The Boundary Element Method - Mechanics of Structures: Variational and Computational Methods - page 549

For each node j, the element matrices H ij and G ij are assembled to form the global matri-
ces H j and G j such that

M

H j =

1 H ij

[ H j 1 H j 2 ··· H jj ··· H jN ]

=

i =

(9.35)

M

G j

1 G ij

[ G j 1 G j 2 ··· G jj ··· G jN ]

= =

i =

The summation process is similar to that given in Chapter 5 for the stiffness matrices, i.e.,
the entries in H ij or G ij corresponding to a common node are added to form an entry in
H j or G j for the node. Since the second term on the left-hand side and the first term on the
right-hand side of Eq. (9.34) involve all the nodal values of u and q, H j and G j are of the
order of 1

N, where N is the total number of nodes. Introduce global vectors containing
nodal values of u and q, V

×

v 1 v 2 ··· v j ··· v N ] T , P q N ] T . The assembly

= [ = [ q 1 q 2 ··· q j ···

process results in a linear equation for node j

v j + H j V = G j P c + B j

Here, c can be replaced by c j because the value of c depends on the smoothness of the
boundary at node j . Put these equations together according to the global node order for all
N nodes to form the global equation

+ HV = GP CV + B (9.36)

where

H 1
H 2
.
H j
·
H N

G 1
G 2
.
G j
·
G N

B 1
B 2
.
B j
·
B N

H G

= , = , and B =

Matrices H and G are of order N × N and B is a vector of order N × 1 . Matrix C = diag ( cc ··· c )

+ H and G = G so that this equation becomes is an N × N diagonal matrix. Let H = C

= +

HV GP B (9.37)

Note that the diagonal elements of H involve the constant c, which is equal to απ in

Eq. (9.29) when ξ is on a smooth boundary and equal to another value when the boundary

at is not smooth. The constant c is often cumbersome to handle. A strategy to circumvent

the treatment of this constant is to avoid the explicit evaluation of the diagonal elements of H .
Examine Eq. (9.37) in some detail. It can be seen that matrices H and G depend on the
fundamental solution, the shape functions, and the contour of the boundary. They do not
depend on V , P , and B . The diagonal entries H ii of H , which involve the cumbersome
c , can be determined by assigning judiciously chosen values to V , P , and B in Eq. (9.36).
Suppose we have V

ξ

I , which is a vector with unit entries. When the variable V is unity
over the whole region, its derivatives are zero. Thus, P , which contains nodal values of q,
is zero. Also set b

=

= 0 , leading to B = 0 . Substitute these assumptions into Eq. (9.37) to

obtain

=

HI 0 (9.38)