Two-scale Convergence Method Research Articles

Homogenization of some degenerate pseudoparabolic variational inequalities

Multiscale analysis of a degenerate pseudoparabolic variational inequality, modelling the two-phase flow with dynamical capillary pressure in a perforated domain, is the main topic of this work. Regularisation and penalty operator methods are applied to show the existence of a solution of the nonlinear degenerate pseudoparabolic variational inequality defined in a domain with microscopic perforations, as well as to derive a priori estimates for solutions of the microscopic problem. The main challenge is the derivation of a priori estimates for solutions of the variational inequality, uniformly with respect to the regularisation parameter and to the small parameter defining the scale of the microstructure. The method of two-scale convergence is used to derive the corresponding macroscopic obstacle problem.

Homogenization of a Large Eddy Simulation Model for Turbulent Fluid Motion Near a Rough Wall

In this paper, we are interested in the numerical simulation of a turbulent viscous flow in the vicinity of a periodic rough wall. To this aim, we consider a large eddy simulation (LES) model, and perform its asymptotic analysis, letting the period $$\varepsilon $$ of the roughness tend to zero. Relying on an homogenization method closely related to the two-scale convergence method (Allaire in SIAM J Math Anal 23(6):1482–1518, 1992; Nguetseng in SIAM J Math Anal 20(3):608–623, 1989), we exhibit a critical scaling associated with parameter $$\lambda :=\lim _{\varepsilon \rightarrow 0}{a_\varepsilon }/{\varepsilon ^{5/3}}$$ , where $$a_\varepsilon $$ stands for the amplitude of the roughness. In the critical regime $$0<\lambda <\infty $$ , the roughness effect is described by an additional, nonlinear friction term, that depends locally on the solution to a nonlinear auxiliary problem of boundary layer type. In the particular case of riblets, which is of major practical interest, we show that the directional invariance of the geometry results in a simplified expression of this extra term, and we propose a numerical method to simulate the homogenized LES model, based on that observation. Finally, we study the influence of parameter $$\lambda $$ on the flow approximated by the homogenized model, using different commonly used riblet shapes. Our simulations demonstrate that the extra friction term can be handled numerically, and may have an important influence on the results of LES in presence of roughness.

Stochastic homogenisation of high-contrast media

ABSTRACTUsing a suitable stochastic version of the compactness argument of [Zhikov VV. On an extension of the method of two-scale convergence and its applications. Sb Math. 2000;191(7–8):973–1014], we develop a probabilistic framework for the analysis of heterogeneous media with high contrast. We show that an appropriately defined multiscale limit of the field in the original medium satisfies a system of equations corresponding to the coupled ‘macroscopic’ and ‘microscopic’ components of the field, giving rise to an analogue of the ‘Zhikov function’, which represents the effective dispersion of the medium. We demonstrate that, under some lenient conditions within the new framework, the spectra of the original problems converge to the spectrum of their homogenisation limit.

Homogenization of Cahn–Hilliard-type equations via evolutionary $$\varvec{\Gamma }$$-convergence

In these notes we discuss two approaches to evolutionary \(\Gamma \)-convergence of gradient systems in Hilbert spaces. The formulation of the gradient system is based on two functionals, namely the energy functional and the dissipation potential, which allows us to employ \(\Gamma \)-convergence methods. In the first approach we consider families of uniformly convex energy functionals such that the limit passage of the time-dependent problems can be based on the theory of evolutionary variational inequalities as developed by Daneri and Savaré 2010. The second approach uses the equivalent formulation of the gradient system via the energy-dissipation principle and follows the ideas of Sandier and Serfaty 2004.We apply both approaches to rigorously derive homogenization limits for Cahn–Hilliard-type equations. Using the method of weak and strong two-scale convergence via periodic unfolding, we show that the energy and dissipation functionals \(\Gamma \)-converge. In conclusion, we will give specific examples for the applicability of each of the two approaches.

Effective interface conditions for processes through thin heterogeneous layers with nonlinear transmission at the microscopic bulk-layer interface

In this paper, we consider a system of reaction-diffusion equations in a domain consisting of two bulk regions separated by a thin layer with thickness of order $e$ and a periodic heterogeneous structure. The equations inside the layer depend on $e$ and the diffusivity inside the layer on an additional parameter $γ ∈ [-1, 1]$. On the bulk-layer interface, we assume a nonlinear Neumann-transmission condition depending on the solutions on both sides of the interface. For $\epsilon \to0 $, when the thin layer reduces to an interface $Σ$ between two bulk domains, we rigorously derive macroscopic models with effective conditions across the interface $Σ$. The crucial part is to pass to the limit in the nonlinear terms, especially for the traces on the interface between the different compartments. For this purpose, we use the method of two-scale convergence for thin heterogeneous layers, and a Kolmogorov-type compactness result for Banach valued functions, applied to the unfolded sequence in the thin layer.

