Programming the boundary element method : an introduction for engineers

Providing an easy introduction to the boundary element method, this book is ideal for any reader wishing to work in this field or use this method for the solution of engineering problems. From the beginning, the emphasis is on the implementation of the method into computer programs which can be used to solve real problems. The book covers two-andthree-dimensional linear and non-linear analysis in potential flow (heat flow and seepage) and static elasticity. Several computer programs are listed in the book and may be downloaded free of charge via the Internet. They include programs and subroutines for:
* 2-D analysis of potential problems using the Trefftz method
* 2-D and 3-D linear analysis of potential and static elasticity problems using isoparametric elements (single and multiple regions)
* implementation of non-linear problems
* coupling to finite elements
The programs (written in FORTRAN 90) are well documented, and can be employed by the user to gain experience with the method through the solution of small test examples. Furthermore, readers may use them as a starting point for developing their own boundary element package. In addition, exercises are included in most chapters involving the use of the programs with answers given in an Appendix, and a number of interesting industrial applications in the areas of mechanical, civil and geotechnical engineering are presented.

Author Notes

GERNOT BEER is Professor of Structural Analysis at the Universtiy of Technology Graz, Austria. he has worked in the area of numerical modelling for more than two decades and has done some pioneering work in the coupling of the FEM and BEM. He is the author of the BEFE program, still the only commercial software available which completely integrates both these methods. This is his second book on the topic.

