Preliminaries - The Boundary Element Method with Programming - page 4

There has been a general misconception that because a fundamental solution of the
problem must exist for the BEM to work, the method can only be applied to linear
problems with homogeneous material. As will be shown in this topic, non-linear
problems can almost as easily be solved as with the FEM, by the repeated solution of
linear problems and special methods may be employed to solve problems with
heterogeneous material properties.

1.2 OVERVIEW OF TOPIC

This topic is designed to be used as basis for a course on the BEM or for self study.
It is recommended that chapters be read consecutively as later chapters build on material
discussed earlier. Throughout the topic, the reader will build a suite of subprograms,
which perform the various tasks needed for the numerical implementation of the BEM.
Various exercises are included which allow the reader to test the programs written and
experience how the method works.
We start with an introduction to the FORTRAN 95 programming language.
FORTRAN, which stands for FORMula TRANslation is still the most widely used
language for programming engineering applications and is easier to learn and more
efficient than other high level languages such as C++. However, there is no reason why
the procedures outlined in some detail in this topic could not be implemented in another
language.
The next chapter deals with the way in which we can describe the geometrical
boundary of a problem and boundary conditions in a numerical way. This is done by
subdividing the surface into small elements and by interpolating between nodal values.
This is essential for the later treatment of integral equations. With the aid of the
examples we can not only test the subroutines developed but also get an understanding
of the error introduced by the approximations used to describe boundaries.
Another fundamental building block is the description of the material response. In
Chapter 4 we introduce basic concepts of elasticity and potential flow and develop
fundamental solutions, that is, simple solutions which satisfy the governing differential
equations. These will be central to our subsequent deliberations.
Next we introduce the concepts of boundary element methods using the method
originally proposed by Trefftz. Although this very simple method cannot be used for
general purpose programs, it serves very well to explain the fundamental ideas of the
method. A small computer program can be developed to solve some simple problems.
Again, this will serve as a tool for learning by experience.
The direct boundary element method used in the majority of BEM software is
introduced next. Here we will use the reciprocal theorem by Betti, which is well known
to engineers to obtain an integral equation. The major task in the implementation
however, is to solve the integral equations numerically.
The next chapter on numerical implementation therefore deals with the evaluation of
integrals using numerical integration. Those familiar with isoparametric finite elements
will recognise the Guass Quadrature method used. However, in contrast to its use in the