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* = j = 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:


Mk + 1
2
I + mNk

f fi + mvi +

Lk + m(Mkt - 1
2
I)

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



Mk + 1
2
I + mNk

^
f
 
= fi + mvi +

Lk + m(Mkt - 1
2
I)

^
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

j = 1 
{Mk ~
e
 
}DSj(p) ^
v
 

j 
+ n

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 +

Lk + n(Mk - 1
2
I)

sn      and
v vi +

Mkt + 1
2
I +nNk

sn .

Applying the boundary condition at the collocation points as in the direct method gives the equation

Da f + Db v = f .
(4.20)
Substituting the approximations for f and v given above into equation (4.18) gives the following:


Da { Lk + n(Mk - 1
2
I)} +Db { Mkt + 1
2
I +nNk }

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:



Da { Lk + n(Mk - 1
2
I)} +Db { Mkt + 1
2
I +nNk }

^
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

j = 1 
( { Lk ~
e
 
}DSj + n{ Mk ~
e
 
}DSj ) ^
s
 

nj 
 .

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