4.2 Boundary Element Method
By approximating the operators in the boundary integral equations
in the way described in Section 3.3, the equations can each be
reduced to a linear system of equations,
as demonstrated in Section 1.2.
The first step is to approximate the boundary S by a set
of n panels S* = åj = 1n DS*j,
as described in Chapter 2.
The integral operators are approximated by the technique
outlined in Chapter 3
and the
resulting
linear system of equations is solved. The overall method
is equivalent to the
solution of the integral equations by collocation.
4.2.1 Direct Boundary Element Method
The application of collocation to the integral equation (4.11)
reduces it to the following linear system of equations:
|
|
é
ê
ë
|
Mk + |
1
2
|
I + mNk |
ù
ú
û
|
f » fi + mvi + |
é
ê
ë
|
Lk + m(Mkt - |
1
2
|
I) |
ù
ú
û
|
v |
| (4.16) |
The Lk, Mk, Mkt
and Nk are n ×n matrices arising from the discretisation
method outlined in Section 1.2 and Chapter 3; for example the components
of Lk are defined by
[Lk]ij = {Lk [e*] }DS*j(pi),
where [e*] is the unit function.
The vectors f and v represent the
values of the boundary functions f and v at the
collocation points. The method involves finding the solution of the
linear system of equations
|
|
é
ê
ë
|
Mk + |
1
2
|
I + mNk |
ù
ú
û
|
|
^
f
|
= fi + mvi + |
é
ê
ë
|
Lk + m(Mkt - |
1
2
|
I) |
ù
ú
û
|
|
^
v
|
|
| (4.17) |
subject to the boundary condition applied at the collocation points;
|
ai |
^
f
|
i
|
+bi |
^
v
|
i = fi for i = 1,2,...,n or Da |
^
f
|
+ Db |
^
v
|
= f , |
| (4.18) |
with
ai = a(pi),
bi = b(pi),
fi = f(pi)
and the
Da and Db denote diagonal matrices
with
[Da]ii = ai and
[Db]ii = bi,
to find [^(f)] and [^(v)].
In the cases of a pure Dirichlet or pure Neumann boundary condition
the equations can be solved by a standard method such as
LU factorization or Gaussian elimination. However in the general
case the equations
(4.17)
can be applied to rearrange the linear
system of equations (5.27) and the matrix-vector equation
that arises can then be solved by standard methods. The method
used for solving systems of equations of the form
(4.17),
(4.18) is described in Appendix 3.
Once [^(f)] and
[^(v)] are obtained, equation (4.10)
can be used to return an approximation to the
solution at any point p in the
domain;
|
|
^
f
|
(p) = fi(p) - |
n
å
j = 1
|
{Mk |
~
e
|
}DSj(p) |
^
v
|
j
|
+ |
n
å
j = 1
|
{Lk |
~
e
|
}DSj(p) |
^
f
|
j
|
. |
| (4.19) |
4.2.2 Indirect Boundary Element Method
The collocation method reduces the indirect boundary integral equations
(4.13), (4.14) to the linear systems of approximations
|
f » fi + |
é
ê
ë
|
Lk + n(Mk - |
1
2
|
I) |
ù
ú
û
|
sn and |
|
|
v » vi + |
é
ê
ë
|
Mkt + |
1
2
|
I +nNk |
ù
ú
û
|
sn . |
|
Applying the boundary condition at the collocation points as in
the direct method gives the equation
Substituting the approximations for f
and v given above into equation (4.18)
gives the following:
|
|
é
ê
ë
|
Da { Lk + n(Mk - |
1
2
|
I)} +Db { Mkt + |
1
2
|
I +nNk } |
ù
ú
û
|
sn + Da fi + Db vi » f |
|
which is also the discrete analogue of equation (4.15).
In the indirect boundary element method, the first stage is to
find the approximation
[^(s)]n to the
source density function
s. This can be done through the solution of
the following linear system of equations:
|
|
é
ê
ë
|
Da { Lk + n(Mk - |
1
2
|
I)} +Db { Mkt + |
1
2
|
I +nNk } |
ù
ú
û
|
|
^
s
|
n + Da fi + Db vi = f . |
| (4.21) |
The form of this equation is more straightforward than
the corresponding equations for the direct method (4.17),
(4.18).
The equation (4.21) is simply a matrix-vector equation
that can be immediately solved by Gaussian elimination-type methods.
Having obtained the solution to (4.21),
the approximation to sn,
the approximate
solution in the domain can be found using the discrete
equivalent of (4.12);
|
|
^
f
|
(p) = fi(p) + |
n
å
j = 1
|
( { Lk |
~
e
|
}DSj + n{ Mk |
~
e
|
}DSj ) |
^
s
|
nj
|
. |
|
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