9 Concluding Discussion

In this paper three different approaches to the computational acoustic modal analysis of an enclosed fluid are reviewed. The finite element method is now a well established technique in this application. The disadvantages of this method are that the full domain rather than just the boundary require discretization and matrices that arise in the eigenvalue problem (13) are generally much larger than the corresponding system (23) arising from the boundary element method.

Some of the current computational acoustics software is now based on boundary elements rather than finite elements. Unfortunately, for modal analysis, the eigenvalue problem (23) that results directly from the boundary element method is non-linear. The dual reciprocity method, outlined in section 5, was originally developed to resolve this difficulty. This method results in a linear eigenvalue problem (35). However, the success of the method is critically dependent on the suitability of the approximation (33). Although modifications in the method are proposed in Ali et al [17], it is likely that such modifications are difficult to automate, especially for the higher modes and in three dimensional problems.

The multiple reciprocity method, outlined in section 5, was introduced for the same reasons as the dual reciprocity method. However, in this case the need for a finite element approximation in the domain was removed altogether. In fact the method is similar to the interpolation method of section 6 in that it computes the eigenvalues of the polynomial matrix that approximates the boundary element matrix. In Kamiya and Andoh [19] the polynomial matrix resulting from the MRM is solved by re-writing it as a generalised linear eigenvalue problem, in a similar way to that shown in section 6 with the interpolation method.

There is clearly some similarity between the interpolation method described on section 6 and the multiple reciprocity method. However whereas the interpolation method approximates the components of the boundary element matrices by polynomials over a subrange of values of k, the multiple reciprocity approximates the matrix components by a Taylor expansion about k=0 for all values of k. Since interpolation is a more effective method of approximating a function than evaluating a Taylor series then the interpolation method is likely to be more efficient. Furthermore, as the magnitude of the eigenvalue increases a higher degree of polynomial approximation will be required to satisfactorily approximate the matrix elements in the multiple reciprocity mehthod. The interpolation method is superior as the same degree of interpolant is equally suitable for all eigenvalues. For example similar accuracies are obtained with second degree polynomials on a square in Kirkup and Amini [18] as obtained using polynomials of degree twelve in table 1 of Kamiya and Andoh [20].

The method that is derived through frequency interpolation of the matrix, outlined in section 6, is a flexible and consistent method for modal analysis of the interior acoustic problem. The most straightforward way of applying the method is to choose a degree for the polynomial approximation and the width of the subinterval [k_A, k_B], then to step through the intervals, computing the successive solutions. An implementation of a particularly straightforward form of the method is described in section 6. The results from the test problem in section 7 demonstrate the method. Further results from test problems, including two-dimensional problems and with a higher degree of interpolation are given in Kirkup and Amini [18].

In section 8, the method based on frequency interpolation and a typical finite element method were applied to the practical problem of the modal analysis of the interior of a loudspeaker. As discussed in section 8, the results show good agreement.

No particular effort was made to optimise the efficiency of the frequency interpolation method in this work. Techniques for improving the method are discussed in Kirkup and Amini [18]. However, the computational efficiency of the interpolated boundary element method applied to the loudspeaker compared well with the finite element method.

Acknowledgements. The first author (SMK) was partially funded through SERC/MOD grant GR/G01416. The second author (MAJ) would like to thank B&W Loudspeakers Ltd, Elm Grove Lane, Steyning, West Sussex, UK and the SERC for their support of this work. The authors are grateful to Dr. S. Amini of the Department of Mathematics and Computer Science, University od Salford, Dr. D. J. Henwood of the Department of Mathematical Sciences, University of Brighton and to Dr P. A. Fryer of B&W Loudspeakers Ltd, Worthing for their support and encouragement in this work.