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A brief introduction to homogenization and miscellaneous applications*

Grégoire Allaire (2012)

ESAIM: Proceedings

This paper is a set of lecture notes for a short introductory course on homogenization. It covers the basic tools of periodic homogenization (two-scale asymptotic expansions, the oscillating test function method and two-scale convergence) and briefly describes the main results of the more general theory of G−  or H−convergence. Several applications of the method are given: derivation of Darcy’s law for flows in porous media, derivation of the porosity...

A comparison of homogenization, Hashin-Shtrikman bounds and the Halpin-Tsai equations

Peter Wall (1997)

Applications of Mathematics

In this paper we study a unidirectional and elastic fiber composite. We use the homogenization method to obtain numerical results of the plane strain bulk modulus and the transverse shear modulus. The results are compared with the Hashin-Shtrikman bounds and are found to be close to the lower bounds in both cases. This indicates that the lower bounds might be used as a first approximation of the plane strain bulk modulus and the transverse shear modulus. We also point out the connection with the...

A general homogenization result of spectral problem for linearized elasticity in perforated domains

Mohamed Mourad Lhannafi Ait Yahia, Hamid Haddadou (2021)

Applications of Mathematics

The goal of this paper is to establish a general homogenization result for linearized elasticity of an eigenvalue problem defined over perforated domains, beyond the periodic setting, within the framework of the H 0 -convergence theory. Our main homogenization result states that the knowledge of the fourth-order tensor A 0 , the H 0 -limit of A ε , is sufficient to determine the homogenized eigenvalue problem and preserve the structure of the spectrum. This theorem is proved essentially by using Tartar’s method...

A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction

Yves Capdeboscq, Michael S. Vogelius (2003)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We establish an asymptotic representation formula for the steady state voltage perturbations caused by low volume fraction internal conductivity inhomogeneities. This formula generalizes and unifies earlier formulas derived for special geometries and distributions of inhomogeneities.

A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction

Yves Capdeboscq, Michael S. Vogelius (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We establish an asymptotic representation formula for the steady state voltage perturbations caused by low volume fraction internal conductivity inhomogeneities. This formula generalizes and unifies earlier formulas derived for special geometries and distributions of inhomogeneities.

A generalized strange term in Signorini’s type problems

Carlos Conca, François Murat, Claudia Timofte (2003)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

The limit behavior of the solutions of Signorini’s type-like problems in periodically perforated domains with period ε is studied. The main feature of this limit behaviour is the existence of a critical size of the perforations that separates different emerging phenomena as ε → 0 . In the critical case, it is shown that Signorini’s problem converges to a problem associated to a new operator which is the sum of a standard homogenized operator and an extra zero order term (“strange term”) coming from the...

A Generalized Strange Term in Signorini's Type Problems

Carlos Conca, François Murat, Claudia Timofte (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The limit behavior of the solutions of Signorini's type-like problems in periodically perforated domains with period ε is studied. The main feature of this limit behaviour is the existence of a critical size of the perforations that separates different emerging phenomena as ε → 0. In the critical case, it is shown that Signorini's problem converges to a problem associated to a new operator which is the sum of a standard homogenized operator and an extra zero order term (“strange term”) coming from...

A note on bounds for non-linear multivalued homogenized operators

Nils Svanstedt (1998)

Applications of Mathematics

In this paper we study the behaviour of maximal monotone multivalued highly oscillatory operators. We construct Reuss-Voigt-Wiener and Hashin-Shtrikmann type bounds for the minimal sections of G-limits of multivalued operators by using variational convergence and convex analysis.

A reduced model for domain walls in soft ferromagnetic films at the cross-over from symmetric to asymmetric wall types

Lucas Döring, Radu Ignat, Felix Otto (2014)

Journal of the European Mathematical Society

We study the Landau-Lifshitz model for the energy of multi-scale transition layers – called “domain walls” – in soft ferromagnetic films. Domain walls separate domains of constant magnetization vectors m α ± ∈ 𝕊 2 that differ by an angle 2 α . Assuming translation invariance tangential to the wall, our main result is the rigorous derivation of a reduced model for the energy of the optimal transition layer, which in a certain parameter regime confirms the experimental, numerical and physical predictions: The...

Alternative approaches to the two-scale convergence

Luděk Nechvátal (2004)

Applications of Mathematics

Two-scale convergence is a special weak convergence used in homogenization theory. Besides the original definition by Nguetseng and Allaire two alternative definitions are introduced and compared. They enable us to weaken requirements on the admissibility of test functions ψ ( x , y ) . Properties and examples are added.

An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations

Antoine Gloria, Stefan Neukamm, Felix Otto (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We establish an optimal, linear rate of convergence for the stochastic homogenization of discrete linear elliptic equations. We consider the model problem of independent and identically distributed coefficients on a discretized unit torus. We show that the difference between the solution to the random problem on the discretized torus and the first two terms of the two-scale asymptotic expansion has the same scaling as in the periodic case. In particular the L2-norm in probability of the H1-norm...

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