The Boundary Element Method - Mechanics of Structures: Variational and Computational Methods - page 548

FIGURE 9.6
Boundary elements for two- and three-dimensional problems.

where, for a 2D problem

ln 1
r

N i dS

r k a k
r 2

H ij G ij

N i dS =− =

S i S i

b ln 1
r

dV

S

B j =−

V s

s = 1

and for a 3D problem

r k a k
r 3

1
r

H ij G ij

N i dS N i dS =− =

S i S i

dV

S

1
r

B j =− b

V s

s = 1

Note that H ij , G ij and B j involve r which is the distance from the nodal point j to points
on S i . For different nodal points, r takes on different values and, hence, H ij , G ij and
B j vary as j is changed. Also, the second term on the left-hand side of Eq. (9.34) involves
all of the nodal values of u on the boundary. Furthermore, the first term on the right-hand
side of Eq. (9.34) involves all of the nodal values of q on the boundary.