# Computational Aspects

## Paperback(Softcover reprint of the original 1st ed. 1987)

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## Overview

Computational Aspects by Carlos A. Brebbia

Past volumes of this series have concentrated on the theoretical and the more formal aspects of the boundary element method. The present book instead stresses the computational aspects of the technique and its applications with the objective of facilitating the implementation of BEM in the engineering industry and its better understanding in the teaching and research environments. The book starts by discussing the topics of convergence of solutions, application to nonlinear problems and numerical integration. This is followed by a long chapter on the computational aspects of the method, discussing the different numerical schemes and the way in which influence functions can be computed. Three separate chapters deal with important techniques which are related to classical boundary elements, namely the edge method, multigrid schemes and the complex variable boundary element approach. The last two chapters are of special interest as they present and explain in detail two FORTRAN codes which have numerous applications in engineering, i.e. a code for the solution of potential problems and another for elastostatics. Each sub­ routine in the programs is listed and explained. The codes follow the same format as the ones in the classical book "The Boundary Element Method for Engineers" (by C. A. Brebbia, Computational Mechanics Publications, first published in 1978) but are more advanced in terms of elements and capabilities. In particular the new listings deal with symmetry, linear elements for the two dimensional elasticity, some mixed type of boundary conditions and the treatment of infinite regions.

## Product Details

ISBN-13: 9783642826658 Springer Berlin Heidelberg 12/15/2011 Topics in Boundary Element Research , #3 Softcover reprint of the original 1st ed. 1987 296 6.14(w) x 9.61(h) x (d)

