论文标题
超级材料的多型和均质化：高阶非局部模型和外表面的散射
Multiband homogenization of metamaterials in real-space: Higher-order nonlocal models and scattering at external surfaces
论文作者
论文摘要
这项工作为超材料开发了动态均质化方法。它发现一个近似的宏观匀浆方程，并在空间和时间上构成恒定系数。但是，所得的均质方程在时空上是更高的阶段。均质方程可用于解决具有宏观异质性的任意非周期性宏观宏观几何形状，例如由几个不同的非材料或外部边界组成的物体。 首先，考虑到一个频带，高阶空间衍生物在均匀材料和超材料之间的边界处导致额外的连续性条件。这些提供了与精确的细尺度溶液匹配的1-D和2-D中波散射的预测。与替代方法相比，它们提供了一个单一方程式，该方程式在广泛的频率上易于应用，并且计算速度更快。接下来，考虑了两个带有带隙的频段的设置。均质方程还具有高阶时间导数。值得注意的是，均质模型提供了一个单一方程式，该方程在频带和带隙上都是有效的。连续性条件应用于边界处的波散射，并与精确的细尺度解决方案显示出良好的一致性。 使用最高时间导数的顺序与所考虑的频带的数量成正比，在无限频带极限的同质化方程中猜想了一个非本地时间结构。这表明，在分散关系中考虑较高的频带进行的时间均匀的长度和时间尺度（通过较高的时间均匀化）是一种出现宏观空间和时间非偏置性的机制，而时间非局部性的范围与考虑频段的数量相关。
This work develops a dynamic homogenization approach for metamaterials. It finds an approximate macroscopic homogenized equation with constant coefficients posed in space and time; however, the resulting homogenized equation is higher order in space and time. The homogenized equation can be used to solve initial-boundary-value problems posed on arbitrary non-periodic macroscale geometries with macroscopic heterogeneity, such as bodies composed of several different metamaterials or with external boundaries. First, considering a single band, the higher-order space derivatives lead to additional continuity conditions at the boundary between a homogeneous material and a metamaterial. These provide predictions of wave scattering in 1-d and 2-d that match well with the exact fine-scale solution; compared to alternative approaches, they provide a single equation that is valid over a broad range of frequencies, are easy to apply, and are much faster to compute. Next, the setting of two bands with a bandgap is considered. The homogenized equation has also higher-order time derivatives. Notably, the homogenized model provides a single equation that is valid over both bands and the bandgap. The continuity conditions are applied to wave scattering at a boundary, and show good agreement with the exact fine-scale solution. Using that the order of the highest time derivative is proportional to the number of bands considered, a nonlocal-in-time structure is conjectured for the homogenized equation in the limit of infinite bands. This suggests that homogenizing over finer length and time scales -- with the temporal homogenization being carried out through the consideration of higher bands in the dispersion relation -- is a mechanism for the emergence of macroscopic spatial and temporal nonlocality, with the extent of temporal nonlocality being related to the number of bands considered.
