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We study the representative volume element (RVE) method, which is a method to approximately infer the effective behavior a hom of a stationary random medium. The latter is described by a coefficient field a(x) generated from a given ensemble ⟨ · ⟩ and the corresponding linear elliptic operator - ∇ · a ∇ . In line with the theory of homogenization, the method proceeds by computing d = 3 correctors (d denoting the space dimension). To be numerically tractable, this computation has to be done on a finite domain: the so-called representative volume element, i.e., a large box with, say, periodic boundary conditions. The main message of this article is: Periodize the ensemble instead of its realizations. By this, we mean that it is better to sample from a suitably periodized ensemble than to periodically extend the restriction of a realization a(x) from the whole-space ensemble ⟨ · ⟩ . We make this point by investigating the bias (or systematic error), i.e., the difference between a hom and the expected value of the RVE method, in terms of its scaling w.r.t. the lateral size L of the box. In case of periodizing a(x), we heuristically argue that this error is generically O (L - 1) . In case of a suitable periodization of ⟨ · ⟩ , we rigorously show that it is O (L - d) . In fact, we give a characterization of the leading-order error term for both strategies and argue that even in the isotropic case it is generically non-degenerate. We carry out the rigorous analysis in the convenient setting of ensembles ⟨ · ⟩ of Gaussian type, which allow for a straightforward periodization, passing via the (integrable) covariance function. This setting has also the advantage of making the Price theorem and the Malliavin calculus available for optimal stochastic estimates of correctors. We actually need control of second-order correctors to capture the leading-order error term. This is due to inversion symmetry when applying the two-scale expansion to the Green function. As a bonus, we present a stream-lined strategy to estimate the error in a higher-order two-scale expansion of the Green function. [ABSTRACT FROM AUTHOR]

Recent stochastic homogenization results for the Francfort–Marigo model of brittle fracture under anti-plane shear indicate the existence of a representative volume element. This homogenization result includes a cell formula which relies on Dirichlet boundary conditions. For other material classes, the boundary conditions do not effect the effective properties upon the infinite volume limit but may have a strong influence on the necessary size of the computational domain. We investigate the influence of the boundary conditions on the effective crack energy evaluated on microstructure cells of finite size. For periodic boundary conditions recent computational methods based on FFT-based solvers exploiting the minimum cut/maximum flow duality are available. In this work, we provide a different approach based on fast marching algorithms which enables a liberal choice of the boundary conditions in the 2D case. We conduct representative volume element studies for two-dimensional fiber reinforced composite structures with tough inclusions, comparing Dirichlet with periodic boundary conditions. [ABSTRACT FROM AUTHOR]

This article develops an efficient uncertainty propagation framework for stochastic multiscale linear elasticity. Stochastic microscale problems are solved on the RVE with random material properties and random geometries. A stochastic homogenization approach is then used to calculate equivalent macroscale random material properties. According to different spatial correlations at the macroscale, random variables, random fields and high-dimensional random inputs are generated to model macroscale randomness. Stochastic finite element equations at both micro and macro scales are solved by using a unified and efficient numerical algorithm, which relies on a unified stochastic solution construction and an efficient iterative algorithm. It is efficient and accurate even for very high-dimensional problems due to its insensitivity to stochastic dimensions. Numerical results demonstrate the promising performance of the proposed framework, especially its high efficiency without loss of accuracy. • An efficient framework is proposed for stochastic multiscale linear elasticity. • The RVE with random material properties and random domains is considered. • A random geometric mapping is used to handle random domains. • Macroscale random inputs integrate macroscale correlations and microscale randomness. • High-dimensional macroscale stochastic problems can be solved efficiently. [ABSTRACT FROM AUTHOR]

