Multifield Problems

State of the Art

By: Anna-Margarete SÃ¤ndig (Editor), W. Schiehlen (Editor), W.L. Wendland (Editor)

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Hardcover | 15 June 2000

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308 Pages

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24.13 x 16.51 x 1.91

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This book on the state of the art in "Multifield Problems" consists of selected articles based on a conference on this topic at the University of Stuttgart in 1999. The first two articles are contributions to the general modelling of multifield problems. S.S. Antman presents the important role of viscoelastic dissipation in the mathematical modelling of bifurcation analysis of nonlinear elasticity for large deformations. G.A. Maugin presents the basic theoretical foundations for the combination of three scales - the microscopic lattice of crystals, the mesoscopic thermomechanical model and the macroscopic con tinuum mechanics model for describing the propagation of phase transition fronts in terms of the Landau-Ginzburg theory and the modelling of nonlinear waves. The other contributions are associated with five main areas of multifield modelling such as two and multiphase flows, the mechanics of materials in terms of multiscaling, the interaction of solids and fluids,efficient solution methods of the discrete equations including adaptivity, and the modelling of contact and fracture.

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The Many Roles of Viscosity in Solid Mechanics	p. 1
Introduction	p. 1
A two-dimensional shearing problem	p. 1
Self-sustained oscillations of a viscoelastic layer	p. 4
Longitudinal motions	p. 5
Longitudinal motions of incompressible rods	p. 6
The motion of rods in space	p. 8
Shocks and viscosity	p. 9
References	p. 10
Multiscale Approach to a Basic Problem of Materials Mechanics (Propagation of Phase-Transition Fronts)	p. 11
Introduction	p. 11
Microscopic condensed-matter-physics approach: solitonics	p. 12
Macroscopic engineering approach: singular surface and thermody-namic criterion	p. 13
Mesoscopic applied-mathematics approach: structured front	p. 16
Theoretical-physics approach: quasi-particle and transient motion	p. 16
Numerics: from finite-differences to continuous cellular automata	p. 17
Finite-Difference Method	p. 17
Finite-Element Method	p. 18
Finite-Volume Method and cellular automaton	p. 18
Conclusion	p. 19
References	p. 21
Two and Multiphase Flows	p. 23
Analysing Particle Sedimentation in Fluids byMeasure-Valued Stochastic Processes	p. 25
Introduction	p. 25
System analysis by measure-valued stochastic processes	p. 27
Sedimentation in 1-d	p. 30
References	p. 33
Phenomenological Model of Sedimentation-Consolidation Processes	p. 34
Introduction	p. 34
Phenomenological sedimentation-consolidation model	p. 34
Analysis of the initial-boundary value problem	p. 37
Comparison with experimental data	p. 38
Conclusions and directions of future research	p. 40
References	p. 40
The Structure of Bidisperse Suspensions at Low Reynolds Numbers	p. 42
Introduction	p. 42
Numerical method	p. 42
Bidisperse suspensions	p. 44
An advection-diffusion model for polydisperse suspensions	p. 46
References	p. 49
Lattice Boltzmann Simulations of Complex Multiphase Flows	p. 50
Lattice-Boltzmann simulation of gas-particle flow in complex geometry	p. 50
Introduction	p. 50
Method and results	p. 51
Direct calculation of hysteresis by Lattice-Boltzmann simulations	p. 54
Introduction	p. 54
Hysteresis	p. 54
Method and results	p. 55
Discussion	p. 56
Acknowledgements	p. 57
References	p. 57
Interface Tracking in Multiphase Flows	p. 58
Introduction	p. 58
The multiphase flow equations	p. 59
Discretization	p. 60
The segment projection method	p. 62
Comparison of three methods for a buoyant bubble	p. 63
References	p. 65
