创建具有期望折射系数的材料：数值实验
Creating materials with a desired refraction coefficient: numerical experiments
AI总结：
通过数值实验实现一种方法，利用嵌入小球的阻抗边界条件，以可控误差逼近给定域内材料的期望折射系数。
摘要：
一种创建具有期望折射系数的材料的配方被数值实现。使用以下假设： \[ \zeta_m = h(x_m)/a^\kappa, \quad d = O(a^{(2-\kappa)/3}), \quad M = O(1/a^{2-\kappa}), \quad \kappa \in (0,1), \] 其中 $\zeta_m$ 和 $x_m$ 分别是第 $m$ 个小球的边界阻抗和中心，$h(x) \in C(D)$，Im$h(x) \leq 0$，$M$ 是嵌入立方体 $D$ 中的小球数量，$a$ 是小球的半径，$d$ 是相邻小球之间的距离。 对于小散射体情形下的多体散射问题，给出了近似解的误差估计。该结果用于估计为获得一种材料而需嵌入给定域 $D$ 中的最小粒子数，该材料的折射系数以不超过期望小量的相对误差逼近期望折射系数。
英文摘要：
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.
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