Multiscale Modeling &Simulation, Volume 19, Issue 2, Page 921-950, January 2021.
This paper presents an efficient multiscale approach for solving the time-domain Maxwell's equations with rapidly oscillating coefficients coupled to a wave-like equation, which arise in the simulation of light interaction with composite materials containing metallic nanostructure arrays with spatial nonlocality embedded in a dielectric host. A key feature of this system is that the new PDE describing the evolution of the polarization current is only defined in the domain of metallic nanostructures. We first extend the equation satisfied by the polarization current into the domain of the dielectric medium in a novel way. The well-posedness of the original system and the extended system are proved and the extension error between these two systems is analyzed rigorously with an exact transparent boundary condition. Next we apply the asymptotic homogenization method to derive the homogenized system and define the multiscale approximate solution for the extended system. Under realistic assumptions, we prove that the homogenized polarization current is negligible and that the homogenized coupled system can be replaced by the homogenized Maxwell's equations with guaranteed accuracy to reduce the computational cost. Finally, to recover the spatial nonlocal effects, the original system is solved in each isolated metallic nanostructure with boundary conditions given by the multiscale approximate solution. Numerical examples demonstrate that our multiscale approach is capable of yielding accurate results with significant computational savings compared to solving the original system directly.

中文翻译：

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非局部金属纳米结构阵列光相互作用瞬态模拟的多尺度方法和分析

多尺度建模与仿真，第 19 卷，第 2 期，第 921-950 页，2021 年 1 月。
本文提出了一种有效的多尺度方法来求解时域麦克斯韦方程，该方程具有耦合到波状方程的快速振荡系数，该方程出现在模拟光与复合材料的光相互作用中，复合材料包含具有空间非局域性的金属纳米结构阵列嵌入介电主体中. 该系统的一个关键特征是描述极化电流演变的新 PDE 仅在金属纳米结构域中定义。我们首先以一种新颖的方式将极化电流满足的方程扩展到电介质域中。证明了原始系统和扩展系统的适定性，并在精确透明边界条件下严格分析了这两个系统之间的扩展误差。接下来我们应用渐近均匀化方法来导出均匀化系统并定义扩展系统的多尺度近似解。在现实假设下，我们证明了均质化极化电流可以忽略不计，并且均质化耦合系统可以用均质化麦克斯韦方程组代替，并保证精度以降低计算成本。最后，为了恢复空间非局部效应，原始系统在每个孤立的金属纳米结构中求解，边界条件由多尺度近似解给出。数值例子表明，与直接求解原始系统相比，我们的多尺度方法能够产生准确的结果，并显着节省计算量。