Mechanics of Materials and Multiscaling	p. 67
Finite Element Computation of Macroscopic Quantities in Nonconvex Minimisation Problems and Applications in Materials Science	p. 69
Introduction	p. 69
Scalar double-well problem and its numerical solution	p. 71
Numerical analysis of linearised phase transitions in elastic solids	p. 76
References	p. 79
Homogenization of an Initial-Boundary Value Problem Describing Evolving Microstructure	p. 80
Introduction	p. 80
A mathematical model with sharp phase interfaces	p. 80
Equations for the stress, displacement and internal variables	p. 80
Evolution equation for the phase interface, dissipation inequality	p. 82
Homogenization of the equations for materials with evolving microstructure	p. 85
References	p. 87
Aspects of Homogenization Techniques and Multigrid Solving	p. 88
Introduction	p. 88
The basic multigrid concept	p. 88
A numerical homogenization procedure	p. 90
Transfer operators for heterogeneous structures	p. 91
Smoothing adapted transfer operators	p. 91
Transfer operators associated with unit displacements	p. 92
Algebraical motivated transfer operators	p. 92
Homogenization based transfer operators	p. 93
Comparative study of model problems	p. 93
Conclusion	p. 95
References	p. 95
Estimates on the Mixture Function for Multiphase Problems in Elasticity	p. 96
Introduction	p. 96
Definition of the mixture function	p. 97
The <$>{cal H}<$>-measure associated with	p. 99
A lower estimate	p. 100
Upper bounds	p. 101
A special case	p. 102
References	p. 103
A Two-Scale Micro-Macro-Approach to Anisotropic Finite Plasticity of Polycrystals	p. 104
Introduction	p. 104
A class of mirco-macro-transitions at large strains	p. 105
A computational model of finite crystal plasticity	p. 108
Texture evolution in homogeneous compression test	p. 110
References	p. 111
Solid-Fluid-Interaction	p. 113
Boundary Conditions at the Viscous Sliding Interface of Incompressible Porous Deformable Media	p. 115
Introduction	p. 115
Formulation	p. 116
Kinematic conditions	p. 117
Interface tractions	p. 118
Limiting cases	p. 120
Solid-solid interface	p. 121
Fluid-fluid interface	p. 121
Solid-fluid interface	p. 121
Biphasic-fluid interface	p. 121
Biphasic-solid interface	p. 121
Inviscid fluid phase	p. 122
Governing equations	p. 122
Discussion	p. 123
References	p. 123
Computational Experience from the Solution of Coupled Problems in Ship Dynamics	p. 125
Introduction	p. 125
Physical assumptions and governing equations	p. 125
Numerical method and computational implementation	p. 127
Computational results	p. 127
Fluid-structure interaction	p. 128
Structural model adaptation	p. 130
Structural mesh refinement	p. 132
Conclusions	p. 132
References	p. 134
On the Adaptive Computation of Shear Bands in Frictional Geomaterials	p. 135
Introduction	p. 135
Governing equations	p. 136
Spatial discretization and numerical tools	p. 138
Numerical example	p. 141
References	p. 142
Intrinsic Viscoelasticity of Porous Materials	p. 143
Introduction	p. 143
Governing equations	p. 144
Finite viscoelasticity law	p. 145
Examples	p. 147
Conclusions	p. 149
References	p. 150
Numerical Simulation of Fluids Interacting with Moving Rigid Bodies	p. 151
Introduction	p. 151
Governing equations and solution methods	p. 152
Navier-Stokes equations for moving grids	p. 152
Linear rigid body dynamics	p. 153
Coupling algorithm	p. 154
Numerical example	p. 155
Conclusions	p. 157
References	p. 158
Partitioned Analysis of Transient Nonlinear Fluid Structure Interaction Problems Including Free Surface Effects	p. 159
Introduction	p. 159
CFD - Computational Fluid Dynamics	p. 159
Arbitrary Lagrangean Eulerian (ALE) formulation	p. 160
Stabilized finite element method	p. 160
Description of the free surface	p. 161
CSD - Computational Structural Dynamics	p. 161
CMD - Computational Mesh Dynamics	p. 162
Partitioned staggered analysis approach	p. 162
Algorithmic setup	p. 163
A robust iterative substructuring scheme	p. 164