We are given a uniformly elliptic coefficient field that we regard as a realization of a stationary and finite-range ensemble of coefficient fields. Given a right-hand side supported in a ball of size ℓ ≫ 1 and of vanishing average, we are interested in an algorithm to compute the solution near the origin, just using the knowledge of the given realization of the coefficient field in some large box of size L ≫ ℓ . More precisely, we are interested in the most seamless artificial boundary condition on the boundary of the computational domain of size L. Motivated by the recently introduced multipole expansion in random media, we propose an algorithm. We rigorously establish an error estimate on the level of the gradient in terms of L ≫ ℓ ≫ 1 , using recent results in quantitative stochastic homogenization. More precisely, our error estimate has an a priori and an a posteriori aspect: with a priori overwhelming probability, the prefactor can be bounded by a constant that is computable without much further effort, on the basis of the given realization in the box of size L. We also rigorously establish that the order of the error estimate in both L and ℓ is optimal, where in this paper we focus on the case of d = 2 . This amounts to a lower bound on the variance of the quantity of interest when conditioned on the coefficients inside the computational domain, and relies on the deterministic insight that a sensitivity analysis with respect to a defect commutes with stochastic homogenization. Finally, we carry out numerical experiments that show that this optimal convergence rate already sets in at only moderately large L, and that more naive boundary conditions perform worse both in terms of rate and prefactor. [ABSTRACT FROM AUTHOR]

We prove homogenization for a class of viscous Hamilton-Jacobi equations in the stationary & ergodic setting in one space dimension. Our assumptions include most notably the following: the Hamiltonian is of the form G (p) + β V (x , ω) , the function G is coercive and strictly quasiconvex, min ⁡ G = 0 , β > 0 , the random potential V takes values in [ 0 , 1 ] with full support and it satisfies a hill condition that involves the diffusion coefficient. Our approach is based on showing that, for every direction outside of a bounded interval (θ 1 (β) , θ 2 (β)) , there is a unique sublinear corrector with certain properties. We obtain a formula for the effective Hamiltonian and deduce that it is coercive, identically equal to β on (θ 1 (β) , θ 2 (β)) , and strictly monotone elsewhere. [ABSTRACT FROM AUTHOR]

This paper presents the computational stochastic homogenization of a heterogeneous 3D-linear anisotropic elastic microstructure that cannot be described in terms of constituents at microscale, as live tissues. The random apparent elasticity field at mesoscale is then modeled in a class of non-Gaussian positive-definite tensor-valued homogeneous random fields. We present an extension of previous works consisting of a novel probabilistic model to take into account uncertainties in the spectral measure of the random apparent elasticity field. A probabilistic analysis of the random effective elasticity tensor at macroscale is performed as a function of the level of spectrum uncertainties, which allows for studying the scale separation and the representative volume element size in a robust probabilistic framework. [ABSTRACT FROM AUTHOR]

We give an example of the failure of homogenization for a viscous Hamilton-Jacobi equation with non-convex Hamiltonian. [ABSTRACT FROM AUTHOR]

We establish a continuity theorem for non-local functionals indexed by Young measures, that we use to deal with homogenization of stochastic non-diffusive reaction differential equations. Non-local effects induced by homogenization of such stochastic differential equations are studied. [ABSTRACT FROM AUTHOR]

This paper deals with the homogenization problem for convolution type non-local operators in random statistically homogeneous ergodic media. Assuming that the convolution kernel has a finite second moment and satisfies certain symmetry and uniform ellipticity conditions, we prove the almost sure homogenization result and show that the limit operator is a second order elliptic differential operator with constant deterministic coefficients [ABSTRACT FROM AUTHOR]

In this paper, we prove the stochastic homogenization of certain nonconvex Hamilton–Jacobi equations. The nonconvex Hamiltonians, which are generally uneven and inseparable, are generated by a sequence of quasiconvex Hamiltonians and a sequence of quasiconcave Hamiltonians through the min-max formula. We provide a monotonicity assumption on the contact values between those stably paired Hamiltonians so as to guarantee the stochastic homogenization. [ABSTRACT FROM AUTHOR]

