Boundary Integral Equations - The Boundary Element Method with Programming - page 107

The reader may verify that

J J

c 1 for 2 D and c 1 for 3 D

(5.30)

2 S 4 S

where J is defined as the angle subtended at P by s H .

S H

Q

J

n

H
I

P

S H

Figure 5.8

Limiting value of integral when P is located on a corner

For two and three-dimensional elasticity problems we may write a more general form
of equation (5.25)

ª º

³ ³

« »

c Iu P lim U PQ , t Q dSQ ( ) ȉ PQ , u Q dSQ ( )

(5.31)

« »

H o 0

¬ ¼ SS H SS H

where c is as previously defined and I is a 2x2 or 3x3 unit matrix.

5.4.3 Solution of integral equations

Using the direct method, a set of integral equations has been produced that relates the
temperature/potential to the normal gradient, or the displacement to the traction at any
point Q on the boundary. Since we are now able to place the source points coincidental
with the points where the boundary conditions are to be satisfied, we no longer need to
be concerned about these points. Indeed, in the direct method, the fictitious sources no
longer play a role.
To use integral equations for the solution of boundary value problems we consider
only one of the two regions created by cutting along the dotted line in Figure 5.4: the
interior or the exterior region, as shown in Figure 5.9. With respect to the integral
equations, the only difference between them is the direction of the outward normal n ,
which is assumed to point away from the solid. The interior region is a finite region, the
exterior an infinite region.