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

Hargreaves, JA; Lam, YW; Langdon, S

Abstract

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.

• Journal Article Type: Article
• Acceptance Date: Jan 4, 2016
• Online Publication Date: Jan 9, 2016
• Publication Date: Sep 8, 2016
• Deposit Date: Jan 11, 2016
• Publicly Available Date: Apr 12, 2016
• Journal: International Journal for Numerical Methods in Engineering
• Print ISSN: 0029-5981
• Electronic ISSN: 1097-0207
• Publisher: Wiley
• Volume: 108
• Issue: 2
• Pages: 93-115
• DOI: https://doi.org/10.1002/nme.5204
• Publisher URL: http://dx.doi.org/10.1002/nme.5204
• Related Public URLs: http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1097-0207