We propose a new Γ-convergent discrete approximation of the Mumford–Shah functional. The discrete functionals act on functions defined on stationary stochastic lattices and take into account general finite differences through a non-convex potential. In this setting the geometry of the lattice strongly influences the anisotropy of the limit functional. Thus we can use statistically isotropic lattices and stochastic homogenization techniques to approximate the vectorial Mumford–Shah functional in any dimension. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of a discrete-in-time minimizing movement scheme for square lattice interfaces when both the lattice spacing and the time step vanish. The motion is assumed to be driven by minimization of a weighted random perimeter functional with an additional deterministic dissipation term. We consider rectangular initial sets and lower order random perturbations of the perimeter functional. In case of stationary, α-mixing perturbations we prove a stochastic homogenization result for the interface velocity. We also provide an example which indicates that only stationary, ergodic perturbations might not yield a spatially homogenized limit velocity for this minimizing movement scheme. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We use the notion of stochastic two-scale convergence introduced in [32] to solve the problem of stochastic homogenization of the elastic plate in the bending regime. [ABSTRACT FROM AUTHOR]

In the context of infinitesimal strain plasticity with hardening, we derive a stochastic homogenization result. We assume that the coefficients of the equation are random functions: elasticity tensor, hardening parameter and flow-rule function are given through a dynamical system on a probability space. A parameter ↦ > 0 denotes the typical length scale of oscillations. We derive effective equations that describe the behavior of solutions in the limit ↦ → 0. The homogenization procedure is based on the fact that stochastic coefficients "allow averaging": For one representative volume element, a strain evolution [0, T] ∍ t ↦ ξ(t) ∊ Rd×ds induces a stress evolution [0, T] ∍ t ↦ Σ(ξ)(t) ∊ Rd×ds. Once the hysteretic evolution law Σ is justified for averages, we obtain that the macroscopic limit equation is given by -∇ · Σ(∇su) = f. [ABSTRACT FROM AUTHOR]

We present an introduction to periodic and stochastic homogenization of elliptic partial differential equations. The first part is concerned with the qualitative theory, which we present for equations with periodic and random coefficients in a unified approach based on Tartar's method of oscillating test functions. In particular, we present a self-contained and elementary argument for the construction of the sublinear corrector of stochastic homogenization. (The argument also applies to elliptic systems and in particular to linear elasticity). In the second part we briefly discuss the representation of the homogenization error by means of a two-scale expansion. In the last part we discuss some results of quantitative stochastic homogenization in a discrete setting. In particular, we discuss the quantification of ergodicity via concentration inequalities, and we illustrate that the latter in combination with elliptic regularity theory leads to a quantification of the growth of the sublinear corrector and the homogenization error. [ABSTRACT FROM AUTHOR]

The current work describes a stochastic framework to investigate the static response characteristics of periodic and random fiber arrays of reinforced three-phase composites with aleatory mechanical and thermal constituent properties. These properties were randomly modeled via Gaussian distribution using the statistical quantities of first and second moments. The framework is based on Variational Asymptotic Method (VAM) based homogenization technique coupled with Stochastic Reduced Order Model (SROM). When compared against the framework of VAM coupled with the traditional Monte Carlo Simulation (MCS) method, the computational cost of the SROM model with VAM was significantly less. Good agreement with solutions from traditional MCS combined with VAM was also achieved. To demonstrate applicability of the framework, a multivariate stochastic analysis was performed to determine the effect of fiber arrangements on the randomness of mechanical and thermal properties in a three-phase composite. Additionally, univariate stochastic analysis was performed, and the sensitivity of each constituent parameter corresponding to the stochastic homogenized response variable was investigated using the bivariate correlation and linear regression analysis methods. [ABSTRACT FROM AUTHOR]

We continue the study of the homogenization of coercive non-convex Hamilton–Jacobi equations in random media identifying two general classes of Hamiltonians with very distinct behavior. For the first class there is no homogenization in a particular environment while for the second homogenization takes place in environments with finite range dependence. Motivated by the recent counter-example of Ziliotto, who constructed a coercive but non-convex Hamilton–Jacobi equation with stationary ergodic random potential field for which homogenization does not hold, we show that same happens for coercive Hamiltonians which have a strict saddle-point, a very local property. We also identify, based on the recent work of Armstrong and Cardaliaguet on the homogenization of positively homogeneous random Hamiltonians in environments with finite range dependence, a new general class Hamiltonians, namely equations with uniformly strictly star-shaped sub-level sets, which homogenize. [ABSTRACT FROM AUTHOR]

