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Summary
Summary
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.
Table of Contents
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 |