5.3 Improved Formulations
The term improved formulations is used to include the
integral equation representations that were introduced in the
1960s and early 1970s for which the integral operator
over which a solution was sought was always nonsingular.
Hence the formulations were intended to form
a firm foundation for the boundary element method
solution of the exterior Helmholtz equation. The general drawback
with these methods is the inclusion of the N_{k} operator, which
is very difficult to discretise. However, for the elements
considered in this manual, the discrete N_{k} operator is
available through the subroutines outlined in Chapter 3.
Hence it is feasible to implement the improved methods
and, because of their robustness in comparison with the
methods outlined earlier, boundary element methods that
result from them are implemented in the
accompanying software and considered in the next Sections.
In this Chapter a particular implementation of an improved
method is described and this forms the basis of the
subroutines HEBEM2, HEBEM3 and HEBEMA
that are introduced in Section 5.5.
5.3.1 Improved Indirect Formulation
The equations for the improved indirect formulation that are
directly applicable to the solution of the Dirichlet problem
were apparently introduced independently by Brakhage and
Werner [12], Leis [57] and Panich [66].
The formulation involves writing f(p) for
p Î E as a linear
sum of single and doublelayer potentials;
f(p) = { ( L_{k} + nM_{k} ) s_{n} }_{S}(p) (p Î E) . 
 (5.13) 
For points on the boundary the equation becomes
f(p) = { ( L_{k} + n( M_{k}  
1
2

I ) ) s_{n} }_{S}(p) (p Î S) . 
 (5.14) 
For a given Dirichlet boundary condition
the boundary integral equation (5.14) has a unique solution
provided the parameter n is such that Im(n) ¹ 0.
Having computed s_{n}(p) for p Î S,
the solution in the domain can be computed through
substituting it into an approximation based on (5.13).
An integral equation that is suitable for solving the exterior
Neumann problem can be derived through differentiating (5.13)
with respect to the outward normal to the boundary
and taking the limit as p approaches the boundary
at the base of the normal. This
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) , 
 (5.15) 
which is attributed to Kussmaul [54].
In correspondence with the earlier formulation
(5.14), the equation (5.15) has a unique solution provided
Im(n) ¹ 0. The solution of the Neumann problem can be
obtained from finding an approximation to s_{n}
by solving the boundary integral equation (5.15). The
solution in the domain can then be obtained by
approximating (5.13).
For the general Robin problem the boundary condition takes
the form (5.1).
Substituting the expressions for f(p) and v(p)
obtained earlier into the boundary condition gives the following
integral equation:
{ (a(p) ( L_{k} + n( M_{k}  
1
2

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

I + nN_{k} ) )s_{n} }_{S}(p;n_{p}) = f(p) . 
 (5.16) 
Given a suitable Robin condition, the functions a(p),
b(p) and f(p), an approximation to
s_{n}(p) for p Î S can be obtained
from the numerical solution of the integral equation (5.16).
The solution at points of the domain E can then be found by
the substitution of the result into (5.13).
5.3.2 Improved Direct Formulation
The improved direct formulation originates in the
paper by Burton and Miller [15]. The formulation
is a hybrid of the elementary direct formulation (5.3)
and equation that arises through differentiating that
equation with respect to the normal to the boundary;
{ N_{k} f}_{S} (p;n_{p}) = { ( M_{k}^{t} + 
1
2

I ) v }_{S} (p;n_{p}) (p Î S) . 

The improved direct integral equation formulation is simply a
linear combination of (5.3) with this equation, giving
the following:
{ (M_{k}  
1
2

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

I ) )v }_{S} (p;n_{p}) (p Î S) . 
 (5.17) 
The boundary integral equation can be used to solve both the
Neumann and Dirichlet problems for the
exterior Helmholtz equation; the numerical solution of
(5.17) gives both functions f and v on S and
approximation of (5.2) gives the solution at any point
in the exterior. The integral equation (5.17) has a unique
solution provided Im(m) ¹ 0 [15].
For the general Robin problem the integral equation (5.17)
must be solved alongside the specified Robin condition (5.1).
A method for carrying this out will be given in the next Section.
5.3.3 Scattering
When an incident Helmholtz field is modified by an obstacle
the result is termed the scattered field.
The concept of scattering is analagous
to that of field modification considered in
the previous Chapter, and the inclusion of the incident
field term in the integral equation formulations is
entirely similar.
In scattering problems there is an incident field
in the domain, termed f^{i}, which is the
field that would exist if there were no boundaries,
or the freespace Helmholtz field.
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 resulting boundary element methods.
To complete this Section the improved integral reformulations of the
Helmholtz equation are generalised. The improved methods
based on these formulations are given in the next Section.
The generalisation of the elementary methods and the Schenck method
in order to include the scattering term is not explicitly
carried out in this text, although the development of such methods
should be clear from the formulations given.
Indirect formulation
The inclusion of the scattering term generalises the
integral equations (5.13), (5.14), (5.15) as follows:
f(p) = f^{i}(p) +{ ( L_{k} + nM_{k} ) s_{n} }_{S}(p) (p Î E) , 
 (5.18) 
f(p) = f^{i}(p) +{ ( L_{k} + n( M_{k} + 
1
2

I ) ) s_{n} }_{S}(p) (p Î S) , 
 (5.19) 
v(p) = v^{i}(p) +{ ( M_{k}^{t}  
1
2

I + nN_{k} ) s_{n} }_{S}(p;n_{p} ) (p Î S) 
 (5.20) 
where v_{i} = [(¶f_{i})/( ¶n_{p})].
For a Dirichlet (Neumann) boundary condition, the solution can be found
by solving (5.19) ((5.20))
to find s_{n} and then substituting
the result into (5.18) to find f in E.
Substituting the expressions (5.19) and (5.20)
for f and v into the equation for the more general
Robin boundary condition (5.1) gives
{ ( a(p) { f^{i}(p) + (L_{k} + n( M_{k} + 
1
2

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

I + nN_{k} ) s_{n} }_{S}(p; n_{p}) = f(p) . 
 (5.21) 
Direct formulation
The solution of the exterior Helmholtz problem on the
boundary S can be determined through solving the
following integral equation:
{ (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}) 
 (5.22) 
for (p Î S),
a generalisation of equation (5.17),
subject to the boundary condition (5.1).
Once f(p) and v(p) are obtained through
solving the above equation, the solution in the domain
can be obtained by the integration
f(p) = f^{i}(p) +{ M_{k} f}_{S} (p) { L_{k} v }_{S} (p) (p Î E) . 
 (5.23) 
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