Homogenization of a fully coupled thermoelasticity problem for a highly heterogeneous medium with a priori known phase transformations

We investigate a linear, fully coupled thermoelasticity problem for a highly heterogeneous, two‐phase medium. The medium in question consists of a connected matrix with disconnected, initially periodically distributed inclusions separated by a sharp interface undergoing an a priori known interface movement because of phase transformations. After transforming the moving geometry to an ? ‐periodic, fixed reference domain, we establish the well‐posedness of the model and derive a number of ? ‐independent a priori estimates. Via a two‐scale convergence argument, we then show that the ? ‐dependent solutions converge to solutions of a corresponding upscaled model with distributed time‐dependent microstructures. Copyright © 2017 John Wiley & Sons, Ltd.     

Homogenization of a non‐linear degenerate parabolic problem in a highly heterogeneous periodic medium

We consider the homogenization of a time‐dependent heat transfer problem in a highly heterogeneous periodic medium made of two connected components having comparable heat capacities and conductivities, separated by a third material with thickness of the same order ε as the basic periodicity cell but having a much lower conductivity such that the resulting interstitial heat flow is scaled by a factor λ tending to zero with a rate λ=λ(ε). The heat flux vectors a_j, j=1,2,3 are non‐linear, monotone functions of the temperature gradient. The heat capacities c_j(x) are positive, but may vanish at some subsets such that the problem can be degenerate (parabolic–elliptic). We show that the critical value of the problem is δ=lim_ε→0ε^p/λ and identify the homogenized problem depending on whether δ is zero, strictly positive finite or infinite. Copyright © 2005 John Wiley & Sons, Ltd.     

Homogenization of the variational inequality corresponding to a problem with rapidly varying boundary conditions

In the present paper, we investigate the asymptotic behavior of the solution of a variational inequality with one-sided constraints on ?-periodically located subsets G _ε belonging to the boundary ?Ω of the domain Ω ? ?³. We construct a limit (homogenized) problem and prove the strong (in H ₁(Ω)) convergence of the solutions of the original inequality to the solution the limit nonlinear boundary-value problem as ? → 0 in the so-called critical case.     

Two-scale Convergence and Homogenization for a Class of Quasilinear Elliptic Equations

By use of Fourier analysis techniques, we obtain some new properties of the almost-periodic functions and extend the two-scale convergence method in the homogenization theory to the case of almost-periodic oscillations. Then, we use some new techniques to study the homogenization for quasilinear elliptic equations with almostperiodic coefficients: div a(x,x/ε, u, Du) = f(x) in Ω and obtain the weak convergence and corrector result.     

Homogenization of the layered medium equation

A new partial differential equation to be called the layered medium equation is introduced, and it is proved that certain relevant initial or periodic boundary conditions give well-posed problems. Then, the homogenized limit of the layered medium equation is studied. It is shown to be preserved in limit in the limit in the physical problem in which the coefficients that arise from the dielectric layer are both proportional to thickness. Otherwise, a non-local problem is obtained as the limiting form     

Homogenization of a nonlinear degenerate parabolic differential equation

This paper is devoted to the homogenization of a nonlinear degenerate parabolic problem ɑtu∈-div(D(x/∈, u∈,▽u∈)+ K(x/∈, u∈))= f(x) with Dirichlet boundary condition. Here the operator D(y, s,s) is periodic in y and degenerated in ▽s. In the paper, under the two-scale convergence theory, we obtain the limit equation as ∈→ 0 and also prove the corrector results of ▽u∈ to strong convergence.     

Homogenization of time harmonic Maxwell equations: the effect of interfacial currents

We consider the Maxwell equations for a composite material consisting of two phases and enjoying a periodical structure in the presence of a time‐harmonic current source. We perform the two‐scale homogenization taking into account both the interfacial layer thickness and the complex conductivity of the interfacial layer. We introduce a variational formulation of the differential system equipped with boundary and interfacial conditions. We show the unique solvability of the variational problem. Then, we analyze the low frequency case, high and very high frequency cases, with different strength of the interfacial currents. We find the macroscopic equations and determine the effective constant matrices such as the magnetic permeability, dielectric permittivity, and electric conductivity. The effective matrices depend strongly on the frequency of the current source; the dielectric permittivity and electric conductivity also depend on the strength of the interfacial currents. Copyright © 2016 John Wiley & Sons, Ltd.     

Homogenization of a viscoelastic matrix in linear frictional contact

The paper is devoted to study of acoustic wave propagation in a partially consolidated composite material containing loose particles. Friction of particles against the consolidated part of the material causes mechanical energy dissipation. This situation is modelled by assuming that the medium has a periodic microstructure changing rapidly on the small scale ε. Each of the periodic microscopic cells is composed of a viscoelastic matrix containing a rigid particle in frictional contact with the matrix. We use the methods of two‐scale convergence to obtain effective acoustic equations for the homogenized material. The effective equations are history‐dependent and contain the body force term, reminiscent of the well‐known Stokes drag force. Copyright © 2004 John Wiley & Sons, Ltd.     

Homogenization of a stationary navier-stokes flow in porous medium with thin film

The article studies the homogenization of a stationary Navier-Stokes fluid in porous medium with thin film under Dirichlet boundary condition.At the end of the article,＂Darcy＇s law＂is rigorously derived from this model as the parameter ε tends to zero,which is independent of the coordinates towards the thickness.     

On the Convergence of a Non-incremental Homogenization Method for Nonlinear Elastic Composite Materials

In order to simulate the nonlinear behaviour of elastomer composite materials, we use a homogenization technique which induces nonlinear problems. This paper presents a non-incremental solving method which allows the reduction of computational costs. In this paper the equivalence between the proposed solving method and a Newton-type method is proved, which allows us to prove the convergence under realistic assumptions. Numerical results on a composite illustrate the performances of this method.     

