Deterministic Homogenization Research Articles

Derivation and analysis of a nonlocal Hele–Shaw–Cahn–Hilliard system for flow in thin heterogeneous layers

We derive, through the deterministic homogenization theory in thin domains, a new model consisting of Hele–Shaw equation with memory coupled with the convective Cahn–Hilliard equation. The obtained system, which models in particular tumor growth, is then analyzed and we prove its well-posedness in dimension 2. To achieve our goal, we develop and use the new concept of sigma-convergence in thin heterogeneous media, and we prove some regularity results for the upscaled model.

Approximation of homogenized coefficients in deterministic homogenization and convergence rates in the asymptotic almost periodic setting

For a homogenization problem associated to a linear elliptic operator, we prove the existence of a distributional corrector and we find an approximation scheme for the homogenized coefficients. We also study the convergence rates in the asymptotic almost periodic setting, and we show that the rates of convergence for the zero-order approximation, are near optimal. The results obtained constitute a step towards the numerical implementation of results from the deterministic homogenization theory beyond the periodic setting. To illustrate this, numerical simulations based on finite volume method are provided to sustain our theoretical results.

Deterministic homogenization under optimal moment assumptions for fast-slow systems. Part 1

Nous étudions un problème d’homogénéisation déterministe (avec convergence vers une équation différentielle stochastique) pour un système multi-échelle de la forme suivante : xk+1=xk+n−1an(xk,yk)+n−1/2bn(xk,yk),yk+1=Tnyk, où la dynamique rapide est donnée par une famille Tn de transformations non uniformément dilatantes. La partie 1 prolonge nos travaux récents sur l’approximation par des martingales pour des familles de transformations non uniformément dilatantes. Nous montrons un principe d’invariance faible itéré, et établissons des bornes optimales sur les moments dans ce cadre (ces bornes sont nouvelles même pour une transformation non uniformément dilatante T fixée). En combinant ceci et des développements parallèles sur la théorie des chemins rugueux par Chevyrev, Friz, Korepanov, Melbourne et Zhang, nous obtenons les résultats d’homogénéisation dans la partie 2.

Deterministic homogenization under optimal moment assumptions for fast–slow systems. Part 2

We consider deterministic homogenization for discrete-time fast-slow systems of the form $$ X_{k+1} = X_k + n^{-1}a_n(X_k,Y_k) + n^{-1/2}b_n(X_k,Y_k)\;, \quad Y_{k+1} = T_nY_k\;$$ and give conditions under which the dynamics of the slow equations converge weakly to an It\^o diffusion $X$ as $n\to\infty$. The drift and diffusion coefficients of the limiting stochastic differential equation satisfied by $X$ are given explicitly. This extends the results of [Kelly-Melbourne, J. Funct. Anal. 272 (2017) 4063--4102] from the continuous-time case to the discrete-time case. Moreover, our methods (c\`adl\`ag $p$-variation rough paths) work under optimal moment assumptions. Combined with parallel developments on martingale approximations for families of nonuniformly expanding maps in Part 1 by Korepanov, Kosloff & Melbourne, we obtain optimal homogenization results when $T_n$ is such a family of maps.

Homogenization Theory: Periodic and Beyond

The objective of the workshop has been to review the latest developments in homogenization theory for a large category of equations and settings arising in the modeling of solid, fluids, wave propagation, heterogeneous media, etc. The topics approached have covered periodic and non-periodic deterministic homogenization, stochastic homogenization, regularity theory, derivation of wall laws and detailed study of boundary layers,...

Simulation of Non-Lipschitz Stochastic Differential Equations Driven by $\alpha$-Stable Noise: A Method Based on Deterministic Homogenization

Simulation of Non-Lipschitz Stochastic Differential Equations Driven by $\alpha$-Stable Noise: A Method Based on Deterministic Homogenization

Deterministic homogenization of Vlasov equations

We address the homogenization of the Vlasov equations using the sigma‐convergence method. Assuming that the electromagnetic field is strong and is highly oscillating in both space and time, we derive the homogenization result. We then study some special cases leading to already known results.

