Two-scale Convergence Research Articles

Two-Scale Convergence of Unsteady Stokes type Equations

In this paper we study the homogenization of evolution Stokes type equations in the periodic setting. The usual Laplace operator involved in the classical Stokes equations is here replaced by a linear elliptic differential operator of divergence form with periodically oscillating coefficients. Our mean tool is the well known two-scale convergence method.

Analysis of the thin film flow in a rough domain filled with micropolar fluid

Inspired by the lubrication framework, in this paper a micropolar fluid flow through a rough thin domain is studied. The domain’s thickness is considered as the small parameter ε, while the roughness is defined by a periodical function with period of order ε2. Starting from three-dimensional micropolar equations and using asymptotic analysis with respect to ε, we formally derive the macroscopic model clearly detecting the effects of the specific rugosity profile and fluid microstructure. We provide the rigorous justification of our formally obtained asymptotic model by deriving the effective system by means of the two-scale convergence.

Homogenization of the nonlinear bending theory for plates

We carry out the spatially periodic homogenization of nonlinear bending theory for plates. The derivation is rigorous in the sense of $$\Gamma $$ -convergence. In contrast to what one naturally would expect, our result shows that the limiting functional is not simply a quadratic functional of the second fundamental form of the deformed plate as it is the case in nonlinear plate theory. It turns out that the limiting functional discriminates between whether the deformed plate is locally shaped like a “cylinder” or not. For the derivation we investigate the oscillatory behavior of sequences of second fundamental forms associated with isometric immersions of class $$W^{2,2}$$ , using two-scale convergence. This is a non-trivial task, since one has to treat two-scale convergence in connection with a nonlinear differential constraint.

Asymptotic Analysis of Acoustic Waves in a Porous Medium: Microincompressible Flow

This is the second in a series of three papers that studies acoustic waves governed by the linearized compressible Stokes equations in a porous medium. In particular, we want to analyze the simultaneous inviscid and high frequency limits of fluid flows in a porous medium. The presence of time-space boundary layers decouples the flow into an incompressible (that we call microincompressible) and an acoustic part (that we call micro-acoustic) on the microscopic scale. While this paper employs the two-scale methods used in our first paper [10], the present boundary layer phenomenon requires additional weak convergence tools. Using the Bloch decomposition, we introduce modified Helmholtz operators, enabling us to split the flow into its microincompressible and microacoustic parts. Closed equations for the microincompressible flow are obtained using two-scale convergence, while closed equations for the microacoustic flow are given in our forthcoming paper.

Homogenization of thermal-hydro-mass transfer processes

In the repository, multi-physics processes are induced due to the long-time heat-emitting from the nuclear waste, which is modeled as a nonlinear system with oscillating coefficients. In this paper we first derive the homogenized system for the thermal-hydro-mass transfer processes by the technique of two-scale convergence, then present some error estimates for the first order expansions.

Stochastic Two-Scale Convergence and Iterated Function System

We will use the random self-similar structure to generalize the concept of stochastic two-scale convergence extending some ideas elaborated in the deterministic case by J. Kolumban.

A one-population Amari model with periodic microstructure

We review the derivation of the homogenized one-population Amari equation by means of the two-scale convergence technique of Nguetseng in the case of periodic microvariation in the connectivity function. A key point in this derivation is Visintin's theorem for two-scale convergence of convolution integrals. We construct single bump solutions of the resulting homogenized equation using a pinning function technique for the case where the solutions are independent of the local variable and the firing rate function is modelled as a unit step function. The parameter measuring the degree of heterogeneity plays the role of a control parameter. The connectivity functions are periodically modulated in both the synaptic footprint and in the spatial scale. A framework for analysing the stability of these structures is formulated. This framework is based on spectral theory for Hilbert–Schmidt integral operators and it deforms to the standard Evans function approach for the translational invariant case in the limit of no heterogeneity. The upper and lower bounds of the growth/decay rates of the perturbations imposed on the bump states can be expressed in terms of the operator norm of the actual Hilbert–Schmidt operator. Intervals for which the pinning function is increasing correspond to unstable bumps, while complementary intervals where the pinning function decreases correspond to stable bumps, just as in the translational invariant case. Examples showing the properties of the bumps are discussed in detail when the connectivity kernels are given in terms of an exponential decaying function, a wizard hat function and a damped oscillating function.

Homogenization and correctors for the wave equation with periodic coefficients

Using the two-scale convergence method, we study the asymptotic behavior of a wave problem in ℝN with periodic coefficients in the space variable and almost-periodic coefficients in the time one. We obtain a nonlocal corrector and show how this implies that the limit problem is nonlocal in general.

Homogenization of immiscible compressible two-phase flow in highly heterogeneous porous media with discontinuous capillary pressures

This paper presents a study of immiscible compressible two-phase, such as water and gas, flow through highly heterogeneous porous media with periodic microstructure. Such models appear in gas migration through engineered and geological barriers for a deep repository for radioactive waste. We will consider a domain made up of several zones with different characteristics: porosity, absolute permeability, relative permeabilities and capillary pressure curves. Consequently, the model involves highly oscillatory characteristics and internal nonlinear interface conditions. The microscopic model is written in terms of the phase formulation, i.e. where the wetting (water) saturation phase and the non-wetting (gas) pressure phase are primary unknowns. This formulation leads to a coupled system consisting of a nonlinear parabolic equation for the gas pressure and a nonlinear degenerate parabolic diffusion-convection equation for the water saturation, subject to appropriate transmission, boundary and initial conditions. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. Moreover, the transmission conditions are nonlinear and the saturation is discontinuous at interfaces separating different media. Under some realistic assumptions on the data, we obtain a nonlinear homogenized coupled system with effective coefficients which are computed via a cell problem and give a rigorous mathematical derivation of the upscaled model by means of the two-scale convergence.

