Stochastic Homogenization Problem Research Articles

PDE acceleration: a convergence rate analysis and applications to obstacle problems

This paper provides a rigorous convergence rate and complexity analysis for a recently introduced framework, called PDE acceleration, for solving problems in the calculus of variations and explores applications to obstacle problems. PDE acceleration grew out of a variational interpretation of momentum methods, such as Nesterov’s accelerated gradient method and Polyak’s heavy ball method, that views acceleration methods as equations of motion for a generalized Lagrangian action. Its application to convex variational problems yields equations of motion in the form of a damped nonlinear wave equation rather than nonlinear diffusion arising from gradient descent. These accelerated PDEs can be efficiently solved with simple explicit finite difference schemes where acceleration is realized by an improvement in the CFL condition from $$\mathrm{d}t\sim \mathrm{d}x^2$$ for diffusion equations to $$\mathrm{d}t\sim \mathrm{d}x$$ for wave equations. In this paper, we prove a linear convergence rate for PDE acceleration for strongly convex problems, provide a complexity analysis of the discrete scheme, and show how to optimally select the damping parameter for linear problems. We then apply PDE acceleration to solve minimal surface obstacle problems, including double obstacles with forcing, and stochastic homogenization problems with obstacles, obtaining state-of-the-art computational results.

On periodic boundary conditions and ergodicity in computational homogenization of heterogeneous materials with random microstructure

Due to high computational costs associated with stochastic computational homogenization, a highly complex random material microstructure is often replaced by simplified, parametric, ergodic, and sometimes periodic models. This replacement is often criticized in the literature due to unclear error resulting from the periodicity and ergodicity assumptions. In the current contribution we perform a validation of both assumptions through various numerical examples. To this end we compare large-scale non-simplified and non-ergodic models with simplified, ergodic, and periodic solutions. In addition we analyze the Hill–Mandel condition for stochastic homogenization problems and demonstrate that for a stochastic problem there are more than three classical types of boundary conditions. As an example, we propose two novel stochastic periodic boundary conditions which possess a clear physical meaning. The effect of these novel periodic boundary conditions is also analyzed by comparing with non-ergodic simulation results.

Stochastic Homogenization of Nonconvex Discrete Energies with Degenerate Growth

We study the continuum limit of discrete, nonconvex energy functionals defined on crystal lattices in dimensions $d\geq 2$. Since we are interested in energy functionals with random (stationary and ergodic) pair interactions, our problem corresponds to a stochastic homogenization problem. In the non-degenerate case, when the interactions satisfy a uniform $p$-growth condition, the homogenization problem is well-understood. In this paper, we are interested in a degenerate situation, when the interactions neither satisfy a uniform growth condition from above nor from below. We consider interaction potentials that obey a $p$-growth condition with a random growth weight $\lambda$. We show that if $\lambda$ satisfies the moment condition $\mathbb E[\lambda^\alpha+\lambda^{-\beta}]<\infty$ for suitable values of $\alpha$ and $\beta$, then the discrete energy $\Gamma$-converges to an integral functional with a non-degenerate energy density. In the scalar case it suffices to assume that $\alpha\geq 1$ and $\beta\geq\frac{1}{p-1}$ (which is just the condition that ensures the non-degeneracy of the homogenized energy density). In the general, vectorial case, we additionally require that $\alpha>1$ and $\frac{1}{\alpha}+\frac{1}{\beta}\leq \frac{p}{d}$. Recently, there has been considerable effort to understand periodic and stochastic homogenization of elliptic equations and integral functionals with degenerate growth, as well as related questions on the effective behavior of conductance models in degenerate, random environments. The results in the present paper are to our knowledge the first stochastic homogenization results for nonconvex energy functionals with degenerate growth under moment conditions.

