Multiscale Computational Methods for Wave Propagation in 2D Phononic Crystals and Acoustic Metamaterials

Abstract

Periodic composites with tailored microstructures and material properties such as phononic crystals and acoustic metamaterials exhibit extraordinary capabilities in controlling elastic waves by manipulating band gaps that forbid waves to propagate within targeted frequency ranges. Modeling these phenomena using direct numerical simulations by resolving all relevant scales is computationally prohibitive for structural design and analysis. This dissertation presents multiscale models for wave propagation in periodic elastic and viscoelastic composites, towards more efficient modeling of their dynamic behaviors. Multiscale models for heterogeneous materials are broadly classified in two categories: (1) scale-separation assumption dependent and (2) scale-separation assumption independent. The scale-separation assumption assumes that the deformation wavelength is much larger than the size of microstructures. In this dissertation, both scale-separable and scale-inseparable multiscale models are formulated for wave propagation in periodic elastic and viscoelastic composites in order to capture short-wavelength phenomena and band gaps.

A spatial-temporal nonlocal homogenization model has been developed for transient wave propagation in periodic elastic and viscoelastic composites. The homogenization model is derived by employing high order asymptotic expansions, extending the applicability of asymptotic homogenization to the short-wavelength regime. A gradient-type spatial-temporal nonlocal macroscopic governing equation is consistently derived from the momentum balance equations of successive asymptotic orders. This homogenization model is valid in the frequency regime that scale-separation assumption is weakly satisfied.

A spectral variational multiscale model is proposed for the scale-inseparable problems. This model is based on the variational multiscale enrichment method. The proposed approach employs a two-scale additive split of the displacement field, and does not make assumptions on the relative size of microstructures and the macroscopic wavelength. The two important ingredients of the proposed spectral variational multiscale approach in achieving both accuracy and numerical efficiency are: (1) higher-order spectral representation of the coarse-scale solution, and (2) material-phase-based modal basis reduction to the fine-scale solution.