Homogenization theory and multiscale numerical approaches for disordered media: some recent contributions

Abstract

Abstract. We overview a series of recent works related to some multiscale problems motivated by practical problems in Mechanics. The common denominator of all these works is that they address multiscale problems where the geometry of the microstructures is not periodic. Random modelling, as well as other types of nonperiodic modelling, can then be used to account for the imperfections of the medium. The theory at play is that of homogenization, in its many variants (stochastic, general deterministic, periodic). The numerical methods developed and adapted are finite element type methods. A special emphasis is laid on situations where the amount of randomness is small, or, put differently, when the disorder is limited. Then, specific, computationally efficient techniques can be designed and employed.Résumé. Nous présentons un panorama d'une série de travaux récents sur des problèmes multiechelles motivés par la science des matériaux. Le dénominateur commun de ces travaux est qu'ils traitent tous de problèmes où la géométrie des microstructures n'est pas périodique. La modélisation aléatoire, ainsi que d'autres types de modélisations non periodiques, sont alors utilisées pour rendre compte des imperfections du matériau. Le cadre théorique est celui de l'homogénéisation dans ses nombreuses variantes (stochastique, déterministique générale, périodique). Les méthodes numériques développées et adaptées sont des méthodes de typeéléments finis. Une attention toute particulière est portée sur les problèmes où la "quantité de hasard" est petite ou, autrement dit, le désordre est limité. Des techniques spécifiques, efficaces pour le calcul, peuvent alorsêtre définies et utilisées.

Quantifying the Influence of the Crowded Cytoplasm on Small Molecule Diffusion

“…In fact, this is the basis for a homogenization scheme valid for nonperiodic materials, wherein it is assumed that periodic spherical inclusions are remapped to nonperiodic positions that remain strictly confined within the unit cell. Additionally, for examination of systems with periodicity-breaking defects, such as the presence of intracellular organelles in a cytoplasmic protein lattice, periodic correctors for localized perturbations may be appropriate. , …”

“…Additionally, for examination of systems with periodicity-breaking defects, such as the presence of intracellular organelles in a cytoplasmic protein lattice, periodic correctors for localized perturbations may be appropriate. 77,78 There are a number of additional model improvements that may be considered, particularly for describing highly charged systems. For the crowder surface potentials considered in this study (|ψ| ≤ 25 mV), the linearized PB was sufficient to describe the diffuse layer about charged crowders as well as their overlapping DLs.…”

Mechanical metamaterials

“…The actual homogenized coefficients are only captured in the asymptotic regime. Important theoretical questions about the quality and the rate of the convergence in terms of the truncation size arise and are addressed in [93][94][95][96]. One should bear in mind that the rate of convergence in L 2 norm of u η to the leading term u 0 of the asymptotic expansion (36) scales as η in the periodic one-dimensional (1D) case and as √ η in the disordered case 1D.…”

Homogenization approximations for unidirectional transport past randomly distributed sinks

“…For proper uncertainty quantification, it is necessary to determine corrections that capture the stochastic variability in the quantity of interest. Such corrections are likely to be entangled with the discrete-to-continuous corrections of the classical homogenization approximation and are typically nonlocal (Heitzinger & Ringhofer, 2014;Le Bris, 2014;Wood & Valdés-Parada, 2013).…”