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Substitute Eqs. (9.103) and (9.104) into (9.34) and (9.92), respectively, to find the expres-
sions for
H
ij
and
G
ij
in terms of
ξ
and
η.
The integrations can be performed using the
Gauss quadrature described in Chapter 6.
For constant elements, the shape function for the displacement is a constant. Use of a
constant shape function for the geometry of the element may cause inaccuracies in forming
the element matrices. For these elements, the geometry of the element may be represented
by the shape functions for linear or quadratic elements. This is similar to the use of super-
parametric elements in the finite element method.
At corners where two elements meet, the derivatives
q
may be discontinuous. If there
is a node at the corner, the determination of
q
may be quite difficult. To alleviate this
problem, introduce a
discontinuous element.
Split the corner node into two nodes and shift
each of these two nodes a small distance into one of the two adjacent elements forming
the corner. The two elements are now discontinuous. Each of these two nodes belongs
to different elements and the values
u
and
q
can be determined using Eq. (9.32). When
the discontinuous elements are assembled into a system with the conventional continuous
elements, for each discontinuous element, add one to the total number of nodes.
There are a variety of alternatives available for treating the boundary element problem.
For example, return to Eq. (9.88)
p
∗
u
dS
u
∗
p
dS
u
∗
cu
+
−
−
p
V
dV
=
0
(9.105)
S
S
V
If an exact solution can be found, it must satisfy Eq. (9.88) exactly. For the boundary element
solution, the substitution of the approximate shape functions of Eqs. (9.89) and (9.90) into
Eq. (9.105) does not make the left hand side of the equation zero but results in a residual.
Let this residual be
R
i
for node
i
, then
R
i
=
H
i
V
−
G
i
P
−
B
i
(9.106)
where
H
i
,
G
i
and
B
i
are obtained after assembling the element matrices
H
ij
,
G
ij
and
B
i
of Eq. (9.93). Various techniques of treating this residual are given in Chapter 7. For ex-
ample, the collocation method is quite suitable. The shape functions can be employed as
approximate solutions to the problem, and the nodes as the collocation points. The standard
collocation method introduced in Chapter 7 can be employed to find the solution at the
collocation (nodal) points. Also, the least square collocation method can be used here.
Another possibility is the use of the minimax method. The objective of the minimax for-
mulation of this problem is to make the maximum residual max
|
R
i
|
a minimum. In Chapter
7, Eq. (7.21) the maximum residual is denoted by
φ
, a scalar. In Eq. (9.106), however, the resid-
ual is a vector, so
φ
is replaced by the norm of a vector, i.e.,
φ
=
max
|
R
i
|
Then the minimax problem is stated as: Find the unknown elements in
V
and
P
such that
the vector
φ
is minimized
−
φ
≤
+
φ
≥
.
under the conditions
R
i
This is in the form of Chapter 7, Eq. (7.23).
Minimizing
φ
means that the components of
φ
are minimized. A linear programming
computer program can be used to solve this problem.
The minimax formulation is especially useful when the problem statement includes con-
straints on certain displacements and tractions. For example, when the prescribed boundary
conditions contain the condition that at some points the displacements are required to be
0 and
R
i
0
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