4.1 Integral Equation Formulation
In this Section we consider the integral equation fomulations
of the interior Helmholtz equation. The range of papers
described earlier each consider the solution with a
particular type of boundary condition such as Dirichlet or Neumann.
An incident field along with
a general boundary condition
(4.1) is included and this leads to
more generalised boundary integral equations.
4.1.1 Direct Formulation
The application of Green's second theorem to the Helmholtz equation
gives the following equations:
{ M_{k} f}_{S} (p) + f(p) = { L_{k} v }_{S} (p) (p Î D) , 
 (4.2) 
{ M_{k} f}_{S} (p) + 
1
2

f(p) = { L_{k} v }_{S} (p) (p Î S) , 
 (4.3) 
where the Helmholtz integral operators,
L_{k} and M_{k}, are defined in Section 3.1
and v(p) = [(¶f)/( ¶n)].
The equations have the same structure as those given in Section 1.2
for the interior Laplace equation. Note that the normals
to the boundary are taken to be in the outward direction.
The above equations can be utilised to solve the
interior Helmholtz equation in the manner outlined in Section 1.2;
equation (4.2) gives (approximations to) both f and
v on the boundary S, equation (4.3) yields
an approximation to f(p) for any point p in the domain.
There is only one small difficulty with this approach and that
is in the case of a Dirichlet boundary condition equation (4.3)
is a Fredholm integral equation of the first kind.
In general
first kind equations are found to be difficult to solve and the
matrices that arise in their equivalent linear systems are
illconditioned (see [26], for example).
Even though first kind equations like (4.2), having
singular kernels, do not present the severe numerical problems
that those with smooth kernels do, it is found that their solution
can lead to a marginal loss of accuracy since the matrices
that arise have higher condition numbers and hence magnify
any numerical error [4].
Boundary integral equation reformulations of the interior
Helmholtz equation provide us with a selection from
which the possibility of having to solve a
first kind equations can be avoided.
Differentiating each term of equation (4.2) with respect to
any vector u_{(}p) gives

¶
¶u_{p}

{ M_{k} f}_{S} (p) + 
¶f(p)
¶u_{p}

= 
¶
¶u_{p}

{ L_{k} v }_{S} (p) (p Î D) , 

or
{ N_{k} f}_{S} (p;u_{p}) + 
¶f(p)
¶u_{p}

= { M_{k}^{t} v }_{S} (p;u_{p}) (p Î D) , 

using the notation of Section 3.1.
By taking the limit as the point p approaches the boundary
with the vector u_{p} being the unit outward normal to the boundary
at p (that is n_{p}),
and taking into account the jump properties of Section 3.1
the following boundary integral equation is obtained:
{ N_{k} f}_{S} (p; n_{p}) = { (M_{k}^{t}  
1
2

I) v }_{S} (p; n_{p}) (p) (p Î S) , 
 (4.4) 
where v(p) = [(¶f(p))/( ¶n_{p})].
Since the operators M_{k}^{t} and N_{k} are available through the
subroutines in Section 3, then it is straightforward to base the
boundary element method on equation (4.4). However, for
the Neumann problem we need to solve over the hypersingular
operator N_{k}, which can lead to some loss of accuracy,
similar to that experienced with the solution of the first
kind equation discussed earlier.
Since neither of equations (4.3) and (4.4) are
universally acceptable for solving the interior Helmholtz equation
for all boundary conditions of the form (4.1), a
hybrid equation is proposed that couples the two original
equations into a single equation
{ (M_{k} + 
1
2

I + mN_{k}) f}_{S} (p; n_{p}) = { (L_{k} + m(M_{k}^{t}  
1
2

I) ) v }(p; n_{p}) (p Î S) , 
 (4.5) 
where m( ¹ 0) is the coupling parameter.
The equation (4.5) provides a suitable basis of
a method for the solution of interior Helmholtz equation for all
boundary conditions and it is the equation employed in the
subroutines associated with this Chapter.
4.1.2 Indirect Formulation
Following on from the ideas in Subsection 1.2.4,
the corresponding indirect integral equation formulations to
(4.3) and (4.4) can be obtained by writing f
as a single or double layer potential;
f(p) = {L_{k} s_{0} }_{S}(p) or f(p) = {M_{k} s_{¥} }_{S}(p) (p Î D) 

where the s_{0} and s_{¥} are source density
functions defined on S. For points on the boundary
the equations become boundary integral equations;
f(p) = {L_{k} s_{0} }_{S}(p) or f(p) = {(M_{k} 
1
2

