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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.



Chapter 1. Numerical Convergence of Boundary Solutions in Transient Heat Conduction Problems

Mathematical theorems on uniform convergence of the boundary solution and stability of the computing scheme are proved for the approximation of a singular boundary integral equation using a time-dependent fundamental solution for two-dimensional isotropic heat conduction problems with non-isothermal boundary conditions. Discussions are extended to efficient numerical techniques developed for the transient boundary solution of problems having singular points and for the solution of external problems. The zoning technique and the double node technique are used together with linear boundary elements. Specific examples involving re-entrant corners and the external region of a circle are considered to show high accuracy of the boundary element solution.
Y. Iso, S. Takahashi, K. Onishi

Chapter 2. New Integral Equation Approach to Viscoelastic Problems

Computer-oriented numerical methods of solution have been successfully applied to various engineering problems. At the first stage of such developments, finite difference and finite element methods have attracted the attention of scientists and engineers. While rapid developments of these numerical methods in the last thirty years have stimulated a tremendous amount of work in computational techniques and engineering software, important research in basic physical principles such as variational techniques or method of residual was originated. It would be one of the most important consequences in the latter research that the integral equation method was re-considered through finite element techniques and a new numerical method of solution was innovated by some pioneering groups in England and also the U.S. (1). Brebbia’s boundary element book (2) and the International Conference on this subject (3–8) have much contributed to recent years’ rapid advances of the boundary element methods and stimulated a wide variety of boundary element applications in engineering. It can be seen that among various integral equation formulations the direct formulation is most successful and promising for engineering analysis.
M. Tanaka

Chapter 3. Numerical Integration

The Boundary Element Method (BEM) requires the evaluation of integrals of different kinds. It is therefore important to have a sound knowledge of the tools available to perform this task as efficiently as possible. By this we mean that numerical integration should provide sufficiently accurate values without incurring in excessive computing time. In this expository chapter we review the fundamentals of numerical integration and describe how to apply this technique to BEM problems.
H. Pina

Chapter 4. Computational Aspects of the Boundary Element Method

In this chapter, some specific aspects of the Boundary Element Method (B.E.M.) will be examined from a computational perspective. The most common techniques, used in the implementation of a computer B.E. program will also be briefly discussed.
M. Doblaré

Chapter 5. The Edge Function Method (E.F.M.) for Cracks, Cavities and Curved Boundaries in Elastostatics

The following paper represents a considerable development of the paper, Quinlan and O’Callaghan (1984), presented at the Sixth International Conference of B.E.M. in Engineering and published in the Conference proceedings. The section on curved boundaries has been greatly expanded with appropriate examples.
P. M. Quinlan, M. J. A. O’Callaghan

Chapter 6. Theoretical and Practical Aspects of Multigrid Methods in Boundary Element Calculations

In this paper multigrid methods are advocated for the fast solution of the large nonsparse systems of equations that occur in boundary-element methods. Multigrid methods combine relaxation schemes and coarse-grid corrections. Ample attention is given to the decomposition of the system matrix in order to obtain a relaxation scheme that reduces the high-frequency components of the iteration error. It is shown that the decomposition should take the edges of the boundary into account, because they have a strong influence on the smoothing property of the relaxation scheme. The practical aspects of the multigrid method are concerned with the use of the method in boundary element calculations. The choice of the coarse-grid operators, the interactions between the grids and the implementation of the algorithm are discussed. The theoretical investigations show that the multigrid method converges more rapidly as the number of boundary elements increases. This is illustrated for two plane problems: (1) potential flow around an aerofoil and (2) interior fundamental problem of elasticity.
H. Schippers

Chapter 7. Complex Variable Boundary Elements in Computational Mechanics

A new and exciting numerical approach to solving two-dimensional potential problems is obtained by use of the Cauchy integral equation for analytic functions. The resulting integral equation is readily solvable by computer, and produces a pair of two-dimensional conjugate harmonic functions which satisfy the Laplace equation over the problem domain.
T. V. Hromadka

Chapter 8. Potential Problems

A large number of engineering problems can be mathematically described by Laplace’s equation. Since functions satisfying Laplace’s equation are generally called potential functions, these engineering problems are referred to as potential problems. Under this classification, we can include heat conduction problems, where the potential function is the temperature; flow of ideal fluids, formulated either with a velocity potential or a stream function; groundwater flow, where the function is the piezometric head; torsion of prismatic shafts, with a warping function; and many others.
L. C. Wrobel

Chapter 9. Elastostatic Problems

In this chapter a simple, yet efficient, FORTRAN computer program for two-dimensional (plane Strain/Stress) problems is described. The code is implemented with linear boundary elements, i.e. linear interpolation functions for boundary tractions and displacements. In order to keep the chapter self-contained, a summary of the theory required is presented together with a complete description of each subroutine, so that the interested reader will find useful means of getting started with the technique and in the future will also be able to modify or adapt the code according to his/her own needs.
J. C. F. Telles


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