Assembly and Solution - The Boundary Element Method with Programming - page 182

As already mentioned previously for the solution of equations involving many
subtractions, it is necessary to use REAL (KIND=8) for the arrays, to avoid an
accumulation of round-off error. For a 3-D elasticity problem involving 1000 nodes, the
space required for storing the coefficient matrix in REAL (KIND=8) is 72 Mbytes. For
the solution on small computers this space may not be available and special algorithms
must be devised, where part of the matrix is written onto disk. Methods for the
partitioned solution of large systems are presented, for example by Beer and Watson 2 .
For the reduction of the system of equations we need three implied DO-loops. In the
implementation the innermost DO-loop is written implicitly using the new feature
available in FORTRAN 90. The innermost DO-loop involves one multiplication and one
subtraction and is executed (M - n) times, where M is the number of unknowns. The DO
loop above it involves a division and is also executed (M - n) times. Finally the
outermost DO-loop is executed M - 1 times. It can be shown, therefore, that the total
number of operations required is 2/3M 3 + ½M 2 + 1/6M. For large systems the first term
is dominant
This means that, for example, for a problem in three-dimensional elasticity involving
1000 nodes, approx. 2x10 10 operations are necessary for the reduction. If we want to
analyse these problems in a reasonable time there is clearly a need for more efficient
solvers. Recently there has been a resurgence of iterative solvers 1 . The advantage of
these solvers is that the number of operations and hence the solution time is only
proportional to M 2 and that they can be adapted easily to run on parallel computers. This
will be discussed in the next chapter.

7.3.2 Scaling

When we look at the fundamental solutions for elasticity we note that kernel U contains
the modulus of elasticity whereas T does not. Depending on the chosen units used we
expect a large difference in values. As we have seen at the beginning of this chapter, if
there is a mixed boundary value problem then there is a mixture of U and T terms in the
assembled coefficient matrix. This may cause problems in the solution of equations,
since very small terms would be subtracted from very large ones. Additionally, we note
that for 2-D problems kernel U varies with ln (1/r) which gives f value as

o r .
For the above reasons scaling of the data is recommended. Scaling is applied in such
a way that all tractions are divided by E and all coordinates by the largest difference
between coordinates (which results in a scaled problem size of unity).

f

7.4 PROGRAM 7.1: GENERAL PURPOSE PROGRAM,

DIRECT METHOD, ONE REGION

We now have developed all necessary tools for writing a general purpose computer
program for computing two and three-dimensional problems in potential flow and
elasticity. The first part of the program reads input data. There are three types of data: