Evolutional contact with Coulomb's friction on a periodic microstructure

We consider the elasticity problem in a heterogeneous domain with an ε-periodic micro-structure, ε ≪ 1, including a multiple micro-contact in a simply connected matrix domain with inclusions completely surrounded by cracks, which do not connect the boundary, or a textile-like material. The contact is described by the Signorini and Coulomb-friction contact conditions. In the case of the Coulomb friction, the dissipative functional is state dependent, like in [2]. A time discretization scheme from [2] reduces the contact problem to the Tresca one (with prescribed frictional traction or state independent dissipation) on each time-increment. We further look for the spatial homogenization. The limiting energy and the dissipation term in the stability condition were obtained for the contact with Tresca's friction law in [4] for closed cracks and can be extended to textile-like materials. Using these results and the concept of energetic solutions for evolutional quasi-variational problems from [2], for a uniform time-step partition, the existence can be proved for the solution of the continuous problem and a subsequence of incremental solutions weakly converging to the continuous one uniformly in time. Furthermore, the irreversible frictional displacements at micro-level lead to a kind of an evolutional plastic behavior of the homogenized medium. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homogenization of the Maxwell Equations: Case I. Linear Theory

The Maxwell equations in a heterogeneous medium are studied. Nguetseng's method of two-scale convergence is applied to homogenize and prove corrector results for the Maxwell equations with inhomogeneous initial conditions. Compactness results, of two-scale type, needed for the homogenization of the Maxwell equations are proved.     

Homogenization of the Maxwell Equations: Case II. Nonlinear Conductivity

The Maxwell equations with uniformly monotone nonlinear electric conductivity in a heterogeneous medium, which may be non-periodic, are homogenized by two-scale convergence. We introduce a new set of function spaces appropriate for the nonlinear Maxwell system. New compactness results, of two-scale type, are proved for these function spaces. We prove existence of a unique solution for the heterogeneous system as well as for the homogenized system. We also prove that the solutions of the heterogeneous system converge weakly to the solution of the homogenized system. Furthermore, we prove corrector results, important for numerical implementations.     

Homogenization of Richardsʼ equation of van Genuchten–Mualem model

This paper is devoted to the homogenization of Richards? equation of van Genuchten–Mualem model, which is a nonlinear degenerate parabolic differential equation. It is usually used to model the motion of saturated–unsaturated water flow in porous media. We firstly apply the Kirchhoff transformation to the equation and obtain a simpler equivalent equation with a linear oscillated diffusion term. Then under the real assumption for van Genuchten–Mualem model, we obtain the homogenized equation based on the two-scale convergence theory. Some results on the first order corrector are also presented.     

Analysis of the heterogeneous multiscale method for elliptic homogenization problems

A comprehensive analysis is presented for the heterogeneous multiscale method (HMM for short) applied to various elliptic homogenization problems. These problems can be either linear or nonlinear, with deterministic or random coefficients. In most cases considered, optimal estimates are proved for the error between the HMM solutions and the homogenized solutions. Strategies for retrieving the microstructural information from the HMM solutions are discussed and analyzed.

     

A Hilbert space approach to homogenization of linear ordinary differential equations including delay and memory terms

We present an abstract approach to homogenization in a Hilbert space setting. Related compactness results are obtained. Moreover, the homogenized equations may be computed explicitly, if periodicity is imposed. Examples for the applicability of our homogenization result for linear ordinary (integro‐)differential equations are given. Copyright © 2012 John Wiley & Sons, Ltd.     

MULTISCALE HOMOGENIZATION OF NONLINEAR HYPERBOLIC EQUATIONS WITH SEVERAL TIME SCALES

We study the multiscale homogenization of a nonlinear hyperbolic equation in a periodic setting.We obtain an accurate homogenization result.We also show that as the nonlinear term depends on the microscopic time variable,the global homogenized problem thus obtained is a system consisting of two hyperbolic equations.It is also shown that in spite of the presence of several time scales,the global homogenized problem is not a reiterated one.     

Polynomial approximations of a class of stochastic multiscale elasticity problems

We consider a class of elasticity equations in \({\mathbb{R}^d}\) whose elastic moduli depend on n separated microscopic scales. The moduli are random and expressed as a linear expansion of a countable sequence of random variables which are independently and identically uniformly distributed in a compact interval. The multiscale Hellinger–Reissner mixed problem that allows for computing the stress directly and the multiscale mixed problem with a penalty term for nearly incompressible isotropic materials are considered. The stochastic problems are studied via deterministic problems that depend on a countable number of real parameters which represent the probabilistic law of the stochastic equations. We study the multiscale homogenized problems that contain all the macroscopic and microscopic information. The solutions of these multiscale homogenized problems are written as generalized polynomial chaos (gpc) expansions. We approximate these solutions by semidiscrete Galerkin approximating problems that project into the spaces of functions with only a finite number of N gpc modes. Assuming summability properties for the coefficients of the elastic moduli’s expansion, we deduce bounds and summability properties for the solutions’ gpc expansion coefficients. These bounds imply explicit rates of convergence in terms of N when the gpc modes used for the Galerkin approximation are chosen to correspond to the best N terms in the gpc expansion. For the mixed problem with a penalty term for nearly incompressible materials, we show that the rate of convergence for the best N term approximation is independent of the Lamé constants’ ratio when it goes to \({\infty}\). Correctors for the homogenization problem are deduced. From these we establish correctors for the solutions of the parametric multiscale problems in terms of the semidiscrete Galerkin approximations. For two-scale problems, an explicit homogenization error which is uniform with respect to the parameters is deduced. Together with the best N term approximation error, it provides an explicit convergence rate for the correctors of the parametric multiscale problems. For nearly incompressible materials, we obtain a homogenization error that is independent of the ratio of the Lamé constants, so that the error for the corrector is also independent of this ratio.     

