The boundary element form of the Helmholtz equation is derived
through recasting the partial differential equation
as a boundary integral equation
and applying an integral equation method. The integral
equation that is selected for the purposes of this work
is the one derived through Green's second
theorem. Collocation is applied to derive its discrete form.
4.1 Formulation of the Helmholtz Eigenvalue Problem
The application of Green's second theorem to the Helmholtz equation
gives the following equations:
{ Mk j}S (p) + j(p) = { Lk
¶j
¶n
}S (p) (p Î D) ,
(14)
{ Mk j}S (p) + c(p) j(p) = { Lk
¶j
¶n
}S (p) (p Î S) ,
(15)
where the Helmholtz integral operators,
Lk and Mk, are defined as follows:
{ Lk m}P(p) º
ó õ P
Gk(p,q) m(q) dSq (p Î D ÈS ),
(16)
{ Mk m}P(p) º
ó õ P
¶Gk
¶nq
(p,q) m(q) dSq (p Î D ÈS ),
(17)
where nq is a unit outward normal to the boundary S at
q and p,
P Í S ,
and m(q) is a bounded function defined for
q Î P. Gk(p,q) is the free-space Green's
function for the Helmholtz equation:
Gk(p,q) =
eikr
4 pr
inthreedimensions,
where r=p - q, r=|r|.
The function c(p) is defined so that
4 pc(p) (p Î S)
is the solid angle subtended by the exterior at p.
Note that if
S is smooth at p then c(p) = [1/2].
In order to derive the integral equation form of the eigenvalue problem,
the homogeneous boundary condition is
enforced on the integral equation formulations (14), (15).
The ensuing formulation for the Neumann eigenproblem is
as follows:
{ Mk j}S (p) + c(p) j(p) = 0 (p Î S) ,
(18)
j(p) = - { Mk j}S (p) (p Î D) .
(19)
The strategy for solving the eigenvalue problem is thus that of computing
the solutions k* and j*(p) (p Î S)
to the integral equation (18).
The substitution of j*(p) (p Î S) into
(19) will then yield the domain solution j*(p) (p Î D).
Collocation is applied to derive the boundary element form of
the eigenvalue problems. Let z1(p), z2(p),
..., zn(p) (p Î S) be a linearly independent set
of basis functions with the property
n å
j=1
zj(p) º 1 (p Î S).
(20)
Let p1, p2,..., pn be the collocation points so that
zi(pj) = dij (i,j = 1,2,...,n) .
(21)
For the methods considered in this paper, the boundary
is smooth at the collocation points, so that c(pj) = [1/2]. This reduces
the integral equation formulation (18)
of the eigenvalue problems to the following
Ak j » 0 ,
where the matrix Ak is defined as follows
[Ak]ij = {Mk zi }S (pj) +
1
2
dij .
(22)
Approximations to the solutions can now be found through the
solution of the following non-linear eigenvalue problem
For a given problem and a given level of required accuracy, the
system of equations (23) will be much smaller than the
system of equations arising from the finite element method (13).
On the other hand, the straightforward application of the boundary element method
results in a non-linear eigenvalue problem (23)
and the components of the matrix Ak are defined in terms of
integrals and hence may be costly to evaluate.
A typical way of computing the approximations to the eigenfrequencies
is to compute the zeros of |Ak|, as considered in [9-12,14].
In Ya Yan Lu and Shing-Tung Yau
[27] solutions are sought through finding the values of k at which
Ak Ak* has a zero eigenvalue (the * denotes the
Hermitian transformation)
and in Bai [28] the values of k are sought for which
Ak has a zero singular value.
Each of these method
is likely to be costly since several matrix evaluations (each component
computed through the evaluation of an integral) will be necessary
(although the matrix components could be approximated by a simple
formula such as a polynomial in k, which is cheaper to evaluate).
A further drawback of these methods is that the solutions are not
computed simultaneously, a starting point is required for each
eigenfrequency.
Finally, the methods may not readily yield the eigenfunction.
The matrix Ak in (23) has the useful property that its
components are continuously differentiable complex-valued functions of
the wavenumber k. This means that the components can be satisfactorily
approximated by a piecewise polynomial. The approximation of
Ak by a matrix polynomial allows us to replace the
eigenvalue problem (23) with a standard generalised
eigenvalue problem. This method will be described fully in section 6.