Nonlinear Problems - The Boundary Element Method with Programming - page 409

with L, M and K being the number of integration points in [ , K and ] directions.
The matrix E is given by

E !

# %#
"

E

§ ·

xxx xxz

¨ ¸

(15.35)

E ¨

¸

¨ ¸

E E

© ¹

zxx zxz

with the coefficients

C

ª º

EPQ , Cr G r G r G Crrr

(15.36) ijk

¬

3, k ij , j ik , i jk 4, i , j , k

¼

n

r

where x,y,z may be substituted for i,j,k and the constants are given in Table 15.2

Table 15.2 Constants for fundamental solution E

Plane strain Plane stress 3-D

n 1 1 2

C

1/8SGQ (1+QSG 1/16SGQ

C 3

1-2Q (1-QQ 1-2Q

C 4 2 3

The above formulae are valid for the case where none of the cell nodes is the
collocation point. The special case where one of the cell nodes coincides with a
collocation point, P i , the kernel

c
ni

' E tends to infinity with o(1/r) for 2-D problems and
o(1/r 2 ) for 3-D problems. To evaluate the volume integral for this case we subdivide a
cell into sub cells, as shown in Figure 15.7.

[

K

Subcell

K

P i

[

Boundary Element

Figure 15.7 Cell subdivision for the case where cell point is a collocation point (plane

problems)