Assembly and Solution - The Boundary Element Method with Programming - page 194

3 4
4 5
5 6
6 7
7 8
8 1
0 ! no Dirichlet BC's
8 ! Neuman BC'c
1 1.000 0.707
2 0.707 0.000
3 0.000 -0.707
4 -0.707 -1.000
5 -1.000 -0.707
6 -0.707 0.000
7 0.000 0.707
8 0.707 1.000

7.5 CONCLUSIONS

In this chapter we have developed a general purpose program, which can be used to
solve any problem in elasticity and potential flow, or if we substitute the appropriate
fundamental solutions, any problem at all. This versatility has been made possible
through the use of isoparametric elements and numerical integration. In essence, the
boundary element method has borrowed here ideas from the finite element method and,
in particular, the ideas of Ergatoudis, who first suggested the use of parametric elements
and numerical integration.
Indeed, there are also other similarities with the FEM in that the system of equations
is obtained by assembling element contributions. In the assembly procedure we have
found that the treatment of discontinous boundary conditions, as they are encountered
often in practical applications, needs special attention and will change the assembly
process.
The implementation of the program is far from efficient. If one does an analysis of
runtime spent in each part of the program, one will realise that the computation of the
element coefficient matrices will take a significant amount of time. This is because, as
pointed out in Chapter 6, the order of DO loops in the numerical integration is not
optimised to reduce the number of calculations. Also in the implementation, all matrices
must be stored in RAM, and this may severely restrict the size of problems which can be
solved.
We have noted that the system of equations obtained is fully populated, that is, the
coefficient matrix contains no zero elements. This is in contrast to the FEM, where
systems are sparsely populated, i.e., containing a large number of zeroes. The other
difference with the FEM is that the stiffness matrix is not symmetric. This has been
claimed as one of the disadvantages of the method. However, this is more than
compensated by the fact that the size of the system is significantly smaller.
The output from the program consists only of the values of the unknown at the
boundary. The unknown are either the temperature/displacement or the flow normal to