A reduced-order computational homogenization framework for locally resonant metamaterial structures

Abstract

A computational homogenization framework is presented to study the dynamics of locally resonant acoustic metamaterial structures. Modelling the resonant units at the microscale as representative volume elements and building on well-established scale transition relations, the framework brings as a main novelty a reduced-order macroscopic homogenized continuum whose governing equations involve no additional variables to describe the microscale dynamics unlike micromorphic homogenized continua obtained by alternative computational homogenization approaches. This model-order reduction is obtained by formulating the governing equations of the micro- and macroscale problems in the frequency domain, introducing a finite-element discretization of the two problems and performing an exact dynamic condensation of all the degrees of freedom at the microscale. An appropriate inverse Fourier transform approach is implemented on the frequency-domain equations to capture transient dynamics as well; notably, the implementation involves the Exponential Window Method, here applied for the first time to calculate the time-domain response of undamped locally resonant acoustic metamaterial structures. The framework may handle arbitrary geometries of micro- and macro-structures, any transient excitations and any boundary conditions on the macroscopic domain.

Current developments in elastic and acoustic metamaterials science

“…Based on this observation, homogenized constitutive relations for the macroscopic stress were obtained in Sridhar et al [63], which depend on additional kinematic degrees of freedom representing the internal dynamics of a representative volume element at the micro-scale and enriching the macroscopic continuum with micro-inertia effects in a micro-morphic sense. Further developments of this approach led to the formulation of a reduced-order macroscopic homogenized continuum whose governing equations involve no additional variables to describe the micro-scale dynamics [65]. Other computational homogenization methods conceived for elastic metamaterials can be found in Roca et al [66,67].…”