Integral equation methods for acoustic scattering by fractals
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866918379034509312 |
|---|---|
| author | Caetano, A. M. Chandler-Wilde, S. N. Claeys, X. Gibbs, A. Hewett, D. P. Moiola, A. |
| author_facet | Caetano, A. M. Chandler-Wilde, S. N. Claeys, X. Gibbs, A. Hewett, D. P. Moiola, A. |
| contents | We study sound-soft time-harmonic acoustic scattering by general scatterers, including fractal scatterers, in 2D and 3D space. For an arbitrary compact scatterer $Γ$ we reformulate the Dirichlet boundary value problem for the Helmholtz equation as a first kind integral equation (IE) on $Γ$ 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 $Γ$ is the boundary of a bounded Lipschitz open set, a screen, or a multi-screen. When $Γ$ 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 $Γ$ 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 $Γ$ is the fractal attractor of an iterated function system of contracting similarities we prove convergence rates under assumptions on $Γ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_02184 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Integral equation methods for acoustic scattering by fractals Caetano, A. M. Chandler-Wilde, S. N. Claeys, X. Gibbs, A. Hewett, D. P. Moiola, A. Numerical Analysis 28A80, 65R20, 78A45 We study sound-soft time-harmonic acoustic scattering by general scatterers, including fractal scatterers, in 2D and 3D space. For an arbitrary compact scatterer $Γ$ we reformulate the Dirichlet boundary value problem for the Helmholtz equation as a first kind integral equation (IE) on $Γ$ 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 $Γ$ is the boundary of a bounded Lipschitz open set, a screen, or a multi-screen. When $Γ$ 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 $Γ$ 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 $Γ$ is the fractal attractor of an iterated function system of contracting similarities we prove convergence rates under assumptions on $Γ$ 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. |
| title | Integral equation methods for acoustic scattering by fractals |
| topic | Numerical Analysis 28A80, 65R20, 78A45 |
| url | https://arxiv.org/abs/2309.02184 |