Abstract/Summary

Time harmonic multiple scattering problems are of interest for a number of applications, from understanding the high frequency scattering of high altitude clouds to modelling outdoor noise propagation. Standard numerical methods, using a polynomial approximation space, have a computational cost which grows as the frequency increases. This can lead to unfeasible computational costs to model high frequency scattering problems using standard methods. One promising solution to this problem is to use an enriched approximation space, specifically in this thesis the Hybrid Numerical Asymptotic (HNA) Boundary Element Method (BEM), which uses carefully selected oscillatory basis functions. This approach has been successfully applied to a range of problems in 2D including scattering by single convex polygons or screens, and some specific non-convex and multiple scattering problems, with a computational cost that is almost frequency independent. In this thesis we extend the HNA BEM to solve scattering by two screens in any orientation, provided they do not intersect, using an iterative method, which takes into account all interactions between the screens. When modelling scattering by screens in the HNA approximation space, previously only single screens or multiple co-linear screens were considered. Here we present an iterative method which at each step solves a single screen scattering problem for some incident field and uses the resulting field in the next iteration as the incident field for the other screen. The iterations continue, allowing the computation of all interactions between the screens, capturing direct reflections and diffraction due to the tips of the screens. Theoretical results are presented which show, under certain assumptions, that this iterative method is a Neumann iteration and converges towards the solution of the scattering problem. Numerical results show that this method can be applied to a range of problems, including direct illumination and the effects of shadowing and partial shadowing, and we make computations comparing both a standard BEM approximation space with the HNA BEM space. It is hoped that this method can be used as a building block to capture multiple scattering for more complex geometries in a computationally efficient way.