Interior Modal Analysis

| (6.1) |

| (6.2) |

The Helmholtz eigenvalue problem is amenable to solution via finite element or finite difference methods. In these cases, the problem reduces to that of solving a generalised linear eigenvalue problem of the form

| (6.3) |

In cases where it is applicable, it is well known that the boundary element method has an important advantage over the finite element and finite difference methods: the partial differential equation governing the domain is reduced to an integral equation relating values of f and [(¶f)/( ¶n)] on the boundary only. Hence the dimension of the problem is effectively reduced by one. However, the application of the boundary element method reduces the Helmholtz eigenvalue problem to that of solving an eigenproblem of the form

| (6.4) |

Because of the main advantage of the boundary element method
over finite element and finite difference methods stated earlier,
the matrix in (6.4) is generally much smaller than the
matrices in (6.3), for any given modal analysis problem and a given
level of required accuracy. The disadvantages of this approach
are that the eigenvalue problem (6.4) is non-linear and the
components of the A_{k} matrix are defined in terms of integrals
and hence may be costly to evaluate.
The solution of non-linear eigenvalue problems are
considered in references [55], [73]
and [84]. Unfortunately, standard algorithms for solving
non-linear eigenvalue problems are not generally available.

The problem of solving the Helmholtz eigenvalue problem
via boundary element-type methods have been given some consideration by
researchers. For example iterative methods such as the secant method
are applied to the problem of finding the roots of the
equation det (A_{k} ) = 0
in references [81], [27]
and [1].
However, this is not a satisfactory
method when the matrix A_{k} is large [84].
A similar method, based on finding the values of k for which the
smallest eigenvalues of A_{k} is zero is considered in [58].
Unfortunately, these methods are
unwieldy since they do not compute the solutions simultaneously;
they require a starting point to be chosen for each required eigenfrequency.

In reference [8] a hybrid of the boundary element and finite element method is introduced. The method seems to have the advantage of the finite element method in that a linear eigenvalue problem results and the advantage of the boundary element method in that a solution on the boundary only is sought in the main computation. The method is considered further in references [24], [2].

In general, both eigenfrequencies and eigenfunctions of the Helmholtz
problem will be of interest
The method considered in this Chapter was introduced
in Kirkup and Amini [46]. The method involves
approximating each
component of the matrix A_{k} by a polynomial in k in some
given sub-range of the full wavenumber range. This allows us to
re-write the non-linear eigenvalue problem (6.4) in the form
of a standard generalised eigenvalue problem. Thus all of the
eigenvalues in the sub-range are computed simultaneously.