Some remarks on the homogenization of immiscible incompressible two-phase flow in double porosity media

This paper presents a study of immiscible incompressible two-phase flow through fractured porous media. The results obtained earlier in the pioneer work by A. Bourgeat, S. Luckhaus, A. Mikelic (1996) and L. M. Yeh (2006) are revisited. The main goal is to incorporate some of the most recent improvements in the convergence of the solutions in the homogenization of such models. The microscopic model consists of the usual equations derived from the mass conservation of both fluids along with the Darcy-Muskat law. The problem is written in terms of the phase formulation, i.e. the saturation of one phase and the pressure of the second phase are primary unknowns. We will consider a domain made up of several zones with different characteristics: porosity, absolute permeability, relative permeabilities and capillary pressure curves. The fractured medium consists of periodically repeating homogeneous blocks and fractures, the permeability being highly discontinuous. Over the matrix domain, the permeability is scaled by \begin{document}${\varepsilon }^θ$\end{document} , where \begin{document}$\varepsilon$ \end{document} is the size of a typical porous block and \begin{document}$θ>0$\end{document} is a parameter. The model involves highly oscillatory characteristics and internal nonlinear interface conditions. Under some realistic assumptions on the data, the convergence of the solutions, and the macroscopic models corresponding to various range of contrast are constructed using the two-scale convergence method combined with the dilation technique. The results improve upon previously derived effective models to highly heterogeneous porous media with discontinuous capillary pressures.

Homogenization of a viscoelastic model for plant cell wall biomechanics

The microscopic structure of a plant cell wall is given by cellulose microfibrils embedded in a cell wall matrix. In this paper we consider a microscopic model for interactions between viscoelastic deformations of a plant cell wall and chemical processes in the cell wall matrix. We consider elastic deformations of the cell wall microfibrils and viscoelastic Kelvin–Voigt type deformations of the cell wall matrix. Using homogenization techniques (two-scale convergence and periodic unfolding methods) we derive macroscopic equations from the microscopic model for cell wall biomechanics consisting of strongly coupled equations of linear viscoelasticity and a system of reaction-diffusion and ordinary differential equations. As is typical for microscopic viscoelastic problems, the macroscopic equations governing the viscoelastic deformations of plant cell walls contain memory terms. The derivation of the macroscopic problem for the degenerate viscoelastic equations is conducted using a perturbation argument.

Homogenization and Electronic Polarization Effects in Dielectric Materials

This paper is devoted to the electronic polarization effect of dielectric materials in plasma physics. Using the two-scale convergence method introduced by G. Nguetseng and G. Allaire, we study the electronic polarization effect induced by two-scale homogenization. The microscopic properties of electrons in dielectric materials are characterized by Vlasov-Poisson system. The homogenized equations describing the mean behaviors of the microscopic equations are obtained. We also derive the modified Gauss law through the polarization effect. From the homogenized Poisson equation, the dielectric function is also acquired.

О модели изотермической акустики для двухкомпонентной среды

A mathematical model describing the processes of isothermal acoustics in a heterogeneous medium with two components separated by a common boundary is studied. One of the components is an elastic body, and the other one is a poroelastic medium (for example, it may be a liquid-saturated soil). The poroelastic medium is permeated with a system of pores filled with viscous weakly compressible liquid. The differential equations of the model describing the motion of an elastic body and the joint motion of a solid skeleton and liquid in the pores are based on the classical laws of continuous medium mechanics and adequately reflect the physical processes. However, these equations contain rapidly oscillating coefficients that depend on a small parameter equal to the ratio of the mean pore size to the size of the region under study. The existence of such coefficients prevents the use of the model for carrying out numerical calculations. The generalized solution of the initial boundary-value problem is given; and the theorem for existence and uniqueness of the generalized solution is presented together with its a priori estimates. For performing the homogenization procedure, the standard assumption about the periodicity of the pore space and solid skeleton is adopted. The obtained a priori estimates and the N. Nguetseng's two-scale convergence method were used as a basis for deriving the averaged equations and the initial boundary conditions (that is, the limit equations with the small parameter tending to zero). Different limiting modes depending on the continuous medium parameters are obtained. An averaged model for a special case that does not contain rapidly oscillating coefficients and can be used for numerical calculations is presented.

