Boundary Element Methods–Numerical Implementation - The Boundary Element Method with Programming - page 126

potential flow, or any other problem for which we can supply a fundamental solution
(see Chapter 18).
The idea of using isoparametric concepts for boundary elements seems to have been
first introduced by Lachat and Watson 3 and this prompted a change of name of the
method to Boundary Element Method . This chapter is about the numerical
implementation of isoparametric boundary elements, using the basic concepts that were
already discussed in detail in Chapter 3.

6.2 DISCRETISATION WITH ISOPARAMETRIC ELEMENTS

We consider the numerical solution of the boundary integral equations using
isoparametric elements where linear or quadratic functions are assumed for the variation
of the known and the unknown boundary values. Recalling from Chapter 3, we have for
a one-dimensional isoparametric element and for potential problems the following
interpolations

¦

e
n

x [ N [ x Geometry

n

¦

e
n

u [ N [ u Temperatur e / Potential (6.1)

n

¦

e
n

Flux

t [ N [ t

n

Consider the example in Figure 6.1, where the boundary of a two-dimensional
potential problem is divided into linear isoparametric elements.

1
1

u

8 1

8

1 1

2

2
1

7

u u

2

7

2

6

6

3

5 3

2
2

4

5 u

4

Figure 6.1 Discretisation of two-dimensional problem into linear boundary elements