc.). The adopted approach allows designing such devices in a CAD like environment, meshing them and simulating their dynamics; materials constitutive equations modeling is operated by a neural network, working directly on numerical data supplied by experimentation [2]. The peculiar properties of rubber, in terms of non-linearity, visco-elasticity as well as frequency, temperature and static strain dependence [3], are taken into account in the numerical analysis. The proposed method avoids the implementation of finite elements models (as proposed by many authors [4]) in order to produce a light and easy-to-use modeling and simulation tool, to be adopted by industrial technicians in the design optimization of the mentioned devices. To achieve this point, discrete models, constructed by a multi-body particle based approach, are applied. Several models of such a kind have been taken into account during this research project, from discrete concentrated parameters models [1] to a chosen hybrid particle based model, where local finite elements stiffness is computed directly by the continuum constitutive equations. Simulative results are successfully compared with experimental data for some existing devices, provided by CF Gomma S.p.A.

Acknowledgements The authors are grateful to CF Gomma S.p.A. (Passirano, Brescia, ITALY) for funding and supporting this research.

References [1] A. Magalini, D. Vetturi, Concentrated parameters modeling for the numerical simulation of the mechanical behaviour of rubber to metal vibration dampers. Adaptive Modeling and Simulation, N. E. Wiberg and P.Diéz Editors, International Center for Numerical Methods in Engineering (CIMNE), Barcelona, Spain, 2003 [2] G. Ramorino, D. Vetturi, D. Cambiaghi, A. Pegoretti, T. Riccò, Developments in dynamic testing of rubber compounds: assessment of non-linear effects. Journal of Polymer Testing 22, 681-687, 2003 [3] A. R. Payne, R.E. Whittaker, Low strain dynamic properties of filled rubbers. Rubber Chemistry and Technology, 44, 340, 1971 [4] D. J. Charlton, J. Yang, K. K. Teh, A review of methods to characterize rubber elastic behaviour for use in finite elements analysis. Rubber Chemistry and Technology, 67, 481-503, 1994

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

145

Contact Analysis of Impeller-Shaft assembly and Reasonably Designing the Amount Interference of Turbocompressors A. H. Liao,

H. W. Zhang,

C. H. Wu

State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian 116024,China [email protected]

ABSTRACT The interference fit between impeller and shaft assembly is one of the most important factors influencing the performance of the turbo unit in the design of turbocompressors. The assembly creates a complicated frictional contact problem the interface of the mating parts when the rotational speed is high. Therefore, stress analysis and suitable amount of interference need to be considered in the structural design. A locomotive-type turbocompressor with 24 blades is used for the present analysis. The FE parametric quadratic programming (PQP) method which was developed based on the parametric variational principle (PVP) is used for the analysis of the stress distribution of 3-D frictional contact problem of impeller-shaft sleeve-shaft, makes it possible to precisely simulate complicated geometrical shapes of impeller and considerably enhances accuracy in computation. The advantages of the parametric programming method as compared with the conventional ones are that the penalty factors can be cancelled and the solutions can be obtained directly without tedious iterative procedures such as general incremental iterative method[1]. The effects of fit tolerance and rotational speed, the displacement and the contact stress on the interference-fitting surfaces are discussed in detail in this paper. A range of friction coefficients in the impeller-shaft sleeve and shaft sleeve-shaft contact interfaces is investigated in order to represent the mechanical behaviors of the structure due to the change of the interface characteristic. To decrease the difficulty of the assembling process and make sure the safety of the working state, the suitable amount of the interference is studied so that the interface can have adequate contact pressure under full load. It is found that nonuniform initial interference value in the structural design would avoid relative displacement generated and ensure uniformity of the contact stress. To assure quality of press-fitting, the amount of interference between the shaft sleeve and shaft by press-fitting should be controlled strictly to avoid the rapid increase of the contact stress. The study provides an effective approach which achieves more reliable interference-fitted connections and more precise assembly accuracy with lower manufacturing cost in the structural design.

References

[1] H W Zhang, W X Zhong, Y X Gu. A combined parametric quadratic programming and iteration method for 3D elastic-plastic frictional contact problem analysis. Comput. Meths. Appl. Mech. Engrg, ELSEVIER, 155:307-324, 1998.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

146

Study of the stress induced granular consolidation process by 3D DEM simulation Yong Sheng *, C.J. Lawrence †, B. J. Briscoe † *

†

School of Civil Engineering, University of Leeds, Leeds, LS2 9JT U.K. [email protected]

Department of Chemical Engineering, Imperial College, London SW7 2AZ U.K. {c.lawrence, b.briscoe}@imperial.ac.uk

ABSTRACT Today’s powder processing technology requires a satisfactory explanation for the evolution of internal structure of powder compacts during the entire consolidation process. Many industries, where processing problems occur, can make use of such an explanation for the process development as well as for their problem solving techniques. Pharmaceutical industry, which tackles with “capping” phenomenon, and, metal and ceramic industries that deal with “near-net shape forming” problems are amongst a few examples of such industries. Many experimental and theoretical studies based on continuum mechanics have been carried out to study the macroscopic mechanical responses of compacts [1]. In recent years, the finite element method has also been adopted in simulations of powder compaction [2]. However in continuum models, the microscopic properties that strongly influence the macroscopic behavior cannot be taken account of. To get optimal mechanical properties of green compacts, better knowledge of the relation between powder characteristics and mechanical behavior of the material during compaction is required. The Discrete Element Method, developed by Cundall and Strack [3], is an effective numerical tool to investigate the micro mechanics of granular materials. In DEM, each individual particle of an assembly is modeled separately and its motion is defined from the interactions with neighboring particles. Then microscopic evolution of the internal stress and the structure of the assembly can be obtained. In the mean time, the macro behavior of the whole assembly can be obtained by statistical sum and average the properties of individual particles. In this paper, a discrete element model, which is capable of modeling friction, cohesion and local plastic deformation at the inter-particle contacts, is used to simulate quasi-static uniaxial compaction of particle assembly. The evolution of microscopic properties of the particle assembly, are presented correlated with the bulk behaviors of the compacts. The relations between particle properties in micro level and macroscopic mechanical properties of the compacts are demonstrated by the comparisons of the numerical results. These results may be further incorporated to an appropriate continuum constitutive model, and used to generate the material parameters of the powder compacts which are sometime very difficult to measure in the experiments.

References [1] Walker,D.M., “An approximate theory for pressure and arching in hoppers”, Chem. Eng. Sci., 21, pp. 975-997, 1966. [2] I. Aydin, B. Briscoe, K. Y.Sanliturk, “The internal form of compacted ceramic components: a comparison of finite element modelling with experiments”, Powder Technology, 89, pp.239254, 1996 [3] P.A. Cundall and O.D.L. Strack., “A discrete numerical model for granular assemblies”, Géotechnique , 29, pp. 47¯65, 1979

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

147

Numerical Analysis of Passive Earth Pressures with Interfaces Jim Shiau and Catherine Smith Faculty of Engineering and Surveying University of Southern Queensland, Toowoomba, QLD, 4350, Australia [email protected]

ABSTRACT Elasto-plastic analysis for classical lateral earth pressures is presented in this paper by applying the explicit finite difference method using FLAC. The numerical model presented here consists of a rigid elastic structure for the gravity wall, a zero thickness interface for modelling the sliding and separation, and a Mohr-Coulomb soil model for the backfill. The rigid wall was pushed into the backfill soil to induce passive failure and the ultimate load required for the failure is calculated. Numerical results are compared with other available solutions and a number of modelling issues in creating an accurate model are discussed. It is hoped that, through these experience learnt, some numerical pitfalls can be avoided by practicing engineers in their future analysis.

References [1] Shiau, J. S., Lyamin, A. V. and Sloan, S. W. (2003). "Rigorous Solutions of Classical Lateral Earth Pressures" Proceedings of the 6th Australia-New Zealand Young Geotechnical Professionals Conference, pp162-167. [2] Shiau, J. S., Lyamin, A. V. and Sloan, S. W. (2003). "Bearing capacity of a sand layer on clay by finite element limit analysis" Canadian Geotechnical Journal, 40, 900-915. [3] Shiau, J. S., Merifield R. S. , Lyamin A. V. , and Sloan, S. W. (2005). "Undrained stability of footings on slopes." Journal of Geotechnical and Geoenvironmental Engineering, ASCE (Submitted).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

148

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

149

The computer analysis of the temperature fields arising in bearing node at rotation of a rotor M. Zhuravkov *, S. Bosiakov *, S. Pronckevich † *Belarusian State University [email protected] [email protected] †

Belarusian National Technical University [email protected]

ABSTRACT Thermal calculation for modern high-speed bearings has crucial importance and is reduced to definition of temperature of the bearing, and also those operating conditions of a turbomachine necessary that the temperature in the bearing was kept in allowable limits. It is essential to simplify and lower complexity input of designing bearing nodes of turbomachines in view of the requirements showed to a temperature mode, the computing experiment based on use of engineering packages of the finite-element analysis allows. Authors developed technology of such calculation on the basis of finite-element package LS-DYNA, and computing experiment by definition of a field of temperatures of a rotating rotor arising at friction about the bearing is carried out. As base model the threedimensional model of contact pair the rotor - bearing turbocompressor with rigidly fixed bearing is accepted, between shaft of a rotor and the bearing the contact simulating liquid friction is set. During calculation for different high-speed modes of rotation of a rotor and factors of liquid friction propagations of fields of temperatures, thermoelastic pressure fields and deformations in a rotor and the bearing are obtained, quantitatively described thermoelastic stress-deformed condition of bearing node. The obtained results can be directly used at designing turbocompressors and their modular nodes.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

150

Updating 3D acoustic models with the Constitutive Law Error method. A two-step approach for absorbing material characterization V. Decouvreur*, P. Ladevèze†, Ph. Bouillard* *

Université Libre de Bruxelles, Structural and Material Computational Mechanics, CP 194/5 Roosevelt Av. 50, 1050 Brussels, Belgium [email protected] †

Laboratoire de Mécanique et Technologie, ENS-Cachan Université Paris VI/CNRS, 61 av. Président Wilson, 94235 Cachan, France [email protected]

ABSTRACT In the global framework of improving vibro-acoustic numerical simulations together with the need to diminish the prototyping stages, getting more accurate acoustic models becomes increasingly important for many industries such as automotive companies, for instance. One way to achieve greater accuracy in numerically simulated pressure fields is to make use of parametric updating techniques enabling the tuning of the vibro-acoustic model parameters inside physically meaningful boundaries. The improved model is re-used in the future, allowing more accurate results within reduced simulation times. The updating technique used in this paper is based on the theory of P. Ladevèze [1], and follows from recent works on the constitutive law error method (CLE) applied to acoustics [2]. The updating process focuses on the improvement of the acoustic damping matrix related to the absorbing properties of acoustic materials covering the borders of the acoustic domain. The present part of the paper proposes a 2-step optimization process. It will be shown that this 2-step updating method presents many advantages, especially by diminishing the computation time and allowing the frequency interpolation of the absorbing properties of the materials outside the studied frequency range. While previous works addressed purely numerical setups without experimental data, this study aims at comparing 3D measured pressure fields with numerically updated ones. The test-case is the TRICARMO setup engineered by the LMS company in Leuven, Belgium. The TRICARMO setup is a simplified car cabin with rigid (concrete) walls and car seats inside. Initially roughly described seat material properties are tuned by running the updating simulation process using a library of acoustic absorbing material models.

References [1] P. Ladevèze, A modelling error estimator for dynamic model updating, in: New Advances in Adaptative Computational Methods in Mechanics, Elsevier, 1998, pp. 135–151. [2] V. Decouvreur, Ph. Bouillard, A. Deraemaeker, P. Ladevèze, Updating 2d acoustic models with the constitutive relation error, in: Journal of Sound and Vibration, Vol. 278(4/5), 2004, pp. 773787.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

151

Analysis of Fluid-Structure coupling by Statistical Energy Analysis J.C. García-Andujar*, L. Fritz†, J. López-Díez† * GAMESA Eólica [email protected] †

ETSI Aeronáuticos. Universidad Politécnica de Madrid Plaza del Cardenal Cisneros 3, 28040 Madrid Spain [email protected]

ABSTRACT Fluid-structure interaction has been of high interest during years. A number of different approaches have been developed to estimate the response of structures to acoustic field. The method of analysis fall in two basic categories [1]: conventional and probabilistic. The most popular approach using conventional methods are based in predicting free vibrational modes and then obtain the synthesized forced solution by modal superposition. Within this approach the finite element method also allows direct solution to be calculated. Coupling with a surrounding fluid can be obtained either by BEM or FEM. In the first case, the fluid behavior is formulated by the boundary element method, using the structure as part of the boundary of the fluid field. In the FEM approach, the fluid volume is also studied bay fem, and the coupling with the structural model results obvious. The inconvenient of this approach is the analysis of the response in high frequency ranges. With this approach, both structure and fluid filed can be studied. The probabilistic approach is applied in the cases where a system’s dynamic behavior is not known with sufficient certainty, or the size of the analysis will be to large for a deterministic approach. Statistical Energy Analysis [2] is one of this method and tries to overcome the limitations of the finite element methods. The acoustic problems in building constructions [3] and spacecraft [1] has been successfully studied with this method. SEA presents advantages over the finite element method for studying phenomena in a high frequency band range, but unsteady phenomena require a new formulation; TSEA introduces a new approach to these problems. In the SEA, the vibration loading spectrum is considered to be known. SEA is able to provide estimates of complex system response because its ability to deal with groups of modes on a statistical basis. But in traditional SEA the interest is focused in the response of the structure, and fluid field has no feedback. In the present work, a formulation of SEA considering the fluid as an additional subsystem is presented in order to analyze the problem of fluid-structure interaction. Therefore, response of the structure and the fluid can be studied simultaneously. The presented approach considered the energy flow between the structure and the fluid. The work presents the problem formulation, and then, a plate simply supported inside an acoustic space is solved. Comparison with solution by FEM and FEMBEM are presented, and benefits of SEA in high frequencies range is showed.

References [1] Structural Acoustic Design Manual. ESA PSS-03-204, 1988 [2] R.H. Lyon and G. Maidanik, Statistical Methods in Vibration Analysis. AIAA J., 2, 1015-1024, 1964. [3] R.J.M. Craik, Sound Transmission through Buildings using Statistical Energy Analysis. Ashgate Publishing Ltd, Aldershot england, 1996.

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Efficient Iterative Solution of Time-harmonic Scattering by Objects in Layered Fluid Kazufumi Ito*, Jari Toivanen†

* Center for Research in Scientific Computation North Carolina State University Box 8205, Raleigh, NC 27695-8205, USA [email protected] †Department of Mathematical Information Technology, Agora University of Jyv¨askyl¨a FI-40014 Jyv¨askyl¨a, Finland [email protected].fi ABSTRACT We consider the computation of time-harmonic acoustic scattering by sound-soft or elastic objects in layered media. An example of such problem is the scattering by a mine buried in sediment. The computational domain can be tens or hundreds of meters long while the target requires modeling of details smaller than one centimeter. A discretized problem can have several billion degrees of freedom. We decompose the computational domain into far-field and near-field domains and then we perform a finite element discretization. For the vastly larger far-field domain we use an orthogonal mesh and a preconditioner based on a fast direct solver [5]. For the near-field domain a standard preconditioner can be used. The combination of these two defines a preconditioner for the GMRES method. An essential implementation detail is the possibility to reduce the iterations into a small sparse subspace [3, 4]. Due to this the memory usage is essentially reduced and the GMRES method can be used without restarts. For two-dimensional problems this approach has been considered in [2]. A related approach for non elastic targets in three-dimensional domains has been described in [1]. We present numerical results with two-dimensional and three-dimensional problems demonstrating the efficiency of the proposed approach. For example, we show that it is possible to solve problems with more than a billion degrees fo freedom on a modern PC.

References [1] E. Heikkola, T. Rossi, and J. Toivanen, A parallel fictitious domain method for the threedimensional Helmholtz equation, SIAM J. Sci. Comput., 24, 1567–1588, 2003. [2] K. Ito and J. Toivanen, A fast iterative solver for scattering by elastic objects in layered media, Appl. Numer. Math. To appear. [3] K. Ito and J. Toivanen, Preconditioned iterative methods on sparse subspaces, Appl. Math. Letters. To appear. [4] Y. A. Kuznetsov, Numerical methods in subspaces, Vychislitel’nye Processy i Sistemy II, Nauka, Moscow, 265–350, 1985. In Russian. [5] T. Rossi and J. Toivanen, A parallel fast direct solver for block tridiagonal systems with separable matrices of arbitrary dimension, SIAM J. Sci. Comput., 20, 1778–1796, 1999.

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Computational Aeroacoustics by Coupling the Finite-Element and the Lattice-Boltzmann-Method Barbara Neuhierl*, Ernst Rank† * Dipl. Ing. Barbara Neuhierl Siemens AG, CT PP 6, Otto-Hahn-Ring 6, D-81730 München [email protected] † Prof. Dr.rer.nat. Ernst Rank Lehrstuhl für Bauinformatik, Technische Universität München, Arcisstr. 21, D-80290 München [email protected]

ABSTRACT Computational aeroacoustics is a relatively young field of engineering. Characteristic application domains are for example the area of aviation and railway, internal flows in pipes or HVAC systems, or even consumer products and household devices, where aerodynamic sound is generated e.g. by fans. Sometimes not only the fluid itself causes sound phenomena, but also vibrations of structures excited by the flow like for example the walls of a pipe or components outside cars, trains or airplanes (mirrors, antennas, pantographs, etc.) lead to further generation of noise. This paper deals with the computation of such fluid-structure coupling effects, considering the interaction of fluid flow, acoustic wave propagation and the structural vibrations of a body lying within or enclosing the flow area. As computational aeroacoustics requires an accurate, time-dependent calculation of pressure and density fluctuations, it was decided to use the lattice-Boltzmann method, a ‘mesoscopic’ formulation based on a strongly simplified kinetic theory nevertheless approximating the Navier-Stokes equations, for the CFD part of the problem. In contrary to “classical CFD”, where a macroscopic model is used by solving the discretized Navier-Stokes equations, the lattice-Boltzmann method enables a time explicit and – as far as our experience goes - less time expensive computation of compressible flows. It is thus well suited for representing acoustic phenomena, allowing to concurrently approximate both flow and acoustic field. On the structural side, the finite element method was applied to represent the structural dynamics, where flow pressures are applied as loads onto surfaces of structures. The vibrations in turn lead to acoustic wave propagation within the fluid. Data (i.e. flow pressure and vibration velocity) is exchanged via interfaces programmed for this special purpose, taking into account typical differences of finite element mesh and CFD grid. In this way, a bi-directional coupling procedure was generated enabling the calculation of aeroacoustic effects in connection with coupling effects that can occur if structure and fluid influence each other mutually. As a suitable model for the verification of computational aeroacoustics as well as coupling effects, the flow around a circular cylinder has been chosen. The occurring effects on the fluid as well as on the structure side are well-known and have been the subject of many experiments and simulations. In the wake of the cylinder, for example, the “v. Kármàn vortex streets” are produced which cause narrow-banded sound, the so-called “Aeolian Tones”. This effect, the structural dynamic vibration excited by the flow, as well as the wave propagation caused by the vibrating body were simulated simultaneously. The application of the method to industrial problems will show the suitability of the approach.

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Analysis of a perforated panel for the correction of low frequency resonances in domestic rooms Andrea Panteghini*, Francesco Genna, Edoardo Piana *

Department of Civil Engineering – University of Brescia Via Branze, 38 – 25123 Brescia, Italy {andrea.panteghini,francesco.genna,piana}@ing.unibs.it

ABSTRACT Perforated panels can be used as devices for the selective acoustic correction of rooms. Unlike panels made by porous materials, they work principally at low and middle frequency, and are acoustically inert at high frequency. For this reason, their main application is for the correction of modal resonances of rooms. Perforated panels consist of one layer of porous (dissipative) material and one of air, closed by a perforated plate and placed against a rigid wall. When this device is invested by a sound wave, the inside air resonates at a specific frequency, and a much energy is absorbed near this resonance frequency. Although perforated panels are used at both low and middle frequencies, there seem to be no studies in the literature about their modeling at low frequency, partly because it is very difficult to make experiments in this part of the audio spectrum. In particular, classical literature models, developed thinking to middle frequency behavior [1], [2], [3], do not perform well, in comparison with experimental and numerical results, when applied to study such low frequency devices. Here, we present an analytical and a numerical approach (this last based on the finite element method), developed in [4], for predicting the sound absorption of a perforated absorber system at low frequency. After validating the capabilities offered by the finite element technique, comparisons are made between the results of numerical analyses performed on both complete coupled acousticstructural and simplified models. A new, simple analytical algorithm for the design of these panels, based on the acoustic impedance method, is also presented. Finally, we briefly discuss the results of a method for the experimental measure of the acoustic impedance of perforated panels at low frequency, a method that has never been used to such a purpose, so far, to the best of our knowledge. Our numerical and analytical results are finally compared both with these experimental data and with the results of literature models [1], [2].

References [1] R. H. Bolt, On the design of perforated facing of acoustic materials, Journal of the Acoustical Society of America, 19(5), 917-921, 1947. [2] K. A. Velizhanina, Design calculation of sound absorbers comprising a porous material with a perforated facings, Soviet Physics Acoustics, 14(1), 37-41, 1968. [3] D. Takahashi, A new method for predicting the sound absorption of perforated absorber systems, Applied Acoustics, 51(1), 71-94, 1997. [4] A. Panteghini, Analisi numerica di pannelli per lo smorzamento delle risonanze a bassa frequenza in ambienti di piccole-medie dimensioni, Laurea Thesis (in Italian), Department of Civil Engineering, University of Brescia, Italy, 2005.

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Sound Insulation Provided by a Multi-layer System Containing a Heterogeneity: a BEM Approach Andreia Pereira*, António Tadeu† Department of Civil Engineering, University of Coimbra Pinhal de Marrocos, Polo II, 3030-290 Coimbra, Portugal [email protected]

*

†

Department of Civil Engineering, University of Coimbra Pinhal de Marrocos, Polo II, 3030-290 Coimbra, Portugal [email protected]

ABSTRACT The sound insulation provided by a multi-layer partition with an infinite plane, composed of sets of fluid and elastic layers, dividing an infinite acoustic medium is analysed in this paper. A circular cylindrical fluid-filled heterogeneity is placed inside one of the elastic layers of the partition and the resulting airborne sound insulation and the impact sound pressure level are calculated. The solution is obtained using a Boundary Element Method (BEM) formulation in the frequency domain, where only the discretization of the heterogeneity’s surface is required, since Green’s functions are used for the multi-layered media. The model is excited by cylindrical line loads placed in either the acoustic or the elastic medium. Material losses are taken into account by means of complex velocities. A similar model has been developed by the authors in previous work, in that case to predict the sound insulation provided by single partitions in the presence of rigid, free, fluid-filled and elastic-filled heterogeneities. It is extended here to handle multi-layer systems. Several simulations are performed to see if the presence of fittings such as pipes, which are often housed inside building partitions, may affect the sound insulation. The analysis assumes multi-layer partitions, such as those built with two elastic layers sandwiching an air gap, and floating systems composed of a concrete screed resting on a cork layer laid over a structural concrete layer. An air-filled heterogeneity is placed inside one of the elastic layers. Responses are compared with those provided by the solution without the presence of the heterogeneity. The analysis allows the conclusion that the sound insulation is slightly affected by the presence of the heterogeneities inside multi-layer systems.

References [1] A. Tadeu and J. António, 2.5D Green functions for elastodynamic problems in layered acoustic and elastic formations, Journal of Computer Modeling in Engineering and Sciences CMES, 2, 477-495, 2002. [2] M. H. Aliabadi, The Boundary Element Method, John Wiley & Sons Ltd, England, 2002. [3] A. Pereira and A. Tadeu, Sound pressure reduction provided by an infinite elastic layer containing heterogeneities via BEM, Proceedings of the 6th congress on Boundary Element Techniques, 193-198, 2005. [4] A. Pereira and A. Tadeu, Sound pressure level differences due to the presence of heterogeneities inside an elastic partition, Proceedings of the 12th International Congress on Sound and Vibration, 2005.

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An efficient wave based method for steady-state vibro-acoustic transmission calculations B. Pluymers∗, W. Desmet, D. Vandepitte, P. Sas ∗K.U.Leuven, Department of Mechanical Engineering, division PMA

Celestijnenlaan 300B, B-3001 Leuven, Belgium. [email protected] ABSTRACT Most commonly, element based methods like the finite, the infinite and the boundary element method are applied for analyzing steady-state vibro-acoustic problems in unbounded domains. The finite element method and the infinite element method discretize the considered problem domain and approximate the dynamic response variables with approximating shape functions. The boundary element method also applies approximating shape functions, but has the advantage that only the boundary of the considered problem needs to be discretized. Due to the selection of approximating shape functions, the discretization has become more dense with increasing frequency, which restricts the applicability of the element based methods to the lower frequency range. Transmission calculations make out an important class of problems in unbounded domains. An aperture or a structure is positioned in an infinitely large rigid baffle plane, which divides the unbounded space into two semi-infinite subdomains. This allows determination of the transmission characteristics of the considered aperture or structure. Transmission calculations are most commonly performed with the boundary element method and the infinite element method. The wave based method [1] has proven to be an efficient alternative simulation technique for steadystate vibro-acoustic problems. In recent years, both bounded [2] and unbounded [3] problems have been successfully tackled. The wave based method is based on the indirect Trefftz approach, in that the dynamic response variables are expanded in terms of wave functions, which are exact solutions of the underlaying differential equations -contrary to the element based methods where approximating shape functions are applied- such that no fine discretization of the problem domain or its boundary is required. The resulting numerical wave models are much smaller than the element based models and they exhibit an enhanced computational efficiency. These properties allow the wave based method to tackle problems at higher frequencies than the element based methods. In this paper, the wave based method is applied for transmission calculations. In the first part of the paper, the selection of a set of converging wave functions and the subsequent construction of the numerical model are discussed. In a second part, the method is applied to a two-dimensional numerical example and compared to the element based techniques.

References [1] W. Desmet A wave based prediction technique for coupled vibro-acoustic analysis, KULeuven PhD. Thesis 98D12, Leuven (1998). [2] W. Desmet, B. Van Hal, P. Sas, D. Vandepitte, A computationally efficient prediction technique for the steady-state dynamic analysis of coupled vibro-acoustic systems, Advances in Engineering Software, Vol. 33, (2002), 527-540. [3] B. Pluymers, W. Desmet, D. Vandepitte, P. Sas On the use of a wave based prediction technique for steadystate structural-acoustic radiation analysis, Journal of Computer Modeling in Engineering & Sciences (CMES), 7(2), 173-184, (2005).

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Modeling Sound Radiation by Structures Caused by a Ground Impact Load: a BEM Approach Paulo Santos*, António Tadeu* Dep. of Civil Eng., University of Coimbra Pólo II – Pinhal de Marrocos, 3030-290 Coimbra, Portugal [email protected] *

ABSTRACT This paper is part of research work whose the main objectives are to model ground vibration induced by rail traffic on the surface or underground, and to predict the acoustic radiation caused by vibration in building structures, using the Boundary Elements Method (BEM). This paper proposes a BEM formulation which is implemented to model the propagation of waves in an elastic homogeneous halfspace confined by a fluid (air), where the waves are produced by steady state, spatially sinusoidal, harmonic line structure-borne loads at low frequencies. The BEM model is formulated in the frequency domain and fully accounts for the air-solid interaction. The required two-and-a-half-dimension fundamental solution (Green’s functions) and stress functions in Cartesian co-ordinates for the elastic and fluid media can be found in [1]. The BEM integrations are performed analytically for the loaded element [2, 3], whereas a Gaussian quadrature scheme is used when the element to be integrated is not the loaded element. This paper describes the analysis as carried out in the frequency domain, for waves that travel perpendicularly to the z-axis (pure 2D problem). The results show the influence of: the impact source position; the wall material stiffness, and the impact source direction - vertical (y) or horizontal (x). The relationship between the normal vibration velocity of the vibrating structure and the acoustic energy radiated by this same structure (vertical wall) is also analyzed. At this early stage of the research the authors modeled a homogeneous elastic flat half-space, excited by an applied impact line load, and studied the acoustic radiation provided by a simple structure (vertical wall). The future developments will include a railway tunnel and also a more realistic (complex) building structure. Comparison of the numerical results with some experimental data is another future objective.

References [1] A. Tadeu and J. António, 2.5D Green’s functions for elastodynamic problems in layered acoustic and elastic formations. Computer Modeling in Engineering and Sciences, 2(4), 477-495, 2001. [2] A. Tadeu, P. Santos, E. Kausel, Closed-form integration of singular terms for constant, linear and quadratic boundary elements – Part I: SH wave propagation. Eng Anal Boundary Elements, 23(8), 671–681, 1999. [3] A. Tadeu, P. Santos, E. Kausel, Closed-form integration of singular terms for constant, linear and quadratic boundary elements – Part II: SV–P wave propagation. Eng Anal Boundary Elements, 23(9), 757–768, 1999.

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Cohesive Model of Electromechanical Fatigue for Ferroelectric Materials and Structures Irene Arias∗, Santiago Serebrinsky, Michael Ortiz Division of Engineering and Applied Science, California Institute of Technology 1200 E. California Blvd, Pasadena, CA 91125, USA [email protected], [email protected] ∗Dep. de Matem`atica Aplicada III, Universitat Polit`ecnica de Catalunya

Jordi Girona 1-3, Barcelona 08034, Spain [email protected] ABSTRACT Ferroelectric materials are extensively used in a variety of sensor and actuator applications, where the non-linear coupling between mechanical and electrical fields are of primary interest. They are also a promising set of materials for improved dynamic as well as non-volatile memory devices, where only the electrical properties are directly exploited. However, ferroelectrics are brittle, and their low fracture toughness (in the order of 1 MPa m1/2 ) makes them susceptible to cracking. In addition, ferroelectric materials exhibit electrical fatigue (loss of switchable polarization) under cyclic electrical loading and, due to the strong electro-mechanical coupling, sometimes mechanical fatigue as well. Conversely, the propagation of fatigue cracks hinders the performance of the devices and raises serious reliability concerns. Despite recent experimental and modelling advances, the precise nature of the interactions between fracture, deformation and defect structures underlying ferroelectric fatigue is in need of further elucidation, and a physics-based multiscale model enabling the prediction of the fatigue life of ferroelectric devices is yet to emerge. Therefore, there remains a need for phenomenological and empirical models that can be experimentally validated and used in engineering design. We present a model of electro-mechanical ferroelectric fatigue based on the postulate of a ferroelectric cohesive law that: couples mechanical displacement and electric-potential discontinuity to mechanical tractions and surface-charge density; and exhibits a monotonic envelope and loading-unloading hysteresis [1]. The model is applicable whenever the changes in properties leading to fatigue are localized in one or more planar-like regions, modelled by the cohesive surfaces. We validate the model against experimental data for a simple test configuration consisting of an infinite slab acted upon by an oscillatory voltage differential across the slab and otherwise stress free. The model captures salient features of the experimental record including: the existence of a threshold nominal field for the onset of fatigue; the dependence of the threshold on the applied-field frequency; the dependence of fatigue life on the amplitude of the nominal field; and the dependence of the coercive field on the size of the component, or size effect. Our results, although not conclusive, indicate that planar-like regions affected by cycling may lead to the observed fatigue in tetragonal PZT.

References [1] I. Arias, S. Serebrinsky, M. Ortiz, A phenomenological cohesive model of ferroelectric fatigue, Acta Materialia, 54, 975–984, 2006.

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Simulation of active systems in a NVH full car model H. Atzrodt*, S. Herold , D. Mayer Fraunhofer Institute for Structural Durability and System Reliability, LBF Bartningstr. 47, 64289 Darmstadt, Germany [email protected]

ABSTRACT Vibrations in the higher frequency range in cars can result in disturbing noise and reduction of comfort. An active interface or a force actuator mounted between suspension and chassis can reduce these vibrations. The simulation of the full car model with the active interface or a force actuator is essential for the dimensioning of the active system, development of the controller, testing of the functionality, estimation of the performance and monitoring of the system reliability. An active interface has been developed and is currently improved in the Fraunhofer FASPAS project [1]. With the simulation of the active Interface first experiences can be gathered even before the interface is integrated in the real structure. In this case the force actuator is composed of smart actuators based on piezoceramic materials. The application of a controller should reduce disturbing vibrations. The simulation is a helpful software tool for selection of the smart actuators and for construction of the force actuator. In this paper a NVH full car model based on FEM and MBS will be conducted. The suspen-sion will be designed in the MBS and embedded in the chassis, which is imported with its elastic eigenmodes from the FEM into the MBS. After integration of the active systems [2] into the MBS model, the controller [3] will be imported from Computer Aided Control Engineering (CACE). The potential of the active systems will be examined by means of a complete system simulation of the active NVH full car model.

References [1] T. Melz, M. Matthias, The Fraunhofer MaVo FASPAS for smart systems design for automotive and machine tool engineering. 12th SPIE International Symposium, San Diego, California, 2005. [2] M. Thomaier, H. Atzrodt, S. Herold, M. Mayer, Simulation of a Complete System Using the Example of an Active Interface for Vibration Reduction. Virtual Product Development in Automotive Engineering, Mürzzuschlag, Österrreich, 2004. [3] A. Preumont, Vibration Control of Active Structures - An Introduction 2nd edition. Kluwer Academic Publishers, Dordrecht Boston London, 2002.

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Toward an exhaustive modeling of the macroscopic behaviour of shape memory alloys Ferdinando Auricchio∗ , Lorenza Petrini† , Alessandro Reali∗ ∗

Dipartimento di Meccanica Strutturale, Universit`a degli Studi di Pavia Via Ferrata 1, 27100 Pavia (PV), Italy [email protected] [email protected] † Dipartimento

di Ingegneria Strutturale, Politecnico di Milano Piazza Leonardo da Vinci 32, 20133 Milano (MI), Italy [email protected]

ABSTRACT The employment of shape memory alloys (SMA) in a large number of applications in the fields of aeronautical, biomedical and structural engineering constitutes the basis for an increasing interest of researchers in the direction of an exhaustive modeling of their macroscopic behaviour. However, many SMA models present in the literature consider a perfect pseudo-elastic behaviour (i.e. no residual strains), which is proved by experiments to be only an approximation. This work takes its basis from the one-dimensional version of a model proposed by Auricchio et al. [1] which is able to describe main SMA macroscopic effects as pseudo-elasticity and shape-memory effect as well as to include permanent inelasticity and degradation effects. Such a model is here improved with the introduction of an elastic modulus which depends on the reached transformation strain level. Finally, some numerical tests are presented in order to show the good performance of the obtained model.

References [1] F. Auricchio, A. Reali, U. Stefanelli, A three-dimensional model describing stress-induced solid phase transformation with permanent inelasticity. To appear on International Journal of Plasticity, 2006.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Vibration Damping Using Resonant Shunted Shear-Mode Piezoceramics Ayech Benjeddou *, Julie A. Ranger † * Institut Supérieur de Mécanique de Paris 3 rue Fernand Hainault, 93407 Saint Ouen CEDEX, France [email protected] †

Institut Supérieur de Mécanique de Paris 3 rue Fernand Hainault, 93407 Saint Ouen CEDEX, France [email protected]

ABSTRACT Vibration damping using shunted piezoceramics was the focus of many researches, in particular during the last decade, as attested by the frequent surveys [1-3]. Most investigations have concerned the classical transverse-mode of piezoceramics which uses the material electromechanical coupling coefficient (EMCC) k31. However, the longitudinal-mode, exploiting the highest EMCC k33, has been much less investigated. Both modes provide the so-called Extension Shunted Damping (ESD) [4]. The shear-mode, which uses the EMCC k15 with a value very close to k33, has been studied only recently [5]. It has been shown that it provides the so-called Shear Shunted Damping (SSD). The ESD has been investigated with resistive, resonant or capacitive shunts; while the SSD has been evaluated only with a resistive shunt. It is then the objective of this proposal to present, for the first time, the theoretical formulation and finite element investigation of the use of the shear-mode piezoceramics for resonant shunted vibration damping. Preliminary results for the first two modes of a vibrating aluminium beam with sandwiched (axially polarized) piezoceramic patches indicate that, unlike ESD, resonant shunting does not enhance much the SSD compared to the resistive shunting. Moreover, it is found that resistive SSD is almost as efficient as a resonant ESD. The latter result is very interesting since the resistive shunting is easier to implement (no tuning and no huge unphysical inductance) than the resonant one.

References [1] G. A. Lesieutre, Vibration damping and control using shunted piezoelectric materials. The Shock and Vibration Digest, 30, 187-195, 1998. [2] M. Ahmadian and A. P. DeGuilio, Recent advances in the use of piezoceramics for vibration suppression, The Shock and Vibration Digest, 33, 15-22, 2001. [3] S. O. R. Moheimani, A survey of recent innovations in vibration damping and control using shunted piezoelectric transducers, IEEE Transactions on Control Systems Technology, 11, 482494, 2003. [4] A. Benjeddou, J.-A. Ranger, Simple finite element modelling and performance evaluation of passive vibration damping using shunted piezoceramics. In First International Congress on Design and Modelling of Mechanical Systems, March 23-25, 2005, Hammamet, Tunisia. [5] A. Benjeddou, J.-A. Ranger, Use of shunted shear-mode piezoceramics for structural vibration passive damping, Computers and Structures, accepted.

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Optimal control of piezoelectric anisotropic plates I. N. Figueiredo∗, G. Stadler ∗ ∗Department of Mathematics, University of Coimbra

Apartado 3008, 3001-454 Coimbra Portugal [email protected] & [email protected]

ABSTRACT Piezoelectric materials deform when they are subject to an electric field. This behavior, known as inverse piezoelectric effect, can be used to design smart materials. In order to obtain the desired behavior of these materials, one can either combine numerous simulations with physical intuition, or utilize an optimization-based method. We apply the latter approach to a recently derived model for piezoelectric anisotropic plates (see [1]). To be precise, we formulate the problem as optimal control problem, i.e., we search for an applied electric field that generates a deformation of the plate, which is as close as possible to the given desired deformation. This is done by introducing a certain cost functional involving both the electric field and the corresponding deformation. The optimal electric field is then characterized as minimizer of that functional. To model physical limitations we additionally impose bound constraints on the electric field. The resulting nonlinear system of coupled partial differential equations is discretized using finite elements and solved using a generalized Newton method (see [2]). In a comprehensive numerical study we consider, among others, plates composed of two layers of different piezoelectric materials. We interpret the derived optimal electric fields, investigate the influence of constraints imposed on these fields and discuss the performance of our algorithms.

References [1] I. N. Figueiredo and C. F. Leal: A Piezoelectric Anisotropic Plate Model. Asymptotic Analysis, 44, 3–4, 327–346, 2005. [2] M. Hinterm¨uller, K. Ito and K. Kunisch, The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim, 13(3), 865–888, 2003.

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Residual Internal Forces in Stiffened Thermal-Bimorph Actuator after Forming Process Wiktor L. Gambin*, A. Zarzycki† *

Warsaw University of Technology Poland [email protected]

†

Warsaw University of Technology Poland [email protected]

ABSTRACT

Thermal-bimorph (bimaterial) actuators are basic elements of MEMS ciliary arrays used for moving and positioning of small objects [1-3]. The arrayed actuators are deformable microstructures that may curl into and out of the substrate plane. Objects placed on the array are moving on the tips of actuators cyclically going down and up. Motion of the ciliary actuators is due to different thermal expansion of two polyimide layers with an integrated heater resistor between them. When an electric current is passed through the heater resistor, the temperature of the actuator increases and the structure, initially deflected out-of-plane, deflects downward. The above process requires of the actuators to generate not only large deflections but also a considerable force. The triangle-shaped actuator proposed in the paper [3] is such a structure. Two perpendicular bars rigidly connected at the moving tip and firmly fixed to the substrate plane at the unconnected ends are a stiffened frame structure with additional kinematical constrains. These constrains generate interaction forces between the connected beams during the forming process. In turn, the interaction forces have influence on the internal forces generated during the loading process. In the presented article, the triangle-shape actuator is considered as initially flat, layered frame structure, which becomes the curved 3-D stiffened structure during the cooling process. The cooling down beams are bent in two orthogonal planes not only due to change of the temperature, but also due to additional kinematical constrains. Relations between bending moments, curvature variations and temperature changes enable to find the residual internal forces on the purely analytical way.

References [1] N. Takeshima and H. Fujita, “Polyimide bimorph actuators for a ciliary motion system”, ASME WAM Symp. Micromech. Sensors, Actuators, and Systems, DSC-32, 203-209, 1991. [2] J. W. Suh, S. F. Glander, R. B. Darling, C. W. Storment, and G. T. A. Kovacs, “Organic thermal and electrostatic ciliary microactuator array for object manipulation”, Sensors and Actuators, A: Physical, 58, 51-60, 1997. [3] J. W. Suh, R. B. Darling, K. –F. Boehringer, B. R. Donald, H. Baltes, and G. T. A. Kovacs, “CMOS integrated ciliary actuator array as a general-purpose micromanipulation tool for small objects”, J. Micromech. Syst., 3, 4, 483-496, 1999.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Spectral Level Set Methodology in the Optimal Design of Adaptive Aeroelastic Structures Alexandra A. Gomes∗, Afzal Suleman ∗ ∗ IDMEC, Instituto Superior T´ecnico Department of Mechanical Engineering Av. Rovisco Pais 1049-001 Lisbon, Portugal [email protected], [email protected]

ABSTRACT This paper proposes a topology optimization approach to the design of adaptive aeroelastic structures. We describe the optimal design of a morphing airfoil for increased roll maneuvering performance using the spectral level set methodology. Structural topology optimization is considered to be the next step in achieving better designs, following size and shape optimization, because the topology of the set constituting the structure is allowed to change. That is, holes may appear or disappear, and breakages or merges of the structure may occur. The level set methods are now an established framework to formulate problems in topology optimization. These methods consider the structural boundary to be an interface, which is modelled as the zero level set of a function. The evolution of this function during the optimization implicitly describes the evolution of the structural interface. The main advantage of this approach is the easy description of topology changes. However, the classic level set function evolves through a partial differential equation which requires specific numerical techniques to yield the correct solution, given the nodal values of the function on a spacial grid. Here, we take a frequency-domain approach to circumvent this disadvantage: the function is expanded into a Fourier series and the first few Fourier coefficients become the design variables assigned to structural definition. Another advantage of this approach, known as spectral level set methodology, compared to classic level set methods is the nucleation of new holes in the interior of the interface. For completeness, we provide an upper bound for the total error, which includes an estimate of the error committed in truncating the Fourier series. Benchmark examples in structural topology optimization, such as the design of short and long cantilevers for maximum stiffness given an amount of material, have already validated the proposed methodology. In this work, we study the problem of aileron control reversal. In particular, we are interested in generating additional lift to counteract the loss of aileron effectiveness due to the elastic properties of the wing. We consider an airfoil with a system of actuators set along its chord. These actuators operate on the airfoil mean camber line to produce additional lift. The airfoil’s morphing capability arises from the changes in its camber line. The optimization problem consists in determining the state of actuation which minimizes the actuation power while generating a prescribed amount of lift. The domain of the level set function is the airfoil’s chord: if the function is positive, the actuator is active and, at that location, the camber is increased by a positive constant; if the function is negative, the actuator is off. We show the topology of the optimum set of activated actuators is distinct from the topology of the initial guess and that the spectral level set methodology can adequately handle the occurrence of spike-like functions with a very small number of Fourier coefficients.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Nondestructive identification of defects for smart plates in bending using genetic algorithms Daniela G. Marinova†, Dimitar H. Lukarski †, Georgios E. Stavroulakis *, Emmanuel C. Zacharenakis& *

Dept. of Production Engineering and Management, Technical University of Crete, Chania, Greece [email protected] †

Faculty of Applied Mathematics and Informatics, Technical University, Sofia, Bulgaria [email protected], [email protected]

&

Department of Civil Engineering, Technological Educational Institute of Crete, Heraklion, Greece [email protected]

ABSTRACT A defect identification problem for elastic plates with damages is studied in this paper. The classical theory of thin plates in bending is applied for the modelling of the mechanical structure. Bonded piezoelectric sensors and actuators are considered in a simplified way. Within classical finite element discretization, cracks and other defects are modelled in a smeared-crack sense by reducing the stiffness of the corresponding element. According to the simplicity of the model, only the position of the possible crack is considered to be the unknown. The most general case of dynamical excitations is considered. The identification is based on dynamic non-destructive loading. A suitable error norm is used to transform the defect identification problem to a nonconvex output error optimisation problem [1], [2]. The difference between the dynamic response of the plate in bending with and without cracks demonstrates the areas of the plate where the influence of the cracked element is higher. The position of the defect, which is unknown in real applications, can be computed from the solution of the arising optimisation problem. This problem has been solved by a genetic algorithm. The investigation includes the effect of different types of boundary conditions. In all cases the error function, which depends on the boundary conditions and the position of the defect, may have local minima and one global minimum, exactly at the position of the real crack. As a general observation, the boundary conditions, the loading and the shape of the structure, as well as the position of the measurement points and the duration of the considered time interval influence the shape of the error function. Numerical results show that by using the genetic algorithm the crack position can be found. Moreover, for a relatively high population size (about 15-20) the probability of finding the crack position on the first few steps is very large.

References [1] G.E. Stavroulakis, Inverse and crack identification problems in engineering mechanics. Kluwer Academic Press, Dordrecht, Boston, London, 2000. [2] G. Liu, X. Han, Computational Inverse Techniques in Nondestructive Evaluation. CRC Press, Boca Raton, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Modeling Shape Memory Alloy Plane Truss Structures using the Finite Element Method Juliana Hoyer Insaurrauld Pereira*, Mauricio Rangel Pacheco*, Pedro Manuel Calas Lopes Pacheco*, Ricardo Alexandre Amar de Aguiar*, and Marcelo Amorim Savi† * 1 CEFET/RJ - Department of Mechanical Engineering Av. Maracanã, 229, 20271-110 - Rio de Janeiro - RJ - Brazil [email protected], [email protected], [email protected], [email protected] †

UFRJ/COPPE – Department of Mechanical Engineering 21.945.970 – Rio de Janeiro – RJ, P.O. Box 68.503 - Brazil [email protected]

ABSTRACT The remarkable characteristics of shape memory alloys (SMA) have been responsible for the increasing interest in different applications varying from biomedical to aerospace industry. During the recovering process of a SMA component, large loads and/or displacements can be generated in a relatively short period of time making this component an interesting mechanical actuator. The thermomechanical behavior of SMA is related to phase transformations induced by stress and/or temperature variations. This article deals with the modeling and simulation of SMA plane truss structures using an anisothermal constitutive model with internal variables that includes three macroscopic phases in the formulation: two variants of martensite and an austenitic phase. A numerical procedure is developed based on the operator split technique associated with an iterative numerical scheme in order to deal with nonlinearities in the formulation. With this assumption, coupled governing equations are solved considering three uncoupled problems: thermal, thermoelastic and phase transformation behaviors. The thermal problem comprises a one-dimensional conduction problem with surface convection. Finite element method is employed for spatial discretization. The thermo-elastic problem evaluates stress and displacement fields from temperature distribution, employing the nonlinear finite element method. Finally, the phase transformation problem determines the phase transformation fields considering the phase evolution in the process. Numerical simulations treats SMA plane truss structures subjected to thermomechanical loadings. The analysis considers results obtained for two models: uncoupled and coupled. The uncoupled model neglects the thermomechanical couplings, corresponding to the rigid body energy equation. The coupled model considers the latent heat associated with phase transformation as a source in the energy equation. Results show that the proposed model captures the general behavior of SMAs, allowing the description of adaptive trusses with large displacements. Moreover, they show the some situations where the thermomechanical coupling plays an important role in the description of SMA behavior.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Evaluation of NiMnGa Magnetic Shape Memory Alloys Using Cellular Neural Networks Gursev Pirge*, Niyazi Kılıç†, Osman N. Uçan†, Sabri Altıntaú* *

†

Department of Mechanical Engineering, Bo÷aziçi University 34342, Bebek, Istanbul, Turkey [email protected], [email protected]

Department of Electrical&Electronics Engineering, Istanbul University 34320, Avcılar, Istanbul, Turkey [email protected], [email protected]

ABSTRACT In this study, CNN–RPLA (Cellular Neural Network-Recursive Perception Learning Algorithm) was used to enhance the Scanning Electron Microscope (SEM) microstructural images of NiMnGa magnetic shape memory (MSM) alloys. Since their introduction in 1988 by Chua, CNN has attracted a lot of attention. Not only from a theoretical point of view these systems have a number of attractive properties, but also furthermore, there are many well-known applications like image processing, motion detection, pattern recognition, simulation. Here, CNN approach was used to enhance SEM images using CNN approach to analyse the various phases in the austenitic phase. MSM alloys are a class of actuator materials which produce strain via the magnetic field induced reorientation of twin variants in response to an applied magnetic field. MSM effect depends on the success of parameters like mobility of twin boundaries, magnetic properties of the alloy, proper specimen orientation and specimen shape. In order to produce a strong MSM effect, it is necessary to obtain a uniform martensite in the entire sample. The martensite transformation temperature is very sensitive to the alloy composition. A single phase having a homogeneous composition is thus desirable in order to avoid having different components of the microstructure undergoing the martensite transformation at different temperatures, or not undergoing the transformation at all in the case of the eutectic structures and Mn-rich particles. The results showed that as solidified, off-stoichiometric alloys had three distinct microstructural features—a Heusler phase, a Mn Rich phase and a eutectic or eutectoid region. The latter could be removed by prolonged annealing at elevated temperatures, but that the coarse Mn-rich particles are much more difficult to remove. This phase is likely harmful to the mechanical properties of the alloy and should be eliminated in the future.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Modelling of fibre Bragg grating sensor plates C.A. Ramos†, R. de Oliveira*, R.D.S.G. Campilho‡, A.T. Marques‡ †

Instituto Superior de Engenharia do Porto (ISEP), Rua Dr. António Bernardino de Almeida 431, 4200-072, Porto, Portugal [email protected]

*

Instituto de Engenharia Mecânica e Gestão Industrial (INEGI), Rua do Barroco, 174, 4465-591, Leça do Balio, Portugal. [email protected] ‡

Departamento de Engenharia Mecânica e Gestão Industrial, Faculdade de Engenharia da Universidade do Porto (FEUP), Rua Dr. Roberto Frias, s/n 4200-465 Porto, Portugal [email protected]

ABSTRACT In this paper is presented the design of a composite sensor plate, based on fibre Bragg grating (FBG) sensors, to be used as alternative to the conventional electric strain gages. The use of such composite sensor plate will permit to profit of FBG sensors advantages compared to conventional electrical strain gages increasing its safety considering that they are commonly simply surface mounted in structure using adhesives. Here, contrarily to what is traditionally performed, FBG sensors are embedded in thin (0.4 mm of thickness) weaved carbon fibre reinforced composite plates. The FBG sensor is hence not oriented parallel to the surrounding reinforcement. Such configuration is known to induce changes on the reflected optical power spectrum of the FBG sensor resulting in an inaccurate answer. A methodology was then developed to guarantee that the resulting FBG sensor optical power spectrum didn’t exhibit any aspect change. The resulting composite sensor plate was mounted on the surface of a composite specimen which was submitted to tensile loading. The axial strain answer of the composite plate sensor is compared to the answer of an electrical strain gage. A three dimensional finite element study is proposed to understand the relation between the strain measured by the FBG sensor and the strain imposed to the host plate and especially the consequence of using such composite sensor plate on the behaviour of tensile test specimen. The benefits and drawbacks of the use of composite sensor plate are then discussed in light of these results.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Refined Finite Element Model for Vibration Analysis of Sandwich Beams with Shear Piezoelectric Actuators and Sensors Marcelo A. Trindade*, Ayech Benjeddou† *

Department of Mechanical Engineering, São Carlos School of Engineering, University of São Paulo, Av. Trabalhador São-Carlense, 400, 13566-590 São Carlos, Brazil [email protected] †

Laboratoire d'Ingénierie des Systèmes Mécaniques et des Matériaux, Institut Supérieur de Mécanique de Paris, 3 rue Fernand Hainault, 93407 Saint Ouen, France [email protected]

ABSTRACT High stresses and possible impacts of surface-mounted piezoelectric actuators can be alleviated by embedding axially poled piezoceramic actuators between two elastic layers. In this case, the application of a transverse electric field induces shear deformation of the actuator thus generating the desired sandwich structure deflection. Although this so-called shear actuation mechanism seems quite promising for structural control, its modelling is still an open issue. For piezoelectric shear actuated sandwich beams, either the classical sandwich theory (CST) or first and higher-order equivalent single layer (ESL) theories were used. However, ESL theories fail to model localized core shear deformations, which are determinant to correctly evaluate the shear actuation of the sandwich beam. This work extends a previously presented refined sandwich beam finite element (FE) model [1] for vibration analysis. The mechanical model is a refinement of CST, for which the core is modelled with a third-order shear deformation theory [2]. The FE model is developed considering, through the beam length, electrically: constant electric difference of potentials for the piezoelectric facing and core layers and quadratic third-order variable of the electric potential in the core, while mechanically: linear axial displacement and quadratic bending rotation of the core, and cubic transverse displacement of the sandwich beam. Despite the refinement of mechanical and electrical behaviours of the piezoelectric core, the model leads to the same number of degrees of freedom (dof) as the previous CST one due to a two-step static condensation of the internal dof (bending rotation and core electric potential third-order variable). The proposed FE model is validated through the comparison with numerical and experimental results [3]. Results confirm that the TSDT and the induced cubic electric potential yield an extra stiffness to the sandwich beam.

References [1] Trindade, M.A. and Benjeddou, A. Refined sandwich finite element model for smart beams with shear piezoceramic actuators and sensors, in II Eccomas Thematic Conference on Smart Structures and Materials, C.A. Mota Soares et al. (eds.), Lisbon, July 2005. [2] Reddy, J.N., A simple higher-order theory for laminated composite plates, Journal of Applied Mechanics, 51(4):745-752, 1984. [3] Baillargeon, B.P. and Vel, S.S. Active vibration suppression of sandwich beams using piezoelectric shear actuators: experiments and numerical simulations. Journal of Intelligent Material Systems and Structures, 16(6):517–530, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Thermoelectromechanical Response of a Parallel Crack in a Functionally Graded Piezoelectric Strip S. Ueda*, H. Kondo* *

Department of Mechanical Engineering, Osaka Institute of Technology 5-16-1 Omiya, Asahi-ku, Osaka, 535-8585, JAPAN [email protected]

ABSTRACT Possible use of distributed actuators and sensors are discussed in the light of smart or intelligent material technology. An area where smart material may have an advantage is to sense thermally induced distortions and to adjust for adverse thermomechanical conditions. As a result, several analytical studies concerned with piezothermoelasticity of homogeneous materials were reported. Recently, with the help of the development in modern material processing technology, functionally graded piezoelectric materials (FGPMs) have been developed to improve their reliability, and the electromechanical fracture of the FGPM has received much attention. Moreover, so much attention has been focused on the thermoelectromechanical fracture analysis of homogeneous piezoelectric materials. However, because the FGPMs are just an emerging class of piezoelectric materials, there are still very few articles considering the fracture problem in an FGPM under the thermoelectromechanical loadings. For example, Wang and Noda investigated the thermally induced fracture of a smart functionally graded composite structure, and the present author also studied the fracture of a functionally graded piezoelectric strip due to the thermal load. In this paper, the mixed-mode thermoelectromechanical fracture problem for a functionally graded piezoelectric material strip is considered. It is assumed that the thermoelectroelastic properties of the strip vary continuously along the thickness of the strip, and that the strip is under the thermoelectric loadings. The crack faces are supposed to be insulated thermally and electrically. By using the Fourier transform, the thermal and electromechanical problems are reduced to a singular integral equation and a system of singular integral equations, respectively, which are solved numerically. Numerical calculations are carried out, and detailed results are presented to illustrate the influence of the crack length, the crack location and the material nonhomogeneity on the stress and electric displacement intensity factors. The temperature-stress distributions are also presented. The following facts can be found from the numerical results. Firstly, the normalized intensity factors are under the great influence of the geometric parameters. Secondly, the intensity factors of crack near the free surfaces due to the thermal load are not so large. This phenomenon is quite different from the intensity factors under the pure electric load. Finally, the effect of the material nonhomogeneity on the intensity factors depends on the geometric parameters.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Multi-Scale Finite Element Modeling of Piezoelectric Materials by A Crystallographic Homogenization Method Yasutomo Uetsuji*, Eiji Nakamachi *

Osaka Institute of Technology 5-16-1 Omiya, Asahi-ku, Osaka 535-8585 [email protected]

ABSTRACT Polycrystalline piezoelectric materials have been used for actuators or sensors as a component of various electronic and mechanical devices. Higher performance is becoming required for piezoelectric materials as they are applied to new technological fields such as micro/nano machines. Each crystal grain in piezoelectric materials shows strong anisotropy in mechanical and electrical behaviors. Macroscopic properties of polycrystalline piezoelectric materials are dominated by microscopic inhomogeneous crystal morphology. Therefore, polycrystalline piezoelectric materials have a large possibility to exhibit higher performance in a macroscopic scale by design of crystal morphology in a microscopic scale. In this study, a multi-scale finite element modeling by “crystallographic homogenization method” is proposed to estimate macroscopic properties considering microscopic inhomogeneous crystal morphology, and to evaluate microscopic behaviors in response to macroscopic external loads as shown in Figure 1. The computational examples of homogenization and localization are presented for a typical piezoelectric material, barium titanate (BaTiO3) with various distributions of crystal orientations. And then, its crystal orientation distribution has been optimized by steepest decent method to maximize macroscopic piezoelectric strain constants. Computational results indicated that piezoelectric strain constants d33 and d31 increase 34% and 180% respectively, compared with conventional piezoelectric polycrystals. It should be emphasized that the optimized polycrystals have higher piezoelectric performance than single crystals.

Figure 1 Hierarchical structure of polycrystalline piezoelectric materials.

References [1] Y. Uetsuji, Y. Nakamura, S. Ueda and E. Nakamachi, Numerical investigation on ferroelectric properties of piezoelectric materials using a crystallographic homogenization method. Modelling Simul. Mater. Sci. Eng., 12, S303-S317, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Vibration Control of A Laminated Composite Plate Subjected to Blast Loading H. Uyanık*, Z. Mecitoğlu† * Turkish Air Force Academy Yeşilyurt, 34149 İstanbul, Turkey [email protected] † İstanbul Technical University Maslak, 34469 İstanbul, Turkey [email protected]

ABSTRACT In this numerical study, vibrations of a cantilevered composite plate subjected to blast loading are suppressed by the use of piezoelectric actuators. Friedlander’s exponential decay function is used for expressing the blast load model. Some parameters of the function are obtained from the experimental studies. A semiloof shell element is developed in order to account for piezoelectric effects. The composite plate is discretized by using the semiloof elements and stiffness and mass matrices of the plate are obtained from the finite element model. In order to reduce the degrees of freedom of the finite element model, mode summation method is used with weighted modal vector including dominant modes in the dynamic behavior. Free vibrations, static behavior and transient vibrations under the blast load are solved by using the full and reduced matrices as well as by using ANSYS software. All results are in a good agreement. State-space equations are obtained from the reduced finite element model. It is proved that the system equations satisfy the controllability condition in order to control the system. Since the system is also desired to be controlled by using a state observer, the observability condition is satisfied for the system. Since the system is on the stability boundary due to neglecting structural damping, the feedback gain matrix of the system is obtained by using optimal linear quadratic regulator (LQR) in order to guarantee the stability of the system. Structural vibrations have been suppressed successfully by using a state observer control system which estimates system states using displacements measured from sensors.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Experimental Identification of GHM and ADF Parameters for Viscoelastic Damping Modeling C. M. A. Vasques∗, R. A. S. Moreira† and J. Dias Rodrigues∗ ∗ Faculdade de Engenharia da Universidade do Porto Departamento de Engenharia Mecˆanica e Gest˜ao Industrial Rua Dr. Roberto Frias s/n, 4200-465 Porto, Portugal [email protected] and [email protected] †Universidade de Aveiro Departamento de Engenharia Mecˆanica Campus Santiago, 3810-193 Aveiro, Portugal [email protected]

ABSTRACT Viscoelastic materials can be used as an effective means of controlling the dynamics of structures, reducing and controlling the structural vibrations and noise radiation. They can be used as surface mounted or embedded damping treatments, utilizing passive viscoelastic materials alone, the so-called passive treatments, or in an unified way with active materials such as piezoelectrics, the so-called hybrid treatments. The use of these materials in damping treatments provides high damping capability over wide temperature and frequency ranges. The extensive use of passive or hybrid treatments using viscoelastic materials has motivated the development of damping models to be used and integrated into commercial or home-made finite element (FE) codes. The implementation of the Golla-HughesMcTavish (GHM) and anelastic displacement fields (ADF) models in a general FE model with viscoelastic damping is presented and discussed in this paper. Additionally, a direct frequency analysis (DFA) is also described and employed. A methodology to identify the complex shear modulus of viscoelastic materials is described. Thus, the identified complex shear modulus of the viscoelastic material 3M ISD112 is curve-fitted in order to obtain the modeling parameters of the GHM and ADF models. A sandwich plate with a viscoelastic core and elastic skins is analyzed. Measured and FE-based predicted FRFs based on a DFA, GHM and ADF models, were compared in order to assess the damping models and validate the experimental procedure for the material properties identification and the curve fitting process. It was found that the application of a discrete dynamic system, describing a SDoF analytical model, provides a reliable identification methodology, since it is based on the direct (in opposition to indirect measuring approaches based on vibrating beams) characterization of the complex stiffness of a viscoelastic material sample in shear deformation, providing results similar to those published by the material manufacturer. Regarding the internal variables models under analysis here, which were implemented at the FE model level, the ADF model leads to an augmented model of the damped structural system with a lower size than the GHM model. To the opinion of the authors, the ADF model represents the best alternative to accurately model the damping behavior since it yields good trade-off between accuracy and complexity. One major disadvantage in using internal variables models such as the GHM or ADF is the creation of additional dissipation (or anelastic) variables increasing the size of the coupled damped FE model. However, an alternative based on the DFA using the FE spatial model and re-calculating the complex viscoelastic stiffness matrix for each discrete frequency value might be used with the outcome of being simpler to implement and the drawback of being time-consuming and not providing directly the modal parameters of the damped structural system. All the models showed similar accuracy and yielded representative results of viscoelastically damped structural systems.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Numerical Aspects of Modelling Thermo- Mechanical Wave Propagation With Phase Transformations L. X. Wang∗, R. V. N. Melnik† ∗

MCI, Faculty of Science and Engineering, University of Southern Denmark, Sonderborg, DK-6400, Denmark [email protected]

† Mathematical Modelling and Computational Sciences,

Wilfrid Laurier University, 75 University Avenue West, Waterloo, ON, Canada, N2L 3C5 [email protected]

ABSTRACT Due to their unique thermo-mechanical properties, the range of existing and potential applications of shape memory alloys (SMA) continues to grow. Many of these applications take advantage of the dynamic response of SMA under dynamic loading conditions. From the engineering design point of view, this requires a better understanding of the effect of phase transformations and thermo-mechanical coupling on wave propagation in the material. In achieving this goal, a fundamental task is to analyse the dynamics of first order phase transformations induced by shock loadings. In this contribution, a mathematical model and its numerical discretization are constructed to analyse the wave propagation in shape memory alloy rods. The first order martensitic transformations and the associated effects of thermo-mechanical coupling are accounted for by employing the modified Ginzburg-Landau-Devonshire. The Landau-type free energy function characterizes different phases, while a Ginzburg term is introduced to account for the internal friction during phase transformations. The effect of the Ginzburg term on wave propagation patterns is analysed under shock loadings implemented via stress boundary conditions. For practical numerical simulations of SMA samples, the constructed model of coupled nonlinear system of PDEs is reduced to a system of differential-algebraic equations, where the Chebyshev collocation method is employed for the spatial discretization, while the backward differentiation is used for the integration in time. A series of numerical experiments is carried out on copper-based SMA samples. Propagation of stress waves induced by shock loadings is analysed for different initial temperature. It is demonstrated that the patterns of wave propagation is complicated at low temperatures by phase transformations, while more regular patterns are observed for high temperature distributions. These observations are in agreement with experiments. Finally, the influence of viscosity effects on the overall thermo-mechanical behaviour of rods is analysed numerically by evaluating the contribution of the Ginzburg term responsible for the internal friction during phase transformations. Key words: Thermo-mechanical coupling, martensitic transformations, shock waves, Landau-Ginzburg theory, Chebyshev collocation method.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Analysis and Design Optimization of Smart Laminated Composite Beams using Layerwise Theory A. Zabihollah*, R. Ganesan†, R. Sedaghati+ *†+

Dept. of Mechanical and Industrial Engineering Concordia University 1515 St. Catherine W., Montreal, Quebec, Canada H3G 2W1, * [email protected], † [email protected], + [email protected]

ABSTRACT Many of the developed models for smart laminated composite beams are based on the single-layer theories. Smart laminated composite beams contain strong inhomogenities through the thickness which lead to erroneous results when these beams are analyzed using single-layer theories. However, accurate results can be achieved by using three-dimensional model by setting up computationally expensive refined meshes. It has been demonstrated that layerwise theory can be applied for the analysis of laminated beam with integrated piezoelectric layers as sensors and actuators with acceptable accuracy and computational efforts [1]. The present work contains two major aspects; first, a finite element model based on the layerwise theory has been developed to investigate the actuation and sensing mechanism of extensional piezoelectric surfaces bonded and embedded as continuous or discrete patches in a composite laminate. Also the effect of shear piezoelectric patches embedded in a composite laminate has been investigated using the developed layerwise model. The elctro-mechanical coupling effect is included. The second aspect concerns with design optimization. Most of the developed models of smart laminated composite beams which have been used for optimization are based on single-layer theories. One of the most recent publications in this field has been the work of Soares et al [2]. From literature survey, it is concluded that smart laminated beam model based on the layerwise theory has not been used for design optimization. In the present work, such a model is developed and used for design optimization. In order to determine the optimal design of the beam for selected purposes, a gradient based optimization procedure, namely, Sequential Quadratic Programming (SQP) is used where the objective is to minimize the mass of the structure while various constraints including layer stress and displacements for static problem as well as frequency constraints for eigenvalue problems are considered. Illustrative examples are given to validate the formulation.

References [1] J.N., Reddy, On laminated composite plates with integrated sensors and actuators. Engineering Structures, 21, 568-593, 1999. [2] C.M.M., Soares, C.A.M., Soares, V.M.F., Correia, Optimal design of piezo-laminated structures. Composite Structures, 47(1-4), 625-634, 1999.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A thermo-viscoplastic model for bituminous materials Hervé Di Benedetto*, Brice Delaporte*†, Cédric Sauzéat *, Mondher Neifar+ *

“Departement Génie Civil et Bâtiment” laboratory of ENTPE, Rue Maurice Audin, 69518 Vaulx-enVelin, France [email protected] [email protected] *†

“Departement Génie Civil et Bâtiment” laboratory of ENTPE, Rue Maurice Audin, 69518 Vaulxen-Velin, and TOTAL France, Centre de Recherche de Solaize, France [email protected] + Ecole Supérieure des Sciences et Techniques de Tunis 5 avenue Taha Hussein, 1008 Monfleury Tunis, Tunisie [email protected]

ABSTRACT This paper introduces the 3D formalism of the DBN (Di Benedetto, Neifar) model developed at ENTPE (Ecole Nationale de Travaux Publics de l’Etat). This model is an attempt to describe with a unique formalism the different types of behavior observed for bituminous materials. The DBN model is very versatile. The proposed formulation can be simplified or adapted following the required property to be introduced. The law can then be very simple and easy to use (linear viscoelastic or even elastic) or more complicated (introduction of non linearity, permanent deformation or fatigue). Considering the thermo-sensitivity of bituminous materials, the temperature influence is always considered. The presented developments focus on the modeling of linear behavior, which is observed for bituminous mixes in the small strain domain (i.e. strain amplitudes less than some 10-5 m/m). The non linear behavior up to failure, which can be ductile or brittle, the rutting and the fatigue phenomena are only evoked due to the lack of space. These phenomena have been treated in others publications. In this paper, the three-dimensional (3D) linear viscoelastic (LVE) behavior is investigated. Complex Poisson’s ratio (Q*) is measured and introduced. Its evolution with temperature and frequency is studied for different bituminous materials. Experimental results show that the Time Temperature Superposition Principle is applicable in the 3D case. The same shift factor applies for E* and Q*. Comparisons between simulations in the linear and non-linear domains and experimental data are proposed. Thermal Stress Restrained Specimen Tests (TSRST) which introduce thermo-mechanical coupling are also simulated.

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From Asphalt to the Arctic: New Insights into Thermo-Mechanical Ratchetting Processes James G A Croll* *

Department of Civil and Environmental Engineering, University College London Gower Street, London WC1E 6BT, UK [email protected]

ABSTRACT Observation of upward bulges forming on the asphalt overlays applied to the pavements of London (and incidentally many other Cities), have recently been re-explained as a form of thermal ratchet buckling process. At many orders of magnitude greater temporal and spatial scales, a similar process is suggested to be taking place in the seasonal growth of the characteristic feature within permafrost regions, particularly in the Arctic, often referred to as pingos. Consideration of this specific example of analogous behaviour between asphalt and permafrost has led to a greater awareness that further close relationships between the mechanics of periglacial processes and those occurring in the morphology of asphalt might exist. This paper will explore some of these close mechanical analogies and in the process challenge in both fields some of the accepted wisdom as to the reasons for the formation of these analogous features. For example, it will be suggested that the thermo-mechanical process responsible for the development if ice-wedges within areas of permafrost are closely related, in terms of the thermo-mechanical processes at work, to the development of certain forms of cracking within sheets of asphalt. Extending this analogous relationship between periglacial processes and asphalt failure phenomenon, it will be argued that the formation of ice-wedge-polygons in permafrost and the development of “alligator cracking” in asphalt also involve very similar mechanics. Furthermore, it will be suggested that the hysteretic effects arising from alternations of heating and cooling are responsible for many other related morphological features of asphalt and permafrost. Certain forms of motion of either ice or asphalt, that are difficult to explain in terms of the effects of gravity alone, become explicable in terms of a form of thermo-mechanical ratchetting. While in the case of periglacial features the interest is largely scientific curiosity, although even here there are significant implications for climate change, for asphalt the mechanics of failure have serious economic implications. By drawing-out the analogies between the mechanics of processes within these two materials it is argued that we will be in a better position to understand and mitigate the worst consequences from these various phenomena.

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An Object-Oriented System for Finite Element Analysis of Pavements Aurea Silva de Holanda*, Evandro Parente Junior†, Teresa Denyse Pereira de Araújo†, Lucas Tadeu Barroso de Melo*, Francisco Evangelista Junior*, Jorge Barbosa Soares* * Federal University of Ceará Department of Transportation Engineering, Campus do Pici, Bl. 703, 60455-760, Fortaleza, Ceará, Brazil {aurea, lucas, fejr, jsoares}@det.ufc.br † Federal University of Ceará Department of Structural Engineering, Campus do Pici, Bl. 710, 60455-760, Fortaleza, Ceará, Brazil {evandro, denyse}@ufc.br

ABSTRACT The calculation of displacements, stresses and strains caused by vehicle loads in pavements is not a simple task even when considering all layers formed by linear elastic materials. In reality, surface and granular layers present a complex constitutive behavior, with nonlinear and time-dependent effects. Such effects should be considered in mechanistic pavement design methodologies which make use of the pavement structural response into specific distress models [1]. Today, there is a trend in the pavement academic community to substitute pavement analysis based on the Multilayer Elastic Theory by analysis based on the Finite Element Method – FEM [2]. There are several different finite element programs for pavement analysis [2, 3]. Most of these programs consider only axisymmetric models and the three-dimensional stress state due to vehicle loads is computed using superposition, which is not correct for nonlinear materials. Moreover, the existing pavement specific programs were developed for design purposes and do not allow the modeling of damage evolution (e.g., crack propagation in bituminous mixtures), which is an important topic in the pavement research community. In this paper, a new computational system developed to be used in both pavement design and research is presented. The system is based on the FEM and is implemented using Object-Oriented Programming (OOP) techniques to make it easily extendable. It contains both 2D (axisymmetric, plane-strain and plane stress) and 3D analysis models and works with different element shapes (triangular, quadrilateral, bricks, etc.) and interpolation orders (linear and quadratic). It also provides an efficient and accurate modeling of different loading types, including time varying loads. Finally, the system provides different numerical algorithms to nonlinear and time-dependent analysis, as well as a set of constitutive models. In this work, the class hierarchy of the system is presented and its main features are thoroughly discussed.

References [1] Y. H. Huang, Pavement Analysis and Design, Second edition, Prentice Hall, Inc., Englewood Cliffs, New Jersey, USA, 2004. [2] NCHRP/TRB, Guide for Mechanistic-Empirical Design of New and Rehabilitated Pavement Structures, Appendix RR: Finite Element Procedures for Flexible Pavement Analysis. 2004. [3] Harichandran, R.S., Yeh, M.-S., and Baladi, G.Y. “MICHPAVE: A Nonlinear Finite Element Program for the Analysis of Flexible Pavements,” Transportation Research Record, 1286, 1990.

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On the treatment of convective terms in coupled hydro-mechanics for porous media subject to dynamic loading Bernd Lenhof, Per Kettil, Kenneth Runesson, Nils-Erik Wiberg Department of Applied Mechanics, Chalmers University of Technology S-41296 G¨oteborg, Sweden [email protected]

ABSTRACT The physical problem of interest is that of a road structure that is subjected to dynamic loading from repeated over-rolling of car wheels. Primarily, we are interested in the mechanical response of bitumenbased porous granular media (such as the asphalt layer). The porous medium is modelled as a mixture of a solid phase, in the sense of a porous solid skeleton, and a fluid phase that represents both liquid and air in the pores. The solid particles are assumed to be intrinsically incompressible, i.e. the compressibility of the solid bulk is brought about by a rearrangement of the particles. In the presence of air, the fluid is assigned a finite intrinsic compressibility; hence, it suffices to consider a binary mixture. An important practical problem is internal erosion of the solid material by the fluid transport, which means that mass of the individual phases is not conserved; however, this effect is ignored here. A major issue, which is targeted in this contribution, is the relative importance of the various acceleration terms for both the solid and the fluid. In particular, we are interested in assessing the convective contributions in comparison with the intrinsic time derivative. In fact, the convective transport terms are normally ignored without any quantitative analysis supporting such a simplification. In order to simplify the computation without loss of the main characteristic, we restrict the formulation to small solid deformations and linear constitutive behavior (constant elastic stiffness and constant permeability). The results show that the introduction of the convective terms into the linearized balance equations can lead to significant differences in the solution of the seepage velocity field, depending on the material parameters. A simplified loading situation is considered: A pulsating stationary load replaces the moving load. A space-time finite element formulation is used that is based on the Galerkin method in time of zero order (which is basically the backward Euler method). In order to avoid using stabilization technique (such as SUPG), we consider very fine meshes in both space and time. The results of the assessment will be presented at the conference.

References [1] Reint deBoer, Theory of Porous Media - Highlights in the Historical Development and Current State. Springer Verlag, Berlin, 2000. [2] R. W. Lewis and B. A. Schrefler, The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media. John Wiley & Sons, Chichester, 1998. [3] Kettil, P. and Engstr¨om, G. and Wiberg, N.E., Coupled Hydro-Mechanical Wave Propagation on Road Structures. Computers and Structures, 83, 1719-1729, 2005.

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Asphalt Mechanics, a Key Tool for Improved Pavement Performance Predictions A.A.A. Molenaar* * Delft University of Technology P.O. Box 5048, 2600 GA Delft, the Netherlands [email protected]

ABSTRACT Pavement design has relied for a very long time on simplified mechanistic methods and experience. The need for long life maintenance structures, the use of new, costly, materials with enhanced characteristics, the need to spend the maintenance euro as effective as possible, the need to quantify risks etc, have given a boost to a renewed interest in the characterization and modeling of pavement materials and structures. This includes proper characterization of materials obeying their stress dependent, rate and temperature dependent, non linear elasto-visco-plastic behavior. These advanced models are necessary to obtain a better understanding in why pavement materials and structures behave like they do and in order to be able to quantify e.g. the beneficial effect of costly new materials with enhanced characteristics. This paper describes a number of developments made in this field at the Delft University of Technology and the power and importance of these models is illustrated by means of two realistic examples. As an introduction a short description is given of the design methodologies as currently used.

References [1] A. Scarpas, CAPA-3D finite element system. User’s manual parts I, II and III. Delft University of Technology, Delft, 1992. [2] C.S. Desai, S. Somansundaram and G. Frantziskonis, Hierarchical approach for constitutive modeling of geologic materials. Int. Journal of Numerical and Analytical Methods in Geomechanics, Vol 10, No 3, 1986. [3] S.M.J.G. Erkens, Asphalt concrete response (ACRe) –determination, modeling and prediction-. PhD Thesis, Delft University of Technology, 2002. [4] T.O. Medani, Design principles of surfacings on orthotropic steel bridge decks. PhD Thesis, Delft University of Technology, Delft, 2006.

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Sensitivity of Blood Flow Patterns to the Constitutive Law of the Fluid ´ A. Feij´oo ∗ Pablo J. Blanco∗, Ignacio Larrabide∗, Santiago A. Urquiza∗∗, Raul ∗Laborat´orio Nacional de Computac¸a˜ o Cient´ıfica - LNCC Rua Getulio Vargas 333, 25651-057, Petropolis, RJ, Brazil pjblanco, nacho, [email protected] ∗∗Laboratorio de Bioingenier´ıa, Universidad Nacional de Mar del Plata

Av. J.B. Justo 4302, 7600 Mar del Plata, Argentina santiagourquiza@fi.mdp.edu.ar ABSTRACT It is well known that hemodynamic factors are strongly influenced by the arterial geometry. Combining computational fluid dynamics with three-dimensional medical data makes possible the study of blood flow in real geometries. By this means we are able to analyze the sensitivity of hemodynamic factors due to shape changes in vascular districts. It is also well known that the fluid’s constitutive law very much influences the flow structure in singularities, such as the carotid sinus. In this work we quantify the sensitivity of blood flow for a model of a carotid bifurcation due to different constitutive behavior. In this case, the flow pattern for the Casson constitutive equation for the fluid is compared with its Newtonian counterpart. To this end a multidimensional 3D-1D FEM model of the whole arterial tree is implemented. It comprises a 3D compliant model of the carotid bifurcation coupled with a 1D model for the remaining part of the arterial tree. With this approach, difficulties arising from the treatment of boundary conditions for the 3D model are naturally handled. In particular, two carotid bifurcation geometries were analyzed. One of them corresponds to a standard model taken from literature. The second geometry was acquired from a patient-specific angiography using image segmentation and reconstruction techniques. Detailed hemodynamic factors and flow patterns in the carotid bifurcation are provided. Finally, this data is analyzed in order to determine how these results are affected in the different cases under study.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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The influence of stem design on strains and micromotion in revision Total Knee Arthroplasty: Finite element analysis A. Completo*, F. Fonseca†, J. A. Simões* * Departamento de Engenharia Mecânica Universidade de Aveiro, 3810-193 Aveiro [email protected] † Faculdade de Ciências da Saúde Universidade da Beira Interior, 6201-001 Covilhã [email protected]

ABSTRACT The main objective of this study was to compare the effect of tibial stem design on load transfer, strains and relative micromotion between the tibial tray and bone. Attempts were also made to correlate FE results with clinical findings, like bone resorption, fibrous layer formation, radiolucencies and pain. The effect of the medial position of the tibial tray to obtain good alignment with press-fit long stems provoked higher strains when compared with the model where the stem was shifted medially. Cemented stems reduce load transfer by the tibial tray to proximal bone while pressfit stems transmit higher percentage of load. The idea that orthopaedic surgeons have that this kind of stems reduce load transmitted to the proximal bone is according to this study a wrong one. The hypothesis that load transferred at the distal tip of the press-fit stems occurs is true in an immediately pos-operative situation if micromotion is minimised due to an adequate press-fit. However, in the middle and long term, the share of load at the distal bone cannot not achieved. FE results demonstrated potential mechanical-clinical relations relative to bone resorption, radiolucencies and pain.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Optimization of targeted movements Anders Eriksson KTH Mechanics Royal Institute of Technology, Osquars backe 18, SE-100 44 Stockholm, Sweden [email protected] ABSTRACT An algorithm was developed for the evaluation of targeted movements, between an initial and a final configuration, possibly with some intermediate configurations. The algorithm was primarily aimed at robotic and musculoskeletal movement simulations, and has been found efficient and reliable for problems with moderate numbers of displacement coordinates, even for complicated dynamic formulations, [1, 2]. The result from the algorithm contains the configuration as function of time, but also a set of a priori unknown control forces needed to create the desired motion. The algorithm is based on a temporal finite element interpolation of displacements and controls, with a third order Hermitian form for the displacement components and a linear interpolation for the control forces. This allows the governing differential equation of the movement to be satisfied at two collocation points in each of the time intervals. With sufficient freedom in the description of the control forces, they are chosen to optimize some measure of the movement or the controls. A common criterion is to minimize the needed forces, by formulating a cost function, which is the sum of square-integrated force components. Another interesting possibility is to seek the smoothest movement, which leads to a cost function based on the jerk values for the displacements, following an idea by Flash and Hogan, [3]. Both criteria were introduced in the algorithm. Examples have shown that the introduced optimization criteria strongly influence the obtained solutions. Even for a simple movement of, e.g., the arm seen in a sagittal view, the obtained movements, and therefore the needed forces, are evaluated as rather different, with different criteria. This observation is independent of whether the muscular forces are seen as a redundant set of individual forces, or are summed to resultant acting moments at the considered joints. The paper will describe the developed algorithm, the studied optimization criteria, and the conclusions from a set of examples concerned with bio-mechanical targeted movement.

References [1] A. Eriksson, Analysis methodology based on temporal FEM for bio-mechanical simulations. In: M. Ursino, C. A. Brebbia, G. Pontrelli, E. Magosso (Eds.), Modelling in Medicine and Biology VI, WIT Press, Southampton, 2005. [2] A. Eriksson, Temporal finite elements for target control dynamics of mechanisms, Comp. Struct., 2005 (submitted). [3] T. Flash, N. Hogan, The coordination of arm movements: an experimentally confirmed mathematical model, J. Neurosci. 5, 1688–1703, 1985.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Persistence of Axial Rotation in Idiopathic Scoliosis Due to the Structural Changes of the Intervertebral Disc Behnam Heidari*, David FitzPatrick*, Damien McCormack†, Keith Synnott † *

School of Electrical, Electronic & Mechanical Engineering, University College Dublin, Dublin, Ireland [email protected] †

National Spinal Centre, Mater Misericordiae Hospital, Dublin, Ireland

ABSTRACT There is some research about the morphological changes, spinal instability and causal factors influencing the mechanism of idiopathic scoliosis, however, there remains some controversy about its aetiology and pathogenesis [1,2] and complementary research is required to better identify the causal factors and pathologic mechanism(s) involved in the initiation and development of idiopathic scoliotic deformities of spine. In this study, a validated three-dimensional, anatomically accurate, mathematical model of the mid-thoracic motion segment (T7-T8), which consists of all bony element, soft tissues and articular contact surfaces is utilized, enabling investigation the influence of collagen deficiency in the initiation and progression of idiopathic scoliosis deformity. Alteration in the structure of the intervertebral disc is applied to the model and the resulting geometrical changes together with the coupling behaviour are computed. The results of the model demonstrate the significance of annulus fibrosus collagen fibre imbalance in developing a rotational effect in a motion segment, such as occurs in spinal deformities of idiopathic scoliosis. The structural changes of the annulus fibrosus alter the load displacement pattern of the motion segment and substantially provide the spine with a pattern of deformity under various loading conditions. It is observed that the subtle change in the pattern of collagen network structure results in a flexional rotation (coupled rotational and lateral displacements) due to the involvement of the posterior elements of the thoracic spine particularly during axial rotation. This continuous coupling effect, mainly flexional rotation can, in the long term, cause the morphological changes of the spine with characteristics of scoliotic deformity (axial rotation and lateral bending). The enhanced knowledge of aetiology of the idiopathic scoliosis, throughout understanding the functional characteristics of the annulus fibrosus and articular facets of the thoracic spine, together with the underlying clinical biomechanics and deformation mechanisms, may result in improved treatment of the spinal deformity through less invasive means.

References [1] Ahn U.M., et al., The etiology of adolescent idiopathic scoliosis. Am J Orthop. 31(7), 387-395, 2002. [2] Roberts S., et al., Structure and composition of the cartilage endplates and intervertebral disc in scoliosis. In: Etiology of Adolescent Idiopathic Scoliosis: Current Trends and Relevance to New Treatment Approaches. Hanley and Belfus Inc., Philadelphia, 2000.

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Numerical Simulation of Hemodynamics in a Cerebral Artery Iwase, H., Himeno, R. RIKEN 2-1, Hirosawa Wako-city, Saitama, 351-0198, JAPAN {iwase、himeno}@riken.jp

ABSTRACT We have developed the simulation system of hemodynamcs based on voxel model method for modeling and the new developed VOF-VOF corresponding to voxel model for the purpose of advance medical surgery. In image-based simulations, the pre-working such as the modelings of blood vessels and mesh generation require many cost, labour and time. To solve these problems, we adopted voxel mode which is constructed from medical image obtained from MRI, CT. Furthermore, the input system has been developed to efficiently input boundary conditions and calculation conditions. This system makes it possible for users to input boundary conditions and calculation conditions, confirming the location of a boundary to be set through the friendly interface. We examined the precise of simulation system of hemodynamics in the case of the linear-tube. The numerical results were in agreement with theory solutions. As an example of hemodynamics simulation, we handled the blood flow in a cerebral artery with large scale problem. Our developed system could succeed in performing the simulation of hemodynamics. Our developed system enables users to reduce the cost and time on total simulation process. Particularly, the cost and time of modeling and mesh generation are drastically decreased by adopting voxel model method and our developed new scheme. Input system of boundary conditions permits user to smoothly define the boundary conditions through the friendly interface, resulting in the reduction of the load of input boundary conditions.

References [1]C.A. Taylor, M.T. Draney, J.P. Ku, D. Parker, N. Steele, K. Wang, C.K. Zaris, Predictive medicine:Computational Techniques in Therapeutic Decision-Making. Computer Aided Surgery, 4, 231-247, 1999. [2]J.R. Cerebral, P.J. Yim, R. Lohner, O. Soto, P.L. Choyke, Blood flow modeling in carotid arteries using computational fluid dynamics and magneitc resonance imaging, Academic Radiology, 9, 1286-1299, 2002.

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Analysis of some Contact Problems in Human Joints after Arthroplasty 2 ´ A. John1, M. Mazdziarz , J. Rojek2 , J.J.Telega 2 , P. Małdyk3 1 Faculty

2

of Mechanical Engineering,Silesian Technical University,Gliwice,Poland [email protected] Institute of Fundamental Technological Research,Warsaw,Poland [email protected] 3 Institute

of Rheumatology,Warsaw,Poland [email protected] ABSTRACT

A general anisotropic model of unilateral contact with adhesion and friction is proposed. The model is applied to the numerical analysis of contact between pelvis and acetabulum after arthroplasty. Clinical evidence confirms the role of not only friction but also of adhesion at the bone-implant interface. We have elaborated a general model of unilateral contact with friction and adhesion applicable to the study of such an interface. Numerical implementation has been performed using the finite element method preceded by a regularization of unilateral conditions. Numerical solutions pertain to the development of zone of adhesion loss. For a discussion of phenomena occurring at the bone-implant interface see [1, 2] The performed numerical analysis pertains to the case of isotropy of bone adhesion but the general anisotropic model presented here can also be used. Also, our approach applies to any joint after arthroplasty and fixation with or without bone cement. Also, one may envisage applications to dental biomechanics. According to clinical data failure of the hip prosthesis is mainly caused by loosening of the acetabulum and not the stem. Surprisingly, in the relevant biomechanical literature not the acetabulum but stem loosening is most often analyzed. Hence the importance of study like the one performed in this paper. Our next goal is to take into account the influence of wear debris, on prostheses loosening. To this end the model proposed by Shillor et al. [3] will be generalized. Acknowledgment: The authors were supported by the Ministry of Science and Information Technology through the grant No. 4 T11F 003 25.

References [1] Rojek J., Telega J.J. (2001) Contact problems with friction, adhesion and wear in orthopaedic biomechanics. Part I: General developments. J. Theor. Appl. Mech. 39:655-677 [2] Rojek J., Telega J.J., Stupkiewicz S. (2001) Contact problems with friction, adhesion and wear in orthopaedic biomechanics. Part II: Numerical implementation and application to implanted knee joints. J. Theor. Appl. Mech. 39:679-706 [3] Shillor M., Sofonea M., Telega J.J. (2003) Analysis of viscoelastic contact with normal compliance, friction and wear diffusion. C.R. Mecanique 331: 395-400 [4] John A. (2004) Identification and analysis of geometrical and mechanical parameters of human pelvis bone Habilitation thesis, in Polish, Silesian Technical University

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Cardiovascular Disease Diagnosis Before Birth by Means of Chaotic Analysis on the Herat Rate Signal Pardis Khayyer*, Alireza Zolghadrasli*, Farhang Daneshmand† and Ali Najafi† *

†

Department of EE, Shiraz University Shiraz, 71345, Iran [email protected]

Department of Mechanical Engineering, Shiraz University Shiraz, 71345, Iran [email protected]

ABSTRACT The heart rate signal contains information about the condition of the heart [1]. One of the features which information can be obtained from the heart rate signal is its chaotic behavior and it may be useful in disease diagnosis of the heart. It is known that a healthy heart, like many other biological systems, has chaotic behavior. This fact is also manifest in the heart rate signal [2]. Among all heart related problems, one of the major problems, facing doctors, has always been the heart related diseases which happen before birth and are not accurately diagnosed until after the birth. Sooner diagnosis will definitely lead to better decision making by doctors. In the present research study, the main purpose is introducing a method for disease diagnosis of fetus before birth. In the procedure of the present research, first the data should be obtained from the heart rate signal and second, the analysis on the signal should be done. Through our work the chaotic analysis are done by means of Matlab programming based on Box Dimension method which is a sub method of fractal dimension and also analysis based on Lyapunov Exponent. A mathematical introduction to Fractal Dimension and Lyapunov exponent is also given in the main paper.

References [1] A. L. Goldberger and et.al. . Fractal dynamics in physiology: Alterations with disease and aging. Colloquium, 99, 2466-2472, 2002. [2] G. Sugihora, W. Allan, D. Sobel and K. D. Allan, Nonlinear control of heart rate variability in human infants. Medical Sciences, 93, 2608-2613, 1996.

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Configurational Derivative as a Tool for Image Segmentation ´ A. Feij´oo∗, Edgardo Taroco∗, Andr´e A. Novotny∗ Ignacio Larrabide∗, Raul ∗ Laboratório Nacional de Computação Científica - LNCC

{nacho, feij, etam, novotny}@lncc.br ABSTRACT The introduction to medicine of techniques coming from areas like Computational Fluid Dynamics, Structural Analysis, and Inverse Problems, made the use of imaging data such us Computed Tomography (CT), Magnetic Resonance Imaging (MRI), Single Photon Emission Tomography (SPECT), Positron Emission Tomography (PET) and Ultrasound (US) mandatory in order to apply this techniques to patient specific data. The process of identification of different tissues and organs, called segmentation, is a maior concern in this analysis. This process can be tedious and time consuming when done by hand, so its been an early concern in image processing to automatize it. Many contributions have been made to the area since the introduction of the Mumford and Shah functional. This functional is endowed to quantify the cost associated to a specific segmentation. Our aim in this paper is to present an image segmentation method based on the configurational derivative of the cost functional F endowed to the image data. The configurational derivative can be viewed as an extension of the well established concept of topological derivative when, instead of a hole, a small inclusion is introduced at a point in the domain. More specifically, for an appropriate functional F(u, v) where u is the given image data, a segmentation algorithm is proposed with the following objective: find the segmented image v that minimizes F. Finally, some results are presented in order to show the computational performance of this methodology.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Numerical Simulations to Analyze and Optimize the Human Substitute Voice G. Link, M. Kaltenbacher, R. Lerch Department of Sensor Technology University of Erlangen-N¨urnberg Paul-Gordan-Str. 3/5, 91054 Erlangen, Germany [email protected]

ABSTRACT After a larynx excision, as consequence of e.g. laryngeal cancer, the base of the human voice gets lost and a substitute voice is needed to communicate. A possible base of such a substitute voice represents the upper part of the esophagus, called the pharyngeal-esopohageal segment (PE segment). Via a valve, which the surgeon places between the esophagus and the trachea, the patient can guide air from the trachea into the esophagus. By virtue of that air stream the PE segment starts to vibrate and sound is generated, which can be used as base of a substitute voice. The geometry of this PE segment has major influence on the engaging voice quality. During surgery the surgeon is able to form the geometry of this PE segment. But until now the surgeon has no precise guidelines or knowledge of how to shape it in an optimal way. The target of this project is therefore to built a simulation tool which is capable to value different PE segment geometries in order to improve the quality of the substitute voice. Within this contribution the mathematical models of the involved physical fields as well as their interactions will be discussed. The arising partial differential equations will be solved by using the finite element method (FEM). For the fluid field the weighted least-squares method (LSFEM) and for the mechanical field the Galerkin method is applied. The numerical schemes are finally used to perform a geometry variation of the PE segment. Three different geometries of the PE segment opening (pseudo glottis) -a round, an elliptic and a triangularare compared with each other in respect to their dynamical behavior under fluid loads.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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The thaw time of frozen cancellous bone for mechanical testing C. Nabais*, R.M. Guedes†, J.A. Simões† *

Instituto de Engenharia Mecânica e Gestão Industrial Rua do Barroco 174, 4465-591 Leça do Balio, Portugal [email protected] †

Departamento de Engenharia Mecânica e Gestão Industrial, FEUP Rua Dr. Roberto Frias s/n, 4200-465 Porto, Portugal [email protected]

†

Departamento de Engenharia Mecânica da Universidade de Aveiro 3850-193 Aveiro, Portugal [email protected]

ABSTRACT The objective of this study was to estimate the temperature profile at the centre of bovine cancellous bone specimens in order to establish the thaw time prior to testing. We assumed that the heat transfer mechanism inside the cancellous bone specimen is mainly of conduction nature. The thermal properties of bone (i.e. thermal conductivity and specific heat) and its density were assumed constant, over time and temperature. It was verified that thawing is a very fast phenomenon and the theoretical model proved to be accurate enough. Given the experimental results, we conclude that it is reasonable to thaw the specimens 1 hour prior to their use in testing.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Modeling of Passive Behavior of Soft Tissues Including Viscosity and Damage Tobias Olsson∗, Jo˜ao A. C. Martins † ∗ Department of Mechanical Engineering

Link¨opings universitet, SE–581 83 Link¨oping, Sweden. [email protected] †Departamento de Engenharia Civil e Arquitectura e I.C.I.S.T.

Instituto Superior T´ecnico, 1049–001, Lisboa, Portugal. [email protected]

ABSTRACT The mechanical properties of soft tissues depend strongly on the orientation of their fibers, and usually they have a highly nonlinear behavior: their stiffness increases as they are stretched. We are interested here in the passive behavior of soft tissues, when subjected to significant stretches, possibly leading to damage. In this paper we develop a model for a transversely isotropic material that has a damageable viscoelastic behavior. This model is then used to simulate the damage evolution of the tissue. The model is developed with the underlying framework of hyperelasticity, and the corresponding strain energy has different parts associated to different contributions to the material behavior: volumetric, isotropic, anisotropic and dissipative contributions. Since soft tissues are almost incompressible we use a multiplicative split of the deformation gradient into a volume preserving part and a part with (small) volume changes. The anisotropic behavior is characterized by the existence of a family of fiber directions within the tissue. The viscoelastic behavior associated with the non–equilibrium stress is treated as a standard solid material with M Maxwell elements simulating the fact that the response of soft tissues is almost independent of the loading frequency. The total damage is modeled by splitting the energy degradation into one isotropic part and one anisotropic part. That is, we can have fiber degradation independently of the damage of the surrounding matrix. The model is implemented in the commercial Finite element software ABAQUS and the tissue behavior is described by an user subroutine (UMAT). Qualitative features of the model are illustrated and discussed.

References [1] G. A. Holzapfel and T. Gasser, A viscoelastic model for fiber–reinforced composites at finite strains: Continuum basis, computational aspects and applications, Computer Methods in Applied Mechanics and Engineering. 190, 4379–4430, 2001. [2] A. N. Natali et al, Anisotropic elastodamage constitutive model for the biomechanical analysis of tendons, Medical Engineering and Physics, 27, 209–214, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Nano and macro structure of cortical bone: numerical investigations M. Racila*,†, J. M. Crolet* *

University of Franche-Comté, Besançon, France 16 Route de Gray, 25000, Besançon [email protected] [email protected] †

University of Craiova, Romania 13 A. I. Cuza, Craiova, Romania

ABSTRACT Our last studies proposed a model of cortical bone [1, 2] with five architectural levels macroscopic, osteonal, lamellar, fibrous and fibrillar. In order to take into account the complex process of mineralisation, a new entity has been introduced, the EVMC (Elementary Volume of Mineral Content), which is made by hydroxyapatite cristals and fluid containing mineral ions. Each cristal is made on a solid part and on surounding gel. The modeling of the EVMC architecture is very important for the construction of bony properties at each level of its architecture and of course at the macroscopic level. In this paper, investigations are pursued on the possible geometrical organisation of these EVMC and their elastic properties are computed. Model is made by taking successively into account spatial organization of hydroxyapatite crystals, different mineralizations, a new behaviour’s law, motion of fluid containing ions in each of architectural levels and homogenization of complex composite structures (lamellae, osteons, and cortical bone). On a mathematically point of view, asymptotic developments method in a new piezoelectric framework (with threshold) is used. Developed computational methods have been packed into the software called SiNuPrOs (Numerical Simulations of Cortical Bone’s Properties). On a biomechanical point of view, it has been established that human cortical bone is a non piezoelectric orthotropic medium for which anisotropy is essentially involved by the nanoscopic architecture. For a given organisation of EVMC, mechanical properties are found at macroscopic level and under a standard loading, the obtained mechanical fields induce nanoscopic fields (strain, stress, electrical potentiel) in these EVMC. This process could be seen as an iterative process or a feedback phenomenon and represents an interesting approach of bone remodeling.

References [1] Racila M., Crolet J. M., "Human cortical bone: A tool for numerical simulation of fluid motion in osteonal architectures", Proceedings of 2nd International Conference on Computational Bioengineering, Vol. 2, IST Press, pp. 711-718, 2005 [2] Crolet J. M., Racila M., Mahraoui R., Meunier A., "New numerical concept for hydroxyapatite in human cortical bone", Computer Methods in Biomechanics and Biomedical Engineering, Vol. 8 (2), pp. 139-143, 2005

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Calculation of Muscle and Joint Forces in the Masticatory System Stefan Rues1, Hans J. Schindler1, Karl Schweizerhof 2, Jürgen Lenz1 1

Research Group Biomechanics, University of Karlsruhe Kaiserstraße 12, 76131 Karlsruhe [email protected] 2

Institute for Mechanics, University of Karlsruhe Kaiserstraße 12, 76131 Karlsruhe [email protected]

ABSTRACT The aim of this study was, among other questions, to disclose the direction and magnitude of the forces acting on the condyles of a mandible. The masticatory system is highly redundant, i.e., there are 12 muscle and 2 joint forces in contrast to 6 equilibrium conditions. To reveal the occurring force patterns, in 10 healthy male subjects (31 ± 2.3 yrs.) simultaneous EMG- and force measurements have been performed [1] using surface as well as intramuscular electrodes and a force measuring device. With the help of horizontal and frontal MRTs a 3D-model of the masticatory musculature was generated for each test person. On the basis of these data the lines of action of all muscles were calculated using a) the muscles’ geometry data, i.e., the centroids of origin and insertion areas [2], and b) FE-models based on an appropriate element which reflects the basic behavior of contractile tissue. The FE-models also take into account the muscle architecture (division into tendon and contractile tissue). The force law relating the muscle force to the corresponding electrical activity was determined on the basis of vertical clenching tasks performed at different resultant bite force magnitudes. The experimental results can be well approximated by a second order polynomial. With the knowledge of the lines of action and the force law, the muscle and joint forces can finally be calculated for the tested resultant bite forces.

References [1] H.J. Schindler, S. Rues, J.C. Türp, J. Lenz, Activity patterns of the masticatory muscles during feedback-controlled simulated clenching activities. Eur. J. Oral Sci., 113, 469-478, 2005. [2] T.M.G.J. van Eijden, J.A.M. Korfage, P. Brugman, Architecture of the human jaw-closing and jaw-opening muscles. The Anatomical Record, 248, 464-474, 1997.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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The algorithms of mathematical programming in muscle recruitment and muscle wrapping problems V´ıt Vondr´ak∗, John Rasmussen†, Michael Damsgaard‡ and Zdenˇek Dost´al∗ ∗ Technical University of Ostrava 17. listopadu 15, CZ-70833 Ostrava-Poruba, Czech Republic [email protected], [email protected] †Institute of Mechanical Engineering, Aalborg University Pontoppidanstræde 101, DK-9220 Aalborg East, Denmark [email protected] ‡AnyBody Technology A/S Niels Jernes Vej 10, DK-9220 Aalborg East, Denmark [email protected]

ABSTRACT In spite of the enormous variety of life on Earth, muscle work in all animals is accomplished by the same physiological mechanism in which an electro-chemical process causes contraction of muscle tissue. Furthermore, muscle systems across species are equipped with a significant level of redundancy, in the sense that the number of muscles significantly exceeds the number of degrees of freedom of the body. Hence, infinitely many combinations of muscle force can balance the external loads, and selecting the best one is an optimization problem that the central nervous system solves instantly when a certain movement is called for. The mathematical modelling of these optimization processes is of vital importance to our basic understanding of human physiology and planning of surgical procedures, design of prostheses and implants, and planning of rehabilitation procedures. In our presentation, we shall consider in particular optimization problems which carry out muscle recruitment in the inverse dynamic simulations of body movements. Namely, we shall introduce those problems which could be formulated as linear or quadratic programming problems and which could be therefore solved very efficiently. The several different algorithms for the solution of such problems will be presented and results will be compared on model and realistic simulations. Another important part of the muscle recruitment problem is an identification of the path of a muscle which wraps over bones or other tissues. Therefore we will also present a geometric model of a muscle that wraps over a set of obstacles. The muscle is represented by a thin elastic string and the obstacles are represented by rigid surfaces such that the muscle wrapping problem is identified as a contact problem. The finite element (FE) method is used to discretize the string and the FE model is solved iteratively with the efficient scalable FETI based contact solver that was originally proposed for solution of the contact problems of elastic bodies as described in [1]. The several practical models built in AnyBody Modeling System will be presented.

References [1] Z. Dost´al, Inexact semi-monotonic augmented Lagrangians with optimal feasibility convergence for convex bound and equality constrained quadratic programming, SIAM J. Numerical Analysis 43, 1, 96-115, 2005. [2] Anybody Modeling System. http://www.anybodytech.com

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Three-Dimensional Numerical Simulation of Airflow and Vibration Analysis for Upper Airway of Humans Chi Yu *, Yuefang Wang †, Yingxi Liu†, Xiuzhen Sun‡ P

* P

P

P

P

P

P

P

P

State Key Lab. of Struct. Anal. for Ind. Equip., Dalian University of Technology 2 Linggong Road, Dalian 116024, China [email protected] † P

P

Department of Engineering Mechanics, Dalian University of Technology 2 Linggong Road, Dalian 116024, China [email protected]; [email protected] ‡ P

P

Department of Otorhinalaryngology, Dalian Medical University 465 Zhongshan Road, Dalian 116027, China [email protected] P

ABSTRACT Obstructive sleep apnea syndrome (OSAS) may cause apneas or hypopneas, both physically and emotionally harmful to their sufferers. It has been realized that the periodic intermittent cessations of breathing or reductions in airflow resulted from OSAS is closely related to the developed pathological change in upper airways of the patients. In this paper, the authors present numerical simulations of airflows and fluid-solid interaction analysis for human upper airways. The objective of the research is to investigate airfield characteristics of the human upper airway by means of computational fluid dynamics (CFD) and the finite element (FE) method. The authors reconstruct three-dimensional models of the upper airway from the nostril to the epiglottis based on CT scanning images collected from two clinic volunteers. Based on the reconstruction three-dimensional CFD models that precisely preserve original configuration of upper airways are created. The CFD analysis is carried out by the FE method with boundary conditions of pressure at the nostril and of velocity at the top of vocal cord. The non-slip boundary conditions are used on the interior walls of the upper airway. With the CFD results the pressure and velocity distributions in the airflow field are quantitatively determined. For fluid-solid interaction analyses, the upper airway in the vicinity of the pharyngeal cavity is meshed using the reconstructed model. The fluid-solid interactive computations are performed for the healthy person and the OSAS patient. The results show that the hypertrophy of the soft palate remarkably escalates both the pressure and the deformation levels of the upper airway and hinders the airflow in the cavity channels.

References [1] R.J. Schwab and A.N. Goldberg, Upper airway assessment: Radiographic and other imaging techniques. Otolaryngoly-nhol Clin North Am, 31, 931-968, 1998. [2] X.L. Sheng, Pathophysiology and epidemiology of obstructive sleep apnea-hypopnea syndrome. The Journal of Practical Medicine, 21, 1973-1975, 2005. [3] T. Yong, M. Palta, J. Dempsey, et al, The occurrence of sleep disordered breathing among middle-aged adults. N Engl J Med, 328, 1230-1235, 1993.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Computer Simulation of Anisotropic Damage and Residual Stresses in Atherosclerotic Arteries D. Balzani∗, J. Schr¨oder∗, D. Gross† ∗ Institute of Mechanics, Department of Civil Engineering, University of Duisburg-Essen

Universtit¨atsstr. 15, 45117 Essen, Germany [email protected] †Institute of Mechanics, Department of Mechanics, Technical University of Darmstadt

Hochschulstr. 1, 64289 Darmstadt, Germany ABSTRACT The goal of this contribution is to propose a method for the numerical simulation of residual stresses in arterial walls. Generally, when axial segments of arterial walls are sliced in radial direction they spring open, which has been observed firstly in Vaishnav & Vossoughi [1]. Thus, residual stresses have to exist in the unloaded configuration. As often stated in the literature an open artery governed by a radial cut can be assumed to be stress-free. In this contribution we firstly close the gap between the opened transmural surfaces of a sliced artery by a displacement-driven procedure introducing some kind of interface elements formulated in the relative displacement of associated nodal points. Then we construct a new mesh of the closed artery and apply a method to incorporate the residual stresses without using the interface elements. This is advantageous for further numerical simulations as e.g., applying an internal pressure. Due to the special composition, orientation and weak interaction of particular fibers within arterial walls we consider the superposition of two transversely isotropic hyperelastic stored energy functions for the description of the anisotropic hyperelastic behavior in the physiological range of deformations. In order to guarantee the existence of solutions of underlying boundary value problems we use the functions proposed in Balzani, Neff, Schr¨oder & Holzapfel [2], which fit into the concept of polyconvexity. When arteries are overstretched, e.g., during a balloon-angioplasty, then deformations occur which are not in the physiological range anymore, and a discontinuous damage effect is observed. Therefore, the thermodynamically consistent anisotropic damage model introduced in Balzani, Schr¨oder & Gross [3] is applied to the polyconvex stored energy. The basic assumption of the damage model is that the damage occurs mainly in fiber direction. As a numerical example we consider the cross-section of a diseased artery in order to give an impression of the performance of the model.

References [1] R.N. Vaishnav & J. Vossoughi, Estimation of Residual Strains in Aortic Segments, C.W. Hall (Ed.): Biomedial Engineering II: Recent Developments, Pergamon Press, 330–333, 1983 [2] D. Balzani, P. Neff, J. Schr¨oder & G.A. Holzapfel, A Polyconvex Framework for Soft Biological Tissues. Adjustment to Experimental Data, International Journal of Solids and Structures, in press, 2005 [3] D. Balzani, J. Schr¨oder & D. Gross, A Simple Model for Anisotropic Damage with Applications to Soft Tissues, Proceedings in Applied Mathematics and Mechanics, Vol. 4, 236–237, 2004

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Computational study on stability and bone remodeling for a hip replacement using a “minimal invasive” femoral stem João Folgado, Rui P. Andrade, Paulo R. Fernandes IDMEC - IST Av. Rovisco Pais, 1049-001 Lisboa, Portugal [email protected]; [email protected]; [email protected]

ABSTRACT In a total hip artroplasty, femoral head is removed and the stem component is inserted into the femoral canal removing a considerable amount of hosted bone. Recently some manufactures have developed the design of smaller stems for “minimally invasive” hip replacement surgery. These smaller stems require minimal bone removal, leaving intact many elements of fixation that would otherwise be lost in a traditional primary arthroplasty. Furthermore, the maintenance of bone mass is of major importance particularly in the case of a future revision surgery. However, one aspect of major concern with these stems is the fixation and the stability of the stem. Actually, the stability of the stem plays a role in the success of a cementless total hip artroplasty. In order to assure the long term stability, most of the stems are design to promote biologic fixation, i.e, the bone attachment into the stem surface. The osseointegration (or bone ingrowth) is achieved coating the surface with hydroxyapatite (HA) or simply with a porous coated layer. However, even with such a special coated surfaces, several factors can inhibit or destroy the biologic fixation. Among these factors are mechanical ones such as large displacements and high stresses in the bone/implant interface. In this work a computational model is developed to study the osseointegration in a cementless femoral stem. The bone ingrowth process is modeled based on the relative displacement between bone and stem as well as on interface stress level. The biological fixation is not dissociated to the remodeling of surrounded bone due to the stress shield effect. Thus the model combines the osseointegration analysis with a bone remodeling model. The law of bone remodeling is derived from a material optimization problem, via the minimization of a function that takes into account structural stiffness and the metabolic cost related with bone mass maintenance, where bone is modeled as a porous material with variable density. The problem considers contact conditions on the interface between bone and implant. During the remodeling process, the mechanical interface conditions are updated according with the ingrowth algorithm: if the displacement and stress conditions necessaries for bone attachment are satisfied, then a connection between bone and implant is established. Consequently, the bone behavior is fully simulated from the immediate post operative condition until a long term condition. The osseointegration process emphasizes the behavior of the bone/stem interface, addressing the problem of stability of the prosthesis. The model is used to analyzed a “minimal invasive” stem (Mayo, Zimmer Inc.) comparing its performance with a conventional one (Trilock, Depuy Orthopaedics, Johnson&Johnson). Results allow us to validate the computational model and evaluate the performance of these two distinct stems.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Porous polycrystals built up by uniformly and axisymmetrically oriented needles: Homogenization of elastic properties Andreas Fritsch∗ , Luc Dormieux† , Christian Hellmich∗ ∗

Vienna University of Technology (TU Wien), Institute for Mechanics of Materials and Structures Karlsplatz 13/202, A-1040 Wien (Vienna), Austria [email protected], [email protected] † Laboratory

of Materials and Structures, National School of Civil Engineering (ENPC) 6 et 8 Avenue Blaise Pascal, 77455 Marne-la-Vall´ee, France [email protected]

ABSTRACT Porous polycrystal-type microstructures built up of needle-like platelets or sheets are characteristic for a number of biological materials, such as bone [2] or eggs [5]. Herein, we consider (i) uniform, (ii) axisymmetrical orientation distribution of linear elastic, isotropic as well as anisotropic needles. The latter requires derivation of the Hill tensor for arbitrarily oriented ellipsoidal inclusions with one axis tending towards infinity, embedded in a transversely isotropic matrix; this is accomplished by a new semi-analytical technique based on the work of Laws [3]. For a porosity lower 0.4, the elastic properties of the polycrystal with uniformly oriented needles are quasi-identical to those of a polycrystal with solid spheres. However, as opposed to the sphere-based model, the needle-based model does not predict a percolation threshold. As regards axisymmetrical orientation distribution of needles, two effects are remarkable: Firstly, the sharper the cone of orientations the higher the anisotropy of the polycrystal. Secondly, for a given cone, the anisotropy increases with the porosity. These results confirm the very high degree of orientation randomness of crystals [4] building up mineral foams [2, 1] in bone tissues.

References [1] Ch. Hellmich, J.-F. Barth´el´emy, and L. Dormieux. Mineral-collagen interactions in elasticity of bone ultrastructure – a continuum micromechanics approach. European Journal of Mechanics ASolids, 23:783 – 810, 2004. [2] Ch. Hellmich and F.-J. Ulm. Are mineralized tissues open crystal foams reinforced by crosslinked collagen? – some energy arguments. Journal of Biomechanics, 35:1199 – 1212, 2002. [3] N. Laws. The determination of stress and strain concentrations at an ellipsoidal inclusion in an anisotropic material. Journal of Elasticity, 7(1):91 – 97, 1977. [4] F. Peters, K. Schwarz, and M. Epple. The structure of bone studied with synchrotron x-ray diffraction, x-ray absorption spectroscopy and thermal analysis. Thermochimica Acta, 361:131 – 138, 2000. [5] H. Silyn-Roberts and R.M. Sharp. Crystal growth and the role of the organic network in eggshell biomineralization. Proceedings of the Royal Society of London, Series B, 227(1248):303 – 324, 1986.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A Poroviscoelastic Overlay Model for Finite Element Analyses of Articular Cartilage at Large Strains † ¨ Uwe-Jens G¨orke∗ , Hubert Gunther , Markus A. Wimmer‡ ∗

Institute of Mechanics, Chemnitz University of Technology Str. d. Nationen 62, D-09111 Chemnitz, Germany [email protected] † TBZ-Pariv, Chemnitz and AO Research Institute, Davos Liddy-Ebersberger-Str. 41, D-09127 Chemnitz, Germany [email protected]

‡ Department

of Orthopedics, Rush University Medical Center 1653 W. Congress Pkwy, Suite 761, Chicago, IL 60612, USA Markus A [email protected] ABSTRACT

Articular cartilage represents a complex inhomogeneous, multiphase material. Due to its sophisticated structure articular cartilage distinguishes itself by exceptional load-bearing properties under a wide variety of loading conditions. Currently, it is generally recognized that articular cartilage is subjected to large strains under in vivo mechanical conditions, which can be described as a first approximation with nonlinear elastic material models. Furthermore, rate dependent phenomena and osmotic effects caused by the presence of fixed charges can be observed. We propose to model structural effects of articular cartilage load response within the context of a phenomenological biphasic material approach resulting from an overlay concept (cf. [1]). The basic idea of this superposition methodology is the additive decomposition of the stress tensor according to specific mechanical properties whose underlying physics can partly be illustrated on rheological models adapted to large strain conditions (see [2] and others). Recent investigations have shown that due to the varying orientation of the collagen fibers the solid matrix of the cartilage tissue can be considered as an anisotropic hyperelastic material with tensioncompression nonlinearities. Frequently, the rate-dependency of a biphasic material behavior is exclusively attributed to the fluid flow through the solid matrix. However, recent observations demonstrate that viscoelastic properties of the solid phase must not be neglected. Therefore, viscous overstresses have been considered by the authors as well as an osmotic pressure model to simulate swelling states. The theoretical background and the numerical algorithms of all the parts of a suitable material model are presented. This model has been implemented into a commercially available FE-code. Some numerical examples showing several structural effects are discussed within the context of experimental results.

References [1] S. Olsen and A. Oloyede, A Finite Element Analysis Methodology for Representing the Articular Cartilage Functional Structure. Comp. Meth. Biomech. Biomed. Eng., 5(6), 377–386, 2002. [2] A. Lion, On the Large Deformation Behaviour of Reinforced Rubber at different Temperatures. J. Mech. Phys. Solids, 45, 1805–1834, 1997.

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A Finite Element Study of Strain Distribution in an Instrumented Knee Prosthesis for Full Force Measurement in Vivo. Rui Guimaraes, Stephen JG Taylor, Gordon W Blunn Centre for Biomedical Engineering, Institute of Orthopaedics and Musculoskeletal Science, (University College London), Brockley Hill, Stanmore HA7 4LP, UK. [email protected], [email protected], [email protected]

ABSTRACT Data on forces applied across joints in the body are needed to know how to better design and test implants, define rehabilitation regimes, and provide basic data for other biomechanics studies. The resultant knee load vector will vary in vivo in magnitude and position due to the joint laxity: the knee has six degrees of freedom and very irregular geometry. The overall aim is to develop an instrumented knee implant that can measure the full force vector (compression, anterio-posterior and medio-lateral shear forces) applied across each compartment of the tibio-femoral joint during activity [1]. Each tibial compartment was hollowed out to allow insertion of electronics and strain gauges, and was enclosed by a solid rim. This rim, although providing good mechanical support, had the effect of cross-coupling the strains around the periphery. A finite element model of the modified tibial tray was developed in order to study the behaviour of strains to applied loads and to determine the optimum locations and angles for the strain gauges in vivo. In this model, single and combination loads were applied to one compartment, inside the gauged periphery, using a modified plastic liner. Twelve strain gauge half bridges (24 sites) were simulated around the compartment periphery, and 3D strains for each found from the FEA. With individually applied axial and shear forces, the strain sensitivity at each of these 24 sites was found for a small range of positions, forming an important base dataset. This dataset was used to develop methods of transforming the measured strains back to the applied loads. The dataset shows a nonlinear, but gentle, variation of load sensitivity with position. Therefore it is possible to use multidimensional interpolation to extend the dataset with little loss of accuracy. With this extended dataset, algorithms were tested to assess how accurately the loads and positions could be calculated. The general procedure was to derive force/strain sensitivity matrices, using the 12 half bridge strains, for many different calibration positions, and to use a least squares minimisation method to determine the closest match between calibration and input data. This technique quickly iterates to the correct positions and axial force if the shear forces are low, and the remaining challenge is to predict the positions and shear forces together, and this is now the focus of our analysis. Using FEA it was possible to verify the design of the implant structurally and devise methods for determining the axial force and both antero-posterior (A-P) and medio-lateral (M-L) shear force magnitudes, and the point of application of those forces in each separate knee compartment.

Reference SJG Taylor, J Gorjon, A Gorjon. Development of an instrumented tibial tray for knee force measurement in vivo. Proc. 12th Conf. European Society of Biomechanics, Dublin, p110, 2000.

[1]

Acknowledgement: This work was funded by the EPSRC.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Topological Derivative Applied to Image Enhancement ´ A. Feij´oo∗ , Edgardo Taroco∗ , Andr´e A. Novotny∗ Ignacio Larrabide∗ , Raul ∗ Laborat´ orio

Nacional de Computac¸a˜ o Cient´ıfica - LNCC nacho, feij, etam, [email protected]

ABSTRACT Medical imaging techniques like Computed Tomography (CT), Magnetic Resonance Imaging (MRI), Single Photon Emission Tomography (SPECT), Positron Emission Tomography (PET) and Ultrasound (US) have introduced a formidably powerful tool in medicine. The poor quality or high signal-to-noise ratio (SNR) of this kind of images is maior limitation for image analysis. For this reason image enhancement takes an important roll in the segmentation and analysis process. Although imaging techniques (e.g., contrast agents, biological markers) should improve the image quality this is not always enough to give a good result. Much effort has been put into the area of image enhancement. Our aim in this paper is to present a method for medical image enhancement based on the well established concept of topological sensitivity analysis and borrowing image processing techniques like anisotropic diffusion. More specifically, an appropriate functional F is associated to the image indicating the cost endowed to an specific image. Let us assume that the image being segmented is characterized by a scalar field u representing the image data. Then, the segmentation algorithm can be cast as: given the evolution equation, ∂u = div(k∇u), (1) ∂t find u such that minimizes the functional F(k, u). This functional is associated to an energy norm of the image intensity u with diffusion coefficient k = k(∇u) selected using the topological derivative associated to this functional. Thus, the topological derivative, denoted as DT F, is used as an indicator which allow us to find the best k that will be used in the evolution equation (eq. ??) ensuring that the functional decreases in every iteration. Finally, several numerical examples are presented in order to show the computational performance of this methodology.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Determination of Contractile Forces Generated by Actin Fibre Networks J. Patrick McGarry*†, Amit Pathak†, Lorenzo Valdevit†, Anthony G. Evans†, Peter E. McHugh*, Robert M. McMeeking† †

University of California, Santa Barbara Santa Barbara, Ca 93106, U.S.A. [email protected] [email protected]

*

National University of Ireland, Galway University Road, Galway, Ireland [email protected]

ABSTRACT Recent experimental studies entail the seeding of cells on an array of microneedles, with the formation of focal adhesions being restricted to the tips of the microneedles. Contractile stresses generated in the cell actin network are transmitted to the microneedle array via such discrete focal adhesions. In order to obtain an enhanced understanding of cytoskeletal contractility from such experiments novel computational models are developed. A constitutive material law to describe cytoskeletal contractile behaviour is formulated. Finite element simulations using experimentally observed cellular geometries and actin network connectivity allow for the determination of the relationship between sub-cellular stresses and microneedle displacements. Parameter studies are performed leading to an accurate calibration of the constitutive formulation based on experimentally measured microneedle deflections. Stresses and strains in the actin network are computed and the evolution of actin fibre properties and geometries is also uncovered. Another interesting phenomenon observed in the aforementioned experiments is the detaching of certain microneedles from the actin network under large contractile forces. Cohesive zone modelling is used to simulate such a debonding of discrete focal adhesions. Such modelling provides quantitative information concerning the strength of focal adhesion bonds. Simulation of the stress redistribution in the actin network and consequent changes in microneedle deflections yields valuable information regarding the contribution of individual actin fibres to cellular deformation.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Hip Prosthesis Design Using a Multi-Criteria Formulation Rui B. Ruben*, Paulo R. Fernandes†, João Folgado†, Helder C. Rodrigues† * ESTG – I. P. Leiria Morro do Lena Alto do Vieiro – 2411-901 Leiria, Portugal [email protected] † IDMEC - IST Av. Rovisco Pais, 1049-001 Lisboa, Portugal [email protected]; [email protected]; [email protected]

ABSTRACT Just after a Total Hip Arthrosplasty, the fixation of the cementless stem mainly depends on mechanical factors. If a good initial mechanical stability is achieved, new bone growths into the stem porous coating, leading to a desirable long-term biologic fixation. On the other hand, an inefficient primary stability could be painful for the patient and promotes aseptic loosening, the main cause of cementless stem failure. In fact, “small” relative displacements between bone and stem as well as “small” contact stresses promote bone ingrowth, while larger displacement and stress values can destroy it. Thus, the primary stability can be improved controlling the displacement and stress level at the bone-implant interface. Such improvement can be performed actuating on stem geometry, friction coefficient between bone and stem, surgery technique and biologic response of bone to stem material. This interface phenomenon occurs simultaneously with bone adaptation process due to stress shield effect. The bone remodeling around the stem is also decisive for prosthesis success and it is itself related with initial conditions, such as primary stability and host-bone quality. To address the problem of hip prosthesis initial stability, a three-dimensional shape optimization procedure is developed to obtain the optimal stem geometry. The model uses a multi-criteria formulation that permits simultaneous minimization of tangential displacement and contact stress on the bone-stem interface. Design variables are geometric parameters describing successive stem sections. These parameters are constrained in order to obtain clinically admissible shapes. The bone-stem set is considered a structure in equilibrium with contact conditions. The contact formulation with friction allows the analysis of different porous coating lengths. Coated surfaces are modeled as contact with friction (friction coefficient + = 0.6 ) and uncoated surfaces as frictionless contact surfaces. A multiple load case, with three load cases, is also considered to simulate different daily life activities. The optimization problem was solved numerically by the method of moving asymptotes (MMA). The three-dimensional stem geometries obtained show the relation between geometry, porous coating length, relative tangential displacement and contact stress. This information leads to a better understanding of initial stability of a hip prosthesis and the relation between stem design and stability.

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Global Dynamical Model of the Cardiovascular System Sergey S. Simakov1, Alexander S. Kholodov2, Yaroslav A. Kholodov3, Alexey A. Nadolskiy4, Alexander N. Shushlebin5 1-3

Department of Applied Mathematics, Moscow Institute of Physics and Technology 9, Instituskii Lane, Dolgoprudny, Moscow Region, Russia, 141700 4-5 All-Russian Institute of Technical Physics P.O.Box 245, Snezhinsk, Chelyabinsk Region, Russia, 456770 1 2 [email protected], [email protected], [email protected], 4 [email protected], [email protected]

ABSTRACT Blood system functions are very diverse and important for most processes in human organism. One of its primary functions is matter transport among different parts of the organism including tissue supplying with oxygen, carbon dioxide excretion, drug propagation etc. Forecasting of these processes under normal conditions and in the presence of different pathologies like atherosclerosis, loss of blood, anatomical abnormalities, pathological changing in chemical transformations and others is significant issue for many physiologists. In this connection should be pointed out that global processes are of special interest as they include feedbacks and interdependences among different regions of the organism. At the modern level of computer engineering the most adequate physical model for the dynamical description of cardiovascular system is the model of non-stationary flow of incompressible fluid through the system of elastic tubes. Mechanics of such flow is described by nonlinear set of hyperbolic equations including mass and momentum conservation joined with equation of state that determines elastic properties of the tube [1]. As we interested in global processes the models of the four vascular trees (arterial and venous parts of systemic and pulmonary circulation) must be closed with heart and peripheral circulation models. Heart operation is described by the model of fluid flow averaged by volume through the system of extensible chambers that results in the set of stiff ordinary differential equations [1]. When combined these models allow us to consider functional changes and responses as during one cardiac cycle and at a longer periods upon 10 minutes that relates to the matter transport and satiation processes.

Figure 1. Pressure wave propagation through the large pulmonary arteries during one cardiac cycle. Grayscale designates divergence from the minimum pressure in each vessel.

References [1] A.S. Kholodov, Some Dynamical Models of External Breathing and Blood Circulation Regarding to Their Interaction and Substances Transfer, Computational Models and Medicine Progress, Science, Moscow, 127-163, 2001.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Computational Study of the Vibrating Disturbances to the Lung Function Sergey S. Simakov1, Alexander S. Kholodov2, Yaroslav A. Kholodov3, Alexey A. Nadolskiy4, Alexander N. Shushlebin5 1-3

Department of Applied Mathematics, Moscow Institute of Physics and Technology 9, Instituskii Lane, Dolgoprudny, Moscow Region, Russia, 141700 4-5 All-Russian Institute of Technical Physics P.O.Box 245, Snezhinsk, Chelyabinsk Region, Russia, 456770 1 2 e-mail: [email protected], [email protected], [email protected], 4 [email protected], [email protected]

ABSTRACT Frequently during its lifetime a human organism is subjected to the acoustical and similar to them vibrating impacts. Under the certain conditions such influence may cause physiological changes in the organs functioning. Thus the study of the oscillatory mechanical impacts to the organism is very important task of the numerical physiology. It allows to investigate the endurance limits of the organism and to develop protective measures in order to extend them. The noise nuisances affects to the most parts of the organism disrupting their functions. The vibrating disturbances caused to the lung function as one of the most sensitive to the acoustical impacts is considered in this work. The model proposed to describe the air motion in trachea-bronchial tree is based on the one dimensional no-linear theory including mass and momentum conservation for the air flow in flexible tubes similar to the model of blood flow in large vessels [1]. It combined with the single-component model of alveole [1, 2]. Two types of vibrating impacts were simulated that affect the thorax and the nasopharynx. The conducted simulations allowed us to detect two resonance frequencies that lay in the ranges from 3 Hz to 8 Hz and from 40 to 70 Hz when the thorax was affected (fig.1). For the nasopharynx disturbances no resonance states were found.

Figure 1. Dependencies of the integral volume and pressure of the lungs from oscillatory impacts.

References [1] A.S. Kholodov, Some Dynamical Models of External Breathing and Blood Circulation Regarding to Their Interaction and Substances Transfer, Computational Models and Medicine Progress, Science, Moscow, 127-163, 2001. [2] M. J. Jaeger, A. B. Otis, Effects of compressibility of alveolar gas on dynamics and work of breathing, J. Appl. Physiol., 19, 83-91, 1964.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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ABAQUS-Based, Coupled Porohyperelastic Transport Finite Element Models for Soft Hydrated Biological Structures Bruce R. Simon*, Paul H. Rigby†, Tyler P. Newberg †, Robert I. Park †, Stewart K. Williams† * The University of Arizona Aerospace and Mechanical Engineering Tucson, Arizona USA 85721-0119 [email protected] † The University of Arizona Biomed. Engr., Aero. & Mech. Engr., Ophthalmology, Biomed. Engr.

ABSTRACT General theoretical, experimental, and finite element models (FEMs) based on porous media formulations were developed to study complex normal and pathological mechanical-transport phenomena in biological structures and soft tissues. These continuum models view soft tissues as a solid (fibrous matrix) in which mobile fluid (water) and mobile species (ions, molecules, drugs) are transported during finite deformation. Two specific models were considered, i.e. the mixed porohyperelastic transport swelling (MPHETS) or equivalent porohyperelastic transport swelling (PHETS) [1] and the “porohypereleastic mass transport (PHEXPT)” ABAQUS models [2]. The theoretical formulations are the basis for FEMs and identify necessary material property functions. A suite of experiments is described that allows measurement of material parameters (elasticity, permeability, diffusivity, etc). If the PHETS “osmotic coefficient” is relatively small, then a novel partially coupled PHEXPT model can be implemented using an extended version of ABAQUS CAE [3]. Mathematical relations were identified that relate PHETS and ABAQUS constitutive equations. A specialized FORTRAN program transfers PHE FEM (hyperelastic and pore fluid element) material parameters and transient response (deformation, pore fluid pressure, water flux, etc.) to the mass transport XPT FEM. This second XPT FEM then provides the transient diffusion-convection mass transport solution (concentration, relative species flux, etc.). The extensive capabilities of ABAQUS CAE (finite element library, anisotropic nonlinear materials, automated model generation, inputoutput display, etc.) can be used in PHEXPT simulations of complex soft tissue mechanics where finite deformations are coupled to fluid and species transport. Example applications of this new PHEXPT FEM procedure will include coupled structural-transport in large arteries, (drug eluting) stented arteries, tissue engineered vascular grafts, and the eyeball.

References [1] B. R. Simon, S. K. Williams, G. A. Radtke, Z. P. Liu, P. H. Rigby, Coupled ‘EMPMTH’ FEMs for transport in soft biological structures. 7th U.S. Natl Congr Comp Mechs, Omnipress CD, Albuquerque, NM, July 27-31, 2003

[2] B. R. Simon, P. H. Rigby, R. I. Park, S. K. Williams, T. P. Newberg, ABAQUS Porohyperelastic transport (PHEXPT) finite element models of soft tissues. Submitted J. Biomech. Engr., 2006 [3] Hibbitt, Karlsson & Sorensen, Inc, ABAQUS Version 6.3. 2003

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Theoretical Modeling of Cyclically Loaded, Biodegradable Cylinders João S. Soares†, James E. Moore, Jr.*, Kumbakonam R. Rajagopal† †

Department of Mechanical Engineering, Texas A&M University 3123 TAMU, College Station, TX, 77843, USA [email protected]

*

Department of Biomedical Engineering, Texas A&M University 3120 TAMU, College Station, TX 77843, USA [email protected], [email protected]

ABSTRACT The adaptation of fully biodegradable stents, thought to be the next revolution in minimally invasive cardiovascular interventions, is supported by recent findings in cardiovascular medicine concerning human coronaries and the likelihood of their deployment has been made possible by advances in polymer engineering. The two main potential advantages of biodegradable polymeric stents are: (1) the stent can degrade and transfer the load to the healing artery wall which allows favorable remodeling, and (2) the size of the drug reservoir is dramatically increased. The in-stent restenotic response usually happens within the first 6 months, thus a fully biodegradable stent can fulfill the mission of restoring flow while mitigating the probability of long-term complications. However, it is a key concern that the stent not degrade away too soon, or develop structural instabilities due to faster degradation in key portions of the stent. We present here a preliminary model of the mechanics of a loaded, biodegradable cylindrical structure. The eventual goal of this research is to provide a means of predicting the structural stability of biodegradable stents. As a first step towards a fully non-linear model, biodegradable polymers are modeled as a class of linearized materials. An inhomogeneous field that reflects the degradation, which we shall henceforth refer to as degradation, and a partial differential equation governing the degradation are defined. They express the local degradation of the material and its relationship to the strain field. The impact of degradation on the material is accomplished by introducing a time-dependent Young’s modulus function that is influenced by the degradation field. In the absence of degradation, one recovers the classical linearized elastic model. The rate of increase of degradation was assumed to be dependent on time and linearized strain with the following characteristics: (1) a material degrades faster when it is exposed to higher strains, and (2) a material that is strained for a longer period of time degrades more rapidly than a material that has been strained by the same amount for a shorter period of time. The initial-boundary value problem considered is that of an isotropic, nearly incompressible, and strain-degradable cylindrical annulus subjected to radial stresses at its boundaries. A semi-inverse method assuming a specific form of the displacement field was employed and the problem reduced to two coupled non-linear partial differential equations for a single spatial coordinate and time. These equations were solved simultaneously for the displacement and degradation fields using a time marching finite element formulation with a set of non-linear iterations for each time step. The three main features that were observed were: (1) strain induced degradation showed acceptable phenomenological characteristics, i.e., progressive failure of the material and parametric coherence with the defined constants, (2) an inhomogeneous deformation leads to inhomogeneous degradation and therefore in an initially homogeneous body the properties vary with the current location, and (3) the linearized model, in virtue of degradation, exhibits creep, stress relaxation, and hysteresis, but this is markedly different from the similar phenomena exhibited by viscoelastic materials.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Validation of a non-linear wear model for UHMWPE Kanagaraj S, Mónica SA Oliveira, José AO Simões Mechanical Engineering Department, University of Aveiro, 3810-193 Aveiro, Portugal [email protected], [email protected], [email protected]

ABSTRACT Since 1960s, Ultra High Molecular Weight Polyethylene (UHMWPE) has been used in orthopedics as a bearing materials, namely for acetabular component of hip prostheses. Despite the success of this material, implants made of this material have limited lifespan due to gradual accumulation of wear debris and loosening of the components. Thus, the prediction of wear plays an important role to predict the life span of the implant materials. It is known that wear dynamics of polymer-metal systems determined by the properties of polymers such as surface energy, modulus of elasticity, specific heat, thermal conductivity, shear strength and other operating variables such as load, speed, sliding duration and sliding length. The hardness and roughness of the counter face also play a role. Though there are different models available for the wear of the polymer, it was not extended to include molecular weight of the polymer, in case of UHMWPE Thus an attempt has been made to develop a non-linear wear model correlating polymer material properties and operating variables which influence the wear behavior of UHMWPE. In this attempt, molecular weight of polymer has been taken into account along with other materials properties. Thus, it can also be extended to HDPE and LDPE. The main aim of this work is to extend a dimensional analysis of wear of polymer to include molecular weight of the polymer. An empirical model in the form of a wear equation is developed based on a non-linear relationship between wear volume and other operating variables. Wear of UHMWPE is tested on pin-on-disk machine. The comparison of results with linear and nonlinear model is also made.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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FEA of Human Knee Joint Replacement Using Real Bone Models Lukas Zach*, Svatava Konvickova*, Pavel Ruzicka* * Laboratory of Biomechanics Czech Technical University in Prague Faculty of Mechanical Engineering Technická 4, 166 07 Praha Czech Republic [email protected]

ABSTRACT The aim of a project of our Laboratory of Human Biomechanics, CTU in Prague engaging finite element analyzes of a human knee joint is to create a finite element model as detailed as possible of this joint including its total replacement. On the basis of previous simplified models [1] and all existing mechanical tests, including those using special films for contact pressure measurement, new model was developed using a Walter Modular (WM) knee endoprosthesis. Bone three-dimensional models were created by common means using freely available CT scans (Visible Human Project [2]). Since it was necessary to be able to edit the bone models in a CAD software a procedure to obtain such models was executed. In these form, they can be used also during a development of new types of WM system. As for the analysis it self, it was prepared for the knee in a full flexion as a static contact problem loaded by a single force equal to three-times body weight. Since the static problem does not correspond to the real joint, another work has to be made and it is a goal of one of the next model. Results of the presented work are useful for a comparison with the results of another authors and our former ones.

References [1]
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    + ,-.,/0 1,--/"2(* ++"$1 ,--/ [2] Visible Human Project. http://www.nlm.nih.gov/research/visible/visible_human.html


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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Subregion boundary element method for piezoelectric structures G. Dziatkiewicz, P. Fedelinski Department for Strength of Materials and Computational Mechanics Silesian University of Technology Konarskiego 18A, 44-100 Gliwice, Poland [email protected], [email protected]

ABSTRACT Piezoelectric materials generate an electric field when they are subjected to strain fields and they deform when an electric field is applied. This phenomenon is widely uitilized in many devices , for example, sensors and actuators, micro-electro-mechanical systems (MEMS), transducers [4]. An analysis of piezoelectric devices requires a solution of coupled mechanical and electrical partial differential equations. In this paper the boundary element method (BEM) is implemented to solve the coupled field problem in piezoelectrics. The method allows the analysis by discretization of the boundary only. The piezoelectric material is modelled as two-dimensional: homogenous, transversal isotropic, linear elastic and dielectric [4]. The numerical solution by the BEM requires fundamental solutions, which have very complicated forms even for a simplified transversal isotropic model of piezoelectric material [2], [3]. In the present work the Stroh formalism is used to obtain fundamental solutions [3]. In many applications piezoelectrics are connected with other materials: conductors, dielectrics and also other piezoelectrics. To analyze this problem the subregion boundary element technique is implemented [1]. Special boundary conditions must be applied on the interfaces, between different materials. The computer code is developed for several connected piezoelectric materials. The connection with other nonpiezoelectric materials is obtained by assuming particular material properties. The following connections are considered: piezoelectric – piezoelectric, piezoelectric – dielectric (for example a typical composite) and piezoelectric – conductor. Numerical examples will be presented and they will show that the subregion boundary element formulation allows to analyze efficiently multimaterial piezoelectric structures.

References [1] C.A. Brebbia, J. Dominguez, Boundary elements. An introductory course. Computational Mechanics Publications, McGraw – Hill Book Company, Southampton – Boston, 1992. [2] G. Dziatkiewicz, P. Fedelinski, Boundary element method for analysis of 2D piezoelectric solids. In: A. Garstecki, B. Mochnacki, N. Sczygiol, eds., 16th International Conference on Computer Methods in Mechanics CMM - 2005, CD-ROM Proceedings, Częstochowa University of Technology, Częstochowa, 2005. [3] E. Pan. A BEM analysis of fracture mechanics in 2D anisotropic piezoelectric solids. Engineering Analysis with Boundary Elements, 23, 67-76, 1999. [4] H.F. Tiersten, Linear piezoelectric plate vibrations: Elements of the linear theory of piezoelectricity and the vibrations of piezoelectric plates, Plenum Press, New York, 1969.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Vibrations of system of plates immersed in fluid by BEM Michał Guminiak, Ryszard Sygulski Institute of Structural Engineering, Poznan University of Technology Piotrowo 5, 60-965 Poznan [email protected] [email protected]

ABSTRACT The vibrations problem of system of plates immersed in fluid is considered using the boundary element method. The vibrations of the system are transmitted by a fluid. This problem in hand is a coupled problem of the fluid – structure type [1], [4], [5]. The boundary integral equation is used for describing the plate motion and the hydrodynamic pressure of the surrounding liquid. The set of constant boundary elements for the plate and internal collocation points associated with lumped masses [3] and rectangular surfaces are used. To avoid calculation of singular integrals the source points are located slightly outside the plate boundary [2], [6], [7], [8], [9]. The direct version of BEM and the static fundamental solution of the Kirchhoff plate problem are used in the paper. The Betti’s theorem is used to derive the boundary integral equations for a plate in bending. This approach avoids the development of Kirchhoff forces at a plate corners and equivalent shear forces at a plate boundary [6], [7], [8], [9]. A system of plates is surrounded from all sides by the infinite fluid which is incompressible and inviscid. A fully populated hydrodynamic matrix is obtained [4], [5], [7], [8], [9]. The dimension of the problem is not increased by the addition of the fluid interaction. Free and harmonic vibrations of system of two plates were considered. Harmonic excitation force was located at the centre of single plate. Resonance curves were made for both of plates.

References [1] Brebbia, C.A., Telles, J.C.F. and Wrobel, L.C., Boundary Element Techniques, Theory and Applications in Engineering, Springer-Verlag, Berlin Heidelberg, New York, Tokyo, 1984. [2] Hartley, G.A., Development of plate bending elements for frame analysis, Engineering Analysis with Boundary Element, 17, pp. 93-104, 1996. [3] Shi, G., Flexural vibration and buckling analysis of orthotropic plates by the boundary element method, Int. J. Solids Structures, vol.26, No.12, pp. 1351-1370, 1990. [4] Jones, W.P., Moore, J.A., Simplified aerodynamic theory of oscillating thin surfaces in subsonic flow, J. Am. Inst. Aeronaut. Ass, No. 11, 9, pp. 1305-1307, 1973. [5] Sygulski, R., Free vibrations of string meshes including the surrounding air mass (in Polish), Archiwum In˙zynierii L¸adowej , tom XXIX, nr 4/83, pp. 3-47, 1983. [6] Guminiak, M., Okupniak, B., Sygulski, R., Analysis of plate bendig by the boundary element method, ECCM-2001, 2nd European Conference on Computational Mechanics, June 26-29, Cracow, Poland, Vol. 1, pp. 176-177, 2001. [7] Guminiak, M., Sygulski, R., Vibration of plate immersed in fluid by BEM, CMM-2003 15th International Conference on Computer Methods in Mechanics, June 3-6, Gliwice-Wisła, Poland, pp. 143-144, 2003. [8] Guminiak, M., Sygulski, R., Vibration of plate immersed in fluid considering a bottom and a free surface by BEM, CMM-2005 16th International Conference on Computer Methods in Mechanics, June 21-24, Cz¸estochowa, Poland, pp. 85-86, 2005. [9] Guminiak, M., Thin plates analysis by the boundary element method with new formulation of boundary condition, PhD Thesis, Poznan University of Technology, Poland, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A coupled periodic FE-BE model for ground-borne vibrations from underground railways S. Gupta1 , G. Degrande1 , H. Chebli2 , D. Clouteau2 , M.F.M. Hussein3 and H. Hunt3 1 K.U. Leuven Department of Civil Engineering, K.U. Leuven B-3001 Leuven, Belgium [email protected] 2 Ecole Centrale de Paris LMSSMat, Ecole Centrale de Paris F-92295 Chˆatenay-Malabry, France [email protected] 3

University of Cambridge Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, United Kingdom [email protected] ABSTRACT An efficient and modular numerical prediction model is presented to predict vibrations in the free field due to metro trains in the tunnel. The three-dimensional dynamic tunnel-soil interaction problem is solved with a subdomain formulation, using a finite element formulation for the tunnel and a boundary element method for the soil. The periodicity of the tunnel and the soil in the longitudinal direction is exploited using the Floquet transform, limiting the discretization effort to a single bounded reference cell. The Craig-Bampton substructuring technique is used to efficiently incorporate a track in the tunnel. The track-tunnel-soil interaction problem is solved in the frequency-wavenumber domain and the wave field radiated into the soil is computed. The numerical model can also account for moving loads and various excitation mechanisms, including quasi-static loads, random loads due to the rail and wheel unevenness, impact excitation due to the rail joints and wheel flats, and parametric excitation excitation due to the sleeper periodicity. In this paper, only the excitation due to rail unevenness and a moving harmonic load are considered. A general analytical formulation is discussed to compute the response of three-dimensional invariant and periodic media that are excited by moving loads. To demonstrate the efficiency of the approach, an invariant tunnel with a concrete lining, embedded in a homogeneous full space is considered. The system is excited by a vehicle moving on an uneven rail. It is emphasized that the wheel/rail interaction strongly depends on the dynamic response of the wheel, the rail and the contact spring. The free field vibrations are predicted, by first computing the contact forces generated by the wheel-track interaction and then solving the dynamic track-tunnel-soil interaction problem. This numerical model provides a better understanding of wave propagation in the track, the tunnel and the surrounding soil and enables to investigate the inherent physics of underground railway vibrations and to control the vibrations propagating out of the tunnel.

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On the Application of the BEM to Rubber-Elastic Materials Antolín L. Ibán, José M. García-Terán, Ignacio P. Rico Solid Mechanics and Structures GIR group. University of Valladolid Paseo del Cauce s/n. 47011 Valladolid. Spain {ali,teran,ignpri}@eis.uva.es

ABSTRACT A BEM approach for the static analysis of the elastic problem with geometrical and material nonlinearities is proposed. The geometrical non-linearities that are considered are both finite strains and large displacements. Material non-linearities are also considered in this paper and the non-linear Mooney-Rivling constitutive law is compared with the generalysed Hooke law. In order to deal with the intricate non-linear equations that govern the problem, an incremental-iterative method is proposed. The equations are linearised and a Total Lagrangian Formulation is adopted. The integral equations of the BEM are developed in an incremental form. The iterative process is necessary in order to achieve a good approximation to the governing equations. The problem of a slab under homogeneous deformation is solved and the results obtained agree with the analytical solution. The problem of a hollow cylinder under internal pressure is also solved and its solution compared to that obtained by a standarised FEM code.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Modeling of Darcy’s Flow in Generally Anisotropic Porous Media Containing Discontinuity Surface by SGBEM-FEM Coupling J. Rungamornrat∗ ∗ Center

for Subsurface Modeling, ICES The University of Texas at Austin Austin, Texas, USA 78712 [email protected]

ABSTRACT In this paper we present a computational procedure for modeling steady-state, Darcy’s flow in a generally anisotropic porous medium containing the discontinuity surface. This computational technique provides a useful means to gain an insight into certain real problems, e.g. flow in a porous reservoir containing seals and/or fractures. The technique employs a weakly singular, symmetric Galerkin boundary element method (SGBEM) to model fluid flow within a (local and homogeneous) region containing the discontinuity surface while employs a standard Galerkin finite element method (FEM) to treat fluid flow in the remaining (possibly very complex and nonhomogeneous) region. The SGBEM is based on a pair of weakly singular, weak-form fluid pressure and fluid flux integral equations which contains only weakly singular kernels of O(1/r) and is applicable to both isotropic and generally anisotropic permeability [1,2]. The formulations of the two methods are properly combined to obtain a final formulation which is in a symmetric form. In the numerical implementation, the region modeled by the SGBEM and that by the FEM are discretized such that meshes on the interface of the two regions are conforming. The important features of the current technique include those: 1) standard C o elements can be employed everywhere in the discretization of the region modeled by the SGBEM since the integral formulation in only weakly singular; 2) special tip elements are utilized along the boundary of the discontinuity surface to accurately capture asymptotic behavior of the jump of the fluid pressure; 3) an efficient interpolation strategy is adopted to evaluate the kernels for generally anisotropic permeability; and 4) the coupling formulation gives rise to a symmetric system of algebraic equations. To demonstrate the accuracy and capability of the coupling technique, two example problems are presented.

References [1] J. Rungamornrat and M.F. Wheeler, Weakly-singular integral equations for steady-state flow in isotropic porous media. Report No. 05-30, The University of Texas at Austin, Texas, 2005. [2] J. Rungamornrat and M.F. Wheeler, Weakly-singular integral equations for Darcy’s flow in anisotropic porous media. Engineering Analysis with Boundary Elements, Article in press, 2006.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

A Displacement Solution to Transverse Shear Loading of Beams by BEM E.J. Sapountzakis* and V.M. Protonotariou *

School of Civil Engineering, National Technical University, Zografou Campus, GR-157 80, Athens, Greece [email protected]

ABSTRACT In this paper the boundary element method is employed to develop a displacement solution for the general transverse shear loading problem in beams of arbitrary simply or multiply connected constant cross section. The shear loading is applied at the shear center of the cross section, avoiding in this way the induction of a twisting moment. A boundary value problem is formulated with respect to a warping function and solved employing a pure BEM approach, that is only boundary discretization is used. The evaluation of the transverse shear stresses at any interior point is accomplished by direct differentiation of this function, while the coordinates of the shear center are obtained from this function using only boundary integration. The shear deformation coefficients are obtained from the solution of two boundary value problems with respect to warping functions appropriately arising from the aforementioned one, using again only boundary integration. The essential features and novel aspects of the present formulation compared with previous ones are summarized as follows. i. The proposed displacement solution constitutes the first step to the solution of the non uniform shear problem avoiding the use of stress functions. ii. All basic equations are formulated with respect to an arbitrary coordinate system, which is not restricted to the principal axes. iii. The shear deformation coefficients are evaluated using an energy approach instead of Timoshenko’s and Cowper’s definitions, for which several authors have pointed out that one obtains unsatisfactory results or definitions given by other researchers, for which these factors take negative values. iv. The present formulation is also applicable to multiple connected domains without fulfillment of further constraints. v. The developed procedure retains the advantages of a BEM solution over a pure domain discretization method since it requires only boundary discretization. Numerical examples illustrate the efficiency, the accuracy and the range of applications of the developed method. The accuracy of both the thin tube theory and the engineering beam theory is examined through examples with great practical interest.

Fig.1. Prismatic beam of an arbitrary cross-section occupying the 2-D region : .

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Non-conforming Coupled Time Domain Boundary Element Analysis † ¨ Martin Schanz∗ , Thomas Ruberg ∗ Graz

†

University of Technology, Institute of Applied Mechanics Technikerstr. 4, A-8010 Graz, Austria [email protected]

Graz University of Technology, Institute of Structural Analysis Lessingstr. 25, A-8010 Graz, Austria [email protected]

ABSTRACT The time domain Boundary Element Method (BEM) has been found to be well suited for modeling wave propagation phenomena in large or unbounded media. Nevertheless, material discontinuities or local non-linear effects are beyond the scope of classical BEM and require special techniques. Here, a (possibly hybrid) Domain Decomposition Method is proposed in order to circumvent these limitations and to obtain an efficient solution procedure at the same time. In time domain analysis it is preferable to have a coupling technique which is able to couple domains with different grids. If e.g. two domains with different materials are considered, in each domain a different suitable spatial and temporal grid is necessary. In the finite element community, several methods which provide such techniques are known, for example the Mortar-Method (see [1, 2]). Such type of method is also proposed here for a BE time domain formulation. By means of local Dirichlet-to-Neumann maps and a weak statement of the interface conditions a condensed abstract formulation is obtained. The global problem is given in a variational principle without a specification of the discretization method (e.g., BEM or FEM). Whereas this methodology has been fully established for elliptic partial differential equations, the transfer to hyperbolic initial boundary value problems is still an open question. Here, the formal procedure is given followed by some exemplary numerical tests.

References [1] O. Steinbach, Stability Estimates for Hybrid Domain Decomposition Methods. Springer, 2003. [2] B.I. Wohlmuth, A mortar finite element method using dual spaces for the Lagrange multiplier, SIAM Journal on Numerical Analysis, 38, 989–1012, 2000.

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Control of Thermal Stress in a Piezoelectric Composite Disk by a Stepwise Applied Electric Potential Distribution Fumihiro Ashida*, Sei-ichiro Sakata*, Kouhei Matsumoto† *

†

Department of Electronic and Control Systems Engineering, Shimane University 1060 Nishikawatsu-cho, Matsue, Shimane 690-8504, Japan [email protected] [email protected]

Interdisciplinary Graduate School of Science and Engineering, Shimane University 1060 Nishikawatsu-cho, Matsue, Shimane 690-8504, Japan [email protected]

ABSTRACT By utilizing the direct and converse piezoelectric effects inherent in piezoelectric materials, functions called "self-monitoring" and "self-control", which are essential for smart structures, can be achieved. For smart structures operating in thermal environments, materials must have heat resistance, high strength and high stiffness. Based on such a background, displacement control problems of layered piezoelectric composite plates have been discussed in the previous paper [1]. But, a problem of controlling thermal stress seems not to have been analyzed within the authors’ knowledge. When a structure is subjected to a heating temperature beyond the allowable thermal load, the maximum thermal stress must be controlled for safety of the structure. This paper deals with a stress control problem of a layered composite circular disk. The disk consists of a transversely isotropic structural layer onto which a piezoelectric layer of crystal class 6mm is perfectly bonded. Some electrodes are arranged concentrically on the top surface of the piezoelectric layer. The cylindrical boundary of the composite disk is taken to be thermally insulated, constrained against radial deformation, and electrically charge free. The top and bottom surfaces of the composite disk are considered to be stress-free. When a heating temperature distribution acts on the bottom surface of the composite disk and heat convection occurs over the top surface, we wish to control the maximum thermal stress induced in the structural layer by applying a stepwise electric potential distribution to the piezoelectric layer. First, by solving Fourier’s heat conduction equations, the temperature distributions in the layers which satisfy the thermal boundary conditions are derived. Next, letting the magnitude of each stepwise applied electric potential be Heaviside’s unit step function, the stresses in the structural layer as well as the stresses, electric potential and electric flux densities in the piezoelectric layer are obtained by means of the potential function methods. Based on the analytical results, the voltage applied to each electrode is determined by optimization using the BFGS quasi-Newton method so that the maximum thermal stress in the structural layer is minimized subject to constraints on the stresses in the piezoelectric layer. Numerical results have been obtained for a composite disk consisting of a transversely isotropic CFRP layer and a cadmium selenide layer of crystal class 6mm.

Reference [1] T. R. Tauchert, F. Ashida, Control of Transient Response in Intelligent Piezothermoelastic Structures. Journal of Thermal Stresses, 26, 559-582, 2003.

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Mitigation of Free-Edge Effects by Meso-Scale Structuring Jared N. Baucom*, Muhammad A. Qidwai†, and James P. Thomas* *

Naval Research Laboratory, Multifunctional Materials Branch 4555 Overlook Avenue SW, Washington, DC 20375, USA [email protected], [email protected] † SAIC, NRL Operations 1220 12th Street SE, Suite 140, Washington, DC 20003, USA [email protected]

ABSTRACT We are developing a new class of fiber-reinforced polymer composite materials to facilitate the embedment of multifunctional features and devices in material systems and to manage interlaminar stresses at the external free edges and internal free surfaces of holes and cut-outs in composite laminates. The idea is centered on the introduction of one or more additional dimensions of design space by a tessellation of individual laminae into sets of discrete tiles, each possessing the same levels of design freedom normally associated with an entire lamina (material constituents, fiber orientation, and so on). In this work, we have focused on the development of tiling schemes that will allow blending of disparate laminates (lay-ups), where a lay-up suitable for suppressing interlaminar stresses could be substituted at necessary locations in place of another lay-up that may be more suitable for the global structural loads. This technique results in the inclusion of possibly detrimental matrix-rich tile-to-tile interface pockets in the plane of each lamina. Mechanical testing has shown that uniaxially reinforced tiled composites maintain stiffness levels near those of their traditional continuously reinforced counterparts, yet with a potential degradation of strength. We have used the finite element method to investigate the effects of resin-rich pocket size, the use of supporting continuous layers, tile size, and tile overlapping schemes (interface stacking geometry) on the distribution of stress and transfer of load around interfaces in uniaxially reinforced tiled composites. This was done with the aim to identify parameters controlling overall strength. We discovered that alignment of the resin-rich pockets through the thickness exacerbates stress-concentration and that outer continuous layers on the composite may help in better load transfer and more efficient material utilization. Failure analyses of the finite element results using three-dimensional Hashin-Rotem failure criteria [1] have shown the concept to be effective in the suppression of free-edge delamination in traditional quasi-isotropic and angle-ply laminates under tensile loading. Although each meso-scale structured solution must be tailored to the exact structural geometry and anticipated loads, the technique shows promise to have broad application

References [1] Z. Hashin, Failure Criteria for Unidirectional Fiber Composites. Journal of Applied Mechanics, 47, 329-334, 1980.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A Quasi-2d Finite Element Formulation for Static and Dynamic Analysis of Sandwich Beams Jean-Jacques R. Boileau Bekuit , Donatus C.D. Oguamanam , Olatunde Damisa†  Ryerson

University Dept. of Mech. & Industrial Engineering, 350 Victoria Street, Toronto, Ontario Canada M5B 2K3 {jboileau,doguaman}@ryerson.ca † University of Lagos Dept. of Mechanical Engineering, Akoka, Yaba, Nigeria [email protected]

ABSTRACT Sandwich structures are typical found in designs that require high stiffness to weight ratio. Their use has also been demonstrated in the enhancement of acoustic properties, attainment of specific thermal properties, and for mere aesthetics. The classical three-layer sandwich structures have a core that is sandwiched between two face-sheets. The inadequacy of the classical assumptions in light of the growing trend towards flexible-core structures has led to the development of higher-order theories and layer-wise theories. With regard to sandwich beams, the more prevalent assumptions are that the face-sheets and the core are adequately modelled by Euler-Bernoulli beam theory and Timoshenko beam theory, respectively. This is very limiting, especially in situations where the core is soft. The proposed quasi-two-dimensional finite element formulation expresses the through-thickness dependency of the field variables as polynomials while their span dependency across a finite element is cubically interpolated. The advantage of this formulation is demonstrated via parametric static and dynamic studies that includes the ratio of the modulus of elasticity of the core and face-sheets, and the ratio of their heights. The fidelity of the formulation in relation to sandwich beams with viscoelastic core is also examined. The viscoelastic damping is represented using fractional derivatives. The Gr¨unwald approximation is adopted and the resulting set of governing equations is integrated using the Newmark time-integration scheme. The quasi-2D formulation permits the clamping of one end of the beam (i.e. fully cantilevered) or the the clamping of only one end of the face-sheets (i.e. partially cantilevered). The beam transverse displacement, but not the through-the-thickness stresses, is observed to be independent of the boundary condition type.

References [1] D.J. Mead, A comparion of some equations for the flexural vibration of damped sandwich beams. J. of Sound and Vibration, 83, 363-377, 1982. [2] A.C. Galucio, J.-F. De¨u, R. Ohayon, Finite element formulation of viscoelastic sandwich beams using fractional derivative operators. Computational Mechanics, 33, 282-291, 2004. [3] S. Oskooei and J.S. Hansen, Higher-Order finite element for sandwich plates. AIAA J., 38, 525533, 2000.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

221

A New Damage Identification and Quantification Indicator for Piezoelectric Advanced Composites Ayech Benjeddou *, Sahadevan Vijayakumar, Imad H. Tawfiq Institut Supérieur de Mécanique de Paris 3 rue Fernand Hainault, 93407 Saint Ouen CEDEX, France * [email protected]

ABSTRACT Vibration-based damage identification and health monitoring are well established techniques for conventional structures [1]. However, their application for composites has attracted researches only recently [2]. Corresponding investigations have concerned mainly optical fibres as sensors; while the use of piezoceramics, as sensors or/and actuators, was mainly limited to Lamb waves- and impedance-based high frequency approaches. It is then the objective of the present contribution to propose an innovative low-frequency vibration-based damage identification (presence) and quantification (location, length and depth) method for piezoelectric advanced composites. It suggests, for the first time, the use of the so-called generalized or effective (structural) electromechanical coupling coefficient (EMCC) as a damage indicator instead of the classical frequency one. The EMCC is a measure of the conversion efficiency of electrical energy to mechanical one and viceversa. Since any change in the host structure stiffness, due to the damage, considerably affects the energy conversion of the piezoelectric devices, the structural EMCC is shown to be a good candidate as a damage indicator. A structural EMCC change factor (ECF), from the healthy to the damaged state, is introduced for the damage identification and compared to the corresponding classical frequency change factor (FCF). It is found that the magnitudes provided by the former are higher than those from the latter. Hence, they can be used efficiently in artificial neural network (ANN) nonmodel based damage quantification in piezoelectric laminated beam structures [3]. Parametric analyses on the damage variables under various mechanical boundary conditions are conducted for the education of the ANN. It is found that the damage can be quantified within a maximum error of 6.5%. It is also evidenced that the ECF predictions are much better than the FCF ones.

References [1] S. W. Doebling, C. R. Farrar, M. B. Prime, A summary review of vibration-based damage identification methods. The Shock and Vibration Digest, 30, 91-105, 1998. [2] Y. Zou, L. Tong, G. P. Steven, Vibration-based model-dependent damage (delamination) identification and health monitoring for composite structures – a review. Journal of Sound and Vibration, 230, 357-378, 2000. [3] S. Vijayakumar, A. Benjeddou, I. Tawfiq, Damage quantification in smart laminated beams using neural networks and static signatures. In C.A. Mota Soares et al. (Eds.), II ECCOMAS Thematic Conference on Smart Structures and Materials, Lisbon, Portugal, July 18-21, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Design of Variable-Stiffness Conical Shells for Maximum Fundamental Frequency ∗ ¨ Adriana W. Blom∗,† , Shahriar Setoodeh∗ , Jan M. A. M. Hol∗ , Zafer Gurdal ∗

Faculty of Aerospace Engineering, Delft University of Technology Kluyverweg 1, 2629 HS Delft, The Netherlands † [email protected] ABSTRACT

A truncated conical shell is a typical configuration that is used in aerospace applications. Many of these applications experience severe dynamic loading during their lifespan, so that optimization for maximum fundamental frequency is often required. Hu and Ou [1] optimized fiber-reinforced composite conical shells by finding the optimum (constant) fiber orientation for certain layers in a laminate stack. Traditionally, the fibers within fiber-reinforced composite laminae are straight, resulting in constantstiffness laminates. New production techniques such as tow placement enable the fibers to curve within the individual planes of the laminate, thereby allowing for variable stiffness. Design of flat plates with variable stiffness while taking into account the manufacturing limits of tow-placement machines is done by Tatting and G¨urdal [2]. A design procedure for tow-placed, variable-stiffness conical shells has been developed by Blom et al. [3]. In this paper, three basic paths defining the angle variation developed by Blom are used to design variable-stiffness conical shells with different geometries for maximum fundamental frequency. These are the geodesic path, the constant angle path and the constant curvature path. The latter is used for a single-stage and a two-stage angle variation. During the design, manufacturability is taken into account in the form of a curvature constraint on the fiber path. Optimization is performed using sequential quadratic programming, where the frequency is calculated by finite element analysis and sensitivity is obtained by finite differences. The manufacturability constraints do not only apply to variable-stiffness shells, but also to constantstiffness shells, because on conical shells these require steering as well. Numerical examples show that manufacturability can have a large influence on the value of the maximum fundamental frequency and thus that it is necessary to take the manufacturing constraints into account in the design phase. Also it is shown that varying the stiffness can increase the maximum fundamental frequency up to 20 percent when compared to the best constant-stiffness laminate. For small cone angles the differences between the four fiber path definitions are relatively small, while the differences increase for larger cone angles. For all cone geometries the two-stage constant curvature laminates perform best, and especially for cones with large axial lengths the improvements with respect to the constant-stiffness laminates become substantial.

References [1] H.-T. Hu and S.-C. Ou, Maximization of the fundamental frequencies of laminated truncated conical shells with respect to fiber orientations. Composite Structures, volume 52, 265–275, 2001 [2] B. F. Tatting and Z. G¨urdal, Automated Finite Element Analysis of Elastically-Tailored Plates, NASA/CR-2003-212679, December 2003 [3] A. W. Blom, B. F. Tatting, J. M. A. M. Hol and Z. G¨urdal, Path Definitions for Elastically Tailored Conical Shells, accepted for AIAA SDM conference 2006

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A Micromechanics-Based Damage Model for the Strength Prediction of Composite Laminates Pedro P. Camanho*, Joan A. Mayugo†, Pere Maimí†, Carlos G. Dávila** *

DEMEGI, Faculdade de Engenharia, Univerisdade do Porto Rua Dr. Roberto Frias, 4200-465 Porto, Portugal [email protected] †

Escola Politècnica Superior, Universitat de Girona Campus Montilivi, 17071 Girona, Spain {pere.maimi,ja.mayugo}@udg.es **NASA Langley Research Center Hampton, VA 23681, U.S.A. [email protected]

ABSTRACT The prediction of the failure of composite laminates subjected to multi-axial loading in regions with mild stress gradients is of considerable interest for some aerospace applications such as pressurized vessels. In such components, which are devoid of notches or other stress concentrations, matrix cracks can accumulate prior to the localization of damage along a narrow fracture path. The prediction of the onset of ply damage can be accomplished by using ply-level failure criteria [1]-[2]. To predict ultimate failure of a laminate the ‘ply-discount’ method is normally used. The ply discount method reduces by an empirical amount the elastic properties of a damaged ply as a function of the type of damage predicted. Such an approach is not rigorous for the simulation of the progressive accumulation of transverse matrix cracks. The main objective of the current work is to develop an alternative method to predict the onset of ply damage and ultimate failure of a laminate. The failure criteria used to predict damage onset are the LaRC03 failure criteria [2]. A micromechanical model of a laminate containing matrix transverse cracks under multi-axial loading is proposed. The model proposed can predict the increased strength of a ply when it is embedded in a multidirectional laminate, and it relates the multi-axial strain state to the density of transverse matrix cracks. Based on the micromechanical analyses, a new model is formulated in the framework of damage mechanics. The characteristics of the constitutive model are based on the micromechanical model. Several examples relating the applied loads to the stiffness of a multidirectional laminate are presented.

