The Boundary Element Method with Programming

This is a sequel to the book “Programming the Boundary Element Method” by G. Beer published by Wiley in 2001. The scope of this book is different however and this is reflected in the title. Whereas the previous book concentrated on explaining the implementation of a limited range of problems into computer code and the emphasis was on programming, in the current book the problems covered are extended, the emphasis is on explaining the theory and computer code is not presented for all topics. The new topics covered range from dynamics to piezo-electricity. However, the main idea, to provide an explanation of the Boundary Element Method (BEM), that is easy for engineers and scientists to follow, is retained. This is achieved by explaining some aspects of the method in an engineering rather than mathematical way. Another new feature of the book is that it deals with the implementation of the method on parallel processing hardware. I. M. Smith, who has been involved in programming the finite element method for decades, illustrates that the BEM is “embarrassingly parallelisable”. It is shown that the conversion of the BEM programs to run efficiently on parallel processing hardware is not too difficult and the results are very impressive, such as solving a 20 000 element problem during a “coffee break”.

Preface.- Acknowledgements.- Preliminaries: Introduction; Overview of book; Mathematical preliminaries; Conclusions; References.- Programming: Strategies; FORTAN 90/95/2000 features; Charts and pseudo code; Parallel programming; BLAS libraries; Pre- and Postprocessing; Conclusions; Exercises; References.- Discretisation and Interpolation: Introduction; One-dimensional boundary elements; Two-dimesional elements; Three-dimensional cells; Elements of infinite extent; Subroutines for shape functions; Interpolation; Coordinate transformation; Differential geometry; Integration over elements; PROGRAM 3.1: Calculation of surface area; Concluding remarks; Exercises; References.- Material Modelling and Fundamental Solutions: Introduction; Steady state potential problems; Static elasticity problems; Conclusions; References.- Boundary Integral Equations: Introduction; Trefftz method; PROGRAM 5.1: Flow around cylinder, Trefftz method; Direct method; Computation of results inside the domain; PROGRAM 5.2: Flow around cylinder, direct method; Conclusions; Exercises; References.- Boundary Element Methods – Numerical Implementation: Introduction; Discretisation with isoparametric elements; Integration of kernel shape function products; Conclusions; Exercises; References.- Assembly and Solution: Introduction; Assembly of system of equations; Solution of system of equations; PROGRAM 7.1: general purpose program, direct method, one region; Conclusions; Exercises; References.- Element-by-element techniques and Parallel Programming: Introduction; The Element by Element Concept; PROGRAM 8.1 : Replacing direct by iterative solution; PROGRAM 8.2 : Replacing assembly by element-by-element procedure; PROGRAM 8.3 : Parallelising the element-by-element procedure; Conclusions; References.- Postprocessing : Introduction; Computation of boundaryresults; Computation of internal results; PROGRAM 9.1: Postprocessor; Graphical display of results; Conclusions; Exercises; References.- Test Examples : Introduction; Cantilever beam; Circular excavation in infinite domain; Square excavation in infinite elastic space; Spherical excavation; Conclusions; References.- Multiple regions: Introduction; Stiffness matrix assembly; Computer implementation; Program 11.1: General purpose program, direct method, multiple regions; Conclusions; Exercises; References.- Dealing with corners and changing geometry: Introduction; Corners and edges; Dealing with changing geometry; Alternative Strategy; Conclusions; References.- Body Forces: Introduction; Gravity; Internal concentrated forces; Internal distributed line forces; Initial strains; Initial stresses; Numerical integration over cells; Implementation; Sample input file and results; Conclusions; Exercises; References.- Dynamic Analysis: Introduction; Scalar wave equation, frequency domain; Scalar wave equation, time domain; Elastodynamics; Multiple regions; Examples; References.- Nonlinear Problems: Introduction; General solution procedure; Plasticity; Contact problems; Conclusions; References.- Coupled Boundary Element/ Finite Element Analysis: Introduction; Coupling theory; Example; Dynamics; Conclusion; References.- Industrial Applications: Introduction; Mechanical engineering; Geotechnical Engineering; Geological engineering; Civil engineering; Reservoir engineering; Conclusions; References.- Advanced topics: Introduction; Heterogeneous Domains; Linear inclusions; Piezo-electricity; Conclusions; References.- Appendix.