A transformation approach for efficient evaluation ofoscillatory surface integrals arising in three-dimensionalboundary element methods

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