6  Test problem - the Burton and Miller Equation

A direct or indirect integral equation formulation of the Helmholtz equation will usually consist of some or all of the Helmholtz integral operators introduced in section 2. The solution of the Helmholtz equation (1) in the exterior E to a closed boundary S with the appropriate radiation condition can be reformulated as the Burton and Miller integral equation [7],

a{ Mk f}S (p) - a
2
 f(p)+ b
2
 { Nk f}S (p;np) =                                             

a{ Lk f
n
 } S (p)+ b{ Mkt f
n
 } S (p;np)+ b
2
 f(p)
np
     for p Î S  ,
(41)
where S is smooth at p, a and b are complex numbers, I is the identity operator and np is the unit outward normal to S at p. The solution in E can be related to the solution on S through the following equation

f(p) = { Mk f}S (p) - {Lk f
n
 }S (p)    for p Î E
(42)
The equation (41) is a useful problem with which the subroutines given in the following section can be tested as it contains all four integral operators and test solutions can easily be devised. Only the solution of the Neumann problem will be considered in this paper, that is f/ nq = v(p) given for all p Î S.

The solution of the Helmholtz problem consists of two stages. The primary stage involves the solution of the integral equation (41). This yields the solution f(p) for points on S. The secondary stage involves computing the solution f(p) for points in E using (42). Methods of this type, known as the Boundary Element Method (see, for example Banerjee and Butterfield [4]), for solving the exterior Helmholtz equation have been considered in references [32], [8], [26], [9], [29], [36], [34], [31], [10], [1], [28], [14], [2], [17], [18], [19], [22] [24].

Applying collocation to (41) generally requires that the closed boundary S is replaced by an approximate boundary S made up of a set of n elements DSj (j = 1,..,n) in the way described in section 3. Let the points pj (j = 1,..,n) with pj Î DSj be the collocation points. In this paper we consider only the approximation of the boundary functions by a constant on each elment. It is helpful to adopt the following notation:

[ Lk ]ij = { Lk ~
e
 
 }DSj(pi) ,
(42)

[ Mk ]ij = { Mk ~
e
 
 }DSj(pi) ,
(43)

[ Mkt ]ij = { Mkt ~
e
 
 }DSj(pi; npi) ,
(44)

[ Nk ]ij = { Nk ~
e
 
 }DSj(pi ; npi) .
(45)
where npi is the unit outward normal to S at pi. This gives the four n ×n matrices Lk, Mk, Mkt and Nk. The approximate boundary functions can be approximated by a vector

m = [ ~
m
 
 (p1),..., ~
m
 
 (pn) ]T.

The application of collocation to the Burton and Miller equation give the following linear systems of approximations

[ a( Mk - 1
2
 I ) + bNk ]  f   »  [ aLk + b( Mkt + 1
2
 I ) ]  v
(46)
where vj = v(pj) for j = 1,...,n. Hence the primary stage of the boundary element method entails the solution of the following linear system of equations:

[ a( Mk - 1
2
 I ) + bNk ]   ^
f
 
 =  [ aLk + b( Mkt + 1
2
 I ) ]  v
(47)
which yields an approximations to f(pj), for j = 1,...,n.

The secondary stage of the boundary element method requires the calculation of the approximation to f(p) where p is a point in the approximate exterior domain E. For this the discrete forms are substituted into (42) to give

^
f
 
 (p) = n
å
 j = 1 
 [ { Mk ~
e
 
 }DSj(p) ^
f
 
 j -{ Lk ~
e
 
 }DSj(p) vj]  (p Î ~
E
 
 ).
(48)
Note that the secondary stage requires the evaluation of only two integral operators in contrast with the primary stage which requires all four. Note also that the special evaluation techniques of subtracting out the singularity are required only for the diagonal components of the matrices in (43), (46). This latter point is a typical property of integral equation methods, the outcome of which is that the generally greater cost of evaluating the discrete forms when p lies on the element is not important when assessing the overall computational cost.

Test problems can easily be devised by setting f(p) = Gk(p, p*) where p Î E ÈS, p* Î D (D is the interior to S) with f/ vp (p) = Gk/ vp (p,p*) for p Î S.