Springer.Press.The.Boundary.Element.Method.With.Programming.Sep.2008: The.Boundary.Element.Method.With.Programming.Sep.2008.part2.rar

Contents
Preface xiii
Acknowledgements xiv
1 Preliminaries 1
1.1 Introduction 1
1.2 Overview of book 4
1.3 Mathematical preliminaries 6
1.3.1 Vector algebra 7
1.3.2 Stress and strain 10
1.4 Conclusions 11
1.5 References 11
2 Programming 13
2.1 Strategies 13
2.2 FORTAN 90/95/2000 features 14
2.2.1 Representation of numbers 14
2.2.2 Arrays 15
2.2.3 Array operations 16
2.2.4 Control 20
2.2.5 Subroutines and functions 21
2.2.6 Subprogram libraries and common variables 23
2.3 Charts and pseudo code 24
2.4 Parallel programming 25
2.5 BLAS libraries 27
2.6 Pre- and Postprocessing 27
2.7 Conclusions 27
2.8 Exercises 28
2.9 References 29
3 Discretisation and Interpolation 31
3.1 Introduction 32
vi The Boundary Element Method with Programming
3.2 One-dimensional boundary elements 32
3.3 Two-dimesional elements 36
3.4 Three-dimensional cells 44
3.5 Elements of infinite extent 44
3.6 Subroutines for shape functions 46
3.7 Interpolation 47
3.7.1 Isoparametric elements 47
3.7.2 Infinite elements 49
3.7.3 Discontinuous elements 50
3.8 Coordinate transformation 53
3.9 Differential geometry 54
3.10 Integration over elements 59
3.10.1 Integration over boundary elements 59
3.10.2 Integration over cells 59
3.10.3 Numerical integration 60
3.11 PROGRAM 3.1: Calculation of surface area 64
3.12 Concluding remarks 65
3.13 Exercises 65
3.14 References 67
4 Material Modelling and Fundamental Solutions 69
4.1. Introduction 69
4.2. Steady state potential problems 70
4.3. Static elasticity problems 76
4.3.1 Constitutive equations 82
4.3.2 Fundamental solutions 85
4.4. Conclusions 94
4.5. References 94
5 Boundary Integral Equations 95
5.1 Introduction 95
5.2 Trefftz method 96
5.3 PROGRAM 5.1: Flow around cylinder, Trefftz method 99
5.3.1 Sample input and output 102
5.4 Direct method 104
5.4.1 Theorem of Betti and integral equations 104
5.4.2 Limiting values of integrals as P coincides with Q 107
5.4.3 Solution of integral equations 110
5.5 Computation of results inside the domain 116
5.6 PROGRAM 5.2: Flow around cylinder, direct method 118
5.6.1 Sample input and output 122
5.7 Conclusions 125
5.8 Exercises 127
5.9 References 127
CONTENTS vii
6 Boundary Element Methods – Numerical Implementation 129
6.1 Introduction 129
6.2 Discretisation with isoparametric elements 130
6.3 Integration of kernel shape function products 133
6.3.1 Singular integrals 134
6.3.2 Rigid body motion 135
6.3.3 Numerical integration 139
6.3.4 Numerical integration over one-dimensional elements 142
6.3.5 Subdivision of region of integration 146
6.3.6 Implementation for plane problems 148
6.3.7 Numerical integration for two-dimensional elements 155
6.3.8 Subdivision of region of integration 159
6.3.9 Infinite elements 160
6.3.10 Implementation for three-dimensional problems 161
6.4 Conclusions 166
6.5 Exercises 167
6.6 References 168
7 Assembly and Solution 169
7.1 Introduction 169
7.2 Assembly of system of equations 170
7.2.1 Symmetry 176
7.2.2 Subroutine MIRROR 180
7.2.3 Subroutine Assembly 183
7.3 Solution of system of equations 184
7.3.1 Gauss elimination 185
7.3.2 Scaling 187
7.4 PROGRAM 7.1: general purpose program, direct method, one region 187
7.4.1 User’s manual 195
7.4.2 Sample input file 198
7.5 Conclusions 199
7.6 Exercises 200
7.7 References 202
8 Element-by-element techniques and Parallel Programming 203
8.1 Introduction 203
8.1 The Element by Element Concept 204
8.1.1 Element-by-element storage requirements 206
8.2 PROGRAM 8.1 : Replacing direct by iterative solution 206
8.2.1 Sample input file 211
8.2.2 Sample output file 212
8.3 PROGRAM 8.2 : Replacing assembly by element-by-element procedure 213
8.3.1 Sample input file 219
8.3.2 Sample output file 219
viii The Boundary Element Method with Programming
8.4 PROGRAM 8.3 : Parallelising the element-by-element procedure 220
8.4.1 Sample input file 227
8.4.2 Sample output file 227
8.4.3 Results from larger analyses 228
8.5 Conclusions 229
8.6 References 229
9 Postprocessing 231
9.1 Introduction 231
9.2 Computation of boundary results 232
9.2.1 Potential problems 232
9.2.2 Elasticity problems 236
9.3 Computation of internal results 241
9.3.1 Potential problems 241
9.3.2 Elasticity problems 245
9.4 PROGRAM 9.1: Postprocessor 250
9.4.1 Input specification 258
9.5 Graphical display of results 259
9.6 Conclusions 261
9.7 Exercises 262
9.8 References 262
10 Test Examples 263
