5.1 Elementary Formulations and Methods
The elementary formulations are those that are derived from Green's
second theorem or from simple layer potential representation of
the solution. Although the methods are not suitable for
general computation, it is important to introduce them as the
alternative methods originate with these formulae.
5.1.1 Elementary Direct Formulations
The application of Green's second theorem to the Helmholtz equation
gives the following equations:
|
{ Mk f}S (p) - f(p) = { Lk v }S (p) (p Î E) , |
| (5.2) |
|
{ Mk f}S (p) - |
1
2
|
f(p) = { Lk v }S (p) (p Î S) , |
| (5.3) |
where Lk and Mk, are defined in Section 3.1
and v(p) = [(¶f)/( ¶n)]. The equations are
also termed the Helmholtz formula or the Helmholtz-Kirchoff
equation. The equation (5.3) is often also termed the
surface Helmholtz equation. In this text the equations
will be known as the
elementary direct formulation.
The most straightforward approach to solving
the exterior Helmholtz equation via
the elementary formulations is to first find f and v
on the boundary from equation (5.3). The value
of f(p) for points p in the domain
can then be obtained through the discretisation of equation
(5.2).
5.1.2 Elementary Indirect Formulations
The elementary indirect integral equation formulations
can be obtained by writing f
as a single- or double-layer potential;
|
f(p) = {Lk s0 }S(p) or f(p) = {Mk s¥ }S(p) (p Î E) |
| (5.4) |
where the s0 and s¥ are source density
functions defined on S. For points on the boundary
the equations become boundary integral equations;
|
f(p) = {Lk s0 }S(p) or f(p) = {(Mk+ |
1
2
|
I) s¥ }S(p) (p Î S), |
| (5.5) |
where the jump conditions of Section 3.1 have been taken into account.
In order to solve the exterior Helmholtz equation with a
Dirichlet boundary condition using the above equations
the first step is to solve one of the equations
to obtain an approximation to
either s0 or s¥. The solution
at any point p in the domain can
then be obtained by approximating the relevant equation of (5.4).
For the Neumann problem, further equations must be derived.
Differentiating the equations (5.4)
with respect to np, the normal to a point p Î S,
and taking the limit as the point approaches S returns the
following equations:
|
v(p) = { (Mkt - |
1
2
|
I) s0 }S(p; np) or v(p) = { Nk s¥ }S(p; np) (p Î S) |
| (5.6) |
The solution of one of the integral equations (5.6) returns
either s0 or s¥ and the solution in the domain
can then be obtained from (5.4).
The solution of the Robin problem by elementary indirect
methods can be achieved by substituting the expressions
for f or v on S into the boundary condition (5.1):
|
a(p) {Lk s0 }S(p) +b(p) { (Mkt - |
1
2
|
I) s0 }S(p;np) = f(p) or |
|
|
a(p) {(Mk+ |
1
2
|
I) s¥ }S(p;np) +b(p) { Nk s¥ }S(p) = f(p) . |
|
Once an approximation to the solution s0 is obtained from
the first of these equations or an approximation to
s¥ is obtained from
the second, the solution in the domain can be obtained
through approximation of the relevant equation of
(5.4) .
5.1.3 Elementary Methods
Methods that are derived straightforwardly from the integral
equations of the previous Section are termed elementary methods.
A typical example is the solution of the exterior Neumann
problem using equations (5.2), (5.3). The application
of collocation to the integral equation (5.3) reduces
it to the following system of equations:
|
|
é
ê
ë
|
Mk - |
1
2
|
I |
ù
ú
û
|
|
^
f
|
= Lk |
^
v
|
|
| (5.7) |
The Lk, Mk are n ×n matrices arising from the discretisation
method outlined in Sections 1.1 and 2.3; 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 and
[^(f)] and [^(v)] are
their approximations.
The solution at any point p in the domain can then
be approximated using
|
|
^
f
|
(p) = |
n
å
j = 1
|
{ Mk |
~
e
|
}DSj(p) |
^
f
|
j
|
- { Lk |
~
e
|
}DSj(p) |
^
v
|
j
|
, |
| (5.8) |
the discrete equivalent of (5.2).
Unfortunately, such methods have been found to be very
unsuitable for the general computational solution of the exterior
Helmholtz equation. The underlying reason for this is that
for each formulation on each boundary with a given form
of boundary condition the operator over which we solve
is singular for certain values of k*, often termed
the characteristic wavenumbers.
For example the direct solution of the exterior Neumann problem
requires the solution of equation (5.3),
obtaining f from v. However at
a set of real values k*, the eigenfrequencies of the interior
Dirichlet problem, the operator Mk* - 1/2 I is
singular.
If k takes the
value of any one of the values k* then a solution is impossible.
It might be argued that the wavenumbers for which a solution
to the Helmholtz equation is sought do not coincide with
any of the characteristic wavenumbers and so the problem does
not arise. However, the operator over which we solve is
not only singular at k = k* it is ill-conditioned
for values of k in the neighbourhood of k*; the
condition of the operator being approximately proportional to
[1/( |k-k*|)] [4].
The condition
of the operator over which a solution is obtained in integral
equation methods is one of the most important factors governing the
numerical error. In this case the numerical error can be
characterised by [(e)/( |k-k*|)], following
the profile of the condition of the operator. The
e is determined by the nature of the problem and
the accuracy of the boundary representation and the
boundary function approximation and it tends to increase
gently with k.
Given also the fact that the characteristic wavenumbers
tend to cluster more and more at higher real wavenumbers,
the elementary methods are generally unsatisfactory.
Reports on results of implementations of elementary methods
first appeared in the 1960s: Banaugh and Goldsmith [6],
Chen and Schweikert [18], Chertock [21]
and Brundrit [14]. All but the last of these
references seem to have been unaware of the difficulties
with the methods at the characteristic wavenumbers.
The computational performance of elementary methods is
compared with various alternative methods in
Schenck [75], Meyer et al [60],
Sayhi et al [74], for example. Because
of the perceived computational difficulties, research has
generally moved away from the elementary methods and towards alternative
methods. A formal analysis of the solution of the exterior
Helmholtz equation by elementary methods is given
in Amini and Kirkup [4]. The convergence of the error
with respect to element size and the accuracy of
the representation of the boundary and boundary
functions is considered in Juhl [38].
Return to boundary-element-method.com/helmholtz
Return to boundary-element-method.com