Preface	p. xi
Acknowledgements	p. xiii
1 Preliminaries	p. 1
1.1 Introduction	p. 1
1.2 Overview of book	p. 4
1.3 Mathematical preliminaries	p. 6
1.3.1 Vector algebra	p. 7
1.3.2 Stress and strain	p. 10
1.4 Conclusions	p. 11
1.5 References	p. 11
2 Programming	p. 13
2.1 Strategies	p. 13
2.2 FORTRAN 90/95 features	p. 14
2.2.1 Representation of numbers	p. 14
2.2.2 Arrays	p. 15
2.2.3 Array operations	p. 16
2.2.4 Control	p. 19
2.2.5 Subroutines and functions	p. 21
2.2.6 Subprogram libraries and common variables	p. 22
2.3 Charts and pseudo-code	p. 23
2.4 Pre- and postprocessing	p. 24
2.5 Conclusions	p. 25
2.6 Exercises	p. 25
2.7 References	p. 26
3 Discretisation and interpolation	p. 27
3.1 Introduction	p. 27
3.2 One-dimensional elements	p. 28
3.3 Two-dimensional elements	p. 32
3.4 Elements of infinite extent	p. 37
3.5 Subroutines for shape function	p. 39
3.6 Description of physical quantities	p. 40
3.7 Coordinate transformation	p. 42
3.8 Differential geometry	p. 43
3.9 Integration over elements	p. 48
3.9.1 Numerical integration	p. 48
3.10 Program 3.1: Calculation of surface area	p. 51
3.11 Conclusions	p. 53
3.12 Exercises	p. 54
3.13 References	p. 55
4 Material modelling and fundamental solutions	p. 57
4.1 Introduction	p. 57
4.2 Steady state potential problems	p. 58
4.3 Static elasticity problems	p. 64
4.3.1 Constitutive equations	p. 70
4.3.2 Fundamental solutions	p. 73
4.4 Conclusions	p. 82
4.5 References	p. 82
5 Boundary integral equations	p. 83
5.1 Introduction	p. 83
5.2 Trefftz method	p. 84
5.3 Program 5.1: Flow around cylinder, Trefftz method	p. 87
5.3.1 Sample input and output	p. 90
5.4 Direct method	p. 93
5.4.1 Theorem of Betti and integral equations	p. 93
5.4.2 Limiting values of integrals as P coincides with Q	p. 96
5.4.3 Solution of integral equations	p. 99
5.5 Computation of results inside the domain	p. 106
5.6 Program 5.2: Flow around cylinder, direct method	p. 108
5.6.1 Sample input and output	p. 112
5.7 Conclusions	p. 116
5.8 Exercises	p. 117
5.9 References	p. 118
6 Boundary element methods - numerical implementation	p. 119
6.1 Introduction	p. 119
6.2 Discretisation and isoparametric elements	p. 120
6.3 Integration of kernel shape function products	p. 123
6.3.1 Singular integrals and rigid body motions	p. 123
6.3.2 Numerical integration	p. 128
6.3.3 Numerical integration over one-dimensional elements	p. 132
6.3.4 Numerical integration for two-dimensional elements	p. 143
6.4 Conclusions	p. 152
6.5 Exercises	p. 153
6.6 References	p. 154
7 Assembly and solution	p. 155
7.1 Introduction	p. 155
7.2 Assembly of system of equations	p. 156
7.2.1 Symmetry	p. 160
7.2.2 Subroutine MIRROR	p. 165
7.2.3 Subroutine ASSEMBLY	p. 167
7.3 Solution of system of equations	p. 169
7.3.1 Gauss elimination	p. 170
7.3.2 Conjugate gradient solver	p. 173
7.3.3 Scaling	p. 174
7.4 Program 7.1: General purpose program, direct method, one region	p. 174
7.4.1 User's manual	p. 182
7.4.2 Sample input file	p. 184
7.5 Conclusions	p. 185
7.6 Exercises	p. 186
7.7 References	p. 188
8 Postprocessing	p. 189
8.1 Introduction	p. 189
8.2 Computation of boundary results	p. 190
8.2.1 Potential problems	p. 190
8.2.2 Elasticity problems	p. 194
8.3 Computation of internal results	p. 199
8.3.1 Potential problems	p. 200
8.3.2 Elasticity problems	p. 203
8.4 Program 8.1: Postprocessor	p. 209
8.4.1 Input specification	p. 216
8.5 Conclusions	p. 216
8.6 Exercises	p. 216
8.7 References	p. 217
9 Test examples	p. 219
9.1 Introduction	p. 219
9.2 Cantilever beam	p. 220
9.2.1 Problem statement	p. 220
9.2.2 Boundary element discretisation and input	p. 220
9.2.3 Results	p. 222
9.2.4 Comparison with FEM	p. 227
9.2.5 Conclusions	p. 227
9.3 Circular excavation in infinite domain	p. 228
9.3.1 Problem statement	p. 228
9.3.2 Boundary element discretisation and input	p. 229
9.3.3 Results	p. 231
9.3.4 Comparison with FEM	p. 232
9.3.5 Conclusions	p. 233
9.4 Square excavation in infinite elastic space	p. 234
9.4.1 Problem statement	p. 234
9.4.2 Boundary element discretisation and input	p. 234
9.4.3 'Quarter point' elements	p. 237
9.4.4 Comparison with finite elements	p. 239
9.4.5 Conclusions	p. 240
9.5 Spherical excavation	p. 240
9.5.1 Problem statement	p. 241
9.5.2 Boundary element discretisation and input	p. 241
9.5.3 Results	p. 245
9.5.4 Comparison with FEM	p. 245
9.6 Conclusions	p. 245
9.7 References	p. 246
10 Multiple regions	p. 247
10.1 Introduction	p. 247
10.2 Multi-region assembly	p. 248
10.3 Stiffness matrix assembly	p. 252
10.3.1 Partially coupled problems	p. 255
10.3.2 Example	p. 257
10.4 Computer implementation	p. 262
10.4.1 Subroutine stiffness_BEM	p. 263
10.5 Program 10.1: General purpose program, direct method, multiple regions	p. 269
10.5.1 User's manual	p. 279
10.5.2 Sample problem	p. 281
10.6 Conclusions	p. 284
10.7 References	p. 285
11 Edges and corners	p. 287
11.1 Introduction	p. 287
11.2 Potential problems	p. 288
11.3 Two-dimensional elasticity	p. 290
11.3.1 Region assembly with corners	p. 296
11.4 Three-dimensional elasticity	p. 301
11.5 Implementation	p. 304
11.1.1 Subroutine for detecting corners	p. 305
11.1.2 Subroutine for computing auxiliary equation coefficients	p. 307
11.6 Conclusions	p. 309
11.7 References	p. 310
12 Body Forces	p. 311
12.1 Introduction	p. 311
12.2 Gravity	p. 312
12.2.1 Postprocessing	p. 314
12.3 Initial strains	p. 317
12.3.1 Volume cells	p. 320
12.3.2 Numerical evaluation of volume integrals	p. 322
12.3.3 Postprocessing	p. 323
12.4 Initial stresses	p. 326
12.4.1 Numerical evaluation of volume integrals	p. 329
12.4.2 Postprocessing	p. 330
12.5 Implementation	p. 333
12.6 Example	p. 333
12.7 Conclusions	p. 335
12.8 References	p. 336
13 Non-linear problems	p. 337
13.1 Introduction	p. 337
13.2 General solution procedure	p. 338
13.3 Plasticity	p. 339
13.3.1 Elastoplasticity	p. 339
13.3.2 Viscoplasticity	p. 342
13.3.3 Method of solution	p. 343
13.3.4 Evaluation of singular integrals	p. 345
13.3.5 Computation of internal stresses	p. 347
13.3.6 Example	p. 349
13.4 Contact problems	p. 351
13.4.1 Method of analysis	p. 352
13.4.2 Solution procedure	p. 355
13.4.3 Example of application	p. 356
13.5 Conclusions	p. 358
13.6 References	p. 358
14 Coupled boundary element/finite element analysis	p. 359
14.1 Introduction	p. 359
14.2 Coupling theory	p. 360
14.2.1 Coupling to finite elements	p. 360
14.2.2 Coupling to boundary elements	p. 367
14.3 Examples	p. 367
14.4 Conclusion	p. 370
14.5 References	p. 370
15 Industrial applications	p. 373
15.1 Introduction	p. 373
15.2 Mechanical engineering	p. 375
15.2.1 A cracked extrusion press causes concern	p. 375
15.3 Geotechnical engineering	p. 379
15.3.1 Instability of slope threatens village	p. 379
15.3.2 Analysis of tunnel advance in anisotropic rock	p. 382
15.3.3 Tunnel approaching fault	p. 384
15.3.4 CERN caverns	p. 386
15.4 Geological engineering	p. 390
15.4.1 How to find gold with boundary elements	p. 390
15.5 Civil engineering	p. 393
15.5.1 Arch dam	p. 393
15.6 Conclusions	p. 395
15.7 References	p. 395
Appendix A Program libraries	p. 397
A.1 Utility_LIB	p. 397
A.2 Geometry_LIB	p. 409
A.3 Integration_LIB	p. 416
A.4 Elast_LIB	p. 421
A.5 Laplace_LIB	p. 426
A.6 Postproc_LIB	p. 428
A.7 Stiffness_LIB	p. 430
Appendix B Answers to exercises	p. 437
Index	p. 453

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