[Paper Review] Scattering of finite-size anisotropic metastructures via the relaxed micromorphic model

Alexios Aivaliotis, Domenico Tallarico+3 more|arXiv (Cornell University)|May 27, 2019

Acoustic Wave Phenomena Research35 references4 citations

TL;DR

This paper introduces a relaxed micromorphic model to accurately simulate broadband scattering of elastic waves in finite-sized anisotropic meta-structures, incorporating enriched continuum mechanics with generalized boundary conditions. It achieves excellent agreement with full finite element simulations across a wide frequency range and incidence angles, while reducing computational cost by orders of magnitude.

ABSTRACT

The conception of new metamaterials showing unorthodox behaviors with respect to elastic wavepropagation has become possible in recent years thanks to powerful dynamical homogenization techniques. Such methods effectively allow to describe the behavior of an infinite medium generated by periodically architectured base materials. Nevertheless, when it comes to the study of the scattering properties of finite-sized structures, dealing with the correct boundary conditions at the macroscopicscale becomes challenging. In this paper, we show how finite-domain boundary value problems canbe set-up in the framework of enriched continuum mechanics (relaxed micromorphic model) by imposing continuity of macroscopic displacement and of generalized traction when non-local effects areneglected.The case of a metamaterial slab of finite width is presented, its scattering properties are studied viaa semi-analytical solution of the relaxed micromorphic model and compared to numerical simulationsencoding all details of the selected microstructure. The reflection coefficient obtained via the twomethods is presented as a function of the frequency and of the direction of propagation of the incidentwave. We find excellent agreement for a large range of frequencies going from the long-wave limitto frequencies beyond the first band-gap and for angles of incidence ranging from normal to nearparallel incidence. The case of a semi-infinite metamaterial is also presented and is seen to be areliable measure of the average behavior of the finite metastructure. A tremendous gain in termsof computational time is obtained when using the relaxed micromorphic model for the study of theconsidered metastructure.

Motivation & Objective

• To address the challenge of modeling wave scattering in finite-sized meta-structures using homogenized models.
• To develop a robust framework for enforcing physically consistent boundary conditions at interfaces between macroscopic continua and micromorphic media.
• To enable accurate, computationally efficient simulation of wave scattering in finite anisotropic meta-structures beyond the long-wave limit.
• To validate the relaxed micromorphic model against detailed finite element simulations for a metamaterial slab.
• To demonstrate the model's capability for large-scale meta-structure analysis by showing strong agreement with high-fidelity simulations.

Proposed method

• The relaxed micromorphic model is employed as an enriched continuum framework to describe the effective behavior of periodic microstructures.
• Generalized boundary conditions are enforced, including continuity of macroscopic displacement and generalized traction, along with conditions on microdistortion and double-traction.
• A semi-analytical solution is derived for wave scattering at a finite slab of relaxed micromorphic material embedded between two Cauchy continua.
• The model incorporates tetragonal symmetry and accounts for anisotropic elastic and microinertia properties via a full set of constitutive parameters.
• The reflection coefficient is computed analytically and compared with finite element simulations of the actual microstructure.
• Energy flux and power balance are rigorously derived using complex harmonic wave decomposition and time-averaged formulations.

Experimental results

Research questions

• RQ1 How can consistent boundary conditions be formulated for finite-sized meta-structures in the relaxed micromorphic framework?
• RQ2 To what extent does the relaxed micromorphic model accurately predict scattering properties of finite anisotropic meta-structures compared to full microscale simulations?
• RQ3 Can the model maintain accuracy across a broad frequency band, including beyond the first band-gap and for oblique incidence?
• RQ4 How does the semi-analytical solution compare with high-fidelity finite element simulations in terms of computational efficiency and accuracy?
• RQ5 Is the semi-infinite relaxed micromorphic model a reliable proxy for the average behavior of a finite meta-structure?

Key findings

• The relaxed micromorphic model achieves excellent agreement with finite element simulations for the reflection coefficient across a broad frequency range, from the long-wave limit to frequencies beyond the first band-gap.
• The model accurately captures scattering behavior for incidence angles ranging from normal to near-parallel incidence, demonstrating robustness across diverse wave propagation directions.
• A significant reduction in computational time is observed—on the order of orders of magnitude—when using the relaxed micromorphic model instead of full microscale finite element modeling.
• The semi-infinite relaxed micromorphic model provides a reliable approximation of the average scattering behavior of the finite meta-structure.
• The enforcement of generalized boundary conditions, including microdistortion and double-traction continuity, ensures energy conservation and physical consistency at interfaces.
• The analytical derivation of energy flux in the anisotropic relaxed micromorphic model confirms consistency with physical principles and supports the validity of the scattering solution.