Dispersive computational continua

Abstract

Please cite this article as: V. Filonova, D. Fafalis, J. Fish, Dispersive computational continua, Comput. Methods Appl. Mech. Engrg. (2015), http://dx. ABSTRACTThe two primary objectives of the present manuscript are: (i) to develop a variant of the computational continua formulation (C 2 ) with outstanding dispersive properties, and (ii) to conduct a rigorous dispersion analysis of it. The ability of the C 2 formulation to capture dispersive behavior stems from its underlying formulation, which does not explicitly assume scale separation and accounts for microstructures of finite size. The dispersion study in heterogeneous elastic media with periodic microstructure has been conducted using both analytical and numerical approaches. The so-called analytical dispersion analysis is based on the Floquet-Bloch wave solution, while the numerical dispersion analysis is based on the modal analysis of the discrete coupled fine-coarse-scale equations. The dispersive curves obtained from the dispersive C 2 formulations were compared with the classical exact Floquet-Bloch wave solution, hereafter referred to as the reference dispersive curve. It has been observed that in the case of the unit cell sizes being either half of the coarse-scale element size or equal to it, the dispersive curves obtained by the dispersive C 2 formulation are practically identical to the reference solution. For other cases, the dispersive C 2 solution is in good agreement with the reference solution provided that the wavelength is resolved by at least two coarse-scale quadratic elements. The dispersion analysis results have been further verified by the wave propagation problem in a periodic heterogeneous medium with a wavelength comparable to the microstructural size.

Nonlocal Homogenization Model for Wave Dispersion and Attenuation in Elastic and Viscoelastic Periodic Layered Media

“…The authors studied the transient response of acoustic metamaterials in two dimensions by concurrently solving a set of fully coupled macroscale and microscale balance equations. Filonova et al [32] and Fafalis and Fish [33] investigated wave dispersion in one dimensional periodic structures using the concept of computational continua, where the quadrature rules for the numerical integration is adjusted as a function of the size scale ratio. While the above-mentioned methods show good accuracy in capturing the transient wave dispersion and attenuation, solving fully coupled momentum balance equations leads to prohibitive computational cost.…”

Bloch theorem with revised boundary conditions applied to glide, screw and rotational symmetric structures

“…Contrary to the inverse approach, the transfer-matrix method has the advantage of providing evanescent modes, but its implementation in multiple dimensions requires more efforts [12,13]. Besides the Bloch theorem, other approaches can be used to investigate wave propagation in infinite periodic structures, such as for example the homogenization methods for low [14,15] and moderated [16] frequencies. However, this paper focuses on the use of the Bloch theorem.…”

Computational certification under limited experiments

“…Methods in this category include but not limited to the following: higher‐order homogenization(); higher grade continua, characterized by higher‐order spatial derivatives of displacements(); higher‐order continua, endowed with additional degrees of freedom independent of the usual translational degrees of freedom ranging from 3 rotational degrees of freedom in the Cosserat or polar continuum(– to 12 degrees of freedom in the micromorphic continuum(–) and more in so‐called multiscale micromorphic continuum(–); and finally, computational continua, which does not require higher‐order continuity and is free of the classical scale‐separation assumption. (–) Coarse‐scale element enrichment methods, which enhance the kinematics of coarse‐scale elements with microstructural deformation modes. Methods is this category include but not limited to are as follows: the multiscale finite‐element method, the variational multiscale method, the discontinuous enrichment method, and the multiscale enrichment based on partition of unity method.…”