## 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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