Periodic Homogenization Problems Research Articles

Uniform integrability in periodic homogenization of fully nonlinear elliptic equations

This paper is devoted to the study of uniform W^{1,frac{np}{n-p}}- and W^{2,p}-estimates for periodic homogenization problems of fully nonlinear elliptic equations. We establish sharp, global, large-scale estimates under the Dirichlet boundary conditions. The main novelty of this paper can be found in the characterization of the size of the “effective” Hessian and gradient of viscosity solutions to homogenization problems. Moreover, the large-scale estimates work in a large class of non-convex problems. It should be stressed that our global estimates are new even for the standard problems without homogenization.

Homogenization of stochastic conservation laws with multiplicative noise

We consider the generalized almost periodic homogenization problem for two different types of stochastic conservation laws with oscillatory coefficients and multiplicative noise. In both cases the stochastic perturbations are such that the equation admits special stochastic solutions which play the role of the steady-state solutions in the deterministic case. Specially in the second type, these stochastic solutions are crucial elements in the homogenization analysis. Our homogenization method is based on the notion of stochastic two-scale Young measure, whose existence is established here.

On the limit spectrum of a degenerate operator in the framework of periodic homogenization or singular perturbation problems

On the limit spectrum of a degenerate operator in the framework of periodic homogenization or singular perturbation problems

A periodic homogenization problem with defects rare at infinity

<p style='text-indent:20px;'>We consider a homogenization problem for the diffusion equation <inline-formula><tex-math id="M1">\begin{document}$ -\operatorname{div}\left(a_{\varepsilon} \nabla u_{\varepsilon} \right) = f $\end{document}</tex-math></inline-formula> when the coefficient <inline-formula><tex-math id="M2">\begin{document}$ a_{\varepsilon} $\end{document}</tex-math></inline-formula> is a non-local perturbation of a periodic coefficient. The perturbation does not vanish but becomes rare at infinity in a sense made precise in the text. We prove the existence of a corrector, identify the homogenized limit and study the convergence rates of <inline-formula><tex-math id="M3">\begin{document}$ u_{\varepsilon} $\end{document}</tex-math></inline-formula> to its homogenized limit.</p>

The geometric average of curl-free fields in periodic geometries

Abstract In periodic homogenization problems, one considers a sequence ( u η ) η {(u^{\eta})_{\eta}} of solutions to periodic problems and derives a homogenized equation for an effective quantity u ^ {\hat{u}} . In many applications, u ^ {\hat{u}} is the weak limit of ( u η ) η {(u^{\eta})_{\eta}} , but in some applications u ^ {\hat{u}} must be defined differently. In the homogenization of Maxwell’s equations in periodic media, the effective magnetic field is given by the geometric average of the two-scale limit. The notion of a geometric average has been introduced in [G. Bouchitté, C. Bourel and D. Felbacq, Homogenization of the 3D Maxwell system near resonances and artificial magnetism, C. R. Math. Acad. Sci. Paris 347 2009, 9–10, 571–576]; it associates to a curl-free field Y ∖ Σ ¯ → ℝ 3 {Y\setminus\overline{\Sigma}\to\mathbb{R}^{3}} , where Y is the periodicity cell and Σ an inclusion, a vector in ℝ 3 {\mathbb{R}^{3}} . In this article, we extend previous definitions to more general inclusions, in particular inclusions that are not compactly supported in the periodicity cell. The physical relevance of the geometric average is demonstrated by various results, e.g., a continuity property of limits of tangential traces.

Finite element approximation of elliptic homogenization problems in nondivergence-form

We use uniform W2,p estimates to obtain corrector results for periodic homogenization problems of the form A(x/ε):D2uε = f subject to a homogeneous Dirichlet boundary condition. We propose and rigorously analyze a numerical scheme based on finite element approximations for such nondivergence-form homogenization problems. The second part of the paper focuses on the approximation of the corrector and numerical homogenization for the case of nonuniformly oscillating coefficients. Numerical experiments demonstrate the performance of the scheme.

FFT based numerical homogenization method for porous conductive materials

The Fourier series method is used to solve the periodic homogenization problem for conductive materials containing voids. The problems involving voids are special cases of infinite contrast whose full field solution is not unique, causing convergence issues when iteration schemes are used. In this paper, we reformulate the problem based on the temperature field in the skeleton and derive an equation where the temperature field is connected to values on the pore boundary. Iteration schemes based on the new equation show that the convergence is fast, yielding good results both in terms of local fields and effective conductive properties.

An Artificial Neural Network based Solution Scheme to periodic Homogenization

AbstractArtificial neural networks (ANNs) have aroused research's and industry's interest due to their excellent approximation properties and are broadly used nowadays in the field of machine learning. In the present contribution, ANNs are used for finding solutions of periodic homogenization problems. The construction of ANN‐based trial functions that satisfy the given boundary conditions on the microscale allows for the unconstrained optimization of a global energy potential. Goal of the present approach is a memory efficient solution scheme as ANNs are known to fit complicated functions with a relatively small number of internal parameters. The method is tested for a three‐dimensional example using a global trial function and is qualitatively compared to a fast Fourier transform (FFT) based simulation.