References [1] Soden, P.D.; Hinton, M.J., and Kaddour, A.S. A comparison of the predictive capabilities of current failure theories for composite laminates, Composites Science and Technology, 58, 12251254, 1998. [2] Dávila, C.G., Camanho, P.P., and Rose, C.A. Failure criteria for FRP laminates, Journal of Composite Materials, 39, 323-345, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

224

Finite Element Analysis of Geometrically Nonlinear Thin-Walled Composite Laminated Beams J. E. Barradas Cardoso*, Nuno M. B. Benedito†, and Aníbal J. J. Valido† *

Instituto Superior Técnico, Departamento de Engenharia Mecânica Av. Rovisco Pais, 1049-001 Lisboa , Portugal [email protected] †

Escola Superior de Tecnologia, Instituto Politécnico de Setúbal Campus do IPS, Estefanilha, 2914-508, Setúbal, Portugal [email protected] , [email protected]

ABSTRACT The purpose of the present work is to develop a finite element model for structural analysis of composite laminated thin-walled beam structures, with geometrically nonlinear behavior, including torsional warping deformation. The structural deformation is described by an Updated Lagrangean formulation. To define the load-deflection path, a generalized displacement control method has been implemented. We have considered a thin-walled section made from an assembly of flat layered laminated composite elements. The parts like a flange and a web in a section will be referred to as a panel. The cross-section bending-torsion properties are integrals based on the cross-section geometry, on the warping function and on the individual stiffness of the laminates that constitute the crosssection. The cross-section geometry is discretized by quadratic isoparametric finite elements in order to determine its bending-torsion properties. The structural discretization is formulated throughout three-dimensional two-node Hermitean finite beam elements, with seven degrees of freedom per node. In numerical examples a thin-walled cross-section cantilever beam is considered. The influence of the lamina orientation on the critical load is studied.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

225

Design Sensitivity Analysis of Composite Thin-walled Profiles including Torsion and Shear Warping J. E. Barradas Cardoso *, Aníbal J. J. Valido † *

Instituto Superior Técnico, Departamento de Engenharia Mecânica Av. Rovisco Pais, 1049-001 Lisboa , Portugal [email protected] †

Escola Superior de Tecnologia, Instituto Politécnico de Setúbal Campus do IPS, Estefanilha, 2914-508, Setúbal, Portugal [email protected]

ABSTRACT The present work deals with the design sensitivity analysis of thin-walled laminated composite beam cross-section bending-torsion properties. These properties are expressed as integrals based on the cross-section geometry, on the warping functions for torsion, shear bending and shear warping, and on the individual stiffness of the laminates that constitute the cross-section. Among these properties, we may emphasize the location of the shear center, the torsion stiffness, the warping stiffness and the shear coefficients. A variational continuum formulation for the design sensitivity analysis of crosssection bending-torsion properties is presented. The formulation is based on the concept of adjoint system, which suffers a specific adjoint warping for each of the properties. The developed formulation can be applied in a unified way to open, closed or hybrid cross-sections, with simply connected or multi-connected domains. The lamina orientation and the laminate thickness are selected as the de-sign variables. For the sensitivity calculations, the cross-section may be modeled throughout design elements to which the element sensitivity equations correspond. Geometrically, the de-sign elements may coincide, with the laminates that constitute the cross-section.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Numerical approach for the design of adhesively-bonded assemblies J.Y. Cognard*, R. Créac’hcadec* * Mechanics of Naval and Offshore Structures ENSIETA - 2 rue F. Verny, 29806 Brest Cedex 09, France {Jean-Yves.Cognard, romaincr}@ensieta.fr

ABSTRACT This paper presents contributions of numerical modelling for the optimization of adhesivelybonded assemblies for marine and underwater applications. Difficulty in modelling the failure of even simple joints (lab shear specimens) highlighted the need for more reliable constituent input data [1 ; 2]. Therefore, in order to be able to study the behaviour of thin adhesive films up to failure, as a function of the normal stress, a modified Arcan fixture has been developed. Numerical simulations in linear elasticity, for bi-material structures show that the use of special geometry for the substrate (a beak close to the adhesive joint) makes it possible to eliminate the contribution of edge effects [3]. Non linear simulations taking into account the fixing system of the substrates on the supporting fixture were used to optimize the design of the complete fixture in order to prevent pre-loading of the adhesive joint. Moreover, in order to have more precise data for design, a study of the evolution of the stresses through the thickness of the adhesive joint under elastic assumption is proposed with respect to the shape of the edge of the adhesive joint. Non-contact extensometry and optimisation techniques have been used to determine the kinematics of the bonded joints deformation. For the epoxy resin Vantico™ Redux 420 the viscoplastic behaviour has been analyse for different radial loadings with the following variables: the applied load and the displacement of both extremities of the adhesive joint. This paper mainly presents the use of these experimental data, to develop a numerical tool to predict the behaviour of adhesively-bonded assemblies. To limit the numerical cost, we propose an approach using joint type elements which allows us to work directly with the relative displacement and the stress. As under elastic assumption, a non uniform stress field is observed in the adhesive joint, an inverse technique has been developed to identify the parameters of the model. For monotonic loadings, a plastic behaviour with isotropic hardening gives good results and allows us to analyse the non-isotropic behaviour of the adhesive with respect to tension-compression. In order to analyse the possibilities of this numerical approach, comparisons with experimental data are presented. A simple lab shear type specimen with thick substrates and beaks is proposed to analyse the influence of different parameters.

References [1] F.E. Penado, Singular intensity factors at bimaterial anisotropic interfaces. Composite Structures, 52, 323-333, 2001. [2] A.G. Magalhães, M.F.S.F. De Moura, J.P.M. Goncalves, Evaluation of stress concentration effects in single-lap bonded joints of laminate composite materials. Inter. J. Adhesion and Adhesives, 25, 313-319, 2005. [3] J.Y. Cognard, P. Davies, B. Gineste, L. Sohier, Development of an improved adhesive test method for composite assembly design. Composites Science and Technology, 65, 359-368, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Enhancement of Blast Resistance of Sandwich Plates George J. Dvorak * and Yehia A. Bahei-El-Din† *

Rensselaer Polytechnic Institute Department of Mechanical, Aerospace and Nuclear Engineering, Troy, NY 12180, USA [email protected] †

Rensselaer Polytechnic Institute Department of Mechanical, Aerospace and Nuclear Engineering, Troy, NY 12180, USA [email protected]

ABSTRACT Conventionally designed sandwich plates for marine structures typically consist of thin fibrous laminate face sheets bonded to a structural foam core. Vinyl ester matrices reinforced by glass or carbon fibers or fabrics, and PVC foam cores are the commonly used materials. Contact with floating objects or moorings, or impact by a stiff indenter in laboratory experiments, may cause local deflection of the face sheet and permanent deformation or crushing of an underlying volume of the foam core, and finally to nucleation and extension of interfacial cracks. Sandwich plates may also be exposed to blast loading, a pressure pulse imparted to the top face sheet, which causes extensive crushing of the foam core and interfacial delamination in the conventionally designed plates. Our work is concerned with development and evaluation of modified sandwich plate designs for improvement of impact and blast resistance. This improvement involves use of a polyurea interlayer, bonded between the impacted face sheet and foam core. The polyurea exhibits significant stiffening at high deformation rates, and thus inhibits overall deformation of the plate and crus hing of the foam core. Finite element models of the sandwich plate designs were developed for explicit dynamic solution with the LS-Dyna software. The models capture the nonlinear deformation and damage modes observed in sandwich plates under dynamic loads. Contact algorithms were invoked to model intermittent separation and coalescence of the facesheets and the inner core following delamination. Comparisons of overall bending deflections and damage resistance show that the proposed design modifications reduce both peak kinetic energy, the maximum longitudinal strain in the facesheets and foam core compression by more than 50%. Moreover, a significant reduction was observed in the overall deflection at mid-span, and in curvature at the supports.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Mechanics and Acoustics of Viscoelastic Composites by Micro-Macro Mean-Field Approach ∗




C. Friebel , I. Doghri †, V. Legat

Universit´e catholique de Louvain (UCL), CESAME, Bˆatiment Euler, 4-6 av. G. Lemaˆıtre, B-1348 Louvain-la-Neuve, Belgium ∗ [email protected], †[email protected],
[email protected]

ABSTRACT We develop mean-field homogenization schemes for the prediction of the effective mechanical and acoustical properties of viscoelastic inclusion-reinforced materials. For mechanical applications, the overall behavior (e.g. stiffness tensor) of composite materials is estimated using Eshelby-based homogenization models. The effective acoustical properties (e.g. attenuation factor) are obtained in a similar way, based on the one particle scattering problem. Viscoelastic materials are known to be mechanical dampers characterised by their loss factor. The latter has also an impact on sound propagation through these materials. For elastic composites the scattering of waves on the inclusions is responsible for sound attenuation. For materials made of viscoelastic components both phenomena, material damping and particle scattering, must be taken into account. We present general Eshelby-based procedures able to predict the viscoelastic mechanical properties of multiphase – more than two – composites. There is the two-step homogenization procedure which besides the usual assumptions of Eshelby-based models, does not suffer any restriction in terms of material properties, aspect ratio or orientation. We also propose a two-level recursive scheme for matrix materials with coated inclusions. We implemented the Mori-Tanaka (M-T) model and a non-trivial interpolation between M-T and inverse M-T estimates to achieve the stages of those procedures. We made (see [1]) extensive confrontations with experimental data or unit cell finite element (FE) simulations. For a large variety of viscoelastic composite materials, the proposed schemes perform much better than other existing homogenization methods. We propose homogenization schemes to predict the effective acoustical properties of viscoelastic composites. It is an extension of the effective medium approach for wave propagation in inclusion-reinforced materials widely studied by various authors (e.g. [2] and references therein) in the framework of linear elasticity. In their approach, every inclusion behaves as isolated in a medium with the effective properties of the composite. We wish to avoid the self-consistent nature of the formulation and study adaptations of the M-T and interpolative models to the scattering problem.

References [1] C. Friebel, I. Doghri, V. Legat,General mean-field homogenization schemes for viscoelastic composites containing multiple phases of coated inclusions, Int. J. Solids and Struc., in press 2005. [2] S.K. Kanaun, V.M. Levin, F.J. Sabina, Propagation of elastic waves in composites with random set of spherical particles (effective medium approach), Wave Motion 40, 69-88, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Numerical Evaluation of Upper and Lower Bounds to the Collapse Limit Load for Composite Laminates P. Fuschi, A. A. Pisano DASTEC, University Mediterranea of Reggio Calabria, Via Melissari, I-89124 Italy {paolo.fuschi, aurora.pisano}@unirc.it

ABSTRACT This paper proposes a procedure based on the limit analysis theory to define the load bearing capacity of laminates made of composite material. To apply the limit analysis theory on structures made of materials, which can, in general, exibit limited ductility and often characterised by a failure criteria rather than by a yield one, the actual material is effectively replaced with a perfectly plastic one, making the failure criteria playing the role of yield condition [1]. Grounding on these assumptions, the kinematic approach for limit analysis of anisotropic composite laminates under plane stress conditions is applied for the direct evaluation of an upper bound on the collapse load multiplier, moreover the static approach is utilized to define a lower bound to the same limit load. In the present formulation the limit analysis is performed via a numerical procedure known as Elastic Simulation Method [2]. The numerical procedure, successfully applied to Von Mises [2,3] and DruckerPrager materials [4], is here suitably modified to ac- count for the specific form of the yield function assumed to deal with anisotropic materials. The effectiveness of the proposed procedure is verified by a few simple numerical examples.

References [1] A. A. Pisano, P. Fuschi, A numerical approach for limit analysis of composite structural elements. Proc. of Complas 2005, VIII Int. Conf. on Computational Plasticity Fundamentals and Applications, Barcelona, 5-8 September 2005, II, pp. 810-813. [2] D. Mackenzie, T. Boyle, A method of estimating limit loads by iterative elastic analysis. I-simple examples. Int. J. Press. Ves. & Piping., 53, 77-97, 1993. [3] A.R.S. Ponter, K.F. Carter, Limit state solutions based upon linear elastic solutions with a spatially varying elastic modulus. Comp. Meth. Appl. Mech. Eng., 140, 237-258, 1997. [4] A.R.S. Ponter, P. Fuschi, M. Engelhardt, Limit analysis for a general class of yield conditions. European J. of Mech. A/Solids, 19, 401-421, 2000.

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Material degradation assessment for stiffened composite shells using metamodelling approach. Kaspars Kalnins*, Janis Auzins†, Rolands Rikards* *

Institute of Materials and Structures, Riga Technical University 1 Kalku St. LV-1658, Riga, Latvia [email protected]; [email protected] †

Institute of Mechanics, Riga Technical University 6 Ezermals Str. LV-2006, Riga, Latvia [email protected]

ABSTRACT The intense interest coming from the aerospace industry indicates the need of safe exploitation of composite materials in stiffened shell structures. Since stiffened shells are far most consumed structural component, it is important to study the behaviour of material degradation to evaluate the safe design guidelines. Moreover, current numerical procedures cannot simulate the collapse of stiffened shells with sufficient reliability and efficiency, leading to over-conservative designs. One can assume that great potential exist for future increase of effectiveness of stiffened composite structures by allowing of post-buckling of skin to occur during the exploitation [1]. An assessment of material degradation in terms of stiffness reduction in the skin stringer zone [2] is carried out to estimate the material degradation influence over post-buckling stiffness of the axially loaded stiffened shells. The presented procedure is based on the building of metamodels employing experimental design and response surface methodology. Metamodels are built using stiffened shell geometrical variables adding material degradation variables as: degradation region length and material elastic property reduction coefficient. The numerical responses, obtained from explicit finite element simulations of composite stiffened shells subjected to buckling and post-buckling, are used for the building of metamodels. The acquired metamodels are built both for approximation of the dependence of load-shortening curve on material degradation and for the inverse problem – the determination of material degradation degree from essential response parameters of the load shortening curve. The resulting design procedure provides an effective optimal design tool for the safe preliminary study of composite stiffened shells under axial compression [3].

References [1] R. Degenhardt, H. Klein, A. Kling, H. Temmen and R. Zimmermann, Buckling and PostBuckling Analysis of Shells under Quasi-Static and Dynamic Loads. DLR report, 2002 [2] R. Rikards, K. Kalnins and O. Ozolinsh, Delamination and Skin-Stringer Separation Analysis in Composite Stiffened Shells. In Proceedings of the 7th International Conference on Computational Structures Technology, B.H.V. Topping and C.A. Mota Soares, (Editors), CivilComp Press, Stirling, United Kingdom, paper 47, 2004. [3] R. Rikards, H. Abramovich, J. Auzins, A. Korjakins, O. Ozolinsh, K. Kalnins and T. Green, Surrogate Models for Optimum Design of Stiffened Composite Shells. Composite Structures, Elsevier, 63, 243-251, 2004

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A new model for the behaviour of the multi-layer material interfaces M. Karama*, K. S. Afaq*, S. Mistou* *

Laboratoire Génie de Production, Equipe CMAO, Groupe M2SF, ENIT, BP 1629, 65016 Tarbes Cedex, France. {moussa, kamran, mistou}@enit.fr

ABSTRACT One of the current problems connected with multiplayer composite structures concerns the analysis of the distribution of the stresses around peculiarities (free edge and loaded edge) and at the interfaces of each layer. This work presents a new shear stress function in the form of the exponential function, to predict the mechanical behaviour of multi-layered laminated composite structures. As a case study, the mechanical behaviour of laminated composite beam (90°/0°/0°/90°) is examined. The results are compared with the model “Sinus” and 2D finite element method studied. Results show that this new model is more precise than older ones as compared to the results by the finite element analysis [Abaqus]. To introduce continuity on the interfaces of each layer, the kinematics defined by Ossadzow is used with new exponential model. The equilibrium equations and natural boundary conditions are derived by the principle of virtual power.

Figure 1: Variation of the stress σ13 through the thickness for a clamped free beam under distributed uniform load Present (-*-), Sine7(-), Abaqus7(- - -)

References [1] C. Ossadzow, P. Muller, M. Touratier, Une Théorie Générale des Coques Composites Multicouches. Deuxième Colloque National en Calcul des Structures, Tome 1, Hermes,1995. [2] M. Karama, K. S. Afaq, S. Mistou, Mechanical Behaviour of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity. International Journal Solide and Structures, 40 N°6 , 1525-1546, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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FE-Tool CODAC for an Efficient Simulation of Low-Velocity Impacts on Composite Sandwich Structures Luise K¨arger, Jens Baaran, Jan Teßmer DLR, Institute of Composite Structures and Adaptive Systems Lilienthalplatz 7, 38108 Braunschweig, Germany [email protected], [email protected], [email protected]

ABSTRACT The finite element based damage tolerance tool, CODAC, has been developed for efficiently simulating the damage resistance of sandwich structures subjected to low-velocity impacts. The considered double shell structures consist of two thin composite face sheets separated by a lightweight core. Due to their high mass specific stiffness and strength, a very weight efficient design is achievable. Moreover, the core can provide damping and insulation, while the outer face sheet can act as an impact detector. However, impact damage in sandwich structures can provoke a significant strength and stability reduction. Therefore, the objective of CODAC is to provide methodologies which reliably simulate impact events and accurately predict impact damage sizes. Since frequent design loops require a quick analysis, efficient deformation and failure models are needed. To achieve a rapid and accurate stress analysis, a three-layered finite shell element has recently been developed [1]. The element accounts for shear deformation and transverse compressibility. Since an accurate approximation of the transverse stresses is an important requirement for detecting impact damage, transverse stresses are improved by the so-called Extended 2D-Method, which is an equilibrium approach that has been applied to a three-layered shell theory [2]. To predict damage growth during the impact event, a progressive damage mechanics approach is applied. Stress-based failure criteria are used to detect damage initiation. Subsequently, material resistance is reduced by applying appropriate degradation models. An experimental impact test program on honeycomb sandwich panels is used to validate the impact simulation of the FE-tool CODAC. Force-time histories and damage sizes are examined, and the influence of distinct failure models on the impact response is analyzed [3]. Comparisons between impact tests and simulations showed that CODAC is capable of accurately and rapidly simulating impact events, which induce barely visible damage.

References [1] A. Wetzel, L. K¨arger, R. Rolfes and K. Rohwer, Evaluation of two finite element formulations for a rapid 3D stress analysis of sandwich structures. Computers and Structures, 83, 1537-1545, 2005. [2] L. K¨arger, A. Wetzel, R. Rolfes and K. Rohwer, A three-layered sandwich element with improved transverse shear stiffness and stresses based on FSDT. Computers and Structures, in press, 2006. [3] L. K¨arger, J. Baaran and J. Teßmer, Rapid Simulation of Impacts on Composite Sandwich Panels Inducing Barely Visible Damage. Composite Structures, in press, 2006.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Preliminary Investigation for Optimization of Fiber Reinforced Cementitious Composite Structures J. Kato*, A. Lipka[, and E. Ramm[ Institute of Structural Mechanics, University of Stuttgart Pfaffenwaldring 7, D – 70569 Stuttgart / Germany {kato, lipka, ramm}@statik.uni–stuttgart.de

ABSTRACT The present study addresses the preliminary steps for the optimization of fiber reinforced cementitious composites (FRC) with respect to brittle failure behavior of a single component. FRC is one of the advanced materials that consist of a concrete matrix and glass, carbon, or aramid fibers. This kind of reinforcement is corrosion free and highly durable when used in concrete, which makes it possible to produce very thin structural concrete elements. It is well known that the properties of such composites depend on the layout of the material at the microscopic level (e.g. fiber size, fiber coating, impregnation, and surface roughness among other material characteristics). The investigation of the sensitivity of the structural response of the composite material with respect to the microscopic material parameters is part of this work. To improve the structural behavior of FRC, the significant parameters of the material models for the interface and the fibers are introduced as design parameters and adjusted by a controlling optimization process. Two different objectives are investigated: strength and ductility of FRC. To perform the structural analysis, an isotropic gradient–enhanced damage model for concrete as well as fibers is used. Numerically integrated interface elements are applied for modeling the debonding between the fibers and the matrix. To obtain a reliable composite by applying structural optimization, the nonlinear failure behavior of matrix material, fiber, and interface models are considered within the optimization process [1]. Gradient–based optimization algorithms are used to adjust these parameters to improve both strength and ductility of such composite structures [2], [3].

References [1] M. Krüger, J. Ozbolt, H.–W. Reinhardt, A discrete bond model for 3D analysis of textile reinforced and prestressed concrete elements, Otto–Graf–journal, vol. 13, 111–128, 2002. [2] K. Maute, S. Schwarz and E. Ramm, Adaptive topology optimization of elastoplastic structures, Struct. Optim. 15, 81–91, 1998. [3] A. Lipka, S. Schwarz, and E. Ramm, Topology optimization of three–dimensional structures with consideration of elastoplastic structural response, European Conference on Computational Mechanics, ECCM, Cracow, Poland, 2001.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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On Post-Buckling Analysis and Experimental Correlation of Cylindrical Composite Shells with Reissner-Mindlin-Von K´arm´an Type Facet Model Petri Kere∗ and Mikko Lyly† ∗ Tampere University of Technology Institute of Applied Mechanics and Optimization P.O.Box 589, 33101 Tampere, Finland petri.kere@tut.fi † Center for Scientific Computing P.O.Box 405, 02101 Espoo, Finland mikko.lyly@csc.fi

ABSTRACT Designing composite structures involving geometric nonlinearities is often very time-consuming with so-called high-fidelity models, which in some cases severely limits the application of these methods. However, to get realistic solutions as a result of the design optimization process, use of the high-fidelity analysis is necessary. With the computationally efficient nonlinear analysis, realization of local failure prediction and imperfection sensitivity analysis of shells is possible. For thin laminate structures the buckling load is low and there is a long postbuckling behavior, which illustrates well the importance of being able to design shells in the postbuckling region to take advantage of the load carrying capability. In this study, post-buckling analysis and experimental correlation of cylindrical carbon fibre reinforced plastic (CFRP) shells is considered. Modeling the shell structure is based on a facet approximation of the undeformed mid-surface for a thin or moderately thick laminated composite shell. The potential energy functional where the linearized strain tensor has been replaced by the nonlinear functions, i.e., the Von K´arm´an model for large deformation was formulated [1]. The kinematical unknowns are determined from the condition that they minimize the potential energy of the shell. In the computations we solve these nonlinear equations iteratively by Riks’ method with Crisfield’s constraint for arc-length. The linearized equations are then discretized by the finite element method. In the FE implementation we use bilinear stabilized MITC elements. A computational model is based on specimens that were used in experimental tests [2]. The numerical results are compared to the experimental observations. Results show that use of the geometric perfect shell model causes a discrepancy between numerically predicted and experimentally observed buckling and post-buckling behavior. However, bringing a diamond shape geometric imperfection in the model significantly improves the correlation. Our aim is to find imperfection shape and amplitude suitable for use in structural optimization such that the computational time remains reasonable still offering accurate results for post-processing results like local failure prediction of composite.

References [1] P. Kere and M. Lyly, Reissner-Mindlin-Von K´arm´an type plate model for nonlinear analysis of laminated composite structures. Composite Structures, 71, 289–292, 2005. [2] C. Bisagni and P. Cordisco, An experimental investigation into the buckling and post-buckling of CFRP shells under combined axial and torsion loading. Composite Structures, 60, 391–402, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Application of Failure Criteria to Short Fiber Reinforced Composites and Experimental Validation 

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Tow-Placed, Variable-Stiffness Composite Panels: Damage Tolerance Improvements over Traditional Straight-Fibre Laminates Cláudio S Lopes1, Zafer Gürdal1, Pedro P. Camanho2 1

Delft University of Technology Kluyverweg 1, 2629 HS Delft, The Netherlands {c.lopes, z.gurdal}@lr.tudelft.nl 2

DEMEGI, Faculdade de Engenharia da Universidade do Porto Rua Dr. Roberto Frias, 4200-465 Porto, Portugal [email protected]

ABSTRACT One of the primary advantages of using fibre-reinforced laminated composites in structural design is the ability to change the stiffness and strength properties of the laminate by designing the laminate stacking sequence in order to improve its performance. This procedure is typically referred to as laminate tailoring. Traditionally, tailoring is done by keeping the fibre orientation angle within each layer constant throughout a structural component. Allowing the fibres to follow curvilinear paths within the plane of the laminates constitutes an advanced tailoring option to account for non-uniform stress states (e.g. plate with a hole) in a continuous manner. In addition, laminates with curved fibres will have stiffness tailoring possibilities that can lead to modification of load paths to result in favourable stress distributions within the laminate and improve the laminate performance. Based on numerical simulations, the present work demonstrates the advantages, of variable-stiffness over straight-fibre laminates in terms of strength. A physically-based set of failure criteria, able to predict the various modes of failure of a composite laminated structure, is implemented in finite element models of straight and variable-stiffness panels under compression. Nonlinear analyses are carried since first-ply failure is expected at loads higher than buckling load.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

238

A Mixed-Formulation Four-Node Rectangular Element in the Modeling of Laminate Composite Beam Structures Francisco Q. de Melo1, Rui A. S. Moreira1, José F. D. Rodrigues2 1

Departamento de Engenharia Mecânica. Universidade de Aveiro Campus Santiago. Aveiro. Portugal {fqm, rmoreira}@mec.ua.pt 2

Faculdade de Engenharia da Universidade do Porto DEMEGI. R. Dr. Roberto Frias. Porto. Portugal [email protected]

ABSTRACT A four node quadrangular element having a mixed formulation for the stress and displacement field is formulated to model the structural behavior of laminate beams. The finite element approach for this type of structures faces the difficulty of a discontinuous shear stress field along the transverse section when different mechanical properties are inherent to the layers integrating the element. The element hereby proposed has four nodes and is assumed incompressible along the transverse beam direction; this issue contributes for a stabilized hourglass-free displacement mode when a one-point reduced gauss integration is used in the evaluation of the stiffness matrix. The stress field is then computed independently of the deformation expressions, avoiding stress-smoothing algorithms in the post-processing step. The element is applied in the analysis of homogeneous and sandwich beams and provides a valuable tool in the simulation of damped sandwich structures by employing viscoelastic layers in the core.

REFERENCES [1] R.A.S. Moreira and J.D. Rodrigues, Passive Damping Control of Plates with Viscoelastic Layers: Experimental Study on Dynamic Characteristics and Radiated Noise - Keynote Lecture, IRF'99-Congress on Integrity, Reliability and Failure, Porto, Portugal, 1999. [2] E. Onãte, Cálculo de Estructuras por el Método de Elementos Finitos (Structural Analisys with the Finite Element Method; in spanish), CIMNE 1st edition, Barcelona 1992. [3] R. Cook, D. Malkus and M. Plesha, Concept and Applications of Finite Element Analysis, Wiley International Edition 3rd Ed, 1989. [4] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York, 1991. [5] T.H.H. Piam and K-Y. Sze, Hybrid Stress Finite Element Methods for Plate and Shell Structures, Advances in Structural Engineering, 4(1), 13-18, 2001.

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Optimal design of stiffened composite underwater hulls Tanguy Messager *, Pierre Chauchot**, Benoit Bigourdan** *

Institut de Recherche en Génie civil et Mécanique (GeM), UMR CNRS 6183 Ecole Centrale de Nantes, BP 92101, 44321 Nantes cédex3, France [email protected]

**

Institut Français de Recherche pour l'Exploitation de la Mer (IFREMER), Service Matériaux et Structures, BP 70, 29280 Plouzané, France [email protected], [email protected]

ABSTRACT This numerical study is devoted to the stiffened composite vessel design for deep submarine exploration housings and autonomous underwater vehicles [1]. The structures under investigation are lengthy laminated cylinders with circumferential and longitudinal (rings and stringers, respectively) unidirectional composite stiffeners and rigid end closures. Structural buckling induced by the high external hydrostatic pressure is considered as the major risk factor under service conditions. The objective of this work is the development of an optimization tool allowing the search of the reinforcement definition (lamination and stiffener characteristics) that maximize the limit of stability. An analytical model of cylindrical composite shell buckling, with transverse shear effects, has been developed. The contribution of the stiffeners is taken into account by correcting the overall laminate stiffness coefficients. The search of optimal design solutions is achieved by coupling this model to a developed genetic algorithm procedure: it manipulates directly integer parameters and, according to results taken from the literature and preliminary tests, the tournament selection, the whole arithmetical crossover and the random uniform mutation are applied. Numerical tests have been performed: multi-parameter (number of composite plies and stacking sequence of the cylinder, numbers of rings and stringers) optimization calculations have been carried out applying user’s fixed amounts of material. The results showed substantial buckling pressure increases measured with respect to reference design solutions. FEM calculations have confirmed the corresponding gains. Besides, the results of the present work corroborate design tendencies validated previously by experiments [2]: the optimized laminations exhibit typical patterns and the rings play a major role.

References [1] P. Davies, P. Chauchot, Composites for marine applications – part 2: underwater structures. Mechanics of composite materials and structures, Kluwer Academic Pub., 249-260, 1999. [2] T. Messager, M. Pyrz, B. Gineste, P. Chauchot, Optimal laminations of thin underwater composite cylindrical vessels. Composite Structures, 58-4, 529-537, 2002.

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Higher Order Model for Analysis of Magneto-Electro-Elastic Plates José S. Moita∗, Cristóvão M. Mota Soares†, Carlos A. Mota Soares† ∗

Universidade do Algarve, Escola Superior de Tecnologia, Campus da Penha, 8000 Faro, Portugal [email protected]

† IDMEC-Instituto de Engenharia Mecânica-Instituto Superior Técnico

Av. Rovisco Pais,1096-Lisboa Codex, Portugal. [email protected], [email protected]

ABSTRACT In the recent years the study of smart structures has attracted significant researchers. The use of smart materials, such as piezoelectric and/or piezomagnetic materials, in the form of layers or patches embedded and/or surface bonded on laminated composite structures, can provide the so-called adaptive structures. In these cases, the structure behaviour is not defined by the geometry and material properties, but also by the electric and magnetic fields that are applied to the structures, because the piezomagnetic materials have the ability of converting energy from one form to the other (among magnetic, electric, and mechanical energies). In this paper we present a finite element model, based in the third-order shear deformation theory, for static and free vibration analysis of plate structures integrating piezoelectric/piezomagnetic layers. A simple and efficient three-node triangular fat plate element is used. The formulation introduces one electric potential and one magnetic potential degree of freedom for each piezoelectric and piezomagnetic layer of the finite element. The solutions given by the proposed model for illustrative numerical examples, in static and free vibration, are presented and discussed.

elastic plate Mode B only 1 12600.01 2 24926.81 3 25147.20 4 34176.64 5 39204.94 6 39963.98

F only 14882.24 27965.02 28198.62 37478.54 42568.46 43475.44

B/F/B 12751.45 25211.44 25433.88 34566.57 39657.88 40422.48

magneto-electro-elastic plate F/B/F 14738.18 27664.41 27895.27 37048.92 42059.00 42960.06

B only 12631.77 24964.69 25185.42 34210.58 39234.02 39995.27

F only 14905.91 27987.31 28221.14 37496.29 42582.32 43490.76

B/F/B 12783.17 25249.50 25472.28 34601.21 39687.79 40454.58

Table 1. Natural Frequencies (rad/s) : L=1 m ; H=0.3 m …HSDT

F/B/F 14761.57 27686.12 27917.20 37065.91 42072.00 42974.49

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Mixed Least-Squares Finite Element Model for the Static Analysis of Laminated Composite Plates F. Moleiro*, C. M. Mota Soares*, C. A. Mota Soares*, J. N. Reddy† *

IDMEC/IST – Department of Mechanical Engineering, Instituto Superior Técnico Av. Rovisco Pais, 1049-001 Lisboa, Portugal [email protected] [email protected] [email protected] †

Department of Mechanical Engineering, Texas A&M University College Station, TX 77843-3123, U.S.A. [email protected]

ABSTRACT A mixed finite element model for the static analysis of laminated composite plates is presented. The formulation is based on least-squares variational principles, which is an alternative approach to the mixed weak form finite element models. The least-squares-based finite element model considers the first-order shear deformation theory, with the generalized displacements and the stress resultants as independent variables. Moreover, high-order C0 Lagrange interpolation functions and full integration are used to develop the discrete finite element model, which by the least-squares formulation, results in a symmetric and positive-definite system of algebraic equations. The predictive capability of the proposed mixed plate finite elements is demonstrated by numerical examples of the static analysis of rectangular laminated composite plates with a variety of boundary conditions and side-to-thickness ratios. Particularly, the high-order mixed plate elements based on this formulation are shown to be insensitive to shear-locking.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Applications of Distributed Piezoelectric Electrode Patches for Active Noise and Vibration Control M. Nader*, H.-G. v. Garssen†, H. Irschik‡ *

Linz Center of Competence in Mechatronics Altenbergerstrasse 69, 4040 Linz, Austria [email protected]

† Siemens AG Munich, CT PS 8 Otto-Hahn-Ring 6, 81729 Munich, Germany [email protected] ‡

Institute of Technical Mechanics, Johannes Kepler University Linz Altenbergerstrasse 69, 4040 Linz, Austria [email protected]

ABSTRACT In the present contribution the concept of dynamic shape control is applied to compensate disturbing vibrations in a composite structure. Particularly, a three-layer cantilever beam is investigated as a representative example, where displacements due to a superimposed support motion are compensated by means of two piezoelectric actuator layers. In a first step, it is assumed that the piezoelectric actuation can be applied continuously along the actuating layers of structure. As an analytic solution it follows that the deflections can be compensated all over the beam at every time instant, if the piezoelectric actuating moment coincides with a quasi static bending moment due to the rigid body parts of the inertia forces that are produced by the support motion. This solution afterwards is realized by using discrete electrode patches attached to the continuous piezoelectric actuator layers. The amount of electrical voltage applied to the discrete electrode patches is chosen such that the area under the analytical solution curve is equalized. Mohr’s analogy for the computation of beam deflections is used in order to show that this equal-area rule enforces the slope of the displacement curve to vanish at several places of the beam in a quasi static sense. In a further step, we apply the electric voltages derived from the equal-area rule of beam theory to a refined model of the composite beam derived by means of the Finite Element method. A harmonic resonant motion of the cantilever is considered, driving the beam into resonant vibrations. The cantilever is modeled by means of plane-stress Finite Elements, taking into account electromechanical coupling. In this numerical study, the goal of shape control is reached with a high accuracy. This gives excellent evidence for the usefulness of the proposed method of shape control by discrete electrode patches also for active control.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Computational Elastoplastic Modeling of Multi-phase Fiber-reinforced Composites Ernest T.Y. Ng∗ , Afzal Suleman† ∗ Department

of Mechanical Engineering, University of Victoria PO Box 3055, STN CSC, Victoria, British Columbia, Canada V8W 3P6 [email protected] † Instituto

de Engenharie Mechanica, Instituto Superior Tcnico (IDMEC-IST) Instituto Superior Tcnico (IDMEC-IST), Lisbon, Portugal [email protected]

ABSTRACT The paper presents the development of a computational model to predict the overall elastoplastic behavior of n-phase fiber-reinforced composites. The micromechanics model used to predict the overall elastoplastic behavior of fiber composite is based on the Transformation Field Analysis (TFA) which allows the combination of various micromechanics models and plasticity models [1]. In the present work, the Eshelby-Mori-Tanaka model is used to predict the concentration factor of the composite medium [2]. Next, the overall governing TFA equations are integrated implicitly using the Governing Parameter Method (GPM) and the model is suited for analyzing a wide range of metal and polymer matrix composites [3]. The use of implicit integration on the TFA equations considerably reduces the computational cost. The reason of using implicit integration on integrating the TFA equations instead of explicit integration is not only the intrinsic advantages of implicit integration over explicit integration, more importantly, the use of explicit integration on integrating TFA equations will generate a system of 6 × n algebraic-differential equations for n-phase fibre composites for the von-Mises case. However, the implicit integration will only need to solve one nonlinear algebraic equation. This certainly reduces the computational cost of the analysis.

References [1] G.J. Dvorak, Transformation field analysis on inelastic composites materials. Proceedings of the Royal Society of London, A437, 311–327, 1992. [2] T. Mori and K. Tanaka, Average stress in matrix and average elastic energy of materials with misfitting inclusions, Acta Metallurgica, 21, 571–574, 1973. [3] M. Koji´c, The governing parameter method for implicit integration of viscoplastic constitutive relations for isotropic and orthotropic metals, Computational Mechanics, 19, 49-57, 1996.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Evolution and analysis of stresses in thixoforged metal matrix composites Konstantin von Niessen1, José A. Moreno2 and Rainer Gadow1 1

University of Stuttgart, Institute for Manufacturing Technologies of Ceramic Components and Composites Allmandring 7 b, Stuttgart, Germany, D-70569. 2

Polytechnic University of Cartagena, Dept. of Mechanical Engineering C/. Doctor Fleming, s/n, Cartagena, Spain, E-30203 [email protected]

ABSTRACT Light metal matrix composites with tailor-made fiber reinforcements are promising materials for lightweight design in structural applications for the automotive and aerospace industry. Lightmetal MMCs manufactured by thixoforging of thermally sprayed prepregs additionally exhibit superior mechanical properties of the metal matrix material as well as low fiber damage during infiltration of the reinforcement fabric and consolidation of the composite. However, one of the difficulties during manufacturing of these materials is the difference in the thermophysical properties of matrix and fiber material. Different thermal expansions lead to the development of residual stresses during the cooling process that can deform the reinforcement fibers, damage the fiber-matrix interface, and hence, lead to a decrease of the mechanical properties of the reinforced component. In the case of carbon fibers, a distinct anisotropy of the thermophysical properties leads to an even increased problem complexity. A finite element model has been developed for the analysis of the plastic behavior of the aluminum alloy during cooling from semi-solid temperature to room temperature, and the orthotropic behavior of a woven carbon fiber fabric. Simulation results of the stress development inside a MMC component during the thixoforging process are introduced.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Piezoelectric Control of Composite Plate Vibration: Effect of Electric Field Distribution M. Pietrzakowski* *

Institute of Machine Design Fundamentals Warsaw University of Technology Narbutta 84, 02-524 Warszawa, Poland [email protected]

ABSTRACT The paper deals with the vibration analysis of active rectangular plates. The plates considered are composites containing piezoelectric sensor/actuator layers, which operate in a closed loop control acting to suppress transverse vibration. PVDF polymeric materials characterized by a relatively low stiffness are used for the sensors while the actuators are made of PZT ceramics because of their high electromechanical efficiency. The piezoelectric layers are poled in the transverse direction and equipped with traditional electrodes on both surfaces. In order to satisfy the Maxwell electrostatics equation the widely used simplification of the electric potential distribution in the actuator layer (linear across the thickness) is replaced by a combination of a half-cosine and linear distribution in the transverse direction. The non-linear potential term, which relates to shortly connected electrodes and a uniform bending moment applied, was analysed in [1] and [2] among others. The in-plane spatial variation of the potential instead of applying uniform distribution is determined by the solution of the coupled electromechanical governing equations with the natural boundary conditions corresponding to both the flexural and electric potential fields. The assumed potential distribution improves the model of interaction between the actuator layer and the main structure. The analysis is performed for simply supported plates. The plats are modelled as laminate or sandwich structures. In the first case the displacement field is based on the Kirchhoff hypothesis. For the sandwich plates the Mindlin model with the effect of shear and rotary inertia is applied. External load acting in the transverse direction is distributed over the limited or total area of the plate surface. The steady-state behaviour of the plate is considered. Therefore, the loading is assumed to be harmonic in time single frequency function. The velocity feedback strategy is applied to reduce the plate transverse vibration. The governing coupled equations describing the active plate behaviour are derived. The influence of the electric potential distribution and also the thickness of piezoelectric layers on the plate dynamics including the natural frequency modification is numerically investigated and discussed.

References [1] M. Krommer, H.A. Irschik, On the influence of electric field on free transverse vibrations of smart beams, Smart Materials and Structures, 8, 401-410, 1999. [2] S.T. Quek, Q. Wang, On dispersion relations in piezoelectric coupled-plate structures, Smart Materials and Structures, 9, 859-867, 2000.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

246

A Finite Element Model for the Analysis of 3D Axisymmetric Laminated Shells with Embedded Piezoelectric Sensors and Actuators H. Santos1, Cristóvão M. Mota Soares1, Carlos A. Mota Soares1 and J. N. Reddy2 1

IDMEC/IST – Instituto de Engenharia Mecânica – Instituto Superior Técnico Av. Rovisco Pais, 1049-001 Lisboa, Portugal [email protected], [email protected], carlosmotasoares@ dem.ist.utl.pt 2

Department of Mechanical Engineering, Texas A&M University College Station, TX 77843-3123 USA [email protected]

ABSTRACT This paper deals with the bending of multilayered cylindrical shells with piezoelectric properties using a semi-analytical axisymmetric shell finite element model with piezoelectric layers which can be used as sensors and/or actuators. The model is developed using the 3D linear elastic theory. The displacement components, electric potential and load are expanded in truncated Fourier series following the approach described in [1,2] by taking advantage of the orthogonal properties of the trigonometric functions, to obtain the equilibrium equations for the element and system for each harmonic. Special emphasis is given to the coupling between symmetric and anti-symmetric terms for laminated materials with piezoelectric rings. Numerical results obtained with the present model are found to be in good agreement with other finite element solutions.

References [1] P. Correia I.F, Mota Soares C.M., Mota Soares C. A., Herskovits J., Development of semianalytical axisymmetric shell models with embedded sensors and actuators, Composite Structures, 47, 531-541, 1999. [2] H. Santos, Mota Soares C.M., Mota Soares C. A., Reddy J.N., A semi-analytical finite element model for the analysis of laminated 3D axisymmetric shells: bending, free-vibration and buckling. Composite Structures, 71, 273-281, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Damage Localization in Laminated Composite Plates using Double Pulse-Electronic Holographic Interferometry ´ dos Santos∗ , H. Lopes† , M. Vaz‡ , C. M. Mota Soares∗ , C. A. Mota Soares∗ , J. V. Araujo M. J. M. de Freitas§ ∗ IST - Instituto Superior T´ ecnico Av. Rovisco Pais, 1049-001 Lisboa, Portugal [email protected] [email protected] [email protected] † ESTIG - Instituto Polit´ecnico de Braganc¸a Campus de Sta Apol´onia, Apartado 134, 5301-857 Braganc¸a, Portugal [email protected] ‡ DEMEGI

- Faculdade de Engenharia do Porto Rua Dr. Roberto Frias, 4200-465 Porto, Portugal [email protected] § ICEMS/UME, Instituto Superior T´ ecnico Av. Rovisco Pais, 1049-001 Lisboa,Portugal [email protected]

ABSTRACT One method for the damage localization of impact damage in laminated composite plates, based on their vibrational characteristics, is presented in this paper. This method uses double pulse-electronic holographic interferometry for mode shapes acquisition and the differences in curvatures. The rotations and curvatures are numerically obtained. The method is applied to a carbon fibre reinforced epoxy rectangular plate, free in space, subjected to two cases of impact damage. It is shown that the method based on curvatures allows for the localization of both cases of damage, which can be undetected by visual, X-ray or C-Scan inspections. The best localizations are achieved by selecting and applying the method to the most changed mode.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Shear Deformation Effect in Nonlinear Analysis of Spatial Composite Beams in Variable Axial Loading by BEM E.J. Sapountzakis* *

School of Civil Engineering, National Technical University, Zografou Campus, GR-157 80, Athens, Greece [email protected]

ABSTRACT In this paper a boundary element method is developed for the nonlinear analysis of composite beams of arbitrary doubly symmetric constant cross section, taking into account shear deformation effect. The composite beam consists of materials in contact each of which can surround a finite number of inclusions. The beam is subjected in an arbitrarily concentrated or distributed variable axial loading, while the shear loading is applied at the shear center of the cross section, avoiding in this way the induction of a twisting moment. To account for shear deformations, the concept of shear deformation coefficients is used. Five boundary value problems are formulated with respect to the transverse displacements, the axial displacement and to two stress functions and numerically approximated employing a pure BEM approach, that is only boundary discretization is used. Application of the boundary element technique yields a system of nonlinear equations from which the transverse and axial displacements are computed by an iterative process. The evaluation of the shear deformation coefficients is accomplished from the aforementioned stress functions using only boundary integration. The essential features and novel aspects of the present formulation compared with previous ones are summarized as follows. i. The beam is subjected in an arbitrarily concentrated or distributed variable axial loading. ii. The beam is supported by the most general linear boundary conditions including elastic support or restrain. iii. The analysis is not restricted to a linearized second – order one but is a nonlinear one arising from the fact that the axial force is nonlinearly coupled with the transverse deflections (additional terms are taken into account). iv. Shear deformation effect is taken into account. v. The shear deformation coefficients are evaluated using an energy approach, instead of Timoshenko’s and Cowper’s definitions, for which several authors have pointed out that one obtains unsatisfactory results or definitions given by other researchers, for which these factors take negative values. vi. The effect of the material’s Poisson ratio Ȟ is taken into account. vii. The proposed method employs a pure BEM approach (requiring only boundary discretization) resulting in line or parabolic elements instead of area elements of the FEM solutions (requiring the whole cross section to be discretized into triangular or quadrilateral area elements), while a small number of line elements are required to achieve high accuracy. Numerical examples with great practical interest are worked out to illustrate the efficiency, the accuracy and the range of applications of the developed method. The influence of both the shear deformation effect and the variableness of the axial loading are remarkable.

1

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Constitutive ply damage modeling, FEM implementation, and analyses of laminated structures Clara Schuecker∗ , Heinz E. Pettermann† Austrian Aeronautics Research (AAR) / Network for Materials and Engineering at the Institute of Lightweight Design and Structural Biomechanics Vienna University of Technology Gusshausstr. 27–29/E317, A-1040 Vienna, Austria ∗ [email protected] † [email protected]

ABSTRACT The present work is concerned with the modeling of progressive damage in fiber reinforced polymer (FRP) laminates and the FEM implementation as constitutive material law. The objective is to predict damage evolution and material degradation due to matrix dominated failure modes (‘matrix cracking’). Secondary failure modes (e.g. micro-crack induced delamination) and their interactions are not considered. In a previous work [1], a ply-level continuum damage model based on ply failure mechanisms postulated by the Puck failure hypothesis [2] has been presented. In its original version, the model is restricted to loading scenarios where the fracture plane orientation predicted by Puck’s failure criterion does not change during loading. In the current work, the damage model is adapted for arbitrary loading paths to be used as a constitutive law with the finite element method (FEM). The model uses a phenomenological scalar evolution law to describe the increase of damage with load. The effect of damage on ply stiffness is imitated by a 4 th order tensor relation derived from a mean field method. By this approach, the complete elasticity tensor of a damaged ply is predicted in a thermodynamically consistent way, reflecting the non-isotropic nature of damage in FRPs and coupling of different components. Additionally, effects like stiffness recovery and degradation under in-plane transverse compression due to slanted matrix cracks are captured. At the same time only a relatively small number of model parameters is required, which can be identified from standard test methods. The damage model is implemented as constitutive material law into the FEM package ABAQUS (ABAQUS Inc., Pawtucket, RI). This way, complex structures can be analyzed and their damage behavior including load redistribution due to damage can be studied. To demonstrate the utilization of the damage model in structural analysis, it is applied to some example problems. Based on comparisons between modeling results and experimental data the validity of model assumptions is discussed. It is shown that the agreement is very good at low and moderate loads where secondary failure mechanisms have not yet developed. The analysis of such loading conditions is the main objective of the presented model.

References [1] C. Schuecker and H.E. Pettermann, A continuum damage model for fiber reinforced laminates based on ply failure mechanisms. Compos.Struct, ICCM-15 special issue, 2005 (submitted). [2] A. Puck and H. Sch¨urmann, Failure analysis of FRP laminates by means of physically based phenomenological models. Compos.Sci.Tech., 58, 1045–1067, 1998.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Simulation of Delamination in Stringer Stiffened Fiber-Reinforced Composite Shells Werner Wagner∗ , Claudio Balzani† ∗

Universit¨at Karlsruhe (TH), Institute for Structural Analysis Kaiserstr. 12, D - 76131 Karlsruhe, Germany [email protected]

† Universit¨ at

Karlsruhe (TH), Institute for Structural Analysis Kaiserstr. 12, D - 76131 Karlsruhe, Germany [email protected] ABSTRACT

Uni-directional fiber-reinforced composite laminates are widely used in aerospace industry for a great variety of structural parts. In order to enhance the exploitation of material reserves, there is a need for the integration of progressive damage scenarios in the design phase. Due to their hazardous effects on the load-carrying capacity of composite structures, this work focusses on the simulation of delaminations. 2D and 3D finite element formulations are developed which are based on a cohesive zone approach. The constitutive law is characterized by an exponential softening after the onset of delamination. A consistent tangent will be provided allowing for robust nonlinear iteration behaviour. The damage process is history-dependent leading to an irreversible stiffness degradation in damaged zones. The FE-formulations are compared to experimental results for typical standard tests like DCB or MMB of American Society for Testing and Materials (ASTM). Furthermore more complex structures like stringer stiffened shells made of fiber reinforced composite material, often used in aerospace industry, are investigated. As one example we present a double cantilever beam test (DCB). Geometrical and material data as well as numerical solutions in comparison to experiments and an analytical solution are depicted in Fig. 1.

Figure 1: DCB test: geometrical and material data (AS-4/3501-6) and load deflection behaviour.

References [1] A.A. Aliyu & I.M. Daniel, Effects of Strain Rate on Delamination Fracture Toughness of Graphite/Epoxy, in: Johnson W.S. (editor). Delamination and Debonding of Materials. ASTM STP 876, 336–348, (1985), Philadelphia.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Development of methods of numerical solution of singular integrodifferential equations for solid mechanics problems Andrey V. Andreev Federal State Unitary Enterprise “Electrogorsk Research and Engineering Center on Nuclear Power Plants Safety” Bezymyannaya St. 6, 142530, Moscow region, Electrogorsk, Russia [email protected]

ABSTRACT The paper is dedicated to the development of methods for solving one-dimensional singular and hypersingular integro-differential equations (SIDE and HSIDE) with generalized Cauchy kernels. Using the means of the theory of special functions, constructive methods of direct (without regularization) numerical solution of such equations are suggested. Approaches to solution HSIDE with complex or real asymptotic are distinguished qualitatively. The former is based on expansion of the solution on the finite system of orthogonal polynomials (with explicit account for the end points asymptotic), analytical calculation of singular and hypersingular integrals, and replacement generalized kernel by confluent one with further analytical integration of this term (or direct numerical calculation of the latter). Then obtained functional equation by means of the collocation method is reduced to linear algebraic system, which used for evaluation of the expansion coefficients. The direct approach is used for real asymptotic, which is based on Lagrangian approximation of the unknown function (with explicit account for the end points asymptotic), interpolation-type quadrature formulae, and the collocation method for reduction HSIDE to linear algebraic system relative to values of unknown function in the discrete points. Explicit analytical account of asymptotic on numerical solution allows the estimation dominant terms of asymptotic expansion of the solution in vicinity of the end points of integration interval with high accuracy. It is important, particularly, in elasticity problems for bodies with crack, since a dominant term of asymptotic expansion of the solution in vicinity of crack tips control the stress intensity factor, which has prime importance in brittle fracture. The numerical results and their comparison to analytical solution of some crack problems are presented in the paper. The developed methods allow to solving SIDE arising in different applied problems of physics and mechanics, particularly, in mixed elasticity problems, contact problems of solid mechanics, and two-dimensional problems of fracture mechanics.

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Validation of large scale simulations of dynamic fracture Irene Arias∗ , Jaroslaw Knap§ , Vijaya B. Chalivendra‡ , Soonsung Hong† , Michael Ortiz† , Ares J. Rosakis† † Division

of Engineering and Applied Science, California Institute of Technology 1200 E. California Blvd, Pasadena, CA 91125, USA

∗ Dep.

de Matem`atica Aplicada III, Universitat Polit`ecnica de Catalunya Jordi Girona 1-3, Barcelona 08034, Spain [email protected] § Lawrence Livermore Natonal Laboratory P.O. Box 808 L-367, Livermore, CA 94550, USA

‡ Department

of Mechanical Engineering, University of Massachusetts 285 Old Westport Road, North Dartmouth, MA 02747, USA ABSTRACT

A novel integrated approach is developed for a systematic validation of large-scale finite element simulations on dynamic crack propagations along a weak plane [1]. A set of well-controlled experimental scheme is specifically designed to provide accurate input data for the numerical simulations as well as to provide metrics for quantitative comparisons between experimental and numerical results. Dynamic photoelasticity with high-speed photography is used to capture experimental records of dynamic crack propagations along a weak plane and to provide the crack propagation history. In the dynamic experiments, a modified Hopkinson bar setup with a notch-face loading configuration is used to obtain controlled loading conditions for the dynamic fracture problem. Also an inverse-problem approach of cohesive zone model is employed to obtain a realistic cohesive law, i.e. a traction-separation law, of the weak plane, from independently measured crack-tip deformation fields using speckle interferometry technique. The experimentally collected data, the loading conditions and the cohesive law, are considered as input for the finite element simulations [2]. We employ finite-deformation cohesive elements to account for crack initiation and growth in bulk finite-element discretizations of the experimental sample. As it is well know, the cohesive elements introduce an additional material-dependent length-scale into the finite element model. The demand of accurately resolving this length-scale by the finite-element discretization, as required for truly mesh-independent results, may often lead to discretizations containing several millions of elements. We therefore resort to massively parallel computing. A comparison of the metrics from the numerical simulations with those from the experimental measurements is performed to validate the large-scale simulations. The numerical results show good agreements with the experimental results, leading to a successful validation of the large scale simulations of the dynamic crack propagations along the weak plane.

References [1] V. B. Chalivendra, S. Hong, J. Knap, I. Arias, M. Ortiz, A. J. Rosakis, Validation of large scale simulations of dynamic fracture along weak planes - A new integrated philosophy, In prep., 2005. [2] I. Arias, J. Knap, V. B. Chalivendra, S. Hong, M. Ortiz, A. J. Rosakis, Numerical modeling and experimental validation of dynamic fracture events along weak planes. Submitted for publication.

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Crack growth in fretting-fatigue problems using the extended finite element method E. Giner, A. Vercher, O.A. González, J.E. Tarancón and F.J. Fuenmayor E.T.S. de Ingenieros Industriales, Dpto de Ingeniería Mecánica y de Materiales Universidad Politécnica de Valencia, Camino de Vera s/n, 46022 Valencia, Spain. [email protected]

ABSTRACT The aim of this work is to apply the extended finite element method (X-FEM) to the analysis of crack orientation and propagation in a 2D fretting-fatigue problem. The X-FEM approach features some important advantages in the computational modelling of crack growth problems. The main advantage is that the mesh generation does not need to conform to the geometric discontinuity caused by the crack presence. The discontinuous behavior is introduced via special functions that enrich the classical finite element formulation. In this way, the initial mesh can be used for any further variation in crack length and orientation. On the other hand, a fretting-fatigue problem generally exhibits large stress gradients. These steep gradients are due to the existence of frictional contact between two bodies, which undergo relative displacements. They eventually lead to crack nucleation and subsequent propagation that may cause the failure of the mechanical component. In addition, if the contact is complete (as is the case in this work) the stress fields may be singular. Then, the singular stress fields caused by the crack tip and the edge of the contact coexist. The problem modelled in this work consists of a specimen with rectangular cross-section subjected to variable tensile loading. This specimen is pressed by two square-ended punches on two opposite sides. A small nucleated crack is assumed to exist by the contact edge. Then, benefiting from the advantages of the X-FEM approach, the orientation of the eventual propagation is analyzed. The objective is to evaluate the extent of the influence of the contact elastic fields on the crack orientation. Several cases are considered, with different starting cracks and different load ratios. The use of the X-FEM enables an efficient evaluation of the contact influence on the crack propagation. Such study would be prohibitive using a standard finite element implementation, because it would imply the generation of a different mesh for each crack growth increment.

References [1] N. Moës, J. Dolbow, T. Belytschko. A finite element method for crack growth without remeshing. Int J Numer Meth Engng, 46, 131-150, 1999. [2] N. Sukumar, J.-H. Prévost. Modeling quasi-static crack growth with the extended finite element method. Part I: Computer implementation. Int J Solids Struct, 40, 7513-7537, 2003 [3] D.A. Hills, D. Nowell. Mechanics of Fretting-Fatigue. Kluwer Academic Publ. Dordrecht, 1994.

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Aspects of crack propagation and hygro-mechanical coupling using X-FEM ¨ Stefan Jox, Peter Dumstorff and Gunther Meschke Institute for Structural Mechanics Ruhr-University Bochum, D-44780 Bochum, Germany [email protected], [email protected]

ABSTRACT Numerical prognoses of durability of structures made of cementitious materials such as concrete requires the consideration of environmental loads in addition to external loading [1]. For problems such as cracking of concrete structures, joints in rocks or shear bands in soft soils, the moisture transport in the opening discontinuities has to be taken into account in durability oriented analyses. The paper is concerned with a concept for coupled hygro-mechanical analyses considering discontinuities in the context of the extended finite element method [2, 3]. The spatial discretization of the displacement field as well as the moisture field, represented by the fluid pressure, are additively decomposed into a regular part and an enhanced part representing possible jumps in the primary variable. The extension to hygromechanical couplings involves the consideration of jump conditions across discontinuities as well as fluid transport in the intact part and the discontinuity, respectively. The present formulation is restricted to fully saturated conditions. The coupled model is investigated by means of an academic benchmark example, characterized by an artificially introduced crack, is analyzed numerically considering a hygromechanical loading scenario. An additional aspect addressed in the paper is concerned with the evaluation of different crack propagation criteria for the determination of the crack growth direction in cementitious materials. A benchmark example characterized by Mixed-Mode fracture is presented to study the performance and robustness of these criteria.

References [1] O. Coussy and F.-J. Ulm, Elements of durability mechanics of concrete structures. In: F.-J. Ulm, Z. P. Baˇzant and F. H. Wittmann (Eds.): Creep, Shrinkage and Durability Mechanics of Concrete and other Quasi-Brittle Materials, Elsevier Science, Oxford, 393-409, 2001. [2] N. Mo¨es, J. Dolbow and T. Belytschko, A finite element method for crack growth without remeshing. International Journal for Numerical Methods In Engineering, 46, 131-150, 1999. [3] M.-A. Abellan, R. de Borst and J.-M. Bergheau, Analysis of discontinuities in fluid-saturated porous media. In A. Carpinteri (Ed.), International Conference on Fracture (ICF 11), Turin, 2005.

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Destruction of rocks by directional hydraulic fracturing on the basis of models of plasticity with internal variables Sergey V. Klishin Mining Institute of Siberian Division of Russian Academy of Sciences Krasny Prospekt, 54, 630091, Novosibirsk, Russia [email protected]

ABSTRACT The present work is aimed at studying of mechanisms of deformation of the structural-non-uniform massif, which is weakened by directed cracks of hydraulic fracturing. The problem about spreading of an artificial crack to the set direction is one of classical in mechanics of rocks. In various statements it is investigated in many tasks (both experimentally, and theoretically). In the majority of tasks this problem is investigated with using of models of the theory of elasticity and plasticity in continued wording. However, heterogeneity of a material plays an essential role on behavior of a rock mass. Materials of rocks always contain the big number of defects of a various origin and the sizes at the several scale levels. I.e. distribution of cracks to rocks is determined by the processes occurring both at a macro-level, and on micro-levels of various scales. Therefore the problem of destruction of rock cannot be solved with application of only models of the deformable solid body in continued wording. In present work the new approach to a problem is offered to use, which will allow describing adequately the behavior of structurally non-uniform rock mass, which possesses of hierarchy of structural levels [1]. The internal structure of a material is modeled by rather rigid-jointed framework, which consists of the packing grains and a bonding material. The bonding material fills pores between grains. Fields of micro-speeds and micro-pressures are introduced, operations of averaging which allow passing from micro- to macro-parameters of model are determined. Structural elements are elastic mediums with various constants. There is a plastic nonlinear law of sliding on contacts between grains. Changing the properties of structural elements under various laws, it is possible to build models of rocks with various fissuring, for sands, and also for dry and water-sated soils. In the work the analytical models of the theory of plasticity with internal variables, and also modern computing methods — Finite Elements Method and Boundary Integral Equations Method (BIEM) are used. Numerical experiments on research of deflected mode and stability of a rock weakened by cracks of directional hydraulic fracturing have been carried out. The numerical evaluation of parameters of tasks has been carried out. Situations, in which there is a steady deformation (when small modifications of external loading result in adequate insignificant reaction) or unstable (catastrophic) deformation when small modifications result in uncontrollable dynamic releasing of the stored energy, are clearly recognized.

References [1] Revuzhenko A.Ph., Lavrikov S.V., Klishin S.V. Structural-unhomogeneous rock massif as environment with internal sources and drains of energy. Problems and Prospects of Mining Sciences, 214–219, Novosibirsk, 2005.

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The finite-element method in linear fracture mechanics problems Natalia V. Krukova*, Igor M. Lavit† *

Department of methimatical modeling, Tula State University Tula 300600, Russia [email protected]

†

Department of methimatical modeling, Tula State University Tula 300600, Russia [email protected]

ABSTRACT While solving problems of linear fracture mechanics applying numerical methods, it is necessary to consider root singularity of stress and strain fields in the crack tip. Commonly they use specific finite elements (Barsoum [1]). However such approach is not very suitable in solving problems of crack growth since we need to transform finite-element mesh in this case. In this research alternative approach using ordinary elements is offered. Some functions are added to the ordinary coordinate function system of the finite-element method to take into consideration the root singularity of stresses and strains. The additional functions result from the linear fracture mechanics problem asymptotic solution, proportional the stress intensity factors. The stress intensity factors are included in the basic unknown quantities. The order of the resulting linear equation system is increased by number of nonzero stress intensity factors. Thus each of equation system matrix additional columns is proportional only one unknown quantity, it become possible to reduce the solution to the determination of the linear equation system with a banded matrix. In the paper several test problems justifying the efficiency the method are solve in comparison with the ordinary finite-element solution where the stress intensity factor for the normal fracture is found by determination of J-integral. The problem of a short crack in a solid specimen with real notch geometry is considered. The obtained results are compared with ones of other researches [2].

References [1] R. Barsoum, Application of quadratic isoperimetric finite elements in linear fracture mechanics. Int. J. Fract., 10, 603-605, 1974. [2] N. Dowling, W. Wilson, Resalts of elastic analysis of bluntly notched compact spesimens. Eng. Fract. Mech., 20, 569-572, 1984.

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Finite deformation fracture modelling of a thermo-mechanical cohesive zone Ragnar Larsson* Martin Fagerström* * Department of Applied Mechanics Chalmers University of Technology 412 96 Gothenburg, Sweden [email protected]

ABSTRACT In the present contribution, the development of a thermo-mechanical cohesive zone model is discussed in the context of a strong discontinuity formulation according to the concept of partitions of unity [1]. On the basis of our previous work, [2, 3], and also the contributions, [4, 5], the deformation map is thereby defined in terms of mutually independent continuous and discontinuous portions of the displacement. As an extension, we also introduce a similar subdivision of the temperature field in one continuous and one discontinuous part, separated by the internal crack surface. As a result, we consider the weak formulation of the equation of motion and the energy equation as consisting of four coupled equations on the structure level. In the paper, we will focus on the conditions for onset as well as continued (coupled) discontinuity development within a thermo-hyperelastoplastic continuum with isotropic plastic hardening as well as thermally softening response. In the case of continued discontinuity development, a cohesive zone law is specified for the representation of the decay of the traction vector combined with heat flux across the interface zone. Apart from the crucial fracture modelling, we also discuss the numerical treatment and aspects of computational implementation of the proposed approaches. A couple of numerical examples that illustrate the capabilities of the proposed approach to the modelling of the thermo-mechanical cohesive zone are included.

References [1] J.M. Melenk and I. Babuška , The partition of unity finite element method: Basic theory and applications, Computer Methods in Applied Mechanics and Engineering, 139, 289-314, 1996 [2] R. Larsson and M. Fagerström, A framework for fracture modelling based on the material forces concept with XFEM kinematics, International Journal for Numerical Methods in Engineering., 62, 1763-1788, 2005 [3] M. Fagerström and R. Larsson, Theory and numerics for finite deformation fracture modelling using strong discontinuities, International Journal for Numerical Methods in Engineering, Accepted 2005 [4] T. Belytschko and T. Black, Elastic crack growth in finite elements with minimal remeshing, International Journal for Numerical Methods in Engineering, 45, 601-620, 1999 [5] G.N. Wells, R. de Borst and L.J. Sluys, A consistent geometrically non-linear approach for delamination, International Journal for Numerical Methods in Engineering, 54, 1333-1355, 2002

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Critical Behavior and Energy Dependence of Mass Distribution in HighVelocity Impact Fragmentation Nikolay N. Myagkov, Timofey A. Shumikhin Institute of Applied Mechanics, Russian Academy of Sciences, Leninsky prospect 32a, Moscow 119991, Russian Federation [email protected]

ABSTRACT Experiments on the high-velocity impact fragmentation for the projectile-bumper system show: (i) the threshold nature of fragmentation [1]; (ii) the similarity of the debris cloud structure [2]. However the current experimental resources cannot give a more detailed picture of the critical and scaling effects of the fragments distribution in a cloud and investigate the fragments structure “in-situ”. These gaps may be filled up by computer simulation [3]. We study fragmentation during high-velocity impact for the projectile-bumper system numerically. To obtain a statistically considerable number of fragments as well as to avoid huge amount of calculations we use two-dimensional particle-based simulations and describe the interparticle interaction by the pair Lennard-Jones potential. Moreover, using the simple model under minimum number of initial assumptions we clarify the understanding of universal properties of the impact fragmentation. The two essential parameters of modeling are: (i) the impact velocity; (ii) the system’s size at condition that ratio of the bumper thickness to the projectile diameter is a constant. The evolution of fragments after impact is considered. It is found that there are regions of fast and slow changes of mass distribution with characteristic time tc, when the fragmentation becomes critical. For t>tc fast relaxation of the distribution to a power-law steady state for the intermediate masses and a weak drift in the region of the small and large masses is observed. For the steady state distributions the control parameter is the impact velocity. The transition to the fragmentation also occurs here as the phase transition. The critical impact velocity is practically independent on the size of the projectile-bumper system. It is found that the power-law exponent increases with energy imparted to the projectile-bumper system non-monotonically. For impact velocities about the sound speed of material it may be approximately considered as a constant. It is considered the mass-size relation estimating the typical size of fragments at the steady-state stage of the fragmentation. For the middle and large masses the fragments can be regarded as fractal objects. The fractal dimension does not change much with the variation of the projectile-bumper system size and impact velocity, and its mean value is close to fractal dimension of percolation.

References [1] D.E. Grady and M.E. Kipp. Int. J. Impact Engng., 20, 293-308, 1997. [2] A. J. Piekutowski. Int. J. Impact Engng., 20, 639-650, 1997. [3] N.N. Myagkov and T.A. Shumikhin. Physica A, 358, 423-436, 2005.

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One 3D adaptive fragmentation procedure for the explicit simulation of brittle material cracking S. H. Reese∗ , P. Wriggers∗ ∗ Institute

for Mechanics and Computational Mechanics University of Hannover Appelstrasse 9a 30167 Hannover GERMANY [email protected] ABSTRACT

Realistic modeling of fragmentation procedures is one of the fundamental tasks in modern computational engineering simulations. In particular, 3D crack propagation with minimal incorporation of restrictions to the crack path is still a challenge. The approach presented in this work will be used to model failure and cracking of brittle materials like concrete structures, applying an explicit finite element integration scheme. For efficient modeling of regions with highly localized strains, enriching the standard Galerkin finite element approximation has become an established method. One of these continuous approaches is the Strong Discontinuity Approach [1]. Here, enhanced assumed strains enrich the standard Galerkin part of the finite element interpolation. This ensures, that the resulting discontinuities in the displacement field can be traced in a realistic way, which allows the non-geometrical representation of crack discontinuities. Another task is to obtain a significant and decisive value for the development of the crack. When using the Strong Discontinuity Approach, we can avail the discontinuity bandwidth as the crucial parameter for the determination of the element splitting initiation. By means of a mixed continuous-discontinuous model, the element representation is subsequently transfered from a non-geometrical to a geometrical one. Therefore, full adaptive procedures have to be incorporated in the 3D finite element model. In order to obtain suitable surfaces for secondary-loading contact computations and for discrete element coupling formulations, an continuous surface condition has to be guaranteed by the adaptive fragmentation algorithm. Furthermore, the physically founded ability of crack unification of the arising fracture planes has to be incorporated. Some general procedures for the generation of these fracture surfaces, while assuring the element sizes remaining as big as possible, will be presented within this work. Also, some advancements to minimize the mesh dependent identification of the material instabilities and to determine significant fracture plane normals will be investigated. This work will show how to handle adaptive procedures in the framework of an explicit finite element code including brittle material behavior. Finally, some comparisons of numerical results emerging from the Strong Discontinuity Approach with standard exponential damage failure criteria for concrete structures will be presented.

References [1] Oliver, J. and Huespe, A.E. (2004): Theoretical and computational issues in modeling material failure in strong discontinuity scenarios. Computer Methods in Applied Mechanics and Engineering, Vol. 193 , pp. 2987-3014

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Finite Element Analysis of Cracked Plates With Circular Stress Raisers Used For S.I.F. Reduction G. J. Tsamasphyros*, Th. K. Papathanassiou† *

Dpt of Mechanics, National Technical University of Athens 9 Iroon Polytechniou, Zografou 15773, Athens, Greece [email protected]

†

Dpt of Mechanics, National Technical University of Athens 9 Iroon Polytechniou, Zografou 15773, Athens, Greece [email protected]

ABSTRACT The Finite Element method is used in this paper in order to compute Stress Intensity Factors for complex 2D domains. These domains consist of centrally cracked rectangular plates, in which circular holes of various diameters are drilled above and below the crack line. These holes can produce a serious decrease of the S. I. F., leading to stress relief in the area of the crack tip. This S. I. F. reduction is found to be depended on both the diameter of the two holes and the distance of them from the crack plane. The lack of analytical solutions in such problems makes the use of numerical methods, such as the Finite Element method practically unavoidable. Optimum placing of the circular holes with respect to the crack is found for certain realistic values of the radius of the holes. The effect that internal pressure applied in these circular holes has on the S. I. F. is also considered. Regression polynomials describing the dependence of the S. I. F. on several parameters of the problem, such as the width of the plate and the radius of the stress relieving holes, are extracted from the Finite Element results with the use of Least Square Procedures. S. I. F. reduction of more than 20% was found for some of the above mentioned cases. Comparison is made between the Finite Element solution and other existing formulae, where available, in order to verify the validity of the results. Declinations of value less than 1% were found.

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Simulation of Materials Damage in the Field of Internal Stresses N.M. Vlasov1, I.I. Fedik2 1

Scientific Research Institute, Scientific Industrial Association "Luch", Zheleznodorozhnaya 24, Podolsk Moscow region, Russia , 142100, [email protected]

2

Scientific Research Institute, Scientific Industrial Association "Luch", Zheleznodorozhnaya 24, Podolsk Moscow region, Russia , 142100, [email protected]

ABSTRACT Internal stresses occur within a material in the presence of non-uniform deformation. The main types of the internal stresses are the thermal and residual ones and fields of structural defects as well. These stresses have an essential effect on the diffusion processes kinetics. In this case change of the strength material properties takes place. The properties degradation is accompanied by damaging and failing the material. The physical mechanisms underlying changes of properties include, for example, decreasing of surface fracture energy, stress corrosion cracking, and hydrogen embrittlement. The diffusion process is described by a non-stationary equation of a parabolic type under both initial and boundary conditions. The purpose of this paper is simulating the material damage as a result of running the diffusion processes. Triple grain boundaries are considered as structural defects. They serve as stress concentrators under dynamic and temperature loadings. This is caused by the orientation dependence of elastic and thermophysical characteristics of the contiguous grain material. The dilatation field of considered defects depends logarithmically on the radial coordinate. Such a dependence enables one to obtain an exact analytical solution for the task on hydrogen segregation kinetics. Analytical relations for the field of atomic hydrogen concentration near the triple grain boundaries are given. If the concentration of hydrogen atoms exceeds the solubility limit at a given temperature, hydride phases are formed in some metals (e.g., Zr). Hydride growth kinetics in the stress field of structural defects under study is considered. The changes of the volume hydride are accompanied by microcrack formation along the grain boundaries. The results of theoretical analysis are employed to explain the hydrogen embrittlement of metals.

References [1] N.M. Vlasov, V.A. Zaznoba, Diffusion processes in the area of threefold junctions of special grain boundaries. Fiz. Tverd. Tela 41 (1999) 64–67 [Phys. Solid State 41 (1999) 55–58.]

[2] B.A. Kolachev, Hydrogen Brittleness of Metals, Metallurgy, Moscow, 1985 (in Russian). [3]

N.M. Vlasov, I.I. Fedik, Hydrogen segregation in the area of threefold junctions of grain boundaries. Int. J. Hydrogen Energy 27 (2002) 921–926.

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Computational methods for the fast boundary stabilization of flexible plates Frederic Bourquin∗ ∗ French

Public Works Research Laboratory 58 bd Lefebvre, 75015 Paris, France [email protected] ABSTRACT

The numerical approximation of an efficient active control strategy [6, 7, 3] for undamped flexible plates of arbirary shape controlled from the boundary is considered. A non-standard computational framework is proved to be relevant both for simulation and control synthesis. It enables one to achieve an arbitrarily large decay rate of some weak norm of the system. The proposed formulation proves necessary in so far as a standard numerical approach is shown to fail in a simple case. The paper focusses on general strategies to compute the boundary control that ensure convergence and uniform stability of the closed-loop system. A Galerkin approximation of both the dynamics of the structure and the controller equation is introduced and further numerical integration schemes for modal stresses along the region of controlled are proposed. To this end, branch modes or Poincar- Steklov modes are shown to be good candidates in view of a high-level implementation in a general purpose finite element softwares. The method extends the work explained in [1, 2, 4, 5].

References [1] F. Bourquin. Approximation theory for the problem of exact controllability of the wave equation with boundary control. In Ralph et al. Kleinman, editor, Mathematical and numerical aspects of wave propagation. Proceedings of the 2nd international conference held in Newark, DE, USA, June 7-10, 1993, pages 103112, Paris, 1993. SIAM. [2] F. Bourquin. Control of flexible structures : control theory and approximation issues. In Advances in Structural Control (J. Rodellar, A.H. Barbat and F. Casciati editors), CIMNE, Barcelona, pages 3150, 1998. [3] F. Bourquin. Approximation for the fast stabilization of the wave equation from the boundary. In Proceedings of MMAR2000, Poland, 08-2000, pages 2532, 2000. [4] F. Bourquin. Numerical methods for the control of flexible structures. Journal of Structural Control, 8(1):83103, 2001. [5] F. Bourquin. On the computation of B ∗ . In Proceedings of the European meeting on intelligent systems, EMIS2001, Ischia, Italy, 09-2001, 2001. [6] V. Komornik. Stabilisation frontiere rapide de systemes distribues lineaires. C.R.Acad. Sci. Paris, Serie 1,321:433437, 1995. [7] V. Komornik. Stabilisation rapide de problemes devolution lineaires. C.R.Acad. Sci. Paris, Serie 1, 321:581586, 1995.

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Sensor and Actuator Capabilities of a Laminated Piezoelectric Plate Model Lino Costa∗ , Isabel N. Figueiredo† , Pedro Oliveira∗ ∗ Departamento

†

de Produc¸a˜ o e Sistemas, Escola de Engenharia, Universidade do Minho Campus de Azur´em, 4800-058 Guimar˜aes, Portugal [email protected], [email protected]

Departamento de Matem´atica, Faculdade de Ciˆencias e Tecnologia, Universidade de Coimbra Apartado 3008, 3001-454 Coimbra, Portugal [email protected]

ABSTRACT In a recent paper (cf. [1]) a new nonhomogeneous anisotropic piezoelectric plate model was derived, using the asymptotic analysis method, and for a plate with nonconstant piezoelectric and dielectric coefficients. In the present paper we adopt this new plate model to analyze the sensor and actuator effects (cf. [2]) of a thin laminated plate formed by stacking two or more layers of different piezoelectric materials. Moreover, we also justify the expression of the electric potential of the laminated plate, as a quadratic polynomial of the plate’s thickness variable. Several numerical results are reported. They illustrate the influence of the location of the applied electric potential and mechanical forces, and the effect of the boundary conditions, on the actuator and sensor capabilities of the thin laminated plate. The test problems are formulated as multi-objective optimization problems, which are solved through genetic algorithms (cf. [3]).

References [1] I. N. Figueiredo and C. F. Leal, A piezoelectric anisotropic plate model. Asymptotic Analysis, 44, 3-4, 327–346, 2005. [2] R. Smith, Smart material systems: model development. (Frontiers in applied mathematics ; 32) Philadelphia: SIAM, 2005. [3] D. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning. Reading, Mass. Addison-Wesley, 1989.

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Numerical Analysis of Stabilized Finite Element Methods for Darcy Flow Abimael F. D. Loula ∗ , Maicon R. Correa† Laborat´otio Nacional de Computac¸a˜ o Cient´ıfica Av. Getulio Vargas, 333, Quitandinha CEP: 25651-075, Petr´opolis, RJ - Brasil ∗ [email protected] † [email protected] ABSTRACT The flow of an incompressible homogeneous fluid in a rigid saturated porous media leads to the classical Darcy’s problem which basically consists of the mass conservation equation plus Darcy’s law, that relates the average velocity of the fluid in a porous medium with the gradient of a potential field through the hydraulic conductivity tensor. Darcy flow is the starting point for more complex flow in porous media and despite its apparent simplicity it has some properties like incompressibility, anisotropy and heterogeneity of the medium that make the search of accurate numerical solutions an important and active research area of numerical analysis and computational modeling. Finite element formulations applied to this problem are essentially based on two different approaches: one involves a single-field formulation for potential [1] and the other employs a mixed formulation in potential and velocity fields. The main characteristic of the mixed finite element methods is the use of different spaces for velocity and potential, requiring a compatibility (LBB condition [2]) between the finite element spaces to ensure existence and uniqueness of solution, which reduces the flexibility in the choice of stable finite element spaces. One well known successful approach is the dual mixed formulation developed by Raviart and Thomas [3] using divergence based finite element spaces for the velocity field combined with discontinuous Lagrangian spaces for the potential. To overcome the compatibility condition, typical of mixed methods, stabilized finite element methods have been developed. A non-symmetrical stabilized mixed formulation for Darcy flow is presented in [4] by adding and adjoint residual form of Darcy’s law to the mixed Galerkin formulation of this problem. We propose a symmetrical and stable mixed finite element method for Darcy’s problem by combining least-squares residual forms of the conservation of mass equation and Darcys law with the classical dual mixed formulation, so that equal-order classical Lagrangian finite element spaces can be adopted for both velocity and pressure fields with continuous or discontinuous pressure interpolations. Stability, convergence and error estimates are proved and numerical experiments are presented to demonstrate the flexibility of the proposed finite element formulation and to confirm the predicted rates of convergence.

References [1] A. F. D. Loula and F. A. Rochina and M. A. Murad, Higher-order gradiente post-processings for second-order elliptic problems, Comp. Meth. in App. Mech. and Eng., 128 (1995) 361-381 [2] F. Brezzi, On the existence, uniqueness and approximation of saddle point problems arising from Lagrange multipliers, RAIRO Analyse num´erique/Numerical Analysis. 8(R-2) (1974) 129-151 [3] P. A. Raviart and J. M. Thomas, A mixed finite element method for second order elliptic problems, in: Math. Asp. of the FEM, Lecture Notes in Mathematics 606, Springer, New York, 1977. [4] A. Massud and T. J. R. Hughes, A Stabilized Mixed FEM for Darcy Flow, Comp. Meth. in App. Mech. and Eng., 191 (2002) 4341-4370

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An Error Estimator for the Reissner-Mindlin Plate Problem C. Lovadina , R. Stenberg† Dipartimento di Matematica dell’Universit`a di Pavia, and IMATI-CNR Via Ferrata 1, I-27100 Pavia, Italy [email protected] †

Institute of Mathematics, Helsinki University of Technology P.O. Box 1100, 02015 HUT, Finland rolf.stenberg@hut.fi

ABSTRACT We present an a posteriori error analysis for some mixed finite element methods to approximate the solution of the Reissner-Mindlin plate problem. More precisely, we focus our attention on a low-order triangular mixed element which takes advantage of the so-called ‘Linked Interpolation Technique’ (see [1] and [6], for instance). For such a scheme we introduce a suitable residual-based error estimator (cf. [5]), and we discuss its reliability and efficiency. In particular, we show that the error estimator is robust with respect to the choice of the thickness parameter. Even though we will consider only the clamped plate case, our analysis can be easily adapted to cover other relevant boundary conditions. We notice that despite its importance, only few results on the a posteriori error analysis for plate finite element methods have been previously developed (see [2], [3] and [4]).

References [1] F. Auricchio, C. Lovadina: Analysis of kinematic linked interpolation methods for ReissnerMindlin plate problems, Comput. Methods Appl. Mech. Engrg., 190, 2465–2482, 2001. [2] C. Carstensen: Residual-based a posteriori error estimate for a nonconforming Reissner–Mindlin plate finite element, SIAM J. Numer. Anal., 39, 2034–2044, 2002. [3] C. Carstensen, J. Sch¨oberl: Residual-based a posteriori error estimate for a mixed Reissner– Mindlin plate finite element, Preprint. [4] E. Liberman, A posteriori error estimator for a mixed finite element method for Reissner-Mindlin plate, Math. Comp., 70, 1383–1396, 2000. [5] C. Lovadina, R. Stenberg: A posteriori error analysis of the linked interpolation technique for plate bending problems, to appear in SIAM J. Numer. Anal. [6] M. Lyly: On the connection between some linear triangular Reissner-Mindlin plate bending elements, Numer. Math., 85, 77–107, 2000.

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A Convergence Study of the Numerical Solution of Two Bi-directionally Coupled Partial Differential Equations in Thermoelectricity Nando Troyani*, Edgar . Gutiérrez† *Centro de Métodos Numéricos en Ingeniería, Escuela de Ingeniería y Ciencias Aplicadas, Universidad de Oriente, Venezuela [email protected] †

Centro de Estudios Energéticos, Departamento de Ingeniería Mecánica, Universidad Nacional Experimental Politécnica Antonio José de Sucre, Puerto Ordaz, Venezuela [email protected]

ABSTRACT Thermal and electrical aspects in the complex multiphysics of Aluminum Reduction Cells (ARC), the major worldwide industrial component to produce primary aluminum, represent issues that define, to a great extent, the performance of the ARC’s and the efficiency in energy consumption of the system. Given the intense energy usage of ARC’s there is a continuous wide interest in accurate and reliable modeling of these systems [1]. In this regard, and more specifically, we present the three dimensional mathematical model that describes the thermoelectric behavior of ARC’s consisting of two, nonlinear, bi-directionally coupled partial differential equations (PDE’s) for both the cell voltage field and the cell temperature field. The non-linearity arises as a result of the thermal dependence of both the electrical and thermal physical properties of the cell materials. A numerical solution of the stated system of PDE’s was developed and founded on the Finite Elements Method in an doubly iterative scheme that resolves both the non-linearity and the coupling of the PDE’s. The so called side ledge position, the ARC solid-liquid interface formed between both the liquid aluminum and liquid electrolytic bath with the solidified bath [2], is determined through a procedure framed in the fixed mesh scheme (FMS) in order to computationally track the position of the stated interface that strongly affects the ARC’s performance. The particular form of the FMS was initially described in the work reported in [3], and represents an alternative scheme to procedures that adjust the position of an initial estimate for the location of the solid-liquid interface without changing the element properties. Lastly, in this work, we present actual results of a convergence study for the described solution of the stated system of PDE’s showing the convergence characteristics of the proposed scheme for the stated iterative solution strategy.

References [1] A. Bermudez, P. Salgado, Domain decomposition/finite element method for the numerical simulation of electrolytic cells, Computer Methods in Applied Mechanics and Engineering, v 188, n 1, 2000. [2] H. Sun, O. Zikanov, D. Ziegler, Non-linear two-dimensional model of melt flows and interface instability in aluminium, Fluid Dyn. Res., v 35, n 4, 2004. [3] E. Gutiérrez, N. Troyani, Determinación de la distribución de temperatura mediante el método de sustitución de dominio en el lecho de las celdas de reducción de aluminio del tipo Hall-Heroult, ENIEF, Memorias de Mecánica Computacional Vol. XXIII, Bariloche, Argentina, Nov. 2004.

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Numerical simulations of rubber-like materials under changing directions M. B¨ol, S. Reese Institute of Mechanics Carolo-Wilhelmina University of Braunschweig [email protected]

ABSTRACT It is well-known that especially filled rubber-like materials exhibit under cyclic loading distinct stresssoftening phenomena commonly known as Mullins effect. Another important effect associated with the Mullins effect is that the material behaves anisotropic after having undergone large strain in a certain direction. These two effects can be explained by chain breakage and reconnection inside the material which is simulated in the present contribution by means of an innovative approach. This approach allows us to transfer information from the micro level to the macro level and contrariwise. Therefore we define special finite element unit cells consisting of one tetrahedral element and six truss elements attached to each edge of the tetrahedron. Putting arbitrary configurations of such unit cells randomly together allows us to simulate complex structures of unfilled elastomers, e.g. rubber boots or seals. Filler particles are added by replacing a certain part of the tetrahedrons by another type of tetrahedrons including linear-elastic material behaviour to represent the filler material. In this way we account for the increase of the stiffness and the strength of the composite material. Based on comparisons with experimental results the breakage and reformation of polymer chains is simulated. We obtain a satisfactory correlation between the numerical results and the experimental data, especially for the large strain regime. A further main focus lies in the studies of the anisotropic effects in combination with the Mullins effect. Here we also arrange validations with experimental data in order to show the advantages of the proposed approach.

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About an Efficient and Consistent Numerical Strategy for the Solution of the Initial-Boundary Value Problem for Anisotropic Finite Elastoplasticity Anke Bucher∗ , Uwe-Jens G¨orke∗ , Reiner Kreißig∗ ∗ Institute of Mechanics Chemnitz University of Technology Str. d. Nationen 62, D-09111 Chemnitz, Germany [email protected]

ABSTRACT Geometrically and physically nonlinear field problems of solid mechanics are numerically processed using suitable spatial discretization methods solving the boundary value problem and time discretization procedures treating the embedded initial value problem (IVP). In case of anisotropic elastoplasticity the IVP is usually defined by a system of differential and algebraic equations (DAE): a stress-strain relation in rate formulation, evolutional equations for the internal variables, and a yield condition. Within the context of a thermodynamically consistent material law considering finite elastic as well as plastic strains we substitute the usual stress-strain rate formulation by an associated yield function (cf. [1]). Therefore, we include the plastic strain tensor instead of the stress tensor into the set of unknown variables of the IVP. Stresses are directly computed from the elastic strain tensor by means of a hyperelastic material law. This strategy is advantageous for more reliable stress results at large load steps. Additionally, a more stable convergence behaviour has been observed. For the numerical treatment of the IVP within a finite element approach we prefer a simultaneous solution of the complex DAE which distinguishes itself by a higher efficiency and accuracy compared with classical methods (cf. [2, 3]). An additional benefit of this strategy results from the efficient numerical determination of the consistent material matrix. The time discretization of the DAE is realized based on the generalized implicit single-step scheme yn+1 = yn + (α fn+1 + (1 − α) fn ) ∆t

(1)

for an ordinary first order differential equations. The time increment ∆t = tn+1 − tn represemts the current load step, and the parameter α ∈ [0, 1] controls the accuracy and convergence rate of the solution of the IVP. Applying the scheme (1) to the DAE representing the material law a homogeneous system of nonlinear algebraic equations can be derived. This system is solved by means of Newtons method. The presented numerical strategy has been implemented into an in-house FE-code. Some examples illustrating its efficiency are discussed.

References [1] A. Bucher, U.-J. G¨orke, R. Kreißig, A material model for finite elasto-plastic deformations considering a substructure. Int. J. Plast., 20, 619–642, 2004. [2] S. Hartmann, G. L¨uhrs, P. Haupt, An efficient stress algorithm with applications in viscoplasticity and plasticity. Int. J. Num. Meth. Eng., 40, 991–1013, 1997. [3] A. Meyer, D. Michael, A modern approach to the solution of problems of classic elastoplasticity on parallel computers. Num. Lin. Alg. Appl., 4, 205–221, 1997.

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Vibrations of composite beams with multiple delaminations Helle Hein Institute of Computer Science, University of Tartu Liivi 2, 50409, Tartu, Estonia [email protected]

ABSTRACT Dynamic response of composite laminates has received a great deal of attention. One of the most common defects in composite laminates is delamination. The presence of delamination may cause changes to the vibration characteristics of the structure and can be the most damaging failure mode of composite materials. The free vibrations of composite beams with multiple delaminations have extensively studied by Shu and Della [1, 2]. Stepped beams and beams resting on elastic foundation are increasingly used in various fields in structural engineering and their dynamic properties have been investigated by many authors [3-8]. In the present work the basic ideas of [1, 2] have been extended to the following cases: (i) delaminated composite beams resting on elastic foundation, (ii) delaminated composite beams with piece-wise constant thickness. The through-width delaminations are parallel to the beam surface and located arbitrarily in both the span-wise and thickness-wise direction. The beam is modeled as consisting of separate component segments each being analysed as an Euler beam. The effect of elastic foundation to the vibration characteristics and mode shape of delaminated beam has been considered. The influence of delamination size and position on the natural frequencies of the stepped beam has been investigated.

References [1] D. Shu, C. N. Della, Vibrations of multiple delaminated beams, Composite Structures, 64, 467477, 2004. [2] D. Shu, C. N. Della, Free vibration analysis of composite beams with two non-overlapping delaminations. International Journal of Mechanical Sciences, 46, 509–526, 2004. [3] M. Eisenberger, Vibration frequencies for beams on variable one- and two- parameter elastic foundations. Journal of Sound and Vibration, 176, 577-584, 1994. [4] M. A. De Rosa, The influence of concentrated masses and Pasternak soil on the free vibrations of Euler beams – exact solution. Journal of Sound and Vibration, 212, 573-581, 1998. [5] W. Q. Chen, C. F. Lü, Z. G. Bian, A mixed method for bending and free vibration of beams resting on a Pasternak elastic foundation. Applied Mathematical Modeling, 28, 877-890, 2004. [6] S. K. Jang, C. W. Bert, Free vibration of stepped beams: Exact and numerical solutions. Journal of Sound and Vibration, 130, 342-346, 1989. [7] S. K. Jang, C. W. Bert, Free vibration of stepped beams: Higher mode frequencies and effects of steps on frequency. Journal of Sound and Vibration, 132,164-168, 1989. [8] M. A. De Rosa, Free vibrations of stepped beams with elastic ends. Journal of Sound and Vibration, 173, 563-567, 1994.

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Computational aspects of anisotropic finite strain plasticity based on the multiplicative decomposition Igor Karˇsaj∗ , Carlo Sansour† , Jurica Sori´c∗ ∗

Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb I.Luˇci´ca 5, HR-10000 Zagreb, Croatia [email protected], [email protected] † The

University of Nottingham, School of Civil Engineering University Park, Nottingham NG7 2RD, UK [email protected] ABSTRACT

The paper is concerned with the computational aspects of two different models for anisotropic multiplicative finite strain plasticity. The first model is based on the idea that Fp , which defines the plastic part of the multiplicative decomposition, is a material tensor. The free energy function is then formulated using the elastic strain measure Ce together with structural tensors defined at the reference configuration [1]. In the second formulation, non-symmetric modified structural tensor are utilized together with the strain measure Ce , following [2]. The second formulation prove to be invariant with respect to rigid body rotations superimposed on Fp . In both formulations a Hill-type yield criterion, described by the material non-symmetric stress tensor and further structural tensors, is utilized. However, there is difference in the form of the stress tensor which the help of which, the yield function is defined. While in the first formulation we use Eshelby-like stress tensor, in the second, a modified stress tensor is applied. The integration of evolution equations is performed using the exponential map which preserves plastic incompressibility. However, the multiplicative nature of the formulations makes the numerical procedures quite involved. Nonetheless, consistent linearisations of the formulations are achieved by recognising various implicit dependencies of the variables involved. Corresponding expressions related to the local iteration and the tangent operator are presented. The interrelation of the two different formulations is demonstrated by numerical examples. In details, the formulations differ from those recently reported in the literature ([3], [4], [5]).

References [1] Carlo Sansour, Igor Karˇsaj, and Jurica Sori´c. A formulation of continuum anisotropic elastoplasticity at finite strains. Part I: Modelling. submitted for International Journal of Plasticity. [2] Carlo Sansour and Jozef Bocko. On the numerical implications of multiplicative inelasticity with an anisotropic elastic constitutive law. Int. J. Num. Methods Engrg., 58:2131–2160, 2003. [3] B. Eidel and F. Gruttmann. Elastoplastic orthotropy at finite strains: multiplicative formulation and numerical implementation. Computational Materials Science, 28:732–742, 2003. [4] A. Menzel and P. Steinmann. On the spatial formulation of anisotropic multiplicative elastoplasticity. Comput. Methods Appl. Mech. Engrg., 192:3431–3470, 2003. [5] Bob Svendsen. On the modelling of anisotropic elastic and inelastic material behaviour at large strain. Int. J. Solids and Structures, 38:9579–9599, 2001.

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Adaptation of biological tissues – a fibre reorientation model for orthotropic multiplicative growth A. Menzel University of Kaiserslautern, Chair of Applied Mechanics Department of Mechanical and Process Engineering P.O. Box 3049, D-67653 Kaiserslautern, Germany [email protected]

ABSTRACT Hard and soft biological tissues posses various substructures on different levels of observation. Even on a macroscopic scale the behaviour of these materials exhibits highly anisotropic response. The modelling of the adaptation of these tissues and the evolution of underlying substructures is of cardinal interest for numerical simulations of for instance hip implants, wound healing, balloon angioplasty, tissue engineering and so forth. A classical example is represented by (extracellular) connective tissue which consists of criss-crossed collagen. This fibre network carries most of the applied stresses and its orientation adapts with respect to the loading directions. The proposed phenomenological constitutive model is essentially based on the introduction of two fibre families which are assumed to remain orthogonal. This approach is well-established for fibre reinforced materials but, for the problem at hand, embedded into the theory of open system thermodynamics. Consequently, volumetric growth together with evolution of the referential density field are addressed. The introduced fibre families enable to additionally incorporate anisotropic growth. On one hand, evolution equations for the fibre diameters, which apparently represent the strength of the fibres, are taken into account. On the other hand, a reorientation of the (referential) fibre directions is proposed. Their evolution is modelled as an (viscous) alignment of the fibres with respect to the principal stretch directions. Both fibre families are thereby restricted to follow the same proper orthogonal transformation. The biological interpretation for this particular ansatz consists in the idea that the free energy takes an extremum if stress and strain fields commute so that the load capacity of the tissues increases according to the predominant loading directions. Moreover, the developed remodelling framework nicely fits into finite element codes so that the applicability of the proposed growth model will be highlighted by means of representative numerical examples.

References [1] A. Menzel, Modelling of anisotropic growth in biological tissues – A new approach and computational aspects. Biomechan. Model. Mechanobiol., 3, 147–171, 2005. [2] A. Menzel, Anisotropic remodelling of biological tissues. In G.A. Holzapfel and R.W. Ogden, editors, Mechanics of Biological Tissue, pages 91–104. Springer, 2006.

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Polyconvex anisotropic hyperelastic energies J. Schr¨oder∗ , P. Neff† ∗

Institute of Mechanics, Department of Civil Engineering, University of Duisburg-Essen Universtit¨atsstr. 15, 45117 Essen, Germany [email protected] † Department

of Mathematics, University of Duisburg-Essen, 45117 Essen, Germany

ABSTRACT The mathematical treatment of boundary value problems is mainly based on the direct methods of variations, i.e. to find a minimizing deformation of the elastic free energy subject to the specific boundary conditions. Existence of minimizers of some variational principles in finite elasticity is based on the concept of quasiconvexity, introduced by Morrey (1952), which ensures that the functional to be minimized is weakly lower semi-continuous. This integral inequality condition is rather complicated to handle. Thus, a more important concept for practical use is the notion of polyconvexity in the sense of BALL [1]. Polyconvex functions are quasi- and rank-one convex. The latter condition ensures the ellipticity of the corresponding acoustic tensor for all deformations. For isotropic material response functions there exist some models, e.g. the Ogden-, Mooney-Rivlin- and Neo-Hooke-type models, which satisfy this concept. For transversely-isotropic materials a variety of polyconvex functions has been proposed in [2, 3]. A detailed analysis in view of materially stability and the adjustment to experimental data for biological tissues can be found in [4, 5], respectively. In this talk we recapitulate the notion of generalized convexity conditions and discuss the modeling of polyconvex anisotropic hyperelastic energies in the framework of the invariant theory. The main attention is turned on orthotropic functions, especially the invariant modeling in the concept of structural tensors and some essential steps of the mathematical analysis are presented. We conclude with some numerical examples.

References [1] BALL , J.M., “Convexity Conditions and Existence Theorems in Non–Linear Elasticity”, Archive of Rational Mechanics and Analysis, Vol. 63, 337–403, 1977 ¨ [2] S CHR ODER , J. & N EFF , P., “On the Construction of Polyconvex Anisotropic Free Energy Functions”, in Proceedings of the IUTAM Symposium on Computational Mechanics of Solid Materials at Large Strains, Ed. C. Miehe, Kluwer Academic Publishers, Dordrecht, 171–180, 2001 ¨ [3] S CHR ODER , J. & N EFF , P., “Invariant Formulation of Hyperelastic Transverse Isotropy Based on Polyconvex Free Energy Functions”, Int. J. of Solids and Structures, Vol. 40, 401–445, 2003 ¨ [4] S CHR ODER , J.; N EFF , P. & BALZANI , D., “A Variational Approach for Materially Stable Anisotropic Hyperelasticity”, Int. Journal of Solids and Structures, Vol. 42/15, 4352–4371, 2004 [5] D. Balzani, P. Neff, J. Schr¨oder & G.A. Holzapfel, A Polyconvex Framework for Soft Biological Tissues. Adjustment to Experimental Data, Int. Journal of Solids and Structures, in press, 2005

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Micromechanically motivated phenomenological modeling of induced flow anisotropy and its application to sheet forming processes with complex strain path changes J. Wang*, V. Levkovitch†, B. Svendsen† * Chair of Mechanics, University of Dortmund Leonhard-Euler-Str. 5, D-44227, Dortmund, Germany [email protected] †

Chair of Mechanics, University of Dortmund Leonhard-Euler-Str. 5, D-44227, Dortmund, Germany [email protected], [email protected]

ABSTRACT Sheet metal forming involves large strains and severe strain path changes. Large plastic strains lead in many metals to the development of persistent dislocation structures resulting in strong flow anisotropy. This induced anisotropic behavior manifests itself in the case of a strain path change by very different stress-strain responses depending on the mode of the strain path change. While many metals exhibit a drop of the yield stress (Bauschinger effect) after a load reversal, some metals show an increase of the yield stress after an orthogonal strain path change (i.e., so-called cross hardening). To model the Bauschinger effect, kinematic hardening has been successfully used for years. However, the usage of the kinematic hardening leads automatically to a drop of the yield stress after an orthogonal strain path change contradicting tests exhibiting the cross hardening effect. So already this example demonstrates that the concept of the combined isotropic-kinematic hardening used in the conventional plasticity has to be extended in order to better simulate processes with complex strain path changes. However, an extension on the phenomenological basis only is a formidable task since one has to make theoretical assumptions with regard to the rather abstract concept of the yield surface. On the other hand, since the deformation mechanisms on the grain level are understood fairly well, polycrystalline simulations yield in a natural way more accurate prediction of the hardening behavior. Admittedly, due to very high numerical costs they cannot be applied to complex industrial simulations. But polycrystalline modeling can be used to better understand the effective macroscopic behavior and can help to develop more realistic phenomenological models. In this work we present such a phenomenological material model whose structure is motivated by polycrystalline modeling that takes into account the evolution of polarized dislocation structures on the grain level – the main cause of the induced flow anisotropy on the macroscopic level. The model considers besides the movement of the yield surface and its proportional expansion as it is the case in conventional plasticity also the changes of the yield surface shape (distortional hardening) and is able to better describe the induced anisotropic behavior with regard to complicated loading histories. The capability of the model is demonstrated on sheet forming processes with complex strain path changes.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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3D homogenization procedure for load bearing masonry columns A. Barbieri1, A. Cecchi1, A. Di Tommaso1 1

Università IUAV di Venezia, dCa Dorsoduro 2206, Venice [email protected] [email protected] [email protected]

ABSTRACT This work proposes a sensitivity study respect to several geometrical and mechanical parameters to evaluate the behaviour of masonry columns - typical of historical buildings and churches - subject to eccentrical loads. The difficulty in modelling masonry lies in its heterogeneous character, since it is composed by blocks between which mortar joints are laid. Moreover the block stiffness is very higher than the mortar one. Here a linear elastic analysis is performed that is significant under service loads. For this reason the determination of homogenised elastic properties for in-plane loaded and out-of plane loaded masonry walls has, in regent years, been the object of a number of studies [1, [2], [3]. In this work, masonry, made of clay bricks and mortar joints, has been identified with a standard elastic continuum by means of a homogenisation method. This method allows to determine values of homogenised axial and bending moduli, for different brick pattern, such as to obtain, starting from a 3D heterogeneous model a beam homogeneous 1D model that takes into account the effective microstructure of masonry column. The 1D masonry column constants are defined as function of geometrical parameters (size of block and mortar thickness) and as function of mechanical parameters (Young modulus and Poisson ratio of block and mortar). An extensive numerical analysis has been carried out to investigate the capacity of the homogenisation method to grasp the effect of geometrical and mechanical parameters in the analysis of masonry columns with equal cross section and different textures. The sensitivity of displacement field to masonry texture is investigated on a meaningful case such as a masonry column loaded by a horizontal force.

References [1] A. Anthoine, Derivation of in plane elastic characteristics of masonry through homogenisation theory. International Journal of Solids and Structures, 32, 137-163, 1995. [2] A. Cecchi and K. Sab, A multi-parameter homogenisation study for modelling elastic masonry. European Journal of Mechanics A/Solids, 21, 249-268, 2002. [3]] A. Cecchi and K. Sab, Out of plane model for heterogeneous periodic materials: the case of masonry. European Journal of Mechanics A/Solids, 21, 715-746, 2002.

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Non Linear Modelling of Masonry Structures Under Cyclic Loads Chiara Calderini* , Sergio Lagomarsino † *

Department of Structural Engineering and Geotechnics – University of Genoa Via Montallegro 1, 16145 Genoa [email protected]

†

Department of Structural Engineering and Geotechnics – University of Genoa Via Montallegro 1, 16145 Genoa [email protected]

ABSTRACT A continuum damage model for historical masonry, able to describe its non-linear behaviour under both static and dynamic cyclic loads, is presented. The model is based on a micromechanical approach. The hypothesis of plane stress is considered. The finite element method is adopted as a framework for numerical implementation. Masonry is considered as a composite material, constituted by a periodic assembly of blocks connected by mortar joints. Mortar joints are described as interfaces. Bed joints only are considered to be mechanically resistant. Both mortar damage and block-mortar decohesion are taken in account; moreover, in order to describe the hysteretic response of the material to cyclic shearing strains, frictional slidings in the interface are considered. Head joints are treated as geometrical discontinuities, neglecting their mechanical properties. Considering the full set of possible in-plane damage mechanisms involving mortar joints [1], an emisymmetric condition on the inelastic strains of bed joints is imposed. Block damage, for normal and shear stresses, is considered. The constitutive equations consider the non linear stress-strain relation in term of mean stresses and mean strains. These latter are produced by an elastic strain contribute, associated with a homogenized elastic continuum, and by an inelastic strain contribute depending on damage. The damage evolution is described by mean of an energetic approach (Rough-Curve approach). The hysteretic behaviour under cyclic loads is described by considering a Coulomb-type friction law on bed joints [2]. In particular, two damage variables are considered for emisymmetric sets of bed joints. This choice let the model able to describe the inelastic strains of mortar head joints in terms of difference between the tangential inelastic strains of bed joints. A compatibility condition is imposed in order to avoid the interpenetration of head joints. The model is implemented in a general purpose finite element code (ANSYS). The constitutive model is used in a finite elements analysis of the lateral response of brick masonry shear walls in-plane loaded by cyclic horizontal actions superimposed on vertical loads. The capabilities and the validity limits of the finite elements analysis are indicated from the simulation of experimental results concerning rectangular slender and squat walls.

References [1] Alpa, G. and Monetto, I., Microstructural model for dry block masonry walls with in-plane loading. J. Mech. Phys. Solids, 42(7), 1159-1175, 1994. [2] Gambarotta, L. and Lagomarsino, S., Damage models for the seismic response of brick masonry shear walls. Part II: the continuum model and its applications. Earthquake Engng. Struct. Dyn., 26, 441-462, 1997.

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Macroscale modelling of structured materials with damage by a specific rigid element model Siro Casolo∗ , Carlo A. Sanjust∗ ∗

Dipartimento di Ingegneria Strutturale. Politecnico di Milano P.zza Leonardo da Vinci 32 - 20133 Milano, Italia [email protected] [email protected] ABSTRACT

The subject of the present contribution is the post-elastic response of an heterogeneous material assembled according with a “masonry-like” texture. For this composite material, a process of loading out of the linear field causes the strain to localize into the mortar joints in which it is essentially concentrated the main part of the mechanical degradation. Since the texture geometry can play a significant role in this process, then it seems of interest to develop a computational model with the feature of retaining some memory of the internal structure also when describing the damage process at the macroscale level. Our attention is focused on this particular aspect by proposing a specific in-plane “mechanistic” approach named rigid element model [1]. As first, a typical masonry-like texture has been studied when subjected to two different loading conditions that caused high shear strains in the mortar joints. This was done by means of a finite element model that described the masonry-like texture with a microscale level of detail, in the frame of a standard continuum approach, assuming a Drucker-Prager plasticity model for the material components. The material parameters were chosen with the main criterion of performing numerical analyses which emphasized the impact of the composite texture on the development of the failure mechanisms. Then, two walls with the corresponding boundary and loading conditions were modelled by the proposed mechanistic rigid element model. The mesh was made by square elements connected each to the other by two normal springs and one shear spring, whose elastic-plastic characteristics were calibrated in order to retain memory of the main texture effects. In particular, the objective was to distinguish the in-plane shear response along the directions parallel and perpendicular to the mortar bed joints, somewhat in analogy with the behaviour of an orthotropic Cosserat continuum [2]. The preliminar numerical results are promising since they show the capacity of the proposed model to include some micro-structure features of the damage process. This notwithstanding, it is apparent that for application to masonry walls it is necessary to improve the present numerical implementation by considering also the effect of the Coulomb internal friction on the shear strengths along the two orthogonal directions.

References [1] S. Casolo, Modelling in-plane micro-structure of masonry walls by rigid elements. International Journal of Solids and Structures, 41(13), 3625–3641, 2004. [2] S. Casolo, Macroscopic modelling of structured materials: relationship between orthotropic Cosserat continuum and rigid elements. International Journal of Solids and Structures, 43(3-4), 475–496, 2006.

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Plate micromechanical models for 3D periodic brickworks A. Cecchi∗ ∗ Universit` a IUAV di Venezia - D.C.A. Dorsoduro 2206- 31000 Venezia - ITALY [email protected]

ABSTRACT This paper proposes a critical approach on some procedures for constructing plate models to describe the out of plane mechanical behaviour of ”masonry” panels with a regular texture. The word ”masonry” is here used to describe a system of blocks which interact with elastic interfaces (i.e. mortar joints). Identification procedures between the 3D discrete model and the 2D plate continuum model, have been here reported based on standard homogenisation methods and based on direct identification method. i.e. by equating the internal work in the 3D discrete model and in the 2D plate model for a class of regular motions - equivalent compatible model -. A crucial problem, with the choice of an identification procedure, is how the kinematic, the dynamic and the constitutive prescriptions of the discrete system are transferred to the continuous one. Hence, constitutive functions of the plate 2D models may be different. A Love-Kirchhoff plate model based on standard homogenisation for linear elastic periodic brickwork has been already proposed by Cecchi and Sab [1]. This model has been also developed in the case both of infinitely rigid blocks and of elastic blocks connected by elastic interfaces taken into account shear effects leading to the identification of a new Reissner-Mindlin homogenised plate model [2]. In this case the identification between the 3D block discrete model and the 2D plate model is based on an identification at the order 1 in the displacement and at the order 0 in the rotation. A Mindlin-Reissner model when block are rigid blocks based on a compatible identification at the order 1 both in the displacement and in the rotation has been performed by Cecchi and Rizzi [3],[4]. Here this model has been implemented also in the case of elastic blocks. All these models permit to obtain the Mindlin-Reissner 2D plate constitutive functions in an explicit symbolic form. The idea is to compare the different constitutive function trough a critical analysis of the identification procedure and to evaluate the accuracy of these identification models by comparison with a 3D F.E. model for a meaningful case.

References [1] A. Cecchi, K. Sab, Out of plane model for heterogeneous periodic materials: the case of masonry. European Jour. of Mechanics. A.-Solids, 21, 249–268, 2002b. [2] A. Cecchi, K. Sab, A comparison between a 3D discrete model and two homogenised plate models for periodic elastic brickwork. Int. J. Solids Structures, vol.41/9-10, 2259–2276, 2004. [3] A. Cecchi A., N.L. Rizzi, Analisi in piu’ parametri perturbativi per murature a struttura regolare: identificazione 3D con modelli 2D di piastra,Proc: XVI congresso AIMETA di meccanica teorica e applicata, Ferrara 9-12 settembre 2003. [4] A. Cecchi, N.L. Rizzi, Modelli 2D con microstruttura per pannelli di muratura in 3D, Proc: XVII congresso AIMETA di meccanica teorica e applicata, Firenze 11-15 settembre 2005.

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Limit analysis of out-of-plane loaded running bond masonry walls under Mindlin-Reissner plate hypotheses A. Cecchi*, G. Milani†, A. Tralli† * IUAV University of Venice Dorsoduro 2206, ex convento Terese, 30123 Venice, Italy [email protected] †

Department of Engineering, University of Ferrara Via Saragat 1 44100 Ferrara, Italy [email protected] [email protected]

ABSTRACT Earthquake surveys have demonstrated that the lack of out-of-plane strength is a primary cause of failure in many traditional forms of masonry. Moreover, bearing walls are relatively thick and, as a matter of fact, many codes of practice impose a minimal slenderness for them, as for instance the recent Italian OPCM 3274 2003, in which the upper bound slenderness is fixed respectively equal to 12 for artificial bricks and 10 for natural blocks masonry. In this context, it seems particularly attractive a formulation at failure for regular assemblages of bricks based both on homogenization and Mindlin-Reissner theory. Starting from a compatible identification, already developed in the framework of linear elasticity by Cecchi and Rizzi [1], in which a 3D system of blocks connected by elastic interfaces is identified with a 2D Mindlin-Reissner plate, in this paper a limit analysis approach for deriving the homogenized failure surfaces for masonry out-of-plane loaded is presented. On the other hand, in a previous paper by Milani et al. [2] failure surfaces for out-of-plane loaded masonry were obtained by means of a static limit analysis approach under Love-Kirchhoff plate hypotheses. In this paper, a kinematic approach is proposed under the hypotheses of Mindlin-Reissner plate theory, infinitely resistant blocks connected by interfaces (joints) with a Mohr-Coulomb failure criterion. In this way, the macroscopic masonry failure surface is obtained as a function of the macroscopic curvatures and shear strains by means of a constrained minimization of the internal power dissipated, once that a subclass of possible deformation modes is a priori chosen in order to characterize out-of-plane masonry behavior. Several examples of technical relevance have been developed with the model at hand and comparisons both with previously developed Love-Kirchhoff kinematic limit analyses [3] and standard 3D FE elasto-plastic procedures on the homogenized failure surfaces are reported in detail.

References [1] A. Cecchi, N. L. Rizzi, Analisi in più parametri perturbativi per murature a struttura regolare: identificazione 3D con modelli 2D di piastra. Proc: XVI congresso AIMETA di meccanica teorica e applicata, Ferrara (Italy) 9-12 September 2003. [2] G. Milani, P. B. Lourenço, A. Tralli, Homogenised limit analysis of masonry walls. Part I: failure surfaces. Comp. & Struct., in press. [3] K. Sab, Yield design of thin periodic plates by a homogenisation technique and an application to masonry walls. C.R. Mechanique, 331, 641-646, 2003.

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Nonlinear Analysis of Brittle Materials Marialaura Malena∗ , Antonio Bilotta∗ and Antonio D. Lanzo† ∗ Dipartimento

di Strutture, Universit`a della Calabria Via P. Bucci, cubo 39/b - 87030 Rende (CS) Italy [email protected], [email protected] † DiSGG, Universit`a della Basilicata Contrada Macchia romana, 85100 Potenza (PZ) Italy [email protected]

ABSTRACT The problem of the evaluation of the nonlinear response of damaged structures is considered. In order to circumvent the numerical difficulties which can affect standard numerical strategies because of the high sensitivity of the stress response to small changes of displacements, a specific implementation of the standard path–following procedure is presented. The approach is based on a parametrization of the equilibrium curve which takes into account also the variables used in the modelling of the damage process. Two damage models, an isotropic one and an orthotropic one, are implemented in order to verify the numerical procedure in the analysis of 2D continua subjected to plane stress condition.

References [1] Lemaitre J., A course on damage mechanics, Springer-Verlag, Heidelberg-Germany, 1996. [2] Chaboche J.L., Continuum damage mechanics:part I–general concepts, Journal of Applied Mechanics 55 (1988), 59–64. [3] Scotta R., Vitaliani R., Saetta A., O˜ nate E., Hanganu A., A scalar damage model with a shear retention factor for the analtsis of reinforced concrete structures:theory and validation , Computers & structures 79 (2001), 737–755. [4] Berto L., Saetta A., Scotta R., Vitaliani R., An orthotropic damage model for masonry structures, Int. J. Num. Meth. Eng. 55 (2002), 127–157. [5] Riks E., An incremental approach to the solution of snapping and buckling problem, International Journal of Solids and Structures 15 (1979), 529–551. [6] Fried I., Orthogonal trajectory accession to the nonlinear equilibrium curve, Computer Methods in Applied Mechanics and Engineering 47 (1984), 283–298. [7] Garcea G., Trunfio A. G., Casciaro R., Mixed formulation and locking in path-following nonlinear analysis, Computer Methods in Applied Mechanics and Engineering 165 (1998), 247–272.

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Damage localisation in computational homogenisation of masonry and its incorporation in a two-scale computational framework. T.J. Massart∗ , R.H.J. Peerlings† , M.G.D. Geers† ∗ Structural and Material Computational Mechanics Departement CP 194/5 Universit´e Libre de Bruxelles, Avenue F.D. Roosevelt 50, 1050 Brussels, Belgium [email protected] † Materials

Technology Institute, Eindhoven University of Technology P.O. Box 513, 5600 MB Eindhoven, The Netherlands [email protected], [email protected] ABSTRACT

This paper presents a multi-scale framework for the computational study of damage development in masonry walls, based on computational homogenisation techniques. In order to overcome the troublesome formulation of closed-form constitutive equations, a first-order computational homogenisation framework is applied to infer the non-linear material behaviour of brick masonry in the presence of quasi-brittle damage of its constituents [1]. A localisation analysis is carried out based on the macroscopic homogenised tangent stiffness to capture the occurrence of localisation at the macroscopic scale [2]. It is shown that localisation is detected along preferential orientations, which are consistent with the underlying mesostructural failure patterns and with the applied loading. An enhanced first-order computational multi-scale solution scheme is then outlined, which allows to incorporate microstructurally-based damage localisation orientations in structural computations. This model uses a first-order computational homogenisation technique enhanced with a finite width embedded damage band model in order to allow the treatment of macroscopic localisation resulting from damage growth in the constituents. The implementation of the multi-scale solution scheme using an nested finite element method is outlined. In particular, as a result of the use of homogenisation techniques on finite volumes in the presence of quasi-brittle constituents, mesostructural snap-back may occur in the homogenised material response. A methodology is proposed to introduce this response in the originally strain-driven multi-scale technique [3], based on energy dissipation control at the mesoscopic scale [4]. Its impacts on the implementation of the framework as well as on the path following techniques needed to trace complete load-deflection paths are discussed. The results obtained by the framework are illustrated by means of a strutural computation example.

References [1] T. J. Massart, R. H. J. Peerlings and M. G. D. Geers. Mesoscopic modeling of failure in brick masonry accounting for three-dimensional effects. Eng. Fract. Mech., 72, 1238–1253, 2005. [2] J. R. Rice and J. W. Rudnicki. A note on some features of the theory of localisation of deformation. Int. J. Sol. Struct., 16, 597–605, 1980. [3] V. G. Kouznetsova, W. A. M. Brekelmans and F. T. P. Baaijens. An approach to micro-macro modelling of heterogeneous materials. Comp. Mech., 27, 37–48, 2001. [4] T. J. Massart, R. H. J. Peerlings and M. G. D. Geers. A dissipation-based control method for the multi-scale modelling of quasi-brittle materials. C.R. M´ecanique , 333, 521–527, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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3D homogenized limit analysis of masonry buildings subjected to horizontal loads G. Milani†, P. B. Lourenço*, A. Tralli† *

Department of Civil Engineering, University of Minho, School of Engineering 4800-058 Azurem, Guimarães, Portugal [email protected] †

Department of Engineering, University of Ferrara Via Saragat 1 44100 Ferrara, Italy [email protected] [email protected]

ABSTRACT The current confidence in the ability to provide buildings with adequate resistance to horizontal actions does not extend back to historic and existing masonry structures. Furthermore, it has been shown that the high vulnerability of historical centers to horizontal actions is mostly due to the absence of adequate connections between the various parts, especially when wooden beams are present both in the floors and in the roof [1]. This characteristic leads to overturning collapse of the perimeter walls under seismic horizontal acceleration and combined in- and out-of-plane failures. Even if limit analysis is not sufficient for a full structural analysis under seismic loads, it can be profitably used in association with more sophisticated techniques (as for instance non-linear FE with damaging procedures) in order to obtain a simple and quick estimation of collapse loads and failure mechanisms. Up to now, simplified limit analysis methods are at disposal to the practitioners both for safety analyses and design of strengthening [2]. Nevertheless, in some cases these methods are based on several simplifications, one of which is an a-priori assumption of the collapse mechanics combined with the separation of in- and out-of-plane effects. In this paper, the micro-mechanical model presented by the authors in [3] and [4] for the limit analysis of respectively in- and out-of-plane loaded masonry walls is utilized in presence of coupled membrane and flexural effects. In the model, the elementary cell is subdivided along its thickness in several layers. For each layer, fully equilibrated stress fields are assumed, adopting polynomial expressions for the stress tensor components in a finite number of sub-domains. The continuity of the stress vector on the interfaces between adjacent sub-domains and suitable anti-periodicity conditions on the boundary surface are further imposed. In this way, linearized homogenized surfaces in six dimensions (polytopes) for masonry in- and out-of-plane loaded are obtained. Such surfaces are then implemented in a FE limit analysis code for the analysis at collapse of entire 3D structures. Several examples of technical relevance are discussed in detail and comparisons with standard FE codes are provided.

References [1] L. Ramos, P.B. Lourenço, Modeling and vulnerability of historical city centers in seismic areas: a case study in Lisbon. Engineering Structures, 26, 1295-1310, 2004. [2] A. Giuffrè (editor), Safety and conservation of historical centers: the Ortigia case. Editore Laterza, Roma - Bari, 1993 [in Italian]. [3] G. Milani, P.B. Lourenço, A. Tralli, Homogenised limit analysis of masonry walls. Part I: failure surfaces. Comp. Struct., in press. [4] G. Milani, P.B. Lourenço, A. Tralli, A homogenization approach for the limit analysis of out-ofplane loaded masonry walls. Submitted.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Modeling the Dynamic Behavior of Masonry Walls as Rigid Blocks Fernando Peña*, Paulo B. Lourenço†, José V. Lemos†† *

Departamento de Engenharia Civil, Universidade do Minho Campus de Azurem, 4800-058, Guimarães, Portugal [email protected]

†

Departamento de Engenharia Civil, Universidade do Minho Campus de Azurem, 4800-058, Guimarães, Portugal [email protected] ††

Laboratório Nacional de Engenharia Civil Av Brasil 101, 1700-066 Lisboa, Portugal [email protected]

ABSTRACT Due to the low tensile resistance of historical constructions formed by stone blocks, these structures are particularly vulnerable objects under lateral seismic loads. In this way, the study based upon the assumption of continuum structures would not be realistic for many cases. On the other hand, models based on rigid-block assemblies provide a suitable frame work for understanding their dynamic behavior under seismic actions. In this context, the problem is primarily concerned with Rocking Motion (RM) dynamics. Some authors have been used successfully the Distinct Element Method in the study of block structures. However, they have used experimental test to calibrate their models and to obtain the parameters used in the DEM; because, these parameters can not be obtained in a direct form from the parameters of the stones. Special attention regards about the damping factor, since in the DEM a viscous damping is considered, but in the reality, the damping is due by impact that can be considered as a type of Dirac-delta forces. This paper is divided in two parts, in the first one, a methodology for the modeling the dynamic behavior of rigid-block structures under seismic loadings by means of a Distinct Element Model is proposed. The commercial code UDEC was used. The methodology proposed links the parameters used by the UDEC and the theoretical parameters used by the classical theory of RM, since these last parameters can be obtained by means of theoretical expressions or by experimental test. In the second part, the paper presents the modeling of the block structures tested at the Laboratory of Civil Engineering (LNEC) of Portugal. Single block and bi-block structures were studied. The experimental tests were carried out at the seismic table of the LNEC. Free rocking, harmonic and random motions were used in the experimental tests. Very good agreement between the numerical model and experiment is achieved, for both free and forced regimens.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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On the Improvement of Monumental Structure Safety: a Case Study G. Zingone1, G. De Canio2, L. Cavaleri3 1

Centro di Prevenzione e Istruzione Sismica, Dipartimento di Ingegneria Strutturale e Geotecnica Università di Palermo, Italy [email protected] 2

3

ENEA Materiali e Nuove Tecnologie Casaccia (Roma) Italy decanio@ casaccia.enea.it

Dipartimento di Ingegneria Strutturale e Geotecnica Università di Palermo, Italy [email protected]

ABSTRACT The present paper deals with the problem of the preservation of the monumental patrimony and discusses a practical application. The analysis methodology based on modern dynamic identification techniques was used for a monumental church in the historic area of Catania (Italy). The dynamic response of the church was measured in terms of accelerations, by means of vibration tests in situ. The numerical analyses highlighted the vulnerability of the drum dome system. Hence interest focused on this part of the building. A reduced scale model of the above system was built to be subjected to dynamic tests on a shaking table at the ENEA laboratory at Casaccia (Rome). Different kinds of reversible devices for the improvement of the structural behavior experienced. The tests gave useful information about the effectiveness of several types of seismic devices inserted into the drum windows. The theoretical and the experimental results showed the importance of employing identification techniques for the improvement of the safety of monuments and of verifying the effectiveness of the reinforcing devices adopted in connection to the stiffness, the strength and the ductility.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Homogenization of masonry using a micro-mechanical model: Compressive behaviour Alberto Zucchini*, Paulo B. Lourenço† * ENEA, FIS.MET, C.R.E. “E.Clementel” v. Don Fiammelli, 2, I – 40129 Bologna, Italy [email protected] † University of Minho Department of Civil Engineering, Azurém, P – 4800-058 Guimarães, Portugal [email protected]

ABSTRACT The present paper aims at further discussing the mechanics of masonry under compression and at proposing a homogenisation tool that is able to reproduce the results of advanced non-linear finite element computations, at a marginal fraction of the cost. For this purpose, a homogenisation approach previously developed by the authors [1,2], is extended to the case of masonry under compression. The proposed simplified approach provides results almost equal to very complex non-linear finite element analysis of a masonry representative volume. It has been generally accepted that the difference in elastic properties of the unit and mortar is the precursor of failure. Uniaxial compression of masonry leads to a state of tri-axial compression in the mortar and of compression/biaxial tension in the unit. Nevertheless, the problem of reproducing the experimental response of the masonry composite from the behaviour of masonry components is rather difficult due to the number of influencing parameters and the complex micro-structure. Models such as the one proposed in the paper allow to better understand the failure of masonry under compression.

(a) (b) (c) Figure 1: Validation of non-linear micro-mechanical homogenization model: (a) failure criterion for masonry; (b) uniaxial tensile loading; (c) uniaxial compressive loading.

References [1] A. Zucchini and PB. Lourenço, A micro-mechanical model for the homogenisation of masonry. International Journal of Solids and Structures, 39, 3233–3255, 2002. [2] A. Zucchini and PB. Lourenço. A coupled homogenisation-damage model for masonry cracking. Computer and Structures, 82, 917–929, 2004.

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Uncertainty characterization and settlement analyses: the importance of distribution types R. Jimenez-Rodriguez∗ , L. M. Lacoma∗ ∗ E.T.S.

Ingenieros de Caminos, Canales y Puertos Universidad Polit´ecnica de Madrid 28040 Madrid, Spain [email protected]; [email protected]

ABSTRACT In recent years, significant effort has been put into considering advanced probabilistic soil models that represent soil variability [see e.g., 1, 2]. In particular, the sensitivity of probability results to changes in the mean and variance of properties, and to changes in the structure of spatial correlation of soil properties has received wide attention [3–5]. We study the influence of using different types of statistical distributions to characterize the Young’s modulus of the soil for computation of settlements on spatially random soil. We use a linear elastic finite element model with a rigid footing founded on elastic soil. Poisson’s ratio of the soil is considered constant, and Young’s modulus is characterized using random fields with two limiting extreme values of their scales of fluctuation. We perform a number of simulations in which finite element program IRIS is used to compute the settlement of the foundation for each realization of the Young’s modulus random field. The mean Young’s modulus of the soil was considered constant in all cases, and standard deviations were varied among different simulations. As expected, computed settlements are observed to be very similar when lognormal and gamma distributions are similar. Similarly, an “averaging” effect is observed for small scales of fluctuation; such “averaging” effect makes settlement results to be similar among different realizations of the random field. The election between lognormal or gamma distributions is significant, however, for high values of σE . Differences between settlements computed with both distributions are particularly significant for large values of the scale of fluctuation. That is, our results suggest that, in some cases, the type of distribution considered for characterization of the random field of Young’s modulus can have a significant impact on computed settlement results. Such observation should be taken into account when performing simulations of random fields in the context of settlement analyses.

References [1] G. A. Fenton (1999). “Estimation for stochastic soil models.” Journal of Geotechnical & Geoenvironmental Engineering, 125(6), 470–485. [2] R. Rackwitz (2000). “Reviewing probabilistic soils modelling.” Computers and Geotechnics, 26(34), 199–223. [3] D. Griffiths and G. Fenton (2004). “Probabilistic slope stability analysis by finite elements.” Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 130(5), 507–518. [4] G. Fenton and D. Griffiths (2002). “Probabilistic foundation settlement on spatially random soil.” Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 128(5), 381–390. [5] D. Griffiths and G. Fenton (1997). “Three-dimensional seepage through spatially random soil.” Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 123(2), 153–160.

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Adaptive probabilistic modeling of localization, failure and size effect of quasi-brittle materials Miroslav Voˇrechovsk´y, Rostislav Chudoba and Jakub Jeˇra´ bek Chair of Structural Statics and Dynamics, Aachen University of Technology Mies-van-der-Rohe-Str. 1, 52056 Aachen, Germany [email protected], [email protected] ABSTRACT Objective simulation of response of nonlinear structures must reflect the spatial variability of local material properties. The main target of the paper is to present ideas behind a computational tool oriented toward adaptive nonlinear simulation driven by spatially (and randomly) varying model properties which is currently under development by authors. In particular, we focus on detailed tracing of the evolution of damage or other nonlinear phenomena during loading of structure with varying properties by nonlinear finite element method with mesh refinement/coarsening in highly/low stressed or damaged regions. We have developed the major ingredients of the algorithm and we present the current stage of progress on the computational platform in this paper. The computations is illustrated on a simple one-dimensional example involving a bar made of plastic material with hardening under uniaxial tension. The spatial random fluctuation of material properties (eg. strength, modulus of elasticity, fracture energy, etc) is modelled by mean of random fields. We aim at solving the problem involving such an uncertainty by means of simulation. In particular, we suppose to have the generated the sample functions of random properties and the task is to accurately reproduce the processes during loading of structures by a numerical method. The presented work relies heavily on combining several concepts and methods from the fields of adaptive finite element modeling and simulation of random fields. Their combination is supposed to yield innovative tools for analyzing and describing e.g. size effect phenomena. The established innovative computational platform (based on the noncommercial platform ORFEUS being developed at RWTH Aachen, Germany) will make it possible to simulate the failure processes for levels of complexity that cannot be tackled or is difficult to model with the currently available models and tools. The adaptive finite element method is effectively combined with the simulation of random fields in order to capture damage and failure phenomena in quasi-brittle materials. In contrast to the usual adaptivity, driven by error minimization or crack propagation, we aspire to control the adaptations of the numerical model by the resolution scale needed to capture the evolution of the damage. This will allow us to zoom only into the critical regions of the model and to significantly limit the computational effort of a nonlinear iteration process (while ensuring that nothing important has been overlooked). The main goals of the development of such an adaptive computational strategy are: • to capture the complex phenomena associated with heterogeneous material behavior including damage localization, • to significantly reduce the computational cost that arises for any fine resolution of the discretization, • to study the size effect in all its complexity with both the statistical and energetic components, including their interaction, and • to provide a robust numerical platform for future development and validation of alternative analytical statistical methods.

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On the use of infinite random sets for bounding the probability of failure in the case of parameter uncertainty Diego A. Alvarez Institut f¨ur Technische Mathematik, Geometrie und Bauinformatik Leopold-Franzens Universit¨at Innsbruck, Austria [email protected]

ABSTRACT This contribution presents a novel technique for estimating the bounds of the probability of failure of structural systems when there is aleatory and epistemic uncertainty in the representation of the basic variables. The proposed methodology allows the designer to model parameter uncertainty without making further suppositions that would be reflected in the estimated value of the probability of failure; since the method takes in consideration all possible variation due to uncertainty in the representation of the basic variables, it gives as an answer upper and lower bounds on the probability of failure. In particular, the methodology allows to represent parameter uncertainty as a possibility distribution, cumulative distribution function, probability box or family of intervals provided by experts. These four representations are special cases of the theory of random sets, which is a generalization of the theories of probability, possibility and interval analysis. Up to the best of the author’s knowledge, until now, all the papers that employ random sets in the case of uncertainty analysis have been confined to a finite random set representation, or to its analog a Dempster-Shafer body of evidence. This implies that the information provided by the experts must be discretized. In this paper, an infinite random set representation is introduced. With this new approach, the information provided does not have to be discretized. Also, a new geometrical representation of the space of basic variables is given, where the methods already existing for estimating the probability of failure and that only require the sign of the evaluations of the basic variables vector on the limit state function may work without additional overhead. Using this kind of methods, the computational cost required for the estimation of the bounds of the probability of failure could decrease notably compared with the discrete approach. Furthermore, the proposed method allows the analyst to model the lack of information about the dependence of the basic variables, and in consequence, provides a new methodology to avoid the misused assumption of independence between the basic variables and the myth that varying the correlation coefficients constitutes a sensitivity analysis for uncertainty about dependence. A benchmark example is used to demonstrate the usefulness of the method.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Stochastic analysis of coupled nonlinear thermo-mechanical problems: SFEM Model Jean-Baptiste Colliat∗, Martin Krosche†, Markus Krosche†, Martin Hautefeuille∗ †, Adnan Ibrahimbegovic´∗, Rainer Niekamp†, Hermann G. Matthies† ∗ ENS

Cachan, Laboratoire de Me´canique et Technologie 61 avenue du pre´sident Wilson, 94235 Cachan, France [email protected]

† TU

Braunschweig, Institut fu˝r Wissenschaftliches Rechnen 38092 Braunschweig, Germany [email protected]

ABSTRACT Dealing with the modelling of brittle and quasi-brittle materials requires to be able to take into account of some special features of such materials, among them is the so-called size-effect. This effect leads to an important uncertainty about the yield or maximum stress in a structure, depending on its size. Moreover this effect is mostly due to the heterogeneity of these materials, at a microscopic scale. Here our cause of concern are structures under fire-loading, for which it is important to be able to predict the behaviour in time in order to asses the safety. Normally this is done in standardised tests at full scale, involving large experimental tools and so leading to important costs. This study is essentially dealing with the case of masonry walls made from clay hollow bricks and submitted to fire. In previous works, a numerical procedure for such masonry walls is proposed and analysed. The purpose is here to investigate the uncertainty that is inherent to several points in this kind of problems. We focuse here on the variations of the yield stress. This uncertainty is directly linked to the so-called size-effect which is an inherent and experimentaly observed property of brittle and quasi-brittle materials class to which the clay is belonging to. Different ways are existing to model this size-effect. Most of them are introducing a ”characteristic” length, which is claimed to be an intrisic property of the considered material, and use non-local constitutive laws formulations. We presented in this work a different point of view in which the non-local approach is replaced by a strong discontinuity formulation. Moreover, the heterogeneity of the material is introduced by the use of correlated random fields for the stress yield and the maximum stress. This stochastic model is solved by direct Monte-Carlo integration provided by a C++ framework named PLATON, which delivers simple-to-use parallelisation features. The results showed that the correlation length might be viewed as a way to introduce a scale into the model and then to reproduce the sizeeffect. Hence we can conclude that stochastic models are of a great importance in the view to provide reliable and predictive structural models.

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Fuzzy Frequency Response Function of a Composite Floor Subject to Uncertainty by Application of the GĮd Algorithm Daan Degrauwe∗ , Guido De Roeck, Geert Lombaert ∗ Division

of Structural Mechanics, KULeuven Kasteelpark Arenberg 40, B-3001 Leuven [email protected] ABSTRACT

During the last decade, the application of fuzzy numbers in the civil engineering domain has evolved from small academic applications to large scale industrial cases. The most popular algorithm to perform the involved fuzzy calculations is the Transformation Method [1], although the application area of this method is limited by the requirement of monotonic input-output relationships on the one hand, and by the number of uncertain variables on the other hand: in case of non-monotonic input-output relationships, the Transformation Method may lead to an underestimation of the uncertainty on response variables, and the computational cost increases exponentially with the number of uncertain variables. In the present paper, the Gradual α-level Decreasing (GαD) algorithm is presented as an alternative to the Transformation Method. This algorithm is based on the observation that the set of extrema with varying α-level is formed by continuous curves. In order to obtain the membership function of the output, the α-level is lowered step-by-step, whereby at each level, the extrema are searched in the vicinity of the extrema of the previous α-level. The GαD-algorithm is applied to the analysis of the fuzzy frequency response function of a composite floor. A total of six uncertain variables is considered, including the material properties of the floor, the damping ratio and the boundary conditions. Comparison of the fuzzy FRF obtained with the Transformation Method and with the GαD algorithm (figure 1) shows that the resonance response uncertainty is underestimated by the Transformation Method, while it is evaluated correctly by the GαD-algorithm.

Figure 1: Fuzzy FRF of a composite floor

References [1] M. Hanss, The transformation method for the simulation and analysis of systems with uncertain parameters, Fuzzy Sets and Systems 130 (3) (2002) 277–289.

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Comparative Numerical Evaluation of Angra I Auxiliary Feedwater System Reliability by the Method of Suplementary Variables José Luiz Fernandes, Marcos Oliveira de Pinho, Antonio Carlos Marques Alvim, Antonio Pithon Federal Center of Technological Education Av. Maracanã, 229, BlocoE, Sala505.7 [email protected] [email protected] [email protected] [email protected]

ABSTRACT This paper proposes a methodology for predicting failure probabilities using the method of supplementary variables for nuclear safety systems subject to aging. Particularly, the failure probability of the auxiliary feedwater (AFSW) system of a typical PWR plant was calculated. In this context, it has been necessary to study systems of partial and ordinary first order differential equations in order to represent a system subject to aging. Numerical solutions were obtained by the method of finite differences using Euler’s implicit to solve the equations related to the concepts “as good as new” and using numerical integration for “as bad as old” concept. These numerical solutions were used to model extension of useful life in a nuclear power plant. For testing the proposed model constant failure rate was used to evaluate the failure probability of the auxiliary feedwater (AFSW). This paper proposes a new method which named is T-method the results obtained with “as good as new” and “as bad as old” models for 240, 400, 720, 1000, 2000, 3000 and 4000 hours of operations, with rate constant failure. With these data, the failure probability for 5000 hours by was estimated the T-method. This numerical solution is very helpful for considering the life extension of nuclear plants, an issue currently under discussion.

References [1] Amaral Netto, J. D., “Simulation of discrete events guided to objects for the analysis of the unavailability of dynamic systems”, PhD Thesis (in Portuguese), COPPE – UFRJ, Nuclear Engineering Program – COPPE/UFRJ- Brazi, 199l.

[2] Pinho, M. O., Fernandes , J.L., Alvim, A. C. M., Frutuoso e Melo, P. F. F. F., “A numerical reliability model evaluation of angra I auxiliary feedwater system by the method of suplementary variables”, International Nuclear Atlantic Conference - INAC 2005, Santos, SP, Brazil, August 28 to September 2, 11p., 2005. [3] Pinho, M. O., Fernandes , J.L., Alvim, A. C. M., Frutuoso e Melo, P. F. F. F., “A new method for reliability evaluation of auxiliary feedwater system by the by the method of suplementary variables”, International Nuclear Atlantic Conference - INAC 2005, Santos, SP, Brazil, August 28 to September 2, 10p., 2005.

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Lifetime Estimation of Vertical Bridge Tie Rods Exposed to Wind-induced Vibrations M´ozes G´alffy, Matthias Baitsch, Andr´es Wellmann-Jelic, Dietrich Hartmann Department of Computational Science in Civil Engineering, Ruhr-University Bochum Universit¨atsstr. 150, D–44780 Bochum, Germany {galffy,hartus}@inf.bi.rub.de, {matthias.baitsch,andres.wellmann}@rub.de ABSTRACT The vortex-induced across-wind vibrations of vertical bridge tie rods, often dominated by the so-called lock-in effect, have been proved as a prominent load factor in the dimensioning of arched steel bridges. In order to realistically predict the oscillation amplitudes of the bridge hangers, an improved load model has been developed. This model is used to estimate the lifetime of the hanger connection plate, which represents the most sensitive part of the bridge. The stresses are computed on an a 3D finite element model of the structure, according to the notch stress approach. The distribution function of the damage accumulated during an excitation process can be evaluated by using the Palmgren-Miner accumulation rule, applied to stochastically defined, uniform Woehlercurves. Analysing several statistically equivalent realisations of the wind process, a distribution function of the damage can be established. With this and the Weibull function describing the occurrencefrequency of mean wind velocities, the time-dependent failure probability of the plate can be estimated. As a reference-system, the hanger-plates of an arched bridge in Germany have been chosen. The form of the cut at the end of the flattened hangers, originally a semi-circle, has been optimized using evolution strategies. The estimated lifetime of the welding and of the bulk plate have been compared for the original and for the optimized geometry. The comparison shows that the expected lifetime of the hanger has been substantially increased, from 0.025 to 1.5 years, and that not the welding (lifetime 1040 years), but the bulk plate (lifetime 1.5 years) is the most sensitive part of the optimized structure.

Figure 1: Failure probabilities as a function of time, computed using Woehler-curves given in Eurocode 3 (bulk plate) and in the IIW-guidelines (weldings)

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Stochastic Static Analyses of FE Models by means of Newton’s series expansions G. Giunta∗ , E. Carrera† ∗ Space

and Aeronautic Engineering Department, Politecnico di Torino Corso Duca degli Abruzzi 24, 10129, Torino, Italy [email protected]

† Space

and Aeronautic Engineering Department, Politecnico di Torino Corso Duca degli Abruzzi 24, 10129, Torino, Italy [email protected] ABSTRACT

A deterministic design approach is not always profitable, and sometimes not even suitable, when adopted in frontier engineering fields that lead to high technology level and high performances products. ”A backward step” in the analyses proceeding is required in order to consider sic et simpliciter the Nature behavior: every natural phenomenon is characterized by an intrinsic scattering. A Monte Carlo Simulation Stochastic Approach (MCSSA) represents an attempt to better reproduce the actual natural behavior. That approach has the clear merit to bring a deeper view of the design problem. On the other hand it implies high computational costs since it is basically a random collection of deterministic System output data. In order to overtake that limitation the coupling of the MCSSA to a System Response Surface (SRS) was suggested in an ESA research work [1]: the output data collection can be performed by random explorations of the SRS itself. This work is articulated in different parts. In the first one the Newton’s series approximation is proposed inside the FEM analyses environment in order to compute the SRS. Newton’s series approximation has to be intended as a generalization of what addressed in [1]. Its mathematical definition and the touching point with the FEM theory are reported. Also, an analytical example is showed in order to provide a better insight of the methodology. In the second section the attention is focused on the application of the Newton’s series based SRS to stochastic static FEM analyses. Three analyses of truss models are taken into account. All of the results are obtained via a FEM software developed ad hoc by the authors. The leitmotiv of this section is the comparison of the obtained statistic results: means values, percentage coefficients of variation, skewness, kurtosis and correlation coefficients and the required cpu time to those computed by means of the classic Monte Carlo Simulation Method (MCSM). Also, from the opposite point of view, the statistic data provided by the MCSM analyses are used to perform a point by point deterministic verification of the SRS approximation quality. In a conclusive section all of the considerations and comments about the obtained results are reported. It is showed how the Newton’s series expansion provides high analysis flexibility, good deterministic and statistic result approximation and how it reduces the computational costs when a suitable number of input stochastic parameters are taken into account. As a final consideration it is important to underline that this work has to be thought as an encouraging starting point, since the suggested methodology has to be tested with stochastic analyses that take into account more complex FE models.

References [1] G. Giunta, E. Carrera, A. Calvi, Monte Carlo Simulation Method coupled to Higher Order Derivatives method Response Surface applied to Static Stochastic FEM Analyses. XVIII AIDAA Naztional Congress , Volterra (Pi), Italy.

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A Galerkin solution for stochastic algebraic equations Mircea Grigoriu Cornell University Ithaca, NY, USA [email protected]

ABSTRACT Galerkin solutions are developed for linear stochastic algebraic equations, that is, linear algebraic equations with random coefficients. Two classes of Galerkin solutions, referred to as optimal and sub-optimal Galerkin solutions, are constructed. It is shown that optimal Galerkin solutions for a linear stochastic algebraic equation are given by conditional expectations of the exact solution of this equation with respect to σ-fields that are coarser than the σ-field relative to which the exact solution is measurable. The σ-fields needed to defined optimal Galerkin solutions can be constructed from, for example, σ-fields generated by measurable partitions of the sample space. Galerkin solutions that are not optimal are called sub-optimal. Optimal and sub-optimal Galerkin solutions are unbiased and biased approximations of the exact solution. The optimal and sub-optimal Galerkin methods developed in the paper are applied to find moments and distributions for the displacement field of a rectangular plate subjected to a uniform load and supported by a linear random foundation. The plate random foundation is modeled by a homogeneous translation field taking values in a bounded interval. Translation fields are memoryless transformations of homogeneous Gaussian fields, and can follow any marginal distribution. Since generally random functions consist of an uncountable number of random variable, they need to be approximated for numerical calculations by parametric random functions, that is, deterministic functions of space argument depending on a finite number of random variables. It is shown that both the optimal and the sub-optimal Galerkin solutions are satisfactory if based on accurate representations of the random field used to represent the plate foundation.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Approximate method for probability density of the response of a linear oscillator to a non-Poisson impulse process Radoslaw Iwankiewicz∗ , Marcello Vasta† ∗ University

of the Witwatersrand Johannesburg, South Africa [email protected]

† University

of Chieta-Pescara Chieta-Pescara, Italy [email protected]

ABSTRACT State vector of a dynamic system under a Poisson train of impulses is a non-diffusive Markov process and its joint probability density function satisfies an integro-differential generalized Fokker-PlanckKolmogorov equation which is also called Kolmogorov-Feller equation. If the train of impulses is driven by non-Poisson, for example renewal, counting processes, the state vector is not a Markov process. Non-Markov pulse problems can be converted into Markov ones by augmenting the state vector of the dynamic system by auxiliary variables driven by Poisson processes. Exact techniques of this kind have been developed for trains of impulses driven by Erlang renewal processes [1] or by a generalized Erlang renewal process [2]. Techniques of equations for response moments have been developed, but no explicit forms of equations governing the response probability density have been given. The excitation considered in the present paper is a renewal random train of impulses with interarrival times being a sum of two independent, negative exponential distributed random variables with different parameters. The original impulse process is exactly converted into a Poisson driven one with the aid of the jump process regarded as an auxiliary state variable. Based on the general integro-differential forward Chapman-Kolmogorov equation the equations governing the joint probability density-distribution function of the response are derived. One of these equations is integro-differential and the other one partial differential. These equations are transformed to two first-order partial differential equations and the approximate solution technique is devised by considering the evolution of the response during small time intervals. The method of characteristics is used to find the explicit solution to these equations. The analytical results are verified against numerical simulations.

References [1] R.Iwankiewicz and S.R.K.Nielsen Advanced methods in stochastic dynamics of non-linear systems, Vibration Theory. Aalborg University Press, Denmark, 1999. [2] R.Iwankiewicz, Dynamic systems under random impulses driven by a generalized Erlang renewal process, Proc. of 10th IFIP WG 7.5 Working Conference on Reliability and Optimization of Structural Systems, 25-27 March 2002, Kansai University, Osaka, Japan. Eds. H.Furuta, M.Dogaki and M.Sakano, Balkema,103-110, 2003.

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Comparative Study of RBDO Algorithms Based on FORM and FAMM H. Y. Kang*, Y. H. Lee†, J. S. Huh†, and B. M. Kwak† *Korea Advanced Institute of Science and Technology 373-1 Guseong-dong, Yusoeng-gu Daejeon, 305-701 Republic of Korea [email protected] † Korea Advanced Institute of Science and Technology 373-1 Guseong-dong, Yusoeng-gu Daejeon, 305-701 Republic of Korea [email protected], [email protected], [email protected]

ABSTRACT In reliability based design optimization (RBDO), one of main concerns is extracting more information from fewer calculations because the computational effort in RBDO is very time consuming. General RBDO problems consist of a double loop, containing, sensitivity analysis of probabilistic constraints and an optimization of the given RBDO problems. To reduce the computational effort, there have been studies to decouple a double loop into single loops. SLSV and SORA belong to a single loop approach. The authors have tried to reduce computational effort developing several new methods. FAMM is one of theses methods and based on the first four moments estimated from a polynomial interpolant of the system response function. The function approximation is based on a specially selected experimental region for accuracy, and the number of function evaluations taken is equal to that of the un-known coefficients for efficiency. In this work, four RBDO methodologies using FORM are compared based on some criteria and a modified SORA method is presented. FAMM is also compared with SORA. The criteria selected are accuracy, efficiency, and generality. Three numerical examples are considered.

References [1] J. S. Huh, K. H. Kim, D. W. Kang, D. G. Gweon, and B. M. Kwak, Performance evaluation of precision nanopositioning devices caused by uncertainties due to tolerances using function approximation moment method, Review of Scientific Instruments, published online January 2006.

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Probabilistic analysis and optimization of a fully composite cylinder M. Olivier-Mailhé*, S. Ben Chaabane*, F. Léné†, G. Duvaut*, Stéphane Grihon†† * ESILV, Dept MS, PULV 92916 Paris La Defense Cedex [email protected] †

LM2S 4 place Jussieur, 75252 Paris Cedex 5 [email protected] ††

Airbus France, ESANT 61060 Toulouse Cedex 4 [email protected]

ABSTRACT Due to their high stiffness and low mass, composite materials take more and more importance in the design of plane structures (wings, tail, …). Their extension to the design of fully composite fuselages is under development. The great increase in computer performance contributes to the generalization of the optimization methods to composite structures. However these optimization methods lead to deterministic optimum designs, and do not include the variability of the design parameters. These configurations ignoring the different uncertainties can result in unreliable designs. In our paper we combine a reliability analysis with an optimization one, using the response surface method. A composite cylinder, with longitudinal and transversal stiffeners is considered. The first buckling load is considered as the dimensioning criteria. Firstly a sensitivity analysis with some parameters is performed (material parameters, geometrical parameters, etc …). The number, position and geometry of the longitudinal stiffeners appear to be the one of the influent parameters and are considered in the further calculations. The continuous parameters are supposed to be uncertain with a normal distribution. In order to be consistent with the manufacturing process and to limit the number of parameters, we define 4 zones in the fuselage. The number, position and geometry of the stiffeners are constant by zone. The objective of the optimization problem is to set the failure probability lower than an acceptable value ε. The repartition of the longitudinal stiffeners, given by the number of stiffeners in each region are the main parameters. The algorithm proposed uses the inertial properties of the omega profiles. A maximization of the first critical load will automatically lead to maximize the height of the stiffeners in the buckled zones. The main interest of this methodology is to replace the discreet parameters by continuous ones, which permits to easily use the response surface method. All the optimization and statistical analysis are computed on the response surface obtained by a Latin Hyper-cube sampling and quadratic approximation. The Monte-Carlo simulation and the optimization are immediate as the first buckling load is analytically defined. The work developed in this paper is a practical way to reach a reliable design for the buckling behavior of the structure without enormous cost nor increase of the mass.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Assessment of dynamic behaviour of spot welds with uncertain parameters using genetic algorithms application Adam Martowicz, Lukasz Pieczonka, Tadeusz Uhl AGH University of Science and Technology al. Mickiewicza 30, 30-059 Krakow, Poland [email protected] [email protected] [email protected]

ABSTRACT Since the topics concerning quantifying dynamic behaviour of different mechanical systems are still of engineers’ concern, there is a significant need of developing new numerical approaches dealing with this research area. While discussing those issues, problems of effective modelling of uncertainties can not be neglected. One should know what is the influence of product variability and uncertainty of design parameters on natural frequencies and normal modes of considered systems. In the paper, dynamic behaviour of spot weld joints is studied. This way of metal linking is mostly used in automotive industry and it is very important to find out as much as possible about vibrations of car body in order to constantly improve comfort of travelling [1]. Finite element method is used for spot welds modelling. A several different variants of model displacements are considered. Different model meshes are used in order to quantify the convergences of the solutions. As far as modelling of uncertainties is concerned, a number of varying parameters are taken into account. Thicknesses of joined shell parts as well as material parameters are assumed to be uncertain. Positions of spot welds and their sizes are also not constant and can change their values randomly within specified ranges. There is a quite large set of existing methods which allow to solve dynamic problems and to introduce uncertainties into models. Some of them are being still developed like the transformation method and new approaches appear as combinations of existing ones. In the paper, genetic algorithms application is being tested as the main tool for mentioned research area. It is based on the optimization of global mass and stiffness matrices of mechanical system [2]. The first natural frequency is evaluated and its relationship with the ranges of uncertain parameters is discussed. The reference results are obtained with Monte Carlo Simulation approach, the vertex method, the transformation method and genetic algorithms used directly for optimizing the first natural frequency. Advantages and drawbacks of tested method are presented and conclusions about its applicability for different types of mechanical structures are described.

References [1] P. Lardeur, E. Lacouture, E. Blain, Spot weld modelling techniques and performances of finite element models for the vibrational behaviour of automotive structures. Proc. of Int. Conf. on Noise and Vibration Engineering ISMA2000, Leuven, September 13-15, 2000, pp 387-394. [2] A. Martowicz, J. Pieczara, T. Uhl, Application of genetic algorithm within fuzzy FE analysis. Proc. of NNSC-2005 – Neural Networks and Soft Computing 2005, Krakow, Poland, June 30 – July 2, 2005, pp. 69-70.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Optimisation algorithms for non-deterministic dynamic finite element analysis of imprecisely defined structures Maarten De Munck, David Moens, Wim Desmet and Dirk Vandepitte Katholieke Universiteit Leuven Department of Mechanical Engineering Celestijnenlaan 300B, B-3001 Leuven, Belgium [email protected]

ABSTRACT The finite element method is a useful tool to predict the behaviour of a structure under static and dynamic loads. Reliable finite element analyses can reduce the need for prototype testing and thus reduce the design validation cost and time. In many real life situations however, a deterministic analysis is not sufficient to assess the quality of a design. In a design stage, some physical properties of the model may not be determined yet. But even in a design ready for production, design tolerances and production inaccuracies introduce variability and uncertainty. In these cases, a non-deterministic analysis procedure is required, either using a probabilistic or a possibilistic approach. In the former case, Monte Carlo simulation is best known. In the latter case, interval arithmetic and global optimisation can be used. Unless precautions are taken, the conservatism of interval arithmetic approaches and the computational cost of global optimisation approaches are prohibitively high for practical applications. The authors developed a hybrid (global optimisation and interval arithmetic) interval finite element procedure [1] to predict the bounds on frequency response functions (FRFs) of problems with interval inputs. In a first step, the bounds on the modal parameters are determined using a global optimisation approach. In a second step, the bounds on the FRF are calculated using an interval arithmetic approach. This hybrid approach reduces the conservatism compared to a full interval arithmetic approach and reduces the computational cost compared to a full global optimisation approach. Still, the optimisation of the modal parameters is by far computationally the most expensive step of the hybrid algorithm. Therefore, highly efficient optimisation algorithms are necessary to perform analyses on industrial sized applications. Response surface based optimisation algorithms take advantage of the fact that – using a standard modal FE solver – the computational cost to calculate all modal parameters for all modes of interest is almost equal to the computational cost to calculate only one modal parameter for only one mode. This paper discusses the application of a response surface optimisation algorithm in the context of interval and fuzzy finite element analysis and compares them to classical optimisation algorithms (line search and trust region) on a reference model.

References [1] David Moens and Dirk Vandepitte. An interval finite element approach for the calculation of envelope frequency response functions. International Journal for Numerical Methods in Engineering, 61(14):2480–2507, 2004.

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Relative Importance of Uncertain Parameters in Aerospace Applications Manuel F. Pellissetti, Helmut J. Pradlwarter, Gerhart I. Schuëller Institute of Engineering Mechanics, Leopold-Franzens University Technikerstr. 13, 6020 Innsbruck, Austria, EU [email protected]

ABSTRACT Sophisticated numerical models play a crucial role in the design of aerospatial structures. While there is general agreement that many parameters of these complex finite element models are affected by uncertainty, it is typically not apparent which of the parameters are the most important ones, in terms of their impact on the response quantity of interest. The bruteforce approach, i.e. analysing the impact of each uncertain parameter individually, is unacceptable due to its exorbitant computational cost. A recently developed algorithm, which makes use of the cross-correlation between the uncertain parameters and the response, is capable of quickly filtering out the most important parameters and delivers a parsimonious estimate of the relative importance of the uncertain parameters. In the present contribution, the application to a satellite structure with 120,000 DOF’s and 1,300 uncertain parameters is demonstrated, along with the full verification using the brute-force approach. It is shown that with the proposed algorithm, the required computational efforts are reduced by one order of magnitude.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Stability of Elastic and Viscoelastic Systems Under Stochastic Non-Gaussian Excitation Vadim D. Potapov Department of Structural Mechanics Moscow State University of Means Communication Obraztsov str., 15, Moscow, 127994, RUSSIA [email protected]

ABSTRACT Stability problems of elastic and viscoelastic systems under the action of random loads, first of all of columns, subjected to the longitudinal force, which is a stochastic stationary process, were considered by many authors. A sufficiently thorough survey of these works is contained in the monograph [1]. The greatest number of results were obtained for that case if the stationary process is proposed as a Gaussian white noise. If parametric forces are arbitrary random stationary processes, then the solution of the stability problem becomes significantly more complicated. In such a case mainly sufficient conditions of the almost sure stability were obtained. It should be underlined that the estimation of stability boundaries, which are obtained with help of these criterions, are rather rough. In the present work an effective method for the investigation of the stability of elastic and viscoelastic systems under parametric excitation is suggested. Parametric forces are assumed in the form of stationary non-Gaussian processes. The proposed method is based on the simulation of random processes, on the numerical solution of differential equations, describing the perturbed motion of the considered system, and on the calculation of top Lyapunov exponents. The considered method makes it possible to estimate the almost sure stability and the stability with respect to statistical moments of the different order. Since the closed system of equations for moments of unknowns yj (t) in the case of filtered noise could not be obtained, the method of statistical data p processing is applied. The estimation of moments yj  for the instant tn can be obtained as a result of p statistical average of values yj  derived from the solution of equations, describing the behavior of the considered system, for the enough large number of realizations. Using the procedure, suggested in the work [1], the estimation of the top Lyapunov exponent can be obtained. Results, obtained for Gaussian and non-Gaussian processes, are compared in the case of the almost stability, stability in the mean and in the mean-square. It is important to underline, that results, found for filtered processes, are principally differ from results, corresponding to stochastic processes in the form of Gaussian white noises.

References [1] V. D. Potapov, Stability of Stochastic Elastic and Viscoelastic Systems. Wiley, Chichester, England, 1999.

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Modelling Uncertainty in Mechanical Joint Parameters using Component Modal and Fuzzy Approaches Jos´e M. C. Dos Santos∗ , and Brian R. Mace† ∗ State University of Campinas - UNICAMP Caixa Postal 6122, 13083-970 Campinas, SP - Brazil [email protected] † Institute

of Sound and Vibration Research - University of Southampton Southampton, SO17 1BJ, UK [email protected]

ABSTRACT Built-up structures consist of substructures connected through mechanical joints such as spot welds, rivets, bolts, etc., whose physical properties (stiffness, damping, thickness, etc.) can vary significantly from one structure to another. Uncertainties in these parameters generate uncertainties in the dynamic behaviour of the structure and there is an interest in predicting the variability of the response given the variability of the joint parameters. In this paper, modelling and identification of a system consisting of two substructures connected by mechanical joints modelled by stiffness parameters is evaluated. Each substructure is modelled by a classical finite element method formulation and a modal solution obtained. The resulting modal solutions for the substructures are assembled in a Craig-Bampton component mode synthesis approach, which includes the joint parameters. The stiffness properties of the joints are described by fuzzy-valued parameters. Fuzzy-parameterized models are obtained using advanced fuzzy arithmetic based on the transformation method. Simulation results for this fuzzy model are obtained for different scenarios. Experimental results for a structure comprising two aluminum beams connected by steel wires are presented and compared with the simulated ones.

References [1] R. R. Craig Jr, Substructure Methods in Vibration, Transaction of the ASME, 117, 207–213, 1995. [2] M. Hanss, Applied Fuzzy Arithmetic. Springer, Berlin, 2005.

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Bayesian Model Updating Approach for Ground-Motion Attenuation Relations Enrico Sibilio∗ , James L. Beck† , Matthew Muto† , Marcello Ciampoli‡ ∗ University

of Florence Via S. Marta, 3, 50139, Florence (Italy) [email protected] † California Institute of Technology 1200 E. California Blvd, MC 104-44, Pasadena, CA 91125 (USA) [email protected], [email protected] ‡ University of Rome ”La Sapienza” Via Eudossiana, 18, 00184, Rome (Italy) [email protected]

ABSTRACT The Bayesian model updating procedure is a powerful and general approach to update the uncertainties in a model response by using the information from available data. It is based on the well-known theorem of Bayes, which states that a posterior (updated) probability distribution for the model parameters conditioned on the data available is proportional to the product between the prior probability distribution and the likelihood function. Herein, the problem of establishing empirical earthquake groundmotion attenuation laws is addressed; in particular, two kinds of regression models are considered. The model parameters are represented by the coefficients of the regression models, and the data consist of a database of strong-motion records from actual earthquakes, where the magnitudes and epicentral distances are known. The computational aspects related to Bayesian updating and predictions are discussed. Two Markov Chain Monte Carlo (MCMC) simulation techniques are used to sample from the posterior distribution. They are based on some modifications of the Metropolis-Hastings procedure. The first one is a resampling adaptive scheme, whereas the second one is a Hybrid Monte Carlo algorithm with Simulated Annealing. The problem of the identifiability of two ground-motion relations is discussed as well. A Bayesian model comparison for the two attenuation relationships is carried out by evaluating the evidence of the two model classes based on the strong-motion data. Finally, the samples obtained by employing the MCMC procedures are used to perform a robust predictive analysis that takes into account all the uncertainties involved in the evaluation of each class of attenuation laws. The objective is to establish a probability distribution for an intensity measure representing the ground motion, conditioned on the magnitude and epicentral distance. The knowledge of such conditional distributions is an important step for the seismic hazard definition in a seismic risk assessment. The Bayesian updating procedure presented has a wide domain of application, including structural model updating.

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Iterative solution of the random eigenvalue problem Clemens V. Verhoosel∗ , Miguel A. Guti´errez∗ , Steven J. Hulshoff∗ ∗ Faculty

of Aerospace Engineering, Delft University of Technology P.O. Box 5058, 2600 GB Delft, The Netherlands [email protected] [email protected] [email protected] ABSTRACT

Algebraic eigenvalue problems play an important role in a variety of fields. In structural mechanics, eigenvalue problems commonly appear in the context of, e.g., vibrations and buckling. In the case that the considered structure as well as the considered loading conditions are exactly known, i.e. deterministic, efficient and robust methods for the computation of eigenvalues and corresponding eigenvectors exist. In the more realistic case that the structure and the loading conditions are uncertain (described by random fields), eigenvalues and eigenvectors will also be uncertain. These random eigenvalues and random eigenvectors can then be determined by solving the random eigenvalue problem, for which the availability of efficient and robust methods is limited. In this contribution an uncertainty analysis is performed for the random eigenvalue problem. The purpose of uncertainty analysis is to determine the statistical moments (mean, standard deviation, etc.) of the eigenvalues and eigenvectors. A projection on a spectral basis of Hermite polynomials is used to determine the statistical moments of the eigenvalues. The Galerkin method [1] as used for linear algebraic problems turns out to yield an ill-posed system of equations. To avoid this problem, the deterministic inverse power method for the computation of eigenvalues is extended to yield the stochastic inverse power method. The stochastic inverse power method consists of two steps. In the first step the eigenvector is updated using inverse iteration. In this step Galerkin’s method is used to determine the projection of the eigenvector on a spectral basis. In the second step the stochastic eigenvalue is updated using the Rayleighquotient. This sequence is repeated until both the stochastic eigenvalue and stochastic eigenvector have converged. The deterministic eigenvalue (with a small variation to avoid singularity problems) can be used as an initial setting for this iterative process. The convergence of the iterative procedure can be controlled by partially updating the eigenvector. Numerical damping is then added to the system, preventing the solution from oscillating. In the case of moderate variations, the stochastic inverse power method turns out to be a robust method for the computation of the eigenvalues and eigenvectors of symmetric matrices. In the case of non-symmetric matrices, possible convergence problems can appear due to the fact that the spectral basis is not capable of spanning the (possibly non-differentiable) exact solution of the random eigenvalue problem. The accuracy of the stochastic inverse power method (compared to a Monte-Carlo simulation) is demonstrated using numerical examples for the symmetric and non-symmetric problem.

References [1] R.G. Ghanem and P.D. Spanos, Stochastic Finite Elements: A Spectral Approach. Springer-Verlag, New York, 1991.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Thermo-electro-mechanical coupling in beam-to-beam contact Daniela P. Boso*, Przemysław Litewka†, Bernhard A. Schrefler*, Peter Wriggers‡ *

Department of Structural and Transportation Engineering, University of Padua Via Marzolo 9, 35131 Padua, Italy [email protected] [email protected] †

‡

Institute of Structural Engineering, Poznan University of Technology ul. Piotrowo 5, 60-965 Pozna´n, Poland [email protected]

Institute of Structural and Computational Mechanics, University of Hanover Appelstr. 9A, 30167 Hannover, Germany [email protected]

ABSTRACT This paper is a further step towards the full analysis of the complex behaviour of contacting beams in the coupled thermo-electro-mechanical field. Now we add to our previous work the coupling with the thermal field. The coupling between electric, mechanical and thermal fields is manifested in: dependence of material parameters on the temperature, frictional heating, heat generation by the electric current flow, dependence of the contact area on the temperature, dependence of thermal and electric fields on the change of the contact area and the contact point position. In the present preliminary analysis we take into account only the last aspect. We also consider the indirect influence of thermal contact on the mechanical field using the thermo-mechanical beam element. The contact constraints are enforced with the penalty method within the finite element technique. It is assumed that the heat flow in the contact area is unlimited which leads to the equalling of temperatures between two contacting bodies. This constraint is also introduced by the penalty method. The set of governing equations including the coupling is solved by the monolithic scheme. The problem is nonlinear, hence the linearized version of equations is derived. The consistent linearization leading to the consistent tangent stiffness matrix and the corresponding residual vector is performed to apply efficiently the Newton-Raphson method and to ensure the quadratic convergence. Some numerical examples are presented to show the efficiency of the suggested approach.

