The Choice of Representative Volumes in the Approximation of Effective Properties of Random Materials

Fischer, Julian

doi:10.1007/s00205-019-01400-w

Cited by 22 publications

(27 citation statements)

References 82 publications

(221 reference statements)

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“…

𝒜 ϕ : = - \nabla \cdot A (x) \nabla ϕ = f (x), x = (x_{1}, \dots, x_{d}) \in Ω,

with the diagonal d × d uniformly elliptic coefficient matrix

A (x)

\infty > β_{0} I \geq A (x) \geq α_{0} I > 0

. In this article, we consider the case d =2 and focus on the special class of elliptic problems arising in stochastic homogenization theory for the corrector problem, where the highly varying coefficient matrix and the right‐hand side (RHS) are defined by a sequence of stochastic realizations as described in References , see details in Sections 3 and 4.…”

Section: Elliptic Equations In Stochastic Homogenizationmentioning

confidence: 99%

“…Homogenization methods allow to derive the effective mechanical and physical properties of highly heterogeneous materials from the knowledge of the spatial distribution of their components . In particular, stochastic homogenization via the representative volume element (RVE) methods provide means for calculating the effective large‐scale characteristics related to structural and geometric properties of random composites, by utilizing a possibly large number of probabilistic realizations . The numerical investigation of the effective characteristics of random structures is a challenging problem since the underlying elliptic equation (the corrector problem) with randomly generated coefficients should be solved for many thousands realizations and for domains with substantial structural complexity to obtain sufficient statistics.…”

Section: Introductionmentioning

confidence: 99%

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Numerical study in stochastic homogenization for elliptic partial differential equations: Convergence rate in the size of representative volume elements

Khoromskaia

Khoromskij

Otto

2020

Numerical Linear Algebra App

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We describe the numerical scheme for the discretization and solution of 2D elliptic equations with strongly varying piecewise constant coefficients arising in the stochastic homogenization of multiscale composite materials. An efficient stiffness matrix generation scheme based on assembling the local Kronecker product matrices is introduced. The resulting large linear systems of equations are solved by the preconditioned conjugate gradient iteration with a convergence rate that is independent of the grid size and the variation in jumping coefficients (contrast). Using this solver, we numerically investigate the convergence of the representative volume element (RVE) method in stochastic homogenization that extracts the effective behavior of the random coefficient field. Our numerical experiments confirm the asymptotic convergence rate of systematic error and standard deviation in the size of RVE rigorously established in Gloria et al. The asymptotic behavior of covariances of the homogenized matrix in the form of a quartic tensor is also studied numerically. Our approach allows laptop computation of sufficiently large number of stochastic realizations even for large sizes of the RVE.

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“…

𝒜 ϕ : = - \nabla \cdot A (x) \nabla ϕ = f (x), x = (x_{1}, \dots, x_{d}) \in Ω,

with the diagonal d × d uniformly elliptic coefficient matrix

A (x)

\infty > β_{0} I \geq A (x) \geq α_{0} I > 0

Section: Elliptic Equations In Stochastic Homogenizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Numerical study in stochastic homogenization for elliptic partial differential equations: Convergence rate in the size of representative volume elements

Khoromskaia

Khoromskij

Otto

2020

Numerical Linear Algebra App

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“…There exist various options to define the RVE approximation, see for example [22,Section 1.4] for a discussion in the linear elliptic setting. In the present work, we shall consider the case of periodic RVEs: One considers an L-periodic approximation P L of the probability distribution P of the random field ω ε , that is an L-periodic variant ω ε,L of the random field ω ε (see below for a precise definition), and imposes periodic boundary conditions on the RVE boundary ∂…”

Section: Optimal-order Error Estimates For the Approximation Of The Homogenizedmentioning

confidence: 99%

Optimal Homogenization Rates in Stochastic Homogenization of Nonlinear Uniformly Elliptic Equations and Systems

Fischer

Neukamm

2021

Arch Rational Mech Anal

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We derive optimal-order homogenization rates for random nonlinear elliptic PDEs with monotone nonlinearity in the uniformly elliptic case. More precisely, for a random monotone operator on $$\mathbb {R}^d$$ R d with stationary law (that is spatially homogeneous statistics) and fast decay of correlations on scales larger than the microscale $$\varepsilon >0$$ ε > 0 , we establish homogenization error estimates of the order $$\varepsilon $$ ε in case $$d\geqq 3$$ d ≧ 3 , and of the order $$\varepsilon |\log \varepsilon |^{1/2}$$ ε | log ε | 1 / 2 in case $$d=2$$ d = 2 . Previous results in nonlinear stochastic homogenization have been limited to a small algebraic rate of convergence $$\varepsilon ^\delta $$ ε δ . We also establish error estimates for the approximation of the homogenized operator by the method of representative volumes of the order $$(L/\varepsilon )^{-d/2}$$ ( L / ε ) - d / 2 for a representative volume of size L. Our results also hold in the case of systems for which a (small-scale) $$C^{1,\alpha }$$ C 1 , α regularity theory is available.

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“…Let us just mention a few directions here: Quantitative error estimates for computation of effective coefficients in stochastic homogenization have been studied in [BP04,Glo12,GM12,EGMN15], where strategies using different boundary conditions or massive terms have been studied. Representative volume method, a popular approach used by engineers, are systematically analyzed in [Fis19,KKO20]. Iterative multigrid methods have been studied in [Mou19,HMS21,AHKM21].…”

Section: Related Workmentioning

confidence: 99%

Optimal Artificial Boundary Condition for Random Elliptic Media

Otto

2021

Found Comput Math

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We are interested in numerical algorithms for computing the electrical field generated by a charge distribution localized on scale in an infinite heterogeneous medium, in a situation where the medium is only known in a box of diameter L around the support of the charge. We propose a boundary condition that with overwhelming probability is (near) optimal with respect to scaling in terms of and L, in the setting where the medium is a sample from a stationary ensemble with a finite range of dependence (set to be unity and with the assumption that 1). The boundary condition is motivated by quantitative stochastic homogenization that allows for a multipole expansion [BGO20].This work extends [LO21] from two to three dimensions, and thus we need to take quadrupoles, next to dipoles, into account. This in turn relies on stochastic estimates of second-order, next to first-order, correctors. These estimates are provided for finite range ensembles under consideration, based on an extension of the semi-group approach of [GO15].

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The Choice of Representative Volumes in the Approximation of Effective Properties of Random Materials

Cited by 22 publications

References 82 publications

Numerical study in stochastic homogenization for elliptic partial differential equations: Convergence rate in the size of representative volume elements

Numerical study in stochastic homogenization for elliptic partial differential equations: Convergence rate in the size of representative volume elements

Optimal Homogenization Rates in Stochastic Homogenization of Nonlinear Uniformly Elliptic Equations and Systems

Optimal Artificial Boundary Condition for Random Elliptic Media

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