Numerical examples	p. 165
Floating vertical cylinder in viscous fluid	p. 165
Cavity with oscillating top plate and flexible bottom	p. 165
References	p. 166
Efficient Solvers and Adaptivity	p. 167
Domain Decomposition Methods in the Design of High Power Electronic Devices	p. 169
Introduction	p. 169
High power electronic devices and systems	p. 170
Eleetrothermomeehanieal couplings in IHV-Modules	p. 170
Minimization of parasitic effects in Converter Modules	p. 172
The mortar element methods	p. 173
Mortar methods for Lagrangian finite elements	p. 173
Mortar methods for curl-conforming edge elements	p. 176
Numerical results	p. 178
References	p. 181
Matrix Compression for the Radiation Heat Transfer in Exhaust Pipes	p. 183
Introduction	p. 183
Mathematical model	p. 184
Approximation of the full dense matrices	p. 187
Adaptive cross approximation	p. 188
Matrix partitioning	p. 188
Low-rank approximation	p. 189
Numerical examples	p. 190
References	p. 191
An Iterative Substructuring Method with Lagrange Multipliers for Elasticity Problems Using Approximate Neumann Subdomain Solvers	p. 193
Introduction	p. 193
The FETI method	p. 194
The new domain decomposition method with approximate subdomain solves	p. 196
Numerical results	p. 198
References	p. 200
A New a Posteriori Error Estimator in Adaptive Direct Boundary Element Methods. The Neumann Problem	p. 201
Introduction	p. 201
An a posteriori error estimator	p. 203
Implementation	p. 205
Numerical example	p. 207
References	p. 208
Efficient Elasto-Plastic Simulation	p. 209
Introduction	p. 209
A general framework for quasi-static plasticity	p. 210
Return mapping algorithms for incremental plasticity	p. 211
Standard materials	p. 212
An example with nonlinear hardening	p. 213
The computation of the return parameter	p. 214
A numerical experiment	p. 215
Conclusion	p. 216
References	p. 216
Contact and Fracture	p. 217
Sensitivity and Optimal Control in Contact Mechanics	p. 219
Introduction	p. 219
Setting of the optimal control and sensitivity problem	p. 219
Adjoint system	p. 225
Conclusions	p. 228
References	p. 228
On the Treatment of Contact Problems in Elasto-Plasticity	p. 229
Introduction	p. 229
Problem setting	p. 230
Boundary element method	p. 234
Numerical example	p. 236
References	p. 236
On Contact Problems for Linear and Nonlinear Elastic Plates: Existence of Solutions and Application of Augmented Lagrangian Method	p. 237
Geometrically linear plates	p. 237
Thin plates	p. 237
Reissner's plate model	p. 239
Von Kármán plates	p. 240
Augmented Lagrangians methods for a class of nonconvex problems	p. 241
Example	p. 243
References	p. 245
Longitudinal Wave Propagation in Conical Rods Subject to Impacts	p. 246
Introduction	p. 246
Symbolical computation for a conical rod	p. 247
Experimental investigation of longitudinal waves	p. 250
Conclusions	p. 252
References	p. 253
A Survey on Dynamic Contact Problems with Coulomb Friction	p. 254
Introduction	p. 254
The dynamic contact problem with Coulomb friction	p. 254
Frictional contact problems with heat transfer	p. 257
References	p. 261
Numerical Simulation of Noise Radiation from Rolling Tires	p. 262
Introduction	p. 262
The simulation procedure	p. 263
Numerical simulation of stationary rolling wheels	p. 264
Numerical analysis of sound radiation	p. 265
Example: A simple wheel	p. 267
Conclusions	p. 267
References	p. 269
Stress Singularities in Composites	p. 270
Introduction	p. 270
Linear examples: Laplacian and Lamé system	p. 271
Linear elliptic boundary transmission problems in Sobolev spaces with detached asymptotics	p. 274
Semilinear problems	p. 275
Continuity and Fréchet-differentiability of the semilinear operator	p. 276
Asymptotic behaviour of the solution of the semilinear problem	p. 276
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Multifield Problems

State of the Art

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