Rate of Convergence in the Weak Invariance Principle for Deterministic Systems

We obtain the first results on convergence rates in the Prokhorov metric for the weak invariance principle (functional central limit theorem) for deterministic dynamical systems. Our results hold for uniformly expanding/hyperbolic (Axiom A) systems, as well as nonuniformly expanding/hyperbolic systems such as dispersing billiards, Hénon-like attractors, Viana maps and intermittent maps. As an application, we obtain convergence rates for deterministic homogenization in multiscale systems.

Corrector problem in the deterministic homogenization of nonlinear elliptic equations

ABSTRACTWe carry out the deterministic homogenization of nonlinear elliptic operators beyond periodicity. To proceed with, we prove the existence of nonlinear correctors in the usual distributional sense. This lays the foundation for the study of regularity results in the nonlinear deterministic homogenization theory beyond the periodic setting.

Computation of effective electrical conductivity of composite materials: A novel approach based on analysis of graphs

In this work we continue the investigation of different approaches to conception and modelling of composite materials. The global method we focus on, is called ‘stochastic homogenization’. In this approach, the classical deterministic homogenization techniques and procedures are used to compute the macroscopic parameters of a composite starting from its microscopic properties. The stochastic part is due to averaging over some series of samples, and the fact that these samples fit into the concept of RVE (Representative Volume Element) in order to reduce the variance effect.In this article, we present a novel method for computation of effective electric properties of composites – it is based on the analysis of the connectivity graph (and the respective adjacency matrix) for each sample of a composite material. We describe how this matrix is constructed in order to take into account complex microscopic geometry. We also explain what we mean by homogenization procedure for electrical conductivity, and how the constructed matrix is related to the problem. The developed method is applied to a test study of the influence of micromorphology of composites materials on their conductivity.

Deterministic homogenization for fast–slow systems with chaotic noise

Consider a fast–slow system of ordinary differential equations of the form x˙=a(x,y)+ε−1b(x,y), y˙=ε−2g(y), where it is assumed that b averages to zero under the fast flow generated by g. We give conditions under which solutions x to the slow equations converge weakly to an Itô diffusion X as ε→0. The drift and diffusion coefficients of the limiting stochastic differential equation satisfied by X are given explicitly.Our theory applies when the fast flow is Anosov or Axiom A, as well as to a large class of nonuniformly hyperbolic fast flows (including the one defined by the well-known Lorenz equations), and our main results do not require any mixing assumptions on the fast flow.

Probabilistic effective characteristics of polymers containing rubber particles of Gaussian random diameter

This study is focused on the problem of statistical distribution of the size of rubber particles as fillers in elastomeric composites. This distribution (average diameter of the injected particles) is assumed to be Gaussian and uniquely defined by its mean value as well as standard deviation. The basic probabilistic parameters of the effective elasticity tensor of the entire elastomer are under consideration by using of the homogenization method. The basic computational ideology is based on strain deformation of the Representative Volume Element under uniaxial and biaxial loads. This deterministic method is enriched with the generalized stochastic perturbation technique and also by semi-analytical strategy, which are used together with the system ABAQUS® as the Stochastic Finite Element Method (SFEM) serving for a solution of the homogenization problem for such a composite. The basic stochastic characteristics of the homogenized elasticity tensor and its deterministic sensitivity coefficients are verified with such coming from analytical deterministic homogenization method extended towards random case in the computer algebra system MAPLE®. The computational study contains additionally computational error analysis as the homogenization problem is solved here with tetrahedral and hexahedral 3D solid finite elements with linear as well as with parabolic shape functions and their meshes with different densities.

Homogenization of a generalized Stefan problem in the context of ergodic algebras

We address the deterministic homogenization, in the general context of ergodic algebras, of a doubly nonlinear problem which generalizes the well known Stefan model, and includes the classical porous medium equation. It may be represented by the differential inclusion, for a real-valued function u(x,t),∂∂t∂uΨ(x/ε,x,u)−∇x⋅∇ηψ(x/ε,x,t,u,∇u)∋f(x/ε,x,t,u), on a bounded domain Ω⊆Rn, t∈(0,T), together with initial–boundary conditions, where Ψ(z,x,⋅) is strictly convex and ψ(z,x,t,u,⋅) is a C1 convex function, both with quadratic growth, satisfying some additional technical hypotheses. As functions of the oscillatory variable, Ψ(⋅,x,u),ψ(⋅,x,t,u,η) and f(⋅,x,t,u) belong to the generalized Besicovitch space B2 associated with an arbitrary ergodic algebra A. The periodic case was addressed by Visintin (2007), based on the two-scale convergence technique. Visintin's analysis for the periodic case relies heavily on the possibility of reducing two-scale convergence to the usual L2 convergence in the Cartesian product Π×Rn, where Π is the periodic cell. This reduction is no longer possible in the case of a general ergodic algebra. To overcome this difficulty, we make essential use of the concept of two-scale Young measures for algebras with mean value, associated with bounded sequences in L2.

