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An efficient frequency-independent numerical method for computing the far-field pattern induced by polygonal obstacles

A. Gibbs, S. Langdon

Abstract

For problems of time-harmonic scattering by rational polygonal obstacles, embedding formulae express the far-field pattern induced by any incident plane wave in terms of the far-field patterns for a relatively small (frequency-independent) set of canonical incident angles. Although these remarkable formulae are exact in theory, here we demonstrate that: (i) they are highly sensitive to numerical errors in practice, and (ii) direct calculation of the coefficients in these formulae may be impossible for particular sets of canonical incident angles, even in exact arithmetic. Only by overcoming these practical issues can embedding formulae provide a highly efficient approach to computing the far-field pattern induced by a large number of incident angles. Here we address challenges (i) and (ii), supporting our theory with numerical experiments. Challenge (i) is solved using techniques from computational complex analysis: we reformulate the embedding formula as a complex contour integral and prove that this is much less sensitive to numerical errors. In practice, this contour integral can be efficiently evaluated by residue calculus. Challenge (ii) is addressed using techniques from numerical linear algebra: we oversample, considering more canonical incident angles than are necessary, thus expanding the set of valid coefficient vectors. The coefficient vector can then be selected using either a least squares approach or column subset selection.

An efficient frequency-independent numerical method for computing the far-field pattern induced by polygonal obstacles

Abstract

For problems of time-harmonic scattering by rational polygonal obstacles, embedding formulae express the far-field pattern induced by any incident plane wave in terms of the far-field patterns for a relatively small (frequency-independent) set of canonical incident angles. Although these remarkable formulae are exact in theory, here we demonstrate that: (i) they are highly sensitive to numerical errors in practice, and (ii) direct calculation of the coefficients in these formulae may be impossible for particular sets of canonical incident angles, even in exact arithmetic. Only by overcoming these practical issues can embedding formulae provide a highly efficient approach to computing the far-field pattern induced by a large number of incident angles. Here we address challenges (i) and (ii), supporting our theory with numerical experiments. Challenge (i) is solved using techniques from computational complex analysis: we reformulate the embedding formula as a complex contour integral and prove that this is much less sensitive to numerical errors. In practice, this contour integral can be efficiently evaluated by residue calculus. Challenge (ii) is addressed using techniques from numerical linear algebra: we oversample, considering more canonical incident angles than are necessary, thus expanding the set of valid coefficient vectors. The coefficient vector can then be selected using either a least squares approach or column subset selection.
Paper Structure (19 sections, 8 theorems, 53 equations, 10 figures, 3 tables, 3 algorithms)

This paper contains 19 sections, 8 theorems, 53 equations, 10 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. The embedding formula for rational polygons
  3. Aims and outline of this paper
  4. Reformulating the embedding formula
  5. Evaluation of residues in finite precision arithmetic
  6. Proof of Theorem \ref{['thm:semidiscrete_error']}
  7. Computing the embedding coefficients
  8. Strategy One
  9. Strategy Two
  10. The algorithm
  11. Standard $hp$ Boundary Element Method
  12. Hybrid Numerical-Asymptotic Boundary Element Method (HNA BEM)
  13. Default parameters
  14. Numerical examples and experiments
  15. Example applications of Algorithm \ref{['alg:main']}
  16. ...and 4 more sections

Key Result

Theorem 1.2

\newlabelth:embedding0 Suppose that $\Omega$ is a rational polygon and that there exist distinct 'canonical incident angles' $\alpha_1,\ldots,\alpha_M$, satisfying Assumption ass:bexists (given below), where $M:=\sum_{j=1}^{S}(q_j-1)$ and $q_j$ is as in Definition def:RationalPolygons. Then there where and $p$ is as in Definition def:RationalPolygons. If $\Lambda(\theta,\alpha)=0$ then one or t

Figures (10)

  • Figure 1: Full far-field characterisation $\log|D(\theta,\alpha)|$ for $(\theta,\alpha)\in\mathbb{T}$, where $\Omega$ is a square of diameter 2 and $k=10$. This plot was computed using the approach described in this paper; see §\ref{['sec:numerics']} for details.
  • Figure 1: Rectangular contour, covering both cases one and two of the proof of Theorem \ref{['thm:semidiscrete_error']}. The arrows are labelled with values (or bounds) of the length of the regions to which the arrows are parallel.
  • Figure 1: The errors in the naive approximation \ref{['eq:Dnaive']} (dotted) and the output of Algorithm \ref{['alg:main']} (solid), for the right-angled triangle with $k=10$ and $\alpha=5\pi/4$. In all cases, the spikes of the naive approximation do not appear using our method.
  • Figure 2: Example of unbounded errors which occur when applying the naive embedding approximation \ref{['eq:Dnaive']} directly. Here $\Omega$ is a square of diameter 2 and $k=10$. The numerical solver used is described in §\ref{['sec:stdbem']}.
  • Figure 2: $\log|D(\theta,\alpha)|$ for right-angled isosceles triangle $k=10$ (left), and corresponding pointwise relative error (right).
  • ...and 5 more figures

Theorems & Definitions (18)

  • Definition 1.1: Rational polygons
  • Theorem 1.2
  • Lemma 2.1
  • Proof 1
  • Definition 2.2: Nearby poles
  • Theorem 2.3
  • Remark 2.4: Analyticity of numerical solutions
  • Definition 2.5: Coalescence points
  • Lemma 2.6
  • Proof 2
  • ...and 8 more