Two-scale Convergence Research Articles

Effective acoustic properties of a meta-material consisting of small Helmholtz resonators

We investigate the acoustic properties of meta-materials that are inspired by sound-absorbing structures. We show that it is possible to construct meta-materials with frequency-dependent effective properties, with large and/or negative permittivities. Mathematically, we investigate solutions $u^\varepsilon: \Omega_\varepsilon \rightarrow \mathbb{R}$ to a Helmholtz equation in the limit $\varepsilon\rightarrow 0$ with the help of two-scale convergence. The domain $\Omega_\varepsilon$ is obtained by removing from an open set $\Omega\subset \mathbb{R}^n$ in a periodic fashion a large number (order $\varepsilon^{-n}$) of small resonators (order $\varepsilon$). The special properties of the meta-material are obtained through sub-scale structures in the perforations.

Homogenization of Biomechanical Models for Plant Tissues

In this paper homogenization of a mathematical model for plant tissue biomechanics is presented. The microscopic model constitutes a strongly coupled system of reaction-diffusion-convection equations for chemical processes in plant cells, the equations of poroelasticity for elastic deformations of plant cell walls and middle lamella, and Stokes equations for fluid flow inside the cells. The chemical process in cells and the elastic properties of cell walls and middle lamella are coupled because elastic moduli depend on densities involved in chemical reactions, whereas chemical reactions depend on mechanical stresses. Using homogenization techniques we derive rigorously a macroscopic model for plant biomechanics. To pass to the limit in the nonlinear reaction terms, which depend on elastic strain, we prove the strong two-scale convergence of the displacement gradient and velocity field.

Computational networks and systems-homogenization of self-adjoint differential operators in variational form on periodic networks and micro-architectured systems

Micro-architectured systems and periodic network structures play an import role in multi-scale physics and material sciences. Mathematical modeling leads to challenging problems on the analytical and the numerical side. Previous studies focused on averaging techniques that can be used to reveal the corresponding macroscopic model describing the effective behavior. This study aims at a mathematical rigorous proof within the framework of homogenization theory. As a model example, the variational form of a self-adjoint operator on a large periodic network is considered. A notion of two-scale convergence for network functions based on a so-called two-scale transform is applied. It is shown that the sequence of solutions of the variational microscopic model on varying networked domains converges towards the solution of the macroscopic model. A similar result is achieved for the corresponding sequence of tangential gradients. The resulting homogenized variational model can be easily solved with standard PDE-solvers. In addition, the homogenized coefficients provide a characterization of the physical system on a global scale. In this way, a mathematically rigorous concept for the homogenization of self-adjoint operators on periodic manifolds is achieved. Numerical results illustrate the effectiveness of the presented approach.

Convergence of monotone operators with respect to measures

This paper deals with monotone operators in the theory of homogenization. The operators introduced may depend on a small parameter δ. We establish some convergence results with respect to singular measures. Our result is valid without using the two-scale convergence. A comparison between the operators for the singular structure and the classical ones is obtained when the thickness δ of the structure vanishes. Finally, under some suitable assumptions, we show the commutativity of the corresponding diagram.

Periodic homogenization of integral energies under space-dependent differential constraints

A homogenization result for a family of oscillating integral energies u_{\epsilon} \mapsto \int_{\Omega} f(x,\frac{x}{\epsilon},u_{\epsilon}(x))\,dx,\quad \epsilon \to 0^+ is presented, where the fields u_{\epsilon} are subjected to first order linear differential constraints depending on the space variable x . The work is based on the theory of \mathscr A -quasiconvexity with variable coefficients and on two-scale convergence techniques, and generalizes the previously obtained results in the case in which the differential constraints are imposed by means of a linear first order differential operator with constant coefficients. The identification of the relaxed energy in the framework of \mathscr A -quasiconvexity with variable coefficients is also recovered as a corollary of the homogenization result.

Multiscale Modelling and Analysis of Signalling Processes in Tissues with Non-Periodic Distribution of Cells

In this paper, a microscopic model for a signalling process in the left ventricular wall of the heart, comprising a non-periodic fibrous microstructure, is considered. To derive the macroscopic equations, the non-periodic microstructure is approximated by the corresponding locally periodic microstructure. Then, applying the methods of locally periodic homogenization (the locally periodic (l-p) unfolding operator, locally periodic two-scale (l-t-s) convergence on oscillating surfaces and l-p boundary unfolding operator), we obtain the macroscopic model for a signalling process in the heart tissue.

