Publication 21-CNA-004

Multiband Homogenization of Metamaterials in Real-Space: Higher-Order Nonlocal Models and Scattering at External Surfaces

Kshiteej Deshmukh
Department of Civil and Environmental Engineering
Carnegie Mellon University
Pittsburgh, PA
kjdeshmu@andrew.cmu.edu

Timothy Breitzman
Air Force Research Laboratory
timothy.breitzman.1@us.af.mil

Kaushik Dayal
Center for Nonlinear Analysis
Department of Civil and Environmental Engineering
Department of Materials Science and Engineering
Carnegie Mellon University
Pittsburgh, PA 15213
Kaushik.Dayal@cmu.edu

Abstract: 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 towave scattering at a boundary, and showgood 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.

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