5.4 Improved Boundary Element Methods
The first step is to approximate the boundary S by a set
of n panels S* = å_{j = 1}^{n} DS*_{j},
as considered 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.
5.4.1 Improved Indirect Method
The collocation method reduces the indirect boundary integral equations
(5.19), (5.20) to the linear systems of approximations
f » f^{i} + 
é ê
ë

L_{k} + n(M_{k} + 
1
2

I) 
ù ú
û

s_{n} and 

v » v^{i} + 
é ê
ë

M_{k}^{t}  
1
2

I +nN_{k} 
ù ú
û

s_{n} . 

Applying the boundary condition at the collocation points as in
the direct method gives the equation
where D_{a}, D_{b} are diagonal matrices with
[D_{a}]_{ii} = a(p_{i}),
[D_{b}]_{ii} = b(p_{i}).
Substituting the approximations for f
and v given above into equation (5.24)
gives the following:

é ê
ë

D_{a} { L_{k} + n(M_{k} + 
1
2

I)} +D_{b} { M_{k}^{t}  
1
2

I +nN_{k} } 
ù ú
û

s_{n} » f  D_{a} f^{i} D_{b} v^{i} , 

which is also the discrete equivalent of (5.21).
In the indirect boundary element method, the first stage is to
find the approximation
[^(s)]_{n} to the
representation of the source density function
s_{n}. This can be done by solving
the following linear system of equations

é ê
ë

D_{a} { L_{k} + n(M_{k} + 
1
2

I)} +D_{b} { M_{k}^{t}  
1
2

I +nN_{k} } 
ù ú
û


^ s

_{n} = f D_{a} f^{i} D_{b} v^{i} . 
 (5.25) 
The equation (5.25) is simply a matrixvector equation
that can be immediately solved by Gaussian eliminationtype methods.
Having obtained [^(s)]_{n},
the solution to (5.25),
the approximate
solution in the domain can be found using the discrete
equivalent of (5.18):

^ f

(p) = f^{i}(p) + 
n å
j = 1

( { L_{k} 
~ e

}_{DSj} +n{ M_{k} 
~ e

}_{DSj} ) 
^ s

nj

. 

5.4.2 Improved Direct Method
The application of collocation to the integral equation (5.22)
reduces it to the following linear system of equations:

é ê
ë

M_{k}  
1
2

I + mN_{k} 
ù ú
û

f » f^{i}  mv^{i}+ 
é ê
ë

L_{k} + m(M_{k}^{t} + 
1
2

I) 
ù ú
û

v . 
 (5.26) 
The L_{k}, M_{k}, M_{k}^{t}
and N_{k} are n ×n matrices arising from the discretisation
method outlined in Sections 1.2 and 3.3; for example the components
of L_{k} are defined by
[L_{k}]_{ij} = {L_{k} [e*] }_{DS*j}(p_{i}) ,
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

M_{k}  
1
2

I + mN_{k} 
ù ú
û


^ f

= f^{i}  mv^{i}+ 
é ê
ë

L_{k} + m(M_{k}^{t} + 
1
2

I) 
ù ú
û


^ v


 (5.27) 
subject to the boundary condition applied at the collocation points
a_{i} 
^ f

i

+b_{i} 
^ v

_{i} = f_{i} for i = 1,2,...,n or D_{a} f + D_{b} v = f , 
 (5.28) 
with
a_{i} = a(p_{i}),
b_{i} = b(p_{i}) and the
D_{a} and D_{b} denote diagonal matrices
with
[D_{a}]_{ii} = a_{i} and
[D_{b}]_{ii} = b_{i},
to find [^(f)] and [^(v)],
the approximations to f and v.
In the cases of a pure Dirichlet or pure Neumann boundary condition
then the equations can be solved by a standard method such as
LU factorization or Gaussian elimination. However in the general
case the equations (5.28) can be used to rearrange the linear
system of equations (5.27) and the matrixvector equation
that arises can then be solved by standard methods. The method
is carried out by subroutine
CGLS used for solving systems of equations of the form
(5.27), (5.28) is described in Appendix 3.
Once [^(f)] and
[^(v)] are obtained, equation (5.23)
can be used to return an approximation to the
solution at any point p in the
domain:
f(p) » f^{i}(p) + 
n å
j = 1

{M_{k} 
~ e

}_{DSj} 
^ v

j

 
n å
j = 1

{L_{k} 
~ e

}_{DSj} 
^ f

j



or

^ f

(p) = f^{i}(p) + 
n å
j = 1

{M_{k} 
~ e

}_{DSj} 
^ v

j

 
n å
j = 1

{L_{k} 
~ e

}_{DSj} 
^ f

j

. 
 (5.29) 
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