Plane wave interpolation in direct collocation boundary element method for radiation and wave scattering: numerical aspects and applications

The classical boundary element formulation for the Helmholtz equation is rehearsed, and its limitations with respect to the number of variables needed to model a wavelength are explained. A new type of interpolation for the potential is then described in which the usual boundary element shape functions are modified by the inclusion of a set of plane waves, propagating in a range of directions. This is termed the plane wave basis boundary element method. The modifications needed to the classical procedures, in terms of integration of the element matrices, and location of collocation points are described. The well-known Singular Value Decomposition solution technique, which is adopted here for the solution of the system matrix equation in its complex form, is briefly outlined. The conditioning of the system matrix is analysed for a simple radiation problem. The corresponding diffraction problem is also analysed and results are compared with analytical and classical boundary element solutions. The CHIEF method is adopted to enhance the quality of the solution, particularly in the vicinity of irregular frequencies. The plane wave basis boundary element method is then applied to two problems: scattering of plane waves by an elliptical cylinder and the multiple circular cylinder plane wave scattering problem. In both cases results are compared with analytical solutions. The results clearly demonstrate that the new method is considerably more efficient than the classical approach. For a given number of degrees of freedom, the frequency for which accurate results can be obtained, using the new technique, can be up to three or four times higher than that of the classical method. This makes the method a powerful new addition to our tools for tackling high-frequency radiation and scattering problems.