5.4  Improved Boundary Element Methods

The first step is to approximate the boundary S by a set of n panels S* = åj = 1n 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 » 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

Da f + Db v = f
(5.24)
where Da, Db are diagonal matrices with [Da]ii = a(pi), [Db]ii = b(pi). Substituting the approximations for f and v given above into equation (5.24) gives the following:
é
ê
ë 		Da { Lk + n(Mk + 1
2
 I)} +Db { Mkt - 1
2
 I +nNk } ù
ú
û 		sn  » f - Da fi- Db vi ,
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 sn. This can be done by solving 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  = f -Da fi- Db vi .
(5.25)

The equation (5.25) is simply a matrix-vector equation that can be immediately solved by Gaussian elimination-type 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) = fi(p) + n
å
 j = 1 
 ( { Lk ~
e
 
 }DSj +n{ Mk ~
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 		Mk - 1
2
 I + mNk ù
ú
û 		f » -fi - mvi+ é
ê
ë 		Lk + m(Mkt + 1
2
 I) ù
ú
û 		v .
(5.26)
The Lk, Mk, Mkt and Nk are n ×n matrices arising from the discretisation method outlined in Sections 1.2 and 3.3; 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
 
(5.27)
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 ,
(5.28)
with ai = a(pi), bi = b(pi) and the Da and Db denote diagonal matrices with [Da]ii = ai and [Db]ii = bi, 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 matrix-vector 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) » fi(p) + n
å
 j = 1 
 {Mk ~
e
 
 }DSj ^
v
 
j 
 - n
å
 j = 1 
 {Lk ~
e
 
 }DSj ^
f
 
j 
or
^
f
 
 (p) = fi(p) + n
å
 j = 1 
 {Mk ~
e
 
 }DSj ^
v
 
j 
 - n
å
 j = 1 
 {Lk ~
e
 
 }DSj ^
f
 
j 
  .
(5.29)
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