Numerical aspects of enriched and high-order boundary element basis functions for Helmholtz problems.

Abstract

In this thesis several aspects of the Partition of Unity Boundary Element Method (PUBEM) are investigated, with novel results in three main areas:
1. Enriched modelling of wave scattering from polygonal obstacles. The plane waves are augmented by a set of enrichment functions formed from fractional order Bessel functions, as informed by classical asymptotic solutions for wave fields in the vicinity of sharp corners. It is shown that the solution accuracy can be improved markedly by the addition of a very small number of these enrichment functions, with very little effect on the run time.

2. High-order formulations. Plane waves are not the only effective means of introducing oscillatory approximation spaces. High-Order Lagrange polynomials and high-order Non-Uniform Rational B-Splines (NURBS) also exhibit oscillation and these are tested and compared against PUBEM. It is found that these high-order functions significantly outperform the
corresponding low-order (typically quadratic) polynomials and NURBS that are commonly used, and that for large problems the highest order tested (11th) has potential to be competitive with PUBEM without the associated ill-conditioning.

3. Integration. The accuracy of PUBEM traditionally comes at the cost of the requirement to evaluate many highly-oscillatory integrals. Several candidate integration strategies are investigated with the aim of find-
ing a robust, accurate and efficient approach. Schemes tested include the Filon and asymptotic methods, as well as the Method of Stationary Phase (MSP). Although these schemes are found to be spectacularly successful for many cases, they fail for a sufficient number of situations to cause a complete PUBEM analysis based on these methods to lack
robustness. Conclusions are drawn about the effective use of more traditional quadrature for robust implementations.