10.1. Introduction 263
10.2. Cantilever beam 264
10.2.1 Problem statement 264
10.2.2 Boundary element discretisation and input 264
10.2.3 Results 266
10.2.4 Comparison with FEM 269
10.2.5 Conclusions 271
10.3. Circular excavation in infinite domain 271
10.3.1 Problem statement 271
10.3.2 Boundary element discretisation and input 272
10.3.3 Results 274
10.3.4 Comparison with FEM 275
10.3.5 Conclusions 276
10.4. Square excavation in infinite elastic space 276
10.4.1 Problem statement 276
10.4.2 Boundary element discretisation and input 277
10.4.3 “Quarter point” elements 281
10.4.4 Comparison with finite elements 282
10.4.5 Conclusions 282
10.5. Spherical excavation 283
10.5.1 Problem statement 283
10.5.2 Boundary element discretisation and input 283
CONTENTS ix
10.5.3 Results 289
10.5.4 Comparison with FEM 290
10.6. Conclusions 290
10.7. References 291
11 Multiple regions 293
11.1 Introduction 293
11.2 Stiffness matrix assembly 294
11.2.1 Partially coupled problems 296
11.2.2 Example 299
11.3 Computer implementation 304
11.3.1 Subroutine Stiffness_BEM 306
11.4 Program 11.1: General purpose program, direct method, multiple regions 311
11.4.1 User’s manual 321
11.4.2 Sample problem 323
11.5 Conclusions 326
11.6 Exercises 327
11.7 References 328
12 Dealing with corners and changing geometry 329
12.1 Introduction 329
12.2 Corners and edges 330
12.2.1 Discontinuous elements 331
12.2.2 Numerical integration for one-dimensional elements 331
12.2.3 Numerical implementation 335
12.2.4 Test example – single region 343
12.2.5 Test example – multi region 344
12.3 Dealing with changing geometry 346
12.3.1 Example 348
12.4 Alternative Strategy 351
12.5 Conclusions 353
12.6 References 353
13 Body Forces 355
13.1 Introduction 355
13.2 Gravity 356
13.2.1 Post-processing 358
13.3 Internal concentrated forces 361
13.3.1 Post-processing 363
13.4 Internal distributed line forces 363
13.4.1 Post-processing 365
13.5 Initial strains 365
13.5.1 Post-processing 369
13.6 Initial stresses 372
x The Boundary Element Method with Programming
13.7 Numerical integration over cells 373
13.8 Implementation 374
13.8.1 Input data specification for Body_force 377
13.9 Sample input file and results 378
13.10 Conclusions 381
13.11 Exercises 382
13.12 References 383
14 Dynamic Analysis 385
14.1 Introduction 385
14.2 Scalar wave equation, frequency domain 385
14.2.1 Fundamental solutions 387
14.2.2 Boundary Integral Equations 388
14.2.3 Numerical Implementation 389
14.3 Scalar wave equation, time domain 390
14.3.1 Fundamental solutions 390
14.3.2 Boundary integral equations 392
14.3.3 Numerical implementation 395
14.4 Elastodynamics 398
14.4.1 Fundamental solutions 399
14.4.2 Boundary integral equations 399
14.4.3 Numerical implementation 400
14.5 Multiple regions 401
14.6 Examples 403
14.6.1 Test example 403
14.6.2 Practical application 405
14.7 References 406
15 Nonlinear Problems 407
15.1 Introduction 407
15.2 General solution procedure 408
15.3 Plasticity 410
15.3.1 Elasto-plasticity 410
15.3.2 Visco-plasticity 413
15.3.3 Method of solution 415
15.3.4 Calculation of residual {R} 417
15.3.5 Computation of stresses at cell nodes 421
15.3.6 Computation of boundary stresses 423
15.3.7 Example 425
15.4 Contact problems 427
15.4.1 Method of analysis 427
15.4.2 Solution procedure 430
15.4.3 Example of application 431
15.5 Conclusions 433
15.6 References 433
CONTENTS xi
16 Coupled Boundary Element/ Finite Element Analysis 435
16.1 Introduction 435
16.2 Coupling theory 436
16.2.1 Coupling to finite elements 436
16.2.2 Coupling to boundary elements 443
16.3 Example 444
16.4 Dynamics 446
16.4.1 Example 447
16.5 Conclusion 447
16.6 References 449
17 Industrial Applications 451
17.1 Introduction 451
17.2 Mechanical engineering 453
17.2.1 A cracked extrusion press causes concern 453
17.3 Geotechnical Engineering 457
17.3.1 CERN Caverns 457
17.4 Geological engineering 461
17.4.1 How to find gold with boundary elements 461
17.5 Civil engineering 464
17.5.1 Masjed-o-Soleiman underground power house 464
17.6 Reservoir engineering 470
17.6.1 Borehole stability 470
17.7 Conclusions 472
17.8 References 473
18 Advanced topics 475
18.1 Introduction 475
18.2 Heterogeneous Domains 476
18.2.1 Theory 476
18.2.2 Example 477
18.3 Linear inclusions 479
18.3.1 Theory 479
18.3.2 Example 484
18.4 Piezo-electricity 485
18.4.1 Changes required in General_Purpose_BEM 487
18.5 Conclusions 488
18.6 References 489
Appendix 491

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