3 From boundary integral formulation to boundary element equations
The boundary element method is derived from the integral equation formulation
by discretizing the boundary and through the application of the method of
collocation. The term `element' includes both the manner in which it
approximates the geometry of the boundary and how the boundary function
is modelled on the element. In this section the stages in the
development of the direct method are given and the analagous
equations for the indirect method are stated.
The boundaries involved in the test problems are those of a circle
and a square. The boundary of a square is always exactly represented.
The circle is exactly represented in some tests and in other tests it
is approximated by straight line elements and quadratically curved
elements as illustrated in figure 1.
Figure 1.
The approximation of the boundary requires that equation (1)
is replaced by
{ M
~
j
}[S\tilde] - { L
~
v
}[S\tilde] »
~
c
(p)
~
j
(p)
where [(j)\tilde], [v\tilde] are the equivalent of
j, v projected from the true boundary onto the approximate boundary.
The other equations (2) and (3) are approximated similarly.
The approximation to the boundary function on S (this is either the potential
[(j)\tilde], its derivative [v\tilde], or the source density
[(s)\tilde])
is considered in three forms.
(1) being constant over each element, determined by its value at the
mid-point (constant elements),
(2) varying linearly over each element, being determined by its value at
the ends of the element (hat elements), and
(3) varying linearly over each element, being determined by its values
at two interior points. These are chosen as the Gauss-Legendre
points, ±[1/(Ö3)], for the standard interval [-1,1]
(Gauss elements).
The related basis functions are shown in figure 2.
Figure 2.
approximated as follows
~
m
(p) »
N å
j=1
mj cj(p)
(4)
where the cj(p) are the basis functions and
mj = m(pj) for j=1,2,...,N
and the p1, p2, ... , pn are the collocation
points.
For the constant and hat elements the number of collocation
points is equal to the number of elements. For the Guassian elements
we have twice the number of collocation points as elements.
The substitution of approximations like (4) into the
integral equation (1) gives the following
[M -
1
2
I] f » Lv
(5)
where L and M are matrices with
[L]ij = { L cj }[S\tilde]
with the matrix M being defined similarly and
j = [ j1, j2, ..., jN]T
and similarly for v.
Since the intention is to solve the Neumann problem then v
is given and by solving
[M - C]
^
f
= Lv
(6)
an approximation [^(f)] to f is obtained.
The matrix C = diag { [^c]1, [^c]2, ..., [^c]N }.
Similarly, for the indirect method, an approximation s
to [^(s)] on the boundary can be obtained through
solving the following equation
[Mt -
1
2
I]
^
s
= v .
(7)
The approximation [^(j)] to j
can be obtained through the following summation
^
f
= L
^
s
.
(8)
The solution in the domain can be obtained through discretising the
integrals (1) and (2) as described above and
susbtituting the results [^(f)] or
[^(s)].
Note that the indirect methods require the determination of the
normal at the collocation points, thus the boundary must be smooth
at these points. As a result c(pi)=[1/2] for
i=1,2,...,N and hence the matrix [1/2] I in
(7) rather than the more general matrix C in (6).
Figure 1.
Figure 2.