A transformation approach for efficient evaluation of oscillatory surface integrals arising in three-dimensional boundary element methods

We propose a method for efficient evaluation of surface integrals arising in boundary element methods for three-dimensional Helmholtz problems (with real positive wavenumber k), modelling wave scattering and/or radiation in homogeneous media. To reduce the number of degrees of freedom required when k is large, a common approach is to include in the approximation space oscillatory basis functions, with support extending across many wavelengths. A difficulty with this approach is that it leads to highly oscillatory surface integrals whose evaluation by standard quadrature would require at least O(k2) quadrature points. Here, we use equivalent contour integrals developed for aperture scattering in optics to reduce this requirement to O(k), and possible extensions to reduce it further to O(1)are identified. The contour integral is derived for arbitrary shaped elements, but its application is limited to planar elements in many cases. In addition, the transform regularises the singularity in the surface integrand caused by the Green’s function, including for the hyper-singular case under appropriate conditions. An open-source Matlab™ code library is available to demonstrate our routines