Deterministic homogenization of weakly damped nonlinear hyperbolic-parabolic equations

Deterministic homogenization is studied for nonlinear hyperbolic-parabolic equations with a linear damping term. It is shown by the sigma-convergence method that the sequence of solutions to a class of highly oscillatory nonlinear evolution problems converges to the solution to a homogenized quasilinear hyperbolic-parabolic problem.     

Approximation numérique d'une classe de problèmes en homogénéisation stochastique

In this Note, we study a numerical strategy for the computation of the homogenized matrix for a stochastic elliptic problem that is a small perturbation of a periodic problem. We adapt the analysis introduced in [X. Blanc, C. Le Bris, P.-L. Lions, Stochastic homogenization and random lattices, J. Math. Pures Appl. 88 (2007) 34–63] to the case when the corrector problems are numerically solved, and we computationally assess the interest and the accuracy of the approach.     

Homogenization and Asymptotics for Small Transaction Costs: The Multidimensional Case

In the context of the multi-dimensional infinite horizon optimal consumption investment problem with small proportional transaction costs, we prove an asymptotic expansion. Similar to the one-dimensional derivation in our accompanying paper, the first order term is expressed in terms of a singular ergodic control problem. Our arguments are based on the theory of viscosity solutions and the techniques of homogenization which leads to a system of corrector equations. In contrast with the one-dimensional case, no explicit solution of the first corrector equation is available and we also prove the existence of a corrector and its properties. Finally, we provide some numerical results which illustrate the structure of the first order optimal controls.     

Aspects of computational homogenization in magneto-mechanics

In the present work, the behavior of heterogeneous magnetorheological elastomers undergoing large deformations under the action of magnetic fields is studied. First-order computational homogenization is used to derive the homogenized stresses and magnetic inductions of the macro-structure from the response of the underlying micro-structure. Different types of boundary conditions are applied to solve the micro problem where the constitutive law is assumed to be known. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homogenization approach to the multi-compartment model of perfusion

The paper deals with modelling of the coupled diffusion-deformation processes in biological tissues with potential applications in describing the blood perfusion, or fluid filtration phenomena in general. The micromodel to be homogenized is based on the Biot type model for the incompressible medium. Due to the strong heterogeneity in the permeability coefficients associated with three compartments of the representative microstructural cell (RMC), the homogenization of the model leads to the double diffusion phenomena. The resulting homogenized equations, involving the stress-equilibrium equation and other two equations governing the mass redistribution, describe the parallel diffusion in two high-conducting compartments (arterial and venous sectors) separated by the low conducting matrix which represents the perfused tissue. To obtain the homogenized model, the method of two scale convergence is applied. The homogenized coefficients are defined in terms of the characteristic response of the RMC. It is possible to identify the instantaneous and fading memory viscoelastic coefficients; other effective parameters, controlling the fluid redistribution between the compartments, are involved also in time convolutions. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Water waves over a rough bottom in the shallow water regime

This is a study of the Euler equations for free surface water waves in the case of varying bathymetry, considering the problem in the shallow water scaling regime. In the case of rapidly varying periodic bottom boundaries this is a problem of homogenization theory. In this setting we derive a new model system of equations, consisting of the classical shallow water equations coupled with nonlocal evolution equations for a periodic corrector term. We also exhibit a new resonance phenomenon between surface waves and a periodic bottom. This resonance, which gives rise to secular growth of surface wave patterns, can be viewed as a nonlinear generalization of the classical Bragg resonance. We justify the derivation of our model with a rigorous mathematical analysis of the scaling limit and the resulting error terms. The principal issue is that the shallow water limit and the homogenization process must be performed simultaneously. Our model equations and the error analysis are valid for both the two- and the three-dimensional physical problems.     

Homogenization of the two-dimensional Hall effect

In this paper, we study the two-dimensional Hall effect in a highly heterogeneous conducting material in the low magnetic field limit. Extending Bergman's approach in the framework of H-convergence we obtain the effective Hall coefficient which only depends on the corrector of the material resistivity in the absence of a magnetic field. A positivity property satisfied by the effective Hall coefficient is then deduced from the homogenization process. An explicit formula for the effective Hall coefficient is derived for anisotropic interchangeable two-phase composites.     

Computational Homogenization of Piezoelectric Materials using FE2

Due to the growing interest in determining the macroscopic material response of inhomogeneous materials, computational methods are becoming increasingly concerned with the application of homogenization techniques. In this work, two-scale classical (first-order) homogenization of electro-mechanically coupled problems using a FE²-approach is discussed. We explicitly formulated the homogenized coefficients of the elastic, piezoelectric and dielectric tensors for small strain as well as the homogenized remanent strain and remanent polarization. The homogenization of the coupled problem is done using different representative volume elements (RVEs), which capture the microstructure of the inhomogeneous material, to represent the macro material response. Later this technique is used to determine the macroscopic and microscopic configurational forces on certain defects. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Towards the computational homogenization of discrete microstructures

The main objective of the current work is the extension of a computational homogenization scheme towards the simulation of discrete micro-structures. On the micro-level so-called multiple particle systems (MPS) are introduced for which a continuization scheme, resulting in a formulation analogous to a homogenization scheme, is developed. Furthermore, three types of constitutive behavior are applied to these particle systems and their influence on the overall macro-structure is studied. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)