7 Concluding Discussion
In this paper a new numerical method for the solution of the Helmholtz
eigenvalue problem is described. Essentially, the method is derived
from a polynomial approximation in the wavenumber k of the matrix
that arises when the boundary element method is applied to the
Helmholtz equation. The resulting matrix polynomial eigenvalue
problem is recast as a
generalised eigenvalue problem with companion matrix structure (11),
which is solved through the use of standard methods. An implementation
of a particular straightforward form of the method is described
and results from the application of the method to test problems
are presented.
The results show the method working effectively,
with clear convergence to the exact solution as the number of elements
and the accuracy of the polynomial interpolation increases.
Inspection of the results reveals that the error is characterised by
the sum of a component resulting from the approximation of the
boundary functions and a component from the polynomial approximation of the
matrix. The results indicate that the approximation
to the boundary function transmits an error of O(n-2), except
for the computation of the eigenvalues of Mkt + [1/2] I
on the square, where the error seems to be O(n-1). This
reduction in the order of convergence could be attributed to the
lack of smoothness of the solution of the indirect
formulation (6). The error
with respect to the polynomial approximation of the matrix components
is more difficult to characterise. It seems that the error is
governed by the proximity of the eigenfrequency to the nearest
interpolation point as well as the general accuracy of the polynomial
approximation.
No particular effort was made to optimise the efficiency of the overall
method. Clearly, there is
some scope for such improvements. For example the number of quadrature
points used to compute the matrix components could be reduced. The efficiency
could be improved by adjusting the polynomial approximation method
so that the error it contributes is approximately the same as the
error resulting from the collocation method. An alternative to the
QZ algorithm for solving equation (11), taking advantage
of its structure, could also be of benefit. In the tests constant elements
have been used, in general higher order elements would result in a more
efficient method.
In practical situations, a set of eigenfrequencies and corresponding
mode shapes will be required. The most straightforward way
of applying the method is to choose a degree for the
polynomial approximation and the width of subinterval [kA, kB],
then to step through the intervals,
computing the successive solutions. It is difficult to compare this
method with the existing methods for computing Helmholtz eigenvalues
outlined in the introduction on the grounds of computational efficiency.
However, the method described in this paper is reasonably straightforward to
implement and could prove a useful addition to existing boundary element
software libraries.
Acknowledgement. The work in this paper was funded by a SERC/MOD grant
(GR/G01416). The Cray Supercomputer at
Rutherford Appleton Laboratory was used to obtain the results
from the test problems (Supercomputing Grant GR/F61639). The authors
are grateful to the reviewers for their helpful suggestions.