Creating materials with a desired refraction coefficient: numerical experiments

TL;DR

This paper presents a numerical method for designing materials with specific refraction properties by embedding small particles, providing error estimates and minimal particle counts for desired accuracy.

Contribution

It introduces a numerical recipe for creating materials with targeted refraction coefficients, including error bounds and particle count estimates.

Findings

Error estimates for the many-body scattering problem with small scatterers

Minimal number of particles needed for desired refraction approximation

Numerical implementation of material design with controlled accuracy

Abstract

A recipe for creating materials with a desired refraction coefficient is implemented numerically. The following assumptions are used: \bee \zeta_m=h(x_m)/a^\kappa,\quad d=O(a^{(2-\kappa)/3}),\quad M=O(1/a^{2-\kappa}),\quad \kappa\in(0,1), \eee where $\zeta_m$ and $x_m$ are the boundary impedance and center of the $m$-th ball, respectively, $h(x)\in C(D)$, Im$h(x)\leq 0$, $M$ is the number of small balls embedded in the cube $D$, $a$ is the radius of the small balls and $d$ is the distance between the neighboring balls. An error estimate is given for the approximate solution of the many-body scattering problem in the case of small scatterers. This result is used for the estimate of the minimal number of small particles to be embedded in a given domain $D$ in order to get a material whose refraction coefficient approximates the desired one with the relative error not exceeding a desired small quantity.