References [1] G. Zavarise, P. Wriggers, Contact with friction between beams in 3-D space, International Journal for Numerical Methods in Engineering, 49, 977-1006, 2000. [2] P. Litewka, P. Wriggers, Frictional contact between 3D beams. Computational Mechanics, 28(1), 26-39, 2002. [3] D.P. Boso, P. Litewka, B.A. Schrefler, P. Wriggers, A 3D beam-to-beam contact finite element for coupled electric-mechanical fields: International Journal for Numerical Methods in Engineering, 64(13), 1800-1815, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Numerical analysis of a dynamic frictional viscoelastic contact problem with damage M. Campo , J.R. Fern´andez , K.L. Kuttler† and M. Shillor‡ Departamento de Matem´atica Aplicada, Universidade de Santiago de Compostela, Facultade de Matem´aticas, Campus Sur s/n, 15782 Santiago de Compostela, Spain macampo,[email protected] † Department

‡ Department

of Mathematics, Brigham Young University, Provo, UT 84602, USA [email protected]

of Mathematics and Statistics, Oakland University, Rochester, MI 48309, USA [email protected]

ABSTRACT In this talk we present the numerical analysis of a dynamic problem which models the bilateral contact between a viscoelastic body and a foundation, taking into account the damage and the friction. The damage, which measures the density of the microcracks in the material and results from tension or compression ([1]), is then involved in the constitutive law (see [2] for details), and modelled using a nonlinear parabolic inclusion. The variational problem is formulated as a coupled system of evolutionary inequalities for which we state the existence of a unique weak solution. Then, we introduce a fully discrete scheme using the finite element method to approximate the spatial variable and the Euler scheme to discretize the time derivatives. Error estimates are derived and, under suitable regularity assumptions, the linear convergence of the numerical scheme is deduced. Finally, numerical results are presented for some two-dimensional examples in order to show the accuracy of the algorithm.

References [1] M. Fr´emond and B. Nedjar, Damage, gradient of damage and principle of virtual work. Internat. J. Solids Structures, 33 (8), 1083–1103, 1996. [2] M. Shillor, M. Sofonea and J. J. Telega, Models and analysis of quasistatic contact. Lecture Notes in Physics, 655, Springer, Berlin, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

308

hp-mortar Boundary Element Method and FE/BE coupling for multibody contact problems with friction. A. Chernov∗ , M. Maischak∗ , E. P. Stephan∗ ∗ Institute

of Applied Mathematics, University of Hanover Welfengarten 1, D-30167 Hannover, Germany {chernov,maischak,stephan}@ifam.uni-hannover.de ABSTRACT

The hp-methods is a very efficient and accurate tool in modern computational mechanics. For a wide range of problems hp-techniques provide the exponential convergence rate of the discrete solution to the exact solution, while h−version and p−version give only an algebraic convergence rate. The finite element method is used commonly for numerical simulations of contact problems [3]. The boundary element techniques are relatively seldom used, despite some significant advantages [1]. In the boundary element method only the boundaries of the bodies are discretized, which reduces the dimension of the problem by one. This simplifies e.g. mesh generation significantly. Also, the number of unknowns in the problem is reduced greatly, but in contrast to finite elements, the Galerkin matrix will be dense due to nonlocal boundary integral operators. We introduce a new hp-BEM mortar technique for multibody contact problems with friction. Often, it is very convenient to use independent discretizations of the bodies, subjected to their particularities. In case of nonmatched meshes on the contact interface, independent refinement of the bodies can be applied. Following the approach of [1] we impose contact constraints on the discrete global set of affinely transformed Gauss-Lobatto points on the single elements. The data transfer is realized in terms of mortar projection. The problem is reformulated as a variational inequality with the Steklov-Poincar´e operator of the second kind over a convex cone of admissible solutions. We obtain an upper error bound in the energy norm. Due to nonconformity of our approach, the error is decomposed to the approximation error and the consistency error. Finally we show that the discrete solution converges to the exact solution as O((h/p)1/4 ). We solve the discrete problem employing the Dirichlet-to-Neumann algorithm. The original two-body formulation is rewritten as a one-body contact problem and a one-body Neumann problem [2]. Then the global problem is solved employing fixed point iterations. We give numerical examples for the two-body contact problem with friction which show efficiency and accuracy of our approach.

References [1] M. Maischak, E. P. Stephan A FEM-BEM coupling for a nonlinear transmission problem modelling Coulomb friction contact. Comput. Methods Appl. Mech. Engrg. 194, 453–466, 2005 [2] A. Chernov, S. Geyn, M. Maischak, E.P. Stephan, Finite Element/Boundary Element Coupling for Two-body Elastoplastic Contact Problems with friction. Proceedings of 4th Contact Mechanics International Symposium, Hannover, July 4-6, 2005, submitted. [3] P. Wriggers, Computational Contact Mechanics. Wiley, Chichester, 2002.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Thermoelastic Wheel - Rail Contact Problem with Temperature Dependent Friction Coefficient ´ † Andrzej Chudzikiewicz∗ , Andrzej My´slinski ∗ Institute

of Transport, Warsaw University of Technology ul. Koszykowa 75, 00 - 662 Warsaw, Poland [email protected] † Systems Research Institute ul. Newelska 6, 01 - 447 Warsaw, Poland [email protected]

ABSTRACT The paper deals with the numerical solution of wheel - rail rolling contact problems. The contact of a rigid wheel with an elastic rail resting on a rigid foundation is considered. The friction between the bodies is assumed to be described by the Coulomb law. Moreover, due to friction, the heat generation and as well as the occurrence of wear are assumed. The wear phenomenon in the contact zone is governed by Archard’s law. This contact problem either as an elastic static problem or a dynamic viscoelastic problem has been investigated by many authors (see [1, 3]). In literature this problem was solved assuming constant friction coefficient. Numerous experiments indicate [2] that at least in the range of higher temperatures the friction coefficient decreases with the increase of the temperature. This paper is concerned with this rolling contact problem where the friction coefficient is assumed to be piecewise linearly dependent on the temperature. The quasistatic approach [1] to solve this contact problem is used. This approach is based on the assumption that for the observer moving with the rolling body the displacement of the supporting foundation is independent on time. The bodies in contact are described by the coupled elliptic variational inequality governing the displacement field and Poisson equation governing the heat flow. To solve numerically this system we will decouple it into mechanical and thermal parts. Using duality theory approach and the regularized relation between tangent and normal components of the contact stress we formulate the mechanical part of this problem as an optimization problem with respect to the normal contact stress. Conjugate gradient method combined with Newton method are used to solve numerically this optimization problem. Next for the calculated displacement field the wear zone is calculated and the thermal part of the system is solved. Numerical examples are provided and discussed. The changes of the temperature distribution and the wear zone due to change of the friction coefficient are reported.

References [1] A. Chudzikiewicz, A. My´sli´nski, Comparison of Numerical Methods for Solution of Thermoelastic Wheel Rail Contact Problems, In: Proceeding of the Conference ”Computer Methods in Mechanics”, Cze¸stochowa, June 21 25, Poland, 2005. [2] V.L. Popov, S.G. Psakhie, E.V. Shilko, et al., Friction Coefficient in Rail - Wheel Contacts as a Function of Material and Loading Parameters. Physical Mesomechanics, 5, No 3, 17 – 24, 2002. [3] M. Ertz, K. Knothe, A Comparison of Analytical and Numerical Methods for the Calculation of Temperatures in Wheel/Rail Contact. Wear, 253, pp. 498 – 508, 2002.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Analysis of a dynamic contact problem with adhesion and friction in viscoelasticity Marius Cocou , Michel Raous, Mathieu Schryve Laboratoire de M·ecanique et d Acoustique CNRS 31 chemin Joseph Aiguier 13402 Marseille Cedex 20, France [email protected] [email protected] [email protected] and Universit·e de Provence, UFR - MIM Marseille, France ABSTRACT In this paper, the interface law coupling adhesion, friction and unilateral contact, proposed in [5] for elastic bodies in a quasistatic evolution, is extended and considered in the case of dynamic contact for viscoelastic bodies. We give continuum thermodynamic and mathematical formulations of the dynamic contact problem with adhesion and nonlocal friction between two viscoelastic bodies of Kelvin-Voigt type. Its variational formulation is written as the coupling between an implicit variational inequality and a differential equation describing the evolution of the intensity of adhesion, that represents the transition from a total adhesive condition to a pure contact condition [3, 4]. The technique used to study some dynamic contact problems with nonlocal friction for viscoelastic bodies [1] and for a cracked viscoelastic body [2] is developed in order to analyze this new class of coupled problems. Finally, some numerical examples are presented.

References [1] M. Cocou, Existence of solutions of a dynamic Signorini s problem with nonlocal friction in viscoelasticity. Z. Angew. Math. Phys., 53, 1099–1109, 2002. [2] M. Cocou, G. Scarella, Analysis of a dynamic unilateral contact problem for a cracked viscoelastic body. Z. Angew. Math. Phys., to appear. [3] M. Fr·emond, Equilibre des structures qui adherent a leur support. C.R. Acad. Sci. Paris, Ser. II, 295, 913–916, 1982. [4] M. Fr·emond, Adh·erence des solides. Journal de M´ecanique Th´eorique et Appliqu´ee, 6, 383–407, 1987. [5] M. Raous, L. Cang·emi, M. Cocou, A consistent model coupling adhesion, friction, and unilateral contact. Comput. Methods Appl. Mech. Engrg., 177, 383–399, 1999.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Influence of the frp strengthening, the shape and the movement of abutments on the collapse of arch stone bridges Georgios A. Drosopoulos∗ , Georgios E. Stavroulakis† , Christos V. Massalas∗ ∗ Department

of Material Science and Technology, University of Ioannina GR-45100 Ioannina, Greece [email protected] (G. A. Drosopoulos), [email protected] (C. V. Massalas) † Department

of Production Engineering and Management, Technical University of Crete GR-73132, Chania, Greece and Department of Civil Engineering, Technical University of Braunschweig Braunschweig, Germany [email protected]

ABSTRACT The ultimate failure load of stone arch bridges is calculated in this paper by using finite element analysis. Contact interfaces simulating potential cracks are considered. Opening or sliding of a number of the potential interfaces indicates crack initiation [1]. Fiber Reinforced Plastic (FRP) strips are then applied to the stone bridge and the ultimate load is recalculated. The failure modes of the reinforced arch are compared well with the ones received from relevant experiments published in the literature. The analysis of the unreinforced arch shows that, under the most critical quarter span loading, a four - hinges collapse mechanism arises. The occurrence of this type of collapse is indicated in the classical work of Heyman [2] and has been observed in experiments. Three types of FRP reinforcement are applied in the arch. In particular, FRP is attached to the whole extrados, to the whole intrados and both to the extrados and the intrados of the arch. A cap model is used in order to model the failure of the masonry due to compression, while a v. Mises yield criterion is used for the FRP yielding. The possible failure modes of the reinforced structure are sliding of the masonry, crushing, debonding of the reinforcement and FRP rupture. Identical failure modes arise from the computer simulation and from experiments on reinforced arches published in the literature. A parametric investigation of the influence of the geometry of the unreinforced stone arch on the mechanical behavior of the structure is briefly described. Finally the effect of support settlement (vertical or horizontal movement) on the limit behavior is investigated. Common remarks with Heyman’s work arise. Further results and more details on the theory and the algorithms are given in the PhD Thesis of the first author [3].

References [1] B.P. Leftheris, M.E. Stavroulaki, A. Sapounaki, G.E. Stavroulakis, Computational Methods for Heritage Structures. WIT Press, Southampton, 2006. [2] J. Heyman, The Masonry Arch. Ellis Horwood, England, 1980. [3] G.A. Drosopoulos, Innovatige Methods for the Analysis and Reinforcement of Masonry Bridges. PhD Dissertation, University of Ioannina, Greece, (to appear).

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Sweeping process for vibro-impact problem with a general inertia operator Raoul Dzonou ∗ , Manuel Monteiro Marques † , Laetitia Paoli ∗ ∗

LaMUSE Universit´e Jean Monnet, 23 rue du Docteur Paul Michelon, 42023 Saint Etienne Cedex 2, France, {raoul.dzonou,laetitia.paoli}@univ-st-etienne.fr. †

CMAF and Faculdade de Ciencias Universidade de Lisboa, Av Prof. Gama Pinto 2 1649-003 Lisboa, Portugal, [email protected]. ABSTRACT

We consider a mechanical system with a finite number of degrees of freedom submitted to a perfect unilateral constraint. We suppose that the contact is frictionless and the local impact law consists in the transmission of the tangential component of the impulsion and the reflexion of the normal component which is multiplied by the restitution coefficient e ∈ [0, 1]. We assume that the inertia operator is state-dependent in a Lipschitz continuous way and we follow J.J. Moreau’s description of the dynamics as a measure differential inclusion:
q(t) ∈ L = {q ∈ Rd : g(q) ≤ 0} (1) q˙+ (t)+eq˙− (t) ) f (t, q, M (q)q)dt ˙ − M (q)dq˙ ∈ ∂IV (q(t)) ( 1+e  V (q) =  ∂IV (q) (y) =

{v ∈ Rd : ∇g(q).v ≤ 0} Rd otherwise.

if

g(q) ≥ 0

{x ∈ Rd ; < x, z − y >≤ 0 ∀z ∈ V (q)} if y ∈ V (q) ∅ otherwise,

(2)

where q denotes the representative point of the system in generalized coordinates and M (q) is the inertia operator. We propose the following numerical scheme: for given initial data (q0 , u0 ) ∈ L × V (q0 ), we let qn,0 = q0 , un,0 = u0 , h = nτ , n ∈ N ∗ and for all i ∈ {0, ..., n} ⎧ ⎨ qn,i+1 = qn,i + hun,i (3) u = projqn,i+1 (un,i + hM −1 (qn,i+1 )f˜n,i+1 , V (qn,i+1 )) ⎩ ˜n,i+1 fn,i+1 = f (tn,i+1 , qn,i+1 , M (qn,i+1 )un,i+1 ), where projq (v, V (q)) denotes the projection of v on V (q) with respect to the kinetic metric at q defined by the inertia operator M (q). We prove the convergence of the scheme to a solution of the Cauchy problem by using the ’sweeping process technique’. Some numerical simulations for a double pendulum model are also presented.

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Spectral element simulations of rupture dynamics along planar and kinked frictional faults Gaetano Festa1 and Jean-Pierre Vilotte1 1

Institut de Physique du Globe de Paris 4, Place Jussieu, 75252 Paris Cedex 05, France [email protected],[email protected]

ABSTRACT Understanding earthquake source dynamics is a major topic in seismology. Earthquake faulting is mainly controlled by non-regular friction, describing the dissipation within the fault interface, by the geometrical complexity of the fault and by the off-fault damage interactions. Recent observations during the Denali and Izmit earthquakes have shed evidence for supershear propagation in relation with the fault geometry. Numerical simulations of earthquake rupturing can bridge the gap between laboratory experiments and observations of large earthquakes. The simulations are expected to capture the different space and time scales involved in the nucleation phase, the rutpure front propagation and the short wave radiation, owing to the fault heterogeneities and geometrical complexities. Non-smooth spectral element method allows for efficient simulations of dynamic rupture along planar and non-planar faults. We present here recent numerical results on sub- and supershear propagation along bending and kinked faults. In particular, we focus the attention on the subshear-to-supershear transition and on the interaction between fault geometry and high-frequency radiation emitted by the fault. This work is a part of the SPICE European Project

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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r-Factor Strategies for the Augmented Lagrangian Approach in Multi-Body Contact Mechanics Martin Foerg∗ , Thomas Geier∗ , Lutz Neumann∗ , Heinz Ulbrich∗ ∗

Institute for Applied Mechanics Technical University of Munich Boltzmannstr. 15 85748 Munich, Garching [email protected] ABSTRACT

Multi-body system theory including unilateral constraints is now well established by means of setvalued force laws in connection with measure equations of motion. The crucial point consists in the numerical solution of such equations, especially when dealing with large systems as they often appear in practical problems of industrial relevance. Therefore the improvement of numerical algorithms is a focus of ongoing research. In the meantime there are different approaches and algorithms in order to formulate and compute unilateral constrained mechanical systems. Besides (N)LCP-formulations the Augmented Lagrangian approach [1] becomes more and more popular in contact mechanics. Within this approach the equations of motion are augmented by projection equations representing the physical constraints. The overall set of non-smooth, nonlinear equations can be solved by a root-finding algorithm, e.g. a fixed-point iteration scheme. The projection equations depend on a non-negative auxiliar parameter r. Though this parameter r is arbitrary from the mathematical point of view, it plays a crucial role in view of the rootfinding method. In particular, the problem of finding an optimal r-factor turns out to be a constrained optimization problem: on the one hand the parameter r may be bounded to ensure the convergence of the algorithm, on the other hand an appropriate choice improves the rate of convergence. In the present paper two different r-factor strategies are presented considering a fixed-point iteration scheme in order to find the root of the Augmented Lagrangian. The first strategy proposes one global r-factor for all constraint equations. The second strategy considers local r-factors, i.e. a different parameter for each constraint. In both cases the conditions for convergence are given and an optimal choice of r is proposed. The paper discusses the treatment of planar and spatial contacts as well as systems that are statically indeterminate, where a unique solution for the constraint forces does not exist. The presented r-factor strategies are applied to several non-smooth systems including a push belt CVT [2]. This large industrial problem is characterized by a hybrid multi-body model with a large number of unilateral and bilateral constraints.

References [1] A. Klabring, Mathematical Programming and Augmented Lagrangian Methods for Frictional Contact Problems. Proceedings Contact Mechanics International Symposium, October 7-9, EPFL, Lausanne, Switzerland, 409–422, 1992. [2] T. Geier, M. Foerg, R. Zander, H. Ulbrich, F. Pfeiffer, A. Brandsma, A. Van der Velde Modeling of contacts in a push belt CVT. Second International Conference on Nonsmooth/Nonconvex Mechanics with Applications in Engineering, Thessaloniki, Greece, 2006 (to appear)

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A simple smoothing procedure of 3D surfaces for accurate contact analysis: application to metal forming problems. L. Fourment*, S. Guerdoux* *

CEMEF, Ecole des Mines de Paris, UMR CNRS n°7635 rue Claude Daunesse, B.P. 207, 06 904 Sophia Antipolis Cedex, France [email protected]

ABSTRACT In most finite element software, it is very convenient to discretise complex obstacles with segments in 2D and facets in 3D. It results in C0 continuity of the obstacle and consequent discontinuity of its normal, which is at the origin of convergence difficulties of algorithms, of numerical oscillations with exaggerated stresses or unjustified loss of contact. Several methods have been proposed in literature to smooth the contact obstacle. They are based on Bezier surfaces [1], Gregory patches [2], or similar methods, which are not easy to implement into an existing finite element code. In [3] a more simple method was proposed for concave parts of the obstacle. For a point M sliding along the obstacle surface, there is a discontinuity of the normal between two segments (or two facets). If we allow a small penetration of M inside the obstacle, as in the following figure (left), and if the normal is defined by the direction MP where P is the orthogonal projection of M onto the obstacle, then the normal varies continuously when M slides on the obstacle surface; the obtained smoothing is proportional to the authorized penetration. In order to avoid this numerical penetration, the discretised contact obstacle is first shifted in the opposite direction, as shown in the following figure (right), with the same amplitude. This way, when M “penetrates” the shifted surface, it actually moves along the actual contact surface, except in the smoothing area.

Symmetrically, this method can be applied to convex parts of the surface. The focus of this paper is on how to combine the smoothing of concave and convex parts. It actually requires shifting the obstacle discretisation in two opposite directions and carrying out a contact analysis with each. It so provides two possible normals, which combination gives an almost continuously varying normal on the contact surface. The method is applied to some metal forming problems with large deformations, such as machining, forging and friction stir welding.

References [1] L. Krstulovi-Opara, P. Wriggers, J. Korelec, A C1-continuous formulation for 3D finite deformation frictional contact, Comput. Mech. 29 n° 1, (2002) 27-42. [2] M. A. Puso, T. A. Laursen, A 3D contact smoothing method using Gregory patches, Int.J.for Numerical Methods in Engineering 54 n° 8, (2002) 1161-1194. [3] L. Fourment, J. L. Chenot, K. Mocellin, Numerical formulations and algorithms for solving contact problems in metal forming simulation, Int J Numer Methods Eng 46 n° 9, (1999) 14351462.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Symmetry preserving algorithm for a dynamic contact-impact problem Duˇsan Gabriel∗ , Jiˇr´ı Pleˇsek∗ , Frantiˇsek Valeˇs† , and Miloslav Okrouhl´ık∗ ∗ Institute of Thermomechanics Academy of Sciences, Praha, Czech Republic [email protected], [email protected], [email protected] † Institute of Thermomechanics Academy of Sciences, Plzen, Czech Republic [email protected]

ABSTRACT In the finite element method, the contact constraints can be introduced either before or after the finite element discretization has been performed, leading to the so-called pre-discretization or postdiscretization techniques [1]. In the paper [2] we focused on the pre-discretization approach, showing this technique to lead naturally to the use of surface integration points as contactors. It was shown that the proposed method preserved the symmetry of the algorithmic approximation with respect to contact boundaries. On the outcome there was nothing like a master or slave definition of contact surface. In this work the expected symmetry of algorithm was tested. The observation or loss of symmetry is clearly manifested in numerical step-by-step procedures, for instance, in the direct integration methods for the equations of motion. Three-dimensional face to face impact of two cylinders was considered as in the Taylor test. Comparisons of the proposed algorithm with the analytical solution and the finite element code MARC were made. The analytical solution of this problem using the Laplace transform is quite complicated [3]. The distribution of displacements and stresses are expressed in the form of an infinite series of improper integrals which are evaluated numerically. A good agreement between the numerical and analytical solution derived for the summation of the first 150 terms of the series was observed. Next, the results were compared to the output of the finite element code MARC, in which the implemented contact algorithm [4] was based on the node-to-segment procedure. The numerical results of the proposed method and MARC s computation, with the central difference scheme, were almost the same. However, a lack of symmetry in all MARC s kinematic and stress quantities was observed if the implicit Newmark integration method had been used. It should be emphasized that the symmetry is perfectly preserved in the proposed algorithm.

References [1] N. Kikuchi, J.T. Oden, Contact problems in elasticity: A study of variational inequalities and finite element methods. SIAM, Philadelphia, 1988. [2] D. Gabriel, J. Plesek, M. Ulbin, Symmetry preserving algorithm for large displacement frictionless contact by the pre-discretization penalty method. Int. J. Num. Met. Engng, 61, 2615–2638, 2004. [3] F. Vales, S. Mor·avka, R. Brepta, J. Cerv, Wave propagation in a thick cylindrical bar due to longitudinal impact. JSME Int. J., Ser.A, 39(1), 60–70, 1996. [4] MARC Analysis Research Corporation. Volume A, Theory and User Information, version K7.3, 1998.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Problems of Concentrated Loads in Microstructured Solids Characterized by Dipolar Gradient Elasticity H.G. Georgiadis * , D.S. Anagnostou Mechanics Division, National Technical University of Athens, GR-15773, Greece (* Corresponding author) [email protected]

ABSTRACT This work studies the response of bodies governed by dipolar gradient elasticity to concentrated loads. The use of the theory of gradient elasticity is intended here to model material microstructure and incorporate size effects into stress analysis in a manner that the classical theory cannot afford. A simple but yet rigorous version of the generalized elasticity theories of Toupin [1] and Mindlin [2] is employed that involves an isotropic linear response and only one material constant (the so-called gradient coefficient) additional to the standard Lame constants. This theory, which can be viewed as a first-step extension of the classical elasticity theory, assumes a strain-energy density function, which besides its dependence upon the standard strain terms, depends also on strain gradients [3]. Twodimensional configurations in the form of either a half-space (Flamant-Boussinesq type problem) or a full-space (Kelvin type problem) are treated and the concentrated loads are taken as line forces. The problems enjoy important applications in various areas, e.g., in Contact Mechanics and Tribology. Also, the Flamant-Boussinesq and Kelvin solutions serve as pertinent Green’s functions in a multitude of problems analyzed by the Boundary Element Method. Our main concern here is to determine possible deviations from the predictions of classical linear elastostatics when a more refined theory is employed to attack the problems. Of special importance is the behavior of the new solutions near to the point of application of the loads where pathological singularities exist in the classical solutions. The solution method is based on integral transforms and is exact. The present results show departure from the ones of the classical elasticity solutions. Indeed, bounded displacements are predicted even at the points of application of the loads. Such a behavior of the displacement fields seems to be more natural than the singular behavior present in the classical solutions. Acknowledgment: This paper is a partial result of the Project PYTHAGORAS II / EPEAEK II (Operational Programme for Educational and Vocational Training II) [Title of the individual program: “Micro-mechanics of contacts and diffusion of humidity in granular geomaterials”]. This Project is co-funded by the European Social Fund (75%) of the European Union and by National Resources (25%) of the Greek Ministry of Education.

References [1] R.A. Toupin, Elastic materials with couple-stresses. Arch. Rational Mech. Anal., 11, 385-414, 1962. [2] R.D. Mindlin, Micro-structure in linear elasticity. Arch. Rational Mech. Anal., 16, 51-78, 1964. [3] H.G. Georgiadis, I. Vardoulakis and E.G. Velgaki. Dispersive Rayleigh-wave propagation in microstructured solids characterized by dipolar gradient elasticity, J. Elasticity, 74, 17-45, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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An Experimentally Validated Model for Unsteady Rolling F. Gutzeit∗ , M. Wangenheim∗ , and M. Kr¨oger∗ ∗ Institute of Dynamics and Vibrations Appelstr. 11, 30167 Hannover, Germany [email protected]

ABSTRACT The description of the tyre road contact is important for brake or drive stability systems working in modern vehicles. Steady models loose accuracy, if slip or normal force contain higher frequencies. This is the case during an ABS brake process. At the Institute of Dynamics and Vibrations, a mobile test rig has been built up to investigate unsteady rolling. A small solid rubber wheel, which is applied for wear experiments in the tyre industry, serves as the specimen. Any given time characteristics of slip and normal force can be realized by the control modules. The forces and moments, as well as the temperatures of the wheel and the road are recorded. Based on the experiments, a numerically efficient model for the unsteady rolling contact was developed, see [1]. The mechanical model describes the dependency on the slip ν(t) for constant normal force FN . The numerical efficiency is achieved by applying a modal condensed formulation of the wheel structure. Furthermore, the contact area is discretized by point contact elements representing the local contact behavior. The model will be revised in order to be able to include time depending normal forces FN (t). The HurtyCraig-Bampton reduction is applied to access the contact nodes during the simulation. The method allows to condense modally the remaining nodes. The model divides the contact zone into a sticking and a sliding area. Within the sticking area, the displacements of the contact nodes are forced according to the kinematic conditions of the wheel. Within the sliding area, the local friction force is calculated for every node. After deformations and external forces have been calculated, appropriate sticking and sliding conditions are proved to compute the division of the contact patch for the next time step. In many respects, rubber is a highly nonlinear material. The strong temperature dependency must be considered in many applications. By means of an infrared camera, thermal images of the contact zone are taken. An analytical approach is adapted to the case of a solid rubber wheel and finally compared to the experimental data. The temperature dependence on the rolling contact behavior is discussed. The results of the modified model are shown and compared to experimental data recorded by the friction robot. The influence of the global slip on the sticking to sliding transition is shown.

References [1] Gutzeit, F., Sextro W., Kr¨oger M.: Unsteady Rolling Contact of Rubber Wheels, 4th Contact Mechanics International Symposium (CMIS), Hannover, Germany, 2005, in press.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A primal-dual active set strategy for unilateral non-linear dynamic contact problems of thin-walled structures Stefan Hartmann ∗, Stephan Brunssen†, Ekkehard Ramm∗ , Barbara Wohlmuth† ∗ Institute of Structural Mechanics University of Stuttgart, Pfaffenwaldring 7, D-70550 Stuttgart, Germany hartmann/[email protected]

† Institut f¨ur Angewandte Analysis und Numerische Simulation Universit¨at Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany brunssen/[email protected]

ABSTRACT The efficient modeling of 3D contact problems is still a challenge in non-linear implicit structural analysis. Most of the existing contact algorithms use penalty methods to satisfy the contact constraints, which necessitates a user defined penalty parameter. As it is well known, the choice of this additional parameter is somehow arbitrary, problem dependent and influences the accuracy of the analysis. We use a primal-dual active set strategy [1], based on dual Lagrange multipliers [4] to handle the nonlinearity of the contact conditions. This allows us to enforce the contact constraints in a weak, integral sense without any additional parameter. Due to the biorthogonality condition of the basis functions, the Lagrange multipliers can be locally eliminated. We perform a static condensation to get a reduced system for the displacements. The Lagrange multipliers, representing the contact pressure, can be easily recovered from the displacements in a variationally consistent way. For our application to thin-walled structures we adapt a three-dimensional non-linear shell formulation, including the thickness stretch of the shell to contact problems. A reparametrization of the geometric description of the shell body gives us a surface oriented shell element, which allows to apply the contact conditions directly to nodes lying on the contact surface. The discretization in time is done with the implicit Generalized Energy-Momentum Method [2]. To conserve the total energy within our contact framework, we follow an approach from Laursen and Love [3], who introduce a discrete contact velocity to update the velocity field in a post processing step. Various examples show the good performance of the primal-dual active set strategy applied to the implicit dynamic analysis of thin-walled structures.

References [1] M. Hinterm¨uller, K. Ito, K. Kunisch, The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim., 13: 865–888, 2003. [2] D. Kuhl, E. Ramm, Generalized Energy-Momentum Method for non-linear adaptive shell dynamics. Computer Methods in Applied Mechanics and Engineering, 178: 343–366, 1999. [3] T.A. Laursen, G.R. Love, Improved implicit integrators for transient impact problems - geometric admissibility within the conserving framework. International Journal for Numerical Methods in Engineering, 53: 245–274, 2002. [4] B. Wohlmuth, A mortar finite element method using dual spaces for the Lagrange multiplier. SIAM J. Numer. Anal., 38: 989–1012, 2000.

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Arc-Length Method for Frictional Contact with a Criterion of Maximum Dissipation of Energy Yoshihiro Kanno∗ , Jo˜ao A.C. Martins† ∗ Department

of Urban and Environmental Engineering, Kyoto University Sakyo, Kyoto 606-8501, Japan [email protected]

† Departamento

de Engenharia Civil and ICIST, Instituto Superior T´ecnico Avenida Rovisco Pais, 1049-001 Lisboa, Portugal [email protected]

ABSTRACT In this paper, we propose an arc-length equilibrium path-following method for quasi-static frictional contact problems incorporating a criterion of maximum dissipation of energy, which is applicable to cases in which there exist critical points along the equilibrium path. The Coulomb friction law and the unilateral contact condition are considered. It is well known that the frictional contact problems may have limit points and successive stable and/or unstable bifurcation points [1], even if small rotations and small strains are assumed. This implies that the corresponding incremental problem does not have unique solution in general. Moreover, this problem often has bifurcated paths such that most sliding contact nodes become stuck and the loading parameter decreases, which are referred to as trivial unloading paths. Our aim is to propose a path-following method that can automatically avoid tracing trivial unloading paths, which seem to be uninteresting from the practical point of view. To this end, we attempt to follow the path with the maximum dissipation of energy when the corresponding incremental problem has some solutions. At each loading stage, the incremental displacements and the reactions are obtained by solving a mathematical program with complementarity constraints (MPEC). Algorithms that do not have any criterion to select among multiple solutions may compute trivial unloading solutions. A regularization scheme of the MPEC is also proposed. In contrast with the fact that the original MPEC fails to satisfy any standard constraint qualification, it is shown that the regularized problem satisfies the linear independence constraint qualification at a feasible solution. This implies that, in the inner iteration of the arc-length method, we can solve the proposed regularized problem by using the conventional nonlinear programming approach. It has been shown in the numerical examples that the proposed method can automatically avoid tracing trivial unloading paths, even when there exist successive bifurcation points due to friction and/or some limit points.

References [1] J.A.C. Martins, A. Pinto da Costa, and F.M.F. Sim˜oes, Some notes on friction and instabilities. In: Friction and Instabilities, J.A.C. Martins and M. Raous (eds.), Springer–Verlag, pp. 65-136, 2002.

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A energy conserving approximation for elastodynamic contact problems H. Khenous∗ , P. Laborde† , Y. Renard∗ ∗ MIP-INSA Complexe scientifique de Rangueil, 31077 Toulouse cedex4, France {khenous, renard}@insa-toulouse.fr † MIP, UPS 118 route de Narbonne, 31062 Toulouse cedex4, France [email protected]

ABSTRACT We are interested in the numerical solution of contact problem in elastodynamics. We present the strong formulation of problem and give the finite element discretization [2]. We prove that the last formulation is not a well posed problem. In order to overcome this difficulty, we propose then an original method based on a redistributed mass matrix. This new mass matrix is assembled conserving the total mass, the gravity center and the momentum inertia. The discrete elastodynamic contact problem expressed with the redistributed mass matrix is well posed, is energy conserving and has a Lipschitz continuous solution[1]. Finally, some numerical results are presented to coroborate the theoritical results. Simulations are done with and without the redistributed mass matrix for a Newmark scheme. We remark that the behaviour of the energy and the normal stress are improved using this new method. The energy is quasi-conserved and will be strictly conserved when time parameter goes to zero [1].

References [1] H.B. K HENOUS , P. L ABORDE & Y. R ENARD. Comparison of two approaches for the discretization of elastodynamic contact problems. Accepted for CRAS paris, 2006. [2] H.B. K HENOUS , J. P OMMIER & Y. R ENARD. Hybrid discretization of the Signorini problem with Coulomb friction. Theoretical aspects and comparison of some numerical solvers, Applied Numerical Mathematics, 2006, Vol 56/2 pp 163-192.

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On models of contact surfaces including anisotropy for friction and adhesion and their experimental validations. Alexander Konyukhov∗ , Karl Schweizerhof† , Peter Vielsack
∗ Institut f¨ur Mechanik Englerstrasse 2, D-76131, Karlsruhe, Germany [email protected] † Institut f¨ ur Mechanik [email protected]
Institut f¨ ur Mechanik [email protected]

ABSTRACT Smoothness and isotropy of contacting body surfaces can vary considerably for different contact problems. Classifying the surfaces roughness two types can be distinguished: a) surfaces with randomly distributed asperities, and b) asperities with algorithmic structure, e.g. the considered surface shows different macro properties in different directions. Mechanical characteristics for the associated contact problems of the first type a) are obtained via statistically distributed asperities. Constitutive modeling is applied for problems of the second type b). Such models are based on the generalization of Coulomb’s friction law into the anisotropic domain, see Zmitrowicz [1] and Curnier [2]. When looking at practical problems concerning friction there are some situations in which the tangential elasticity of the contact surfaces should be taken into account. Such a model including anisotropy for both friction and adhesion has been developed and analyzed numerically in Konyukhov and Schweizerhof [3]. In the current contribution we discuss the validation of this model with a particular experimental test. The contact surfaces are chosen to possess elastic properties, thus a corrugated rubber mat is taken. The results of the experiments show the necessity to use the model including anisotropy for both friction and adhesion. Thus, some originally surprising experimental phenomena, as e.g. geometrical isotropy despite obvious physical anistropies can be explained only within the proposed model, though the latter shows rather qualitative correlations then quantitative ones. It was shown in experiments that the classical model of orthotropic friction does not lead to the good correlation and cannot describe a particular phenomena when a sliding block shows isotropic behavior. A good qualitative result can be achieved with the model involving both orthotropy for adhesion and friction.

References [1] Zmitrowicz, A. A theoretical model of anisotropic dry friction. Wear. 73 (1981) 9–39. [2] Curnier, A. A theory of friction. International Journal of Solids and Structures. 20 (1984) 637– 647. [3] Konyukhov A., Schweizerhof K. Covariant description of contact interfaces considering anisotropy for adhesion and friction. Part 1. Part 2. Computer Methods in Applied Mechanics and Engineering. (submitted).

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Fast and Robust Solution Methods for Dynamic Contact Problems Rolf H. Krause∗ ∗ Institute

for Numerical Simulation, University of Bonn Wegelerstraße 6, D-53115 Bonn [email protected] ABSTRACT

For the numerical solution of dynamic contact problems, often implicit time discretization methods are used for stability reasons [2], giving rise to a nonlinear contact problem to be solved in each time step. We use a globally convergent non-smooth multigrid method which allows for solving multibody contact problems in linear elasticity as fast as linear elliptic problems. This multigrid method can be extended to the case of frictional contact problems leading to a nonlinear method for frictional contact problems with multigrid efficiency [1]. Using nonconforming domain decomposition methods, a stable discretization for the transfer of displacements and stresses at the interface between the bodies coming into contact can be developed, which is also capable of handling complicated geometries. In the framework of a Newmark-based time discretization scheme, this method is used for the construction of an efficient implicit time discretization scheme. Due to the inequality constraints at the contact interface, the time integration of dynamic contact problems often gives rise to oscillations in displacements and stresses. To remove these oscillations, we consider a stabilization based on a global L2 -projection which can be interpreted as a solution dependent correction of the velocities. The stabilization leads to an additional variational inequality to be solved in each time step, which can be done efficiently using our monotone multigrid method. We discuss the properties of the resulting method and illustrate its performance for a contact problem in biomechanics.

References [1] K. Fackeldey, R. Krause, Solving Frictional Contact Problems with Multigrid Efficiency Proceedings of the 16th International. Conference on Domain Decomposition Methods, to appear [2] A. Pandolfi, C. Kane, J. E. Marsden, M. Ortiz, Time–discretized Variational Formulation of non-smooth Frictional Contact International Journal for Numerical Methods in Engineering, 53, 1801–1829, 2002

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A Velocity-Based Time-Stepping Method for Frictional Dynamics Manuel D. P. Monteiro Marques∗ , Laetitia Paoli† ∗ C.M.A.F.,

Faculdade de Ciencias da Universidade de Lisboa Av. Prof. Gama Pinto, 2; 1649-003 Lisboa; Portugal [email protected]

† Universite de Saint-Etienne 23, rue Michelon; 42023 Saint-Etienne cedex 2; France [email protected]

ABSTRACT The dynamics of a mechanical system (a particle or a rigid body) submitted to one unilateral constraint possibly with friction may be formulated as a so-called measure differential inclusion. This was shown by J. J.Moreau (see e.g. [2]), who also introduced and used with great success several numerical timestepping methods for a variety of progressively more complex problems. A convergence proof and existence result may be found in [1], which concerns the case of a constant inertia matrix, equal to the identity. Here, we extend that study, by considering general state-dependent inertia matrices as well as anisotropic Coulomb friction. A convergence result for the corresponding time-stepping scheme, velocity-based ”a la Moreau”, is obtained. A thorough analysis of the limit friction law, including the situations leading to the well-known Painleve’s ”paradoxes”, is also given. For the single constraint case, the present study also extends and completes the results of D. E. Stewart [3].

References [1] M.D.P. Monteiro Marques, Differential inclusions in non-smooth mechanical problems: shocks and dry friction. Birkhauser, Boston, 1993. [2] J. J. Moreau, Unilateral contact and dry friction in finite freedom dynamics. Nonsmooth Mechanics and Applications (J. J. Moreau and P. D. Panagiotopoulos, eds.), Springer, New York, 1988. [3] D. E. Stewart, Convergence of a time-stepping scheme for rigid body dynamics and resolution of Painleve’s paradoxes. Arch. Rational Mech. Anal., 145, 215–260, 1998.

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Mechanical Modelling of Friction and Adhesion of Elastomers at Rough Interfaces T. Meyer, A. Le Gal, M. Klüppel Deutsches Institut für Kautschuktechnologie e. V., Eupener Straße 33, D-30519 Hannover [email protected]

ABSTRACT An advanced model of dynamic contact and sliding friction of elastomers at rough, self-affine interfaces is presented. It describes the frictional force via the dissipated energy, resulting from sliding stochastic excitations of the rubber by surface asperities on various length scales. The effect of surface roughness is considered by three surface descriptors: the fractal dimension and two cut-off lengths, which are obtained from a fractal analysis of the surface via stylus- or laser measurements. The hysteresis response of the rubber enters through viscoelastic master curves of the complex modulus up to high frequencies. Based on this concept stationary friction curves are estimated numerically over a broad velocity scale depending upon surface roughness and temperature. They are compared to experimental friction data found for filled elastomer systems. The obtained results provide a deeper insight into the role of adhesion forces under different contact conditions, e. g. by applying various lubricants. The investigations are found to be useful for a better understanding of the traction behavior of tires on dry and wet roads during ABS-braking of passenger cars. References [1] [2] [3] [4] [5] [6]

M. Klüppel and G. Heinrich, Rubber Chem. Technol. 73, 578 (2000); ibid. Paper No. 43, ACS Rubber Division Meeting, Chicago, 13.-16. May (1999) A. Le Gal, X. Yang and M. Klüppel, Journal of Chemical Physics, 123, 014704, 2005 M. Klüppel, A. Müller, A. Le Gal and G. Heinrich, “Dynamic contact of tires with road tracks“, Paper No. 49, ACS Meeting, San Francisco, April 28-30 (2003) J.A. Greenwood, J.B.P. Williamson, Contact of nominally flat surfaces, Proc. Of the Royal Soc. London, A 295, 300, 1966 H. Hertz, “Miscellaneous Papers”, Macmillan, London, 1896, p. 146 A. Le Gal, M. Klüppel, Kautschuk Gummi Kunststoffe, submitted (2006)

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A 9m Drop Test Simulation of a Dual Purpose Cask for Nuclear Research Reactors Spent Fuel Elements Carlos A. J. Miranda*, Miguel Mattar Neto†, Gerson Fainer†† CNEN-IPEN/SP - Nuclear Engineering Center Av. Prof. Lineu Prestes, 4492 - São Paulo, SP, Brazil * [email protected] † [email protected] †† [email protected]

ABSTRACT The qualification of casks for transportation or storage of nuclear spent fuel elements involves the evaluation of some conditions that simulate possible accidents. The cask should maintain its safety functions through its structural and functional integrity (in any condition, there should be the containment of the radioactive products inside it, the integrity of its biological shielding and assurance against criticality). The main conditions the cask should satisfy, mainly by test, to be qualified are: a 9m drop test against a rigid surface, a penetration test, 30min of fire under 800 oC and 200m immersion during one hour. The first condition is the most critical one. The regulatory bodies stress the qualification “by test” instead of “by analysis”. However, numerical simulations are important to determine, for instance, the most critical position for the free drop tests, saving a lot of money without reducing the project degree of safety. There is a multi-country project, sponsored by the IAEA, with the participation of Latin American countries with research reactors, to develop and qualify a shipping cask for their spent fuel elements. It involves, in its first phase, the project, construction, test and numerical simulation of a half scale model to establish parameters for the tests (mostly the 9m drop test). The cask is a stainless steel cylinder with flat heads, the bottom one is welded while the upper one has flanged threaded connections, and internal structures for the fuel elements. An external stainless steel cylinder contains the biological lead shielding. There are two impact limiters contained by steel shells, which are planned to be filled with a reconstituted wood. This work describes the cask project in details, the main hypothesis and some results obtained with the 9m drop test numerical simulation. Its purpose is to develop a modeling and results evaluation methodology to help the field tests, in order to be applied in future prototype design. In the simulations all non-linearities, mostly associated with the contacts among the cask several parts, the mechanical properties of the materials and the geometric changes, were considered.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A Discontinuous Galerkin Approach for the Numerical Treatment of Tractive Rolling Udo Nackenhorst∗ , Matthias Ziefle† ∗ Institute

of Mechanics and Computational Mechanics, University of Hanover Appelstr. 9A, 30167 Hannover, Germany [email protected]

† Institute

of Mechanics and Computational Mechanics, University of Hanover Appelstr. 9A, 30167 Hannover, Germany ziefl[email protected] ABSTRACT

Rolling contact problems are described in an Arbitrary Lagrangian Eulerian (ALE) kinematics for a numerically efficient treatment, see e.g. [1]. By this approach the motion is split into a pure rigid body motion, which is described in Eulerian coordinates, and into the deformation measured in Lagrangian coordinates relative to the formerly obtained reference configuration. One advantage of this relative kinematics framework is, that steady state rolling is described independent of time. An additional advantage concluded from the spatially fixed mesh is that a local mesh refinement can be introduced to the contact region as needed for a detailed contact analysis. However, one difficulty arises when inelastic material behavior has to be taken into account. Because within the ALE–framework the mesh points are neither fixed in space nor associated with the material particles, the path–lines of the particles have to be traced. This is solved within a fractional step approach, by which in a first (Lagrangian) step the local evolution of the inelastic variables is solved and in a second step the history is advected onto the motion of the particles. I has been proven that Time Discontinuous Galerkin methods possess of optimal stability and convergence behavior for the numerical solution of the related advection equations [2]. A further problem lies in the numerical solution of frictional contact, which yet seems not been solved satisfactory as more recently discussed in [3]. In this presentation on the basis of the experience made for the treatment of inelastic material properties within the ALE–framework of rolling, a fully implicit scheme for the computation of the spatial slip distribution is suggested. By a weak formulation of the stick conditions the spatial distribution of the slip–distances is computed directly. This enables the application of established implicit schemes for frictional contact. Numerical studies on rather simple examples so far show optimal convergence rated.

References [1] U. Nackenhorst, The ALE-Formulation of Bodies in Rolling Contact – Theoretical Foundations and Finite Element Approach. Computer Methods in Applied Mechanics and Engineering, 193(39–41), 4299–4322, 2004. [2] M. Ziefle and U. Nackenhorst, A new update procedure for internal variables in an ALEdescription of rolling contact. Proceedings in Applied Mathematics and Mechanics (PAMM), 5, 71–74, 2005. [3] T. A. Laursen and I. Stanciukescu, An algorithm for incorporation of frictional sliding conditions within a steady state rolling framework, Communications in Numerical Methods in Engineering, in press

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An iterative method with BEM discretization for the friction contact problems P. Neittaanm¨aki∗ , A.S.Kravchuk† , I.G.Goryacheva† ∗ Department of Mathematical Information Technology P.O. Box 35 (Agora), FIN-40014 University of Jyv¨askyl¨a, Finland [email protected].fi † Department

of Mechanics and Mathematics, Moscow State University 119899 Moscow State University, Moscow, Russia {kravchuk biocom, goryache}@mail.ru ABSTRACT

A new formulation for the friction contact problem is proposed. The feature of this formulation consists of using of the relative velocities in the friction law, and of taking into account the reciprocal influence of the normal and tangential components of contact stresses. This require a new constructions of the impenetrability condition and, secondly, to use the step–by–step type methods for the modelling of evolution of the contact forces. With a new definition of the kinematically admissible displacements and velocities including the unilateral constraints the local problem is transformed to a variational one, which is revealed as a quasi– variational inequality. To solve this quasi–variational inequality, a new iterative method is given. This method is based on the calculation of a solution increment using the calculation of the contact pressure from the previous iteration. Such a idea permits to reduce the problem to the sequence of a saddle–point problem. The convergence theorem is demonstrated. Numerical algorithm is based on the boundary elements approach. Numerical results, obtained for the 2D problems, give an estimates for the difference between the solutions corresponding to the different impenetrability conditions, and to different quantity of the steps with respect to a parameter defining the evolution of the system of contacting bodies. In particular, it is demonstrated that the normal displacements and contact pressure are defined with a good precision for all the formulations of the impenetrability condition, and for one step of the evolution parameter. But the analysis of the friction forces and relative sliding phenomena requires the non–linear (strict) impenetrability condition, and several steps for loads. This conclusion play a very important role for the fretting–wear investigation and for the reliability and life–time prediction. Analytical solution was obtained with the method developed in [1]. The work was partly supported by the grant No 05-01-00591a of the Russian Foundation for the Fundamental Researches, and TEKES MASI program.

References [1] I.G.Goryacheva, Contact Mechanics in Tribology. Kluwer, Dordrecht, 1998. 99

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Frictional Contact of Elastomer Materials on rough rigid Surfaces Jana Nettingsmeier∗ , Peter Wriggers∗ ∗

Institut f¨ur Baumechanik und numerische Mechanik Universit¨at Hannover Appelstraße 9a 30167 Hannover [email protected] ABSTRACT

Within the analysis of many technical problems with frictional contact, Coulombs law is used, which implies a constant friction coefficient µ. This assumption is sufficient for many applications in structural mechanics; however in the special case of rubber friction on rough surfaces the resulting simplification cannot be accepted. The physical interactions between tire and road surface are very complex and still widely unknown. As it is apparent from experiments, the friction coefficient depends heavily on various parameters like sliding velocity, surface roughness, normal forces and temperature change. Our aim is now to derive a realistic friction law based on micromechanical observations. It can be shown that the energy dissipation in a rubberlike viscoelastic material implies a frictional behaviour for the whole specimen, even if we neglect any predetermined local friction on the microscale. The friction coefficient advances to zero for both very high and low sliding velocities, but reaches a maximum for middle speed. This effect is known as hysteretic friction and represents the main part of rubber friction. Due to the multiscale character of the surface roughness, we have to model the problem on different length scales. Therefore the frictional behaviour on the microscale is analyzed and the homogenized characteristics result in a friction law, which is projected onto the macroscopic problem in a staggered procedure. The fractal road surface is approximated by a superposition of several harmonic functions. The three-dimensional numerical problem will be modeled with a four-node contact element. Its G AUSS points are projected to the closest point on the rigid surface, which is given as an analytical function z = f (x, y). Even if we can use frictionless contact on the finest scale, we need to transport the frictional constitutive behaviour to the next scale. The friction coefficient will be adapted to the local conditions (normal stress, sliding velocity, etc.) obtained in the contact element. In addition to the described approach for pure hysteretic friction, adhesional effects between the surfaces will be included into the contact model. It has to be analyzed how the transmission of tensile stresses influences the global frictional behaviour of rubber. The properties of elastomer materials are sensitive to temperature changes. Rising temperatures have the same effect as a decreasing load frequency and vice versa. Because of this relation it is necessary to determine the magnitude of material heating due to internal energy dissipation.

References [1] B. Persson Theory of Rubber Friction and Contact Mechanics. Journal of Chemical Physics, 115,8, 2001. ´ [2] M. Raous, L. Cangemi, M. Cocu, A consistent model coupling adhesion, friction and unilateral contact. Computer Methods in Applied Mechanics and Engineering 177: 383-399, 1999. [3] P. Wriggers, Computational Contact Mechanics. Wiley, 2002.

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Optimizing the Description of Forming Tools with Bézier Surfaces in the Numerical Simulation of the Deep Drawing Process M.C. Oliveira*, J.L. Alves †, L.F. Menezes * *

CEMUC, Department of Mechanical Engineering, University of Coimbra Pinhal de Marrocos, Polo II, 3030 Coimbra, Portugal [email protected]; [email protected] †

Department of Mechanical Engineering, University of Minho Campus de Azurém, 4810 Guimarães, Portugal [email protected]

ABSTRACT

In the simulation of the deep drawing process one of the challenges is the correct prediction of the actual contact surface and kind of contact established between the tools and the blank sheet, since they determine the boundary conditions. In the numerical simulation of the deep drawing process the contact conditions change continuously, increasing the importance of a correct evaluation of these parameters in each load step. In the finite element implicit code DD3IMP [1], devoted to the simulation of the deep drawing process, the tools are modeled with Bézier parametric surfaces, for which a frictional contact algorithm has been developed and continuously improved [1]. The description of the tools by parametric surfaces allows the direct use of the information provided by CAD software. However, in order to guarantee the contact algorithm convergence it is necessary to impose some continuity conditions between the surfaces that define the forming tools. In many situations the tools are defined by a set of plane surfaces connected by fillings with surfaces of constant or variable radius. To guarantee the correct tools’ modeling it is necessary to correctly model the curve that defines the radius. Previous works had shown that, to assure a correct correlation between tools’ geometry and Bézier description at least cubic segments should be used to define the radius [2]. The Bézier tools description must accurately represent the tools’ geometry, but also satisfy the continuity conditions that guarantee the contact algorithm convergence. In this work the influence of the degree of the polynomial functions used in the Bézier surfaces on the contact algorithm is evaluated. Also the error in the and continuity of the Bézier surfaces obtained with the CAD system is estimated, and its influence on the convergence of the contact algorithm is studied.

References [1] M.C. Oliveira, J.L. Alves and L.F. Menezes, Improvement of a Frictional Contact Algorithm for Strongly Curved Contact Problems. International Journal for Numerical Methods in Engineering, 58, 2083-2101, 2003. [2] M.C. Oliveira e L.F. Menezes, Optimização da Descrição das Ferramentas por Superfícies de Bézier na Simulação do Processo de Estampagem. Proceedings of the V Congresso Métodos Numéricos en Ingeniería, Ed. J.M. Goicolea, C. Mota Soares, M. Pastor and G. Bugeda, 2002 (CD-ROM edition).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Exploring the dynamics of a simple system involving Coulomb friction Elaine Pratt, Alain L´eger CNRS, laboratoire de M´ecanique et d’Acoustique 31, chemin Joseph Aiguier, 13402 Marseille cedex 20, France [email protected] [email protected]

ABSTRACT The work presented here consists in an exhaustive study of a simple mass-spring system involving Coulomb friction. The aim was to gain some insight into the behaviour of a chain of masses in frictionnal contact. If it is simple to explicit the analytical solution of a single mass system, the analytical solution for a two mass system is already far more complicated. We thus consider two masses linked by a spring in bilateral contact with Coulomb friction and submitted to an external force applied onto one of the masses. The existence, uniqueness and regularity of the dynamics of the system is established through a recent paper [1]. Once the uniqueness is ensured it is simple to exhibit the explicit solution for certain values of the external force (i.e. when the amplitude of the force is either small or large). When the amplitude of the external force belongs to a certain intermediate range the dynamics turns out to be more interesting. The solution can be calculated analytically, however as the computation becomes rapidly tiresome, we use a symbolic calculus tool to compute a solution corresponding to a given external force. We thus observe that: • for a given value of the amplitude of the external force, the two masses oscillate for a certain time before coming to rest (these oscillations are obviously not periodic because of the non linearity due to the Coulomb friction), • for increasing values of the amplitude of the external force, the number of oscillations that the masses carry out before coming to a halt, grows each time that the amplitude of the external force passes through a critical value, • these critical values of the amplitude of the external force accumulate as the amplitude of the force reaches the upper bound of the interesting range. Extending these results to more than two masses may prove to be not such a simple task but some progress has been made and shall be presented.

References [1] P. Ballard et S. Basseville, Existence and uniqueness for dynamical unilateral contact with Coulomb friction: a model problem. Mathemetical Modelling and Numerical Analysis, Vol. 39, 1, 57-77, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Multibody Modeling of Pantographs for Catenary-Pantograph Interaction Frederico Grases Rauter1, João Pombo2, Jorge Ambrósio2, Manuel Seabra Pereira2 1

2

SNCF – Direction de l’Innovation et de la Recherche 45 Rue de Londres, 75008 Paris, France [email protected]

IDMEC – Instituto Superior Técnico, Technical University of Lisbon Av. Rovisco Pais, 1049-001, Lisboa, Portugal {jpombo,jorge,mpereira}@dem.ist.utl.pt

ABSTRACT In the great majority of railway networks the electrical power is provided to the locomotives by the pantograph-catenary system. From the mechanical point of view, the single most important feature of this system consists in the quality of the contact between the contact wire(s) of the catenary and the contact strips of the pantograph. Therefore not only the correct modeling of the catenary and of the pantograph must be achieved but also a suitable contact model to describe the interaction between the two systems must be devised. The work proposed here aims at enhancing the understanding of the dynamic behavior of the pantograph and of the interaction phenomena in the pantograph-catenary system. The potential contribution of this work to the railway community includes the decrease of the number of incidents related to this system and the reduction of the maintenance and interoperability development costs. The catenary system is described by a detailed finite element model of the complete subsystem while the pantograph system is described by a detailed multibody model. The dynamics of each one of these models requires the use of different time integration algorithms. In particular the dynamics of the finite element model of the catenary uses a Newmark type of integration algorithm while the multibody model uses a Gear integration algorithm, which is variable order and variable time step. Therefore, an extra difficulty that arises in study of the complete catenary-pantograph interaction concerns the need for the co-simulation of finite element and multibody models. As the gluing element between the two models is the contact model, it is through the representation of the contact and of the integration schemes applied for the finite and multibody models that the co-simulation is carried on. The work presented here proposes an integrated methodology to represent the contact between the finite element and multibody models based on a continuous contact force model that takes into account the co-simulation requirements of the integration algorithms used for each subsystem model. The discussion of the benefits and drawbacks of the proposed methodologies and of its accuracy and suitability is supported by the application to the real operation scenario considered and the comparison of the obtained results with experimental testing data. In the process future developments that include representing the pantograph model with flexible bodies, non-linear force elements and the inclusion of wear and cross-wind effects are also discussed.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Comparisons and coupling of algorithms for collisions, contact and friction in rigid multibody simulations Mathieu Renouf† and Vincent Acary∗ †

LaMCoS - TMI team CNRS/INSA de Lyon - UMR5514 18-20, rue des sciences F69621 VILLEURBANNE cedex - France [email protected] ∗ Bipop Project, INRIA Rhone–Alpes ZIRST Montbonnot, 655, avenue de l’Europe, 38664 Saint ISMIER, France [email protected]

ABSTRACT Numerous works in computational mechanics are dedicated to multi-body systems [1, 2]. This leads to the use of various methods to simulate the static or dynamic evolution of complex systems. The case of dense multi-contact assemblies is one of the more complex one: the problem have often a large number of unknown and have a infinity of solution due to the definition of the matrix of the system. Moreover this problem become harder when friction or more complex laws are introduced in the system. Thus we need fast and robust solvers to perform mechanical studies. These performances can be increased when the special problem structure is considered (sparse matrices, block structured problem). Our work is based on the Non Smooth Contact Dynamic framework introduced by Moreau [3]. The method uses a time-stepping integrator without explicit event-handling procedure and an unilateral contact impact formulation associated to Coulomb’s friction. In this paper we use and compare different iterative algorithms such as Gauss-Seidel, projected conjugate gradient and direct ones as Lemke and Quadratic programming solvers [4]. The efficiency of the methods is compared in terms of complexity, convergence criterion and of CPU time. To illustrate the results, we focus on granular assemblies. 3D frictional contact simulations are performed with ConF&TiS and the Numerics library of the siconos project.

References [1] M. Jean and J. J. Moreau, Unilaterality and dry friction in the dynamics of rigid bodies collection. In: Contact Mechanics International Symposium, A. Curnier ed., 1992. [2] C. Glocker and F. Pfeiffer. Multibody dynamics with unilateral contacts. John Wiley and Sons, 1996. [3] J.-J. Moreau, Numerical aspects of the sweeping process. Comp. Meth. Appl. Mech. Engrg, 177:329–349, 1999. [4] R. W. Cottle, J.-S. Pang and R. E. Stone, The linear complementarity problem. Academic Press, Inc., 1992.

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Third body flow during wheel-rail interaction Mathieu Renouf, Aur´elien Saulot, Yves Berthier TMI Team - LaMCoS/INSA de Lyon - UMR5514 18-20, rue des sciences, F69621 VILLEURBANNE cedex - France {Mathieu.Renouf,Aurelien.Saulot,Yves.Berthier}@insa-lyon.fr ABSTRACT In a mechanism, when a contact occurs between two contactors, third body [1] is the generic name used to describe the material generated as a result of the contact interaction. Thus the wear phenomenom can be considered as the third body flow Qw definitely ejected from the contact area. Because the wear phenomenom involves as well the global scale (wheel and rail) as the local one (contact interface), the study of this phenomenom via numerical tools needs both continuum and discrete approaches [2, 3]. We propose here a two-dimensional analysis of the third body flow induced by the relative transverse

sliding motion between wheel and rail [2](Vp represents the micro transerve periodic velocity and F the normal load). Both Finite Element and Discrete Element Methods [4] are used. At the continuum level we take into account an elastoplastic behaviour for the two bodies. At the local level we use the Non Smooth Contact Dynamic method developped by Moreau and Jean [5], using non-smooth contact laws to describe body interactions. We use a non-smooth cohesive law to describe the behaviour of the third body.

References [1] M. Godet. The third-body approach : a mechanical view of wear. Wear, 100:437–452, 1984. [2] A. Saulot, S. Descartes, D. Desmyter, D. Levy and Y. Berthier. A tribological characterization of the ”damage mechanism” of low rail corrugation on sharp curved track. Wear, In Press. [3] N. Fillot, I. Iordanoff, and Y. Berthier. Simulation of wear through a mass balance in a dry contact. ASME J. Tribol.,127(1):230–237 [4] A. Munjiza The combinated finite-discrete element method. John Wiley and Sons, 2004. [5] M. Jean and J. J. Moreau. Unilaterality and dry friction in the dynamics of rigid bodies collection. In: Contact Mechanics International Symposium, A. Curnier ed., 1992.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Modelling thermal contact resistance on glass forming processes with special interface finite elements José César de Sá†, Sébastien Grégoire*, Philippe Moreau*, Dominique Lochegnies* †

Faculty of Engineering of the University of Porto Rua Dr. Roberto Frias, s/n, 4200-465 Porto, Portugal [email protected] *

Laboratoire d’Automatique, de Mécanique et d’Informatique Industrielles et Humaines – UMR CNRS 8530 Université de Valenciennes – Le Mont-Houy – Jonas 2 59313 Valenciennes Cedex 9 – France [email protected] , [email protected] , [email protected]

ABSTRACT The forming process of glass containers is a complex coupled thermal/mechanical problem with interaction between the heat transfer analysis and the viscous flow of molten glass. The transfer of heat and change of viscosity are fundamental phenomena in this process. The changes in temperature influence the very process of heat transfer since the thermal properties of glass change with temperature. On the other hand the great dependence of glass viscosity with the temperature influences dramatically the flow of the material and therefore the final product. The successive changes in shape produced by gravity and blow pressure, which depend on the actual properties that are influenced by temperature, affect subsequently the heat transfer process. The thermal contact between the glass and the metal moulds influences dramatically the glass thickness distribution of the final product. The heat flux at the interface is a function of the contact pressure, of the temperatures of both the glass and the mould, of the glass viscosity and of the presence of asperities and lubricants in the mould. Another decisive aspect for obtaining a good final product is the chosen procedure for the cooling of the moulds. In the modelling of theses processes with the finite element method a moving mesh, attached to the deforming glass, deals with the mechanical and thermal problems in the glass. Due to the low pressures involved only the heat transfer problem is addressed in the moulds that are therefore discretized with a fixed mesh. Consequently the thermal contact is dealt with non-matching meshes. A classical master/slave strategy may not be always adequate as there may be zones in the mould, where, due to large curvatures, the mesh is more refined than in the glass (typically the slave). A heat transfer contact element is therefore proposed, inspired on the contact “mortar method” developed for mechanical contact. Its formulation is obtained from a variational formulation in which the thermal contact is imposed in a penalised form, in which the penalty term is a function of the heat transfer coefficient. In the interface elements for some nodes the temperatures are constrained to have the same temperatures as the corresponding ones in the mesh of the glass and the surface temperatures are interpolated from the temperatures in the nodes of the mould. In other nodes of the interface elements the temperatures are interpolated from the node temperatures in the glass and the surface temperatures are the corresponding ones node in the mould. The heat transfer coefficient is evaluated for each node from the contact pressure and viscosity evaluated at the glass nodes. As a result an over constrained problem is avoided and a more effective thermal contact is obtained.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Numerical Investigation of Shakedown Residual Stresses Under Moving Surface Loads Jim S. Shiau Lecturer, Faculty of Engineering and Surveying University of Southern Queensland, Toowoomba, QLD, 4350, Australia [email protected]

ABSTRACT It can be shown theoretically that there is a load magnitude below which a protective residual stress will develop in a rolling and sliding contact of continuum structure, and above which it will undergo an incremental failure. This load is known as the `shakedown limit load' and the protective residual stresses associated with this shakedown limit load are the optimal residual stresses for the life of the structure. In his "Contact Mechanics" book, Professor Johnson described an analytical shakedown approach to predict the shakedown load limit and the associated residual stress distribution. This problem will be revisited in this paper using a numerical method proposed in the author's PhD Thesis. A numerical formulation based on Bleich-Melan shakedown theorem will be discussed by making use of finite element techniques and mathematical programming. The proposed numerical procedure can be used to solve for the shakedown limit load and the associated developed residual stresses of structures subjected to repeated moving surface loads. A series of results on shakedown residual stresses will be examined.

References [1] Jim S. Shiau (2001) “Numerical methods for shakedown analysis of pavements under moving surface loads". Ph.D. Thesis. The University of Newcastle, NSW, Australia [2] K. L. Johnson (1985) “Contact Mechanics”. Cambridge University Press.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Frictional Contact/Impact between a Hyperelastic Body and Moving Rigid Obstacles N. Str¨omberg∗ ∗ J¨ onk¨oping

University, Department of Mechanical Engineering P.O. Box 1026, 551 11 J¨onk¨oping, Sweden [email protected]

ABSTRACT In this paper a method for frictional contact/impact between a hyperelastic body and moving rigid obstacles is suggested and investigated. The work is a further development of the suggested method in [1]. The motion of an obstacle is defined by a prescribed translation vector and a prescribed rotation matrix. The geometry of the obstacles are defined by smooth functions. Each function is formulated in a moving frame which is governed by the translation vector and the rotation matrix. These functions are then included in new formulations of Signorini’s conditions and Coulomb’s law of friction. Instead of using contact forces, the mean value impulses are utilized in these formulations, which also are adopted in the law of motion which is given on velocity form. By following this approach, no search algorithm is needed, the normal and tangential directions are well defined and the treatment of non-constant transformation matrices in the law of motion is straight-forward. A total Lagrangian formulation of the system is given. The elastic properties of the body are defined by coupling the second Piola-Kirchhoff stress to the Green-Lagrange strain via the Kirchhoff-St.Venant law. The governing equations are solved by a nonsmooth Newton method. This is performed by following the augmented Lagrangian approach and deriving the consistent stiffness matrix as well as the contact stiffness matrices. The method is implemented in TriLab. TriLab is a user-friendly finite element toolbox for simulating contact and impact problems. TriLab is developed using Matlab and Visual Fortran. The Fortran code is linked to Matlab as mex-files. The code is vectorized and the sparsity is utilized. By using Trilab, the presented method will be demonstrated by solving two-dimensional problems.

References [1] N. Str¨omberg, An implicit method for frictional contact, impact and rolling, European Journal of Mechanics, A/Solids, 24, 1016–1029, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Incorrect Contact of Screw Surfaces and its Consequences Jaromír Švígler University of West Bohemia in Pilsen, Department of Mechanics, Czech Republic [email protected]

ABSTRACT The contribution deals with the analysis of the correct and incorrect contact of the screw surfaces which are represented with two-dimensional manifolds. One surface is defined by parametric equation and the conjugate surface is created in the direct envelope way according to the Distelli theorem. The solution which is presented in this contribution is founded on kinematical principles. The correct contact of both conjugate surfaces takes place in a contact line ch32, see Figure, for instantaneous time.

Correct line contact of conjugate screw surfaces σ 2 and σ 3 Temperature and force deformation bring about displacement of surfaces and the correct contact changes into the incorrect contact. Instead of the line contact of surfaces the contact at isolated points takes place. In this contribution the incorrect contact of surfaces is caused by a large parallel displacement of the axis of one of the surfaces. The inner deformation of screw surfaces is not involved into solution and likewise general position of rotor axes that arises by real deformation is not accepted. This simplification was necessary for the first step of the solution of this problem. The work is solved as three-dimensional problem. Theory of incorrect contact of screw surfaces is applied to the teeth of rotors of the screw machines. One of the most important parts of screw machines, i.e. screw compressors or expanders, is a work space with its boundary surface which is produced by the screw surfaces of teeth and an inner cylinder surface of a machine box. In course of working process the work space has a complicated, non-stationary shape. The tooth screw surfaces, which consist of different coupled surfaces, create the most important part of the work space. The original line contact between surfaces changes into the contact in four points along a length of teeth of rotors. The rise of a technically undesirable gap between both conjugated surfaces and disturbance of gear ratio are consequences of this change. The arisen gaps caused the increase in inner loss of screw machines. Geometrical visualization of the conjugated teeth with the arisen gap is performed in instantaneous time.

References [1] [2]

J. Švígler, Incorrect Contact of Screw Machine Rotors. International Conference on Compressors and their Systems, 3-12, John Wiley &Sons London Ltd., 2005. N. Stosic, I.K. Smith, A. Kovacevic, Rotor interference as a criterion for screw compressor design. Journal of Engineering Design, volume 14, No 2, 209-220, Taylor&Francis Ltd, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Three-Dimensional Rupture Instability of a Displacement-Softening Interface under Nonuniform Loading Koji Uenishi* *

Research Centre for Urban Safety and Security, Kobe University 1-1 Rokko-dai, Nada, Kobe 657-8501 Japan [email protected]

ABSTRACT In previous studies [1, 2], we investigated rupture instability on two-dimensional interfaces that follow a nonlinear displacement-softening (slip-weakening) relation and are subjected to a loading stress that is locally peaked spatially, the level of which changes quasi-statically in time. We showed that for the case in which the interface strength weakens linearly with slip (i.e., displacement gap), there exists a universal length of the slipping region at instability, independent of any length scales entering into the description of the shape of the loading stress distribution. Here, we study slip development and its (in)stability for three-dimensional planar interfaces that follow the linear displacement-softening relation. We employ an energy approach to give a Rayleigh-Ritz approximation for the dependence of size of the rupture zone and maximum slip on the level and shape of the loading stress distribution. The zone is assumed to be elliptical, x2/a2 + z2/b2 ≤ 1, with unknown axes a and b, where the interface coincides with the x-z plane (y = 0). A loading stress τp + Rt − κ (x2 + m2z2)/2 is assumed on that plane; κ and m are positive constants and Rt is the stress change from that for which the peak in the loading stress distribution equals the interface strength τp. The results, again independent of the curvature of the loading stress distribution κ, indicate that the loading induces a rupture zone whose aspect ratio a(t)/b(t) changes with time t except for the cases of circular cracks (m = a/b = 1) for the opening (tensile) mode and m = a/b ≈ 1/(1 − ν) for the sliding mode (unidirectional shear in the x-direction). Here, ν is Poisson’s ratio. These values of m minimize the area of critical rupture zone at instability (πacbc), and for the opening mode, the corresponding critical diameter is 2ac = 2bc ≈ 1.960 µ/[W(1 − ν)], with µ being shear modulus and W linear displacement-softening slope. For the sliding mode, the critical rupture size is written in terms of µ, ν and W (and the complete elliptic integrals of the first and second kinds) [3]. When, for example, ν = 0.25, those critical lengths are 2ac ≈ 2.598µ/W and 2bc ≈ 1.951µ/W. Comparison of these results with the two-dimensional ones (1.158µ/[W(1 − ν)] for modes I and II; 1.158µ/W for mode III) shows that the critical lengths are of the same order for all two- and three-dimensional cases.

References [1] K. Uenishi and J. R. Rice, Universal nucleation length for slip-weakening rupture instability under nonuniform fault loading. J. Geophys. Res., 108(B1), cn:2042, doi:10.1029/2001JB001681, pp. ESE 17-1 to 17-14, 2003. [2] K. Uenishi, Rupture instability on a displacement-softening interface under heterogeneous loading. In: Proc. ECCOMAS 2004 (edited by P. Neittaanmäki et al.), 14 pages, Department of Mathematical Information Technology, University of Jyväskylä, Finland, 2004. [3] K. Uenishi and J. R. Rice, Three-dimensional rupture instability of a slip-weakening fault under heterogeneous loading. Eos Trans. AGU, 85(46), Fall Meet. Suppl., Abstract S13E-04, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Contact of rough surfaces - A comparison of experimental and numerical results Kai I. Willner∗ , Daniel B. G¨orke∗ ∗ Universit¨ at Erlangen-N¨urnberg Egerlandstraße 5, 91058 Erlangen, Germany {willner,goerke}@ltm.uni-erlangen.de

ABSTRACT Measurements of rough surfaces show that the roughness topography can be described as a fractal over several length scales. A suitable description is then given by a discrete structure function. For a large class of typical surfaces measured structure functions can be approximated by a three-parameter function, employing the rms-value of the roughness, a transition length between fractal behaviour at high wavenumbers and stationary behaviour at low wavenumbers, and the fractal dimension in the fractal region, respectively, as intrinsic parameters to describe an isotropic rough surface. To study the normal contact behaviour of such fractal surfaces numerically, the first author presented in [?] a numerical model which allows to describe the elasto-plastic normal contact of isotropic fractal surfaces. This model is now tested against experimental data which are obtained from contact tests of several aluminum specimens. The paper gives a short review of the numerical model and describes than the experimental set-up for the contact tests. Numerical and experimental data for several aluminum specimens with different surface properties are shown and compared.

References [1] K. Willner, Elasto-plastic contact of three-dimensional fractal surfaces using halfspace theory. Journal of Tribology, 126, 28-33, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Continuum Mechanics Modelling and Simulation of Carbon Nanotubes Marino Arroyo1 and Ted Belytschko2 1

` Universitat Politecnica de Catalunya E-08034 Barcelona, Spain [email protected] 2

Northwestern University Evanston, IL 60208, USA [email protected]

ABSTRACT The understanding of the mechanics of atomistic systems greatly benefits from continuum mechanics. One appealing approach aims at deductively constructing continuum theories starting from models of the interatomic interactions. This viewpoint has become extremely popular with the quasicontinuum method. The application of these ideas to carbon nanotubes presents a peculiarity with respect to usual crystalline materials: their structure relies on a two-dimensional curved lattice. This renders the cornerstone of crystal elasticity, the Cauchy-Born rule, insufficient to describe the effect of curvature. We discuss the application of a theory which corrects this deficiency to the mechanics of carbon nanotubes [1,2,3]. We review recent developments of this theory, which include the study of the convergence characteristics of the proposed continuum models to the parent atomistic models, as well as large scale simulations based on this theory. The latter have unveiled the complex nonlinear elastic response of thick multiwalled carbon nanotubes, with an anomalous elastic regime following an almost absent harmonic range.

References [1] Marino Arroyo and Ted Belytschko, An atomistic-based finite deformation membrane for single layer crystalline films, Journal of the Mechanics and Physics of Solids 50, 1941-1977, 2002. [2] Marino Arroyo and Ted Belytschko, Nonlinear mechanical response and rippling of thick multiwalled carbon nanotubes, Physical Review Letters, 91, 215505, 2003. [3] Marino Arroyo and Ted Belytschko, Finite element analysis of the nonlinear mechanics of carbon nanotubes, International Journal for Numerical Methods in Engineering, 59, 419-456, 2004.

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Comparison of computational efficiency of modeling approaches to prediction of damping behavior of nanoparticle-reinforced materials Liya V. Bochkareva*, Maksim V. Kireitseu†, Geoffrey R. Tomlinson† * UIIP National Academy of Sciences of Belarus Address: Filatova 7 – 28, Minsk 220026, Belarus [email protected] †

Department of Mechanical Engineering, the University of Sheffield Address: Mappin Street, Sheffield S1 3JD, the United Kingdom [email protected]

ABSTRACT Carbon nanotube-reinforced polymer-matrix composite materials (CNT-PMC) are now intensively studied; however, CNT-PMC damping behavior is rather contradictory result than plausible information. Therefore, it requires urgent investigations from multi-disciplinary viewpoint. The CNTreinforced material damping phenomenon is complex because of friction between nanotube and a matrix and the variety of other energy dissipation/fracture mechanisms involved, and because of the complex nature of the nanoparticles themselves, multi-walled structure etc. that are affect a damping. All of these mechanisms may be beneficial for dumping and/or add multifunctionality to engineering structures. It is worth noting that interfacial fracture energy is important and may play a great role for a total energy dissipated by the damping material. Particular advanced energy dissipation phenomena of CNT-reinforced polymeric materials can be explained by considerable interfacial fracture mechanics and bonging energy between CNT and polymeric molecular chains. Quantitative prediction of toughness would require a coupled and detailed modeling of the various damping / dynamic mechanisms and criteria for the different modes, which is at present still not feasible. Computational simulation and modeling tools called as a Virtual Reality Environment (VRE) can help to understand many physical effects and predict the behavior of materials and machine components via computer-generated media. In the present paper, multiscale computational approaches to modeling of nanoparticle-reinforced composite materials and virtual reality engineering tools have been used to describe/model an intuitive interface of some CNT-reinforced materials to enable efficient design and synthesis of next generation materials and nanoscale devices. The paper presents a comparison between computational approaches to modeling of damping/dynamics of CNTreinforced composite materials so as to estimate a validity of proposed methods. The underlying mechanics of material has been partially simulated by the use of energy dissipation mechanisms and programmed by using fast multipole method FMM-BIEM [1] accordingly. In the virtual working environment, the user can naturally grab and steer a nanoparticle, matrix and composite because the information flow between the user and the VRE is bidirectional and the user can influence the environment.

References [1] Y.J. Liu, N. Nishimura, Y. Otani, T. Takahashi et al., Fast multipole approach to modeling of nanocomposites, ASME J. of Appl. Mech., 72 (1), 140-160, 2005

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A Hybrid Atomistic-Continuum Finite Element Modelling of Nanoindentation – Test on Copper 1 , M. Mazdziarz 1 , G. Jurczak1 , P.Traczykowski 1 , ˙ ´ P. Dłuzewski S. Nagao2 , R. Nowak2 3 and K. Kurzydłowski 1 Institute

2 Nordic

3 Warsaw

of Fundamental Technological Research,Warsaw,Poland [email protected]

Hysitron Laboratory, Dept. Materials Sci. and Engg., Helsinki University of Technology (TKK/HUT), Vuorimiehentie 2A, Helsinki FIN-02015, Finland [email protected].fi University of Technology, Faculty of Materials Science and Engineering (InMat), Wooska 141, Warszawa 02-507, Poland [email protected] ABSTRACT

Problem of locally disordered atomic structure is solved by using a hybrid formulation in which nonlinear elastic finite elements are linked with discrete atomic interaction elements. The continuum approach uses nonlinear hyperelasticity based upon the generalized strain while the atomistic approach employs the Tight-Binding Second-Moment Approximation potential to create new type of elements. The molecular interactions yielding from constitutive models of TB SMA were turned into interactions between nodes to solve a boundary value problem by means of finite element solver. Atomistic psudoelements are noting more than two-node atomic interaction. The application is used to deal with the problem of edge dislocation and its dissociation into two Schockley’s dislocation in Cu crystals. Three examples are shown. In the first, the whole considered crystal region has been discretized and solved by means of the nonlinear elastic FEs. In the second example, the same FE region has been discretized by means of the molecular lattice and solved by molecular dynamics and statics. Finally, in the third example the regions around the dislocations’ cores have been replaced by discretized atomic structure and linked on the boundary with corresponding nodes of continuum FEs. In this way, a single boundary value problem with two different types of discretization of the crystal structure has been solved in this example. A transition on the continuum/atomic interface is assured by taking into account a crystallographic data in the finite element mesh preparation. In this example the regions of dislocation cores were modelled using molecular interaction mesh while the ordered lattice was discretized by FEs. The obtained MD-FE model has been applied to simulate the nanoindentation test on nanocrystalline copper in order to conclude on the singularities observed in P-h curves.

References [1] P.Dłu˙zewski, G.Maciejewski, G.Jurczak, S.Kret and J.-Y.Laval, Nonlinear finite element analysis of residual stresses induced by dislocations in heterostructures. Computational Materials Science , 29, 379–395, 2004. [2] P.Dłu˙zewski, G.Maciejewski,Nonlinear finite element calculations of residual stresses in dislocated structures.Computational Materials Science, 30, 44–49, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Continuum Models for Composites Reinforced by Micro/Nano-Fibers Vladimír Kompiš*, Mário Štiavnický*, Michal Kaukiþ† *

Academy of Armed Forces of General M. R. Štefánik, Demänovská 393, 031 19 Lipt. Mikuláš, Slovakia [email protected], [email protected] †

University of Žilina, Faculty of Control and Informatics, VeĐký Diel, 010 24 Žilina [email protected]

ABSTRACT We will present a method of continuum models for composites reinforced by micro/nano-fibers. The fibers are supposed to be much stiffer than the matrix and their diameter can be much smaller than their length. The ideal cohesion between fiber and matrix is supposed in the present models. Interactions of a fiber with the matrix, with the other fibers and with the domain boundaries, are important parameters, which are decisive for the stiffening effect and for the bearing capacity of the composite material. Both near and far field effects are important for the material behavior. The near fields are important for simulation of the local effects like fracture and the far field effects determine the stiffening. For modeling of such problems the methods like FEM and BEM are not very efficient. BIE using distributed forces, dislocations and dipoles along the fiber axis (the source points) enable to model these effects much more efficiently. The intensity of the source functions simulates the interaction of the fiber with the other subjects. Because of very close distance of the source points to the fiber boundary and the quasi-singular form of the integrals, the integration is performed analytically in the fiber direction. The proposed models can be used in the multi-scale simulation of such composite structures.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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First Principles Evaluation of Ideal Strength of Cu Nanowire A. Kushima∗ , Y. Umeno∗ , T. Kitamura∗ ∗ Graduate

School of Engineering, Kyoto University Yoshidahonmachi Sakyo-ku Kyoto, 606-8501, Japan [email protected] ABSTRACT

Studying the ideal strength of the materials with high-symmetric structures can provide us with the fundamental mechanical property of the nano-structured materials. In this study, tensile simulations for cylindrical-shaped Cu nano-wires with different diameters are conducted and their ideal strengths are explicitly evaluated by the first principles calculation. And the results are compared with that of the atomic chain and the sheet, the elements constructing the wire. The ideal strengths of the wire is lower than that of the chain and the sheet because the fracture of the wires is caused by the formation of gap in the surface, while the direct breaking of inter-atomic bonds is observed in the case of the chain and the sheet. The ideal strength and the strain at which the fracture takes place of the wire with larger diameter are smaller than those of the wire with smaller diameter. This is due to the existence of the minute gaps in the surface of the wire, caused by the difference in the curvature between the surface layer and the layer inside it. In the tensile process, the gaps become larger and finally the fracture occurs to the wire. Fig.1 denotes the change in electronic structure of the wire under tension. The isosurface of 0.16×103 nm−3 is shown, and the contours of the surface charge distributions are extracted for detailed observation.

Figure 1: Change in atomic structure and charge density of the Cu nanowire under tension.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Variation Descriptions of Nano-Structured Media Svetlana A. Lisina1, Alexander I. Potapov 1,2 1 Mechanical Engineering Research Institute of the Russian Academy of Sciences 85 Belinsky str., 603024, Nizhny Novgorod, RUSSIA [email protected] 2

Nizhny Novgorod State Technical University, 24 Minina str., 603600, Nizhny Novgorod, RUSSIA [email protected]

ABSTRACT Medium microstructure (in particular, size and morphology of a grains) is the most important property of materials that directly influences their strength and other physical and mechanical characteristics. Complex dynamic behavior of materials is determined by the existence of intrinsic space-time scales - size of grain, lattice period, relaxation time, etc. The orientational effects that cannot be described by equations of the classic theory of continuum mechanics occur in nanostructured material [1]. In the phenomenological description of such media it is supposed that its representative volume contains discrete material microvolumes (structural elements). Two vector fields can describe the kinematics of oriented medium: field of particles displacements and field of microrotations. In this paper it has been shown that the variational approach allows one to construct effectively the nonlinear mathematical models of nanostructured media in terms of both Lagrange's and Euler's variables. The variational principle for oriented medium has been formulated, from which the variational equations of dynamics and their integrals of motion have been found. The first sets of variational equations describe macromotions (i.e. motions of mass centers of the particles), and the second one describe microrotations of structural elements [2]. Equivalence of the variational equations and local laws of conservation of the energy, momentum and angular momentum in terms of Euler's variables has been proved using an example of liquid crystal. It has been shown that the Ericsen's stress tensor and the molecular field in liquid crystal are defined as partial derivatives of free energy. Research was done under the Presidential Program of Support for Leading Scientific Schools of Russia, and at partial support by grants of RFBR (project N 04-02-17156). REFERENCES [1] Eringen, A.C., Microcontinuum Field Theories. 1: Foundation and solids. Springer Verlag, New York Inc. 1999. [2] V.B.Lisin, A.I.Potapov, Variational principle in mechanics of liquid crystals, Int. J. Non-Linear Mech., 1997, 32, ʋ 1, pp. 55-62.

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Homogenization of single-walled carbon nanotubes Tanguy Messager, Patrice Cartraud Institut de Recherche en Génie civil et Mécanique (GeM), UMR CNRS 6183 Ecole Centrale de Nantes, BP 92101, 44321 Nantes cédex3, France [email protected], [email protected]

ABSTRACT This work deals with the computation of the overall axial elastic behavior of single-walled carbon nanotubes (SWCNTs). The SWCNTs are modeled as space-frame structures, using beam elements to represent atomic bonds [1]. The application of homogenization theory [2] enables to derive rigorously the macroscopic anisotropic beam behavior of the SWCNT, from the solution of three-dimensional basic cell problems. Moreover, taking benefit of the two helical symmetries [3] of the microstructure, the basic cell can be reduced to only one half of an hexagon, as depicted in Fig.1 for a zigzag SWCNT. Therefore, the overall stiffness coefficients can be computed efficiently using very concise FE models (including only 3 beam elements): the helical symmetry properties of the displacement field lead to a set of linear relationships expressed in local cylindrical axes between the opposite nodes m and n (see Fig.1) then acting as boundary conditions [3]. The accuracy of this approach has been assessed with respect to reference solutions of the literature [1] and also from comparison with results given by large FE model (left part of Fig.1). This method has been applied for the computation of zigzag and armchair SWCNTs: as shown in Fig.2, the developed procedure allows to study the scale effects.

Fig.1: Overall structure and microscopic FE model

Fig.2: tensile stiffness evolution

References [1] K.I. Tserpes, P. Papanikos, Finite element modeling of single-walled carbon nanotubes. Composites Part B, 36, 468-477, 2005. [2] P. Cartraud, T. Messager, Computational homogenization of periodic beam-like structures. Int. J. of Solids and Structures, in press, 2005. [3] T. Messager, P. Cartraud, Homogenization of helical beam-like structures. Finite Elements in Analysis and Design, submitted, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Phonon scattering by perturbed multichannel waveguides M. S. Rabia Laboratoire de Mécanique, Structures et Energétique, Département de Génie Mécanique, Université M. Mammeri, Tizi-Ouzou 15000, Algérie. [email protected]

ABSTRACT During the last two decades, scattering and localization phenomena in disordered mesoscopic systems have been well established as well experimentally as theoretically. They are actually of renewed interest owing to advances in nanotechnologies, the basic motivation being the need to understand the limitations that structural disorder may have on the physical and mechanical properties of nanocrystalline materials. Our present knowledge of the related phenomena has been given by the work of Landauer who has related the conductance of the sample to its scattering matrix. His approach reveals further the essential difference between elastic and inelastic scattering regimes: the latter is responsible of dissipation of energy, whereas the no dissipative and phase conserving elastic process introduce quantum interference effects due to the coherent scattering between defects. This interpretation has stimulated many researchers to look for the effects of quantum coherence in dc transport particularly. Recently, several authors have shown that multiple scattering and quantum interference become very important to describe transport phenomena, localization of electron states in disordered media, coherent magnetotransport and to investigate structural properties of lowdimensional samples. In the present work, we concentrate on the influence of local defect on scattering properties of elastic waves in perturbed crystalline quasi-three-dimensional structure in the harmonic approximation. Our model consists of three infinite atomic plans, assimilated to a perfect waveguide in which a scatterer (or defect) is inserted in bulk or in surface. The numerical treatment of the problem, based on the Landauer approach, resorts to the matching method initially employed for the study of surface localized phonons and resonances. Numerical results show that the interferences between the multiple scattered waves give rise to a broad variety of structures in transmission (or conductance) spectra which can be regarded as identifying features of the specific defect structures and may therefore be used for their characterization. Some of these structures, resulting from the interferences between incidental and reflected waves, correspond to Fabry-Pérot oscillations and others, due to the coupling between propagating modes and a defect induced states, are identified to Fano resonances. The interaction between defect-induced states and propagating waveguide eigenmodes could provide an interesting alternative to investigate structural properties.

References [1] B. Kramer, Quantum Coherence in Mesoscopic Systems, (plenum, New York, 1991). [2] R. Landauer, Z. Phys. B 68, 217 (1987); J. Phys. Condens. Matter 1, 8099 (1989). [3] A Fellay, F. Gagel, K. Maschke, A. Virlouvet and A. Khater, Phys.Rev., B 55, 1707 (1997). [5] M. S. Rabia, H. Aouchiche and O. Lamrous, Eur. Phys. J. – A. P. 23, 95-102 (2003).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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An Atomistic-Information-Based Continuum Inhomogeneous Material Model for Metal Nanorod H. A. Wu, X. X. Wang CAS Key Laboratory of Materials Behavior and Design, Department of Mechanics and Mechanical Engineering, University of Science and Technology of China, Hefei, Anhui 230026, China [email protected]

ABSTRACT Mechanical behaviors of materials and structures at nano scale are essentially different from those at macro scale, resulting from surface effect, size effect and time scale effect. To correctly predict the structure-property relations of elemental nanocomponents are very important for the design of nanodevices. Atomistic simulations have been widely used to investigate nanomechanics. The equivalent elastic modulus of a copper nanorod under extension can be obtained using molecular dynamics simulation [1]. In our previous work, it was found that the correct deflection could not be obtained when above equivalent elastic modulus was used to predict the bending behaviors of nanorod [2]. The error was even up to 50%, compared with the direct atomistic simulation. The aim of this work is to investigate the original mechanism of above discrepancy and to present a novel continuum model to predict the bending modulus correctly. We owe this difference mainly to the surface effect. The ratio of surface atoms to the totality is about 60% for copper nanorod with cross-section size of 2nm. In some approximation, the nanorod can be considered as continuum. However, it is not homogeneous across the section. In our continuum model, the nanorod is considered as inhomogeneous material. The material constants of surface atoms and inner atoms are calculated from atomistic simulation, so our material parameters for continuum model of metal nanorod are based on the atomistic information. A finite element analysis of bending is carried out. The result agrees with direct atomistic simulation well, which validates the continuum nanorod model. Further work is to incorporate the research output as a new material model into commercial finite element software. With the development of accurate inter-atomic potentials for a range of materials, classical MD simulations have become prominent as a tool for elucidating the mechanical behaviors of nano-structures. However, the length and time scales that can be probed using MD are still fairly limited. Our continuum metal nanorod model includes nano-effects and supplies another way to study nanomechanics.

References [1] H. A. Wu, Molecular dynamics study on mechanics of metal nanowire. Mechanics Research Communications, 33(1), 9-16, 2006. [2] H. A. Wu, Molecular dynamics simulation of loading rate and surface effects on the elastic bending behavior of metal nanorod. Computational Materials Science, 31(3-4), 287-291, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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On the Modeling of Deformation-Diffusion-Damage Coupling in Elastic Solids Fernando P. Duda∗ , Leonardo J. Guimar˜aes† , Angela C. Souza‡ , Jos´e M. Barbosa† ∗ Universidade Federal do Rio de Janeiro PEM-COPPE, Cidade Universit´aria, Rio de Janeiro, RJ, Brasil [email protected] † Universidade Federal de Pernambuco CTG-Cidade Universit´aria, Recife, PE, Brasil {leonardo,jmab}@ufpe.br ‡ Universidade

Federal Fluminense LMTA-PGMEC-TEM, Niteroi, RJ, Brasil [email protected]

ABSTRACT This paper deals with the formulation and numerical implementation of a fully coupled continuum model for deformation-diffusion-damage in elastic solids. The formulation is carried out within the framework of continuum mechanics, where, in addition to the standard fields, extra fields are introduced in order to describe diffusion and damage processes. The governing equations are then obtained after supplementing the basic balances with a thermodynamically consistent constitutive theory. The couplings are implemented via the free energy response and include both deformation and damage assisted diffusion. It is worth mentioning that a gradient damage theory is obtained, which allows the modeling of fracture problems. The numerical implementation is based on the finite element method and a Euler implicit scheme for spatial and temporal discretizations, respectively. A numerical algorithm is presented to solve the discrete system of equations. In order to illustrate the potentiality of the proposed model, applications in the context of hydrogen embrittlement are presented.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Frost Growth on Cold Flat Plate: a Correlation for the Diffusion Resistance Factors Jorge A. Gatica*, Vicente A. Pita*, Nisio de C. Lobo B.† *

Universidad del Bío-Bío Concepción, Chile [email protected] [email protected]

†

Universidade Federal do Rio de Janeiro Rio de Janeiro, Brasil [email protected]

ABSTRACT The condensation of the vapor of water contained in the humid air on low temperature surfaces is a very frequent phenomenon. When the surface is below zero degrees, an ablimation of the vapor of water phenomenon happens, depositing as frost on the cold surface. In this work a numeric computational model is developed for to simulate the frost formation and his growth process on low temperature plates. The frost formation is a common operational problem in fins of evaporators used in low temperature warehouses. This deposit affects the heat transfer reducing the refrigeration capacity; therefore, one better understanding of this phenomenon is of great interest. During this work several analytic-experimental investigations were revised. Of particular interest they were the proposals [1, 2] relative to the effective diffusion coefficient. Based on a control volumes analysis a non-linear system of differential partial equations is obtained: a mass diffusion equation and energy equation besides a group of auxiliary relationships that allow to close the system and defining interface and contour conditions. The finite differences method is used to solve the system. The equations are transformed to linear equations and resolved iteratively using the Thomas' algorithm for tridiagonal systems. The program allows obtaining in transient form the distribution of temperatures in the frost layer, its variation of thickness and growth, also the influence on the same process of the Reynolds' number , the humid air temperatures, etc. A strong influence of the diffusion resistance factors used in the reproduction of experimental data was observed. Then, is generated a correlation to determine these factors that it turns out to be function of the humid air and the cold flat plate temperatures.

References [1] Y.X. Tao, R.W. Besant and K.S. Rezkallah, A mathematical model for predicting the densification and growth of frost on a flat plate, Int. Journal of Heat and Mass Transfer, 36, n. 2, 353-363, 1993. [2] R. Le Gall, J.M. Grillot and C. Jallut, Modeling of frost growth and densification. Int. Journal of Heat and Mass Transfer, 40, n. 13, 3177-3187, 1997.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Discontinuous Space-Time Galerkin Finite Element Method in Linear Dynamic Fully Coupled Thermoelastic Problems with Strain and Heat Flux Discontinuities Dinara K. Khalmanova∗ , Francesco Costanzo† The Pennsylvania State University Engineering Science and Mechanics Department, 212 EES Building, University Park, PA, 16802 ∗ [email protected] † [email protected]

ABSTRACT A discontinuous space-time Galerkin finite element method has been developed by the authors for the study of linear elasto-dynamic and fully coupled thermoelastic problems with discontinuities in the displacement and temperature gradients. The method is proven to be unconditionally stable and capable of automatic adaptive mesh refinement. The developed formulation has been implemented to solve a number of model solid-solid phase transition problems in one and two dimensions. The results of numerical study are presented to illustrate the convergence to an analytical solution of a problem with smooth coefficients. The presented method is particularly suitable for the study of thermoelastic and elasto-dynamic moving boundary problems and can be applied to dynamic fracture problems in heterogeneous thermoelastic materials using cohesive zone models.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Pore gas interaction in polymeric foams with respect to energy absorption Mats Landervik and Ragnar Larsson Department of Applied Mechanics Chalmers University of Technology SE-412 96 G¨oteborg, Sweden [email protected], [email protected]

ABSTRACT The main object of the present paper is to investigate the significance of gas-solid interaction in foamed polymers subjected to large deformations, particularly in compression, in combination with high rates thereof. The present development thus represents an extension of previous work in [1], where we developed a phenomenological model representing the response of the sole cellular network on the basis of a viscoplastic Perzyna model to account for the rate dependence of the cellular network response. In this context, a main feature of the paper concerns the establishment of gas-filled foams in the context of a two-phase porous material within the Theory of Porous Media [2], where the interaction between the phases is modeled in terms of a deformation dependent Darcian filter law [3]. Thereby, a continuum mechanical coupling between the gas pressure and the deformation of the cellular network is obtained. We propose to resolve this coupling in terms of a staggered solution procedure similar to the staggered handling of the thermo-mechanically coupled problem [4]. ”Constant pressure” decoupling, where the pressure is considered constant during the mechanical step, is used. The model is implemented in the finite element code LS-DYNA, and the paper is concluded by representative numerical simulations. As main feature of the paper, a parametric study in terms of the permeability properties of the foam is carried out to display the significance of the gas-solid interaction for polymeric foam used for energy absorption in vehicles.

References [1] M. Landervik and R. Larsson. Modeling of large inelastic deformations of foams with respect to energy absorption, Submitted for international publication, 2006. [2] R. de Boer. Highlights in the historical development of the porous media theory: Toward a consistent macroscopic theory. Appl. Mech. Rev., 49, 201–262, 1996. [3] W. Ehlers and B. Markert. A macroscopic finite strain model for cellular polymers. Int. J. Plast., 19, 961-976, 2003 [4] E. Armero and J.C. Simo. A priori stability estimates and unconditionally stable product formula algorithms for nonlinear coupled thermoplasticity, Int. J. Plast., 9, 749–782, 1993

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A thermo-hydro–damage model for the dehydration creep of concrete subjected to high temperature Fekri Meftah* and Hassen Sabeur*,† * Laboratoire de Mécanique – Université Marne La Vallée 5 boulevard Descartes – 77454 Marne La Vallée Cedex 2 – France e-mail: [email protected]

† LAMI – Ecole Nationale des Ponts & Chaussées 8 avenue Blaise Pascal – 77455 Marne La Vallée Cedex 2 – France [email protected]

ABSTRACT When concrete is ascribed to combined mechanical loads and high temperature distributions, it exhibits strains which are conventionally [1] split to a set of additive components: - Stress-free components, referred to thermo-hygral strains, which include thermal expansion and hygral shrinkage due to both drying and dehydration. - Stress induced thermal strains which mainly consist in a temperature depend elastic strain, a micro-cracking strain and an additional component, commonly referred to as transient creep [1,2,3]. This additional component is generally related to the fact that physical transformations, such as drying and dehydration, are occurring under sustained stress fields, which lead to a rearrangement of the evolutionary microstructure and give rise to this macroscopically measured strain. In this contribution, a full coupled thermo-hydro-mechanical model is proposed for the modeling of the transient creep in the range of 105°C-400°C, which is referred here to as dehydration creep. In this model, a dehydration variable is introduced to describe chemical transformations due to the temperature increase. It also allows to govern the occurrence of the dehydration creep when the stress level does not exceeds 40% of the ultimate strength. In addition, the proposed model considers that the dehydration creep occurs with a kinetics controlled by the relaxation time of the dehydration process. Further, the model allows to describe an irrecoverable transient creep upon cooling or during a second heating to the same maximum dehydration level. This model have be implemented in a the finite element code CAST3M. The algorithm for the update, at the constitutive level, of the mechanical behavior is presented. A particular attention is the given to the treatment of the transient creep component. Numerical simulations are performed to assess the capability of the model to predict transient load induced thermal behavior of concrete.

References [1] G.A. Khoury, P.J.E. Sullivan and B.N. Grainger, Strain of concrete during first heating to 600°C under load. Magazine of Concrete Research, 37 (133), 1985. [2] Y. Anderberg, Fire-exposed Hyperstatic Concrete structures- An experimental and Theorical Study, Div. of Struct. Mech. And Concrete Constr., Inst. Of Tech., Lund,1976. [3] U. Schneider, Concrete at high temperature: A general review. Fire Safety Journal, 13, 55-68, 1988.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A time-space framework suitable for the LATIN computational strategy for multiphysics problems David Néron , Pierre Ladevèze

,∗

and Bernhard A. Schrefler2

LMT-Cachan (ENS de Cachan/CNRS UMR8535/Paris 6 University) 61, avenue du Président Wilson, F-94235 CACHAN CEDEX, France {neron, ladeveze}@lmt.ens-cachan.fr 2

Department of Structural and Transportation Engineering (University of Padova) Via Marzolo 9, I-35131 PADOVA, Italy [email protected] ∗

EADS Foundation Chair “Advanced Computational Structural Mechanics” ABSTRACT

Since the last few decades, the simulation of multiphysics phenomena has become one of the major issue in the design of computational methods. Indeed, these problems often lead to computationally intensive analyses and then strategies to keep these problems affordable are of special interest. In this context, a strategy based on the LArge Time INcrement (LATIN) method [1] was introduced in [2] and also defined in the general framework of discretized problems in [3]. The main idea of this new method was the concept of interface ‘between physics’, which can be viewed as an extension of the ‘material’ interface classically introduced between two substructures [4]. This concept made it possible to introduce in a simple way several improvements (use of different discretizations for space and time, radial loading approximation, treatment of nonlinearities [3, 5]). The proposed test case concerned the consolidation of saturated porous soils and the LATIN strategy was compared to ISPP, a standard partitioning scheme, showing a significant decrease in computational cost [2, 5]. The aim of this paper is to introduce a general time-space framework suitable for the LATIN computational strategy for multiphysics problems. This framework is based on the concept of generalized radial loading [6], namely on the representation of all fields in an algebra generated by sums of products of time functions per space functions. The treatment of multiphysics problems in this framework enables a important gain in terms of computational and storage costs.

References [1] P. Ladevèze. Nonlinear Computational Structural Mechanics – New Approaches and NonIncremental Methods of Calculation. Springer Verlag, 1999. [2] D. Dureisseix, P. Ladevèze and B. A. Schrefler. A computational strategy for multiphysics problems – application to poroelasticity. International Journal for Numerical Methods in Engineering, 56(10):1489–1510, 2003. [3] P. Ladevèze, D. Néron and B. A. Schrefler. A computational strategy suitable for multiphysics problems. Proceedings of the First International Conference on Computational Methods for Coupled Problems in Science and Engineering, 2005. [4] P. Ladevèze, D. Néron and P. Gosselet. On a mixed and multiscale domain decomposition method. Submitted to Computer Methods in Applied Mechanics and Engineering. [5] D. Dureisseix, P. Ladevèze, D. Néron and B. A. Schrefler. A multi-time-scale strategy for multiphysics problems: application to poroelasticity. International Journal of Multiscale Computational Engineering, 1(4):387–400, 2003. [6] P. Ladevèze. A technique relative to the LATIN method for the calculation of time and space integrals. Internal Report of LMT-Cachan, 193, 1997.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Finite Element Analysis of the Thermomechanical Coupling in Quenching of Steel Cylinders Using a Constitutive Model with Diffusional Phase Transformations Wendell P. de Oliveira*, Marcelo A. Savi*, Pedro M. C. L. Pacheco†, and Luís F. G. de Souza† * UFRJ/COPPE – Department of Mechanical Engineering 21.945.970 – Rio de Janeiro – RJ, P.O. Box 68.503 - Brazil [email protected], [email protected] † 1 CEFET/RJ - Department of Mechanical Engineering Av. Maracanã, 229, 20271-110 - Rio de Janeiro - RJ - Brazil [email protected], [email protected]

ABSTRACT Quenching is a heat treatment usually employed in industrial processes. It provides a mean to control mechanical properties of steels as toughness and hardness. Phenomenological aspects of quenching involve couplings among different physical processes and its description is unusually complex. Basically, three couplings are essential: thermal, phase transformation and mechanical phenomena. This article deals with the modeling and simulation of quenching in steel cylinders using a multiphase constitutive model with internal variables formulated within the framework of continuum mechanics and the thermodynamics of irreversible processes. The energy equation thermomechanical coupling terms are exploited, considering two different models. The first one is an uncoupled model where thermo-mechanical couplings are neglected, corresponding to the rigid body energy equation. The second model considers the latent heat associated with phase transformation in order to represent thermomechanical coupling. A numerical procedure is developed based on the operator split technique associated with an iterative numerical scheme in order to deal with non-linearities in the formulation. With this assumption, the coupled governing equations are solved to obtain the temperature, stress and phase fields from four uncoupled problems: thermal, phase transformation, thermo-elastic and elastoplastic. Finite element method is employed for spatial discretization. The proposed general formulation is applied to the through hardening of steel cylinders. Numerical simulations present a good agreement with experimental data for temperature and phase transformation distributions, indicating some situations where it is important to consider the thermomechanical coupling in the description of quenching process.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A combined fracture-micromechanics model for tensile strain-softening in brittle materials, based on propagation of interacting microcracks Bernhard Pichler∗ , Christian Hellmich ∗ , Herbert Mang∗ ∗

Vienna University of Technology (TU Wien), Institute for Mechanics of Materials and Structures Karlsplatz 13/202, A-1040 Wien (Vienna), Austria [email protected], [email protected], [email protected]

ABSTRACT Strain-softening is the decline in stress at increasing strain. Although microcracking is a commonly accepted reason for strain-softening, the majority of theoretical developments involve macroscopic damage evolution laws [3, 2]. To improve this situation, we propose a micromechanics-based damage evolution law by coupling two seemingly separated scientific fields, i.e. by combining (i) the propagation criterion for a single penny-shaped crack embedded in an infinite matrix subjected to remote stresses (taken from linear-elastic fracture mechanics) and (ii) stiffness estimates for representative material volumes comprising interacting microcracks (taken from continuum micromechanics [4, 1]). This combination allows for modeling tensile strain-softening as a result of propagation of interacting microcracks, i.e. as a microstructural effect. The initial degree of damage, i.e. the initial microcrack size and the number of microcracks per unit volume, implies two different types of model-predicted tensile strain-softening behavior under strain control: (i) continuous strain-softening, which occurs in case of initial damage beyond a critical value, and (ii) an instantaneous stress drop at the peak load (”snap-back”), which occurs in case of initial damage below a critical value.

References [1] Y. Benveniste. On the Mori-Tanaka’s method in cracked bodies. Mechanical Research Communications, 13(4):193–201, 1986. [2] V. Pens´ee, D. Kondo, and L. Dormieux. Micromechanical analysis of anisotropic damage in brittle materials. Journal of Engineering Mechanics (ASCE), 128(8):889 – 897, 2002. [3] P.C. Prat and Baˇzant Z.P. Tangential stiffness of elastic materials with systems of growing or closing cracks. 45(4):611–636, 1997. See also: Addendum (with Errata). J. Mech. Phys. Solids 45(8):1419–1420, 1997. [4] A. Zaoui. Continuum micromechanics: Survey. Journal of Engineering Mechanics (ASCE), 128(8):808 – 816, 2002.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Transient Dynamic Response of Thermoelastic Cylindrical Layered Media Ö. ùEN*, D. TURHAN† *

The Scientific & Technical Research Council of Turkey Defense Industries Research & Development Institute 06261 Ankara, Turkey [email protected] †

Department of Engineering Sciences Middle East Technical University 06531 Ankara, Turkey [email protected]

ABSTRACT In this study, the transient dynamic response of thermoelastic, hollow circular cylindrical composites consisting of n-different isotropic, homogeneous and elastic layers is investigated. Thermomechanical behavior of each layer is governed by the equations of generalized thermoelasticity with two relaxation times predicting finite wave speeds for thermal disturbances [1-2]. The body is subjected to uniform dynamic mechanical and thermal inputs at inner and/or outer surfaces. The time dependence of the dynamic inputs may be arbitrary. The cylindrical composite is of finite thickness in the radial direction and extends to infinity in the axial direction. The layers of the body are assumed to be perfectly bonded to each other. Furthermore, the layered medium is assumed to be initially at rest. The governing field equations of generalized thermoelasticity with two relaxation times are applied to each layer and the solutions are required to satisfy the continuity conditions at the interfaces of the layers, the boundary conditions at the inner and outer surfaces, and quiescent initial conditions. Method of characteristics is employed to obtain the solutions [3]. The convergence and numerical stability of the method are well established, and different interface, boundary and initial conditions can be handled conveniently. The solutions reveal the existence of two wave fronts. The numerical results are displayed in curves denoting the variations of circumferential and radial stresses and temperature deviation with time at different locations and variations of stresses and temperature deviation along the thickness of the bodies at different times. The solutions reveal clearly the thermal and geometric dispersions in the wave profiles and the effects of refractions and reflections at the interfaces and at inner and outer surfaces of the body. The curves further display the severe variations in the field variables at the wave fronts.

References [1] A. E. Green and K. A. Lindsay, Thermoelasticity. Journal of Elasticity, 2, 1-7, 1972. [2] E. S. Suhubi, Thermoelastic Solids In Continuum Physics (A. C. Eringen, editor). New

York: Academic Press, 2, 279-265, 1975. [3] R. Courant and D. Hilbert, Methods of Mathematical Physics. New York: Interscience, 2, 1966.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Novel nonlocal continuum formulations. Part 1: gradient elasticity based on nonlocal displacements and nonlocal strains Harm Askes∗ , Miguel A. Guti´errez† , Antonio Rodr´ıguez-Ferran‡ ∗ University

of Sheffield Department of Civil and Structural Engineering Sheffield S1 3JD, United Kingdom h.askes@sheffield.ac.uk † Delft University of Technology Faculty of Aerospace Engineering Delft, The Netherlands [email protected] ‡ Universitat Polit` ecnica de Catalunya Department of Applied Mathematics III Barcelona, Spain [email protected]

ABSTRACT Nonlocal continuum formulations exist in integral formats and differential formats. The latter, also known as gradient-enriched continua, have successfully been applied in elasticity, plasticity and damage and provide a robust framework to analyse size effects and dispersive waves. Moreover, gradient enhancement can be used to remove singularities from elastic fields as well as guarantee well-posedness in the modelling of post-peak phenomena. In this paper, the focus will be on novel formulations for gradient elasticity. Aifantis and coworkers suggested a simple format of gradient elasticity in which the usual stress-strain is augmented with an additional term, namely the Laplacian of the strain. This then leads to a fourthorder differential equation in terms of the displacement and, hence, C1 continuity requirements in case of a numerical implementation. However, it is possible to split the fourth-order equations into two sets of second-order equations, whereby the first set coincides with the equations of classical elasticity and the second set comprises a set of diffusion-type equations that introduce the gradient enrichment. Hence, a staggered solution algorithm is obtained, whereby the displacements of classical elasticity are computed first and then used as input for the second set of equations in order to compute the gradient-enriched displacements. The staggered, displacement-based approach will be scrutinised together with two alternative formulations of gradient elasticity: (i) a staggered, strain-based approach, and (ii) another strain-based approach that can be derived from Pade approximations of the previous method. The three approaches will be presented with their boundary conditions, and it will be verified whether all singularities are removed from the strain field. Also, size effect tests will be reported. In the accompanying paper, the displacement-based approach will be used in the formulation of nonlinear gradient models.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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On the use of a damage model based on non-local displacements in the Element-Free Galerkin method Terry Bennett∗ , Sivakumar Kulasegaram† ∗ Cardiff

School of Engineering Cardiff University, UK [email protected]

† Cardiff

School of Engineering Cardiff University, UK [email protected] ABSTRACT

One of the ingredients of Element-Free Galerkin (EFG) shape functions are the weight functions as employed in Moving Least Squares interpolations [1]. They are similar to the weight functions used in non-local damage [2]. Therefore, it seems a natural choice to use non-local damage as a localisation limiter within the EFG method. This contribution investigates the need for regularisation techniques within the EFG method as meshfree approximations can be already considered to possess intrinsic non-local properties [3]. However, one must make a distinction between the mathematical non-local properties of the interpolation method (whether it be Finite Elements, EFG etc), and the mechanical non-local properties of the constitutive model. The recently developed displacement based non-local damage model [4] is utilised here to regularise damage within the EFG framework and provide further computational efficiency over strain based nonlocal methods due to (typically) less nodes being employed than integration points and less components for the displacements compared to the strains. The formulation of the displacement based non-local damage within the EFG framework is elucidated and examples of the new model’s regularisation capabilities is compared to a local damage formulation within the EFG method.

References [1] T. Belytschko, Element-Free Galerkin Methods. International Journal for Numerical Methods in Engineering, 37, 229–256, 1994. [2] G. Pijaudier-Cabot and Z.P. Bazant. Nonlocal damage theory. Journal of Engineering Mechanics (ASCE), 118(10), 1512–1533, 1987. [3] J-S. Chen, C-T. Wu and T. Belytchko, Regualrisation of material instabilities by meshfree approximations with intrinsic length scales. International Journal for Numerical Methods in Engineering, 47, 1303-1322, 2000. [4] A. Rodriguez-Ferran, I. Morata and A. Huerta, A new damage model based on non-local displacements. International Journal for Numerical and Analytical Methods in Geomechanics, 29, 473–493, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Modelling of Reinforced Concrete Beams Strengthened With Pre-Stressed CFRP 1

2

Thiago Domingues , J. Alfaiate

1 Instituto Superior Técnico and ICIST Av. Rovisco Pais 1049-001 Lisbon, Portugal [email protected] 2 Instituto Superior Técnico and ICIST Av. Rovisco Pais 1049-001 Lisbon, Portugal [email protected]

ABSTRACT In this work a numerical simulation on the behaviour of reinforced concrete beams, strengthened with pre-stressed CFRP at the lateral faces of the beams is presented. The numerical results are compared to experimental results obtained from a testing campaign made in 2004 by França [1]. In this simulation, several hypotheses are adopted related to the material and numerical models, namely: i) cracked concrete is modelled using a discrete cracking approach and strong discontinuities embedded in the finite elements; ii) an elastoplastic behaviour is adopted for concrete under compression; iii) reinforced bars and CFRP are also modelled using an elastoplastic stress-strain relationship; iv) the bond-slip relationship adopted between the concrete and reinforcement in tension is based on the MODEL CODE 1990 and v) a mode-II bond-slip relationship is adopted between the concrete and CFRP, ac-cording to the work presented by Costa[2]. Four beams with T section are studied. Two of them are reference beams, without CFRP pre-stressed and the other two beams are reinforced with pre-stressed CFRP, before and after the support, respectively. From this simulation some conclusions are obtained, related to the verification of the above hypotheses and the behaviour to the ultimate limit and serviceability states. From the numerical analysis it has been possible to verify the failure mechanism of each beam studied, such as crushing of concrete, bond slip and/or yielding of the tension reinforcement, as well as cracking of the interface between the concrete and the CFRP pre-stressed laminate. Cracking patterns and the deformed mesh are also an output of the analysis, allowing to determine the localization of the main cracks.

References [1] P. França, A. Costa and J. Appleton, Reforço de Estruturas com Laminados de CFRP Préesforçados. Encontro Nacional de Betão Estrutural, FEUP, Porto, Portugal, 759-766, 2004. [2] R. Costa, Modelação de Vigas de Betão Armado Reforçadas com Chapas Metálicas. Dissertação de Mestrado em Engenharia de Estruturas, IST, Lisboa, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Numerical and experimental studies of damage in porous materials ∗, ¨ T. Fiedler∗ , L. Cunda† , A. Ochsner , G.J. Creus‡ , J. Gr´acio∗ ∗ Centre

†

for Mechanical Technology and Automation, University of Aveiro, Aveiro, Portugal Department of Mechanical Engineering, University of Aveiro, Portugal tfi[email protected], [email protected], [email protected]

Departamento de Materiais e Construc¸a˜ o, Fundac¸a˜ o Universidade Federal do Rio Grande, Rio Grande, Brasil. [email protected]

‡ Programa

de P´os-Graduac¸a˜ o em Engenharia Civil, Universidade Federal do Rio Grande do Sul, Porto Alegre, Brasil [email protected]

ABSTRACT Metallic foams comprise a steadily growing group of porous materials and are gaining more and more importance in industrial applications. Main advantages of this innovative material are high ability of energy adsorption and high specific stiffness. Accordingly, porous materials are utilised in lightweight construction and passive safety components which requires exact prediction of the evolution of damage and its impact on the mechanical behaviour of metallic foams. Two different approaches, namely finite element investigations and experiments, are applied. The effect of damage is mathematically described by the Gurson-Tvergaard model. This formulation is based on the isotropic damage parameter volumetric void fraction. The evolution of microscopic damage, respectively increase of volumetric void fraction, can be monitored by macroscopic parameters. Therefore, e.g. Young’s modulus is utilised for the determination of damage parameters. In order to compare the numerical findings to experimental results, the finite element analysis is confined on easy to manufacture, periodic porous geometries. The intention of this work is the development of computational and experimental procedures to analyse the damage in metallic foams. In the last instance, the determination of adequate damage parameters for foams is projected.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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The Development of a Continuum Damage Model for Fibre Metal Laminate Structures Ronan M. Frizzell, Conor T. McCarthy, Daire Cronin, Michael A. McCarthy, Ronan M. O’Higgins Composites Research Centre, Dept. of Mechanical & Aeronautical Engineering, University of Limerick, Limerick, Ireland. {ronan.frizzell, conor.mccarthy, daire.cronin, michael.mccarthy, ronan.ohiggins}@ul.ie

ABSTRACT Fibre Metal Laminates (FMLs) are a family of materials consisting of alternating layers of thin metal sheets and fibre-reinforced plastic. Glass composite based FMLs, commercially available under the name GLARE, have recently found application in the aircraft industry due to their excellent fatigue performance and impact properties. This work aims to develop a computational damage model for GLARE for use in finite element simulations. Unique challenges arise in modelling this material since it possesses both the elastic-plastic characteristics common to metals and the more brittle behaviour of glass based composite. Of particular interest to this project is the behaviour of GLARE in jointed configurations. The project proposes to develop a methodology for modelling damage initiation and growth in jointed GLARE structures and to validate the results against experimental data. An experimental investigation has been conducted on simple yet representative jointed GLARE structures. A pin-bearing test setup, without lateral constraints, has been used in order to model the central lap of a double-lap joint. Information has been gathered on the progression of damage and the failure mechanisms present in these joints. Specimens that promote bearing, net-tension and shear-out failure modes were examined. Damage in failed specimens and specimens tested to percentages of failure have been studied using microscopy. Results demonstrate that delamination between plies, matrix cracking and fibre failure are the dominant failure modes. In order to capture these complex failure modes, three-dimensional models of the pin-loaded GLARE specimens were developed in the non-linear finite element code ABAQUS. An in-house developed delamination model was used to predict the initiation and growth of delamination between the plies. This model uses three-dimensional failure criteria to predict delamination onset and a user-defined decohesion contact interface to model delamination growth. The model successfully predicted the initiation and growth of delamination in the joint configurations tested. A global-local sub-modelling approach was used to increase computational efficiency, and this is described. Current work is aimed at developing a continuum damage mechanics model to better capture the complex damage mechanisms that have been seen to occur in the composite plies. As a starting point, the Ladeveze [1] damage model has been implemented using a user-defined material subroutine UMAT, available in ABAQUS. This model is being extended to the three-dimensional case and its development and application will be described.

References [1]

P. Ladeveze, E. Le Dantec, Damage Modelling of the Elementrary Ply for Laminated Composites, Composites Science and Technology, 43, 257-267, 1992.

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Residual Strength of the Frost-Damaged Reinforced Concrete Beams Manouchehr Hassanzadeh* and Göran Fagerlund† *

Lund Institute of Technology, Division of Building Materials P.O. Box 118, SE22100, Lund, Sweden [email protected]

†

Lund Institute of Technology, Division of Building Materials P.O. Box 118, SE22100, Lund, Sweden [email protected]

ABSTRACT The most severe types of destruction mechanisms are those causing internal cracking and thereby loss of cohesion of the concrete, i.e. internal expansive attacks. The internal frost damage belongs to this category of destructive mechanisms. The frost attack causes a random system of cracks in the heart of the concrete together with cracks parallel to the surface of the concrete. In many cases there are also extended cracks parallel to joints and edges of the concrete, or emanating from corners. The amount of damage can vary from place to place in the same structure. Therefore, in most cases, series of data from many different places are required. Only in special cases data taken from one place can be used for the entire structure. Internal frost damage appears as loss in compressive and tensile strength, loss in E-modulus, and loss in the bond between the concrete and the reinforcement. Reductions in tensile and bond strength can be as high as 70%, or more. The effect on compressive strength is often limited to about 30% for normal grade concrete. The effect on E-modulus can be extremely high. Internal damage will affect the moment and shear capacity of slabs and beams, and the compression capacity of columns. It might seriously affect the structural capacity of pre-stressed concrete by significantly lowering the E-modulus of the concrete. It also changes the moment and force distribution in the structure by changing the stiffness in parts of the structure. In order to study the structural effects of the internal frost-damage large reinforced concrete beams were subjected to frost attack. The frost in combination with high degree of saturation induced internal cracks in concrete. The internal cracks reduced the strength of the reinforced beams and in some cases also changed the designed failure mode of the beams. For instance, beams which in an undamaged condition would fail due yielding of the reinforcement failed as a result of compression fracture of concrete caused by the loss of strength due to internal frost-damage. Results of this investigation show that internal frost damage, besides causing loss of strength, also causes reduction of the stiffness and extensive visible cracking. Furthermore, this investigation shows that the remaining load bearing capacity of the beams is remarkably high despite the big concrete destruction.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Numerical and experimental evaluation of damage parameters for textile reinforced concrete under cyclic loading Martin Konrad∗ , Rostislav Chudoba ∗ , Bong-Gu Kang † ∗ Chair

†

of Structural Statics and Dynamics, RWTH-Aachen University Mies-van-der Rohe-Strae 1, 52074 Aachen, Germany [email protected]

Institute of Building Materials Research, RWTH-Aachen University Schinkelstr. 3, 52062 Aachen, Germany ABSTRACT

The textile reinforced concrete (TRC) has emerged in the last decade as a new composite material combining the textile reinforcement with cementitious matrix. Its appealing feature is the possibility to produce filigree high-performance structural elements that are not prone to corrosion as it is the case for steel reinforced concrete. In comparison with other composite materials, textile reinforced concrete exhibits a high degree of heterogeneity and imperfection that requires special treatment in the development of numerical models. The prerequisite for correct modeling of the bonding behavior is the proper representation of the damage process in the crack bridges. In a way the crack bridge can modelled in matters of a tensile test on yarn with an extremely short length extended with a shear-lag-like clamping of filaments. Due to the varying penetration profile along the yarn, the quality of the shear lag clamping exhibits high scatter. In the applied deterministic and stochastic bond layer models the scatter in the interface layer and the disorder in the filament bundle are reflected by distributions (1) of the bond quality ϕ, (2) of the bond free length λ and (3) of the delayed activation θ of filaments within the bond free length. The failure process in the bond layer next to the crack bridge involves both the damage of filaments and of the bond between the filaments and the matrix. In order to quantify the separate damage laws the bond layer model has been equipped with a material model that combines plastic softening with damage evolution.

The damage laws are derived using the measured load-displacement curves of the double-sided pull-out test with cyclic loading. In combination with the FILT-test (Failure Investigation using Light Transmission) these tests yield additionally the curve representing the instantaneous fraction of the broken filaments during the loading process. This important information will be utilized to verify the damage laws.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Modeling the behavior of reinforced concrete beams strengthened with FRP P. Neto*, J. Alfaiate†, J. Vinagre† *

Escola Superior de Tecnologia do Barreiro/IPS Rua Stinville, nº 14, 2830-144 Barreiro [email protected] † Instituto Superior Técnico/UTL Av. Rovisco Pais, 1049-001 Lisboa [email protected] [email protected]

ABSTRACT The strengthening of reinforced concrete structures with fiber reinforced polymers (FRP) is particularly attractive due to their mechanical properties. The understanding of the premature failure modes is of great importance. Since rupture is frequently found to occur at the interface FRP-concrete, there is a clear need to study the nature of the bonding so as to develop techniques to permit its design modeling. The stress distribution in shear test models does not precisely match the one obtained in flexural reinforcement; in the latter, according to various authors [1], [2], in addition to the stresses tangential to the interface, normal stresses are also important. In this paper, a numerical model is presented to describe the behavior of reinforced concrete beams strengthened with FRP. This model is based on previous studies focused both on: i) the distribution of shear stresses at the interface FRP-concrete and on ii) the stress concentration at the plate ends in flexural models. Furthermore, the importance of the flexural cracks in the premature rupture of the element is also analyzed. The behavior of reinforced concrete beams strengthened with both FRP laminates and sheets is considered. The FRP-epoxy-concrete arrangement and the flexural cracks are modeled with interface elements with initial zero thickness, using a discrete approach and a localized damage model. A softening behavior is adopted to simulate the stress transfer along the FRP-concrete interface. The importance of considering the mixed mode of fracture is discussed. Mention is also made to some of the main mathematical models found in the literature.

References [1] A. M. Malek, H. Saadatmanesh, R. E. Mohammad, Prediction of failure load of r/c beams strengthened with FRP plate due to stress concentration at the plate end, ACI Structural Journal, 95(2), 142-152, 1998. [2] Z. S. Wu, H. D. Niu, Shear transfer along FRP-concrete interface in flexural members, Journal of Material, Concrete Structures and Pavements, JSCE, 49(662), 231-245, 2000.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Cracking Analysis in Concrete Dams using Isotropic Damage Models. Objectivity of Numerical Solutions Sérgio Oliveira1, Nelson Gaspar2, Pedro Dinis3 1

Laboratório Nacional de Engenharia Civil Av. do Brasil, 101, 1700-066, Lisboa, Portugal [email protected] 2 Instituto Superior de Engenharia de Lisboa R. Conselheiro Emídio Navarro, 1, 1950-062, Lisboa, Portugal

[email protected] 3 Instituto Superior Técnico Av. Rovisco Pais nº1 1049-001 Lisboa [email protected]

ABSTRACT The numerical simulation of cracking in large concrete structures can be made, in many cases, adopting the smeared cracking approach and using constitutive laws of continuous damage (with softening), in order to simulate the material tension ruptures. The consideration of a tension softening branch that depends on the value of the material fracture energy, implies the localization phenomena and requires the use of some specific numerical procedures in finite element analysis. Namely, consistent formulations evolving the energy dissipated during the rupture process must be used in order to obtain numerical results that do not dependent on the mesh discretisation – mesh objectivity. In this paper, a 3D finite element formulation and a constitutive law of isotropic damage, with two independent variables, conceived to model the tension and compression softening effects (independently), are presented. The finite element model is used in Fig A - Comparison of tensile the analysis of the Cabril Dam (the largest Portuguese arch dam) damages for the two meshes. when submitted to the self-weight and the hydrostatic pressure (water at crest level). Numerical results related with the cracks propagation for dif-ferent 3D finite element discretisations are presented, in order to analyze the solutions objectivity. These results consist of (i) the radial displacements, (ii) the principal stresses and (iii) the tensile damages at the dam (i) central cantilever and/or (ii) upstream and downstream faces (Fig.A).

References [1] Bazant, Z.P., Planas, J., Fracture and size effect in concrete and other quasi-brittle materials. CRC Press, USA, 1998. [2] Oliver, J., A consistent characteristic length for smeared cracking models. Int. J. of Numerical Methods in Engineering, Vol.28, pp. 461-474, 1989. [3] Oliveira, S.B., Modelos para análise do comportamento de barragens de betão considerando a fissuração e os efeitos do tempo. Formulações de dano (in Portuguese). Tese de doutoramento, FEUP, 2000.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Continuous-discontinuous modelling of dynamic failure of concrete using a viscoelastic viscoplastic damage model R.R. Pedersen, A. Simone and L.J Sluys Delft University of Technology, Faculty of Civil Engineering and Geosciences P.O. Box 5048, 2600 GA Delft The Netherlands [email protected]

ABSTRACT Concrete is a highly rate-dependent material at loading rates exceeding 15 GPa/s. This means that the apparent macroscopic mechanical properties of concrete depend on the applied loading rate. This has been determined, experimentally, for material strength and, to a smaller extent, for stiffness and fracture energy. The physical mechanisms responsible for the rate-dependency in high-rate dynamics are mainly inertia effects; moisture in nano- and micro pores contributes to an increase of the material parameters for moderate loading rates. Dynamic fracture of concrete is time dependent due to (i) viscoelastic material behaviour in the bulk material, and (ii) rate processes including inertia effects in the fracture process zone. In order to take the rate-dependency of concrete in dynamics into account we present a material model with viscous contribution to the bulk material in the elastic response and a viscous contribution to the cracked material. We elaborate the viscoelastic plastic model described in [2, 3] by coupling it to a viscoplastic damage model [4]. This model accounts for the strengthening effect associated with the viscous phenomenon due to moisture and includes retardation of micro-crack growth with an increase of strain rate. In the combined continuous-discontinuous approach, a crack opening is inserted after some degradation of the continuum material stiffness. Displacement discontinuities are incorporated via the partition of unity concept. The viscosity in the elastic bulk material is related to porosity and saturation level, while in the viscoplastic model the viscosity is linked to the width of the micro cracked zone. With this computational tool we examine the influence of the loading rate on the shape and size of the process zone as well as the crack velocity, and the increase in fracture energy for higher loading rates due to micro branching instabilities for crack velocities exceeding a critical velocity [1].

References [1] S.P. Gross E. Sharon and J. Fineberg. Energy dissipation in dynamic fracture. Physical Review Letters, 76:2117–2120, 1996. [2] J. Sercombe, F.-J. Ulm, and H.-A. Mang. Consistent return mapping algorithm for chemoplastic constitutive laws with internal couplings. International Journal for Numerical Methods in Engineering, 47:75–100, 2000. [3] J. Sercombe, F.-J. Ulm, and F. Toutlemonde. Viscous hardening plasticity for concrete in high-rate dynamics. Journal of Engineering Mechanics, 124:1050–1057, 1998. [4] A. Simone and L. J. Sluys. The use of displacement discontinuities in a rate-dependent medium. Computer Methods in Applied Mechanics and Engineering, 193:3015–3033, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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On the Formulation of Damage Constitutive Models for Bimodular Anisotropic Media José Julio de Cerqueira Pituba* *

Engineering Department, University of Western Sao Paulo - UNOESTE Rodovia Raposo Tavares, km 572, 19067-175, Presidente Prudente, São Paulo, Brazil [email protected]

ABSTRACT This work deals with the formulation of constitutive laws for elastic media that start to present different behaviors in tension and compression as well as some anisotropy degree when damaged. Initially, a formulation for bimodular and anisotropic elastic media proposed by [1] is reviewed. In this formulation, for the modeling of a bimodular hyperelastic material, the elastic potential energy density must be once continuously differentiable (whole wise), but only piecewise twice continuously differentiable. The stress-strain relationship derived from this potential is piecewise continuously differentiable leading to an elasticity tensor discontinuous referred to a hypersurface that contains the origin and divides the strain space into a compression and tension sub-domains. In this way, the modeling is able to reproduce different response in tension and compression. Soon after, the formulation proposed by [1] is extended to take into account the damage process. Accordingly with, the coefficients named bulk and shear modulus are considered as functions of the damage state, so that the stress-strain relationship would be influenced by damage variables. Moreover, the hypersurface taken as the criterion for the identification of the constitutive responses in compression or tension would be also influenced by the damage variables. It must be observed that the condition no tangential discontinuity of the elasticity tensor is also valid in this formulation. Fourth-order anisotropic tensors are requested by the formulation depending on the class of anisotropy that is assumed. The matricial form of those tensors is presented in the end of the work. Some damage models [2, 3] found in the literature are written according to the formalism proposed here in order to show its potentiality. It must be observed that the objective of this work is to show the potentialities of the formalism proposed here in way to be used as a proper tool in the formulation of the damage models. Therefore, some aspects related to criteria and evolution laws of damage were not discussed. This work also presents contributions related to matricial forms of the fourth-order tensors involved in the formulation of damage models. Those forms are quite interesting to the computational implementations. Finally, the proposal of the formulation for the damaged media can be used in future works for the development of many constitutive models depending of the involved phenomena.

References [1] A. Curnier, Q. He and P. Zysset, Conewise linear elastic materials. Journal of Elasticity, 37, 138, 1995. [2] J. J. C. Pituba, On the formulation of a damage model for the concrete (in portuguese). Sao Carlos School of Engineering, University of Sao Paulo, 2003. [3] C. Comi, A nonlocal damage model with permanent strains for quasi-brittle materials. A. Benallal ed. Continuous Damage and Fracture, Cachan, France, 221-232, 2000.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Soil-Structure Interaction - Case History Analysis Involving Structural Damage Luciana M. P. Rosa1, Eliane M. L. Carvalho1, Bernadete R. Danziger2 1

Universidade Federal Fluminense Niterói, RJ - Brazil [email protected] and [email protected] 2

Universidade do Estado do Rio de Janeiro Rio de Janeiro, RJ - Brazil [email protected]

ABSTRACT This thesis presents a case history contemplating a soil x structure interaction analysis of a construction that revealed unsatisfactory foundation performance with time, showing cracks due to excessive distortional settlements. The analysis includes settlement predictions with and without due consideration of soil x structure interaction with the foundation soil. The analyses contain the prediction of settlement development with time, its uniformization tendency, the columns load redistribution and the relevant changes in the stresses of some structure sections. The predicted distortional settlements with and without due consideration of soil x structure interaction were compared and confronted in conformity with the documented structure damages in different periods of the construction life time. The structure has been analysed with aid of a three-dimensional linear elastic FEM model, without considering the rheological deformation of concrete. The Kelvin model has been employed in order to properly represent the soil behaviour with time. It has been found that, notwithstanding the simplifications adopted in the analysis and tentatively justified in various phases of the study, the results obtained in the numerical analysis were capable of duplicating the damages that occurred. The conclusions emphasized the importance of a more real design conception of the structure, including the foundation soil, possible to be implemented with the computational tools available at present.

References [1] F. A. B. Danziger, Parecer Geotécnico sobre os Problemas Verificados no Edifício de Vitória, ES. COPPE/UFRJ, Rio de Janeiro, 21p, 2002. [2] F. E. Barata and B. R. Danziger, Compressibilidade de Argilas Marinhas Moles Brasileiras. Anais do VIII Congresso Brasileiro de Mecânica dos Solos e Engenharia de Fundações. Porto Alegre, 14p, 1986. [3] J. H. C. Reis and N. Aoki, Análise de interação solo-estrutura em maciço de argila mole. Seminário de Interação Estrutura-Solo em Edifícios. São Carlos, SP, 14p, 2000. [4] L. M. P. Rosa, Interação Solo-Estrutura – Análise de um Caso de Obra Envolvendo Danos Estruturais. Tese de M.Sc, Engenharia Civil, Universidade Federal Fluminense, Niterói, RJ, 117p, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Evolution Equation of Creep Damage Under Stress Variation Yukio Sanomura*, Kzutaka Saitoh† * Tamagawa University 6-1-1 Tamagawa-gakuen, Machida, Tokyo, 194-8610 Japan [email protected] †

Honda R&D Co. Ltd 4630 Simotakanezawa, Haga-machi, Haga-gun, Tochigi, 321-3321 Japan [email protected]

ABSTRACT Design and assessment of structural components at elevated temperature are very significant for ensuring the safety. Lear damage accumulation (summation of creep time fraction) is widely used to predict creep rupture time under stress and temperature variation. Life prediction of creep under stress variation by creep damage mechanics of Kachanov-Rabotnov concides with that of linear damage accumulation model. However, creep rupture time under stress variation is essentially the nonlinear problem. A modified evolution equation of creep damage by Kachanob-Rabotnov is formulated by memory effect of the previous stress. The evolution equation of creep damage consists of tow terms as follows:(1) the power damage law of Kachanov-Rabotnov, (2) acceleration anad deceleration by memory effect of the previous stress. The memory effect of the previous stress gradually disappears under the present stress. Additional internal state variable describing this effect is defined and the evolution equation is formulated in order to approach the present stress value from previous stress. The evolution equation of creep damage can be extended to the multiaxial state of stress with isotropic creep damage (scalar) and anisotropic creep damage (2nd symmetric tensor) proposed by Murakami and Ohno. Creep constitutive equation (McVetty type) for damaged material is formulated by the conventional creep damage theory. The model is lacking in the representation of the transient creep after increased stress. Micrographs of specimen ruptured under constant stress and stress variation are observed and creep damage mechanisms are discussed. After material constants are identified by describing creep curves at constant uniaxial stress, the validity of the proposed theory is discussed by the theoretical predictions with the corresponding experiments on tough pitch copper under nonsteady uniaxial stress at 250℃. The prediction of creep rupture time by the present theory is fairly good with the experiment results under stress variation.

References [1] S. Murakami, Y. Sanomura and K. Saitoh: Formulation of Cross-Hardening in Creep and its Effect on the Creep Damage Process of Copper, Trasactions of ASME, Journal of Engineering Materials, 108, 167-173, 1986.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Hybrid and Mixed Finite Element Formulations for Softening Materials Cristina M. Silva, Luís M.S.S. Castro Departamento de Engenharia Civil e Arquitectura, Instituto Superior Técnico Avenida Rovisco Pais, 1049-001 Lisboa {cmsilva,luis}@civil.ist.utl.pt ABSTRACT The conforming displacement elements are nowadays dominant in standard finite element applications. Nevertheless, they present some known limitations, particulary in what concerns accuracy and safety in stress estimates. With computational development and the motivation to model increasingly more complex structural problems, several alternative numerical techniques have been proposed to substitute or complement the tradicional displacement formulation, e.g. the boundary elements, meshless models and hybrid and mixed formulations. The hybrid and mixed finite element formulations adopted in this work [2] are developed from first-principles of Mechanics, namely, equilibrium, compatibility and constitutive relations. Recently, these formulations have been tested with continuum damage models in order to correctly simulate the behavior of softening materials such as concrete [3, 5, 6, 4]. The most promising formulations are the hybrid-mixed stress formulation, with an independent approximation of the effective stress field instead of an approximation of the stress filed [6], and the hybrid displacement formulation [4]. The objective of this communication is to compare the numerical performance of these two alternative numerical techniques with each other and also with the usually adopted displacement finite element formulation. A simple isotropic integral nonlocal damage model is adopted [1] and all the approximation functions of the hybrid-mixed formulations are chosen as complete sets of orthogonal Legendre polynomials. A set of benchmark tests are presented and discussed. It is shown that the alternative techniques may be competitive, namely in terms of stress estimates and computational effort.

References [1] C. Comi and U. Perego. Nonlocal aspects of nonlocal damage analyses of concrete strucutres. European Journal of Finite Elements, 10:227–242, 2001. [2] J. A. T. Freitas, J. P. M. Almeida, and E. M. B. R. Pereira. Non-conventional formulations for the finite element method. Computational Mechanics, 23:488–501, 1999. [3] C. M. Silva and L. M. S. S. Castro. Hybrid-mixed stress model for the nonlinear analysis of concrete structures. Computers & Structures, 83:2381–2394, 2005. [4] C. M. Silva and L. M. S. S. Castro. Modelos híbridos de deslocamento com dano contínuo. In J. L. P. Aparicio, J. César de Sá, R. Delgado, R. Gallego, J. Martins, M. Pasadas, and A. R. Ferran, editors, Congreso de Métodos Numéricos en Ingeniería. SEMNI, 2005. [5] C. M. Silva and L. M. S. S. Castro. Hybrid-displacement (trefftz) formulation for softening materials. Computers & Structures, accept for publication, 2006. [6] C. M. Silva and L. M. S. S. Castro. Hybrid-mixed stress formulation using continuum damage models. Communications in Numerical Methods in Engineering, accept for publication, 2006.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Theoretical and computational aspects of an elastoplastic damage gradient non local model Denis.L.S.Sornin1 , Kh´ema¨ıs.Saanouni1 1

Institut Charles Delaunay. Universit´e de Technologie de Troyes. FRE CNRS 2848. LASMIS 12, rue Marie Curie- BP 2060- 10000 TROYES cedex FRANCE [email protected] , [email protected] ABSTRACT

FEM results of softening material problems are known to show a pathological mesh dependency. To avoid this problem, the mechanical behavior in each integration point must take into account the immediate vicinity. Models including the influence of neigbourhood give rise to non local formulation in the early 70th. Nowadays, numerous formulations have been proposed to ensure mesh independency in post peak zone. The majority of non local models have been initially developed for concrete or brittle materials and can’t be easily generalized to ductile elastoplasticity. However, the case of damaging ductile materials have been treated in some papers ([1],[4]).These models introduce simples higher gradient formulations, able to provide mesh independent FEM results. This paper proposes a non local elastoplastic model fully coupled to damage . A thermodynamically consistent formulation involving a scalar damage field and its first gradient in the Helmholtz free energy is used. The formulation is based on a classical material theory accounting for a strong damage-behavior coupling. The strong coupling to the elastic and isotropic and kinematic hardening modulus is based on the energy equivalence assumption [2]. The local damage evolution law depends on a non local variable associated to the damage driving force [3]. This variable is the solution of a non local condition solved in a coupled fashion with the standard equilibrium equation [5]. This gives rise to a new finite element with additional DOFs. The relevant numerical aspects related to the FEM development and material integration scheme are presented. The damage gradient model is coded using both UEL and UMAT subroutines of ABAQUS/standard. Comparison between the standard local and non local formulations are discussed.

References [1] B.Nedjar, Elastoplastic-damage modelling including the gradient of damage: formulation and computational aspects, Int Journal of solids ans structures 38 (2001), 5421–5451. [2] K.Saanouni and J.L.Chaboche, Comprehencive structural integrity, ISBN: 0-08-043749-4, 2003. [3] T. Liebe P.Steinmann and A.Benallal, Theoretical and computational aspects of thermodynamically consistent framwork for geometrically linear gradient damage, Comput. Methods Appl. Mech. Engrg 190 (2001), 6555–6576. [4] M.Geers R.L.J.M.Ubachs and R.A.B.Engelen, Strongly non-local gradient-enhanced finit strain elastoplasticity, Int Journal for Numerical Methods in Engineering 56 (2003), 2039–2068. [5] R.H.J.Peerlings T.J.Massart and M.G.D.Geers, A thermodynamical motivated implicit gradient damage framework and its application to brick masonry cracking, Compt Methods Appl. Mech. Engen. 193 (2004), 3403–3417.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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On a New Framework for Anisotropic Damage Model Jian-Ying Wu∗ , Jie Li† ∗ Department

†

of Civil Engineering, South China University of Technology Guangzhou, 510640, P.R. China [email protected]

Department of Building Engineering, Tongji University Shanghai, 200092, P.R. China [email protected] ABSTRACT

Damage induced anisotropy is crucial for those initially isotropic materials, e.g., quasi-brittle materials such as concrete, geomaterials, ceramics, etc. The modeling of anisotropic damage is not a straightforward task as that of isotropic one. Despite the substantial research efforts and the noteworthy recent contributions, this problem still remains a challenging issue, among which two major shortcomings of the existing damage models are to be resolved: (i)The damage variables are to a large extent arbitrarily selected without considering their physical meanings, regarding both the nature (scalar, vector, secondorder tensor, etc.) and the number; (ii)Directly using the concept of effective stress and the hypothesis of strain equivalence generally can not guarantee the major symmetry of the derived secant stiffness, leading to non-existence of an elastic potential. Introducing the energy equivalence hypothesis partially solves this problem but the physical definition of the damage variable is lost. In this paper a novel and rigorous theoretical framework for anisotropic damage model is developed to remedy the foregoing shortcomings. This framework rests on an improved version of representation theorem[1] on (virgin and damaged) secant stiffness tensors and the equivalent thermodynamical considerations. To be more specific, damaged elastic properties are represented in terms of Fourier series expansion of the shear and bulk modulus orientation distribution functions, where two damage variables respectively characterizing the damage mechanisms in the deviatoric and volumetric spaces are inherently selected. For different degree of approximation, the geometric characters and macroscopic effects of the microdefects (microcracks and microvoids) can be described and well controlled by the selected damage variables. Corresponding to the above method, the equivalent thermodynamical formulations are established based on the concept of effective stress and the hypothesis of strain equivalence, where the Helmholtz free energy is decomposed into its deviatoric and volumetric components respectively influenced by the selected damage variables. The problems of lacking uniformity and rigor in the selection of damage variables and the incompatibility between physics and thermodynamics resulted from introducing the hypothesis of energy equivalence are thus solved. To illustrate the proposed framework in modeling anisotropic (orthotropic) and isotropic damage, the special model with a second-order tensor for the deviatoric damage and a scalar for the volumetric damage, as well as the one with two damage scalars are exemplified. Then by considering the deviatoric-volumetric coupling Helmholtz free energy, a restrictive orthotropic damage model with a single second-order damage tensor is presented, demonstrating its capability of describing the shearbulk coupling effects experimentally evidenced in those quasi-brittle materials.

References [1] Q.-C. He and A. Curnier, A more fundamental approach to damaged elastic stress–strain relations. International Journal of Solids and Structures, 32, 1433–1457, 1995.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Shell Optimization under Constraint on Damage Accumulation Nikolay V. Banichuk*, Svetlana Yu. Ivanova*, Evgeniy V. Makeev*, Alexsander V. Sinitsin* *

Institute for Problems in Mechanics, Russian Academy of Scienses Prospect Vernadskogo 101, 119526 Moscow Russia {banichuk,ivanova,makeev,sin}@ipmnet.ru

ABSTRACT The optimization problems taken into consideration consist in finding the shape and thickness distribution of brittle or quasibrittle elastic shells with cracks, loaded by fixed statical forces in such a way that the cost functional (volume of the shell material or weight) reaches a minimum, while satisfying some fracture mechanics constraints. As constraints we use bounds on stress intensity factors or structural longevity. In the case of cycling loadings we consider the number of cycles before fracture as the constraint for optimization problem. Considered optimization problems are characterized by incomplete information concerning initial crack size, crack location and its orientation. In this context the paper presents some possible formulations of optimal structural design problems, based on guaranteed (minimax) and probabilistic approaches. Some examples of analytical and numerical solutions are presented.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Design of Acceptance-Sampling Plans under Bayesian Risk João M. Casaca*, A. Silva Gomes† National Laboratory for Civil Engineering Av. do Brasil 101, 1700-066 Lisbon, Portugal * [email protected], † [email protected]

ABSTRACT The paper deals with generic situations where a consumer acquires a product which he wants to be conforming to certain specifications. The product is delivered by lots and, on the reception of every lot, to be sure that the product is conforming to the specifications, the consumer inspects a sample of the lot and accepts or rejects it according to the results of the inspection. The consumer and the producer must negotiate previously an acceptance-sampling plan [1], contemplating the sampling strategy, the size of the sample and the acceptance rules. As the acceptance of a defective lot implies a loss for the consumer and the rejection of a lot, whether conform or defective, implies a loss for the producer, the acceptance plan must minimize simultaneously expected losses for both the consumer and the producer. The paper presents a model to design acceptance-sampling plans, taking into account the quality level of the lots and using Bayes risk [2] as a criterion to balance the expected losses for the consumer and for the producer. The model is based on an operational characteristic function [1] derived from the binomial distribution and uses acceptance rules similar to a statistical test of hypothesis. The quality level of the lot is modelled by a prior probability density function so that the significance level and the power of the acceptance-sampling plan, may be computed and incorporated in consumer and producer risk functions, for different quality backgrounds. The model has been previously applied to design acceptance-sampling plans for the objects and attributes of Geographic Information Systems and for positional errors of large scale Topographical Maps. The application of the model to the construction quality control of large fill dams is currently being studied. The fill dams are built by placing successive thin layers of compacted materials, each layer being inspected before placing the next one. The rejected layers must be removed. The consumer (the dam owner) does not want defective layers to be accepted. The producer (the contractor) does not want any layers to be rejected. The design of an acceptance-sampling plan minimizing the consumer’s and the producer’s expected losses is of utmost importance for both. References [1] D. Montgomery, Statistical Quality Control. John Wiley & Sons, New York, 1991. [2] V. Barnett, Comparative Statistical Inference. John Wiley & Sons, New York, 1975.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Optimal design for the worst case scenario Elena Cherkaev∗ , Andrej Cherkaev† ∗ University of Utah, Department of Mathematics 155 South 1400 East, JWB 233, Salt Lake City, UT 84112 [email protected] † University of Utah, Department of Mathematics 155 South 1400 East, JWB 233, Salt Lake City, UT 84112 [email protected]

ABSTRACT The talk discusses a problem of robust optimal design of elastic structures when the loading is unknown, and only an integral constraint for the loading is given [1]. The optimal design problem is formulated as minimization of the principal compliance of the domain equal to the maximum of the stored energy over all admissible loadings. The principal compliance is the maximal compliance under the extreme, worst possible applied force [2]. The robust optimal design is a min-max problem for the energy stored in the structure. The minimum is taken over the design parameters, while the maximum of the energy is chosen over the constrained class of loadings. It is shown that the problem for the extreme loading is reduced to an elasticity problem with mixed nonlinear boundary conditions; the last problem may have multiple stationary solutions. The optimization takes into account the possible multiplicity of extreme loadings and designs the structure to equally resist all of them. Continuous change of the loading constraint causes bifurcation of the solution of the optimization problem. It is shown that an invariance of the constraints under a symmetry transformation leads to a symmetry of the optimal design. Examples of optimal design are investigated; symmetries and bifurcations of the solutions are discussed.

References [1] A. Cherkaev and E. Cherkaev, Optimal design for uncertain loading conditions. In: Homogenization, V. Berdichevsky, V. Jikov, and G. Papanicolaou, eds., World Scientific, 193–213, 1999. [2] E. Cherkaev and A. Cherkaev, Principal compliance and robust optimal design, Journal of Elasticity, 72, 1-3, 71–98, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

381

Dimension Reduction Method for Reliability-Based Robust Design Optimization K.K. Choi*, Ikjin Lee*, and David Gorsich† *

Department of Mechanical and Industrial Engineering The University of Iowa, Iowa City, IA 52242, U.S.A. [email protected] [email protected] † AMSTA-TR-N (MS 263) US Army National Automotive Center Warren, MI 48397-5000, U.S.A. [email protected]

ABSTRACT In reliability-based robust design optimization formulation, the product quality loss function is minimized subject to probabilistic constraints. Since the quality loss function is expressed in terms of the first two statistical moments, mean and variance, several methods have been proposed to accurately and efficiently estimate the statistical moments. However, it is computationally expensive to calculate the statistical moments of the output performance function using the multidimensional integral, especially when the number of the random input variables is relatively high. To overcome the shortcomings, three methods have been recently proposed: univariate dimension reduction method (DRM) [1], performance moment integration (PMI) method [2], and percentile difference method (PDM) [3]. In this paper, a reliability-based robust design optimization method is developed using DRM and compared to PMI and PDM for the robust design part. It is found that PDM cannot estimate the statistical moments of the performance function accurately. The PMI and DRM are also compared in terms of accuracy and efficiency in estimation of statistical moments of the performance function. Several numerical examples are used to compare accuracy and efficiency of these methods. The numerical results show that DRM is effective when the number of random variables is small, whereas PMI is more effective when the number of random variables is relatively large. For the inverse reliability analysis, the enhanced hybrid mean value (HMV+) method is used, whereas the enriched performance measure approach (PMA+) is used for reliability-based design optimization.

References [1] Xu, H., and Rahman, S., “A Univariate Dimension-Reduction Method for Multi-dimensional Integration in Stochastic Mechanics,” Probabilistic Engineering Mechanics, Vol. 19, No. 4, pp. 393-408, 2004 [2] Youn, B. D., Choi, K. K., and Yi, K., "Performance Moment Integration (PMI) Method for Quality Assessment in Reliability-Based Robust Optimization," Mechanics Based Design of Structures and Machines, Vol. 33, No. 2, pp. 185-213, 2005. [3] Du, X., Sudjianto, A., and Chen, W., “An Integrated Framework for Optimization Under Uncertainty Using Inverse Reliability Strategy,” ASME Journal of Mechanical Design, Vol 126, No. 4, pp. 562-570, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A Fatigue Life Reliability-based Design Optimization of a Slat Track using Mesh Morphing Roberto d’Ippolito1, Stijn Donders1, Luc Hermans1, Michael Hack2, Joost Van de Peer3 and Nick Tzannetakis3 1

LMS International Interleuvenlaan 68, B-3001 Leuven, Belgium [email protected], [email protected], [email protected] 2

LMS Deutschland GmbH Kaiserslautern, Germany [email protected] 3 Noesis Solutions Interleuvenlaan 68, B-3001 Leuven, Belgium [email protected], [email protected]

ABSTRACT Although the aerospace production process is much better controlled than the process in other industries, it remains true that very small manufacturing tolerances exist in the geometrical parameters (flange thicknesses, hole diameters, …). In the current design process, the effect of this manufacturing variability on the structural durability and safety cannot be accurately assessed and is hence compensated for by applying safety factors. This is not an ideal situation, as it may lead to slightly over-designed structures. A much more promising approach is to include probabilistic models of design variables into the mechanical simulation process. Then, with a new methodology based on reliability analysis, engineers can obtain a better understanding of the actual effect of the manufacturing tolerances. Based on the analysis results, the robustness and reliability of the design can be assessed and improved if needed. In this paper, the above-mentioned probabilistic approach is demonstrated on a slat track structure. Measurements of different geometrical properties have been collected during the manufacturing process and their variability has been characterized probabilistically with statistical models. Then, a reliability analysis has been carried out using morphing technology and fatigue life predictions with an industrial-sized FE model of the slat track to assess the reliability of the structure in terms of fatigue life. The outcome of the analysis consists of a probabilistic model of the fatigue life, given the variability in the geo-metrical parameters. This analysis not only provides a better insight in the effect of variability in the fatigue life prediction, but also provides sensitivity measurements of the design parameters on the final performance of the structure. These results provide guidelines to improve structural designs and manufacturing tolerances, by using a reliability-based design optimization procedure. A powerful tool is thus obtained to reduce design conservatism while maintaining and even improving structural safety.

References [1] R.E. Melchers, Structural Reliability Analysis and Prediction, 2nd Edition, John Wiley & Sons, UK, 1999. [2] B.D. Youn, K.K. Choi, Liu Du, Adaptive Probability Analysis Using An Enhanced Hybrid Mean Value Method, Journal of Structural and Multidisciplinary Optimization, vol. 29, no. 2, 2004, pp. 134-148, Springer Berlin Heidelberg.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

383

A New Approach of Robust Design Based on the Concept of Allowable Load Set Byung Man Kwak*, Jinho Chang*, Jae Hyun Kim† *Korea Advanced Institute of Science and Technology 373-1 Guseong-dong, Yusoeng-gu Daejeon, 305-701 Republic of Korea {bmkwak,jhchang}@khp.kaist.ac.kr † LG Electronics Co., Digital Appliance Research Lab. 327-23 Gasan-dong, Geumcheon-gu, Seoul, 153-802, Korea [email protected]

ABSTRACT A concept called “Allowable load set (ALS)” developed by the authors allows designers to view a design process in a completely different direction. In the usual structural design, a load is given at a point and the size or shape of a structure is to be found to support the given load. Now an allowable load set denotes the set of loads that are safe to a given structure. The design problem is to find the most suitable allowable load set by adjusting the size or shape. The ALS can be visualized graphically for simple cases, and the integrity of the structure for a given design load can be seen visually. To quantify the structural integrity, two measures are devised: one is a relative safety index (RSI) denoting the distance from the mean design load vector to the boundary of the ALS, and another is a normalized safety index (NSI) which is just the value of a performance function or a constraint function. This latter involves no optimization of finding a distance as required in the usual reliability index approach. Secondly ALS can be applied easily to multi-body mechanical systems, especially efficiently for linear elastic material. Examples thus include illustrations from three bar truss and torque arm design. A robust design is to obtain a design which is most insensitive to uncertainties. In robust design optimization, no probability information of uncertainties is assumed given. The ALS design approach is a new method of achieving this goal. Another challenging example is applying the ALS design method to find trajectory of motions of a biomechanical system. The philosophy is that a man will take a motion such that its configuration at any instant is as far as possible from danger when taken naturally. As will be shown, this gives natural motions of lifting a heavy object with or without low back pain. Especially the NSI formulation is very efficient. The ALS design concept is innovative and well applicable to real world problems. The RSI formulation is physically more meaningful but requires much larger computational cost than the NSI. One disadvantage of NSI is that a suitable scaling may be necessary sometimes to obtain right proportions among constraints, but no simple way is found yet. Comparative study of ALS and other methods of robust design will be given in addition to the theory and numerical applications.

References [1] B. M. Kwak and J. H. Kim, “Concept of allowable load set and its application for evaluation of structural integrity,” Journal of Mechanics of Structures and Machines, 30, 213-247, 2002

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

384

A sampling technique enhancing accuracy and efficiency of metamodel-based RBDO: constraint boundary sampling Tae Hee Lee*, Jae Jun Jung*, Do Hyun Jung† * Hanyang University School of Mechanical Engineering, Sungdong-ku, Seoul 133-791, Korea [email protected] [email protected] † Korea Automotive Technology Institute Body and Chassis Engineering Center, Chonan, Chungnam 300-912, Korea [email protected]

ABSTRACT Reliability-based design optimization (RBDO) has been developed to consider uncertainty of input design variables during optimization process. To provide the reliability, reliability index approach (RIA) and performance measure approach (PMA) are often used.[1, 2] However, these reliability analyses usually require extremely expensive computational costs due to many simulation runs. Thus, it is necessary to reduce significantly the number of actual simulation runs during RBDO. Metamodels such as response surface model and kriging model are investigated for this purpose [3]. Metamodel for computer simulation is often built from space-filling sampling that evenly locates sample points within whole design domain. However, it requires considerably many sample points to approximate probabilistic constraints throughout whole design region when constraints reveal nonlinearity and when feasible region is small compared to whole design region. In this research, constraint boundary sampling technique is proposed to maximize accuracy and efficiency of metamodel-based RBDO. Constraint boundary sampling is sequentially to locate sample points around constraint boundary by using kriging metamodel and its mean squared error. To verify the proposed method, mathematical examples are performed and their accuracy and efficiency are compared to those obtained from classical space-filling design. Through this study, we learn that RBDO using kriging model under the constraint boundary sampling technique coincides precisely with the exact solutions. Moreover, the efficiency of RBOD is improved so that RBDO using constraint boundary sampling technique can reduced by about 50% compared to conventional RBDO in the number of actual response analysis.

References [1] Yu, X., Chang, K.H., and Choi, K.K., Probabilistic Structural Durability Prediction, AIAA Journal, 36, 628-637, 1998. [2] Tu, J. and Choi, K.K., A New Study on Reliability Based Design Optimization, Journal of Mechanical Design, ASME, 121, 557-564, 1999. [3] Choi, K. K., You, B.D., and Yang, R.J., Moving Least Square Method for Reliability-Based Design Optimization, 4th World Congress of Structural and Multidisciplinary Optimization, Dalian, China, June 4-8, 2001.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

385

Interval sensitivity analysis of dynamic response envelopes for uncertain mechanical structures David Moens∗ , Dirk Vandepitte∗ ∗ K.U.Leuven,

Dept. Mechanical Engineering, PMA Celestijnenlaan 300B, B-3001 Heverlee, Belgium [email protected]

ABSTRACT Non-deterministic approaches are gaining momentum in the field of finite element analysis. The ability to include non-deterministic properties is of great value for a design engineer. It enables realistic reliability assessment that incorporates the uncertain aspects of the design. Furthermore, the design can be optimised for robust behaviour under varying external influences. Recently, criticism arises on the general application of the probabilistic concept in this context. Especially when objective information on the uncertainties is limited, the subjective probabilistic analysis result proves to be of little value, and does not justify its high computational cost. Consequently, alternative non-probabilistic concepts are used for non-deterministic finite element analysis, as e. g. the fuzzy and interval concept. Recently, a fuzzy finite element methodology to calculate a fuzzy frequency response function (FRF) of uncertain undamped structures was developed by the authors. The procedure consists of the solution of a sequence of interval problems. The goal of each interval analysis is to calculate the envelope of the FRF taking into account that the input uncertainties can vary within the bounded space defined by their combined intervals. The resulting envelope response function gives a clear view on the possible variation of the response in the frequency domain. The applicability of this interval response analysis was proven on realistic case studies. In this result, all uncertain parameters are considered to act simultaneously. While this is often the most realistic representation of the physical condition of the actual product, for design purposes, it can be of great value to know the contribution of the individual uncertainties to the response range obtained from the interval analysis. This enables a designer to distinct between the non-deterministic influences that have an important contribution to the fuzziness on the dynamic behaviour of the design, and those that have little or no influence. This distinction can be very valuable in the definition of e. g. tight design tolerances or realistic allowable working conditions, and as such, could lead to less conservative designs. This paper introduces an interval sensitivity procedure that calculates the sensitivity of the envelope response function in the outcome of the interval FRF analysis to each individual interval model uncertainty. The procedure focusses on the calculation of the sensitivity of the bounds defining the FRF response range to the width of each individual uncertain input parameter. The approach differs from the classical sensitivity analysis in the fact that it does not calculate local changes on the output resulting from local changes in the input. The interval sensitivity result describes the change of the response interval width, taking into account a change in the parameter interval width. The paper first describes the theoretical background of the fuzzy and underlying interval FRF procedure. Next, it introduces the methodology for interval sensitivity analysis. Finally, the method is illustrated on a numerical example.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

386

Multi-Objective Robust Design Optimization of an Engine Crankshaft Carlo Poloni*, Paolo Geremia†, and Alberto Clarich† *

Università di Trieste-Dipartimento di Energetica via Valerio 10, 34100 Trieste [email protected]

† Esteco srl Area Science Park, Padriciano 99, 34100 Trieste {geremia, clarich}@esteco.it

ABSTRACT When designing a commercial product, engineers have to meet several requirements which boil down to finding the better performances and the higher reliability as possible. Another significant factor that determines product quality is its sensitivity to external or uncontrollable variations. This methodology of design is generally called Robust Design [1]. This paper shows an application of Robust Design methodology to a multi-disciplinary optimization of an engine crankshaft by considering uncertainties in terms of manufacturing errors over the shaft dimensions as well as dynamic loads variability. The application is run using ANSYS Workbench solver and modeFRONTIER [2], through a direct interface between the two codes that has been recently developed. A full Robust Design analysis is applied in order to check the stability of the best candidate solutions according to uncertainties in terms of both manufacturing errors and forcing loads The results obtained are very encouraging, and the procedure described can be applied, in principle, to even more complex problems.

References [1] Clarich A., Pediroda V., Poloni C., A competitive Game Approach for Multi Objective Robust Design Optimization, AIAA 2004-6511, Chicago, 20-22 September 2004 [2] www.esteco.com

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

387

Development and Application of A New Metropolis GA for the Structural Design Optimization Yeon-Sun Ryu Dept of Ocean Engineering, Pukyong National University, Busan, Korea [email protected]

ABSTRACT A Metropolis genetic algorithm (MGA) is developed and applied for the structural design optimization. In MGA, favorable features of Metropolis criterion in simulated annealing (SA) are incorporated in the reproduction operations of simple genetic algorithm (SGA). This way, the MGA maintains the wide varieties of individuals and preserves the genetic information of early generations. Consequently, the proposed MGA alleviates the disadvantages of finding imprecise solution in SGA and time-consuming computation in SA. Performances of MGA are compared with those of conventional algorithms such as Holland's SGA, Krishnakumar's micro genetic algorithm (µGA), and Kirkpatrick's SA. Typical numerical examples are used to evaluate the favorable features and applicability of MGA. The effects of population sizes and maximum generations are also evaluated for the performance reliability of MGA. From the theoretical evaluation and numerical experience, it is concluded that the proposed MGA is a reliable and efficient tool for structural design optimization.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

388

Stochastic Response Surface Using the Enhanced Dimension-Reduction (eDR) Method for Reliability-Based Robust Design Optimization Byeng D. Youn1, Zhimin Xi1, Lee J. Wells1, and David A. Lamb2 1

2

Department of Mechanical Engineering and Engineering Mecha Michigan Technological University Houghton, MI 49931 {bdyoun,zxi,ljwells}@mtu.edu

National Automative Center (NAC), U.S. Army RDECOM-TARDEC Warren, MI 48397-5000 [email protected]

ABSTRACT As the reliability analysis and design methodology has been advanced, its implementation becomes more complicated to improve computational efficiency and stability. Furthermore, most reliability analysis methods in RBDO require gradient (or sensitivity) information [1]. Therefore, this paper attempts to develop a stochastic response surface method. The method makes it possible to perform sensitivity-free RBDO using any deterministic optimizer. Recently, the dimension reduction (DR) method has been proposed [2]. Although the DR method is known to be an accurate and efficient method for the uncertainty quantification (UQ) of system responses, it may produce a relatively large error for the second-order or higher moments of nonlinear responses. Thus, this paper first proposes the enhanced dimension-reduction (eDR) method [3] by incorporating two alternative integration schemes and one-dimensional response approximations. Both moment based quadrature rule and an adaptive Simpson integration rule are alternatively used for numerical integration. The stepwise moving least squares (SMLS) method is proposed for response approximation. The SMLS is based on a moving least squares (MLS) method. Secondly, the paper proposes a stochastic response surface method. The stochastic response surface is built using the SMLS method with the results of the eDR method at sampled designs. In aid of the stochastic response surface method, RBDO or robust design optimization can be performed with commercial (deterministic) optimization softwares (e.g., Microsoft Excel, Matlab, etc.). In this paper, some examples are used to demonstrate the eDR method and further the stochastic response surface method for RBDO.

