Discretisation and Interpolation - The Boundary Element Method with Programming - page 59

We find that with increasing degree of polynomial p , we need an increasing number
of Gauss points. Table 3.4 gives an overview of the number of Gauss N points needed to
integrate a polynomial of degree p up to degree 5. The computed location of the
sampling points and the weights are given in Table 3.5 for one to three Gauss points
(data for up to 8 Gauss points are given in the program listing). It should be noted here
that in the application of numerical integration later in this topic the integrands can not
be replaced by polynomials. However, it can be assumed that as the rate of variation of
the functions is increased more integration points will be required.

K

1

2

K 1 3 0 . 57 ...

[

4 3

K 1 3

[ 1 3 [ 1 3

Figure 3.29 Gauss integration points for a two-dimensional element

If we apply the numerical integration to two-dimensional elements or cells then a
double sum has to be specified

I J

11

| ¦¦

³³

I f [K[K , d d W W f [K ,

(3.66) i j i j

11

i 11 j

The Gauss integration points for a two-dimensional element and a 2x2 integration are
shown in Figure 3.29. For the integration over 3-D cells we have:

I J K

111

| ¦¦¦

³³³

I f [K[K , d d W W W f [K] , ,

(3.67)

i j k k i k

111

i 111 j k

A subroutine can be written which returns the Gauss point coordinates and weights
depending on the number of Gauss points for an integration order of up to 8.