The Boundary Element Method - Mechanics of Structures: Variational and Computational Methods - page 567

where C = diag ( cc ··· c ) is a diagonal matrix which is of the order of 3 N × 3 N for three-

dimensional problems and 2 N 2 N for two-dimensional problems, where N is the total

number of nodes for the system. Let H

×

+ H and G = C = G , so that the above equation

becomes

HV = GP + B (9.94)

for the whole system, where V and P contain all the nodal displacements and tractions,

H 11 H 12 ··· H 1 N

H N 1

H =

H N 2 ··· H NN

and

G 11 G 12 ··· G 1 N

G N 1

G =

G N 2 ··· G NN

Note that the matrices H and G depend on the fundamental solution, the shape func-
tions and the contour of the boundary only and are independent of the applied forces and
boundary conditions. In other words, they will not be changed if the applied forces and
boundary conditions are changed. The diagonal submatrices H ii of H , which involves the
cumbersome c of Eq. (9.88), can be evaluated by imposing specific displacement, force, and
boundary conditions on the body. This is in the same situation as that of Eq. (9.37). Hence,
the diagonal elements of H can be evaluated using

N

H ii =− H ij

j = 1

j = i

Impose the boundary conditions of displacements and tractions on Eq. (9.94) and move
all the unknown quantities to the left-hand side and all the known quantities to the right.
The final system of linear equations can appear as

AX = F (9.95)

The solution of Eq. (9.95) provides all the nodal displacements and tractions on the
boundary.
The characteristics of the global equations of the boundary element method have been
compared to those of the finite element method in Section 9.2.2. The comparisons apply to
linear elasticity problems as well.
See the references for more details of the boundary element formulation for linear elas-
ticity problems.

9.3.6 Displacements and Stresses Inside the Body
With all the boundary displacements and tractions known, the displacements at a point ξ

inside the body can be calculated using Eq. (9.91). Note that in this case, from Eq. (9.88),
when ξ is inside the body, c i = 1 and hence c = I , an identity matrix. Thus the displacements

u (ξ ) are

u ∗ N dS p j p ∗ N dS v j u ∗ p V dV

M M S

u (ξ ) = − + (9.96)

S j S j V S

j = 1 j = 1 s = 1