References [1] Youn, Byeng D., Choi, K. K., and Du, L., “Enriched Performance Measure Approach (PMA+) for Reliability-Based Design Optimization,” AIAA Journal, Vol. 43, No. 4, pp. 874-884, 2005. [2] Rahman, S. and Xu, H., "A Univariate Dimension-Reduction Method for Multi-Dimensional Integration in Stochastic Mechanics," Probabilistic Engineering Mechanics, Vol. 19, pp. 393-408, 2004. [3] Youn, B.D., Zhmin, X., Wells, L., and Lamb, D., "The Enhanced Dimension Reduction (eDR) Method for Reliability-Based Robust Design Optimization," 11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Accepted, Portsmouth, Virginia, Sep. 6-8, 2006.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

389

DEM analysis of granular flow in pyramidal hoppers R. Baleviìius*, R. Kaìianauskas* *

Laboratory of Numerical Modelling, Vilnius Gediminas Technical University, Vilnius, Lithuania [email protected] [email protected]

ABSTRACT The processes of granular material handling in the hoppers are of a great importance in pharmaceutical, chemical, food and other industries. Theoretical treatments of such problems are usually based on simplified continuum models, which are useful to predict the stress field within the hopper, especially, on the walls at the end of filling process. The Continuum approach has some drawbacks for discharge state modeling when description of transient flow is required. Consequently, discrete element method (DEM) based on the application Newton’s and contact mechanics laws predicting dynamical parameters, such as position, velocity, etc., of individual particles, have been adopted in numerical analysis of granular flow in hoppers. Filling and discharge flow in pyramidal hoppers of different shape is considered by the discrete element method. Non-cohesive frictional visco-elastic spherical particles are applied in modelling. The boundary conditions are described by using locally oriented planes of a finite size enabling to handle different shapes of hoppers. Evolution of the system kinetic energy, discharge mass fraction as well as distribution of particle velocities and material porosity fields is considered. Geometry of the hopper, particle contact forces and the velocity fields during discharge are presented in Fig. 1.

a)

b)

Fig. 1 Hopper geometry, particle flow (a) and velocity fields (b) during discharge at t=1.5 s

The DEM concept is implemented into original software code DEMMAT [1], where the 5th-order Gear’s predictor-corrector scheme is used for numerical integration of equations of motion. Particular postprocessors for evaluation of various field variables are also developed.

References [1] Baleviìius R, Kaìianauskas R, Džiugys A, Maknickas A, Vislaviìius, DEMMAT code for numerical simulation of multi-particle dynamics. Information Technology and Control, 34(1), 71-78, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

390

DQEM and DQFDM for Computational Mechanics Problems Chang-New Chen Department of Systems and Naval Mechatronic Engineering National Cheng Kung University, Tainan, Taiwan, ROC [email protected]

ABSTRACT Because only problems having simple regular domains and under simple external environments can be solved by using DQ, the application of this method is very limited. The author has proposed the DQEM for solving a generic engineering or scientific problem having an arbitrary domain configuration. Like the FEM, in this method, the analysis domain of a problem is first separated into a certain number of subdomains or elements. Then the DQ, GDQ or EDQ discretization is carried out on an element-basis. The governing differential or partial differential equations defined on the elements, the transition conditions on inter-element boundaries, and the boundary conditions on the analysis domain boundary are in computable algebraic forms after the DQ, GDQ or EDQ discretization. By assembling all discrete fundamental equations an overall algebraic system can be obtained which is used to solve the problem. The DQFDM has also been proposed by the author. The finite difference operators are derived by DQ. They can be obtained by using the weighting coefficients for DQ discretizations. The derivation is straight and easy. By using different orders or the same order but different grid DQ discretizations for the same derivative or partial derivative, various finite difference operators for the same differential or partial differential operator can be obtained. Finite difference operators for unequally spaced and irregular grids can also be generated through the use of GDQ. DQEM and DQFDM have been used to develop solution algorithms for computational mechanics. In this paper, numerical results are presented to demonstrate these two discrete analysis methods.

References [1] R.E. Bellman and J. Casti, Differential quadrature and long-term integration. Journal of Mathematical Analysis and Applications, 34, 234-238, 1971. [2] C.N. Chen, A differential quadrature element method. Proceedings of the First International Conference on Engineering Computation and Computer Simulation, Changsha, China, 1, 25-34, 1995. [3] C.N. Chen, A differential quadrature finite difference method. Proceedings of the First International Conference on Advanced Computational Methods in Engineering, Gent, Belgium, 1, 713-720, 1998. [4] C.N. Chen, DQEM and DQFDM for the analysis of composite two-dimensional elasticity problems.Composite Structures, 59, 3-13, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

391

An innovative truly–mixed method for cohesive–crack propagation problems C. Cinquini∗ , M. Bruggi∗ , Paolo Venini∗ ∗ University

of Pavia, Department of Structural Mechanics via Ferrata 1, I27100 Pavia, Italy {cinquini,matteo.bruggi,paolo.venini}@unipv.it ABSTRACT

We propose a novel approach for the analysis of cohesive crack propagation in elastic media. Unlike all existing methods that move from continuous displacement formulations that are properly enriched to handle the discontinuity, see e.g. the extended finite element method (XFEM) [Mo¨es et al., 1999] or the embedded discontinuity [Jir´asek, 2000] approaches, inherently discontinuous displacements and H(div) stresses in a truly mixed setting are herein proposed. The formulation, originally introduced to handle incompressible materials in plane elasticity, is herein extended to the analysis of propagating cohesive cracks in elastic media thanks to a novel variational formulation that is enriched with an interface energy term. Notably, no edge element is introduced but simply the inherent discontinuity of the displacement field is taken advantage of. Furthermore, stress flux continuity is imposed in an exact fashion within the formulation and not as an additional weak constraint as classically done. Extensive numerical simulations are presented to complete the theoretical framework.

References [Jir´asek, 2000] Jir´asek, M., 2000. Comparative study on finite elements with embedded cracks, Computer Methods in Applied Mechanics and Engineering, 188, 307–330. [Mo¨es et al., 1999] Mo¨es, N., Dolbow, J., Belytschko, T., 1999. A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering, 46, 131– 150.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

392

Compactly Supported Fundamental Functions for Spline-Based Differential Quadrature ˜ P´erez† Domingo Barrera Rosillo∗ , Francisco Ib´anez ∗ Departamento de Matem´ atica Aplicada ETS de Ingenieros de Caminos, Canales y Puertos Universidad de Granada Campus Universitario de Fuentenueva, 18071-Granada, Spain [email protected] † Grupo Sacyr-Vallehermoso Paseo de la Castellana, 83, 28046-Madrid, Spain fi[email protected]

ABSTRACT The Differential Quadrature Method (DQM) is a numerical discretization technique for the approximation of derivatives by means of weighted sums of function values. It was proposed by Bellman and coworkers in the early 1970’s, and it has been extensively employed to approximate spatial partial derivatives (cf. [1], [4] for instance). The classical DQM is polynomial-based, and it is well known that the number of grid points involved is usually restricted to be below 30. Some spline based DQMs have been proposed to avoid this problem, but the construction of these schemes depends strongly on the degree of the considered B-spline (see for instance [2] and [5]). In this work we present a general DQM based on interpolation and quasi-interpolation. Firstly, we consider the construction of compactly supported cardinal functions L that interpolate the Kronecker sequence. They are linear combinations of translates of a B-spline Mn centered at the origin. Then, we revise some spline discrete quasi-interpolants defined from the same B-splines. We are interested in some recently defined and analyzed quasi-interpolants (cf. [3]). They are constructed by minimizing an error constant appearing in a particular expression of the quasi-interpolation error for regular enough functions. Finally, both the interpolants and the quasi-interpolants are used to define new interpolants having compactly supported fundamental functions again, and the maximal order of approximation, and the quintic case is described and compared with the results obtained in [5].

References [1] C. W. Bert and M. Malik, Differential quadrature method in computational mechanics: a review. Appl. Rev., 49, 1–27, 1996. [2] Q. Guo, H. Zhong, Non-linear vibration analysis y a spline-based differential quadrature method. Journal of Sound and Vibration, 269, 413–420, 2004. [3] M. J. Ib´an˜ ez-P´erez, Quasi-interpolantes spline discretos de norma casi m´ınima. Teor´ıa y aplicaciones. Doctoral Dissertation, University of Granada, 2003. [4] C. Shu, Differential Quadrature and its applications in Engineering. Springer-Verlag, London, 2000. [5] H. Zhong, Spline-based differential quadrature for fourth order differential equations and its applications to Kirchhoff plates. Applied Mathematical Modelling, 28, 353–366, 2004.

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Differential Quadrature Solution for Parabolic Structural Shell Elements Francesco Tornabene*, Erasmo Viola† *

DISTART - Department, Faculty of Engineering, University of Bologna Viale Risorgimento 2, 40136 Bologna, Italy [email protected]

†

DISTART - Department, Faculty of Engineering, University of Bologna Viale Risorgimento 2, 40136 Bologna, Italy [email protected]

ABSTRACT This work deals with the dynamical behaviour of complete parabolic shells of revolution and parabolic shell panels. The First-order Shear Deformation Theory (FSDT) is used to analyze the above moderately thick structural elements. The treatment is conducted within the theory of linear elasticity, when the material behaviour is assumed to be homogeneous and isotropic. The governing equations of motion, written in terms of internal resultants, are expressed as functions of five kinematic parameters, by using the constitutive and the congruence relationships. The boundary conditions considered are clamped (C) and free (F) edge. Numerical solutions have been computed by means of the technique known as the Generalized Differential Quadrature (GDQ) Method. The solution is given in terms of generalized displacement components of the points lying on the middle surface of the shell. At the moment it can only be pointed out that by using the GDQ technique the numerical statement of the problem does not pass through any variational formulation, but deals directly with the governing equations of motion. Referring to the formulation of the dynamic equilibrium in terms of harmonic amplitudes of mid-surface displacements and rotations, in this paper the system of second-order linear partial differential equations is solved, without resorting to the onedimensional formulation of the dynamic equilibrium of the shell. The discretization of the system leads to a standard linear eigenvalue problem, where two independent variables are involved. Several examples of parabolic shell elements are presented to illustrate the validity and the accuracy of GDQ method. The convergence rate of the natural frequencies is shown to be very fast and the stability of the numerical methodology is very good. The accuracy of the method is sensitive to the number of sampling points used, to their distribution and to the boundary conditions. The effect of the distribution choice of sampling points on the accuracy of GDQ solution is investigated. GDQ results, which are based upon the FSDT, are compared with the ones obtained using commercial programs such as Ansys, Femap/Nastran, Abaqus, Straus, Pro/Engineer.

References [1] E. Reissner, The effect of transverse shear deformation on the bending of elastic plates. Journal of Applied Mechanics ASME 12, 66-77, 1945. [2] E. Viola and E. Artioli, The G.D.Q. method for the harmonic dynamic analysis of rotational shell structural elements. Structural Engineering and Mechanics 17, 789-817, 2004. [3] C. Shu, Differential Quadrature and Its Application in Engineering. Springer, Berlin, 2000.

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Multiple Crack Growth failure in Cortical Bone under Tension by the eXtended Finite Element Method Elisa Budyn∗, Laurent Henry† , Thierry Hocr† ∗ University of Illinois at Chicago 842 West Taylor Street, Chicago, IL 60607, USA [email protected] † Ecole Centrale Paris Grande Voie des Vignes, 92295 Chatenay Malabry, France [email protected], [email protected]

ABSTRACT A multi-scale analysis for multiple crack growth in unit cell of cortical bone is presented. The cracks are grown until complete failure of the cell. The initial cracks are placed in maximum strain locations. The stress intensity factors are computed at each crack tip and a load parameter is adjusted so that the stress intensity factors remain at the critical value. In the case of competitive crack tips, a stability analysis is performed by computing the second derivative of the potential energy for each crack. The load deflection behavior of the representative volume element is obtained until the point of complete failure. The model is fed with experimental geometrical, mechanical and damage parameters and validated through a comparison with experimental samples. The discretization utilizes the eXtended Finite Element Method and requires no remeshing as the cracks grow. The crack geometries are arbitrary with respect to the mesh, and are described by a vector level set. Special boundary conditions and the algorithm for detecting crack bridging and crack entering Haversian canals which allows the cracks to grow until maximum failure and/or percolation is presented.

References [1] E. Budyn, G. Zi, N. Mo¨es and T. Belytschko, A Method for Multiple Crack Growth in Brittle Materials without Remeshing, Int. J. for Num. Meth. in Eng., 61, Number 10, pp. 1741-1770,2004. [2] T. Belytschko and T. Black, Elastic Crack Growth in Finite Elements With Minimal Remeshing, Int. J. for Num. Meth. in Eng., 45, Number 5, pp. 601-620.1999.

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Subdivision Shells Fehmi Cirak California Institute of Technology Center for Advanced Computing Research, Pasadena, CA 91125, U.S.A. [email protected] ABSTRACT Subdivision elements, as originally introduced by Cirak, Ortiz, and Schr¨oder [5], provide a new paradigm for thin-shell finite-element analysis based on the use of subdivision surfaces for (i) describing the geometry of the shell in its undeformed configuration, and (ii) generating smooth interpolated displacement fields. The displacement fields obtained by subdivision are H 2 and, consequently, have a finite Kirchhoff-Love energy. The displacement field of the shell is interpolated from nodal displacements only and no nodal rotations are used. The interpolation scheme induced by subdivision is nonlocal, i.e. the displacement field over one element depends on the nodal displacements of the three element nodes and all nodes of immediately neighboring elements. However, the use of subdivision schemes ensures that all the local displacement fields combine conformingly to define one single limit surface. Numerical tests, demonstrate the high accuracy and optimal convergence of the method even in highly nonlinear problems [4]. Furthermore, because of the unification of representations for mechanics and geometric modeling (i.e. CAD: Computer Aided Design), subdivision elements are ideally suited to applications in shape optimization [3]. Recently, specialized cohesive elements have been developed that account for in-plane tearing, shearing, and hinge modes of shell fracture [1]; and methods for coupling subdivision shells to gas dynamics [2].

References [1] F. Cirak, M. Ortiz and A. Pandolfi. A Cohesive Approach to Thin-Shell Fracture and Fragmentation. Computer Methods in Applied Mechanics and Engineering, 194, 2604–2618, 2005. [2] F. Cirak and R. Radovitzky. A Lagrangian-Eulerian Shell-Fluid Coupling Algorithm Based on Level Sets. Computers & Structures, 83, 491–498, 2005. [3] F. Cirak, M.J. Scott, E.K. Antonsson, M. Ortiz and P. Schr¨oder. Integrated Modeling, Finite-Element Analysis, and Engineering Design for Thin-Shell Structures Using Subdivision Computer-Aided Design, 34, 137–148, 2002. [4] F. Cirak and M. Ortiz. Fully C 1 -Conforming Subdivision Elements for Finite Deformation ThinShell Analysis International Journal for Numerical Methods in Engineering, 51, 813–833, 2001. [5] F. Cirak, M. Ortiz and P. Schr¨oder. Subdivision Surfaces: A New Paradigm for Thin-Shell FiniteElement Analysis International Journal for Numerical Methods in Engineering, 47, 2039–2072, 2000.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Capturing Slip Weakening and Variable Frictional Response in Localizing Geomaterials Using an Enhanced Strain Finite Element Craig D. Foster*, Ronaldo I. Borja† * Stanford University Blume Earthquake Engineering Center MC 4020 Stanford, CA 94305 [email protected] † Stanford University Terman Engineering Center MC 4020 Stanford, CA 94305 [email protected]

ABSTRACT The formation and propagation of faults, cracking of concrete structures, and progressive fracture in ceramics are examples of localized deformation in quasi-brittle geomaterials. In these materials, localization tends to take the form of a fractured surface rather than a deformation band of finite width. In order to capture the propagation and post-localization slip of these surfaces, we must properly model their behavior. The constitutive response along the surface exhibits two distinct phases. The first is slip weakening, in which the shear strength degrades in an approximately linear fashion with slip displacement. This degradation corresponds to a loss of cohesive strength as a coherent macrocrack forms, and takes place over small displacements, on the order of 0.5 mm. After the complete loss of cohesive strength, the response is purely frictional. The frictional response may vary with slip speed, wear on a changing population of contacts, temperature, and other factors. For many materials and applications, this response is captured well using a frictional model developed by Dieterich [1], Ruina, Rice, and others. Coupling this friction law to the slip-weakening model becomes challenging since it is difficult to predict the shear stress at the end of the slip-weakening phase while the coefficient of friction is changing. In this work, we embed this coupled slip weakening-frictional response into an enhanced strain element with an embedded strong discontinuity, similar to the one described in [2]. The weakening and frictional models are successfully coupled by making some simplifying assumptions, and embedded numerically in the slip response by means generalized trapezoidal method. The slip speed and state variable are solved via an element-level Newton iteration.

References [1] J.H. Dieterich and M.F. Linker, Fault stability under conditions of variable normal stress. Geophysical Research Letters, 19, 1691-1694, 1992. [2] R.I. Borja and R.A. Regueiro, Strain localization of frictional materials exhibiting displacement jumps. Computer Methods in Applied Mechanics and Engineering 190, 2555-2580, 2001.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A partition of unity finite element method applied to the study of viscoelastic sandwich structures L.Hazard*, Ph. Bouillard*, J.-Y. Sener† *

Structural and Material Computational Mechanics Department, CP 194/5 Université Libre de Bruxelles, Av. F.D. Roosevelt 50, 1050 Brussels, Belgium [email protected] †

Arcelor Innovation, Boulevard de Colonster, B57, 4000 Liège, Belgium

ABSTRACT The scope of this research concerns the passive damping of vibrations of structures by the use of viscoelastic layers. It is motivated by the need for efficient numerical tools to deal with the medium frequency behaviour of industrial viscoelastic sandwich products. The sandwich modelling technique is based on the use of an interface element: the two deformable plates are modelled by special plate elements while the intermediate dissipative layer is modelled with interface elements. This interface element is based on the first-order shear deformation theory and assume constant peel and shear stresses in the polymer thickness. This element couple the lower and upper layers without additional degrees of freedom. The partition of unity finite element method (PUFEM) is applied to the development of enriched Mindlin plate elements. The element shape functions are obtained as the product of partition of unity functions with arbitrary chosen enrichment functions. Polynomial enrichment leads to the generation of high-order polynomial shape functions and is therefore very similar to a p-FEM technique. Numerical examples illustrate the use of both PUFEM Mindlin plate elements and interface elements for the simulation of viscoelastic sandwich structures.

References [1] I. Babuška, J. Melenk, The partition of unity method, International Journal for Numerical

Methods in Engineering, 40, 727-758, (1997) [2] E. De Bel, P. Villon and Ph. Bouillard, Forced vibrations in the medium frequency range

solved by a partition of unity method with local information, International Journal for Numerical Methods in Engineering, 62, 1105-1126, (2005)

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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An enriched space-time finite element method for fluid-structure interaction – Part II: Thin flexible structures A. K¨olke∗ and A. Legay† ∗ Institut f¨ur Statik Technische Universit¨at Braunschweig, Beethovenstraße 51, 38106 Braunschweig, Germany [email protected] † Chaire de M´ecanique Conservatoire National des Arts et M´etiers, 2 rue Cont´e, 75003 Paris, France [email protected]

ABSTRACT A new numerical approach for fluid-structure interaction of viscous fluid flow and flexible structures of negligible thickness (e.g. membranes, plates) on a topologically fixed fluid discretization is presented. The linear elastic and geometrically nonlinear structure is embedded [1] into the flow field described by the incompressible Navier-Stokes equations using locally enriched space-time (EST) finite elements [2, 3] to consider resulting strong and weak discontinuities in the fluid field appropriately [4]. Since the formulation of fluid, structure and coupling conditions uniformly uses velocities as unknowns and the integration of governing equations is perfomed on the deformed space-time mesh, the simulation of the strongly coupled physical domains becomes very comfortable and results in a monolithic system [5]. Numerical examples of fluid-stucture interaction processes show the ability of the developed numerical method to describe those situations of coupled systems for that common mesh moving strategies are not useful applicable and would require the introduction of time-consuming remeshing algorithms.

References [1] A. Legay, J. Chessa and T. Belytschko. An Eulerian-Lagrangian Method for Fluid-Structure Interaction Based on Level Sets. Computer Methods in Applied Mechanics and Engineering, in press, 2005 [2] A. K¨olke and D. Dinkler. Extended Space-Time Finite Elements for Two-Fluid Flows in FluidStructure Interaction. Proceedings of Sixth World Congress on Computational Mechanics, China, 2004 [3] A. K¨olke and D. Dinkler. Extended Space-Time Finite Elements for Boundary-Coupled MultiField Problems on Fixed Grids. Proceedings of International Conference on Computational Methods for Coupled Problems in Science and Engineering, Greece, 2005 [4] T. Belytschko, T.N. Mo¨es, S. Usui, and C. Parimi. Arbitrary Discontinuities in Finite Elements. International Journal of Numerical Methods in Engineering, 50:993–1013, 2001. [5] B. H¨ubner, E. Walhorn, and D. Dinkler. A Monolithic Approach to Fluid-structure Interaction using Space-time Finite Elements. Computer Methods in Applied Mechanics and Engineering, 193(23-26):2069–2086, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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An enriched space-time finite element method for fluid-structure interaction – Part I: Prescribed structural displacement A. Legay∗ and A. K¨olke† ∗

Chaire de M´ecanique Conservatoire National des Arts et M´etiers, 2 rue Cont´e, 75003 Paris, France [email protected] † Institut f¨ur Statik Technische Universit¨at Braunschweig, Beethovenstraße 51, 38106 Braunschweig, Germany [email protected]

ABSTRACT This contribution introduces a new approach to treat fluid-structure interaction problems. This presentation (part one) focuses on applications with prescribed and a priori known displacement of thin structures. The extension of the presented numerical method to flexible structures enables the approach to handle fully coupled fluid-structure interaction situations (part two). A velocity-pressure-based weak formulation of the governing equations of viscous and incompressible fluid flow (Navier-Stokes-Equations) is discretized by finite space-time elements using a discontinuous Galerkin-scheme for time integration. The resulting space-time slabs are computed sequentially. The location of infinite thin structures in the fluid domain is represented by the zero level set of a space-time defined level set function [1]. To capture the occuring moving discontinuities from embedding a thin solid body into the flow field, a locally enriched space-time (EST) finite element method [2] is applied to ensure a fluid mesh independent from the current configuration of the structure. Based on the concept of the extended finite element method [3] the space-time approximation of the pressure is enriched to represent strongly discontinuous solutions at the position of the structure. The velocity approximation is properly enriched to capture discontinuities in the gradient. The presentation concludes with several numerical examples including flow around flaps with large displacements and rotating blades.

References [1] A. Legay, J. Chessa and T. Belytschko. An Eulerian-Lagrangian Method for Fluid-Structure Interaction Based on Level Sets. Computer Methods in Applied Mechanics and Engineering, in press, 2005 [2] A. K¨olke and D. Dinkler. Extended Space-Time Finite Elements for Boundary-Coupled MultiField Problems on Fixed Grids. Proceedings of International Conference on Computational Methods for Coupled Problems in Science and Engineering, Greece, 2005 [3] T. Belytschko, T.N. Mo¨es, S. Usui, and C. Parimi. Arbitrary Discontinuities in Finite Elements. International Journal of Numerical Methods in Engineering, 50:993–1013, 2001.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Hybrid-Trefftz Finite Element Models for Bounded and Unbounded Elastodynamic Problems Ionut¸ D. Moldovan∗ , Jo˜ao A. Teixeira de Freitas† ∗† Departamento

de Engenharia Civil e Arquitectura, Instituto Superior T´ecnico, Avenida Rovisco Pais, 1049-001 Lisboa ∗ [email protected], † [email protected] ABSTRACT

The displacement and stress models of the hybrid-Trefftz finite element formulation are applied to the spectral analysis of bounded and unbounded elastodynamic problems [4, 3]. The displacement model is derived by constraining the approximation of the displacement field to satisfy locally the governing Navier differential equations and by approximating independently the tractions on the Dirichlet and inter-element boundaries and used to enforce, on average, the local displacement continuity conditions. Conversely, the stress model is derived by constraining the approximation on the stress field to satisfy locally the governing Beltrami differential equations. The displacements are approximated independently on the inter-element and Neumann boundaries and used to enforce, on average, the local flux continuity conditions. Two alternative approaches are used to extend the proposed formulations to semi-infinite (half-space) media, namely a finite element with absorbing boundary and an infinite element. The (fully localised) absorbing boundary condition is built for the displacement model through Dirichlet-to-Neumann (Neumann-to-Dirichlet, in the stress model) mapping [2]. The tractions (displacements) are approximated independently on the absorbing boundary and used to enforce the asymptotic approximation of the Sommerfeld radiation condition. The domain approximation used in the displacement (stress) model infinite element satisfies explicitly the Sommerfeld radiation condition, thus eliminating the uncertainties regarding possible spurious reflections in the vicinity of the absorbing boundary. The alternative stress and displacement models are used to model the response of a fluid saturated porous media, using the Biot’s theory of porous media [1]. Their performance in terms of convergence of the mechanical energy, stresses and displacement estimates and sensitivity to mesh distortion is emphasised.

References [1] M. A. Biot, Theory of propagation of elastic waves in a fluid saturated porous solid. I. Low frequency range. J. Acoust. Soc. America, 28, 168–178, 1956. [2] S. V. Tsynkov, Numerical solution of problems on unbounded domains. A review. Appl. Num. Math., 27, 465–532, 1998. [3] J. A. T. Freitas, C. Cismas¸iu, Hybrid-Trefftz displacement element for spectral analysis of bounded and unbounded media. Int. J. Sol. Struct., 40, 671–699, 2003. [4] J. A. T. Freitas, I. D. Moldovan and M. Toma, Trefftz spectral analysis of biphasic media. Proc. VI World Conf. on Comp. Mech. in conjunction with Asia-Pacific Conf. on Comp. Mech., Sept. 5-10, Beijing, China, Tsinghua University Press & Springer-Verlag, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Explicit dynamic with X-FEM to handle complex geometries P. Rozycki1, E. Bechet 2 , N. Moës1 1 Institut de Recherche en Génie Civil et Mécanique UMR CNRS 6183 Ecole Centrale de Nantes, 1 rue de la Noë, BP 92101, F-44321 Nantes Cedex 3 [email protected]; [email protected] 2

Laboratoire de Physique et Mécanique des Matériaux UMR CNRS 7554 Université de Metz, Ile de Saulcy, F-57045 Metz Cedex 1 [email protected]

ABSTRACT Although the calculation capacities have considerably increased these last years, the complexity of the numerical simulations in dynamic fields still induce many problems, essentially due to CPU time calculation. For instance, the use of explicit scheme yields a critical time step. It depends on the greatest structure eigenvalue [1], [2]. Commonly, rather than to identify this eigenvalue, an upper approximation computed corresponds to the smaller characteristic size (if all elements share the same behavior). These critical time steps are usually induced by mesh constraints. For complex geometries, very small sized elements may arise. One approach is then to optimize the mesh by removing elements or by using mass scaling to improve the critical time step. The work is based on the developments suggested for static problems, which are using the eXtended Finite Element Method [3], [4]. Thanks to the unity partition theory, it is possible to add some specific functions of enrichment to the conventional approximation field of the displacement. These added functions allow, for example, the treatment of cracks, material interfaces, holes, etc. Consequently, this approach authorizes the non-conformity between mesh and discontinuities. This paper presents the work carried out about the X-FEM finite elements, which are dedicated to the dynamic explicit problems including holes or external surfaces. In a first part, the developments are presented for 1D cases and for mono-material structures. The objective is to propose the theoretical framework of the X-FEM finite element: a reformulation of the stiffness matrix and an adapted mass matrix offer the possibility to increase time step calculation in the case of non-meshed surfaces. A generalization of the method is then proposed for the 2D and 3D cases. Validations for different type of structure are exposed. The results in comparison with ABAQUS software are relevant and allows us to present some outlooks.

References [1] M.N. Newmark, A method of computation for structural dynamics, Proc. ASCE 85, EM3, 1959. [2] T. Belytschko, T.J.R. Hughes, Computational methods for transient analysis, North-Holland, 1986. [3] N. Sukumar, D. L. Chopp, N. Moës and T. Belytschko, Modeling Holes and Inclusions by Level Sets in the Extended Finite–Element Method, Computer Methods in Applied Mechanics and Engineering, Vol. 190, Number 46–47, pp. 6183–6200, 2001 [4] C. Daux, N. Moës, J. Dolbow, N. Sukumar, and T. Belytschko, Arbitrary branched and intersecting cracks with the eXtended Finite Element Method, International Journal for Numerical Methods in Engineering, 48:1741-1760, 2000.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Prediction of macroscopic material failure based on microscopic cohesive laws Lidija Stankovi´c, J¨orn Mosler Institute of Mechanics Ruhr University Bochum, Universit¨atstr. 150, D-44780 Bochum, Germany {lidija,mosler}@tm.bi.ruhr-uni-bochum.de ABSTRACT A three–dimensional finite element formulation is applied to the process of determination of macroscopic material properties based on constitutive relationships characterising a microscale. More specifically, a macroscopic failure criterion is computed numerically. The adopted finite element model captures the localised fully nonlinear kinematics associated with the failure on the microscale by means of the Strong Discontinuity Approach (SDA). In contrast to classical continuum mechanics, the deformation gradient is additively decomposed into a conforming part corresponding to a smooth deformation mapping and an enhanced part reflecting the final failure kinematics of the microscale. The implementation of the Enhanced–Assumed–Strain (EAS) concept leads to the elimination of the additional degrees of freedom (displacement jump) on the material point level. More precisely, the applied numerical implementation is similar to that of standard (finite) plasticity. The model does not require any assumption regarding neither the type of the finite elements, nor the constitutive behaviour. Any traction–separation law, connecting the displacement jump to the traction vector, can be chosen. Based on the proposed finite element formulation, microscopic material properties (traction–separation laws) are then used for the computation of the macroscopic material failure. The applicability of the presented numerical model is demonstrated by means of rather academic examples.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Strict, sharp and practical bounds of computed outputs of interest for evolution problems L. Chamoin∗ , P. Ladev`eze∗

†

∗ LMT-Cachan

(ENS-Cachan / CNRS / Paris 6 University) 61, avenue du Pr´esident Wilson - 94235 Cachan Cedex - France [email protected] † EADS

Foundation Chair Advanced Computational Structural Mechanics ABSTRACT

A constant concern both in industry and in research has been the verification of the models used for simulation of physical phenomena. We particularly need to assess the quality of the numerical solutions we get using approximate methods such as the FEM. Effective tools had appeared for thirty years [1] [2], allowing to assess global error (in the energy norm) then local error for quantities of interest which are relevant data for design. For this latter topic, most of the works concern linear problems and yield relatively good bounds of the error for this kind of problem. However, local error estimation for more complex problems has not been mastered yet. Some works dealing with this issue do not yield guaranteed bounds, which is a serious drawback. Others tend to give strict upper and lower bounds but with prohibitive computer ressources. This paper focuses on a method that yields strict bounds for quantities of interest resulting from a finite element analysis of linear evolution problems. The method, developped over the time-space domain, leans on an extraction technique [3] leading to the solution of an adjoint problem. We use dissipation error which is a practical tool developped at the LMT-Cachan for more than ten years [4]. An important feature is the solution of the adjoint problem by means of techniques introducing numerical or analytical functions in space (Partition of Unity Method) and time. Thus, one gets good quality for the bounds on the error with a reasonable numerical cost and without changing the framework of finite element codes. Another aspect of the method is the way to deal with quantities of interest more or less sensitive to history. We take cumulative error effects into account to reach a reliable assessment of the local errors. First results are presented in this paper for linear viscoelasticity problems in 2D. In conclusion, this work which can be extended to non linear problems shows that we can get both good and strict bounds and seems to be therefore a new step forward for the issue of model verification.

References [1] I. Babus˘ka and T. Strouboulis, The finite element method and its reliability, Oxford university press, 2001. [2] P. Ladeve`ze and J-P. Pelle, Mastering calculations in linear and nonlinear mechanics, Springer NY, 2004. [3] R. Becker and R. Rannacher, An optimal control approach to shape a posteriori error estimation in finite element methods, A. Isereles (Ed.), Acta Numerica, 10, 1–120, Cambridge Uni. Press, 2001. [4] P. Ladeve`ze, Nonlinear Structural Mechanics - New Approaches and Non-Incremental Methods of Calculation, Springer NY, 1998.

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Error bounds on outputs of interest for linear stochastic problems. ´ . Florentin, P. Ladeve`ze(
), J. Bellec E LMT-Cachan ENS-Cachan/CNRS/Paris VI University 61, avenue du Pr´esident Wilson - 94235 Cachan Cedex - France e-mail: [email protected] - Web page: http://lmt.ens-cachan.fr (
) EADS Foundation Chair Advanced Computational Structural Mechanics. ABSTRACT This work deals with outputs of interest for linear elastic F.E. analysis in the presence of uncertainties (material, loads...). The objective of this paper is precisely to develop tools for the assessment of linear stochastic models. Our approach relies on an extension of the constitutive relation error method, which is a very effective verification tool in the deterministic case. In order to obtain bounds of outputs of interest, one must solve an adjoint problem. In order to do that, one must build for the direct and adjoint problems an associated admissible displacement-stress pair. Then, bounds corresponding to a given level of certainty can be calculated. Theses bounds take into account the errors due to the finite element discretization as well as the errors due to the stochastic approximation method. The method is illustrated through numerical tests. These tests demonstrate the capabilities of this new tool in providing bounds which can be of direct use to the designer. With such bounds, calculation can lead to certification, even in the case of uncertain loading cases.

References [1] P. Ladev`eze and J.P. Pelle Mastering calculations in linear and nonlinear mechanics Springer NY, 2004. [2] I. Babu˘ska and T. Strouboulis The finite element method and its reliability. Oxford university press, 2001. [3] R. Ghanem and P. Spanos Stochastic Finite Element : A special approach. Springer, 1991. [4] M.K. Deb, I. Babu˘ska, and J.T. Oden Solution of stochastic partial diffenrential equations using galerkin finite element techniques. Comp. Meth. in Applied Mech. and Engrg., 190:6359–6372, 2001. [5] Ghanem R. and Pelissetti M. Error estimation for the validation of model-based predictions. In Proc. of 5th World Congress on Computational Mechanics, 2002.

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An hp-adaptive analysis of some linear free vibration problems ´ ∗ , G. Zboinski ´ † M. Jasinski ∗ Institute

of Fluid-Flow Machinery of PASci Fiszera 14, 80-952 Gda´nsk, Poland [email protected] † [email protected]

ABSTRACT This paper concerns an hp-adaptive finite element analysis of the free vibration problems of linear elasticity. We have implemented the 3D formulation of the Reissner-Mindlin shell theory, higher order hierarchical shell models and the 3D-elasticity to analyse complex (shell-solid) structures. The idea is to use modified methods derived for elastostatics adaptive solutions i.e. Equilibrated Residual Method to estimate local error and Texas 3-step strategy (solutions for an initial mesh, h-refined mesh and p-enriched mesh) to reach the finite element space of desired properties. Basically the procedure for error estimation is as follows. Firstly, we solve the eigenproblem for a given h and p. Then, for a given frequency: we calculate equilibrated interelement stresses and solve local (elemental) problems with these stresses as boundary conditions with h, p+1. With the latter we estimate local error in the energy norm. Having local errors one can perform finite element space update. In this paper we show how the hp-adaptive technique for linear elastostatics can be used in case of free vibrations. Numerical examples conclude the paper.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Moving Mesh Adaptivity applied to Railway Dynamics Håkan Lane, Per Kettil, Nils-Erik Wiberg Department of Applied Mechanics, Chalmers University of Technology 412 96 Gothenburg, Sweden [email protected]

ABSTRACT Elastic wave propagation induced by the passage of high-speed trains can be the cause of major nuisances for people living close to railway lines. Because of resonance, the phenomenon is especially visible in configurations with a soft material in the soil. When the speed of the train matches propagation velocities in the ground, the vehicle catches up with the wave, leading to greatly magnified displacements. A compound multi body – finite element model has been created to simulate the interaction between the train, the track and the soil [1]. A fixed mesh with a length of 89 m of straight track was used for these analyses. More comprehensive simulations in various conditions require a more flexible model, where the train and the track/soil mesh domain move together across long distances [2]. It will be possible to move the domain in the tangent direction representing straight track, in a linear or quadratic transition followed by a circular elevated curve or along a constant direction, e.g. a slope. As less degrees of freedom are used compared to a large, static mesh, the approach will lead to faster calculations and lower memory demands. Moving an entire FE domain has been used with success in other applications [3]. The magnitude of waves in the different geometries will be analysed and evaluated. It will also be investigated whether vibrations change over long distances in any configuration and/or velocity. Another issue is the question of how many elements are needed for accuracy.

References [1] H. Lane: Rail Vehicle – Track Structure – Subgrade Computational Analysis. Thesis for the degree of Licentiate Engineering. Department of Applied Mechanics, Chalmers University of Technology, Gothenburg, Sweden, 2005. [2] P. Kettil, H. Lane and N.-E. Wiberg: Moving Mesh Domain Adaptation Technique – Application to Train Induced Wave Propagation. Proceedings of the Eighth International Conference on Computational Plasticity held in Barcelona, Spain, 5th – 7th September 2005. [3] R. Gwynllyw, A.R. Davies and T. Phillips: A moving spectral element approach to the dynamically loaded journal bearing problem. Journal of Computational Physics, 2, pp. 476-494, 1996.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Finite strain r-adaption based on a fully variational framework J. Mosler∗ , M. Ortiz† ∗ Institute of Mechanics Ruhr University Bochum, Universit¨atsstr. 150. D-44780 Bochum, Germany [email protected] † Graduate

Aeronautical Laboratories California Institute of Technology, Pasadena, CA 91125, USA [email protected] ABSTRACT A novel r-adaptive finite element strategy based on a fully variational framework is presented. Provided the underlying physical problem is characterized by means of a minimization principle, the proposed method seeks, for a fixed number of nodes, for the best finite element interpolation depending on the nodal positions with respect to the deformed (x) as well as the undeformed (X) configuration, cf. [1]. The existence of a minimization problem does not represent a very strong restriction, since for many physical problems such as standard dissipative media an incremental potential can also be recast, cf. [2]. While minimizing the potential considered by fixing the nodes within the undeformed configuration corresponds to classical N EWTONian mechanics, a variation with respect to (X) is associated with E SHELBY mechanics, cf. [3]. However, in contrast to the simplicity of the concept, its numerical implementation is far away from being straightforward. According to [4], the resulting system of equations is highly singular and hence, standard optimization strategies cannot be applied. In this paper, a viscous regularization is used. This approach is designed to render the minimization problem well-posed while leaving its solutions unchanged. Obviously, relocating the nodes within the undeformed configuration by fixing the triangulation (the connectivity) may lead to strong topological constraints. As a consequence, an energy based re-meshing strategy is advocated. Contrary to classical mesh-improvement methods based on geometrical quality measures, the novel concepts identifies local energy minimizers. That is, the energy of the new triangulation is always lower than that of the initial discretization. The performance of the resulting finite element model is demonstrated by fully three-dimensional examples.

References [1] P. Thoutireddy and M. Ortiz, A variational r-adaption and shape-optimization method for finitedeformations elasticity. International Journal for Numerical Methods in Engineering, 61, 1-21, 2004. [2] M. Ortiz and L. Stainier, The variational formulation of viscoplastic constitutive updates. Computer Methods in Applied Mechanics and Engineering, 171, 419-444, 1999. [3] M. Braun, Configurational forces induced by finite element discretization. Proc. Estonian Acad. Sci. Phys. Math., 46, 24-31, 1997. [4] J. Mosler and M. Ortiz, On the numerical implementation of Variational Arbitrary LagrangianEulerian (VALE) formulations. International Journal for Numerical Methods in Engineering, 2005 (accepted).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Extension processes, adaptivity and remeshing for elasto-plastic problems Ernst Rank*, Vera Nübel†, Alexander Düster* *

Lehrstuhl für Bauinformatik, Technische Universität München D-80290 München, Germany {rank,düster}@bv.tum.de †

Numerical Simulation, New Business & Technology Hilti Entwicklungsgesellschaft mbH D-86916 Kaufering, Germany [email protected]

ABSTRACT Convergence of the finite element method is obtained by a systematic extension of the Ansatz spaces approximating a given mathematical model. The classical h-version extends the approximation by (uniform or adaptive) mesh refinement using a fixed polynomial degree in each element. The pversion keeps the mesh fixed and increases the element polynomial degree, where again uniform or adaptive methods can be applied. The r-method reallocates nodes and elements and adjusts (if high order elements are used) the geometric shape of element edges and faces to certain criteria. All of the three methods (h-, p-, r-extension) can be combined in order to achieve optimized control over approximation error and computational resources. We will characterize these methods in this paper and compare their performance on benchmark problems for elasto-plastic computation. Rate independent as well as rate dependent elastoplastic problems will be investigated in two as well as three dimensions and it will be shown that an exponential rate of convergence can be obtained by a combination of r- and p-methods. Finally, we will give guidelines for practical computation using lower order elements as they are available in commercial finite element codes.

References [1]

[2]

[3]

[4]

A. Düster, E. Rank. The p-version of the finite element method compared to an adaptive h-version for the deformation theory of plasticity. Computer Methods in Applied Mechanics and Engineering. 190:1925-1935, 2001. A. Düster, E. Rank. A p-version finite element approach for two- and threedimensional problems of the J2 flow theory with non-linear isotropic hardening. International Journal for Numerical Methods in Engineering, 53:49-63, 2002. B. Szabo, A. Düster, E. Rank. The p-version of the finite element method. In: E. Stein, R. de Borst, T.J.R. Hughes, Editoren: Encyclopedia of Computational Mechanics, Volume 1: Fundamentals, Chapter 5, pp. 119-139, John Wiley & Sons, 2004. V. Nübel, A. Düster, E. Rank. An rp-adaptive finite element method for the deformation theory of plasticity. To appear in: Computational Mechanics (2006)

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Efficient implementation of domain decomposition methods using a hierarchical h-adaptive finite element program Juan J. Ródenas1, José Albelda1, Cristina Corral2 and José Mas2 1

Centro de Investigación en Tecnología de Vehículos Departamento de Ingeniería Mecánica y de Materiales Universidad Politécnica de Valencia Camino de Vera s/n. 46022-Valencia, Spain {jjrodena,jalbelda}@mcm.upv.es 2

Instituto de Matemática Multidisciplinar Universidad Politécnica de Valencia Camino de Vera s/n. 46022-Valencia, Spain {ccorral,jmasm}@imm.upv.es

ABSTRACT A previous contribution[1] showed the hierarchical relationships between parent and child elements that come out if these elements are geometrically similar. Under this similarity condition, the terms involved in the evaluation of element stiffness matrices (ke=³BtDB|J|dV), corresponding to parent and child elements, are related by a constant which is a function of the ratio of the element sizes (scaling factor). These relations were used in the basic implementation of a hierarchical h-adaptive Finite Element program based on element subdivision for the resolution of the 2-D linear elasticity problem. The program makes use of a hierarchical data structure to carry out the h-adaptive process, which significantly reduces the amount of calculations required for the evaluation of the problem stiffness matrix, element stresses, element error estimation,… The h-adaptive refinement process based on element splitting produces a natural decomposition of the domain which, together with the hierarchical data structure of the program directly produces an arrowhead stiffness matrix allowing for a decomposition of the global problem into smaller problems. Thus, a domain decomposition solver has been used in this paper to efficiently solve the linear system of equations arising during the analysis process. The numerical test presented in the paper clearly show a considerable improvement in memory requirements and solution times and suggest the use of recursive domain decomposition into the original subdomains.

References [1] J.J. Ródenas, J.E. Tarancón, J. Abelda, A. Roda, J. Fuenmayor, Hierarquical properties in elements obtained by subdivision: a hierarquical h-adaptivity program. In P. Díez and N.E. Wiberg, editors, Adaptive Modeling and Simulation 2005. CIMNE, Sept. 2005.

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Adaptive remeshing in transient impact processes with large deformations and nonlinear material behavior Wolfgang A. Wall *, Tobias Erhart†, Ekkehard Ramm†† * Chair for Computational Mechanics, TUM Boltzmannstr. 15, 85748 Garching (b. München), Germany [email protected] †

††

DYNAmore GmbH, Industriestr. 2, 70565 Stuttgart, Germany [email protected]

Institute of Structural Mechanics, Paffenwaldring 7, 70550 Stuttgart [email protected]

ABSTRACT The present study is concerned with impact processes that appear in civil and military security technology, dynamic soil compaction, vehicle crash or fastening and demolition technology. They are characterized by varying non-linearities, as e.g. large deformations and strains, highly non-linear material behavior, frictional contact between multiple bodies and stress wave propagation. A combination of different methods in adaptivity, constitutive modeling, element technology, efficient time discretization and contact are essential for the reliable computation of practical relevant engineering tasks and for predictions in industrial applications. Accuracy, robustness and efficiency are the authoritative requirements for the solution of those complex problems. In this contribution, we will mainly focus on two issues, namely adaptive remeshing along with subcycling strategies and constitutive modeling aspects especially prepared for impact loading. Since large deformations occur in impact simulations and a Lagrangean description is used, repeated remeshing of individual domains is essential [1,2]. To achieve quality controlled solutions and an optimal distribution of used computational resources at the same time, an adaptive strategy is applied. The core of this strategy is the assessment of discretization errors by adequate error indicators. For this purpose, different possibilities are presented and new methods are developed, which are appropriate for the simulation of transient impact processes. Based on the theory of finite plasticity, constitutive models for thermoviscoplastic metals and cohesive as well as non-cohesive frictional materials are presented and developed. Here, the main focus will be on a formulation for loose, granular media under high pressure loadings [3]. Therefore, a Drucker-Prager-Cap model [4] is modified and enhanced. The properties and effects of the developing powder will be examined. The proposed methods are verified for model problems and their performance in practical relevant applications is evaluated.

References [1] G.T. Camacho and M. Ortiz, “Adaptive Lagrangian modelling of ballistic penetration of metallic targets”, Comp. Meth. Appl. Mech. Engrg., Vol. 142, pp. 269-301, (1997). [2] T. Erhart, L. Taenzer, R. Diekmann and W.A. Wall, “Adaptive remeshing issues for fast transient, highly nonlinear processes”, Proc. of ECCM 2001, Cracow, Poland, (2001). [3] T. Erhart, W.A. Wall and E. Ramm, “A robust computational approach for dry powders under quasi-static and transient impact loadings”, CMAME, 194, pp. 4115-4134, (2005). [4] G. Hofstetter, J.C. Simo and R.L. Taylor, “A modified cap model: Closest Point Solution Algorithms”, Comput. & Struct., Vol. 46, pp. 203-214, (1993).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Particle Swarms in Engineering Design Problems Bogdan Bochenek, Paweá ForyĞ Institute of Applied Mechanics, Cracow University of Technology Jana Pawla II 37, 31-864 Krakow, Poland [email protected], [email protected]

ABSTRACT Many modern computational techniques are inspired by biological systems. For example, artificial neural network is a simplified model of the human brain whereas genetic algorithm is inspired by the evolution of living forms. Here we discuss another type of biologically based intelligence system called particle swarm. It is an artificial intelligence technique based on the study of social behaviour in the systems of self-organized population. Such a system is made of a group of simple agents which interact with one another and with their environment. The sum of local interactions results in a global behaviour of the population. Ant colonies, bird flocking and fish schooling can serve as examples of swarm intelligence systems found in nature. Nowadays, the most popular swarm inspired method in computational intelligence area is Particle Swarm Optimization (PSO). PSO shares many similarities with evolutionary computation techniques such as genetic algorithms. The system is initialized with a population of random solutions. Taking into account the best positions of particles at subsequent iteration steps the algorithm searches for optima by updating generations. However, unlike genetic algorithms, PSO has no evolution operators such as crossover and mutation. In PSO, particles being potential solutions fly through the problem space searching for the optimum configuration. In past several years, PSO has been successfully applied in many application areas being the basis of the most commonly used non-gradient based stochastic search algorithms. It is demonstrated that in many cases PSO leads to better results obtained in a faster, less expensive way compared with other methods. Another reason that makes PSO attractive is that there are only few parameters to adjust. Usually one version, with slight modifications, works well in a wide range of applications. In this paper a new improved algorithm based on the Particle Swarm Optimization concept is developed and its application to engineering optimization is presented. Many extensions to the original version of the Particle Swarm method introduced by Kennedy and Eberhart in 1995 are proposed. The amendments regard constraint handling as well as modification of the rules of velocities updating. The two-state version of the algorithm is developed in which two weighting factors are used and their values are selected according to the swarm performance. If a particle moves to a better position a history information is disregarded and the free move in this direction is allowed for - state 1, otherwise the new position is calculated based on the swarm performance history - state 2. This new switching technique allows natural creation of swarm leaders. Their behaviour has then great impact on other swarm members what finally speeds up search process convergence. In classic PSO algorithm velocities are limited by arbitrarily selected values what should be treated as a weakness of the algorithm. For example, if many local optima exist and the distance between them is larger than move limits imposed, finding global optimum among them may be even impossible. Here more flexible approach to limiting velocities values is proposed. The algorithm starts with large kinetic energy of particles, which is then limited while exploring search space. As for engineering optimization the implementation of mixed integer/continuous design variables is discussed in detail, and the effective application technique is proposed. The paper is illustrated by numerical results of selected engineering design problems.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Particle Swarm Optimization: efficient globally convergent modifications Emilio F. Campana∗ , Giovanni Fasano∗† , Daniele Peri∗ , Antonio Pinto∗ ∗ The Italian Ship Model Basin (INSEAN) via Di Vallerano 139, 00128 Rome, ITALY {e.campana,g.fasano,d.peri,a.pinto}@insean.it †

Istituto di Analisi dei Sistemi ed Informatica “A. Ruberti” (IASI) - CNR viale Manzoni 30, 00185 Rome, ITALY [email protected]

ABSTRACT In this paper we consider the Particle Swarm Optimization (PSO) algorithm [1, 2], in the class of Evolutionary Algorithms, for the solution of global optimization problems. We analyze a couple of issues aiming at improving both the effectiveness and the efficiency of PSO. In particular, first we recognize that in accordance with the results in [3], the initial points configuration required by the method, may be a crucial issue for the efficiency of PSO iteration. Therefore, a promising strategy to generate initial points is provided in the paper. Then, we address some very preliminary aspects of PSO global convergence towards stationary points, for some Ship Design problems. To this purpose observe that the class of Ship Design applications includes several challenging smooth problems, where expensive simulations provide information to the optimizer, and each function evaluation may require up to hours of CPU-time. In addition, the final solution provided by the optimization method is also required to be a stationary point.

References [1] M.Clerc, J.Kennedy, The Particle Swarm - Explosion, Stability, and Convergence in a Multidimensional Complex Space, IEEE Transactions on Evolutionary Computation, 6, 58–73, 2002. [2] J.Kennedy, R.C.Eberhart, Particle swarm optimization, Proceedings of the 1995 IEEE International Conference on Neural Networks (Perth, Australia), IEEE Service Center, Piscataway, NJ, IV, 1942–1948, 1995. [3] E.F.Campana, G.Fasano, A.Pinto, Dynamic system analysis and initial particles position in Particle Swarm Optimization, IEEE Swarm Intelligence Symposium (SIS2006), May 12-14, 2006, Indianapolis, Indiana, USA.

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Evolutionary Optimization of Chemistry of Bulk Metallic Glasses George S. Dulikravich1, Igor N. Egorov2, Nenad Jelisavcic1 1

Florida International University, Department of Mechanical & Materials Engineering 10555 West Flagler Street, Room EC 3474, Miami, Florida 33174, U.S.A. [email protected] 2 IOSO Technology Center Kasatkina 13, Moscow, 129301, Russia [email protected]

ABSTRACT Metallic glass is basically an alloy whose metallic species are “frozen” in amorphous glassy state rather than forming a standard crystalline structure. Metallic glasses have no grain boundaries and no dislocations and stacking faults. They are several times stronger than steel and considerably harder and more elastic. Formation of metallic glasses by extremely high cooling (~105 K/sec) of the melt was first accomplished in 1960s. The resulting metallic glass thickness was limited to extremely thin ribbons. In the 1990s, researchers formed new classes of metallic glasses in bulk. The bulk metallic glasses (BMGs) are composed of three or more metals in the alloy melt and a few diatomatous earth ingredients in order to lower the cooling rate. Cooling rates of the new alloys are from 100 K/s to 1 K/s. The possible thickness of these newer metallic glasses increased from micrometers to centimeters. One of the keys to lowering the cooling speed and creating larger specimens is that bulk metallic glasses should have ingredients with atomic species having large size and chemical differences. Thus, multiple thermo-mechanical properties and the cooling speed of bulk metallic glass alloys depend strongly on the concentrations of each of the chemical elements in a given alloy. The proposed methodology for accurately determining concentration of each of the important alloying elements is based on the use of a combination of a robust multiobjective optimization algorithm and on traditional experimentation. Specifically, the proposed alloy design method combines an advanced stochastic multi-objective evolutionary optimization algorithm based on self-adapting response surface methodology and a relatively very small data set of thermo-mechanical properties and the corresponding concentrations of alloying elements. During the iterative computational design procedure, new metallic glass alloys need to be manufactured and experimentally evaluated for their properties in order to continuously verify the accuracy of the entire design methodology. This metallic glass alloy design optimization method thus minimizes the need for costly and time-consuming experimental evaluations of new metallic glass alloys to fewer than 200 new alloys. References 1. I. N. Egorov-Yegorov, G.S. Dulikravich, Chemical Composition Design of Superalloys for Maximum Stress, Temperature and Time-to-Rupture Using Self-Adapting Response Surface Optimization. Materials and Manufacturing Processes, 20 (3) (2005), 569-590. 2. G.S. Dulikravich, I.N. Egorov, Optimizing Chemistry of Bulk Metallic Glasses for Improved Thermal Stability. Symposium on Bulk Metallic Glasses. TMS 2006 Annual Meeting & Exhibition, eds: Liaw, P. K. and Buchanan, R. A., San Antonio, TX, March 12-16, 2006.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Introduction of Control Points in Splines for Synthesis of Optimized Cam Motion Program Tarun K. Naskar Department of Mechanical Engineering, Jadavpur University Kolkata 700032, India [email protected]

ABSTRACT One of the basic objectives of the synthesis of cam motion program is to minimize the peak values of the kinematic parameters ( KP ) – acceleration ( AP ) and jerk ( J P ) -- of the follower for ensuring smooth and noiseless follower action especially in high-speed machines using cams. Higher order polynomials are combined piecewise at knots for constructing splines and B -splines for cam displacement functions. In this work classical splines of 6th, 7th and 8th orders are taken as cam displacement functions. Multiple control points ( CP s), characterized by parameters like angular position ( AP ) and follower displacement ( FD ), are introduced. The AP and J P are minimized by manipulating the CP parameters. Two parameters of a CP are varied first independently and finally simultaneously with a view to minimizing the AP and J P . The acceleration and jerk of a cam follower are so interrelated that lowering of the value of one gives rise to the value of the other. A method is suggested for minimizing both by varying the values of AP and FD of each CP . A comparative study of AP and

J P obtained from different order spline functions is made. A searching procedure is adopted, based on genetic algorithm (GA) and fuzzy membership function, for obtaining goal function. CP s are also introduced in 6-order and 8-order B -splines. Minimization of the AP and J P is done for two situations – jerk finite and ping finite. CP s in B -splines are characterized by AP . For jerk finite AP and jerk at end position ( J e ) are manipulated, while for ping finite AP and ping at end position ( Pe ) are manipulated. -- first independently and finally simultaneously -- to minimize the

JP .

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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On the use of Differential Evolution in the Trajectory Modeling of Parallel Architecture Robot Giovana T. S. Oliveira*, Sezimária F. P. Saramago*, Plínio J. Oliveira† *

Federal University of Uberlândia 2160 João Naves de Ávila Av., CEP 38400-902, Uberlândia - MG, Brazil saramago @ ufu.br †

Federal University of Goias 1120 Lamartine Pinto Avelar Av., Catalão (GO), Brazil [email protected]

ABSTRACT The advance of the computational recourses has encouraged the utilization of the optimization techniques in the solution of complex problems. Thus, become very attractive the possibility of to join the feature of natural optimization methods to one algorithm which allow to work with small populations and large reduction of computational time. The Differential Evolution (DE) is a simple evolutionary algorithm and it has these advantages. The most distinct feature of DE is to perturb individuals of a population by weighted difference between random individuals of the population. The simplicity, efficiency and robustness of the Differential Evolution in terms of easy implementation are demonstrated by an engineering problem. Thus, to demonstrate the algorithm potentiality the trajectory optimization of a parallel structure is shown and discussed. The procedure is used for optimize the trajectory of a parallel manipulator named as CaPaMan (Cassino Parallel Manipulator) by applying a performance criterion that includes the mechanical energy and total traveling time. The multi-objective function is minimizing by using Differential Evolution. The trajectory is modeling by cubic B-splines. The results are compared with those obtained using Genetic Algorithms.

References [1] S. F. P. Saramago, G. Carbone, M. Ceccarelli, P. J. Oliveira, J. C. M. Carvalho, Optimum

path Planning of Capaman(Cassino Parallel Manipulator) by Using Inverse Dynamics. In: 2nd International Symposium On Multibody Systems And Mechatronics, Musme2005. IFToMM, 1, 332-343, 2005. [2] R. Storn, K. Price, Differential Evolution: a simple and efficient adaptive scheme for global optimization over continuous spaces. Technical Report TR-95-012, International Computer Science Institute, Berkeley, 1995.

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Evolutionary Topologic Optimization using the Finite Element Method Mário M. R. Teixeira*, Maurício P. Brandão† *

Instituto Tecnológico de Aeronáutica Praça Marechal Eduardo Gomes, 50 Campus do CTA - Vila das Acácias 12228-901 São José dos Campos - SP – Brasil [email protected] †

Comando Geral de Tecnologia Aeroespacial Avenida Brigadeiro Faria Lima, 1941 Jardim da Granja 12227-000 São José dos Campos - SP - Brasil [email protected]

ABSTRACT Darwin’s Species Evolution Theory is presented as a suitable method to determine a structures optimum topology for a given loading and prescribed boundary conditions. Among other existents methodologies for this task, Genetic Algorithms are discussed with more detail, evidencing its difficulty in simulating biological or physical evolution processes. The Topological Evolutionary Method is applicable to every class of problems, without knowing the crystalline structure or any intrinsic material characteristics, just by the elimination of the elements chosen by the system itself as less useful according to a given criterion. The Finite Elements Method (FEM) is examined as a Solids Mechanics good solution finder. Its main characteristics, as accuracy, modelling capacity, and userfriendliness, are discussed. The applications use a hexahedral finite element to define the domain under analysis and a Von Mises stress value as criterion to select the to-be-eliminated elements. The operator defines approximated, but not necessarily exact, domain, exact loads, and nodes restrictions positioning. Then, the operator searches for a solution that can be achieved in two ways: best topological shape (positive-defined global stiffness matrix) or desired specific mass. The application, software developing, auxiliary computational tools, and microcomputer operational system are all based on free software. Some classical examples (truss structure, cantilever and Michel beam) in 3D and 2D formulations have solutions obtained by the application. The results are discussed and compared with solutions available in the literature. Checkerboards, as a side effect of approximated methods implementations, like the FEM, are discussed. Filters are applied for 2D formulations to decrease checkerboard effects and also to accelerate execution. Applications can be used to solve any structural topologic optimization problem with suitable geometric domain, actual loading, and nodes restrictions, by using adequate finite element grids.

References [1] C. Darwin. On the origin of species by means of natural selection, or the preservation of favoured races in the struggle for life. John Murray, Albemarle Street, London, 1859. [2] Technical Committee on Optimal Structural Design of the ASCE. Recent Advances in Optimal Structural Design. Chap. 1. Edited by Scott A. Burns. May, 2002. 384p.

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A Multilevel Domain Decomposition Methodology for Solving Coupled Problems in Fluid-Structure-Thermal Interaction Eugenio Aulisa∗ , Sandro Manservisi∗,† , Padmanabhan Seshaiyer∗ ∗

Texas Tech University Department of Mathematics and Statistics, Lubbock, 79409-1042, TX {eugenio.aulisa, sandro.manservisi, padmanabhan.seshaiyer}@ttu.edu † University of Bologna DIENCA, Laboratory of Montecuccolino, via dei Colli 16, Bologna, 40136, Italy [email protected]

ABSTRACT Engineering analysis is constantly changing to develop novel techniques to solve coupled processes that arise in multi-disciplinary applications. Efficient solutions to complex coupled processes involving fluid-structure-thermal applications are still a challenging problem in computational sciences and engineering. Currently there exist numerous public-domain and commercial codes available for Computational Fluid Dynamics (CFD), Computational Structural Dynamics (CTD) and Computational ThermoDynamics (CTD). Different groups specializing in modeling individual process such as CSD, CFD, CTD often come together to solve a complex coupled application. The coupling of these solvers provide an insight for predictive capability for simulating complex nonlinear interactions that arise in several applications such as, hypersonic flight, where the structural deformation due to the aerodynamics and thermal loads leads to a significant flow field variation; MAVs (Micro Air Vehicles) where the geometry changes possibly due to thermal effects may lead to a transient phase in which the structure and the flow field interact in a highly non-linear fashion. Direct numerical simulation of the highly non-linear equations, governing even the most simplified fluid-structure-thermal interaction models depend on the convergence of iterative solvers which in turn relies heavily on the properties of the system coupling. Domain decomposition techniques have become increasingly popular in this regard, for obtaining fast and accurate solution. The the global domain (on which the coupled process evolves) is partitioned into several sub-domains over each of which, local problems are solved. The solution of the global problem is then constructed by suitably piecing together solutions obtained locally from independently modeled sub-domains. During this assembly process, it is often necessary to guide and coordinate non-matching grids arising over separate sub-domains. The purpose of this paper is to introduce a flexible multilevel algorithm with finite elements that can be used to study a coupled fluid-structure-thermal interaction (FSTI). The method relies on decomposing the complex global domain, into several local sub-domains; solving smaller problems over these subdomains and then gluing back the local solution in an efficient and accurate fashion to yield the global solution. Our numerical results suggest that the proposed solution methodology is robust and stable.

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Moving Mesh Algorithm for Unstructured Grids Based on Interpolation with Radial Basis Functions A. de Boer, M. S. van der Schoot, H. Bijl Delft University of Technology Kluyverweg 2, 2629 HS, Delft, The Netherlands [email protected] ABSTRACT Fluid-structure interaction computations typically involve moving boundaries for the flow due to the deformation of the structure. Examples can be found in: flutter simulation of wings, blood flow through veins and stability analysis of bridges and tall buildings subjected to windloads. Because of these moving boundaries a fast and reliable method for deforming the computational grid is needed to be able to perform the unsteady flow computations accurately and efficiently. Structured grids can be deformed by fast and accurate algebraic techniques, but for the meshing of complex domains and grid adaptation the greater flexibility of unstructured grids is required. For unstructured grids two different mesh movement strategies are known. The first exploits the connectivity of the internal grid points. The connection between the grid points is represented for example by springs or as solid body elasticity. These methods involve solving a system of equations as large as the number of flow points involved and are therefore very expensive. Also special treatment is required for hanging nodes. The other strategy moves each grid point individually based on its position in space and are the so called point-by-point schemes. Hanging nodes are no problem and when radial basis function interpolation is used, a much smaller system, only involving the nodes on the boundary, has to be solved. Also the implementation for partitioned meshes, occuring in parallel flow computations, is straightforward. However, untill now point-by-point schemes are only applied to the boundary nodes of multi-grid blocks [1] (the structured interior mesh of the blocks is adapted with algebraic techniques) or the data transfer over the fluid-structure interface [2, 3]. In this paper a new point-by-point mesh movement algorithm based on interpolation with radial basis functions (RBF’s) is developed, which interpolates the displacements of the boundary nodes to the whole flow mesh, instead of over the fluid-structure interface only as is the case in [2] and [3]. The algorithm is tested with several RBF’s for a variety of problems. The new method can handle large translations, rotations and deformations, depending on the used RBF. The best accuracy and robustness are obtained with the thin plate spline [3]. However, when efficiency is more important, the C 2 RBF with compact support [2] is the best choice. Further research includes comparing the new method with existing methods on accuracy and efficiency and applying it to a real fluid-structure interaction problem.

References [1] M. A. Potsdam, G. P. Guruswamy, A parallel multiblock mesh movement scheme for complex aeroelastic applications, Tech. Rep. AIAA-2001-0716, 2001. [2] A. Beckert and H. Wendland, Multivariate interpolation for fluid-structure-interaction problems using radial basis functions, Aerospace Science and Technology, 0, 1–11, 2001. [3] M. J. Smith, C. E. S. Cesnik, D. H. Hodges, Evaluation of some data transfer algorithms for noncontiguous meshes, Journal of Aerospace Engineering 13 (2), 52–58, 2000.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

419

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Mechanical Engineering Department, School of Engineering, Shiraz 71345, Iran e-mail: [email protected] Ŕ

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Mechanical Engineering Department, School of Engineering, Shiraz 71345, Iran 5KCOCMPA"[CJQQEQWM

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

420

Fluid-Structure Interaction in FEM Journal Bearing Simulations Alex de Kraker∗ , Daniel J. Rixen† , Ron A.J. van Ostayen∗ ∗ Delft University of Technology, Faculty of Mechanical, Maritime and Marine Engineering, Department of Precision & Microsystems Engineering, Mechatronic Design, laboratory of Tribology Mekelweg 2, 2628 CD Delft, The Netherlands [email protected] † Delft

University of Technology, Faculty of Mechanical, Maritime and Marine Engineering, Department of Precision & Microsystems Engineering, Mechatronic Design, laboratory of Dynamics [email protected] ABSTRACT

This paper describes a numerical method solving the mixed lubrication problem for stationary running elastic journal bearings. The problem is described by a coupled set of 2D Reynolds - and 3D structure deformation equations. An asperity contact model, relating the contact pressure to the film height, has been used to account for partial contact in mixed lubrication. With increasing loads, the bearing deformation becomes more important to the performance of the bearing system, especially with the introduction of polymers as a bearing material. This is due to the high sensitivity of the pressure solution from the Reynolds equation with respect to a variation in film height. Therefore, a strong coupling exists between the fluid and structure equations. The fluid and structure equations are solved seperately by finite elements. A direct iterative algorithm with various under relaxation strategies has been found insufficient to find a converged solution to the coupled journal bearing problem [1], even in full film lubrication where no contact occurs. For softer surfaces or higher loads, the bearing deformation becomes larger and the coupling between the equations becomes stronger. In the method presented in this paper, artificial dynamics have been added to the stationary structure deformation equations by the introduction of a damping term to the discretised set of equations. To enforce convergence of the problem, the ratio between the damping coefficient and the artificial time step can chosen. The so called Stribeck curve, depicting the bearing coefficient of friction at constant load as a function of the rotational frequency of the journal - or shaft -, is a useful tool to evaluate the bearing performance. Therefore, an additional computing loop iterating for the correct journal position that results into the target load, is needed. To calculate the complete Stribeck curve, at least 50 points are necessary and hence, the fluid structure problem including the constant load constraint has to be solved about 50 times at different journal frequency. Hence, three nested loops are necessary: a time-loop, solving the fluid structure equilibrium, a second loop, iterating for the target load and an outer loop that takes a number of velocity steps, computing the Stribeck curve. Reduction of computing time can be obtained by scaling the intermediate solutions in the fluid-structure - or time - loop. As we know that the final solution will meet the load constraint, intermediate solutions for the fluid - and contact pressure distribution in the time loop are scaled with respect to the target load. Hence, the total load that is applied to the bearing surface is constant and that we only iterate for the correct balance between fluid - and contact pressure. As a result, no large differences in the solution for the bearing deformation between successive iterations occur and convergence is obtained faster.

References [1] Oh, K.P., Huebner, K.H., Solution of the Elastohydrodynamic Finite Journal Bearing Problem. Journal of Tribology, July, 342–352, 1973.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Numerical simulation of wind-structure interaction for thin shells and membranes Alexander Kupzok, Roland Wüchner and Kai-Uwe Bletzinger Technical University Munich, Arcisstr. 21, 80290 München, Germany { kupzok, wuechner, kub}@bv.tum.de

ABSTRACT Modern architecture promotes light and efficient structures. With the use of innovative constructions and materials, the realization of wide-spanned and creative buildings is possible. However, increasing lightness and slenderness bring along a higher susceptibility to wind effects, which can become the decisive design factor. An accurate assessment of these wind effects with deterministic tools is complicated, in particular in the case of aeroelastic phenomena. In this regard, numerical multiphysics simulations are a promising complement and enhancement to elaborate experimental approaches. The long-term aim of this research is to propose a methodology for the analysis and improvement of light, thin-walled structures, such as thin shells and membrane roofs towards wind effects. The focus is on the appropriate combination of different physical and numerical disciplines to account for the relevant factors inherent to the simulation of light, thin-walled structures as well as highly turbulent air flows. To fulfill these requirements the occurring wind-structure interaction is accessed by a surface-coupled fluid-structure interaction (FSI) method. This is realized in a modular and flexible software environment with the use of a partitioned coupling approach: the structural field is solved by the in house finite element program CARAT using several finite element types and advanced solution strategies for form finding, nonlinear and dynamical problems. The fluid field is solved by the CFD software package CFX-5 of ANSYS Inc. Additional care towards the realistic modeling of physical wind is taken. A prerequisite to allow for the assessment of aeroelastic problems, beyond the mere exchange of data between the two physical fields, is the utilization of stable as well as efficient coupling strategies. Moreover, the comprehensiveness of this approach opens the possibility for multiphysics optimization. The contribution will present theory and realization of an implementation enhanced by illustrative examples. Strategies for the extension of the approach towards multiphysics optimization will be presented.

References [1] K.-U. Bletzinger, Roland Wüchner, Alexander Kupzok: “Towards FSI for light-weight structures subjected to wind“, in: Conference Proceedings of Int. Conf. on Computational Methods for Coupled Problems in Science and Engineering COUPLED PROBLEMS 2005 (M. Papadrakakis, E. Oñate and B. Schrefler ,eds) CIMNE, Barcelona, Spain, 2005 [2] M. Kuntz, J.Carregal Ferreira, F. R. Menter and G. N. M. Oudendijk, “Analysis of FluidStructure Interaction with an improved coupling strategy”, ECT Conference, Prague, (2002). [3] D.P. Mok, W.A. Wall, “Partitioned analysis schemes for the transient interaction of incompressible flows and nonlinear flexible structures”, in Trends in Computational Structural Mechanics (W.A. Wall, K.-U. Bletzinger, K. Schweizerhof, eds.), CIMNE, Barcelona, 689- 698, (2001).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

422

Reliability Analysis of Prestressed Egg-shaped Digester Jie LI *, Hua-ming CHEN †, Jian-bing CHEN † *

School of Civil Engineering, Tongji University Siping Road 1239, Shanghai, P. R. China [email protected]

† Shanghai Municipal Engineering Design Institute Zhong Shan North Second Road 901, Shanghai, P. R. China [email protected] †

School of Civil Engineering, Tongji University Siping Road 1239, Shanghai, P. R. China [email protected]

ABSTRACT The fluid-solid interface of prestressed egg-shaped digester is conical, so the dynamic liquid pressure and the resulting reactions are rather difficult to gain[1]. Through the computer program ANSYS, seismic evaluation of the digester is performed using a 3D finite-element model that includes the effect of fluid-solid interaction[2]. Comparisons of calculated responses and those of shaking table tests show reasonably good agreement. By using the probability density evolution method for random vibration analysis of stochastic structures which has been developed in recent years [3,4], reliability of the prestressed egg-shaped digester is investigated. The results demonstrate that the proposed method is efficiency for larger structures.

References [1] Sutter,Gerhard; Hanskat, Charles S. World’s largest egg-shaped digesters. Water Environment and Technology, 1990, 29(40):52-55 [2] Mehdi S. Zarghamee, Atis A. Liepins, etc. Egg-shaped digester design and seismic evaluation. Restructuring: America and Beyond Structures Congrees - Proceedings. ASCE, New York, NY, USA. 1995:1766-1780 [3] Li J. Stochastic Structural System: Analysis and Modeling. Beijing: Science Press, 1996. [4]

Li J, Chen JB. Probability density evolution method for analysis of stochastic structural dynamic response. Actamechanica sinica. 2003, 35(6):716-722.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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The Vortex Structures in the Sphere Wakes in the Wide Range of the Reynolds and Froude Numbers Paul V. Matyushin, Valentin A. Gushchin Institute for Computer Aided Design of the Russian Academy of Sciences (ICAD RAS) 19/18, 2nd Brestskaya str., Moscow 123056, Russia [email protected], [email protected]

ABSTRACT At the present paper the homogeneous (at 200 ” Re ” 106, fig. 1 (a-b)) and stratified (at 50 ” Re ” 1000, 0.005 ” Fr ” 1, fig. 1 (c)) viscous incompressible fluid flows around a sphere are investigated by means of the direct numerical simulation (DNS) and the visualization of the 3D vortex structures in the wake (Reynolds number Re = Ud/v, where U is the free-stream velocity, d is the diameter of the sphere, and v is the kinematic viscosity; Froude number Fr = U/(N·d), where N is the buoyancy frequency). For DNS the Splitting on physical factors Method for Incompressible Fluid flows (SMIF-MERANGE) with hybrid explicit finite difference scheme (second-order accuracy in space, minimum scheme viscosity and dispersion, monotonous) is used [1]. For the visualization of the 3D vortex structures in the sphere wake the isosurfaces of Im(ı1,2) are drawing, where Im(ı1,2) is the imaginary part of the complex-conjugate eigen-values of the velocity gradient tensor (fig. 1). In spite of the set of the papers devoted to the homogeneous fluid flows around a sphere the detailed formation mechanisms of vortices (FMV) in the sphere wake are still unclear [2]. At the present paper for the homogeneous fluid flows the six basic FMV have been selected; the detailed FMV for the different unsteady periodical flow regimes are explained (270 < Re d 290, 290 < Re d 320, 320 < Re ” 400, 400 < Re < 700 and Re > 700); at 290 < Re d 320 a new flow regime has been found; the laminar-turbulent transition in the boundary layer on the sphere is discussed (9·104 < Re < 4·105). The numerical studies of the stratified fluid flows past a sphere are very rare. At the present paper four different flow regimes of the stratified fluid flows have been simulated: 1) “Two steady twodimensional attached vortices” (Re < 120, Fr < 0.2, fig. 1 (c)); 2) “Unsteady two-dimensional attached vortices” (Re > 120, Fr < 0.15); 3) “Lee-wave instability” (0.2 < Fr ” 0.4); 4) “Nonaxisymmetric attached vortex” (Re < 500, Fr > 0.4). The high gradient sheets of density have been observed near the poles of the resting sphere [3] and of the moving sphere (Fr ” 0.02). The lee waves, the recirculating zone and other vortex structures of the wake have been visualized (fig. 1 (c)). This work is supported by Russian Foundation for Basic Research (grant ʋ 05-01-00496); by the program “Mathematical Modeling” of the Presidium of the Russian Academy of Sciences (RAS); by the program ʋ 3 for Basic Research of the Department of the Mathematical Sciences of RAS.

a)

b)

c)

Fig. 1. a) Re = 350, Im(ı1,2) = 0.05; b) Re = 4.1·105, Im(ı1,2) = 2; c) Re = 100, Fr = 0.08, Im(ı1,2) = 0.005.

References [1] V. A. Gushchin, V. N. Konshin, Computational aspects of the splitting method for incompressible flow with a free surface. J. of Computers and Fluids, 21, ʋ 3, 345-353, 1992. [2] V. A. Gushchin, A. V. Kostomarov, P. V. Matyushin, 3D Visualization of the Separated Fluid Flows. Journal of Visualization, 7, ʋ 2, 143-150, 2004. [3] V.G. Baydulov, P.V. Matyushin, Yu.D. Chashechkin, Structure of a diffusion-induced flow near a sphere in a continuously stratified fluid. Doklady Physics, 50, ʋ 4, 195-199, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Transient Analysis Methods for Hypersonic Applications with Thermo-Mechanical Fluid-Structure Interaction Reinhold Niesner*, Matthias Haupt*, Peter Horst* *

Institute of Aircraft Design and Lightweigh Structures, TU Braunschweig Hermann-Blenk-Str. 35, 38108 Braunschweig, Germany [email protected]

ABSTRACT A numerical framework developed for coupled fluid-structure interaction problems is presented, with emphasis on the numerical methods employed for time integration and equilibrium iteration. The framework has been succesfully used in the German IMENS project for the simulation of hypersonic applications. The framework uses a modular concept with stand-alone solvers for the disciplines involved (fluid and structure) with a thoroughly designed data interexchange interface [1]. The solution approach used for mechanical quasi-stationary and thermal transient coupled analyses is presented, taking into account different time scales of the physical domains and efficiency aspects, which are essential when dealing with complex models and solution strategies (e.g. Navier-Stokes codes). For time integration both iterative and simple staggered methods have been used to account for accuracy and efficiency demands of the different problem cases. Several means to accelerate the time integration have been studied. For iterative staggered methods the acceleration methods for the equilibrium iteration presented in [2] have been adopted and improved for the present study case. For example, the gradient method proposed in [2] has been modified to suit the modular concept approached here, where neither the Schur complements can be explicitly computed nor is it possible to easily switch off boundary conditions, as demanded in [2]. The control theory approach for adaptive time stepping presented in [3] has been tested and adapted for the present application area of hypersonic fluid-structure interaction (thermo-mechanical interaction with compressible NavierStokes flows and geometrically and physically non-linear structures). The benefits of this approach over the classical time stepping strategies have been evaluated on the basis of selected parameter studies. The different numerical methods have been tested, evaluated and compared using some generic example models from the IMENS project, e.g. a flap-gap configuration or a nosecap model, which are the subject of this presentation.

References [1] M. Haupt, R. Niesner, P. Horst, Coupling Techniques for Aero-Thermo-Elasticity. Proc. of Int. Conf. on Computational Methods for Coupled Problems in Science and Engineering, Coupled Problems 2005, Santorin, 2005. [2] W.A. Wall, D.P. Mok, E. Ramm, Interactive Substructuring Schemes for Fluid Structure Interaction. Analysis and Simulation of Multifield Problems (W.L. Wendland, M. Efendiev, eds.), 349-360, Springer, 2003. [3] A.M.P. Valli, G.F. Carey, A.L.G.A Coutinho, Control strategies for timestep selection in finite element simulation of incompressible flows and coupled reaction-convection-diffusion processes. International Journal for Numerical Methods in Fluids, 47, 201-231, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Stall Induced Vibration & Flutter In A Symmetric Airfoil Sunetra Sarkar∗ , Hester Bijl∗ ∗

Delft University of Technology [email protected] ABSTRACT

This paper investigates the aeroelastic stability of a wind turbine rotor in the dynamic stall regime. Increased flexibility of modern turbine blades make them more susceptible to aeroelastic instabilities. Further, complex oscillation modes like flap/lead-lag are of particular concern, which give way to potential structural damage [1], [2]. We study the stall induced oscillations in pitching direction and in combined flapwise, lead-lag wise directions. The aerodynamic loads acting on the rotor body in the stall regime are nonlinear. We consider a wide ranging parametric variation and underline their effect on the aeroelastic instability and overall nonlinear dynamical behavior of the system. An engineering dynamic stall model (Onera) [4], [3] has been used to calculate the aerodynamic loads. They represent the aerodynamic loads well in the dynamic stall regime and captures the bifurcation behavior and the chaotic routes of the aeroelastic system under study. The aerodynamic loads are given in terms of differential equations which are combined with the governing equations of the aeroelastic system; the resulting system of equations are solved by a 4/5th order variable step Runge-Kutta method. Parameters considered for the pitching oscillation case are nondimensional airspeed (U ), mean angle of attack (αm ), initial condition (αinit ), structural nonlinearity (Knl ) and reduced frequency (k) and amplitude (F¯0 ) of external forcing. Both the self excited and forced system reveal existence of routes to chaos. For different αm , period doubling routes to chaos have been obtained with different initial conditions. A cubic structural nonlinearity has been seen to alter the bifurcation pattern of the above system. Varying k as a bifurcation parameter in the forced system shows presence of period-3 orbits near chaos. The second case of flap/edgewise oscillation in the stall regime identifies nondimensional rotational speed of the rotor along with structural stiffnesses and nonlinearity as most important parameters of the self excited system. However, no chaotic response has been obtained. External forcing shows presence of higher harmonics and quasi-harmonics in the response. Once again, no chaotic attractor has been found.

References [1] Chaviaropoulos, P., Flap/lead-lag aeroelastic stability of wind turbine blade sections. Wind Energy, 2, 99–112, 1999. [2] Chaviaropoulos, P. , et al., Viscous and aeroelastic effects on wind turbine blades. Wind Energy, 6, 387–403, 2003. [3] Dunn, P. and Dugundji,J, Nonlinear stall flutter and divergence analysis of cantilevered graphite/epoxy wing. AIAA Journal, 30, 153–162, 1992. [4] Tran, C.T. and Petot, T., Semi-empirical model for the dynamic stall of airfoils in view of the application to the calculation of responses of a helicopter blade in forward flight. Vertica, 5, 35– 53, 1981.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Application of Lagrange Multipliers for Computational Aeroelasticity Ralf Unger∗ , Matthias C. Haupt∗ , P. Horst∗ ∗ Institute

of Aircraft Design and Lightweight Structures Technical University Braunschweig Hermann-Blenk-Str. 35 38108 Braunschweig, Germany [email protected]

ABSTRACT The prediction of aeroelastic effects is one of the key problems during the design process of an aircraft. One challenging aspect of this goal is to compute space and time-accurate fluid and structural interactions. In the loose coupling approach, well-established CFD and CSD codes are taken and integrated in a flexible software environment. The main focus of the present work is on the state and load transfer over nonconforming grids on the coupling interface. To fulfill conservation in the overall solution process, a weak formulation of the continuity conditions on the common interface is used and Lagrange multipliers are introduced. This approach is based on a variational formulation of the scalar energy functional of the full system, which utilize Hamilton’s principle and which will be given here. Using an intermediate frame between the interfaces to be joined leads to the so-called three-field formulation, where an additional interface state variable and two Lagrange multipliers are used, [1]. The simplified and more common two-field formulation is utilized which is equivalent with a weighted residual formulation of the continuity condition. Using Galerkin’s method leads to a transfer scheme, which minimizes the L2 error norm. An extended transfer approach, which minimizes the more general Sobolev norm, [2], will be discussed and applied to aeroelastic problems and further the use of dual-Lagrange multipliers will be presented. Numerical results obtained from simulation of an oscillating one-dimensional plate in transonic flow and three-dimensional wing example will be presented to demonstrate the applicability and performance of the concepts and to compare the properties of the different coupling techniques and transfer methods.

References [1] K.C. Park, C.A. Felippa, R. Ohayon, Partitioned formulation of internal fluid-structure interaction problems by localized Lagrange multipliers. Comput. Meth. Appl. Mech. Eng., 190, 2989–3007, 2001. [2] X. Jiao, M.T. Heath, Common-refinement-based data transfer between non-mathing meshes in multiphysics simulations. Int. J. Num. Meth. Engng., 61, 2402–2427, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

427

Computer Simulation of Diffraction Technique Applied for Measurements of Surface Stress Gradients J.T. Assis, V.I. Monine, S.A. Philippov, and S.M. Iglesias

Instituto Politécnico/UERJ Nova Friburgo, Rio de Janeiro, Brazil [email protected]

ABSTRACT Modern surface treatment technologies, like laser treatment or surface modifications by ion beams, introduce high residual stresses characterized by strong gradients of stress distribution in the depth of treated surface. The diffraction methods of stress gradient determination based on analysis of nonlinearity of dϕ,ψ - sin2ψ dependency is not sufficient to practical using because experimental criterions predicting the existence and level of stress gradient are absent. Experimental attempts to develop the methodology of stress gradient determination are not successful because of the difficulty to prepare the samples with known parameters of stress gradient. Computer simulation of diffraction profile formed by reflection from surface layers with stress gradient allows to simulate experimental dependency dϕ,ψ = f(sin2ψ) permitting to obtain the relationships between the stress gradient parameters, non-linearity of dϕ,ψ = f(sin2ψ) and broadening of diffraction line caused by stress.

References [1] P. Nikravesh, Computer-aided analysis of mechanical systems. Prentice-Hall, Englewood Cliffs, New Jersey, 1988. [2] W. Schiehlen, Multibody system dynamics: Roots and perspectives. Multibody System Dynamics, 1, 149–188, 1997. [3] F. Armero and S. Glaser, Enhanced strain finite element methods for finite deformation problems. M. Doblaré, J.M. Correas, E. Alarcón, L. Gavete and M. Pastor eds. III Congreso de Métodos Numéricos en Ingeniería, SEMNI, Barcelona, Spain, 423-437, 1996.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Numerical Assessment of a Micromorphic Model of Ductile Rupture Koffi Enakoutsa∗ , Jean-Baptiste Leblond∗ , Gilles Perrin† ∗ Universit´ e Pierre et Marie Curie (Paris VI) Laboratoire de Mod´elisation en M´ecanique, Tour 65-55, 4 place Jussieu, 75252 Paris Cedex 05, France [email protected], [email protected] † Institut Franc ¸ ais du P´etrole Division de M´ecanique Appliqu´ee, 1-4 avenue de Bois-Pr´eau, 92852 Rueil-Malmaison Cedex, France [email protected]

ABSTRACT All constitutive models involving softening predict unlimited strain localization, and the famous Gurson [1] model of ductile rupture is no exception. An improved variant of this model aimed at solving this problem was derived by Gologanu et al. [2] from some refinement of Gurson’s original homogenization procedure. They obtained a new model of “micromorphic” nature, involving the second gradient of the macroscopic velocity and generalized macroscopic stresses of “moment” type. In this paper, the practical relevance of this new model is investigated through study of its numerical predictions. Two criteria are used for this critical assessment: absence of mesh size effects in finite element computations and agreement of numerical and experimental results for some typical ductile fracture tests. The necessary implementation of the model into some finite element code raises two main problems. The first one is the apparent need for elements of class C 1 . This need is obviated through introduction of some new nodal variables representing the components of the strain rate. The second difficulty lies in the necessary operation of “projection” onto the sophisticated yield locus. An implicit algorithm similar in principle to the classical Nguyen [3] algorithm for the von Mises criterion, although much more complex in detail, is adopted for this purpose. Numerical simulations of the fracture of axisymmetric pre-notched and pre-cracked specimens reveal satisfactory independence with respect to mesh size and reasonable agreement with experimental results. This shows that the model of Gologanu et al. [2] may be regarded as a viable solution to the problem of unlimited localization in Gurson’s model of ductile rupture.

References [1] A.L. Gurson, Continuum theory of ductile rupture by void nucleation and growth: Part I - Yield criteria and flow rules for porous ductile media. ASME Journal of Engineering Materials and Technology, 99, 2-15, 1977. [2] M. Gologanu, J.B. Leblond, G. Perrin and J. Devaux, Recent extensions of Gurson’s model for porous ductile metals. P. Suquet ed., Continuum Micromechanics, CISM Courses and Lectures 377, Springer, 61-130, 1997. [3] Q.S. Nguyen, On the elastic plastic initial-boundary value problem and its numerical integration. International Journal for Numerical Methods in Engineering, 11, 817-832, 1977.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

429

Fatigue Crack Trajectory Analysis of Single-Side Repaired Thin Aluminum Panels with Various Composite Patch Lay-up Configurations H. Hosseini-Toudeshky, B. Mohammadi, S. Bakhshandeh Aerospace Engineering Department, Amirkabir University of Technology Hafez Ave., 424, Tehran, Iran {Hosseini, Bijan_Moh, Bakhshandeh}@aut.ac.ir

ABSTRACT In this paper experimental and numerical finite elements fatigue crack propagation analysis of the single-side repaired thin aluminium panels containing an initial inclined flaw of 450 are studied. These panels are repaired with the 4 layers glass/epoxy composite materials with the lay-up configurations of [90]4, [105]4, [-45]4, [-45/+45]2, and [902/02]. In the performed three dimensional analyses it was assumed that the crack-front remains perpendicular to the panels’ surfaces during its propagation. The effects of the patch lay-up configurations on the crack trajectories at patched and un-patched surfaces of the panels are presented. It will be shown that crack trajectories at patched surface is different from the un-patched surface and therefore leads to the existence of a three dimensional fatigue-fracture surface for all repaired panels. A typical fracture surface is shown in Figure 1-(a). Figure 1-(b) compare the typical crack trajectories of un-patched surface of the repaired panel with the patch layup of [-45]4 obtained from experiment with that obtained from finite elements analyses. The finite elements results show a comparable agreement with those obtained from the experiments. The finite elements results also show that using various lay-up configurations may lead to various crack propagation orientation with 50 to 100 differences.

2B FEM

Exp.

(a) (b) Figure 1. (a) Typical fracture surface; (b) Typical crack trajectories of un-patched surface of the repaired panel with the patch lay-up of [-45]4 obtained from experiment and FEM

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

430

Prediction of the Crack Initiation Life of Turbine Blade Sharadchandra D. Jog and Rajeshwar. Baddam Indian Institute of Technology, Bombay Mechanical Engineering Department IIT, Powai, Mumbai, 400076. [email protected] Indian Institute of Technology, Bombay Mechanical Engineering Department IIT, Powai, Mumbai, 400076. [email protected]

ABSTRACT High Cycle Fatigue of turbo machinery blades is a significant design problem because one of the turbine stages may operates very close to the resonant condition and lead to fatigue failures. In order to assess the crack initiation life of a turbine blade, it is essential to correlate vibration to fatigue. Often a crack initiates from the material imperfections under the combination of steady stresses and fluctuating stresses in high cycle fatigue phenomena. This work models a turbine blade as an untwisted, non tapered cantilever beam with asymmetric cross section. The natural frequencies of the turbine blade were determined by using modal analysis in ANSYS. Nozzle excitation frequencies and forces were determined from the analysis of flow path field between stator and rotor blades. The critical condition at which natural frequencies are coincident with nozzle excitation frequencies were spotted from the Campbell diagram. Steady stresses and dynamic stresses were calculated in ANSYS using excitation forces corresponding to the resonance condition. The stress results obtained were compared with the analytical approach. The true stresses in the vicinity of the defect were calculated by Neuber’s rule with dynamic stresses as input. Local strain around defect was calculated through the formulae given by Martin et al. Crack initiation life was predicted by solving strain life equation.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

431

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

432

Computed Analysis to Determine Service Life Criteria of Special Elements and Applications M. Kopecky Dept.of Physical Material Engineering Faculty of Industrial Technologies, University of Trencin SK-020 01 Puchov, Slovak Republic [email protected]

ABSTRACT A characteristic feature of new trends in development of new aggregates of mobile machinery is a continuous increase in manufacturing and operating costs. Simultaneously, transmitted outputs are also higher and a sufficient reliability has to be maintained. There is a tendency towards a higher use of materials, i.e. a relatively higher stress on particular parts of the aggregate. At the same time, a real safety of operation against the maximum admissible stress decreases. This all requires a further improvement of the method of designing and strength checking of a construction. The problem of fatigue strength and service-life, as the most important phenomena of strength reliability under those conditions, is connected more or less with a certain degree of uncertainty. The methods described in this paper are the ways to reach the solution goals by means of a characteristic curve of fatigue strength and reduced fatigue curve with the maximum use of computer technology.

References [1] M. Kopecky, F. Peslova. Assessment methodology of elements and constructions, reliability criteria for mobile machines and equipment. In: ISTLI special publication 2: Teaching and Education in Fracture and Fatigue, Imprint: E & FN SPON , London, England, pp.325-330, (1996). [2] V. Cuth, J. Tvaruzek, J. Vavro, S. Husar, B. Varkolyova. The stress analysis and the service life prediction on the low-power motorcycle. In: 4th Mini Conf. on Vehicle System Dynamics, Identification and Anomalies, Budapest, Hungary, pp.171-177, (1994). [3] I. Letko, O. Bokuvka, G. Nicoletto, M. Janousek, P. Palcek. Fatigue resistance of two tool steels. In: 11th Danubia-Adria Symposium on Experimental Methods in Solid Mechanics, Baden, Austria, pp.139-140, (1994). [4] J. Vavro. Optimisation of the Design of Cross-Sectional Quantities in Transport Machines and Equipments. In: Studia i materialy, Technika, Zelena Gora, Poland, pp.187-194, (1998). [5] W. Weibull. A statistical distribution function of wide applicability. In: Journal of Appl. Mechanics, No.3, (1951).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

433

Analysis of Crack Initiation and Propagation in Polycrystalline Meso and Microstructures of Metal Materials Torsten Luther*, Carsten Könke† *

Institute of Structural Mechanics Marienstrasse 15, 99421 Weimar, Germany [email protected] † Institute of Structural Mechanics Marienstrasse 15, 99421 Weimar, Germany carsten.kö[email protected]

ABSTRACT Durability and life cycle analysis of engineering structures is often based on numerical simulations of macroscopic damage behaviour using phenomenological damage and fracture models. Therewith the true mechanisms of crack initiation and various crack propagation can not be covered. In order to integrate the physical material effects which are leading to crack initiation as well as crack propagation, simulations on the meso- or microstructure have to be performed. For metallic polycrystals we can assume that crack propagation on mesoscale (10-3m – 10-6m) occurs mainly along grain boundaries and depends strongly on atomic debonding on microscale (10-6m – 10-10m). The mutual dependence can be investigated by a multiscale analysis obtaining a reasonable damage model based on micro mechanical features. The current work is focused on the investigation of micro structural damage behavior on mesoscale using a two dimensional polycrystal model. The geometry of our mesoscale model describing the polycrystal material structure was first based on a Voronoi cell diagram, wherein each cell was assigned to a single crystal. A comparison of grain size distribution in generated Voronoi structures with grain size distribution measured in natural thin layers of metal materials has shown significant differences. Hence, a Voronoi geometry is not flexible enough to represent realistic grain size distributions. More suitable representations can be obtained by a Weibull or Lognormal distribution. In the current work an advanced algorithm is used to generate polycrystal material structures based on arbitrary distribution functions for a more realistic simulation on mesoscale. In order to take into account the dependency of material properties on crystal orientation we assign an orthotropic linear elastic material model to the single crystals. Extension to elastic plastic material model with realistic plasticity properties has shown no relevant improvements against the linear elastic model. In the analysis both, crystal orientation and material properties of each crystal are distributed in a statistical manner. To simulate crack propagation we apply a coupled cohesive zone model (CCZM) on the interface along grain boundaries. Therein, the peak strength of the CCZM depends directly on the missorientation between neighbouring single crystals. The polycrystal model is applied to analyse the crack initiation and propagation in static loaded representative volume elements (RVE) of metal materials on mesoscale without necessity of initial damage definition. The future research work is focused on the determination of constitutive relations for the CCZM from quasicontinuum (QC) simulations performed on a RVE on microscale and homogenized to the mesoscale. Therefor, we show first investigations of atomic debonding along grain boundaries based on the QC method.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

434

Analysis of displacement and stress distributions in riveted joints E. Szymczyk*, A.Derewońko*, J.Jachimowicz† *

Military University of Technology Kaliskiego Str, Warsaw, Poland [email protected] † Institute of Aviation Al. Krakowska, Warsaw, Poland [email protected]

ABSTRACT This paper deals with the displacement and stress analyses in riveted joint. This is a stage of study of the local physical phenomena in riveted joint. The aim of the investigation is to improve the prediction method for the joint failure mode associated with a crack initiation and propagation at a rivet hole. Riveting is the most commonly used method of joining sheet metal components of the aircraft structures. The residual stress and strain state appears in the joint after the riveting process. Furthermore in service condition aircraft structures are subjected to variable fatigue loads. The riveted joints are critical areas of the aircraft structure due to severe stress concentrations, plastic strain and effects such as surface damage (fretting wear) and secondary bending. Therefore the fatigue crack initiation will start at the rivets holes. Fretting fatigue is recognised as a surface damage phenomenon and describes situation where microslip between contacting surfaces appears to give rise to reduction in fatigue life [2]. The object of the analysis is a tensile loaded lap joint. Fatigue tests are performed for the riveted joint specimens consisting of a steel rivet (shank diameter 5mm) and two D-16 aluminium alloy plates (thickness 2.8 mm). Steel and aluminium interaction due to their different properties causes fretting corrosion and decreasing in fatigue life. This feature is convenient to experimental tests. Numerical FE simulations are carried out with the MARC code for a single lap riveted joint specimen. Three-dimensional numerical model is used to analyse the resulting displacement fields at a hole and a rivet in the neighbourhood of a contact interface. Relative displacements between the rivet and the hole are investigated for various friction coefficients. The local numerical model describes a single rivet, a hole and its neighbourhood in two plates. This type of problem requires the use of contact between the elements assembled and non-linear geometric and elasto-plastic multilinear material models to simulate the behaviour of the rivet and plates [1]. The double sided contact for deformable bodies is defined between the two metal plates and between the rivet and the hole. Although the literature on the fatigue behaviour of riveted joints is quite abundant, many aspects are still not sufficiently understood and investigated and, therefore, they require a further study. The advantage of the numerical simulation is to limit development costs and to improve analysis by giving more complete information about contact stress compared to the pure experimental way.

References [1] B. Langrand at al., An alternative numerical approach for full scale characterisation for riveted joint design. Aerospace Science and Technology 6 (2002) 343–354 [2] D.Nowell, Recent developments in the understanding of fretting fatigue. EFM 73 (2006), 207222

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

435

The synergetic effects of hybrid crossover operators in structural optimisation Carlos Conceição António * *

Faculty of Engineering, University of Porto 4200-465 Porto, Portugal [email protected]

ABSTRACT The development of the crossover operator is based on three different mechanisms: mating selection mechanism, offspring generation mechanism and offspring selection mechanism. Most of the crossover operators are able to get exploration or exploitation of the domain depending on the way in which they handle the current diversity of the population. Each crossover operator directs the search towards a different zone in the neighbourhood of the parents. The quality of the elements that belong to the visited region depends on the particular problem to be solved. This is confirmed by the well known no free lunch theorems. The simultaneous use of diverse crossover operators on the population will induce more efficient algorithms. The aiming of this paper is to analyse and to study the complementary properties resulting from the synergetic effects using several crossover operators in genetic algorithms. The improvements reached in structural optimisation using hybrid crossover operators will be analysed through some standard examples.

References [1] C.A.C. António and I.A. Lhate, A hybrid crossover operator for structural optimization based on commonality in genetic search. Engineering Computations, International Journal for ComputerAided Engineering and Software, 20(4), 390–408, 2003. [2] C.A. Conceição António, A hierarchical genetic algorithm with age structure for multimodal optimal design of hybrid composites. Structural and Multidisciplinary Optimization, SpringerVerlag, Editors: G. Rozvany (Germany) and J. Sobieski (USA), in print, online January 2006.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

436

An adaptive mesh generation strategy for the solution of structural shape optimization problems using evolutionary methods Gabriel Bugeda‡, Juan José Ródenas †, Elke Pahl* , Eugenio Oñate * ‡

Escola Universitària d’Enginyeria Tècnica Industrial de Barcelona (EUETIB-UPC) C/ Comte d'Urgell, 187; 08036 Barcelona (Spain) [email protected] *

International Center for Numerical Methods in Engineering (CIMNE-UPC) C/ Gran Capitán s/n; Campus Nord UPC; Módulo C1; 08034 Barcelona (Spain) {elkepahl,onate}@cimne.upc.edu †

Departamento de Ingeniería Mecánica y de Materiales; Universidad Politécnica de Valencia (UPV) Camino de Vera s/n; 46022 Valencia (Spain) [email protected]

ABSTRACT Evolutive methods are a powerful and robust tool for the resolution of structural shape optimization problems. Nevertheless, the use of these methods requires the analysis of an important number of different designs. The computational cost and the quality of the solutions are very much dependent on the quality of the finite element meshes used for the analysis. One important ingredient of the numerical analysis is the strategy for the generation of a proper mesh for each design. Here we can see two types of strategies: 1. To adapt a single existing mesh to the geometries of all different designs. Some existing strategies allow adapting an existing mesh for very big modifications of the boundary shape preventing the elements from being too much distorted. Nevertheless, despite the fact that this type of strategies provides a valid mesh for each design, there is no control of the discretization error contained in the results of each analysis. 2. To perform a classical adaptive remeshing procedure for the analysis of each different design. Of course, this procedure ensures good quality results in the numerical analysis of each design, but the total computational cost grows significantly because each design is computed more than once. This work presents a new strategy that allows generating an adapted mesh for each design without the necessity of performing a full adaptive remeshing procedure. It is based on the use of sensitivity analysis of all magnitudes related with adaptive remeshing (location of nodes, error estimation,…) with respect to the design variables. This sensitivity analysis is performed only once using a geometry of reference and it is used to project the results of the corresponding analysis to all other designs to be analyzed. The projected information allows generating an appropriate adapted mesh for each new design usually in one shot, greatly reducing the computational cost compared with the described strategy 2. This method was developed and used in the context of the solution of shape optimization problems using deterministic methods.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

437

Multiobjective Optimization of Multibody Systems with Genetic Algorithms João P. Dias*, Ricardo M. Corrêa* *IDMEC - Instituto de Mecânica - Pólo IST Instituto Superior Técnico, Technical University of Lisbon, 1049-001 Lisbon, Portugal [email protected]; [email protected]

ABSTRACT The technological breakthroughs in the transportation industry have made it necessary to simultaneously develop vehicle’s structures, in order to achieve better results in impact situations. In the last few years, the passive safety of vehicles is becoming more important in society, and, in the specific case of automobiles, there are entities responsible for testing, analyzing and reporting impact studies conducted over these vehicles. In this context, engineers are required to achieve better results, in some cases with restricted design schedules. The problem of designing a vehicle’s structure is, in fact, a multiobjective optimization problem in which factors like deformations, accelerations, costs and others, conflict amongst themselves. The most rigorous way to test and study impact scenarios – excluding experimental tests because of the high costs they involve – is to use finite elements methods. However, finite element methods require not only big modeling periods, but also high computation times. This way, using finite elements to study multiobjective optimization problems is not the better approach, mainly in the first stages of the development of a vehicle’s structure, when only its major properties are to be determined. One of the alternatives to finite element methods is to use simplified models, based on multibody dynamics. These models have been capable in the past to produce good results in optimization problems, requiring at the same time small computation times [1]. Mainly in the 90s, genetic algorithms have been used in several engineering problems, and have shown to have numeral advantages over classical methods of optimization, specifically in cases of complex mathematical problems. In this work a methodology for the multiobjective optimization of general structures, with application to railroad vehicles, is presented. This methodology uses simplified 1D and 2D models of the structures, in association with several genetic and classical optimization algorithms [2-4], and a simple graphical interface to help defining the optimization problems. One of the implemented genetic algorithms has been developed specifically to deal in a more efficient way with impact problems. Several 1D and 2D problems are approached with the proposed methodology and presented in this work, showing satisfactory results when compared with past studies.

References [1] M. S. Pereira, J. P. Dias, Analysis and Design for Train Crashworthiness Using Multibody Models, Vehicle System Dynamics Supplement, 40, 107-120, 2004. [2] G. Vanderplaats, DOT – Design Optimization Tools, Version 3.0, VMA Engineering, Colorado Springs, 1992. [3] K. Deb, A. Pratap, S. Agarwal e T. Meyarivan, A Fast and Elitist Multi-Objective Genetic Algorithm: NSGA-II, Kanpur Gen. Algorithms Laboratory Report 200001, India, 2001. [4] Dias, J. P., Corrêa, R. e Antunes, F., “Crashworthiness Optimization of Train Structures with Evolutionary Algorithms”, EUROGEN 2003, 83, Barcelona, Spain, 15-17 September, 2003. .

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

438

Buckling Optimization of Grid Structures via Genetic Algorithms L. Iuspa*, V. Minutolo†, E. Ruocco† *

Dept. Aerospace and Mechanical Engineering Second University of Naples [email protected]

† Dept. Civil Engineering Second University of Naples [email protected], [email protected]

ABSTRACT Optimization methods are very useful tools in the design process of complex structural systems. Use of fiber-reinforced composites in mechanical, aerospace, advanced civil structures and other branches of engineering, all require some numerical technique to find the best configuration that satisfies the assumed goal and meets assigned constraints. In the present work a topological optimization of a generalised 2D grid structure is shown. Specifically, the minimum weight design of homogeneous/composite plates and stiffened panels subjected to buckling load is herein addressed. A closed-form solution was implemented in a numerical procedure and then used to analyze arbitrary geometries. That procedure has proved to solve efficiently the governing equations using coarse meshes with a high-speed convergence [1]. Starting from a bounded, variously arranged orthogonal grid, a prismatic shape is obtained by extrusion and then solved to give the main critical load under assigned boundary conditions. Grid assembly is based on a hybrid (continuous-discrete) parametric description. Discrete parameters are used to control the number of cells in both x and y planar directions. Continuous parameters are then used in NURBS based spatial distributions to alter locally grid points, leaving untouched the topology. Additional parametric distributions are finally added to define local thickness. To perform the optimization task, a bit-masking oriented genetic algorithm has been used. The evolutionary engine of this implementation improves efficiency and flexibility of the optimisation process, and provides also specific capability for handling properly the discretecontinuous domain[2]

References [1] V. Minutolo, E. Ruocco, On Initial Postbuckling of Composite plate Assemblies by semianalytical Procedure, Mechanics of Composite Materials and Structures, 8, 1-14, 2001. [2] L. Iuspa, F. Scaramuzzino, P. Petrenga, Optimal Design of an Aircraft Engine mount via bitmasking oriented Genetic Algorithms. Advances in Engineering Software, 34, 707-720, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

439

Multiscale Multiresolution Genetic Algorithm Using Diverse Population Groups Dae Seung Kim*, Yoon Young Kim† *

School of Mechanical and Aerospace Engineering, Seoul National University, Seoul, Korea Shillim-Dong San 56-1, Kwanak-Gu, Seoul, 151-744, Korea

[email protected] †

National Creative Research Initiatives Center for Multiscale Design, School of Mechanical and Aerospace Engineering, Seoul National University, Seoul, Korea Shillim-Dong San 56-1, Kwanak-Gu, Seoul, 151-744, Korea

[email protected]

ABSTRACT When globally optimal designs are important and required sensitivity analysis is complicated, genetic algorithms can be effective alternatives to gradient-based optimization methods. Genetic algorithms are being used more widely in various engineering fields. However, when the design variables become large, standard genetic algorithms become difficult to use because of excessive computation time needed to obtain converged solutions. In this investigation, we consider two-dimensional structural topology optimizations based on finite element models as a large-size optimization problem. To expedite the solution convergence by orders of magnitude, the genetic algorithm in the multiscale multiresolution setting is proposed. By the multiscale setting, standard single-scale binary design variables are represented in multiple scales by the so-called Binary Wavelet Transform. Note that all genetic operators such as crossover and mutation must be redefined or modified for multiscaled design variables. A key advantage of using multiscale design variables is that the exploration of solution becomes more effective with them. If the multiresolution setting based on the multiscale variables is used, the design optimization can proceed from low to high resolution over several resolution stages where the information passing over resolutions is very effective. In this multiresolution approach, a converged population at a certain resolution level is reused as a part of the initial population at the next higher resolution level. In fact, we form the initial population in three groups: the converged population in the previous resolution, a randomly generated population and the mixture of the converged population and the random population. The detailed technique to generate the three groups is presented, which is an important part of the developed multiscale multiresolution genetic algorithm. The convergence improvement by the multiscale multiresolution approach was verified numerically with several numerical case studies. G

References [1] D. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning. AddisonWesley, 1989 [2] Y.Y. Kim and G.H. Yoon. Multi-resolution multi-scale topology optimization - a new paradigm. International Journal of Solids and Structures, 37, 5529-5559, 2000.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

440

An Adaptive Correction Function for Structural Optimization with Genetic Algorithms Pascual Martí-Montrull*, Osvaldo M. Querin†, Concepción Díaz-Gómez* *

Department of Structures and Construction, Technical University of Cartagena Campus Muralla del Mar, 30202 Cartagena (Murcia), Spain [email protected] [email protected] †

School of Mechanical Engineering, The University of Leeds Leeds LS2 9JT, United Kingdom [email protected]

ABSTRACT This paper presents a new self adaptive correction function for the optimisation of size, geometry and topology of space truss structures using the Genetic Algorithm (GA) method, applied to both continuous and discrete design variables. This function guarantees the diversity of the population at the early stages of the optimisation process. In addition, this function moves the final solution to the feasible region. The adaptive correction function proposed is the product of two independent functions. The first is an individual correction function that corresponds to the increase of the objective function that will be necessary to move an unfeasible individual into the feasible region. The second is a penalty function that increases or decreases the imposed correction, achieving this based on the feasibility or infeasibility of the population members during recent generations. The application of this self adaptive correction function to structural optimization was made using a very simple GA algorithm [1] with binary codification, standard crossover, mutation and elitism. The adaptive correction function proposed was implemented in the optimal design system DISSENY [2]. Numerical experiments were carried out with different structural optimization problems (i.e. crosssectional size, topology and shape optimization of 3-D trusses), with continuous and/or discrete variables, stress, displacement, slenderness and buckling constraints. The obtained results demonstrate that this self adaptive correction function is effective and robust, relieving the user from the burden of having to determinate the penalty parameters for each new problem. The results produced using this self adaptive function are equal or better than those produced using penalty functions.

References [1] D.E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning. AddisonWesley, Reading, Massachusetts, 1989. [2] P. Martí and P. Company, An Integrated System for the Structures and Structural Elements Optimal Design, B.H.V. Topping (Ed.). Advances in Structural Engineering Optimization, CivilComp Press, 1996.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

441

Optimization of the Topology of Masonry Units from the Thermal Point of View using a Genetic Algorithm Luísa C. Sousa, Catarina C. Castro and Carlos C. António Faculty of Engineering, University of Porto Rua Dr. Roberto Frias s/n, 4200-465 PORTO {lcsousa, ccastro, cantonio}@fe.up.pt

ABSTRACT Masonry enclosures subjected to environment conditions have a considerable economic weight in building construction cost, and are also important when considering thermal and structural behaviour. Also, more accurate, methods of analysis of wall systems create an opportunity to design buildings where masonry walls can be more efficient. In Mediterranean countries like Portugal the use of thick single leaf envelope walls can be an interesting alternative to cavity walls because thermal insulation inserts are expensive components of masonry walls [1]. The use of single leaf external walls is acceptable, as long as the correct material and construction techniques are used and a satisfactory structural behaviour is attained [2]. The use of masonry blocks made with lightweight concrete with expanded clay aggregates is increasing in Portugal. Lightweight concrete expanded clay aggregates exhibit particular properties as good thermal and acoustic behaviour provided by the volume of voids. Unfortunately, compressive strengths of lightweight concretes are lower than normal density concretes and low compressive strength reduces the load that can be carried by walls made of lightweight concretes. Moreover the quantity of lightweight concrete must be limited so the production cost is proportional to its quantity and the cost of transport and laying increases with weight. In this paper we present a computational method to optimize the masonry unit topology according to thermal normative requests. Current techniques for the evaluation of the wall thermal performance are focused on the thermal resistance value of the clear wall area. Finite element twodimensional computer simulations are used to optimize the topology of vertically perforated lightweight concrete masonry units. A developed numerical evolutionary algorithm [3] iterates over the direct analysis performed by the commercial code ABAQUS. The optimal solution presented in this paper exhibits a thermal transmittance of the masonry unit U = 0.54 W/(m2 ºC) with an air percentage equal to 42.2% .

References [1] R.Veiga, F. Carvalho and H. Sousa, Experimental evaluation of watertightness of single leaf walls. 12th Int. Brick/Block Masonry Conf. Proc. Vol. IV, Madrid, 25-28 June, 2201-2214, 2000. [2] S.Alves e H.Sousa, Paredes exteriores de edifícios em pano simples, Lisboa - PortoCoimbra, Lidel – Edições Técnicas, Lda, 2003. [3] C.F. Castro, C.A.C. António e L.C. Sousa, Optimization of shape and process parameters in metal forging processes using genetic algorithms. Journal of Materials Processing Technology, 146, 356-364, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Topology Optimization of bidimensional continuum structures by genetic algorithms and stress iso-lines Mariano Victoria, Pascual Martí Department of Structures and Construction, Technical University of Cartagena Campus Muralla del Mar, 30202 Cartagena (Murcia), Spain

[email protected] [email protected]

ABSTRACT In these last years, the algorithms based on the biological process of natural evolution have been confirmed as a potent and robust search procedure. Presently work is introduced a new algorithm for the topology optimization of bidimensional continuum structures. The topology and the external shape of the design depend on a genetic algorithm, which, through the stress iso-lines of Von Mises defines the number, forms and distribution of the contours. The analysis of the structure is carried out by a fixed mesh of finite elements. The genetic algorithm (GA) uses the operators: selection (binary tournament), crossover (single point), and mutation (multibit). The procedure has been implemented in the programming language FORTRAN 95, the versatility and flexibility of the algorithm has been proven through several examples, using for it different fitness functions (Fully Stressed Design, compliance, weight, strain energy, etc). The results have been contrasted with the obtained of the most recent bibliography. Due to the scheme of the procedure, the number of evaluations of the fitness function is inferior to the needful for other procedures of similar characteristics, as: Multi-GA, VCL-GA. The produced results confirm the robustness and efficiency of the procedure.

References [1] H. Kim, M.J. García, O.M. Querin, G.P. Steven and Y.M. Xie, Introduction of fixed grid in evolutionary structural optimisation. Eng. Comp., 17(4), 427–439, 2000. [2] S.Y. Woon, L. Tong, O.M. Querin, G.P. Steven, Optimising Topologies through a Multi-GA System. WCSMO 5, Venecia, 2003. [3] L.Y. Kim, O.L. Weck, Variable chromosome length genetic algorithm for progressive refinement in topology optimization. Struc. Multidisc. Optim., DOI 10.1007/s00158-004-0498-5, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Dynamic Analysis of Folding Patterns for Multi-Folding Structures Ichiro Ario∗ , Piotr Pawlowski† and Jan Holnicki-Szulc† ∗ Dept.

† Institute

of Civil & Environmental Engineering, Hiroshima University Higashi-Hiroshima, 739-8527, Japan [email protected]

of Fundametal Technological Research, Polish Academy of Sciences Swietokrzyska 21, 00-049 Warsaw, Poland {ppawl,holnicki}@ippt.gov.pl

ABSTRACT We present the basic mechanisms for the folding of a multi-layered truss, such as a combination of pantographs under dynamic loading, from a post-buckling perspective. This problem is considered in terms of the large-deflection range that the truss is allowed to fold. It develops several element forms to be stronger for dynamic control in a mechanism reminiscent of a deployable structure. Although there are several kinds of folds at the critical points, we need to develop a new concept for a multifolding mechanism (MFM) that allows various different folds even through a simple structure. We explore dynamic critical and post-buckling effects through the concept of energy minimization and hidden symmetries. For comparisons with the final large-deflection folded patterns, we use the original dynamic program for truss analysis. We demonstrate that, as final buckling develops, the mode patterns must change depending on both the velocity of the dynamic loading and some of the imperfections of the geometry of structure through the behavior of the post-buckling aspects of the fold pattern.

References [1] J. Holnicki-Szulc, P. Pawlowski and M. Wiklo, High-performance impact absorbing materials the concept, design tools and applications, Smart Materials and Structures, 12 (2003), 461-467. [2] J.M.T. Thompson and H.B. Stewart, Nonlinear dynamics and chaos geometical methods for engineers and scientists, John Wiley & Sons Ltd., 1986. [3] J.M.T. Thompson & G.W. Hunt, A general theory of elastic stability, London: Wiley, 1973.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Mechanical Systems Design and Control Optimization with Varying Time Domain J. E. Barradas Cardoso *, P. Pires Moita †, Aníbal J. J. Valido † *

Instituto Superior Técnico, Departamento de Engenharia Mecânica Av. Rovisco Pais, 1049-001 Lisboa , Portugal [email protected] †

Escola Superior de Tecnologia, Instituto Politécnico de Setúbal Campus do IPS, Estefanilha, 2914-508, Setúbal, Portugal [email protected] , [email protected]

ABSTRACT This paper presents an integrated methodology for optimal design and control of nonlinear flexible mechanical systems. A design and control sensitivity analysis and multicriteria optimization formulation is derived. This formulation is implemented in an optimum design code and it is applied to the nonlinear behavior response. Damping and stiffness characteristics plus control driven forces are considered as decision variables. A conceptual separation between time variant and time invariant design parameters is presented, this way including the design space into the control space and considering the design variables as control variables not depending on time. By using time integrals through all the derivations, the design and control problems are unified. In the optimization process we can use both types of variables simultaneously or by interdependent levels. Total time is also considered as varying. For treating time domain variation, a unit time interval is mapped onto the original time interval, then treating equally time variant and time invariant problems. The dynamic response and its sensitivity are discretized via space and time finite elements, and are integrated either by at-once integration or step-by-step. Adjoint system approach is used to calculate the sensitivities. The response analysis and corresponding DSA are implemented into an optimal design code.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Modelling for the determination of the interaction force of impacted Structures. S. Pashah∗ , E. Jacquelin∗ , J.P. Lain´e† , M. Massenzio∗ ∗

LBMC - Universit´e Claude Bernard Lyon I - IUT B - INRETS 17 rue de France - 69627 Villeurbanne - France sulaman.pashah,[email protected], [email protected] † Ecole

Centrale de Lyon - LTDS - 36 Av. Guy de Collongue - 69134 Ecully - France [email protected] ABSTRACT

The design of impacted structures requires the determination of the interaction force between the structures involved. This requires to take the local problem of contact into account and the global description of the structures as well: the cost of numerical calculations can be very high. Hence, a modelling with few degrees of freedom (dof) is required to minimize the durations of calculations. Usually, to determine the interaction force, structures are modelled either by a single degree-of-freedom system , or by a modal description [1], [2]. The modal description is not adapted for non-linear simulations and is slowly convergent: hence a lot of eigenmodes are required for a good accuracy. This interaction problem may also be studied by describing a structure with its “anti-oscillators” : they are an alternative of the traditional eigenmodes. In fact, the ideas which lead to anti-oscillators come from the component modal synthesis method, that is from Craigh and Bampton [3]. Indeed, they are based on: 1- the constraint modes of a structure; they are the eigenmodes of the structure with an extra boundary condition: the displacement at the impact location vanishes; 2- the static mode: it is the shape caused by a static load applied at the impact point in the direction of the impact, such that the displacement at the impact location is equal to one. It is possible to show that these modes lead to a single dof system: the mass of this latter is connected to a set of single dof systems referred to as the “anti-oscillators” because their natural frequencies are some antiresonances of the structure. This modelling is based on a modal approach, but its philosophy is very different because it uses the antioscillators which compel the structure to be motionless: the anti-oscillators have a physical meaning. Then this model allows not only a simulation of the impact with few dof, but also a better understanding of the phenomena involved. An application of cylinder to cylinder impact is given.

References [1] S Abrate Impact on Composite Structures. Cambridge University Press, Cambridge, 1998. [2] W. J. Stronge, Impact mechanics. Cambridge University Press, Cambridge, 2000. [3] R. Roy and J. Craig and MCC Bampton, Coupling of substructure for dynamic analysis. AIAA Journal, 6(7), 1313–1319, july 1968.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Evolutionary Identification and Optimization of Composite Structures W. Beluch Department for Strength of Materials and Computational Mechanics, Silesian University of Technology, Konarskiego 18a, 44-100 Gliwice, Poland [email protected]

ABSTRACT Composite materials, especially composite laminates, play a significant role in the modern industry. Laminate is a material built by joining two materials and it usually consists of two phases: the matrix and the reinforcement. The laminate is typically build of many plies (laminas) having different ply angles. Laminates are popular due to two main reasons: i) the high weight-strength ratio (in comparison with the conventional materials); ii) the possibility to tailor the material properties to the designer requirements by manipulating several parameters like: components material, stacking sequence, fibres orientation or layer thickness [1]. If laminas are composed of the different materials the laminate is called a hybrid one. Two aims of the present paper are: i) to identify the material constants of the laminates; ii) to find the optimum stacking sequence of the laminates. The standard laminates as well as the hybrid ones are considered [2]. Different optimization criteria connected with the modal analysis and free vibrations are taken into account. To solve global optimization tasks and discrete optimization tasks as well as to avoid difficulties with the objective function gradient computation, the evolutionary algorithm (EA) is employed as the optimization procedure. To reduce the computation time, the distributed version of the evolutionary algorithm is used [3]. The finite element method (FEM) professional software package with the laminate modeller is used to solve a direct eigenfrequency problem for the laminate plates. The numerical examples presenting the efficiency of the proposed attitude are attached. As it can be seen from the numerical examples, the proposed identification and optimization method gives positive results. Consequently, this method can be applied to different laminated structures in order to identify the material constants or to find the desired laminate properties for a given criteria.

References [1] S. Venkataraman, R.T. Haftka, Optimization of Composite Panels - A Review. Proc. of the 14th Annual Technical Conf. of the American Society of Composites, Dayton, OH. Sep. 27-29, 1999. [2] L. Grosset, S. Venkataraman and R. Haftka, Genetic optimization of two-material composite laminates. Proc. of ASC conference, Blacksburg, 2001. [3] T. BurczyĔski, W. KuĞ, Distributed and parallel evolutionary algorithms in optimization of nonlinear structures. 15th International Conference on Computer Methods in Mechanics CMM-2003, Wisáa, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Parallel evolutionary optimization of heat radiators by using MSC MARC/MENTAT software Adam Dđugosz Department for Strength of Materials and Computational Mechanics, Silesian University of Technology, Konarskiego 18a, 44-100 Gliwice, Poland [email protected]

ABSTRACT The paper deals with the application of parallel evolutionary algorithms [1] (PEA) and the finite element method [3] (FEM) in shape optimization of heat radiators. The fitness function is computed by means of the thermoelsticity problem modeled by MSC MARC/MENTAT software. In order to create mesh, boundary conditions and material properties of the model a preprocessor MENTAT is used. Internal script language implemented in MENTAT allows avoiding external mesher procedure. Another benefit of this approach is that MENTAT takes into account shadowing effect in radiation [2]. Figure 1a shows the main steps of evaluation of the fitness function. The aim of the optimization is to find the optimal shape of the heat radiator shown in Figure 1b. This problem is solved by the minimization of the different types of functionals.

Figure 1. a) Evaluation of the fitness function b) The geometry of the heat radiator In order to reduce the number of design parameters in evolutionary algorithms the shape of the structure is modeled by Bezier curves. Numerical examples for some shape optimization are included.

References [1] Michalewicz Z. Genetic Algorithms+Data Structures = Evolutionary Programs. Springer Verlag, Berlin and New York, 1996. [2] Siegel H., Howell J.R., Thermal Radiation Heat Transfer, 3rd ed. Hemisphere, Washington, 1992. [3] Zienkiewicz O.C., Taylor R.L. The Finite Element Method, Vol. 1-3, Butterworth, Oxford 2000.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Evolutionary algorithm and boundary element method for solving inverse problems of piezoelectricity G. Dziatkiewicz, P. Fedelinski Department for Strength of Materials and Computational Mechanics Silesian University of Technology Konarskiego 18A, 44-100 Gliwice, Poland [email protected], [email protected]

ABSTRACT The piezoelectric phenomenon is widely uitilized in many devices, for example, sensors and actuators, micro-electro-mechanical systems (MEMS), transducers [5]. An analysis of piezoelectric devices requires a solution of coupled mechanical and electrical partial differential equations. In this paper the boundary element method (BEM) is implemented to solve the coupled field problem in piezoelectrics. The method allows the analysis by discretization of the boundary only [1]. The piezoelectric material is modelled as two-dimensional: homogenous, transversal isotropic, linear elastic and dielectric [3]. In most boundary value problems, the governing equations have to be solved with the appropriate boundary conditions [3]. These problems are called direct. However, when the boundary conditions are incomplete on a certain boundary part, the boundary value problems are generally ill-posed, then the existence, uniqueness and stability of the solution is not always guaranteed. These problems are inverse problems. In this paper the Cauchy problem is considered [3]. Another inverse problem is identification of the material constants. For considered two-dimensional case, the physical properties of the piezoelectric material depend on the nine material constants: four elastic constants, three piezoelectric constants and two dielectric constants. A relatively big number of the constants and difficulties in obtaining the gradient information, cause, that the identification problem of the piezoelectric material constants is quite complicated. To solve the inverse problems the distributed evolutionary algorithm is used [2], [4]. Numerical examples will be presented and they will show that the boundary element formulation with evolutionary algorithm gives an efficient computational intelligence tool for solving inverse problems of piezoelectricity.

References [1] C.A. Brebbia, J. Dominguez, Boundary elements. An introductory course. Computational Mechanics Publications, McGraw – Hill Book Company, Southampton – Boston, 1992. [2] T. Burczyński, W. Kuś, Optimization of structures using distributed and parallel evolutionary algorithm, Lecture Notes on Computational Science, 3019, 572-579, 2004. [3] G. Dziatkiewicz, P. Fedelinski, Boundary element method for solving direct and inverse problems of piezoelectricity. In 10th International Conference on “Numerical Methods in Continuum Mechanics NMCM – 2005, CD- ROM Proceedings, University of Zilina, Zilina 2005. [4] Z. Michalewicz, Genetic Algorithms + Data Structures = Evolutionary Programs, Springer Verlag, Berlin, 1992. [5] H.F. Tiersten, Linear piezoelectric plate vibrations: Elements of the linear theory of piezoelectricity and the vibrations of piezoelectric plates, Plenum Press, New York, 1969.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Evolutionary Optimization of Preform and Die Shape in Forging Using Computational Grid Wacáaw KuĞ * *

Silesian University of Technology, Department for Strength of Materials and Computational Mechanics, Konarskiego 18a, 44-100 Gliwice, Poland wacá[email protected]

ABSTRACT The paper is devoted to shape optimization of perform and die in forging process[1]. The idea is to use evolutionary optimization in computational grid environment[2]. The shape optimization of structures can be solved using methods based on sensitivity analysis information or non-gradient methods based on genetic algorithms. Applications of evolutionary algorithms in optimization need only information about values of an objective (fitness) function. The fitness function is calculated for each chromosome in each generation by solving the boundary - value problem by means of the Finite Element Method. This approach does not need information about the gradient of the fitness function and gives the great probability of finding the global optimum. The main drawback of this approach is the long time of calculations. The applications of the distributed evolutionary algorithms can shorten the time of calculations. The computational grids allows to use distributed computational resources. The use of grid techniques in optimizations can lead to improvements in hardware and software utilization. The other advantages of grids are simple and uniform end user communication portals and programs.

References [1] S. Badrinarayanan, Preform and die design roblems in metalforming, PhD. thesis,

Cornell University, 1997. [2] KuĞ W., BurczyĔski T., Grid-based evolutionary optimization of structures, Proc. PPAM 2005, PoznaĔ, 2005 (the full paper will apper in Springer LNCS series)

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Optimization of mechanical structures using serial and parallel artificial immune systems Wacáaw KuĞ *, Tadeusz BurczyĔski *† *

Silesian University of Technology, Department for Strength of Materials and Computational Mechanics, Konarskiego 18a, 44-100 Gliwice, Poland wacá[email protected] †

Cracow University of Technology, Institute of Computer Modeling, Artificial Intelligence Division, Warszawska 24, 31-155 Cracov, Poland [email protected]

ABSTRACT A shape optimization problem of structures can be solved using methods based on sensitivity analysis information or non gradient methods based on genetic algorithms or on artificial immune systems. This paper is devoted to the method based on the serial and parallel artificial immune system. Artificial immune systems are developed on the basis of mechanism discovered in biological immune systems [?9]. An immune system is a complicated, distributed group of specialized cells and organs. The main purpose of the immune system is to recognize and destroy pathogens – funguses, viruses, bacteria and improper functioning cells. The artificial immune systems (AIS) [1] take only few elements from the biological immune systems. The most frequently used are mutation of the B cells, proliferation, memory cells, and recognition using the B and T cells. The artificial immune systems are used to optimization, classification and also computer viruses recognition. A parallel artificial immune system (PAIS) was introduced in [2] for classification problems. The applications of an artificial immune system in optimization need only information about values of an objective function. The objective function is calculated for each B cell in each iteration by solving the boundary – value problem of elasticity by means of the finite element method (FEM). The main drawback of this approach is the long time of calculations. The applications of the parallel artificial immune system can shorten the time of calculations but additional requirements are needed: a multiprocessor computer or a cluster of computers are necessary. The message passing paradigm of parallel computations is used in presented approach. An artificial immune system is implemented as one master process, other processes – workers evaluate objective functions for B cells.

References [1] L. N. de Castro, F. J. Von Zuben, Learning and Optimization Using the Clonal Selection Principle, IEEE Transactions on Evolutionary Computation, Special Issue on Artificial Immune Systems, 6, n. 3, 239-251, 2002. [2] A. Watkins, X. Bi, A. Phadke, Parallelizing an Immune-Inspired Algorithm for Efficient Pattern Recognition. Intelligent Engineering Systems through Artificial Neural Networks: Smart Engineering System Design, 13, 225-230, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Experiments of Damage Detection in Strips Based on Soft Computing Methods and Wave Propagation ´ ∗ Piotr Nazarko∗ , Leonard Ziemianski ∗

Rzesz´ow University of Technology, Department of Structural Mechanics, W.Pola 2, 35-959 Rzesz´ow, Poland [email protected], [email protected]

ABSTRACT All industry branches like aerospace, mechanical and civil engineering are interested in less intrusion and more accuracy failure assessment techniques. They are mostly interested in damages like cracks, delaminations, disbanding, corrosion, etc. Damage detection and assessment technique was developed in this paper. It uses variations in structural wave propagation for undamaged and damaged structure. This Structural Health Monitoring (SHM) method is useful especially in large, complex and inaccessible structures [1, 2]. Based on earlier promising results with this approach [3, 4] a set of laboratory tests were carried out on simple elements like strips. Two kind of materials were used: steel and plexy. Several failure cases were introduced by cutting or drilling the samples. Piezoceramics (PZT) elements were served as transmitters and receivers of elastic waves trough the monitored specimens. During these experiment different groups of excitation signals (continuous sine wave, one, four and six sine wave impulses) and frequency (frequency range from 2 to 50 kHz) were applied to introduce wave to the structure. The numerical models were also created using Finite Element Method (FEM). Defects in the form of a notch were simulated by the removal of selected finite elements from the model. This simulation gave possibility to extend set of damages cases and improved nets generalization properties. In both laboratory and numerical experiments advanced signal processing techniques were adopted. The measured signals were preprocessed by wavelet transform in order to remove noise. Frequency analysis was carried out by Fast Fourier transform (FFT). Replication technique was adopted to experimental data. To realize dependences between input (harmonic frequencies) and output data (height, width and localization of damage) Artificial Neural Networks (ANNs) were used. Several input combinations and nets architectures were tested. Results presented in this paper proved reliability and usefulness of proposed approach.

References [1] Doebling S.W., Farrar C.R., Prime M.B., A summary review of vibration-based damage identification methods, The Shock and Vibration Digest, 30(2), 91-105, 1998 [2] W. Ostachowicz, M. Krawczuk, M. Cartmell, M. Gilchrist, Wave propagation in delaminated beam, Computers and Structures 82, 475483, 2004 [3] S. Bhalla, C.K. Soh and Z. Liu, Wave propagation approach for NDE using surface bonded piezoceramics, NDT& International, 38, 143-150, Elsevier, 2005. [4] P. Nazarko, L. Ziemia´nski, The Use of Neural Networks in planning of experimental research on identification in rods. Proceeding of AI-METH, Gliwice, 2002.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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The Optimization and Identification Problems of Structures with Fuzzy Parameters Piotr Orantek Silesian University of Technology, Department for Strength of Materials and Computational Mechanics Konarskiego 18a, 44-100 Gliwice, Poland [email protected]

ABSTRACT This paper is devoted to optimization and identification problems of structures with fuzzy parameters. The elasticity problem is considered in the paper. The fuzzy shape of the body, boundary conditions and material parameters are assumed. The optimization and identification problems concern on finding the above parameters. The objective functions of optimization and identification problems are proposed. The objective function includes: (i) mass, (ii) stresses and (iii) displacements of the structure in the case of the optimization problems. The objective function includes the measured and computed displacements in selected sensor points placed on the boundary of the structure in the case of identification problem. The fuzzy numbers are expressed as the set of interval values. To solve the boundary value problem the interval finite element method (IFEM) is used. The optimization and identification problems are carried out using the Two-Stages Fuzzy Strategy (TSFS). The Fuzzy Evolutionary Algorithm is used in the first stage of TSFS. The fuzzy gradient method with neuro-computing of the sensitivity is applied in the second stage. Several tests are performed, and will be presented in the full paper.

References [1] H.D.Bui, Inverse Problems in the Mechanics of Materials: An Instroduction. CRC Pres, Bocca Raton 1994. [2] T. Burczynski, W. Beluch, A.Długosz, P. Orantek, M. Nowakowski. Evolutionary methods in inverse problems of engineering mechanics. In: Inverse Problems in Engineering Mechanics II (eds. M. Tanaka and G.S. Dulikravich), Elsevier, 2000, pp. 553-562. [3] P. Orantek, Fuzzy evolutionary algorithms and neural networks in uncertain optimization problems. International Symposium on Neural Networks and Soft Computing NNSC 2005, Cracow 2005. [4] W. Pedrycz, Fuzzy evolutionary computing. Soft Computing 2 (1998), Springer-Verlag 1998. [5] R. Schaefer. The basis of the genetic global optimization. WUJ, Krakow, 2002. (in Polish)

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Topology optimization of the 3-D structures for various criteria using evolutionary algorithm Arkadiusz Poteralski Silesian University of Technology, Department for Strength of Materials and Computational Mechanics, Konarskiego 18a, 44-100 Gliwice, Poland [email protected]

ABSTRACT Evolutionary methods have been applied in mechanics, especially in structural optimization. The main aim of these methods is the simulation of biological processes based on heredity (genetics) and on the natural selection (the theory of evolution) to create the optimal individuals (solutions) presented by single chromosomes. Evolutionary models of computation can be performed by using genetic algorithms. Evolutionary computations are performed on a population of individuals. The main advantage of the evolutionary algorithm is the fact that this approach does not need any information about the gradient of the fitness function and gives a strong probability of finding the global optimum. The main drawback of this approach is the long time of the calculations. In order to eliminate this disadvantage the distributed evolutionary algorithm can be used to speedup the computations. The shape, topology and material of the structure are generated for various optimization criteria like: the minimization of mass with constraints imposed on equivalent stresses and resultant displacements of the structure, the maximization of the first eigenfrequency, the maximization of difference between first, second and third eigenfrequency and the maximization of the difference between first, second, third eigenfrequencies and forced vibration frequency with constraint imposed on mass of the structure. The interpolation based on the neighborhood of elements which aim at the appropriate selection of mass densities is used. For this interpolation several kinds of number and distribution of control points is considered. Dependence between Young’s modulus and mass density is used in this paper [1]. The optimization process is controlled by evolutionary changing of the mass density. After optimization the procedure, which smoothes an external and internal boundary of threedimensional structure, is used. As a tool for solving the direct problems concerning displacement and stress analysis problems of 3-D elastic structures the Finite Element Method is chosen.

References [1] M. P. Bendsøe: Optimal shape design as a material distribution problem, Struct. Optim. Vol. 1, s. 193-202, 1989

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Computational intelligence system in non-destructive identification of internal defects Antoni Skrobol*, Tadeusz Burczyński*† *

†

Silesian University of Technology, Department for Strength of Materials and Computational Mechanics, Konarskiego 18a, 44-100 Gliwice, Poland [email protected]

Cracow University of Technology, Institute of Computer Modeling, Artificial Intelligence Division, Warszawska 24, 31-155 Cracov, Poland [email protected]

ABSTRACT There are many ways to identify the internal defects in the body, however the most interesting are non-destructive methods of identification. Using these methods, the internal defect can be found only on the basis of the knowledge that can be acquire without destroying of the considered structure. The application of the computational intelligence system (CIS) in non-destructive identification of the internal defects on the basis of the knowledge about displacements in several sensor points on the boundary of the structure is presented. The CIS is composed of an evolutionary algorithm (EA) coupled with the adaptive fuzzy inference systems with pseudo-gaussian membership functions (PGFISs) [2]. The aim of the EA is to minimize the fitness function that is formulated as a weighted sum of difference between the measured boundary displacements of the considered structure and displacements computed for the numerical model of the body with assumed number and kind of defects. The displacements are approximated by the PG-FISs. The values of the position and size of defects are put on the inputs of the systems, the approximated values of the boundary displacements are obtained on the outputs of the PG-FISs. During the tests the CIS had to identify the number, position and size of defects in a two-dimensional elastic body under dynamic load. It had also to select the actual kind of defect (circular void, crack or circular inclusion). The PG-FISs were trained using the steepest descent method and conjugate gradient method. The training and testing data were obtained by the boundary element method (BEM). The CIS is able to identify the kind, position, size and the number of the internal defects in the 2D body on the basis of the knowledge about boundary displacements in elastodynamics [1][2]. The time of identification is very short and depends very slightly on the geometry of the body [1].

References [1] T. Burczyński , P. Orantek , A. Skrobol, Fuzzy-neural and evolutionary computation in identification of defect, Journal of Theoretical and Applied Mechanics, Vol. 42, No. 3, 445-460, 2004. [2] T. Burczyński, A.Skrobol, Internal defect identification by the computational intelligence system. Recent Developements in Artificial Intelligence Methods, Gliwice, 4 pages, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Optimization of topology and stiffeners locations in 2-D structures using evolutionary methods Mirosław Szczepanik Silesian University of Technology, Department for Strength of Materials and Computational Mechanics, Konarskiego 18a, 44-100 Gliwice, Poland [email protected]

ABSTRACT The paper deals with an application of the distributed evolutionary algorithm and the finite element method to the optimization problems of 2-D structures in respect of topology and stiffeners arrangement. Recently, evolutionary methods [2] have been wide applied in mechanics, especially in structural optimization [1]. The main feature of those methods is to simulate biological processes based on heredity principles (genetics) and the natural selection (the theory of evolution) to create optimal individuals (solutions) presented by single chromosomes. The main advantage of the evolutionary algorithm is the fact that this approach does not need any information about the gradient of the fitness function and gives a strong probability of finding the global optimum. The main drawback of this approach is the long time of calculations. The applications of distributed evolutionary algorithms can shorten the time of calculations. The fitness function is calculated for each chromosome in each generation by solving the boundary-value problem by means of the finite element method. The main feature of the first proposed optimization method is an evolutionary distribution of the material in the construction by the change of its material properties (Youngs moduli, densities) or by the change of thickness. This process leads to the elimination of the part of material from the construction and in effect the new shape and topology of the construction emerges. Using interpolation surfaces reduces the number of design variables and shortens the time of the computation. The main feature of the second proposed optimization method is an evolutionary change of the stiffeners arrangement and their shape. The application of the professional program of the finite element method MSC NASTRAN in both methods enables optimization of the complex mechanical systems. The numerical examples confirm the efficiency of the proposed optimization method and demonstrate that the method based on evolutionary computation is an effective technique for solving computer aided optimal design problems.

References [1] Burczyński T., Osyczka A. (eds): Evolutionary Methods in Mechanics. Proc. IUTAM Symposium, Kluwer, Dordrecht 2003. [2] Michalewicz Z. : Genetic Algorithms + Data Structures = Evolutionary Programs. Springer Verlag, Berlin 1992.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A Structural Optimization Problem Formulation for Design of Compliant Gripper Using a Genetic Algorithm Nianfeng Wang , Kang Tai School of Mechanical and Aerospace Engineering Nanyang Technological University 50 Nanyang Ave Singapore 639798 [email protected] , [email protected]

ABSTRACT This paper demonstrates the automatic design of a compliant gripper by structural optimization using genetic algorithms. Compliant mechanisms are single-piece jointless structures that use compliance (elastic deformation) as a means to achieve motion. As such, they have many advantages compared to conventional rigid-link mechanisms and so can be created as a replacement for their rigid-link counterparts, especially when the applications are in the micro-dimensional scale. Recently, relatively simple compliant mechanisms have been successfully synthesized by applying structural optimization methods because these methods automatically determine the topology and shape of structures based on any given desired structural criteria. The mechanism designed in this paper is meant to be able to grip an object and convey it from one point to another. Such a mechanism has useful applications in MEMS and various automation devices, but it is relatively complex. They are difficult to design mainly because their motion has to be analyzed by finite element methods and the relationship between their geometry and their elastic behavior is highly complex and non-linear. The synthesis of such a mechanism is to be achieved by a structural optimization approach using a genetic algorithm as the optimizer and a special morphological representation for defining the design geometry. The problem is formulated as a discrete multiobjective constrained optimization problem. The genetic algorithm is a multiobjective algorithm with constraint handling, based on maintaining separate non-domination (Pareto) rankings for objectives and constraints satisfaction, thus enabling an intelligent selection of solutions for cooperative mating which eliminates the need to prescribe penalty function parameters commonly used for constraint handling. In addition, a recently developed morphological geometric representation scheme is used to define the topology and shape of the structure via an arrangement of skeleton and surrounding material. This technique facilitates the transmission of topological/shape characteristics across generations in the evolutionary process, and will not render any undesirable geometric features such as disconnected segments, checkerboard patterns or single-node hinge connections. A non-linear finite element program has also been used for the large-displacement analysis of the structure, and the program is integrated with the genetic algorithm to form an overall working framework for structural optimization. Some tentative formulations of the optimization problem needed to achieve the required elastic/structural behavior of the design have been developed. The resulting geometries obtained here are clearly defined due to the discrete nature of the geometry representation, unlike in the homogenization or material density methods of topology optimization which require the prescription of some threshold point to interpret whether the resulting material density values in the elements indicate solid material or void. Optimal resulting designs have been obtained that are valid and practical for realization/fabrication.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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The prediction of bankruptcy using weighted fuzzy classifiers R. J. Almeida∗ , S. M. Vieira∗ , J. M. C. Sousa∗ and U. Kaymak† ∗ Technical

University of Lisbon, Instituto Superior T´ecnico Dept. of Mechanical Engineering GCAR - IDMEC Av. Rovisco Pais, 1049-001 Lisbon, Portugal [email protected], [email protected]

† Erasmus

University Rotterdam, Faculty of Economics Department of Computer Science P.O. Box 1738, 3000 DR Rotterdam, The Netherlands, Portugal [email protected] ABSTRACT In real-world databases sometimes one of the classes is more difficult to classify then the others. This can happen, for instance, when one of the class is much bigger than the other. To cope with this problem, this paper proposes to assign specific weights to each class in the model evaluation criterion of the feature selection algorithm. The proposed technique is applied to a real world classification problem: the prediction of bankruptcy. The data set used in this study has missing values and extreme values. The data set also presents a much smaller bankruptcy class than the not bankruptcy class. Feature selection is used to choose the number of relevant features, using the so called correctly classified criterion. In the case study, less features are selected using this criterion, consequently less computational time is taken. Moreover, this paper compares five different fuzzy clustering algorithms in terms of model accuracy and computational burden. These clustering algorithms are also used and compared during the feature selection procedure. For this bankruptcy data set, the classification rate with only 16 features is 80% for the companies that bankrupt, whereas the percentage of companies that do not bankrupt is 95.4%. If we take in consideration that the number of obtained rules is 6, only 16 features are used, as opposed to 70 rules and 35 features as in [?], where the accuracy is 81%, with all of the interpretability issues that this carries, then it can be considered that these are promising results.

References [1] Jerzy Stefanowski and Szymon Wilk, Evaluating business credit risk by means of approachintegrating decision rules and case-based learning. International Journal of Intelligent Systems in Accounting, Finance and Management, 10(2), 97114, June 2001.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Optimization of a logistic process by ant colonies, wasp swarms and genetic post-optimization Pedro C. Pinto∗,† , Thomas A. Runkler∗ , Jo˜ao M. C. Sousa† ∗ Siemens

AG, Corporate Technology, Information and Communications, CT IC 4 81730 Munich - Germany {pinto.pedro.ext,Thomas.Runkler}@siemens.com † Technical

University of Lisbon, Instituto Superior T´ecnico Av. Rovisco Pais, 1049-001 Lisbon - Portugal [email protected]

ABSTRACT This paper addresses the problem of system optimization using meta-heuristical algorithms inspired by biological processes. In particular, we focus on Ant Colony Optimization (ACO), Wasp Swarm Optimization (WSO) and Genetic Algorithms (GA), where GA is used alone and for post-optimization. The meta-heuristics inspired by social insects share a common background, and it is interesting to see how they compare, how they can be applied to a problem, and to which problem each algorithm is more adequate. The table shows the main structure of both ACO and WSO algorithms. As it can be seen, the processes behind them have many similarities, with the pheromones of the ants being the equivalent to the force based hierarchy of the wasps. x is the system variable, computed in both algorithms from a probability matrix. ACO and WSO basic algorithm Initialization of the pheromone matrix τ Computation of the forces f (x) Computation of the probability matrix p(τ ) Computation of the probability matrix p(f ) Computation of the solution x(p) Computation of the solution x(p) Update of the pheromone matrix τ Update of the forces f (x) ACO is based on how unsupervised colony agents cooperate to achieve a common goal, and as such it is a natural way of optimizing systems where cooperation is advantageous. With some bigger or smaller modifications it can be applied to several different problems. WSO is based on how wasps compete between themselves, and as the optimization of logistic systems or network routing usually imply some kind of competition, for resources, power, etc, WSO can be applied successfully in many cases. The paper has four main sections. First, we do a brief introduction. In the second part we present a point by point comparison between ACO, WSO and GA, analyze their potential in optimizing different kinds of example systems and how to determine which one is the best option for a given situation. We follow this study with a comparison of the five variants ACO, WSO, GA, ACO-GA and WSOGA when applied to two systems, a theoretical benchmark problem and a real world logistic system at Fujitsu-Siemens Computers. We end the paper with a global conclusion of the matters discussed, and a summary of the future work. We conclude that WSO produces better results than ACO and GA for the considered logistic problem. We also conclude that the hybrid WSO/ACO-GA performs better than their stand-alone counterparts, and that WSO/GA performs better than ACO/GA due to the GA’s part of the algorithm having a more diversified population.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Kriging-based estimation with noisy data S. Sakata*, F. Ashida*, M. Zako† *

Department of Electronic Control Systems Engineering, Interdisciplinary Faculty of Science and Engineering, Shimane University, 1060 Nishikawatsu-cho, Matsue, Shimane, Japan

[email protected] [email protected] †

Department of Management of Industry and Technology, Graduate School of Engineering, Osaka University, 2-1, Yamada-Oka, Suita, Osaka, Japan

[email protected]

ABSTRACT Several approximate optimization methods will be effective for a recent engineering optimization, especially more flexible methods such as neural network, radial basis function or Kriging method are applicable to complex problem, for example, a non-convex and multi-peaked solution space. These methods will generate an estimated surface which is fit well to the sampling results It is sometimes difficult, however, that these methods are applied to noisy data because of its flexibility. Especially the ordinary Kriging will give an exact interpolation [1], therefore some improvement will be required to be used for approximate optimization with noisy data. In this study, the ordinary-type Kriging method will be originally improved to apply to both of precise and noisy data. A formulation of Kriging estimation and empirical semivariogram will be arranged from the viewpoint of dispersion of sampling results. We choose different types of semivariogram function for sampling data and estimated surface in the case of using the semivariogram model, and the Gaussian type semivariogram model is adopted in this study. The nugget effect is included in the semivariogram model to consider some noises. The nugget effect will cause discontinuity of estimated surface, but this approach enables to eliminate the discontinuity. To take a dispersion of sampling results into account, the empirical semivariogram and Cressie’s weighted least squares criterion is re-defined. The effect of changing empirical semivariogram on estimated results will be also investigated. As a test problem, a surface is estimated with using noisy data, which are generated by giving random noises to a known mathematical function. By comparing the results obtained by the proposed Kriging system with the exact ones or the results obtained by other approximation method, estimation quality of the proposed method is investigated. Influence of noises in sampling results on estimated results is investigated. The proposed method will give a better estimation, and numerical examples illustrate validity and effectiveness of the proposed approach.

Reference [1] Hans Wackernagel, Multivariate Geostatistics, Springer Verlag, 1995 (Japanese Translated edition).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Distributed Optimization using ACO for Concrete Delivery C. A. Silva*, J. M. Faria*, D. Naso† *

Instituto Superior Técnico IDMEC / GCAR [email protected]

† Politecnico de Bari Dep. Electrotechnics and Electronics [email protected]

ABSTRACT The timely production and distribution of rapidly perishable goods such as concrete is a complex combinatorial optimization problem in the context of supply chain management [1]. The problem involves several tightly interrelated scheduling and routing problems that have to be solved considering a trade-off of production and delivery costs. Different approaches have been developed for this problem: a hybrid meta-heuristic method combining genetic algorithms with constructive heuristics [1]; a hybrid approach combining genetic algorithms and ant colony optimization [2]. However, all these approaches consider the optimization problems as separate problems. This paper introduces a novel approach, based on he distributed optimization paradigm proposed in [3], which obtained good results for other supply chains examples. In the distributed optimization framework, both problems are optimized in parallel and exchange information during the optimization process through the pheromone matrix. In this way, it is possible to bias the solution of one of the system in order to improve the performance of the other and thus achieve a better global solution for the supplychain. The simulation results show that this approach globally improves the supply chain results.

References [1] D.Naso, M. Surico, B. Turchiano, U. Kaymac. Justi-in-Time Production and Delivery in Supply Chains: a hybrid evolutionary approach. Proceedings ofIEEE SMC 2004. [2] C.A Silva, J.M.Faria, P. Abrantes, J.M. Sousa, M.Surico, D. Naso, Concrete Delivery using GA and ACO. Proceedings of the 44th IEEE Conference on Decision and Control and European Control Conference ECC 2005, Seville, Spain. [3] C.A Silva, J.M. Sousa, T. Runkler, J.M.G. Sá da Costa, Distributed Optimization of a Logistic System using Ant Colonies. Accepted in International Journal of Systems Science, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

461

Tuning a Vibrating Blade Dynamic Vibration Absorber by using Ant Colony Optimization and Finite Element Modeling Felipe Antonio Chegury Viana *, Giovanni Iamin Kotinda*, Valder Steffen, Jr* *

Federal University of Uberlândia, School of Mechanical Engineering 2121 João Naves de Ávila Av., Campus Santa Mônica, CEP 38400-902, P.O. Box 593, Uberlândia, Brazil [email protected], [email protected], [email protected]

ABSTRACT The present contribution deals with the optimal tuning of a vibrating blade dynamic vibration absorber (VBDVA). To achieve this aim, the natural optimization technique named Ant Colony Optimization (ACO) is applied to the finite element model of the system. Dynamic vibration absorbers (DVAs) are systems constituted by mass, spring and damping elements (secondary structure), which are coupled to a mechanical system (primary structure). The main idea behind the DVAs is the generation of a force, which has the same intensity of the excitation force but in the opposite phase [1]. This phenomenon is known as anti-resonance. The tuning of the DVAs is the procedure that sets the anti-resonance frequency to a given value by changing the DVA parameters (mass, spring and damping values). VBDVA was studied in this work, which is composed by a blade that is subjected to an initial tension and fixed lumped mass. These three parameters (the mass value and its position and the initial tension) are responsible for the VBDVA tuning [2]. Supported by this theory, the optimization problem is described as the minimization of the objective function that relates the difference between the resonance frequencies of the primary system and the VBDVA. The optimum tuning defines the minimum difference respecting the design constraints. To solve the optimization problem it was used ACO [3]. This population-based technique is inspired in the behavior of real ants and their communication scheme by using pheromone trail. A moving ant lays some pheromone on the ground, thus marking the path. The collective behavior that emerges from the participating agents is a form of positive feedback where the more the ants follow a trail, the more attractive that trail becomes for being followed. In the early nineties, when the Ant Colony algorithm was first proposed, it was used as an approach for the solution of combinatorial optimization problems, such as the traveling salesman problem. However, the extension for continuous variables is recent and it is still under development. In this context, this paper presents an engineering application of a continuous domain problem of ACO. Numerical results are reported, illustrating the success of using the methodology presented, as applied to mechanical systems.

References [1] J. P. Den Hartog, Mechanical Vibrations, 4th edition. McGraw-Hill, New York, 1956. [2] G. I. Kotinda, Vibrating Blade Dynamic Vibration Absorber (in Portuguese), M. Sc. Dissertation, Federal University of Uberlândia, Brazil, 2005. [3] K. Socha, ACO for Continuous and Mixed-Variable Optimization, Proc of 4th International Workshop on Ant Colony Optimization and Swarm Intelligence – ANTS’2004, Brussels, Belgium, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Energy pumping of systems connected to a nonlinear energy sink device. C.H. Lamarque∗ , E. Gourdon∗ ∗ LGM / DGCG / ENTPE Rue Maurice Audin 69518 Vaulx en Velin Cedex FRANCE [email protected], [email protected]

ABSTRACT The present study deals with energy pumping phenomenon which consists in passive irreversible transfer of energy from a linear system to a strong nonlinear attachment [1, 2, 3]. The aim is to be able to design efficient nonlinear energy sink devices (for example with cubic nonlinearity [4]) in particular to attenuate modal responses for transient and steady vibrations. The main point is the strong nonlinear coupling. Contrary to the case of standard tuned mass dampers, energy transfer is irreversible because of modal localization which prevents the energy from being released back to the main structure. Various results (theoretical, numerical and experimental) about energy pumping based on recent works are given. Thus, the phenomenon is studied for different excitations: transient and periodical. For transient excitations, the case of seisms is also considered with an indicator of efficiency (i.e. the Arias Intensity). Moreover, advantages of such a system are carried out. The theoretical findings are tested and verified experimentally using appropriately designed reduced scale buildings with one or four floors. In particular, a comparison with an optimized classical linear tuned mass damper (like Frahm damper) can be done. With strong cubic coupling, the features (in particular the modes) of the system are not modified. Indeed, experimental verification has shown the efficiency of energy pumping compared to classical linear tuned mass damper. Not only is the phenomenon robust theoretically but it is possible to implement it practically with a small realistic building model.

References [1] O. Gendelman, Transition of Energy to a Nonlinear Localized Mode in a Highly Asymmetric System of Two Oscillators Nonlinear Dynamics, 25, 237-253, 2001. [2] A.F. Vakakis, Inducing passive nonlinear energy sinks in linear vibrating systems. ASME J. Vibr. Acoust., 123, 324-332, 2001. [3] A.F. Vakakis, O. Gendelman, Energy Pumping in Nonlinear Mechanical Oscillators II: Resonance Capture. ASME J. Appl. Mech., 68, 42-48, 2001. [4] E. Gourdon, C.H. Lamarque, Energy Pumping with various Nonlinear Structures : Numerical Evidences. Nonlinear Dynamics, 40, 281-307, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Stress analysis of curved elastic bar and elastic wedge under bending load; infinite systems and asymptotic Vyacheslav V. Lyakh, Vyacheslav V. Meleshko Kyiv National Taras Shevchenko University 01033 Kyiv, Ukraine [email protected] ABSTRACT The paper addresses the classical problem of plane elasticity – bending of a curved elastic bar a≤r≤b, −α≤ϑ≤α (r, ϑ – polar coordinates, α – the opening angle), and an infinite elastic wedge a≤r<∞, −α≤ϑ≤α. Among various mathematical and engineering approaches, the method of superposition is effective [1]. The problems amount to infinite systems of algebraic or integro-algebraic (for the case of an infinite domain) equations, which appear to be regular at least. The existence and uniqueness of their solutions are vastly elucidated. The question is how to find the solution. The classical approach of simple reduction by Fourier assumes that coefficients with subscripts higher than a chosen value may be neglected. However, it can provide unsatisfactory results whatever high we choose this value. The more abruptly physical quantities will change in corner points of the boundary, e.g. tangential stresses, the less satisfactory results of fulfilling boundary conditions will be. The circumstances change when the algorithm of so-called improved reduction is used. According to this technique, one should take into account the asymptotic behaviour of unknowns at infinity. Providing appropriate change of unknowns, e.g. Xn = G + xn where Xn →G, n→∞, we derive a transformed infinite system, as a rule fully regular, which may be readily solved by simple reduction retaining a few equations only. As for numerical techniques, this procedure is nothing more than improving of convergence by extracting principal terms of series or integrals. For mechanics, it is a way to fulfill boundary conditions with any accuracy – comparative analysis of results is presented – as well as to analyze mechanical effects thoroughly, since every equation determining the asymptotic has a clear mechanical sense. The peculiar aspect of finding a necessary asymptotic is also discussed. It happens when no currently available finite-element or boundary-element code allows that may be done by virtue of asymptotic. It especially concerns physical fields near corner points of the boundary, e.g. edge reactions in a plate bending under normal load, or fundamental stress singularities in mixed problems [2]. Mechanical conclusions are original and significant. Thus, the solution on bending of a curved bar shows that corresponding results by Timoshenko and his notion of Saint-Venant’s principle as regards closed bodies should be considerably refined and corrected. As to bending of a truncated infinite wedge, we significantly advanced in clarification of Carothers paradox of plane elasticity. The results are presented in graphs and tables.

References [1] V. V. Meleshko, Selected topics in the history of the two-dimensional biharmonic problem. Appl. Mech. Reviews., 56, 33–85, 2003. [2] A. F. Ulitko, V. V. Lyakh, Torsional load transfer from a rigid shaft to an elastic plane with a slit. J. Eng. Math., 46, 395–408, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Nonlinear Oscillations in Discretely Continual System Alexander I. Potapov 1,2, Irina V. Miloserdova 2, Ivan A. Potapov 1 1

Mechanical Engineering Research Institute of the Russian Academy of Sciences 85 Belinsky str., 603024, Nizhny Novgorod, RUSSIA [email protected] 2 Nizhny Novgorod State Technical University, 24 Minina str., 603600, Nizhny Novgorod, RUSSIA

ABSTRACT Mechanical structures, for example, rods of vibro-shock and ultrasound devices, frequently work in limits of linear-elastic behavior of a rod material, and nonlinear effects take place only on their boundaries due to constructive features of fixture (presence of splits, restraints in detail connections and etc.). Pulse periodic waves with wide spectrum of frequencies can exist as in such systems as in systems with a distributed nonlinearity [1]. The longitudinal oscillations of the rod, one extremity of which is rigidly fixed, and another one is fixed by non-linearly elastic spring are consider. The periodic and quasiperiodic regimes of oscillations described by a wave equation with nonlinear boundary conditions are investigated. Non-sinusoidal processes can be studied using the spectral method as analysis of interaction of many harmonics, or by means of spatio-temporal description of the process. A second approach enabling to reduce investigation of wave processes to analysis of an ordinary differential equation with a deviating argument is employed below. Methodology for analysis of nonlinear oscillations is suggested for a weak dispersion of boundaries that leads to appearance of a small parameter at higher-order derivative in the equation. Amplitude-frequency characteristics of the system and dependences of the oscillation shape on the amplitude have been obtained in the system with a cubic nonlinearity. Non-sinusoidal oscillations of the relaxation type are shown to exist in the system when amplitudes are large. The stability of periodic oscillations with respect to slow variations of amplitude and phase is considered. Variations of amplitude and phase can vary essentially, but only for time much more than period of wave. It allows to distract from study of fast motions and, using procedure of averaging, to evolve the equations for slow processes describing wave modulation. Research was done under the Presidential Program of Support for Leading Scientific Schools of Russia (project N 1638.2003.8), and at partial support by grants of RFBR (project N 04-02-17156).

References [1] Ostrovsky L.A., Potapov A.I., Modulated Waves. Theory and Applications. The Johns Hopkins Univ. Press, Baltimore and London, 1999.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Propagation properties of bi-coupled nonlinear oscillatory chains Francesco Romeo, Giuseppe Rega Dipartimento di Ingegneria Strutturale e Geotecnica, Universit`a di Roma “La Sapienza” Via Gramsci 53, 00197 Rome, Italy [email protected], [email protected]

ABSTRACT Free wave propagation properties in one dimensional chains of nonlinear oscillators are investigated by means of nonlinear maps. The study extends to bi-coupled oscillators previous results obtained for mono-coupled ones [1]. In this realm, the governing difference equations are regarded as symplectic nonlinear transformations relating the amplitudes in adjacent chain sites ( n, n+1) thereby considering a dynamical system where the location index n plays the role of the discrete time [2]. Thus, the regions of propagating wave solutions are equivalent to the regions of linearly stable map solutions. The bi-coupled model refers to a chain of linearly coupled oscillators characterized by on-site cubic nonlinearities in both the longitudinal and rotational degrees of freedom. Pass, stop and complex regions are analytically determined for period-q orbits as they are governed by the eigenvalues [3] of the linearized map arising from linear stability analysis of periodic orbits. By varying the parameters governing both the coupling between the two d.o.f. and the nonlinearity, a variable scenario of propagation regions can be obtained and the transition between the mono- and bi-coupled behaviour can be described. In both cases the nonlinearity causes amplitude dependent propagation regions. The analytical findings concerning the propagation properties are then compared with numerical results obtained through nonlinear map iteration. The latter are mainly focused on the bounded solutions occurring within the pass-pass band, where, besides periodic orbits, quasiperiodic and chaotic orbits may occur in each d.o.f. with a possibly rich variety of overall response. Parametric investigations show good agreement between the analytical approximation of the propagation regions and the numerical evidence. In particular, it is worth underlying that, regardless of the periodicity q, the bounded orbits region coincides with that of the period-1 case. However, the loss of stability, or orbits’ unboundedness, occurring at the upper bound involves different bifurcations if even-period or odd-period orbits are considered. Namely, while odd-period ones lose stability via period-doubling bifurcation, the even-period orbits lose stability via saddle-node bifurcation.

References [1] F. Romeo, G. Rega, Wave propagation properties of chains of oscillators with cubic nonlinearities via nonlinear map approach, Chaos, Solitons and Fractals , 27, 606–617, 2006. [2] D. Hennig, G.P. Tsironis, Wave transmission in nonlinear lattices, Physics Reports, 307, 333–432, 1999. [3] F. Romeo, A. Luongo, Invariant representation of propagation properties for bi-coupled periodic structures, Journal of Sound and Vibration , 257, 869–886, 2002.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

466

Numerical Analysis of the Estimation of Three Boundary Conditions in Two Dimensional Inverse Heat Conduction Problem 1

2

S. Abboudi , E. Artioukhine

1 FEMTO-ST, UMR 6174 CNRS, CREST Department, UTBM, site de Sévenans, 90010 Belfort, France [email protected] 2 FEMTO-ST, UMR 6174 CNRS, CREST Department, site IGE, 2 avenue Jean Moulin, 90000 Belfort, France [email protected]

ABSTRACT A simultaneous estimation of three boundary conditions of heat conduction problem is proposed by numerical approach. A finite-difference method is used to discretize the governing equations. The aim is to estimate the evolution of the distributions of the unknown heat fluxes from the transient temperature histories taken with several sensors sensors inside a two-dimensional sheet. The temperatures are known at three lines in the finite body. The estimation algorithm of this inverse heat conduction problem is based on the iterative regularization method and on the conjugate gradient method. The alternative direction implicit method is used to solve the direct, the adjoin and the variation problems. For each boundary condition, a descent parameter is computed. An optimal choice of the vector of the descent parameters is used in this study and shows an increase in the convergence rate. All numerical simulations are performed for two-dimensional linear heat conduction problem. The temperatures are given with measurement errors and the iterative process is stopped in accordance with the residua criterion. Numerical results are presented in this study. The temperature and heat fluxes graphs are smoothed. The accuracy and efficiency of the inverse analysis for simultaneously estimating the heat fluxes and the temperature is examined by several cases. Finally, the effects of sensor position and the magnitude of measurement error on the inverse solutions are discussed. Numerical results for some representative cases prove that heat fluxes and temperature can be predicted well by this method. In this paper, simulation results are presented and numerical performance of the method will be discussed.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Metamodeling for the Identification of Composite Material Properties Janis Auzins, Sandris Ruchevskis, Rolands Rikards, Andris Chate Institute of Materials and Constructions, Riga Technical University 6 Ezermalas Str., LV-2006, Riga, Latvia [email protected]

ABSTRACT The problem of identification of elastic properties E={Ex, Ey, Exy, µ} of composite structural elements from vibration tests is considered. The metamodels built on basis of FEM computer experiments and natural measurements of eigenfrequencies are used for the identification [1, 2]. Two methods are compared. First – the minimization of the discrepancy between calculated and measured frequencies. The numerical frequencies are calculated by finite element model using numerical experiment – a set of trial values for the unknown material parameters. The numerical frequencies are compared with the measured frequencies, and material properties are found by minimizing the relative discrepancy m ⎛ f exp − f calc min ∑ ⎜⎜ i exp i E fi i =1 ⎝

⎞ ⎟⎟ ⎠

2

Second – the direct creating of the inverse metamodel

E = E ( f1 , f 2 , ..., f m )

In this case the calculated frequencies are taken as inputs, and material parameters as outputs of the metamodel. For the second case the input variables are highly correlated [3], but the analysis of significance gives the possibility of the best choice of eigenmode numbers for the first method. The combination of both methods is demonstrated on the layered curved carbon/epoxy panel example. Identification of elastic properties of a panel has been carried out with satisfactory results. However the errors of frequency determinations, caused by manufacturing errors (the dimensions, density and others differ from it’s nominal values) and measurement errors (caused e.g. by discretization of frequency band using fast Fourier transformation) can produce unacceptable error of identification, therefore all parameters, not only frequencies, must be measured with split-hair accuracy. In the case of simultaneous identification of elastic and density parameters the result can be indefinite.

