Chapter 6
Interior Modal Analysis

In this Chapter it is shown how the boundary element method can be used to obtain the resonant wavenumbers and the mode shapes of an enclosed region; the computational solution of the interior Helmholtz eigenvalue problem. The problem is that of finding the values of the wavenumber k and a non-trivial scalar function f such that the Helmholtz equation
2 f(p) + k2 f(p) = 0    (p D)
(6.1)
is satisfied in an interior domain D with boundary S and subject to a homogeneous boundary condition of the form
a(p) f(p) + b(p) f(p)
np
= 0    (p S)
(6.2)
where a(p) and b(p) are known complex-valued functions of p ( S) and np is the unit outward normal to the boundary at p. The non-trivial solutions k = k* and f(p) = f*(p)  (p D S) are termed the characteristic wavenumbers and eigenfunctions and they are dependent on the boundary S and the boundary functions a(p) and b(p). The characteristic wavenumbers are all real numbers and they correspond to the resonant wavenumbers of the enclosed region. The eigenfunctions are equivalent to the mode shapes.

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

(K - k2 M) x = 0
(6.3)
where the matrices K and M (termed the stiffness and mass matrices) in (6.3) are sparse and structured and are independent of k. Standard computational algorithms are available for solving generalised linear eigenvalue problems. Indeed special techniques (such as iterative methods) are available for solving the general problem (6.3), given the special structure of the matrices and the fact that only a fraction of the full set of eigenvalues are generally required [69]. Hence eigenfrequency analysis of the Helmholtz problem via the finite element or finite difference method is straightforward.

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

Ak m = 0
(6.4)
where the matrix Ak is generally full, having no particular structure but with each component being a continuously differentiable complex-valued function of k.

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 Ak 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 (Ak ) = 0 in references [81], [27] and [1]. However, this is not a satisfactory method when the matrix Ak is large [84]. A similar method, based on finding the values of k for which the smallest eigenvalues of Ak 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 Ak 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.