This work develops a dynamic linear homogenization approach in the context of periodic metamaterials. By using approximations of the dispersion relation that are amenable to inversion to real-space and real-time, 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. The approach is applied here to problems with scalar unknown fields in one and two spatial dimensions.

First, considering a single band, the dispersion relation is approximated in terms of rational functions, enabling the inversion to real space. The homogenized equation contains strain gradients as well as spatial derivatives of the inertial term. Considering a boundary between a metamaterial and a homogeneous material, the higher-order space derivatives lead to additional continuity conditions. The higher-order homogenized equation and the continuity conditions 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 for the higher-order spatio-temporal homogenized equation are applied to wave scattering at a boundary, and show good agreement with the exact fine-scale solution. The method is also applied to a problem with multiple scattered propagating waves for which the classical jump conditions cannot provide even approximate solutions, and the results are shown to match reasonably well with the exact fine-scale solutions.

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.

这项工作在周期性超材料的背景下开发了一种动态线性均匀化方法。利用可反演到实空间和实时的色散关系的近似，找到了一个在空间和时间上具有常系数的近似宏观均匀化方程；然而，由此产生的均匀化方程在空间和时间上是更高阶的。均质化方程可用于解决在具有宏观异质性的任意非周期宏观几何形状上提出的初始边界值问题，例如由几种不同的超材料组成或具有外部边界的物体。该方法在这里应用于一维和二维空间中标量未知场的问题。

首先，考虑到单个波段，色散关系用有理函数近似，可以反演到真实空间。均质方程包含应变梯度以及惯性项的空间导数。考虑到超材料和均质材料之间的边界，高阶空间导数导致额外的连续性条件。高阶均匀化方程和连续性条件提供了与精确精细尺度解很好匹配的一维和二维波散射的预测；与其他方法相比，它们提供了一个在广泛的频率范围内有效、易于应用且计算速度更快的单一方程。

接下来，考虑设置具有带隙的两个频带。同质化方程也有高阶时间导数。值得注意的是，同质化模型提供了一个对波段和带隙都有效的单一方程。将高阶时空均匀化方程的连续性条件应用于边界处的波散射，与精确的精细尺度解具有良好的一致性。该方法也适用于具有多个散射传播波的问题，经典跳跃条件无法提供近似解，结果与精确的精细尺度解相当匹配。

利用最高时间导数的阶数与所考虑的带数成正比，推测无限带极限的均匀化方程具有非局部时间结构。这表明，在更精细的长度和时间尺度上的均匀化——通过考虑色散关系中的更高波段来进行时间均匀化——是宏观空间和时间非局域性出现的一种机制，与时间非局域性的程度相关到考虑的频段数量。