References [1] A.L. Araújo, C.M. Mota Soares, J. Herskovits, P. Pedersen Development of a finite element model for the identification of mechanical and piezoelectric properties through gradient optimisation and experimental vibration data. Composite Structures, 58, 2002, 307–318 [2] R. Rikards and J. Auzins, Response Surface Method For Solution Of Structural Identification Problems, Inverse Problems in Science and Engineering, Vol. 12, No. 1, Taylor&Francis, February 2004, 59–70. [3] Y. B. Lim, J. Sacks, W.J. Studden and W.J. Welch, Design and Analysis of Computer Experiments When the Output is Highly Correlated Over the Input Space, Canadian Journal of Statistics, 30, 2002, 109-126.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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An energy approach for a Cauchy problem in elasticity T. N. Baranger* and S. Andrieux† *

LDMS UMR CNRS 5006 INSA, ISTIL-UCBL 18 rue des Sciences, 69621 Villeurbanne, France. [email protected] †LaMSID, UMR CNRS-EDF 2832 1 avenue du Général de Gaulle, 92141, Clamart, France. [email protected]

ABSTRACT We are interested in the problem of data completion, which consists in recovering the data on an inaccessible part of the boundary of a solid using overspecified data measured on another part of it. This is an old problem mathematically known as the Cauchy problem. This kind of problem arises in many industrial, engineering or biomedical applications under various forms: identification of boundary conditions or expansion of measured surface fields inside a body. But also it can be the first step in general parameters identification problems and damage detection where only partial boundary data are under control. Hence, robust and efficient data completion method is an essential and basic tool in structural identification. In this paper we present a method for data completion based on the minimization of an energy-like error functional which depends on lacking data. We consider homogeneous elastic solid ȍ with smooth boundaries, where *c is the part of the boundary where the data Uc and Fc, respectively the displacement and the pressure fields, are known, and *i is the part of the boundary where the data have to be recovered. Following, (V, H and u) will denotes the stress, strain and displacement fields. The problem can be stated as follows, find (Ui, Fi) on *i such that: ­°div (V ( u )) 0 in :, V ( u ).n Fc , u U c on *c ® °¯V ( u ) ^:H ( u ) in :, V ( u ).n Fi , u U i on *i

(1)

where ^ is the fourth-order elasticity tensor. In the approach presented here, we consider, for a given pair (f, d), the following two mixed and well-posed problems, whose solutions are denoted by u1 and u2:

­div (V (u1 )) 0 in :, u1 U c ® ¯V (u1 ) ^ : H (u1 ) in :, V (u1 ).n

on * c f on *i

­div (V (u2 )) 0 ® ¯V (u2 ) ^ : H (u2 )

in :, V (u2 ).n in :, u2 d

Fc on * c on *i

The displacements fields u1 and u2 are equal when the pair (f, d) meets the real data (Ui, Fi) on the boundary *i. Hence, we propose to solve the data completion problem via the minimization of the energy functional:

( Fi , U i )

arg min ³ (V (u1 )  V (u2 )) : (H (u1 )  H (u2 )) ( f ,d )

:

To explore the efficiency of this method, several numerical examples are presented for 2D and 3D situations. The results are in good agreement with the actual ones. The method turns out to be very efficient with respect to the precision of the solution but also with respect to the amount of computation needed. The formulation is very general and can be used with heterogeneous materials and other physical phenomena. Some variant of the problem (1) will also be addressed illustrating the flexibility of the approach and potential applications in experimental methods.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Flat-jack tests and parameter identification for diagnostic analysis of dams R. Fedele, G. Maier, L. Marazza Department of Structural Engineering, Technical University (Politecnico) of Milan, Piazza Leonardo da Vinci n. 32, 20133 Milan, Italy [email protected], [email protected], [email protected]

ABSTRACT In many existing concrete dams chemical-physical processes, such as the alkali-aggregate reaction (AAR) [1] and/or past extreme loadings, have given rise to reductions of local stiffness and strength during decades of service life. Moreover, concrete expansion due to AAR and possible geological motions in the foundation may have generated self-equilibrated stress states, additional to stresses caused by external loadings. In this communication an experimental-computational method centred on flat-jacks is outlined for the assessment of local stress states and possibly deteriorated properties of concrete in existing dams. Herein the synergistic combination of a novel experimental pattern, computer simulation of the tests (by conventional finite elements) and inverse analysis techniques allows to exploit the experimental data more effectively than in the current practice of flat-jack tests [2], without recourse to traditional semi-empirical formulae relating measurable quantities to material parameters. At suitably chosen locations on the free surface of the monitored dam, the proposed technique identifies all the components of local (plane) stress state, the Young moduli in vertical and horizontal direction (often different due to the compacting process), tensile strength and fracture energy of the dam concrete. The inverse problem in point is formulated as a sequence of parameter estimations: the constrained minimization of least-square objective functions is performed by means of a gradientbased, interior-point Trust-Region algorithm (see e.g. [3]). As an alternative to conventional identification techniques, the application of artificial neural networks to the present inverse analysis problem is investigated, in view of their routine, cost-effective use in situ as a “black-box”, by a procedure similar to the one recently proposed in [4] for dilatometric in-depth tests.

References [1] T. Ahmed, E. Burley, S. Rigden, A.Abu-Tair, The effect of alkali reactivity on the mechanical properties of concrete, Construction and Building Materials, 17, 123-144, 2003. [2] American Standard Test Methods, In situ measurements of masonry deformability properties using flat-jack method, ASTM Standard C 1197-91, 1991. [3] Z. Mróz, G. Stavroulakis, G. (Eds.), Parameter identification of Materials and Structures, CISM book, Vol. 469, Springer Verlag, Wien, 2005. [4] R. Fedele, G. Maier, B. Miller, Identification of elastic stiffness and local stresses in concrete dams by in situ tests and neural networks, Structure and Infrastructure Engineering, 1(3), 165180, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Convective Boiling in Mini-Channels: Flow Visualization and Inverse Thermal Characterization H. Louahlia-Gualous, D. Zibret, and E. Artioukhine FEMTO ST, CNRS UMR 6174, CREST 2 avenue Jean Moulin, 90000 Belfort [email protected]

ABSTRACT The electrical and thermal performances of Fuel Cells depend on the temperature distribution inside them. The operating temperature of the Fuel Cell is an important parameter that influences extremely the lifetime of the system. In the previous works, some authors showed that the performance of the Fuel Cells can be increased when the temperature distribution inside the system is uniform. Proton Electrolyte Membrane Fuel Cells (PEMFC) have been considered for transport applications because they operate at low temperatures, between 50°C and 80°C. In order to carry out the uniformity of the temperature and the dissipated heat flux, the two phase flow cooling system is required for this application. Boiling in minichannels is used in the compact heat exchanger for many interesting applications such as: the transport industry, electronics, chemical industry, etc. We are conducted an experimental study for convective boiling inside a minichannels which have the hydraulic diameter of 750µm. The test section is a compact heat exchanger consisting of a set of five parallel, rectangular minichannels machined on the top of a copper block. Some thermocouples are placed inside the wall in order to measure the temperatures. The inverse heat conduction problem is solved to evaluate the thermal boundary conditions: the surface temperature, the heat flux and the local heat transfer coefficient. The Iterative Regularization Method is used to solve the inverse problem. Experiments are conducted by varying the electrical heat flux used for visualizing the different flow patterns inside the parallel minichannels. The influence of temperature errors and the position of thermocouples on the estimated local thermal characteristics are studied.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

471

Solution of Shape Identification Problems on Thermoelastic Solids Eiji Katamine∗ , Hideyuki Azegami† , Masahiro Hirai‡ ∗ Department

of Mechanical Engineering, Gifu National College of Technology 2236-2, Kamimakuwa, Motosu, Gifu 501-0495, Japan [email protected]

† Department

of Complex Systems Science, Nagoya University Furo-cho, Nagoya, Aichi 464-8601, Japan [email protected]

‡ Department

of Mecanical Engineering, Toyohasi University of Technology Tempaku-cho, Toyohashi, Aichi 441-8580, Japan ABSTRACT

Boundary shape determinations, in which the distributions of state functions, such as displacement or stress on linear elastic bodies and temperature on heat-conduction fields, are specified with prescribed distributions for the purpose of improving the performance of machines and structures, are important in mechanical and structural design. In the present study, we consider a shape identification problem in which the thermal deformation distribution at the sub-boundary are specified by prescribed distributions on thermoelastic solids. The shape identification is a very important problem in the development of machine tools that are subject to thermal deformation, and the shape design of the equipment for the purpose of improving machining accuracy by decreasing the thermal deformation is a problem that is directly related to this study. This paper presents a numerical analysis method for solving the shape identification problems of the thermoelastic fields. The square error integral between the actual thermal deformation distributions and the prescribed thermal deformation distributions on the sub-boundaries is used as the objective functional. The shape gradient of the shape identification problems is derived theoretically using the adjoint variable method, the Lagrange multiplier method and the formulae of the material derivative. Reshaping is accomplished using a traction method [1] that was proposed as a solution to domain optimization problems. In the traction method, the domain variations that minimize the objective functional were obtained as solutions to the pseudo-linear elastic problems of continua, defined based on the design domain and loaded with pseudo-distributed traction in proportion to the shape gradient function of the design domain. In addition, a numerical procedure using the finite element method for the shape identification problem is presented. The validity of the proposed method is confirmed based on the results of 2D numerical analysis.

References [1] H. Azegami, S. Kaizu, M. Shimoda and E. Katamine, Irregularity of shape optimization problems and an improvement technique, C om puter A ided O ptim um D esign of Structures V , S. Hernandez and C. A. Brebbia eds, Computational Mechanics Publications, Southampton, 309-326, 1997.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

472

Stochastic And Hybrid Methods For The Solution Of An Inverse Mass Transfer Problem Luciana F. Lage*, Ana Paula C. Cuco*, Flávio M. Folly*, Francisco J.C.P. Soeiro†, Antônio J. Silva Neto* * Departamento de Engenharia Mecânica e Energia, Instituto Politécnico, IPRJ, Universidade do Estado do Rio de Janeiro P.O. Box 97282, CEP 28601-970, Nova Friburgo, RJ, Brazil. [email protected], [email protected], [email protected] †

Departamento de Engenharia Mecânica, Universidade do Estado do Rio de Janeiro Rua São Francisco Xavier 524 s/5024-A, CEP 20550-013, Rio de Janeiro, RJ, Brazil [email protected]

ABSTRACT Recent developments in the pharmaceutical industry have led to the discovery of increasingly complex substances, and a large number of drugs going to market at the moment are chiral substances with optical isomers. Even though the physical properties of optical isomers are very similar, their effects on the human organisms may be drastically different, possibly causing harmful side effects. Therefore, there is a growing demand for efficient methods to purify optical isomers, with one of the most promising alternatives being the simulated moving beds (SMB) chromatography. For the full understanding of the operation of SMBs, and a possible scale-up to industrial production, it is required a better knowledge of the mass transfer mechanisms involved and their dependence on the physico-chemical and process parameters. The first step in that direction consists of the characterization of adsorption columns. In the present work a simple model with an analytical solution is used for the solution of the direct problem of biomolecules adsorption in fixed resin beds. Such model yields good solutions when the axial dispersion coefficient is negligible. The inverse problem is formulated implicitly in which we seek to minimize the norm of the squared residues between calculated and measured values for the breakthrough curves, and for its solution stochastic and hybrid methods are used. Test case results are presented, taking into account real experimental data available in the literature.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

473

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

474

Parameter Identification of Elastic Modulus at Futatsuishi Site Haruki Nakamura∗ , Mututo Kawahara† ∗ Department

of Civil Engineering, Chuo University Kasuga 1-13-27, Bunkyou, Tokyo 112-8551,Japan [email protected]

† Department

of Civil Engineering, Chuo University Kasuga 1-13-27, Bunkyou, Tokyo 112-8551,Japan [email protected]

ABSTRACT This paper presents a parameter identification of the elastic modulus of the ground using the finite element method. In this research, the elastic modulus is identified to minimize the performance function with consists of the square residual between observed and computed result. Using the adjoint method, which is one of the techniques of inverse analysis. The actual site is assumed as an elastic body. To calculate the velocity of wave transmitted through the stratum, the equilibrium of stress equation, the strain-displacement equation, and the stress - strain equation are employed. To identify the elastic modulus, the gradient of the performance function is calculated using the adjoint equation. The SakawaShindo method is employed for the minimization algorithm. The Galerkin method and the Newmark β method are employed to the space and temporal discretizations, respectively. The elastic modulus at the Futatsuishi site can be solved by this analysis.

References [1] A. Hikawa, M. Kawahara, and N. Kaneko: Parameter Identification of Ground Elastic Modulus at Excavation site of Tunnel, Vol.1. Research Report of Kawahara Lab, 16–39, 2004. [2] N.Koizumi: Development of bedrock features inquiry technology that uses numerical analysis, Vol.1. Report of Satokogyo, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

475

Parameter Identification of the Attenuation using First Order Adjoint Method Kazuhiro Ogura *, Mutsuto Kawahara† * Institution of First Author Department of Civil Engineering, Chuo University, Kasuga 1-13-27, Bunkyo-ku, Tokyo 112-8551, Japan [email protected] † Institution Second Author Department of Civil Engineering, Chuo University, Kasuga 1-13-27, Bunkyo-ku, Tokyo 122-8551, Japan [email protected]

ABSTRACT This paper presents a research of Parameter Identification of the Attenuation using finite element method. This parameter is identified so as to minimize the performance function. In addition, this parameter is also identified using the technique of the first order adjoint method. The first order adjoint method is one of the inverse analysis techniques. As the basic equation, to calculate velocity and acceleration in the ground, the balance of stress equation, the strain – displacement equation and stress – strain equation are applied. These are so-called order analysis. As the algorithm, (1) get the observation value, (2) calculate the initial value including displacement, velocity and acceleration (3) iterate the calculation inverse analysis and order analysis (4) calculate performance function (5) convergence. It is necessary to apply the Sakawa-Shindo method to minimize performance function. As the numerical study, at the large factory, there are large press machine that effect the surrounding environment. To reduce the vibration, put into the some kind of material in the ground. This coefficient of attenuation must be calculated. In fact, we observed velocity and acceleration at the factory. These values are applied for this research.

References [1] A, Hikawa, M, Kawahara, and N. Kaneko : Parameter Identification of Ground Elastic Modulus at Excavation Site of tunnel, Vol.1, Research report of professor M.Kawahara Lab, pp72-83, (2004). [2] J, Matsumoto and M. Kawahara : Shape Identification for Fluid-Structure Interaction Problem using Improved Bubble Element, International Journal of Computational Fluid Dynamics, 15, pp.33-45,(2001). [3] H, Sakurai and M. Tanigawa : Study of some problems on the back analysis of measured displacements in tunneling, Vol.1, Journal Geothechnical Engineering, pp85-94,(1990).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Parameters identification of a nonlinear viscoelastic model via an energy error functional Khaled Hadj Sassi, Stéphane Andrieux LaMSID, UMR CNRS-EDF 2832, 1, avenue du Général de Gaulle, 92141 Clamart, France [email protected], [email protected]

ABSTRACT The purpose of this work is to derive and use a variational method to identify the material parameters of a model of creep for the concrete of the containment of nuclear power plants from extensometric experimental measurements. The three-dimensional nonlinear viscoelastic selected model is recast into the framework of generalized standard materials, and highlights two distinct evolutions : reversible evolution and irreversible evolution. An identification of the parameters is designed, grounded on the minimization of an energy like error functional, extending preceding approaches [1][2][3]. The energy error incorporates simultaneously a free-energy error and a dissipation error. This original feature, with respect with other errors in constitutive equations that are sometimes used for parameter identification, is obtained by using general convexity properties of the incremental total energy that can be derived for the evolution problem when using a total implicit time discretization scheme for the standard generalized materials. Control, via this error functional, of the free energy and the dissipation makes possible to obtain a better behavior of the functional when the material parameters to be identified concern both the reversible and the irreversible parts of the constitutive equation. The resolution of the optimization problem requires the efficient and accurate computation of the gradient of the cost function with respect to the parameters. For that purpose, the adjoint state approach is employed. This method, contrary to direct methods and finite differences, is independent of the number of parameters to identify and its computation costs are proportional to the number of cost functions only. Here the formulation being based of the minimization of the error in total constitutive equation between two fields of state variable, each one meeting one of the overspecified boundary conditions on the extensometers, the resolution of four evolution problems is needed for each iteration (2 direct problems and 2 (backward) adjoint problems). Some numerical results are shown, using the finite elements code Code-Aster. In these simulations, one mainly focuses on determining the seven parameters of rigidity and viscosity of the model of creep of the concrete in an axisymmetric containment, where three different homogeneous zones have been defined, taking into account the possible heterogeneity of the building.

References [1] S. Andrieux, A. Ben Abda, T. Nouri Baranger, Data completion via an energy error functional. C.R. Méca 333, pp 171-177, 2005. [2] P. Ladevèze, M. Reynier, N. M. Maia, Error on the constitutive relation in dynamics : Theory and Applications for model updating. Inverse Problems in Engineering and Mechanics, Edited by H. D. Bui, M. Tanaka et al., 1994. [3] L. Rota, An inverse approach for identification of dynamic constitutive equations. Inverse Problems in Engineering and Mechanics, Edited by H. D. Bui, M. Tanaka et al., 1994.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Inverse Radiative Transfer Problems In Two-Layer Participating Media Francisco J.C.P. Soeiro* , Antônio J. Silva Neto† * Departamento de Engenharia Mecânica, Universidade do Estado do Rio de Janeiro Rua São Francisco Xavier 524 s/5024-A, CEP 20550-013, Rio de Janeiro, RJ, Brazil [email protected] †

Departamento de Engenharia Mecânica e Energia, Instituto Politécnico, IPRJ, Universidade do Estado do Rio de Janeiro P.O. Box 97282, 28601-970, Nova Friburgo, RJ, Brazil. [email protected]

ABSTRACT In recent years an increasing number of researchers have been investigating inverse radiative transfer problems due to its relevant applications in engineering and medicine, as well as in several areas of research and technological development. Different formulations and solution techniques have been proposed for such inverse problems. They may be formulated either explicitly or implicit, with the implicit formulations often involving the solution of optimization problems. In the present work we focus on the implicit formulation and solution of an inverse radiative transfer problem in a two-layer plane-parallel medium. For the direct problem solution we use the well known Chandrasekhar´s discrete ordinates method combined with the finite difference method, and for the solution of the inverse problem we use stochastic methods and a hybridization of such methods with a deterministic method. As experimental data we use synthetic data on the radiation intensity with dependence on the polar angle. The formulation and solution of the direct and inverse problems are presented, as well as test case results.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

478

Computation of Sensitivity Coefficients and Estimation of Thermophysical Properties with the Line Heat Source Method Nicolau Hess Thomson* and Helcio R. B. Orlande* * Federal University of Rio de Janeiro, UFRJ Department of Mechanical Engineering, POLI/COPPE Caixa Postal: 68503, Cidade Universitária Rio de Janeiro, RJ, 21941-972 Brasil [email protected] and [email protected]

ABSTRACT This work deals with the identification of thermophysical properties with the line heat source probe developed by Blackwell in 1954 [1]. Such method, which is now a standard ASTM method [2,3], relies on large time transient temperature measurements to identify the thermal conductivity of granular materials or viscous liquids. In this work we present a mathematical model that enables the estimation of the material volumetric heat capacity, in addition to the thermal conductivity, by using temperature measurements taken at short and long times. The complex step method [4] is utilized in order to verify the accuracy of the sensitivity coefficients computed with finite differences. Actual temperature measurements taken with a line heat source probe are used for estimating the properties of quartz sand. Results obtained with the developed approach are compared to those obtained with the classical method.

References [1] J. H. Blackwell, A transient-flow method for determining thermal constants of insulating materials in bulk, J. Appl. Physics, 25, 137-144, 1954 [2] ASTM Standard D5334-00, Standard test method for the determination of thermal conductivity of soil and soft rock by thermal needle probe procedure, 2000. [3] ASTM Standard D5930-97, Standard test method for the determination of thermal conductivity of plastic by means of transient line source technique, 1997. [4] B. F. Blackwell and K. J. Dowding, Sensitivity and uncertainty analysis for thermal problems, Inverse Problems in Engineering: Theory and Practice (Ed. Helcio R. B. Orlande), Vol. I., 6172, e-papers, Rio de Janeiro, 2002.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Combining a Parallel Genetic Algorithm with Variational Approach for Assessing Structural Damage Haroldo F. Campos Velho∗ , Sabrina B. M. Sambatti∗ , Leonardo D. Chiwiacowsky∗ ∗ Instituto Nacional de Pesquisas Espaciais Av. dos Astronautas, 1.758 - S˜ao Jos´e dos Campos/SP, Brasil [email protected]; [email protected]; [email protected]

ABSTRACT Considerable research and effort over the last few decades has taken place in the field of system identification problem for different reasons. One of the most interesting applications involves the monitoring of structural integrity through the identification of damage. The basic idea remains that measured modal parameters (notably frequencies, mode shapes, and modal damping) are functions of the physical properties of the structure (mass, damping, and stiffness). Therefore, changes in the physical properties, such as reductions in stiffness resulting from the onset of cracks, loosening of a connection or more in general due to the aging of material, will cause detectable changes in these modal properties. Among the methods developed for solving the damage identification problem, the use of the conjugate gradient method with the adjoint equation, also named Variational Approach, has recently been presented as a satisfactory choice to face this inverse problem, when small structures are considered [2]. Considering slightly bigger structures, the variational approach can be trapped at local minima [3], i.e., it does not work. Stochastic approaches, such as Genetic Algorithms (GA) represent a powerful choice for solving non trivial problems. By conducting the search in a global domain, the GA approach reduces the chance of converging to local optima, however this approach is very CPU time-consuming. Therefore, the structural damage assessment problem will be solved through the use of a hybrid method where the stochastic approach, a Parallel Genetic Algorithm (PGA), is firstly employed, reducing the CPU time required and providing an appropriate initial guess for the deterministic approach, the Variational Method. Concerning the PGA, it is codified considering the island model, and the parallel code is generated using calls to the message passing communication library MPI (Message Passing Interface). Moreover, the damage estimation has been evaluated using noiseless and noisy synthetic experimental data, and the reported results are concerned with simple mechanical structures.

References [1] S.W. Doebling, C.R. Farrar, M.B. Prime and D.W. Shevitz, Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: a literature review, Los Alamos National Laboratory, report LA-13070-MS, USA, 1996. [2] C.H. Huang, An inverse vibration problem for simultaneously estimating the time-dependent stiffness coefficients. Proceedings of the 4th International Conference on Inverse Problems in Engineering (ICIPE), Angra dos Reis(RJ), Brazil, 26-31 May, 2002. [3] Chiwiacowsky LD, Campos Velho HF, Gasbarri P. The damage identification problem: a hybrid approach. Proceedings of the 2nd Thematic Congress on Dynamics and Control, DINCON-2003, S˜ao Jos´e dos Campos, Brazil, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Design of Variable-Stiffness Composite Panels for Maximum Buckling Load ∗ ¨ Mostafa M. Abdalla∗ , Shahriar Setoodeh∗,1 , Zafer Gurdal ∗

Chair of Aerospace Structures, Delft Technical University Kluyverweg 1, Delft, The Netherlands 1 [email protected] ABSTRACT

Design of constant-stiffness composite laminates for maximum buckling load is well studied in the literature. Using gradient based optimization methods, laminate thickness and total number of layers are often kept fixed and optimum fiber angles are sought such that the buckling load is maximized. Although significant increase in the buckling load can be obtained through tailoring composite panels with straight fibers, the potential of fibrous composites is not fully exploited. A new concept for the design of fibrous composites, known as variable-stiffness panels, was introduced in the late eighties [2]. Instead of straight fiber paths, the fiber paths are allowed to be arbitrary curves. Ideally, by varying the fiber steering geometry, the stiffness properties at each point in the panel can be independently varied. The additional freedom in locally tailoring the stiffness properties means that the performance of variablestiffness panels can be highly improved over constant-stiffness (straight fibers) panels. However, this additional freedom comes at the price of having significantly enlarged design space. In the present study, we extend the generalized reciprocal approximation approach of Abdalla et al. [1] to buckling design of variable-stiffness panels. In the standard reciprocal approximation, the objective function is expanded in a Taylor series in terms of reciprocal variables. Reciprocal variables are traditionally defined as the reciprocals of the design variables. The present generalized reciprocal approximation is obtained by expanding the buckling load in terms of the inverse tensor of the stiffness tensor, commonly known as the compliance tensor. Such an approximation has the important property of being a separable approximation. In order to update the design for the next iteration, we maximize the reciprocal approximation of the buckling load. Because of the separability, the maximization can be carried out at each node separate from the others. This makes the algorithm particularly suited to parallel computations. The sensitivity analysis is performed exactly using an adjoint method which requires only one back substitution using an already factored left hand side (the inplane stiffness used in the buckling analysis) with a different right hand side to compute the sensitivities for all design variables. A conforming CLPT finite element is used for the buckling analysis of rectangular plates and the proposed reciprocal approximation is used to update fiber orientation angles at each FE node. Numerical results obtained for rectangular plates show that improvements up to 40% can be achieved in the buckling load using a variable-stiffness design.

References [1] M. M. Abdalla, S. Setoodeh, and Z. G¨urdal. Design of variable stiffness composite panels for maximum fundamental frequency using lamination parameters. In European Conference on Spacecraft Structures, Materials & Mechanical Testing, Noordwijk, The Netherlands, May 2005. [2] M. W. Hyer and H. H. Lee. The use of curvilinear-fiber format to improve buckling resistance of composite plates with central holes. Composite Structures, 18(3):239–261, 1991.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Optimum Structure with Homogeneous Optimum Truss-like Material Gengdong Cheng, Ling Liu, Jun Yan State Key Lab of Structural Analysis for Industrial Equipment Dalian University of Technology, Dalian, 116024, China [email protected]

ABSTRACT This paper presents a new formulation of concurrent topology design of structure and material, in which the micro-structure of material is assumed uniform in macro-scale to meet the manufacturing requirement. Throughout the development history of structural topology optimization, the conception of micro-structure of material is introduced to implement structural topology optimization as well as concurrent optimization of structure and material. For structural topology optimization, researchers aim at black-white design, and the area of material with complicated micro-structure is suppressed by various penalty approaches. For concurrent optimization of structure and material, material microstructures are different from point to point in the final design obtained by optimization, which gives rise to insurmountable manufacturing difficulty. Since that, the micro-structure of material in structural topology optimization is merely a mathematical treatment for the ill-posed problem. However, in engineering practice there is a need for concurrent topology design of structure and material, in which the micro-structure of material is assumed uniform in macro-scale to meet the manufacturing requirement. In order to simultaneously implement the topology design for macrostructure and homogeneous micro-structure of material, SIMP is utilized in both scales with the definition of two classes design variables, i.e. macro-density distribution within the design domain and micro-density distribution within the homogeneous unit cell. The two scales are linked by performing the homogenization procedure in the micro-structure which determines the elastic material properties used in macro-scale analysis. Numerical experiments for a number of examples demonstrate the advantage of truss-like material. Furthermore, another case is also discussed where the direction of the principal axis for material is taken as an additional design variable. Minimum compliance topology optimization of an elastic continuum is considered in the formulations, and SQP (Sequential Quadratic Programming) method is adopted. To avoid checkerboard patterns, a variant perimeter constraint is used in the micro-scale design. The attached figure from one example shows that an optimum structure with homogeneous optimum truss-like material is obtained under prescribed boundary conditions and material volume. The assumption of uniform configuration of micro-structure may lead to an easier manufacturing process, thereby greatly increasing the possibility of engineering application.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Non-Parametric Large Scale Structural Optimization for Industrial Applications Peter M. Clausen and Claus B. W. Pedersen FE-DESIGN GmbH Haid-und-Neu-Str. 7, 76131 Karlsruhe /Germany, {Peter.Clausen, Claus.Pedersen}@FE-Design.de

ABSTRACT The optimization problems in the industry, here mainly the automotive industry, are closely related to the current state and capabilities of the CAE-tools in the industry. The user expects that the functionalities of the commercial FE-solvers can be applied directly in the optimization definitions and executions. The FE-models may contain more than a million elements, thus the same number of design variables in e.g. topology optimization. The aim of this work is primarily to focus on the current and future industrial optimization problems, which are encountered in this context and often overlooked in the academic world. To clarify these problems a general introduction is outlined. Afterwards, several specific examples are shown and discussed. These examples are produced by FE-Design and their industrial partners using the optimization program TOSCA in combination with commercial CAE-software. The modeling possibilities of today’s CAE-environments are numerous and can solve a variety of problems. A standard industrial CAE model often contains different element types (solids, shells, beams, etc.) in an irregular mesh of a geometrically complex structure having sophisticated boundary conditions, such as springs, kinematic couplings and contacts. Normally, such models are difficult to parameterize. Consequently, non-parametric optimization approaches are generally preferred. The present work deals with three kind of non-parametric optimization strategies which are briefly discussed in the following. Topology optimization is typically used in the early design phase for producing new design suggestions. Topology optimization can include several different constraints imposed on both the static responses and the dynamic responses of the structure. Today, industrial topology optimization often deals with large-scale optimization including more than a million elements.

Shape optimization is often done later in the design process compared to topology optimization. Applying a non-parametric shape optimization approach allows one to deal with arbitrary surfaces and manufacturing constraints because no splines are defined. Shape optimization may also deal with the responses of non-linear implicit or explicit FE-simulations. Bead optimization has its roots in the deep drawing manufacturing of sheet metal. Again, the nonparametric approach is used, allowing complex geometric structures and advanced shell elements to be utilized.

Several examples of topology, shape and bead optimization for large-scale CAE-models will be shown.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Optimal Joint Placement and Modal Disparity in Control of Flexible Structures Alejandro R Diaz*, Ranjan Mukherjee† Michigan State University Mechanical Engineering Department East Lansing MI 48824 USA *

†

[email protected]

[email protected]

ABSTRACT Stiffness variation can be generated in a flexible, vibrating structure using a number of different mechanisms, including cables, variable stiffness members, and variable stiffness joints. In such structures stiffness variation results in modal disparity, a property that allows the flow of vibration energy from one set of modes of the structure to another. The purposeful introduction of modal disparity in a structure using stiffness variation can be viewed as an effort to enhance controllability and observability of the structure, whereby vibration suppression is enabled by sensing and controlling only a few select modes. In previous work by the authors ([1,2]) modal disparity was introduced by means of cables and non-structural masses, placed strategicaly on the structure. In this work we explore other mechanisms to introduce modal disparity, including various implementations of variable stiffness joints. As in [1,2], the objective is to facilitate the transfer of energy from modes that are not controlled to modes that are, and this is achieved by maximizing modal disparity. This work presents a computational scheme to maximize modal disparity by placing a finite number of variable stiffness joints on a flexible frame. The scheme is based on standard methods of topoplogy optimization. The method is illustrated by an example.

References [1] Diaz, A. R.; Mukherjee, R. A topology optimization problem in control of structures using modal disparity. Proceedings of the 2005 ASME Design Engineering Technical Conference, Long Beach, CA, 2005. Also, J. of Mechanical Design (to appear). [2] Diaz, A. R.; Mukherjee, R. Modal disparity enhancement through optimal insertion of nonstructural masses. Structural and Multidisciplinary Optimization (to appear)

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Topology optimization of 3D Stokes flow problems A. Gersborg-Hansen∗ ∗ Department

of Mathematics, Technical University of Denmark,DK-2800 Lyngby, Denmark [email protected] ABSTRACT

Topology optimization has been applied to a multitude of physical systems and is now a mature technology used in industrial practice, see [1] for an overview. Borrvall and Petersson [2] introduced topology optimization of Stokes flow problems which initiated works on extending topology optimization to different flow problems. However, this research has focused on 2D fluid modelling, which limits the practical impact of the computed designs. The explanation of the limitation is that the finite size domain used in topology optimization problems ensures that the velocity components couples, even for Stokes flow [3]. Furthermore, it is questionable if such a coupling can be captured by a 2D model especially in non-trivial geometries as typically seen in topology design. This statement is widely accepted in the fluid mechanics community, i.e. that planar fluid models are useful for academic test problems only. The motivation for considering topology optimization in 3D Stokes flow originates from micro fluidic systems. At small scales the Stokes equations are a reasonable mathematical model to use for the fluid behavior. Physically Stokes flow is an exotic inertia free flow, which in practice complicates mixing by passive devices. Passive mixing devices are relevant particular at micro scales since they are manufacturable (without moving parts) and maintenance free. In order to tackle such a challenging problem a robust method is needed which we approach by this contribution. It contains fundamental aspects of the topology optimization method applied to the Stokes equations as described below. This work consists of two parts. The main part elaborates on effects caused by 3D fluid modelling on the design. Numerical examples are shown where the design is planar - relevant to micro fluidic fabrication techniques - and where the designs are 3D. The second part focuses on post–processing by a comparison of shape optimization and topology optimization in 2D Stokes flow. This investigates if topology design is sufficient for creating optimal designs. This question is addressed in the setting of standard analysis software which enables a credible performance check relevant before design manufacturing. Note that this requires a proper interpretation of a computed design used to generate a body fitted mesh. In addition issues related to the parallel solution of the linear algebra problems are discussed which is the critical bottleneck for the 3D problem. The implementation uses semi–analytical sensitivities to drive a gradient based optimization algorithm.

References [1] M. P. Bendsøe and O. Sigmund, Topology Optimization - Theory, Methods and Applications, Springer Verlag, Berlin Heidelberg, 2003. [2] T. Borrvall and J. Petersson, Topology optimization of fluids in Stokes flow, Int. J. Num. Meth. Fluids 41, 77-107, 2003. DOI:10.1002/fld.426 [3] E. Lauga, A. D. Stroock, and H. A. Stone, Three–dimensional flows in slowly varying planar geometries. Physics of fluids 16(8), 3051-3062, 2004. DOI: 10.1063/1.1760105

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Efficient Optimization of Dynamic Systems using Pad´e Approximants Jakob S. Jensen Department of Mechanical Engineering, Solid Mechanics Building 404, Nils Koppels All´e, Technical University of Denmark [email protected]

ABSTRACT A new efficient approach is proposed for optimization of the dynamic response of structures subjected to a time-harmonic load. Optimization of the dynamic response of large-scale systems is usually based on the modal approach, with only the lowest modes included in the expansion, see e.g. Tcherniak (2002). Especially if the response at more than a few frequencies is to be optimized, the direct approach in which the system equations are solved in the frequency domain will be computationally expensive. In Jensen & Sigmund (2005) it was demonstrated how the direct method can be used to optimize the response in a frequency range with reduced computational effort. Only a few target frequencies were considered, but these were repeatedly updated to match the most ”critical” frequencies in the frequency range. The feasibility of this approach relies on the possibility for computing fast frequency response curves using Pad´e approximants (see e.g. Jin, 2002). In this work this approach is extended. Instead of using Pad´e approximants only to evaluate the frequency response to locate the critical frequencies, the response and the sensitivity analysis in the entire frequency domain of interest is now directly based on the Pad´e expansion. With this approach the direct system needs only be solved for a single frequency and the expansion to the entire frequency range is computed simply as a number of extra r.h.s.’s to the system equation and the solution of a few additional low-rank matrix equations. The new approach is exemplified by the design of a bi-material 2D elastic structure subjected to a time-harmonic load using topology optimization. The response of the structure is minimized in a finite frequency range based on a direct solution of the system equation for a single frequency only.

References [1] D. Tcherniak, Topology optimization of resonating structures using simp method. International Journal for Numerical Methods in Engineering, 54, 1605–1622, 2002. [2] J. S. Jensen and O. Sigmund, Topology optimization of photonic crystal structures: a highbandwidth low-loss T-junction waveguide. Journal of the Optical Society of America B, 22, 1191– 1198, 2005. [3] J. Jin, The Finite Element Method in Electromagnetics, 2nd Ed.. Wiley, New York, 2002.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Ground Structure Based Joint Stiffness Controlling Method for Joint Compliant Mechanism Design Myungjin Kim*, Gang-Won Jang** and Yoon Young Kim† *

Ph.D. Candidate National Creative Research Initiatives Center for Multi Scale Design and School of Mechanical and Aerospace Engineering, Seoul National University, Korea [email protected] **

†

Assistant professor, School of Mechanical Engineering, Kunsan National University, Korea [email protected]

Director, National Creative Research Initiatives Center for Multi Scale Design and Professor of School of Mechanical and Aerospace Engineering, Seoul National University, Korea [email protected] ABSTRACT

The object of this study is to develop a topology optimization method for the design of joint compliant mechanism structures. Joint compliant mechanisms are compliant mechanisms consisting of rigid or sufficiently rigid one-dimensional elements and elastic hinge joints. Although continuumbased methods have been successful to configure compliant mechanisms, the optimized mechanisms are usually difficult to manufacture because of their geometric complexities. The ground structure based topology optimization method using beam elements can alleviate the geometric complexity issue, but it is still difficult to actually fabricate the optimized layouts. Existing topology optimization results on compliant mechanisms indicate that optimized compliant mechanisms have very localized eastic deformations to wok as mechanisms. This means that most elastic deformations in compliant mechanisms occur in very localized hinge regions. Therefore, it will be advantageus to find directly the elastic hinge joint locations and stiffnss values by a topology optimization formulation while ground beams are treated only as joint-connecting elements without going through much deformation. Motivated by this observation, we propse a joint stiffness controlling method where gournd beams of given thickness are connected through joint springs at beam joints. The main characteristics of the proposed method are: 1) beam elements are connected by elastic joints, not by rigid ones, 2) instead of geometric dimensions of beam elements, the stiffness of the joints is varied during optimization, 3) the maximum translational and rotational stiffnesses of the joints are assumed to be proportional to those of neighboring beams, and 4) the final topology of a structure is obtained by considering the connectivity of beam elements at the joints. To take manufacturability into account, joint springs are allowed to take on only a few discrete values; a set of pre-manufactured elastic joints having different rotational stiffness are assumed to be available in designing compliant mechanisms. The validity and effectiveness of the proposed design method is investigated by solving a couple of numerical examples including the popular design problems of a micro force converter and micro gripper.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Topology optimization of structures subject to random excitations with fatigue life constraints S.Lambert*, E.Pagnacco, L.Khalij, A.El Hami *

INSA de Rouen, Laboratoire de Mécanique de Rouen, Avenue de l’Université, 76801 St Etienne du Rouvray, France [email protected]

ABSTRACT In this paper a topology optimization strategy for randomly excited linear elastic structures with fatigue life constraints is presented. In random vibration it is often convenient to describe the excitation and response in term of function in the frequency domain. The random vibration theory was developed to deal with random excitations and is based on the assumption of excitations fully described by their second order statistical properties: their mean value, their autocorrelation function and also their power spectral density function (PSD). In the light of this, specific frequency formulations [1] of commonly used damage criteria are chosen as fatigue life assessment techniques in the optimization procedure. These frequency formulations are well suited to random vibration problems and give a fast and accurate estimation of the structural fatigue life from the response PSD (stress PSD). In structural optimization, sensitivity analyses are frequently used in order to obtain useful information for directing the optimization procedure. However sensitivity analysis considering fatigue life of structures subject to random excitations is rarely attempted due to computational cost reasons. Thus based on the finite element method, a method for eigenderivatives evaluation and some approximate process a fatigue life sensitivity quantity adapted to each of these frequency formulations is presented. These quantities allow a good estimation of the fatigue life change in an element due to the removing of another element from the discretized structure. Moreover the estimation strategy appears to be computationally efficient because only one modal analysis is needed to obtain the overall sensitivity quantities. An evolutionary structural optimization algorithm [2] is used for its simplicity and effectiveness within application. The method uses the finite element analysis results and the fatigue life sensitivity quantity concept to select and remove the most inefficiently used material until reaching an optimal topology design. Finally the resulting procedure for topology optimization is implemented and applied to several numerical examples with different fatigue life requirements. The frequency formulation of the Crossland’s damage criterion is chosen as fatigue life assessment technique. The fatigue life sensitivity computation technique is then applied to this frequency formulation. Informations provided by the fatigue life sensitivity quantities allow the convergence of the optimization procedures toward an optimal topology design for a reduced computational cost.

References [1] X. Pitoiset, A. Preumont, Spectral methods for multiaxial random fatigue analysis of metallic structures, International Journal of Fatigue, vol. 22, p.541-550, 2000. [2] Y. M. Xie, G. P. Steven, Evolutionary structural optimisation, Berlin: Springer-Verlag, 1997

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Improvement of pull-in voltage of electromechanical microbeams using topology optimization Etienne Lemaire*, Pierre Duysinx*, Véronique Rochus*, Jean-Claude Golinval* *

University of Liège Department of Mechanics and Aerospace Institut de mécanique, Bat B52 Chemin des chevreuils 1, 4000 Liège Belgium {E.Lemaire, P.Duysinx, V.Rochus, JC.Golinval}@Ulg.ac.be

ABSTRACT The electrostatic actuation devices used in MEMS are generally based on capacitive systems in which one electrode is mobile and the other one is fixed. Applying voltage between the electrodes generates an electrostatic force which tends to reduce the gap between the electrodes. Due to the non-linearity of the electrostatic force in function of the distance between electrodes, there exists a limit voltage from which there is no equilibrium between the electrostatic and mechanical forces leading to the pull-in phenomenon. In some applications, the pull-in instability is undesirable and maximizing pullin voltage is searched. The pull-in behavior involves a strong coupling between mechanical and electrostatic phenomena. Therefore the computation of the pull-in voltage for a given system requires multiphysics finite element simulations [1]. In addition, to compute efficiently the pull-in conditions, the multiphysics finite elements method is combined with a Riks-Crisfield algorithm [2]. The considered design problem consists in maximizing the pull-in voltage of a microbeam. Indeed, microbeam is the simplest example of electrostatically actuated MEMS exhibiting pull-in and consequently it is suited to serve as test to develop topology optimization of similar devices. Topology optimization is formulated as the research of the optimal distribution of a fixed volume of material. To avoid important modification of the electric field by the optimization process, this first study considers a non design electrode and uses topology optimization to design an optimal suspension structure. In this way, the structural optimization domain is separated from the electrical domain. The solution procedure of the optimization problem is based on CONLIN optimizer using a sequential convex linear programming. On each step of the optimization process, the sensitivity analysis is performed with the formulation of eigenvalue topology optimization problem on the basis of the computed pull-in conditions [3]. Two applications of this new method are finally proposed.

References [1] V. Rochus, D.J. Rixen, J.-C. Golinval, Monolithic modeling of electro-mechanical coupling in micro-structures, International Journal for Numerical Methods in Engineering, JOHN WILEY & SONS LTD, In press 2005. [2] J.E. Warren (January 1997), Nonlinear stability analysis of frame-type structures with random geometric imperfections using a total-lagrangian finite element formulation, Doctoral Thesis, Virginia Polytechnic Institute and State University. [3] Mostafa M. Abdalla, Chevva Konda Reddy, Waleed F. Faris, Zafer Gurdal , Optimal design of an electrostatically actuated microbeam for maximum pull-in voltage, Computers and Structures 83 (2005) 1320–1329.

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Large Scale Optimization of Compression Loaded Composite Structures Erik Lund Department of Mechanical Engineering, Aalborg University Pontoppidanstraede 101, Dk-9220 Aalborg East, Denmark [email protected] ABSTRACT The size of wind turbines has increased dramatically in the last decade, and today a standard wind turbine blade has a length between 40 and 60 meters. These high performance, multi-material structures are lightweight and may exhibit maximum tip displacements of up to about 25% of the length before they fail due to local buckling on the compressive side of the blade. Thus, in order to improve the structural performance the design objective is to increase the buckling load factor, taking weight considerations into account. This paper deals with this design problem for wind turbine blades specifically, but the methodology can be applied to any laminated multi-material composite shell structure. The outer shape of a wind turbine blade is determined by aerodynamic considerations and therefore in general not subject to change. These structures consist of stiff fiber reinforced polymers such as Glass or Carbon Fiber Reinforced Polymers (GFRP/CFRP) together with foam and different types of wood stacked in a number of layers and bonded together by a resin. The design problem is then to determine the best stacking sequence by proper choice of material and fiber orientation of each FRP layer in order to obtain the desired structural performance. For complicated geometries like wind turbine blades this is a very challenging design problem that calls for use of sophisticated structural optimization tools, and in this paper the so-called Discrete Material Optimization (DMO) approach is used. The DMO method is based on ideas from multi-phase topology optimization where the material stiffness (or density) is computed as a weighted sum of candidate materials. In this way the discrete problem of choosing the best material (with the right orientation) is converted to a continuous formulation where the design variables are the scaling factors (or weight functions) on each candidate material. The method has been successfully applied for compliance problems using as many as 13 candidate materials at each point and several hundred thousands of design variables in total, see [1, 2]. The analysis of the compression loaded multi-material composite structure is based on a linearized buckling problem for the undeformed geometry. Layered shell finite elements are used for the analysis, and the sensitivities of the buckling load factor are determined analytically. Multiple eigenvalues are taken into account, and the optimization problem is solved using the Method of Moving Asymptotes. Several examples involving thousands of design variables will illustrate the potential of the approach for buckling optimized designs of multi-material composite structures.

References [1] J. Stegmann and E. Lund, “Discrete Material Optimization of General Composite Shell Structures”. International Journal for Numerical Methods in Engineering, 62:14, 2009–2027, 2005. [2] E. Lund and J. Stegmann, “On Structural Optimization of Composite Shell Structures Using a Discrete Constitutive Parameterization”. Wind Energy, 8:1, 109–124, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Topology Synthesis of Compliant Mechanisms Using the Hybrid Cellular Automaton Method with an Efficient Mass Control Strategy Carlos A. Narváez, Andrés Tovar, Diego A. Garzón Department of Mechanical and Mechatronic Engineering, National University of Colombia Cr. 30 45–03, Of. 453–401, Bogotá, Colombia {canarvaezt, atovarp, dagarzona}@unal.edu.co

ABSTRACT Cellular automaton (CA) models have been used to simulate biological phenomena for over sixty years. In previous research, CA principles were applied to solve topology optimization problems using the hybrid cellular automaton (HCA) method. This method combines local evolutionary design rules with finite element analysis. Different control strategies, incorporated into the design rules, seek to minimize the error between a local mechanical signal and its optimum value. The HCA algorithm has demonstrated to be an efficient computational technique since optimal (black and white) topologies are obtained after a few iterations without checker-board patterns [4]. With a change in the objective function, the HCA method was also applied to compliant mechanisms design in a continuum design domain [3]. Nonlinear finite element analysis was incorporated into the algorithm [2]. Typically, a mass constraint is required to obtain slender flexible structures [1]. This constraint is satisfied by calculating the exact value of the associated Lagrange multiplier in every iteration. The objective of this work is to develop a new mass control strategy. In the proposed approach, the value of the Lagrange multiplier is corrected gradually during the iterative process and its final value is found at convergence. This strategy is implemented in both two- and three-dimensional models. The results are compared with the ones obtained using other approaches.

References [1] M. P. Bendsøe and O. Sigmund, Topology Optimization Theory, Method and Applications. Springer, 2003. [2] C. A. Narváez, A. Tovar, D. A. Garzón N. M. Patel J. E. and Renaud, Topology synthesis of pathfollowing compliant mechanisms using hybrid cellular automata. In Proceedings of 6th World Congress of Structural and Multidisciplinay Optimization. Rio de Janeiro, Brazil, 2005. [3] N. M. Patel, J. E. Renaud, H. Agarwal, and A. Tovar, Compliant mechanism design using the hybrid cellulat automaton model- In Procedings of the 1st AIAA Multidisciplinary Design Optimization Specialist Conference, Austin, Texas, 2005. [4] A. Tovar, N. M. Patel, A. K. Kaushik, G. A. Letona, and J. E. Renaud, Hybrid Cellular Automata: a biologically-inspired structural optimization technique. In Proceedings of the 10th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization. Albany, New York, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Topological Design for Minimum Sound Radiation from Structures Subjected to Forced Vibration Niels Olhoff*, Jianbin Du† * Institute of Mechanical Engineering, Aalborg University, Pontoppidanstraede 101, DK-9220 Aalborg East, Denmark [email protected] † Institute of Mechanical Engineering, Aalborg University, Pontoppidanstraede 101, DK-9220 Aalborg East, Denmark [email protected]

ABSTRACT Problems of passive topological design optimization of structures against vibration and noise have only been undertaken during the last decade, cf. [1] and papers cited therein. The problems have dealt with maximization of intrinsic properties like fundamental eigenfrequencies, higher order eigenfrequencies and eigenfrequency gaps of freely vibrating structures, and minimization of the dynamic compliance of structures subjected to forced vibration [1]. Unlike earlier work on topological design against vibration and noise, the present paper takes into account the interaction between the structure and the acoustic medium. Thus, our paper deals with passive topological design optimization of vibrating elastic continuum structures with the objective of minimizing the total sound power radiation from the structural surfaces into a surrounding acoustic medium. The volume, admissible design domain and boundary conditions of the structure are prescribed. The structural vibrations are assumed to be excited by a time-harmonic mechanical loading with prescribed forcing frequency and amplitude, and structural damping is not considered. A bi-material model, which is an extended form of the SIMP model, is employed for the topology optimization. This implies that the boundary shape of the structure is not changed during the design process, which leads to a great simplification of the sensitivity analysis since the calculation associated with the shape gradients of the acoustic pressure loading is avoided. It is assumed that air is the acoustic medium and that a feedback coupling between the acoustic medium and the structure can be neglected. Certain conditions are assumed, under which the sound power radiated from the structural surface can be estimated by using a simplified approach [2] instead of solving the Helmholz integral equation. This implies that the computational cost of the structural-acoustical analysis can be considerably reduced. Numerical results are presented for plate and pipe-like structures with different sets of boundary conditions.

References [1] N. Olhoff, J. Du, Topology Optimization of Structures against Vibration and Noise. Proc.12th International Congress on Sound and Vibration, Lisbon, Portugal, 20pp., 2005. [2] D.W.Herrin, F. Martinus, T.W. Wu, A.F. Seybert, A New Look at the High Frequency Boundary Element and Rayleigh Integral Approximations, Society of Automotive Engineers, Inc., 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Aspects of 3D Shape and Topology Optimization with Multiple Load Cases Pauli Pedersen Dept. of Mechanical Engineering, Solid Mechanics Technical University of Denmark Nils Koppels All´e, Building 404, DK-2800 Kgs. Lyngby, Denmark [email protected]

ABSTRACT In the present study we concentrate on the possibilities for three dimensional design of density distributions. Simple recursive optimizations are performed and the optimal solutions are normally obtained within 10 iterations. For single load cases, being static or dynamic, the necessary optimality criterion for compliance or frequency optimization is loosely stated uniform energy density. For static problems uniform elastic energy and for dynamic problems uniform difference between elastic and kinetic energy, in the active design domains. For size optimization in all non-restricted design domains and for the shape optimization along the non-restricted boundaries. For these single load cases we study the difference between optimizing for compliance (stiffness), and optimizing for von Mises stress or alternative reference stress (strength). For multiple load cases the formulation with linear combination of the individual cases is in realty just as simple as for the single load cases, but the design solution naturally depends on the chosen linear combination factors. We illustrate this by examples. Linear combinations of static and dynamic load cases are treated in the same manner. By adjusting the linear combination factors this may lead to extremize eigenfrequencies or eigenfrequency gaps with constraint on compliance, and we can still obtain solutions by optimality criterion iterations. An alternative to the linear combination of different load cases is to optimize with focus on fully stressed design, that is, each part of the non-restricted design domain should be used to the same extent in at least one of the load cases. Examples based on this strategy are compared to the alternative optimizations, and in general the goal of the study is to gain more knowledge related to the effect of the chosen optimality objective. Even when an actual problem is so complex that mathematical programming must be involved, it is often a good strategy to use optimality criterion optimization for simplified versions of the problem in order to get a good starting design for the mathematical programming.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

493

On Shape, Material and Orientational Design of Plates in Relation to Dynamics Niels Leergaard Pedersen Dept. of Mechanical Engineering, Solid Mechanics Technical University of Denmark Nils Koppels All´e, Building 404, DK-2800 Kgs. Lyngby, Denmark [email protected]

ABSTRACT The focus of this work is on simultaneous shape, material and orientational design of plates. The shape optimization is related to the shape and position of a hole in the plate. The material design is the design of an orthotropic material that can be considered to be a fiber-net within each finite element. This fiber-net is then oriented in the individual elements of the finite element discretization. The object of the optimization is dynamics, specifically the eigenfrequencies. We maximize eigenfrequencies or gaps between eigenfrequencies to avoid resonance. Generally optimization of eigenfrequencies is also applied in case of minimizing the possibility of internal resonance or inverse dynamics where specific demands on eigenfrequencies are meet. Using composite material we can tailor the behavior of a structure to meet specific demands. Many of the early papers dealing with the subject of orienting material to maximize eigenfrequencies considered only one orientation variable for each element. We assume that we are dealing with an orthotropic material and align part of the material in one direction and the rest of the element material in a direction perpendicular to the first direction. The integrated constitutive relation is then given as the sum of the two constitutive relations, so that a fiber-net is oriented individually in each finite element. The fibernet can be seen as a regularization of the formulation with unidirectional fibers, the basic advantage is the possibility to use fibers in both of the principal stress directions. That this is an advantage can be argued from the case of an isotropic stress state or a pure shear state where the choice of direction using unidirectional fibers is arbitrary. The optimizations are performed using the finite element method for analysis and the optimization approach is a two-step method, where both a recursive optimization procedure based on an optimality criterion (OC) is used and mathematical programming (MP) with sensitivity analysis is applied to find the final optimized design. The examples show that the two-step approach of using both OC and MP gives fast and robust convergence of the design. The paper shows that we can manipulate the eigenfrequencies of a plate by varying the material(fiber-net) and the orientation of the fiber-net together with the shape of a hole in the plate (within the physical limits of the problem).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

494

Optimal Layouts of Stiffeners for Plates in Bending – Topology Optimization Approach Nathan Perchikov and Moshe B. Fuchs School of Mechanical engineering, Tel-Aviv University 69978 Ramat-Aviv, ISRAEL [email protected]

ABSTRACT This paper applies optimum structural topological design methodologies to the stiffening of plates in bending. The purpose of the design is to position a given amount of material, in the form of orthogonal stiffeners, minimizing the compliance of a given plate. Cheng and Olhoff have shown that if left unchecked, this formulation includes dense sets of infinitely thin stiffeners – an optimal yet obviously impractical solution. In order to circumvent the obstacle, the present formulation assumes a plate meshed by an orthogonal lattice with rectangular elements allowing the stiffeners to be positioned only along the lines of the mesh. The basic problem is to add uniform beams, of generally variable stiffness, in an optimal fashion along the gridlines, having a constant amount of material at hand. A beam stiffness parameter, usually the width or the height, is taken as the design variable xi of the problem, having i=1,..,l+t, assuming the lattice has l longitudinal and t transversal lines. In this form, we have stated a standard topological design problem that calls for the minimization of the compliance, subjected to a constraint on the amount of material, which can be solved by classical optimization techniques, not unlike the erstwhile topology optimization for 2D membrane problems, using density design variables. It is agreed that when limiting the positions of the stiffeners to prescribed gridlines, one does not necessarily obtain the best possible layouts. However, the discretization of the problem allows for using topological optimization methodologies that produce noteworthy results. In particular, we treat the problem of getting 0/1 solutions by using rigidity interpolation and/or a norm constraint. Herein, the issue of the convexity of the problem is addressed. The optimal designs obtained exhibit interesting features. Influence of the plate-to-beam stiffness ratio is discussed. The technique is illustrated by clamped rectangular plates under different lateral loading conditions. Below is a typical example of a result produced by the presented formulation.

Figure 1: Optimal layout of orthogonal stiffeners for hydrostatic loading

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

495

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

496

Structural Shape Optimisation by using Multi-direction Boundary Points Movement Method Y.Y. Sia, O.M. Querin School of Mechanical Engineering University of LEEDS, LS2 9JT, UK [email protected] [email protected]

ABSTRACT Structural shape optimisation is a process of changing the structural boundary to reach to an optimum design (i.e. minimum weight, low cost, etc.) while satisfying the structural geometry (i.e. height, radius, etc.) and behaviour (i.e. stress, natural frequency, etc.) constraints. These problems can be solved by either calculus or heuristic-based methods. Although the calculus based methods are by far the best in terms of computational efficiency, however, they may not reach to the global optimum. A Heuristic-based methods with the distinctive feature of local and global search is more robust to obtain the global optimum. There are several heuristic-based methods in existence, however the most common, robust and reliable one is the Genetic Algorithms (GA). It works by mimicking the evolution process found in natural to obtain the optimum solution [1]. The aim of this research is to apply GA to the problem of shape optimisation. Past research has concentrated in applying binary GA to shape optimisation by carrying out step-wise movements of the boundary of the structure in a specific direction in 2D space [2,3]. The research presented here extends the method by allowing the movements of boundary in any direction. For all of the methods mentioned, the structural boundary under design condition is represented by a finite number of segments with two boundary points (BP) connected at each segment ends. In fact, the change of the structural boundary is governed by the movement of these points. This study looks at what effect multi-directional movement of these points have on the optimum design. Several examples with different movement cases including the stepwise, recursive step-wise and multi-direction were considered. The results obtained by the latter two methods indicated an improvement over the step-wise method, thus allowing the structure to adapt itself closer to the global optimum.

References [1] D. E. Goldberg, Genetic algorithms in search, optimization, and machine learning. AddisonWesley Publishing Company, Inc., 1989. [2] S.Y. Woon, O.M. Querin, G.P. Steven, Application of the fixed grid FEA method to step-wise GA shape optimization. Proceedings of the 2nd ASMO-UK/ISSMO conference on Engineering Design Optimization Swansea UK, Swansea, UK, 265-272, 2000. [3] S.Y. Woon, O.M. Querin, G.P. Steven, Structural application of a shape optimization method based on a genetic algorithm. Structural and Multidisciplinary Optimization, 1(2), 57-64, 2001.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

497

On topology Optimization with Manufacturing Constraints O. Sigmund Department of Mechanical Engineering, Solid Mechanics, Technical University of Denmark Nils Koppels Alle, Building 404, DK-2800 Lyngby, Denmark [email protected] ABSTRACT A standing problem in topology optimization problems is the efficient enforcement of manufacturing constraints such as minimum or maximum feature sizes. A large number of methods have been proposed including filtering techniques, perimeter control and gradient control. In this paper we propose some new filtering schemes that may alleviate the problem of grey transition regions in optimized topologies and ensure a minimum length scale for structural features and holes. The new filter techniques are based on ideas borrowed from image morphology operators. It is shown that operators like “dilation”, “erosion”, “opening” and “closing” can be used to obtain black and white solutions with minimum feature size control using a continuation scheme. The schemes are implemented based on efficient analytical sensitivity calculations and tested on standard benchmark problems.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

498

Sequential integer programming methods for stress-constrained shape and topology optimization Krister Svanberg and Mats Werme Optimization and Systems Theory, Royal Institute of Technology (KTH), SE-10044 Stockholm, SWEDEN. {krille,werme}@math.kth.se

ABSTRACT This presentation deals with stress-constrained shape and topology optimization problems of loadcarrying structures. The structure is approximated by a finite element model, where each element is either filled with material or void. The starting point of the optimization is a nonlinear integer programming formulation in which the binary design variable vector (x1 , . . . , xn ) describes completely the shape and topology of the discretized structure: xj = 1 if the j:th element is filled with material, while xj = 0 if it is void. For nonlinear optimization problems with continuous design variables instead of binary, a fundamental algorithmic approach is to generate and solve a sequence of approximating subproblems. In each subproblem, the original objective- and constraint functions are replaced by explicit, relatively simple, approximating functions which are based on calculated derivatives of the original functions at the current iteration point. This is the framework for several well-known optimization methods like sequential linear programming, method of moving asymptotes, and sequential quadratic programming. In our considered problems, the design variables are binary and not continuous. Then there are no derivatives, but there are corresponding natural definitions of sensitivities of a function fi with respect to either one binary variable xj or with respect to two binary variables xj and x
. In a recent work we developed efficient methods to calculate these discrete first and second order sensitivities if the function fi stands for e.g. the von Mises stress in the ith element. In the current work we investigate several “sequence of subproblem” approaches, based on these discrete sensitivities, in particular a sequential integer all-quadratic programming method.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

499

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

500

Topological design of acoustic-structure interaction structures with the mixed finite element method Gil Ho Yoon*, Jakob Søndergaard Jensen†, Ole Sigmund†† Department of mechanical engineering, Technical University of Denmark, DK-2800, Lyngby, Denmark * [email protected], †[email protected] , †† [email protected]

ABSTRACT We use the mixed finite element procedure with the displacement field and the pressure as primal variables to find out optimal topologies of coupled acoustic-structure systems. Changing topologies, it is challenging to alternate between the governing equations (Helmholtz equation and the linear elasticity equation) and impose the coupling boundary conditions. Thus this coupled acoustic-structure systems are usually optimized using the shape optimization scheme with the explicit boundary representation. In this paper, in order to perform topology optimization for the coupled systems, we adopt the mixed finite element procedure called a u/p-formulation [1]. Compared to the standard displacement formulation, this mixed formulation is suitable to analyze the response of imcompressible media. Moreover, in the frequency domain, assigning zero shear modulus and proper values of bulk and structural density, we can derive the Helmholtz equation (Wave equation). Hence, by spatial variation of the mass density, shear and bulk moduli, we are able to simulate the coupled system in the single governing equation. For optimization, we interpolate these material properties based on the SIMP (Solid Isotropic Material with Penalization) [2]. Compared to shape optimization, this procedure does not require the shape parameterization. Several two-dimensional acoustic-structure problems are optimized in order to verify the proposed method. The detail numerical behavior during optimization are also studied.

References [1] Yoon GH, Jensen JS, Sigmund O. Topology Optimization of Acoustic-Structure Interaction Problems using a Mixed Finite Element Formulation, Submitted. [2] Bendsøe MP, Sigmund O. Topology Optimization Theory, Methods and Applications. Springer-Verlag: New York, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

501

Kinematic influence on the vibrations of stratified plates First A. Nour*, Y. Chevalier‚, S. Aguib* ,N. Chikh* *

Dynamic Motors and Vibroacoustic laboratory (L.D.M.V.), University of M.B. Boumerdes 35000, Algeria. Tél/Fax: (213)24 81 14 78. [email protected] [email protected] [email protected] ‚

Institut Supérieur de la Mécanique de Paris (ISMEP), Paris, France. [email protected]

ABSTRACT In this paper, we present the simulation results of the vibratory behaviour for an orthotropic stratified plate (Graphite/epoxy) embedded on his four sides, with dimensions [00/900/00/900/00]. The work presented consists of the study of free vibrations of laminated plates thus worked out by two analytical and numerical approaches and an experimental validation. A tool intended for the calculation of the eigen frequencies of thin sections was used. In the first time, the problem of elasticity in small displacements discretized by the method of Rayleigh-Ritz has allowed the writing of a system of algebraic linear equations depending on the dimensional parameters. By the application of the algorithm of Graeff one carries out the construction of the nearest solutions (eigen frequencies). The developed method of Rayleigh-Ritz, valid whatever the number of terms retained in the series of approximation, highlighted an appreciable flexibility of handling especially in the case of free vibrations. The simulations carried out show the extent of the possibilities offered by the proposed technique Using the last results, we evaluated the coupling flexion torsion on the dynamic analysis of the laminated plates and its influence on the normal frequencies, the orientation of fibers and the order of stratification on vibratory behaviour of the plate, the geometry of the plate and the conditions at the borders in the case of the plates with embedded borders, the matter of composition of the layers as well as the transverse shearing which has a great effect on the vibratory behaviour of the laminated plates in particular for embedded plates.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

502

Optimization of resign transfer molding process by a virtual manufacturing and a genetic algorithms D.H. Bassir*, W.H. Zhang†, S. Guessasma** *

Institut FEMTO-ST, Dept. LMARC, UMR 6174, 24, rue de l’épitaphe, 25000 Besançon, France [email protected]

†

Sino-French Laboratory of Concurrent Engineering Dept. of Aircraft Manufacturing Engineering Northwestern Polytechnical University 710072 Xi'an, Shanxi , China [email protected] **

INRA, B.I.A. Group Rue de la Géraudière, 44316 Nantes Cedex 03, France [email protected] ABSTRACT An advantage of using fibre-reinforced composites over conventional materials is that they can easily reduce the mass of structures by adapting the stiffness to the requirements of many practical applications. Thus, they are often used in the various fields related to aeronautics, space, or cars manufacturing. Depending on the size of the structure we want to manufacture, different techniques exist. Among those techniques, we have the resign transfer molding (RTM) process which is more suitable for low to medium volume [1]. In the RTM process some parameters have a significant impact on the production. In general, optimization of the RTM process, use an iterative procedure based on the RTM simulation by finite elements method and an optimization tool. This procedure converge when the error between the response of the simulated model and the experimental test is very low. This procedure is easy to implement but, it’s time-consuming and expensive even with a high computing machines. In this article, an hybrid strategy based on the genetic algorithms (GA) and an artificial neural networks (ANN) will be introduced to reduce the time-consuming in the RTM process. This approach starts by a mapped solutions (created by simulation results) using a small niching parameter of the GAPS program (genetic algorithm with parallel selection [2]), then an ANN is trained to create a RTM process model. Once this Meta-model is created (in general nonlinear) we run the GAPS with the objective function evaluated from the Metamodel response in order to locate the global solution that optimize the RTM process design. In some cases, design variables can be continuous or discrete variables. In such case binary and real coding are used simultaneously. The efficiency of the above strategy will be illustrated through a numerical examples.

References [1] S.G. Advani, Flow and rheology in polymer composites manufacturing. Amsterdam :

Elsevier Science, 1994. [2] D.H. Bassir, S. Carbillet, M.L. Boubakar, Algorithme génétique à sélection parallèle : Application à l’identification paramétrique de CMO. Revue des Matériaux Composites et Avancés (Ed. Hermes), 15, 53-70, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

503

Post-processing Techniques Suitability for Mesolevel Free Boundary Flows Zuzana Dimitrovová* * UNIC, Department of Civil Engineering, New University, Monte de Caparica, Portugal [email protected]

ABSTRACT Reliable flow simulation software is inevitable to determine an optimal injection strategy in Liquid Composite Moulding processes. Void formation during the injection phase can be explained as a consequence of the non-uniformity of the flow front progression. Origin of this fact lies in the dual porosity of the fibre preform and therefore the best explanation can be provided by mesolevel analysis. In the mesolevel analysis, single scale porous media (fibre tows) and open spaces are presented in the flow domain and therefore different flow regimes must be considered and linked together in one analysis, at each time step. In such simulation it is extremely important to account correctly for the surface tension effects, which can be modelled as capillary pressure applied at the free flow front, [1-2]. Post-processing a finite element solution is a well-known technique, which consists in a recalculation of the originally obtained quantities such that the rate of convergence increases without the need for expensive remeshing techniques [3]. Post-processing is especially effective in problems where better accuracy is required for derivatives of nodal variables in regions where Dirichlet essential boundary condition is imposed strongly [4]. Consequently such an approach can be exceptionally good in modelling of resin infiltration under quasi steady-state assumption, because only free-front normal velocities are necessary to advance the resin front to the next position. The new contribution lies in the post-processing free-boundary velocities implementation in the mesolevel infiltration analysis. Such implementation ensures better accuracy on even coarser meshes, which in consequence reduces the computational time also by the possibility of employing larger time steps. Suitability of the known technique is analysed for Darcy’s flow, new procedure is suggested for Stokes’ flow, a methodology for regions with different permeabilities is introduced and conclusions are taken.

References [1] S.G. Advani, Z. Dimitrovová, Role of Capillary Driven Flow in Composite Manufacturing, in Surface and Interfacial Tension: Measurement, Theory and Applications, ed. S. Hartland, Surfactant Science Series, 119, 263-312, 2004, Marcel Dekker, Inc., NY. [2] Z. Dimitrovová, S.G. Advani, Mesolevel analysis of the transition region formation and evolution during the liquid composite molding process, Computers & Structures, 82, 1333-1347, 2004. [3] I. Babuška, A. Miller, The post-processing approach in the finite element method - Part 1: Calculation of displacements, stresses and other higher derivatives of the displacements, International Journal for numerical methods in Engineering, 20, 1085-1109, 1984. [4] T.J.R. Hughes, G. Engel, L. Mazzei, M.G. Larson, The continuous Galerkin method is locally conservative, Journal of Computational Physics, 163, 467-488, 2000.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

504

Adaptive Simulation of cohesive interface debonding for crash and impact analyses M. Nossek*, M. Sauer*, K. Thoma* *

Fraunhofer Institute for High-Speed Dynamics -Ernst-Mach-InstitutEckerstraße 4 D-79104 Freiburg Germany [email protected] [email protected]

ABSTRACT In many applications, e.g. crushing of CFRP structures in automotive applications, modelling of interface debonding is of crucial importance. Although the amount of energy absorbed by the delamination process itself might be negligible, it changes the integrity of a cross section and might trigger specific failure modes of entire parts [1]. On way of modelling debonding is the use of cohesive zone models. Such models are currently being implemented in some commercial finite element codes in the form of either particular element formulations [3] or special contact algorithms. A traction-separation-relation describes the crack opening behaviour. It comprises an initial elastic behaviour and an energy based damage formulation. In such models, the initial rigid connection before crack growth starts is not reproduced. Particularly when wave propagation perpendicular to possible crack areas shall be investigated, initial elastic models have some drawbacks. In this paper, an alternative adaptive initial rigid cohesive zone model is presented for the simulation of delamination between material boundaries in three-dimensional bodies. It is in parts similar to the approach by Sam [4]. The model contains on one hand algorithms for initiation and crack propagation, on the other hand procedures to handle the mesh adaptation when crack propagation has been identified. Both Mode I and Mode II failure behaviour are being considered. The paper summarizes the model and its implementation in the FE-code Sophia [2] including the required remeshing tool. Convergence of the approach is being examined by comparison with analytical results. The properties of the implemented model are analysed by comparison of numerical results with results of the commercial software ABAQUS, where initial elastic cohesive elements were used. The test case is an in-plane impact on a laminated plate.

References [1] [2]

[3]

[4]

Peter, J., Experimentelle und numerische Untersuchungen zum Crashverhalten von Strukturbauteilen aus kohlefaserversärkten Kunststoffen, ISBN 3816767486, 2005. Sauer, M.; Hiermaier, S.; Thoma, K., Modeling the Continuum/Discrete Transition Using Adaptive Meshfree Methods, Proceedings of the Fifth World Congress on Computational Mechanics (WCCM V), July 7-12, 2002, Vienna, Austria, http://wccm.tuwien.ac.at. Camanho, P.P., Dávila, C.G., Mixed-Mode Decohesion finite elements for the Simulation of Delamination in Composite Materials, NASA-LRC, TM-2002-211737. C-H. Sam, K.D. Papoulia, S.A. Vavasis, Obtaining initially rigid cohesive finite element models that are temporally convergent. EFM, in press, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

505

Failure analysis of cement-treated soil by FEM implemented with particle discretization Shin-ich Shigehisa*, Yoshihisa Miyata*, Hidetoshi Ochiai† *

National Defense Academy 1-10-20 Hashirimizu, Yokosuka 239-8686 {sigehisa,miyamiya}@nda.ac.jp † Kyushu University 6-10-1 Hakozaki, Higasiku. Fukuoka 812-8581, Japan [email protected]

ABSTRACT The authors conducted failure analysis of cement-treated soil by using the finite element method (FEM) implemented with particle discretization. The analysis results were compared with laboratory test results. To predict the tensile failure of the cement-treated soil, the material properties of cementtreated soil dependent on confining stress dependent were introduced into the FEM. The results of the numerical simulation were in good agreement with the value that had been obtained by laboratory tests.

References [1] Hori.M, Oguni.K, Sakaguchi.H, Proposal of FEM implemented with particle discretiza-tion for analysis of failure phenomena. J. Mech. Phys. Solids, 53, 681–703, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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An Homogenization Procedure for Cardboard and Stitched Sandwiches using Respectively Analytical and Numerical Simulation First Z. Aboura*, N. Talbi†, R. Ayad†, S. Allaoui†*, M. Benzeggagh†* * L3M.IUT de Tremblay en France Paris 8, Rue de la Râperie 93290 – Tremblay-en-France [email protected] † Groupe Mécanique Matériaux et Structures GMMS EA 2617 ESIEC, Université de Reims, Esplanade de Roland Garros BP 1029, 51686 – Reims Cedex 2 [email protected], [email protected] †* Université de Technologie de Compiègne Centre de Recherche Royallieu B.P 20529 F – 60205 Compiègne Cedex [email protected], [email protected]

ABSTRACT The interest of the use of sandwich structures is not any more to show. Their performances go from pairs with their complexities. The structural analysis using this type of materials becomes very quickly expensive (mesh, CPU). The idea used through this study consists: working out a procedure of homogenization to obtain an equivalent homogeneous material and then reproduce the mechanical behaviours of these structures by simulation. Two types of sandwich structures were studied: corrugated cardboard and a stitched sandwich with PU core and whose skins are out of woven glass. The method of homogenization of these structures differs according to studied material. For corrugated cardboard, the procedure consists to discretize the unit cell and apply to it the theory of the stratification and to go up to the global rigidities [1]. For stitched sandwich, the analytical model consists in determining the global rigidity starting from the basic rigidities of the components [2]. In each configuration, two new elements were developed the DMTS and the SFR [3]. The 3-node multilayered shell element DMTS has 6 dof per node. It’s obtained by combining the membrane element CST and a bending/shear plate element DDMT (Displacement Discrete Mindlin Triangle). The new 8-node solid element SFR, is based on kinematical concept based on the rotation of a space fibre. This concept allows to improve the accuracy of the displacement field {U} and to get a higher element with comparison to the 20-node quadratic hexahedral element. Results of the two finite element approaches are compared with experiments related to the corrugated cardboard structures and the stitched sandwiches

References [1] [2] [3]

Z. Aboura, N. Talbi, S. Allaoui, A. Benzeggagh, Elastic behaviour of corrugated cardboard: Experimental and modelling. Composite Structures 2004; 63:53-62. D. Scida, Z. Aboura, M.L. Benzeggagh, Prediction of elastic behaviour of hybrid and nonhybrid woven composite, Composite, Science and Technology 1997, 54, 1727-1740. R. Ayad, Contribution a la modelisation numerique pour l’analyse des solides et des structures et pour la mise en forme des fluids non newtoniens. Application a des materiaux d’embalage, Habilitation a Diriger les Recharches, URCA, Reims, 2002.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Modelling of woven fabrics with the Discrete Element Method Dirk Ballhause, Manfred Ko¨ nig, Bernd Kr¨oplin Institute of Statics and Dynamics of Aerospace Structures (ISD) University of Stuttgart Pfaffenwaldring 27, 70569 Stuttgart, Germany [email protected] ABSTRACT The mechanical behaviour of woven fabrics is dominated by the kinematics of the constituents on the microscopic scale. Their macroscopic response usually shows non-linearities which are due to the mobility of the interlaced yarns. The major deformation mechanisms of fabrics, i.e. the crimp interchange in case of biaxial tension and the trellising motion of the yarns in case of shear, reflect the dependency of the macroscopic material behaviour on the microstructural deformation mechanisms. The adaption of crimp and yarn directions according to the loading state leads to a significantly different material behaviour than observed for common homogeneous materials. We present a novel modelling approach for woven fabrics which is capable to represent directly and locally the microstructure and its kinematics. With only a small set of assumptions on the micro-scale the complex macroscopic material behaviour can be directly obtained. The proposed model uses the Discrete Element Method (DEM) [1] for the representation of the fabric s microstructure. It is modelled by discrete point masses and force interactions between them. These interactions can be rheological elements like springs and dashpots or any arbitrary function relating a reaction force to the kinematic state variables of the nodes. The model is intrinsically dynamic since the equations of motion are solved numerically for every mass point using a predictor-corrector algorithm taking into account the changing interaction forces. It thus can cover the full mobility of the fabric s microstructure while being efficiently enough to model macroscopic patches of the material. With this model we can study the influence of the different material features of the microscale on the macroscopic material behaviour. With some further extensions accounting for coatings or embeddings, the range from pure fabrics to fabric reinforced membranes and composites can be covered. Problems related to large deformations and localization as well as damage can be addressed with this modelling approach. Numerical results for the constitutive material behaviour of fabrics are presented in order to demonstrate the capabilities of this kind of approach.

References [1] N. Bi·cani·c. Discrete Element Methods. In: Encyclopedia of Computational Mechanics, E. Stein, R. de Borst, T.J.R. Hughes (Eds). John Wiley & Sons, (2004).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A new approach for the simulation of damage effects in rubber-like materials using chain statistics M. B¨ol, S. Reese Institute of Mechanics Carolo-Wilhelmina University of Braunschweig [email protected]

ABSTRACT The enormous development of computer technology in the last years has caused an increased demand for numerical investigations of industrially relevant elastomeric building components. In particular the prediction of the fatigue behaviour and a realistic life time estimation are of high interest. In order to establish a powerful damage modelling it is essential to understand the micro mechanical behaviour of rubber-like polymers. The material behaviour of rubber-like materials can be characterised as follows. Under static loading the material exhibits strongly non-linear elasticity. Dynamic testing conditions show that the material behaviour is rate-dependent including hysteresis effects. These hysteresis effects are reinforced when the material is filled for example with carbon black. If the stress level is increased in every cycle stresssoftening known as Mullins effect becomes evident. When the specimen is overloaded damage occurs inside of the rubber-like material, the specimen collapses. The aim of this paper is to present a new method to simulate the general Mullins effect as well as the idealised one, both can be seen as a kind of damage effects. The latter one is characterised by the fact that there is no remaining strain when the specimen is completely unloaded. First ideas of damage simulations will follow using the transfer of information from the micro to the macro level. To simulate rubber-like material behaviour a special finite element unit cell is introduced which consists of one finite tetrahedral element and six truss elements lying on each edge of the tetrahedron. Each truss element represents the micro mechanical material behaviour of a bundle of polymer chains. By help of a random assembly of such unit cells it is possible to simulate the non-linear material behaviour of unfilled elastic rubbers. To simulate the Mullins effect we look in more detail at the physics of rubber. During cyclic loading of a rubber specimen the chains at the micro level start to break and to reconnect. In the case of macro mechanical damage whole parts of the network structure break. The comparison with experimental data shows that the proposed method leads to satisfying agreement.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Numerical modelling of Nomex® honeycomb cores : Failure and effective elastic properties L. Gornet, S. Marguet, G. Marckmann

École Centrale de Nantes, GeM, UMR CNRS 6183, 1 Rue de la Noë, BP 92101, 44321 Nantes Cedex 3, France {laurent.gornet, steven.marguet, gilles.marckmann}@ec-nantes.fr

ABSTRACT The aim of the present study is to propose and develop the numerical determination of the effective stress–strain behaviour of Nomex® honeycomb cores made from aramid paper material. This study highlights the determination of the hexagonal and rectangular over-expanded core materials. These honeycombs are extensively used in the manufacturing of aeronautic structures and of oceanic multihull sailing race boats. These sandwich structures are made of carbon-fiber epoxy-matrix composite laminate skins and Nomex® cores. The understanding of the behaviour and eventually failure of honeycombs are extremely important for the design of these engineering composite sandwich structures. Honeycomb cores predictions are directly related to the structural integrity and safety requirements of the entire composite structure. Since the pioneering work numerous studies on the effective properties of cellular sandwich cores have been published [1-2]. In the past, core behaviours were studied under strength of material assumptions. In this context a software dedicated to Nomex® honeycombs was developed at the Laboratory in order to predict the failure conditions of these cores [2]. Our software NidaCore has been developed to determine the three dimensional mechanical core properties. The elastic mechanical properties have been determined by a three-dimensional Finite Element model that involves periodic homogenization techniques. For the homogenisation of the honeycomb microstructure, a strain energy-based concept is used which assumes macroscopic mechanical equivalence of a Representative Volume Element for the given microstructure with a similar homogeneous volume element. The software has been developed using the Finite Element structure analysis program Cast3M-CEA. Numerical predictions are compared with the mechanical properties given by the Euro-Composites company. The present study confirms that for honeycombs under consideration the Representative Volume Element symmetries lead to orthotropic homogenized mechanical properties. The key point of the modeling is that the RVE buckling modes conduct to determine the ultimate stresses of the homogenized core. Based on buckling modes, numerical analysis reproduces the ultimate stresses experimentally observed on standard test methods. This approach strengthens by experimental results leads to a failure criteria based on the mechanical understanding of local damage effect. In order to go further, the skin effect on the core properties is discussed for T700/M10 carbon-fiber epoxymatrix cross-ply and angle-ply laminates skins. The honeycomb mechanical properties and ultimate stresses are used to model three dimensional reinforcements that we used for the study of the Oceanic sailboat structures.

References [1] Hohe, Beschorner C., Becker W., Effective elastic properties of hexagonal and quadrilateral grid structures, Composite Structures, 46, 73-89, 1999. [2] Gornet L., Marckmann G., Lombard M., Détermination des coefficients d’élasticité et de rupture d’âmes nids d’abeilles Nomex® : homogénéisation périodique et simulation numérique, Mécanique Industrie, to appear 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Multi-Scale Modelling and Simulation of Textile Reinforced Materials G. Haasemann∗ , M. K¨astner† , V. Ulbricht† ∗ TU Dresden Institute of Solid Mechanics [email protected] † TU Dresden Institute of Solid Mechanics [email protected]

ABSTRACT Novel textile reinforced composites provide an extremely high adaptability and allow for the development of materials whose features can be adjusted precisely to certain applications. A successful structural and material design process requires an integrated simulation of the material behaviour, the estimation of the effective properties which need to be assigned to the macroscopic model and the resulting features of the component. In this context two efficient modelling strategies - the Binary Model [1] and the Extended Finite Element Method (X-FEM) [2] - are used to model materials which exhibit a complex structure on the meso-scale. For these investigations the focus is set on composites made of glass fibers, thermoset or thermoplastic matrices and on the application of commingled thermoplastic and glass fibers. Homogenization techniques are applied to compute effective macroscopic stiffness parameters. Problems arising from a complex textile reinforcement architecture, e.g. bi- or multiaxial weft-knit, woven and braided fabrics, in combination with a high fiber volume fraction will be addressed and appropriate solutions are proposed. The obtained results are verified by experimental test data. The macroscopic stress and strain fields in a component are used for optimisation of the construction and the material layout. These distributions are computed in a global structural finite element analysis. Based on the global fiber orientation the required macroscopic material properties obtained from homogenization on the meso-scale are mapped to the model of the structural part. The configuration of the fiber-orientation and textile shear deformation in complex structural components caused by the manufacturing process is determined by a three-dimensional optical measurement system.

References [1] Carter W.C., Cox B.N., Fleck N.A., A Binary Model of Textile Composites - I. Formulation, Acta metall. mater. 42(10), 3463–3479, 1994. [2] Mo¨es N., Cloirec M., Cartraud P., Remacle J.-F., A computational approach to handle complex microstructure geometries. Comput. Method. Appl. M. 192, 3163–3177, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Enforcing Boundary Conditions in Micro-macro Transition for Second Order Continuum Łukasz Kaczmarczyk∗ , Zenon Waszczyszyn Cracow University of Technology Institute of Computer Methods in Civil Engineering e-mail: [email protected] web page: http://www.twins.pk.edu.pl ABSTRACT In recent years the multiscale computational homogenization has been extensively developed. Such macro-modelling does not require any constitutive assumptions at the macro-level. The multi-scale computational homogenization has also been extended for the second order continuum at the macro level. The second-order framework is based on incorporation of the gradient of macroscopic deformation in micro to macro multiscale transition. The introduction of the second-order continuum at macro-scale takes into account the size effect and gives more accurate results in case of insufficient scale separation. This paper concentrates on some issues of the fully coupled second order homogenization scheme. Attention is focused on micro-macro transitions of the discretized microstructure. In the presented paper a new approach is proposed which can handle any type of boundary conditions (i.e. displacement, periodic and static). The boundary conditions enforce the deformation of representative volume element (RVE) according to given gradient and second gradient of displacements in average sense. After expansion of the displacement vector in geometric centre of RVE and truncation after second order term the continuum boundary conditions can be written in integral form as    δt · r dΓ = 0, n ⊗ r dΓ = 0, n ⊗ x ⊗ r dΓ = 0, (1) Γ

Γ

Γ

where n is the normal vector, r is the microfluctuation of displacement field and δt is statically admissable variation of tractions on boundary. If the first integral satisfies Hill-Mandel theorem, the second and third integral enforce deformation of RVE according to given strain tensor and given gradient of deformation in average sense, correspondingly. After FE discretization of (1), the constraint equation is formulated in matrix from Cq = g, where q is the vector of displacemet-type degrees of freedom. To enforce the constraints the projection matrices are formulated as Q = I − CT (CCT )−1 )C

(2)

and applied in the computation of stiffness matrix and load vector. One of advantages of the proposed approach is that the rigid translation and rotation, i.e. 1st order and 2nd order deformations can be applied individually by a sequence of operations. Moreover, enforcing constraints can be performed element by element subassembly procedure. In the paper analytical and numerical results are presented. Equivalent stiffness operators are derived analytically for the linear and homogeneous material in RVE. The results are compared with the Mindlin constitutive model. In the case of heterogeneous non-linear material illustrative examples was numerically computed. The shear layer problem and plate bending in plane strain for a non-linear homogeneous material with an intristic length scale are also presented.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Micro-Structure Based Modeling of Elastomer Materials Manfred Klüppel1, Jens Meier2 and Markus Ramspeck1 1

Deutsches Institut für Kautschuktechnologie e. V., Eupener Straße 33, D-30519 Hannover, FRG

2 Henniges Automotive GmbH & Co. KG Am Buchholz 4, D-31547 Rehburg-Loccum, FRG

ABSTRACT A micro-mechanical model of hyperelasticity and stress softening of filler reinforced elastomer materials is presented. It is based on recent investigations of filler morphology in elastomers and considers an advanced concept of rubber elasticity of bulk polymer networks together with a micromechanical model of stress induced filler cluster breakdown. The polymer network is described by a non-affine tube model of rubber elasticity with highly entangled chains, which takes into account that fluctuations in bulk networks are strongly suppressed by packing effects. The evaluation of stress softening is obtained via a pre-strain dependent hydrodynamic amplification of the rubber matrix by a fraction of rigid filler clusters with virgin filler-filler bonds. The filler-induced hysteresis is described by a cyclic breakdown and re-aggregation of the residual fraction of more soft filler clusters with already broken filler-filler bonds. From the simulations of stress-strain cycles at small and medium strain it can be concluded that the model of cluster breakdown and re-aggregation for pre-strained samples represents a fundamental micro-mechanical basis for the description of non-linear viscoelasticity of filler reinforced rubbers. Thereby, the mechanisms of energy storage and dissipation are traced back to the elastic response of the polymer network as well as the elasticity and fracture properties of flexible filler clusters. It is shown that the developed concept is in fair agreement with experimental stress-strain data of carbon black and silica filled elastomers. The obtained microscopic material parameter appear reasonable, providing information on the mean size and distribution width of filler clusters, the tensile strength of filler-filler bonds and the polymer network chain density. In particular it is shown that the model fulfils a “plausibility criterion” important for FEM applications. Accordingly, any deformation mode can be predicted based solely on uniaxial stress-strain measurements, which can be carried out relatively easily.

References [1] [2] [3] [4] [5] [6] [7] [8]

M. Klüppel and G. Heinrich, Rubber Chem. Technol. 68, 623, 1995. M. Klüppel, R. H. Schuster and G. Heinrich, Rubber Chem. Technol. 70, 243, 1997. M. Klüppel and J. Schramm, Macromol. Theory Simul. 9, 742, 2000 G. Heinrich and M. Klüppel, Adv. Polym. Sci. 160, 1, 2002 M. Klüppel, Adv. Polym. Sci. 164, 1, 2003 H. Luo, M. Klüppel und H. Schneider, Macromolecules 37, 8000, 2004 M. Klüppel und G. Heinrich, Kautsch. Gummi Kunstst. 58, 217, 2005 M. Klüppel, J. Meier, M. Dämgen, "Modeling of Stress Softening and Filler Induced Hys-teresis of Elastomer Materials", S. 171 in P.-E. Austrell und L. Kari (Eds.), "Constitutive Models for Rubber IV", A. A. Balkema Publishers, Lisse, Abingdon, Exton, Tokyo, 2005

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Analysis and effective properties of honeycombs with non-symmetric unit cells Fabian Lipperman, Michael Ryvkin, Moshe B. Fuchs School of Mechanical Engineering, Tel Aviv University Tel Aviv 69978, Israel {fabian, arikr, fuchs}@eng.tau.ac.il

ABSTRACT Planar cellular materials, or honeycombs, are considered as two-phase materials where the properties of one of it phases (voids) degenerates to zero. They are characterized by a low relative density and herein we assume that the solid phase of the material can be modeled as an assembly of axially loaded rods or general flexural elements in infinite and repetitive truss or frame structures, respectively. For the analysis and evaluation of the effective mechanical properties of a 2D cellular material the repetitive structure representing the material is assumed to be in a repetitive stress state. This allows for the analysis to be performed on a single repetitive cell under appropriate boundary conditions. However, all published results refer to cellular materials with repetitive modules possessing at least two orthogonal axes of symmetry. It seems that no general method of analysis of cellular material construed as infinite repetitive structures under periodic stresses for arbitrary repetitive cells is available, unless, as said, the repetitive cell possesses at least two perpendicular axes of symmetry. This is the subject matter of this study - to present a general exact technique to analyze infinite repetitive structures, with nonsymmetrical repetitive cells, under periodic stresses. We will be using a displacement based approach with a direct assembly of the stiffness matrix of a single repetitive cell. The repetitive infinite structures are loaded by stresses applied at infinity. The global displacements of the unit cell are formed by both micro displacements, which are the displacements of the nodes of the unit cell, and macro displacements, which arise from the distortion of the unit cell. Elements inside the cell are assembled in the classical manner. Elements which cross the boundaries are construed as being connected to an internal node and to a subset of the distortional displacements. This leads to a simple assembly procedure followed by the solution of a small set of equilibrium equations. The technique enables to find the effective elastic properties of any general periodic topology. As a consequence, several optimal low density honeycombs with symmetric and non-symmetric topologies whose effective bulk modulus and conductivity coincide with the Hashin-Shtrikman upper bounds, were found.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Influence of defects and perturbations on the performance of 3D open cell structures M.H. Luxner ∗, J. Stampfl †, A. Woesz††, P. Fratzl††, H.E. Pettermann∗ ∗ Institute of Lightweight Design and Structural Biomechanics, Vienna University of Technology

Gusshausstr. 27-29, 1040 Vienna, Austria [email protected] †Institute of Materials Science and Technology, Vienna University of Technology

Vienna, Austria †† Max Planck Institute of Colloids and Interfaces

Potsdam, Germany ABSTRACT Regular and irregular highly porous open cell structures with a relative density of 12.5% are investigated by the Finite Element Method. The three-dimensional models are based on beam elements and account for the material distribution and the constrained deformation in the vertices [1]. Six generic cell structures with regular arrangements of struts (Simple Cubic, Body Centered Cubic, Reinforced Body Centered Cubic, Gibson Ashby, Kelvin, and Weaire Phelan) are modeled by a unit cell approach for predicting the entire tensors of elasticity. Out of the six the two cell architectures with the highest and the lowest elastic anisotropy are selected for further nonlinear studies. Cuboidal and cylindrical samples consisting of a given number of base cells and different cell orientations are generated. Irregular modifications thereof are created by randomly shifting the position of the vertices within a spherical domain. Defects are introduced by randomly removing base cells. The nonlinear overall mechanical behavior is predicted for uniaxial compression in different directions under consideration of an elastic-plastic strut material and large deformations. Also localization of the deformation is considered [2]. The mechanical performance of the structures with different defects and different magnitudes of perturbations is compared in terms of the absorbable energy, the maximum bearable load, and deformation localization. The computational predictions are compared to results from experiments. The latter are performed on samples with corresponding architecture, fabricated by rapid prototyping [3].

References [1] M. H. Luxner, J. Stampfl, and H. E. Pettermann. Finite element modeling concepts and linear analyses of 3D regular open cell structures. J. Mat. Sci., 40:5859–5866, 2005. [2] M. H. Luxner, J. Stampfl, and H. E. Pettermann. Numerical simulations of 3D open cell materials – Influence of structural perturbations on elasto-plasticity and deformation localization. Mech. Mat. submitted (Oct. 2005). [3] A. Woesz, J. Stampfl, and P. Fratzl. Cellular solids beyond the apparent density - an experimental assessment of mechanical properties. Advanced Eng. Mat., 6:134–138, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Mechanics of Elastometer-Shim Laminates A.H.Muhr TARRC, Brickendonbury, Hertford, SG13 8NL, UK [email protected]

ABSTRACT The mechanics of laminates of elastomer and shims of high modulus material are reviewed. Such structures are often built to provide engineering components with specified, and quite different, stiffnesses in different modes of deformation. The shims may either be rigid or flexible, flat or curved, but are usually close to inextensible, being made of a high modulus material such as steel. On the other hand, rubber has an exceptionally low shear modulus, about one thousandth of its bulk modulus, so that shear of the rubber layers and flexure of the high modulus layers (if thin) are the dominant mechanisms of deformation of the composite. In comparison, extension of the layers and changes to their separation are highly constrained. Similar considerations apply also to cordreinforced elastomers, and are exploited in products such as tyres and air springs.Modes of failure are addressed as well as force-deformation behaviour. For compression normal to the laminations, the shear in the rubber results in in-plane tension in the shims, possibly leading to tensile failure. For tension normal to the laminations, the elastomer can cavitate, which would relieve the shear in it and hence the in-plane compressive stress applied to the shim. In flexure, shear in the rubber can apply inplane compressive stress to the shims and cause buckling failure. The relevance of such macromechanics to the behaviour of elastomers with nanofillers is discussed.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Application of the Mori-Tanaka method to analysis of woven composites with imperfections ˇ Jan Skoˇcek, Jan Zeman, Michal Sejnoha Department of Mechanics, Faculty of Civil Engineering, Czech Technical University in Prague Th´akurova 7, 166 29, Praha 6, Czech Republic [email protected], [email protected], [email protected] ABSTRACT Woven composites, i.e. composites reinforced with mutually interlaced systems of unidirectional fiber tows, belong to the most progressive material systems with applications ranging from strengthening of civil engineering structures to design of biocompatible medical implants. Therefore, understanding of overall behavior of these materials is a problem of considerable practical importance. Modeling of woven composites, however, presents a significant challenge due to their complex threedimensional geometry. A variety of approaches to simulation of these material systems has been proposed in the last four decades (see, e.g., [1] for an exhaustive review), including simple rule of mixtures, dimensionally-reduced models and detailed three-dimensional numerical studies. It is generally accepted that the last class of models is the most realistic one at the expense of higher computational cost, especially when taking into account microstructure imperfections [2]. To circumvent this limitation, the Mori-Tanaka method was selected due to its successful application to diverse heterogeneous material systems. In the present contribution, we employ the reformulation of the Mori-Tanaka method introduced in [3]. Compared to the original setting, the generalization ensures the major symmetry of the effective stiffness tensor and incorporates a two-step homogenization to capture the interaction between reinforcements. Application of the method to analysis of real-world material systems involves three steps. In the first step, we determine the parameters of the adopted scheme in order to optimally approximate results of a FEM-based first order homogenization study. In the next step, we take into account imperfections of fiber tow paths by incorporating an experimentally observed orientation distribution. Finally, the “effective” porosity of the matrix can be included on the basis of available experimental data. Acknowledgments. This work was supported by research project No. MSM 6840770003 of the Ministry of Education of the Czech Republic.

References [1] P. W. Chung and K. K. Tamma, Woven fabric composites - developments in engineering bounds, homogenization and applications, International Journal for Numerical Methods in Engineering, 45, 1757–1790, 1999. ˇ [2] J. Zeman and M. Sejnoha, Homogenization of balanced plain weave composites with imperfect microstructure: Part I - Theoretical formulation, International Journal of Solids and Structures, 41, 6549-6571, 2004. [3] J. Schjødt-Thomsen and R. Pyrz, The Mori-Tanaka stiffness tensor: Diagonal symmetry, complex fibre orientations and non-dilute volume fractions, Mechanics of Materials, 33, 531–544, 2001.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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An homogenization iterative process for nonlinear materials applied to compacted clays Salma Smaoui∗ , Abdelwahed Ben Hamida∗ , Irini Dj´eran-Maigre† and H´el`ene Dumontet∗ ∗

Laboratoire de Mod´elisation, Mat´eriaux et Structures (LM2S), Universit´e Paris VI, CC 161, 4 place Jussieu, 75252 Paris Cx 05, France. [email protected] †

Unit´e de Recherche de G´enie Civil (URGC), INSA de Lyon, Domaine Scientifique de la Doua, 69621 Villeurbanne, France. ABSTRACT In previous works an homogenization iterative approach has been successfully proposed to predict the linear behavior of reinforced and porous materials. This homogenization process consists to construct the Representative Elementary Volume of the heterogeneous media, by adding gradually to the matrix, low heterogeneity proportions until reaching the final rate of heterogeneity of the material following a process closed to the differential scheme method [1]. At any intermediate step of this process, an homogenization is carried out by any classical explicit method and the obtained effective behavior becomes the matrix of the following step. The equivalent homogeneous behavior is reached after convergence of the succession of intermediate homogeneous media. A significant result shows that the application of this iterative process to different homogenization approaches like dilute approximation, Hashin’s bound, self consistent method, and even the morphological representative pattern leads to the same behavior even for significant rates of heterogeneities or porosities [2]. In this work we extend this homogenization iterative approach in the nonlinear domain. After a secant linearization of the celular problem the iterative process is coupled to the ”comparison linear material” with the classical and modified secant homogenization methods [3]. We show that these two methods coupled to any explicit homogenization and iterative process lead to the same behavior for all rates of heterogeneities or porosities. This approach is here applied to study the hydro-elastoplastic behavior of compacted clays. The model parameters quantification is based on œdometric experimental results for different clays [4].

References [1] R.W. Zimmerman, Elastic moduli of a solid with spherical pores : New self-consistent method. International Journal of Rock Mechanics, 21(6), 339-343, 1984. [2] A. Benhamida, I. Djeran-Maigre , H. Dumontet, S. Smaoui, Clay compaction modeling by multiscale homogenization theory. International Journal of Rock Mechanics & Mining Sciences, 42, 996-1005, 2005. [3] P. Suquet, Effective properties of nonlinear composites. Continuum Micromechanics, CISM Lecture Notes, Springer Editor, 197-264, 1997. [4] I. Djeran-Maigre, D. Tessier, D. Grunberger, V. Velde, G. Vasseur, Evolution of microstructures and of macroscopic properties of some clays during experimental compaction. Marine and Petroleum Geology, 15, 109-128, 1998.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A Micromechanical Approach for the Simulation of Rubberlike Materials with Damage Mario Timmel*, Michael Kaliske*, Stefan Kolling†, Ralf Mueller†† *

Institute for Structural Mechanics, University of Leipzig Marschnerstrasse 31, D-04109 Leipzig, Germany [email protected] † DaimlerChrysler AG, Sindelfingen HPC X411, D-71059 Sindelfingen, Germany [email protected] †† Institute of Mechanics, TU Darmstadt Hochschulstrasse 1, D-64289 Darmstadt, Germany [email protected]

ABSTRACT The numerical simulation of rubberlike materials subjected to high strain rates requires the consideration of damage to take softening into account. In this context, phenomenological damage approaches are widespread in commercial finite element codes. In our presentation, we discuss a micromechanical approach to compute damage in rubbers. Based on Eshelby’s work [1], an analytical treatment of embedded inclusions in a matrix material is suggested. Following [2], the description of the temporal evolution of the inclusions’ shape is pointed out. In our work, we use these basics to describe the evolution of inclusion growth. Due to the assumption of a low stiffness of the embedded particles, which is equivalent to damaged regions, the growth of voids is approximated. As a useful tool to evolve suitable evolution laws, we use the method of configurational forces. In the consequence, the information how evolution should take place to maximize the dissipation is obtained. Configurational forces will be calculated on the phase boundary between the inclusion and the surrounding matrix. To allow an analytical description, the embedded particles will be approximated by an ellipsoidal shape. Due to the usability of Eshelby’s work in the case of small strains only, we apply the aforementioned procedure in an incremental way to describe rubberlike materials at finite strains. Based on a hyperelastic constitutive law, the entire loading process will be subdivided in a finite number of small strain problems. The damage behavior subsequently follows from the incremental decrease of tangential stiffness tensors.

References [1] J.D. Eshelby, The determination of the elastic field of an ellipsoidal inclusion and related problems. Proceedings of the Royal Society London, 241, 376-396, 1957. [2] R. Mueller and D. Gross, A time dependent constitutive law for materials with microstructural evolution, Mechanics of Materials, 33, 63-76, 2001.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Influence of grains misorientation on material hardening on example of aluminum bicrystals deformed in channel die W. Wajda, H. Paul Institute of Metallurgy and Materials Science of the Polish Academy of Sciences 25. Reymonta St, 30-059 Kraków, Poland [email protected]ów.pl [email protected]

ABSTRACT During deformation of polycrystalline aggregate hardening of the material is affected by microstructural phenomena in each grain of the material. In addition important is interaction between grains which depends among others on misorientation angle. However, investigation of grains misorientation on hardening in polycrystal material is not an easy task due to large number of grains with unknown orientations. In this paper results of bicrystal material compression are presented. Use of bicrystal material allows investigating influence of grains misorientation on hardening and at the same time examining the phenomena occurring at the grains boundary and within the grains (e.g. structure and texture development). In the paper experiments with aluminum bicrystals are presented with the following grains orientations: ‘cube’/‘hard’ ({100}<001>/{110}<011>); ’cube’/’Goss’ ({100}<001>/{110}<001>); ‘shear’/’Goss’ ({100}<011>/{110}<001>). In addition channel die compression tests on aluminum single crystals with ‘cube’, ‘Goss’ and ‘shear’ orientation were performed. Based on the experiments stress – starin curves for single- and bicrystals were established. The stress – strain curves comparison allows describe influence of grain misorientation on material hardening. An additional analysis was preformed by linking obtained stress – strain curves to microstructure development [1, 2, 3]. In the work computer simulation of bicrystals compression in channel die was performed. For simulation purposes the bicrystals behavior was described using stress – strain curves obtained from single crystals compression tests in channel die with selected orientations. The simulation of the test was simplified to 2D plain strain problem. It was possible due to lubricant application during the experiment which allows to approach very close to plain strain conditions. Results of the simulation of the strain distribution were verified by comparison with experimental data of the substructure development.

References [1] H. Paul, J.H. Driver, C. Maurice and M. Bijak, Large strain deformation substructures and local crystallography in {100}<001>/{110}<001> aluminium bicrystals. in press. [2] H. Paul, J.H.Driver, Deformation behaviour of channel-die compressed Al bicrystals with {100}<001>/{110}<011> orientation, Archives of Metallurgy and Materials 50, 209-218, 2005. [3] H. Paul, J.H. Driver, A. Lens, Mechanisms of new orientation formation during recrystallization of cold deformed aluminium bicrystals, Materials Science Forum, 495-497, 1249-1254, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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The Method of Fundamental Solutions applied to the calculation of eigensolutions for simply connected plates Carlos J. S. Alves∗ , Pedro R. S. Antunes∗ ∗ CEMAT,

Department of Mathematics, Instituto Superior T´ecnico, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal. {calves,pant}@math.ist.utl.pt ABSTRACT

In this work we study the application of the Method of Fundamental Solutions (MFS) to the numerical calculation of the eigenvalues and eigenfunctions for the 2D Bilaplacian in simply connected plates. This problem was considered in [5] using wave-type functions and in [4] using radial basis function for a circular and rectangular domain. The MFS is a meshfree method that was already applied to the calculation of the eigenvalues and eigenfunctions associated to the Laplace operator (cf. [6], [2] and [1]). The application of this method to the Bilaplacian was already considered in [3], but only for simple shapes. Here we apply an algorithm for the choice of point-sources, as in [1], that leads to very good results for a fairly general class of domains.

References [1] Alves C J S and Antunes P R S, The method of fundamental solutions applied to the calculation of eigenfrequencies and eigenmodes of 2D simply connected shapes., Computers, Materials & Continua 2(4), 251–266 (2005). [2] Chen J T, Chen I L and Lee Y T, Eigensolutions of multiply connected membranes using the method of fundamental solutions., Engineering Analysis with Boundary Elements, Vol. 29, no. 2, 166–174, 2005. [3] Chen J T and Lee Y T, True and spurious eigensolutions for membrane and plate problems by using the method of fundamental solutions., ECCOMAS Thematic Conference on Meshless Methods 2005. [4] Chen J T, Chen I L, Chen K H, Lee Y T and Yeh Y T, A meshless method for free vibration analysis of circular and rectangular campled plates using radial basis function., Engineering Analysis with Boundary Elements, Vol. 28, 535–545, 2004. [5] Kang S W and Lee J M, Free vibration analysis of arbitrary shaped plates with clamped edges using wave-type functions., J. Sound Vibr. 242(1), 9–16, 2001. [6] Karageorghis A, The method of fundamental solutions for the calculation of the Eigenvalues of the Helmholtz Equation., Appl. Math. Letters 14(7), 837–842, 2001.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Comparison between Meshfree and Boundary Element methods applied to BVPs in domains with corners Carlos J. S. Alves∗ , Svilen S. Valtchev† ∗ CEMAT,

Instituto Superior T´ecnico 1049-001 Lisbon, Portugal [email protected]

† CEMAT,

Instituto Superior T´ecnico 1049-001 Lisbon, Portugal [email protected]

ABSTRACT A Dirichlet boundary value problem (BVP) for the Laplace equation will be considered in a bounded domain with corners. Two distinct types of numerical methods will be applied for the solution of this problem. A modification of the Boundary Element Method (BEM), as presented in [1], based on the double-layer potencial representation of the solution will be applied. Both piecewise constant and piecewise linear spline collocation will be used for the approximate solution of the resulting Fredholm integral equation of the second kind. Quadratic order of convergence is achieved in this case. On the other hand, the classical Method of Fundamental Solutions (MFS, e.g. [2]) will be applied. It is claimed that this meshfree method exhibits exponential rate of convergence when regular data (and domain) is considered, e.g. [3], [4]. Numerical results for polygonal domains will be presented. The importance of the location of the artificial boundary and the choice of the source points will be discussed for the MFS. The numerical methods will be compared in terms of absolute error of the approximate solution for a fixed number of boundary knots/boundary elements.

References [1] R. Kress, Linear Integral Equations, volume 82 of Applied Mathematical Sciences, Springer, 2nd edition, 1999. [2] V. D. Kupradze, M. A. Aleksidze, The method of functional equations for an approximate solution of certain boundary value problems. U.S.S.R. Comput. Math. Comput. Phys. 4, 82-126, 1964. [3] M. Katsurada, Asymptotic error analysis of the charge simulation method in a Jordan region with an analytic boundary. J Fac. Sci. Univ Tokio, 37, 635-657, 1990. [4] X. Li, On convergence of the method of fundamental solutions for solving the Dirichlet problem of Poisson’s equation. Adv. Comput. Math., 23, 265-277, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

522

The Method of Fundamental Solutions applied to a heat conduction inverse problem Carlos J. S. Alves∗ , Nuno F. M. Martins† ∗ Instituto Superior T´ ecnico Avenida Rovisco Pais, 1096 Lisboa Codex, Portugal [email protected] † Faculdade de Ciˆ encias e Tecnologia Quinta da Torre, 2829-516 Caparica, Portugal [email protected]

ABSTRACT The Method of Fundamental Solutions (MFS) is a meshless method that is used to solve an inverse heat conduction problem. The inverse problem consists in finding the shape of a cavity inside a domain Ω ⊂ R2 with prescribed temperature on the boundary of the cavity. On the exterior boundary of the domain Ω the temperature is imposed and we measure the induced heat flux. We prove that this problem has a unique solution, under certain hypothesis and use MFS to solve it numerically. Density results for the MFS justify the adequacy of this method as a direct solver and suggest its use as an inverse method. However the straightforward application of the MFS as an inverse method lead to poor numerical reconstructions. Therefore we also introduce an alternative use of the MFS with a minimization Quasi-Newton method that uses the MFS as forward solver. Using this procedure, the numerical reconstructions are much better and stable to the introduction of random noise in the measured data. Recovery of the cavities with partial access for measurements is also considered (see Fig. 1).

Figure 1: Left plot - Accessible parts of the exterior boundary. Right plot - Results obtained with the iterative scheme. Full line: starting curve; dashed lines: intermediate curves; dotted line: final curve; full bold line: cavity shape.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

523

Local Maximum-Entropy Approximation Schemes Marino Arroyo1 and Michael Ortiz2 1 Universitat

` Politecnica de Catalunya E-08034 Barcelona, Spain [email protected]

2

California Institute of Technology Pasadena, USA [email protected]

ABSTRACT We present a new approach to construct approximation schemes from scattered data on a node set, i.e. in the spirit of meshfree methods. The rational procedure behind these methods is to harmonize the locality of the shape functions and the information-theoretical optimality (entropy maximization) of the scheme, in a sense to be made precise in the paper. As a result, a one-parameter family of methods is defined, which smoothly and seamlessly bridges meshfree-style approximants and Delaunay approximants. Besides an appealing theoretical foundation, the method presents a number of practical advantages when it comes to solving partial differential equations. The non-negativity introduces the well-known monotonicity and variation-diminishing properties of the approximation scheme. Also, these methods satisfy ab initio a weak version of the Kronecker-delta property, which makes essential boundary conditions straightforward. The calculation of the shape functions is both efficient and robust in any spacial dimension. The implementation of a Galerkin method based on local maximum entropy approximants is illustrated by examples.

References [1] Marino Arroyo and Michael Ortiz, Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods, International Journal for Numerical Methods in Engineering, (in press) 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

525

Solving the Chloride Diffusion Equation in Concrete Structures for Prediction of Initiation Time of Corrosion Using The Finite Point Method M. Bitaraf*, S. Mohammadi† School of Civil Engineering, University of Tehran, Tehran, Iran * Graduate Student, [email protected] † Assistant Professor, [email protected]

ABSTRACT Reinforced concrete structures exposed to sea environments suffer from corrosion of steel bars due to the chloride ingress. This corrosion may lead to serious damages to concrete structures and cost of repair, inspection and maintenance activities for these structures could reach a level comparable to the cost of construction of new structures. Therefore, the chloride penetration is a major factor that affects the durability of concrete structures. Diffusion of chloride ions is generally assumed to follow the Fick’s second law [1]. In this paper, the finite point method is adopted for solving the chloride diffusion equation for prediction of service life of concrete structures and initiation time of corrosion of reinforcement. Finite point method (FPM) is a truly meshless method which uses a moving least square approximation (MLS) within a collocation strong form for solving the governing differential equation [2]. To calculate the derivatives of variables with respect to time, both forward and backward difference methods are used, the former proved to be more accurate and satisfactory. An optimum value for radius of support domain in MLS approximation is obtained in order to achieve more precise results. Several 1D and 2D problems of chloride diffusion are solved using FPM and the results are compared with the analytical solution, classical finite element and finite difference methods, and weak form meshless based Element Free Galerkin method. 1D tests demonstrated that FPM and FDM provide very close predictions whereas for 2D problems, if regular distribution of nodes are used, the FPM and FDM remain close, while FPM can also be efficiently used for accurate simulation using irregular distribution of nodes and solving complex geometries. Although, it is expected that prediction of the initiation time of corrosion by EFG be much closer to the exact initiation time, nevertheless, it should be noted that the FPM procedure is more straightforward in comparison to EFG and it does not need any integration procedures, making it a simple approach to implement as well as being computationally inexpensive. Furthermore, the comparison of computing time for solving the 2D equation in different methods illustrate that FPM is the fastest computational approach for solving general complex geometries.

References [1] P. Goltermann, Chloride ingress in concrete structures; Exptrapolation of Observations. ACI Material Journal, 100: 2, 114-119, 2003.

[2] E. Onate, F. Perazzo, J. Miquel. A finite point method for elasticity problems. Computers and Sructues, 70, 2151-263, 2000.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

526

Solution of the Stationary Three-dimensional Navier-Stokes Equations by Using Radial Basis Functions and Reducing the Number of Governing Equations Somchart Chantasiriwan Faculty of Engineering, Thammasat University Rangsit Campus, Khlong Luang, Pathum Thani 12121, Thailand [email protected]

ABSTRACT The stationary three-dimensional Navier-Stokes equations consisting of four equations and four unknowns have been solved by conventional methods using the primitive-variable approach. By getting rid of pressure and one of the velocity components, the Navier-Stokes equations can be reduced to two higher-order partial differential equations with the remaining two velocity components as the only two unknowns. In this paper, a meshless method based on a radial basis function, known as multiquadrics, is proposed to solving such equations. Unknown velocity components are approximated as linear combinations of multiquadrics centered at domain nodes and boundary nodes. Unknown coefficients are solved by a Picard iterative scheme. The proposed method is used to solve a test problem, for which exact solution is known. It is found that the number of iterations required for a converged solution and the accuracy of the solution depend on the shape parameter of multiquadrics. A small value of the shape parameter results in a low number of required iterations, but the resulting solution may not be accurate. On the other hand, a large value of the shape parameter can yield a very accurate solution provided that a converged solution is obtained. There appears to be an upper limit to the value of the shape parameter for which the proposed method is capable of yielding a converged solution.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

527

An enhanced Moving Least Squares interpolation for the Element-free Galerkin Method Thomas Most and Christian Bucher

Institute of Structural Mechanics Bauhaus-University Weimar, Germany {thomas.most,christian.bucher}@uni-weimar.de

ABSTRACT The Element-free Galerkin Method has become a very popular tool for the simulation of mechanical problems with moving boundaries. The internally applied Moving Least Squares interpolation uses in general Gaussian or cubic weighting functions and has compact support. Due to the approximative character of this interpolation the obtained shape functions do not fulfill the interpolation conditions, which causes additional numerical effort for the application of the boundary conditions. In this paper a new weighting function is presented, which was designed for meshless shape functions to fulfill these essential conditions with very high accuracy without any additional effort. Furthermore this interpolation gives much more stable results for varying size of the influence radius and for strongly distorted nodal arrangements than existing weighting function types. One of the most time consuming step in the MLS interpolation is the assembling of the coefficient and weighting matrices and the inversion, which is the base of the least square approach  −1 PW(x)u. uh (x) = pT (x) PW(x)PT

(1)

This effort increases even more for higher dimensions and higher order base polynomials. Since in the Element-free Galerkin Method generally a large number of integration points is used, where these expensive operations have to be performed, the numerical effort is much higher than in the Finite Element Method. In this paper a new concept is presented, where the least squares coefficients are calculated only at the nodes  −1 a(xI ) = PW(xI )PT PW(xI )u.

(2)

At each interpolation point the nodal coefficients are weighted using the improved weighting function. The final interpolation reads

[a(xI )wI (x)] / wI (x). (3) uh (x) = pT (x) Finally a concept will be presented, which enables an efficient analysis of systems with strongly varying node density. In this concept the nodal influence domains are adapted depending on the nodal configuration by interpolating the influence radius for each direction from the distances to the natural neighbor nodes. This approach requires a Voronoi diagram of the domain, which is available in this study since Delaunay triangles are used as integration background cells. In the numerical examples it will be shown, that this enhanced method leads to similar results as the classical approach, but will reduce the numerical effort significantly.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

528

Enforcement of boundary conditions in meshfree methods using Interpolating Moving Least Squares Hennadiy Netuzhylov Scientific assistant Technical University of Brunswick MSc Programme “Computational Sciences in Engineering” B¨ultenweg 17, D-38106 Brunswick, Germany [email protected]

ABSTRACT This paper presents a meshfree Interpolating Moving Least Squares (IMLS) method based on singular weights for the solution of partial differential equations. Due to the to the specific singular choice of the weight functions, which is needed to guarantee the interpolation, there arises a problem of the finding the inverse of the singular matrix. We extend the perturbation technique originally presented in [5] to allow the correct evaluation of all necessary derivatives in interpolation points at a reasonable cost. The inverse is carried out using the regularized weight function. It turns out that a stable inverse is obtained when the vanishing regularization parameter is considered. Unlike standard kernel funktions used in EFGM, RKPM etc., the singular kernel functions lead to truly interpolating functions which satisfy the Kronecker-delta property. They can be used for enforcement of Dirichlet boundary conditions when solving boundary value problems. Solution to a model BVP as well as the experimental convergence study of the method to analytical solutions, confirming the mathematical derivations, are given.

References [1] Ted Belytschko, Y Krongauz, D Organ, M Fleming, and Petr Krysl. Meshless methods: An overview and recent developments. Comput. Methods Appl. Mech. Engng, 139:3–47, 1996. [2] Jiun-Shyan Chen and Hui-Ping Wang. New boundary condition treatments in meshfree computation of contact problems. Comput. Methods Appl. Mech. Engrg., 187(3-4):441–468, 2000. [3] Thomas Peter Fries. Classification and overview of meshfree methods. TU Branshweig, 2002. [4] Xiaozhong Jin, Gang Li, and N. R. Aluru. Positivity conditions in meshless collocation methods. Comput. Methods Appl. Mech. Engrg., 193(12-14):1171–1202, 2004. [5] Matthias Kunle. Entwicklung und Untersuchung von Moving Least Square Verfahren zur numerischen Simulation hydrodynamischer Gleichungen. Dissertation, Fakult¨at f¨ur Physik, Eberhard-Karls-Universit¨at zu T¨ubingen, page 123, 2001. ˇ [6] Peter Lancaster and Kestutis Salkauskas. Surfaces generated by moving least squares methods. Math. Comp., 37(155):141–158, 1981.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A simple and less-costly integration of meshless Galerkin weak form Victoria E. Rosca*, Vitor M.A. Leitão † *

I DECivil-ICIST, Instituto Superior Técnico Av. Rovisco Pais 1049-001 Lisboa, Portugal [email protected] †

DECivil-ICIST, Instituto Superior Técnico Av. Rovisco Pais 1049-001 Lisboa, Portugal [email protected]

ABSTRACT Computational efficiency and reliability are the most important requirements for the success of a meshless numerical technique. The efficiency of these methods depends on the proper choice of the interpolation scheme, numerical integration procedures and techniques of imposing the boundary conditions. From all these difficulties, the integration is the most time-consuming part in meshless calculation due to the large number of integration points needed for a sufficiently accuracy of the integration of the weak form. Also, insufficiently accurate numerical integration may lead a deterioration and rank-defieciency in the numerical solution. The difficulty is due to the complexity of the MLS shape functions in an integration domain. The purpose of the present paper is to alleviate the difficulty in the numerical integration of the weak form in the MLPG [1] and EFG [2] methods. For this aim we implement a 3D integration technique for the evaluation of the stiffness matrix that does not rely on a partition of the domain into cells. This is made by using Quasi-Monte Carlo techniques [3]. The integration domain can be the entire computational domain, without the need to use background meshes. For comparison of the efficiency and accuracy for these two meshless formulations based on Galerkin weak form the study is based on various Quasi-Monte Carlo sequences. The method is applicable to any type of problem with any number of dimensions. Here, we have presented a numerical test for a 3D elasticity problem. The test demonstrates the efficiency of the integration techniques for both meshless formulations. We find this new method simultaneously simple and efficient because the integration scheme is related to the set of quasi random points used for the approximation. We also expect that our method will be found especially simple for 3D problems with complicated shapes.

References [1] S.N Atluri, S. Shen, The Meshless Local Petrov-Galerkin (MLPG) Method: A Simple & Lesscostly Alternative to the Finite Element and Boundary Element Methods, CMES, 3,(1), 11-51, 2002 [2] B. Fox, Implementation and relative efficiency of quasirandom sequence generators. ACM Trans. on Mathematical Software, 12, 362–376, 1986. [3] J. Dolbow, T. Belytschko, An Introduction to Programming the Meshless Element Free Galerkin Method, Archives of Computational Methods in Engineering, 5, 3, 207-241, 1998.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Geometrically exact analysis of shells by a meshless approach Carlos Tiago∗ , Paulo M. Pimenta† ∗ Instituto

Superior T´ecnico, Universidade T´ecnica de Lisboa Av. Rovisco Pais, 1049-001, Lisboa, Portugal [email protected]

† Escola Polit´ ecnica, Universidade de S˜ao Paulo Av. Prof. Almeida Prado, trav. 2, 83, S˜ao Paulo, Brazil [email protected]

ABSTRACT There is a growing interest in the geometrically exact analysis of structures. The innate elegance of this king of formulations arises from the exact representation of the rotations. In this case, the rotation vector is parameterized by the Euler-Rodrigues formula. The internal power arises from the first PiolaKirchhoff stress tensor and the deformation gradient. A consistent plane stress condition is imposed in a hyperelastic material to derive the appropriate (symmetric) constitutive operator [?]. In the present work a hybrid method of analysis is proposed where the solution is obtained by the approximation of the generalized internal displacement fields through the Moving Least Squares (MLS) scheme and the generalized boundary tractions are interpolated by Lagrange polynomials. To completely eliminate shear-locking phenomenon a consistency requirement is imposed to the generalized internal displacement fields: the exact reproduction of the Kirchhoff-Love constraints. An extension of the arc-length method that includes the generalized internal displacement fields, the generalized boundary tractions and the load parameter in the constraint equation of the hyperellipse is proposed to solve the resulting nonlinear problem. A consistent linearization procedure is performed, resulting a semi-definite system matrix which, for hyperelastic materials and conservative loadings, is always symmetric (even for configurations far from a equilibrium trajectory). Differently from the standard Finite Element Methods (FEM), the resulting solution are (arbitrary) smooth generalized displacements and stress fields. Also, the representation of the initial configuration is exact, contrary the usual FEM, where a C 0 approximation of the original problem is made (usually by the assembly of flat elements).

References [1] E. M. B. Campello, P. M. Pimenta and P. Wriggers, A triangular finite shell element based on a fully nonlinear shell formulation, Computational Mechanics, 31(6), 505–518, 2003. [2] C. Tiago and P. Pimenta, Geometrically exact analysis of space frames by a meshless method, ECCOMAS Thematic Conference on Meshless Methods, C44.1–C44.8, Lisbon, Portugal, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

531

Weight Functions Analysis in Elastostatic Problems for Meshless Element Free Galerkin Method O.F. Valencia*, Francisco J. Gómez-Escalonilla†, F. Urbinati ** , J. López-Díez‡ * ATOS Origin SAE Albarracín 25, 28037 Madrid; (Spain) [email protected] † EADS MTAD Avda John Lennon s/n; 28906 Getafe (Spain) [email protected] ** ATOS Origin SAE Albarracín 25, 28037 Madrid; (Spain) [email protected] ‡

ETSI Aeronáuticos. Universidad Politécnica de Madrid Plaza del Cardenal Cisneros 3, 28040 Madrid Spain [email protected]

ABSTRACT Weight functions of the Moving Least Square (MLS) approximation are the key to solve EDPs’ problems by Element Free Galerkin Method (EFGM). That is the reason why it is so important to use the best weight function for each case. This paper contains a systematic analysis of the most common weight functions used in solving elastostatics problems by meshless EFGM. This analysis comprises the influence of the main parameters and their effect on the accuracy of the weight functions on the solution of the problems studied. The referred accuracy is set by calculating the error of the numerical solution versus the actual result, in terms of standard energy norm. However, it is not always the best function the one that solves the problem with the minimum error, but the one that does it with the lowest computational cost. A computational cost analysis method is set to let it be quantifiable. This way, an analysis of the error, as previously said, in terms of standard energy norm, versus computational cost will be done, by minimizing this relationship. Therefore, the accurate weight function is determined at each case analyzed. In the spirit of letting the essential concepts be explained, some numerical examples are provided in the field of Elastostatics.

References [1] T. Belytschko, Y. Y. Lu, I. Gu, Element-free Galerkin Methods. Element-free Galerkin Methods. 37, 229-256, 1994. [2] L. Gayate, B. Alonso, J. J. Benito, Error approximation in EFG method, Fifth World Congress on Computacional Mechanics, 2002. [3] J. Dolbow, T. Belytschko, An Introduction to Programming the Meshless Element Free Galerkin Method, Archives of Computational Mechanics in Engineering, 5, 207-241, 1998.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Prediction of Sheet Metal Formability (FLD) By Using Diverse Method F. Abbassi *†, O. Pantale †, A. Zghal*, R. Rakotomalala † *

Laboratory of Mechanics, Solids, Structures and Technological Development, ESSTT 5, Av. Taha Hussein, 1008 Tunis, Tunisia [email protected]

† Laboratoire de Génie Productique C.M.A.O., ENIT, Avenue d’Azereix-BP 1629, 65016 Tarbes Cedex, France {abbassi.fethi, olivuer.pantale, roger}@enit.fr

ABSTRACT In this paper we propose a procedure to evaluate the material formability based on the use of a Forming Limit Diagram. FLD method is proven to be a useful tool in the analysis of forming severity, it has been shown to be valid only for cases of proportional loading, where the ratio between the principal stresses remain constant throughout the forming process. In this works, quadratic and non quadratic Hill’s yield criteria are used to study the material formability, we determinate the limit strain by using Hill’s 93 connecting with the Swift and Hill necking condition in the case of material assuming a swift hardening low. And we compare an empirical model proposed by NADDG between theatrical results.

References [1] D. Banabic “Limit strains in the sheet metals by using the new Hill’s yield criterion (1993)” journal of Materials Processing Technology 92-93 429-432, 1999. [2] R.Hill “A user-friendly theory of orthotropic plasticity in sheet metals” Int. J. Mech. Sci. 35, 1925, 1993. [3] M J Painter and R. Pearce "Instability and fracture in sheet metal" j. phys. D: appl. Phys. Vol 7, 1974.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Modeling Drawbeads in Deep Drawing Simulations J.L. Alves *, M.C. Oliveira†,L.F. Menezes† *

†

Department of Mechanical Engineering, University of Minho Campus de Azurém, 4810 Guimarães, Portugal [email protected]

CEMUC, Department of Mechanical Engineering, University of Coimbra Pinhal de Marrocos, Polo II, 3030 Coimbra, Portugal [email protected]; [email protected]

ABSTRACT The numerical simulation has become an indispensable tool in the design, production and process set up of deep drawn parts. The advantages of these tools are nowadays well known. If the numerical model correctly describes both the technological procedure and process parameters, the use of numerical simulation allows to save time, money and effort. One of the challenges to guarantee this correlation between the numerical models and the real process conditions is the drawbead modeling. Drawbeads are often used in sheet metal forming processes to provide a better control of the material flow. Numerically there are still numerous difficulties to accurately model and describe drawbeads’ geometries and actions. Due to such difficulties, most of the numerical strategies used in finite element codes simply replace the real drawbead by an “equivalent drawbead model”. The majority of these models are based on the analysis of a 2-D model of the physical drawbead. The drawbead geometry is simplified to an artificial line where a restraining and a lift force are associated with. More complex models are based on an algorithm that acts on the finite elements that cross the artificial line in order to update both thinning and strain and stress states. The first drawback of these strategies is associated with its accuracy if in the real process the drawbead is not linear or the deformation process is not close to plain strain state. The second drawback is associated with the fact that the clamping phase disappears, and in this phase the drawbeads can induce some state variables change in nodes that aren’t close to the artificial line. In this work, two different kinds of forming processes that require the modeling of a drawbead are analyzed. In the first, a bulge test example, the drawbead acts to eliminate any gliding between the blank and the drawbead. The second example is the U shape geometry according to the Benchmark#3 (Stage 1) proposed in the framework of Numisheet’2005 conference. In this example the drawbead acts only to control the material inflow, but the material passing through the drawbead will be included in the final formed part [1]. Simulation results of these examples with the physical drawbead model, with an “equivalent drawbead model” and with no drawbead are analyzed. All simulations were performed with the implicit finite element code DD3IMP.

References [1] J.L. Alves, M.C. Oliveira e L.F. Menezes, Drawbeads: to be or not to be. Proceedings of the NUMISHEET’2005, 6th International Conference and Workshop on Numerical Simulation of 3D Sheet Forming Processes – On the Cutting Edge of Technology, Ed. L.M. Smith, F. Pourboghrat, J. W. Yoon e T.B. Stoughton, Part A, 655-660, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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On a 2-phase finite element model for the coherent mushy zone in casting applications S. Benke, G. Laschet ACCESS e. V. Materials and Processes, RWTH Aachen, D-52072 Aachen, Germany Address [email protected] [email protected]

ABSTRACT During the solidification procedure in casting processes, metallic alloys undergo a continuous transition from liquid to solid by passing a semi-solid or mushy state. Depending on the morphology of the microscale the rheological behaviour of the mush is divided into a more fluid-like and a more solid-like behaviour. In the fluid-like regime at high temperatures, only isolated germs exist in the melt and the mush can be described as a non-Newton fluid. After the temperature falls below a critical value which is defined as the coherency temperature, the solidified dendrite skeletons get connected and form a continuous skeleton. This skeleton is able to sustain tensile strains and behaves more solid-like. During the whole solidification process the solid and the liquid interact in the mushy regime due to momentum, energy, mass and solute exchange. In the present contribution we focus on the modelling of the thermo-mechanical behaviour of the mush below the coherency temperature. We present a continuum-mechanical model describing the deformation and solidification phenomena, based on the thermodynamically consistent theory of porous media (TPM). The model is based on two constituents, namely an elasto-viscoplastic solid skeleton which is incompressible on the microscale and a viscous pore fluid. The balance equations are formulated on the macroscale in a general setting in the material space and the exchange of mass, momentum and energy between the constituents is considered. Between both constituents a local thermal equilibrium is assumed. Restrictions and dependencies for the constitutive equations describing the exchange terms and the material laws are obtained form the evaluation of the entropy balance following the proposal of Coleman and Noll [Arch.Rat.Mech.Anal.13 (1963) 167]. This procedure leads to a definition of the mass exchange term as a function of the free energy of the constituents. The non-isothermal model takes thermal deformations into account and achieves a direct coupling to the thermal solidification analysis. The initial boundary value problem is solved by a numerical solution scheme based on mixed Galerkin finite elements. The velocity of the solid skeleton, the pressure of the pore fluid and the temperature are taken as the primary unknown field variables. Using the implementation of the solution algorithm in the casting software CASTS the solidification progress as well as the isothermal and non-isothermal visco-plastic deformation is studied.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Modeling of ductile behavior of metals under a wide range of loading rates: semi-empirical approach Ch. Husson*, S. Ahzi†, L. Daridon‡ *

Institut Supérieur d’Ingénierie de la Conception – Equipe ERMeP 27 rue d’Hellieule, F-88100 Saint-Dié-des-Vosges [email protected] †

‡

Université Louis Pasteur, IMFS-UMR 7507 2 rue Boussingault, 67000 Strasbourg [email protected]

Laboratoire de Mécanique et Génie Civil, UMR 5508 Place Eugène Bataillon, F-34095 Montpellier [email protected]

ABSTRACT For a large number of metal forming processes and impact applications, the loading conditions are so severe. Therefore, it is necessary to develop robust tools to predict structural response of metals under these loading conditions. In this paper, we propose a semi-empirical elastic-viscoplastic material model combined with a non-linear isotropic damage evolution law. The effect of strain hardening, strain-rate hardening, pressure and thermal softening have been incorporated in the response of material for a wide range of loading rates. Like in the mechanical threshold stress model, the flow stress is decomposed as the sum of an effective stress and a thermally-activated component. The proposed model is a semi-empirical description of the plastic deformation behaviour of ductile metals where some of the physical aspects are taken into account via the mechanical threshold stress and an Arrhenius type expression relating strain rate to activation energy and temperature. The damage evolution law is based on the theory of continuum damage mechanics by assuming the existence of new ductile damage dissipation potential. The models have been implemented in the form of a user material subroutine (VUMAT) in the commercial finite element code ABAQUS/Explicit. We simulated the high speed blanking process of thin copper sheets and uniaxial loading tests of copper. Results from our model are in a good agreement with the existing experimental results for stressstrain behaviour, damage evolution and blanking profile.

References [1] Ch. Husson, S. Ahzi, L. Daridon, T. Courtin, Continuum damage modeling for ductile under high strain rate deformation, J. Phys. IV, 110, 63-68, 2003. [2] Ch. Husson, C. Poizat, N. Bahlouli, S. Ahzi, T. Courtin, L. Merle. On plasticity and damage evolution during sheet metal forming in Multiscale modeling and characterization of elasticinelastic behavior of engineering materials, Marrakech (IUTAM symposium) S. Ahzi et al. (eds.), Kluwer Academic Publishers, 105-112, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A New Relaxation Method for Roll Forming Problems Shichao Ding, William J.T. Daniel, Paul A. Meehan Mechanical Engineering Division, School of Engineering, The University of Queensland St Lucia, QLD 4072, Australia [email protected], [email protected], [email protected]

ABSTRACT Finite element analysis (FEA) of nonlinear problems in solid mechanics is a time consuming process, but it can deal rigorously with the problems of both geometric, contact and material nonlinearity that occur in roll forming. The simulation time limits the application of nonlinear FEA to these problems in industrial practice, so that most applications of nonlinear FEA are in theoretical studies and engineering consulting or troubleshooting. Instead, quick methods based on a global assumption of the deformed shape have been used by the roll-forming industry. These approaches are of limited accuracy. This paper proposes a new form-finding method – a relaxation method to solve the nonlinear problem of predicting the deformed shape due to plastic deformation in roll forming. This method involves applying a small perturbation to each discrete node in order to update the local displacement field, while minimizing plastic work. This is iteratively applied to update the positions of all nodes. As the method assumes a local displacement field, the strain and stress components at each node are calculated explicitly. Continued perturbation of nodes leads to optimisation of the displacement field. Another important feature of this paper is a new approach to consideration of strain history. For a stable and continuous process such as rolling and roll forming, the strain history of a point is represented spatially by the states at a row of nodes leading in the direction of rolling to the current one. Therefore the increment of the strain components and the work-increment of a point can be found without moving the object forward. Using this method we can find the solution for rolling or roll forming in just one step. This method is expected to be faster than commercial finite element packages by eliminating repeated solution of large sets of simultaneous equations and the need to update boundary conditions that represent the rolls.

References [1] C.K. McClure and H. Li, Roll Forming Simulation Using Finite Element Analysis, Manufacturing Review, vol.8, 114-119, 1995. [2] S.M. Panton, S.D. Zhu and J.L. Duncan, Geometric Constraints on the Forming Path in Roll Forming Channel Section, Proc. Instn Mech. Engrs, Part B, vol.206(B2), 113-118, 1992. [3] N. Duggal, M. A. Ahmetoglu, G. L. Kinzel and T. Altan, Computer Aided Simulation of Cold Roll Forming - A Computer Program for Simple Section Profiles, Journal of Materials Processing Technology, vol.59, 41-48, 1996. [4] C. Liu, Y. Zhou and W. Lu, Numerical Similation of Roll Forming by B-spline Finite Strip Method, Journal of Materials Processing Technology, vol.60, 215-218, 1996. [5] N. Kim, B. Kang and S. Lee, Prediction and Design of Edge Shape of Initial Strip for Thick Tube Roll Forming Using Finite Element Method, Journal of Materials Processing Technology, vol.142, 479-486, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Computer Modelling of Deformation of Steel Samples with Mushy Zone M. Glowcki*, M. Hojny* *

AGH-University of Science and Technology Mickewicza 30, 30-059 Kraków, Poland [email protected]

ABSTRACT The paper presents the results of experimental and theoretical work leading to mathematical modelling of phenomena accompanying the deformation of steel which is still subjected to the last phase of the solidification process. Physical simulations of deformation of steel at such extra high temperature performed with the aid of advanced testing machines (for example GLEEBLE 3800 simulator) are very expensive and extremely difficult due to extra high temperature, simultaneous deformation and solidification of the metal and its very unstable behaviour. The Authors has presented dedicated computer program taking into consideration the density changes and analytical form of the compressibility condition. The aim of the modelling was the reconstruction of the temperature, strain and stress changes inside steel samples subjected to both deformation and final stage of solidification. The presented program enables the right interpretation of very high temperature tests. It is equipped with inverse analysis described in the current paper. Application of the system makes possible the identification of strain-stress curve parameters and lowering testing cost. In Fig. 1 the example flow stress versus strain curve (temperature 1460 °C, tool velocity – 20 mm/s) is presented. Fig. 2 shows comparison of measured and predicted loads for the discussed case.

Figure 1: Flow stress vs strain at temperature 1460°C and tool velocity 20 mm/s.

Figure 2: Comparison of measured and predicted loads at temperature 1460°C and tool velocity 20 mm/s.

REFERENCES [1] M. Glowacki, Z. Malinowski, M. Hojny, D. Jedrzejczyk, The Physical and Mathematical Modeling of Plastic Deformation of Samples with Mushy Zone, Proc. Int. Conf. Inverse Problems, Design and Optimization Symposium, Rio de Janeiro, Brazil, p. 79-86, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Prediction of strain localisation in forming process using advanced elasticplastic behaviour models coupled with damage B. Haddag, F. Abed-Meraim, T. Balan Laboratoire de Physique et Mécanique des Matériaux, CNRS-UMR 7554, ENSAM of Metz 4, rue Augustin Fresnel, 57078 Metz Cedex 3, France badis.haddag@ metz.ensam.fr [email protected] [email protected]

ABSTRACT This work aims to study the strain localisation in the sheet metals during the deep drawing process. This phenomenon is precursor for the failure of drawing parts, so its prediction with more sophisticated behaviour models is an active field. Using elastic-plastic behaviour models is not sufficient to predict the final failure of sheet components, although the localisation zones are often well located in finite element simulations. To improve the prediction capability, it is important to consider in the behaviour models, the degradation of the material properties due to damage and to use a criterion for plastic flow instability. For this purpose, an elastic-plastic model coupled with the classical damage model of Lemaitre and Chaboche [1] is developed. In the elastic-plastic constitutive laws, two different hardening models are considered to reproduce more accurately the work-hardening phenomenon at moderate and large strain. The first one is the classical cyclic hardening model of Chaboche [1] and the second one is the dislocation-based microstructural model of Teodosiu and Hu [2]. These two models reproduce the transient hardening phenomena at strain path change; a better description is nevertheless obtained with the microstructural model. The coupling with the damage is carried out in the frame of continuum damage mechanics thanks to the introduction of a scalar variable giving the degradation of the elastic proprieties. In order to define the critical value of damage during sheet forming, the Rice’s localisation criterion [3] is introduced giving the limit of formability of sheet metals. The coupled elastic-plastic damage model and the Rice’s criterion are implemented in the Abaqus software. Simulations of rheological tests are realised on two steel grades, a mild steel and a dual phase steel, showing a very different softening behaviour when damage occurs. Indeed, the flow stress decreases more rapidly for the dual phase steel at the final stage than for the mild steel. To show the capability of this model to predict the strain localisation on industrial applications, the Nakazima tests are considered and computational results are discussed.

References [1] J. Lemaitre, J. L. Chaboche, Mécanique des matériaux solides. Editions Dunod, Paris, 1988. [2] C. Teodosiu, Z. Hu, Evolution of the intragranular microstructure at moderate and large strains: Modelling and computational significance. Simulation of Material Processing: Theory, Methods and Applications. Shen & Dawson editors, 173-182, 1995. [3] J. Rice, The localisation of plastic deformation. Theoretical and Applied Mechanics, 1976.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Symulation of Hydroforming on Tailor-Welded Tubular Blanks Using solid-shell Finite Elements André P. Roque1, R.M. Natal Jorge1, M.P.L. Parente1, R.A.F. Valente2, A.A. Fernandes1 1 IDMEC – Pólo FEUP Faculdade de Engenharia da Universidade do Porto Rua Dr. Roberto Frias, 4200-465 Porto {andré.roque,rnatal,mparente,[email protected] 2

Dep. Mechanical Engineering, University of Aveiro Campo Universitário de Santiago, 3810-193 Aveiro [email protected]

ABSTRACT Tube hydroforming is nowadays widely used in automotive and aircraft industries. The technology presents strong advantages when compared to conventional stamping forming. Some of them include: less parts in process and, therefore, less operations and lower costs; the use of more complex geometries, with implications on reducing weight and increasing mechanical properties; the need for fewer secondary operations; the obtaining of reduced dimensional variations; better surface finish due to the reduced friction and, finally, reduced scrap [1]. Nevertheless, the process also has some disadvantages, such as: the larger cycle time for some components; the need for relatively expensive equipment and the lack of extensive knowledge base for both process and tool designs. In this sense, the use of a reliable computational-based design tool can be highly desirable. Tailor-welded blank processes, on the other hand, consist on two or more sheets that have been welded together in a single plane prior to forming [2]. The sheets can be identical or, alternatively, can have different thickness, mechanical properties or surface coatings, leading to a versatile production process, from the mechanical properties standpoint. This type of blanks has several advantages, namely the low cost; the reduction in part weight and the flexibility in mass production. Numerical investigations of tubular hydroforming processes, starting from tailor-welded tubes based on one butt welded joint, have been carried out [3]. The need to understand this process, mainly to reduce simulation time and to accurately predict the process, leads to the present study. For this purpose, using solid-shell finite elements [4], the inclusion of the geometry of the weld line will be studied.

References [1] M. Ahmetoglu, K. Sutter, X. J. Li, T. Altan, Tube hydroforming: current research, applications and need for training, J. of Mat. Proc. Technology, 98, 224-231,2000. [2] Ana Reis, Pedro Teixeira, J. Ferreira Duarte, Abel Santos, A. Barata da Rocha, A.A. Fernandes, Tailored welded blanks––an experimental and numerical study in sheet metal forming on the effect of welding, Computers and Structures, 82, 1435-1442, 2004. [3] R.M. Natal Jorge, A.P. Roque, M.P.L. Parente, R.A. Fontes Valente, A.A. Fernandes, Simulation of Tubular Hydroforming, The Fourth International Conference on Engineering Computational Technology, Lisbon, Portugal, 2004. [4] R.A. Fontes Valente, R.J. Alves de Sousa, M. Parente, R.M. Natal Jorge, J.M.A. César de Sá, J.J. Grácio, Enhanced Assumed Strain Shell and Solid-Shell Elements: Application in Sheet Metal Forming Processes, Numiform 2004, Ohio, USA, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Fast method to predict an earing profile based on Lankford’s coefficients and yield locus T. Lelotte*, L. Duchêne†, A.M. Habraken* *

Department of Mechanics of materials and Structures, University of Liege Chemin des Chevreuils 1, 4000 Liège, Belgium thomas.lelotte, [email protected] † COBO department, Royal Military Academy Avenue de la Renaissance 30, 1000 Brussels, Belgium [email protected]

ABSTRACT Appearance of an earing profile during the circular cup deep-drawing of a metal sheet is due to the anisotropy of the material. FEM simulations of deep-drawing process can predict an earing profile but it takes a long computation time and a lot of memory storage. This article presents a texture based method to calculate Lankford’s coefficients for different orientations from the rolling direction (RD) of the metal sheet. It is very fast and, knowing that the variation of the height of the cup at the end of the deep-drawing is linked to the variation of the Lankford’s coefficient by Yoon’s formula [1], we can predict the earing profile appearing during a deep-drawing. We will make three simulations with a steel sheet and compare the results. x FEM simulations of tensile tests are done using a texture based constitutive law (MINTY [2]) without texture updating. This law is implemented in the finite element code LAGAMINE developed at the M&S department. Seven simulations are done, each 15° from 0° to 90° from RD. For each simulation, the Lankford’s coefficient is calculated by the ratio r = İTD/İND where İTD is the deformation along the transverse direction of the sheet and İND is the deformation along the thickness direction. Using Yoon’s formula, we can deduce the earing profile. x FEM simulation of the deep-drawing test is done using the same constitutive law. At the end of the deep-drawing process, the height of the cup for different angles from RD is provided. x We have developed a program to represent some cuts of the yield locus computed from the initial texture of the material. Each cut is in the ı11 – ı22 plane or in a plane parallel to it. The principle will be described in the article. On these curves, the points representing the tensile tests for any angle from RD can easily be identified. The normals of the yield locus at these points are related to the Lankford’s coefficients corresponding to these orientations from RD. We can transform these coefficients into cup heights by Yoon’s formula and finally obtain the earing profile. This approach is much faster than the two others and requires less memory storage. The goal of the present article is to explain how works the texture method (third one) and validate it by comparison with results of the two other methods and experimental tests on a steel sheet.

References [1] J.W. Yoon, F. Barlat, R.E. Dick, M.E. Karabin, Prediction of six or eight ears in a drawn cup based on a new anisotropic yield function, Int. Journal of Plast., 22, 174-193, 2006. [2] A.M. Habraken, L. Duchene, Anisotropic elasto-plastic finite element analysis using a stress strain interpolation method based on a polycrystalline model, Int. Journal of Plast., 20, 1525-1560, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Possibilities of application of the multi scale strain localization CAFE model L. Madej*, A. Zmudzki*, P.D. Hodgson†, M. Pietrzyk* *

Akademia Gorniczo Hutnicza UST, Kraków, Polska lmadej, azmudzki, [email protected] †

Deakin University, Victoria, Australia [email protected]

ABSTRACT Complex information about material behavior during deformation can be achieved by using finite element numerical simulations based on a realistic rheological model. However, when more complicated deformation process is being modeled, several phenomena, which are not accounted for in the reological model, can occur inside the sample. It is the reason why on going research in several laboratories focus on development of advanced rheological models, which are capable of including stochastic and discontinuous processes occurring in materials under thermo mechanical conditions. One of the solutions may be the rheological model based on the combination of the Cellular Automata (CA) technique and the Finite Element (FE) method [1]. Developed multi scale CAFE model for prediction of initiation and propagation of micro shear bands and shear bands in metallic materials subjected to plastic deformation is presented in this work. Such model was applied to real industrial processes, such as extrusion and rolling, and has proved its good predictive capabilities (Figure 1). As it is seen in the figure, in the CAFE approach a material flow in the area of high strain localization is higher than in the conventional FE approach. It is due to the simultaneous development of micro shear and shear bands in the material. It agrees well with the experimental observatios.

Figure 1. Strain distribution obtained from the FE (right) and CAFE (left) approach in the extrusion process.

References [1] L. Madej, J. Talamantes-Silva, I.C. Howard, M. Pietrzyk, Modeling of the initiation and propagation of the shear band using the coupled CAFE model. Arch. Metall. Mater., 50, 563-575, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A 3D Arbitrary Lagrangian Eulerian formulation for the numerical simulation of forming processes. Application to cold roll forming and metal cutting. R. Boman, L. Papeleux, Q.V. Bui, J.-P. Ponthot University of Li`ege LTAS-MC&T, Chemin des Chevreuils, 1, B4000 Li`ege, Belgium [email protected] ABSTRACT The paper deals with the numerical simulation of forming processes by the finite element method. In order to find the solution of steady state processes by numerical simulation with the classical Lagrangian formulation, very large and useless meshes have to be considered. For example, when dealing with rolling simulation, a large part of the sheet has to be discretized even if the results in the first finite elements, which are introduced between the rolls, are not important. However, these finite elements cannot be removed because they are required in order to reach the steady state solution. Consequently, the CPU time is very large. Another approach is the well-known Eulerian formulation: the media flows through the mesh, which is fixed in space. However, boundary conditions are rather difficult to handle particularly frictional contact and free surfaces. As an alternative, the Arbitrary Lagrangian Eulerian (ALE) formulation was introduced to overcome these problems, see e.g. [1] for a general reference. The mesh can be automatically handled by the software, irrespective to the material motion, so that both previous classical formulations can be obtained as particular cases if the mesh sticks to the body or is fixed in space. Recent developments in our implementation of the ALE method will be presented, including some original 3D aspects of the formulation. To illustrate the possibilities of the formulation, some 3D applications will be presented, including metal cutting and cold roll forming simulations. In order to validate the reliability of the simulations, numerical results will also be compared to physical results, including springback simulation, a major concern for cold roll forming of high-strength materials.

References [1] J. Don´ea, A. Huerta, J.P. Ponthot & A. Rodriguez-Ferran, Arbitrary Lagrangian-Eulerian methods. Chapter 14, Vol. 1: Fundamentals of Encyclopaedia of Computational Mechanics, Edited by E. Stein, R. de Borst and T.J.R. Hughes, Wiley & sons, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A study for the Constitutive Equations of 1.4021 and 1.4841 Stainless Steels in Hot Deformation M. Poursina*, H.Ebrahimi† and J.Parvizian† * Assistant Prof. Mechanical Engineering Group, Faculty of Engineering, University of Shahre-Kord, Iran. [email protected] † PhD Student, Assistant Prof. Mechanical Engineering Department, Amir-Kabir University, Iran. Department of Industrial Engineering, Isfahan University of Technology, Isfahan, Iran, [email protected], [email protected]

ABSTRACT Analysis of deformation processes depends on many factors such as initial shape of material and dies, rate of deformation, temperature, constitutive equation, friction and so on. In this study, constitutive equations were developed for 1.4021 and 1.4841 stainless steels and the best one was selected. Compression test was used to depict the flow stress behavior of material vs. strain. These tests were done by Dilatometer. Accuracy of results depends on the specimen geometry and lubricants in Dilatometery test. Upper bound method was used to estimate the coefficient of friction between dies and work piece. Homogenous compression is accomplished by eliminating friction at the contact surfaces or changing the initial shape of the work piece. Under homogenous compression conditions, height reduction and resulting radial and circumferential expansion are uniform throughout the test specimen. Furthermore, under these conditions, radial and circumferential stresses are zero, and the only stress acting is the uniform compressive stress in the axial direction. In this study work piece has a cylindrical shape with 5 mm diameter and 7.5 mm height and two ends were tapered with an angle 22°. Other initial shapes of work piece can be used are cylindrical, cylindrical with 16° conic angle and cylindrical with Rastegaev model. Several constitutive equations were used to describe the flow stress curves. The validity of selected constitutive equations was verified by comparing the free boundary of material after Dilatometery test and numerical simulation of the compression test using Msc.Super forge software.

References [1] P. Haupt, Continuum Mechanics and Theory of Materials, Germany, 2000. [2] T. Belytschko, K. L. Wing, and M. Brian, Nonlinear Finite Elements for Continua and Structures, John Wiley, England, 2000.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

545

Modelling of Spread and Side-Form Function in Hot Rolling by Different Upper-Bound Approaches W. Rasp*, A. Yusupov† *

Max-Planck-Institut für Eisenforschung Max-Planck-Str. 1, 40237 Düsseldorf, Germany [email protected] † Max-Planck-Institut für Eisenforschung Max-Planck-Str. 1, 40237 Düsseldorf, Germany [email protected]

ABSTRACT Physically based models for predicting the spread behaviour in hot rolling are developed. The energy minimisation method the upper-bound technique is used to take advantage of an analytical formulation of the problem and its fast calculation compared to numerical methods like the finite-element method. It is shown that the formulation of the velocity field, the equation for the transition function from rollgap entry to exit, and the accurateness of the consideration of the boundary conditions for the velocities influence the result considerably. Moreover, the model shall fulfil the practical needs in industry. Here, the conditions of a roughing stand are of highest interest. This means that with several passes the roll-gap aspect ratio and the reduction are varied. Additionally the model shall describe the situation of different thickness-width ratios. The application of this model to industrial hot rolling demands a sufficiently good assumption for the flow stress with respect to the material and the strain, strain rate and temperature. In this paper the highest stress is laid on the comparison of the different approaches. Velocity discontinuities at the entry to the plastic zone as well as at the exit should be avoided or at least it must be judged how big their influence is. First attempts are carried out by introducing a weighted function in the equation for the longitudinal velocity. All results are compared to experiments performed in the laboratory mill of Max Planck Institute for Iron Research. A special attention is laid on a good accordance of the computer results with those gained in an industrial plant.

References [1] W. Rasp, A. Yusupov, A Newly Developed Upper-Bound Approach for Calculating the Width Flow in Hot Rolling. steel research international 1/2, 99-105, 2005 [2] W. Rasp, A. Yusupov, Upper-Bound Model for Prediction of Spread and Side-Form Function in Hot Rolling and its Experimental Verification. STEEL GRIPS 3, 120-125, 2005 [3] W. Rasp, A. Yusupov, 3D Modelling of Metal Flow in Hot Rolling. Proc. VIII Int. Conf. on Computational Plasticity (COMPLAS VIII), Barcelona 392-395, 2005

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Non-local models and length scale effects on metal forming processes José M. A. César de Sá 1, Cai Zheng 2 1

FEUP- Faculty of Engineering of the University of Porto Rua Dr. Roberto Frias, s/n, 4200-465 Porto, Portugal [email protected] 2

IDMEC-Institute for Mechanical Engineering Faculty of Engineering of the University of Porto Rua Dr. Roberto Frias, s/n, 4200-465 Porto, Portugal [email protected]

ABSTRACT The classical theories of plasticity or continuum damage mechanics do not include scale effects associated to a characteristic length of the material, which is often present in deformation processes, and which results in a limitation of the localization of the plastic deformation or damage in narrow bands. The numerical models based in these theories suffer from size and orientation pathological dependence of the geometrical discretization and are unable to accommodate that limitation of the localization. This is true for both the classical finite element method as well as the novel meshless methods, for example. In fact the homogenization hypotheses usually admitted for the continuum and that are the basis of those numerical methods do not accommodate large gradients of the internal variables like the plastic deformation or damage in the localization zones and therefore second order constitutive laws are needed. Length scale effects can be explained by micro mechanics theories but their numerical implementation is still computational very expensive. One way of bridging the gap between the classical continuum s and micro mechanics theories is the recourse to non-local theories. In the nonlocal approaches the history of the internal variables at a material point is influenced by the surrounding material points. Non-local counterparts of those variables are then defined by integral averaging techniques or by including gradients of the internal variables in the formulation. These nonlocal formulations can also be generalised to deal, for example, with micro forming processes where geometric and microstructure size effects are present. The softening induced by the standard implementation of damage models in finite element solutions leads to mesh and orientation dependence, as the localization effects are not correctly dealt with by mesh refinement. Non-local models are a solution to solve this pathological behaviour. They include some length scale information, related with localization effects due to microstructure heterogeneity, to average an internal variable associated with dissipative process. Two types of models are usually adopted for this purpose: integral and gradient models. In this work, non-local damage models, in which the internal dissipative non-local variable adopted is the damage, are implemented. Damage definition is based on the Lemaitre´s work, which is enhanced by the introduction of a crack closure scalar parameter, allowing to treat differently the damage evolution in either traction or compression stress states. The idea is to compare the performance of some of those models and to assess the performance of some novel methods, particularly meshless methods, that some claim that may solve the numerical pathologies referred above. Comparisons are performed with non-local integral, implicit and explicit gradient methods implemented in two different frameworks, namely the Finite Element Method and the Reproducing Kernel Particle Method.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

547

Analysis of Thermal Cracking of an Industrial Duct Using Finite Element Simulation M. Salmani Tehrani * and M. Poursina† *

PhD Student. Department of Mechanical Engineering, Isfahan University of Technology, Iran. salmanitehrani@ me.iut.ac.ir † Assistant Prof. Mechanical Engineering Group, Faculty of Engineering, University of Shahre-Kord, Iran. [email protected]

ABSTRACT Thermal stresses can play an important role in failure of mechanical parts, when the body is nonuniformly warmed with some portions of high gradient of temperature variation. Thermal cracking is a special case in which cracks are opened and grown in the body, as a result of thermal stresses. These ducts, which are used in a steel complex factory, transfer the exhaust gas of induction furnaces to the duster. The temperature of the exhaust gas is changed during each period of operation. Therefore the ducts are warmed because of high temperature of exhaust gas, but not uniformly due to precipitation of dusts inside them. After eight years of putting into operation, some peripheral or hoop cracks were appeared on the surface of the ducts. Because of low pressure of exhaust gas, thermal stresses were taken into consideration. A thermo-vision camera was used to record the distribution of temperature on the surface of the duct. It was observed that most of the duct was highly warmed, while a small zone with lower temperature was evident, under the precipitated dusts. For applying to the finite element model, an analytical expression was fitted on recorded data. Making use of ANSYS software, a static analysis was done with thermal loading to compute resulted thermal stresses distribution on the duct surface. The results of simulation are shown that: a) Maximum axial stress was greater than maximum peripheral stress; b) The maximum axial stress was located on the inner surface of duct, near the verge of low-temperature zone; c) The maximum axial stress was greater than endurance limit. Since the maximum temperature of exhaust gas varies during each operation period, these results confirm that peripheral cracks may occur under high-cycle fatigue. Furthermore, it was reported that cracks were appeared on the inner surface near the verge of precipitated dusts. This also agrees with the location of maximum axial stress in simulation results.

References [1] S.P. Timoshenko and J.N. Goodier, Theory of Elasticity. McGraw-Hill Book Company, New York, 1970. [2] R.Kuguel, “A Relation between Theoretical Stress Concentration Factor and Fatigue Notch Factor Deduced from the Concept of Highly Stressed Volume” Proc.ASTM, Vol. 61, pp.732-748, 1961.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Analysis of forming processes with efficient finite element procedures Marco Schwarze∗ , Stefanie Reese Institute of Mechanics Technical University of Braunschweig, Germany {m.schwarze, s.reese}@tu-bs.de ABSTRACT Forming technologies are widely used in the manufacturing processes of industries. The numerical simulation of such processes makes high demands on the finite element technology. Element formulations which do not show the undesirable effect of locking in the cases of nearly incompressible material behaviour like during the plastification and in large deformations with extreme bending, are required. Unfortunately classical low order isoparametric element formulations show the effect of locking. An underestimation of the deformation associated with an overestimation of the stress state can be observed. To overcome this problem several autors [1, 4] propose finite element formulations based on the concept of reduced integration with hourglass stabilization by using the enhanced strain method. Especially for the efficient numerical simulation of sheet forming processes so-called solid-shell elements are developed [2, 4]. The starting point of the present formulation is the same three-field variational functional on which many three-dimensional enhanced strain concepts are based. A new aspect is the Taylor expansion of the first Piola-Kirchhoff stress tensor with respect to the normal through the center of the element. Together with a constant Jacobi matrix due to the computation in the center of the element this concept leads to a powerful element formulation with only two Gauss points over the thickness. Furthermore continuum mechanical laws can be implemented without additional assumptions about the kinematics or the stress state. A further important aspect is the modelling of the contact behaviour. Advantageous is that the extension of the structure in thickness direction is taken into account by the solid-shell formulation. However working with contact surfaces discretized by low order finite elements leads to discontinuities of the normal vector and to a non-smooth sliding [3]. That is why a smoothing of the surface description has to be implemented. For the efficient and user-friendly analysis the solid-shell formulation has been implemented into the commercial code ABAQUS. Examples for comparison of the numerical results with realistic forming processes are given.

References [1] T. Belytschko & L.P. Bindemann, Assumed strain stabilization of the eight node hexahedral element. Computer Methods in Applied Mechanics and Engineering, 105, 225–260, 1993. [2] R. Hauptmann, K. Schweizerhof & S. Doll, Extension of the solid-shell concept for application to large elastic and large elastoplastic deformations. International Journal for Numerical Methods in Engineering, 49, 1121–1141, 2000. [3] P. Wriggers, L. Krstulovic-Opara & J. Korelc, Smooth C1-interpolations for two-dimensional frictional contact problems. International Journal for Numerical Methods in Engineering, 51, 1469– 1495, 2001. [4] S. Reese, A large deformation solid-shell concept based on reduced integration. submitted, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

549

Numerical investigation of fracture onset in sheet metal forming P. Teixeira *, F.M. Andrade Pires † , A.D. Santos †, J. César de Sá † *

INEGI- Institute for Mechanical Engineering and Industrial Management Rua do Barroco 174, 4465-591 Leça do Balio, Portugal [email protected] †

FEUP- Faculty of Engineering of the University of Porto Rua Dr. Roberto Frias, s/n, 4200-465 Porto, Portugal [email protected]

ABSTRACT Sheet metal forming processes involve finite inelastic strains that are mainly restricted by the occurrence of strain localization and instability due to necking. Therefore the ability to predict the formability limit is of paramount importance in order to optimize the process, examine the influence of each parameter on the necking occurrence and consequently to improve the press performance. Forming limit diagrams, associated with finite element simulations are currently performed by powerful commercial codes. Nevertheless, when complex strain paths are involved these predictions may fail to give the right answer. Localization is a result of internal degradation described throughout metallographic observations by the nucleation, growth and coalescence of voids and micro-cracks, that may lead to macroscopic failure. The fact that internal degradation (or damage) accompanies large plastic deformation suggests that these two dissipative processes, although different in nature, influence each other and should, therefore, be coupled at the constitutive level. Consequently the theory of Continuum Damage Mechanics (CDM) may provide a better insight to the physical phenomenon and play a significant role in the study of the formability and fracture onset in sheet metal forming processes. In the framework of CDM the damage variable represents, in an average sense, the effect internal degradation which is reflected on the local load carrying capacity of the material and its evaluation may play an important role in the definition of a criterion for predicting localized necking and fracture initiation. In this work Lemaitre’s ductile damage model is implemented in the Abaqus/Explicit code system in order to compare the performance of this coupled approach with the traditional use of forming limit diagrams. These are usually employed as an a posteriori analysis of the finite element solution in which the necking phenomenon is carried out in the framework of Marciniak-Kuczinsky (M-K) analysis coupled with the conventional theory of plasticity. The previous strategies are compared with a set of experimental results of sheet forming of rails with different stamping conditions.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

550

Effect of Surface Geometry and Local Mechanical Environment on Periimplant Tissue Differentiation: A Finite Element Study. Andriy Andreykiv∗ , Fred van Keulen∗ ∗ Delft

University of Technology, The Netherlands Mekelweg 2, 2628 CD, Delft, The Netherlands [email protected], [email protected] ABSTRACT Implant surface geometry, the amount of the soft tissue at the bone-implant interface and the type of loading can influence the course of bone ingrowth processes. These factors need to be taken into account during the design of a new prosthesis coating. However even animal experiments can not give a good insight into the environment at the bone-implant interface. One of the ways to understand why certain coatings perform better than the others is to perform a numerical simulation of the bone ingrowth process. Detailed three-dimensional finite element models of the interface tissue adjacent to the implant surface are developed (Fig.1). Two types of implant surfaces are considered: a smooth surface and a surface covered with sintered beads. A numerical model for tissue differentiation is developed. The model simulates migration of mesenchymal cells and fibroblasts, proliferation, differentiation and replacement of cells and production of tissues depending on the local mechanical environment. The model is tested on a bone fracture healing application. The main assumption of the present study is that the bone ingrowth process can be modelled the same way as bone fracture healing. A set of boundary conditions, aiming at replicating the mechanical environment at the interface, was applied to the models. In order to make the models representative for a large area of the interface, symmetric and periodic boundary conditions were applied. The results of the simulation show a Figure 1: Derivation of the geometry and FE mesh of the higher rate of bone ingrowth into the sur- interface representative volume element face with porous coating comparatively to the smooth surface. It is also shown that a thicker interface does not increase the chance of the fixation failure, as the local strain of the periimplant tissue for the thicker interface is lower than for the thin interface.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Identification of bone structure from effective measurements Carlos Bonifasi-Lista∗ , Elena Cherkaev† ∗ University of Utah, Department of Bioengineering 155 South 1400 East, JWB 233, Salt Lake City, UT 84112 [email protected] ah.edu † University of Utah,Department of Mathematics 155 South 1400 East, JWB 233, Salt Lake City, UT 84112 [email protected]

ABSTRACT The talk deals with the recovery of the microstructural parameters characterizing the microgeometry of bone from measured effective torsional complex modulus. We assume that the bone is modeled as a porous cylinder filled with a viscoelastic material as in case of trabecular porous bone filled with bone marrow. The torsional moments are applied at the opposite ends of the cylinder. We assume that the microstructure is finely scaled. The response to the harmonically oscillating moments can be calculated using homogenized equations [1]. The effective torsional modulus µ∗ has a Stieltjes analytic representation [1,3] with the spectral measure ν. Structural information about the microgeometry is incorporated in the moments νn of the measure ν. For instance, the volume fraction of one of the components is given by the zero-order moment ν0 . Hence, reconstructing ν0 we can estimate the porosity of the bone, which is one of the most important structural parameters for osteoporosis monitoring. We show that the reconstruction of the spectral function ν has a unique solution, however, the reconstruction problem is very ill-posed [2, 3]. We discuss several approaches to regularization of the inverse problem for the function ν based on different classes of constraints, such as quadratic , total variation, and non-negativity constraints. Reconstructed function ν is used to recover geometric information about the bone structure. We show results of computation of the function ν f rom the effective shear modulus for several examples of the microstructure, and use the first moment ν0 to evaluate the bone porosity.

