Boundary Element Methods–Numerical Implementation - The Boundary Element Method with Programming - page 139

where U(P i , [ ) and T(P i , [ ) are the fundamental solutions at Q( [ ) for a source at point P i ,
J( [ ) is the Jacobian and N n ( [ ) are linear or quadratic shape functions.
When point P i is not one of the element nodes, both integrals can be evaluated by
Gauss Quadrature and the integrals in equation (6.33) can be replaced by two sums

M

¦

e
ni

' T | N [ T P , [ J [ W

n m i m m m

(6.34)

m 1

M

¦

m

e
ni

' U | N [ U P , [ J [ W

n m i m m

m 1

where the number of integration points M is determined as a function of the proximity of
P i to the integration region as explained previously. If P i is close to the integration region
a subdivision will be necessary.

R

P i

L

2

[

3

1

Figure 6.8 One dimensional element, integration where P i is not one of the element nodes

When P i is one of the element nodes, functions U and T tend to infinity within the
integration region. Consider the two cases in Figure 6.9:

(a) P i is located at point 1 and n in the equation (6.33) is 2:

This means that although Kernels T and U tend to infinity as point 1 is approached,
the shape function tends to zero, so the integral of product N n ( [ )U(P i [ ) and
N n ( [ )T(P i [ ) tend to a finite value. Thus, for the case where P i is not at node n of the
element, the integral can be evaluated with the formulae (6.34) without any
problems.

(b) P i is located at point 2 and n in the equation (6.33) is 2:

In this case, Kernels T and U tend to infinity and the shape function to unity and
products N n ( [ )U(P i [ ) and N n ( [ )T(P i [ ) also tend to infinity. Since Kernel U has a
singularity of order ln(1/r), the first product cannot be integrated using Gauss