I) s_{¥} }_{S}(p) (p Î S) 

where the jump condition of Section 3.1 has been taken into account
in the second equation.
The integral equations arrived at in this way have the same
difficulties as the corresponding direct equation (4.3);
the Dirichlet problem is replaced by a first kind equation.
Again the problem can be circumvented by
using a hybrid formulation;
writing f as a weighted sum of single and double layer
potentials
f(p) = {(L_{k} + nM_{k}) s_{n} }_{S}(p) (p Î D) , 
 (4.6) 
where n is a weighting parameter. This gives rise to the
following boundary integral equation:
f(p) = {(L_{k} + n(M_{k}  
1
2

I)) s_{n} }_{S}(p) (p Î S) . 
 (4.7) 
The parameter n should be chosen in a similar way to the
parameter m in the direct formulation.
The equation (4.7) is only suitable for solving the Dirichlet
problem since it refers to f and not v on the boundary.
Differentiating equation (4.6) with respect to n_{p}
and taking the limit as
the point p approaches a point on the boundary gives
the following boundary integral equation:
v(p) = { (M_{k}^{t} + 
1
2

I + nN_{k} ) s_{n} }_{S} (p;n_{p}) (p Î S) , 
 (4.8) 
which can be used for the solution of the Neumann problem.
Both equations (4.7) and (4.8)
are required in the indirect solution of the Helmholtz equation with a
Robin boundary condition. In this case the relevant integral equation
is obtained through the substitution of the forms
(4.7) and (4.8) into the boundary condition (4.1)
to give
a(p) {(L_{k} + n(M_{k}  
1
2

I)) s_{n} }_{S}(p)+b(p) { (M_{k}^{t} + 
1
2

I + nN_{k} ) s_{n} }_{S} (p;n_{p}) = f(p) (p Î S) . 
 (4.9) 
4.1.3 Field Modification
The Helmholtz field
need not be a result of the boundary and boundary condition
alone; the surface
may simply act to modify an existing field. A simple example
of this is that of a loudspeaker in a room; the loudspeaker
produces an Helmholtz field that is modified by the
walls of the room.
In such cases there is an incident field
in the domain, termed f^{i}, which is the
field that would exist if there were no boundaries.
Such problems can also be solved by the boundary element method,
it only requires a generalisation of the integral equations
and the corresponding alteration
of the boundary element methods.
In the simplest case, the equation (4.2) may be
generalised as follows:
f(p) = f^{i}(p){ M_{k} f}_{S} (p) + { L_{k} v }_{S} (p) (p Î D) ; 
 (4.10) 
the solution f(p) is equated to the
incident field f^{i}(p) and modified by
the other terms. The boundary integral equation that
arises from the formulation (4.10) is as follows:
{ M_{k} f}_{S} (p) + 
1
2

f(p) = f^{i}(p) + { L_{k} v }_{S} (p) (p Î S) . 

The corresponding generalisation of (4.4) is
{ N_{k} f}_{S} (p; n_{p}) = v^{i}(p) +{ (M_{k}^{t}  
1
2

I) v }_{S} (p; n_{p}) _{S} (p) (p Î S) , 

where
v^{i}(p) = [(¶f^{i})/( ¶n_{p})](p).
The formulation employed in subroutines HIBEM2, HIBEM3
and HIBEMA is a hybrid of these equations
{ (M_{k} + 
1
2

I + mN_{k}) f}_{S} (p; n_{p}) = f^{i}(p) +mv^{i}(p) + { (L_{k} + m(M_{k}^{t}  
1
2

I) ) v }_{S}(p; n_{p}) (p Î S) . 
 (4.11) 
The equation (4.11) is the generalisation of (4.5)
and the equations are equivalent when there is no incident
field (f^{i}(p) = 0, v^{i}(p) = 0
for all p Î D ÈS).
Indirect formulation
Generalising equations (4.6)(4.8) to include the incident
field gives rise to the following integral equations:
f(p) = f^{i}(p) + {(L_{k} + nM_{k} ) s_{n} }_{S}(p) (p Î D) , 
 (4.12) 
f(p) = f^{i}(p) + {(L_{k} + n(M_{k}  
1
2

I)) s_{n} }_{S}(p) (p Î S) , 
 (4.13) 
v(p) = v^{i}(p) + { (M_{k}^{t} + 
1
2

I + nN_{k}) s_{n} }_{S} (p;n_{p}) (p Î S) . 
 (4.14) 
The indirect boundary integral equation for the solution of the
interior Helmholtz equation with the general Robin boundary
condition (4.1) and with an incident field is as follows:
a(p) { f^{i}(p) + (L_{k} + n(M_{k}  
1
2

I)) s_{n} }(p)+b(p) { v^{i}(p)+ (M_{k}^{t} + 
1
2

I + nN_{k} ) s_{n} }_{S} (p;n_{p}) = f(p) 
 (4.15) 
for (p Î S).
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