References [1] S. Tokarzewski, J.J. Telega, and A. Galka, Torsional rigidities of bone filled with marrow: The application of multipoint Pade approximants, Engineering Transactions, 49, 135–153, 2001. [2] E. Cherkaev, Inverse homogenization for evaluation of effective properties of a mixture, Inverse Problems, 17, 1203–1218, 2001. [3] C. Bonifasi-Lista and E. Cherkaev, Identification of bone microstructure from effective complex modulus, submitted.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Computational modeling of mechanical environment within tissue engineered cartilage Margherita Cioffi, Fabio Galbusera, Manuela T. Raimondi, Federica Boschetti, Gabriele Dubini Laboratory of Biological Structure Mechanics, Politecnico di Milano Piazza Leonardo da Vinci 32, 20133 Milano (Italy) [email protected]

ABSTRACT Bioreactors allowing culture medium perfusion overcome diffusion limitations associated with static culturing, and provide flow-mediated mechanical stimuli. Currently, one of the main issues in bioreactors design is to build a set up which may provide the required nutrients, optimizing the mechanical stimulation on the cells. However, in most of the direct-perfusion systems currently available, the hydrodynamic environment imposed to cells is neither characterized nor controlled. The former target of this study was to numerically predict the hydrodynamic environment which develops in a microporous polymeric scaffold (Degrapol®), during the first period of perfused culture experiments. The latter aim was to develop and validate a methodology which will allow to characterize the mechanical stimulation on the cells also in the last period of the culture, when extracellular matrix may grow. To address the first target, two CFD models were developed: the first one (Model 1) was built from micro-computed tomography reconstruction of the actual scaffold geometry while the second one (Model 2) was based on a simplification of the actual scaffold microstructure. Both the CFD models were studied using the commercial finite-volume code Fluent. The two models showed comparable results in terms of the distribution of the shear stresses acting on the inner surfaces of the scaffold walls. Model 1 and Model 2 gave a median shear stress of 3 mPa, at a flow rate of 0.5 cm3 min-1 through a 15 mm diameter scaffold. A third model (Model 3), based on the simplified scaffold geometry employed for Model 2, was built using the Lattice-Boltzmann method. For the same boundary conditions applied to Model 1 and Model 2, Model 3 produced results in terms of wall shear stress comparable with those obtained with the finite-volume code. A preliminary tissue growth law, simulating the ECM growing inside the scaffold during the culture, was also added to Model 3. Computational modeling can be successfully used to quantify, on a microscopic level, the shear stress artificially applied to cells in three-dimensional engineered cell systems in bioreactors. Our results provide a basis for the completion of more exhaustive quantitative studies to further assess the relationship between perfusion, at known micro-fluid dynamic conditions, and tissue growth in vitro.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

553

Stress-Based Optimum and Bone Architecture Hyunsun Kim*, Paul J. Clement*, James L. Cunningham* * University of Bath Department of Mechanical Engineering, Bath BA2 7AY United Kingdom [email protected]; [email protected]; [email protected]

ABSTRACT The well defined internal architecture of bone is thought to represent an optimum biological response to its mechanical environment. What is not clearly established however, is to what problem it is an optimal response. A common understanding is that the mechanical loading environment is a dominant factor that influences bone architecture. Many of the bone remodelling algorithms therefore employ the material strain as the stimulus, e.g. [1] and efforts have been seen in understanding the bone remodelling by applying structural optimisation, e.g. [2]. The structural optimisation problem is often defined as minimisation of the total compliance subject to a volume constraint, and considerable research efforts in recent decades have developed topology optimisation methods based on a discretised elastic finite element formulation. The continuum topology optimisation problem is often relaxed by using a continuous material property as the design variable, which is penalised by a power-law to obtain a 0/1 discrete solution. Whilst this mathematically rigorous approach provides an optimum solution with some confidence, the specification of such a problem requires a volume constraint which is often difficult to define. An alternative approach is Evolutionary Structural Optimisation (ESO) based on the fully stressed design concept. Unlike the minimum compliance design which uses the local strain energy density as the sensitivity, ESO uses Von Mises stress as the local response to modify the structure, and iteratively adds and removes finite elements to minimise the volume and achieve an optimum topology [3]. In this paper, we investigate the optimality of bone’s internal architecture by applying stress-based ESO. Bones in a relatively simple mechanical environment such as the os-calcis (heel bone), were used as test problems. The solution is compared with x-ray and CT images and is demonstrated to agree favourably with the bone’s typical trabecular orientations. The optimality of bones is further investigated by applying a perturbation during optimisation and comparing the optimum topology with in-vivo experimental results from a model in which the same perturbation was applied. The findings indicate that there may exist more than one equilibrium state for the constant mechanical conditions, and structural optimisation can be used to investigate the bone remodelling mechanism.

References [1] L.E. Claes, C.A. Heigele, Magnitudes of local stress and strain along bony surfaces predict the course and type of fracture healing. Journal of Biomechanics, 32, 255-266, 1999. [2] J.E. Taylor, Optimal modification and evolution of elastic continuum structures. IUTAM Symposium and Synthesis in Bio Solid Mechanics, 115-128, 1999 [3] H. Kim, O.M. Querin, G.P. Steven, Y.M. Xie, Improving efficiency of evolutionary structural optimisation by implementing fixed grid mesh. Structural and Multidisciplinary

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Parameterized orthotropic cellular microstructures as mechanical models of cancellous bone Piotr Kowalczyk Institute of Fundamental Technological Research, Polish Academy of Sciences wi˛etokrzyska 21, 00-049 Warsaw, Poland [email protected]

ABSTRACT Constitutive properties of cancellous bone depend on microstructural geometry. Evaluation of the interrelationship is a crucial issue in analysis of stresses and strains in bone tissues and simulation of their remodelling. Known limitations of experimental methods as well as of the micro-FE techniques make the analysis and homogenization of ‘equivalent’ trabecular microstructures an advantageous tool for this task. In this study, parameterized orthotropic constitutive models of cancellous bone are derived from finite element analysis of repeatable microstructure cells. The models are fully three-dimensional, have realistic curvilinear shapes and are parameterized with four shape parameters. Variation of the parameters allows to imitate most of the typical microstructure patterns observed in real bones, along with variety of intermediate geometries. The models are a geometrically enhanced version of those presented in the author’s previous work [1]. Static numerical tests are performed with the finite element method for an exhaustive number of parameter value sets (microstructure instances). Repeatability of the microstructure allows to test only a sigle cell with appropriate boundary conditions. Values of computed stresses and strains allow to determine all coefficients of elastic orthotropic stiffness matrix. Results have a form of tabularized functions of elastic constants versus the shape parameters. Comparison of the results with micro-FE data obtained for a large set of cancellous bone specimens [2] proves a good agreement.

References [1] P. Kowalczyk, Elastic properties of cancellous bone derived from finite element models of parameterized microstructure cells. Journal of Biomechanics, 36, 961–972, 2003. [2] B. van Rietbergen, R. Huiskes, Elastic constants of cancellous bone. In: Cowin S.C. (Ed.), Bone Mechanics Handbook, 2nd Ed., Chapter 15. CRC Press, 2001.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Modeling of cancellous bone surface adaptation based on the 3-dimensional trabeculae topology evolution Michał Nowak Poznan University of Technology ul. Piotrowo 3, 60-965 Pozna´n, Poland [email protected]

ABSTRACT In the paper the computer simulation model of the trabecular bone surface adaptation is presented. The base of algorithm formulation was the bone remodeling phenomenon leading to optimization of the trabecular network in the bone [1] ruled by the the strain energy density. In contrast to approaches used so far, the system proposed in this study mimics the real 3-dimensional bone geometry evolution, where not only the volumetric Finite Element Method mesh, but also the surface of trabecular network is controlled during the simulation. The computer system assumptions were presented on the ICTAM Congress last year [2]. In contrary to other voxel models, the remodeling phenomenon is simulated exactly on the analyzed structure without recalling the voxels. Adaptation to the mechanical stimula-tion results in adaptation of the surface position in the virtual space. In Figure 1 the remodeling simu-lation of the trabecular bone sample under compression is depicted.

Figure1. Remodeling simulation - bone sample under compression. From the left to the right side structure adaptation. Such approach allows to mimic in details the real biological process of bone formation and resorption. The model including the 3-dimensional trabeculae topology is simpler and closer to the real trabecular bone properties. The complex mesh control during the simulation eliminate the difficulties with the Finite Element Method computations. The possibility of use of the system in mechanical design is discussed. Some results of computations are presented.

References [1] R. Ruimerman, Modeling and remodeling in bone tissue. Eindhoven, ISBN 90-386-2856-0, 2005. [2] M. Nowak , M. Morzy´nski, Simulation of Trabecular Bone Adaptation - Creating the optimal structure. ICTAM04 Proceedings, p.360, 21st Congress of Theoretical and Applied Mechanics, August 15-21, Warsaw, Poland, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Theoretical Analysis of Remodelling Processes in Bony Tissue Engineered Implants Julien Pierre, Christian Oddou B2OA UMR CNRS 7052 & Université Paris 12, Faculté des Sciences, 61 avenue du Général de Gaulle, 94010 Créteil, France. [email protected]

ABSTRACT In order to design adequate and performing artificial tissue implants, porous biocompatible and biodegradable substrates have to be cells seeded. Moreover they have to be submitted to perfusion inside a bioreactor, during few weeks, with a nutritive fluid carrying solute ingredients necessary for the active cells to grow, proliferate, differentiate and produce extra-cellular matrices. From the understanding and control of the processes leading to the substrate degradation and extra cellular matrix remodelling taking place during the in vitro culture phase, depends widely the success in the realization of new orthopaedic biomaterials. Within this context, the analysis of the interactions between convective phenomena of hydrodynamic origin and chemical reaction of biological order which are associated to these processes is a fundamental challenge in the framework of the bone tissue engineering. In order to better account for the different intricate processes taking place in such a sample and to design a relevant experimental protocol leading to the definition of an optimal tissue implant, we proposed a theoretical model based on transport phenomena in porous active media. For these "opened" systems in state of permanent imbalance created by the local convection induced in the medium, the analysis of such complex interactions remains relatively coarse. The adopted approach is based on a formulation similar to that of the active porous media in which all the biochemical processes related to the cellular metabolism are described in terms of the physicochemical processes taking place at the fluid-solid interface, leading to the degradation and remodelling of the solid matrix. The model includes the effects of the complex microscopic architecture of the substrate and the specificity of the biological processes such as extra cellular matrix production and resorption. This leads to a numerical solution of the coupled fluid dynamics, transport equations and biochemical reaction inside the porous medium. The fundamental parameters on which depends the evolution of the structure of the medium are thus put in evidence and evaluated in the framework of bone tissue engineering. One of the revealed outcomes is the fact that local and minor interaction can lead to long-range imbalance and substantial changes in the initial architecture of the substrate even to engender phenomena of chaotic instability..

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Bone remodeling description based on micro mechanical/biological effects G. Sciarra∗ , T. Lekszycki† ∗ Department

† Institute

of Chemical and Material Engineering University of Rome “La Sapienza” via Eudossiana 18 00184 Rome Italy [email protected]

of Fundamental Technological Research IPPT PAN Polish Academy of Sciences ul. Swietokrzyska 21 Warsaw Poland [email protected]

ABSTRACT This paper is devoted to formulate a model for bone remodeling capable to account for the microscopical phenomena which drive resorption and rebuilt of cancellous bone tissues, by merging micro-mechanics and optimization. The behaviour of the effective bone is described by means of a coarse model which catches the main topological, mechanical and biological characteristics of the structure via a micromechanical analysis of the stress state in a properly defined Reference Elementary Volume (RVE) and provides a description of remodeling evolution according to some optimality conditions for properly defined control functions. Adopting a macroscopic approach bone remodeling arises from admissible variations of the constitutive parameters and is associated to a non-trivial dissipation inequality; our goal here is to identify the basic set of control parameters, associated to cancellous bone architecture, which affect the macroscopic constitutive relations, and to describe their time evolution according to suitable control laws. The control parameters are chosen in such a way to be related to local mechanical and biological phenomena. From the point of view of micro-mechanics the idea is to describe the architecture of the bone by suitably oriented RVEs inside which ellipsoidal inclusions aligned along the same main axes can be detected; the centers of the ellipsoid are equally spaced along the common main directions. It is therefore the distribution of orientation of the RVEs which gives the grade of anisotropy of the problem together with the shape of the inclusions. This simplified description of bone architecture allows us to account for biology, since it seems almost well grounded that remodeling occurs on the surface of the inclusions; on the other hand no periodicity of the structure is considered, which means that no homogenization techniques are going do be developed. Every RVE is uniformly strained (or stressed) through its boundary by the corresponding macroscopic fields; this means that the characterization of the microscopic state of strain in the RVE (micro problem) provides the macroscopic constitutive law when identifying the average strain energy of the RVE with the energy associated to the average strain. In the description of the micro problem we assume the material to be wherever spread inside the RVE with a non-homogeneous distribution of the constitutive parameters: the RVE is replaced by a kind of grey material whose constituive coefficients are not uniform over the domain but fit the effective spatially discontinuous distribution of the elastic moduli which characterized the effective RVE. The size of the inclusions, their spatial distribution inside the RVE and the orientation of the RVE itself are regarded as the control parameters of remodeling which affect the behaviour of the macroscopic constitutive relations. A properly defined mechanical constraint for the fourth order elasticity tensor guarantees via an optimization procedure to determine the missing evolution equations for the control functions.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

558

Two-dimensional elastodynamic wave propagation in graded structures L. Aebi∗ , J. Vollmann∗ , J. Dual∗ ∗ Center

of Mechanics, (Swiss Federal Institute of Technology) ETH Zentrum, CLA J35 CH 8092 Zurich [email protected]

ABSTRACT A two-dimensional simulation of elastodynamic waves propagating in a continuum with graded elastic properties in one dimension is presented and the refraction/reflection properties are discussed. Graded materials or functionally graded materials (FGM) are defined as materials featuring gradual spatial transitions in microstructure and/or composition thus having gradually varying mechanical properties like Youngs moduli, shear moduli, and/or densities. In engineering applications the development of such material systems is motivated so far by the demand to produce objects which are hard and wear resistant at the surface while having a ductile tough body like cutting tools to give an example. Ceramically coated turbine blades might be mentioned as another application where FGMs are used in order to reduce thermally induced mechanical stress concentrations along the ceramic metal interface. The dynamic response to local disturbances of the equilibria in particular the elastodynamic wave propagation in graded continua is a rarely treated topic. It is of particular interest since the reflection, refraction, and transmission of elastodynamic waves is frequency dependent provided that the spatial area in which the material properties vary is in the order of the elastic wavelengths to be distinguished, a fact which opens a wide field of engineering applications like frequency filters, spectrum analyzers, or acoustic isolation layers. A finite difference method is implemented to solve the time-boundary problem numerically. The frequency dependent refraction of a plane wave front hitting a graded zone under an arbitrary angle as well as the behavior of a longitudinal wave hitting the ’soft’ acoustic interface perpendicularly are discussed. Aspects of accuracies of the numerical simulation are discussed as well. Future directions of the on-going research project are presented.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

559

A Computational Study on Functionally Graded Rotating Solid Shafts: Analysis of Preliminary Results Hakan Argeso* and Ahmet N. Eraslan† *

Department of Mechanical Engineering Baskent University, Ankara 06530, Turkey [email protected] † Department of Engineering Sciences Middle East Technical University, Ankara 06531, Turkey

[email protected]

ABSTRACT A computational model is developed for the analysis of elastic, partially plastic and residual stress states in functionally graded (FGM) rotating solid shafts. The modulus of elasticity, Poisson's ratio, uniaxial yield limit and density of the shaft material are assumed to vary radially in any prescribed functional form. Small deformations, sufficiently long shaft, and a state of generalized plane strain are presumed. Using the von Mises yield criterion, total deformation theory and a Swift-type nonlinear hardening law, a single nonlinear equation describing elastoplastic behavior of rotating shafts is obtained. A shooting technique using Newton iterations with numerically approximated tangents is designed and used for the computer solution of the governing equation. The model is verified comprehensively by comparing predictions with (i) analytical elastic solutions of homogeneous shafts and (ii) elastoplastic solutions of homogeneous shafts available in the literature.

Figure 1. Stresses in a partially plastic rotating FGM solid shaft. Using linear variation as much as 10% for each of the material parameters, the partially plastic stress state for a rotating FGM solid shaft with axially unrestricted ends and traction free surface is computed and displayed in Fig. 1. Formal dimensionless variables are used in drawing this figure. Depending on how the material properties vary in the shaft, plasticization may commence somewhere inside the FGM shaft and the plastic ring expands in both radial directions with increasing angular speeds. It is noted that this behavior is never observed for homogeneous shafts. As shown in Fig. 1, at the speed of rotation considered, the shaft is composed of a plastic region in 0 d r d rep , and an elastic region in rep d r d 1 . The plastic-elastic border rep is expected to move toward the edge if the rotation speed is increased further.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

560

Static Analysis of Thick Functionally Graded Plates by using a HigherOrder Shear and Normal Deformable Plate Theory and MLPG method with Radial Basis Functions 1

2

2,5,7

5,6,7

Romesh C. Batra , Jia-Run Xiao , David F. Gilhooley , Michael A. McCarthy John W. Gillespie Jr.2,3,4 1

2

5

and

Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA [email protected]

Center for Composite Materials, 3Department of Materials Science and Engineering, 4Department of Civil and Structural Engineering, University of Delaware, Newark, DE 19716, USA {xiaoj, gillespie}@ccm.udel.edu

Composites Research Centre, 6Materials and Surface Science Institute, 7Department of Mechanical & Aeronautical Engineering, University of Limerick Limerick, Ireland {david.gilhooley, michael.mccarthy}@ul.ie

ABSTRACT Infinitesimal deformations of a thick functionally graded elastic plate have been analyzed by using the meshless local Petrov-Galerkin (MLPG) method and the higher-order shear and normal deformable plate theory (HOSNDPT). Radial basis functions (RBF) are employed for constructing trial functions, while a spline function is used as the weighting function over a local subdomain. The present method employs a number of randomly located nodes in the domain. It does not require a mesh for either interpolation of the field variables or integration of the weak form and hence is truly meshless. Two types of RBFs, i.e. Multiquadrics (MQ) and Thin Plate Splines (TPS), are used. Effective material moduli of the plate, made of two isotropic constituents with volume contents varying only in the thickness direction, are computed using the Mori-Tanaka homogenization technique. Computed results for a simply supported plate are found to agree well with their analytical solutions. The through-the-thickness distributions of the deflection and stresses in the plate are also presented for three types of boundary conditions: simply supported, clamped and free.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

561

Three-Point-Bending and Indentation Tests for the Calibration of Functionally Graded Material Models by Inverse Analysis Massimiliano Bocciarelli, Gabriella Bolzon, Giulio Maier Department of Structural Engineering, Politecnico di Milano Piazza Leonardo da Vinci 32, 20133 Milano, ITALY {bocciarelli, bolzon, maier}@stru.polimi.it

ABSTRACT Three-point bending (3PB) and indentation tests are often employed for calibrating mechanical models of homogeneous materials. Mechanical calibration means here identification of parameters which are contained in constitutive elastic-plastic and fracture models and turn out to be not directly measurable by traditional elementary tests. Either semi-empirical formulae or inverse analysis can be used for such calibration. Inverse analysis based on suitable finite element simulations of the test allows to exploit wider sets of experimental data and to increase the number of estimated parameters and estimation accuracy. In the two calibration methods proposed and discussed in this paper, traditional experimental information consisting of loading-unloading curves (reaction force versus imposed displacement) are supplemented by further data: in the 3PB tests, by displacements measured through interferometric optical means on the lateral surfaces of the specimen like in [1]; in instrumented indentations, by the mapping of residual imprints as in [2]. The estimation of the mechanical properties dealt with herein concerns metal-ceramic functionally graded materials (FGMs), the mechanical behavior of which is described by modified mixture laws [3,4]. The indentation technique is applied to a thin FGM coating on ductile metallic substrate, in order to assess the volume fraction distribution of the components across the thickness of the FGM layer. The purpose is to establish the effectiveness and robustness arising from the use of imprint mapping data in the identification process. Parameter estimation by 3PB tests is applied to FGM layers of relatively large thickness (few millimeters of magnitude order) and concerns the phase distribution transversal to the specimen (which exhibits vertical symmetry) and the fracture properties of the components, namely tensile strength and fracture energy, consistently with a simple cohesive crack model.

References [1] G. Bolzon, G. Maier, Identification of cohesive crack models for concrete on the basis of threepoint-bending tests. In Computational Modelling of Concrete Structures (R. de Borst, N. Bicanic, H. Mang, G. Meschke eds), Balkema, Rotterdam, 301-310, 1998. [2] M. Bocciarelli, G. Bolzon, G. Maier. Parameter identification in anisotropic elastoplasticity by indentation and imprint mapping. Mechanics of Materials, 37, 855-868, 2005. [3] T. Nakamura, T. Wang, S. Sampath, Determination of properties of graded materials by inverse analysis and instrumented indentation. Acta Materialia, 48, 4293-4306, 2000. [4] Z.-H. Jin, G.H. Paulino, R.D. Dodds, Cohesive fracture modeling of elastic-plastic crack growth in functionally graded materials. Engineering Fracture Mechanics, 70, 1885-1912, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

562

Static Deformations and Natural Frequencies of Functionally Graded Plates by a Hybrid Meshless Method Ant´onio J. M. Ferreira∗ , Greg E. Fasshauer† , Carla M. C. Roque∗ , Renato M. N. Jorge∗ ∗ Departamento

de Engenharia Mecˆanica e Gest˜ao Industrial Faculdade de Engenharia da Universidade do Porto Rua Dr. Roberto Frias, 4200-465 Porto, Portugal ferreira,croque,[email protected]

† Department

of Applied Mathematics, Illinois Institute of Technology, Chicago 60616, USA [email protected]

ABSTRACT In this paper we apply a hybrid numerical scheme based on pseudospectral methods radial basis functions. This is a global collocation method that avoids the tensor-product grids of pseudospectral methods. The static deformations and the free vibration analysis of functionally graded plates is performed. We use the first and third order shear deformation theories. In functionally graded materials we seek the equivalent, homogeneized, material properties. Here the estimation of the elastic material properies is based on the Mori-Tanaka technique which presents a better estimate than the law of mixtures. We compare the present meshless method with exact solutions by Vel and Batra and an alternative meshless solution by Qian and colleagues, based on the meshless local Petrov-Galerkin method. The present radial basis function solution produces highly accurate results, particularly when the third order shear deformation theory is used.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

563

Graded Fourier p-Element Calculation of the Steady State Heat Conduction in Functionally Graded Materials S.M. Hamza-cherif*, A. Houmat†, A. Hadjoui† *

University of tlemcen, department of mechanical engineering BP 230 Tlemcen 13000 Algeria [email protected]

†

University of tlemcen, department of mechanical engineering BP 230 Tlemcen 13000 Algeria [email protected]

ABSTRACT The functionally graded materials (FGM) applications in engineering become more and more important in the future. The FGMs are composite materials where the microstructures are locally varied. The composition and the volume fraction of FGMs constituents vary gradually, giving a non uniform microstructure with continuously graded macro properties. Details of design, processing and applications of FGMs can be found in [1]. In order to be able to predict the damage and strength degradation, an understanding of the temperature distribution is important [2-3]. In this paper, a pversion of the finite element method is considered to determine the steady state temperature distribution in functionally graded materials. The graded Fourier p-element is used to set up the one dimensional heat conduction equations. The temperature is formulated in terms of linear shape functions used generally in FEM plus a variable number of trigonometric shapes functions [4] representing the internals degrees of freedoms. In the graded Fourier p-element, the function of the thermal conductivity is computed exactly within the conductance matrix and thus overcomes the computational errors caused by the space discretisation introduced by the FEM. Explicit and easy programmed trigonometric enriched conductance matrices and heat load vectors are derived by using symbolic computation. The convergence properties of the graded Fourier p-element proposed and the results of the numbers of test problems are in good agreement with the analytical solutions. Also, the effect of the non-homogeneity of the FGM on the temperature distribution is considered.

References [1] Y. Miyamoto, W.A Kaysser, B.H. Rabin, A. Kawasaki, R.G. Ford, Functionally graded materials: design, processing and applications. Kluwer Academic Publishers, 1999. [2] S.S. Vel, R.C. Batra, Three dimensional analysis of transient thermal stresses in functionally graded plates. Int. J. Solids and Structures, 40, 7181-7196, 2003. [3] B. Wang ,Y. Mai, Transient one-dimensional heat conduction problems solved by finite element. Int. J. of Mechanical Sciences, 47, 303-317, 2005. [4] S.M. Hamza-cherif, Free vibration analysis of rotating flexible beams by using the Fourier pversion of the finite element method. Int. J. Comput. Methods, 2(2), 255-269, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

564

A Reissner-Mindlin Plate Model for Functionally Graded Materials Trung-Kien Nguyen1 , Karam Sab1, and Guy Bonnet2 1 LAMI (ENPC/LCPC) 6-8 Avenue Blaise Pascal, Cite Descartes, 77455 Marne-La-Vallee, France [email protected] 2 LaM (Universite de Marne-La-Valle) 5 bd Descartes, 77454 Marne-La-Vallee, France [email protected]

ABSTRACT The concept of functionally graded materials (FGM) was proposed in 1984 by material scientists in the Sendai area of Japan. During twenty years of development, it has been efficiently applied to many industries. Moreover, it is a potential structural material for high-temperature environments. This material is composed of two or more constituents whose volume fractions change continuously as functions of position. Functionally graded plate models have been studied with analytical and numerical methods. The significant advantage of FG plate over a laminated plate is to eliminate failure modes at interfaces. Several researchers have analyzed the behavior of thick FG plates. They proposed models that take into account the transversal shear effect, by using the five sixth correction factor. However, this factor is not appropriate to the FG plate analysis because of the position dependences of the FGM properties. Number of studies used the higher-order shear deformation theory to deal with this problem. In this paper, a Reissner-Mindlin plate model for calculation of functionally graded materials is proposed. Identification of transverse shear factors is investigated through this model. The transverse shear stresses are derived by using energy considerations from the expression of membrane stresses. Using the transverse shear factor thus obtained, a numerical analysis is performed on a simply supported FG rectangular plate whose elastic properties are isotropic at each point and vary through the thickness according to a power law distribution . The numerical results of a static analysis are compared with available solutions from previous studies.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

565

A Review of the Chosen Problems of FEM Modeling of Surface Coatings Wiesław Szymczyk Military University of Technology, Faculty of General Mechanics Kaliskiego Str. 2, 00-908 Warsaw, Poland [email protected]

ABSTRACT The author performed numerical simulations of several different surface coatings with the use of FEM. Several types of materials for the coatings were investigated like thermal barriers made of Al2O3 or ZrO2 and anti wear coatings TiN, TiC or TiCN. There were simulated coatings on the parts of a combustion engine like a piston as well as a valve. The were also performed investigations for the needs of fabrication of surgical tools coated with the use of DLC – diamond like carbon. The surface coatings were established on different substrates like steel, titanium, aluminum alloys or beryllium copper. The aim of the works is to establish the methods which can be useful in analyses of influence of heat treatment (i.e. heat impulse shocks) on the stress distribution in the area of coating-substrate system and on adhesion of the coating to the substrate. On the base of these works the review of surface coatings modeling will be presented: problems concerning the analysis of residual stresses distribution in the area of coating-substrate collaboration, the influence of the substrate shape, modeling in the macro and micro scale connected with the need of use of the different types of models, analyzing the coating-substrate system as a graded or functionally gradient material – FGM, building models with or without simulation of the microstructure imperfections like voids, roughness of the substrate and the coating as well as the bonds between particular surface layers. The preliminary results obtained from micromechanical models of the ceramic FGM surface coatings with the linear function of the coating material volume fraction variation, show out the necessity of consideration in numerical simulations the residual stress relaxation mechanisms like development and joining of microcracks as well as the effect of presence and nucleation of the voids (on the way of plastic deformations). In the distance of barely several model grid domains there were observed the strong gradients of stresses which reach the extreme values from the calculated stress range.

References [1] D. Delfosse et al., Numerical and experimental determination of residual stresses in graded materials, Composites, Part B, 28B, pp. 127-141, 1997. [2] M. Grujicic & H. Zhao, Optimization of 316 stainless steel/alumina functionally graded material for reduction of damage induced by thermal residual stresses, Mat. Sci. Eng., A252, pp. 117-132, 1998. [3] T. Reiter et al., Micromechanical models for graded composite materials, J. Mech. Phys. Solids, 45(8), pp. 1281-1302, 1997.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

566

On the Optimal Scaling of Index Three Daes in Multibody Dynamics Carlo L. Bottasso∗ , Daniel Dopico† , Lorenzo Trainelli∗ ∗ Politecnico di Milano Via La Masa 34, 20156 Milano, Italy [email protected], [email protected] † Escuela

Polit´ecnica Superior, University of La Coru˜na Mendiz´abal s/n, 15403 Ferrol, Spain [email protected]

ABSTRACT Errors and perturbations due to finite precision arithmetic pollute the numerical solution of high index differential algebraic equations. This pollution causes disastrous effects for small values of the time step size. Various remedies have been so far offered in the multibody dynamics literature to this problem. All remedies point to the reduction of the index, which requires a rewriting of the equations of motion. This either increases the cost, since additional constraints and multipliers are introduced, or causes additional problems, like the drift of constraint violations. In References [1, 2], we have shown that the pollution problem can be avoided for BDF schemes by a proper scaling of the index three problem. The procedure involves both a scaling of the equations and a scaling of the unknowns. This way, we were able to achieve perfect time step size independence in the sensitivity to perturbation, as in the case of well behaved ordinary differential equations. In this paper we offer a new theoretical analysis of the perturbation problem, and we extend the scaling to the case of Newmark-type integration schemes. The predictions of the theory are confirmed by numerical experiments.

References [1] C.L. Bottasso, O.A. Bauchau, Time-step-size-independent conditioning and sensitivity to perturbations in the numerical solution of index three differential algebraic equations, SIAM Journal on Scientific Computing, under review. [2] C.L. Bottasso, O.A. Bauchau, Reduced sensitivity to perturbations in the numerical solution of multibody DAEs, IDETC/CIE 2005, ASME 2005 International Design Engineering Technical Conferences, Computers and Information in Engineering Conference, Long Beach, CA, USA, September 24–28, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

567

Redundant Optimal Control of Manipulators along Specified Paths Sylvia Breun∗ , Rainer Callies∗ ∗ Centre

for Mathematical Sciences Munich University of Technology Boltzmannstr. 3, 85748 Garching, Germany {breun, callies}@ma.tum.de

ABSTRACT The time-optimal motion of a four-link manipulator is investigated with its end-effector following a prescribed path in space. The dynamic equations together with the spatial restrictions yield a nonlinear differential-algebraic system of differential index 3. Additionally, the optimal control problem for this DAE system contains multiple restrictions and several interior points. A transformation to minimum coordinates is the most elegant way to eliminate severe mathematical problems arising from the algebraic constraints. The transformation results in a system of linear equations of motion. The remaining controls are the angular acceleration of the fourth joint and the acceleration of the end-effector along the prescribed trajectory, replacing the four actuator torques in the original formulation. The structure of the remaining controls is unknown and too difficult to be estimated in practical applications. Introducing eight highly nonlinear control/state constraints is another price to be paid for the much simpler structure of the overall problem. However, only the state variables are affected by the nonlinearity, whereas the control variables appear linearly in these constraints. The set of admissible controls turns out to be a convex subset of the IR2 . The optimal control minimizes the Hamiltonian which is also linear in this case; at every moment the structure of the controls can be determined by techniques of linear programming; the results are valid for multi-link manipulators with an arbitrary number of joints. With this information, the complete optimal control problem is transferred into a highly nonlinear multi-point boundary value problem and solved numerically by the advanced multiple shooting method JANUS. By backward transformation the actuator torques and their switching behaviour are easily obtained.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

568

Optimal Tool Point Control of Hydraulically Actuated Flexible Multibody System With an Operator-in-the-Loop Morten K. Ebbesen*, Michael R. Hansen*, Torben O. Andersen† *

Institute of Mechanical Engineering, Aalborg University Pontoppidanstræde 101, 9220 Aalborg E., Denmark [email protected], [email protected] † Institute of Energy Technology Pontoppidanstræde 105, 9220 Aalborg E., Denmark [email protected]

ABSTRACT The paper is on automated tool point path and velocity control of a loader crane. A two step approach is used that combines a direct computation of the optimal cylinder velocities with a general procedure that includes the different saturation phenomena encountered in a hydraulically actuated mobile machine. The two step procedure is envoked at each sampling instant during operation and returns the desired valve control signals. The first step of the procedure involves a direct computation of the optimal cylinder velocities for all possible combinations of active and non active cylinders. Among the configurations that may produce the exact velocity reference, the one requiring the least hydraulic power is chosen. Some simulation examples of the proposed procedure are shown and dynamic simulation based on the computed control signals is applied taking into account the finit bandwith of the hydraulic control valves and the flexibility of the hydraulic fluid and the mechanical structure, respectively..

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

569

Slim Elastic Structures with Transversal Isotropic Material Properties under Finite Deformations D. Franitza∗ , T.Lichtneckert† ∗

Institute for Mechatronics and System Dynamics University of Duisburg-Essen, Germany [email protected] † Institute of Solid State Mechanics Dresden University of Technology, Germany [email protected]

ABSTRACT Compliant mechanisms offer numerous advantages compared to their rigid counterparts such as a reduction in part number, the corresponding reduction in assembly time and the avoidance of coulomb friction and the resulting wear. In small scale applications i.e. microelectromechanical systems classical joints cannot be manufactured in a reasonable cost range and compliant solutions usually apply. The existing applications and simulation strategies have in common that the initial and mostly also the deformed configuration is planar and the mechanisms only undergo small deformations. The presented work sketches the idea of modelling compliant mechanisms using beam-like structures which can be subjected to finite deformations. These beam elements are described on velocity level in convected coordinates. The rate description transforms the nonlinear initial boundary value problem (IBVP) into a linear boundary value problem (BVP) which is embedded in a nonlinear initial value problem (IVP). In the case of one dimensional (beam) elements the BVP can be solved accurately using a Runge Kutta (RK) integration scheme. For a sufficiently smooth behavior of the integrated function a 4th order RK integration shows accurate results [1]. In dynamic simulations the smooth behavior is usually not achieved for the deformations and for the kinetic variables as well. Here the use of lower order integration schemes leads to more reasonable results. The description of the constitutive behavior of the beam material follows the above mentioned rate description by using the objective Truesdell stress rate. The corresponding material tensor which connects the Truesdell stress rate with the deformation gradient is determined by differentiating an appropriate strain energy function with respect to the left Cauchy-Green deformation tensor. Hyperelastic isotropic and transversal-isotropic materials [2] are used for the compliant members. The use of transversal-isotropic material leads to a coupling between the bending and the torsional deformation which allows i.e. the generation of complex spatial movements with initially planar geometry and fully planar loads.

References [1] D. Franitza, T.Lichtneckert, Fast simulation of nonlinear compliant mechanisms using convective rate formulation, Proceedings of MSNDC, Long Beach, California, September 24-28, 2005. [2] M. Rueters, E.Stein, Analysis, finite element computation and error estimation in transversely isotropic finite elasticity, Comput. Methods Appl. Mech. Engrg., 190, 519-541, 2000.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

570

Modelling of Flexible Rods Falling in Fluid with Possible Contacts Michal Hajˇzman∗ , Pavel Polach∗ ∗ Section

ˇ of Materials and Mechanical Engineering Research, SKODA RESEARCH Ltd. Tylova 1/57, CZ-31600 Plzeˇn, Czech Republic {michal.hajzman, pavel.polach}@skoda.cz ABSTRACT

Practical problems in engineering demand the solution of a wide range of multidisciplinary tasks. One of these branches is nuclear engineering. During the analysis and control of nuclear reactors in nuclear power plants the complex problems are coming out and connect multibody systems analysis with influences of fluid, possible contacts and impacts, seismic excitations and flexibility [1]. With respect to different types of nuclear reactors, different control systems composed of various mechanical parts and transmissions can be distinguished. Generally they can be simplified to a typical problem of a long thin rod moving through guide tubes and driven by a motor. The simplest way how to analyze the emergency situation, in which the rod has to drop down and cause the stopping of nuclear reaction, neglects the drive system and analyses the control rod as a falling flexible body in a limited space. In the first part the paper is focused on the brief description of complex multibody models of real control assemblies used in various real nuclear reactors, which were created in the alaska simulation toolbox. The multibody models are intended for the numerical simulations of the control rods drops, which are the necessary condition for the stopping of nuclear reaction. The complexity of these tasks was the motivation for a deeper theoretical study of such system multibody modelling. Thus the second part of this paper deals with the preliminary study of the possible approaches to the modelling of control rods influenced by contacts, flowing water and eventually excited by prescribed kinematic seismic excitation. The flexible rod performs a large rigid body motion followed by an elastic deformation. The simplified approach, sometimes called a rigid finite element method, is characterized by the artificial division of the flexible body to the set of rigid bodies coupled by massless spring-damper elements. Mass and couplings parameters can be identified from the modal values of the flexible rod. In the second case flexible beam models [2] are used to approach the behaviour of the thin rod directly in the dynamic model. The nonlinear Hertzian law is used for the modelling of contact forces. Influences of flowing fluid are involved by the fluid resistance force. The presented approach can be easily generalized for the complex multibody systems composed of rigid and flexible bodies connected by joints [2, 3]. Then the whole control assembly system including the motor and the transmissions for the real nuclear reactor can be analyzed.

References [1] M. Hajˇzman and P. Polach, Modelling and Seismic Response of the Control Assembly for the VVER 440/V213 Nuclear Reactor. J.M. Goicolea, J. Cuadrado and J.C. Garc´ıa Orden eds. ECCOMAS Thematic Conference Multibody Dynamics 2005, UPM, Madrid, Spain, CD-ROM P38, 2005. [2] A. Shabana, Dynamics of Multibody Systems. Cambridge University Press, New York, 2005. [3] V. Stejskal and M. Val´asˇek, Kinematics and Dynamics in Machinery. Marcel Dekker, New York, 1996.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

571

Tracking of Displacements in Smart Elastic Beams Subjected to Rigid Body Motions H. Irschik *, M. Nader†, Ch. Zehetner† *

Institute of Technical Mechanics, Johannes Kepler University of Linz, Austria A 4040 Linz-Auhof, Altenbergerstr. 69, Austria [email protected] †

Linz Center of Competence in Mechatronics (LCM) A 4040 Linz-Auhof, Altenbergerstr. 69, Austria [email protected], [email protected]

ABSTRACT The present paper is concerned with displacement tracking of a smart cantilever beam. The beam is subjected to a rigid body motion and to external forces. Additionally, a distributed actuation by piezoelectric layers is considered. The scope of the present work is to design the piezoelectric actuation such that the displacements of the beam relative to the rigid body motion follow a prescribed field of trajectories, despite the presence of the external forces and the inertia forces induced by the rigid body motion. We study slender beams, such that the Bernoulli-Euler theory of beams is applicable. The effect of geometric stiffening is taken into account according to v. Karman. In a first step, the problem of displacement tracking is analytically solved in the framework of the latter beam theory. The problem of tracking zero displacements, i.e. the shape control problem, is included as a special case. The practical relevance of the displacement tracking solution is afterwards validated by means of Finite Element computations for a three-layer beam made of two PZT actuation layers and an aluminum substrate layer. The example beam is subjected to a resonant transverse rigid body translation and is assumed to fall in its own weight initially. A harmonic vibration with limited amplitude about a static deflection is tracked. The structure is modeled by means of plane-stress Finite Elements, taking into account electromechanical coupling, where the electric potential resulting from the beam analysis is applied to the electrodes of the PZT layers. In this numerical study, the goal of displacement tracking is reached with a high accuracy, despite the piezoelectric actuation has been designed in the framework of beam theory, while the numerical computations were performed using a refined FE-model. This gives excellent evidence for the appropriateness of the proposed method.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Dynamic Analysis of Constrained Nonlinear Multibody Systems with Intermittent Contact Elisabet Lens, Alberto Cardona Centro Internacional de Métodos Computacionales en Ingeniería CIMEC-INTEC, Conicet-Universidad Nacional del Litoral Güemes 3450, 3000 Santa Fe, Argentina [email protected]

ABSTRACT This work deals with the dynamic analysis of constrained nonlinear multibody systems undergoing contact between rigid bodies. Contact can occur between two bodies of the system or with an external body. Intermittent contact can be, for example, accidental such as the impact of a member of the system on an unexpected object (obstacle). The different approaches to model intermittent contact can be classified according to the assumed duration of the contact: (a) the duration of contact is assumed to tend to zero (contact is treated as a discontinuity) and (b) the duration of the impact is finite and the history of the forces acting between the contacting bodies is explicitly computed during the simulation. In this work, the contact event is assumed to be of finite duration. The unilateral contact condition is expressed in terms of the relative distance q between the bodies with q t 0 . The contact condition is enforced through a purely kinematic condition q  r 2 0 , where r is a slack variable used to assure the positiveness of q. This approach is shown to yield a discrete version of the principle of impulse and momentum [1]. The use of unconditional stable schemes is relevant in intermittent contact problems whose dynamic response is very complex due to the large and rapidly varying contact forces applied to the system. A scheme based on Time Continuous Galerkin approximations applied to the equations of motion proposed by Lens et al. [2] is used in the frame of a variational formulation. The energy preservation argument is used to prove its unconditional stability [3]. Kinematic constraints are enforced via the Lagrange multipliers technique.

References [1] O. Bauchau, Analysis of Flexible Multibody Systems with Intermittent Contacts, Multibody System Dynamics, 4, 23-54, 2000. [2] E. Lens, A. Cardona and M. Géradin, Energy Preserving Time Integration for Constrained Multibody Systems, Multibody System Dynamics, 11, 41-61, 2004. [3] T.J.R. Hughes, Analysis of Transient Algorithms with Particular Reference to Stability Behavior, Computational Methods for Transient Analysis, T. Belytschko and T.J.R. Hughes eds., North Holland, 1983.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

573

Evolution of Rotation of a Triaxial Satellite under the Action of Gravitational Torque and Light Pressure Torque Leonid D. Akulenko1, Dmytro D. Leshchenko2, Svetlana G. Suksova 1

Institute for Problems in Mechanics Russian Academy of Science Prospect Vernadskogo, 101, building 1, 119526, Moscow, Russia [email protected] 2 Chair of Theoretical Mechanics Odessa State Academy of Civil Engineering and Architecture Didrikhson Str., 4, 65029, Odessa, Ukraine [email protected]

ABSTRACT The attitude evolution of a spacecraft under various torques has been studied extensively over the pass few decades. Analytical models can be of great help in obtaining a qualitative understanding of the dynamical features involved. In the case of axisymmetric spinning satellites a fairly complete theory exists. The classical literature contains many special cases of rigid body motion with unfortunately are hardly ever directly applicable to particular satellite problems. Evolution of rotation of a nearly dynamically spherical triaxial satellite under the action of gravitational and light pressure torques is investigated. We assume that the spacecraft moves around the Sun along an elliptic orbit and that the satellite surface is surface of revolution. The light pressure torque has a force function depending only on the orientation of the symmetry axes of the body. The coefficient of the light pressure torque is approximated by a trigonometric polynomial of an arbitrary order. Assume that the principal central moments of inertia of the satellite are close to each other. Assume also that both the light pressure torques and the gyroscopic torques are of the order of a small parameter . İWe apply the averaging method to investigate the motion of a spacecraft with respect to the center of mass. We perform the averaging with respect to the precession angle and true anomaly independently, as for nonresonance cases. We arrive at the average system of the first approximation. The angular momentum is constant in magnitude and inclined at a constant angle to the normal to the plane of the orbit. The first integral of the system of equations for the nutation and the proper rotation angle are found. We represent the coefficient of the light pressure torque in the approximation taking into account the third and all even harmonics. We qualitatively investigate the phase plane of system for the nutation and the proper rotation angles with the first integral. The critical points of this system are determined. The face portraits of the average system are constructed numerically, the phase curves describe oscillations and rotations. Special cases of motion of the body are investigated. Some qualitative effects have been demonstrated. New qualitative properties of the rotation of the satellite are established.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

574

Mechanical integrators for nonlinear flexible multibody dynamics S. Leyendecker∗ , P. Betsch† , P. Steinmann∗ ∗ Chair

of Applied Mechanics, University of Kaiserslautern P.O. Box 3049, 67653 Kaiserslautern, Germany [email protected]

† Institute for Mechanics, University of Siegen Paul–Bonatz–Str. 9–11, 57068 Siegen, Germany [email protected]

ABSTRACT The modeling of flexible multibody systems in nonlinear structural dynamics as finite dimensional Hamiltonian system subject to holonomic constraints constitutes a general framework for a unified treatment of rigid and elastic components. Internal constraints, which are associated with the kinematic assumptions of the underlying continuous theory, as well as external constraints, representing the interconnection of different bodies by joints, can be accounted for in a likewise systematic way. The formalism provides the possibility to use different methods for the constraint enforcement (cf. [1]), which differ significantly in the following categories: dimension of the system of nonlinear equations, condition number of the iteration matrix during the iterative solution procedure, exactness of the constraint fulfilment and computational costs. The discrete null space method developed in [2] provides an integration scheme for the constrained equations of motion, which has proven to perform excellently in the mentioned categories. It relies on the elimination of the constraint forces (comprising the Lagrange multipliers) from the energy–momentum conserving time stepping scheme, emanating from the underlying DAEs by direct temporal discretisation. This is accomplished via the premultiplication of the discrete scheme by a null space matrix, which spans a basis of the null space of the discrete constraint Jacobian. A six-body-linkage possessing a single degree of freedom, addressed in [3], is analysed as an example. The treatment of this closed loop structure by the discrete null space method results in the solution of only one scalar equation of motion. Furthermore it circumvents the involved investigation of dependent constraints, which lead to rank-deficiency of the constraint Jacobian occurring in the time stepping scheme pertaining to the direct temporal discretisation of the DAEs.

References [1] Leyendecker, S., Betsch, P. and Steinmann, P. (2004): Energy–conserving integration of constrained Hamiltonian systems – a comparison of approaches, Comp. Mech.,33, 174–185. [2] Betsch, P. (2005): The discrete null space method for the energy consistent integration of constrained mechanical systems. Part I: Holonomic constraints, Comput. Methods Appl. Mech. Engrg.,54, 1775–1788. [3] Wittenburg, J. (1977): Dynamics of systems of rigid bodies, Teubner-Verlag, Stuttgart.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Some really simple but useful model of substitutable elasticity modelled as elasticity in six subsequent joints .U]\V]WRI/LSLĔVNL *GDĔVN8QLYHUVLW\RI7HFKQRORJ\)DFXOW\RI0HFKDQLFDO(QJLQHHULQJ ul. Narutowicza 11/12; 80-952 Gdansk; POLAND; tel.xx_58_347-29-96 [email protected]

ABSTRACT The paper devotes its attention to an analysis of dynamics of multibody systems. We assumed as evident that any rigid bodies’ multibody structure could be modelled easily, as a big number of methodologies exist and their descriptions are present in the literature from years. Thus the actual challenge is not a new methodology. The challenge is the effective way of modelling. At this same time, there is not any unique and universal method of modelling. For most of real structures, they can be modelled using few alternative sequences of bodies. Moreover, within the structures we can easily locate some specific substructures that can be modelled with the general, classical methodology or by some shorter algorithm dedicated to these specific substructures. We will focus on one of such substructures. The detail is an elastic support of a rigid body. If motions of such body are small one of the most popular model is an elasticity formula with nonzero diagonal elasticity coefficients. The model of the spring with diagonal matrix becomes inoperative if someone considers significant displacements of the body. Then some alternative could be the concept of elasticity present in six subsequent joints of massless multibody structure. It can be modelled as a closed loop substructure, or as some direct analytical formula. The second is numerically shorter and more effective. However, its equations are not intuitive, and thus it looks interesting to present them in a separate paper. The detail is extracted from the multibody structure and is presented. The paper is divided into seven parts. The first focuses on the idea of substitutable coefficients itself and on the necessity of use of such models. The two following presents most popular ideas of such models. In paragraph number two you could found some brief idea of a model known as spring with diagonal matrix when in the third a model called as elasticity in six subsequent joints. Paragraph forth presents classical methodology used to model supporting structure of six subsequent joints. Fifth presents our proposition of the direct model of such structure and its equations. The next is devoted to numerical validation of equations, base on some 6 elements multibody model [1,2], and to a presentation of an exemplary code of the procedure. The final one presents main conclusions. As it is proofed, the direct model gives exact results of error tolerance better that the classical model. Simultaneously, the use of the direct model drastically reduces the number of multiplications in compare with the classical model. These numbers are calculated and compared.

References [1] Fisette, P. and Samin, J.-C.: Symbolic Modeling of Multibody System, Kluwer Academic Publishers, Dordrecht, 2003

[2] P. Fisette, K. LipLĔVNL-&6DPLQ Symbolic Modelling for the Simulation, Control and Optimisation of Multibody Systems, Advances in Multibody Systems and Mechatronics, Gratz, 139-174, 1999

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

576

Optimal dampers localization for a body under double load and the body behaviour for its intermediate loads .U]\V]WRI/LSLĔVNL *GDĔVN8QLYHUVLW\RI7HFKQRORJ\)DFXOW\RI0HFKDQLFDO(QJLQHHULQJ ul. Narutowicza 11/12; 80-952 Gdansk; POLAND; tel.xx_58_347-29-96 [email protected]

ABSTRACT A multibody dynamics is a well-known tool used in many analyses of mechanisms. It simplifies predictions of any mechanism behaviour, even in long before of time when the final mechanism is set to any physical experiment. Some important step in a project preparation is a selection of elastic and damping coefficients for elements of the system. The process can be simplified if a combination of dynamics and optimisation tools is in a disposition of project manager. But even then, it request in a numbers of numerical calculation, so numerically effective models are necessary. The paper presents researches in an optimal configuration of damping elements. In some previous, publication [2] we had focus on question: whether it is possible to obtain a configuration of dampers that satisfy presumed level of damping of vibrations for a structure with varying load? Especially, does exist it for a single body with double sets of workload. The answer was negative, even if some important improvement of damping was obtained. Within the actual paper some deeper analyse of behaviour of the optimised system is performed, especially its behaviour for intermediate loads is analysed. Short description of rigid bodies multibody dynamics Any multibody system consists of two kinds of elements: rigid bodies and massless joints. Any of the bodies have to be connected to at least one joint. Any of the joints can connect two bodies. To obtain the dynamics equations, we will employ methodology described in [1]. It leads to the second order differential equations expressed in from M (q ) ˜ q  F (q , q )  Q (q , q , f e , t e , t ) 0 (1) The methodology is cited with lot of details in the full version of the article. Optimization The problem is formulated as a multi objective optimization problem. The simplex algorithm is proposed as the optimization method. Tested system A test system is a single body, planar system. The body has form of a rectangle. Position the body is fixed with the use of 4 elastic elements. It is extended with 3 dampers and optimized under double load conditions. To avoid “infinite parameters”, there are restrictions set on dampers localisation. The test The system is tested for intermediate loads. Important deviations from presumed values are observed.

References [1] Fisette, P. and Samin, J.-C.: Symbolic Modeling of Multibody System, Kluwer Academic Publishers, Dordrecht, 2003 [2] /LSLĔVNL . Going into the research of dampers configuration for bodies under double load conditions, Research-Education-Technology, *GDĔVN, p. 137-144, 2005

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

577

Design of Characteristics of Air Pressure Controlled Hydraulic Shock Absorbers in an Intercity Bus Pavel Polach*, Michal Hajžman* *

Section of Materials and Mechanical Engineering Research, ŠKODA RESEARCH Ltd. Tylova 1/57, CZ-316 00 PlzeĖ, Czech Republic {pavel.polach,michal.hajzman}@skoda.cz

ABSTRACT The air pressure controlled hydraulic telescopic shock absorber of axles air suspension is capable of changing its damping force in dependence on air pressure in air springs. Air pressure in springs rises as vehicle loading grows and damping forces in the shock absorbers increase. When reducing vehicle loading, air pressure in springs drops and causes decrease in damping forces in the shock absorbers. Thus the vehicle keeps the constant driving stability and comfort in various operational situations. This performance of the controlled shock absorber can be able use of at all vehicles that use the axles air suspensions. Due to the possibility of improving dynamic properties of those types of vehicles BRANO Inc., the producer of shock absorbers, started to develop air pressure controlled hydraulic telescopic shock absorbers of axles air suspensions. The SOR C 12 intercity bus is the reference vehicle, for which research and development of controlled shock absorbers is done and on which the shock absorbers are verified. Force-velocity characteristics of the controlled shock absorbers of axles air suspension were designed on the basis of results of computer simulations with the bus multibody models created in the alaska simulation toolbox. Multibody models of an empty vehicle, a fully loaded vehicle and four variants of partly loaded vehicle were created. For each weight of the bus three multibody models of various level of complexity were created. Since the bus multibody models should be used especially for designing force-velocity characteristics of air pressure controlled shock absorbers, great attention was paid to the correct interpretation of real behavior of hydraulic shock absorbers and air springs of axles suspension. As a criterion for the design of the optimum forcevelocity characteristics of shock absorbers, maximum similarity of dynamic responses of multibody models of the bus of all the considered weights to dynamic response of multibody model of the bus with the same loading as during the experimental measurements with the real vehicle (approx. 71.5 % of the maximum permissible weight), was chosen. In the course of measurements the shock absorbers characteristics were optimally tuned for this vehicle weight. Time histories of relative deflections of axles air springs determined during the simulations of the vehicle run over the modified standardized obstacle were compared. Approach based on the evaluation of the correlation coefficient of two time series was used for the evaluation of accordance of dynamic responses. Then, the suitability of the designed characteristics of the controlled shock absorbers was verified from the point of view of improving the horizontal dynamic behavior of the vehicle during the simulations.

References [1] M. Blundell and D. Harty, The multibody systems approach to vehicle dynamics. Elsevier Butterworth, Norfolk, 2004. [2] P. Polach and M. Hajžman, Approaches to the creation of the intercity SOR bus multibody models. J. Vimmr ed. Conference Computational Mechanics 2005, 477-484, Hrad Neþtiny, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

578

Robot Path Planning in a Constrained Workspace by using Optimal Control Techniques Rogério R. dos Santos *, Valder Steffen, Jr. *, Sezimária de F. P. Saramago † *

School of Mechanical Engineering, Federal University of Uberlândia Av. João Naves de Ávila, 2160 - Campus Santa Mônica - Bloco 1M Uberlândia/MG - Brazil {rrsantos,vsteffen}@mecanica.ufu.br † Faculty of Mathematics, Federal University of Uberlândia Av. João Naves de Ávila, 2160 - Campus Santa Mônica - Bloco 1F Uberlândia/MG - Brazil [email protected]

ABSTRACT Robot manipulators are programmable devices designed to execute a great variety of tasks in a repetitive way. In industrial environment, while productivity increase, cost reduction associated with robotic operation and maintenance can sometimes be obtained as a result of decreasing the values of dynamic quantities such as torque and jerk, with respect to a specific task. Furthermore, this procedure allows the execution of various tasks that require maximum system performance. By including obstacle avoidance ability to the robot skills, it is possible to improve the robot versatility, i.e., the robot can be used in a variety of working situations. In the present contribution, a study concerning the dynamic characteristics of serial robot manipulators is presented. An optimization strategy that considers the obstacle avoidance index and the dynamic performance index associated with the movement of the robot is proposed. It results an optimal path planning strategy of a serial manipulator over a moving restricted workspace [1].This is achieved by using multicriteria optimization concepts [2], variational calculus tools and optimal control techniques. Numerical results illustrate the interest of the proposed methodology. The present strategy can be useful for the design of robot controllers [3].

References [1] S. F. P. Saramago and V. Steffen, Jr., Optimization of the trajectory planning of robotmanipulators taking into-account the dynamics of the system. Mechanism and MachineTheory, 33 (7), 883–894, 1998. [2] H. Eschenauer, J. Koski and A. Osyczka, Multicriteria Design Optimization. Springer-Verlag, 1990. [3] Y. Nakamura, Advanced Robotics: redundancy and optimization. Addison-Wesley, 1991.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

579

Optimal Control of Multi-Link Manipulators with Rivalling Actuators Gerhard F. Schanzer∗ , Rainer Callies∗ ∗ Centre

for Mathematical Sciences Munich University of Technology Boltzmannstraße 3, 85748 Garching, Germany [email protected] and [email protected] ABSTRACT Miniaturized robotic manipulators are a key element in future high-pricision telesurgery with artificial fingers and hands. This development is supported by the rapidly decreasing size of robotic sensors and actuators. Currently severe limitations are induced by the drives of the micro-joints, which are either slow or can produce only low specific forces. The present paper deals with the optimal control of an advanced six-sectional branched manipulator. Joints are driven by weak, but fast, and strong, but slow, actuators acting in parallel. This results in an optimal control problem of a multibody system subject to rivalling controls. Moreover, not also the control amplitudes, but also the rate of change of the controls is constrained. From the point of view of optimal control theory, the restriction of the first derivative of a control induces a severe problem: By definition, a control is a variable for which no derivatives occur in the complete problem. The straightforward approach – defining the derivatives of the controls as the new control variables and thus transforming the original controls into additional state variables – leads to a transformed optimal control problem with a sensitivity not acceptable for numerical treatment. A new approach is proposed to transform the control problem into a stable one. By a series of transformations, the original problem is converted into a piecewise defined, highly nonlinear multi-point boundary value problem for ordinary differential equations, but now with non-constant dimension. Based on the minimum principle, a strict and careful mathematical analysis of the coupling of the single parts of the problem leads to new interior point conditions at the junction points. As an example, highly accurate optimal manipulator motions are calculated with the respective weak control being able to react about ten times faster than the accompanying strong control, but with a maximum torque which is about ten times smaller than for the strong control. Using the new transformations, a stable and efficient numerical solution becomes possible.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

580

Development of Validated Generic Road Vehicles for Crashworthiness Through Optimization Procedures Luís Sousa, Paulo Veríssimo, Jorge Ambrósio *

IDMEC – Instituto Superior Técnico, Technical University of Lisbon Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal {lsousa, pauloverissimo, jorge}@dem.ist.utl.pt

ABSTRACT The development of passive safety devices for vehicles or setup of virtual tests requires that detailed numerical models are available. Generally, the vehicle manufacturers are unable to release the detailed data for their models due to commercial and legal restrictions. Since finite element models for road vehicles suitable for crashworthiness analysis are very detailed, their complexity involve computations that require long time to perform analysis involving large deformations with plasticity and contact. With the actual computer technology the computational time for these models is measured in terms of days. However, during the design of a new vehicle, the redesign of a component or of a particular safety device, many simulations must be performed to appraise different design solutions. An alternative approach to the use of detailed FEM models is important to reduce calculations time. Multibody models of road vehicles are shown here to be suited for early design phases of such vehicles, components or devices, being the computational time required for any specific crash analysis being measured in same number of minutes than the hours required for FEM models. This work presents a methodology for modeling generic road vehicle models for multibody dynamic analysis that have all the characteristics of a real cars, have crash responses for front and side impact similar to the top of line vehicle of its class for the Euro NCAP test but still are not the exact model of any existing vehicle. The multibody modeling work is based on a previously existing finite elements model of the vehicle. First a strategy to convert the finite element to a multibody system model is presented so that the structural features of the MB model are correlated with those of the complex of the FEM model. The responses of the two models are correlated through selected responses extracted from results of impact simulations regarding occupant safety and energy absorption. The crash scenarios used, according to standards for crash tests, are the same for finite element and multibody models. Selected responses, obtained from the results of the simulations in representative points of the car and the barrier, are used to compare the performance of the models in terms of displacements, velocities and accelerations. The plastic hinge characteristics of the multibody model are redefined, using an optimization procedure to ensure that the selected responses of the multibody and finite element models are similar, thus leading to validated multibody models.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

581

Developing mathematical and computer models for car dynamics using joint co-ordinates and homogenous transformations Marek Szczotka∗ , Iwona Adamiec-W´ojcik∗ , Stanisław Wojciech† ∗ University

of Bielsko-Biała ul. Willowa 2, 43-309 Bielsko-Biała, Poland [email protected], [email protected] † Automotive

Development and Research Centre BOSMAL ul. Sarni Stok 93, 43-300 Bielsko-Biała, Poland [email protected]

ABSTRACT In modelling vehicle dynamics absolute coordinates are mostly used. This approach allows the equations of motion to be obtained by means of independent generalized coordinates which describe the motion of each body in the kinematic chain. Then the equations of motion are completed with constraint equations. The absolute coordinates are used for modelling multibody system dynamics in ADAMS and DADS packages. In the paper a different approach using joint (relative) coordinates is presented. The motion of a body is defined in relation to the preceding body in the kinematic chain. This method allows both the number of degrees of freedom of the system and the number of constraint equations to be minimized. This method can be used to modeling tree-like systems, with and without closed kinematic sub-chains, and has advantages when modelling the dynamics of vehicles. Two models of vehicles, those of a passenger car and an articulated vehicle, are discussed. Both models are formulated using the same approach, namely joint coordinates and homogenous transformations. The model of a passenger car is a dynamic model with over 50 degrees of freedom. It includes structural suspension models, flexible drive and steering systems. The Dugoff-Uffelman and Magic Formulae models of tires are applied. The model of an articulated vehicle is a model of a tractor with a trailer. This model does not include any closed sub-chains and the vehicle is treated as a system of four rigid bodies. In order to derive the equations of motion, the Lagrange equations are used. On the basis of the models computer programmes have been developed, and calculations have been carried out. The results of numerical calculations have been compared with the results of experimental measurements. Good correspondence of results has been achieved. The results of calculations have also been compared with those obtained with the ADAMS/Car system which uses absolute coordinates. In this case, too, good compatibility has been achieved.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

582

Finite element simulation of deformation bands in saturated granular media with inhomogeneous porosities at the meso-scale Jos´e E. Andrade∗ , Ronaldo I. Borja∗ ∗ Department

of Civil and Environmental Engineering, Stanford University Stanford, CA 94305 [email protected]

ABSTRACT The balance of mass and linear momentum of a solid-fluid mixture furnish a complete set of equations from which the displacements of the the solid matrix and the pore pressures can be resolved for the case of quasi-static loading, resulting in the so-called u − p Galerkin formulation. In this work, a recently proposed model for dense sands is utilized to model the effective stress response of the solid matrix appearing in the balance of linear momentum equation [1, 2, 3]. In contrast with other more traditional models, inherent inhomogeneities in the density field at the meso-scale can be easily incorporated and coupled with the macroscopic laws of mixture theory. The hydraulic conductivity is naturally treated as a function of the porosity in the solid matrix, hence allowing for a more realistic representation of the physical phenomenon. The aforementioned balance laws are cast into a fully nonlinear finite element program utilizing isoparametric elements satisfying the Babuˇska-Brezzi stability condition. Numerical simulations on dense sand specimens are performed to study the effects of inhomogeneities on the stability of saturated porous media at the structural level. Supported by National Science Foundation under Grant Nos. CMS-0201317 and CMS-0324674 to Stanford University.

References [1] R. I. Borja and J. E. Andrade. Critical state plasticity, Part VI: Meso-scale finite element simulation of strain localization in discrete granular materials.Computer Methods in Applied Mechanics and Engineering, 2006. In press for the John Argyris Memorial Special Issue. [2] J. E. Andrade and R. I. Borja. Capturing strain localization in dense sands with random density. International Journal for Numerical Methods in Engineering. In review, 2006. [3] M. G. Jefferies. Nor-Sand: a simple critical state model for sand. G´eotechnique, 43:91–103, 1993.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

583

Computational Framework For Multilayer Plasticity Based On Critical State Soil Mechanics Miguel A. Caminero∗ , Francisco J. Mont´ans∗,† ∗ Universidad

de Castilla-La Mancha E.T.S. de Ingenieros Industriales. C/ Camilo Jos´e Cela s/n. 13071-Ciudad Real. Spain [email protected] † Universidad Polit´ ecnica de Madrid E.T.S. de Ingenieros de Minas. C/ Rios Rosas 21. 28003-Madrid. Spain [email protected]

ABSTRACT In this work a computational plasticity model for soils under cyclic loading is presented. The model is based on the Critical State Soil Mechanics Theory. Multiple nested surfaces are employed in order to properly predict the soil behaviour under a large variety of loading types. The first, innermost surface of the model acts always as yield surface, whereas the last, outermost surface of the model acts as consolidation surface, a sort of bounding surface. The remaining surfaces are used only as a tool to compute the effective hardening modulus inside the consolidation surface. In contrast with the classical Cam-Clay plasticity, the model permits the dissipation of some energy for cycles inside the consolidation surface, while preserving all the features of the classical Cam-Clay models under monotonic, consolidating loading. The cycles inside the consolidation surface preserve the Masing extended behaviour at any stress level. The model uses a stored energy function for the elastic strains, so elastic strains do not dissipate energy. This is an important feature in a model for cyclic plasticity, as for example in earthquake engineering. The model is amenable of being implemented via a totally implicit algorithm. Two distinct algorithms are present during the integration procedure. One is for the case of loading/unloading inside the consolidation surface. This algorithm is presented in this work. The other algorithm is for the case of loading on the consolidation surface. In this case, a typical Cam-Clay model may be employed without remarkable changes. This work presents a useful framework for the simulation of cyclic loading of soils using nested surfaces plasticity in the light of Critical State Soil Mechanics.

References [1] R.I.Borja, C. Tamagnini. Cam–Clay plasticity, Part III: Extension of the infinitesimal model to include finite strains. Computer Methods in Applied Mechanics and Engineering. 155, 77-95 (1998). [2] R.I. Borja, C.H.Lin, F.J.Montans. Cam-Clay plasticity, Part IV: Implicit integration of anisotropic bounding surface model with nonlinear hyperelasticity and ellipsoidal loading function. Computer Methods in Applied Mechanics and Engineering. 190; 3293-3323. 2001.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

584

Creep of Geomaterials due to the Coupled Damage and Spontaneous Mineral Dissolution Liang Bo Hu1and Tomasz Hueckel1 1

Duke University, Department of Civil and Environmental Engineering Durham, NC 27708-0287, USA [email protected]; [email protected]

ABSTRACT A basic model of chemical softening, and chemically enhanced deviatoric strain - hardening for saturated geomaterials is presented. The aim is to simulate the material behavior that exhibits characteristics of creep induced by environmental conditions. Chemical softening is postulated to occur as a consequence of mineral dissolution and precipitation enhanced by the material damage. Dissolution of minerals in pore water is assumed to take place at the internal free surfaces not only of the initial pore space but also that generated during damage. The rate dependence of the processes of dissolution and precipitation renders also the resulting chemical softening behavior to be rate dependent. In this paper a closed system is discussed only, in which deformation at constant stress results entirely a from a local compensation mechanism between the chemical softening and strain hardening. Three stages of creep are interpreted in terms of mechanisms of dissolution and precipitation, as well as the variation in the reaction area involved in the mass exchange. An open system, these local mechanisms are enhanced by diffusion of species affecting the mass balance and need to be addressed via a boundary value problem, as described elsewhere.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

585

Validation of an extended cap model for partially saturated soils R. Kohler, G. Hofstetter Institute for Structural Analysis and Strength of Materials, University of Innsbruck Technikerstr. 13, A-6020 Innsbruck, Austria [email protected], [email protected]

ABSTRACT The present paper deals with the development and validation of an elasto-plastic material model for partially saturated soils. A cap model originally proposed by DiMaggio and Sandler [1] for drained conditions, allowing control of dilatancy by means of a strain hardening cap, which intersects a shear failure surface in a non-smooth fashion, serves as starting point for the development of a constitutive model for partially saturated soils. In contrast to constitutive models for fully saturated soils, which are usually formulated in terms of a single effective stress variable, some of the most fundamental features of partially saturated soils can only be taken into account using two stress state variables. In the present case the material model is formulated in terms of the average soil skeleton stress tensor and matric suction, which is consistent with thermodynamic considerations. Accounting for the evolution of both, the shear failure surface and the hardening law of the cap in terms of matric suction as well as including the third invariant of the deviatoric stress tensor in the formulation of the yield surfaces allows to predict the behavior of partially saturated soils. The capability of the developed model is demonstrated by the numerical simulation of a series of suction controlled tests, published in [2] and [3], involving hydrostatic compression, consisting of loading and unloading at several constant values of matric suction, triaxial compression tests, conventional triaxial compression tests, triaxial extension tests and simple shear tests at different values of matric suction and different values of hydrostatic net stress, i.e. total stress in excess of pore air pressure.

References [1] DiMaggio, F.L. & Sandler, I.S. 1971. Material Models for Granular Soils. Journal of the Engineering Mechanics Division. ASCE. [2] Macari, E.J., Hoyos L.R. & Arduino, P. 2003. Constitutive modeling of unsaturated soil behaviour under axisymmetric stress states using a stress/suction-controlled cubical test cell. International Journal of Plasticity 19: 1481-1515 [3] Macari, E.J. & Hoyos, L.R. 2001. Mechanical behavior of an unsaturated soil under multi-axial stress states. Geotechnical Testing Journal 24(1): 14-22

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Partially saturated porous medium vibration induced by an impulsional load A. Mesgouez, G. Lefeuve-Mesgouez, A. Chambarel UMR A 1114 Climate, Soil and Environment Universit´e d’Avignon et des Pays de Vaucluse, 33 rue Louis Pasteur, F-84000 Avignon, France [email protected] ABSTRACT The Laboratory Climate, Soil and Environment of the University of Avignon and the National Institute for Agricultural Research in France has worked for many years on wave propagation in porous soils: the main topics concern the study of transient electromagnetic and poroviscoelastic waves in realistic grounds. Numerical tools have been developed to simulate the behavior of such a complex medium. Concerning the study of mechanical wave propagation, the numerical approach we propose here deals with the spatial macroscopic Biot’s modelling: the medium is considered to be a biphasic continuum composed of a porous deformable viscoelastic solid skeleton and a fluid component corresponding to the viscous fluid which partially or totally saturates the porous space. In such a medium, three body waves exist: the P1 and P2 compressional waves and the S shear wave. Moreover, in the case of a semi-infinite medium, a surface wave also exists denoted as the Rayleigh R wave. In the specific case of transient regimes, numerical results have been presented by several authors, Zienkiewicz and Shiomi [1] or Zhao [2] for instance. These works are restricted to specific configurations of the porous medium; for instance, most of the articles do not take into account all the mechanical couplings of the Biot theory or concern completely saturated grounds. Results using a Finite Element formulation and taking into consideration the whole Biot theory in the case of a saturated medium has been presented by the authors, Mesgouez et al. [3]. A C++ Object Oriented Programming for the Finite Element code has been developed to obtain the solid and fluid horizontal and vertical displacements for a point located on the surface of the semi-infinite medium or in depth. The authors propose here an extension of their work in the case of a partially saturated soil. The influence of the third phase, a gas one in general, on the solid and fluid displacements and velocities are analyzed and developed in this communication. Particularly, we note the strong dependence of the saturation on the celerity of the second compressional wave, even with a low proportion of gas amount. Moreover, for soft grounds, the saturation has also a notable influence on the P1 wave celerity.

References [1] O.C. Zienkiewicz and T. Shiomi, Dynamic behaviour of saturated porous media : the generalized Biot formulation and its numerical solution. International Journal for Numerical and Analytical Methods in Geomechanics, 8,71-96, 1984. [2] C. Zhao and W. Li and J. Wang, An explicit finite element method for Biot dynamic formulation in fluid-saturated porous media and its application to a rigid foundation. Journal of Sound and Vibration, 282, 1169-1181, 2005. [3] A. Mesgouez and G. Lefeuve-Mesgouez and A. Chambarel, Transient mechanical wave propagation in semi-infinite porous media using a finite element approach. Soil Dynamics and Earthquake Engineering, 25, 421-430, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Macroscopic Behavior of Smectitic Clays Derived From Nanostructure M´arcio A. Murad∗ , Christian Moyne† ∗ Laborat·orio

Nacional de Computac‚ao Cient·fica LNCC/MCT Av. Get·ulio Vargas 333, 25651.070 Petro· polis, RJ, Brazil [email protected]

† LEMTA CNRS–INPL–UHP (UMR 7563) 2, av. de la Foret de Haye, BP 160, 54504 Vandoeuvre les Nancy, France [email protected]

ABSTRACT In this article we present a precise correlation between macroscopic electro-chemical properties of swelling clays such as permeabilities (chemico-osmotic and electro-osmotic), partition coefficient and the nanoscale distribution of the electrical double layer potential satisfying the Poisson-Boltzmann problem in the nano-pore domain occupied by an electrolyte solution. This is accomplished within the framework of a dual porosity model which is derived by a formal homogenization asymptotic analysis applied to two levels of averaging to include nano and macro-pores of the swelling medium. When applied to reactive contaminant transport in montmorillonite this framework combined with the threescale picture of the swelling medium allows to build-up numerically a generalized adsorption isotherm of Freundlich type and to provide nanoscopic representations for the partition coefficient in terms of the solution of the Poisson-Boltzmann problem. In addition, by solving this local closure problems posed on a unit periodic cell at the nanoscale we develop accurate constitutive laws for the chemico-osmotic and electro-osmotic permeabilities of the clay. The up-scaling technique proposed herein along with the correlations between properties at different scales allow to bridge the fields of macroscopic Soil Mechanics and nanoscopic Colloid Science.

References [1] C. Moyne and M.A. Murad, Electro-chemo-mechanical couplings in swelling clays derived from a micro/macro homogenization procedure, International J. Solids and Structures, 39, 6159–6190, 2002. [2] C. Moyne and M.A. Murad, Macroscopic behavior of swelling porous media derived from micromechanical analysis, Transport in Porous Media, 50, 127–151, 2003. [3] C. Moyne and M.A. Murad, A two-scale model for coupled electro-chemo-mechanical phenomena and Onsager s reciprocity relations in expansive clays: I. Homogenization analysis, Transport in Porous Media, 62(3), 333–380, 2006. [4] C. Moyne and M.A. Murad, A two-scale model for coupled electro-chemo-mechanical phenomena and Onsager s reciprocity relations in expansive clays: II Computational validation, Transport in Porous Media, To appear, 2006.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Mechanical Modeling of Multi-Layer Sedimentary Rock Folding Pablo F. Sanz1, Ronaldo I. Borja2, David D. Pollard3 1

Department of Civil and Environmental Engineering Stanford University Stanford, CA 94305 [email protected]

2

Department of Civil and Environmental Engineering Stanford University Stanford, CA 94305 [email protected]

3

Department of Geological and Environmental Sciences Stanford University Stanford, CA 94305 [email protected]

ABSTRACT Sedimentary rock folding results from a number of mechanisms, including buckling due to lateral tectonic compression and slip on thrust faults in the underlying strata. Movements experienced by folded layers are typically very large and may include significant rigid body translation and rotation, considerable straining, and relative slip at the interface between layers. In this paper we present a mechanical model for capturing isothermal ductile folding processes of sedimentary rocks using nonlinear continuum mechanics and finite deformation contact kinematics. Folding of rock layers with distinct mechanical properties may result in relative tangential slip at the interface between them. Of particular interest is the formulation and implementation of a finite deformation frictional contact model to account for relative sliding of two adjacent rock layers. Our method considers a Coulomb friction law that is suitable for geomaterials. The penalty method is used to implement the frictional contact model [1]. The formulation of the model includes a consistent linearization of the weak form of the linear momentum balance to enable optimal convergence for Newton-Raphson iterations. To capture the ductile response of the rock layers, we implement an elastoplastic constitutive model; a three-invariant yield criterion is used to define plastic loading and a non-associated flow rule to control inelastic dilatancy. To integrate the stresses we employ a fully Lagrangian approach along with multiplicative plasticity theory for finite deformations. This work enables us to investigate the relationship among folded shapes, internal stress state, and the occurrence of deformation bands and/or relative slip at the layer interfaces. Supported by U.S. Department of Energy, Grant No. DEFG02-03ER15454, and U.S. National Science Foundation, Grant No. CMG-0417521.

References [1] P.Wriggers, T.V. Van, and E. Stein, Finite-element-formulation of large deformation impactcontact-problems with friction, Computers and Structures, 37, 319-333, 1990.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Fast Numerical Simulation of Porous Media Flows Denis A. F. de Souza*, Alvaro L. G. A. Coutinho†, José L. D. Alves* *

Federal University of Rio de Janeiro, Laboratory for Computational Methods in Engineering and Department of Civil Engineering (COPPE/UFRJ) PO Box 68506, Rio de Janeiro, RJ, 21945-970, Brazil [email protected], [email protected]

†

Federal University of Rio de Janeiro, Center for Parallel Computations and Department of Civil Engineering (COPPE/UFRJ) PO Box 68516, Rio de Janeiro, RJ, 21941-972, Brazil [email protected]

ABSTRACT The numerical simulation of multiphase flow through porous media in unstructured grids is a key issue in Petroleum Reservoir Engineering and Basin Analysis. To overcome computational limitations, a set of fast and innovative computer strategies is presented for sequentially implicit in time, 3D 2-phase immiscible or miscible flows in porous media. The procedures contain stabilized finite element technologies for both tetrahedra (T4) and hexahedra (H8). For the T4, is employed an edge-based data structure [1] to accelerate linear systems iterative solutions and a variationally consistent velocity post-processing technique [2]. For the H8, is implemented a one-point integration plus hourglass control [3] and superconvergent velocity approximation at the element centroid [4]. For both elements, a dynamic deactivation [5] procedure is used on the transport equation, which updates solution only on the gradient varying part of the mesh. By combining those techniques, one can achieve solutions up to 2 times faster, and another relative gain is the possibility to run larger and more complex models.

References [1] A.L.G.A. Coutinho, M.A.D. Martins, J.L.D Alves, L Landau, A. Morais, Edge-Based Finite Element Techniques for Nonlinear Solid Mechanics Problems. International Journal for Numerical Methods in Engineering 50 (9), 2053-2068, 2001. [2] S.M.C. Malta, A.F.D. Loula, A.L.M. Garcia, A Post-Processing Technique to Approximate the Velocity Filed in Miscible Displacement Simulations. Matemática Contemporânea 8, 239-268, 1995. [3] C.M. Dias, A.L.G.A. Coutinho, Stabilized Finite Element Methods with Reduced Integration Techniques for Miscible Displacements in Porous Media. International Journal for Numerical Methods in Engineering 59, 475–492, 2004. [4] G. F. Carey, Computational Grids: Generation, Adaptation and Solution Strategies. Taylor & Francis: Philadelphia, PA, USA, 1997. [5] R. Löhner, F. Camelli, Dynamic Deactivation for Advection-Dominated Contaminant Transport. Communications in Numerical Methods in Engineering 20, 639-646, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A micromechanically-based interface model for the periodontal ligament Francesco Genna∗ ∗ Department

of Civil Engineering — University of Brescia Via Branze 38 — 25123 Brescia, Italy [email protected] ABSTRACT

An interface, 3D finite element is presented, specifically designed to simulate the mechanical behavior of the periodontal ligament (hereafter shortened into PDL). The PDL is a thin layer (about 0.25 mm in adult humans) of soft tissue that connects the root of a tooth to the surrounding alveolar bone. From the mechanical viewpoint, it can be considered as a thin interface made by a solid phase, consisting mainly of collagen fibers, immersed into a so-called ground substance. Previous work [1] attempted its finite element modeling in terms of a nonlinear interface description, based, however, on the simple fitting of experimental results. Here we attempt to substantially improve on what done in [1], by developing an interface element whose constitutive behavior is obtained from a simple, yet reasonably complete micromechanical model. This is based on an idea described in [2], but further extended to cover the case of a 3D fiber arrangement. The model is defined by a 2 cable, X-shaped structure, each cable representing a large number of collagen fibers, each of a different length, and presenting a crimped geometry at rest. Until a fiber is not fully uncrimped, it offers no stiffness contribution. The cables, in the model, are inclined, both with respect to the direction of the root main axis, and with respect to the direction orthogonal to the root’s axis, so as to be able to catch all the stiffness contributions provided in reality by the PDL. In each cable, a single collagen fiber is governed by a local (microscopic) non-linear stress-strain relationship, which includes a failure strain in tension. Simple statistical integrations over the length of each fiber, as described in [3], allow us to obtain a macroscopic behavior, coupled in tension and shear, including a toe stiffness, the subsequent quasi-linear stiffness, a peak macroscopic stress and strain, and a post-peak behavior until the complete failure of the element. The behavior in compression, much simpler, is governed by the same phenomenological equations as described in [1]. We can thus take into account, in a complex FE model of the tooth-bone system, several features of the PDL at the microscopic level, as well as study their influence on the overall response, on the basis of a relatively simple model, whose parameters have a physical meaning.

References [1] M. Gei, F. Genna, D. Bigoni, An interface model for the periodontal ligament, A SM E J. B iom echanicalEngineering, 124(5), 538–546, 2002. [2] F. Genna, M. Perelmuter, Micromechanical modelling of the periodontal ligament in tension and shear: closed-form stress–strain relationships, A nnals ofB iom edicalEngineering, submitted, 2005. [3] C. Hurschler, B. Loitz-Ramage, R. Vanderby, Jr., A Structurally Based Stress–Stretch Relationship for Tendon and Ligament, A SM E J.B iom echanicalEngineering, 119, 392– 399, 1997.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Computation of cortical bone macroscopic properties from microscopic elastic data Quentin Grimal*, Kay Raum†, Pascal Laugier* * Laboratoire d'Imagerie Paramétrique Université Pierre et Marie Curie Paris6 / UMR CNRS 7623 15, rue de l'école de médecine - 75006 PARIS, France [email protected], [email protected] †

Q-BAM Group, Dept. of Orthopedics Martin Luther University of Halle-Wittenberg, Magdeburger Straße 22, 06097 Halle, Germany [email protected]

ABSTRACT Introduction. Cortical bone is a multi-scale compact tissue with a hierarchical organization. It exhibits both structural heterogeneities (various levels of porosity) and heterogeneity of the elastic properties of the bone matrix. The scales relevant for applications in the field of biomechanics range from the micrometer to the centimeter. Numerical values of the mechanical properties at various scales are needed as input in numerical models of bone such as finite element models widely used in orthopedics and finite difference models used for the simulation of bone-ultrasound interaction. Furthermore, models are required to assess the impact at the macro level of pathologies, age-related changes, and therapies which act at the microscopic level. The aim of this work is to obtain numerical values for the linear elastic mechanical parameters of cortical bone at a scale of 1mm from a quantitative mapping of structural and elastic data at the microscopic level. The 1mm-scale is considered as macroscopic. In addition, the resolution of 1 mm would possibly be compatible with current computing capacities. Method. Quantitative acoustic impedance images of eight transverse sections of human radii were obtained with 50-MHz scanning acoustic microscopy (SAM). The resolution of the images is 23 µm. The measured acoustic impedance data were converted into isotropic elastic data at each point of the image [1]. For each radius section, between 50 and 60 square regions of 1 mm square were randomly selected and extracted from the impedance images of the cortical shell. Each extracted region was considered as a “virtual” heterogeneous sample. A program was developed to compute the effective stiffness tensor (equivalent homogeneous material) of each virtual sample using displacements boundary conditions. For each virtual sample, the mechanical homogenization problem was solved using the finite element method. Assuming hexagonal symmetry, five engineering elastic constants of an equivalent homogeneous material were derived upon post-treatment of the effective computed stiffness tensors. Results. The dependence of the elastic constants on porosity and mean elastic value of the bone material around the pores was analyzed. This dependence was used to formulate a phenomenological model of the elastic parameters at the 1-mm scale as a function of local elastic stiffness and porosity.

References [1] Raum, K. et al., Derivation of elastic stiffness from site-matched mineral density and acoustic impedance maps. Phys Med Biol, 2005, in press

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

592

Computational Homogenization in Multi-scale Shell Analysis at Large Strains ¨ Rafael Grytz∗ and Gunther Meschke ∗ Institute for Structural Mechanics Ruhr-University Bochum, Bochum, Germany [email protected], [email protected]

ABSTRACT Biological tissues such as those existing in the eye, heart, veins or arteries are heterogeneous on one or another spatial scale and can undergo very large strains in the elastic range (hyperelasticity). Frequently, these tissues are characterized by thin-walled, shell-like structures. The use of computational homogenization schemes [1], [2], [3] together with a formulation of the continua in curvilinear coordinates is a prerequisite for realistic biomechanical simulations of thin-walled soft tissues on different scales. The goal of this contribution is the generalization of the computational homogenization scheme for the formulation of micro-macro transitions in curvilinear convective coordinates. We consider a homogenized macro-continuum with a locally attached representative micro-structure. The deformation and the coordinate system of the micro-structure are assumed to be coupled with the local deformation and the local coordinate system at a corresponding point of the macro-continuum. For a consistent formulation of micro-macro transitions principal material directions are defined on both scales. To formulate the generalized micro-macro transitions in absolute tensor notation the new operations scale-up and scaledown are introduced. The multi-scale analysis discussed in this paper considering arbitrary curvilinear coordinates as well as geometrical and material nonlinearities rests upon a finite-element discretization of the macro-continuum by means of a bilinear finite shell element with a quadratic kinematic assumption in thickness direction [4]. A finite-element discretization of the micro-structure is attached to each integration point of these macro-elements and is modeled with trilinear brick elements.

References [1] C. Miehe, J. Schotte, and J. Schr¨oder, Computational Micro-Macro Transitions and Overall Moduli in the Analysis of Polycrystals at Large Strains. Computational Materials Science, 16, 372– 382, 1999. [2] S. L¨ohnert and P. Wriggers, Homogenisation of Microheterogeneous Materials Considering Interfacial Delemination at Finite Strains. Technische Mechanik, 23, No. 2–4, 167–177, 2003. [3] V.G. Kouznetsova, Computational Homogenization for the Multi-Scale Analysis of Multi-Phase Materials. PhD thesis, Eindhoven University of Technology, Department of Mechanical Engineering, The Netherlands, 2002. [4] Y. Bas¸ar and R. Grytz, Incompressibility at Large Strains and Finite-Element Implementation. Acta Mechanica, 168, 75–101, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

593

Mineral Proximity Influences Protein Unfolding: A Molecular Dynamics Study Dinesh R. Katti, Pijush Ghosh, Kalpana Katti North Dakota State University Department of Civil Engineering CIE 201, NDSU, Fargo, ND, 58105, USA {dinesh.Katti, pijush.ghosh, kalpana.katti}@ndsu.edu

ABSTRACT Hybrid protein mineral interfaces are found in a number of biological materials and are of significant interest to researchers in the biomedical fields and in materials science and engineering. The study of protein unfolding both experimentally and using molecular dynamics has been undertaken by researchers to understand both biochemical aspects as well as mechanics of structural proteins. However, most of the research has been conducted on individual protein molecules without considering the influence of adjacent mineral. Inspired by the protein mineral interface of biological nanocomposite nacre, the shiny inner layer of seashells, we have conducted a comprehensive study of the role of mineral proximity on the unfolding of proteins. Our previous work has shown that the nanoscale organic phase sandwiched between the mineral aragonite in nacre has extraordinary mechanical properties. The organic phase consists of proteins and polysaccharides. In the current work, we are using steered molecular dynamics techniques to understand the influence of aragonite on a domain commonly found in structural proteins and nacre proteins. This research required the development of CHARMm force field parameters for aragonite to allow us to conduct the interaction studies. Our paper describes the modeling techniques and the results of the simulations with and without mineral proximity. Our results indicate that the mineral has a very significant influence on the unfolding response of the protein domain. Our paper also describes the influence of the mineral on the unfolding mechanisms and specific regions of the protein domain influenced by the mineral. Our work gives an insight into protein behavior at mineral protein interfaces. The work highlights the importance of understanding and modeling molecular interactions at interfaces for accurately predicting the behavior of biological composites.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

594

Modeling the Role of Interfaces on Mechanical Response in Composite Bone Biomaterials Kalpana S. Katti, Rahul Bhowmik, Dinesh R. Katti, Devendra Verma North Dakota State University Department of Civil Engineering CIE 201, NDSU, Fargo, ND, 58105, USA {Kalpana.Katti, Rahul.Bhowmik, Dinesh.Katti, Devendra.Verma}@ndsu.edu

ABSTRACT Polymer-hydroxyapatite (HAP) composites have potential use as bone replacement materials and are also the subject of several recent research studies. The molecular interactions between the mineral and polymer are known to have significant role on mechanical response of the composite system. We have used molecular dynamics to model the interaction between the polymers and HAP. Molecular dynamics studies require force field parameters for both molecules. Some force fields are described in literature representing the structure of hydroxyapatite reasonably well. Yet, the applicability of these force fields for studying the interaction between dissimilar materials (such as mineral and polymer) is limited, as there is no accurate representation of polymer in these force fields. We have derived the parameters of CVFF (consistent valence force field) for monoclinic hydroxyapatite. These parameters are validated by comparing the computationally obtained unit cell parameters, vibrational spectra and atomic distances with XRD and FTIR experiments. Using the previously obtained parameters of HAP and available parameters of polymer (polyacrylic acid), interaction study was performed using MD simulations between these dissimilar molecules. The MD simulations indicate that several hydrogen bonds and chelation bonds may form between HAP and polyacrylic acid depending upon the exposed surface of HAP. Also, the favourable planes of HAP where polyacrylic acid is most likely to attach are obtained. We have also simulated the mineralization of HAP using a “synthetic biomineralization”. These modeling studies are supported by photoacoustic spectroscopy experiments on both porous and non porous composite samples for potential joint replacement and bone tissue engineering applications.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

595

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

596

Microstructural model of the viscoelastic behaviour of biological tissues ∗ ¨ Fanny Moravec∗ , Milan Muller ∗ Department

of Mechanics, University of West Bohemia in Pilsen Univerzitni 22, 306 14 Pilsen, Czech Republic [email protected], [email protected] ABSTRACT

A model of viscoelastic material, whose elastic and dissipative potentials are build on a microscopic restructuring, is proposed. The continuum body is viewed as a collection of ‘elementary structures’ whose borders’ deformation is governed by the gradient of deformation F. Each elementary structure is constituted of a central cell floating in a viscous matrix to mimic, in a simple way, the micro-structure of biological tissues. The application of an external load leads to the restructuring of the inner geometry of the elementary structure by stretching or contracting the fibers present in cell and in matrix, and by running the flow of the viscous fluid around the central cell. Simulations are restricted to the two-dimensionnal unidirectional traction test. In this case, the inner configuration of the elementary structure is completely characterized by one size length of the cell, let say c1 . The law governing the time evolution of the internal variable c1 is determined solving an ordinary differential equation, resulting from the compensation between elastic and viscous forces within the elementary structure f (c1 ) + g(c1 , c˙1 ) = 0, where f derives from the elastic potential, and g derives from the dissipative potential. The time dependence is transmitted to the macroscopic behaviour of material. A simulation test applying an instantaneous strain is done and the relaxation properties of the material are discussed.

Figure 1: Traction test and material microstructure.

Figure 2: Response of the microstructure to an instantaneous strain loading

Figure 3: Response of the macro-stress to an instantaneous strain loading (relaxation phenomenon).

References [Holeˇcek 2005] M. Holeˇcek, F. Moravec, Hyperelastic model of a material whose microstructure is formed by “balls and springs”, in submission to the International Journal of Solids and Structures. [Pioletti 2000] D.P. Pioletti, L.R. Rakotomanna (2000), Non-Linear viscoelastic laws for soft biological tissues, Eur. J. Mech. A/Solids 19, 749-759

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Scaling views on strength of soft/hard composites Ko Okumura Department of Physics, Graduate School of Humanities and Sciences Ochanomizu University, 2-1-1, Otsuka, Bunkyo-ku, Tokyo 112-8610, Japan [email protected] ABSTRACT We frequently find composite structures, especially comprised of soft and hard elements, in many strong materials in nature: timber, teeth etc. For example, in nacre, found inside of certain seashells, hard and brittle aragonaite plates are glued together by soft and thin protein layers. Toughness of nacre is about 3000 times as high as that of pure aragonite, although the volume fraction of the soft protein is only 1/100! This lecture concerns, on a macroscopic level of continuum models, fracture properties of nacre and similar layered structures. We show that one of the important mechanisms of the enhancement of strength is a weakened stress concentration around the tip due to the structure [1-4]. Our approach may be quite unusual in the light of computational solid and structural mechanics; we work with the vision of impressionists: ignoring many details to capture simple views from complex systems; in many cases, discussing only on the level of scaling laws (i.e., power laws), which is one of the standard strategies of soft matter physics. The mechanism of the above less effective stress concentration works in the regime of linear elasticity. With the above-mentioned strategy, we also discuss how viscoelastic effects can play a vital role. This predicts an unusual crack shape, different from the conventional parabolic one in the linear elastic fracture mechanics: the shape is like that of a trumpet! [5] In addition, we may discuss some of the following topics: (1) Possible views on an isotropic composite [6]: doublenetwork gel consisting of two independently cross-liked networks, composites especially promising for artificial joints that are synthesized recently [7]. (2) Elastic particle-reinforced composites [6]. We propose a continuum elastic model to extract the controlling parameters of toughness of the composites. (3) Strength of perforated paper [8], related to freezing effects of apples and potatoes in food science [9]. (4) a direct experimental confirmation of Griffith’s scaling law in linear-elastic polymer foam [10].

References [1] P.-G. de Gennes and K. Okumura, C. R. Acad. Sci. Paris t.1, Ser. IV (2000) 257. [2] K. Okumura and P.-G. de Gennes, Eur. Phys. J. E 4 (2001) 121. [3] K. Okumura, J. Phys.: Condens. Matter 17 (2005) S2879. [4] K. Okumura, Eur. Phys. J. E 7 (2002) 303. [5] K. Okumura, Europhys. Lett. 63 (2003) 701. [6] K. Okumura, Europhys. Lett. 67 (2004) 470. [7] J. P. Gong, Y. Katsuyama, T. Kurokawa, and Y. Osada, Advanced Materials 15 (2003) 1155. [8] S. Nakagawa, K. Okumura and J. F.V. Vincent, submitted. [9] A. Khan and J. F.V. Vincent, J. Texture Studies 27 (1996) 143. [10] Y. Shiina, Y. Hamamoto, and K. Okumura, in preparation.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A micro-Macro strategy for ship structural analysis with FETI-DP method A. Mobasher Amini∗ , D. Dureisseix† , P. Cartraud∗ , N. Buannic‡ ∗ G` ´ eM, CNRS UMR 6183 / Ecole Centrale de Nantes 1, rue de la No¨e, BP 92101 - 44321 NANTES CEDEX 3 - France [email protected] [email protected] † LMGC, CNRS UMR 5508 / Universit´ e Montpellier 2 CC048,Place E. Bataillon, 34095 MONTPELLIER CEDEX 5, France [email protected] ‡ Principia

Marine 1, rue de la No¨e, BP 22112 - 44321 NANTES CEDEX 3 - France [email protected] ABSTRACT In the analysis of ship structures at small scale, with structural details heterogeneities and because there is only one prototype produced, which is the final product, the designers rely on finite element simulations. The finite element discretization of such structure, leads to a huge global numerical model, that suffers for computational cost and memory resource that may be unaffordable. In such a case, a multi-scale analysis should be performed. The classical local-global analysis that is used by engineers has several limitations such as:

• structure details are not periodic, therefore classical homogenization methods are not easily applicable; • edge effects are not taken into account; • zooming techniques are not easy to use: the gluing they require with the global scale often introduces artificial edge effects. This paper presents a micro-macro strategy based on the domain decomposition FETI-DP method as the solver in analysis of ship structure. With this approach, the two scales (micro and macro) are coupled during the iterations of the solver and we can consider the structural details in areas of interest, area where the fine mesh is used and a sub-domain is located. Performances are discussed and results in term of convergence are presented for several examples.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

599

On the use of Fourier expansions for the simulation of elastic composite pipes with defects E. Baranger∗ , O. Allix∗ , L. Blanchard† ∗ LMT-Cachan 61 Avenue du Pr´esident Wilson 94235 Cachan Cedex France [email protected], [email protected] † Alcatel Alenia Space 100 Boulevard du midi BP 99 Cannes la Bocca Cedex 60156 France [email protected]

ABSTRACT Due to the manufacturing process, defects such as delamination or matrix cracking are present in composite pipes used in satellite applications. To determine if these parts must be rejected, an experimental approach is used at the present time. The purpose of this study is to provide the Alcatel Alenia Space engineers with a dedicated numerical decision-making tool and, thus, reduce both the cost of verification and the number of rejected pipes. The behavior of the damaged pipes is modeled at the scale of the elementary ply with the model described in [1] following the work of [2]. A major issue at this scale is the computational cost, because the ply thickness is 0.2 mm. This leads to solve a several million degrees of freedom problem to catch edge effects. The proposed strategy is divided in two steps. First, due to the damage location, the exact beam theory developed in [3] is used to build the elastic solution at the middle of the pipe [4] while the ends are treated using a fine non linear modeling. Second, the elastic problem defined on the end zone and used in a non linear resolution scheme is solved using special finite elements based on Fourier expansions in the hoop direction. This allows, by the use of a preconditioned conjugate gradient method, to uncouple the resolution of the non axisymmetric problems [5]. A major question is then tackled, it is concerning the expansion order needed to get a given quality in the hoop description. Industrial cases of several millions of degrees of freedom with defects have been treated in elasticity with a prototype code developed in Matlab.

References [1] C. Hochard, P. A. Aubourg and J. P. Charles, Modelling of the mechanical behaviour of wovenfabric CFRP laminates up to failure, Composites Science and Technology, 61, 221–230, 2001. [2] P. Ladev`eze and E. Le Dantec, Damage modelling of the elementary ply for laminated composites, Composites Science and Technology, 43(3), 257–267, 1992. [3] P. Ladev`eze and J. Simmonds, New concepts for linear beam theory with arbitrary geometry and loading, European Journal of Mechanics and Solids, 17(3),377–402,1998. [4] E. Baranger, O. Allix and L. Blanchard, A dedicated computational strategy for composite pipes: basic principle and illustration, Science and Engineering of Composite Materials, 12(1-2), 2005. [5] O. Allix, E. Baranger and L. Blanchard, An efficient strategy for the calculation of end effects on composite pipes : the thermoelastic case, Composite Structures, accepted.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

600

A multigrid approach for non-linear structural analysis in explicit dynamics B. Bourel∗,† , A. Combescure∗ , L. Di Valentin† ∗ Laboratoire

de M´ecanique des Contacts et des Solides, UMR CNRS 5514, INSA Lyon 20 Av. Albert Einstein, 69621 Villeurbanne, France [email protected], [email protected] † PSA Peugeot Citro¨ en Route de Gisy, F-78943 V´elizy-Villacoublay Cedex, France [email protected]

ABSTRACT This study deals with a method to change the space-time scales for multi-domains calculations in explicit dynamics. The interest of such a method inspired by the techniques of mesh refinement [1], is to improve only when necessary the space (and time) discretization of one or more subdomain, during a significant phase of calculation. The method is based on a mesh refinement or coarsening which is activated according to a predefined criterion. Here, the different meshes used for the same domain are defined before the calculation. This is why we will refer to switch or mesh change rather than remeshing. Although this work is a part of researches on multi-domains approaches [2,3], we will concentrate here on refinement of one of the subdomains omitting the interactions with the other subdomains. After the mesh change, the different mechanical fields associated with the new mesh must be projected from the old mesh and must allow us to continue the calculation on the new mesh. This continuation is correct if the projected fields, associated with the new mesh, satisfy the equilibrium equations as well as possible. The transfer of fields by simple interpolation does not ensure, in general, this condition, in particular for problems with high non-linearities. Moreover, these simple interpolation methods become particularly unstable when used in an dynamic explicit scheme. So, the main difficulty will be to maintain stability and precision of calculation. That is why we will concentrate on the transition step and on the way of equilibrating the solution on the new discretization in the linear and non-linear case (material non-linearity). Finally, the decision to switch from a coarse mesh to a fine mesh is controlled here by a physical criterion based on the maximum plastic strain. Currently the algorithm developed in c has been tested on several simple geometries. These first examples allowed us to validate CASTEM the method and to show its efficiency in the linear and non-linear case.

References [1] P. Cavin, A. Gravouil, T. Lubrecht, A. Combescure. Efficient FEM calculation with predefined precision through automatic grid refinement, Finite Elements Anal. Des. 41 : 1043-1055, 2005. [2] B. Herry, L. Di Valentin, A. Combescure. An approach to the connection between subdomains with non-matching meshes for transient mechanical analysis, Int. J. Numer. Meth. Engng. 55 : 973-1003, 2002. [3] A. Gravouil, A. Combescure. Multi-time-step and two-scale domain decomposition method for non-linear structural dynamics, Int. J. Numer. Meth. Engng. 58 : 1545-1569, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

601

Mechanical behaviour of textile structures: two-scales approach V. Carvelli , C. Corazza, C. Poggi Department of Structural Engineering, Technical University (Politecnico) of Milan Piazza Leonardo da Vinci 32, 20133 Milano (Italy) {valter.carvelli, corazza, carlo.poggi}@polimi.it

ABSTRACT Technical textiles are widely used in different industrial fields for applications like screen printing and filtration. In order to predict the mechanical response of textile structural components, the knowledge of the textile mechanical features is indispensable in the design and the development activities. The prediction of the mechanical properties of textiles has been item of several researchers by different approaches. In particular the investigations can be grouped in three main methodologies: experimental (see e.g. [1]), analytical (see e.g. [2]) and numerical (see e.g. [3]). This paper presents a numerical approach based on two scale modelling. The first scale deals with the prediction of the mechanical behaviour of the plane weave monofilament textile by the numerical analysis of a Representative Volume (RV size of the order 10-5 m) assuming a regular distribution of the fibres in the warp and weft directions (Figure 1a). In the second scale modelling the numerical analysis of the entire textile component (size of the order 1m) is performed employing the homogenized textile mechanical behaviour obtained in the first step. The first scale analysis (textile scale) is validated by experimental results including uniaxial and biaxial tensile tests. The second scale analyses deals with a cylindrical component employed in a screen printing technique (Figure 1b). Different configurations of the cylinder are considered in order to understand the influence of the textile and the geometric parameters on the deformed shape (i.e. the print quality).

(a) (b) Figure 1. (a) First scale modelling, analysis of the RV. (b) Second scale modelling, analysis of the structural component.

References [1] A. Gasser, P. Boisse, S. Hanklar; Mechanical behaviour of dry fabric reinforcements. 3D simulations versus biaxial tests, Comp. Mat. Science, vol. 12, 7-20, (2000). [2] S.V. Lomov, G. Huysmans, I Verpoest; Hierarchy of textile structures and architecture of fabric geometric models, Textile Research Journal, vol. 71, 534-543, (2001). [3] M Tarfaoui, J. Y. Drean; Predicting the stress-strain behaviour of woven fabrics using the finite element method, Textile Research Journal, vol. 71, 790-795, (2001).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

602

On multilevel strategies for nonlinear computations with domain decomposition: application to post-buckling Philippe Cresta†,*, Olivier Allix*, Christian Rey*, Stéphane Guinard† *Laboratory of Mechanics and Technology (LMT) ENS Cachan, 61 av. du Président Wilson, 94235 Cachan Cedex, France [email protected] †EADS Corporate Research Center (CRC) 12 rue Pasteur, BP 76, 92152 Suresnes Cedex, France [email protected]

ABSTRACT
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     [2] P. Le Tallec, J. Mandel, M. Vidrascu, A Neumann-Neumann Domain Decomposition Algorithm for Solving Plate and Shell Problems, J. Numer. Anal. , Vol. 35, No. 2, 836-867, 1998. [3] C. Farhat, K. Pierson, M. Lesoinne, The second generation FETI methods and their application to the parallel solution of large-scale linear and geometrically non-linear structural analysis problems, Comput. Meth. In Appl. Mech. and Engrg., 18, 333-374, 2000. , 833"'9

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

603

Textile fabric simulator: collisions handling at the level of yarns M. Kuprys*, R. Barauskas† *

Department of System Analysis, Kaunas University of Technology Studentu str. 50-313b, LT-51368 Kaunas, Lithuania [email protected]

†

Department of System Analysis, Kaunas University of Technology Studentu str. 50-407, LT-51368 Kaunas, Lithuanian [email protected]

ABSTRACT The paper deals with the modeling of the physical behavior of the woven structures imitating the textile fabrics. The model is focused on the mezzo-layer of the woven fabric, which gives us ability to investigate these structures in more precisely view. The model is based on a combined approach which presents longitudinal elastic properties of each yarn by a system of non-volumetric structural elements (springs), which us note as combined particles (CP-s), while the cross-deformation of the volumetric yarn is evaluated in a 3D space with the help of tight-fitting of oriented bounding boxes. Collision detection and response is an essential part of the simulation process. Dealing with deformable bodies it is the main time consuming stage of the computational model covering both collisions detection and response stages among colliding parts of the yarns. At a present time several techniques suitable for collision handling of deformable objects exist, so the brief discussion of the mostly suitable for level of yarns is presented. The focus of this paper is to develop an efficient model of the fabric as weaved structure. The collision detection algorithm implemented in this work is based on the idea of tracking the closest pairs of colliding elements known as temporal coherence together with the stochastic approach that is used to generate a new pairs of potentially colliding elements anywhere on the two approaching yarns. After local minima of a distance are reached the collision response scheme between two closest CP-s is applied. Collisions are updated at each time step during the simulation process, thus avoiding interpenetrations of the yarns while the cross-sectional deformations of the yarns are assumed not to exceed the half of the initial radius. An empiric model for evaluating deformations is used by assuming the geometrical shape of the cross-section being always elliptic with changing the length of the axes. The approach is a compromise between the simplified uni-dimensional rod system and a fully volumetric model of a yarn in a weave. It enables to achieve good performance along with the possibility to analyze the deformable yarn structure in 3D space. The advantage in comparison with traditional models presenting a yarn as a full volumetric deformable body is the significantly reduced number of degrees of freedom of the structure while preserving the “volumetric” behavior. Numerical examples considering the generation of the initial woven structure by tension of the structure of crimped yarns and the failure at shooting-through the fabric are presented.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

604

A computational strategy for contact simulation Julien Pebrel∗ , Pierre Gosselet∗ , Christian Rey∗ ∗ LMT Cachan ENS Cachan, 61 av du pr´esident Wilson, 94235 Cachan cedex, France {pebrel,gosselet,rey}@lmt.ens-cachan.fr

ABSTRACT We consider the simulation of structures undergoing frictional contact conditions. The chosen formulation [1] coupled with a Newton-like solver, leads to the resolution of a sequence of non-symmetric linear systems with non-invariant matrices. We propose a computational strategy based on non-overlapping domain decomposition method [2, 3, 4] and augmented Krylov iterative solvers [5]. After each Newton iteration, numerical information on the condensed interface problem is stored inside so called Krylov subspace; we propose to reinject most significant part of this information inside a second scale problem in order to accelerate the resolution of following systems. This strategy can be viewed as an extension of previous works [6] to non-symmetric problems: new strategies to build reused information and relevance estimators are assessed.

References [1] P. Alart and A. Curnier, A mixed formulation for frictional contact problems prone to Newton like solution methods Comp. Meth. Appl. Mech. Eng., 92, 353–375, 1991. [2] J. Mandel, Balancing domain decomposition Comm. in Appl. Num. Meth. and Eng., 9, 233–241, 1993. [3] C. Farhat and F.-X. Roux, Implicit parallel processing in structural mechanics Computational Mechanics Advances, 2 (1), 1-124, 1994. [4] P. Alart, M. Barboteux, P. Letallec and M. Vidrascu, M´ethode de Schwartz additive avec solveur grossier pour probl`emes non sym´etriques C.R. Acad. Sci. Paris, t. 331, S´erie I, 399-404, 2000. [5] Y. Saad, Iterative methods for sparse linear systems, 2000. [6] P. Gosselet and C. Rey, On a selective reuse of Krylov subspaces in Newton-Krylov approaches for nonlinear elasticity Proceedings of the 14th conference on domain decomposition methods, 419–426, 2002.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

605

A simulation strategy for life time calculations of large, partially damaged structures Christian Rickelt∗ , Stefanie Reese∗ ∗ Institute

of Solid Mechanics TU Braunschweig, D-38023 Braunschweig, Germany [email protected], [email protected] ABSTRACT The present paper is motivated by the increasing concern to accomplish more realistic lifetime estimations of complex engineering structures. For this purpose the entire system and additional mechanisms like damage evolution or changes in the loading situation as well as in the surroundings have to be incorporated into such a long-term computation. Undoubtedly the finite element method represents a suitable tool to this end. But inspite of the fast development of computer technology the life time computation is still to complex to be carried out without advantageous and effective strategies to reduce computational cost. In this contribution we present a discretisation strategy which takes into account that only small parts of a structure demand a non-linear analysis. Accordingly we strictly decompose our system on the structural level into non-linear and linear subsystems by an exact substructure technique. We are finally able to determine the entire system response by the solution of a number of small non-linear subsystems. Additionally, if it is required, the linear subsystems may be evaluated in a post processing calculation. Further we may reduce the number of degrees-of-freedom of the linear subsystems, because they influence the evolution of damage only indirectly. Hence we join our proposed substructure strategy with projection-based model reduction techniques for linear second order systems, like modal truncation, Ritz vectors and the proper orthogonal decomposition [1]. An alternative approach of partial model reduction is e.g. presented by [2]. In the non-linear substructures we model the evolution of damage for ductile damage behaviour of metals taking into consideration large inelastic strains by the material model of [3]. At the material level we exploit the advantages of a formulation in principle axes in combination with the exponential mapping algorithm. This material model is implemented into the computationally efficient Q1SP finite element formulation of [4], which is based on the concept of reduced integration with hourglass stabilisation.

References [1] P. Holmes, J.L. Lumley and G. Berkooz, Turbulence, coherent structures, dynamical systems and symmetry. Cambridge University Press, 1996. [2] J.E. Barbone, D. Givoli and I. Patlashenko, Optimal modal reduction of vibrating substructures. Int. J. Numer. Meth. Engrg., 57, 341–369, 2003. [3] A. Eckstein and Y. Bas¸ar, Ductile damage analysis of elasto-plastic shells at large inelastic strains. Int. J. Numer. Meth. Engrg., 47, 1663–1687, 2000. [4] S. Reese, On a physically stabilized one point finite element formulation for three-dimensional finite elasto-plasticity. Comput. Methods Appl. Mech. Engrg., 194, 4685–4715, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

606

Finite Element Method In First-principles Calculation Yoshinori Shiihara*, Osamu Kuwazuru†, Nobuhiro Yoshikawa† *

The University of Tokyo 4-6-1, Komaba, Meguroku, Tokyo, 153-8505 Japan [email protected] †

Institute of Industrial Science, The University of Tokyo 4-6-1, Komaba, Meguroku, Tokyo, 153-8505 Japan {kuwa, yoshi}@iis.u-tokyo.ac.jp

ABSTRACT We propose a finite element implementation for the first-principles calculation based on the Density Functional Theory (DFT). The atomic-scale simulation based on the DFT plays an important roll to predict various material properties such as the physical strength. Such simulation seems contribute much to design of new materials of useful functions without loborious compricated experiments. Practical complex atomic systems, such as interfaces of metal/ceramics, contain huge number of atoms and it certainly results in large-scale calculation. The traditional DFT scheme based on the plane-wave basis is not higly recommended for the large-scale calculation because the plane-wave basis scheme requires the Fast Fourier Transforms (FFT). The FFT requires all-node communications, which result in reduction of the parallel-computing performance. An advantage of the DFT scheme based on the Finite Element Method (FEM) is its parallelability. The localization of the finite elements in real space corresponds to the localization of components in the global matrix. This feature is suitable for the massively parallel computation. We formulate the DFT scheme based on the norm-conserving pseudo-potential technique by the FEM. The Kohn-Sham equation as the governing equation of the DFT is discretized by the Galerkin’s weighted residual method. The Kohn-Sham equation is treated as a integral form in the FEM scheme and the nonlocal pseudo-potential can be easily estimated by the integral form. In our finite-element formulation the divergence term in the periodic local potential is treated by the Ewald scheme[1]. Inadequate setting of parameters employed in the Ewald scheme gives wrong potential, yield incorrect results and also causes the inefficient calculation. We carry out the parameter setting for the Ewald scheme in the real-space method. The optimized parameters are systematically obtained and the computational efficiency and the numerical accuracy is conserved. Test calculation for a silicon dimer is performed. Through the calculation, we show the fact that 1) Our FEM formulation follows the variational principle. 2)The free energy obtained by our FEM formulation is consistent with the highly-converged value obtained by the established plane-wavebased package to the meV/atom order. 3) The order of the error in the free energy is O(element size4) in our FEM formulation. 4) The equilibrium bonding length predicted by our FEM code is consistent with that of the plane-wave-based package to the mÅ order.

References [1] M.C. Payne, M. P. Teter, D. C. Allan, T. A. Arias and J. D. Joannopoulos, Iterative minimization techniques for ab-initio total-energy calculations: molecular dynamics and conjugate gradient, Rev. Mod. Phys., 64, 1045-1097, 1992.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

607

Modelling and Simulation of Earthquake Ground Motion via Functional Series TARMA Models with Wavelet Basis Functions Minas D. Spiridonakos and Spilios D. Fassois1 Stochastic Mechanical Systems & Automation (SMSA) Laboratory Department of Mechanical & Aeronautical Engineering University of Patras, GR 265 00 Patras, Greece E-mail: {mspirid,fassois}@mech.upatras.gr Internet: //www.mech.upatras.gr/∼sms ABSTRACT The present study explores non-stationary Functional Series Time-dependent AutoRegressive Moving Average (FS-TARMA) models with wavelet basis functions for the modelling and simulation of earthquake ground motion. FS-TARMA models constitute conceptual extensions of their conventional (stationary) counterparts, in that their parameters are time-dependent belonging to functional subspaces [1]. Wavelets, with their scaling and localization in time, comprise a promising functional basis for “fast” evolutions in the dynamics. The study focuses on the assessment of wavelet based FS-TARMA modelling and simulation for two California earthquake ground motion signals: an El Centro accelerogram recorded during the 1979 Imperial Valley earthquake, and a Pacoima Dam accelerogram recorded during the 1994 Northridge earthquake. A systematic analysis leads to a TARMA(2, 2) model for the El Centro case and a TARMA(3, 2) model for the Pacoima Dam case. Both models are formally validated and their analysis and simulation (synthesis) capabilities are demonstrated via Monte Carlo experiments focusing on important ground motion characteristics.

Figure 1: (a) 2-D plot of the El Centro accelerogram non-parametric STFT-based time-dependent PSD estimate, and (b) 2-D plot of the TARMA(2, 2)[2,2] -based parametric Melard-Tjøstheim time-dependent PSD estimate.

References [1] A. Poulimenos and S. Fassois, Parametric time-domain methods for non-stationary random vibration modelling and analysis – A critical survey and comparison. Mechanical Systems and Signal Processing, 20, 763–816, 2006. 1

Corresponding author.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

608

Identification of Damage in Multispan Beams Using Parameter-dependent Frequency Changes and Neural Networks ´ ∗ Artur Borowiec∗ , Leonard Ziemianski ∗ Rzesz´ ow

University of Technology, Department of Structural Mechanics, W.Pola 2, 35-959 Rzesz´ow, Poland [email protected], [email protected] ABSTRACT

Nowadays the knowledge of the structure condition is considered to be more and more important. The state of the structure and its safety strongly depends on the degradation of the structure elements (beams, connections, etc.). Nondestructive methods predict the location and the extent of damage in existing engineering structures. The publications on the identification of damages present mainly the approach which implies the knowledge of eigenfrequencies and eigenmodes of an undamaged structure. The damage is identified on the basis of the variations of dynamic parameters with respect to the initial values [1]. Some methods require the introduction of external perturbations to the structure. The detection method, which provides the global assessment of damage, is usually not sensitive to the size of the damage. In paper by Dems et al. [2] to increase the accuracy of identification an additional parameter is introduced (namely concentrated elastic or rigid support, additional mass elastically or rigidly attached to the structure, boundary constraint). This paper is intended to provide the analysis of eigenvalues with respect to the additional mass [4] and the application of ANNs [3] to the damage identification. ANN is applied for the analysis of dynamic response of a structure and for the assessment of the structure condition. Herein three examples are discussed, in all of them ANNs are applied to develop a new method of identification. The assessment of the state of the considered structure relies, in the case of application of the proposed extended identification method, on the comparison of structure eigenfrequencies obtained from the systems with additional masses placed in different nodes. The differences in the source of information employed to identify the location and the extent of the damage. The additional parameter introduced to the structure increases the identification accuracy. The Artificial Neural Networks are able to locate the damage and the extent of the structure degradation. The obtained results show that it is possible to identify the damage using the dynamic responses of the structure. The results presented in this paper are very promising, in the next step more complicated structure will be taken into account. Moreover other perturbations should be also considered.

References [1] Deobeling S.W, Farrar C.R, Prime M.B, Sheritz D.W. Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristic: a literature review. Los Alamos Natl. Lab, 1996. [2] Dems K, Mr´oz Z. Identification of damage in beam and plate structure using parameter-dependent frequency changes. Engineering Computations, 18, 96-120, 2001. [3] Waszczyszyn Z., Ziemia´nski L., Neural Networks in Mechanics of Structures and Materials — New Results and Prospects of Applications, Computers&Structures, 79: 2261–2276, 2001. [4] Ziemia´nski L, Pia¸tkowski G. The detections and localizations of an attached mass in plates Proceedings of third European Conference on Structural Control, Vienna, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Microplane model parameters estimation using neural networks Anna Kuˇcerov´a, Matˇej Lepˇs, Jan Zeman Czech Technical University in Prague Th´akurova 7, Prague 6, 166 29, Czech Republic {anicka, leps, zemanj}@cml.fsv.cvut.cz ABSTRACT Concrete is one of the most frequently used material in Civil Engineering. Nevertheless, as a highly heterogeneous material, it shows very complex non-linear behavior, which is extremely difficult to describe by a sound constitutive law. As a consequence, a numerical simulation of response of complex concrete structures still remains a very challenging and demanding topic. One of the most promising approaches to modelling of concrete behavior is based on the microplane paradigm [1]. It is a fully three-dimensional material law that incorporates tensional and compressive softening, damage of the material, supports different combinations of loading, unloading and cyclic loading along with the development of damage-induced anisotropy of the material. As a result, the material model [1] is fully capable of predicting behavior of real-world concrete structures, once provided with proper input data. The major disadvantages of this model are, however, a large number of phenomenological material parameters and a high computational cost associated with structural analysis even in a parallel implementation [2]. The authors of the microplane model proposed a heuristic calibration procedure [1], that is based on the trial-and-error method but is computationally inefficient. Therefore, a new procedure based on artificial neural networks is proposed in the present contribution. In order to asses the reliability of identified material parameters, results of a stochastic sensitivity study based on the Latin Hypercube Sampling (LHS) method are presented first. Different tests, proposed in [2], are simulated numerically and used to determine, which model parameters can be reliably identified from these tests. In the next step, a neural network-based procedure is presented for identification of material parameters. A crucial point is the generation of a training set used to determine weights of individual neurons. To this end, the LHS method is again employed as it allows using a limited number of computational simulations while ensuring the representativeness of the generated training set. The training procedure itself is based on a real-coded genetic algorithm SADE [3]. Finally, the application of the proposed identification procedure to the back analysis of laboratory experiments is presented.

References [1] Z.P. Baˇzant, F.C. Caner, I. Carol, M.D. Adley, S.A. Akers, Microplane model M4 for concrete. Part I: Formulation with work-conjugate deviatoric stress, Part II: Algorithm and calibration, Journal of Engineering Mechanics-ASCE, 126, (2000), 944-953, 954–961. [2] J. Nˇemeˇcek, B. Patz´ak, D. Rypl, Z. Bittnar, Microplane models: Computational aspects and proposed parallel algorithm, Computers and Structures, 80, (2002), 2099–2108. [3] O. Hrstka, A. Kuˇcerov´a, Improvements of real coded genetic algorithms based on differential operators preventing the premature convergence, Advances in Engineering Software, 35, (2004), 237–246.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

610

ANNs and Linguistic Variables in the Analysis of Mine Induced Rockbursts Transmission to the High Building Krystyna Kuzniar Pedagogical University of Cracow ul. Podchorazych 2, 30-084 Krakow, Poland [email protected]

ABSTRACT Rockbursts are one of the negative phenomena accompanying underground mining. A large quantity of energy is released during a tremor. The energy causes propagation of seismic waves that reach the surface of the earth. They induce the building vibrations subsequently. Although these tremors are strictly connected with human activity, they differ considerably from other paraseismic vibrations. They are not subject to human control and they develop in an uncontrolled manner. In Poland, mining tremors resulted from underground raw mineral material exploitations in Legnica-Glogow Copperfield (LGC) induce the surface horizontal vibrations reaching even 0.2 acceleration of gravity (g) and vertical components reaching 0.3g. The large scale of the effects might be shown by the fact that the intensity of surface vibrations is greater than the predicted (and taken into consideration in structural design) intensity of vibrations from earthquakes in neighbouring countries: Slovak Republic, Czech Republic and Germany. Soil-structure interaction is a very important problem from the engineering point of view. The prognosis of vibration influences on structures as well as estimation of the way of ground vibrations transmission to building basements are essential. The comparison of maximal values (amplitudes) of vibrations (accelerations, velocities and displacements) recorded at the same time on the ground and on the basement level is the simplest and very often employed way of estimation of the vibrations transmission from the ground to the building. The paper deals with an application of artificial neural networks (ANNs) for evaluation of soilstructure interaction in case of the transmission of ground vibrations from mining tremors to building basement. The problem is analysed with respect to typical prefabricated eleven-storey building with load bearing walls. The influence of mining tremors parameters as mining tremor energy and epicentral distance on the soil-structure interaction effect is also discussed. The parameters are estimated as approximate values found experimentally. Therefore the linguistic variables associated with the fuzzy character of the parameters are introduced in the neural network analysis. From the obtained results it can be stated that application of simple neural networks enables us to predict the building foundation vibrations with satisfactory accuracy, thus effects of the transmission of ground vibrations to building foundation (soil-structure interaction) may be analysed using neural networks.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

611

Differential Flatness of Aircraft Flight Dynamics and Neural Inversion Wen C. Lu *, †, Lili Duan†, Félix Mora-Camino*, †, and Roger M. Faye# * LAAS du CNRS 7 avenue du Colonel Roche, 31077, Toulouse, France {wclu, mora}@laas.fr † ENAC 7 avenue Edouard Belin, 31055, Toulouse, France {wenchi.lu, lili.duan, felix.mora}@enac.fr #

Ecole Supérieure polytechnique BP5085, UCAD, Dakar, Sénégal [email protected]

ABSTRACT Differential flatness, a property of some dynamic systems, introduced by Fliess et al., has made possible the development of new tools to control complex nonlinear dynamic systems. Many dynamic non linear systems have been proved to be differentially flat. Some authors have investigated the differential flatness of conventional aircraft dynamics, although none of them has considered separately the flatness property of the flight guidance and the attitude dynamics of a rigid aircraft. In this paper, it is shown that the inertial position coordinates of an aircraft can be considered as differential flat outputs for its flight guidance dynamics. Since this differential flatness property is implicit, a neural network is introduced, as a numerical device, to deal with the inversion of the guidance dynamics. It is shown also that, once conveniently structured and trained, the neural network is able to generate in real time directives to conventional autopilot systems concerned with attitude and engine regime control so that the reference trajectory can be tracked. Numerical simulation results are displayed and discussed.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

612

Finite Element Analysis of an Energy Absorbing Crush Zone Using Expanded Metal A. Ayestarán*, C. Graciano† Universidad Simón Bolívar, Departamento de Mecánica, Caracas 1080-A, Venezuela. * [email protected] † [email protected]

ABSTRACT The design of crashworthy structures is a very important issue in automotive design. Structures able to absorb a great amount of energy during impact are a challenge for automotive engineers nowadays. Currently, computer simulation is an efficient and costeffective manner of designing structures and the understanding of their structural behavior can be studied in depth. This paper is aimed at studying the nonlinear behavior of the crush zone of a vehicle used for the Annual Formula SAE competition. The crush zone is made of expanded metal sheets and solid corners joined by welding. The use of expanded metal sheets reduces considerably the weight of the structure. In addition the energy absorbing properties of the crush zone are improved due to the imperfection sensitive nature of structures made of expanded metal sheets. A finite element model is built taking into material and geometrical nonlinearities using the commercial software MSC.Marc. In the numerical analysis, the crush zone is subjected to axial compression in order to obtain the energy absorbing properties of this crush zone. A sensitivity analysis is performed on the influence of the following parameters: material properties, gauge thickness, size of the expanded metal pattern and orientation of the pattern (longitudinal or transversal). The results show the enhancement in the energy absorbed by the crush zone obtained by means of using expanded metal.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

613

Design of Satellite Control System using the Optimal Nonlinear Theory Luiz Carlos Gadelha DeSouza National Institute for Space Research – INPE. Av. dos Astronautas, 1758, 12201-940 - São José dos Campos - SP-Brasil [email protected] †

Institution Second and Third Author Address [email protected]

ABSTRACT Nowadays, attitude control systems of satellites with rigid and flexible components are demanding more and more better performance resulting in the application of new advanced nonlinear control theory. This is the case when the dynamics of the plant that describe the attitude motion of the satellite is nonlinear and the mission involves stringent pointing accuracy. As a result, control designs methods presently available, needs more investigation to know their capability and limitations. In that context, the guaranty of the controller performance depends not only on its good design but also on the knowledge of the nonlinear characteristics of the system in order to improve the overall control system efficiency. In this paper, a new nonlinear control law for satellite attitude control (SAC) is presented. It is based on an extension of the linear quadratic regulator (LQR) theory to the case where the dynamics is described by a nonlinear system of equation (Euler´s equation). The control law performance is investigated in frequency domain evaluating the compromise between the gain level and bandwidth length. In time domain the control law performance is observed by its capability of shifting the overshot to the origin direction. By and large, one observes that the nonlinear terns in the control law are able to deal with the nonlinear term in the model, which reveals that the nonlinear control law is more efficient than the control law based on the linear theory, even in the presence of the parameters variation.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

614

Design for Crashworthiness of Train Structures with Simplified Multibody Models João P. Dias*, Filipe Antunes† and Manuel S. Pereira† *IDMEC - Instituto de Mecânica - Pólo IST Instituto Superior Técnico, Technical University of Lisbon, 1049-001 Lisbon, Portugal [email protected] *IDMEC - Instituto de Mecânica - Pólo IST Instituto Superior Técnico, Technical University of Lisbon, 1049-001 Lisbon, Portugal [email protected]; [email protected]

ABSTRACT In this paper simplified models for the design of vehicle structures under impact are presented. The use of multibody system dynamics based models in structural crashworthiness problems proved to be useful and accurate enough in simulating train collisions and in the optimization of energy absorption devices [1]. The presented simplified computational models are based on multibody rigid-flexible systems, where flexibility is included using the finite element method. Structural damping and contact models are also considered. In structural impact, members can be subjected to axial or bending tensions that usually result in components’ plastic deformation in areas known as plastic elements. Plastic elements can be modeled associating cinematic joints with non-linear springs, whose constitutive relationship correspond to the components’ collapse behavior. The constitutive relationship is computationally defined using parameter identification techniques. Sometimes not all the areas where plastic deformation occurs can be predicted, so, a methodology to detect plasticity and automatically insert new plastic elements in simplified models, called remodelling, is proposed. Since simplified models are based on multibody systems, crashworthiness simulations are much faster when compared to finite element commercial software, enabling the use of genetic algorithms for the design process [2]. Dynamic analysis formulations are integrated with the multiobjective optimization evolutionary algorithm NSGAII [3] resulting in a 2D mechanical systems multiobjective optimization tool, used to perform optimization simulations where parameters most frequently used in crashworthiness problems are studied. The presented methodologies and formulations have been implemented computationally in order to develop a tool for the first stages of vehicle design. Remodelling and multiobjective optimization examples are presented to demonstrate the presented methodologies.

References [1] J.P. Dias and M.S. Pereira, Optimal Design of Train Structures for Crashworthiness using a Multi-load Approach, I. J. of Crashworthiness, 7, 331-343, 2002. [2] Dias, J. P., Corrêa, R. e Antunes, F., “Crashworthiness Optimization of Train Structures with Evolutionary Algorithms”, EUROGEN 2003, 83, Barcelona, Spain, 15-17 September, 2003. . [3] K. Deb, Multi-Objective Optimization Using Evolutionary Algorithms, John Wiley and Sons, LTD, 2001.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

615

Analysis of Stress and Strain in the Absolute Nodal Coordinate Formulation with Nonlinear Material Behavior Johannes Gerstmayr*, Marko K. Matikainen† *

†

Institute for Technical Mechanics, Johannes Kepler University of Linz Altenbergerstr. 69, 4040 Linz, Austria [email protected]

Department of Mechanical Engineering, Institute of Mechatronics and Virtual Engineering Lappeenranta University of Technology, P.O.Box 20, FIN-53851, Lappeenranta, Finland [email protected]

ABSTRACT The present paper deals with the analysis of strain and stress in the absolute nodal coordinate formulation (ANCF). An accurate stress distribution is needed for the evaluation of comparative strains in nonlinear material behavior. The ANCF has been recently developed and studied by many investigators in the field of flexible multibody dynamics. The ANCF focuses on the modeling of beams and plates including large deformation and represents exact rigid body inertia. The derivation of the equations of motion for an ANCF element is usually based on a solid finite element formulation and thus leads to finite elements that show locking behavior. While the problem of locking in the ANCF might be solved by means of different techniques [1, 2], the accuracy of stress and strain quantities within the element is still poor and needs to be improved in order to incorporate nonlinear material behavior. In the present paper, a higher order element is presented where locking is prevented by means of standard selective reduced integration techniques and the improved order and accuracy of stress and strain quantities is shown in comparison with the original formulation. As an example of nonlinear material behavior, Prandl-Reuss plasticity is included to the absolute nodal coordinate formulation. The results of stress and strain components for the improved higher order element are compared to the solution of fully tree-dimensional computations performed with the commercial software ABAQUS. Good agreement of the ANCF is found with the results of ABAQUS as well as with examples of elasto-plastic multibody systems available from the literature.

References [1] J. Gerstmayr, A.A. Shabana, Efficient Integration of the Elastic Forces and Thin ThreeDimensional Beam Elements in the Absolute Nodal Coordinate Formulation, Proceedings of the Multibody Dynamics 2005 ECCOMAS Thematic Conference, Goicolea, Cuadrado, García Orden (eds.), Madrid, Spain, 2005. [2] A.L. Schwab, J.P. Meijaard, Comparison of three-dimensional flexible beam elements for dynamic analysis: finite element method and absolute nodal coordinate formulation, Proceedings of the ASME 5th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, paper number DETC2005-85104, ASME, New York, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

616

Nonlinear Multimode Dynamics of a Moving Microbeam For Noncontacting Atomic Force Microscopy S. Hornstein and O. Gottlieb

Department of Mechanical Engineering Technion – Israel Institute of Technology Haifa 32000, Israel. [email protected], [email protected]

ABSTRACT Atomic force microscopy (AFM) is a modern imaging technique that is used to map surfaces down to atomic resolution and enables a quantitative estimation of atomic interaction forces. This is obtained by measuring a van der Waals like atomic interaction between a sample and a vibrating microcantilever, which has a sharp tip at its free end. Of particular importance are biological and nonconducting materials that cannot be mapped by alternative methods, without destruction of their surfaces by a conducting coating layer. The growing demand for detection of sub-atomic features and industrial use, increases the need for faster and more accurate scanning. Various methodologies have been proposed to speed up the scan rate. These include individual moving microbeam control strategies and use of an array of probes. However, the accuracy of force estimation from measured data crucially depends on the quality of the mathematical model in use. A typically used model is that of a lumped mass system that reduces the microbeam to a linear spring with a nonlinear force, derived from a candidate tip-sample interaction. This model does not incorporate the dynamic boundary condition of the scan process and cannot resolve the rich spatio-temporal dynamic response of the nonlinear dynamical system. Thus, the objectives of this research include theoretical derivation and analyses of a continuous model for the moving and vibrating AFM microbeam that consistently incorporates the nonlinear atomic interaction and the dynamic conditions of the scan process. A nonlinear initial boundary-value problem is derived using the extended Hamilton's principle. The continuum system is then reduced to a multimode dynamical system using a Galerkin procedure. Numerical analysis of a three mode system reveal that below the 'jump-to-contact' stability threshold, there exist a dense set of coexisting bounded periodic (ultrasubharmonic) solutions. This complex bifurcation structure is augmented by quasiperiodic solutions that are found to correspond to a 3:1 internal resonance between the third and second microbeam modes.

‘Jump to contact’ stability threshold

A quasiperiodic time-series (top) and power spectra (bottom).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

617

Model Reduction with Mean-Axes in Deformable Multibody Dynamics Yi-shih Lin and Parviz E. Nikravesh Department of Aerospace and Mechanical Engineering University of Arizona, Tucson, AZ 85721 {yishih, pen}@email.arizona.edu

ABSTRACT Dynamic analysis of multibody systems containing deformable bodies is computationally time consuming due to the large number of deformation degrees-of-freedom. Transforming the equations of motion to modal space and then truncating a significant number of higher frequency modes can greatly reduce the problem size and hence improve the computational time. There is a misconception that if mean-axes are adopted as the floating reference frame for a deformable body, modal truncation will yield inaccurate results. This may be due to the misunderstanding that only free-free modes are compatible with the mean-axes and, therefore, mean-axes cannot be used for constrained systems. This paper attempts to clarify this issue and shows that it is perfectly fine to use the standard static and normal modes, as they are obtained in structural analysis, in conjunction with the mean-axis conditions.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

618

Impacts With Friction In Flexible Multibody Dynamics Juana M. Mayo Department of Mechanical and Materials Engineering University of Sevilla Camino de los Descubrimientos s/n, 41092 Sevilla [email protected]

ABSTRACT Most formulations on normal impacts of flexible multibody systems use the momentum balance equations. Newton’s hypothesis is usually employed, where the restitution coefficient is defined by the relative normal velocities of the impacting bodies before and after the collision. When friction is present in the impact, the process becomes more complicated. Different impact modes are possible: sliding, sticking or reverse sliding. The variables in the momentum balance equations are the changes in the velocity and the two components of the impulse, one in the normal direction to the common tangent of the contact surfaces and other in the tangent direction. To solve the equations two additional conditions are needed, one comes from the Coulomb law, and the other from the definition of the restitution coefficient. The use of the Newton hypothesis for the definition of the restitution coefficient leads to wrong results in the simulation of impacts with friction. Therefore, it is necessary to use the Poisson hypothesis. The restitution coefficient is defined through the normal impulses of the compression and restitution periods. In this paper a formulation of impacts with friction of planar flexible multibody systems is presented. The floating frame of reference formulation is used to model the flexible bodies. The normal and tangential impulses in the contact point are calculated by a computational algorithm based on the graphics techniques developed by Routh. Lankarani and Pereira used this technique to analyse impacts with friction of planar rigid multibody systems.

References [1] E.J. Routh, Dynamics of a system of rigid bodies. MacMillan, London, 1891. [2] H.M. Lankarani and M.Pereira, Treatment of impact with friction in planar multibody mechanical systems. Multibody System Dynamics, 6, 203-227, 2001.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A conservative augmented Lagrangian algorithm for the dynamics of constrained mechanical systems Juan C. Garc´ıa Orden∗ , Roberto A. Ortega∗ ∗ Computational

Mechanics Group. School of Civil Engineering. Universidad Polit´ecnica de Madrid. c/ Profesor Aranguren s/n 28040 Madrid, Spain [email protected], [email protected] ABSTRACT

The motion of many practical mechanical systems is often constrained. An important example is the dynamics of multibody systems, where these constraints arise from the modeling of joints that connect different bodies. The numerical solution of the dynamics of this type of systems faces several difficulties, mainly due to stability problems [1]. Different methods have been proposed in the literature to overcome these problems, based on different strategies for the constraints formulation. One of these strategies is the augmented Lagrange formulation, which allows the use of numerical integrators for Ordinary Differential Equations, combined with an update scheme for the algebraic variables, accomplishing exact fulfillment of the constraints. In this context, this work focuses on the design of a conservative version of this augmented Lagrangian formulation for holonomic constraints, proposing a numerical procedure that exhibits excellent stability, thus providing an interesting alternative for the dynamical analysis of these type of systems. A point of departure is a conservative formulation based in the penalty method [2], which exhibits good stability but does not accomplish exact fulfillment of the constraints.

References [1] K. E. Brenan, S. L. Campbell, and L. R. Petzold. Numerical Solution of Initial-Value Problems in Differential-Algebraic Numerical Solution of Initial-Value Problems in Differential-Algebraic. SIAM, 1996. [2] J. C. Garc´ıa Orden and J. M. Goicolea. Conserving properties in constrained dynamics of flexible multibody systems. Multibody System Dynamics, 4:225–244, 2000.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

620

Non Linear Model for Coupled Axial/Torsional/flexural Vibrations of Drill-strings. Marcelo T. Piovan∗† , Rubens Sampaio† ∗ Mechanical

Systems Analysis Group, Universidad Tecnol´ogica Nacional - Fac. Reg. Bah´ıa Blanca, 11 de Abril 461, Bah´ıa Blanca, BA, 8000, Argentina [email protected]

† Department

of Mechanical Engineering, Pontif´ıcia Universidade Catˆolica - Rio de Janeiro, Rua Marquˆes de S˜ao Vicente 225, Rio de Janeiro, RJ, 22453-900, Brasil [email protected], [email protected]

ABSTRACT In the present work a continuous model is presented to study, by means of finite element discretization, the coupling of extensional, flexural and torsional vibrations under a state of initial stresses on a drillstring, which is described as a vertical slender beam under axial rotation [1]. The structure is subjected to distributed loads due to its own weight, the reaction force and perturbation moments at the lower end. The beam structure is also confined to a move inside a rigid cylinder, which simulates the borehole [2]. The impacts and friction of the drill-string with the borehole are modeled employing simplified forms. It is known that the state of initial stresses (which implies accounting for geometrical non-linearities) affects the dynamics of slender beams. The vibrations of drill-strings are frequently analyzed by means of lumped parameter models [3]. Normally, these models employ equivalent lumped parameters which are obtained from experimental field data or from continuous models assuming one-mode approximation for extensional, flexural and torsional vibrations. However, the lumped parameter models do not include dynamical effects due to geometrical non-linearities. In this context, the objective of present work is to analyze the effects of geometrical non-linearities due to initial stresses in the vibration of drill-strings together with the patterns of vibroimpact and comparing the results with the predictions of linear models. The beam model is discretized using a finite element with 12 degrees of freedom. The results have shown an important influence of the geometric non-linearities (when compared with the predictions of a linear model) in the dynamic responses of the drill-strings, especially when the beam undergoes impact patterns with the borehole or the rock formation. This influence can be observed in the calculation of reaction forces at top position as well as the time histories of radial displacements.

References [1] Sampaio, R., Piovan, M.T. and Venero Lozano, G., 2005, ”Non Linear model for Coupled Axial/Torsional Vibrations of Drill-Strings”, Proceedings COBEM 2005. [2] Trindade, M.A., Wolter, C. and Sampaio, R., 2005, ”Karhunen-Lo`eve descomposition of coupled axial/bending vibrations of beams subjected to impact”, Journal of Sound and Vibration, Vol.279, pp. 1015-1036. [3] Yigit, A.S. and Christoforou, P., 2003, ”Fully coupled vibrations of actively controlled drillstrings”, Journal of Sound and Vibration, Vol.267, pp. 1029-1045.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

621

Identification of Time-Varying Structures Under Unobservable Excitation: An Overview and Experimental Comparison of Parametric Methods Aggelos G. Poulimenos, Minas D. Spiridonakos, and Spilios D. Fassois Stochastic Mechanical Systems & Automation (SMSA) Laboratory Department of Mechanical & Aeronautical Engineering University of Patras, GR 265 00 Patras, Greece E-mail: {poulimen,mspirid,fassois}@mech.upatras.gr Internet: //www.mech.upatras.gr/∼sms ABSTRACT This paper addresses the problem of parametric time-domain identification and dynamic analysis for time-varying mechanical structures under unobservable random excitation. The identification uses Time-dependent AutoRegressive Moving Average (TARMA) models (or state-space equivalents), which are conceptual extensions of their conventional (time-invariant) ARMA counterparts in that their parameters and innovations variance are varying with time. TARMA methods may be classified according to the type of mathematical structure imposed upon the evolution of the model parameters as follows: (a) Unstructured parameter evolution methods, (b) stochastic parameter evolution methods, and (c) deterministic parameter evolution methods [1]. The characteristics and relative merits of each class are outlined. A representative method from each class is then applied to the modelling of a time-varying (moving) laboratory structure. This structure consists of a beam with a cylindrical mass sliding on it at a selected speed. The setup is meant to model a “bridge-like” structure with a heavy vehicle travelling along its length. The beam is subject to broadband random force excitation, while structural identification is solely based upon its vertical acceleration response. The methods applied are: The Recursive Maximum Likelihood TARMA (RML-TARMA) method (unstructured parameter evolution), the Smoothness Priors TARMA (SP-TARMA) method (stochastic parameter evolution), and a Functional Series TARMA (FS-TARMA) method (deterministic parameter evolution) [1]. The models obtained by each method are shown to accurately describe the time-varying structural dynamics in terms of the “frozen-time” power spectral density function and the modal quantities. Modelling accuracy is judged based upon “frozen-configuration” (fixed mass location) characteristics of the experimental structure also extracted through “frozen-configuration” (multiple stationary experiments) identification (baseline modelling). The highest tracking accuracy and model parsimony (economy) is exhibited by the deterministic parameter evolution (FS-TARMA) method, followed by the unstructured parameter evolution (RML-TARMA) method, which is in turn followed by the stochastic parameter evolution (SP-TARMA) method. Overall, the results demonstrate the parametric methods’ applicability, effectiveness, high potential for parsimonious and accurate identification, and model-based dynamic analysis of time-varying structures under unobservable excitation.

References [1] A. Poulimenos and S. Fassois, Parametric time-domain methods for non-stationary random vibration modelling and analysis – A critical survey and comparison. Mechanical Systems and Signal Processing, 20, 763–816, 2006.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

622

Variational integrators for the rigid body dynamics Ignacio Romero ∗ E.T.S.I. Caminos Universidad Polit´ecnica de Madrid Profesor Aranguren s/n, 28040 Madrid, Spain [email protected]

ABSTRACT In recent years, variational integrators have emerged as an original, powerful, and promising family of methods for integrating in time the equations of nonlinear solid dynamics. See, for example, [1, 2, 3, 4]. The idea underlying these methods is strikingly simple: instead of discretizing the equations of motion (as traditional time stepping methods do), variational integrators seek the stationary solution of a discrete action functional. In order to do so, only an approximation of the Lagrangian is required which must be expressed in terms of the discrete positions of the system. The formulation of such a Lagrangian is trivial for mechanical systems with linear configuration spaces. In contrast, if the mechanical system has a nonlinear configuration space the task is not so simple. Rigid bodies and some models of rods and shells fall into the later case and they are of obvious interest in Computational Mechanics. The first two are formulated in SO(3), the space of proper rotation tensors, and the last one in S2 , the unit sphere. In this work we will present variational integrators for the Euler equations of rigid body dynamics formulated directly on SO(3), without equations of restriction. We will discuss the advantages and drawbacks of these methods when compared with traditional ODE solvers, paying special attention to the conservation properties.

References [1] J. M. Wendlandt and J. E. Marsden. Mechanical integrators derived from a discrete variational principle. Phys. D, 106(3-4):223–246, 1997. [2] C. Kane, J. E. Marsden, M. Ortiz, and M. West. Variational integrators and the newmark algorithm for conservative and dissipative mechanical systems. International Journal for Numerical Methods in Engineering, 49:1295–1325, 2000. [3] J. E. Marsden and M. West. Discrete mechanics and variational integrators. Acta Numerica, pages 357–514, 2001. [4] A. Lew, J. E. Marsden, M. Ortiz, and M. West. Asynchronous variational integrators. Archives of Rational Mechanics and Analysis, 167:85–146, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

623

Planning and Optimization of Maneuver Strategy of Large Flexible Space Structures Evtim V. Zahariev Institute of Mechanics, Bulgarian Academy of Sciences Acad. G. Bonchev Sy., bl. 4, Sofia 1113, Bulgaria [email protected]

ABSTRACT The nowadays space missions are implement by large lightweight structures which flexibility causes significant influence over the mission from launching till landing. Space ships, stations and satellites caring large flexible devices, for example, long booms, tethered satellites, solar arrays, antennae and many others implement complex motion in space. The reliable dynamic model and preliminary analysis of the tasks, as well as, of the possible scenarios and casual events are the preconditions for the mission success. Major undesirable phenomena are the large flexible deviations and vibrations exaggerated because of implementation of deployment, folding, maneuvers, etc. Planning of motion strategies for minimization and passive and active damping of flexible deviations and vibrations is of crucial importance. Solution of these problems is a challenging realm for scientific investigations [1]. The concepts developed are based on the modal and eigenvale analysis of the flexible systems. This approach considers small deviations around the equilibrium position and very often it is inapplicable for extremely large flexible systems. Large flexible deflections and vibrations are subject of the contemporary multibody system investigations [2]. The present paper regards the problems of path planning and optimization of maneuver motion of large space flexible structures as long booms and tethered satellites. The approach is based on the multibody system methodology for dynamics simulation and forward analysis. The optimization problem is defined as nonlinear programming problem. Polynomial approximation of the input motion is used, its coefficient being changed using optimization techniques. The principle for minimization of the total energy of the deformable system is applied in an algorithm for planning of controlled motion and suppression of flexible deviations. Admissible velocities of the maneuvers are estimated. The process of exaggeration of high order vibrations is analyzed. Examples of maneuver implementation of extremely flexible structures and damping of vibrations are presented.

References [1] D. Izzo, L. Pettazzi and C. Valente, A comparison between Models Representing Flexible Spacecrafts. 6th International Conference on Dynamics and Control of Systems and Structures in Space, Riomaggiore, Italy, 18-22 July, 2004. [2] W. Schiehlen, Multibody system dynamics: Roots and perspectives. Multibody System Dynamics, 1, 149-188, 1997.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Free plain motion of flexible beams in MBS – A comparison of models R. Zander∗ , H. Ulbrich∗ ∗ Institute of Applied Mechanics · Department of Mechanical Engineering Technical University of Munich · Boltzmannstr. 15 · 85 748 Garching · Germany [zander,ulbrich]@amm.mw.tu-muenchen.de

ABSTRACT In multibody systems (MBS) structural elasticities are usually described using one floating frame of reference for each body. This leads to a compact mathematical formulation for the described physics, but it is restricted to small deflections relative to the floating frame. To overcome this limit, we combine the MBS ideas with concepts of the finite element (FE) method. Single elements are treated in a formulation of hybrid MBS giving exact rigid body movements for single elements. The compact form of equations is maintained and independent discretisations for longitudinal and transversal deformations are possible. To permit the assembly of several elements to one structure, the equations of motion are transformed to a coordinate set motivated by FE, whereas global positions are used as nodal coordinates instead of displacements. This approach allows for large translations and geometrically large deformations of the entire structure. The ‘absolute nodal coordinate formulation’ (ANCF) [2] also uses global coordinates and provides exact rigid body motions for single elements, thus allowing for large structural deflections. Instead of rotations, the derivatives of coordinates are used. In contrast to our approach, the ANCF uses equal discretisations for longitudinal and bending deformations in the case of a one-dimensional continuum and gives a constant mass matrix. Based on two examples, both models are compared with respect to their relative accuracy and numerical efficiency. As a first model a simple crank-shaft with a highly flexible long crank is investigated. To minimise the influence of the time discretisation on the results, an error-controlled time integration is used for these simulations. For fine spatial discretisations both models converge to one solution which is used as a reference for the evaluation of the model. The main focus lays on the number of degrees of freedom and the computational effort that is required for each method to obtain a desired accuracy. Moreover, the relative error using equal numbers of degrees of freedom is studied. The second example – an elastic rocking rod – is built by a highly flexible and initially straight beam bouncing on two point obstacles. Impacts and constraints are treated following modern methods for non-smooth rigid MBS dynamics [1] as an extention to flexible systems. In this context, the computational effort for contactsolving together with the robustness of the formulations are investigated.

References [1] F. Pfeiffer, M. F¨org, and Heinz Ulbrich. Numerical aspects of non-smooth multibody dynamics. Computer Methods in Applied Mechanics and Engineering, 2005. In Press, Available online 10 October 2005. [2] A. A. Shabana. Computer implementation of the absolute nodal coordinate formulation for flexible multibody dynamics. Nonlinear Dynamics, 16:293–306, 1998.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Numerical computation of Non Linear Modes of elastic structures Arquier R.∗ , Cochelin B.† Laboratoire de mecanique et d’acoustique CNRS UPR7051 31 chemin Joseph-Aiguier 13402 Marseille cedex 20 FRANCE ∗ [email protected] † [email protected]

ABSTRACT This paper concerns the computation of nonlinear modes of elastic structures under large displacements. We present a numerical method that we have implemented in a general purpose finite element code. Bifurcation of modes will be also addressed. We begin by introducing a general simple quadratic framework that is suitable for most elastic models (beam, plate, shell) and most classical finite elements. We define the non linear modes as two dimensional invariants of the phase space which are tangent to the eigenspaces of the associated linear system [1]. Theses invariant subsets are determined by making continuation of one dimensional families of periodic orbits. The periodic solutions are computed using the periodic orbit approach [2]. We use the exact energy-conserving Simo scheme [3] to time-discretise the periodic orbits. We do not use the classical shooting method to compute the periodic orbits but another one which consist to write the governing equation at each time step in a whole system. This lead to a large system of algebraic equations containing the displacement of the degres of freedom at each time step. The nonlinear modes (or their approximations) are obtained by making the continuation with respect to adequate parameters. We use the asymptotic-numerical method [4] for this purpose, since it is particularly efficient for such difficult problems with quite complex bifurcation diagrams.

References [1] S.W. Shaw and C. Pierre, Non-linear normal modes and invariant manifolds. Journal of Sound and Vibration, 150(1), pp. 170-173, 1991. [2] R. Seydel, Practical Bifurcation and Stability Analysis, from equilibrium to chaos. SpringerVerlag, second edition , 1994 [3] J. C. Simo and N. Tarnow, The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics. Z angew Math Phys, 43,pp. 757-792, 1992 [4] B. Cochelin, N. Damil and M. Potier-Ferry, Asymptotic numerical methods and Pade approximants for non-linear elastic structures. International journal for numerical methods in engineering, 37, pp 1187-1213, 1994

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Nonlinear modes: amplitude-phase formulation and bifurcation analysis Sergio Bellizzi, Robert Bouc Laboratoire de M´ecanique et Acoustique - CNRS 31 chemin Joseph Aiguier 13402 Marseille France {bellizzi,bouc}@lma.cnrs-mrs.fr

ABSTRACT Nonlinear Modes (NMs) are efficient tools for analysing the behaviour of dynamical mechanical systems[1]. The objective of this contribution is to show how this concept can be used to characterize periodic orbits and limit cycles. Following Shaw and Pierre[2] the concept of NMs is introduced here in the framework of the invariant manifold theory for dynamical systems. A NM is defined in terms of amplitude, phase, frequency, damping coefficient and mode shape with the distinctive feature that the last three quantities are amplitude and phase dependent. An amplitude-phase transformation is performed to give as well the time evolution of the NM motion (through the two first order differential equations governing the amplitude and phase variables) as the geometry of the invariant manifold. The conservative case was considered in [4]. The formulation is extended here to autonomous mechanical systems including gyroscopic and/or nonlinear damping terms. Our approach differs of that in [3] where the amplitude-phase transformation is based on the frequency of the linearized system. Bifurcation analysis, existence and stability of periodic orbits on the associated invariant manifold can be studied from the differential equations governing the amplitude and phase variables.The procedure is illustrated on a 2 DOF van der Pol mechanical system.

References [1] A.F. Vakakis, Nonlinear normal modes (NNMs) and their applications in vibration theory: an overview. Mechanical Systems and Signal Processing, 11(1), 3–22, 1997. [2] S.W. Shaw, C. Pierre, Normal modes for nonlinear vibratory systems. Journal of Sound and Vibration, 164(1), 85–124, 1993. [3] E. Pesheck, C. Pierre, S.W. Shaw, A new Galerkin- based approach for accurate non-linear normal modes through invariant manifolds. Journal of Sound and Vibration, 249(5), 971–993, 2002. [4] S. Bellizzi, R. Bouc, A new formulation for the existence and calculation of nonlinear normal modes. Journal of Sound and Vibration, 287(3), 545–569, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

627

Multi-Modal Non-linear Free Vibration of Thin Isotropic Circular Plates M. Haterbouch1 , P. Ribeiro2 , R. Benamar3 1 LMCS,

Facult´e des Sciences et Techniques d’Errachidia BP 509 Boutalamine, Errachidia, Morocco [email protected]

2

IDMEC/DEMEGI, Faculdade de Engenharia, Universidade do Porto Rua Dr. Roberto Frias, 4200-465 Porto, Portugal [email protected] 3

GDNLS/LERSIM, Ecole Mohammadia d’Ing´enieurs BP 765 Agdal, Rabat, Morocco [email protected] ABSTRACT

The large-amplitude periodic free axisymmetric vibration of clamped immovable thin isotropic circular plates is investigated using the energy method and a multimode approach. Von K´arm´an’s non-linear strain-displacement relationships are employed and the middle plane in-plane displacements are included in the model. The equations of motion are derived by applying Lagrange’s equations. Using the harmonic balance method (HBM), the equations of motion are converted into a non-linear algebraic form and are solved by a continuation method. Interesting results for the plate’s fundamental mode shape such as the occurrence of internal resonance, resulting in a complicated backbone curve, and the variation of the mode shape with the amplitude of vibration and during the period of vibration are obtained.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

628

Computational approaches to prediction of damping behavior of nanoparticle-reinforced coatings and foamy structures Maksim V. Kireitseu†, Geoffrey R. Tomlinson† †

Department of Mechanical Engineering, the University of Sheffield Address: Mappin Street, Sheffield S1 3JD, the United Kingdom [email protected]

ABSTRACT This paper concerns advanced computational engineering approach based on finite element modeling and fundamental physical phenomena of energy dissipation mechanisms related to vibration damping [1]. Nanoparticle/tube-reinforced composite materials are relatively new class of engineering materials and their vibration damping application is commonly unknown from both computational and experimental sides [2]. The novel concept of nanoparticle-based damping technology shows that a molecule-level mechanism can considerably enhance vibration damping and dynamic of aerospace components (fan blades) via enhanced energy dissipation because of large surface-to-volume aspects in nanoparticle-reinforced composite material, large damping energy sources for friction and slipstick motion at interfaces of matrix and nanoparticle. Therefore, to add some knowledge our group is working on computational characterization approach and modeling technique that describe relationships between structure and damping/dynamic properties of the materials, formalize the set of structural mechanical approaches to build a bridge between macro and nanoscales. Structural micro to nanomechanical approach has been developed to predict damping (dynamic) behavior of carbonnanotube-reinforced composite material. The model is based on ‘‘stick-slip’’ frictional motion to address the damping characteristics of SWNT-reinforced composite material. It is worth noting that SWNT can be represented as a shell hollow frame-like structure with a simple nanoscale damping spring characteristics. Thus the developed model can be assembled into entire engineering workbench. A comparison of available modeling strategies is presented. Carbon nanotube-reinforced material is particularly illustrated via advanced numerical codes, using a hollow shell representation of the individual nanotubes. Comparing to the FEM, the new technique may introduce further reduction of both computer time and storage requirement. Thus results of the project will potentially create fundamental basis for investigation and development of 3-D reinforced composite structures with high nanoscale structures volume content, using nano-scale reinforcement architecture to reduce component weight and dimension.

References [1] M. Kireitseu, V. Kompiš, H. Altenbach, L. Bochkareva, D. Hui, S. Eremeev, Continuum mechanics approach and computational simulations of submicrocrystalline and nanoscale composite materials. Fullerenes, Nanotubes and Carbon Nanostructures, Marcel Dekker Press, 13(4), 313-329, 2005. [2] M. Kireitseu, G. Tomlinson, G. Rongong and D. Hui (INVITED), Next generation damping systems: nanoparticle reinforcement design concepts and computational modeling tools, in Proceedings of ICCE-12 Twelve Int.-l Conference on Composites/Nano Engineering, ed. by D. Hui, in Tenerife, Canary Islands, Spain, 280-284, August 1-6, 2005 (WEB: www.uno.edu/~engr/composite/)

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

629

Buckling under conservative and nonconservative load Attila Kocsis , and Gy¨orgy K´arolyi† Department of Structural Mechanics Budapest University of Technology and Economics M˝uegyetem rkp. 3., H-1111 Budapest, Hungary [email protected] † Department of Structural Mechanics, and Center for Applied Mathematics and Computational Physics Budapest University of Technology and Economics M˝uegyetem rkp. 3., H-1111 Budapest, Hungary [email protected]

ABSTRACT Since the first invention of chaos theory, it has been found to play a very important role in many different fields, ranging from physics through biology to engineering, among others. We deal with a phenomenon called spatial chaos which is a special form of spatial complexity, when the governing equations are reminiscent of a chaotic dynamical system, but the role of time is taken over by a spatial coordinate (e.g. arc-length). Many examples of spatial chaos have been addressed recently in general mathematical studies, in fluid dynamics, in the case of buckling of elastic rods or linkages. It also plays an important part in biology where biological filaments – like DNA, (bio)polymers, or tendrils – may exhibit complicated spatial patterns. It has been shown that the elastic linkage provides both a mathematical discretization of Euler’s buckling problem and a mechanical discretization of a continuous rod.The discrete problem is in the state of spatial chaos: it has much more complicated equilibrium shapes than has the continuous Euler-problem. The reason of this is that the governing equations of the continuous problem coincide with a non-chaotic initial value problem, the mathematical pendulum, while the equations of the linkage are the same as the well-known chaotic map, the standard map. We deal with the buckling problem of a cantilever under a quite general set of loads which can be either conservative or non-conservative. We assume that the material behavior can be nonlinear and the rod can be non-prismatic. Using a discrete model, an elastic linkage we show that the static stability is related to a chaotic map, which is conservative both in case of conservative or non-conservative loads. It proves that conservative spatial chaos is not a unique feature of conservative buckling problems. We detail some special examples and construct their global bifurcation diagrams.

References [1] G. Domokos and P. Holmes: Euler’s problem, Euler’s method, and the standard map; or, the discrete charm of buckling. Journal of Nonlinear Science 3 (1993) 109–151. [2] A. Kocsis, Gy. K´arolyi: Buckling under nonconservative load: conservative spatial chaos. Periodica Polytechnica 49/2 (2006) 86–101.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

630

Bifurcation of Periodic Solutions in the Two-Degree-of-Freedom System With Clearances N. Kranjcevic, M. Stegic, N. Vrankovic

Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb P.O. Box 102, 10002 Zagreb, Croatia {nkranjce, mstegic, nvrankov}@fsb.hr

ABSTRACT Clearances exist in many mechanical systems either by design or due to manufacturing tolerances and wear. The characteristics of such systems include abrupt variation of stiffness usually approximated as piecewise linear. It is well-known that the stiffness discontinuity can be a source of the instabilities in the dynamic behavior of the system. In this paper, periodic solutions of the two-degree-of-freedom mechanical system with clearances subjected to periodic excitations are studied. The periodic solution may lose its stability via a static bifurcation (cyclic-fold or flip), or via a Neimark bifurcation. The bifurcation depends on the eigenvalues of the Jacobian matrix of the nonlinear vector field. By applying Hurwitz criterion on the Jacobian matrix, the bifurcation can be classified. For the analyzed dynamical system with clearances, a Neimark bifurcation occurs. The analytical results are compared with the numerical solutions obtained by the finite element in time method. The bifurcation analysis in time finite element procedure is performed by using Poincaré map. The system is assumed to be controlled by the excitation frequency (codimension-one bifurcation). Imposing small increments in the excitation frequency, the critical point is found from which the Neimark bifurcation takes place. The qualitatively different phase portraits, prior to and after the critical point, confirm the Neimark bifurcation.

References [1] J.M.T. Thompson and H.B. Stewart, Nonlinear dynamics and chaos: Geometrical methods for engineers and scientists. John Wiley, Chicester, UK, 1986. [2] A.H. Nayfeh and B. Balachandran, Applied nonlinear dynamics. John Wiley, New York, 1995. [3] N. Kranjcevic, M. Stegic and N. Vrankovic, Stability and bifurcation analysis of a twodegree-of-freedom system with clearances. K.J. Bathe ed. Computational Fluid and Solid Mechanics 2005, Elsevier, Boston, USA, 297-301, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

631

Perturbation Method for Analysis of Strongly Non-Linear Free Vibration of Beams R. Lewandowski* *

Poznan University of Technology, Institute of Structural Engineering 60-695 Poznan, ul. Piotrowo 5, Poland [email protected]

ABSTRACT The perturbation method is one of the oldest methods used to analyse the dynamic behaviour of nonlinear systems. There are many versions of the perturbation method but most of them apply to weakly non-linear cases only. To overcome this limitation, new techniques have been proposed. Cheung et al. [1], Lim et al. [2] and Hu [3] proposed some modifications which make possible the analysis of strongly non-linear systems but with one degree of freedom only. In this paper, the possibility of application of the perturbation method to the dynamic analysis of strongly non-linear free vibrations of beams is discussed. Beams are treated as geometrically nonlinear systems. The von Karman theory is used to describe non-linear effects. The finite element method is adopted to discetize the beam and the motion equation is written in a matrix form.

The first order perturbation equation is solved and the obtained solution is compared with the solution found with the help of the harmonic balance method which is widely used and applicable to the analysis of strongly non-linear dynamic systems. It was proved that both solutions are almost identical and differences are negligibly small. On the basis of similarities discovered in the both solutions, it was concluded that, for the value of small parameter ε = 1 , the solution obtained by means of the perturbation method is almost identical to the one given by the harmonic balance method. The results of typical calculations confirm these observations. Finally, it is concluded that the perturbation solution has also enough accuracy when the strongly non-linear systems are considered. It is believed that the reason of success of the presented perturbation method comes from a feedback which is taken into account when the tangent stifness matrix is introduced. The numerical procedure enabling determination of backbone curves is also briefly described. Theoretical results are supplemented by a description of the results of typical calculations.