Stochastic homogenization of the Keller–Segel chemotaxis system

In this paper, we focus on the Keller–Segel chemotaxis system in a random heterogeneous domain. We assume that the corresponding diffusion and chemotaxis coefficients are given by stationary ergodic random fields and apply stochastic two-scale convergence methods to derive the homogenized macroscopic equations. In establishing our results, we also derive a priori estimates for the Keller–Segel system that rely only on the boundedness of the coefficients; in particular, no differentiability assumption on the diffusion and chemotaxis coefficients for the chemotactic species is required. Finally, we prove the convergence of a periodization procedure for approximating the homogenized macroscopic coefficients.

Homogenization of incompressible generalized Stokes flows through a porous medium

We study the homogenization for families of steady and also unsteady incompressible generalized Stokes systems in a periodic porous medium. We assume that the stress tensor possesses an Orlicz growth and the size of solid parts of the porous medium is comparable to the size of the period. Homogenized systems are established using the two-scale convergence method adopted to Orlicz space setting. We prove the existence and uniqueness of weak solutions of the homogenized systems.

Homogenization of a system of elastic and reaction-diffusion equations modelling plant cell wall biomechanics

In this paper we present a derivation and multiscale analysis of a mathematical model for plant cell wall biomechanics that takes into account both the microscopic structure of a cell wall coming from the cellulose microfibrils and the chemical reactions between the cell wall’s constituents. Particular attention is paid to the role of pectin and the impact of calcium-pectin cross-linking chemistry on the mechanical properties of the cell wall. We prove the existence and uniqueness of the strongly coupled microscopic problem consisting of the equations of linear elasticity and a system of reaction-diffusion and ordinary differential equations. Using homogenization techniques (two-scale convergence and periodic unfolding methods) we derive a macroscopic model for plant cell wall biomechanics.

Homogenization of Reaction--Diffusion Processes in a Two-Component Porous Medium with Nonlinear Flux Conditions at the Interface

In this paper, we are dealing with the mathematical modeling and homogenization of nonlinear reaction-diffusion processes in a porous medium that consists of two components separated by an interface. One of the components is connected, and the other one is disconnected and consists of periodically distributed inclusions. At the interface, the fluxes are given by nonlinear functions of the concentrations on both sides of the interface. Thus, the concentrations may be discontinuous across the interface. For the derivation of the effective (homogenized) model, we use the method of two-scale convergence. To prove the convergence of the nonlinear terms, especially those defined on the microscopic interface, we give a new approach which involves the boundary unfolding operator and a compactness result for Banach-space-valued functions. The model is motivated by metabolic and regulatory processes in cells, where biochemical species are exchanged between organelles and cytoplasm through the organellar membranes. In this context the nonlinearities are given by kinetics corresponding to multispecies enzyme catalyzed reactions, which are generalizations of the classical Michaelis--Menten kinetics to multispecies reactions.

Homogenization of the stochastic Navier–Stokes equation with a stochastic slip boundary condition

The two-dimensional Navier–Stokes equation in a perforated domain with a dynamical slip boundary condition is considered. We assume that the dynamic is driven by a stochastic perturbation on the interior of the domain and another stochastic perturbation on the boundaries of the holes. We consider a scaling ( for the viscosity and 1 for the density) that will lead to a time-dependent limit problem. However, the noncritical scaling () is considered in front of the nonlinear term. The homogenized system in the limit is obtained as a Darcy’s law with memory with two permeabilities and an extra term that is due to the stochastic perturbation on the boundary of the holes. The nonhomogeneity on the boundary contains a stochastic part that yields in the limit an additional term in the Darcy’s law. We use the two-scale convergence method after extending the solution with 0 inside the holes to pass to the limit. By Itô stochastic calculus, we get uniform estimates on the solution in appropriate spaces. Due to the stochastic integral, the pressure that appears in the variational formulation does not have enough regularity in time. This fact made us rely only on the variational formulation for the passage to the limit on the solution. We obtain a variational formulation for the limit that is solution of a Stokes system with two pressures. This two-scale limit gives rise to three cell problems, two of them give the permeabilities while the third one gives an extra term in the Darcy’s law due to the stochastic perturbation on the boundary of the holes.