A direct verification argument for the Hamilton–Jacobi equation continuum limit of nondominated sorting

Nondominated sorting is a combinatorial algorithm that sorts points in Euclidean space into layers according to a partial order. It was recently shown that nondominated sorting of random points has a Hamilton–Jacobi equation continuum limit. The original proof, given in Calder et al. (2014), relies on a continuum variational problem. In this paper, we give a new proof using a direct verification argument that completely avoids the variational interpretation. We believe this may be generalized to apply to other stochastic homogenization problems for which there is no obvious underlying variational principle.

A central limit theorem for fluctuations in 1D stochastic homogenization

In this paper, we analyze the random fluctuations in a 1D stochastic homogenization problem and prove a central limit theorem: the first order fluctuations is described by a Gaussian process that solves an SPDE with an additive spatial white noise. Using a probabilistic approach, we obtain a precise error decomposition up to the first order, which also helps to decompose the limiting Gaussian process, with one of the components corresponding to the corrector obtained by a formal two-scale expansion.

Polynomial-based Approximate Inverse Stochastic Homogenization Analysis of a Particle Reinforced Composite Material Considering Correlated Multiple Microscopic Random Variations

This paper describes an inverse stochastic homogenization analysis of a particle composite material considering correlated multiple microscopic random variations. Since a microscopic random variation will cause a random response of an equivalent material property or mechanical response of a composite structure, the analysis of its influence is important for reliability evaluation of the structure. On the other hand, in general, the probabilistic property of a microscopic quantity is unknown after manufacturing. From this reason, this study aims at development of the analysis method for the inverse stochastic homogenization problem, which identifies the microscopic probabilistic property from that of the macroscopic quantities. In particular, applicability of the polynomial-based approximation to the inverse stochastic homogenization analysis is investigated in this paper.

Stochastic analysis of laminated composite plate considering stochastic homogenization problem

This paper discusses a multiscale stochastic analysis of a laminated composite plate consisting of unidirectional fiber reinforced composite laminae. In particular, influence of a microscopic random variation of the elastic properties of component materials on mechanical properties of the laminated plate is investigated. Laminated composites are widely used in civil engineering, and therefore multiscale stochastic analysis of laminated composites should be performed for reliability evaluation of a composite civil structure. This study deals with the stochastic response of a laminated composite plate against the microscopic random variation in addition to a random variation of fiber orientation in each lamina, and stochastic properties of the mechanical responses of the laminated plate is investigated. Halpin-Tsai formula and the homogenization theory-based finite element analysis are employed for estimation of effective elastic properties of lamina, and the classical laminate theory is employed for analysis of a laminated plate. The Monte-Carlo simulation and the first-order second moment method with sensitivity analysis are employed for the stochastic analysis. From the numerical results, importance of the multiscale stochastic analysis for reliability evaluation of a laminated composite structure and applicability of the sensitivity-based approach are discussed.

Variance reduction using antithetic variables for a nonlinear convex stochastic homogenization problem

We consider a nonlinear convex stochastic homogenization problem, in astationary setting. Inpractice, the deterministic homogenized energy density isapproximated by a random apparent energy density, obtained by solving thecorrector problem on a truncated domain.&nbsp; &nbspWe show that the technique of antithetic variables can be used to reducethe variance of the computed quantities, and thereby decrease thecomputational cost at equal accuracy. This leads to an efficientapproach for approximating expectations of the apparent homogenizedenergy density and of related quantities.&nbsp; &nbspThe efficiency of the approach is numerically illustrated on severaltest cases. Some elements of analysis are also provided.

Rate of convergence of a two-scale expansion for some “weakly” stochastic homogenization problems

We establish a rate of convergence of the two scale expansion (in the sense of homogenization theory) of the solution to a highly oscillatory elliptic partial differential equation with random coefficients that are a perturbation of periodic coefficients.

