Numerical solution of many-body wave scattering problem and creating materials with a desired refraction coefficient

TL;DR

This paper develops an asymptotic and numerical approach to solve many-body wave scattering problems involving small particles, enabling the creation of materials with tailored refraction properties, supported by numerical results for up to one million particles.

Contribution

It introduces a novel asymptotic and numerical method for solving many-body wave scattering problems with small particles, facilitating material design with specific refraction coefficients.

Findings

Numerical solutions for up to 10^6 particles demonstrate scalability.

The method accurately predicts effective refraction coefficients.

The approach enables designing materials with desired wave propagation properties.

Abstract

Scalar wave scattering by many small particles with impedance boundary condition and creating material with a desired refraction coefficient are studied. The acoustic wave scattering problem is solved asymptotically and numerically under the assumptions $ka \ll 1, \zeta_m = \frac{h(x_m)}{a^\kappa}, d = O(a^{\frac{2-\kappa}{3}}), M = O(\frac{1}{a^{2-\kappa}}), \kappa \in [0,1)$, where $k = 2\pi/\lambda$ is the wave number, $\lambda$ is the wave length, $a$ is the radius of the particles, $d$ is the distance between neighboring particles, $M$ is the total number of the particles embedded in a bounded domain $\Omega \subset \RRR$, $\zeta_m$ is the boundary impedance of the m\textsuperscript{th} particle $D_m$, $h \in C(D)$, $D := \bigcup_{m=1}^M D_m$, is a given arbitrary function which satisfies Im$h \le 0$, $x_m \in \Omega$ is the position of the m\textsuperscript{th} particle, and $1 \leq m \leq M$. Numerical results are presented for which the number of particles equals $10^4, 10^5$, and $10^6$.