We study the homogenization of Hamilton–Jacobi equations which arise in front propagation problems in stationary ergodic media. Our results are obtained for fronts moving with possible unbounded velocity. We show, by an example, that the homogenized Hamiltonian, which always exists, may be unbounded. In this context, we show convergence results if we start with a compact initial front. On the other hand, if the media satisfies a finite range of dependence condition, we prove that the effective Hamiltonian is bounded and obtain classical homogenization in this context. [ABSTRACT FROM AUTHOR]

We study the qualitative homogenization of second-order Hamilton-Jacobi equations in space-time stationary ergodic random environments. Assuming that the Hamiltonian is convex and superquadratic in the momentum variable (gradient), we establish a homogenization result and characterize the effective Hamiltonian for arbitrary (possibly degenerate) elliptic diffusion matrices. The result extends previous work that required uniform ellipticity and space-time homogeneity for the diffusion. [ABSTRACT FROM AUTHOR]

We prove stochastic homogenization for a general class of coercive, nonconvex Hamilton–Jacobi equations in one space dimension. Some properties of the effective Hamiltonian arising in the nonconvex case are also discussed. [ABSTRACT FROM AUTHOR]

It is proved that for the 2-dimensional case with random shear flow of the G-equation model with strain term, the strain term reduces the front propagation. Also an improvement of the main result by Armstrong-Souganidis is provided. [ABSTRACT FROM AUTHOR]

The paper aims at analytically exhibiting for the first time the fundamental mechanisms underlying the fact that effective biological tissue electrical properties and their frequency dependence reflect the tissue composition and physiology. For doing so, a homogenization theory is derived to describe the effective admittivity of cell suspensions. A new formula is reported for dilute cases that gives the frequency-dependent effective admittivity with respect to the membrane polarization. Different microstructures are shown to be distinguishable via spectroscopic measurements of the overall admittivity using the spectral properties of the membrane polarization. The Debye relaxation times associated with the membrane polarization tensor are shown to be able to give the microscopic structure of the medium. A natural measure of the admittivity anisotropy is introduced and its dependence on the frequency of applied current is derived. A Maxwell–Wagner–Fricke formula is given for concentric circular cells, and the results can be extended to the random cases. A randomly deformed periodic medium is also considered and a new formula is derived for the overall admittivity of a dilute suspension of randomly deformed cells. [ABSTRACT FROM AUTHOR]

We prove quenched L p -type estimates for the gradient of a solution of a quasilinear elliptic equation with random coefficients. [ABSTRACT FROM AUTHOR]

In our previous papers we have described efficient and reliable methods of generation of representative volume elements (RVE) perfectly suitable for analysis of composite materials via stochastic homogenization. In this paper we profit from these methods to analyze the influence of the morphology on the effective mechanical properties of the samples. More precisely, we study the dependence of main mechanical characteristics of a composite medium on various parameters of the mixture of inclusions composed of spheres and cylinders. On top of that we introduce various imperfections to inclusions and observe the evolution of effective properties related to that. The main computational approach used throughout the work is the FFT-based homogenization technique, validated however by comparison with the direct finite elements method. We give details on the features of the method and the validation campaign as well. [ABSTRACT FROM AUTHOR]

In this work, stochastic homogenization analysis of heterogeneous materials is addressed in the context of elasticity under finite deformations. The randomness of the morphology and of the material properties of the constituents as well as the correlation among these random properties are fully accounted for, and random effective quantities such as tangent tensor, first Piola-Kirchhoff stress, and strain energy along with their numerical characteristics are tackled under different boundary conditions by a multiscale finite element strategy combined with the Montecarlo method. The size of the representative volume element (RVE) with randomly distributed particles for different particle volume fractions is first identified by a numerical convergence scheme. Then, different types of displacement-controlled boundary conditions are applied to the RVE while fully considering the uncertainty in the microstructure. The influence of different random cases including correlation on the random effective quantities is finally analyzed. [ABSTRACT FROM AUTHOR]