Deterministic homogenization of nonlinear degenerate elliptic operators with nonstandard growth

We study the deterministic homogenization of nonlinear degenerate elliptic equations with nonstandard growth. One fundamental in this topic is to extend the classical compactness results of the Σ-convergence method to the Orlicz spaces. We also show that one can homogenize nonlinear Dirichlet problems in a general way by leaning on a simple abstract hypothesis contrary to what has been done in the determinstic homogenization theory.

Deterministic homogenization for media with barriers

Averaging coefficient in a second order elliptic equation is a well known and important model problem. Additional to non-periodic rapid oscillations, the coefficient may contain barriers and channels - long and narrow bodies with low or high values of the coefficient. When the length of such structures is comparable with the problem size - there is no scale separation. &nbsp In this article we consider coefficients with barriers. We show how the averaged coefficient may be inadequate near the barriers and propose a generalization which can detect the potential problems and improve the accuracy of the averaged solution.

Deterministic homogenization on problems of minimal hypersurface type

We study here the deterministic homogenization for a particular family of convex integral functionals, namely problems of minimal hypersurfaces. By the sigma-convergence method relooked especially in appropriate representation of non-periodic functions and in the extension of sigma-convergence on the L1-space, we obtain a general homogenization result, and thanks to the particularity of minimal hypersurfaces, we derive a macroscopic integral functional. We then provide a number of physical applications (including the periodic case) of this result.

Erratum to: Deterministic homogenization of weakly damped nonlinear hyperbolic-parabolic equations

Erratum to: Deterministic homogenization of weakly damped nonlinear hyperbolic-parabolic equations

Deterministic homogenization of weakly damped nonlinear hyperbolic-parabolic equations

Deterministic homogenization is studied for nonlinear hyperbolic-parabolic equations with a linear damping term. It is shown by the sigma-convergence method that the sequence of solutions to a class of highly oscillatory nonlinear evolution problems converges to the solution to a homogenized quasilinear hyperbolic-parabolic problem.

Corrector Theory for MsFEM and HMM in Random Media

We analyze the random fluctuations of several multiscale algorithms, such as the multiscale finite element method (MsFEM) and the finite element heterogeneous multiscale method (HMM), that have been developed to solve partial differential equations with highly heterogeneous coefficients. Such multiscale algorithms are often shown to correctly capture the homogenization limit when the highly oscillatory random medium is stationary and ergodic. This paper is concerned with the random fluctuations of the solution about the deterministic homogenization limit. We consider the simplified setting of the one-dimensional elliptic equation, where the theory of random fluctuations is well understood. We develop a fluctuation theory for the multiscale algorithms in the presence of random environments with short-range and long-range correlations. For a given mesh size , we show that the fluctuations converge in distribution in the space of continuous paths to Gaussian processes as the correlation length . We next derive the limit of such Gaussian processes as and compare this limit with the distribution of the random fluctuations of the continuous model. When such limits agree, we conclude that the multiscale algorithm captures the random fluctuations accurately and passes the corrector test. This property serves as an interesting benchmark to assess the behavior of the multiscale algorithm in practical situations where the assumptions necessary for the theory of homogenization are not met. What we find is that the computationally more expensive methods MsFEM, and HMM with a choice of parameter , correctly capture the random fluctuations both for short-range and long-range oscillations in the medium. The less expensive method HMM with correctly captures the fluctuations for long-range oscillations and strongly amplifies their size in media with short-range oscillations. We present a modified scheme with an intermediate computational cost that captures the random fluctuations in all cases.

Deterministic homogenization of quasilinear damped hyperbolic equations

Deterministic homogenization is studied for quasilinear monotone hyperbolic problems with a linear damping term. It is shown by the sigma-convergence method that the sequence of solutions to a class of multi-scale highly oscillatory hyperbolic problems converges to the solution to a homogenized quasilinear hyperbolic problem.