6.7  Conclusion

It may seem unnecessary to develop the boundary element method for the solution of interior Helmholtz modal analysis or the interior boundary value problem since such problems can satisfactorily solved by the more traditional finite element method. Furthermore, no claim is being made that the BEM is generally more efficient than the finite element method in these applications, although in some cases it is easier to apply since only the boundary requires discretisation. However, in software libraries for solving Helmholtz problems based on the boundary element method, such as the one that accompanies this text, it would be anomalous not to provide a method for solving the modal analysis poblem. Besides, the underlying technique has been applied to the modal analysis of structures in contact with fluids in Kirkup and Amini [41] - such problems can be difficult to solve by other methods.

Underlying the solution of the modal analysis problem by the BEM is a method for solving a non-linear eigenvalue problem. The method that is derived through frequency interpolation of the matrix described in Section 6.3 is a flexible and consistent method for the modal analysis of the interior Helmholtz problem. However, it may be possible to develop more efficient methods than the QZ algorithm for solving the eigenvalue problem (6.12).

The subroutines described in this Section are flexible in that the wavenumber range over which the resonant frequencies are sought and the number of interpolation points within that range are parameters. In general the sequence of resonant frequencies can be obtained by stepping through the full wavenumber range covering a fixed subinterval at each stage. Ideally users need to determine the wavenumber interval and the number of interpolation points used in each interval to obtain satisfactory solutions with minimum processing time. It is beneficial for users to gain some experience with the methods before using them in practical situations.

The results from the test problems show that the boundary element method can be confidently applied to the modal analysis problem. The comparison of numerical and experimental results in the application of the methods to the loudspeaker enclosure show that the BEM is able to extract the resonant frequencies and mode shapes of an enclosure in a practical application. The mode shapes shown in the figures are comparable with those obtained by the finite element method, see Kirkup and Jones [48].

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