Integral equation methods for acoustic scattering by fractals

TL;DR

The paper develops a rigorous framework for acoustic scattering by fractal and irregular scatterers in 2D and 3D by recasting the Dirichlet problem as a first-kind integral equation on the scatterer using the acoustic Newton potential. It extends traditional boundary-integral formulations to general compact sets via a d-set and Hausdorff-measure approach, and introduces a Hausdorff-measure Galerkin discretization with piecewise-constant spaces that converges as the mesh is refined. For IFS attractors, the authors establish convergence rates under structural assumptions and provide a fully discrete quadrature-enabled implementation, including singular-quadrature techniques tailored to fractals. Numerical experiments in 2D and 3D demonstrate the method on Cantor dust, Koch curves, Koch snowflakes, and Sierpinski tetrahedra, with the Julia code released for public use, highlighting the practical feasibility of high-fidelity fractal scattering simulations.

Abstract

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.

Figures (9)

• Figure 1: Examples of $d$-sets in two-dimensional space ($n=2$), with: a) $d=2$; b) $d=1$; c) $d=1$; d) $d=1$; e) $d= \log(2)/ \log(3) \approx 0.63$; f) $d = \log(4)/ \log(3) \approx 1.26$; g) $d=2$. For details see text of §\ref{['sec:intro']}.
• Figure 2: Sierpinski tetrahedron $d$-sets in 3D space, attractors of the IFS \ref{['eq:TetIFS']} for $\rho=1/2$ ($d=2$) and $\rho=3/8$ ($d=\log{4}/\log(8/3)\approx 1.41$).
• Figure 3: Schema of relevant function spaces and operators for $s=1$, $t=t_d:=1-\frac{n-d}{2}\in(0,1]$. The operators $\mathrm{tr}_\Gamma$ and $\mathrm{tr}_\Gamma^*$ are isometries, indeed unitary isomorphisms if Assumption \ref{['ass:TildeCirc']} holds.
• Figure 4: Level 1 (a) and level 2 (b) decompositions of the Koch curve. To make the labelling more compact, in (a) the labels "$1$",…,"$4$" indicate the subsets $\Gamma_1,\ldots,\Gamma_4$, and in (b) the labels "$ij$" and "$ijk$" indicate $\Gamma_{(i,j)}$ and $\Gamma_{(i,j,k)}$. In (b) the insert shows the level 3 decomposition of $\Gamma_{(1,1)}$.
• Figure 5: Scattering by a middle third Cantor dust, (a)-(c), and a Koch curve, (d). See §\ref{['sec:2D']}(\ref{['sec:DustKoch']}).

Theorems & Definitions (43)

• Remark 2.1: Lifting attractors to higher dimensions
• Example 2.2: Cantor set examples of IFS attractors
• Example 2.3: Koch curve
• Example 2.4: Koch snowflake
• Example 2.5: Sierpinski tetrahedron
• Remark 3.2
• Lemma 3.3
• proof
• Theorem 3.4
• proof