Shadow boundary effects in hybrid numerical-asymptotic methods for high-frequency scattering

The hybrid numerical-asymptotic (HNA) approach aims to reduce the computational cost of conventional numerical methods for high-frequency wave scattering problems by enriching the numerical approximation space with oscillatory basis functions, chosen based on partial knowledge of the high-frequency solution asymptotics. In this paper, we propose a new methodology for the treatment of shadow boundary effects in HNA boundary element methods, using the classical geometrical theory of diffraction phase functions combined with mesh refinement. We develop our methodology in the context of scattering by a class of sound-soft non-convex polygons, presenting a rigorous numerical analysis (supported by numerical results) which proves the effectiveness of our HNA approximation space at high frequencies. Our analysis is based on a study of certain approximation properties of the Fresnel integral and related functions, which govern the shadow boundary behaviour.

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Fast hybrid numerical-asymptotic boundary element methods for high frequency screen and aperture problems based on least-squares collocation

SN Partial Differential Equations and Applications ◽

10.1007/s42985-020-00013-3 ◽

2020 ◽

Vol 1 (4) ◽

Author(s):

A. Gibbs ◽

D. P. Hewett ◽

D. Huybrechs ◽

E. Parolin

Keyword(s):

Least Squares ◽

Boundary Element ◽

High Frequency ◽

Singular Integrals ◽

Weighted Least Squares ◽

Computational Cost ◽

Boundary Element Methods ◽

Two Dimensional ◽

Numerical Instabilities ◽

Value Decomposition

Abstract We present a hybrid numerical-asymptotic (HNA) boundary element method (BEM) for high frequency scattering by two-dimensional screens and apertures, whose computational cost to achieve any prescribed accuracy remains bounded with increasing frequency. Our method is a collocation implementation of the high order hp HNA approximation space of Hewett et al. (IMA J Numer Anal 35:1698–1728, 2015), where a Galerkin implementation was studied. An advantage of the current collocation scheme is that the one-dimensional highly oscillatory singular integrals appearing in the BEM matrix entries are significantly easier to evaluate than the two-dimensional integrals appearing in the Galerkin case, which leads to much faster computation times. Here we compute the required integrals at frequency-independent cost using the numerical method of steepest descent, which involves complex contour deformation. The change from Galerkin to collocation is nontrivial because naive collocation implementations based on square linear systems suffer from severe numerical instabilities associated with the numerical redundancy of the HNA basis, which produces highly ill-conditioned BEM matrices. In this paper we show how these instabilities can be removed by oversampling, and solving the resulting overdetermined collocation system in a weighted least-squares sense using a truncated singular value decomposition. On the basis of our numerical experiments, the amount of oversampling required to stabilise the method is modest (around 25% typically suffices), and independent of frequency. As an application of our method we present numerical results for high frequency scattering by prefractal approximations to the middle-third Cantor set.

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