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H
11
H
12
···
H
1
N
1
1
1
.
H
21
H
22
···
=
0
.
.
.
···
H
N
1
H
N
2
···
H
NN
This is a system of homogeneous linear equations. The
i
th equation in this system is
N
H
ij
=
0
j
=
1
Thus, the diagonal entries
H
ii
of the matrix
H
are
N
H
ii
=−
H
ij
(9.39)
j
=
1
j
=
i
Impose the
u
and
q
boundary conditions on Eq. (9.37) and rearrange the equations such
that the unknown quantities are on the left-hand side and all the known quantities are on the
right. The final system of linear equations can appear as
AX
=
F
(9.40)
where
X
contains all of the unknown nodal values of
u
and
q
, and
F
contains the known
quantities. The solution of Eq. (9.40) provides the nodal values of
u
and
q
on the boundary.
It is of interest to compare some characteristics of Eq. (9.40) with the global equations of
the finite element method. The global stiffness matrix
K
of the finite element method is a
banded symmetric matrix for which some special techniques can be employed to solve the
equations. The boundary element method, on the other hand, leads to a full, unsymmetric
matrix
A
. Usually, such matrices are less efficient to solve than the banded symmetric ones.
Since the dimensionality of the problem is reduced by 1 relative to that of the finite element
method, the order of
A
is much smaller than that of
K
for the same problem, and hence the
boundary element method is generally more efficient than the finite element method, espe-
cially for a small surface-to-volume ratio. Also, the unknowns of the displacement method
of finite elements are nodal displacements only. A postprocessing routine is necessary to
compute the reaction forces at the locations where the displacements are prescribed. For
the boundary element method, however,
X
in Eq. (9.40) contains both
u,
often the nodal
displacements, and
q,
often the tractions on the boundary, so that the reaction forces are
calculated at the same time as the boundary displacements. This saves some effort in the
postprocessing computations.
More details of boundary element formulations are provided in Brebbia and Dominguez
(1992), Brebbia, et al. (1984), and Gipson (1987).
EXAMPLE 9.3 Calculation of the Torsional Constant
Use the direct boundary element method to calculate the torsional constant
J
of a 1
×
1
square cross section.
The formulation for the direct boundary element method for the pure torsion of a pris-
matic bar is given in Example 9.2, where the formula to calculate the torsional constant is
provided by Eq. (15).
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