Advanced finite element methods and applications

Cover......Page 1
Front matter......Page 2
Introduction......Page 15
Transputer......Page 16
Available Parallel Computers......Page 18
Some Remarks on Performance......Page 26
The Scientific Environment......Page 27
Summer School 1989......Page 28
Multigrid Seminars......Page 30
FEM Symposia......Page 32
Historic Numerical Examples......Page 33
State of Hardware......Page 34
Performance Numbers......Page 35
References......Page 37
Introduction......Page 41
HDG Discretization......Page 43
Domain Decomposition Preconditioning......Page 47
Schwarz Type Domain Decomposition......Page 48
BDDC Preconditioners......Page 49
Traces and Polynomial Extensions on the Interval......Page 50
Extension from a Vertex......Page 57
Extension from an Edge......Page 59
Numerical Results......Page 67
References......Page 68
Introduction......Page 71
Discrete Analogues of Boundary Norms for Harmonic Functions in Thin Rectangles......Page 77
Finite-Difference Shape Dependent Boundary Norm for Finite Element Functions......Page 81
Preconditioning by Nonoverlapping Domain Decomposition......Page 84
Finite Element Space Decomposition......Page 86
Reducing to Isotropic Discretization......Page 87
Compatible Schur Complement Preconditioner......Page 93
Schur Complement and Domain Decomposition Algorithms......Page 96
References......Page 100
Introduction......Page 104
Setting of the Problem, Discretization......Page 106
The Scalar Elliptic Case......Page 107
Numerical Experiments......Page 112
Scalar Elliptic Problems......Page 113
Lame ´Equations of Linear Elasticity......Page 116
References......Page 119
Introduction......Page 122
Model Problem and FEM-BEM Coupling......Page 124
Subdomain Decomposition......Page 125
Operators Associated to the Potential Equation......Page 126
Boundary Integral Operators......Page 127
Continuous Domain-Skeleton Formulation......Page 128
Symmetric FEM-BEM Coupling......Page 129
MainResult......Page 131
Explicit Bounds for the Constants......Page 132
An Extension Operator for Polytopes......Page 133
Explicit Bounds for Boundary Integral Operators......Page 136
Error Analysis......Page 137
A Lemma a `la Strang......Page 138
Error Estimate for the Dirichlet Data......Page 139
Error Estimate for the Neumann Data......Page 140
Conclusion and Extensions......Page 142
References......Page 143
Introduction......Page 146
Background......Page 147
Refinement Criteria......Page 152
Post-processing and Recovery Techniques......Page 153
Residual Estimates......Page 154
Dual Weighted Residual (DWR) Estimates......Page 155
Node Relocation......Page 156
Splitting of Elements......Page 158
Summary of Properties of the Three Refinement Strategies......Page 161
Conclusions......Page 162
References......Page 163
Introduction......Page 166
The Model Boundary Value Problem and Notations......Page 168
Singular Decomposition of Solution......Page 170
The Finite Element Approximation......Page 172
Approximation on Quasi Uniform Mesh......Page 173
Improving the Accuracy of the Approximation......Page 174
Example 1......Page 176
Example 2......Page 180
Conclusions......Page 184
References......Page 185
Introduction......Page 187
Reinforcement Learning......Page 188
Solution of the Bellman Equation......Page 191
Temporal-Difference Learning......Page 193
Application to Recommendation Engines......Page 194
A Multilevel Preconditioner for the TD Method......Page 197
Evaluation on Real-Life Data......Page 204
References......Page 206
Introduction......Page 208
The General Framework......Page 210
Preconditioners Based on Schur Complements and Interpolation Theory......Page 212
Preconditioners Based on Inexact Schur Complements......Page 216
Application to Optimal Control Problems......Page 217
Distributed Optimal Control for Elliptic Equations......Page 218
The Velocity Tracking Problem for Stokes Flow......Page 220
Numerical Experiments......Page 223
References......Page 225
Introduction......Page 228
Basics of Holomorphic Eigenvalue Problems......Page 230
Nonlinear Generalized Rayleigh Quotient Iteration......Page 233
Augmented Newton Method......Page 237
Nonlinear Rayleigh Quotients and Functionals......Page 241
Examples......Page 243
Conclusions......Page 246
References......Page 247
Introduction......Page 249
Problem Formulation......Page 251
Discretization......Page 253
Algebraic Eigenvalue Computation......Page 256
Sensitivity Analysis......Page 258
Numerical Results......Page 260
References......Page 264
Introduction......Page 267
The Tearing and Interconnecting Method......Page 269
The Light Path Tree......Page 272
Field Tracing Methods......Page 276
Numerical Examples......Page 279
Conclusions......Page 282
References......Page 283
Introduction......Page 284
Short Review of Techniques for Treating Volume Integrals in BEM......Page 285
Description of the Problem......Page 287
Examples of the Body Forces......Page 288
Particular Solution......Page 291
Galerkin Procedure......Page 292
Adaptive Cross Approximation......Page 293
Exact Solution for the Polynomial Body Forces......Page 294
Approximation of the Body Force......Page 295
Radial Basis Functions......Page 296
Exact Solutions......Page 298
Numerical Examples......Page 299
References......Page 304
Towards the Direct and Inverse Adaptive Mixed Finite Element Formulations for Nearly Incompressible Elasticity at Large Strains......Page 307
Introduction......Page 308
Kinematics......Page 310
Thermodynamically Consistent Constitutive Equation......Page 312
Objective Function......Page 313
Weak Formulation of the Coupled Boundary Value Problem of Nearly Incompressible Elasticity at Large Strains......Page 315
Linearization of the Weak Formulation – Direct Problem......Page 316
Parameter Derivatives of the Weak Formulation – Inverse Problem......Page 318
Stable Element Formulations......Page 319
Error Controlled Mesh Adaptation......Page 320
Numerical Examples......Page 321
Adaptive Numerical Solution of Cook’s Membrane Problem......Page 322
Parameter Identification for Cook’s Membrane Problem......Page 325
Conclusions......Page 327
References......Page 329
Introduction......Page 332
The Forward Problem......Page 333
The Inverse Problem......Page 338
Reciprocity Principle and Reciprocity Gap......Page 339
Methods of Identification......Page 341
Stability......Page 343
Determination of the Crack Plane or -line......Page 346
Variants for Determining the Plane Offset......Page 347
Approximative Mid Point Determination of a Single Crack......Page 349
Setup of Examples......Page 350
FE-Meshes and Refinement Strategies in the Forward Computation......Page 351
Results with Uniform Refinement......Page 353
Comparison of Results with Different Refinement Strategies......Page 358
References......Page 359
Introduction......Page 361
Fatigue Crack Growth......Page 362
Calculation of Stress Intensity Factors......Page 365
Determination of Crack Deflection......Page 367
Determination of Crack Growth Increments......Page 368
Program Layout......Page 370
Procedure of Crack Generation......Page 372
Submodel Technique......Page 373
Adaptation of the Crack Discretization......Page 374
Application Examples......Page 375
Crack Growth in a Steering Knuckle......Page 376
References......Page 379