Frequency domain boundary value problem analyses of acoustic metamaterials described by an emergent generalized continuum

Abstract

This paper presents a computational frequency-domain boundary value analysis of acoustic metamaterials and phononic crystals based on a general homogenization framework, which features a novel definition of the macro-scale fields based on the Floquet-Bloch average in combination with a family of characteristic projection functions leading to a generalized macro-scale continuum. Restricting to 1D elastodynamics and the frequency-domain response for the sake of compactness, the boundary value problem on the generalized macro-scale continuum is elaborated. Several challenges are identified, in particular the non-uniqueness in selection of the boundary conditions for the homogenized continuum and the presence of spurious short wave solutions. To this end, procedures for the determination of the homogenized boundary conditions and mitigation of the spurious solutions are proposed. The methodology is validated against the direct numerical simulation on an example periodic 2-phase composite structure.

Metamaterial shields for inner protection and outer tuning through a relaxed micromorphic approach

“…Dynamic homogenization techniques that are able to provide PDEs describing the macroscopic material's response have been extensively studied [13,35,36,41,42,43,48,49,51]. Yet, fundamental difficulties arise when specimens of finite size are considered, especially in the context of establishing well-posed homogenized boundary conditions.…”

“…One important application is that of exploiting metamaterials' band-gaps to create elastic shielding devices protecting objects from elastic waves [8,25,29,37]. Depending on the devices' operating frequencies, these shields can act for seismic [20], acoustic [25,35,38,41,43] or ultrasonic [2,13,38,39,40] protection of objects that need to be isolated from the external environment.…”

“…We show in this paper that, once such a model is established, suitable boundary value problems can be set up allowing an eased simulation of both the inner and outer domain. In fact, even if dynamic homogenization methods have provided important advancements in the modeling of infinite-size metamaterials [13,32,36,41,42,43,48,52], little is known about handling scattering problems at the boundaries of finite-size metamaterials. This lack of knowledge is mainly due to the difficulties arising in establishing pertinent macroscopic boundary conditions formally upscaled from microscopic ones.…”

Boundary and interface conditions in the relaxed micromorphic model: Exploring finite-size metastructures for elastic wave control

“…Sridhar et al [45] proposed an alternative ad hoc up-scaling procedure, only valid for locally resonant metamaterials, leading to a homogenized equation which the authors recognize to be of the micromorphic type. In a similar spirit, in [50] a homogenized continuum with extended kinematics was obtained, classifying it as micromorphic, and proposed its use to study a simple 1D boundary value problem for a periodic metamaterial. However, major concerns are always encountered when trying to establish well-posed homogenized boundary conditions.…”

Boundary and interface conditions in the relaxed micromorphic model: exploring finite-size metastructures for elastic wave control

“…Sridhar et al [53] proposed an alternative ad-hoc upscaling procedure, only valid for locally resonant metamaterials, leading to a homogenized equation which the authors recognize to be of the micromorphic type. In a similar spirit, [52] obtained a homogenized continuum with extended kinematics, classifying it as micromorphic, and proposed its use to study a simple 1D boundary value problem for a periodic metamaterial. However, major concerns are always encountered when trying to establish well-posed homogenized boundary conditions.…”