Creating materials with a desired refraction coefficient: numerical experiments
This work provides a theoretical and numerical framework for designing metamaterials with prescribed refractive properties, which is relevant for materials science and engineering.
The paper numerically implements a recipe for creating materials with a desired refraction coefficient by embedding small balls with specific boundary impedances. An error estimate for the many-body scattering problem is provided, and the minimal number of particles needed to achieve a desired accuracy is estimated.
A recipe for creating materials with a desired refraction coefficient is implemented numerically. The following assumptions are used: \bee ζ_m=h(x_m)/a^κ,\quad d=O(a^{(2-κ)/3}),\quad M=O(1/a^{2-κ}),\quad κ\in(0,1), \eee where $ζ_m$ 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.