Hierarchical stochastic homogenization analysis of a particle reinforced composite material considering non-uniform distribution of microscopic random quantities

This paper discusses a stochastic homogenization problem for evaluation of stochastic characteristics of a homogenized elastic property of a particle reinforced composite material especially in case of considering a non-uniform distribution of a material property or geometry of a component material and its random variation. In practice, some microscopic random variations in composites may not be uniform. In this case, a non-uniformly distributed random variation of a microscopic material or geometrical property should be taken into account. For this problem, this paper proposes a hierarchical stochastic homogenization method. This method assumes that a two-phase composite material can be separated into three-scales, and propagation of the randomness through the different scales can be evaluated with the perturbation-based technique. As an example, stochastic characteristics of homogenized elastic properties of a glass-particle reinforced plastic are estimated using the proposed approach. With the numerical results, importance of the problem and validity of the proposed method are discussed.

Hierarchical Stochastic Homogenization Analysis of a Short-Fiber Reinforced Composite Material Considering a Non-Uniform Microscopic Random Variation

This paper discusses a stochastic homogenization analysis problem of a unidirectionally aligned short fiber reinforced composite material considering a uniform and/or non-uniform random variation of a microscopic quantity such as a material property or volume fraction of a microstructure. The homogenization analysis is performed with the homogenization theory and the equivalent inclusion method, and the stochastic analysis is performed with the perturbation-based method. In order to analyze the stochastic homogenization problem considering a periodic non-uniform microscopic random variation of the microscopic quantity, the hierarchical stochastic homogenization analysis procedure is proposed. As an example, the stochastic homogenization problems considering a non-uniform random variation of Young's modulus of resin and volume fraction of fiber are solved. From the numerical results, influence of the non-uniformity of the microscopic random variation is discussed. Also, from comparison between the results obtained with the proposed method and the Monte-Carlo simulation, accuracy of the proposed approach for the stochastic homogenization analysis of the short fiber reinforced composite material is discussed.

Stochastic Σ-convergence and applications

Motivated by the fact that in nature almost all phenomena behave randomly in some scales and deterministically in some other scales, we build up a framework suitable to tackle both deterministic and stochastic homogenization problems simultaneously, and also separately. Our approach, the stochastic Σ-convergence, can be seen either as a multiscale stochastic approach since deterministic homogenization theory can be seen as a special case of stochastic homogenization theory (see Theorem 3), or as a conjunction of the stochastic and deterministic approaches, both taken globally, but also each separately. One of the main applications of our results is the homogenization of a model of rotating fluids.

Stochastic multiscale stress analysis via identification of microscopic randomness

This paper proposes a concept of a stochastic stress analysis procedure via identification of a microscopic random variation. Some results on a stochastic homogenization problem of composites have been reported, and those illustrate that a microscopic random variation in material property or geometry will have an influence on a homogenized property of composites. Similarly, the microscopic randomness causes a random variation of macroscopic and microscopic stress field. If the microscopic random variation is unknown, a method for estimating the microscopic random variation from a characteristic of a macroscopic random variation will be also needed. In this paper, a stochastic multiscale stress analysis with both approaches is introduced, and necessity of an inverse stochastic homogenization analysis and validity of the proposed approach are discussed.

Identification of a microscopic randomness of a particle reinforced composite material with Monte-Carlo Simulation and inverse homogenization analysis

This paper describes a detailed procedure for identifying a microscopic randomness of a particle-reinforced composite material considering a microscopic uncertainty in elastic property of component materials of a particle reinforced composite material. Some reports on the stochastic homogenization analysis considering a microscopic random variation can be found in literatures. A microscopic stress field will take an influence of the microscopic variation, the stochastic microscopic stress analysis will be also important. In the previous reports, it is assumed the microscopic random variation is known. However, it is sometimes difficult to identify a microscopic random variation in a composite material especially after manufacturing process. An identification process for a microscopic randomness with solving an inverse problem, therefore, will be needed for the stochastic microscopic stress analysis. In this paper, an inverse stochastic homogenization problem is tried to be analysed with the inverse homogenization analysis and the Monte-Carlo simulation for the stochastic homogenization analysis. With numerical results, accuracy of the inverse analysis is discussed.