Modeling electromagnetism in and near composite material using two-scale behavior of the time-harmonic Maxwell equations

The main purpose of this article is to study the two-scale behavior of the electromagnetic field in 3D in and near composite material. For this, time-harmonic Maxwell equations, for a conducting two-phase composite and the air above, are considered. Technique of two-scale convergence is used to obtain the homogenized problem.

Expansion of a singularly perturbed equation with a two-scale converging convection term

In many physical contexts, evolution convection equations may present some very large amplitude convective terms. As an example, in the context of magnetic confinement fusion, the distribution function that describes the plasma satisfies the Vlasov equation in which some terms are of the same order as $\epsilon^{-1}$, $\epsilon \ll 1$ being the characteristic gyrokinetic period of the particles around the magnetic lines. In this paper, we aim to present a model hierarchy for modeling the distribution function for any value of $\epsilon$ by using some two-scale convergence tools. Following Frénod & Sonnendrücker's recent work, we choose the framework of a singularly perturbed convection equation where the convective terms admit either a high amplitude part or a an oscillating part with high frequency $\epsilon^{-1} \gg 1$. In this abstract framework, we derive an expansion with respect to the small parameter $\epsilon$ and we recursively identify each term of this expansion. Finally, we apply this new model hierarchy to the context of a linear Vlasov equation in three physical contexts linked to the magnetic confinement fusion and the evolution of charged particle beams.

Homogenization of nonlinear dissipative hyperbolic problems exhibiting arbitrarily many spatial and temporal scales

This paper concerns the homogenization of nonlinear dissipative hyperbolicproblems\begin{gather*}\partial _{tt}u^{\varepsilon }\left( x,t\right) -\nabla \cdot \left( a\left(\frac{x}{\varepsilon ^{q_{1}}},\ldots ,\frac{x}{\varepsilon ^{q_{n}}},\frac{t}{\varepsilon ^{r_{1}}},\ldots ,\frac{t}{\varepsilon ^{r_{m}}}\right) \nablau^{\varepsilon }\left( x,t\right) \right) \\+g\left( \frac{x}{\varepsilon ^{q_{1}}},\ldots ,\frac{x}{\varepsilon ^{q_{n}}},\frac{t}{\varepsilon ^{r_{1}}},\ldots ,\frac{t}{\varepsilon ^{r_{m}}},u^{\varepsilon }\left( x,t\right) ,\nabla u^{\varepsilon }\left( x,t\right)\right) =f(x,t)\end{gather*}where both the elliptic coefficient $a$ and the dissipative term $g$ areperiodic in the $n+m$ first arguments where $n$ and $m$ may attain anynon-negative integer value. The homogenization procedure is performed withinthe framework of evolution multiscale convergence which is a generalizationof two-scale convergence to include several spatial and temporal scales. Inorder to derive the local problems, one for each spatial scale, the crucialconcept of very weak evolution multiscale convergence is utilized since itallows less benign sequences to attain a limit. It turns out that the localproblems do not involve the dissipative term $g$ even though the homogenizedproblem does and, due to the nonlinearity property, an important part of thework is to determine the effective dissipative term. A brief illustration ofhow to use the main homogenization result is provided by applying it to anexample problem exhibiting six spatial and eight temporal scales in such away that $a$ and $g$ have disparate oscillation patterns.

Homogenization of an incompressible stationary flow of an electrorheological fluid

We combine two-scale convergence, theory of monotone operators and results on approximation of Sobolev functions by Lipschitz functions to prove a homogenization process for an incompressible flow of a generalized Newtonian fluid. We avoid the necessity of testing the weak formulation of the initial and homogenized systems by corresponding weak solutions, which allows optimal assumptions on lower bound for a growth of the elliptic term. We show that the stress tensor for homogenized problem depends on the symmetric part of the velocity gradient involving the limit of a sequence selected from a family of solutions of initial problems.

Homogenization of a pore scale model for precipitation and dissolution in porous media

In this article, we employ homogenization techniques to provide a rigorous derivation of the Darcy scale model for precipitation and dissolution in porous media. The starting point is the pore scale model in van Duijn & Pop (2004 ), which is a coupled system of evolution equations, involving a parabolic equation which models ion transport in the fluid phase of a periodic porous medium, coupled to an ordinary differential equations modelling dissolution and precipitation at the grains boundary. The main challenge is in dealing with the dissolution and precipitation rates, which involve a monotone but possibly discontinuous function. In order to pass to the limit in these rate functions at the boundary of the grains, we prove strong two-scale convergence for the concentrations at the microscopic boundary and use refined arguments in order to identify the form of the macroscopic dissolution rate, which is again a discontinuous function. The resulting upscaled model is consistent with the Darcy scale model proposed in Knabner et al . (1995 ).