We are interested in the averaging behavior of interfaces moving in stationary ergodic environments with oscillatory normal velocity which changes sign. The problem can be reformulated as the homogenization of a Hamilton–Jacobi equation with a positively homogeneous of degree one non-coercive Hamiltonian. The periodic setting was studied earlier by Cardaliaguet, Lions and Souganidis (2009) [16] . Here we concentrate in the random media and show that the solutions of the oscillatory Hamilton–Jacobi equation converge in L ∞ -weak ⋆ to a linear combination of the initial datum and the solutions of several initial value problems with deterministic effective Hamiltonian(s), determined by the properties of the random media. [ABSTRACT FROM AUTHOR]

We study homogenization for fully nonlinear uniformly parabolic equations in stationary ergodic spatio-temporal media from the qualitative and quantitative perspectives. Under suitable hypotheses, solutions to fully nonlinear uniformly parabolic equations in spatio-temporal media homogenize almost surely. In addition, we obtain a logarithmic rate of convergence for this homogenization in measure, assuming that the environment is strongly mixing with a prescribed logarithmic rate. A general methodology to study the stochastic homogenization and rates of convergence for stochastic homogenization of uniformly elliptic equations was introduced by Caffarelli, Souganidis, and Wang [1] , and Caffarelli and Souganidis [2] . We extend their approach to fully nonlinear uniformly parabolic equations, developing a number of new arguments to handle the parabolic structure of the problem. [ABSTRACT FROM AUTHOR]

In [G. Bouchitté and D. Felbacq, C. R. Math. Acad. Sci. Paris 339 (2004) 377-382; D. Felbacq and G. Bouchitté, Phys. Rev. Lett. 94 (2005) 183902; D. Felbacq and G. Bouchitté, New J. Phys. 7 (2005) 159], a theory for artificial magnetism in two-dimensional photonic crystals has been developed for large wavelength using homogenization techniques. In this paper we pursue this approach within a rigorous stochastic framework: dielectric parallel nanorods are randomly disposed, each of them having, up to a large scaling factor, a random permittivity ε(ω) whose law is represented by a density on a window Δh = [a-, a+] × [0, h] of the complex plane. We give precise conditions on the initial probability law (permittivity, radius and position of the rods) under which the homogenization process can be performed leading to a deterministic dispersion law for the effective permeability with possibly negative real part. Subsequently a limit analysis h → 0, accounting a density law of ε which concentrates on the real axis, reveals singular behavior due to the presence of resonances in the microstructure. [ABSTRACT FROM AUTHOR]

In this paper we describe efficient methods of generation of representative volume elements (RVEs) suitable for producing the samples for analysis of effective properties of composite materials via and for stochastic homogenization. We are interested in composites reinforced by a mixture of spherical and cylindrical inclusions. For these geometries we give explicit conditions of intersection in a convenient form for verification. Based on those conditions we present two methods to generate RVEs: one is based on the random sequential adsorption scheme, the other one on the time driven molecular dynamics. We test the efficiency of these methods and show that the first one is extremely powerful for low volume fraction of inclusions, while the second one allows us to construct denser configurations. All the algorithms are given explicitly so they can be implemented directly. [ABSTRACT FROM AUTHOR]

The article discusses a study presenting the first quantitative homogenization results for Hamilton-Jacobi equations in the stochastic setting. It assumes that the Hamiltonian H = H (p,y,w) satisfies a finite range dependence hypothesis in its spatial dependence. Its arguments rely on adaptations of some of the probabilistic techniques based on Azuma's inequality and the martingale method of bounded differences and do not yield sharp error estimates or convergence rates for homogenization.