Homogenization of Acoustic Equations for a Partially Perforated Elastic Material with Slightly Viscous Fluid

In this paper a mathematical model describing small oscillations of a heterogeneous medium is considered. The medium consists of a partially perforated elastic material and a slightly viscous compressible fluid filling the pores. For the given model the corresponding homogenized problem is constructed by using the two-scale convergence method. The boundary conditions connecting equations of the homogenized model on the boundary between the continuous elastic material and the porous elastic material with fluid are found.

On the homogenization of multicomponent transport

This paper is devoted to the homogenization of weakly coupled cooperativeparabolic systems in strong convection regime with purely periodiccoefficients. Our approach is to factor out oscillations from the solution viaprincipal eigenfunctions of an associated spectral problem and to cancel anyexponential decay in time of the solution using the principal eigenvalue of thesame spectral problem. We employ the notion of two-scale convergence with driftin the asymptotic analysis of the factorized model as the lengthscale of theoscillations tends to zero. This combination of the factorization method andthe method of two-scale convergence is applied to upscale an adsorption modelfor multicomponent flow in an heterogeneous porous medium.

Homogenization of oxygen transport in biological tissues

In this paper, we extend previous work on the mathematical modeling of oxygen transport in biological tissues. Specifically, we include in the modeling process the arterial and venous microstructure within the tissue by means of homogenization techniques. We focus on the two-layer tissue architecture investigated in [Matzavinos et al., PNAS; 2009;106:12091] in the context of abdominal tissue flaps that are commonly used for reconstructive surgery. We apply two-scale convergence methods and unfolding operator techniques to homogenize the developed microscopic model, which involves different unit cell geometries in the two distinct tissue layers (skin layer and fat tissue) to account for different arterial branching patterns.

Asymptotic Analysis of a Linear Isotropic Elastic Composite Reinforced by a Thin Layer of Periodically Distributed Isotropic Parallel Stiff Fibres

We present some mathematical convergence results using a two-scale method for a linear elastic isotropic medium containing one layer of parallel periodically distributed heterogeneities located in the interior of the whole domain around a plane surface Σ. The aim of this paper is to study the situation when the rigidity of the linearly isotropic elastic fibres is 1/ε m the rigidity of the surrounding linearly isotropic elastic material. We use a two-scale convergence method adapted to the geometry of the problem (layer of fibres). In the models obtained Σ behaves for m=1 as a “material surface” without membrane energy in the direction of the plane orthogonal to the direction of the fibres. For m=3 the “material surface” has no bending energy in the direction orthogonal to the fibres.

Upscaling nonlinear adsorption in periodic porous media – homogenization approach

We consider the homogenization of a model of reactive flows through periodic porous media involving a single solute which can be absorbed and desorbed on the pore boundaries. This is a system of two convection–diffusion equations, one in the bulk and one on the pore boundaries, coupled by an exchange reaction term. The novelty of our work is to consider a nonlinear reaction term, a so-called Langmuir isotherm, in an asymptotic regime of strong convection. We therefore generalize previous works on a similar linear model. Under a technical assumption of equal drift velocities in the bulk and on the pore boundaries, we obtain a nonlinear monotone diffusion equation as the homogenized model. Our main technical tool is the method of two-scale convergence with drift.

Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition

The evolution Stokes equation in a domain containing periodically distributed obstacles subject to Fourier boundary condition on the boundaries is considered. We assume that the dynamic is driven by a stochastic perturbation on the interior of the domain and another stochastic perturbation on the boundaries of the obstacles. We represent the solid obstacles by holes in the fluid domain. The macroscopic (homogenized) equation is derived as another stochastic partial differential equation, defined in the whole non perforated domain. Here, the initial stochastic perturbation on the boundary becomes part of the homogenized equation as another stochastic force. We use the two-scale convergence method after extending the solution with 0 in the holes to pass to the limit. By Itô stochastic calculus, we get uniform estimates on the solution in appropriate spaces. In order to pass to the limit on the boundary integrals, we rewrite them in terms of integrals in the whole domain. In particular, for the stochastic integral on the boundary, we combine the previous idea of rewriting it on the whole domain with the assumption that the Brownian motion is of trace class. Due to the particular boundary condition dealt with, we get that the solution of the stochastic homogenized equation is not divergence free. However, it is coupled with the cell problem that has adivergence free solution. This paper represents an extension of the results of Duan and Wang (Comm. Math. Phys. 275:1508--1527, 2007), where a reaction diffusion equation with a dynamical boundary condition with a noise source term on both the interior of the domain and on the boundary was studied, and through a tightness argument and a pointwise two scale convergence method the homogenized equation was derived.