Homogenization of degenerate coupled fluid flows and heat transport through porous media

We establish a homogenization result for a fully nonlinear degenerate parabolic system with critical growth arising from the heat and moisture flow through a partially saturated porous media. Existence of a global weak solution of the mesoscale problem is proven by means of a semidiscretization in time, a priori estimates and passing to the limit from discrete approximations. After that, porous material exhibiting periodic spatial oscillations is considered and the two-scale convergence (as the oscillation period vanishes) to a corresponding homogenized problem is rigorously proven.

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.

High-frequency homogenization for travelling waves in periodic media.

We consider high-frequency homogenization in periodic media for travelling waves of several different equations: the wave equation for scalar-valued waves such as acoustics; the wave equation for vector-valued waves such as electromagnetism and elasticity; and a system that encompasses the Schrödinger equation. This homogenization applies when the wavelength is of the order of the size of the medium periodicity cell. The travelling wave is assumed to be the sum of two waves: a modulated Bloch carrier wave having crystal wavevector [Formula: see text] and frequency ω1 plus a modulated Bloch carrier wave having crystal wavevector [Formula: see text] and frequency ω2. We derive effective equations for the modulating functions, and then prove that there is no coupling in the effective equations between the two different waves both in the scalar and the system cases. To be precise, we prove that there is no coupling unless ω1=ω2 and [Formula: see text] where Λ=(λ1λ2…λ d ) is the periodicity cell of the medium and for any two vectors [Formula: see text] the product a⊙b is defined to be the vector (a1b1,a2b2,…,adbd ). This last condition forces the carrier waves to be equivalent Bloch waves meaning that the coupling constants in the system of effective equations vanish. We use two-scale analysis and some new weak-convergence type lemmas. The analysis is not at the same level of rigour as that of Allaire and co-workers who use two-scale convergence theory to treat the problem, but has the advantage of simplicity which will allow it to be easily extended to the case where there is degeneracy of the Bloch eigenvalue.

Error estimates for nonlinear reaction-diffusion systems involving different diffusion length scales

We derive quantitative error estimates for coupled reaction-diffusion systems, whose coefficient functions are quasi-periodically oscillating modeling the microstructure of the underlying macroscopic domain. The coupling arises via nonlinear reaction terms and we allow for different diffusion length scales, i.e. whereas some species have characteristic diffusion length of order 1 other species may diffuse with the order of the characteristic microstructure-length scale. We consider an effective system, which is rigorously obtained via two-scale convergence, and we derive quantitative error estimates.

Homogenization of integral energies under periodically oscillating differential constraints

A homogenization result for a family of integral energies $$\begin{aligned} u_{\varepsilon }\mapsto \int _\Omega f(u_{\varepsilon }(x))\,dx,\quad \varepsilon \rightarrow 0^+, \end{aligned}$$ is presented, where the fields \(u_{\varepsilon }\) are subjected to periodic first order oscillating differential constraints in divergence form. The work is based on the theory of \(\mathscr {A}\)-quasiconvexity with variable coefficients and on two-scale convergence techniques.

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.

Two-scale homogenization of the Poisson equation with friction boundary condition in a perforated domain

We study the homogenization of the Poisson equation in a periodically perforated domain, of period e> 0, with a friction type boundary condition on the holes' boundary. This non-linear condition allows the solution to be non-zero on the periodic boundary if some conditions are satisfied. Using two-scale convergence results we prove that the solution of the mixed variational formulation converges, as e goes to 0, to the solution of a two-scale mixed problem. We also prove that this homogenized problem is well-posed. A numerical test is done, using the Finite Element Method and a quadratic programming algorithm, in order to compare the heterogeneous and homogenized solutions.

Homogenization of compressible two-phase two-component flow in porous media

The paper is devoted to the homogenization of immiscible compressible two-phase two-component flow in heterogeneous porous media. We consider liquid and gas phases, two-component (water and hydrogen) flow in a porous reservoir with periodic microstructure, modeling the hydrogen migration through engineered and geological barriers for a deep repository for radioactive waste. Phase exchange, capillary effects included by the Darcy–Muskat law and Fickian diffusion are taken into account. The hydrogen in the gas phase is supposed compressible and could be dissolved into the water obeying the Henry law. The flow is then described by the conservation of the mass for each component. The microscopic model is written in terms of the phase formulation, i.e. the liquid saturation phase and the 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 liquid saturation, subject to appropriate 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. Under some realistic assumptions on the data, we obtain a nonlinear homogenized problem with effective coefficients which are computed via a cell problem. We rigorously justify this homogenization process for the problem by using the two-scale convergence.