Accessibility navigation


Integral equation methods for acoustic scattering by fractals

Caetano, A. M., Chandler-Wilde, S. N. ORCID: https://orcid.org/0000-0003-0578-1283, Claeys, X., Gibbs, A., Hewett, D. P. and Moiola, A. (2024) Integral equation methods for acoustic scattering by fractals. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. ISSN 1364-5021 (In Press)

[img]
Preview
Text (Open Access) - Accepted Version
· Available under License Creative Commons Attribution.
· Please see our End User Agreement before downloading.

12MB

It is advisable to refer to the publisher's version if you intend to cite from this work. See Guidance on citing.

To link to this item DOI: 10.1098/rspa.2023.0650

Abstract/Summary

We study sound-soft time-harmonic acoustic scattering by general scatterers, including fractal scatterers, in 2D and 3D space. For an arbitrary compact scatterer $\Gamma$ we reformulate the Dirichlet boundary value problem for the Helmholtz equation as a first kind integral equation (IE) on $\Gamma$ involving the Newton potential. The IE is well-posed, except possibly at a countable set of frequencies, and reduces to existing single-layer boundary IEs when $\Gamma$ is the boundary of a bounded Lipschitz open set, a screen, or a multi-screen. When $\Gamma$ is uniformly of $d$-dimensional Hausdorff dimension in a sense we make precise (a $d$-set), the operator in our equation is an integral operator on $\Gamma$ with respect to $d$-dimensional Hausdorff measure, with kernel the Helmholtz fundamental solution, and we propose a piecewise-constant Galerkin discretization of the IE, which converges in the limit of vanishing mesh width. When $\Gamma$ is the fractal attractor of an iterated function system of contracting similarities we prove convergence rates under assumptions on $\Gamma$ and the IE solution, and describe a fully discrete implementation using recently proposed quadrature rules for singular integrals on fractals. We present numerical results for a range of examples and make our software available as a Julia code.

Item Type:Article
Refereed:Yes
Divisions:Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
ID Code:117651
Publisher:Royal Society Publishing

Downloads

Downloads per month over past year

University Staff: Request a correction | Centaur Editors: Update this record

Page navigation