Caetano, António M;
Chandler-Wilde, Simon N;
Claeys, Xavier;
Gibbs, Andrew;
Hewett, David P;
Moiola, Andrea;
(2025)
Integral equation methods for acoustic scattering by fractals.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
, 481
(2306)
, Article 20230650. 10.1098/rspa.2023.0650.
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Abstract
We study sound-soft time-harmonic acousticscattering by general scatterers, including fractalscatterers, in 2D and 3D space. For an arbitrarycompact scatterer Γ we reformulate the Dirichletboundary value problem for the Helmholtz equationas a first kind integral equation (IE) on Γ involvingthe Newton potential. The IE is well-posed, exceptpossibly at a countable set of frequencies, andreduces to existing single-layer boundary IEs whenΓ is the boundary of a bounded Lipschitz open set,a screen, or a multi-screen. When Γ is uniformlyof d-dimensional Hausdorff dimension in a sensewe make precise (a d-set), the operator in ourequation is an integral operator on Γ with respectto d-dimensional Hausdorff measure, with kernel theHelmholtz fundamental solution, and we proposea piecewise-constant Galerkin discretization of theIE, which converges in the limit of vanishing meshwidth. When Γ is the fractal attractor of an iteratedfunction system of contracting similarities we proveconvergence rates under assumptions on Γ and the IEsolution, and describe a fully discrete implementationusing recently proposed quadrature rules for singularintegrals on fractals. We present numerical results fora range of examples and make our software availableas a Julia code.
| Type: | Article |
|---|---|
| Title: | Integral equation methods for acoustic scattering by fractals |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1098/rspa.2023.0650 |
| Publisher version: | https://doi.org/10.1098/rspa.2023.0650 |
| Language: | English |
| Additional information: | © 2025 The Author(s). Published by the Royal Society under the terms of theCreative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/). |
| Keywords: | Helmholtz equation, function spaces, iteratedfunction system, Galerkin method, boundaryelement method |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10204451 |
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