Homogenization of plasticity equations with two-scale convergence methods

We investigate the deformation of heterogeneous plastic materials. The model uses internal variables and kinematic hardening, elastic and plastic strain are used in an infinitesimal strain theory. For periodic material properties with periodicity length scale , we obtain the limiting system as . The limiting two-scale plasticity model coincides with well-known effective models. Our direct approach relies on abstract tools from two-scale convergence (regarding convex functionals and monotone operators) and on higher order estimates for solution sequences.

On periodic homogenization in perfect elasto-plasticity

The limit behavior of a periodic assembly of a finite number of elasto-plastic phases is investigated as the period becomes vanishingly small. A limit quasi-static evolution is derived through two-scale convergence techniques. It can be thermodynamically viewed as an elasto-plastic model, albeit with an infinite number of internal variables.

Introduction to the Theory of Two-Scale Convergence

Basic properties of weak and strong two-scale convergence are established for Lebesgue and Sobolev spaces, in particular, those with periodic measures depending on a small parameter. These results are applied to the homogenization of elliptic operators and spectral problems.

Homogenization of fast diffusion on surfaces with a two-step method and an application to [formula omitted]-cell signaling

In the context of periodic homogenization based on two-scale convergence, a nonlinear system of six coupled partial differential equations is homogenized. The system describes the process of signaling in a T cell (thymus lymphocyte) including the dynamics of calcium and of the molecule Stim1. Two of the six equations are defined on the finely structured surface of the endoplasmic reticulum and to make global diffusion after homogenization possible, we extend the existing theoretical convergence results and introduce the two-step method. Therefore the membrane of the endoplasmic reticulum is given an extent in normal direction such that it has a volume with width 0<δ≪1. For convergence of the functions defined on the membrane we can now use well-known two-scale convergence results and obtain fast diffusion after homogenization. To come back to the original shape of the surface, δ tends to zero in the reference cell, if some compactness results are satisfied, which leads to a non-standard cell problem, and we obtain global diffusion on the surface of the endoplasmic reticulum. The results justify a model for signaling in a T-cell recently proposed heuristically.

Homogenization of a carcinogenesis model with different scalings with the homogenization parameter

In the context of periodic homogenization based on two-scale convergence, we homogenize a linear system of four coupled reaction-diffusion equations, two of which are defined on a manifold. The system describes the most important subprocesses modeling the carcinogenesis of a human cell caused by Benzo-[a]-pyrene molecules. These molecules are activated to carcinogens in a series of chemical reactions at the surface of the endoplasmic reticulum, which constitutes a fine structure inside the cell. The diffusion on the endoplasmic reticulum, modeled as a Riemannian manifold, is described by the Laplace-Beltrami operator. For the binding process to the surface of the endoplasmic reticulum, different scalings with powers of the homogenization parameter are considered. This leads to three qualitatively different models in the homogenization limit.

Homogenization of a thermo-diffusion system with Smoluchowski interactions

We study the solvability and homogenization of a thermal-diffusion reaction problem posed in a periodically perforated domain. The system describes the motion of populations of hot colloidal particles interacting together via Smoluchowski production terms. The upscaled system, obtained via two-scale convergence techniques, allows the investigation of deposition effects in porous materials in the presence of thermal gradients.

Interfacial boundary conditions between a free domain and thin porous layers for non-Newtonian fluid flows

A non-Newtonian fluid flows in a free domain and in a periodically perforated thin layer which are connected through a permeable interface. Two scales are present in the porous layer: one associated to the periodicity of the distribution of the channels which is associated to the thinness of the layer and the other to the diameter of these channels. Using Γ-convergence and two-scale convergence methods, we derive boundary conditions of Beavers–Joseph–Saffman type on the permeable interface between the free domain and the thin layer.

Homogenization of Richardsʼ equation of van Genuchten–Mualem model

This paper is devoted to the homogenization of Richardsʼ 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.

An attempt at classifying homogenization-based numerical methods

In this note, a classification of Homogenization-Based Numerical Methods and (in particular) of Numerical Methodsthat are based on the Two-Scale Convergence is done. In this classification stand: Direct Homogenization-Based Numerical Methods;H-Measure-Based Numerical Methods; Two-Scale Numerical Methods and TSAPS: Two-Scale Asymptotic Preserving Schemes.

Homogenisation and spectral convergence of a periodic elastic composite with weakly compressible inclusions

A two-phase elastic composite with weakly compressible elastic inclusions is considered. The homogenised two-scale limit problem is found, via a version of the method of two-scale convergence, and analysed. The microscopic part of the two-scale limit is found to solve a Stokes type problem and shown to have no microscopic oscillations when the composite is subjected to body forces that are microscopically irrotational. The composites spectrum is analysed and shown to converge, in an appropriate sense, to the spectrum of the two-scale limit problem. A characterisation of the two-scale limit spectrum is given in terms of the limit macroscopic and microscopic behaviours.

Acoustic propagation in a random saturated medium: the biphasic case

Homogenized acoustic properties of a non-periodic porous linear elastic solid filled with a viscous Newtonian fluid are derived. A small parameter of the problem is the ratio between typical size of the microstructural inhomogeneity and the macroscopic length scale. We consider the ratio of elasticity coefficients to viscosity coefficients to be of order which leads to the biphasic macroscopic behaviour. To pass to the limit in the governing equations, we employ stochastic two-scale convergence in the mean. Periodic approximation is then used and elimination of corrector terms is carried out in detail for the periodic geometry. The effective equations for the solid displacement, fluid displacement and fluid pressure are obtained and compared to the classical Biot system.