Sigma-convergence of semilinear stochastic wave equations

We address the homogenization of a semilinear hyperbolic stochastic partial differential equation with highly oscillating coefficients, in the context of ergodic algebras with mean value. To achieve our goal, we use a suitable variant of the sigma-convergence concept that takes into account both the random and deterministic behaviours of the phenomenon modelled by the underlying problem. We also provide an appropriate scheme for the approximation of the effective coefficients. To illustrate our approach, we work out some concrete problems such as the periodic homogenization problem, the almost periodic and the asymptotically almost periodic ones.

Homogenization of Fučík Eigencurves

In this work we study the convergence of an homogenization problem for half-eigenvalues and Fucik eigencurves. We provide quantitative bounds on the rate of convergence of the curves for periodic homogenization problems.

Convergence Rates for the Stratified Periodic Homogenization Problems

Convergence Rates for the Stratified Periodic Homogenization Problems

On the homogenization of partial integro-differential-algebraic equations

We present a Hilbert space perspective to homogenization of standard linear evolutionary boundary value problems in mathematical physics and provide a unified treatment for (non-)periodic homogenization problems in thermodynamics, elasticity, electro-magnetism and coupled systems thereof. The approach permits the consideration of memory problems as well as differential-algebraic equations. We show that the limit equation is well-posed and causal. We rely on techniques from functional analysis and operator theory only.

Slow Convergence in Periodic Homogenization Problems for Divergence-Type Elliptic Operators

We introduce a new constructive method for establishing lower bounds on convergence rates of periodic homogenization problems associated with divergence type elliptic operators. The construction is applied in two settings. First, we show that solutions to boundary layer problems for divergence type elliptic equations set in halfspaces and with infinitely smooth data, may converge to their corresponding boundary layer tails as slow as one wish depending on the position of the hyperplane. Second, we construct a Dirichlet problem for divergence type elliptic operators set in a bounded domain, and with all data being $C^\infty$-smooth, for which the boundary value homogenization holds with arbitrarily slow speed.

Periodic homogenization under a hypoellipticity condition

In this paper we study a periodic homogenization problem for a quasilinear elliptic equation that present a partial degeneracy of hypoelliptic type. A convergence result is obtained by finding uniform barrier functions and the existence of the invariant measure to the associate diffusion problem that is used to identify the limit equation.

Homogenization of Neumann boundary data with fully nonlinear operator

In this paper we study periodic homogenization problems for solutions of fully nonlinear PDEs in half-spaces with oscillatory Neumann boundary data. We show the existence and uniqueness of the homogenized Neumann data for a given half-space. Moreover, we show that there exists a continuous extension of the homogenized slope as the normal of the half-space varies over “irrational” directions.

Periodic homogenization of strongly nonlinear reaction–diffusion equations with large reaction terms

We study in this article the periodic homogenization problem related to a strongly nonlinear reaction–diffusion equation. Owing to the large reaction term, the homogenized equation has a rather quite different form which puts together both the reaction and convection effects. We show in a special case that the homogenized equation is exactly of a convection-diffusion type. This study relies on a suitable version of the well-known two-scale convergence method.

Plane-Like Minimizers and Differentiability of the Stable Norm

In this paper we investigate the strict convexity and the differentiability properties of the stable norm, which corresponds to the homogenized surface tension for a periodic perimeter homogenization problem (in a regular and uniformly elliptic case). We prove that it is always differentiable in totally irrational directions, while in other directions, it is differentiable if and only if the corresponding plane-like minimizers satisfying a strong Birkhoff property foliate the torus. We also discuss the issue of the uniqueness of the correctors for the corresponding homogenization problem.

Homogenizations of integro-differential equations with Lévy operators with asymmetric and degenerate densities

We consider periodic homogenization problems for Lévy operators with asymmetric Lévy densities. The formal asymptotic expansion used for the α-stable (symmetric) Lévy operators (α ∈ (0, 2)) is not directly applicable to such asymmetric cases. We rescale the asymmetric densities and extract the most singular parts of the measures, which average out the microscopic dependencies in the homogenization procedures. We give two conditions, (A) and (B), that characterize such a class of asymmetric densities under which the above ‘rescaled’ homogenization is available.

Characterization of composite material properties using eigenstrain method

A technique to solve the periodic homogenization problem is described systematically in this work. The method is to solve the cell problems by imposing eigenstrains in terms of thermal or piezoelectric strain to the representative volume element. Homogenized coefficients are then calculated from stress solutions of those cell problems. Benefit of the proposed technique is that it is readily applicable for common finite element softwares regardless of using user subroutines. Several numerical examples are examined. The obtained results show good agreements with the published data.

Periodic approximations of homogenization problems

We show that any non‐periodic nonlinear elliptic homogenization problem can be approximated by periodic homogenization problems. Copyright © 2012 John Wiley & Sons, Ltd.