Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Acoustic scattering by fractal inhomogeneities via geometry-conforming Galerkin methods for the Lippmann-Schwinger equation

Joshua Bannister, David P. Hewett, Andrew Gibbs

TL;DR

The paper tackles acoustic scattering by fractal-boundary inhomogeneities and develops a geometry-conforming Galerkin method for the Lippmann-Schwinger volume integral equation. It provides thorough semi-discrete and fully discrete analyses for $h$- and $p$-versions, showing convergence that depends on refractive-index regularity and boundary fractality, and exploits self-similarity via $n$-attractors to generate fractal meshes. The approach uses a piecewise-constant VIM with singular-quadrature rules on $L_h$ meshes, proving well-posedness, stability, and superconvergence for linear functionals such as the near-field and far-field evaluations. Numerical results in 2D with Fudgeflake, Gosper Island, and Koch Snowflake demonstrate the method’s accuracy gains over prefractal methods and confirm the theoretical convergence rates and conditioning advantages of the geometry-conforming strategy.

Abstract

We propose and analyse a numerical method for time-harmonic acoustic scattering in $\mathbb{R}^n$, $n=2,3$, by a class of inhomogeneities (penetrable scatterers) with fractal boundary. Our method is based on a Galerkin discretisation of the Lippmann-Schwinger volume integral equation, using a discontinuous piecewise-polynomial approximation space on a geometry-conforming mesh comprising elements which themselves have fractal boundary. We first provide a semi-discrete well-posedness and error analysis for both the $h$- and $p$-versions of our method for completely arbitrary inhomogeneities (without any regularity assumption on the boundary of the inhomogeneity or of the mesh elements). We prove convergence estimates for the integral equation solution and superconvergence estimates for linear functionals such as scattered field and far-field pattern evaluations, and elucidate how the regularity of the inhomogeneity boundary and the regularity of the refractive index affect the rates of convergence predicted. We then specialise to the case where the inhomogeneity is an ``$n$-attractor'', i.e.\ the fractal attractor of an iterated function system satisfying the open set condition with non-empty interior, showing how in this case the self-similarity of the inhomogeneity can be used to generate geometry-conforming meshes. For the $h$-version with piecewise constant approximation we also present singular quadrature rules, supported by a fully discrete error analysis, permitting practical implementation of our method. We present numerical results for two-dimensional examples, which validate our theoretical results and show that our method is significantly more accurate than a comparable method involving replacement of the fractal inhomogeneity by a smoother prefractal approximation.

Acoustic scattering by fractal inhomogeneities via geometry-conforming Galerkin methods for the Lippmann-Schwinger equation

TL;DR

The paper tackles acoustic scattering by fractal-boundary inhomogeneities and develops a geometry-conforming Galerkin method for the Lippmann-Schwinger volume integral equation. It provides thorough semi-discrete and fully discrete analyses for - and -versions, showing convergence that depends on refractive-index regularity and boundary fractality, and exploits self-similarity via -attractors to generate fractal meshes. The approach uses a piecewise-constant VIM with singular-quadrature rules on meshes, proving well-posedness, stability, and superconvergence for linear functionals such as the near-field and far-field evaluations. Numerical results in 2D with Fudgeflake, Gosper Island, and Koch Snowflake demonstrate the method’s accuracy gains over prefractal methods and confirm the theoretical convergence rates and conditioning advantages of the geometry-conforming strategy.

Abstract

We propose and analyse a numerical method for time-harmonic acoustic scattering in , , by a class of inhomogeneities (penetrable scatterers) with fractal boundary. Our method is based on a Galerkin discretisation of the Lippmann-Schwinger volume integral equation, using a discontinuous piecewise-polynomial approximation space on a geometry-conforming mesh comprising elements which themselves have fractal boundary. We first provide a semi-discrete well-posedness and error analysis for both the - and -versions of our method for completely arbitrary inhomogeneities (without any regularity assumption on the boundary of the inhomogeneity or of the mesh elements). We prove convergence estimates for the integral equation solution and superconvergence estimates for linear functionals such as scattered field and far-field pattern evaluations, and elucidate how the regularity of the inhomogeneity boundary and the regularity of the refractive index affect the rates of convergence predicted. We then specialise to the case where the inhomogeneity is an ``-attractor'', i.e.\ the fractal attractor of an iterated function system satisfying the open set condition with non-empty interior, showing how in this case the self-similarity of the inhomogeneity can be used to generate geometry-conforming meshes. For the -version with piecewise constant approximation we also present singular quadrature rules, supported by a fully discrete error analysis, permitting practical implementation of our method. We present numerical results for two-dimensional examples, which validate our theoretical results and show that our method is significantly more accurate than a comparable method involving replacement of the fractal inhomogeneity by a smoother prefractal approximation.
Paper Structure (27 sections, 37 theorems, 212 equations, 14 figures)

This paper contains 27 sections, 37 theorems, 212 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Problem formulation, well-posedness and regularity
  3. The full-space problem and the Lippmann-Schwinger equation
  4. The restricted LSE
  5. Solution regularity
  6. Scattering problems and total field formulation
  7. The volume integral method (VIM)
  8. Abstract semi-discrete error analysis
  9. Application to the scattering problem
  10. Abstract fully discrete error analysis
  11. Iterated Function Systems and $n$-attractors
  12. Meshing an $n$-attractor
  13. Numerical quadrature - piecewise constant case
  14. The composite barycentre rule
  15. Regular interactions
  16. ...and 12 more sections

Key Result

Theorem 2.4

Suppose that $\Im[\mathfrak{m}] \geq 0$. Then there exists a unique $u^{\rm s}\in H^{1}_{\rm loc}(\mathbb{R}^n)$ satisfying e:FSP-e:SRC.

Figures (14)

  • Figure 1: Scattering of a plane wave by the Fudgeflake, Gosper Island and Koch Snowflake, with a constant refractive index on each inhomogeneity. For details see §\ref{['s:Numer']}.
  • Figure 2: Examples of geometry-conforming meshes of the Fudgeflake, Gosper Island and Koch Snowflake - for details see §\ref{['s:Examp']}. In (a) and (b) the meshes are uniform, because the associated IFS is homogeneous (all similarities have the same contraction factor), while in (c) the meshes are quasi-uniform.
  • Figure 3: Different discretisations of the Koch Snowflake. In this paper we study the "geometry-conforming" approach of (c), using a mesh with fractal elements.
  • Figure 4: The first five prefractal approximations of the Koch Snowflake.
  • Figure 5: Lattice structure of the $L_{h_l}$ meshes for the Koch Snowflake, $l=1,2,3$. The elements corresponding to the index sets $B_l$, $S_l^1$ and $S_l^2$ are coloured green, red and blue respectively.
  • ...and 9 more figures

Theorems & Definitions (84)

  • Definition 2.1: Full-space problem (FSP)
  • Remark 2.2
  • Remark 2.3: Transmission problem in the Lipschitz case
  • Theorem 2.4: FSP well-posedness
  • proof
  • Theorem 2.5: FSP-LSE equivalence
  • proof
  • Corollary 2.6: LSE well-posedness
  • Proposition 2.7: LSE-ILSE equivalence
  • proof
  • ...and 74 more