Preliminaries - The Boundary Element Method with Programming - page 5

FEM, one must be very careful to select the number of integration points, as they are
dependent on how close the singularity is to the integration region. This is the most
difficult and crucial part in the implementation of the BEM. The integration over the
boundary surface is carried out over a boundary element and the contributions of all
elements which describe a boundary are then added. We will see that this is very similar
to the assembly procedure in the FEM.
After the numerical treatment of the integral equations we end up with a system of
equations. In contrast to the FEM, the coefficient matrix is fully populated and
unsymmetrical. Standard Gauss elimination can be used but, for large systems, the
storage requirement and the computation times may be reduced considerably by iterative
solvers, such as conjugate gradient methods. Such special solution techniques are
introduced in the next chapter. Here we also find that the method is “embarrassingly
parallelisable” i.e. that excellent speed up rates can be achieved with special hardware.
The primary results obtained from the analysis are values of displacement or traction
at the boundary depending on the boundary condition specified. In contrast to the FEM,
primary results do not include values in the interior of the domain but these are
computed by post-processing. In Chapter 9 it is explained how the stresses at the
boundary and in the interior can be obtained from boundary displacements and tractions.
This is indeed an advantage of the method, because the user has free choice of the
locations where results are obtained.
We now have all the building blocks together and are able to compile a computer
program that is able to solve two and three-dimensional problems in elasticity and
potential flow, depending on which fundamental solution is used. In Chapter 10 we
apply the program developed to test examples and find out what level of accuracy can be
obtained in comparison with the FEM.
For inhomogeneous domains, where we can not obtain a fundamental solution, we
introduce the concept of multiple regions, where the domain is subdivided into sub-
regions, similar to the FEM. There is an additional advantage in this concept, because
sparseness is introduced in the system of equations. We will also find out in a later
chapter that the multi-region method allows contact and excavation problems to be
solved in an elegant way.
In the next chapter we deal with problems that involve corners and geometry which
changes with time, as is the application to sequential excavation/construction of a tunnel.
Because elements only exist on the boundary the BEM has difficulty dealing with
problems where forces are applied inside the domain. These forces can be loosely
termed “body forces”. It will be shown that an additional volume integral has to be
considered. For body forces that are constant the volume integral can be transformed
into a surface integral. However, if the body forces are not constant throughout the
domain the volume integral needs to be evaluated numerically. This can be done by
using internal cells, which look like finite elements, but do not involve any additional
degrees of freedom, as they are only used for integration. The implementation of this
procedure, discussed in chapter 13 also allows the solution of problems in elasto- and
visco-plasticity. Body forces of a different kind (mass forces) occur in the case of
dynamics, but their treatment with the BEM is quite different to the FEM and this is
discussed in Chapter 14.