References [1] Y.K. Cheung, S,H, Chen, L.S. Lau, A modified Lindsteadt-Poincare method for certain strongly non-linear oscillators, International Journal of Non-Linear Mechanics, 26, 367-378, 1991. [2] C.W. Lim, B.S. Wu, A modified Mickens procedure for certain non-linear oscillators, Journal of Sound and Vibration, 257, 202-206, 2002. [3] H. Hu, A classical perturbation technique which is valid for large parameters, Journal of Sound and Vibration, 269, 409-412, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

632

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

633

Resonant Non-Linear Dynamic Responses of Horizontal Cables via Kinematically Non-Condensed/Condensed Modeling Narakorn Srinil and Giuseppe Rega Department of Structural and Geotechnical Engineering, University of Rome ‘La Sapienza’, via A. Gramsci, 53 Rome 00197, Italy [email protected], [email protected]

ABSTRACT

This paper focuses on a comparison of cable non-linear dynamic responses obtained with the kinematically non-condensed and condensed modeling. Planar non-linear interactions involving simultaneous primary external and internal resonance in horizontal suspended cables are analytically investigated. 1:1 or 2:1 internal resonance is considered. The governing partial-differential non-linear equations of motion of the non-condensed cable model account for the effects of dynamic extensibility, i.e., dynamic tension spatio-temporal variation, and capture the non-linear coupling and contributions of longitudinal/transversal modal displacements. On the contrary, in the condensed cable model, a single integro-partial-differential equation of motion is obtained by neglecting the longitudinal inertia according to a quasi-static stretching assumption of cable in motion. This entails linking the longitudinal displacement to the transversal one and considering a space-independent dynamic tension. This simplified model is typically considered in the literature involving cable nonlinear dynamics. Based on a multi-dimensional Galerkin-based discretization and a second-order multiple scales approach accounting for higher-order non-linear effects and resonant/non-resonant modal contributions, the ensuing dynamic responses and their stability are evaluated by means of force- and frequency-response diagrams with stability analyses. Moreover, the corresponding spacetime non-linear coupled configurations and dynamic tension distributions are analyzed. The numerical explorations highlight that, depending on cable elasto-geometric properties, internal resonance condition and system control parameters, the condensed model may lead to quantitative and/or qualitative discrepancies in the non-linear dynamic responses, with respect to the non-condensed model. The results allow us to point out such meaningful effects of disregarding the system longitudinal dynamics via the kinematic condensation procedure, and to identify cases where the parametric investigation has to be pursued with the more accurate non-condensed model.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

634

Computing Effective Properties of Nonlinear Structures Exposed to Strong High-Frequency Loading at Multiple Frequencies Jon Juel Thomsen Technical University of Denmark MEK - Solid Mechanics Nils Koppels Alle, Building 404 DK-2800 Kgs. Lyngby, Denmark [email protected]

ABSTRACT Effects of strong high-frequency excitation at multiple frequencies (multi-HFE) are analyzed for a class of generally nonlinear systems. The effects are illustrated for a simple pendulum system with a vibrating support, and for a parametrically excited flexible beam. For the latter, theoretical predictions are supported by preliminary experimental results. The main effect of strong multi-HFE is to change the effective or apparent stiffness in a manner similar to that of mono-HFE (e.g. [1-5]), provided the HFE frequencies are well separated and non- resonant. Then the change in effective stiffness is proportional to the mean-square velocity of the excitation velocities, and the corresponding changes in equilibriums, equilibrium stability, and natural frequencies can be computed as for the mono-HFE case. When there are two or more close excitation frequencies, an additional contribution of slowly oscillating stiffness appears. This may cause strong parametrical resonance at conditions that might not appear obvious, i.e. when the difference in two HFE-frequencies is near twice an effective natural frequency of the system, which due to the HFE itself is shifted away from the natural frequency without HFE. Also, it is shown that strong multi-HFE can stabilize otherwise unstable equilibriums, but only if the frequencies are well separated; thus continuous broadband and random HFE does not have a stabilizing effect paralleling that of mono-HFE, or multi-HFE with non-close frequencies. The general results may be used to investigate general effects, or as a short cut to calculate effective properties for specific systems, or to calculate averaged equations of motion that may be much faster to simulate numerically.

References [1] I.I. Blekhman, Vibrational Mechanics - Nonlinear Dynamic Effects, General Approach, Applications. World Scientific, Singapore, 2000. [2] A. Fidlin, Nonlinear Oscillations in Mechanical Engineering. Springer-Verlag, Berlin Heidelberg, 2005. [3] J.J. Thomsen, Vibrations and Stability: Advanced Theory, Analysis, and Tools. Springer-Verlag, Berlin Heidelberg, 2003. [4] J.J. Thomsen, Some general effects of strong high-frequency excitation: stiffening, biasing, and smoothening. Journal of Sound and Vibration, 253, 807-831, 2002. [5] J.J. Thomsen, Slow high-frequency effects in mechanics: problems, solutions, potentials. International Journal of Bifurcation and Chaos, 15, 2799-2818, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Implementation of a Vapour Cavitation into Computational Models of Rotors Supported by Long Journal Bearings Jaroslav ZapomČl VSB-Technical University of Ostrava Centre of Inteligent Systems and Structures - branch of IT ASCR in Ostrava 17.listopadu 15, Ostrava-Poruba, 708 33, Czech Republic [email protected]

ABSTRACT Dynamical behaviour of rotors supported by hydrodynamical bearings is considerably influenced by presence of a gas phase in the lubricating oil. The reason for its occurance is a vapour cavitation. The observations show that pressure of the two-phase medium in cavitated regions remains approximately constant [1]. At long bearings ( most of the bearings ) the pressure gradient in the axial direction is insignificant and the pressure distribution in bearing gap can be described by a simplified Reynolds equation. To be satisfied the continuity of flow and incompressibility of oil the pressure gradient must be zero at the entrance into the cavitated region. To determine edges of the cavitated area a new algorithm has been developed. First the pressure distribution between two nodes corresponding to two adjacent oil inlets into the bearing is calculated. If the minimum pressure drops below the critical value, a vapour cavitation occurs. Then the border nodes are successively chosen from the node of the pressure minimum in the direction opposite to the rotor rotation and the Reynolds equation is solved for the boundary conditions : pressure at the oil inlet, zero pressure gradient at the chosen border node. This process continues until the pressure in the border node is equal to the cavitation one. In the next step the border nodes are chosen from the node of the pressure minimum in the direction of the rotor rotation and the Reynolds equation is calculated for the boundary conditions : pressure in the cavitated area, pressure at the oil inlet. The border node at which the difference between the flow rate and the flow rate through the inlet edge of the cavitation area is minimum is considered to be the outlet edge of the cavitated region. This procedure was implemented into the algorithms for investigation of the transient response of rotors excited by force and kinematic effects ( rotor unbalance, earthquake excitation, etc. ). A modified Newmark method has been chosen [2] for solution of the equation of motion. The modification consists in continuous linearization of the vector of hydraulical forces in the neighbourhood of the current rotor position.

References [1] F.Y. Zeidan, J.M. Vance, Cavitation regimes in squeeze film dampers and their effect on the pressure distribution. STLE Tribology Transactions, 33, 447-453,1990 [2] J.Zapomel, E.Malenovsky, Approaches to numerical investigation of the character and stabilityof the forced and self-excited vibration oof flexible rotors with non-linear supports, IMechE Conference Transactions, 7th International Conference on Vibrations in Rotating Machinery, University of Nottingham, 691-700, 2000.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

636

Multi-criteria optimizations and robustness estimations for problems of crashworthiness, structural dynamics, and acoustics of car bodies. Fabian M. E. Duddeck Queen Mary College, London University Department of Engineering Mile End Road, London E1 4NS, UK [email protected]

ABSTRACT Recent work on optimization for crashworthiness have shown that evolutionary algorithms, e.g. [1] perform superior compared to other optimization strategies when meta-modelling fail in representing the physics of the problems [2]. This is valid especially for simultaneous optimization of several disciplines, i.e. for multi-disciplinary optimization (MDO). These problems are normally driven by the most non-linear problem (e.g. a frontal impact). The high non-linearity of the optimization problems may lead to optimal designs which are loosing their optimality when the design variables are altered only slightly. Unfortunately, this will happen in all real industrial applications – either by additional constraints, by manufacturing irregularities, or by uncertainties inherent to the system. Thus, a robustness analysis should be integrated in the overall optimization scheme. For many of the industrial-sized problems, computing time is not a negligible criterion for the successful implementation of the optimization into the product development process [3]. Evolutionary algorithms require a high amount of CPU time; an additional robustness analysis is thus problematic. Nevertheless, multi-criteria optimization problems have recently been solved successfully for acoustics, structural dynamics and even for crashworthiness. Because a real robustness analysis is often too time demanding, a first estimate of the robustness of the chosen design on the Pareto front can be obtained by regarding the neighbouring points on the front. The variance of the design variables of these points along the Pareto front indicates sensitivities and therefore robustness. This will be demonstrated in first examples taken from industrial studies on acoustics, structural dynamics, and crashworthiness.

References [1] T. Bäck, Evolutionary Algorithms in Theory and Practice. Oxford University Press, New York, 1996. [2] F. Duddeck, K. Volz, Evaluation of optimization algorithms for crash and NVH problems. In: K.J. Bathe (ed.): Computational Fluid and Solid Mech. 2005, Elsevier, 2005. [3] J. Lescheticky, F. Duddeck, L. Willmes, S. Girona, Efficient Product Development of Car Bodies Using Multi-disciplinary Optimization. In: Numerical Analysis and Simulation, VDI Conference, Würzburg, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

637

Process Robustness in Sheet Metal Forming by an Integrated Engineering Strategy K. Grossenbacher*, F. Duddeck†, P. Hora#, M. Ganser* *

BMW Group, Product and Process Development Knorrstrasse 147, D-80788 Munich, Germany [email protected], [email protected] †

#

Queen Mary College, London University Mile End Road, London E1 4NS, UK [email protected]

Institute for Virtual Production, ETH Zurich (Center) CLA F 9 Tannenstrasse 3, CH-8092 Zurich [email protected]

ABSTRACT In the modern automotive industry the development of car bodies depends considerably on the use of computer-aided tools. Herewith one can meet the challenges of rising product complexity and growing number of variants. About fifteen years ago simulation of sheet metal forming was used for the first time in industrial application. However, the calculation times were long and the quality of the results was often unsatisfactory. Today, enabled by improved material models and new numerical methods, those simulations have become essential for the evaluation of press-tools before they are manufactured. Due to the high number of varying influences on the production process of sheet metals, the resulting quality of the parts is not always stable. In most cases, these variances are lying in predefined limits of tolerance. Otherwise additional efforts and costs for testing the parts and for reworking them are required leading to higher reject costs in total. By the means of existing highly qualitative methods for numerical simulation combined with standardized statistical methods one can identify these varying influences, their interconnection and effects on car body parts. On the basis of such an analysis appropriate optimization algorithms will lead to an improved overall part quality along with higher robustness. In this paper, the realization of an integrated engineering strategy as mentioned above within the forming department of the BMW Group, the combination of corresponding engineering tools, and their reasonable cooperation in a planning process will be described. The essential steps of these processes and the methods and tools used are presented illustrated by industrial-sized examples.

References [1]

E.Dietrich, A.Schulze, Statistische Verfahren zur Maschinen- und Prozessqualifikation. Carl Hanser Verlag München Wien, 5. Auflage, 2005

[2]

J. Meinhardt, W. Volk, H. Schmidt, “Virtuelle Prozessentwicklung von Presswerk-zeugen im industriellen Umfeld”, in E. Doege (Editor), Umformtechnik – Erschließung wirtschaftlicher und technologischer Potenziale, 17. Umformtechnisches Kolloquium Hannover, 271-284 (2002).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Topology Optimization of Robots Using Mapping Techniques Michael R. Hansen*, Torben O. Andersen†, John M. Hansen††, Ole Ø. Mouritsen* *

Institute of Mechanical Engineering, Aalborg University Pontoppidanstræde 101, 9220 Aalborg E, Denmark [email protected], [email protected] †

Institute of Energy Technology, Aalborg University Pontoppidanstræde 105, 9220 Aalborg E, Denmark [email protected] ††

Man B&W Diesel Teglholmsgade 41, 2450 Copenhagen SV, Denmark [email protected]

ABSTRACT This work is an extension of previous work that utilizes mapping techniques to handle the discrete nature of robot design. The developed design approach utilizes dimensionless parameters that map data bases of commercially available components. The important properties are derived by applying interpolation on the data bases. The handling of both distinct components sub-groups as well as a method for the gradual conversion of the dimen-sionless variables into integers has already been presented. In this work the approach is further developed by introducing a parameter that maps a set of possible system topologies. The possible topologies include three different spatial mechanisms and the dynamic time domain simulation model that is used to evaluate a design is developed so that it can analyze each of the possible topologies. The mapping of the robot topologies has been done in such a way that the performance of a design varies continuously despite a change in topology during the optimization. The design criteria include costs of drives and structural components, tool point precision, fatigue in welded details, over heating and stalling of the motors and gears as well time of operation. A number of examples are given.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Optimization Strategies for Highly Non-Linear FE-Applications as Crashworthiness Applications ¨ 1 , M. van den Hove2 , and B. Mlekusch2 ¨ H. Mullersch on 1

DYNAmore GmbH - www.dynamore.de [email protected]

2 AUDI AG - www.audi.de {marcel.vandenhove,bernd.mlekusch}@audi.de

ABSTRACT The purpose of this paper is to explore some interesting aspects of optimization for crashworthiness occupant safety applications and to propose optimization strategies for highly nonlinear problems. In the first part of the paper different optimization strategies are discussed and pros and cons are compared. In addition, a methodology to get a reliable surrogate model using neural networks is introduced. The surrogate model (Meta-Model or Response Surface Model) approximates the relationship between design parameters and a physical response and can be used to visualize and explore the design space. In the second part the application of the Successive Response Surface Scheme (SRSM) for the optimization of an adaptive restraint system is conducted. For this, several front crash load cases are considered. This is performed using LS-OPT (Stander et al. [11]) as optimization software and PAM-Crash as solver for the finite element occupant safety simulations. The procedure of generating an advanced meta-model to get an approximation of the global design space using neural networks is demonstrated for this example. Furthermore, the visualization of multi-dimensional meta-models in two- and three-dimensional design space is illustrated

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

640

Local and Global Searches of Approximate Optimal Designs of Regular Frames Makoto Ohsaki∗ ∗

Department of Architecture and Architectural Engineering, Kyoto University Kyotodaigaku-Katsura, Nishikyo, Kyoto 615-8540, Japan [email protected] ABSTRACT

In this paper, methods of local and global searches of approximate optimal designs are presented for regular frames subjected to static loads. Constraints are given for stresses and displacements. Recently, it has been pointed out that there may exist many fully stressed frames with almost the same total structural volume [1]. Therefore, obtaining only one solution will not be enough for practical purpose, where several solutions satisfying stress constraints should be compared in view of other performance measures such as eigenfrequencies and requirements in construction process. Furthermore, the objective function may not be strictly minimized; i.e., it will be helpful for the designers if several approximate solutions with different distributions of cross-sectional areas are obtained. Jog and Haber [2] suggested that the nonuniqueness of the solution to a compliance optimization problem can be detected by the singular values of the matrix defined as the gradients of the equivalent force vector with respect to the design variables. However, they did not show how the singular vectors are used for finding approximate optimal solutions. In this paper, we first demonstrate nonuniquesness of the optimal solution by a continuous beam with periodic boundary conditions for uniform loads. The optimal solutions are locally searched from a solution found from an arbitrary generated initial solution. The search direction is determined by singular value decomposition of the stiffness matrix with respect to the cross-sectional areas or the sensitivity matrix of the constraints. Approximate optimal solutions are next globally and consecutively found so as to maximize the distance from the already found solutions under stress and displacement constraints. The distance between the solutions is defined by the Euclidean norm of the differences in the cross-sectional areas. The constraint is given for the total structural volume, where the specified upper bound is slightly larger than the objective value of an optimal solution. The effectiveness of the proposed methods is demonstrated in application to a 10 × 10 and 3 × 27 plane frames. It is shown that approximate optimal solutions have been successfully found using the singular vectors of the stiffness matrix with respect to the crosssectional areas. However, accuracy of the solutions can be improved using the singular vectors of the sensitivity matrix of the constraints.

References [1] K. M. Mueller, M. Liu and S. A. Burns, Fully stresses design of frame structures and multiple load paths. J. Structural Engineering, 128(6), pp. 806–814, 2002. [2] C. S. Jog and R. B. Haber, Stability of finite element models for distributed-parameter optimization and topology design, Comp. Meth. Appl. Mech. Eng., 130, pp. 203–226, 1996.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

641

Optimisation of car body parts regarding equivalent radiation power using a genetic algorithm and morphing J. Reger*, T. Schneider*, C. Ehlert* * P+Z Engineering München, Germany [email protected] [email protected] [email protected]

ABSTRACT The continuing demand for light weight car body structures and the increasing awareness of acoustic comfort is a conflict of objectives. Excitations introduced by the driveline or the drivetrain can not be avoided. The body structure will radiate sound to the passenger compartment. Consequently, reducing the radiated sound power in car body structures develops towards a main topic in the car body development process. The common approach to stiffen the body structure parts by introducing beads using an empirical approach is not sufficient for reducing the sound pressure level. Stiffening, that decreases the radiated sound power at certain frequencies, often leads to increased radiation at other frequencies. To improve the overall acoustic behaviour the complete frequency response of the structure has to be taken into account. Amplitudes of radiated sound power have to be decreased over the complete frequency range of interest. State-of-the-Art optimisation methods can help to find solutions. Presently in this context optimisation is normally used with shell thickness parameters to increase eigenvalues or to decrease frequency dependent accelerations of dedicated points in the structure. However, the gradient methods used are inadequate for the optimisation of body structures concerning equivalent radiation power. The reasons for the poor performance of gradient methods are highly nonlinear objective functions when minimising the equivalent radiation power. To find a topography of the body structure which radiates significantly less sound a global optimisation has to be run. The introduced method changes the topology of the structure by morphing the FE-Mesh. A genetic algorithm with discrete variable representation is applied for global optimisation. In order to perform successful optimizations with genetic algorithms a large number of function evaluations is required. To reduce the overall computation time the use of substructuring and parallelisation is necessary. The acoustic behaviour of the complete car body is taken into account.

The example shows a successful optimization of the topography of a car body part. The equivalent radiation power can be decreased significantly over the complete frequency range of interest.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

642

Grid-Based Topology Optimization of Rigid Body Mechanisms Using Different Problem Formulations Kai Sedlaczek, Peter Eberhard Institute of Engineering and Computational Mechanics University of Stuttgart Pfaffenwaldring 9, 70569 Stuttgart, Germany {sedlaczek,eberhard}@itm.uni-stuttgart.de

ABSTRACT In the design process of rigid body mechanisms two main development steps can be distinguished, namely type (topology) and dimensional synthesis. Whereas much work has already been done on solving the problem of dimensional synthesis, optimization based approaches to topology design of rigid body mechanisms are rare. Unlike solving the discrete combinatorial problem of optimal mechanism topology by means of genetic algorithms [1], we investigate in this work a ground structure approach similar to [2] but based solely on rigid bars. A relaxed formulation of the kinematic constraint equations allows an almost straightforward kinematic analysis despite the over-determined system of equations due to redundant bars in the ground structure. Similar to cross sectional areas in topology optimization of truss structures, the bars in the ground structure are parameterized by continuous design variables that can have intermediate values between 0 and 1. This continuous description allows a solution with efficient gradient-based optimization methods. However, the problem is of discrete (binary) nature and intermediate values are physically meaningless so that appropriate problem formulations must be found in order obtain a 0-1 design. This work investigates different problem formulations and solution techniques and their ability to solve the intrinsically discrete problem of mechanism topology optimization. All presented formulations are using the continuously parameterized truss-like structure with rigid bars, but they are based on different (separation) constraints in order to achieve a 0-1 design. This includes a simple quadratic penalization as well as the power-law (SIMP) method for the solution of the path generation and output maximization problem. The functionality, the advantages and drawbacks of the grid structure approach with respect to the problem of rigid body mechanism design are discussed and illustrated by example problems.

References [1] K. Sedlaczek, T. Gaugele, P. Eberhard, Topology Optimized Synthesis of Planar Kinematic Rigid Body Mechanisms. In: J.M. Goicolea, J. Cuadrado, J.C. Garcia Orden (Eds.), Advances in Computational Multibody Dynamics. Proceedings of the ECCOMAS Thematic Conference on Multibody Dynamics, Madrid, Spain, 2005. [2] A. Kawamoto, M.P. Bendsøe, O. Sigmund. Articulated Mechanism Design with a Degree of Freedom Constraint. International Journal for Numerical Methods in Engineering, 61, 1520-1545, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

643

Combining Optimization and Robust Engineering Methods in the Engineering Product Design Process Holger Wenzel∗ ∗ Engineous

Software GmbH Lichtenbergstrasse 8 85748 Garching [email protected]

ABSTRACT Many engineers that work in the simulation based product creation process are faced with a multitude of different and often conflicting demands: The product needs a better performance, it should be more reliable, yet less expensive and on top of all, the design time is shortened dramatically. These challenges have driven the use of process integration and design optimization (PIDO) tools into all stages of product development. This trend has been fueled further by the increase of computational power that spreads the application of automatic design improvement methods even to areas that are extremely expensive to simulate. One of the inherent challenges of the use of optimization algorithms in the design of industrial products is the fact that they tend to drive the design to extreme points, where even very small changes in the setup can cause the product to fail. This effect can be counteracted by the application of robust engineering methods, which can analyze and automatically improve the robustness and reliability of the product. Nonetheless, a systematic coupling of optimization and robust engineering methods is a relatively new and still emerging field. This work gives an overview of four recent examples from the aerospace and automotive industry: a blade-disc connection of a gas turbine, the robust optimization of the idle shake of a passenger car and two applications of passenger car occupant safety. Owing to the newness of this approach, there are no established methods or best practices and the presented examples show a large variety in the solution of the problem. The most straightforward way is to combine an optimization with a robust assessment of the optimum, either via a Monte-CarloAnalysis or a robustness-estimation method like FORM (First-Order-Reliability-Method). A very interesting approach is established in the idle shake application. Here a multi objective optimization is performed, using the NCGA method, and then the robustness of the points of the Pareto-front is used to select the design used for production. The other examples show more different methods, the application of the Taguchi robust design method, or the direct use of the output variation as part of the objective function, in order to achieve a robust design. Although the combination of optimization and robust engineering is a too young field to already propose finalized routines and fixed solutions, the examples shown in this work illustrate the great benefit that already can been realized by this approach. Some of the designs are in production and one design system for passenger safety is implemented based on one of the presented applications.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

644

Combining topological and shape derivatives in structural optimization G. Allaire∗ , F. Jouve∗ , F. de Gournay∗ & A.-M. Toader† ∗ Centre

de Math´ematiques Appliqu´ees (UMR 7641), Ecole Polytechnique 91128 Palaiseau, France

† CMAF,

Faculdade de Ciˆencias da Universidade de Lisboa Av. Prof. Gama Pinto 2, 1699 Lisboa, Portugal [email protected] ABSTRACT

Two recent methods in shape and topology optimization of structures are combined in order to obtain an efficient optimization algorithm that benefits of advantages from both methods. The level set method, based on the classical shape derivative, is known to easily handle boundary propagation with topological changes. However, in practice it does not allow for the nucleation of new holes (at least in 2-d). The bubble or topological gradient method of Schumacher, Masmoudi, Sokolowski and their co-workers, is precisely designed for introducing new holes in the optimization process. Therefore, the coupling of these two methods yields a robust algorithm which can escape from local minima in a given topological class of shapes. The method we propose is a logical sequel of our previous work [1], [2] where we proposed a numerical method of shape optimization based on the level set method and on shape differentiation. The novelty is in the coupling and in the robustness of the proposed numerical implementation. Our basic algorithm is to iteratively use the shape gradient or the topological gradient in a gradient-based descent algorithm. The tricks are to carefully monitor the decrease of the objective function (to avoid large changes in shape and topology) and to choose the right ratio of successive iterations in each method. We provide several 2-d and 3-d numerical examples for compliance minimization and mechanism design. The main advantage of our coupled algorithm is to make the resulting optimal design largely independent of the initial guess, although local minima may still exist (even in the class of shapes sharing the same topology). Similar numerical results where discussed in [3].

References [1] Allaire G., Jouve F., Toader A.-M., A level set method for shape optimization, C. R. Acad. Sci. Paris, S´erie I, 334, 1125-1130 (2002). [2] Allaire G., Jouve F., Toader A.-M., Structural optimization using sensitivity analysis and a level set method, J. Comp. Phys., Vol 194/1, pp.363-393 (2004). [3] Allaire G., De Gournay F., Jouve F., Toader A.-M., Structural optimization using topological and shape sensitivity via a level set method, Control and Cybernetics, 34, 59-80 (2005).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

645

A Comparison of Displacement and Mixed Finite Element Formulations for Variational Design Sensitivity Analysis Franz-Joseph Barthold∗ , Karin Wiechmann† ∗ Numerical

Methods and Information Processing, University of Dortmund August-Schmidt-Str. 8, D-44227 Dortmund, Germany [email protected]

† Institute

of Mechanics and Computational Mechanics, University of Hannover Appelstr. 9A, D-30167 Hannover, Germany [email protected] ABSTRACT

The authors formulate design sensitivity analysis in form of a variational approach based on a novel local representation of continuum mechanics, see the publications of the first author, e.g. [1], for a summarising overview on the concept and the publications of the second author, e.g. [3], for the subsequent application to elasto-plastic material behaviour. The central idea is to trace and separate the influence of geometry mappings from the influence of deformation mappings on all field quantities. Thus, a reformulation of continuum mechanics following the intrinsic formulation by Noll [2] but using two independent mappings defined on a local parameter space is advocated. Consequently, the consistent linearisation concept of computational mechanics used to derive tangent stiffness matrices, should also be applied to the geometry mappings, i.e. to compute tangent geometry sensitivity matrices. Firstly, the fundamentals of the advocated treatment are described in general terms. Secondly, the consequences for the finite element development procedure are outlined on the theoretical as well as computational level. Here, the parallelism of sensitivity and stiffness computation are highlighted for different finite elements, i.e. for standard displacement and mixed formulations. Thirdly, hints are given to guarantee correct variational sensitivity information by a general comparison strategy using finite differences. The outlined theoretical and computational framework is seen to be an efficient method to investigate the influence of different element formulations on the solution of the optimisation problem.

References [1] F.-J. Barthold, Zur Kontinuumsmechanik inverser Geometrieprobleme. Braunschweig Series on Mechanics. Report No. 44-2002, Braunschweig, ISBN 3-920395-43-3, 2002. [2] W. Noll, A new mathematical theory of simple materials, Archive of Rational Mechanics, 48(1), 1-50, 1972. [3] K. Wiechmann, Theorie und Numerik zur Berechnung und Optimierung von Strukturen mit elastoplastischen Deformationen. Institute of Mechanics and Computational Mechanics, Report No. F01/8, Hannover, ISBN 3-935732-02-3, 2001.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

646

Inverse acoustic scattering by small-obstacle expansion of misfit function Marc Bonnet∗ ∗ Solid Mechanics Laboratory (UMR CNRS 7649) Ecole Polytechnique, F-91128 Palaiseau cedex, France [email protected]

ABSTRACT The present study is set in the general framework of inverse scattering of scalar (e.g. acoustic) waves. To identify hidden obstacles from external measurements (e.g. overspecified boundary data) associated with the scattering of known incident waves by the unknown object(s), it is customary to invoke iterative algorithms such as gradient-based optimization procedures. The numerical solution of the forward scattering problem associated with an assumed obstacle configuration is often a computationally demanding task. Besides, iterative inversion algorithms are sensitive to the choice of initial “guess” (number of components, initial location, shape and size of obstacle(s)). This has prompted the definition of preliminary probing techniques, which aim at delineating in a computationally fast way the hidden obstacle(s), namely the linear sampling [2], not pursued here, or the concept of topological sensitivity [1, 3]. If J denotes the cost function used for solving the inverse problem, then in 3D situations the topological derivative T3 (xs ) associated with the nucleation of a small obstacle of volume O(ε3 ) and specified shape appears through the expansion J(ε, xs ) − J(0) = ε33 T (xs ) + o(ε3 )

(1)

In this communication, an extension of the topological derivative is presented, whereby J(ε, xs ) is expanded further in powers of ε. Specifically, the expansion to order O(ε6 ) for 3D acoustic scattering by a hard obstacle of size ε is presented. The choice of order O(ε6 ) is important for cost functions J of least-squares format. In particular, the expansion of J for any centrally-symmetric infinitesimal hard obstacle of radius ε centered at xs is found to have the form J(ε, xs ) − J(0, xs ) = ε3 T3 (xs ) + ε5 T5 (xs ) + ε6 T6 (xs ) + o(ε6 ) = J(0, xs ) + J6 (ε, xs ) + o(ε6 ) (2) The previously known topological derivative T3 (xs ) and the new coefficients T5 (xs ), T6 (xs ) have explicit expressions in terms of the relevant acoustic Green’s function. Expansions of the form (2) offer the option of minimizing the approximate polynomial expression J6 (ε, xs ). This is a simple and inexpensive task, which can be performed for locations xs spanning a search grid, thereby defining a (approximate) global search procedure. The values of xs and ε leading to an absolute minimum of J6 (ε, xs ) over the search grid then constitute the best estimate of the hidden scatterer furnished by this procedure, and might provide e.g. a useful initial guess for an iterative inversion algorithm. Results of numerical experiments in 3D conditions based on this idea will be presented at the conference.

References [1] B ONNET, M., G UZINA , B. B. Sounding of finite solid bodies by way of topological derivative. Int. J. Num. Meth. in Eng., 61, 2344–2373 (2004). [2] C OLTON , D., K IRSCH , A. A simple method for solving inverse scattering problems in the resonance region. Inverse Problems, 12, 383–393 (1996). [3] G UZINA , B. B., B ONNET, M. Topological derivative for the inverse scattering of elastic waves. Quart. J. Mech. Appl. Math., 57, 161–179 (2004).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Second Order Topological Sensitivity Analysis J.R. Faria∗ , R.A. Feijo´o∗ , A.A. Novotny∗ , E. Taroco∗ & C. Padra† ∗ Laborat´ orio

Nacional de Computac¸a˜ o Cient´ıfica LNCC/MCT, Av. Get´ulio Vargas 333, 25651-075 Petr´opolis - RJ, Brasil [email protected], [email protected], [email protected], [email protected] †

Centro At´omico Bariloche, 8400 Bariloche, Argentina [email protected] ABSTRACT

The topological sensitivity analysis provides an asymptotic expansion of a given cost function with respect to the insertion of a small hole at an arbitrary point of the domain. This sensitivity results in a real function called topological derivative [3] that has been used as a descent direction to solve several problems, among others: topology optimization and inverse problems. In order to present the basic idea, let us consider an open bounded domain Ω ⊂ R 2 with a smooth boundary ∂Ω and a cost function ψ (Ω) = JΩ (u), where u denotes the solution of a state equation defined in Ω. If the domain Ω is perturbed by introducing a small hole B ε of radius ε at an arbitrary point x ˆ ∈ Ω, we have a new domain Ω ε = Ω − B ε , whose boundary is denoted by ∂Ωε = ∂Ω ∪ ∂Bε . Therefore, the topological asymptotic expansion of the cost function may be expressed as following ψ(Ωε ) = ψ(Ω) + f (ε)DT ψ + g(ε)DT2 ψ + O(g(ε)) ,

(1)

where f (ε) and g (ε) ∈ O(f (ε)) are positive functions that decreases monotonically such that f (ε) → 0 and g(ε) → 0 with ε → 0+ . The term DT ψ is classically defined as the first order topological derivative [1] of ψ. In addition, if we divide eq. (1) by g (ε) and after taking the limit ε → 0, we can recognize the term D T2 ψ as the second order topological derivative of ψ. In fact, DT2 ψ = lim

ε→0

ψ(Ωε ) − ψ(Ω) − f (ε)DT ψ , g(ε)

(2)

which can be used to devise optimality conditions in the context of topology optimization and inverse problems, for instance. In this work, we apply the Topological-Shape Sensitivity Method developed in [2] as a systematic approach to compute first as well as second order topological derivative for the Poisson’s equation, taking into account homogeneous Neumann and Dirichlet boundary condition on the hole.

References [1] J. C´ea, S. Garreau, Ph. Guillaume & M. Masmoudi. The Shape and Topological Optimizations Connection. Computer Methods in Applied Mechanics and Engineering, 188:713-726, 2000. [2] A.A. Novotny, R.A. Feij´oo, C. Padra & E. Taroco. Topological Sensitivity Analysis. Computer Methods in Applied Mechanics and Engineering, 192:803-829, 2003. ˙ [3] J. Sokolowski & A. Zochowski. On the Topological Derivative in Shape Optimization. SIAM Journal on Control and Optimization, 37:1251-1272, 1999.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Sensitivity Analysis of Shape Memory Alloy Shells Matthijs Langelaar*and Fred van Keulen† Department of Precision and Microsystems Engineering, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands * [email protected] † [email protected]

ABSTRACT Shape memory alloys (SMAs) are active materials with a high power density capable of producing comparatively large actuation strains and stresses. However, designing effective multi-dimensional SMA actuators is a challenging task, due to the complex behavior of the material and the fact that often electrical, thermal and mechanical aspects have to be considered simultaneously. For this reason, interest in the application of systematic computational design approaches, such as design optimization techniques, to the design of SMA structures is increasing. To enable efficient SMA design optimization procedures, the availability of sensitivity information is crucial. This paper presents the formulation and computation of design sensitivities of SMA shell structures using the direct differentiation method, in a steady state electro-thermo-mechanical finite element context. Semi-analytical and refined semi-analytical approaches are considered. The SMA constitutive model used in this study is particularly intended for design optimization of SMA structures and actuators. In contrast to the majority of SMA models, the formulation of this model is history-independent, which simplifies the sensitivity analysis considerably. The model is specifically aimed at the description of the superelastic behavior of NiTi alloys, based on the R-phase transformation. This behavior is characterized by its negligible hysteresis, which is very attractive for actuator applications. This research is aimed at SMA shell structures, which can generate large actuator displacements through bending. The most general case of actuation is considered, where the SMA effects are initiated by temperature changes induced by Joule heating. This implies that a coupled electrothermo-mechanical finite element analysis is required. As a consequence, the sensitivity analysis includes coupling terms between three different physical fields. The most challenging aspect of this work lies in particular in the fact that the constitutive model in the considered plane stress setting contains implicit equations, which lead to complications in the sensitivity analysis. This problem manifests itself in the thermo-mechanical sensitivity coupling terms and in sensitivities of derived mechanical responses such as stresses or equivalent strains. Solutions for this difficulty based on finite difference and analytical approaches are discussed. Finally, the accuracy of the sensitivities is evaluated on a representative finite element model of a miniature gripper, as a function of relative design perturbations in thickness, shape, loading and material parameters. A comparison is given between results obtained by finite-difference, semi-analytical and refined semi-analytical methods.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Stress Constrained Optimization using X-FEM and Level Set Description L. Van Miegroet*, T. Jacobs†, E. Lemaire* & P. Duysinx* * University of Liège LTAS - Department of Mechanics and Aerospace Chemin des chevreuils 1, 4000 Liège Belgium [email protected], [email protected] † Université Laval Département de Génie civil, Pavillon Adrien-Pouliot, Québec Canada [email protected]

ABSTRACT Topology optimization has experienced an incredible soar since 1988 and is now available within several commercial finite element (FE) codes. Meanwhile, parametric shape optimization has found few industrial applications. This is may be due to its inherent difficulties to deal with mesh management with boundary modifications. Recently the extended finite element method (X-FEM) has been proposed (see [1] for a review) as an alternative to remeshing methods. The X-FEM method is naturally associated with the Level Set description of the geometry to provide an efficient treatment of problems involving discontinuities and propagations. Up to now the X-FEM method has been mostly developed for crack propagation problems, but the potential interest of the X-FEM method and the Level Set description for other problems like shape and topology optimization was identified very early (see [2]). In this paper, the authors present an intermediate approach between parametric shape and topology optimization by using the X-FEM and Level Set Description. The method benefits from fixed mesh work using X-FEM and from smooth curves representation of the Level Set description. One major characteristic of the approach is to be able to model exactly void and solid structures. The statement of the optimization problem is similar to classical shape optimization: Design variables are the parameters of basic Level Set features (circles, rectangles, super ellipse, etc.) or could be NURBS control points, while various global (compliance) and local responses can be considered in the formulation. Conversely to shape optimization, structural topology can be modified since basic Level Sets can merge or separate from each other. The sensitivity analysis (related to the compliance and/or the stresses) and the way it can be carried out efficiently is detailed. A special attention is paid to stress constrained problems which are often neglected in other Level Set Methods works. Numerical applications revisit some classical 2D (academic) benchmarks from shape optimization and illustrate the great interest of using X-FEM and Level Set description. The paper presents the results of minimum compliance and stress constrained problems using the proposed method.

References [1] T. Belytschko. C. Parimi, N. Moes, N. Sukumar & S. Usui. Structured extended finite element methods for solids defined by implicit surfaces. Int. J. Numer. Meth. in Engng 2003; 56 : 609-635. [2] T. Beltschko, S. Xiao & C. Parimi. Topology optimization with implicit functions and regularization. Int. J. Numer. Meth. in Engng 2003; 57 : 1177-1196.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Level Set Method for Optimization of Contact Problems ´ † Andrzej My´slinski † Systems Research Institute ul. Newelska 6, 01 - 447 Warsaw, Poland [email protected]

ABSTRACT This paper deals with the numerical solution of structural optimization problems of an elastic body in unilateral contact with a rigid foundation. The contact problem with a given friction is described by an elliptic inequality of the second order governing a displacement field. The optimization problem consists in finding, in a contact region, such topology and shape of the boundary of the domain occupied by the body that the normal contact stress is minimized. Level set methods [3, 4] are numerically efficient and robust procedures for the tracking of interfaces, which allows domain boundary shape changes in the course of iteration. The evolution of the level set function is governed by the Hamilton Jacobi equation. The speed vector field driving the propagation of the level set function is given by the Eulerian derivative [2] of an appropriately defined cost functional with respect to the free boundary. In this paper the necessary optimality condition for the shape and topology optimization problem of this contact problem is formulated. The paper extends results of [1] to contact problems with a given friction. The level set method, based on the classical shape gradient, is coupled with the bubble or topological derivative method, which is precisely designed for introducing new holes in the optimization process. The holes are supposed to be filled by weak phase mimicking voids.Since both methods capture a shape on a fixed Eulerian mesh and rely on a notion of gradient computed through an adjoint analysis, the coupling of these two method yields an efficient algorithm. Moreover the finite element method is used as the discretization method. Numerical examples are provided and discussed.

References [1] A. My´sli´nski, Topology and Shape Optimization of Contact Problems using a Level Set Method, Proceedings of 6 WCSMO Conference, Rio de Janeiro, Brasil, 2005. ˙ [2] J. Sokołowski and A. Zochowski, Topological optimization of contact problems. Research Report No 25/2005, Institute Ellie Cartan, 2005. [3] Y. H. Tsai, Y. Giga and S. Osher, A Level Set Approach for Computing Discontinuous Solutions of Hamilton - Jacobi Equations, Mathematics of Computation, 72, 159 - 181, 2002. [4] M.Y. Wang, X. Wang, D. Guo, A Level Set Method for Structural Topology Optimization, Computer Methods in Applied Mechanics and Engineering, 192, 227 - 246, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

651

An Iterative Procedure for Model Updating Based on Selective Sensitivity Hoang Anh Pham∗ , Christian Bucher† ∗ Institute

†

of Structural Mechanics, Bauhaus-University Weimar Marienstr. 15, D-99423 Weimar, Germany [email protected]

Institute of Structural Mechanics, Bauhaus-University Weimar Marienstr. 15, D-99423 Weimar, Germany [email protected]

ABSTRACT Model updating of a structural system may require a large number of parameter to be identified simultaneously. Due to the ill-conditionedness, large errors in identified parameter values will occur when errors are present in the measurements. One solution for this problem is using the concept of selective sencitivity [1]. The method allows to reduce ill-conditioning by providing specific excitations causing model responses sensitive to a small number of model parameters. Thus, only a few parameters are estimated at a time. However, defining such excitations generally involves the knowledge of all parameters to be identified. Therefore, an iterative experiment procedure is suggested (e.g the method of multi-hypothesis testing [2]) which is normally a time-consuming process. This paper presents the theory for an alternative iterative procedure for dynamical excitation of undamped, linear structures. The approach is developed using the concept of predictive control and then is incorporated into a Bayesian updating methodology to reduce the uncertainty in the system parameters. Simulation examples of a multi-storey frame structure and a continuously supported beam under hamonic excitation demonstrate the potential of the proposed method.

References [1] Y. Ben-Haim, Robust reliability in the mechanical sciences. Springer, Berlin-Heidelberg-New York, 1996. [2] U. Prells and Y. Ben-Haim, Selective sensitivity in the frequency domain-II. Applications. Mechanical System and Signal Processing, 7(6), 551–574, 1993.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Extension, bending and torsion of cylindrical Cosserat shells made from a porous elastic material Mircea Bˆırsan ‘A.I. Cuza’ University of Ias¸i, Faculty of Mathematics Bvd. Carol I, no. 11, 700506 Ias¸i, Romania [email protected]

ABSTRACT We consider the relaxed Saint–Venant’s problem in the linear theory of elastic shells made from a porous material. For our purpose, we use the model of Cosserat surfaces and the Nunziato–Cowin theory of elastic materials with voids [1]. We extend the method employed in [2, 3] for the case of Cosserat shells with two porosity fields: one field characterizes the volume fraction variations along the middle surface of the shell, while the other accounts for the changes in volume fraction along the shell’s thickness. We obtain the solution of the extension, bending and torsion problem in closed form, for both open and closed cylindrical shells of arbitrary cross–section. The solutions determined are shown to be minimizers of the strain energy functional on certain classes of solutions to the relaxed Saint–Venant’s problem for shells, by analogy with the well–known characterizations of the classical Saint–Venant’s solutions from the three–dimensional theory of elasticity. We observe that the torsion of cylindrical shells does not affect the porosity fields, so that the torsion problem reduces to the previously known results for the purely elastic case. On the other hand, for the extension and bending problem, we remark the influence of the material’s porosity on the deformation of the shell. In the particular case of porous Cosserat plates, we compute the solution of the extension and bending problem and show that this solution is in agreement with the corresponding results obtained in [1] and [4] using three–dimensional approaches. The solutions determined in this paper are exact and they prove useful in solving many practical problems and for the comparison with related results obtained by various computational methods.

References [1] S.C. Cowin and J.W. Nunziato, Linear elastic materials with voids, Journal of Elasticity, 13, 125– 147, 1983. [2] M. Bˆırsan, The solution of Saint–Venant’s problem in the theory of Cosserat shells, Journal of Elasticity, 74, 185–214, 2004. [3] M. Bˆırsan, Saint–Venant’s problem for Cosserat shells with voids, International Journal of Solids and Structures, 42, 2033-2057, 2005. [4] M. Ciarletta and D. Ies¸an, Non–classical Elastic Solids, Pitman Research Notes in Mathematics, no. 293, Longman Scientific & Technical, London, 1993.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

653

A study of discontinuous Galerkin methods for thin bending problems N.T. Dung∗ , G.N. Wells† ∗

Faculty of Civil Engineering and Geosciences, Delft University of Technology P.O. Box 5048, 2600 GA Delft [email protected]

† Faculty

of Civil Engineering and Geosciences, Delft University of Technology P.O. Box 5048, 2600 GA Delft [email protected]

ABSTRACT In this work, various continuous/discontinuous Galerkin (C/DG) formulations are examined for the analysis of thin plates. The continuous/discontinuous Galerkin method allows fourth-order partial differential equations to be solved using standard C 0 finite element shape functions. The concept was presented by Engel et al. [1], who utilised an interior-penalty type formulation for solving thin beam and plate problems. The interior-penalty method has some drawbacks, such as the conditional stability, the loss of accuracy for large penalty values and ambiguities when making the extension to nonlinear problems. Here, we draw on developments in discontinuous Galerkin methods for second-order elliptic equations, for which several unconditionally stable methods are known [2], and present continuous/discontinuous Galerkin formulations for thin bending problems inspired by these methods. Two important aspects that have been studied are the stability condition and the convergence rate. For each approach, benchmark simulations have been performed and compared. Also, conclusions are drawn on to the computational efficiency of the different methods. The presented numerical examples have been produced using the FEniCS Form Compiler (FFC) [3, 4]. FFC takes the variational form in a format which reassembles standard mathematical notation, and generates optimised finite element code for arbitrary order shape functions. It allows for the rapid development of efficient finite element code and it simplifies significantly the implementation of DG methods, an issue which is often cited as a disadvantage of DG methods. Examples of FFC functionality and use for DG problems are presented.

References [1] G. Engel, K. Garikipati, T. J. R. Hughes, M. G. Larson, L. Mazzei, and R. L. Taylor, Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beam and plates, and strain gradient elasticity. Comput. Methods Appl. Mech. Engrg., 191, 3669-3750, 2002. [2] D. N. Arnold, F. Brezzi and B. Cockburn, and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Number. Anal., 39 (5), 1749-1779, 2002. [3] FEniCS Form Compiler. www.fenics.org. [4] R. C. Kirby, and A. Logg, A Compiler for Variational Forms. Submited.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

654

Discrete Strain Gap (DSG) solid finite elements at large deformations for non-linear analysis of shells and solids Moritz Frenzel∗ , Manfred Bischoff† , Wolfgang A. Wall∗ ∗ Chair

of Computational Mechanics Technical University of Munich Boltzmannstr. 15, 85747 Garching b. M¨unchen, Germany [email protected] † Chair

of Structural Analysis Technical University of Munich [email protected] ABSTRACT We present a non-linear solid finite element for structural analysis, based on the Discrete Strain Gap (DSG) method, involving large-deformations and non-linear material laws. For the linear case, the DSG method has proven to eliminate all geometric locking effects, such as shear locking, membrane locking, and trapezoidal locking, with one unique approach [1]. Satisfaction of the patch test – which is violated in the original DSG formulation for solids – may be ensured via a decomposition of strain modes into a constant part and higher order modes. However, this modification goes along with the appearance of trapezoidal locking (cf. “MacNeal’s Dilemma” [2]). In the particular case of thin shell analysis with solid or solid-shell elements, this is of noticeable relevance. A way out of this dilemma is to choose the original, locking-free formulation for the “out-of-plane” strain components while ensuring consistency via the mesh layout in transverse direction. The proposed method allows for an easy switch from 3d-structures to thin structures within one common element technology, and it is both consistent and locking-free in either case [3]. In order to remedy the material locking phenomenon of volumetric locking the well-established Enhanced Assumed Strain (EAS) method is used. For the presented element only three additional internal parameters are necessary. In the present contribution this element is extended to non-linear dynamic problems. Both geometric and material non-linearities are taken into account. From the material model point of view special attention will be drawn to hyperelastic and viscoelastic materials. Numerical examples involving static and dynamic analysis with both material and geometric non-linearities are given and compared with existing results obtained with other shell elements.

References [1] F. Koschnick, M. Bischoff, N. Camprubi, K.-U. Bletzinger, The discrete strain gap method and membrane locking. Computer Methods in Applied Mechanics and Engineering, 194,2444–2463, 2005. [2] R. H. MacNeal, A theorem regarding the locking of tapered four-noded membrane elements. International Journal for Numerical Methods in Engineering, 24:1793–1799, 1987 [3] M. Frenzel, M. Bischoff, K.-U. Bletzinger, W. A. Wall, Performance of Discrete Strain Gap (DSG) finite elements in the analysis of three-dimensional solids. In: Proceedings of the 5th International Conference on Computation of Shell and Spatial Structures, Salzburg, Austria, June 1-4, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

655

A Simple Co-Rotational Geometrically Non-Linear Membrane Finite Element Wrinkling Analysis E. Gal*, M. Zelkha†, R. Levy† * Ben-Gurion University of the Negev Department of Structural Engineering, 84105 Beer-Sheva, Israel [email protected] †

Technion- Israel institution of Technology Faculty of environmental and Civil Engineering, 32000 Haifa, Israel [email protected]

ABSTRACT Thin pre-tensioned membranes are often used in many technological applications such as fabric constructions, marine and space technology. The advantages of these structures are the lightness of the membrane, which facilitates coverage of large spans and the ability to create a variety of shapes. Fabric membranes by definition have little compression resistance and have no bending stiffness, hence is very easy to wrinkle. This research presents the analysis of wrinkled membranes using a geometrically nonlinear membrane finite element by Levy and Spillers [1] as the core of the proposed analysis procedure. The wrinkle state of a membrane finite element can be defined as follows: Taut V 1 ! 0 and V 2 d 0 and Slack state: state: V 1 ! 0 and V 2 ! 0 ; Wrinkled state: V 1 d 0 and V 2 d 0 where V 1 and V 2 are the maximum and the minimum principle stresses of the element respectively. The physical interpretation of this definition is that elements in the Taut state are fully tensioned in both directions and therefore follow the isotropic constitutive equation, elements in the Wrinkled state are in tension the V 1 direction only and therefore follow the orthotropic constitutive equation and elements in the Slack state are in full compression and therefore inactive (e.g. see Tabarrok and Qin [3] and Miller and Hedgepeth [2]). The proposed analysis is performed in an incremental iterative fashion. At each increment a two step convergence criterion is imposed. The first step handles equilibrium in the final configuration of the membrane using the Newton-Rapson method. The second step takes care of the final wrinkled-status of the elements where stresses are updated (if necessarily) according to the element state i.e. negative principle stress are set to zero when the unbalanced force vector is computed due to these changes. Convergence of the second step is achieved when none of the elements changes its wrinkled-status. Finally validation and verification of the proposed analysis is performed by comparing results to those of several benchmark problems.

References [1] [2] [3]

R. Levy and W.R. Spillers, Analysis of geometrically nonlinear structures. Chapman & Hall, New York:,1994. R.R. Miller and J.M. Hedegepth, An Algorithm for finite element analysis for partly wrinkled membranes. AIAA Journal, 20(12), 1761-1763, 1982. B. Tabbarrok and Z. Qin. Nonlinear analysis of tension structures. Computer and Structures, 45, 973-984, 1992M. Stein and J. M. Hedegepth, Analysis of partly wrinkled membranes. NASA Technical Note D-813 July 1961.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

656

Co-rotational System Definitions for Large Displacement Triangular and Quadrilateral Shell Elements Bassam A. Izzuddin Imperial College Lodnon London SW7 2AZ, United Kingdom [email protected]

ABSTRACT The co-rotational approach offers exceptional benefits for large-displacement structural analysis problems with deformations of the bending type, particularly when accounting for arbitrarily large rigid body rotations. A principal issue in any co-rotational approach is associated with the specific choice of the local reference system in relation to the current deformed element configuration. Whilst an arbitrary choice that closely follows the current element configuration, for example using the current positions of any two of the element sides, does not significantly affect the large displacement response predictions for small strain problems, this often leads to local system definitions which are not invariant to the specified order of the element nodes. It has been previously argued that this invariance characteristic would be desirable for extending the co-rotational approach to large strain problems [1] and, more recently, for identifying the bifurcation points of perfectly symmetric structures [2]. Two approaches were employed to achieve the invariance of the local system to nodal ordering [1,2], but these suffered from complexity associated with application to material points within the element domain, and resulted in an asymmetric consistent tangent stiffness matrix which leads to further computational disadvantages. In this paper, new definitions of the local co-rotational system are proposed for quadrilateral and triangular shell elements, which achieve the invariance characteristic to nodal ordering in a relatively simple manner, and importantly result in a symmetric tangent stiffness matrix. The proposed definitions utilise only the nodal coordinates in the deformed configuration, where two alternative definitions are outlined for each of the quadrilateral and triangular element shapes. The first is a bisector definition utilising alignment along the bisectors of one or more internal element angles, while the second is a zero-‘macro spin’ definition considering the minimisation of the spin for the triangular/quadrilateral shape as described by the element nodes. The paper presents the co-rotational transformations linking the local and global element freedoms for both definitions, and provides a numerical example to demonstrate their relative accuracy in large displacement analysis of plates and shells. It is shown that both definitions are equally accurate for small strain problems, but that the zero-‘macro spin’ definition has more general potential application in large strain problems.

References [1] M.A. Crisfield and G.F. Moita, A Unified Co-rotational Famework for Solids, Shells and Beams, Int. J. Solids Struct., 33, 2969-2992, 1996. [2] J.M. Battini and C. Pacoste, On the Choice of Local Element Frame for Corotational Triangular Shell Elements, Commun. Numer. Meth. Engng., 20(10), 819-825, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Non-Linear Analysis of Composite Plates and Shells Using a New Shell Element Peyman Khosravi∗ , Rajamohan Ganesan† , Ramin Sedaghati‡ Department of Mechanical and Industrial Engineering, Concordia University Montreal, Quebec, H3G 1M8, Canada email∗ : peyma [email protected] email† : [email protected] email‡ : [email protected] ABSTRACT One of the most popular approaches in the finite element analysis of plates and shells is using an assemblage of facet triangular elements built by combining a membrane and a plate bending element to model the curved surface. Due to the lack of a drilling degree of freedom in most triangular membrane elements, these elements may cause rotational singularity in the stiffness matrix. One approach to overcome this problem is using membrane elements with in-plane (or drilling) rotational degree of freedom. Although some elements with drilling degree of freedom have been derived, most of them suffer from aspect ratio locking. Recently, Felippa [1] developed an optimal membrane element with drilling degree of freedom. Its response for in-plane pure bending is not dependent on the aspect ratio. There are several triangular plate bending elements to combine with a membrane element. Batoz et al. [2] studied several triangular Kirchhoff plate bending elements and showed that Discrete Kirchhoff Triangle (DKT) [3], is the most reliable triangular element for analysis of thin plates. Katili [4] developed a discrete Kirchhoff-Mindlin triangular plate bending element called DKMT which is capable to include the transverse shear effects in thick plates, and coincides with the DKT element in case of thin plates. As a result, both thin and thick plates can be modeled with this element. In the present work, a new shell element for both thin and thick plates is developed by combining the DKMT plate bending element and the optimal membrane triangular element (also called OPT). The membrane-bending coupling effect of composite laminates is incorporated in the formulation, and inconsistent stress stiffness matrix and tangent stiffness matrix are formulated. Using co-rotational method and the tangent stiffness matrix, this new shell element is used to solve problems with geometric nonlinearity and the results are compared with analytical solutions or those available in the literature. The behavior and advantages of the new element are studied.

References [1] CA. Felippa, A study of optimal membrane triangles with drilling freedoms. Computer Methods in Applied Mechanics and Engineering, 192(16), 2125–2168, 2003. [2] JL. Batoz, An explicit formulation for an efficient triangular plate-bending element. International Journal for Numerical Methods in Engineering, 18, 1077–1089, 1982. [3] JL. Batoz, KJ. Bathe, LW. Ho, A study of three-node triangular plate bending elements. International Journal for Numerical Methods in Engineering, 15, 1771–1812, 1980. [4] I. Katili, A new discrete Kirchhoff-Mindlin element based on Mindlin-Reinssner plate theory and assumed shear strain fields- Part I: An extended DKT element for thick-plate bending analysis. International Journal for Numerical Methods in Engineering, 36, 1859–1883, 1993.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

658

Total and Updated Lagrangian Geometrically Exact Beam Elements *

Jari Mäkinen , Heikki Marjamäki

†

Tampere University of Technology, Applied Mechanics and Optimization P.O. Box 589, FIN-33101 Tampere, FINLAND * † [email protected], [email protected]

ABSTRACT In the paper [1] is given finite element implementations for a geometrically exact beam element with different updating procedures. The formulations are named Eulerian, total Lagrangian, and updated Lagrangian. The updated Lagrangian formulation can bypass the well-known singularity problem of the total Lagrangian formulation which is singular at the rotation angle 2π and its multiples. The u pdated Lagrangian formulation has additional benefits such as a fully symmetrical stiffness tensor when applying a conservative loading. In addition, any time integration algorithm can be used because the changes of the rotation vector belong to the same tangent space of the rotation manifold SO(3). The updated Lagrangian formulation requires some secondary storage variables for the curvature and rotation vectors, at every spatial integration point. A total Lagrangian formulation in static cases with the consistent stiffness tensor is given in [2]. Generally, a total Lagrangian formulation in a static case with a conservative loading has an important property that is path-independence, whereas an updated Lagrangian formulation is path-dependent. Lagrangian formulations have a consistent interpolation while in Eulerian formulations the interpolation has to apply within an approximate, inconsistent, way. As we have noted earlier, total Lagrangian formulations have singularity at the rotation angle 2π and its multiples that are a remarkable restriction, especially in dynamic cases. We consider a total Lagrangian updating formulation [3] for a material rotation vector and compare it with an updated Lagrangian formulation. The total Lagrangian formulation preserves the pathindependence and it can be regarded as a consistent updating formulation. The major drawback is the singularities at the rotation angle 2π and its multiples, but we overcome this difficulty by the complement rotation vector that is the change of parametrization. Numerical examples are also given.

References [1] A. Cardona, M. Géradin, A Beam Finite Element Non-Linear Theory with Finite Rotations, International Journal for Numerical Methods in Engineering, 26, 2403-2438, 1988. [2]

A. Ibrahimbegović, F. Frey, I. Kozar, Computational Aspects of Vector-Like Parametrization of

Three-Dimensional Finite Rotations, International Journal for Numerical Methods in Engineering, 38, 3653-3673, 1995. [3] Mäkinen, J., A Formulation for Flexible Multibody Mechanics – Lagrangian Geometrically Exact Beam Elements using Constraint Manifold Parametrization, Doctoral Thesis, TUT, Applied Mechanics and Optimization, 2004, 89 pp. http://www.tut.fi/~jmamakin/vk.pdf

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Large Displacement Analysis of Plates using Hybrid Equilibrium Elements Edward A. W. Maunder* , Bassam A. Izzuddin† * University of Exeter Harrison Building, North Park Road, Exeter, EX4 4QF, UK [email protected] †

Imperial College, University of London South Kensington Campus, London SW7 2AZ, UK [email protected]

ABSTRACT Small strains can coexist with large displacements, particularly for structures composed of thin plates. This paper is concerned with finite element modelling of such non-linear behaviour when the material retains its linear elastic constitutive relations and Reissner-Mindlin theory can be invoked. This type of behaviour is of particular interest when for example in-plane stiffening becomes significant in a floor slab, or when elastic buckling due to geometric imperfections can occur. The co-rotational concept is invoked whereby the rigid body translations and rotations of elements are accounted for by the associated movements of element local reference axes, and the linear elastic stiffness matrices of elements are maintained unchanged with respect to current local configurations. Such formulations have been described by Izzuddin [1] where a more conventional conforming type of quadrilateral element is used. Due to the inherent generality of this approach, it has now been extended to hybrid equilibrium flat shell quadrilateral elements, and this is described in the paper. The formulation of this element is also presented and use is made of local element axes which are defined so as to lead to symmetric tangent stiffness matrices. The hybrid equilibrium model [2,3] enables solutions to be determined which satisfy equilibrium in a strong pointwise sense. This feature is believed to be a novel one in the case of modelling geometrically non-linear behaviour with finite elements, and it allows new meaning to be given to the presentation of equilibrium paths. Thus it may be expected that a plot of a quantity of interest versus a load parameter is more accurate when the quantity is a stressresultant. Numerical examples are presented to compare the performance of the equilibrium models with conventional conforming models, and to investigate the potential advantages of having dual solutions to a geometrically non-linear problem. Theoretical questions regarding the significance of fully equilibrated solutions have not yet been answered for such problems, e.g. can such solutions lead to upper bounds to the energy of the errors in a conforming solution? Nevertheless in practice it may be argued that two solutions are always better than one, and useful bounds may be achieved.

References [1] B.A. Izzuddin, An enhanced co-rotational approach for large displacement analysis of plates. International Journal for Numerical Methods in Engineering, 64, 1350-1374, 2005. [2] E.A.W. Maunder, Hybrid elements in the modelling of plates. B.H.V.Topping, ed., Finite Elements: Techniques and Developments, Civil-Comp Press, 165-172, 2000. [3] E.A.W. Maunder, J.P. Moitinho de Almeida, A triangular hybrid equilibrium plate element of general degree, International Journal for Numerical Methods in Engineering, 63, 315-350, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Buckling Modes of Large-Scale Shell Structures Automatically Detected from Linearized Stiffness by Iterative Solvers Hirohisa Noguchi *, Fumio Fujii†, Yoshikazu Ishihara†† *

†

Department of Systems Design Engineering Keio University, Yokohama, Japan [email protected]

Department of Mathematical and Computational Engineering Gifu University, Gifu, Japan [email protected] ††

Safety Engineering and Technology Department Mitsubishi Research Institute, Inc., Tokyo, Japan [email protected]

ABSTRACT This study presents a novel method to automatically detect bifurcation buckling modes of large scale shell structures during solving a set of linearized stiffness equations in geometrically nonlinear problems by iterative solvers, such as the CG method or the Lanczos method. The proposed method is based on the LDLT decomposition method for direct solvers proposed by the authors [1][2] and is extended for the iterative solvers in order to handle large-scale problems. First, the proposed method detects the approximate buckling mode during the simultaneous process of tri-diagonalization and the LDLT decomposition of the stiffness matrix, utilizing the fact that the Lanczos algorithm still preserves the eigenvalue properties of original matrix during the tri-diagonalization. Second, it is also shown that the correction vector in the direction of solution in the CG method can approximate the buckling mode. The proposed method can avoid a time-consuming eigenanalysis, which is usually necessary for detecting bifurcation modes at critical points, and can compute the approximate bifurcation modes closed to the critical points very efficiently and accurately. Several numerical examples of bifurcation buckling of shells demonstrate the potential of the proposed method

References [1] F. Fujii, F. H. Noguchi, The buckling mode extracted from the LDLT-decomposed large-order stiffness matrix, Communication in numerical methods in engineering, 18, 459-467, 2002. [2] H. Noguchi, F. Fujii, Eigenvector-free indicator, pinpointing and branch-switching for bifurcation, Communications in Numerical Methods in Engineering, 19, 445-457, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Finite Element analysis of the wrinkling of orthotropic membranes Igor P. Oliveira, Eduardo M. B. Campello, Paulo M. Pimenta Department of Structural and Foundation Engineering, University of São Paulo P.O. Box 61548, São Paulo, Brazil [email protected], [email protected], [email protected]

ABSTRACT This work presents a fully nonlinear formulation for the analysis of the wrinkling on orthotropic membranes. Our approach describes the membrane kinematics as a thin shell motion, whose bending stiffness comes naturally from the shell assumptions. We combine the geometrically-exact isotropic shell model of [1,2] with an orthotropic constitutive equation for the membrane strains (see [3,4]), so that both bending and typical membrane capabilities are present in a totally consistent way. The strain energy function is split into an isotropic and an orthotropic part, the first one being relative to the shell (hyperelastic) behavior and the latter to the membrane deformations. The model is discretized under the light of the finite element method using the six-node triangular element of [2], and the performance of the formulation is assessed in several numerical examples (see e.g. Fig. 1). Unstructured meshes are deliberately employed whereas small geometrical imperfections are imposed for the wrinkles to be initiated. Experimental data from the membrane tests of [5] are also taken into account for comparison with our results.

Fig.1. Stretching of two orthotropic membranes. Deformed configurations.

References [1] P. M. Pimenta, On a geometrically-exact finite-strain shell model, Proceedings of the 3rd PanAmerican Congress on Applied Mechanics, III PACAM, São Paulo, 1993. [2] E. M. B. Campello, P. M. Pimenta and P. Wriggers, A triangular finite shell element based on a fully nonlinear shell formulation, Computational Mechanics, 31, 505-518, 2003. [3] Oliveira I. P., Campello E. M. B. and Pimenta P. M., “Wrinkling of nonlinear orthotropic membranes by the finite element method”, submitted to Comput. Mech., 2006. [4] Oliveira I. P., Análise não-linear de membranas: ortotropia e enrugamento. Master of Science Dissertation, Department of Structural and Foundation Engineering, University of São Paulo, 2006. [5] Shelter Rite£ by Seaman Corporation, High Performance 9032 – Architectural Fabric, Biaxial Stretch test, available at www.architecturalfabrics.com/9032.html, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Theory and Numerics of a Surface–Related Shell Formulation Rainer Schlebusch∗ , Bernd W. Zastrau† ∗

Technische Universit¨at Dresden 01062 Dresden, Germany [email protected] † Technische

Universit¨at Dresden 01062 Dresden, Germany [email protected] ABSTRACT The solution of structural analysis problems, especially of shell structures, demands an efficient numerical solution strategy. Since one–sided contact problems are investigated, the shell model is formulated with respect to one of the outer surfaces, i.e. the shell formulation is surface–related. In particular the investigation of textile reinforced strengthening layers [2] will be carried out by this approach. Since even shells are three–dimensional structures, i.e. bodies, the basic field equations of continuum mechanics must be the starting point. This set of partial differential equations with pertinent boundary conditions has to be solved. An efficient numerical solution of this problem becomes easier, if the problem is reformulated against a background of variational calculus. The discretization of the resulting variational formulation is, among others, the source of several locking phenomena. The presented shell formulation uses linear shell kinematics with six displacement parameters, circumventing a rotational formulation. This low–order shell kinematics produces parasitical strains and stresses, leading to wrong or even useless results. Therewith an extension and/or adjustment of well– known techniques to prevent or reduce locking like the assumed natural strain (ANS) method [3] and the enhanced assumed strain (EAS) method [4] has to be performed. Using these adapted methods, a reliable and efficient solid–shell element with tremendously reduced locking properties is obtained. This concept comprises the utilization of unmodified three–dimensional constitutive relations by a minimal number of kinematical parameters [1]. With the aid of two nonlinear examples, the reliability and the efficiency of the new solid–shell element is demonstrated.

References [1] N. B¨uchter, E. Ramm, 3d–Extension of Nonlinear Shell Equations Based on the Enhanced Assumed Strain Concept, in C. Hirsch, ed., Computational Methods in Applied Sciences, Elsevier, 55–62, 1992. [2] R. Schlebusch, J. Matheas, B. Zastrau, On Surface–Related Shell Theories for the Numerical Simulation of Contact Problems, Journal of Theoretical and Applied Mechanics, 41(3), 623–642, 2003. [3] J.C. Simo, T.J.R. Hughes, On Variational Foundations of Assumed Strain Methods, Journal of Applied Mechanics, 53, 52–54, 1986. [4] J.C. Simo, M.S. Rifai, A class of mixed assumed strain methods and the method of incompatible modes, International Journal for Numerical Methods in Engineering, 29, 1595 - 1638, 1990.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Modelling and optimization of sails M. Spalatelu-Lazar, F. L´en´e, N. Turb´e Laboratoire Mod´elisation, Mat´eriaux, Structures (LM2S) Universit´e Paris VI, CC 161, 4 Place Jussieu, 75252 Paris Cedex 05, France [email protected] ABSTRACT The sails fabrication is an old activity principally based on the practice and the experience of sailmakers and users in constant search of performances and safety. The sails design registered a significant development under the impulse of sailing races like America’s Cup [1]. These competitions require the use of advanced technologies to increase the sails performances: optimization of the fibres orientation, of the weight, of the sails shapes in navigation. The objective of this paper is to shed some light on how to improve the quality and the performances by a control of the fibres orientation calling, for modelling, numerical experimentation and optimization methods. A sail is a lightweight flexible structure, made up of an assembly of panels, reinforced in its critical points by doublings or straps, and often rigidified by battens [2]. The efforts acting on the sails involve significant deformations according to the force of the wind and the nature of sailcloth. The parameters of the problem are thus very numerous, related to the definition of the sail and the loading cases. The sail is modelled here by a triangular membrane, submitted to large displacements and small strains [3]. Initial pre-tension load is required [4]. The behaviour law of the structure enters in the framework of the linear orthotropic elasticity modelling long fibres materials. The equilibrium equations, formulated on the midsurface, are solved by a modified Newton-Raphson method. The sailmaker can modulate the mechanical behaviour according to three parameters: the nature of the components, their proportion and the orientation of fibres. The presence of strong stresses zones usually leads to a sail cut-out in pieces with rectilinear fibres of constant orientation. Recent progress in sails fabrication allow the realization of structures with curvilinear supports and variable density rates. It then becomes essential to determine with precision which are, in each point, the orientation and the optimal rate of fibres. The present study is focused on the optimal fibres orientation. The mathematical problem of optimization is related to the displacement in the transverse direction of the sail. The optimization method uses the Nelder-Mead algorithm, efficient to solve non-linear problems. The numerical results are always independent of the initial fibres distribution and of the mesh.

References [1] C. A. Marchaj, Aero-hydrodynamics of sailing. Adland Coles limited, Granada Publishing, 1979. [2] V. Boh´e, P. Casari, F. L´en´e, P. Davies Comportement des mat´eriaux a` voiles de bateau. Revue des Composites et des Mat´eriaux Avanc´es , 3, 2003. [3] F. Muttin, Structural analysis of sails. European Journal of Mechanics, 10 (5) 517-534, 1991. [4] M. Quadrelli, S. Sirlin Modeling and control of membranes for gossamer spacecraft. AIAA Journal, 1201, 2001.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Instability Analysis of thin-walled structures using Incompressible Hyperelastic Shell Elements Masato Tanaka*, Hirohisa Noguchi† *

School of Science for Open and Environmental Systems Keio University, Yokohama, Japan [email protected] †

Department of Systems Design Engineering Keio University, Yokohama, Japan [email protected]

ABSTRACT In the present study, a new hyperelastic shell element is proposed which can be applied to the numerical simulation of instability analyses of thin-walled structures made of rubber-like materials. The recent research on the shell problem also involves flexible structures. This refers to many manmade rubber-like structures such as pneumatic membranes, automobile tires, hydraulic hoses and various biological soft tissues such as blood vessels, lung pleura and cardiac muscles. The numerical simulation of such materials is not possible under the assumption of small elastic strains and requires a hyperelastic constitutive formulation. Therefore, in order to analyze these materials, the effect of change in the shell thickness due to the normal strains should be taken into account. In recent years, these finite strain shell elements accounting for thickness change have been presented [1]. The elements use additional degrees of freedom for thickness change and three dimensional constitutive relations. Therefore, their shell elements are addressing the solid models and furthermore introduce one extra degree of freedom for pressure in the case of incompressible materials. However, these additional degrees are a drawback in the recent advances of large scale computational analysis. Contrary to the above theories, the aim of this study is to derive a nonlinear formulation of a shell element for incompressible hyperelastic materials undergoing large elastic strains without any additional degrees of freedom. The uniform deformation in the normal direction due to stretching of the shell middle surface is assumed in the element through the incompressibility condition. The additional state variable, hydrostatic pressure, which occurs for incompressible materials, is eliminated on the element level using the plane stress condition. Thus, the present formulation includes thickness change and hydrostatic pressure implicitly, and uses only 5 d.o.f per node; three translation degrees and two rotation degrees. The presented element provides a consistent tangent stiffness during deformation very efficiently according to this assumed normal strain formulation. The present shell element is based on the method of MITC4[2], and we derive formulation of the assumed shear and normal strains using the mixed variational principle. Several numerical instability analyses such as tube bending and balloon analysis are conducted to illustrate the performance of the shell elements developed herein.

References [1] M. Braun, M. Bischoff, E. Ramm, “Nonlinear Shell Formulation for Complete ThreeDimensional Constitutive Laws Including Composites and Laminates”, Computational Mechanics, 15, 1-18 (1994) [2] D. Chapelle, K.J. Bathe, “The finite element analysis of shells - fundamentals”,

Springer,(2003)

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

665

Stochastic simulation of pitting corrosion J. L´opez De La Cruz∗ , M. A. Guti´errez∗ , L. Koene∗ ∗ Delft

University of Technology, Department of Aerospace Engineering P.O. Box 5058, 2600 GB Delft, The Netherlands [email protected] [email protected] [email protected]

ABSTRACT The simulation of pit initiation is the starting point for the simulation and understanding of stress corrosion cracking (SCC). The corrosion process is a phenomenon where few variables are deterministic. It has been found in previous research that not only the pitting potential is a random variable with a known distribution [1, 2] but also the pit growth rate follows a distribution. The simulation of SCC is a further step. After pit initiation and pit growth are successfully simulated by including their stochastic properties in the analysis , it is possible to obtain reliable insights with respect to the SCC nature. In this paper, the simulation of pit initiation applying the Bernoulli lattice process and the finite elements method (FEM) is shown. As supporting electrolyte, NaCl is selected and iron is the metal undergoing localized corrosion. Anodic and cathodic sites are determined beforehand taking into account the location of the pit and the area surrounding it. In the electrochemical model the reactions considered are the oxidation of iron, oxygen reduction and hydrogen reduction. The Nernst-Planck equation is employed to model the mass transport in the domain (region between the metal and the electrolyte). In the metal boundaries the flux of species is determined by the Butler-Volmer equation. In the electrolyte boundaries the current density in the normal direction is restricted to zero. The current density distribution calculated by the model is used to compute the corrosion intensity (loose of material per unit time). A deterministic point process [3] is used to sample pitting spots and the probability of pitting corrosion is computed assuming that all the points over the metal surface are uniformly distributed with equal probability of becoming anodic. Restrictions are imposed to the metal and physical variables affecting it. The results obtained from this study are a key tool for the analysis of SCC initiation and further behavior.

References [1] Hong, H..P. ”Application of the stochastic process to pitting corrosion”. Corrosion, vol. 55, No. 1, January (1999). [2] Digby D. Macdonald and Mirna Urquidi-Macdonald . ” Distribution functions for the breakdown of passive films”. Electrochimica acta, vol. 31, No. 8, pp. 1079-1086, (1986). [3] Dietrich Stoyan, Wilfrid S. Kendall & Joseph Mecke. Stochastic geometry and its applications. John Wiley & Sons Ltd, Second Print (1995).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

666

Non Gaussian Response of Bridges Subjected to Turbulent Wind – Effect of the non Linearity of Aerodynamic Coefficients Vincent Denoël*, Hervé Degée* *

University of Liège Chemin des chevreuils, 1, B-4000 Liège (Belgium) [email protected]

ABSTRACT Wind loads acting of bluff bodies like bridge decks are complex functions of the components of the turbulence and of the structural displacements and velocities. In order to simplify the representation of these loads, approached models are generally considered. Since a convenient linear approximation gives accurate results in many cases, such a model has been widely used during last decades. Some researchers have however showed that it is possible, and even necessary, to account for the non linearity of this kind of loading. Such a non linearity is likely to come either from the squared velocity or from the shape of the aerodynamic coefficients as functions of the wind angle of attack. Non linear loading of the first kind (squared velocity) have already been studied in rather recent researches. Some of them are referenced at the end of this abstract. It has been showed that this so-called non linear quasi-steady aerodynamic loading leads to a non Gaussian response of the structure although the components of the turbulence are Gaussian. This has of course significant consequences on the structural design. This paper aims at showing that the second origin of non linear loading terms, i.e. the non linearity of the aerodynamic coefficients, can also have significant consequences on the design of wind-loaded structures. Developments are carried out for bi- and tri-dimensional turbulence fields, and the main conclusions are that these effects are of prime importance of course when the non linearity of the aerodynamic coefficients is important, but also when the transverse component of the turbulence is important. In a more detailed way, the proposed paper intends at presenting two main features. The first one consists in the determination of the statistical characteristics of the loading. In particular it is showed that the traditional linearization of the aerodynamic coefficients may lead to a significant inaccuracy of these statistical characteristics. The second feature is the presentation of a simplified method to derive statistical characteristics of the response from those of the loading. This very simple method is compared with a heavy rigorous analysis. It is concluded that this simple approach can give good estimations of the response that could be used, for instance, for the pre-design of a structure.

References [1] P.D. Spanos, Spectral moment calculation of linear system output. Journal of Applied Mechanics ASLE, 50, 901-903, 1983. [2] Kareem A. and al., Modeling and analysis of quadratic term in the wind effects on structures. Journal of Wind Engineering and Industrial Aerodynamics, 74, 1101-1110, 1998. [3] Gurley K. and al., Analysis and simulation tools for wind engineering. Probabilistic Engineering Mechanics, 12, 9-31, 1997.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

667

Simulation of non-Gaussian stochastic processes and fields with applications to structural engineering problems Mircea Grigoriu Cornell University Ithaca, NY, USA [email protected]

ABSTRACT Three classes of non-Gaussian functions are defined and Monte Carlo algorithms are developed for generating samples of the random functions in these classes. These classes consist of translation random functions, diffusion processes and memoryless transformations of these processes, and spectral
representation based non-Gaussian processes. A translation random function X(t) ∈ Rd , t ∈ Rd , is defined by     Xi (t) = Fi−1 ◦ Φ Gi (t) = hi Gi (t) , i = 1, . . . , d,


where Φ is the distribution of the standard Gaussian variable N (0, 1) and G(t), t ∈ Rd , is an Rd valued stationary Gaussian function  with coordinates  Gi (t), i
= 1, . . . , d, of mean 0, variance 1, and covariance functions ρij (τ ) = E Gi (t + τ ) Gj (t) , τ ∈ Rd . Hence, X is specified by its marginal distribution and second-moment properties. Diffusion processes can be viewed as outputs of dynamic systems to Gaussian white noise. For example, X is said to be a diffusion process if it is defined by the stochastic differential equation dX(t) = a(X(t)) dt + b(X(t)) d(t), t ≥ 0, ¯ where a and b denote the drift and diffusion of X and B is a Brownian motion. Spectral representation based non-Gaussian real-valued processes are defined by  ∞   X(t) = cos(ν t) dM1 (ν) + sin(ν t) dM2 (ν) , t ≥ 0, 0

where M1 and M2 are square integrable martingale. Monte Carlo simulation algorithms are developed for all non-Gaussian functions considered in the paper. The algorithms are simple, efficient, and can be based on MATLAB functions. Numerical examples are used to illustrate the implementation of some of the Monte Carlo simulation algorithms presented in the paper. It is shown that translation functions are versatile and their construction involves relatively relatively simple concepts.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A novel approach for the efficient simulation of highly skewed non-Gaussian stochastic fields Nikolaos D. Lagaros, George Stefanou, Manolis Papadrakakis Institute of Structural Analysis & Seismic Research National Technical University of Athens, 9, Iroon Polytechniou Str., Zografou Campus, GR-15780 Athens, Greece {nlagaros,stegesa,mpapadra}@central.ntua.gr

ABSTRACT The problem of simulating non-Gaussian stochastic processes and fields has received considerable attention recently in the field of stochastic mechanics. This is due to the fact that several quantities involved in practical engineering problems (e.g. material and geometric properties of structural systems, soil properties in geotechnical engineering applications, wind loads, waves) exhibit nonGaussian probabilistic characteristics [2]. In this paper, a novel, computationally efficient method is presented for the simulation of homogeneous non-Gaussian stochastic fields with prescribed target marginal distribution F and spectral density function S ff (N ) [3]. The proposed approach is based on the translation field concept [2], but uses the extended empirical non-Gaussian to non-Gaussian mapping f ( x) F 1 ˜ F [ g ( x)] introduced in [1] for the generation of a non-Gaussian field f ( x ) having the prescribed characteristics. In this way, the possible incompatibility between the marginal distribution and the correlation structure of a translation field is surpassed and an algorithm covering a wider range of non-Gaussian fields is produced. The new algorithm is spectral representation-based as it makes use of the spectral representation method in order to generate sample functions of the underlying Gaussian field g ( x ) . It retains the accuracy characteristics of the method proposed in [1] while drastically reduces the computational effort of the simulation by reliably predicting the unknown Gaussian spectrum S gg (N ) . Precisely the function fitting ability of Neural Networks (NN) is employed to approximate the Gaussian spectrum and the resulting methodology matches the prescribed target marginal distribution and spectral density function with remarkable accuracy even in the case of narrow-banded fields with very large skewness. Various features of the new algorithm are demonstrated with the simulation of two stochastic fields having different correlation structure and following a highly skewed lognormal distribution.

References [1] G. Deodatis, R. C. Micaletti, Simulation of highly skewed non-Gaussian stochastic processes. J. of Engineering Mechanics (ASCE), 127, 1284-1295, 2001. [2] M. Grigoriu, Applied non-Gaussian processes. Prentice-Hall, Englewood-Cliffs New Jersey, 1995. [3] N. D. Lagaros, G. Stefanou, M. Papadrakakis, An enhanced hybrid method for the simulation of highly skewed non-Gaussian stochastic fields. Computer Methods in Applied Mechanics and Engineering, 194, 4824-4844, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

669

The Wave Propagation in a Vertically Inhomogeneou Soil with a Random Dynamic Shear Modulus Mattias Schevenels, Geert Lombaert, Geert Degrande, Daan Degrauwe, Brecht Schoors Department of Civil Engineering K.U.Leuven, Kasteelpark Arenberg 40, B-3001 Leuven, Belgium [email protected] ABSTRACT Vibrations induced by road and rail traffic are a common source of discomfort to people. Numerical models have been developed for the prediction of traffic induced vibrations in the free field or in the built environment. These models consist of a finite element formulation for the vehicles and the buildings and a boundary element formulation that accounts for the wave propagation in the soil. The latter is based on the Green’s functions of a horizontally layered halfspace. The experimental validation of these models reveals a discrepancy between the predicted and measured response in the higher frequency range. Given the crucial role of the Green’s functions in the prediction model, the dynamic soil characteristics governing these functions are a possible source of the discrepancy. Common techniques for the in-situ measurement of the dynamic soil characteristics such as the spectral analysis of surface waves (SASW) test and the seismic cone penetration test (SCPT) are based on local averages of the soil characteristics and have a limited resolution. The small scale variations of the soil characteristics are not revealed. In this paper, the influence of the small scale variations of the dynamic shear modulus on the Green’s functions of a vertically inhomogeneous soil is studied. A probabilistic approach is followed where the mean soil is modelled using the results of the aforementioned measurement techniques. Superimposed on the mean profile is a zero mean random process that represents the small scale variations of the dynamic shear modulus. This process is characterized by a marginal probability density function and a correlation function, estimated by means of a cone penetration test (CPT). The resolution of the CPT test is high as compared to the SASW and the SCPT tests. The non-Gaussian random process is discretized by means of a Hermite polynomial expansion and a Karhunen-Loeve decomposition [1]. The methodology of the stochastic finite element method [2] is applied to a hybrid thin layer – direct stiffness formulation [3] in order to assemble the stochastic system equations. These are solved by means of a Monte Carlo simulation to obtain the stochastic Green’s functions. The results of the simulation are in good correspondence with the discrepancy observed in the validation of the deterministic vibration prediction models. In the low frequency range, the small scale variations of the dynamic shear modulus are not resolved by the waves and all realizations of the stochastic Green’s functions tend to the Green’s functions of the mean soil. In the high frequency range, the waves do resolve the small scale variations. As a result, phase shifts and variations of the amplitude occur between different realizations of the stochastic Green’s functions.

References [1] B. Puig, F. Poirion, and C. Soize. Non-Gaussian simulation using Hermite polynomial expansion: convergences and algorithms. Probabilistic Engineering Mechanics, 17:253–264, 2002. [2] R.G. Ghanem. Ingredients for a general purpose stochastic finite elements implementation. Computer Methods in Applied Mechanics and Engineering, 168:19–34, 1999. [3] E. Kausel and J.M. Ro¨esset. Stiffness matrices for layered soils. Bulletin of the Seismological Society of America, 71(6):1743–1761, 1981.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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On the Karhunen-Loeve expansion and spectral representation methods for the simulation of Gaussian stochastic fields George Stefanou, Manolis Papadrakakis Institute of Structural Analysis & Seismic Research National Technical University of Athens, 9, Iroon Polytechniou Str., Zografou Campus, GR-15780 Athens, Greece {stegesa,mpapadra}@central.ntua.gr

ABSTRACT The parameters describing a structure are uncertain quantities and the analysis and safe design of most engineering systems must take into account these uncertainties. Uncertain structural parameters are usually modeled as random fields. Despite the fact that most of the uncertain quantities appearing in practical engineering problems are non-Gaussian in nature (e.g. material and geometric properties, wind loads), the Gaussian assumption is often used due to the lack of relevant experimental data. From the wide variety of methods developed for the simulation of Gaussian stochastic processes and fields, two are most often used in practice: the spectral representation method and the KarhunenLoeve (K-L) expansion. The K-L expansion can be seen as a special case of the orthogonal series expansion where the orthogonal functions are chosen as the eigenfunctions of a Fredholm integral equation of the second kind with the autocovariance as kernel. In this paper, a wavelet-Galerkin scheme is adopted for the efficient solution of the Fredholm equation. A one-dimensional homogeneous Gaussian random field with two types of autocovariance function (power spectrum), exponential and square exponential, is used as test example. The numerical instabilities arising in some cases during the calculation of eigenvalues of both kernels at high wavelet levels ( m ≥ 6 ) are reported. The influence of the scale of correlation on the simulation quality is quantified by using several values of correlation length parameter b. In this work, a comparison of the accuracy achieved and the computational effort required by the K-L expansion and the spectral representation for the simulation of the stochastic field is pursued. The accuracy obtained by the two methods is examined by comparing their ability to produce sample functions that match the target correlation structure and the Gaussian probability distribution or, alternatively, its low order statistical moments (mean, variance and skewness).

References [1] R. Ghanem, P. D. Spanos, Stochastic finite elements: A spectral approach. Springer-Verlag, Berlin, 1991. [2] K. K. Phoon, S. P. Huang S. T. Quek, Implementation of Karhunen-Loeve expansion for simulation using a wavelet-Galerkin scheme. Probabilistic Engineering Mechanics, 17, 293-303, 2002. [3] M. Shinozuka, G. Deodatis, Simulation of stochastic processes by spectral representation. Applied Mechanics Reviews (ASME), 44, 191-203, 1991. [4] G. Stefanou, M. Papadrakakis, Spectral representation versus Karhunen-Loeve expansion for the simulation of Gaussian stochastic fields: A comparative study. Submitted for publication, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Study of Mechanical Properties of Human Skin 1 1 1 1 J.T. Barbosa , R.M. Natal Jorge , M.P.L. Parente , A.A. Fernandes , 2 2 T. Mascarenhas , B. Patrício 1 IDMEC – Pólo FEUP Faculdade de Engenharia da Universidade do Porto Rua Dr. Roberto Frias, 4200-465 Porto {jtrigo,rnatal,mparente,aaf}@fe.up.pt 2 Faculdade de Medicina/Hospital de S. João Alameda Prof. Hernâni Monteiro, 4200-319 Porto [email protected],[email protected]

ABSTRACT Human skin is a complex tissue consisting of several distinct layers, each consisting of their own components and structure. These several layers can be grouped into four layers, namely: stratum corneum dermis, living epidermis, dermis and subcutaneous fat. The main goal of this work is the development of numerical-experimental procedure to evaluate the elasticity of human skin in vivo. Experimental tests are carried out applying a non invasive method, the cutometer aspiration technique, and numerical tests using the ABAQUS software are employed for comparison. Regarding the numerical tests, both three-dimensional and axisymmetric finite element formulations were employed, and the Mooney-Rivlin, Yeoh and Neo-Hookean constitutive material models were found suitable for the present analysis.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Thermophysical Properties of Different Samples of Tissue-Mimicking Materials for Ultrasound Hyperthermia Phantoms 1

2

1

Rodrigo L. Q. Basto , Cláudio S. P. Costa , Marco A.V. Krüger , Wagner C. A. 1 2 2 Pereira , Henrique M. Fonseca and Helcio R. B. Orlande 1

2

Biomedical Engineering Program, COPPE/UFRJ, P. O. Box 68510, 21941-972, Rio de Janeiro, RJ, Brazil [email protected]

Department of Mechanical Engineering - POLI/COPPE/UFRJ P. O. Box 68503, 21941-972, Rio de Janeiro, RJ,Brazil [email protected]

ABSTRACT In the biomedical ultrasonic domain, it is common to use test-objects made of materials that can mimic the main ultrasonic properties of biological tissues (acoustic impedance, wave propagation velocity and attenuation). These devices are called “phantoms” and they are used with several purposes, being the most common the study of image quality control of ultrasonographic equipment. In the last two decades, ultrasound therapy, as a mean to deliver energy to biological tissue in order to produce healing became common practice. This application raises an obvious concern about tissue safety and as a consequence, another kind of phantom is required. Such sort of phantoms must be able to mimic the tissue ultrasound propagation properties and also the heating produced by irradiation of tissues by ultrasound (absorption and thermal conductivity). The present work is aimed towards the development of phantoms for assessment of performance and safety of ultrasonic therapeutic equipments. It consists in measuring and adjusting the thermophysical properties of materials already employed in mimicking acoustic properties of biological tissues. The thermo physical properties of three composition samples measured with a Netzsch Nanoflash LFA 447/1 are presented. These samples are composed of water, glycerin and agar. PVC powder and graphite powder are added to generate the ultrasonic backscattering increasing the attenuation coefficient in order to mimic human soft tissue. The result closest to human tissue was achieved with a mixture containing 5% graphite and 80% PVC.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Different computational approaches in the modeling of wrinkling of biological membranes Cavicchi A.*, Gambarotta L. **, Massabò R. ***

Department of Structural and Geotechnical Engineering, University of Genova, Italy * [email protected] [email protected] *** [email protected]

**

ABSTRACT In [1,2], the authors examined the problem of the wrinkling of plane isotropic biological membranes following the approach of Pipkin [3] and treating the out of plane geometric nonlinearities as constitutive nonlinearities through a modification of the elastic potential. The problem has been solved within the framework of finite strain hyperelasticity for a material characterized by a Fung type constitutive law in biaxial tension. All assumptions of classical Tension Field theory emerge as a result of such formulation. The model formulated in [1,2] is able to identify the distinct regions of behavior that characterize the response of stretched membranes: taut (biaxial tension), wrinkled (uniaxial tension) and slack (inactive). However, the assumption of zero bending stiffness does not allow for detailed predictions of the deformation fields of real membranes where wrinkles with finite magnitude and wavelength develop. This aspect of the problem has been highlighted by Cerda [4] where the limits of the Tension Field Theory [5] in the description of the wrinkling phenomenon have been discussed referring to the problem of a stretched annular membrane. In this paper simulations of reconstructive surgical procedures are presented where wrinkling of the skin occurs during and after the suture of the wound edges leading to unaesthetic scars with dog-ear formations. The effects of the natural tension of the skin on the wrinkling extension and final stress field are highligthed. A critical evaluation of the Tension Field assumptions is then made by considering the problem of the annular membrane subject to inner and outer uniform tractions, already examined by Cerda [4]. The results of the Tension Field theory are compared with the results of a buckling analysis. In the problem studied and under certain loading conditions, the extension of the wrinkled region and the number of circumferential waves predicted by the buckling analysis prove to be independent of the elastic constants and the bending stiffness of the membrane. This allows for a quantitative comparison with the results of the Tension Field theory. The comparison shows analogies and differences between the onset of membrane instability and the simplified description of the post-buckling configuration given by the Tension Field theory.

References [1]

[2] [3] [4] [5]

Gambarotta, L., Massabò, R., Morbiducci, R., Raposio, G., and Santi, P., In vivo experimental testing and model identification of human scalp skin, Journal of Biomechanics, 38, 2237-2247, 2005. Gambarotta, L., Massabò, R., Wrinkling of plane isotropic biological membranes, Journal of Applied Mechanics, in press, 2006. Pipkin, A.C., Relaxed Energy Density for large deformations of membranes, IMA J. Appl. Math., 52, 197-308, 1994. Cerda, E., Mechanics of scars, Journal of Biomechanics, 38, 1598-1603, 2005. Mansfield, E.H., The bending and stretching of plates, Second ed.Cambridge University Press., Cambridge, 1989.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

674

Dynamic study of the middle ear F Gentil(1), RM Natal Jorge(2), AJM Ferreira(3), MPL Parente(4), M Moreira(5), E Almeida(6) (1)

(2) (4)

Escola Superior de Tecnologia da Saúde do Porto Clínica ORL - Dr Eurico Almeida, Widex [email protected]

IDMEC-Polo FEUP, Faculdade de Engenharia, Universidade do Porto (2) [email protected] (4) [email protected] (3) (5)

INEGI, Faculdade de Engenharia, Universidade do Porto (3) [email protected] (6)

Clínica ORL - Dr Eurico Almeida [email protected]

ABSTRACT The human ear is a complex biomechanical system and is divided by three parts: outer, middle and inner ear. When a sound is made outside the outer ear, the sound waves travel down the external auditory canal and strike the eardrum. It vibrates and the vibrations pass through three tiny bones in the middle ear called the ossicles (malleus, incus and stapes). The ossicles amplify the sound and send the sound waves to the inner ear and into the fluid filled hearing organ (cochlea), by the oval window. However, the ossicles can suffer from several damages, for example, the Otosclerosis, being a need the application of mechanical prosthesis, by chirurgic intervention, keeping the right travel of sound wave. In order to study the implementation of prosthesis, it is very important to achieve a correct modelling of the vibro-acoustic behaviour of middle ear. In this work, a finite element modelling of the middle ear will be done. For this proposes, a dynamic study will be presented by using the ABAQUS program. The model will include the different ligaments of the support structure. A hyperelastic behaviour of this ligaments will be taken into account [1] and for the ossicles and eardrum the mechanical properties available in the literature [2] will be considered. The connection between ossicles will be done by using contact formulation. The eigenvalues and the eigenvectors will be carried out.

References [1] P.A.L.S. Martins, R.M. Natal Jorge, A.J.M. Ferreira, A.A. Fernandes, M. Figueiredo, R. Silva, “Modelling the mechanical behavior of soft tissues using hyperelastic constitutive models”, Proc. of II Int. Conf. on Comp. Bioengineering, ICCB2005, H Rodrigues et al. (Eds.), pp.403-410, Instituto Superior Técnico, Lisboa, 2005. [2] PJ Prendergast, P Ferris, HJ Rice, AW Blayney, Vibro-acoustic modeling of the outer and middle ear using the finite element method, Audiol Neurootol, 4, 185-191, 1999.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Image-based Inverse Problems to Identify Three-dimensional Displacement Field Shouji Kuzukami*, Nobuhiro Yoshikawa†, Osamu Kuwazuru† *

Graduate Student, The University of Tokyo 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505 [email protected] †

Institute of Industrial Science, The University of Tokyo 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505 [email protected]

ABSTRACT An identification methodology of three-dimensional displacement within a biological soft tissue is presented in line with non-invasive testing by means of X-ray CT images. The high-compliance and heterogeneity of biological soft tissue induces its complex and nonlinear displacement field hardly measured by conventional methods, although the displacement field is necessary for determinations of its material properties. The full-field digital image correlation method[1] with the LevenbergMarquardt method [2] has been established to identify the displacement field by using a pair of twodimensional digital images with crisp contrast in its intensity distribution. The proposed approach is an extension of the method to three-dimensional problems. A pair of three-dimensional images, which are constituted from multi-slice CT images captured with small intervals, is obtained from the deformed and undeformed states of a soft tissue. To identify the displacement field, the undeformed state image is virtually deformed by a tentative displacement field described by the tri-cubic B-Spline basis functions with unknown parameters initially set to tentative values. The error of this identification is evaluated in terms of intensity difference between actually and virtually deformed images. The unknown parameters are successively modified to minimize the error. Primary obstacle against the three-dimensional image correlation is an explosion of computational cost for solving the huge simultaneous equations successively constructed during the error minimization procedure. We reduce the calculation time to a reasonable one by utilizing the numerical symmetry of the coefficient matrix and the locality of the basis functions. Thus, a fast parallel solver for the inverse problem is proposed to minimize the computational time. In this problem, moreover, an indeterminacy of the displacement field arises from noisy images with obscure contrast in its intensity distribution, as is the common case with the X-ray CT images. In this study, therefore, we propose a formulation stabilized by imposing incompressibility of the materials. By using an experimental specimen under compression load, the validity of the algorithm is checked.

References [1] P. Cheng, et. al. , Full-field Speckle Pattern Image Correlation with B-Spline Deformation Function. Experimental Mechanics, 42-3, 344-352, 2002. [2] W. H. Press, et. al. , Numerical Recipes, The Art of Scientific Computing. Cambridge University Press, U.S.A., 1986.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

676

Experimental Study of the Middle Ear Biological Support Structures Pedro S. Martins∗1 , Renato N. Jorge∗2 , Antonio M. Ferreira† , Fernanda Gentil‡ ∗ IDMEC-Polo

FEUP, Faculdade de engenharia da Universidade do Porto Rua Dr.Roberto Frias, 4200-465 Porto, Portugal 1 [email protected] 2 [email protected]

†

DEMEGI, Faculdade de engenharia da Universidade do Porto Rua Dr.Roberto Frias, 4200-465 Porto, Portugal [email protected] ‡ Escola Superior de Tecnologia da Saude do Porto Rua Joo de Oliveira Ramos, 87, 4000-294 Porto, Portugal [email protected]

ABSTRACT During the last few years, the effort for a better understanding of biological systems from a structural and physiological perspective, lead to the appearance of several scientific works concerning the biomechanical study and modeling of different human, and also animal structures. Biological systems are in general, the assembly of a given number of distinguishable components. In this context, the middle ear can be modeled as a mechanical system composed by several linked elements [1] (ossicles-Stirrup, Anvil and Hammer, and Eardrum). The simulation’s accuracy rely on the material parameters that are fed to the numerical approximation method (ex. FEM, BEM,...). The material parameters however, can only be obtained by a direct or indirect [2] measurement procedure, otherwise, the simulations may lead to results widely away from the physiological reality of the system in focus. In previous studies of the middle ear [3], the mechanical properties have been established and used in linear behavior context, assumption that is questionable due to widely common nonlinearity of biological materials [4]. The authors propose in this paper a experimental approach for the determination of the mechanical properties of middle ear’s support structures. This procedure assumes from a starting point the (possible) nonlinearity of these structures, which is an important step to increase the accuracy of the mechanical simulations, and ultimately may lead to a better understanding of middle ear’s physiology.

References [1] F. Gentil, R. Natal, A. Ferreira, M. Parente, M. Moreira and E. Almeida, Biomechanical Study of Middle Ear. Proc. COMPLAS VIII, 2, 785–788, Barcelona, 2005. [2] J. Fay, S. Puria, W. F. Decraemer and C. Steele, Three approaches for estimating the elastic modulus of the tympanic membrane. Journal of Biomechanics, 38, 1807–1815, 2005. [3] P. Ferris and P. J. Prendergast, Middle-ear dynamics before and after ossicular replacement. Journal of Biomechanics, 33, 581,2000 [4] Y. C. Fung, BIOMECHANICS-Mechanical Properties of Living Tissues, Second Edition. SpringerVerlag, 1993

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

677

A Real-time FEM Simulation for Cutting Operation using Haptic Device Tomoyuki Miyashita*, Hiroshi Yamauchi†, Masatomo Inui†, Hiroshi Yamakawa* *

Department of Mechanical Engineering, WASEDA University in JAPAN 3-4-1, Okubo, Shinjuku-ku, Tokyo 169-8555, JAPAN [email protected], [email protected] †

Department of Systems Engineering, IBARAKI University in JAPAN 4-11-1, Nakanarusawa-cho, Hitachi-shi, Ibaraki 316-8555, JAPAN [email protected], [email protected]

ABSTRACT A Computational simulation for cutting of materials is one of the difficult problems mainly because of time consumable calculation of finite elements method. Especially, the boundary and loading conditions may change during simulation and re-mesh procedures are often necessary to adopt above situations. However, considering haptic device to obtain dynamical information from FEM analysis, it is difficult to apply the formal FEM procedures described above from the point view of response time. There were previous studies to propose the approximation procedures to omit remesh procedures and obtain the approximated dynamical response using FEM analysis within allowable computer resources. Recently, the computer power is gradually improved and then we can propose new approximation procedure to make full use of computer resources to treat the cutting simulation. The cutting simulation is useful to discuss about surgical simulation from the point of view of dynamical properties of human organs or tissue structure. In this study, we will review the previous studies treating same problem using FEM analysis and propose the method to treat the cutting simulation considering real time computation and simulation results are directly transferred to the user through the haptic device. The hex and tetra elements were used to model the structure using FEM analysis and dynamical response was calculated using newmark E method. Here, the elements matrix was normalized according to the distance between a cutting device and nodes to omit the re-mesh procedures. The proposed method was implemented using three threads that handle graphics for display, dynamical calculation and model construction to improve the response for user operation. Then, we have developed the simulation system composed of the haptic device (PHANToM force feedback device) using ToolKit and the proposed method including graphics animation. Using the haptic device, we have been able to discuss about the obtained feeling through the device and obtain experimental results and compared from the points of view of the previous simple calculation method and the proposed method, the obtained reaction forces. We could confirm that the performances of the developed system using thread is very good and effective for the further improvement using detail calculation on FEM, the qualitative feelings and quantitatively obtained reaction forces are different among the compared method. Then, we could confirm the some properties and the effectiveness of the proposed method and the developed system.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

678

The biomechanical behavior of the pelvic floor muscles during a vaginal delivery M. Lages Parente1,2 , R. Natal Jorge1 ,T. Mascarenhas3 A.A.Fernandes1 , J.A.C. Martins4 1 Department

2

of Mechanical Engineering, Faculty of Engineering, University of Porto Rua Dr. Roberto Frias, 4200-465 Porto, Portugal {mparente,rnatal,aaf}@fe.up.pt

Faculty of Architecture and Arts, University Lusiada of Porto Rua Dr. Lopo de Carvalho 4369-006 Porto, Portugal 3 Faculty of Medicine, University of Porto Al. Prof. Hernni Monteiro 4200 - 319 Porto, Portugal [email protected] 4 Department

of Civil Engineering and ICIST Instituto Superior Tcnico Av. Rovisco Pais, 1049-001 Lisboa, Portugal [email protected]

ABSTRACT The women pelvic floor extends itself on the lower part of the pelvic cavity like a net to give support to the organs in the abdominal cavity. One of the most important muscles of the pelvic floor, that gives support to the pelvic organs, namely, to the urethra, vagina and the rectum, is the levator ani. The levator ani muscles actively support the pelvic contents, compressing the urethra and vagina by elevating the pelvic floor. Relaxation of these muscles allows evacuation of the bladder and rectum [1]. Using a finite element model for the pelvic floor, built using the dataset made public in [2], we present the preliminary results for the simulation of a birth trough vaginal delivery, in vertex presentation [3]. We present the results obtained for the stresses that appear on the the ligaments that connect the pelvic floor muscles to the coccyx, before the movement of extension [3]. The objective of this research is to help predict the damage to the pelvic floor that can occur during childbirth.

References [1] P.E. Papa Petros, The Female Pelvic Floor, Function, Dysfunction and Management Accordind to the Integral Theory, Springer Medizin Verlag, 2004. [2] S. Janda, F.C.T. Van der Helm, and S.B. de Blok, Measuring morphological parameters of the pelvic floor for finite elements modelling purposes. Journal of Biomechanics, 36, 749–757, 2003. [3] Alan H. DeCherney, Lauren Nathan, Current Obstretic & Gynecologic, Diagnosis & Treatment, Ninth Edition. Lange Medical Books/McGraw-Hill, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

679

Buckling Analysis of Unbranched Thin-Walled Members: Generalised Beam Theory and Constrained Finite Strip Method Sándor Ádány1, Nuno Silvestre2, Ben Schafer3 and Dinar Camotim2 1

2

Budapest University of Technology and Economics, Department of Structural Mechanics 1111 Budapest, MĦegyetem rkp. 3, Hungary [email protected]

Technical University of Lisbon, Department of Civil Engineering and Architecture, ICIST/IST Av. Rovisco Pais, 1049 Lisboa, Portugal {nunos, dcamotim}@civil.ist.utl.pt 3

Johns Hopkins University, Department of Civil Engineering Latrobe Hall 210, Baltimore, MD 21218, USA [email protected]

ABSTRACT The load-carrying capacity of thin-walled members is often governed by buckling phenomena. Usually, three main families of buckling phenomena/modes are considered: (i) global buckling, in which the member axis deforms (e.g., flexural or lateral-torsional buckling), (ii) local-plate buckling, involving only plate (wall) bending, and (iii) distortional buckling, combining wall bending with crosssection distortion  the last two phenomena are sometimes jointly described as “local buckling”. Although there exist several numerical and/or analytical methods to determine the buckling load/moment values and the associated buckling mode shapes, it is fair to state that only generalised beam theory (GBT) and the constrained finite strip method (cFSM) are able to perform this task for isolated (“pure”) or arbitrarily combined (“coupled”) modes. Although both methods lead to very similar solutions, (i) GBT is a generalisation of classical beam theories that includes additional degrees of freedom to allow for cross-section deformation, whilst (ii) cFSM is a specialisation of the classical plate theory that carefully selects constraints in order to force the member to deform (buckle) according to pre-defined configurations. This paper provides an in-depth comparison between the fundamentals of the two above approaches (GBT / cFSM), focusing on (i) their mechanical assumptions and domains of application, and (ii) the procedures adopted. This will contribute to a better understanding of both methods and the phenomena that they aim to uncover, thus paving the way to the development of more efficient tools for the analysis and design of thin-walled members. In order to illustrate the GBT / cFSM comparison, the local, distortional and global buckling behaviours of lipped channel columns (see Figure 1) and beams are analysed in detail. As one would expect, there is a virtually perfect coincidence between the two sets of buckling results. Figure 1: Variation of the buckling load Pb with the column length L

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

680

Lateral-Torsional Buckling Analysis of Web-Tapered I-Beams Using Finite Element and Spline Collocation Methods Anísio Andrade*, Paulo Providência e Costa*, Dinar Camotim† *

Civil Engineering Department, FCT, University of Coimbra Rua Luís Reis Santos – Pólo II da UC, 3030-788 Coimbra, Portugal {anisio,provid}@dec.uc.pt †

Department of Civil Engineering and Architecture, ICIST/IST, Technical University of Lisbon Av. Rovisco Pais, 1049-001 Lisboa, Portugal [email protected]

ABSTRACT Tapered members are widely used in the steel construction industry, because of their structural efficiency, ability to meet architectural and functional requirements and competitive fabrication costs. According to the current design codes, the load-carrying capacity of laterally unsupported beams (either prismatic or tapered) bent in their stiffer principal plane is estimated on the basis of their cross-sectional and elastic lateral-torsional buckling resistances. Bearing this in mind, two of the authors have recently proposed and validated a one-dimensional model to characterise the elastic lateral-torsional buckling behaviour of tapered thin-walled open beams [1]. This model can be described as a kinematically constrained version of a nonlinear shell model and, from a mathematical viewpoint, it amounts to a self-adjoint eigenvalue problem for a system of ordinary differential equations and separated boundary conditions. Numerous numerical methods are available for the solution of such an eigenvalue problem. Among the structural engineering community, the finite element method (FEM) is unquestionably the most popular one. The FEM is based on a variational (or weak) form of the problem, which involves lower-order derivatives than the classical (or strong) form and, therefore, poses less stringent continuity requirements. Moreover, it allows for a very simple and flexible geometrical description of irregular-shaped domains. However, these two features, which are responsible for the key role played by the FEM in solving boundary value problems for partial differential equations, are not essential in the case of ordinary differential equations: the construction of a high-order spline basis, for instance, is more or less straightforward and special geometrical flexibility is not needed. Therefore, it appears to be worth investigating other numerical approaches and, in this context, the collocation method seems particularly promising. The paper begins by specialising the one-dimensional mathematical model for doubly symmetric webtapered I-section cantilevers acted by tip point loads. Two distinct numerical approaches are then considered, namely (i) the development of a conforming displacement finite element model, tailor-made for this specific problem, and (ii) the use of a general purpose code (COLSYS) based on spline collocation at Gaussian points and designed to solve nonlinear multi-point boundary value problems for mixed-order systems of ordinary differential equations. The paper closes with the presentation of some numerical results and an appraisal of the relative merits of the two approaches.

References [1] A. Andrade and D. Camotim, Lateral-torsional buckling of singly symmetric tapered beams: theory and applications, Journal of Engineering Mechanics (ASCE), 131(6), 586-597, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

681

Stability of telescopic props for temporary structures João André, António M. Baptista Laboratório Nacional de Engenharia Civil (LNEC) Av. Brasil 101 - 1700-066 Lisboa - Portugal [email protected] Laboratório Nacional de Engenharia Civil (LNEC) Av. Brasil 101 - 1700-066 Lisboa - Portugal [email protected]

ABSTRACT Telescopic props represent one of the most common temporary structural elements used in the construction of buildings, to support the formwork. The design of these props is often associated to high safety factors, due to insufficient information about their real behaviour at the construction site, under the influence of load eccentricities and geometric imperfections. A research project is now being developed at the Portuguese National Laboratory of Civil Engineering (LNEC), involving experimental and numerical studies of the props behaviour and, in particular, of the effects of the geometric imperfections and corresponding tolerances on their stability. The numerical studies will take in account the geometric and material nonlinearities affecting the props resistance. The influence of the base plates stiffness will be analysed. This paper describes the results obtained in numerical studies carried out during an initial stage of this project, as well as their interpretation and subsequent conclusions.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

682

Formulation of a GBT-Based Finite Element to Analyse the Global Buckling Behaviour of Plane and Spatial Thin-Walled Frames Cilmar Basaglia, Dinar Camotim and Nuno Silvestre

Department of Civil Engineering and Architecture, ICIST/IST, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisboa, Portugal {cbasaglia, dcamotim, nunos}@civil.ist.utl.pt

ABSTRACT This paper deals with the use of Generalised Beam Theory (GBT) to analyse the global buckling behaviour of plane and spatial thin-walled frames. After a brief presentation of the main concepts and procedures involved in the performance of a GBT buckling analysis, one presents in detail the formulation and numerical implementation of a GBT-based beam finite element that includes only the four rigid-body deformation modes  namely, one describes the determination of the elementary and frame linear and geometric stiffness matrices (the latter incorporate the influence of the frame joints and boundary conditions). Particular attention is paid to issues concerning (i) the quantification of the warping transmission at the frame joints, (ii) effects stemming from the non-coincidence of the member centroidal and shear centre axes (cross-sections without double symmetry), and (iii) the definition of joint elements that relate the connected member GBT degrees of freedom to the joint generalised displacements. Next, one addresses kinematical models to simulate the warping transmission at frame joints connecting two or more non-aligned U and I-section members and exhibiting two different configurations (diagonal-stiffened and box-stiffened). Finally, in order to illustrate the application and capabilities of the proposed GBT-based finite element formulation, one presents and discusses numerical results concerning the global buckling behaviour of (i) an “L–shaped” frame (see Fig. 1), (ii) a pitched-roof plane frame (in-plane and spatial behaviours) and (iii) a three-bar simple spatial frame, acted by loadings that cause only member compression. Both diagonal-stiffened and box-stiffened joints are considered and, for validation purposes, most of the GBT-based results are compared with values yielded by beam finite element analyses carried out in the commercial code ANSYS. An excellent correlation, involving both the frame critical buckling loads and mode shapes, was found in all cases.

Figure 1: “L– shaped” plane frame global buckling: deformed configurations of the member mid-span cross-sections.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

683

GBT-Based Finite Element Formulation to Analyse the Buckling Behaviour of Thin-Walled Members Subjected to Non-Uniform Bending Rui Bebiano, Nuno Silvestre and Dinar Camotim Department of Civil Engineering and Architecture, ICIST/IST, Technical University of Lisbon Av. Rovisco Pais, 1049-001 Lisboa, Portugal e-mails: {rbebiano, nunos, dcamotim}@civil.ist.utl.pt

ABSTRACT In this paper, one investigates the local-plate, distortional and global buckling behaviour (critical bifurcation loads and buckling mode shapes) of thin-walled steel beams subjected to non-uniform bending moment diagrams, i.e., under the presence of longitudinal stress gradients. In order to achieve this goal, one begins by developing and numerically implementing a beam finite element formulation based on Generalised Beam Theory (GBT), which (i) can handle beams with arbitrary open cross-sections and (ii) incorporates all the effects stemming from the presence of longitudinally varying stress distributions. After presenting the main concepts, procedures and assumptions involved in the above formulation, one addresses the derivation of the equilibrium equation system that needs to be solved in the context of a GBT buckling analysis. Particular attention is devoted to the main steps involved in the determination of the elementary linear and geometric stiffness matrices, as they must incorporate the stiffness reduction stemming from the presence of the non-uniform bending moments (longitudinal stress gradients) and also of the pre-buckling shear stresses caused by them – the inclusion of this last effect constitutes an original contribution within the context of GBT buckling analyses. Then, in order to illustrate the application and capabilities of the proposed GBT-based finite element formulation, one presents and discusses numerical results concerning thin-walled steel Ibeams acted by various (uniform and non-uniform) bending moment diagrams. In particular, one analyses (i) cantilevers subjected to uniform major axis bending (Fig. 1(a)), tip point loads (Fig. 1(b)) and uniformly distributed loads (Fig. 1(c)), as well as (ii) simply supported lipped beams subjected to uniform major axis bending, mid-span point loads and uniformly distributed loads í by taking full advantage of the GBT modal features, one is able to acquire a much deeper understanding about the influence of the longitudinal stress gradients and shear stresses on the beam local and global buckling mode shapes. For validation purposes, some GBT-based critical loads/moments and buckling mode shapes are compared with values either (i) yielded by shell finite element analyses, performed in the code ANSYS, or (ii) reported in the literature. Finally, one assesses the computational efficiency of the buckling analyses carried out using the proposed GBT-based beam finite element, by comparing the number of degrees of freedom involved with those required to obtain equally accurate results with discretisations in shell finite elements (note that “uniform stress” GBT-based beam finite elements are no longer applicable).

(a)

(b)

(c)

Figure 1: Local-plate buckling mode shapes of I-section cantilevers subjected to (a) uniform major axis bending, (b) a tip point load and (c) a uniformly distributed load.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

684

A Non-Linear 3-D Beam Finite Element for the Study of Steel Frames with Tapered Members N. Boissonnade*, H. Degée† * University of Liège, M&S Department Chemin des Chevreuils, n°1, Bât B52/3 B-4000 Liège Belgium [email protected] †

University of Liège, M&S Department Chemin des Chevreuils, n°1, Bât B52/3 B-4000 Liège Belgium [email protected]

ABSTRACT In recently build structures, instability problems have become of prime importance, mainly because of the general tendency to increase the structural and members slenderness, coupled with the development of high strength steels. In that way, tapered I-members become more and more usual since they allow significant material savings and a consistent design. Indeed, because such members are built-up from several plates, designers get the possibility to spread out the material in the most efficient way over the cross-section and over the beam length, thus leading to a rather economical distribution of the material. Such members are particularly efficient in steel portal frames of medium and long span with no in-plane bracing system, where the increase in the fabrication cost is more than compensated by the decrease in weight when compared to rolled sections. Despite these advantages, the use of tapered members suffers from the lack of appropriate simple but accurate design formulae in most codes of practice. Then, design solutions resorting to tapered elements are often given up, because the only available approaches and/or recommendations consist in rough elastic design formulae, where taper effects are rather badly accounted for. Consequently, the derivation of successful sets of design formulae for tapered members would be helpful and useful. Checking both accuracy and safety of such rules over a wide range of parameters implies the availability of a specific numerical tool (i. e. through the finite element method). Once such a set of reference results would be made available, then the ability of several proposals to be efficient in daily use could be easily checked. Present paper intends at filling this gap. First, the background of a fully geometrically and materially non-linear tapered 3-D finite element is given. It consists in a 2-nodes beam finite element, with 7 degrees of freedom per node, written in total lagrangian corotational description. Then, as a validation study, results for all types of static analyses are presented, from the basic case of linear elastic behaviour to full non-linear analysis. They show that the use of the new beam element provides accurate results, in close agreement with those provided by shell elements, and that resorting to a so-called “segmentation technique” (stepped beams) can lead to significant mistakes. Finally, a numerical comparison between shell and beam modelling of a tapered steel frame is presented. The use of the beam tapered elements is found to be accurate and convenient: the number of degrees of freedom is limited when compared to a shell model, while accuracy remains reasonable for engineering purposes. The new tapered beam element then allows an easy meshing, a short computation time and an straightforward results interpretation.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Non-Linear Dynamical Response of Steel Portal Frames with Semi-Rigid Connections Rafael A. Castro1, José Guilherme S. da Silva2, Pedro C. G. da S. Vellasco3, Sebastião A. L. de Andrade4, Luciano R. O. de Lima5 and Luis F. da C. Neves6 1

MSc Student in Civil Engineering, Faculty of Engineering State University of Rio de Janeiro, UERJ, Brazil [email protected] 2 Mechanical Engineering Department State University of Rio de Janeiro, UERJ, Brazil [email protected]

3,4,5 Structural Engineering Department State University of Rio de Janeiro, UERJ, Brazil [email protected]; [email protected]; [email protected] 6

Civil Engineering Department University of Coimbra, Portugal [email protected]

ABSTRACT Traditionally, the steel portal frame design assumes that beam-to-column joints are rigid or pinned. Rigid joints, where no relative rotations occur between the connected members, transfer not only substantial bending moments, but also shear and axial forces. Alternatively, pinned joints are characterised by an almost free rotation movement between the connected elements preventing the bending moment transmission. Despite these facts, it is largely recognised that the great majority of joints does not exhibit such idealised behaviour. These joints, called semi-rigid, should be designed according to their actual structural behaviour. Considering all these facts one of the main objectives of this investigation is to propose a modelling strategy to represent the dynamical behaviour of semirigid joints under dynamic actions. The developed finite element model included geometric nonlinearities and considered the influence of non-linear and hysteretic joint stiffness. The updated Lagrangean formulation is used to model the geometric non-linearity. The mathematical model calibration was made based on comparisons to semi-rigid tests and other numerical models [1,2] and proving to be in accordance to them. However, it must be emphasized that cautions should be taken on the direct use of the results in structural design. The main reasons for this affirmative are related to the occurrence of very important distortions due to the consideration of the semi-rigid joints geometric non-linearity effects on the steel portal frames dynamical response.

References [1] P.P.T Chui and S.L. Chan, Transient response of moment-resistant Steel frames with flexible and hysteretic joints. Journal of Constructional Steel Research, Vol 39, pp 221-243, 1996. [2] J.G.S. da Silva, P.C.G. da S. Vellasco, S.A.L. de Andrade, L.R.O. de Lima and R. de K.D Lopes, A dynamical parametric analysis of semi-rigid portal frames, The Ninth Int. Conference on Civil and Structural Engineering Computing, CC 2003, Netherlands, Civil Comp Press, pp. 1-17, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Buckling Analysis of Stiffened Composite Panels Nian-Zhong Chen, C. Guedes Soares Unit of Marine Technology and Engineering, Technical University of Lisbon, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal {chennianzhong, guedess}@mar.ist.utl.pt

ABSTRACT In view of high specific strength and specific modulus, stiffened composite panels have been widely used in modern engineering fields. However, stiffened composite panels can develop buckling failure under service conditions as they are generally very thin. The buckling strength of stiffened composite panels is usually sensitive to the variation of boundary conditions, stacking sequences and lamina thickness. In order to permit stiffened composite panels to be designed efficiently with high reliability and safety against buckling, a parametric study to investigate the effects of boundary conditions, stacking sequences and lamina thickness on buckling strength of stiffened composite panels with various types of stiffeners is presented in the paper. The accurate buckling analysis of stiffened composite panels can be achieved by an incremental nonlinear finite element analysis. However, the complete incremental nonlinear solution of a stiffened composite panel up to buckling is in general expensive and thus a linearized buckling analysis for lowest buckling loads based on an updated Lagrangian formulation with the degenerated shell element and an explicit through-thickness integration scheme is presented in the paper. In this method, it is assumed that the pre-buckling deformations of the structure are small and buckling analysis of stiffened composite panels is considered as an eigenvalue problem. The numerical accuracy of the linearized buckling analysis has been proven to be effective by comparison with the experimental data of three graphite-epoxy stiffened panels with “ blade ” and “ I ” section stiffeners and two GRP hat section stiffened panels. After analysis, the results of the parametric study shows that boundary conditions, stacking sequences and lamina thickness generally have significant and distinct influence on buckling strength of stiffened composite panels. However, the buckling strength of stiffened composite panels will increase significantly with the increase of the lamina thickness while the effects of stacking sequences on buckling strength vary with various boundary conditions and the influence of boundary conditions on buckling strength varies with various stacking sequences.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

687

Interactive buckling of thin-walled rectangular hollow sections – a comparison between modified beam models and shell finite elements Hervé Degée*, Nicolas Boissonnade* *

University of Liège Chemin des chevreuils, 1, B-4000 Liège (Belgium) [email protected]

ABSTRACT In nowadays steel construction, thin-walled members have become very common through the use of both welded and cold-formed profiles. This implies that phenomena such as local buckling of the walls or distortional buckling of the section require a due attention. As soon as it comes to computational approaches, different methods are available to model thinwalled members and study their behavior. Among others, we can point out the finite strip method, the use of shell finite elements or the recently revived Generalized Beam Theory (GBT). All of these have shown their ability to provide accurate results, but exhibit of course typical advantages and drawbacks. In particular, almost none of them is suitable for an efficient analysis of a whole structure, such as for example a bridge or a tall building. Indeed only the use of shell finite elements would allow such an analysis, but with an unavoidable increase of the size of the problem leading to difficulties in the elaboration of the model, very long computation time and great amount of results to manage. In this context, a special beam finite element accounting for a possible deformation of the cross section is developed. In classical beam elements, the cross section is assumed to be fully rigid and the consecutive displacement field of the element is well known. In this proposal, it is assumed that the displacement of any point of the element is described by the classical beam displacement together with an additional local displacement field representing the local and distortional behavior. In this paper, we present an application of this special element to the modeling of thin-walled rectangular hollow profiles, in comparison with the use of shell FE models. In particular, the following topics are discussed: - Computation of critical bifurcation loads; - Geometrically non linear analysis of an elastic member susceptible to local and global buckling; - Geometrically and materially non linear analysis of a member susceptible to local and global buckling. In the case of non linear computations, some simplifications are proposed in order to avoid the need of an explicit description of the second order local membrane displacement field. The following conclusions can be drawn: - The results show a good agreement between the different models until the maximum load regarding both the evaluation of this maximum load but also the evaluation of the loss of axial stiffness. In particular, the modeling of short columns is very accurate; - The proposed simplifications lead to underestimate the coupling between local and global transverse effects in members with high global slenderness in both pre- and post-buckling range.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

688

Design and Analysis of Composite Panels R. Degenhardt, A. Kling, K. Rohwer DLR, Institute of Composite Structures and Adaptive Systems, Lilienthalplatz 7, 38108 Braunschweig, Germany [email protected]

ABSTRACT European aircraft industry demands for reduced development and operating costs, by 20% and 50% in the short and long term, respectively. The European Commission project POSICOSS, which lasted from January 2000 to September 2004 and the 4-year follow-up project COCOMAT, which started in January 2004, contribute to this aim [1]. Both projects are under the co-ordination of DLR, Institute of Composite Structures and Adaptive Systems. They reduce structural weight by exploiting considerable reserves in primary fibre composite fuselage structures through an accurate and reliable simulation of postbuckling and collapse. The POSICOSS team developed fast and reliable procedures for postbuckling analysis of fibre composite stiffened panels, created experimental data bases and derived design guidelines. The COCOMAT project builds up on the POSICOSS results and goes beyond by simulation of collapse. The project improves existing tools as well as design guidelines for stiffened panels taking skin stringer separation and material degradation into account and it creates a comprehensive experimental data base. The improved tools, developed within the POSICOSS and COCOMAT project, have to be validated by test results. Since appropriate test data bases were not available, both projects were constrained to create new experimental data bases for curved stringer stiffend CFRP panels. To that end suitable panels are designed, manufactured, inspected and tested under own project objectives. Each project differentiates between verification panels and industrial panels. The verification panels are designed as to specific limiting aspects of application of the software to be verified, e.g. small or large stiffness reduction in the postbuckling regime. These panels should have a significant postbuckling range up to collapse and have an early onset of degradation. The industrial panels were designed in regard to industrial applications, mainly by existing procedures used in day-to-day industrial design practice. For the analysis of the panels the partners utilized different finite element software tools. This paper focuses on the experience of DLR on the design and analysis of stringer stiffened CFRP panels gained in the frame of the POSICOSS and COCOMAT projects. Geometrical nonlinear computations up to collapse were performed applying the software ABAQUS/Standard. The material was assumed linear elastic. The onset of degradation of the structure and the skin-stringer connection was determined using different failure criteria. Results achieved so far will be presented and an outlook towards future activities will be given.

References [1] Degenhardt R., Zimmermann R., Rolfes R., Rohwer K., “Improved Material Exploitation of Composite Airframe Structures by Accurate Simulation of Postbuckling and Collapse – The projects POSICOSS and COCOMAT”, Proceedings of the 11th Australian International Aerospace Congress“, Melbourne, Australia, 13-17 March, 2005

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

689

On the Use of Shell Finite Element Analysis to Assess the Local Buckling and Post-Buckling Behaviour of Cold-Formed Steel Thin-Walled Members Pedro B. Dinis and Dinar Camotim Civil Engineering Department, ICIST/IST, Technical University of Lisbon Av. Rovisco Pais, 1049-001 Lisboa, Portugal {dinis,dcamotim}@civil.ist.utl.pt

ABSTRACT This paper deals with the use of shell finite element analyses to assess the (i) elastic bifurcation and (ii) elastic and elastic-plastic local-plate and distortional post-buckling behaviours of cold-formed steel thin-walled members (mostly columns, i.e., uniformly compressed members)  all the geometrically and physically non-linear analyses are performed using the code ABAQUS and adopting 4-node isoparametric shell elements to discretise the members. First, one addresses several relevant issues concerning (i) the member discretisation (shell element type and mesh refinement), (ii) the simulation of the member end support conditions (a key aspect in numerical structural analysis), (iii) the modelling of the applied loading and material behaviour, (iv) the incorporation of member initial geometrical imperfections and residual stresses, (v) the assessment of buckling mode interaction effects and (v) the methods employed to solve either the eigenvalue problem or the system of non-linear algebraic equilibrium equations. Then, in order to illustrate the concepts and issues mentioned above and, at the same time, illustrate the power and versatility of the shell finite element analyses, one presents and thoroughly discusses a fairly large number of numerical results concerning the buckling and post-buckling behaviour of lipped channel (mostly), Zed-section and Rack-section cold-formed steel members  some of the post-buckling analyses include interaction effects between local-plate and distortional buckling modes. These results consist of (i) buckling curves providing the variation of the critical stress with the member length (see Fig. 1(a)), (ii) elastic and elastic-plastic non-linear (post-buckling) equilibrium paths (see Figs. 1(b)-(c)), (iii) figures providing the evolution, along those equilibrium paths, of the elastic and elastic-plastic member deformed configurations, and (iv) figures showing the spread of plasticity along the members up to failure (see Fig. 1(d)) and conveying relevant information about the nature of their collapse mechanisms.

Figure 1: Lipped channel simply supported columns: (a) elastic buckling, (b) elastic distortional post-buckling and (c) elastic-plastic distortional post-buckling results, and (d) distortional post-buckling plastic strain evolution.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

690

A Finite Volume Method for Plate Buckling Analysis Nosrat A. Fallah Department of Civil Engineering, Faculty of Engineering, University of Guilan Rasht P. O. Box 3756, Iran [email protected]

ABSTRACT The problem of buckling of plates has been the subject of numerous investigations because of its relevance to structural, mechanical, aeronautical, nuclear, offshore and ocean engineering. The buckling of plates can be solved analytically, which is useful for plates having simply supported edges and simple configurations. The application of the analytical method for plates with complex configurations, which are of practical importance, may be quite tedious and difficult. In such cases the energy method is used to obtain the approximate buckling loads. Alternatively, the numerical methods such as finite difference method, finite element method and finite strip method can be used for solving the problem. However, researchers still present new methods from the viewpoint of computational modeling of the problem. The application of finite volume method to the analysis of structural problems has been growing in recent years. This is due to the simplicity of the method and its capability in accurate predictions of structural behavior of the problems investigated so far. For instance it has been observed that the displacement finite volume formulation based on the MindlinReissner plate theory behaves well in the bending analysis of thin to thick plates [1]. In the other hand, the displacement finite element formulation based on the same theory locks in the thin plate analysis due to the shear locking phenomena. It is well known that treating this form of locking in the displacement finite element formulation is due to a number of works that have been devoted to the problem. In this paper a formulation is developed for computing the buckling loads of isotropic plates. The implementation of finite volume technique for the instability analysis of plates is quite new and has not been reported so far. To obtain the buckling load of a plate, it is idealized by the mesh of elements. The elements are regarded as control volumes or cells. Equilibrium equations of the cells are expressed explicitly and an approximate variation of section rotations and transverse displacements are assumed and introduced to the equilibrium equations. These approximated equations with boundary conditions are such expressed to yield a system of linear equations. The eigenvalue equation is then derived with standard form that is solved to obtain the buckling load of the plate. The formulation is verified on two test problems. This testing demonstrates the capability of the method in terms of accurate predictions and wide range of applicability.

References [1] N. Fallah, Cell vertex and Cell centred finite volume methods for plate bending analysis, Computer methods in applied Mechanics and Engineering, 193, 3457-3470, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

691

Design of slender steel shear panels: a numerical study A. Formisano*, G. De Matteis†, M. Maruzzelli*, F.M. Mazzolani* *

University of Naples “Federico II” - Department of Structural Analysis and Design P.le Tecchio, 80 – 80125 Naples (Italy) [email protected]; [email protected]; [email protected] †

Università of Chieti/Pescara G. d’Annunzio - PRICOS V.le Pindaro, 42 – 65127 Pescara (Italy) [email protected]

ABSTRACT In the framework of passive control devices for the seismic protection of new and existing buildings, in the last years large attention has been focused on shear wall systems. They are based on the use of a series of steel plates which realise a stiffened central nucleus able to absorb the horizontal action effect. These devices, which are obtained by inserting a steel panel within an external reaction steel frame, have a low realization cost and high speedy of erection. They can be classified in two main categories: - panels acting on the stiffness and the strength of the main structure; - panels having a dissipative function. The Steel Plate Shear Walls (SPSW), belonging to the first typology, are characterised by slender steel plates. They have been largely used in the last years in North America and Japan as an effective device against seismic actions. The behaviour of such system is strongly conditioned by buckling phenomena occurring at the early stages of the loading process. Such phenomena may have a significant influence also on the ultimate strength of the system, despite the development of stable post-critical behaviour due to the well known tension field mechanism. The theoretical and numerical studies on the behaviour of these devices confirm the reliability of the structural system when specific geometrical ratios of the panel are respected, i.e. for panels having b/d ratio raging from 0.8 to 2.5 [1]. In this paper, the theoretical behaviour of steel panels in shear, based on existing simplified methodologies (strip model theory) [2] is analysed and then compared with the results obtained by an extensive numerical study carried out using sophisticate finite element models (implemented in the ABAQUS non linear code). The comparison between theoretical and numerical results has been developed with reference to different values of the late thickness and varying the b/d ratio. In addition, the influence of intermediate stiffeners is analysed. In the whole the obtained results provide useful information for the correct design of slender steel plates in shear to be used as stiffening devices in new and existing framed structures.

References [1] Canadian Standards Association (CSA), Limit states design of steel structures. CAN/CSA S1601, Canadian Standards Association, Willowdale, Ont., Canada, 2001. [2] S. Sabouri-Ghomi, C. Ventura, M.H.K. Kharrazi, Shear analysis and design of ductile steel plate walls. Proc. of the 4th International Conference STESSA 2003, Behaviour of Steel Structures in Seismic Areas, Naples, 9-12 June, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

692

Plastic Bifurcation of Thin-Walled Members: Thin Shell Elements vs. GBT-Based Beam Elements Rodrigo Gonçalves*, Philippe Le Grognec† and Dinar Camotim‡ *

†

Escola Superior de Tecnologia do Barreiro, Polytechnic Institute of Setúbal R. Stinville 14, 2830-114 Barreiro, Portugal [email protected]

Ecole des Mines de Douai, Dépt. Technologie des Polymères et Composites et Ingénierie Mécanique 941, rue Charles Bourseul - BP 10838, 59508 DOUAI Cedex, France [email protected] ‡

Civil Engineering Department, ICIST/IST, Technical University of Lisbon Av. Rovisco Pais, 1049-001 Lisbon, Portugal [email protected]

ABSTRACT In this paper, one compares plastic bifurcation results concerning thin-walled members made of non-linear elastic-plastic materials, which are obtained by means of two independent approaches, namely (i) a Total Lagrangian thin shell finite element formulation, developed by the second author, and (ii) a computationally efficient beam formulation based on Generalised Beam Theory (GBT), developed by the remaining two authors  it is worth mentioning that the latter neglects the effect of pre-buckling deflections. Initially, one addresses the fundamental concepts, procedures and underlying assumptions involved in the application of the above two formulations, focusing on the similarities and differences existing between them. Then, one presents and thoroughly discusses a set of numerical results, determined through analyses based on the two alternative approaches and concerning (i) aluminium lipped channel and (ii) stainless steel rectangular hollow section (RHS) thin-walled columns (i.e., uniformly compressed members). In the first case, a very good correlation was found between the results (bifurcation loads/stresses and buckling mode shapes) yielded by the two formulations (e.g., see Fig. 1). In the second case, a non negligible discrepancy was observed, as the bifurcation loads provided by the shell formulation consistently lay below the GBT values and the differences, due to the combined influence of relevant pre-buckling deflection and a high imperfectionsensitivity, reached 19%  however, the RHS buckling mode shapes exhibited again an excellent agreement.

Figure 1: Plastic distortional buckling mode shapes of a lipped channel column of length L=55cm (shell FEA and GBT).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

693

A Large Displacement and Finite Rotation Thin-Walled Beam Finite Element Formulation Rodrigo Gonçalves*, Manuel Ritto-Corrêa† and Dinar Camotim† *

Escola Superior de Tecnologia do Barreiro, Polytechnic Institute of Setúbal R. Stinville 14, 2830-114 Barreiro, Portugal [email protected]

†

Civil Engineering Department, ICIST/IST, Technical University of Lisbon Av. Rovisco Pais, 1049-001 Lisbon, Portugal {mcorrea,dcamotim}@civil.ist.utl.pt

ABSTRACT In this paper, one presents, implements and validates a total Lagrangian beam finite element formulation capable of handling large displacements and finite rotations, as well as cross-section in-plane (distortion and local bending) and out-of-plane (warping) deformations. This formulation can be viewed as a generalisation of the well-known Reissner-Simo beam theory that includes warping/transverse bending deformation modes and in which the member walls can undergo in-plane finite relative rotations. When compared with a shell finite element discretisation, the proposed beam formulation has the distinct advantage of drastically reducing the number of degrees-of-freedom required to achieve equally accurate results. Initially, the kinematical description of the member is addressed, devoting special attention to the assessment of the transverse plate bending effects associated with cross-section distortion. Next, the equilibrium equations and the corresponding symmetric tangent operator are obtained, by assuming an additive update of the rotational parameters. In order to illustrate the application, provide validation and show the capabilities of the proposed formulation, several numerical results are presented, discussed and compared with values yielded by large displacement shell finite element analyses (see Fig. 1)  one obtains a good correlation in all cases, which clearly demonstrates the vast potential of the proposed formulation.

(a)

(b)

Figure 1: Beam deformed configurations yielded by the (a) proposed beam formulation and (b) shell element model

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

694

A new method to assess the rotation capacity of structural hollow sections based in multibody theory *

R. Goñi*, E. Bayo† Assistant Professor. Department of Structural Analysis and Design. School of Architecture. University of Navarra. Spain [email protected] †

Chairman. Department of Structural Analysis and Design. School of Architecture. University of Navarra. Spain [email protected]

ABSTRACT When modern structural codes are used for plastic analysis, it is important to define the class of the section. One of the main issues for the class classification is the rotation capacity of the section, that is, the inelastic rotation that the section can sustain after the plastic hinge is formed. EC3 requires class1 to perform plastic analysis, unless the rotation capacity is known. Sometimes, the election of class 1 is too restrictive and plastic analysis could be performed with class 2 provided that the section has sufficient inelastic rotation [1]. For this reason, it is important to know the inelastic rotation of a section, and it could be provided in property tables of sections. There are two reliable ways to calculate the rotation capacity: experimental tests and simulation by finite element analysis. In general, both of them are expensive. This paper presents a new and reliable method to obtain the rotation capacity of square and rectangular hollow sections. The method expands the original work of Kecman [2], simulating the plastic hinge model by a complete multibody system. The method performs successive static equilibriums, as described [3], for each angle of rotation of the plastic hinge. The forces considered in the model are in agreement with the elasto-plastic theory of materials The work developed by Kecman only considers the post-critical behavior of plastic hinge, however the proposed method, provides the complete moment-rotation curve. This method allows performing a quick simulation to obtain the inelastic rotation instead of large FEM simulations or expensive execution of experiments.

References [1] B. Gil, J. M. Cabrero, R. Goñi, E. Bayo, An Assessment of the Rotation Capacity Required for Structural Hollow Sections for Plastic Analysis, Tubular Structures X. Edit. Swets & Zeiltlinger Publishers. pp. 277 – 284, 2003 [2] D. Kecman, Bending Collapse of Rectangular and Square Section Tubes, Int. J. Mech. Sci. Vol. 25, No. 9-10, pp. 623 – 636, 1983 [3] J. Garcia de Jalon and E. Bayo, Kinematic and Dynamic Simulation of Multiboy Systems – The Real Time Challenge, Springer-Verlag, New York, 1993

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

695

On the Stability Analysis of Thin Walled Shell Structures Containing Gas or Fluid Marc Haßler∗ , Karl Schweizerhof ∗ Institut

f¨ur Mechanik Englerstrasse 2, D-76131, Karlsruhe, Germany {Haßler,Schweizerhof}@ifm.uni-karlsruhe.de ABSTRACT Thin shell or membrane structures containing gas or fluid are widely standard, such as oil and water tanks, gas containers or even atmospheric balloons, pressurized girders or inflatable dams. For such thin walled structures the gas or fluid can be considered either as support or loading. It may have a major influence on the stability behavior under other external loading as for example in the Tensairity-concept [2], where internal air pressure in combination with some external strengthening is used to overcome buckling of thin walled girders. The goal of this contribution is to present some investigation of the influence of such a gas or fluid support on the stability, here the eigenvalues and eigenmodes of the stiffness matrix of shell or membrane-like structures undergoing large displacements. For this purpose an analytical meshfree or lumped parameter description for the fluid/gas (see also [1], [3], [4])is taken, which yields a special structure of the nonlinear equations representing the change of the gas or fluid volume or alternatively the change of the wetted part of the shell surface. Finally this procedure leads first to the so-called load-stiffness matrix [5], to which several rank-one updates depending on the volumes containing either gas or fluid or both are added. These rank updates are a key part in the stability analysis: They describe the different coupling of the fluid or gas volume modification with the structural displacements in addition to the deformation dependence of the standard pressure. The specific rank-one updates allow the derivation of a very efficient algorithm to compute the change of the eigenvalues and eigenmodes of the original stiffness matrix without gas or fluid loading or support.

References [1] Bonet J, Wood RD, Mahaney J and Heywood P. Finite element analysis of air supported membrane structures. Computer Methods in Applied Mechanics and Engineering. 190 (2000) 579–595. [2] Luchsinger RH, Pedretti A, Steingruber A, Pedretti M. The new structural concept Tensairity: Basic principles. Proceedings of the Second Conference of Structural Engineering, Mechanics and Computation; A.A. Balkema/Swets Zeitlinger, Lisse. 2004. [3] Rumpel T and Schweizerhof K. Volume-dependent pressure loading and its influence on the stability of structures. International Journal for Numerical Methods in Engineering. 56 (2003) 211– 238. [4] Rumpel T and Schweizerhof K. Hydrostatic Fluid Loading in Non-Linear Finite Element Analysis. International Journal for Numerical Methods in Engineering. 59 (2004) 849–870. [5] Schweizerhof K and Ramm E. Displacement Dependent Pressure Loads in Nonlinear Finite Element Analyses. Computers & Structures. 18 (1984) 1099–1114.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

Sensitivity Analysis on Ultimate Strength of Stiffened Aluminum Plates under Combined Inplane Compression and Lateral Pressure M.R.Khedmati*, M.R.Zareei† * Faculty of Marine Technology Amirkabir University of Technology, Tehran, Hafez Ave., Iran [email protected] †

Faculty of Marine Technology Amirkabir University of Technology, Tehran, Hafez Ave., Iran

ABSTRACT Aluminum structures for marine applications have normally been built by welding. It is well recognised that welding significantly affects the behaviour of aluminum alloys. In particular, heat affected zone (HAZ) is softened by welding, and this reduces the ultimate strength of welded aluminum structures. It is of vital importance for structural designers to better understand how fabrication by welding affects the aluminum panel ultimate strength characteristics. It is commonly accepted that the collapse characteristics of welded aluminum structures are similar to those of welded steel structures until and after the ultimate strength is reached, regardless of the differences between them in terms of material properties. However, it is also recognised that the ultimate strength design formulae available for steel panels cannot be directly applied to aluminum panels even though the corresponding material properties are properly accounted for. One of the major reasons for this is due the fact that the softening in HAZ reduces the ultimate strength behaviour of welded aluminum structures, whereas it can normally be neglected in welded steel structures. There are some research works on the ultimate strength behaviour of aluminum unstiffened/stiffened panels under longitudinal inplane compression. In spite of that, studies on the collapse behaviour of such panels under the combined action of lateral pressure and axial compressive loads are rarely published. Aluminum stiffened panels applied in the construction of high speed crafts are under big magnitudes of lateral hydrostatic and hydrodynamic loads. It is aimed in this paper to perform numerical collapse simulations on the aluminum stiffened panels under combined lateral pressure and axial inplane compression, applying a series of non-linear finite element analyses on such plate elements. Both material and geometric nonlinearities are taken into account. The values of lateral load, panel geometric properties and HAZ width are changed in a systematic manner. Buckling, post-buckling, ultimate strength and post-ultimate strength characteristics of the panels are investigated in details.

References [1]J.K. Paik, A. Duran, Ultimate strength of aluminum plates and stiffened panels for marine applications, Marine Technology, Vol.41, No.3, 2004.

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

697

Large Deflection Behavior of Functionally Graded Plates Under Pressure Loads, Using Finite Strip Method H. R. Ovesy, S. A. M. GhannadPour Aerospace Engineering Department, Amirkabir University of Technology Tehran, Iran [email protected]

ABSTRACT The application of the same FSM as that developed earlier by the authors in Ref. [1] is extended to the analysis of large deflection behavior of functionally graded plates subjected to the normal pressure loading. The fundamental equations for the plates of functionally graded material (FGM) are obtained by discretizing the plate into some functionally graded strips (FGS), which are developed by combining the Von-Karman theory for large transverse deflection and the concept of functionally graded material. The material properties of the functionally graded strips are assumed to vary continuously through the thickness of the plate, according to the simple power law distribution of the volume fractions of the constituents. The solution is obtained by the minimization of the total potential energy. The Newton-Raphson method is used to solve the resulting non-linear equilibrium equations. Numerical results for square functionally graded plates subjected to normal pressure loading are generated by varying the combination of the constituents. The effects of material properties on the stress field through the thickness and on the variation of the central deflection at a given value of normal pressure loading are determined and discussed. The results are also compared with those available in the literature, wherever possible. Some representative results are given in the table below. The good comparison of the results verifies the current large deflection FSM analysis for the case of functionally graded plates. Non-dimensional central deflection w/h at a non-dimensional normal pressure of Q=-400 for an aluminum-alumina FGM plate with A/h=20. w/h n=0 n=0.5 n=2 n=Infinity Present -1.95 -2.40 -3.05 -4.52 Ref [2] -1.98 -2.44 -3.16 -4.71 In the above table, the non-dimensional normal pressure Q is defined as Q = qA4/(Emh4), where q is a uniformly distributed pressure load, h is the thickness of the plate, A is the length of the plate and Em 70 GPa . The description of parameter n, which is used within simple power law distribution, is

given in reference [2].

References [1] Ovesy H.R., GhannadPour S.A.M. Geometric Nonlinear Analysis of Imperfect Composite Laminated Plates, Under End Shortening and Pressure loading, Using Finite Strip Method. Accepted for publication in special issue of the composite structures (iccs/13-14 November 2005-Australia-Monash University), 2006. [2] Woo J., Meguid S.A. Nonlinear analysis of functionally graded plates and shallow shells. International Journal of Solids & Structures, 38, 7409–7421, 2001.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

698

Buckling Analysis of Laminates with Multiple Through-The-Width Delaminations by Using Spring Simulated Model H. R. Ovesy, H. Hosseini-Toudeshky, M. Kharazi

Aerospace Engineering Department, Amirkabir University of Technology Tehran-Iran {ovesy,hosseini}@aut.ac.ir

ABSTRACT Fiber-reinforced composite materials have been increasingly used over the past few decades in a variety of applications in which a fairly high ratio of stiffness/strength to weight is required. However, these materials are prone to wide range of defects and damages that can cause significant reductions in stiffness and strength. In particular, when the laminated composites are subjected to compressive loads, delamination becomes a constraint in the design process. Various methods have been proposed for the analysis of a plate that contains through-the-width delaminations. In the current paper, a continuous method of analysis is developed for determining the buckling loads of delaminated plates. The system is modeled as a plate on an elastic foundation. The elastic adhesive layer between the buckled sublaminates is represented by some parallel springs. The plate on a discontinuous foundation is treated as a continuous foundation but with added transverse forces at a number of discrete points in the delamination regions to make the net transverse force at each of these points to vanish. The delaminated plates which contain one or two through-the-width delaminations are analyzed. In the cases where two delaminations are located at different depths across the thickness of the laminate, the governing differential equations of each sublaminates become coupled, resulting in a more challenging analysis. Some representative results are shown in the figure below.

Buckling load of a simply supported laminate with a single delamination (Pc: Buckling load of a laminate with a centrally located delamination)

As shown in the figure, the buckling loads obtained in the current study are in good agreement with those available ref. [1]. It is worth noting that the so-called spring simulated model, outlined above, is fairly simple, and it can be used in a variety of applications.

References [1] James Ting-Shun Wang and Shou-Hsiung Cheng, Local buckling of delaminated beams and plates using continuous analysis, Journal of Composite Materials, 29, 10, 1374-1402, 1995.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

Numerical Validation of the Eurocode 3 Design Rules for LateralTorsional Buckling of I-Beams Vila Real P. M. M. †, Lopes N. †, Simões da Silva L. *, Rebelo C.*, † Universidade de Aveiro Dep. Civil Engineering, 3810 Aveiro, Portugal [email protected]; [email protected] *

Universidade de Coimbra Dep. Civil Engineering, 3030 Coimbra, Portugal [email protected]; [email protected]

ABSTRACT This work presents a numerical study of the behaviour of steel I-beams subjected to lateral torsional buckling. The results obtained are compared with the beam stability design curves from Eurocode 3. The EN version of part 1-1 of this Eurocode introduces significant changes in the evaluation of the lateral torsional buckling [1] resistance of unrestrained beams, that improve the too conservative approach of part 1-1 of Eurocode 3 from 1992 in case of non uniform bending. The EN version of the Eurocode 3 [2] presents two methods for the evaluation of the design buckling resistance moment of laterally unrestrained I-beams subject to major axis bending. One of these methods is similar to the one already prescribed in Eurocode 3 of 1992, while the other presents some differences in the buckling curves used. Further, it explicitly allows taking into account the influence of the loading type, through the introduction of a correction factor. The present study aims at evaluating the behavior of these two methods and to analyse the influence of the loading type in the first method through the use of that correction factor. Aiming the safety evaluation of the design formulae, a statistical analysis of the results is performed on the basis of the EN 1990-Annex D [3] in a companion paper [4].

References [1] Boissonnade, N., Greiner, R. and Jaspart, J.P., Rules for member stability in EN 1993-1-1. Background documentation and design guidelines, ECCS Technical Committee 8 – Stability, 5th draft, 2005 [2] CEN, European Committee for Standardisation, Eurocode 3: Design of steel Structures – Part 1-1: General Rules and Rules for Buildings. EN 1993-1-1, Brussels, Belgium, 2005.

[3] CEN, European Committee for Standardisation, Basis of Structural Design, EN 1990, Brussels, Belgium, 2002. [4] Rebelo C., Simões da Silva L., Vila Real P. M. M., Lopes N., Statistical Evaluation of the Eurocode 3 Design Rules for Lateral-Torsional Buckling of IBeams, III European Conference on Computational Mechanics, Solids, Structures and Coupled Problems in Engineering, C.A. Mota Soares et.al. (eds.), Lisbon, Portugal, 5–8 June 2006.

699

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

700

Statistical Evaluation of the Eurocode 3 Design Rules for LateralTorsional Buckling of I-Beams Rebelo C.†, Simões da Silva L.†, Vila Real P. M. M.*, Lopes N.*, †Universidade de Coimbra Dep. Civil Engineering, 3030 Coimbra, Portugal [email protected]; [email protected] * Universidade de Aveiro Dep. Civil Engineering, 3810 Aveiro, Portugal [email protected]; [email protected]

ABSTRACT This work reports the statistical evaluation of resistance design models for lateral-torsional buckling of I-beams according to the EN1993(eurocode 3) [3] and EN1990(eurocode 0) [2]. Aiming at the preparation of the Portuguese National Annex of EC3 part 1-1 and the establishment of the corresponding NDP's (Nationally Determined Parameters) it becomes necessary to define the partial coefficients of safety for beams design formulae when lateral-torsional buckling is a potential failure mode. In this paper the methodology for the resistance evaluation of beam elements subjected to instability is briefly described and the results are compared with FEM numerical results for the same elements obtained in a companion paper [4]. Aiming the safety evaluation of the design formulae, a statistical analysis of the results is performed on the basis of the EN 1990-Annex D. A methodology is proposed for definition of the partial safety factors concerning the uncertainties in the resistance model. Results are presented for a wide set of beam geometries and loading cases. Main conclusions can be summarized as follows: (i) all the design methods give better results for rolled sections; the scatter and the mean value of the results grow with growing slenderness; only the Special Case method can be considered an exception, since the mean diminishes in the range of low slenderness (0,2 – 0,5) assuming values lower then the unity, therefore unsafe; (ii) the safety factor is lower for the General Case and higher when the Special Case is applied; particularly for the most common medium slenderness the safety factor for the Special case is substantially higher than for the other two methods; if the factor 1.1 is applied, it becomes equivalent to the f-modified General Case method..

References [1] Background Document for EC3, Doc. 5.01, Background document for the justification of safety factor gM0 = 1.0 for rolled beams in bending about the strong axis, 1989 [2] CEN, European Committee for Standardization, EN 1990:2002, Basis of Structural Design, April 2002, Brussels, 2002. [3] CEN, European Committee for Standardisation, EN 1993-1-1:2005, Eurocode 3: Design of steel Structures – Part 1-1: General Rules and Rules for Buildings, Brussels, Belgium, 2005. [4] Vila Real P. M. M., Lopes N., Simões da Silva L., Rebelo C., Numerical Validation of the Eurocode 3 Design Rules for Lateral-Torsional Buckling of I-Beams, III European Conference on Computational Mechanics, Solids, Structures and Coupled Problems in Engineering, C.A. Mota Soares et.al. (eds.), Lisbon, Portugal, 5–8 June 2006.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

701

On the Interpolation of Rotations and Rigid-Body Motions in Nonlinear Beam Finite Elements Manuel Ritto-Corrˆea, Dinar Camotim Civil Engineering Department, ICIST/IST, Technical University of Lisbon Av. Rovisco Pais, 1049-001 Lisboa, Portugal {mcorrea,dcamotim}@civil.ist.utl.pt ABSTRACT The formulation of 3D geometrically exact beam finite elements relies heavily on the interpolation scheme used for the unidimensional variables describing the rotations or the rigid-body motions of the beam cross-sections. Since these rotations and positions belong to the non-commutative Lie groups SO(3) and SE(3), they are not easily amenable to a direct discretisation. Several schemes to interpolate the rotations have been suggested in the last two decades, each of them with specific advantages and drawbacks. Although the subject is far from settled, it has reached a stage of maturity which makes it possible to clearly identify several properties that an interpolation scheme should preferably display: it ought to (i) preserve orthogonality, (ii) be independent from the iterative process adopted, (iii) be path-independent, (iv) be frame-invariant, (v) be applicable to an arbitrary number of nodes and (vi) lend itself to a computationally implementable linearisation. Until very recently, none of the interpolation schemes for the rotation field in structural finite elements fulfilled all these requirements. However, the interpolation rule proposed by Buss and Fillmore [1] and Merlini and Morandini [2] seems to gather all the aforementioned desirable properties. This paper discusses the implementation of a (geometrically exact) Reissner-Simo beam element adopting this interpolation scheme for the rotation field. In addition, a growing trend in the development of geometrically exact beam finite elements involves the so-called helicoidal interpolation, in which the displacement and rotation fields are assumed to be coupled – this designation stems from the fact that a linear helicoidal interpolation between two adjacent nodes is an helix. In view of some mathematical analogies between groups SO(3) and SE(3), it is believed that the interpolation scheme applicable to rotations can be extended to the case of rigidbody motions.

References [1] Samuel R. Buss and Jay P. Fillmore. Spherical averages and applications to spherical splines and interpolation. ACM Transactions on Graphics, 20(2):95–126, 2001. [2] T. Merlini and M. Morandini. The helicoidal modeling in computational finite elasticity. Part II: Multiplicative interpolation. International Journal of Solids and Structures, 41:5383–5409, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

702

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

703

Equivalent Geometric Imperfections for Steel Shell Structures Subject to Combined Loading Werner Schneider*, Marco Gettel† University of Leipzig, Institute for Structural Mechanics Marschnerstr. 31, 04109 Leipzig, Germany * [email protected] †[email protected]

ABSTRACT The inevitable deviations from the nominal data of the resistance parameters have to be included in a numerical calculation of the load-bearing capacity of shells, because these structures are very imperfection-sensitive. In addition to simpler methods, the new European code for the resistance verification of steel shell structures EN 1993-1-6:2005 allows a geometrically and materially nonlinear analysis with imperfections included. The assumed imperfections are fundamental for this most sophisticated numerical buckling strength verification, because they have to cover the influence of all accidental imperfections of the structure in a consistent manner. According to the Eurocode, the influence of all various deviations should be included by only geometric equivalent imperfections. In spite of the intensive research efforts in the last decades, many problems are still residual, which have to be solved in order to apply the mentioned most realistic basic principle of the Eurocode to shell buckling cases, which are not yet sufficiently investigated. Equivalent geometric imperfections are called consistent, if nonlinear numerical analyses including these imperfections result in the experimentally based buckling resistance. Consistent equivalent geometric imperfections have been developed during the last years for the basic buckling cases of the circular cylindrical steel shell ([1], [2]). The situation at shells subject to combined loading is more difficult, because not so much experimental data are available. Fundamental problems and previous proposals for assuming equivalent imperfections at combined loading are discussed in the contribution. In particular, it is mooted, if the equivalent geometric imperfections have to be chosen without regard to the loading case, because imperfections of real shells are caused by manufacturing and not by loading. Reasons are given for, why this is not the case at equivalent imperfections of a numerical simulation. The conception of quasi-collapse-affine imperfections [3], which has already been proved at the basic buckling cases, can also be applied to shells under combined loading. General information for the application is given on the basis of two relevant buckling cases under combined loading.

References [1] W. Schneider, Stimulating Equivalent Geometric Imperfections for the Numerical Buckling Strength Verification of Steel Shell Structures. Proc. 6th World Congress on Computational Mechanics - WCCM VI, Beijing, China, 2004. [2] W. Schneider; A. Brede, Consistent equivalent geometric imperfections for the numerical buckling strength verification of cylindrical shells under uniform external pressure. Thin-Walled Structures, 43/2, 175-188, 2005. [3] W. Schneider, I. Timmel, K. Höhn, The Conception of quasi-collapse-affine imperfections. A new approach to unfavourable imperfections of thin-walled shell structures. Thin-Walled Structures, 43/8, 1202-1224, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

704

Shear Buckling of Thin Plates with Constant In-Plane Stresses Igor Shufrin*, Moshe Eisenberger*,** *

Faculty of Civil and Environmental Engineering Technion – Israel Institute of Technology Technion City, 32000, Israel **

Department of Building and Construction City University of Hong Kong, Tat Chee Ave., Kowloon Hong Kong {shufrin, cvrmosh}@techunix.technion.ac.il

ABSTRACT This work presents highly accurate numerical calculations of the buckling loads for thin elastic rectangular plates with known constant uni-axial in-plane loading, and in-plane shear loading that is increased until the critical load is obtained and the plate losses its stability. The solutions are obtained using the multi term extended Kantorovich method. The solution is sought as the sum of multiplications of two one dimensional functions. In this method a solution is assumed in one direction of the plate, and this enables to transform the partial differential equations of the plate equilibrium into a system of ordinary differential equations. These equations are solved exactly by the exact element method [1], and an approximate buckling load is obtained. In the second step, the derived solution is now taken as the assumed solution in one direction, and the process is repeated to find an improved buckling load. This process converges with a small number of solution cycles. For shear buckling this process can only be used if two or more terms are taken in the expansion of the solution. As an example the shear buckling load of a simply supported square plate with different levels of constant compressive load, as shown in Figure 1, is given. In Figure 2 the variation of the normalized shear buckling load as a function of the compression level is shown. Many more new results will be given. 10.00

Lx/Ly=1.0 h=constant Nx=DNcr

9.00 8.00

Nxy O

7.00

SS SS

6.00 5.00

SS

4.00 3.00

SS Nxy

2.00

Nx=DNcr

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 D

References [1] M. Eisenberger, Buckling loads for variable cross-section members with variable axial forces, Int. Jour. Solids Structures, 27, 135-143, 1991.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

705

GBT Formulation to Analyse the Buckling Behaviour of FRP Composite Branched Thin-Walled Members Nuno Freitas Silva, Nuno Silvestre and Dinar Camotim

Department of Civil Engineering and Architecture, IST, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, Portugal [email protected], {nunos, dcamotim}@civil.ist.utl.pt

ABSTRACT In this paper, one develops a Generalised Beam Theory (GBT) formulation to analyse the local and global buckling behaviour of composite thin-walled members with branched cross-sections made of laminate plate FRP (fibre-reinforced plastic), which takes into account the shear deformability effects. After briefly reviewing the most noticeable differences between the GBT formulations applicable to members with branched and unbranched open thin-walled cross-sections, the paper presents in some detail the steps and procedures involved in performing a GBT cross-section analysis of an arbitrarily branched composite (laminate plate) thin-walled member, which include (i) the identification and characterisation of conventional and shear deformation modes (the latter are not relevant in isotropic members) and (ii) the determination of the corresponding modal mechanical properties. Then, one addresses the numerical implementation of the proposed GBT formulation, which is carried out by means of the finite element method (GBT-based beam element)  particular attention is devoted to derivation of the elementary generalised stiffness and geometric matrices, which incorporate all the material coupling effects. Finally, in order to illustrate the application and capabilities of the proposed formulation/implementation, one presents and discusses numerical results concerning the local and global buckling behaviour of shear deformable FRP composite I-section columns with different ply orientations and stacking sequences (see Fig. 1)  in particular, one takes advantage of the modal features of GBT to acquire a deeper insight on complex buckling mode interaction phenomena, such as the ones due to bending-torsion or local-shear coupling effects. For validation purposes, some of the above results are also compared with values recently reported in the literature and obtained by means of numerical implementations of analytical models developed independently by several researchers  with a single exception (addressed in detail in the paper), an excellent correlation was found between the buckling load values (the differences are always below 2%).

Figure 1: Composite I-section columns local buckling mode and variation of global buckling loads with fibre angle ș.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

706

On the Influence of Material Couplings on the Buckling Behaviour of FRP Thin-Walled Columns – a GBT-Based Approach N. Silvestre and N. Freitas Silva Department of Civil Engineering and Architecture, IST, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

[email protected] ; [email protected] ABSTRACT The paper began by describing the main concepts involved in a GBT formulation to analyse the buckling behaviour of thin-walled composite members. Then, a beam finite element is derived in order to solve the system of differential equations. For validation purposes, some GBT-based results are compared with experimental and theoretical values available in literature. Finally, in order to illustrate the application and capabilities of the above formulation, the results of a study concerning the local and global buckling behaviour of fully fixed I-section columns is presented and discussed. Among the several conclusions drawn from this study, the following ones deserve to be specially mentioned: (i) In the context of linear (first order) analyses of I-section beams characterized by material couplings, the GBT-based results agree very well with the experimental and theoretical estimates. (ii) In the context of stability analyses of an I-section column characterized by material couplings, it is found that local buckling modes might be critical. Unlike columns made of isotropic materials, columns characterized by material couplings exhibit buckling modes with very unusual deformed configurations. In fact, it is unveiled that the shear modes play a relevant role in the mechanics of coupling between the different conventional modes (all shear undeformable). (iii) It is found that the incorporation of both matrices H and F, accounting for the modal couplings, is indispensable to achieve reliable results. Thus, neglecting these matrices may lead to nonconservative buckling load values (up to 25%) and to very different buckling mode shapes, as it can be observed from figures 1(a) (buckling mode from the exact analysis) and 1(b) (buckling mode from the analysis without matrices H and F).

(a)

(b)

Figure 1: Buckling mode configuration: (a) exact and (b) approximate (analysis without matrices H and F)

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

707

Ultimate Strength of Plate Assemblies with Localized Imperfection Subjected to Compressive Loads Rui M. Luís, C. Guedes Soares Unit of Marine Technology and Engineering, Technical University of Lisbon, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa [email protected]; [email protected]

ABSTRACT In a ship usually there are many plates, they are load carrying elements and they must be designed to resist the loading. The critical condition is the strength of deck structures under in-plane compressive loads leading to elasto-plastic buckling collapse. Plates are not perfect elements and their imperfections must be taken into account during the design process. In [1] it was shown that the strength of rectangular plates was governed by the amplitude of the buckling mode. In [2] a design process was proposed to take into account the global imperfections. However, localized imperfections can appear in a ship in any number of ways, the most commonly found are caused by welding or local damage. So, it is important for the designer to take into account the ability of the plate to resist to these imperfections. Most of the studies of the effect of imperfections have concentrated on individual plate elements or in stiffened plates. In [3] the effect of localized imperfections on long plates was studied and in [4] the study was extended to smaller plates and to combined imperfections. In welded plates the generated or induced initial imperfections, tend to have similar patterns in adjacent plates. Therefore the study of individual elements may be representative of the behavior of panels made of various plates. Localized imperfections related with local damage are not periodic and one cannot assume that adjacent plates have similar pattern of localized imperfections. This study analyzes the behavior of panels made up of three individual plates with localized imperfections. The effect in the ultimate collapse load of the level of imperfection and of its spatial location was studied. It was found that local imperfections clearly changed the strength of the panel when combined with the global ones. The effects of changing the position of the localized imperfection depend much on the final shape of the imperfections (local plus global). The effect of the localized imperfections can not be ignored by the designer and must be taken into account.

References [1] M. Kmiecik, Behavior of axially loaded simply supported long rectangular plates having initial deformations. Ship Research Institute, Report No. 84, Trondheim, 1971. [2] C. Guedes Soares, Design Equation for Ship Plate Elements under Uniaxial Compression. Journal of Constructional Steel Research, 22: 99–114, 1992. [3] R. S. Dow and C. S. Smith, Effects of Localized Imperfections on Compressive Strength of Long Rectangular Plates. J. Construct. Steel Research 4: 51-76, 1984. [4] C. Guedes Soares, A. P. Teixeira, R. M. Luís, T. Quesnel, P. I. Nikolov, E. Steen, I. A. Khan, C. Toderan, V. D. Olaru, A. Bollero and M. Taczala, Effect of the shape of localized imperfections on the collapse strength of plates. C. Guedes Soares, Y. Garbatov and N. Fonseca eds. Maritime Transportation and Exploitation of Ocean and Coastal Resources, IMAM, Lisbon, Portugal, 429-437, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

708

Higher Order Analysis of a Thin-Walled Beam Vieira, R., Virtuoso, F., Pereira, EBR. DECivil - Civil Engineering and Architecture Department Technical Superior Institute Technical University of Lisbon Av. Rovisco Pais, 1049-001 Lisboa, Portugal {ricardov, fvirtuoso, eduardo}@civil.ist.utl.pt

ABSTRACT A thin walled beam model formulation for the analysis of two dimensional problems is presented in this paper. The underlying concept of the model is to separate the correspondent two-dimensional elasticity problem into two parts: i) an approximation of the displacement field over the cross section and ii) a set of governing differential equations defined along the beam axis. A set of basis functions that uncouples to the most possible form the governing equations of the problem is obtained, which permits to consider explicitly higher order modes of the cross section deformation, in particular, warping and transverse shear effects. This process of uncoupling has the advantage of permitting a better physical understanding of the beam structural behaviour. An implementation of a numerical model for the solution of the orthogonal governing equation is developed within the framework of the finite element method, interpolating the coordinates of the deformation modes basis functions by a set of Hermite functions. Some numerical examples are presented in order to verify the model capabilities in modeling the non classical effects associated with high order deformation modes within the scope of a two dimensional thin-walled beam analysis. Acknowledgements ˜ para a Ciencia ˆ This work has been partially supported by FEDER and FCT (Fundac¸ao e Tecnolo´ gia) through the funding of research unit ICIST (Instituto de Engenharia de Estruturas Territorio e ˜ Construc¸ao).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

709

Structural Optimization Using OPTIMIZER Program M.H. Abolbashari1, M. Majdi, and M.R. Mahpeykar 1

Manufacturing & Automotive Engineering Research Center, Ferdowsi University of Mashhad, PO Box 91775-1111, Mashhad, Iran [email protected]

ABSTRACT OPTIMIZER is a user-friendly design optimization study tool that helps users to optimize almost any optimization problems. There are several optimization algorithms in OPTIMIZER program such as Genetic Algorithm (GA), Constraint Steepest Descent (CSD) and Constraint Quasi-Newton (SQP). The OPTIMIZER can only solve problems that have an explicit mathematical expression both for cost function and constraints. To extend the OPTIMIZER capability for other applications, it is linked with an analysis software like ANSYS. In this paper, several structural optimization problems are solved using OPTIMIZER and the results are compared with other reported solutions. Furthermore, the effectiveness of the above-mentioned methods for the selected problems is presented.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

710

Optimization of dissipative characteristics of structures on the basis of problems on natural vibrations of viscoelastic solids Dmitry V. Babkin*, Eugeny P. Kligman†, Valery P. Matveyenko†, Natalya A. Yurlova† *†

Institute of Continuous Media Mechanics of Ural Branch of RAS Academician Korolev str., 1 Perm, 614013, Russia [email protected]

ABSTRACT The damping ability of a material plays an important role in the dynamical behavior of structures. It is responsible for decay of free vibrations, drastic decrease in amplitudes of the displacements and stresses arising in structures subjected to dynamical actions. To date, a lot of approaches have been developed which describe the mechanism of internal friction of materials, which causes the energy dissipation under vibrations [1]. The Boltzmann-Volterra theory [2] is the most general linear one, reflecting practically all peculiarities of the quasistatic and dynamic behavior of viscoelastic materials. The damping for structures can be positive or negative factor. A quantitative assessment of the dissipative properties of structures is generally based on the results of solving free vibrations. In this case the dissipation of a system leads to the decay of vibrations, and the rate of decay estimates quantitatively the dissipative properties of a system. The higher is the decay rate of vibrations, the greater are the dissipative properties. The problem of natural damped vibrations is formulated using a complex analog to the Boltzman-Volterra equations. The transition to the complex analog is made under some assumptions. To confirm the validity of the algorithm for optimization of the dissipative properties of a construction, the optimization search by solving the problem of forced steady-state vibrations is also performed. The application of the finite element method to the stated problem reduces it to the algebraic problem of eigenvalues for complex matrices. To have assurance that the lowest vibration modes defining the damping properties of the system will be determined and to decrease significantly the volume of calculations, the method is proposed which is based on expansion of the viscoelastic problem solution into finite series with respect to eigenforms of vibrations of the corresponding elastic structure. The optimization problem is solved within the framework of nonlinear programming. Mechanical and geometrical parameters of the system are used as optimization parameters. The performed numerical experiments showed that the optimal structure could be achieved even in the case when the real viscous properties of the material are given in rather rough manner.

