Discretisation and Interpolation - The Boundary Element Method with Programming - page 58

3.10.3 Numerical integration

In numerical integration schemes, the integral is approximated by a sum of values of the
integrand evaluated at certain points, times a weighting function. For the integration of
function

f [ , for example we can write ()

I

1

| ¦

³

(3.63)

I f [ [ d f [

i i

1

i 1

In the above, W i are weights and [ i are the intrinsic coordinates of the integration
( sampling ) points. If the well known trapezoidal rule is used, for example, then I =2, the
weights are 1 and the sampling points are at +1 and -1. That is

³

1

(3.64)

I f [ d [ | f 1 f 1

1

However, the trapezoidal rule is much too inaccurate for the functions that we are
attempting to integrate. The Gauss Quadrature with a variable number of integration
points can be used to integrate more accurately. In this method it is assumed that the
function to be integrated can be replaced by a polynomial of the form

2 p

f [ a a [ a [ ! a [

(3.65)

0 1 2 p

where the coefficients are adjusted in such a way as to give the best fit to f ([). We
determine the number and location of the sampling points, or Gauss points, and the
weights by the condition that the given polynomial is integrated exactly.

Table 3.4

Gauss point and degree of polynomial

No. of Gauss points, I Degree of polynomial p

1 1 (linear)

2 3 (cubic)

3 5 (quintic)

Table 3.5

Gauss point coordinates and weights

I [ i W i

1 0.0 2.0

2 0.57735 , -0.57735 1.0,1.0

3 0.77459, 0.0 , -0.77459 0.55555, 0.88888, 0.55555