Homogenization of a two‐phase incompressible fluid in crossflow filtration through a porous medium

We study the homogenization of a slow viscous two‐phase incompressible flow in a domain consisting of a free fluid domain, a porous medium, and the interface between them. We take into account the capillary forces on the fluid‐fluid interfaces. We construct boundary layers describing the flow at the interface between the free fluid and the porous medium. We derive a macroscopic model with a viscous two‐phase fluid in the free domain, a coupled Darcy law connecting two‐phase velocities in the porous medium, and boundary conditions at the permeable interface between the free fluid domain and the porous medium.     

Homogenization of the Signorini boundary‐value problem in a thick junction and boundary integral equations for the homogenized problem

We consider a mixed boundary‐value problem for the Poisson equation in a thick junction Ω_ε which is the union of a domain Ω₀ and a large number of ε—periodically situated thin cylinders. The non‐uniform Signorini conditions are given on the lateral surfaces of the cylinders. The asymptotic analysis of this problem is done as ε→0, i.e. when the number of the thin cylinders infinitely increases and their thickness tends to zero. We prove a convergence theorem and show that the non‐uniform Signorini boundary conditions are transformed in the limiting variational inequalities in the region that is filled up by the thin cylinders as ε→0. The convergence of the energy integrals is proved as well. The existence and uniqueness of the solution to this non‐standard limit problem is established. This solution can be constructed by using a penalty formulation and successive iteration. For some subclass, these problems can be reduced to an obstacle problem in Ω₀ and an appropriate postprocessing. The equations in Ω₀ finally are also treated with boundary integral equations. Copyright © 2010 John Wiley & Sons, Ltd.     

Homogenization of Optimal Control Problems for Functional Differential Equations

The paper deals with the variational convergence of a sequence of optimal control problems for functional differential state equations with deviating argument. Variational limit problems are found under various conditions of convergence of the input data. It is shown that, upon sufficiently weak assumptions on convergence of the argument deviations, the limit problem can assume a form different from that of the whole sequence. In particular, it can be either an optimal control problem for an integro-differential equation or a purely variational problem. Conditions are found under which the limit problem preserves the form of the original sequence.     

Homogenization of degenerate coupled transport processes in porous media with memory terms

In this paper, we establish a homogenization result for a doubly nonlinear parabolic system arising from the hygro‐thermo‐chemical processes in porous media taking into account memory phenomena. We present a mesoscale model of the composite (heterogeneous) material where each component is considered as a porous system and the voids of the skeleton are partially saturated with liquid water. It is shown that the solution of the mesoscale problem is two‐scale convergent to that of the upscaled problem as the spatial parameter goes to zero.     

Homogenization of some parabolic operators with several time scales

The main focus in this paper is on homogenization of the parabolic problem ∂ _t uɛ − ∇ · (a(x/ɛ,t/ɛ,t/ɛ ^r )∇u ^ɛ ) = f. Under certain assumptions on a, there exists a G-limit b, which we characterize by means of multiscale techniques for r > 0, r ≠ 1. Also, an interpretation of asymptotic expansions in the context of two-scale convergence is made.     

Homogenization of the linearized system of elasticity in anisotropic heterogeneous thin cylinders

This paper is the second of the two announced in our Note (Sili A, [16]). It generalizes to the linearized system of elasticity the results of our previous work (Sili, 2000 [15]) on the heat equation. We study the asymptotic analysis, as ? tends to zero, of the solution u^? of the linearized system of elasticity posed on a composite elastic cylindrical domain Ω^? with radius ? and height L. The heterogeneities of the material are assumed to be periodic with a period ? along the cylinder axis and with a period ?² along the sections of the cylinder. It is shown that the limit problem is a system in which appear two entities: the first one (u, v, w) corresponds to the reduction of dimension 3d–1d while the second one (û, v?, ?) takes into account the homogenization process. Moreover, a corrector result is given. Copyright © 2002 John Wiley & Sons, Ltd.     

Decomposition Method for a Class of Monotone Variational Inequality Problems

In the solution of the monotone variational inequality problem VI(, F), with

非齐次边界条件泛定方程的代换选择

对具有非齐次边界条件的泛定方程齐次化过程中代换的选择进行研究和探讨.基于一些相关结论和齐次化的定义得出新的研究成果,即给出对三类非齐次边界条件齐次化都适用的代换W(x,t)=A(t)x<'3>+B(t).     

Homogenization of contact problem with Coulomb's friction on periodic cracks

We consider the elasticity problem in a domain with contact on multiple periodic open cracks. The contact is described by the Signorini and Coulomb‐friction conditions. The problem is nonlinear, the dissipative functional depends on the unknown solution, and the existence of the solution for fixed period of the structure is usually proven by the fix‐point argument in the Sobolev spaces with a little higher regularity, H^1+α. We rescaled norms, trace, jump, and Korn inequalities in fractional Sobolev spaces with positive and negative exponents, using the unfolding technique, introduced by Griso, Cioranescu, and Damlamian. Then we proved the existence and uniqueness of the solution for friction and period fixed. Then we proved the continuous dependency of the solution to the problem with Coulomb's friction on the given friction and then estimated the solution using fixed‐point theorem. However, we were not able to pass to the strong limit in the frictional dissipative term. For this reason, we regularized the problem by adding a fourth‐order term, which increased the regularity of the solution and allowed the passing to the limit. This can be interpreted as micro‐polar elasticity.