References [1] A.A. Iliushin, and B.Y.Pobedrja, Fundamentals of mathematical theory of thermoviscoelasticity. Nauka, Moscow, 1970. [2] A. Nashif, D. Johnes and J. Henderson, Vibration damping., Mir, Moscow, 1988.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

711

Accuracy of Design Sensitivity Analysis for Optimization of Structures Using Small Strain Theory by Finite Element Method Kiran A. Bhagate1, Prashant Pawar2, Arbind Kumar Singh3, Jianqiao Ye1 1

2

School of Civil Engineering, University of Leeds, UK. [email protected]

Department of Aerospace Eng., IISc, Bangalore, India. 3 Department of Civil Eng, IIT Guwahati, India.

ABSTRACT Optimization of structures is always a fantasizing area for many researchers from past few decades. Many efforts have been taken in reducing the errors from the optimization process especially after advancement in computer faclities. The resulting structures are more efficient, economical and reliable. In traditional optimization techniques, the most important factor affecting on optimization process is the search direction which is the derivative of change in respose of structure due to change in design variables. This derivative is called as sensitivity derivative. Accuracy of sensitivity analysis is very much dependent on the method of structural analysis, technique of sensitivity calculation, computational efficiency etc. In this work, accuracy of sensitivity derivatives in elastic and plastic analyses are investigated on the basis of small strain theory. Combined with the Finite element method, which provides an excellent tool for the analysis of complex structures, the different techniques used for sensitivity calculations are finite difference method, semianalytical method and analytical method. Detail discussion of formulation and implementation of these methods are presented. Comparative study shows the relative error, cost of computation and efficiency of the above methods. From the results obtained, it can be stated that the finite difference method is the simplest technique that does not require access to the finite elemen analysis code and hence requires less efforts. However, this method is inefficient and less accurate. Analytical method is the most accurate method but its formulation and implementation is difficult as compared to other two metods. Semianalytical method is found to be a compromise of the two that results in more accurate solutions than from the finite dif erence method and is easy to implement as compared to the analytical method. The comparisons provide useful information for design engineers to decide a suitable method in the calculation of sensitivity erivatives for different structural optimization problems.

References [1] K. K. Choi and N. Kim, Structural Sensitivity Analysis and Optimization 1, Springer, 2005. [2] R. T. Haftka, Semi-Analytical Static Nonlinear Structural Sensitivity Analysis, AIAA Journal, 31, 1307-1312, 1993. [3] G. Cheng, Y. Gu and Y. Zhou, Accuracy of Semi-Analytical Sensitivity Analysis, Finite Elements in Analysis and Design, 6, 113-128, 1989.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

712

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

713

Optimisation of unsteady aerodynamic forces for aircraft aeroservoelastic studies R. M. Botez*, A.D. Dinu*, I. Cotoi* * École de technologie supérieure Département en génie de la production automatisée 1100 Notre Dame Ouest, Montréal, Québec, Canada, H3C 1K3 [email protected] * École de technologie supérieure Département en génie de la production automatisée 1100 Notre Dame Ouest, Montréal, Québec, Canada, H3C 1K3 [email protected] [email protected]

ABSTRACT One main aspect of the aeroservoelasticity is the optimization of the unsteady generalized aerodynamic forces Q(k, M) from the frequency domain into the Laplace domain Q(s), where k represents the reduced frequency, M is the Mach number and s is the Laplace variable. These forces were calculated at NASA DFRC by use of the finite element software STARS and by use of Doublet Lattice Method DLM and Constant Pressure Method [1]. The optimization yielding until now in the literature the smallest order time-domain state-space model is the MS method. In this paper, we present a new optimization method of the generalized aerodynamic forces for aeroservoelastic studies by use of Chebyshev polynomials and their orthogonality properties. A comparison of results obtained by use of this new method with the flutter results obtained experimentally at NASA DFRC (Dryden Flight Research Laboratory) is presented on the F/A-18 SRA (System Research Aircraft). The Chebyshev optimization method provides the smallest error by comparison with the other method’s given errors. Due to the fact that the Chebyshev polynomials had to be generated using the data provided on the F/A-18 SRA, which involve large differences between the values of the elements contained in the unsteady generalized aerodynamic forces matrices (1e+10), certain restraints regarding the threshold of the error had to be imposed, i.e. for smaller elements we have imposed an error value of 1e-4 and for larger elements an error value of 1e-2. Without these restraints, the Chebyshev polynomials cannot be generated. In case when the approximation order for Chebyshev method is increased, then the overall error will decrease even faster than by use of the LS method.

References [1] K.K. Gupta, STARS – An integrated, multidisciplinary, finite-element, structural, fluids, aeroelastic, and aeroservoelastic analysis computer program. NASA TM-4795, 1-285, 1997.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

714

A multidisciplinary design optimization framework applied to mechanical systems Rui N. Cadete*, João P. Dias*, Manuel S. Pereira* * IDMEC – Instituto de Mecânica – Pólo IST Instituto Superior Técnico, Technical University of Lisbon, 1049-001 Lisbon, Portugal [email protected]; [email protected]; [email protected]

ABSTRACT Most of the design problems in engineering are by nature multidisciplinary and the optimization of these systems are complex and difficult to accomplish due to the dependence level in the disciplines involved. Often the optimum solution can take significant amounts of time just for one single evaluation. This makes a challenge for designers to develop new methodologies and reduce design cycle time. To overcome the computational barrier at primary design stages simplified simulation tools and the integration of approximation techniques based on response surface models and optimization techniques can be applied. In this paper, a multidisciplinary design optimization framework to evaluate the analysis and design of mechanical systems involving the finite element and multibody dynamic disciplines, are presented. The goal of this methodology is to provide an efficient and improved design solution that ensures the safety and the dynamic characteristics of the mechanical system of interest. The optimized system must satisfy structural and mechanical constraints while achiving an optimum design for the overall behavior of the system. The two disciplines involved can be separated easily based on a single-level approach and the optimimun solution compared with the design found by optimizing each disciplines sequentially, since it can exploit the interactions between them. To demonstrate the use of the present approach some examples in the field of the multidisciplinary design of mechanisms and structures are presented.

References [1] [2]

[3]

Lodewijk Franciscus Pascal Etman,“ Optimization of Multibody systems using Approximation Concepts”, Ph. D. Thesis, Technische Universiteit Eindhoven, 1997 Srinivas Kodiyalam, Jaroslaw Sobieszczanski-Sobieski, “Multidisciplinary Design Optimization – Some Formal Methods, Framework, Requirements and Application to Vehicle Design”, International Journal of Vehicle Design (Special Issue) , pp. 3-22, (2001) J. Sobieszczanski-Sobieski, R. T. Haftka, “Multidisciplinary Aerospace Design Optimization: Survey of recent developments”, Structural Optimization, vol. 14, pp. 1-23, (1997)

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

715

Shape Optimization using the Boundary Element Method and a SAND Interior Point Algorithm for Constrained Optimization A. Canelas∗ , P. Mappa∗ , J. Herskovits∗ , J.C.F. Telles† ∗ Mechanical

Engineering Program - COPPE - Federal University of Rio de Janeiro CT, Cidade Universit´aria, Ilha do Fund˜ao, Rio de Janeiro, Brasil [email protected] [email protected] [email protected]

† Civil

Engineering Program - COPPE - Federal University of Rio de Janeiro CT, Cidade Universit´aria, Ilha do Fund˜ao, Rio de Janeiro, Brasil [email protected] ABSTRACT

Simultaneous Analysis and Design technique (SAND) for structural optimization considers the state variables as unknowns of the optimization problem and includes the equilibrium equations as equality constraints. In this way, equilibrium is only obtained at the end of the optimization process. Therefore, it is not necessary to solve the equilibrium equation per iteration of the optimization process. In the literature, some advantages in the application of the SAND technique have been recognized [1]. The shape optimization problem consists of looking for the geometry that minimizes an objective function, like mass or compliance, subject to mechanical constraints. In the present work a discrete model of the geometry that employs spline interpolation curves and defines the shape as a function of nodal coordinates is used. The Boundary Element Method (BEM) is a technique for structural analysis based on an integral form of the equilibrium equations [2]. For linear elasticity problems, the BEM needs only a mesh on the boundary of the structure and its corresponding state variables, displacements and stresses, therein defined. This characteristic makes the BEM a natural method for shape optimization, since only the boundary is needed to define the optimization problem and to carry out the structural analysis. In this paper the BEM formulation is used to define the shape optimization problem, the evaluation of the derivatives of the equilibrium equations is shown and the shape optimization problem is dealt with using an interior point algorithm based on the SAND technique. Numerical results for two-dimensional linear elasticity problems are presented to illustrate the efficiency of the proposed technique.

References [1] J. Herskovits, P. Mappa, E. Goulart, C.M. Mota Soares, Mathematical programming models and algorithms for engineering design optimization. Computer Methods in Applied Mechanics and Engineering, 194, 3244–3268, 2005. [2] C.A. Brebbia, J.C.F. Telles, L.C. Wrobel, Boundary Element Technique: Theory and Aplications in Engineering. Springer - Verlag, Berlin, 1984.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

716

A three-dimensional hierarchical model for topology optimization of structures P. G. Coelho*, P. R. Fernandes†, J. B. Cardoso*, J. M. Guedes† and H. C. Rodrigues†

*

Departamento de Engenharia Mecânica e Industrial, FCT/UNL 2829-516 Caparica Portugal [email protected], [email protected] † IDMEC - IST Av. Rovisco Pais, 1049-001 Lisboa, Portugal [email protected]; [email protected]; [email protected]

ABSTRACT The topology optimization of structures consists of the identification of solid and void regions within a design domain, for a given criterion and a prescribed amount of material. Using a material approach, the optimal topology is achieved by the distribution in space of a nonhomogeneous material of variable density. Regions with high density identify structure while regions with low value are interpreted as holes. This nonhomogeneous material is often assumed as a cellular material with at least two scales: A macro scale apparently homogeneous with a given relative density, and a micro scale defined by a cell with a suitable geometric parameterization. The apparent mechanical properties are often computed by homogenization and depend on the cell parameters, which are the design variables of the optimization problem. For some problems, the question of how broad is the micro-scale parameterization to obtain the optimal structure is not an issue. Actually, in many implementations the material model for topology design is seen mainly as a tool for interpolating between material and void, in search for the optimal structural topology. Here we address not only the structural optimal topology design but also the design of the material used. To achieve this we developed a hierarchical model for topology optimization of threedimensional structures where we underline the scope of the material distribution methods by emphasizing the design of the microstructure, i.e. we seek the general layout of composite structures, where the design of the microstructure is also controlled through computational models. This will allow us to work with a broad class of composites neither restricted to a specific type (cellular materials with square and rectangular holes, laminates), nor only optimal in a special setting (like single load compliance optimization). Moreover, the microstructures employed can, through the optimization model, be optimally selected based on design and manufacturing criteria. Finally note that the use of similar modeling methodologies can give a considerable insight into the fine structure of natural materials, for example in bone or wood. The hierarchical three-dimensional model leads to massive computation. The two-scale model implies the iterative solution of one problem at a global (or macro) scale and many problems at a local (or micro) scale in order to characterize the microstructure. Once local problems are independent it is developed in this work a parallel computation environment to solve them.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

717

A New Hybrid Meta-Heuristic Method for Optimal Design of Space Trusses with Elastic-Plastic Collapse Constraints A. Csebfalvi*, and G. Csebfalvi† *

University of Pecs H-7625 Boszorkany u. 2, Pecs, Hungary [email protected] † University of Pecs H-7634 Rakoczi u.80, Pecs, Hungary [email protected]

ABSTRACT Within the framework of the finite element method, we present in this paper an efficient new hybrid meta-heuristic – named in other context ANGEL – for solving discrete size optimization of truss structures. ANGEL combines ant colony optimization (ACO), genetic algorithm (GA) and local search (LS) strategy. The procedures of ANGEL attempt to solve an optimization problem by repeating the following steps. First time, ACO searches the solution space and generates structure designs to provide the initial population for GA. After that, GA is executed and the pheromone set in ACO is updated when GA obtains a better solution. When GA terminates, ACO searches again by using the new pheromone set. ACO and GA search alternately and cooperatively in the solution space. This study also proposes an efficient local search procedure, which is applied to yield a better solution when ACO or GA obtains a solution. In this paper we applied ANGEL for discrete minimal weight design of space trusses with elastic-plastic collapse constraints. The geometrically and materially nonlinear space trusses are formulated as a large displacement structural model. The method of elastic-plastic collapse analysis is based on a path-following method [6]. The applied method is a combination of the perturbation technique of the stability theory and the non-linear modification of the classical linear homotopy method. With the help of the higher-order predictorcorrector terms, the method is able to follow the load- deflection path even in case of elastic-plastic material law.

References [1] A. Csébfalvi, A non-linear path-following method for computing the equilibrium curve of structures, Annals of Operation Research, 81: 15-23, 199 [2] A. Csébfalvi, A simulated annealing algorithm for discrete minimal weight design of shallow space trusses with stability constraints, WCCM-V Fifth World Congress on Computational Mechanics, July 7-12, 2002, Vienna, Austria, eds. H. A. Mang, F. G. Rammerstorfer, J. Eberhardsteiner, On-line publication (ISBN 3 9501554-0-6) Paper-ID: 81234, Session: RS 207.4, 2002 [3] A. Csébfalvi, Probabilistic Diversification and Intensification in Local Search for Optimal Design of Shallow Space Trusses, Proc., European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2004) P. Neittaanmäki, T. Rossi, K. Majava, and O. Pironneau (eds.) I. Lasiecka (assoc. ed.), Vol.I, p.355 +CD, 2004

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

718

The concept of homogeneous thermodynamical potentials for non linear structural rigidity optimization B. Desmorat LM2S - Universit´e Paris 6 4 place Jussieu, Case 161, 75252 Paris cedex 5, France [email protected] ABSTRACT For structural rigidity optimization under the assumption of small strains and small displacements, a numerically efficient approach is to consider the minimization of the compliance with distributed parameters in the form of a double minimization problem over the design parameters and the statically admissible stress field [1]. Initially introduced in the framework of linear elasticity, it was used with the homogenization method to relax the illness posed problem of topology optimization (repartition of void and material in a fixed domain). Using the concept of homogeneous thermodynamical potentials, we present the general form of the simultaneous extension of this optimization algorithm to a class of nonlinear elastic materials and a class of nonlinear structural analysis. The main idea is to formulate the local equations such that there exists a variational equality for the problem. Class of nonlinear elastic materials: the stress and strain are related by a behavior law deriving from two dual (by the Legendre transform) homogeneous thermodynamical potentials (e.g. piecewise linear elasticity, power law nonlinear elasticity and dissymmetric in tension-compression power law). In this special case, those two potentials are proportional by a factor related to the degree of homogeneity [2]. Class of nonlinear structural analysis: the nonlinear phenomenon is modelled by a behavior law deriving from an homogenous potential. This of course limits the number of nonlinear structural phenomenons that can be taken into account. For example, a frictionless unilateral contact can be modelled as an interface with a behavior law relating the normal component of the stress vector to the normal displacement. The degree of homogeneity is chosen (imposed) identical for every potential considered. The compliance and the complementary energy are proportional by a factor related to the degree of homogeneity. The optimization algorithm is then extended to nonlinear material behaviors and to a structural nonlinearity if we assume that this homogeneity degree is independent of the design parameters. Different numerical examples are given: optimal repartition of composite fibers with different behavior in tensioncompression, topology optimization of a structure with frictionless unilateral contact.

References [1] G. Allaire, R.V. Kh¨on, Optimal design for minimum weight and compliance in plane stress using extremal microstructures. European Journal of Mechanics A/Solids, 12, 839–878, 1993. [2] B. Desmorat, G. Duvaut, Compliance optimization with nonlinear elastic materials. Application to constitutive laws dissymmetric in tension-compression. European Journal of Mechanics A/Solids, 22, 179–192, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

719

6DWHOOLWH$WWLWXGH&RQWURO6\VWHP3DUDPHWHUV2SWLPL]DWLRQ

Luiz Carlos Gadelha DeSouza

National Institute for Space Research – INPE. Av. dos Astronautas, 1758 , 12201-940 - São José dos Campos –SP - Brasil [email protected]

ABSTRACT The design of the attitude control system (ACS) of satellites with flexible structures like panels and antennas as well as of the flexible robotic manipulators become difficult task as long as the dimensions of such structures increase and the degree of pointing is very stringent. This because the ACS needs to carry out many different space maneuvers and at the same time it has to damp out the associated residual vibration remaining to a level such the mission requirements can be performed. Examples of projects that involve great flexible space structures are: the Hubble Space Telescope, the International Space Station (ISS) and the ROKVISS (Robotic Components Verification at the ISS), the former in development at German Space Center (DLR) in cooperation with the Division of Space Mechanics and Control - DMC of the National Instituted for Space Research - INPE. In this paper one presents the results of a rigid-flexible satellite attitude control system design where parameters like the moment of inertia of the reaction wheel and the length of the panel are optimized in order to improve the satellite ACS performance. The results of this investigation have given important information, which can be used in the beginning of the ACS design, like the size of the reaction wheel to be used, the level of panels vibration tolerated to carry out attitudes maneuvers keeping the stability and the static form of the flexible structure. The optimization of the these values and measures are extremely important in order to assure the control system good performance in different space mission phase when it is necessary the micro-gravity environment and/or a great degree of pointing accuracy

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

720

A Limited Memory Quasi-Newton Preconditioner for Large Scale Optimization Veranise Dubeux∗ , Jos´e Herskovits∗ , and Sandro Rodrigues Mazorche† ∗ COPPE

- Federal University of Rio de Janeiro - Mechanical Engineering Program Caixa Postal 68503, 21945 970, Rio de Janeiro - Brazil. [email protected] , [email protected]

† Departamento

de Matem´atica, ICE - Universidade Federal de Juiz de Fora. Cidade Universitria, Cep 36036-330 - Juiz de Fora - MG - Brazil. [email protected]

ABSTRACT We study the appication of the conjugated gradient method preconditioned by a limited memory quasiNewton matrix in the resolution of the FAIPA’s internal linear systems. FAIPA, the Feasible Arc Interior Point Algorithm,[[1],[2],[3]], is an interior-point algorithm that solves nonlinear optimization problems. It makes iterations in the primal and dual variables of the optimization problem to solve the Karush-Kuhn-Tucker optimality conditions. Given an initial interior point, it defines a sequence of interior points with the objetive function monotonically reduced. FAIPA requires the solution of three linear systems whit the same coefficient matrix at each iteration. These systems, when solved in terms of the Lagrange’s multipliers, are in general full, symmetric and positive definite. The conjugated gradient is largely employed for the iterative solution of symmetric and positive definite linear systems. When the system is badly conditioned, a preconditioner matrix con be employed. The preconditioner’s choice is fundamental for the technique’s efficiency. We present a preconditioner for the FAIPA’s linear systems based on the limited memoy BFGS technique and consider problems with full and sparse matrices. Numerical examples show the performance of our methodology.

References [1] Herskovits, J., A View on Nonlinear Optimization,in Advances in Structural Optimization, Edited by J. Herskovits, Kluwer Academic Publishers, Dordrecht, Holland, pp. 71-117, 1995. [2] Herskovits, J., A Feasible Directions Interior Point Technique For Nonlinear Optimization, JOTA, Journal of Optimization Theory and Applications, , Vol.99, N.1, pp.121 - 146, Plenum, London, 1998. [3] Herskovits, J. and Santos, G., Feasible Arc Interior Point Algorithm for Nonlinear Optimization, Computational Mechanics, New trends and Applications, Ed. by S. Idelsohn, E. O˜nate and E. Dvorkin, CIMNE, Barcelona, 1998.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

721

Design Optimization of Conveyor Systems M.K. Ebbesen∗ , M.R. Hansen† , N.L. Pedersen†† ∗ Aalborg

University, Dept. of Mechanical Engineering Pontoppidanstraede 105, 9220 Aalborg East, Denmark [email protected]

† Aalborg

University, Dept. of Mechanical Engineering Pontoppidanstraede 105, 9220 Aalborg East, Denmark [email protected]

†† Technical

University of Denmark, Dept. of Mechanical Engineering 404, Nils Koppels All´e, 2800 Kgs. Lyngby, Denmark [email protected]

ABSTRACT In this paper an approach to the design of conveyor systems and particular the design of tracks suited for luggage handling systems in airports is presented. The considered system consists of a closed track with a closed loop of carts carrying the luggage for different flight destinations. The track is a 3 dimensional structure assembled of both straight and arc segments as well as level changes of different kinds. The loop of carts is considered as a multibody system including joint flexibility and the stationary track as a guiding frame. While the track forms a continuous line the closed loop of carts approximates the same geometry in a piecewise linear discretized form. Due to geometric effects caused by the use of straight and arc segments the required length of the discretized line varies in order to remain closed when the system is in service. Since the length variation is one of the principal factors in the resulting load spectrum for the carts in the system the aim of the presented work is to reduce this length variation and thereby reduce the fatigue load on the carts. The length variation is split into two: a global variation of the length of the entire track and a local variation of the length of smaller parts composed of two curves and one straight segment or two straight segments and one curve. An optimization problem is formulated with a view to minimize the fatigue load on the carts using the radii of the arcs and the length of the straight segments as design variables. The design problem is subjected to kinematic constraints ensuring that both the carts and the tracks remain as closed loops. The approach is applied to an existing design, and both the original design and the optimized design are implemented in an experimentally verified time domain simulation model of the entire system and the load spectra for the designs are compared.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

722

Sparse quasi-Newton Matrices for Large Size Optimization with FAIPA, the Feasible Arc Interior Point Algorithm J. Herskovits, E. Goulart and M. Aroztegui OptimizE, Engineering Optimization Lab Mechanical Engineering Program COPPE, Federal University of Rio de Janeiro Caixa Postal 68503, 21945970 Rio de Janeiro, Brazil [email protected], [email protected], [email protected] ABSTRACT Real life engineering systems involve a very large number of design variables and constraints. Evaluation of functions and of derivatives coming from engineering models is very expensive in terms of computer time. In practical applications, calculation and storage of second derivatives are impossible to be carried out. Then, numerical techniques for engineering optimization must be capable to solve very large problems with a reasonable number of function evaluations and without needing second derivatives. Robustness is also a crucial point for industrial applications. Quasi-Newton techniques for nonlinear optimization construct a full matrix that is an approximation of the second derivative of the function, in the unconstrained case, or of the second derivative of the Lagrangian, when constraints are considered. Usually, numerical algorithms require positive definite quasi-Newton matrices. Classical techniques work with full quasi-Newton matrices requiring a very large storage area and a great number of computations. We present a new updating technique to obtain positive definite sparse quasi-Newton matrices. This technique can be included in the Feasible Arc Interior Point Algorithm (FAIPA),[1, 2, 3], in the Sequential Quadratic Programming Method (SQP) and in Primal-Dual optimization Algorithms. Several very large test constrained optimization problems, employing the present technique within FAIPA, were solved very efficiently.

References [1] Herskovits, J., A View on Nonlinear Optimization, in Advances in Structural Optimization, Edited by J. Herskovits, Kluwer Academic Publishers, Dordrecht, Holland, pp. 71-117, 1995. [2] Herskovits, J., A Feasible Directions Interior Point Technique For Nonlinear Optimization, JOTA, Journal of Optimization Theory and Applications, , Vol.99, 1, pp.121 - 146, Plenum, London, 1998. [3] Herskovits, J. and Santos, G., Feasible Arc Interior Point Algorithm for Nonlinear Optimization, Computational Mechanics, New trends and Applications, Ed. by S. Idelsohn, E. O˜nate and E. Dvorkin, CIMNE, Barcelona, 1998.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

723

Sensitivity and Sizing of Nonlinear Structures Made of Anisotropic Rubber-like Material Sami Holopainen∗ ∗ Institute

of Applied Mechanics and Optimization P.O.Box 589, FIN-33101 Tampere, Finland sami.holopainen@tut.fi ABSTRACT

This research deals with the nonlinear anisotropic material with large strains affected generally from large rotations. Consequently many difficulties arises, as the caring about objectivity or spatial covariance. For that, the theory and the modeling aspects have been researched and published only during couple of the latest decades [1], [2]. Additionally, e.g. the anisotropic composites motivate to develop still more efficient numerical modeling and optimization methods. In that context, some information about the design perturbation should be known, when the design sensitivity analysis (DSA) plays an important role. Here, design denotes some parameter as cross-section area, a material property or an orientation of anisotropy. For novelty, there are scarcely details published concerning the sensitivities of the nonlinear anisotropic constitutive equations. The used material model exhibits anisotropic behavior also in the deformed current configuration and thus the model is valid for fully nonlinear range. No sliding is assumed between the one dimensional fibers and the surrounding matrix material, which leads to the simplified but computationally efficient description of material in averaged sense. Without loss generality, it will be assumed that the free energy of the material response can be decomposed into an isotropic and an anisotropic component [1]. Here, the total Lagrangian (TL) formulation is adopted, i.e. all quantities are referred to the initial configuration and the objectivity requirement is fulfilled. The motion of the body, the reference domain and its configurations are described in the fixed Cartesian coordinate system. The body experiences large deformations with hyperelastic material constraint. The reference domain is design independent and developing the design variations in that configuration often leads to the simpler expressions. Sensitivities are derived in arbitrary material point using the continuum approach without uncertainty associated with the finite-dimensional approximation error [3]. The semi-discretized scheme will be employed, where both the equilibrium and the sensitivity equations are computed straightforward by means of applicable external FE-implementation using precisely the same increments and the algorithmic tangent modulus. Finally, the sizing optimization example of simple truss illustrate the paper. For the optimization a sequential convex approximation method or the method of moving asymptotes (MMA) is used. In context, the methods of that type would seem to be efficient also in large-scale sizing or topology optimization of materially nonlinear structures including much members as design variables.

References [1] Weiss J.A, Maker B.N, Govindjee S. Finite element implementation of incompressible, transversely isotropic hyperelasticity. Comput. Methods in Appl. Mech. Engrg 135, 107–128, 1996. [2] Marsden J.E, Hughes T.J.R. Mathematical foundations of elasticity. Dover: New York, 1993. [3] Choi K.K, Duan W. Design sensitivity analysis and shape optimization of structural components with hyperelastic material. Comput. Methods in Appl. Mech. Engrg, 187, 219–243, 2000.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

724

Optimization of Laminated Composite Structures Using Delamination Criteria and Adaptive Models Leon S. Johansen, Erik Lund Department of Mechanical Engineering, Aalborg University Pontoppidanstraede 101, Dk-9220 Aalborg East, Denmark [email protected], [email protected]

ABSTRACT Laminated composite shell structures are becoming increasingly popular because of good stiffness/strength to weight properties. The matrix material bonding the fibres together is weak compared to the reinforcing fibres, and laminated composite structures thus can fail because of delamination due to high transverse stress levels. In this work the objective is to optimize laminated composite structures against delamination failure. Due to many material layers in laminates, it is necessary to reduce the number of degrees of freedom in finite element models by use of Equivalent Single Layer (ESL) elements, to be able to describe the global response of laminated shell structures. When doing so information about the detailed response of the laminate is lost and it is not possible to predict delamination failure accurately. Adaptively refined analysis models are introduced to overcome this issue. In this paper prediction of delamination onset in laminated composite structures involve stress and/or strain based criteria, i.e., it is assumed that the process of degradation begins when the stresses and/or strains satisfy certain damage initiation criteria. In this work a criterion involving the transverse normal stress and the two transverse shear stresses is used. In order to capture these three transverse stress components, the finite element analysis model is based on continuum based 3D shell elements as described in [1]. Initially, the laminated composite structure is modelled using a finite element model consisting only of ESL solid shell elements. Based on the stress delamination criterion, the most critical area of the structure is adaptively refined by modeling each layer of the laminated structure using 3D continuum based shell elements. At the interfaces between ESL elements and the refined area localized Lagrange multipliers are introduced according to [2]. In this way the interface patch test condition is satisfied a priori. Thus, the analysis model is adaptively refined for the optimization process, such that reliable predictions of delamination onset can be obtained. Having identified critical zones for delamination onset, the refined model is used as reference model for an optimization problem with regard to delamination failure. The design variables of the problem may be fiber angles of each layer. A gradient based optimization approach is applied. Design sensitivity analysis of the linear problem is performed using the direct differentiation approach, and the mathematical programming problem is solved using Sequential Linear Programming. Examples illustrate the mesh refinement, design sensitivity analysis and the optimization approach.

References [1] S. Klinkel, F. Gruttmann and W. Wagner, A continuum based three-dimensional shell element for laminated structures. Computers & Structures, 71, 43–62, 1999. [2] M. M. Denn, Optimization by variational methods., McGraw-Hill, New York, 1969.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

725

Yeast Biomechanics Johan L.F. Kock*, Catrine J. Strauss*, Carolina H. Pohl*, Pieter W.J. van Wyk†, Piet J. Botes* *

Department of Microbial, Biochemical and Food Biotechnology University of the Free State P.O. Box 339 Bloemfontein 9301, South Africa [email protected] †

Center for Confocal and Electron Microscopy University of the Free State P.O. Box 339 Bloemfontein 9301, South Africa [email protected]

ABSTRACT Over million years of evolution, micron-scale structures (capsules) of different shapes (e.g. beanshaped, saturnoid, needle-shaped, hat-shaped, etc.) and nano-scale surface ornamentations (e.g. hairy, warty, hooked ridges, etc.) were optimized in yeasts for effective water driven smart movement [1,2]. Until now the function of these shapes and ornamentations has been speculative [2]. Using immunofluorescence, electron and confocal laser scanning microscopy, physical mechanisms in yeasts are exposed preventing capsules becoming tightly packed (“stuck”) within micron-scale bottleshaped containers (birth sacs) thereby blocking their water driven forced release through narrow birth sac openings. In some cases oxylipin lubricated nano-scale gear-like structures, orientated across the surfaces of bean-shaped capsules, are used to affect unhindered release. In other cases gear-like structures are replaced by lubricated compressible slimy sheaths to affect the same [3]. We also found nano-scale ornamentations on capsule surfaces affecting boomerang spore movement and eventual birth sac rupture through the penetration of turgor-directed spiky ends into birth sac cell walls [4]. This interpretation of the mechanics involved might find application in nano-, aero- and hydrotechnologies with the re-scaling of these structures.

References [1] D. Yarrow, Methods In: The Yeasts – a taxonomic study (4th Ed.) (Kurtzman, C.P. and Fell, J.W., eds), p. 86, Elsevier, Amsterdam, The Netherlands, 1998. [2] J.L.F. Kock, C.J. Strauss, C.H. Pohl and S. Nigam, Invited review: The distribution of 3-hydroxy oxylipins in fungi. Prostaglandins & other Lipid Mediators, 71, 85-96, 2003. [3] A. van Heerden, J.L.F. Kock, P.J. Botes, C.H. Pohl, C.J. Strauss, P.W.J. van Wyk and S. Nigam, Ascospore release from bottle-shaped asci in Dipodascus albidus. FEMS Yeast Research, 5, 1185-1190, 2005. [4] J.L.F. Kock, C.J. Strauss, E.E. Pretorius, C.H. Pohl, A.S. Bareetseng, P.J. Botes, P.W.J. van Wyk, S.W. Schoombie and S. Nigam, Revealing yeast spore movement in confined space. South African Journal of Science, 100, 237-240, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

726

Optimization of postbuckling path for cylindrical shells under external pressure Jacek Krużelecki1, Mariusz Król2 Cracow University of Technology Jana Pawła II 37, 31-864 Kraków, Poland 1 [email protected] 2 [email protected]

ABSTRACT The standard problem of structural optimization under stability constraints is usually formulated as maximization of the instability load for a prescribed volume of a design element. Very often the standard optimal structure has unstable postbuckling behaviour and it is very sensitive to imperfections. That is weakness of the design and it indicates that the combination of geometrically nonlinear analysis with the design becomes necessary, especially from practical point of view. The postbuckling constraints of a special form added to formulation of optimization problem permit to modify the postbuckling path and a stable postbuckling path can be created, even in the case of unstable behaviour of the reference structure. The effect of modification of the postbuckling behaviour in most cases has been obtained by changing sizing variables which are usually dimensions of the design element. In this paper an alternative concept is applied, namely stabilization of the postbuckling path is obtained by application of additional loadings acting on the structure without changing its shape and sizes of the optimized element. These loadings can be either active ones applied to the structure or passive ones (reactions of the additional supports), or both active and passive forces acting simultaneously. In the paper stabilization of the postbuckling path for a simply supported cylindrical shell under radial compressive pressure is considered. All three types of stabilizing forces are investigated. In the case of the active loadings a shell is axially loaded by the active forces at both ends. We look for the minimum value of the axial force which stabilize the postbuckling path. In the case of the passive force an axial movement of the both ends of a shell is constrained by additional elastic elements. In this case we look for the minimum value of the axial stiffness of these additional elements, which causes stabilization of the postbuckling path. The last case consists two types of loadings. First, an axial pretension of a certain value is applied and next the axial movement of the both ends of a shell is blocked. Then, the external pressure is applied. In this case we look for the minimum pretension, which together with the pressure stabilizes the postbuckling path. Calculations were performed using ANSYS code for elastic and elastic-plastic deformations of shells of different length and thickness. It occurred that in a case of the passive force stabilization of the postbuckling path is not possible for any stiffness of additional element. For a case of the active force and for the case of the mixed variant of loadings such stabilization can take place for elastic and elastic-plastic deformations.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

727

Structural Analysis in Continuum Media Using Cellular Automata Rodrigo López, Andrés Tovar, Carlos A. Narváez Department of Mechanical and Mechatronic Engineering, National University of Colombia Cr. 30 45-03, Of. 453-401, Bogotá, Colombia {orlopezv, atovarp, canarvaezt}@unal.edu.co

ABSTRACT The finite element method (FEM) is probably the most popular numerical technique to perform structural analysis. However, one of its major disadvantages is the high computational cost required to solve problems of practical size with sufficient accuracy. This problem has a significant impact on structural optimization techniques that make use of the FEM [2]. Recent efforts have been made to develop alternative computational algorithms appropriate to massive parallel processing. Some of the most efficient approaches make use of the cellular automaton (CA) paradigm. CA models are an idealization of a physical system in which space and time are discrete. They are composed of a regular lattice of identical cells that are defined by their state. The state of each cell is determined through interaction with the state of its immediate neighbors by applying a local evolutionary rule. Locally, the behavior of the cells is rather simple, but on a larger scale a new complex (and sometimes unexpected) collective behavior emerges [5]. Some applications have incorporated CA principles in structural optimization in continuum media [4], structural analysis and design combined with genetic algorithms [3] and simultaneous analysis and design in discrete structures [1]. The goal of this investigation is to develop a new CA computational application suitable for structural analysis in a continuum. The cells in this model are regularly distributed nodes connected to each other by an elastic material. The physical properties of the continuum are defined by its Young’s modulus and Poisson’s ratio. The nodal displacement and internal force define the state of the cell. Local evolutionary rules, derived from the principle of minimum total potential energy, find the optimal kinematic configuration of every cell in the structure. This algorithm is implemented in twodimensional problems and their results are compared with the ones obtained using the FEM.

References [1] M. M. Abdalla and Z. Gürdal, Structural design using optimality based cellular automata. In Proceedings of 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference. Denver, Colorado, 2002. [2] M. P. Bendsøe and O. Sigmund, Topology Optimization Theory, Method and Applications. Springer, 2003. [3] P. Hajela and B. Kim, On the use of energy minimization for CA based analysis in elasticity. Struct Multidisc Optim 23, 24–33, 2001. [4] A. Tovar, J. D. Muñoz, H. Cortés, N. Patel and J. E. Renaud, Hybrid cellular automata with local control rules: a new approach to topology optimization inspired by bone functional adaptation. In Proceedings of 6th World Congress on Structural and Multidisciplinary Optimization. Rio de Janeiro, Brazil, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

728

Multidisciplinary Optimization of Complex Technical Systems Kuzmenko L. Mikhail1, Egorov N. Igor1, Shmotin N. Yuri1, and Fedechkin S. Konstantin2 1

NPO SATURN Leninprospekt, 163, Rybinsk, Jaroslawl Oblast, Russia {mikhail.kuzmenko, yuri.shmotin}@npo-saturn.ru, [email protected] 2 IOSO Technology Center, Kasatkina, 13, Moscow, 129301, Russia fedechkin @iosotech.com

ABSTRACT Modern computer technologies now allow us to conduct rather complex mathematical calculations in a relatively short period of time. Thus, it has become possible to employ optimization methods in the design of complex technical systems, even when calculations require large computational resources (structural, thermal, and gasdynamics calculations). The designer may have to vary more than a hundred design variables and constraints during the optimization process. Therefore the procedure of preparing the initial data for optimization may take a long time. That is why we developed the optimization software for designing technical systems. This software system includes the IOSO optimization procedure and modules of automatic data preparation and handling. The data is represented in the format that is convenient and understandable for a designer. The optimization procedure is based on the response surface methodology, when response surfaces are constructed for objective functions and constraints and then optimized at each iteration in a current search region. The objective function and constraints are then evaluated at the optimal point using the mathematical model of the system under consideration. The paper presents the results of optimizing a three-stage axial compressor. The optimization goal was to improve the compressor efficiency at two flight conditions by optimizing geometry of the 5 compressor rows (62 design parameters). As the analysis tools the well-known commercial software package (FINE/Design3D) is used. The conducted investigations showed that it is possible to perform optimization studies using 3-D method of calculating the flow parameters in the axial compressor. The most important elements in organizing the optimization process, that influence the efficiency of such studies are the following: the capability to automatically generate the mesh for a wide range of blade geometry configurations and an efficient optimization algorithm, that allows for a maximum improvement in objective function with minimum number of flow model evaluations. In our particular case as a result of optimizing compressor blade geometry parameters we got a 3% improvement in maximum efficiency while satisfying all the constraints.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

729

Adaptive Shape Optimization Method Jean R. Roche∗ ∗ IECN,UMR

7502, INRIA Lorraine, Universit´e Henri Poincar´e Nancy I B.P. 239, 54506 Vandoeuvre l`es Nancy [email protected] ABSTRACT

The principal objective of this work is to introduce an adaptive strategy to monitor the convergence rate of a Newton-like method in shape optimization. Shape optimization problems are characterized by a cost function and a partial differential equation (P.D.E.) which depend both of the geometrical domain. Typically we want to compute a shape Ω∗ such that Ω∗ = argmin{G(Ω, ϕΩ ) : Ω ∈ O} The scalar potential ϕΩ is the solution in a Sobolev space H of an elliptic problem, the state equation. The function G(Ω, ϕΩ ) : O × H → IR is the cost function. The set of admissible domains O is characterized by geometrical and regularity constraints, for analytical calculus we consider domains of class C 2 . To obtain a superlinear convergence rate of the shape optimization procedure we introduce Newton’s like methods. As is well known the convergence rate depends on how accurate is the numerical solution of the state equation. More precisely let us denote en the error at iteration n in the optimization procedure, then in an appropriate norm we obtain: ||en+1 || ≤ C1 (||en ||2 + ∆n ||en || + En ) The terms En and ∆n are related to the accuracy tolerance when we compute the first and second order shape derivative of the cost function G. Approximation errors in computation of the shape gradient and Hessian are cumulated with rounding errors and numerical errors in the resolution of the (P.D.E.). If ϕh is the numerical solution of the state equation the term En and ∆n are related to the error ||ϕΩ − ϕh ||L2 . Thanks to a local a posteriori estimation of ||ϕΩ − ϕh ||L2 an adaptive algorithm to obtain mesh refinements of ∂Ω is derived in order to have superlinear convergence rate of the shape optimization Newton-like method. The adaptive procedure consists in a monitoring step and in a mesh refinement step. A contraction factor has been introduced and we give a criterion to decide when we need to refine the mesh. The model problem concerns a cost function depending on the curvature of Γ and the solution of the exterior Dirichlet problem. The equilibrium shape is shown to be the stationary state of the total energy under the constraint that the surface (the volume in 3-d) is prescribed. Numerical results of the adaptive technique applied to the model problem are analyzed.

References [1] A. Novruzi and J.R. Roche, Newton Method in 3-dimensional Shape Optimization Problems. Application to electromagnetic casting. BIT, 40:1, 102–120, 2000.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

730

A PEM Fuel Cell Cathode Model for Gradient-Based Optimization M. Secanell∗ , B. Carnes∗ , A. Suleman∗ , N. Djilali∗ ∗ Institute

for Integrated Energy Systems and Dept. Mechanical Engineering, University of Victoria PO Box 3055 STN CSC, Victoria, BC, V8W 3P6 Canada {secanell,bcarnes,suleman,ndjilali}@uvic.ca

ABSTRACT In order for fuel cells, and in particular, proton exchange membrane fuel cells (PEMFCs) to enter the marketplace, their design has to be improved to achieve increased performance. In particular, the cathode electrode in a PEMFC is a critical component from the view point of both cost and performance. Since, fuel cells and electrodes are complex systems that depend on a large number of coupled physical phenomena and a large number of design parameters, current trial-and-error approaches to design lead to an inefficient and time consuming design process. In recent years, new techniques based on extensive use of computational modeling and numerical optimization algorithms have emerged that lead to shorter design times and optimal designs. In this paper, a cathode electrode PEM fuel cell model is presented that can be readily used with a gradient-based optimization algorithm to perform systematic multivariable optimization. The cathode electrode PEM fuel cell model is two-dimensional and accounts for oxygen and water mass transport in the gas phase, for the transport of protons and electrons, and for the electrochemical reactions that occur at the electrode. Furthermore, since the cathode electrode PEM fuel cell model is to be used for design, the input parameters are selected to be representative of the design parameters used in manufactured an electrode as suggested in [1]. Finally, for any given electrode design, the model can be used to predict the current density which is a good measure of performance, and therefore, can be used as the objective function for the electrode optimization. The modeling equations for the cathode electrode yield a system of three nonlinear partial differential equations which are solved using adaptive finite elements. In particular, to solve this system a C++ program that uses the adaptive finite element deal.ii library [2] was developed and validated. Furthermore, in order to be able to use this model for design and to couple this model with a multivariable gradientbased optimization algorithm, the analytical sensitivity equations of the current density produced at the cathode electrode are obtained with respect to the design parameters using the direct method. These sensitivity equations are also implemented using a finite element formulation and validated by comparing the analytical gradients to numerical gradients obtained using forward differences.

References [1] C. Marr and X. Li. Composition and performance modelling of catalyst layer in a proton exchange membrane fuel cell. Journal of Power Sources, 77(1):17–27, January 1999. [2] W. Bangerth, R. Hartmann, and G. Kanschat. deal.II Differential Equations Analysis Library, Technical Reference. http://www.dealii.org.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

731

Optimisation of a Railway Wheel Profile 1

1

1

2

I.Y. Shevtsov , V.L. Markine , C. Esveld , M.V. Markina 1

Section of Road and Railway Engineering Faculty of Civil Engineering and Geosciences Delft University of Technology Stevinweg 1, 2628 CN, Delft, The Netherlands E-mail: [email protected], [email protected], [email protected] 2

Faculty of Mechanics and Mathematics Nizhny Novgorod State University Gagarin Avenue 23, Nizhny Novgorod, 603600, Russia [email protected]

ABSTRACT During the last decades substantial progress has been made in design of railway vehicles and running gears. Tilting trains, high speed trains, active steering wheelsets and many other sophisticated solutions have been implemented in recent years on the railways. But despite this progress, the mechanics of railway wheelset remains the same and an inappropriate combination of wheel and rail profiles can easily diminish all this technological advances. Besides, many old fashioned vehicles are still in too good condition to be replaced. They have a special need for appropriate combinations of wheel/rail profiles since such vehicles do not have high-tech devices which improving performance. The paper presents a procedure for optimal design of a wheel profile based on geometrical wheel/rail contact characteristics such as the rolling radii difference (RRD). The procedure uses an optimality criteria based on a RRD function. The criteria accounts for stability of wheelset, cost efficiency, minimum wear of wheels and rails as well as safety requirements. The shape of the wheel profile approximated by a piecewise cubic Hermite interpolating polynomial is varied during the optimisation process in order to satisfy the optimality criteria. The optimization problem described above has been solved using the MARS method (Multipoint Approximations based on Response Surface fitting). The method has been specifically developed for problems where multiple response analyses and (time consuming) simulations are involved. Finally dynamic simulations of vehicle with obtained wheel profile have been performed in ADAMS/Rail program package in order to control wheel/rail wear and safety requirements. A solution of the optimization problem is then taken as a new wheel profile. Wheels with such profile have the given contact characteristics, which results in improved wheelset dynamics and in reduction of wheel wear. The proposed optimum design procedure has been applied to improve the performance of the metro trains in Rotterdam, The Netherlands (RET), which were suffering from severe wheel wear and as a result the hunting of the vehicles. The results of the optimisation have shown that the performance of railway vehicle can be improved by improving the contact properties of the wheel and rail. Using the proposed procedure a new wheel profile has been obtained and applied to the RET metro trains. Due to the application of the optimised wheel profile the instability of the metro trains has been eliminated and the lifetime of the wheels has been increased from 15000 km to 114000 km.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

732

Discrete Material Optimization of Laminated Composites – SIMP vs. Global Optimization Jan Stegmann∗ and Mathias Stolpe† ∗

Department of Mechanical Engineering, Aalborg University, Denmark [email protected] † Department

of Mathematics, Technical University of Denmark [email protected]

ABSTRACT Design of laminated composites structures is becoming increasingly important as the use of composite materials steadily increases. This development is driven by the aerospace, automotive and wind turbine industries who need still lighter and stiffer/stronger structures. This presents a very challenging design task that calls upon structural optimization tools for providing basic design ideas. However, existing methods for handling laminated composites suffer from problems with local optima when optimizing the fiber orientation, which is the key to efficient design with laminated composites. To counter this problem Discrete Material Optimization (DMO) was suggested in [1] where an alternative parametrization of the optimization problem is used, inspired by the procedures in topology optimization. The idea is to discretize the problem by using only a limited number of pre-defined candidate fiber orientations, each described by a constitutive matrix, Ci . The optimization problem  is then parameterized on the element level by expressing the constitutive matrix for lamina j as Cj = i xij Ci where ∀ xij ∈ {0, 1} are the design variables for material i in lamina j. The objective of the optimization is then to choose one distinct material from the set of candidates, i.e. i xij = 1, ∀ j. The design variables, xij , may be associated with a specific lamina/element or a patch consisting of several laminae/elements, thereby significantly reducing the total number of design variables. The constitutive matrices, Ci , may represent any type of material, allowing for simultaneously optimization for fiber orientation and material choice. The discrete problem stated above was solved successfully for minimum compliance of large-scale structures in [1] using relaxation and mathematical programming with a SIMP penalization strategy for obtaining 0/1 solutions. Inspired by this and the work in [2] the authors have recently solved smallerscale DMO problems to provable global optimum using the discrete variables directly in a nonlinear branch and bound framework. The relaxed problem is modeled as a convex program with a nonlinear objective function and linear constraints and solved using a Newton method. In this work we compare the results obtained using both the continuous SIMP relaxation and the global optimization strategy in order to evaluate the efficiency of the two methods. Preliminary results show that the SIMP method is efficient at obtaining near-optimal 0/1 solutions for even large-scale problems but that the convergence to global optimum (if possible) is dependent on the chosen penalization strategy. The global optimization method converges successfully but is limited by computational requirements to smaller-scale problems. The global optimization method can provide valuable benchmark solutions and furthermore seems very promising for problems involving patches of design variables. REFERENCES [1] J. Stegmann, E. Lund. Discrete material optimization of general composite shell structures, International Journal for Numerical Methods in Engineering, 62(14), 2009-2027, 2005. [2] M. Stolpe. Global optimization of minimum weight truss topology problems with stress, displacement, and local buckling constraints using branch-and-bound, International Journal for Numerical Methods in Engineering, 61(8), 1270–1309, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

733

First and Second Order Sensitivities of Functions with Respect to Binary Variables and their Application in Topology Optimization Krister Svanberg

,1 ,

Mats Werme

,2

Optimization and Systems Theory, Department of Mathematics, KTH SE-100 44, Stockholm, Sweden 1 [email protected], 2 [email protected]

ABSTRACT This paper deals with topology optimization of load-carrying linearly elastic continuum structures that have been discretized by a finite element model. The topology optimization problem is then to decide which parts of the structure that should be filled with material and which parts that should be made into void in order to minimize/maximize some objective function while at the same time not violate the constraints. The discrete decision between void and non-void gives rise to binary design variables indicating the presence or absence of material in the various finite elements. The method proposed in this paper uses the fact that if only one or two design variables are changed to their opposite values the new stiffness matrix is just a low rank modification of the old one. It turns out that it is possible to compute these first (if one design variable is changed) and second (if two design variables are changed) order sensitivities exactly in a highly efficient way, without the need for a refactorization of the stiffness matrix. The focus of this paper is a treatment of how to compute these first and second order sensitivities under the assumption that the stiffness matrix for the current topology has already been Cholesky factorized. Numerical results obtained by a neighbourhood search method are presented

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

734

Optimal design of smart composite laminates by the polar method and the genetic algorithm BIANCA A. Vincenti*, P. Vannucci† *

Laboratoire de Modélisation, Matériaux et Structures (LM2S) – Université Paris 6 Case 161 – 4, place Jussieu – 75252 Paris Cedex 05 – France [email protected]

†

Laboratoire d’Etudes Mécaniques des Assemblages (LEMA) – Université de Versailles et St.Quentin Bât Descartes – 35, avenue des Etats Unis – 78035 Versailles – France [email protected]

ABSTRACT Smart and adaptive structures are designed to change some of their features in response to variations in the external environment. This is achieved by the introduction of sensors, actuators and control devices that measure these changes, and react by affecting the properties of the structure itself. Therefore, a strong interaction is present between the different parts of the system in order to behave in a smart way, which depends strictly on an integrated and intelligent design of the structure as a whole, starting with the choice of optimal constitutive material properties. In this paper, we consider the case of composite laminates including piezoactive layers, which are among the most popular smart materials, where the application of an electric potential between the piezoelectric layers induces the laminate to change its shape by expansion and/or deflection. For most applications of this kind, it is therefore essential to tailor the behavior of laminates according to the desired response, namely it can be interesting to have laminates having null expansion and/or deflection coefficient in a given direction, or to have an isotropic expansion/deflection response under the application of an electric potential. In a previous paper, the authors showed [1] a statement for this class of problems in the form of an optimization problem, where the objective function takes into account not only the electric response of the laminate but also more general elastic properties, such as elastic uncoupling, orthotropy and so on. Our formulation, which is very general and makes abstraction of any simplifying hypothesis, is developed in the framework on the Classical Laminated Plates Theory, and it is based on the polar method for the representation of plane tensors [2]. In this paper, we show that a general resolution is not possible analytically and we propose a numerical tool to solve the optimization problem, the genetic algorithm BIANCA. We describe the structure of BIANCA, which is adapted to take into account very different design cases, involving various design parameters. We finally show the effectiveness of the combined polar-genetic approach in the design of smart composite laminates, by giving a number of numerical results obtained by BIANCA.

References [1] A.Fernandes, A.Vincenti, P.Vannucci, J.Pouget, Optimal design of composites including piezoelectric active layers. II ECCOMAS Thematic Conference on Smart Materials and Structures, 18-21 July 2005, Lisbon, Portugal. [2] P. Vannucci, The polar method as a tool for analysis and design of plane anisotropic problems. Proceedings of XVII AIMeTA Congress of Theoretical and Applied Mechanics, 11-15 September 2005, Florence, Italy.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

735

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

736

Optimising Buckling Capacities for Composite Shells Hongtao Wang*, James G A Croll*, Nobuhisa Yamamoto†, Seishi Yamada† *

Department of Civil and Environmental Engineering, University College London Gower Street, London WC1E 6BT, UK [email protected]

†

Department of Architecture and Civil Engineering, Toyohashi University of Technology Tempaku-cho, Toyohashi 441-8580, Japan [email protected]

ABSTRACT With their extremely large numbers of independent material and geometric parameters the problem of choosing the appropriate combinations to optimise buckling performance is even greater for advanced composite shells than for equivalent metallic shells. Furthermore, for thin shells constructed from advanced composites it is generally more important to also take account of the reductions in elastic load carrying capacities that result from the severe sensitivities of buckling loads to the effects of initial imperfections. Current approaches of nonlinear numerical analysis combined with optimisation algorithms heavily rely on analytical and computational efforts. Given the immense numbers of independent parameters that might be involved and the complexity of the nonlinear analysis needed to assess the worst effects of initial imperfections, it is unlikely that reliance on just nonlinear numerical solution algorithms will be sufficient for this purpose. This paper will outline an alternative approach based on the so-called “reduced stiffness method” (RSM) [1] that enables an extension of classical buckling analysis to provide safe lower bounds to the imperfection sensitive buckling loads of shells. The RSM has successfully predicted safe lower bounds for a range of isotropic and stiffened shells. Recent carefully controlled numerical analysis shows that RSM can also provide reliable estimates of the lower bounds to composite shells’ buckling loads [2]. Since this method also encourages the delineation of those components of the shell’s membrane and bending stiffness that are important and those that are unimportant within each of the prospective buckling modes it provides an intuitive strategy for better design decision making. In this paper by examining the effect of lamina eccentricities to the imperfection sensitive buckling loads of composite shells it is demonstrated how the RSM can be used as a tool for guiding appropriate combinations of parameters to enhance buckling capacities of fibre reinforced laminated composite shells. The simplicity of this analytically based method enables the prediction of the likely consequences of variations of the many material and geometric parameters that govern the safe resistance to buckling. As a tool for guiding appropriate combinations of parameters to affect enhanced, or even “optimum”, buckling capacities this approach will be shown to have considerable advantage over many of the currently available alternative means for improving design performance.

References [1] J. G. A. Croll, R.C. Batista, Explicit lower bounds for the buckling of axially loaded cylinders. Int. J. Mech. Sci., 23, 333-343, 1981. [2] S. Yamada, N. Yamamoto, J.G.A. Croll, P. Bounkhong, Local buckling criteria of thin-walled FRP circular cylinders under compression. International Colloquium for the Application of FRP to Bridges, Tokyo, Japan, 2006.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

737

Multiaxial Plastic Hardening Models used in Shape Optimization with respect to Fatigue Life B. Wilczynski*, Z. Mróz† *

†

Mechanical Department, Technical University of Koszalin Racáawicka 15/17, 75-620 Koszalin, Poland [email protected]

Institute of Fundamental Technological Research, Polish Academy of Science SwiĊtokrzyska 21, 00-049 Warsaw, Poland [email protected]

ABSTRACT The total fatigue life Nf of notched machine or structural elements is given as a sum of two phases (stages): the initiation stage Ni (fatigue crack initiation), and the second phase, fatigue crack propagation Np. For non sharply notched parts, 50% (some sources report even 85-90%) of the whole lifetime is connected with the initiation phase. Hence, the problem of predicting the critical number of loading cycles corresponding to crack initiation in a structural element is of fundamental importance for rational design with specified service life. When the elastic local stress and strain exceed the elastic limit, an elastic-plastic stress evolution occurs. The crack initiation is then dependent on the plastic dissipated energy and the stress at the notch root. A closely related problem is that of rational design of notch shape in order to maximize the critical number of cycles corresponding to crack initiation. Generally the value of the fatigue life Nf of machine components depends on several factors: geometry of the element, material properties, presence of residual stresses, history of loading and boundary conditions. Today, it is very well known, that by proper modification of the shape of notched parts (shape optimization) we can significantly reduce the peak stress and increase significantly the lifetime of machine parts. In this paper we treat the optimisation problem as follows: for a given boundary conditions, external loading and material properties find a such shape of notched part for which for which Ni(j)(*), j = 1, …, n, reaches maximum with constraints *  **,where * is a boundary shape to be modified, ** is a given variation domain of *, and n is the number of critical points (critical planes). Different uniaxial and multiaxial approximate models are used to evaluate the actual stress-strain behaviour, for instance multi-surface plastic hardening rules accounting for anisotropic response, proposed by Mróz. The shape of the notch * is defined by a special concept of Bezier interpolant (modification of the standard Bezier’s curve), what significantly reduces the number of design variables. Analytical relation between the hypothetical linear elastic stress-strain response at the notch tip and the actual elasto-plastic stresses and strains, called notch stress-strain conversion (NSSC) rule, is evaluated using the Boundary Element Method (BEM). The optimization procedure used is the Sequential Linear Programming Method with move limits. Optimization results obtained with application of the above mentioned NSSC rules for uniaxial and multiaxial loading have been compared with results obtained by the nonlinear elastic-plastic variant of the BEM.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

738

Multi-level optimization of material and structural layout A.J. de Wit∗ , A. Lipka† , E. Ramm† , F. van Keulen∗ ∗ Department

of Precision and Microsystems Engineering, Faculty of Mechanical, Maritime and Materials Engineering (3mE), Delft University of Technology Mekelweg 2, 2628 CD Delft, The Netherlands {A.J.deWit,A.vanKeulen}@tudelft.nl † Institute

of Structural Mechanics, Universit¨at Stuttgart Pfaffenwaldring 7, 70569 Stuttgart, Germany {Lipka,Eramm}@statik.uni-stuttgart.de ABSTRACT

Within the classical design of, for example, bridges and airplane structures, optimization approaches generally focus primarily on adjusting large-scale (macroscopic) parameters and neglect the influence of small-scale (microscopic) parameters on the large-scale behavior. The main reason is because of numerical efficiency. Multi-level optimization techniques rely on a decomposition of the optimization problem into separate levels or subsystems. Thus, it is attempted to incorporate design variables originating from different (multiple) levels in a cost-effective manner. In literature various multi-level optimization approaches are described in which six main approaches can be distinguished, Optimization by Linear Decomposition, Collaborative Optimization, Concurrent SubSpace Optimization, Bi-Level Integrated System Synthesis, Analytical Target Cascading and the method of Quasi-separable Subsystem Decomposition. However, a comparative overview of these methods is still lacking. In this study these multi-level optimization schemes are compared on the basis of treatment of interdisciplinary consistency constraints between the hierarchical levels. Multilevel optimization methods use the capability of an optimization problem to be decomposed into separate levels or subsystems. In order to decompose the coupled analysis, linking variables have to be introduced. A clear distinction between the aspects of optimization and analysis in multi-level design is made using a new multi-level notation. It emphasizes the handling of inconsistencies between subsystems and their solution. The proposed notation enables a clear comparison of the multi-level methods on the basis of their treatment of interdisciplinary consistency constraints and allows classification on the basis of handling of constraints, as previously demonstrated by Alexandrov and Lewis [1] for two multi-level optimization formulations. In this paper, this classification of multi-level optimization approaches is extended to include all six multi-level formulations. Furthermore, the multi-level optimization framework is demonstrated by applying the six multi-level methods to a classical two bar truss example. The work here is part of a larger ongoing effort towards integrating multi-scale mechanics and multi-level optimization. The two bar truss example is in this context a suited example. It contains relevant characteristics of multi-scale mechanics and multi-level optimization and its simplicity helps to capture the essence of multi-level design approaches. In future work we aim to describe multi-scale mechanics in a similar fashion, which is expected to open up possibilities of exploiting synergy effects through further integration of the two formulations.

References [1] N.M. Alexandrov and R.M. Lewis. Comparative properties of collaborative optimization and other approaches to mdo. Technical Report ICASE Report No. 99-24, Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, Virigina, July 1999.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

739

A Stochastic Modelling of the Dynamical Response of Highway Bridge Decks Under Traffic Loads Ricardo S. de Almeida1, José Guilherme S. da Silva2 1

MSc Student in Civil Engineering, Faculty of Engineering State University of Rio de Janeiro, UERJ, Brazil [email protected] 2 Mechanical Engineering Department State University of Rio de Janeiro, UERJ, Brazil [email protected]

ABSTRACT In this paper an analysis methodology is developed to evaluate the dynamic response, displacements and stresses, on highway bridge decks due to vehicles crossing on the rough pavement surfaces [1]. The analysis methodology follows a statistical model running in the time domain. The mathematical model simulates the bridge structure and the vehicle series as a system, the vehicle-bridge system. The bridge deck follows a straight beam model made discrete by finite elements and nodal concentrated masses, with vertical translations and in-plane rotations as degrees of freedom. Rotatory inertia and shear deformations are not considered. The vehicle simulation uses concentrated parameters of mass, stiffness and damping. Four different types of vehicles are modelled as rigid masses connected by springs and dampers with one, two, four or five degrees of freedom. According to each vehicle model, translational and rotational displacements are considered. The deck surface roughness is defined by a weakly stationary, second order and ergodic random process based on a well-known power spectrum density of road pavement profiles [2]. The moving load is modelled by an infinite series of equal vehicles, regularly spaced, and running at constant velocity. Only steadystate response is considered. Response data are produced on reinforced concrete highway bridge decks made of a straight box girder cross section based on several spans and support arrangements. Results of a parametric analysis are presented to verify the extension of the dynamical effects on highway bridge decks, due to vehicles crossing on the irregular pavement surface. Preliminary results have indicated in all cases studied, in the present investigation, for usual vehicle velocities, that the dynamical effects on highway bridge decks due exclusively to the interaction of the vehicle suspension flexibility with an irregular pavement surface represent a significant parcel on the vehiclebridge system response.

References [1] L. Fryba and M. Pirner, Load tests and modal analysis of bridges. Engineering Structures, 23, 102-109, 2001 [2] J.G.S. da Silva, Dynamical performance of highway bridge decks with irregular pavement surface. Computer & Structures, 82 (11-12), 871-881, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

740

Comparison of Two Dimensional Nonlinear Analyses of Integral Abutment Bridge and Simply Supported Bridge Under Near Fault Earthquake Loading With Consideration Soil Structure Interaction Shahin Nayeri Amiri1, Majid Barghian, Mikail Yousefzadeh Fard and Javad Safadoust Department of Civil Engineering, Tabriz University Tabriz, Iran {sh.nayeri, barghian, mikail}@tabrizu.ac.ir [email protected]

ABSTRACT Bridges form important link in the transportation system of a country. Bridges are also considered to be structures of post earthquake importance because of their need for emergency response, relief and rehabilitation measures. It is known from experience that bridges are vulnerable to earthquake damage. Thus the safety and protection of bridges in earthquake is utmost important. There are a number of bridges which are designed according to old codes when modern seismic provisions were not developed. Such bridges are mostly found deficient and may need seismic retrofitting. The occurrence of large earthquakes close to fault is inevitable and one kind of bridges that needs more investigation is integral-abutment bridge. This paper presents results of two-dimensional nonlinear analysis of integral abutment bridges under near fault earthquake loading. The analyses were performed on a nonlinear dynamic finite element (FE) model using the computer program PLAXIS. Near fault earthquake acceleration records with various amplitudes of velocity pulses were used as input ground motions in the analyses. The results of the analyses indicate that the input motions with strong velocity pulses may cause excessive displacements of the bridge super structure and abutments. The objective of this research is to investigate the response of the integral abutment bridge to a near field earthquake loading taking into account the soil-structure interaction effects and compare with simply supported bridge. Further study is needed to investigate the effects of strong velocity pulses on overall response of a integral abutment and fully understand the dynamic behavior of an integral bridge.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

741

Dynamic Stability Analysis of Truss Structures under Nonconservative Constant and Pulsating Follower Forces Jiann-Tsair Chang *, I-Dan. Huang †, Wei-Ming Hou†, Ping-Kun Chang † *

Department of Civil Engineering, Vanung University, No.1 Van-Nung Rd. Chung Li City, Taoyuan Connty 320, Taiwan (R.O.C.) [email protected] † Department of Power Mechanical Engineering, Army Academy, No.113, Sec. 4, Zhongshan E. Rd., Zhongli City, Taoyuan County 320, Taiwan (R.O.C.) [email protected]

ABSTRACT The aim of this study is to investigate the dynamic stability behaviors of truss structures under nonconservative constant and pulsating follower forces through the finite element formulation and the multiple scales perturbation method. In physical phenomena, we may face the problem of truss structures subjected to wind loadings in sinusoidal types. This system can be modeled as truss systems subjected to constant and pulsating follower forces. So the truss system may lost its instability due to the so-called sum type combination resonance or difference type combination resonance just as a liquid-filled projectile under a thrust does [1],[2]. Herein a definition direct approach is used to derive the constant and pulsating follower load stiffness matrix of a truss element. This non-conservative load matrix is contributed by constant and pulsating follower forces acting on the nodes on the truss element. Using Euler finite rotation formula and the principle of virtual work, the constant and pulsating follower forces acting on the nodes of truss element can be related to the degrees of freedom of the truss element in a simple close form, then an explicit form of the nonconservative load stiffness matrix can be derived. A simple matrix from to present the nonconservative force effect in the element is purposed in this paper. Then it is assembled with the associated matrices that contain mass stiffness matrix and elastic stiffness matrix. Finally the equations of motion of the truss systems under pulsating follower forces are formulated and solved by the multiple scales perturbation method. The dynamic stability diagrams that show the resonant frequencies of summation type or the resonant frequencies of difference type of the system are also discussed in this paper.

References [1]J.-H. Kim, and Y.-S. Choo, Dynamic stability of a free-free Timoshenko beam subjected to a pulsating follower force. Journal of sound and vibration, 216, 623-636, 1998. [2]S.-W. Jung, K.-S. Na, and J-H Kim, Dynamic stability of liquid filled projectiles under a thrust. Jurnal of Sound and Vibration, 280, 611-631, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

742

Comparison of concrete tall building behavior using an intermittent shear walls form in one frame with continuous shear walls form. Ali Davaran1, Mikail Yousefzadeh Fard, Shahin Nayeri Amiri and Armin Kashefi Department of Civil Engineering, Tabriz University Tabriz Iran {davaran,mikail,sh.nayeri}@tabrizu.ac.ir [email protected] ABSTRACT

In this study a new approach of using shear walls in the buildings has been introduced. Shear walls are the vertical component of lateral load resisting system of a structure. Shear walls transfer lateral loads from a horizontal diaphragm above to a diaphragm or wall below, or to the foundation. The use of continuous shear walls in height of building from structural viewpoint is very useful. However in some buildings, the continuous shear walls lead to some difficulties, such as architectural limitations, unwanted force interaction effect between frame and walls especially in the upper stories and huge foundation design requirement under the shear walls. In this paper to overcome the above problems, staggered shear walls has been used in the plane of one frame. For this purpose three building with nine, eleven and thirteen stories have been modeled and static and dynamic analysis were performed. All dynamic analyses have been carried out using acceleration spectrum recommended in Iranian building code standard No.2800. Analysis were performed by Etabs2000 software. Finally, the behavior of aforementioned structures such as, the frequencies, the maximum lateral displacements and the internal forces have been compared. The results have shown an increase in the lateral stiffness of the structure using the proposed shear wall pattern. Also the applied maximum moment and axial force exerted to the foundation has been reduced using this approach.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

743

An iterative coupled boundary-finite element method for the dynamic response of structures S. Franc¸ois and G. Degrande Structural mechanics, Department of Civil Engineering K.U. Leuven, Kasteelpark Arenberg 40, B-3001 Heverlee, Belgium [email protected] http://www.bwk.kuleuven.be/bwm

ABSTRACT Dynamic excitations in the built environment as caused by earthquakes, heavy traffic and pile driving may result in structural damage, which is determined by the constitutive behaviour of building materials and foundation soils under cyclic loading. This paper presents a coupled finite element-boundary element approach for the calculation of the dynamic response of structures due to dynamic excitations. Both the non-linear constitutive behaviour and the dynamic interaction between the soil and the structure are accounted for. A time domain finite element formulation is used for the structure, as the non-linear constitutive behaviour of the structural materials requires a direct time integration procedure. The soil is assumed to be linear elastic and a boundary element method is used to fully account for dynamic soil-structure interaction. As both the finite and the boundary element method impose different conditions on the time step regarding stability and accuracy, an iterative coupling scheme is proposed that allows for a different time step in both subdomains. An interface relaxation technique is employed in order to speed up convergence. An optimal relaxation parameter is computed using Aitken’s method instead of selecting a constant value. This results in a non-stationary Richardson iteration. The numerical behaviour of this scheme is studied in detail. The method is applied to the problem of a circular foundation resting on a linear elastic halfspace and to the calculation of the response of a structure due to traffic induced vibrations, where the dynamic interaction between the soil and the structure is fully accounted for.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

744

Pendulum Mode Control in the Dynamic Analysis of Lift-off of Launchers Sebastiaan Fransen*, Daniel Rixen† ∗ European Space Agency - Structures Section PO Box 299, 2200 AG Noordwijk, The Netherlands [email protected] †

Delft University of Technology - Faculty 3mE, Engineering Dynamics Mekelweg 2, 2628 CD Delft, The Netherlands [email protected]

ABSTRACT Predicting the vibrational behavior of a launcher and its payload during lift-off is of prime importance to guarantee the integrity of the system when excited through thrust and acoustics loads. The payload and launcher structures are shell constructions that experience significant prestress due to gravity, acceleration under the action of the thrust and internal pressure in fuel tanks and boosters [1, 2]. The prestress effects are accounted for in vibration simulation through additional stiffness contributions. Some of the prestress contributions (such as the total apparent gravity forces) cause the zero eigenfrequencies of the rigid body modes to become negative and therefore render the model unstable. In practice the launcher is stabilized by a Thrust Vector Control such that the rigid body motion of the launcher has low positive associated eigenfrequencies. In this contribution we will recall a numerical procedure to stabilize the prestressed launcher model [3]. Introducing an explicit rigid degree of freedom and applying a proper change of basis the rigid body motion of the system can be modified. The procedure will then be interpreted as a collocated controller to bring the rigid modes back to a zero eigenvalue. This is not exactly how real thrust vector controllers are implemented in the system. However, based on the analysis of a simplified thrust vector controller, we will show that the numerical procedure affects the vibrational dynamics of the launcher in a way very similar to the a real thrust vector and can therefore be used in numerical simulations. We will discuss the consequence of different choices of sensor and actuator locations in the numerical procedure and give results obtained for the lift-off analysis of an Ariane 5 launcher.

References [1] A. Kreis and M. Klein On the Analysis of Free-Free Prestressed Structures.In proceedings of International FEM Conference, Baden-Baden, Germany, IKISS GmbH, 1992. [2] S. Fransen, A Comparison of Recovery Methods for Reduced Dynamic Substructure Models with Internal Loads. AIAA Journal, 42(10), 2130-2142, 2004. [3] S. Fransen, A. Kreis and M. Klein, Pendulum Mode Control of Free-Free Launcher Structural Models in Gravity Fields. AIAA journal of spacecraft and rockets 42(6), 1109–1121, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

745

Dynamic Response of Long Span Cable-stayed Bridge subjected to Earthquake and Moving Train Mangmang Gao*, Jiaying Pan†, Jianzhen Xiong† * China Academy of Railway Sciences [email protected] †

China Academy of Railway Sciences [email protected], [email protected]

ABSTRACT For a long-span railway bridge with running-through train, the earthquake will cause extra deflection and vibration of the structure and give influences on runnability of the train,thus the dynamic interaction between train and bridge with earthquake effect will differ from that without earthquake effect. In order to study this problem, an earthquake-vehicle-bridge dynamic interaction model of train and long span cable-stayed bridge under earthquake load is established in this paper. For this model , earthquake inertia force of both train and bridge are considered and the geometry nonlinear effects of bridge is included. Seismic responses of the train-bridge system are analyzed using dynamic timehistory analytic method. For cars and locomotives, a model of 27 degree of freedom with two stage suspension is used. By taking a cable-stayed bridge alternative of Tianxinzhou Yangtze-River Bridge with 504m main span and four railway lines as an example, dynamic response of this vehicle/bridge system under earthquake load are analyzed with two kind of train running through the bridge, one is a freight train of one locomotive plus 20 freight cars at 80km/h velocity,the other .is a passenger train of one locomotive plus 18 passenger cars at 200km/h velocity. And the seismic time-history curve is selected according to the principle which the earthquake will occur very possibly and the structure can normally be used under such an earthquake. Finally the dynamic performances of this bridge alternative under earthquake and moving train effects is evaluated.

References [1] Jiaying Pan, Yongqiang Li, Mangmang Gao, Study on train runnability for railway cable-stayed bridge, Proceedings of Symposium on High Speed Railways and Bridge Dynamics, Taibei, (2002). [2] Mangmang Gao et al, Dynamic interaction analysis of Wuhu Yangtze-River Bridge under the action of high-speed train, China Railway Science,Oct.(2001),(in Chinese). [3] M. Shinozuka and C. M. Jan, Digital Simulation of Random Processes and Its Applications, Journal of Sound and Vibration,Vol.25(1) 111-128,1972. [4] BRDI, Design alternative of Tianxinzhou Yangtze-River bridge,(2005),(in Chinese). [5] Chinese Railway Engineering Press,Beijing,2005.(in Chinese)

Aseismatic

Design

Code,

China

communications

[6] George Deodatis, Simulation of Ergodic Multivariate Stochastic Processes, Journal of Engineering Mechanics, Vol. 122, NO. 8, August, 1996. [7] He Xia, Dynamic interaction between vehicle and structure, Science Press,Beijing,2002(in Chinese).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

746

Parametrisation of the Newmark time integrator for non-linear solid dynamics Miguel A. Guti´errez∗ , Harm Askes† ∗ Delft

University of Technology, Faculty of Aerospace Engineering, P.O. Box 5058, 2600 GB Delft, The Netherlands [email protected]

† University

of Sheffield, Department of Civil and Structural Engineering, Mappin Street, Sheffield S1 3JD, United Kingdom h.askes@sheffield.ac.uk ABSTRACT

A popular class of integrators for the time discretisation in finite element algorithms applied to solid dynamics is given by the Newmark family. The basic idea is to express the nodal displacements, velocities and accelerations on a two-point finite difference scheme. By resolving any two of these quantities in terms of the quantity left and substituting them into the spatially discretised equations of motion, the solution can be obtained for the considered time increment. The chosen independent quantity is referred to as parameter in the sequel. One speaks of the d-form, the v-form or the a-form for the parametrisation with the displacements, the velocities or the accelerations respectively [1]. There exists a unique relation between the displacements, velocities and accelerations at the end of a time step and, as a consequence, the choice for a particular parametrisation is apparently arbitrary. Traditionally, the d-form has been preferred in physically non-linear problems because the non-linearity is usually found in the stiffness term. Linearisation of the discretised equations of motion for the purpose of solution through iterative algorithms can then be performed in an elegant way. While the choice of the parametrisation does not alter the results, it is observed to introduce a scaling related to the size of the time step in the algebraic system of equations. This scaling can influence the performance of the iterative solution algorithms. In particular, when a Newton-Raphson algorithm is used it is observed that the asymptotic error constant [2] explicitly depends on the scaling and direct influence on the convergence properties is therefore predicted. For the usual magnitude of material properties in engineering solid dynamics and the corresponding optimal time-step size, the asymptotic error constant is invariably relatively reduced if one switches from the d-form to the v-form and, again, from the v-form to the a-form. From a numerical point of view, it is observed that the average number of iterations required per time step to attain a given relative convergence norm in the residual is smaller for the v-form and smallest for the a-form. From the point of view of efficiency it can thus be concluded that the a-form is preferable, especially when several time steps are considered. The numerical evidence of the influence of the parametrisation is illustrated by means of examples.

References [1] X. Zhou and K.K. Tamma, Design, analysis, and synthesis of generalized single step single solve and optimal algorithms for structural dynamics, Int. J. Numer. Meth. Engng, 59, 597–668, 2004 [2] R.L. Burden and J.D. Faires, Numerical analysis. Brooks/Cole, Pacific Grove, 2001

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

747

Global Formulation of Conservative Time Integration by the Increment of the Geometric Stiffness Steen Krenk Department of Mechanical Engineering, Technical University of Denmark Nils Koppels Alle, Building 403, DK-2800 Lyngby [email protected]

ABSTRACT A momentum and energy conserving time integration algorithm is developed for the motion of elastic bodies described in terms of the quadratic Green strain. Momentum conserving algorithms are formulated from an integral of the equations of motion, and energy conservation has traditionally been obtained by evaluating the internal forces by combining the mean value of stresses and virtual strains at the element level [1]. It is here demonstrated that momentum and energy conservation can be obtained from the classic central difference formulation by including an extra global term in the form of the increment of the geometric stiffness matrix. The geometric stiffness matrix is usually available in assembled form in existing programs, and thus a global form is attained that avoids the need for modifying the classic element implementation. The algorithm is derived by use of a state space formulation. In the state space format the extra term containing the increment of the geometric stiffness matrix is located in the same position as the viscous damping matrix, indicating that the effect of the extra incremental geometric stiffness term in the nonlinear algorithm is equivalent to a variable damping term, depending on the change of the state of stress over a time increment. This explains the observation made in the literature, that the simple mean value algorithm needs artificial damping to be stable, when used in kinematically nonlinear problems. After the derivation of the algorithm in the state space format, the velocity can be condensed out, leaving the resulting algorithm in ‘single step – single solve’ format, where the system of equations is of the same size as the equations of the corresponding quasi-static problem. The implementation and performance of the algorithm are illustrated by numerical examples. The basic properties including a discussion of damping were obtained for non-linear truss elements in [2].

References [1] J.C. Simo & N. Tarnow, The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics. Zeitschrift f¨ur angewandte Mathematik und Physik, 43:757–792, 1992. [2] S. Krenk, Non-Linear Modelling and Analysis of Structures and Solids. Cambridge University Press, Cambridge, UK, 2006.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

748

Response Analysis of Building Loaded by Groundborne Transient Vibration Daniel Makoviþka*, Daniel Makoviþka, Jr.† *

Czech Technical University in Prague, Klokner Institute CZ-16608 Praha 6, Šolínova 7, Czech Republic e-mail: [email protected] † Static and Dynamic Consulting CZ-28401 Kutná Hora, Šultysova 170, Czech Republic e-mail: [email protected]

ABSTRACT By sitting of a building in the vicinity of underground tube structure the effect of train operation excites the groundborne vibration. This vibration as a technical seismicity propagates through subsoil to the building foundations in the vicinity of the source. The solution of vibration transfer from the subsoil environment to the building structure is demonstrated using the example of a multistorey reinforced concrete building, founded on large diameter piles mutually connected with the lower foundation plate by reinforcement. On top of this plate an antivibration layer of rubber has been designed. Above the rubber there is an upper foundation plate in which the cast-in-place skeleton building structure is constrained. The elastic rubber course separates upper and lower parts of foundation plate and single footings. The structure is loaded by the groundborne vibrations from the rail systems of underground. The actual history of dynamic load measured on the pile heads was used as an input data for vibro-base insulation design and dynamic analysis of the structure. The structure model takes into account the individual storeys, broken down into the floor, foundation and roof slabs, columns, load-bearing walls and peripheral and interior girders. The layer of rubber was considered as the elastic subsoil of the Winkler-Pasternak model below the whole area of the upper part of the foundation plate and as the elastic support for columns and walls above the piles on the upper foundation plate level. The rubber stiffness in the theoretical model takes into account the type of plates used as well as the mutual superposition of surface and point support on the upper foundation plate level. The mass of the floor and foundation plates includes the masses of the nonload-bearing components (thin partitions, floorings, etc.) as well as the equivalent of the useful loads of floors, roof and terraces. The prediction of floor vibrations is determined and compared with nonisolated state of the building. The vibro-base isolated structure fulfils the standard requirements for limit vibration level for residential structure.

References [1] E. Booth, Seismic design practice into the next century. A.A.Balkema, Rotterdam, 1998. [2] D. Makoviþka and D. Makoviþka, Insulation of buildings against excessive vibrations from the operation of the underground (in Czech), Stavební obzor, 14, No.1, 8-15, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

749

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

750

Textile reinforced concrete structures under uncertain dynamic loading processes B. Möller*, W. Graf *, A. Hoffmann*, J.-U. Sickert*, F. Steinigen* *

Institute of Statics and Dynamics of Structures Technische Universität Dresden 01062 Dresden, Germany [email protected]

ABSTRACT Concrete constructions with textile reinforcement provide an opportunity of increasing load-bearing capacity of existing structures. New textile technologies permit the effective production of textile surface structures with several layers of filament threads made of glass or carbon. Although this new type of reinforcement applied to strengthening of existing structures is effective, clarification is still required concerning the modified load-bearing behavior and an assessment of structural safety. This paper is mainly devoted to enhanced computational algorithms to account for the load-bearing reserves of reinforced concrete structures with textile reinforcement under dynamic loading processes. A multi-reference-plane model (MRM) was developed for the realistic numerical simulation of the load bearing behavior of reinforced concrete structures with textile strengthening [1]. The equation of motion is set up with a mass matrix of the MRM-element which was developed to compute damage indicators [2]. For the solution of the system of differential equations a modified time step operator (Newmarkt) is applied. Data uncertainty, e.g. of the material, bond and geometry parameters, has a significant influence at textile strengthened concrete. The equation of motion is transferred to a fuzzy random equation of motion in order to consider of data uncertainty. The Fuzzy stochastic finite element method (FSFEM) is used to take into account the data uncertainty in the numerical simulation. The developed algorithms was verified at experiments and applied to real construction.

References [1] B. Möller, W. Graf, A. Hoffmann, F. Steinigen, Numerical simulation of structures with textile reinforcement. Computers and Structures, 83, 1659-1688, 2005. [2] B. Möller, W. Graf, A. Hoffmann, F. Steinigen, Numerical models concerning structures with multi-layered textile strengthening. B. H. V. Topping ed., 7th Int. Conference on Compu-tational Structures Technology, Lisboa, Proceedings, Civil-Comp Press, Stirling, Abstract, pp. 97-98, Paper (21 p.), CD-ROM, 2004. [3] J.-U. Sickert, M. Beer, W. Graf, B. Möller, Fuzzy probabilistic structural analysis considering fuzzy random functions. A. Der Kiureghian, S. Madanat, J.M. Pestana eds. 9th Int. Conference on Applications of Statistics and Probability in Civil Engineering, San Francisco, pp. 379-386, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

751

Shock Response Spectrum Analysis for Measured Earthquake Data Mihai Ursu Technical University of Cluj-Napoca Department of Mechanics and Computer Programming Muncii Bvd. 103-105, 400641 Cluj-Napoca, Romania [email protected]

ABSTRACT Current methods of analyzing signals delivered from vibrations transducers are mainly based on realtime processing in frequency domain. After a shock measurement results a trace that contains a value of a shock’s parameter on the time axis, during its whole action. In earthquake engineering, the parameter usually defines acceleration. Often, this trace is by itself not directly useful for the physical phenomena interpretation and therefore it must be transformed. Regarding the analysis type, these transformations are based on the Fourier transform and on the digital filtering. The principle of the digital filtering method is based on applying the z-transform to the transfer function of a single degree of freedom system, for the relative displacement or for the absolute acceleration model. It is applied by studying the effects on structures when the shock acts like an excitation. To distinguish it from the Fourier spectrum, the digital filtering method is called also the shock response spectrum (SRS) or, simply, shock spectrum. The mathematical model for the SRS analysis is considered in this paper, for the particular case of using accelerograms as input signals. These are applied to a C++ written computer simulator which is able to adapt the damping coefficient of the system, the sampling rate of the digital filter, to plot the shock response spectrum as a function of the natural frequency and to generate the spectrum pattern for each particular accelerogram. The frequency contents of two major earthquakes from 1977 and 1986, with the epicenter in the Vrancea region, Romania, are also analyzed. In contrast with the elastic response spectrum method, where the damping ratio is typically set to 5%, the digital filtering method uses variable damping ratios in order to match the system equivalent viscous damping for a specific acceleration time-history. The digital filtering also has the advantage of a much lower natural frequency (or a higher natural period), for the same sampling rate, when compared to step-by-step procedures. By halving or doubling the sampling rate the filter will operate in a frequency domain one octave lower or upper, and because of its z-transform derivation, the digital filtering is of high practical interest for data acquisition systems with spectrum analyzers.

References [1] C.M. Harris and A.G. Piersol, Harris’ Shock and Vibration Handbook, McGraw-Hill, New York, 2002. [2] D.O. Smallwood, An Improved Recursive Formula for Calculating Shock Response Spectra. The Shock and Vibration Bull., 51 (2), 211-217, 1981. [3] D. Lungu et al, Frequency bandwidth of Vrancea earthquakes and the 1991 edition of seismic Code of Romania. Earthquake Engineering, Tenth World Conference, Balkema, Rotterdam, 5633-5638, 1992.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

752

Histride: an integrated software environment for dynamic structural identification Antonella Frigerio*, Elia Bon and Guido Mazzà† *

CESI S.p.A. via Rubattino, 54 20134 Milan, ITALY [email protected] †

CESI S.p.A. via Rubattino, 54 20134 Milan, ITALY [email protected] [email protected]

ABSTRACT The construction industry is increasingly involved in the survey of existing structures. This is due to limited resources for new construction, a greater environmental awareness and a cultural desire to maintain historic structures. Theoretical models of civil engineering structures are essential for the safety assessment and for the design of alterations. For older structures these models often do not exist. For newer structures “asbuilt” information often does not accurately reflect the work done on site. Given the shortcomings in design data and surveys as a basis for structural models, it is clear that better non-destructive methods are needed for creating reliable models. HISTRIDE greatly extends and simplifies a non-destructive identification approach based on dynamic structural response. Physical parameters of the Finite Element model of a structure are modified to reduce the error between experimental modal features and computed ones. HISTRIDE is a new software environment running on a desktop PC, developed by five European partners within the EC ESPRIT Programme. Main features are: • Data Acquisition Planning: the number of sensors and their locations on the structure are optimized to reduce the costs of testing and to increase data quality. • Unique Set of Updateable Parameters: stiffness, mass and damping parameters can be corrected during the identification process. • Extended Complex Approach: complex modal analysis is available to eliminate restriction on the damping model and to enhance modal matching reliability. • Up-to-date Signal Processing: the State Space Method has been applied and improved to extract modal features from experimental data in the case of forced, random, pull&release and impact tests. Within the frame of the public research funding for the Italian Electric Sector (named "Ricerca di Sistema"), CESI has recently improved the least squares algorithm of the Histride identification process to try to avoid divergent oscillations that move away from the structural response. The Coupled Local Minimizers (CLM) method has been implemented. This method is based on a cooperative search mechanism set up by performing simultaneously a number of local optimization runs, that are coupled by information exchange instead of running independently.

HISTRIDE will be presented with reference to a full scale validation test.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

753

Damage Detection by the Topology Design Formulation Using Modal Parameters Joong Seok Lee*, Jae Eun Kim† and Yoon Young Kim†† *

National Creative Research Initiatives Multiscale Design Center, School of Mechanical and Aerospace Engineering, Seoul National University Shinlim-Dong San 56-1, Kwanak-Gu, Seoul 151-742, Korea [email protected] †

Digital Storage Research Laboratory, LG Electronics Inc. Woomyeon-Dong 16, Seocho-Gu, Seoul 137-724, Korea [email protected]

†† National Creative Research Initiatives Multiscale Design Center, School of Mechanical and Aerospace Engineering, Seoul National University Shinlim-Dong San 56-1, Kwanak-Gu, Seoul 151-742, Korea [email protected]

ABSTRACT The feasibility of using the topology design method for structural damage identification was investigated. For efficient damage identification in a structure, we used not only resonances but also anti-resonances as the damage identifying modal parameters and set up a formulation suitable for damage identification. The use of anti-resonances as well as resonances made it possible to identify damage location more effectively because changes in mode shapes due to any damage can be indirectly represented by the changes in anti-resonances. Point-frequency response functions were used to facilitate the extraction of the anti-resonances of a damaged structure. A finite element model of an undamaged structure was assumed to be given. Since considerably many resonances and antiresonances were used for damage identification, they were stated as constraint equations in the present formulation. An explicit penalty function designed to suppress intermediate design variables was used as the objective function because it may be otherwise difficult to obtain clear black-andwhite images at the end of the identification process. When damage extent is not severe, it may be difficult to identify the damage location only with a single-stage application of the present topology optimization-based method. To overcome this difficulty, a multi-stage damage location identification scheme, which reduces the number of candidate damaged elements over several topology optimization stages, was proposed. With the multi-stage technique, relatively small-sized damages were identified accurately as compared with a single-stage application. Through numerical examples, the identification performance improvement by the additional consideration of anti-resonances was demonstrated and the effectiveness of the multi-stage identification technique was also confirmed.

References [1] M. P. Bendsøe and N. Kikuchi, Generating Optimal Topologies in Structural Design Using a Homogenization Method. Computer Methods in Applied Mechanics in Engineering, 71(2), 197224, 1988. [2] T. Uhl and K. Mendrok, Overview of modal model based damage detection methods. Proceedings of International Conference of Noise & Vibration Engineering, ISMA, 561-575, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

754

Experimental Validation of the Finite Element Modelling of Pinhão Bridge Filipe Magalhães1, Bruno Costa1, Álvaro Cunha1 and Elsa Caetano1 1

Faculty of Engineering of the University of Porto Rua Dr. Roberto Frias, 4200-465 Porto [email protected]

ABSTRACT The Pinhão Bridge is a Portuguese metallic roadway bridge constructed in 1906 over the Douro river, whose state of degradation requires the development of rehabilitation works (Figure 1). In this context, an ambient vibration test of the bridge was recently performed. The collected data was processed using the Peak-Picking method that allowed the identification of the most relevant natural frequencies (Figure 2) and mode shapes. The experimental identified modal parameters were used to improve the numerical model developed by the consultancy office responsible for the rehabilitation design. The updated finite element model provides very well correlated natural frequencies and vibration modes with regard to the corresponding experimental estimates (Figure 3). Amplitude 1E-01

ANPSD - Average of the vertical accelerations 2.771

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12.671 10.608 15.649 13.965

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Figure 1: View of Pinhão Bridge f = 2.771 Hz

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Figure 3: Experimental vs numerical vertical bending mode shapes (lateral view of the bridge)

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

755

Development of a Cabril Dam Finite Element Model for Dynamic Analysis Using Ambient Vibration Tests Results Sérgio Oliveira*, Paulo Mendes† *

Laboratório Nacional de Engenharia Civil Av. do Brasil, 101, 1700-066, Lisboa, Portugal [email protected] † Instituto Superior de Engenharia de Lisboa R. Conselheiro Emídio Navarro, 1, 1950-062, Lisboa, Portugal [email protected]

ABSTRACT This paper discusses the finite element development of a numerical model of 3D elements for Cabril dam, based on the main fundamental parameters of the dynamic response of the dam, obtained on several experimental results on ambient vibration tests campaigns. These experimental results are used to calibrate the numerical model of 3D finite elements considering two hypothesis to simulate the hydrodynamic water pressure: i) first assuming that the is properly simulated through associated water masses, in accordance with Westergaard’s formula, and ii) second considering water finite elements.

References [1] P. Mendes, Observação e Análise do Comportamento Dinâmico de Barragens de Betão sob Excitação Ambiente (in Portuguese). Master Thesis, Instituto Superior Técnico, Universidade Técnica de Lisboa, 2005. [2] S. Oliveira, J. Rodrigues, P. Mendes, A. Campos Costa. Damage Characterization in Concrete Dams Using Output Only Modal Analysis. Proceedings of XXII IMAC, 2004. [2] Y. Toyoda, M. Ueda, H. Shiojiri. Study of Joint Opening Effects on the Dynamic Response of an Existing Arch Dam. 15th ASCE Engineering Mechanics Conference, New York, USA, 2002.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

756

Application of EERA Method for Identification of Modal Parameters of a Simulated Aircraft Daniela Cristina Rebolho*, Luciane F. R. Souza*, Eduardo Morgado Belo* *

Engineering School of Sao Carlos – University of Sao Paulo Laboratory of Aeroelasticity, Flight Dynamics and Control

Av. Trabalhador Sancarlense 400, Cep 13566- 590, Sao Carlos, SP,Brazil. {danielar, lfrs, belo}@sc.usp.br

ABSTRACT System identification is generally the art of mathematical modeling, given input – output measurements from a dynamical system. The problem is of interest in a variety of applications, ranging from chemical process simulation and control to identification of vibrational modes in flexible structures. Some identification methods in the area of flight dynamics have been considered and verified, such as the methods of the error equation [1] and the error equation in the frequency domain [2] and other methods. Systematic modal identification methods such as the Eingensystem Realization Algorithm (ERA) have advanced the complexity of structures that can be modeled by experimental measurements. The extended Eigensystem Realization Algorithm (EERA) method is a modified form of an ERA and calculates the modal parameters by manipulating both input and output time histories. The block Hankel matrices are built directly from the system input and output time history database. The development of these subspace identification methods is motivated by difficulties in estimating modal parameters for multiple-input multiple-output vibratory systems. To verify the performance of the EERA algorithm were realized an assessment for the identification of modal parameters of a Simulated Aircraft. The A4-D aircraft has been simulated using Simulink of Matlab“ for some flight conditions such as varied altitude and Mach number. Amongst some models that describe the flight dynamics of an aircraft, the linear and non-linear mathematical model presented by [3] was adopted. Using the EERA method, the natural frequencies and damping factors of the implemented linear model were calculated from the simulation responses and were compared with the modal parameters obtained analytically. Then, using the simulation responses of the non linear model, the A4-D aircraft modal parameters were identified and compared with those of the linear model. The results obtained in the identification of the modal parameters of the non linear model did not present significant differences from the linear ones. This has shown that the EERA method is efficient enough in modal parameters estimation.

References [1] Iliff, K. W. and Wang, K. C. (1997): “Extration of Lateral-Directional Stability and Control Derivatives for the Basic F-18 Aircraft at High Angles of Attack”, Nasa Technical Memorandum 4786. [2] Schimidt, L.V., 1998, “Introduction to Aircraft Flight Dynamics”. American Institute of Aeronautics, Inc., Reston, Virginia. [3] Etkin, B. and Reid, D. R. (1996): “Dynamics of flight: stability and control”, – 3rd ed, John Wiley and Sons.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

757

Reference-based combined deterministic-stochastic subspace identification for experimental and operational modal analysis Edwin Reynders and Guido De Roeck K.U.Leuven, Department of Civil Engineering Kasteelpark Arenberg 40 3001 Leuven Belgium [email protected] ABSTRACT In this paper, the use of the combined deterministic-stochastic subspace identification algorithm for the experimental modal analysis of mechanical structures is discussed. The algorithm requires artificial forces to be applied to the structure, so it can be used for experimental modal analysis (EMA). The algorithm can also be used for operational modal analysis (OMA), since the excitation level of the artificial force(s) can be low compared to the excitation level of the ambient forces. Both the modes that are artificially excited and those that are excited by the ambient forces are identified. This type of operational modal analysis is called an OMAX analysis (Operational Modal Analysis with eXogenous inputs) [1]. The main advantages of OMAX over OMA are that the modes that are excited by the artificial forces can be scaled to unity modal mass and that a higher number of modes can be identified. An original contribution of the paper is that the time-domain combined deterministic-stochastic subspace (CSI) algorithm of [2] is extended to a reference-based version (CSI/ref), which makes use of a Kalman filter that is constructed with reference outputs only. Advantages of CSI/ref over CSI are a faster performance because of the data reduction and the possibility to lower the influence of noisy channels without loosing their useful information. The capabilities of both algorithms for the determination of the eigenfrequencies, mode shapes, modal damping factors and modal scaling factors of mechanical structures in both experimental and operational conditions are illustrated by simulated tests. With these simulations, it is demonstrated that both the CSI and the CSI/ref identification methods provide accurate estimates for the modal parameters of a structure, even if the measurement errors are relatively high compared to the amplitude of the signals. It is even possible to combine output channels in which only ambient excitation is present with output channels in which only artificial excitation is present, while still obtaining accurate modal parameters. It is also shown that, if the channels with the highest SNR are chosen to be the reference channels, the CSI/ref method gives the most accurate estimates for the modal parameters, because with these reference channels, the CSI/ref method gets rid of the noise faster than the CSI method and so it does not need high model orders to produce a clear stabilization diagram.

References [1] Cauberghe, B.: Applied Frequency-domain System Identification in the Field of experimental and operational modal Analysis. Ph.D. Thesis, Vrije Universiteit Brussel, 2004 [2] van Overschee, P. and De Moor, B.: Subspace identification for linear systems. Kluwer Academic Publishers, Boston, 1996

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

758

Numerical Modelling of Time-Dependent Behaviour of High Strength Concrete Beams Daniel Dias da Costa *, Vítor D. Silva *, and Eduardo N. B. S. Júlio * *

Department of Civil Engineering, Faculty of Science and Technology, University of Coimbra Pólo II - Pinhal de Marrocos 3030-290 Coimbra, Portugal [email protected], [email protected], and [email protected]

ABSTRACT Subjected to constant load, concrete exhibits an instantaneous deformation, designated elastic deformation, and a time-dependent deformation, commonly known as creep. Ever since this phenomenon has been identified, several theories and models have been proposed to explain and to simulate this behaviour. With recent developments in concrete mixing, namely with higher quality cements, pozzolanic additions and third generation admixtures, it became possible to produce concretes with high compressive strength showing, simultaneously, high workability. These new concretes present new possibilities and also new problems associated with these that have to be studied. For instance, very high strengths at very young ages allow the application of loads sooner. This represents a major advantage when prestress is used but it also becomes necessary to verify the applicability of old models in these new situations. With the objective of simulating time-dependent behaviour of innovative prestressed long span beams (40.0 m) precast with high strength concrete (120.0 MPa), several numerical codes, based on the finite element method, were considered. Cast3m open-code was selected and all available creep models were tested considering examples with known analytical results. Several limitations were registered bearing in mind the objective of simulating the creep behaviour of the long span beams: (a) most models do not consider a linear relation between stress and creep strain as observed for serviceability stresses and, therefore, are not applicable; (b) all applicable models are based in only one Kelvin element; and (c) all these models use an algorithm based only on a deviatoric stress tensor, resulting in an incorrect simulation of three-dimensional creep. For all these reasons, it was decided to implement an algorithm allowing the use of a Kelvin-chain with a chosen number of elements. This led to an adequate numerical simulation of uniaxial creep tests. Subsequently, the implemented algorithm was used to simulate the time-dependent deformation of the long span beams. From the comparison between numerical and experimental results, namely displacements and curvatures at different cross-sections, it was concluded that the adopted algorithm can be used to model creep of this new type of prestressed long span beams precast with high strength concrete.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

759

Behaviour of composite steel-concrete beams with longitudinal and transverse partial interaction in fire G. Ranzi1, M.A. Bradford2, and P. Ansourian1 1 The University of Sydney Department of Civil Engineering, The University of Sydney, NSW2006, Australia [email protected], [email protected] 2 The University of New South Wales School of Civil and Environmental Engineering, UNSW, Sydney, NSW2052, Australia [email protected]

ABSTRACT The use of composite construction is widely favoured in engineering applications, where the composite action is provided by means of different types of locking devices or mechanical connectors. In the case of composite steel-concrete beams, the kinematical implications of the deformability of the shear connectors become significant and an adequate model needs to account for the displacement discontinuities at the interface between components. Both longitudinal and transverse deformations can occur between the slab and the steel joist connection, and these are usually referred to as longitudinal and transverse partial interaction. This paper presents a novel analytical model which accounts for both longitudinal and transverse partial interaction in the analysis of steel-concrete composite beams subjected to fire loading. In this instance, the thermal distribution in the steel joist and in the concrete slab is likely to be quite different despite the similarities in their thermal expansion coefficients. The proposed model is well suited to depict the complex stress and deformation state of the connectors, as vertical separation between the slab and the steel joist is likely to occur, and significantly affect the structural response. The analytical model is derived using the principle of virtual work and, for completeness, both strong and weak formulations are presented. All materials are assumed to behave linearly while the analysis accounts for the degradation of the elastic moduli. A finite element formulation is then derived to model the composite behaviour.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

760

Partial interaction analysis of multi-layered composite beams accounting for time effects G. Ranzi*, P. Ansourian*, L. Dezi†, S. Zhang# * The University of Sydney Department of Civil Engineering, The University of Sydney, NSW2006, Australia [email protected], [email protected] † Universita’ Politecnica delle Marche Department of Architecture, Construction and Structures, Universita’ Politecnica delle Marche, Via Brecce Bianche, 60131, Ancona, Italy [email protected] # Harbin Institute of Technology Department of Civil Engineering Harbin Institute of Technology, 92 West Dazhi Street, Harbin 150001, China [email protected]

ABSTRACT Innovative structural solutions have been achieved in the fields of structural and mechanical engineering by combining different materials to produce more economical and efficient elements. The interaction between different materials has been achieved in various manners and this paper is concerned with those situations in which layers of different materials are interconnected by means of flexible mechanical devices. Of particular interest to this paper are composite members with partial shear interaction which have been studied for several decades. One of the first papers dealing with the partial interaction analysis of two-layered composites was the one by Newmark et al. [1] who focussed their attention on steel-concrete composite beams. Due to the wide acceptance of this work, its formulation is simply referred to in the literature as Newmark model. This paper extends the applicability of this model to study the time-dependent behaviour of multi-layered composite beams with partial shear interaction formed by n layers. A generic displacement-based finite element formulation is presented for the derivation of n-layered beams and is then applied to the case of a three-layered element representing the particular case of a composite steel-concrete beam stiffened by a longitudinal steel plate in which the partial interaction occurs between the slab and the steel joist as well as between the joist and the stiffening plate. The accuracy of the proposed procedure is validated against closed form solutions for the two limiting cases in which both shear connection stiffnesses tend to infinity, representing the full interaction condition; and also where only one connection stiffness is infinitely high, thus degenerating to the conventional two-layered composite partial interaction behaviour. Applications are then presented to investigate the effects of the two interface connection stiffnesses on the structural response of the stiffened composite beam. For this purpose, different lengths are considered for the longitudinal plate and the time-dependent behaviour of the concrete is modelled by means of the step-by-step method.

References [1] N.M. Newmark, C.P. Siess, I.M. Viest, Test and analysis of composite beams with incomplete interaction. Proceedings of the Society of Experimental Stress Analysis 1951; 9(1): 75-92.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

761

Numerical Behaviour of a Steel Sub-frame System in Fire Aldina Santiago*, Luís Simões da Silva* and Paulo Vila Real† *

Department of Civil Engineering University of Coimbra, Portugal [email protected]; [email protected] †

Department of Civil Engineering University of Aveiro, Portugal [email protected]

ABSTRACT Steel framed buildings are generally designed with “simple” shear-resisting connections, and lateral forces are resisted by vertical bracing and shear walls. However, when a beam form part of a complete structure, its behaviour is complex and very much dependant on the restraint at the member ends. In order to capture the beam behaviour during a fire, it is therefore required to take a wonder view and examine the behaviour of a representative sub-structure [1]. Using the data and results from the few available sub-structure fire tests, numerical models were developed and validated to try to explain and predict the behaviour of such systems. This paper presents a numerical parametric study by which the behaviour of a system consisting of a pair of fire protected steel columns and an exposed steel beam restrained between the columns, can be simulated. The structural sub-frame is modelled by a sophisticated 3D shell elements, thereby enabling i) the identification of the local failure modes, and ii) the behaviour of the beam-column system during the heating and cooling phases. Due to the complex behaviour of a restrained beam under fire, this model allows for the inclusion of initial geometrical imperfections, non-linear temperature gradient over the cross-section, geometrical and material nonlinearity and temperature dependent material properties. After validation of the LUSAS model against a specialised fire dedicated program – SAFIR, factors as well as beam span/depth ratio, lateral restraint, gradient temperature within the cross-section and mechanical load level are investigated. It is concluded that the local buckling of the beam was one of the main failure mechanisms. It is observed in the bottom beam flange and web adjacent to the connection. On beams with lower span and beams with high loading, shear buckling on the web is also observed. The detailed observation of the internal stresses and strains throughout the analysis shows that: 1) At low temperatures, when the beam behaviour is controlled by bending, a longer beam span gives larger beam deflection, but at high temperatures, when catenary action is in control of the beam behaviour, the beam deflection is mainly dependent of the applied load. 2) The temperature distribution only has minor effect on the beam mid-span deflection. However, some differences are observed on the internal stresses, especially during the cooling phase. 3) All the study cases show that, at high temperatures, the bending moment values are equal to those observed at room temperature. 4) The parameters studied on this paper only have minor effect on the maximum tensile force, after the fire (around 700 kN).

References [1] L. Simões da Silva, A. Santiago, D. Moore and P. Vila Real, Behaviour of steel joints and fire loading. International Journal of Steel and Composite Structures 5(6), pp. 485-513, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

762

Structure Design and Dynamic Analysis of Vehicle using Metamodeling and Optimization Techniques José A. F. Borges1, Marcus F. Leal2, Rômulo R. P. Filho1 and Jean C. C. Rezende1 1 Federal University of Uberlândia Av. João Naves de Ávila 2160 - Campus Santa Mônica – 38400 902 – Uberlândia MG – Brazil [email protected] [email protected] [email protected] 2 Embraer – Empresa Brasileira de Aeronáutica S. A. Av. Brigadeiro Faria Lima 2170 – São José dos Campos SP – Brazil [email protected]

ABSTRACT The design and dynamic analysis of vehicles have been widely improved through numerical simulation techniques, mainly related to the several analysis possibilities applied to complex and representative models. The automotive industry has already used the numerical optimization techniques in product development. In this paper an optimization technique is applied to a full vehicle model, using several modeling and simulation tools, besides experimental tests that have been used in order to get physical properties and validation data. The methodology consists in modeling an offroad reference vehicle chassis using software based on the finite element method. The computational models are built taking into account different approaches, shell and beam elements. A series of experimental tests including modal analysis, bending and torsion measurements are conducted to obtain information for modeling and validation. The validated chassis models allow the shape and dimensional optimization of the structure looking for higher stiffness and maintenance of the total frame mass. The definition of a better vehicle chassis shape leads to the suspension systems characterization. These steps involves the definition of a high fidelity multi-body model to the reference vehicle, vertical dynamic behavior through experimental measurements, stiffness properties for springs, tires and bushings. Inertia properties for the chassis, axles, engine and body are obtained through an experimental procedure. The analysis are performed considering different situations, such as road irregularities and changes in speed, tire pressure and loading. Once defined the validated multi-body model, statistical techniques known as design of experiments are applied to efficiently use the numerical analysis data in order to improve the suspension system configuration. The computational experiments provide the necessary data to build predictive models for the suspension behavior using the response surface method. Once these predictive metamodels are available, an automatic numerical optimization algorithm is used to improve these responses for a given operating condition.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

763

Race driver model: identification of the driver’s inputs F.Braghin1, F. Cheli1, E. Sabbioni1 1

Politecnico di Milano, Department of Mechanics, Via La Masa, 34 20158 Milan, Italy [email protected]

ABSTRACT Once the best trajectory has been determined (both in terms of best track and best speed profile), it is necessary to identify the driver’s inputs to follow the given trajectory. The driver’s inputs are the following: the gear, the clutch, the brake, the accelerator and the steer wheel. In order to simplify this identification problem, it was decided to implement an automatic gear shift mechanism: during acceleration/braking manoeuvres the gear is incremented (in 0.1s) when the engine speed reaches a value related to the characteristic curve of the engine. Accelerator and brake inputs, as well as steer wheel inputs, are instead determined using two PID controllers. The accelerator/brake controller, determines the driver’s inputs on the base of the actual reference speed as well as the reference speed that will be required some distance in advance. This “visual distance” is a function of the vehicle speed (an of the vehicle acceleration): the higher the speed the longer the distance. To improve the promptness, a feed-forward contribution based on a simplified vehicle model is also added. Also the steer wheel controller determines the steer wheel angle adding two contributions: a feed-back PID contribution and a feed-forward contribution. For what concerns the feed-back contribution not only the position of the cog of the vehicle with respect to the ideal trajectory is taken into account but also the orientation of the vehicle with respect to the ideal track tangent. The proposed race driver model has been compared to IPG Driver model on several circuits. The lap time reduction achieved about 10% on any circuit.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

764

Race driver model: trajectory planning F.Braghin1, F. Cheli1, S. Melzi1 1

Politecnico di Milano, Department of Mechanics, Via La Masa, 34 20158 Milan, Italy [email protected]

ABSTRACT The best race driver is the one that, with a given vehicle, is able to drive on a given track in the shortest possible time. Thus, the only target is the lap time. A race driver model has to do the same. The first step towards this target is to decide which trajectory to follow. In fact, the optimal trajectory is the best compromise between the shortest track and the track that allows to achieve the highest speeds (least curvature track). Thus, the problem of trajectory planning is a bounded optimisation problem that has to take into account not only the geometry of the circuit but also the dynamics of the vehicle. A simplified vehicle dynamic model is used for this purpose. Due to the fact that the vehicle will be driven at its limit performances, although simplified the model has to correctly reproduce the maximum possible acceleration, a function of the vehicle speed, the maximum possible deceleration, again a function of the vehicle speed, and the maximum lateral acceleration, a function of both the vehicle speed and the steering angle. Knowing the trajectory, the vehicle model allows to determine the lap time. Through an optimisation algorithm it is therefore possible to determine the best compromise between shortest track and track with the minimum curvature, i.e. the trajectory (in terms of track and speed profile) that allows to minimise the time lap. It is important to point out that the model is able to consider the whole track and not only a portion of it thus allowing to determine the overall best solution.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

765

A Simplified ABS Numerical Model for Actively Controlled Vehicle Dynamic Simulations: Validation with Experimental Data F. Cheli1, E. Giangiulio1, E. Sabbioni1, and A. Concas1 1

Mechanical Engineering Department, Politecnico di Milano Via La Masa 34, 20158 Milano, Italy [email protected] [email protected] [email protected] [email protected]

ABSTRACT Antilock Braking System (ABS) is nowadays a standard equipment for passenger cars. It increases vehicle safety preventing wheels from being locked up in emergency braking, especially on low friction road surfaces, allowing the driver to maintain steering control of the vehicle, to avoid obstacles and reducing vehicle stopping distance. Thus, in order to simulate the vehicle behaviour during braking and to compare the obtained results with the experimental data, is essential to model the ABS system. A flexible mechatronic test bench for ABS Electronic Control Unit (ECU) performance evaluation, based on Hardware-In-the-Loop Simulation (HILS) technique, has been developed. It consists of a passenger car hydraulic braking system (from master cylinder to brake callipers), an ABS ECU integrated with pressure control valves and a flexible real-time platform which simulates vehicle dynamics. A fourteen degrees of freedom optimized vehicle model has been developed in order to reproduce with sufficient accuracy the vehicle dynamics. An ABS model has also been implemented, trying to reproduce the real control system behaviour on the basis of the performed HILS simulations. Besides the well known control algorithms for modulating the braking pressure at each wheel in the cases of high and low adhesion coefficients, in fact, only few information on the other parts of the logic can be found in the literature. Through HILS simulations, the ABS model could be completed. In particular, two strategies to recognize and adapt the control cycle to the jump reductions and increases of the adhesion coefficient between tires and road (Pjump conditions) have been implemented. A control cycle has also been developed to prevent the excessive rising of the vehicle yaw moment in P-split conditions (different adherence coefficient for the left and the right wheels) or during braking when cornering. The ABS model has been integrated within the 14 d.o.f. vehicle model in order to perform limit braking simulations both in straight line and during cornering on different road surfaces (dry and wet asphalt, ice and snow). Simulations results have been compared with the experimental data collected on an instrumented passenger car.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

766

Aerodynamic Sensitivity Analysis Of The New EMUV250 Train To Cross Wind By Wind Tunnel Test And CFD Analysis F. Cheli*, G. Diana*, F. Ripamonti *, G. Tomasini *, G. Zanetti † * Mechanical Department, Politecnico di Milano Via La Masa, 34 20158 - MILANO [email protected] †

AnsaldoBreda Via Ciliegiole [email protected]

ABSTRACT When dealing with high-speed trains, the effects of the aerodynamic forces acting on the vehicle as a consequence of cross wind are one of the most critical problems connected with running safety. Moreover this theme is very actually since a TSI, covering this specific problem, is under definition. As a consequence, also in the design of a new train become necessary an aerodynamic study for the determination of the vehicle stability, when it is subjected to cross wind action. In this work, a sensitivity analysis carried out on the new train AnsaldoBreda EMUV250 for the minimisation of the aerodynamic forces acting on railway vehicle subjected to crosswind will be presented. The research consists in evaluating the aerodynamic forces associated to cross-wind on first vehicle of train, by means of wind tunnel tests on 1:10 scale model and CFD simulations, for different aerodynamic layouts of the vehicle. The experiments have been carried out in the Politecnico di Milano Wind Tunnel. Different releases of the first vehicle have been tested in the wind tunnel and the effects of some aerodynamic elements have been studied. The cross wind aerodynamic behaviour of the train has been analysed in terms of Critical Wind Curves, that represent the limit wind speeds that lead the railway vehicle to safety limit conditions. The Critical Wind Speed have been evaluated by the numerical-experimental methodology developed, during this last years, at Mechanical Department of Politecnico di Milano. The final objective is to evaluate the Critical Wind Curves of train in the different analysed configurations in order to find the better layout in terms of aerodynamic requirements. At the same time, CFD numerical analysis have been carried out in order to understand the effects of some aerodynamic parameters on wake structure and, in general, on flow field around the vehicle. Different train layouts have been reproduced adopting a commercial code (Fluent). 2D simulations have been developed to set up the correct boundary conditions and the simulations parameters, highlighting the running configurations dependence of the global flow field (viaduct or embankment, in upwind and downwind arrangement,…). 2D simulation results have shown the most critical aspects of the cross wind aerodynamics and allowed the preliminary insight on the car body shape and the correct set up of the more demanding 3D simulations. This analysis succeeded in reproducing the three-dimensional flow effects due to the loco front shape for different angles of attack.

References [1] Cheli F., Corradi R., Diana G., Tomasini G. A Numerical-Experimental Approach to Evaluate the Aerodynamic Effects on Rail Vehicle Dynamics. 18th IAVSD; Atsugi, Japan, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

767

Three-dimensional Large Deformation Finite Element Analysis of Belt Drives Kari Dufva1, Kimmo Kerkkänen1, Luis Maqueda2 and Ahmed A. Shabana2 1

Department of Mechanical Engineering Lappeenranta University of Technology Skinnarilankatu 34 FIN-53850 Lappeenranta, Finland [email protected], [email protected] 2

Department of Mechanical Engineering University of Illinois at Chicago 842 West Taylor Street, Chicago Illinois 6067-7022 [email protected], [email protected]

ABSTRACT In this paper, methods for the large deformation finite element analysis of belt drives are presented. The new nonlinear dynamic formulations for belt drives are based on the three-dimensional large deformation absolute nodal coordinate formulation. Two different belt drive models that have different numbers of degrees of freedom and different modes of deformation are presented. Both three-dimensional finite elements are based on a nonlinear elasticity theory that accounts for geometric nonlinearities due to large deformation and rotations. In the first model, a thin plate element that is based on the Kirchhoff plate assumptions and captures both membrane and bending stiffness effects is used. In the second model, a cable element obtained from a more general threedimensional beam element by eliminating degrees of freedom which are not significant in some cable and belt applications is used. Both finite elements used in this investigation allow for systematic inclusion or exclusion of the bending stiffness, thereby enabling one to systematically examine the effect of bending on the nonlinear dynamics of belt drives. The finite element formulations developed in this paper are implemented in a general purpose three-dimensional flexible multi-body algorithm that allows for developing more detailed models of mechanical systems that include belt drives subject to general loading conditions, nonlinear algebraic constraints, and arbitrary large displacements. The plate formulation also allows using a surface distribution of the contact forces; such a distribution can not be obtained using beam elements since this element is represented by its centerline. Contact forces on the surface are compared to analytical results of similar but twodimensional model. The friction and normal force distributions are in agreement with analytical models. Some differences in results between the plate, cable and analytical formulations are obtained and discussed.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

768

Multibody and Finite Element Models for the Design of Motorcyclist’s Roadside Protections Ana P.C. Freitas*, Rui F. M. Silva*, João M. P. Dias* *IDMEC - Instituto de Mecânica - Pólo IST Instituto Superior Técnico, Technical University of Lisbon, 1049-001 Lisbon, Portugal [email protected]; [email protected]; [email protected]

ABSTRACT In Europe, road accidents represent one of major causes of death. Huge number of fatalities happens in two wheel motor vehicles, mainly in those due to crashes against guardrails and others safety structures and obstacles, which represents besides the social problem, an economical issue associated with this type of fatalities that causes a spending of millions of euros by the governments. Because of the main point of reducing in 50% the road accidents fatalities in European Union, it becomes fundamental to develop protections to avoid the major injuries and to increase the protection of the motorcyclists. This development can be taken by using commercial software like Madymo [1] in case of multibody systems [2] or LS-Dyna to finite elements studies. In this article, different kind of road side protections are modeled and analyzed, such as the conventional steel system, “Ecran Motard”, concrete barriers or usual guardrails posts protections [3,4] and they are tested to ensure the safety conditions of each one. In order to determine the injuries risk, biomechanical parameters (such as HIC – Head Injuries Criteria, chest and pelvis accelerations, etc) [5] are calculated and compared with the limit values. The main target of this study is to determine the best geometry to ensure the best protection of motorcyclists in the cases of collisions against roadside protection systems in typical crash characteristics.

References [1] TNO, “MADYMO Theory Manual Version 6.2”, TNO Automotive, Delft, The Nederlands, 2004; [2] Berg F. et all, “Motorcycle Impacts Into Roadside Barriers – Real-World Accident Studies, Crash Tests and Simulations Carried Out in Germany and Australia, DEKRA; [3] Blige, R., “Finite Element Analysis of Roadside Safety Devices”, Project Summary Report 01816-S, Texas Transportation Institute, USA [4] Duncan C. et all, “Motorcycle and Safety Barrier – Crash Testing: Feasibility Study”, Departement of Transport and Regional Services Australian Transport Safety Bureau; [5] Seiffert, U. et all, “Automotive Safety Handbook”, Professional Engineering Publishing, USA, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

769

An Integrated Educational Tool for Vehicle Dynamical Response Studies Jánes Landre Jr., Leonardo J.M. Saturnino, Marcelo Becker, Lúcio F.S. Patrício, and Clovis S. Barcellos Pontifical Catholic University of Minas Gerais – PUC Minas Av. Dom José Gaspar 500 – Belo Horizonte – MG, 30535-610, Brazil {janes, marcelo.becker, clovis}@pucminas.br, [email protected], and [email protected]

ABSTRACT The automotive industry is today highly competitive, globalized, and characterized by continuous efforts to improve its products quality and reduce the products costs and time-to-market. Solutions to this paradox do not permit trial-and-error, necessitating instead the adoption of a more complex developmental paradigm. In this scenario the term “road-to-lab-to-math” describes the effort to reduce the quantity of on-road testing and replaces it with laboratory testing components and subsystems, and to so efficiently by using complex mathematical models that make evaluation of inuse conditions more precise and realistic. Due to this, Engineering Schools are turning attention to the use of simulation tools in the undergraduate courses. There are several commercial software packages developed to support the “road-to-lab-to-math”. If we focus on virtual prototyping tools applied to vehicle dynamic responses, all of the commercial simulation packages implement multi-body models composed of both rigid and flexible parts. It is possible to find on market several examples of these packages, such as ADAMS (MSC - USA), AutoSim (Mechanical Simulation Corp. - USA), DynaFlex (Waterloo - Canada), MECANO (Samtech - Belgium), RecurDym (Function Bay Inc. - Korea) and many others. Frequently these packages use a multi-body system approach to obtain the vehicle dynamic responses during maneuvers. Most of the commercial software packages are prohibitively expensive for mechanical engineering schools and students to buy them. In addition to this, usually these software packages do not have tools to permit more complex analysis, using for example Finite Element Modeling (FEM) and Operating Deflection Shapes (ODS). This forces the users to acquire other software packages to complete the dynamical response study. This paper presents the development and implementation in MatLab of an educational tool developed to help mechanical engineering students to understand and visualize the vehicle chassis vibration under given operating conditions (for example, during ride analysis). It is constituted of two integrated parts: the first one, a multi-body-based handling, ride, and comfort analysis toolbox called as MDV, and the second one, a CAD, ODS, and FEM analysis toolbox called as ADES.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Robust Methods for Detecting Defects in Overhead Contact Line Based on Simulation Results J.P. Massat*†, A. Bobillot*, J.P. Laine† *

SNCF Innovation and Research Department 45, rue de Londres – 75008 Paris - France [email protected] [email protected] †

Ecole Centrale de Lyon 36, Av de Collongue 69134 Ecully Cedex - France [email protected]

ABSTRACT Today, catenary incidents represent one of the principal causes of disturbance of railway traffic: throughout Europe, more than one million minutes of delay are due to overhead line incidents. In a near future, SNCF as a high-speed train operator, targets commercial speed up to 350 Km/h. The question of current collection thus requires a thorough study of the overhead line design and an evolution of the existing infrastructure maintenance strategies. Two aspects of the study are presented in this paper. The first concerns the pantograph-catenary interaction simulation and the second presents methods used to detect defects. Two numerical models allow simulating the catenary-pantograph dynamic behaviour, which consists in a flexible non-linear structure coupled with a moving mechanical system. The first one is a simplified model based on displacement decomposition in sine series, which provides good correlation with test data. Nevertheless, the diversity and complexity of the three-dimensional overhead lines geometries requires the use of the Finite Element method, through a software named OSCAR, which offers greater flexibility. This choice presents some drawbacks due to element passage. In this paper, some solutions allowing reducing discretisation effects on the contact force results are proposed. Moreover, the effects of deteriorated conditions on the pantograph-catenary behaviour are modelled and studied. The main catenary defects (missing droppers, geometry defects…) are included in the simulation models. Localised defects, such as junction claws, are also studied. Finally, signatures of these typical defects are obtained in order to perform diagnostic activity using a measurement train. The second part of this paper proposes methods for automatic and systematic defect detection using an instrumented pantograph, which are developed and validated using simulation results as test cases. The great amount of measured data requires appropriate data analysis in order to perform defect recognition. Because of the lack of measurement reproducibility, the defect detection methods applied must be particularly robust and reliable. In the future, the comparison of these measurements with “nominal” conditions will help to detect the emergence of a defect or malfunction. The inclusion of deteriorated conditions effects into the simulation tool shows the relevance of the simulation. Indeed, the simulation tool thus provides the measurement systems with singularity and defect signatures, allowing them to detect and identify defects and thus to become expert system. This tool could also include the meteorological conditions to allow defect detection in perturbed situations.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

771

Mode Decoupling Vehicle Suspension System Applied to a Race Car Basileios Mavroudakis, Peter Eberhard Institute of Engineering and Computational Mechanics, University of Stuttgart Pfaffenwaldring 9, 70569 Stuttgart, Germany {mavroudakis,eberhard}@itm.uni-stuttgart.de

ABSTRACT A suspension system, consisting of hydraulic rams and a central unit, enabling decoupling of the four vehicle body modes, namely bounce, roll, pitch and warp, is investigated. Allowing independent tune of each mode, this system enables the engineer to enhance the performance envelope of a vehicle by exploiting the full potential of the vehicle configuration and achieves the optimal compromise between ride and handling qualities. Attempts to partially decouple the body modes have already been in use, e.g. anti-roll bars, while suspension systems that fully decouple the four body modes have appeared in experimental state [1]. A main objective of this research project is the definition of the benefits such a system can have when applied to modern track racers by addressing the extremely challenging requirements of suspension design while satisfying motorsport regulations being passive in function. In order to verify the performance gains, a modern Formula 1 race car is modelled as a multibody system. Two variations of the model are investigated, a conventional one featuring a state of the art suspension and one using the proposed mode decoupling system. Both variations feature same tires and take advantage of the ample downforce courtesy of advanced aerodynamics applied in motorsport. They are subjected to identical handling tests and their performance is validated, showing a definite advantage of the mode decoupling system which is able to better exploit the available tire grip, thus enabling higher velocities in critical conditions. Furthermore, proper tuning of the warp mode enables the decoupling suspension system to handle extreme situations such as riding over a kerb (e.g. through a chicane) or passing through track points of uneven ground (e.g. entering or exiting significantly banked curves). The former case is presented here with both vehicles crossing a chicane, simulated as a double lane change manoeuvre, at 290km/h, when the inner front wheel rides over a kerb of 75mm height (sinusoidal road irregularity of 1m length). The investigated system (blue line) presents superior behaviour (low yaw and ride height perturbations) while it should be noted that in case of a kerb 100mm high, the conventionally suspended vehicle model (red line) is unable to complete the test

References: [1] Integral Suspension System for Motor Vehicles Based on Passive Components, Josep Fontdecaba - CREUAT S.L., SAE Technical Papers, Document Number: 2002-01-3105

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

772

Linearized equations for an extended bicycle model J. P. Meijaard∗ , A. L. Schwabℵ ∗

School of Mechanical, Materials and Manufacturing Engineering The University of Nottingham University Park, Nottingham NG7 2RD, United Kingdom [email protected] ℵ Laboratory

for Engineering Mechanics Delft University of Technology Mekelweg 2, NL-2628 CD Delft, The Netherlands ABSTRACT The linearized equations of motion for a bicycle of the usual construction travelling straight ahead on a level surface have been the subject of several previous studies [1, 2]. In the simplest models, the pure-rolling conditions of the knife-edge wheels are introduced as non-holonomic constraints and the rider is assumed to be rigidly attached to the rear frame. There are two degrees of freedom for the lateral motion, the lean angle of the rear frame and the steering angle. In the present paper, the model is extended in several ways, while the simplicity of having only two degrees of freedom is retained. The extensions of the model comprise the shape of the tires, which are allowed to have a finite transverse radius of curvature, the effect of a pneumatic trail and a damping term due to normal spin at the tire contact patch, the gradient of the road, the inclusion of driving and braking torques at the wheels and the aerodynamic drag at the rear frame. Owing to the gradient, the yaw angle of the rear frame is no longer a cyclic coordinate and the kinematic differential equation for its evolution needs to be included. A further consequence is that the stiffness matrix is no longer symmetric, even for zero speed and acceleration. The way of decelerating has a marked influence on the stability characteristics: braking at the rear wheel, braking at the front wheel, aerodynamic drag and riding up an incline influence the lateral dynamics in different ways. The acceleration makes the coefficients of the linearized system time-varying. A comparison of the derived equations and the results obtained by a multibody dynamic program is made, which shows a complete agreement. The equations can be used for several purposes: firstly, they provide a non-trivial example of a non-holonomic system that can be used to illustrate some of the characteristic properties of systems of this kind; secondly, they can be used as a test problem for the verification of multibody dynamic codes; thirdly, the simple model already yields valuable insight in the effects of several system parameters on the dynamics of a bicycle.

References [1] F. J. W. Whipple, The stability of the motion of a bicycle. The Quarterly Journal of Pure and Applied Mathematics, 30, 312–348, Plate, 1899. [2] A. L. Schwab, J. P. Meijaard and J. M. Papadopoulos, Benchmark results on the linearized equations of motion of an uncontrolled bicycle. KSME International Journal of Mechanical Science and Technology, 19, 292–304, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

773

A New Vehicle 3D Model with 7 Degrees of Freedom for Vehicle Dynamical Response Studies Lúcio F.S. Patrício, Marcelo Becker, Jánes Landre Jr., and Clovis S. Barcellos Pontifical Catholic University of Minas Gerais – PUC Minas Av. Dom José Gaspar 500 – Belo Horizonte – MG, Brazil [email protected] and {janes, marcelo.becker, clovis}@pucminas.br

ABSTRACT Last decades offered significant technologic achievements regarding vehicle power, speed, and complexity. The full understanding of vehicle characteristics and their influence on vehicle response behavior during maneuvers became essential for vehicle quality as well as passenger safeness and comfort. On the other hand, due to high competitive and globalized economy, automotive industries underline the necessity of reducing costs and time-to-market while improving the quality of their products. A critical part in the vehicle design process that usually requires specialist knowledge and time consuming work is suspension adjustment. In this scenario an obvious alternative for evaluating dynamically new vehicle projects is the use of mathematical models and simulation tools in order to reduce the need of vehicle prototypes and experimental tests. There are several commercial software packages developed to support the “road-to-lab-to-math”. It is possible to find on market several examples of these packages, such as ADAMS (MSC - USA), AutoSim (Mechanical Simulation Corp. - USA), DynaFlex (Waterloo - Canada), MECANO (Samtech - Belgium), RecurDym (Function Bay Inc. - Korea) and many others. Unfortunately most of these software packages are not flexible enough to satisfy the engineers’ needs and still require a strong knowledge about vehicle dynamics. Due to this, Engineering Schools are turning their attention to the development of simulation tools that are able to provide a link between academy knowledge and industry necessities, preparing engineering students to become automotive engineers. This paper presents a new 3D 7-DOF vehicle model that takes into account the anti-roll bar when it comes to modeling the vehicle suspension. We applied the d’Alembert approach to obtain the model equations. In order to implement the model, we used the MatLab software and its ode45 function for solving the differential equation system. The model developed succeeded in providing time responses, which are coherent with the ones offered by literature. This model is also in accordance with assumptions made by engineers who are involved with vehicle tests in the automotive industry. Regarding the analysis of the suspension system interaction – with and without the anti-roll bar and its influence in the body response – it is positively in accordance with the hypothesis raised. A new version of this software is being designed to add optimization and sensitive analysis tools.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

774

A Hertzian Contact Formulation for the Wheel–Rail Contact Problem in Railway Dynamics João C. Pombo1, Jorge A. C. Ambrósio2 1

Escola Superior de Tecnologia de Abrantes – Instituto Politécnico de Tomar Rua 17 de Agosto de 1808, 2200-370 Abrantes, Portugal [email protected] 2

IDMEC – Instituto Superior Técnico, Technical University of Lisbon Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal [email protected]

ABSTRACT The main difference between railway and other vehicles is that they move on tracks. In conventional rail vehicles, the wheels, assembled in a wheelset, are not free to rotate independently. Hence, their treads are profiled to allow them to negotiate curves without slipping. The accurate prediction of the contact points location between wheel and rail surfaces is fundamental for the accuracy of the dynamic analysis results. A generic contact detection formulation is proposed here in order to determine, online during the dynamic simulation, the coordinates of the multiple contact points between each wheel and rail even for the most general three dimensional motion of the wheelset with respect to the track. The formulation allows the detection of flange contact with lead and lag contact configurations, which are fundamental to evaluate the risk of derailments, to study switches or to investigate the curving performance when dealing with high angles of attack. This methodology is used in conjunction with a general geometric description of the track, which includes the representation of the rails spatial geometry and irregularities. Due to the efficient parameterization, this formulation can be used for any kind of wheel or rail profiles and still be computationally efficient, without requiring the use of lookup tables. The movement of the wheelsets along the rails is characterized by a complex contact with relative motions on the longitudinal and lateral directions and relative rotations of the wheels with respect to the rails. These motions generate tangential creep forces and moments at the wheel-rail interface, especially when the wheelset departs from the center of the tangent track. One of the difficulties of the wheel-rail contact problem arises when the contact point is located in the concave region between the wheel tread and the wheel flange. In this work a Hertzian contact formulation is developed to calculate the contact forces that develop in the wheel-rail interface. The discussion on the benefices and drawbacks of these methodologies is supported by an application to the dynamic analysis of the railway vehicle ML95, which is used by the Lisbon subway company.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

775

Multibody Models for Vehicle Accident Reconstruction Ricardo J.F. Portal*, João M.P. Dias* *IDMEC – Institute of Mechanical Engineering – Pólo IST Instituto Superior Técnico, Technical University of Lisbon Av. Rovisco Pais, 1049-001 Lisbon, Portugal e-mail: [email protected], [email protected] web: http://www.dem.ist.utl.pt/acidentes

ABSTRACT Simplified multibody models can be used to reconstruct accidents involving complex dynamics, particularly, in the first stages of accident investigation, accidents involving motorcycles and pedestrians [1, 2]. In this work, vehicle models based on multibody dynamics formulations [3] are developed to be used in motorcycle accident reconstruction. The models developed are a four-wheel vehicle, a motorcycle and a human biomechanical model. The biomechanical model could be used as a vehicle driver, a pedestrian or a motorcycle rider. The characteristics of the models could be adapted to match real accident conditions, so that the majority of the daily traffic accidents could be investigated. The accidents involving motorcycles are difficult to investigate due to its complex 3D dynamic motion. The motorcycle model developed is based on the multibody motorcycle models proposed by Huyge [2] and Cossalter [4]. To investigate road accidents, contact-impact models are included in the formulations and methodologies. The developed methodology uses the Hertzian contact model [5], of continuous analysis, once it allows obtaining nonlinear forces considering energy dissipation. Examples for different collision scenarios are presented and the benefits of the proposed approach are discussed.

References [1] P. Lima, R. Portal and J. Dias, Modelos Biomecânicos Simplificados Baseados na Dinâmica de Sistemas de Corpos Múltiplos para a Reconstituição de Acidentes com Atropelamento. Congreso de Métodos Numéricos en Ingeniería. Escuela de Ingenieros de Caminos, Canales y Puertos de la Universidad de Granada, Spain, 2005. [2] K. Huyge, Multibody Motorcycle, Modeling and Control, Master Thesis, Instituto Superior Técnico, Technical University of Lisbon, 2005. [3] Nikravesh, P.E., Computer Aided Analysis of Mechanical Systems, Prentice-Hall, Englewood Cliffs, NJ, 1988. [4] V. Cossalter and R. Lot, A motorcycle Multi-body Model for Real Time Simulations Based on the Natural Coordinates Approach, Vehicle System Dynamics, 37(6), 423-447, 2002. [5] Lankarani, H., Contact/Impact Dynamics Applied to Crash Analysis, Proceedings of the Crashworthiness of Transportation Systems: Structural Impact and Occupant Protection, pp. 445-473, 1997.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

776

First Order Tire Dynamics G. Rill University of Applied Sciences Regensburg Galgenbergstr. 30 D-93053 Regensburg [email protected] ABSTRACT For the dynamic simulation of road vehicles, the model-element ’tire/road’ is of special importance, according to its influence on the achievable results. In the interest of a balanced modeling, the precision of the complete vehicle model should stand in reasonable relation to the performance of the applied tire model. Fully nonlinear and dynamic tire models are very complex, [3]. Usually, they are used to investigate and evaluate the stochastic vehicle vibrations occurring during rough road rides and causing strength-relevant component loads. Comparatively lean tire models, like TMeasy [1] or the Magic Formula [2], are based on an analytical approximation of steady-state characteristics. They are widely used with multi-body system programs to investigate the handling properties of vehicles. This handling tire models are characterized by an useful compromise between user-friendliness, model-complexity and efficiency in computation time on the one hand, and precision in representation on the other hand. Within the handling tire models simplified transient tire properties are used to approximate the low frequency tire dynamics. Usually, the dynamic forces and torques are generated by first order differential equations driven by the steady state tire forces and torques. It is a common practice to derive the time constants from so called tire relaxations lengths. However, measurements [4] show that the relaxations lengths cannot be considered as constant but will strongly depend on the wheel load and the slip quantities. In this paper a method is presented where the first order tire dynamics is generated by a TaylorExpansion of the steady state forces and torques. Thus, relaxations lengths which include the wheel load and slip dependencies are automatically generated from the steady state tire properties. Slight model modifications make it possible to simulate stick slip effects during parking maneuvers. The results of this simple but effective approach correspond quite well with measurements.

References [1] Hirschberg, W; Rill, G. Weinfurter, H.: User-Appropriate Tyre-Modeling for Vehicle Dynamics in Standard and Limit Situations. Vehicle System Dynamics, 38/2, 103-125, 2002. [2] Pacejka, H.B., Bakker, E.: The Magic Formula Tyre Model. Proc. 1st Int. Colloquium on Tyre Models for Vehicle Dynamic Analysis, Swets&Zeitlinger, Lisse 1993. [3] Lugner, P.;, Pacejka, H.; Pl¨ochl, M.: Recent Advances in Tyre Models and Testing Procedures. Vehicle System Dynamics, 43/6-7, 413-436, 2005. [4] van der Jagt, P.: The Road to Virtual Vehicle Prototyping; new CAE-models for accelerated vehicle dynamics development. PhD-Thesis, Tech. Univ. Eindhoven, ISBN 90-386-2552-9 NUGI 834, Eindhoven, 2000,

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Experimental Validation of a model of an Uncontrolled Bicycle A. L. Schwabℵ , J. P. Meijaard∗ , J. D. G. Kooijmanℵ ℵ Delft University of Technology Laboratory for Engineering Mechanics Mekelweg 2, NL-2628 CD Delft, The Netherlands [email protected] ∗ School

of MMME, The University of Nottingham, Nottingham, UK ABSTRACT

In this paper, an experimental validation of some modelling aspects of an uncontrolled bicycle is made. In computer models, many physical aspects of the real bicycle are considered negligible, such as the flexibility of the frame and wheels, play in the bearings, and precise tire characteristics. The admissibility of these assumptions can be checked by comparing experimental results with numerical simulation results. The numerical simulations are performed on a benchmarked bicycle model [1]. This model (Fig. 1) consists of four rigid bodies connected by revolute joints. The contact between the knife-edge wheels and the flat level surface is modelled by holonomic constraints in the normal direction and by non-holonomic constraints in the longitudinal and lateral direction. In the absence of a rider we assume no-hands operation. This system has three velocity deFigure 1: Bicycle model. grees of freedom, the roll, the steer, and the forward speed. For the validation we consider the linearized equations of motion for small perturbations of the upright steady forward motion. Apart from flexibility and play, the greatest uncertainty to be verified in this model is the replacement of the tires by ideal rolling knife-edge wheels. front frame, H

rear frame, B

rear wheel, R

front wheel, F

λ

P

z

w

x

Q

c

The experimental system consists of an instrumented bicycle without rider. Sensors are present for measuring the roll rate and the yaw rate, the steering angle and the rear wheel rotation. Trainer wheels prevent the complete fall of the bicycle for unstable conditions. Measurements are recorded for the case in which the bicycle coasts freely on a level surface. From these measured data eigenvalues are extracted by means of curve fitting. These eigenvalues are then compared with the results from the linearized equations of motion of the model.

Figure 2: Instrumented bicycle. As a result, the model appears to be fairly accurate for the low-speed low-frequency behaviour.

References [1] A. L. Schwab, J. P. Meijaard, and J. M. Papadopoulos, (2005). “A Multibody Dynamics Benchmark on the Equations of Motion of an Uncontrolled Bicycle,” in Proceedings of the Fifth EUROMECH Nonlinear Dynamics Conference, ENOC-2005, August 7-12, 2005, Eindhoven University of Technology, The Netherlands, pp. 511-521.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

778

Numerical-experimental methodology for runnability analysis and wind-bridge-vehicle interaction study A. Collina*, E. Leo*, F. Resta*, G. Tomasini* Mechanical Department, Politecnico di Milano Via La Masa, 34 20158 - MILANO [email protected]

*

ABSTRACT The analysis of runnability of railway bridges is one of the tasks related to high speed trains since it involves the dynamic response of a bridge crossed by a train. In the proposed paper the runnability analysis has been carried out by means of numerical simulation based on mathematical model of train-track-bridge interaction developed in previous researches in which also the effect of cross wind acting on the train and the bridge is considered. In particular, two long railway bridges on Beijing - Shangai line have been studied. The following topics are mainly involved in the runnability: x bridge fatigue resistance, with respect to the dynamic amplification of the structure’s stresses; x influence of bridge deformations due to the train itself and other moving and stationary train, on train’s ride safety and comfort, in particular at bridge extremities; x effect of cross wind acting on train and bridge, from the point of view of the train safety; x track performances with respect to transmitted forces and structural noise propagation. In the first part of the work, the runnability study has been developed in absence of cross wind, following the subsequent steps: x first an analysis of the bridge response to constant travelling loads, corresponding to the axle loads, in order to establish the critical speed related to train transit; x then the verification with the full dynamic train model, in correspondence of the found critical speed is carried out; x a more refined modelling is implemented in the bridge, in order to investigate more local problems, such as local behaviour of the bridge deck. In the second part of the work, un algorithm ([1]) that allows to simulate the effects of the cross wind on vehicle and on bridge has been introduced. In particular, first the critical wind curves (CWC), that represent the wind speeds that lead the vehicle to the limit safety conditions, have been evaluated with the train running on a viaduct or a ground, considering a non deformable structure. Then the critical conditions have been verified in the transit of the train on the bridge. In this analysis, the aerodynamic forces on the train are applied dynamically, considering the turbulence of the wind, while the effects of the wind on the bridge are included considering the mean wind speed: in fact, the bridge response is mainly static and the effect of bridge motion due to the effect of turbulent wind is negligible.

References [1] Cheli F., Corradi R., Diana G., Tomasini G. A Numerical-Experimental Approach to Evaluate the Aerodynamic Effects on Rail Vehicle Dynamics. 18th IAVSD; Atsugi, Japan, 2003. [2] Bruni S., Cheli F., Diana G. Railway runnability and train- track in long span cable supported bridges. Advances in structural dynamics, Vol. 1, Elsevier Science, pp 43-54, 2001.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

779

Improved Bushing Models for Vehicle Dynamics Paulo Veríssimo, Jorge Ambrósio IDMEC – Instituto Superior Técnico, Technical University of Lisbon Av. Rovisco Pais, 1, 1049-001 Lisboa, PORTUGAL {pauloverissimo, jorge}@dem.ist.utl.pt

ABSTRACT This work presents the development and computational implementation, on a multibody dynamics environment of a constitutive relation to model bushing elements associated with different mechanical joints used in the models of road and rail vehicles. This kind of elements can be found in a wide number of mechanisms, besides the suspension cars, where they play important roles in absorption of vibrations, for instance due to road irregularities, to prevent misalignment of axes and noise reduction from the transmission and to decrease wear of the mechanical joints. In vehicle dynamics, in particular, the ride and handling of a vehicle are conditioned by the existence of these bushing elements. Therefore, suitable bushing models for vehicle multibody models must be accurate and at the same time computationally efficient so that they can be included in the vehicle models for a more reliable dynamic response obtained from the multibody simulations. Bushings are made of a special rubber, used generally in absorption of energy, which present a nonlinear and viscoelastic relationship between the forces and moments and their corresponding displacements and rotations. In the methodology proposed here a finite element model of the bushing is developed in the framework of the FE code ABAQUS to obtain the curves of displacement and rotation versus force and moment for different loading cases. Then the characteristics obtained from the nonlinear finite element analysis are interpolated through carpet plots in order to have a continuous description of the constitutive relations for each bushing. The bushing is modeled in a multibody code as an arrangement of springs that restrain the motion between the bodies connected. The basic ingredients of the multibody model are the same vectors and points relations used to define kinematic constraints in any multibody formulation. In the process, four types of bushing joints are implemented: spherical, revolution, cylindrical and translational. Finally, the methodology is demonstrated through the simulation of two multibody models of a road vehicle, one with perfect kinematic joints, for the suspension subsystems, and the other with bushing joints. The tests conducted to measure and compare the handling and ride of the two MB models are a ride over a road bump and obstacle avoidance maneuver at different speeds.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

780

Simulation of large deformations on timber joints using 3D FEM models A.M.P.G. Dias * *

Universidade de Coimbra Polo II 3030-290 Coimbra [email protected]

ABSTRACT The seismic behaviour of timber structures is highly dependent on the performance of their joints. Furthermore, the performance of the joints in such an extreme event, is significantly influenced by its plastic deformation capacity. Many times this plastic deformation capacity is low due to the brittle behaviour of timber. There are, however, certain types of joints that show considerable plastic deformation capacities when slender fasteners are used as, for example, the dowel type fasteners. Usually the prediction of the load and deformation characteristics for this type of joints is done based on the embedding tests. Most of the times, the plastic deformation capacity of the joints can be anticipated by the tests. The basis of these tests consists on a rigid fastener that is pushed against timber up to the failure or to a limit deformation of 5mm. This method is, however, expensive and takes much time, becoming worthless in many practical situations. An alternative to the tests are the numerical simulations using various types of models, as for example, 3D FEM models. This paper presents a 3D FEM model developed to simulate the stresses and deformations that occur on such tests. The model mesh is constituted by solid eight node elements. Both materials, timber and steel, are simulated considering non-linear material behaviour. Timber was simulated assuming an orthotropic behaviour and an anisotropic yield criteria, while steel was simulated assuming isotropic behaviour and an isotropic yield criteria. The model was validated using test results available for test specimens made of Spruce. Most of the data necessary to perform the simulations was available from the tests: when it was not available, a range of possible realistic values were assumed and verified by performing extra numerical simulations. Once all these parameters were decided the numerical results, load, deformation and stress distribution were compared with test results. The developed FEM model was also used to perform a parametric analysis considering two of the most relevant properties in terms of timber behaviour: yielding strength and modulus of elasticity. The results are presented and discussed in this paper. From that discussion it is concluded that the model is generally able to simulate the behaviour of timber in embedment tests.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

781

Nonlinear Dynamics of Flexible Partially Collapsed Structures Freire, A.M.S.*, Negrão, J.H. † * Department of Civil Engineering Polytechnical Institute of Castelo Branco, Portugal [email protected] †

Departament of Civil Engineering University of Coimbra, Portugal [email protected]

ABSTRACT The aim of this work is to present an implementation of some of the latest theories and developments in the finite element method involving the beam element and its potential for the analysis of structures undergoing sudden changes during its service, such as local structural collapses and/or instabilities or abrupt actions such as impacts. Geodesic domes, cable structures and cable supported bridges are some of the types of structures which may experiment such effects. The large deflection theory of linear elastic rods is used to model the structural elements, using a procedure that is geometrically exact and strain invariant, in which the Petrov-Galerkin method leads to non-symmetric stiffness matrices preserving the strain measures objectivity and path independence. The non-symmetry is in fact much more pronounced due to the gyroscopic (Coriolis and centrifugal) effects. The formulation uses Euler parameters matrix rotation which does not lead to singular transformation between inertial, material and space frames, undergoing arbitrary large rotations. The problems are solved using the finite element method in an implicit finite difference scheme in time that exactly preserves energy and momentum, using the Cayley transform despite of the Rodrigues formula in a mid-point rule integration. Several examples using the strain-invariant formulation for the static analysis are presented. The nonlinear dynamic using the energy momentum conserving algorithm, that doesn’t preserve the invariance of strains, is used to analyse simple examples as cantilever beams in free vibration, post collapsed cables and a beam-cable system subjected to a quasi-harmonic load. The cases analysed show the efficiency of the algorithmic procedure for both static and dynamics dealing with arbitrary large rotation. The Reissner-Simo finite-strain theory and the Simo-Tarnow-Doblare conserving time integration algorithm have shown to be effective in analyzing highly flexible structures. The robustness of the time stepping procedure and the effectiveness of the strain-invariant approach introduced by Jelenic and Crisfield in the Reissner-Simo rod element seem to allow their use in problems involving sudden changes in structures, such as local structural collapses and instabilities and extreme actions as impacts and shock phenomena. The accurateness of the algorithms in the presence of large displacements and arbitrary rotations as well as finite strains seems to be an advantage in face of other finite element procedures.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

782

Optimal steel frame design for fire resistance K. Jármai1, J.P.C. Rodrigues2 1

University of Miskolc, H-3515 Miskolc, Egyetemváros, Hungary [email protected] 2

University of Coimbra, Polo II, 3030 Coimbra, Portugal [email protected]

ABSTRACT Steel structures have been used in industrial and residential buildings, because they offer a wide range of advantages. However these structures, when unprotected, behave poorly in fire situation [1]. The high thermal conductivity of steel, together with the deterioration of its mechanical properties as a function of temperature, can lead to large deformations of structural elements and the premature failure of the buildings [2]. The fire design of steel structures can be according to Eurocode 1 and 3 part 1.2 [3-6]. The steel can be protected by materials such as mineral fibres, gypsum boards, concrete, intumescent painting and water-filled structures. In this study the optimal fire design of a steel frame structure is investigated. Using a relatively simple frame model it is shown how to apply the optimum design system for the case of fire resistance of a welded steel structure. Hollow sectional columns and beams are designed for minimum volume and weight. Overall and local buckling constraints are considered. A comparison is made using square hollow section (SHS) columns and SHS or rectangular hollow section (RHS) for beams at a pressure vessel supporting frame. The results are discrete ones according to [7]. Optimizing for fire resistance for a given time it shows the prize of safety, the relation between mass and safety. To increase fire resistance we have to put more steel into the structure. In the first design phase the structural mass is used as an objective function. A refined objective function can be the material cost. A final objective function will be the total cost including the steel mass, fabrication, the fire protection technique employed and the erection costs.

References [1.] Kay,T.R.,Kirby,B.R.& Preston,R.R., Calculation of the heating rate of an unprotected steel member in a standard fire resistance test, Fire Safety Journal, Vol. 26, 1996, pp. 327-350. [2.] Rodrigues, J. P. C.; Neves, I. C.; Valente, J.C., Experimental research on the critical temperature of compressed steel elements with restrained thermal elongation, Fire Safety Journal, Vol. 35, 2000, pp. 7798. [3.] European Committee for Standardization (CEN); Eurocode 1 (ENV 1991-1) - Basis of Design and Actions on Structures – Part 1: Basis of Design, Brussels, Belgium, May 2000. [4.] European Committee for Standardization (CEN); Eurocode 1 (ENV 1991-1-2) - Basis of Design and Actions on Structures – Part 2-2: Actions on Structures - Actions on Structures Exposed to Fire, Brussels, Belgium, April 2002. [5.] European Committee for Standardization (CEN); Eurocode 3 (ENV 1993-1-1) - Design of Steel Structures, Part 1 – General Rules and Rules for Buildings, Brussels, Belgium, May 2003. [6.] European Committee for Standardization (CEN); Eurocode 3 (ENV 1993-1-2) - Design of Steel Structures, Part 1.2: General Rules - Structural Fire Design, Brussels, Belgium, December 2003. [7.] Dutta,D.: Hohlprofil-Konstruktionen. Ernst & Sohn, 532 p. 1999, ISBN 3-433-01310-1

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Simulation of Shock Wave Loaded Concrete with Discrete Cracks Martin Larcher∗ , Lothar Stempniewski∗ ∗ Institut

f¨ur Massivbau und Baustofftechnologie Universit¨at Karlsruhe (TH), D-76128 Karlsruhe [email protected] ABSTRACT

This work presents a method to simulate the behavior of concrete under a high dynamic load. A discrete crack model with a cohesive crack zone is used to describe the fragmentation of the concrete. The aim of the presented research is the simulation of blasting of concrete. The results of these calculations will be compared with experimental results. Belytschko [1] proposed the element-free Galerkin method (EFG) which approximates a field by using a moving least-squares interpolation. Cracks can be implemented in EFG by cutting off the shape functions at the location of the crack. The chosen integration method and the influence of the size of support are presented. A discrete crack model is useful to represent the fragmentation of concrete by high dynamic loading. In the presented work discrete cracks are implemented with EFG. The use of discrete cracks with a cohesive zone makes it possible to work with a material model without damage formulation. A Rankine criterion is used to identify the growth of the crack. The numerical results of the implemented static material model are compared with test results of standard beams. The nonlinear volumetric stress-strain relation is the cause for shock waves. After the destruction of the micropores (decreasing stiffness) the stiffness of concrete is getting higher by compacting of the material (Hugoniot). The increased stiffness is the reason for the development of the shock waves. In the presented work a Y-function (Schmidt-Hurtienne [2]) is used to consider the increasing stiffness. The young’s modulus has to be multiplied with this function. The tensile and the compression strength are increasing with the strain rates. This has been shown in a lot of experiments. If concrete is blasted the strain rate reaches values of 106 sec−1 . It is not possible to get experimental data for strain rates over 100 sec−1 . So the strength factor for strain rates over this point is hypothetical and should be limited. The high pressure as a result of the high strain rate cause high temperature and with the high temperature the strength is descreasing. The proposed material model with EFG is implemented in an explicit time integration code. The results show the development of the cracks in shock wave loaded concrete.

References [1] T. Belytschko, Y.Y. Lu, and L. Gu. Element-free galerkin methods. International Journal for Numerical Methods in Engineering, 37:229–256, 1994. [2] Bj¨orn Schmidt-Hurtienne. Ein dreiachsiales Sch¨adigungsmodell f¨ur Beton unter Einschluss des Dehnrateneffekts bei Hochgeschwindigkeitsbelastung. Schriftenreihe des Instituts f¨ur Massivbau und Baustofftechnologie; Dissertation, Universit¨at Karlsruhe, 2001.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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CFD based evaluation of the lock-in phenomenon of a bridge under wind load Lopes A.V. *, Cunha A. †, Simões L.M.C. ‡ * Department of Civil Engineering Faculty of Sciences and Technology of the University of Coimbra, Coimbra, Portugal [email protected] † Department of Civil Engineering Faculty of Engineering of the University of Porto, Porto, Portugal [email protected] ‡ Department of Civil Engineering Faculty of Sciences and Technology of the University of Coimbra, Coimbra, Portugal [email protected]

ABSTRACT The analysis of the dynamic behaviour of long span bridges under wind excitation is usually performed on the basis of experimental tests on physical models in wind tunnels. As an alternative to such procedure, some numerical methodologies have been developed, namely in terms of flutter analysis, though they are still based on some coefficients (flutter derivatives) whose evaluation still involves usually the use of experimental tests. An attempt to overcome such limitations consists in using different algorithms of Computational Fluid Dynamics (CFD) for the evaluation of force and Scanlan coefficients. After recent progress in computer technology, the authors could develop and implement a new numerical methodology for the aeroelastic analysis of slender structures. This computational algorithm is a time incremental approach based on two numerical algorithms working together: one of them determines the fluid flow action and the other one evaluates the structural response. However, most of the studies performed deal with the evaluation of the critical velocity, also known as critical flutter velocity. This procedure can be understood as a verification of the structural safety in terms of ultimate limit state. But, as mentioned by the Eurocodes, it is also needed to verify the serviceability limit states of vibrations caused by wind action, unless the effective span is short enough. In the particular case of very flexible bridges under wind action, this verification can be done in terms of undesirable effects for users (discomfort), comparing the evaluated acceleration (or velocity) peak values of movements with human body acceptance criteria for vibrations. These unknown peak values are achieved by checking the maximum amplitude in the range of synchronized phenomena (lock-in) due to vortex-shedding. In this context, this paper presents the application of the above mentioned computer algorithm to the evaluation of the serviceability conditions of a simply supported bridge with a rectangular cross-section (B/D=6), under wind load considering their fundamental frequency. In particular, it will be evaluated the range of this synchronized phenomena, the peak value of acceleration obtained and: i) the time to start the phenomena; ii) the fully developed time; iii) and the corresponding time step.. Some of the most interesting results associated with the evaluation of the corresponding acceleration peak values of movements are presented, and compared with available human body acceptance criteria for vibrations (comfort evaluation) listed in the bibliography.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

785

CFD Based Aerodynamic Study to Discrete Optimization of Bridge Cross Sections Lopes A.V. *, Carlos D. P. C. Gomes †, Simões L.M.C. † *

Department of Civil Engineering Faculty of Sciences and Technology of the University of Coimbra, Coimbra, Portugal [email protected] †

Department of Civil Engineering Faculty of Sciences and Technology of the University of Coimbra, Coimbra, Portugal [email protected] ; [email protected]

ABSTRACT Aerodynamic instability of bridges should be one of the most concerns for bridge designers. Along with all studies about aerodynamic studies, a few ones are related to improvements of bridge cross sections. Among them, strategies such as grating, edge plates, edge fairing plates, side plates, baffle plates or flaps has been tested. The aim of this study is associated with the efficiency of using some based lateral inside appendages with the purpose of improving the aerodynamic characteristics of a Π cross section (B/D=6). It is turned to the Scanlan model, namely to the A*2 coefficient, with the aim of evaluating the aerodynamic efficiency of the cross section. In order to determine the fluid flow around the obstacle, it is used a numerical algorithm of computational fluid dynamics based on the Finite Volume Method (FVM). Additionally, Forced Oscillation Method (FOM) is adopted for evaluating aeroelastic coefficients.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

786

Progressive Collapse simulation in RC Structures K. Menchel, T.J. Massart and Ph. Bouillard Department of Structural and Material Computational Mechanics, Université Libre de Bruxelles F.D. Roosevelt Av., 50, CP 194/5 – Brussels Belgium {kmenchel, thmassar, pbouilla}@smc.ulb.ac.be

ABSTRACT Throughout recent history, famous records of building failures may be found, unfortunately accompanied by great human loss and major economic consequences. One of the mechanisms of failure is referred to as ‘progressive collapse’: one or several structural members suddenly fail, whatever the cause (accident or attack). The building then collapses progressively, every load redistribution causing in turn the failure of other structural elements, until the complete failure of the building or of a major part of it. The civil engineering community’s attention to this type of event was first drawn by the progressive collapse of the building called Ronan Point, following a gas explosion in one of the last floors. Different methods are proposed in the literature [1-4] in order to simulate this phenomenon, all of them based on different more or less simplifying and debatable assumptions, such as the independence of the method with respect to the cause of the initial failure, or the sequence in which the loads are applied. Since the degree of validity of these basic assumptions is not discussed in these pioneering contributions, the aim of this paper is to assess the degree of approximation that they induce. A more complete yet still simplified approach, avoiding some of these hypotheses, is devised, based on a finite element large displacement code, using plastic hinges associated to beam finite elements. Comparisons are used between these results and those obtained with the methods already present in the literature. A discussion based on relatively simple examples provides an assessment of their degree of validity. This paper focuses on reinforced concrete structure, where, as opposed to steel structures, assembly parts do not play a major role during collapse.

References [1] B. Ellingwood, E. Leyendecker, Approaches for design against progressive collapse. J. Struct. Div. ASCE, 104 (ST3) 413-423 (1978). [2] J. Gilmour, K.S. Virdi, Numerical modelling of the progressive collapse of framed structures as a result of impact or explosion. 2nd Int. PhD Symposion in Civil Engineering 1998 Budapest. [3] H. Choi, T. Krauthammer, Investigation of progressive collapse phenomena in a multy story building. International Symposium of Interaction of Munitions Effects with Structures, May 2003. [4] D.E. Grierson, L. Xu, Y. Liu, Progressive-failure analysis of buildings subjected to abnormal loading. Computer-aided Civil and Infrastructure Engineering, 20 (2005) Pages 155-171.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Vulnerability assessment for pipelines under permanent ground deformations. Comparison between analytical and empirical approaches V. Terzi*, M.Alexoudi†, K. Pitilakis†, Th. Hatzigogos† *

Civil Engineer Department, Aristotle University of Thessaloniki Laboratory of Soil Mechanics, Foundations & Geotechnical Earthquake Engineering, P.O.B. 450, 54124, Thessaloniki, Greece [email protected] †

Civil Engineer Department, Aristotle University of Thessaloniki Laboratory of Soil Mechanics, Foundations & Geotechnical Earthquake Engineering, P.O.B. 450, 54124, Thessaloniki, Greece [email protected], [email protected], [email protected]

ABSTRACT The seismic assessment of pipelines is a trigging issue due to their spatial distribution, their individual characteristics and the several uncertainties that are associated with the seismic response. In order to access pipeline damage, it is necessary to develop methods for seismic risk analysis based on the generation of fragility curves. This paper provides both analytical and empirical fragility curves validated in a specific water system. Although, Lefkas water system was rather well designed and maintained network constructed with modern regulations, faced several failures in water mains and in service connections as a result of Lefkas earthquake (14-8-2003, Ms=6.4, Greece). The main reason of such failures was the large permanent deformations (settlements, lateral spreads) especially in the coastline where continuous pipes where placed. The originality of the present research is a comparison between the obtained analytical fragility curves for continuous pipes with advanced FEM analyses and the empirical fragility curve obtained by the statistical elaboration of real recorded data based on Lefkas earthquake. The proposed fragility curves connect Repair Rate/km, a vulnerability index, with permanent ground deformation for different types of pipeline configuration. The information gathered for Lefkas water network (type, diameter, joint type, type of damage) was analyzed in GIS, creating the first, as far as we know, pipeline damage database and empirical fragility relation in Europe. The damage state of the pipes was defined empirically. Analytically, the modeling of the pipelines as well as the surrounding soil was achieved by the use of the finite element program ADINA. Concerning the pipes, 3D beam elements were used functioning under inelastic material laws and uncertainties issues of loading conditions were faced by different loadings configuration (X.Liu and M.O’Rourke,1997b). Soil – structure interaction effects were accounted by introducing simplified soil impedance functions in the form of spring elements as well as 3D finite elements. The analyses concern static analyses. Using the FEM approach, damage state was defined by the existence of plastic hinges. The final result of this research is the selection of the optimum load configuration that gives the corresponding result in the vulnerability assessment of water pipes compared with empirical data. After the validation with Lefkas earthquake, a series of parametric finite element analyses with different pipelines construction characteristics, diameter and material type took place in order to improve existing or proposed new fragility curves.

References [1] X.Liu, and M.J.O’Rourke, Behaviour of continuos pipeline subject to transverse PGD. Earthquake Engineering and Structural Dynamics,26, 989-1003,1997.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Seismic performance and strengthening of traditional masonry buildings in the city centre of Coimbra Romeu Vicente*, Hugo Rodrigues*, Humberto Varum* *

Civil Engineering Department, University of Aveiro Campus Universitário de Santiago, 3810-193, Aveiro, Portugal [email protected] [email protected] [email protected]

ABSTRACT The increasing concern and consequent appraisal on durability, conservation state and changeable use and function of old buildings in urban centres relies a great deal on the structural safety evaluation of vertical load capacity but also the capable resistance to horizontal forces. The need to assess seismic vulnerability, particularly of the traditional masonry buildings is a key issue. Particular attention has been put upon the building stock of the old city centre of Coimbra, mainly constituted by old masonry load-bearing buildings of significant architectural value. The evaluation of the seismic vulnerability of old buildings is essential in the definition of the strengthening needs and minimization of possible damages due to seismic actions, in safeguarding of built heritage or in the identification of critical buildings. This paper intends to contribute for the assessment of old buildings considering the local seismic risk. A three dimensional model was developed for an aggregate of four buildings. The finite element modelling of these buildings has intended to identify structural fragilities, help understand the damages detected in the existing structures (crack opening) and evaluate the global structural safety of this type of buildings. It will be presented the main results obtained in this study, interpreted the structural damage, stress distribution and verified the global stability and its consequences. The dynamic response of such constructions to seismic actions has allowed studying the structural vulnerability. Different strengthening techniques to improve the global behaviour of these buildings were modelled and analysed. Efficiency comparison of the strengthening strategies is also discussed.

References [1] H. Varum; H. Rodrigues, Avaliação do comportamento estrutural da torre medieval de Vilharigues – Vouzela. Congreso de Métodos Numéricos en Ingeniería 2005, Universidad de Granada, Spain, 2005. [2] R. Cardoso; M. Lopes; R. Bento, Seismic evaluation of old masonry buildings. Part I: Method description and application to a case-study. Journal of Engineering Structures, 27, 2024-2035, 2005. [3] R. Vicente; H. Varum; J.A.R. Mendes da Silva, Seismic vulnerability assessment of buildings in the old city centre of Coimbra. International Conference 250th Anniversary of the 1755 Lisbon Earthquake, 206-213, 2005.