1 Numerical Convergence of Boundary Solutions in Transient Heat Conduction Problems.- Abstract.- 1.1 Introduction.- 1.2 Boundary Element Approximations.- 1.2.1 Motivation.- 1.2.2 Boundary Integral Equations.- 1.2.3 Boundary Element Discretization.- 1.2.4 Interpolation of Functions.- 1.2.5 Discretization of the Equations.- 1.2.6 Problem with Boundary Condition of the Linear Radiation Type.- 1.3 Convergence and Stability.- 1.3.1 Preliminary Estimates.- 1.3.2 Theorem on Convergence and Stability.- 1.3.3 Numerical Demonstrations.- 1.4 Singularities and Unbounded Domains.- 1.4.1 Re-entrant Corners.- 1.4.2 Slit Boundary.- 1.4.3 Interzonal Singularity.- 1.4.4 Infinite Domain.- 1.5 Conclusions.- Acknowledgements.- References.- 2 New Integral Equation Approach to Viscoelastic Problems.- 2.1 Introduction.- 2.2 Preliminaries.- 2.3 Boundary Integral Equation in Space and Time.- 2.4 Alternative Approach — Incremental Formulation.- 2.5 Numerical Results and Discussion.- 2.6 Concluding Remarks.- Acknowledgements.- References.- 3 Numerical Integration.- 3.1 Introduction.- 3.2 Integration over One-Dimensional Domains.- 3.2.1 Basic Rules.- 3.2.2 Gauss Formulas.- 3.2.3 Singular Integrals.- 3.3 Integration for Two-Dimensional Problems.- 3.3.1 Regular Boundary Integrals.- 3.3.2 Singular Boundary Integrals.- 3.3.3 Domain Integrals.- 3.4 Integration for Three-Dimensional Problems.- 3.4.1 Regular Boundary Integrals.- 3.4.2 Singular Boundary Integrals.- 3.4.3 Domain Integrals.- Acknowledgements.- References.- 4 Computational Aspects of the Boundary Element Method.- 4.1 Introduction.- 4.2 Formulation and Numerical Treatment.- 4.3 A Boundary Element Program Organization.- 4.4 Storage and Management of Data.- 4.4.1 Variables in a Boundary Element Program.- 4.4.2 Data Structure.- 4.5 Data Input.- 4.6 Computation of the Matrices.- 4.6.1 Computation of the Integration Coefficients.- 4.6.1.1 Analytical and Semianalytical Integration over an Element Containing the Collocation Point.- 4.6.1.2 Special Numerical Integration Techniques over an Element Containing the Collocation Point.- 4.6.1.3 Adaptive Numerical Integration Schemes.- 4.6.1.4 Computation of the Diagonal Terms of the Matrix B.- 4.6.2 Boundary Conditions and Assemblage of the Matrices.- 4.6.3 Symmetries.- 4.6.4 Piecewise Heterogeneous Bodies.- 4.6.4.1 Condensation Process.- 4.6.4.2 Ordering of the System of Equations.- 4.7 Solution of the System of Equations.- References.- Other Bibliography.- 5 The Edge Function Method (E.F.M.) for Cracks, Cavities and Curved Boundaries in Elastostatics.- 5.1 Introduction.- 5.2 The Edge Function Method (E.F.M.) (a Qualitative Description).- 5.3 Complex Displacement Method for Elastostatics.- 5.4 Displacements and Stresses for Arbitrary Cartesian Axes.- 5.5 Basic Problems.- References.- Appendix A — Harmonic Fitting.- Appendix B — Data Inputs for 2-D Elastostatic Program “EQUINP”.- Appendix C — Program EQUINP.- 6 Theoretical and Practical Aspects of Multigrid Methods in Boundary Element Calculations.- Summary.- 6.1 Introduction.- 6.2 Boundary Integral Equations.- 6.3 Approximation of Boundary Integral Equations.- 6.4 Practical Aspects of Multigrid Methods.- 6.5 Smoothing Property of the Relaxation Process.- 6.6 Theoretical Aspects of Multigrid Methods.- 6.7 Numerical Results.- 6.8 Conclusions and Recommendations.- References.- 7 Complex Variable Boundary Elements in Computational Mechanics.- 7.1 Introduction.- 7.2 A Complex Variable Boundary Element Approximation Model.- 7.3 The Analytical Function Defined by the Approximator $$\hat \omega \left( z \right)$$.- 7.4 A Constant Boundary Element Method.- 7.5 The Complex Variable Boundary Element Method.- 7.6 Approximation Error from the CVBEM.- 7.7 A CVBEM Modeling Strategy to Reduce Approximation Error.- 7.8 Expansion of the Hk Approximation Function.- 7.9 Upper Half Plane Boundary Value Problems.- 7.10 The Approximate Boundary for Error Analysis.- 7.11 Locating Additional Nodal Points on ?.- 7.12 Sources and Sinks.- 7.13 Regional Inhomogeneity.- 7.14 The Poisson Equation.- 7.15 Computer-Aided-Analysis and the CVBEM.- References.- 8 Potential Problems.- 8.1 Introduction.- 8.2 Review of the Theory.- 8.3 Numerical and Computational Aspects.- 8.3.1 Integration.- 8.3.2 Infinite Regions.- 8.3.3 Robin Boundary Conditions.- 8.3.4 Symmetry.- 8.4 Description of the Computer Program.- 8.4.1 Main Program and Data Structure.- 8.4.2 Subroutine INPUT.- 8.4.3 Subroutine GENER.- 8.4.4 Subroutine SYMMET.- 8.4.5 Subroutine GAUSS.- 8.4.6 Subroutine FMAT.- 8.4.7 Subroutine INTE.- 8.4.8 Subroutine SLNPD.- 8.4.9 Subroutine REORD.- 8.4.10 Subroutine INTER.- 8.4.11 Subroutine OUTPT.- 8.5 Applications.- 8.5.1 Pipe of Elliptical Cross-Section.- 8.5.2 Circular Cavity in Infinite Medium.- 8.5.3 Rectangular Concrete Column.- References.- 9 Elastostatic Problems.- 9.1 Introduction.- 9.2 Outline of Theory.- 9.3 Numerical and Computational Procedures.- 9.3.1 Integration.- 9.3.2 External Problems in Infinite Regions.- 9.3.3 Stresses on the Boundary.- 9.3.4 Surface Traction Discontinuities.- 9.3.5 Symmetry.- 9.4 Computer Program.- 9.4.1 Main Program and Data Structure.- 9.4.2 Subroutine INPUT.- 9.4.3 Subroutine MATRX.- 9.4.4 Subroutine FUNC.- 9.4.5 Subroutine SLNPD.- 9.4.6 Subroutine OUTPT.- 9.4.7 Subroutine FENC.- 9.4.8 Examples.- References.