S0303-3-5 非一様な微視的ばらつきを考慮した粒子強化複合材料の確率均質化解析([0303-3]材料機能性と強度に及ぼす環境と場の影響に関する弾性数理解析(3))

This paper proposes a new approach for stochastic homogenization analysis of a particle reinforced composite material considering a random variation of an elastic property of component materials. Especially, a non-uniform microscopic random variation is taken into account. In order to analyze a stochastic homogenization problem considering a non-uniformly distributed random variation, a hierarchical modeling with an intermediate scale is introduced. In order to investigate validity of the proposed approach, stochastic homogenization analysis of a glass particle reinforced composite material is performed. From the numerical results, accuracy of the proposed approach is discussed.

等価介在物法を用いた一次摂動解析による球形粒子強化複合材料の確率均質化解析

This paper describes a method for stochastic homogenization analysis of a particle reinforced composite material. Since it is difficult to apply the homogenization-based perturbation method using the finite element method to a stochastic homogenization problem with considering a microscopic geometrical random variation in a microstructure, the equivalent inclusion method is used for the perturbation-based stochastic homogenization analysis. In using the equivalent inclusion method, each perturbation term for a random variation in a material property, size of inclusions or their volume fraction can be computed analytically. This fact shows applicability to apply the proposed method to a stochastic homogenization problem considering the microscopic geometrical random variation. At first, formulations for the proposed method are introduced. Next, the proposed method is applied to several kinds of numerical examples. From the numerical results, validity and accuracy of the proposed method are confirmed.

Stochastic homogenization of random elastic multi-phase composites and size quantification of representative volume element

By using effective properties derived from homogenization techniques, a heterogeneous medium is transformed into a homogenous one, and the number of the degrees of freedom (DOF) can therefore be significantly reduced. This dimension reduction approach has been a most widely used computational approach for efficient simulation of heterogeneous materials. The success of such a deterministic Representative Volume Element (RVE) practice critically relies on satisfaction of the assumption on scale decoupling, i.e. the size of heterogeneity is significantly smaller than the size of finite elements or RVE. The condition however has been incautiously or unknowingly ignored in quite a number of engineering practices. One typical example is finite element modeling of concrete materials where the element size is often chosen to be comparable to or even smaller than the size of aggregates. In computational modeling of small scale systems such as NEMS and MEMS, the scale decoupling assumption is commonly invalid. For such circumstances, the use of deterministic homogenization to reduce DOF results in large errors and new methodology using stochastic homogenization and non-deterministic material constants in finite elements should be considered. This paper is aimed to quantify the size effect of RVE and therefore provide estimates for the threshold of scale separation. The generalized variational principles developed in Xu [Xu, X.F., submitted for publication. Generalized variational principles for uncertainty quantification of boundary value problems of random heterogeneous materials] are adapted to stochastic homogenization problems, which results in size-dependent energy bounds and Hashin–Shtrikman (HS) bounds. The difference between the size dependent stochastic bounds and the classical deterministic bounds demonstrates the size effect of RVE. Numerical characterization suggests that for three-dimensional problems the size threshold of scale-decoupling, or equivalently the minimum size of deterministic RVE, is approximately between 10 and 20 times of “correlation diameter” based on a range of accuracy criteria between 1% and 0.1%. The threshold for two-dimensional problems is approximately between 15 and 50 times of correlation diameter for the same criteria, which shows stronger size effect than the 3-D cases.

A stochastic computational method for evaluation of global and local behavior of random elastic media

A stochastic computational method is developed to evaluate global effective properties and local probabilistic behavior of random elastic media. The elasticity solution for random media is addressed with a method that combines a fast iterative numerical method with stochastic decomposition techniques. Stochastic homogenization problems are mathematically formulated with an ensemble average approximation scheme based on a concept of stochastic representative volume element (SRVE). Probabilistic descriptors of local strain are generated based on convolution equations derived from the equilibrium equations of elasticity. With the application of the Galerkin formulation in probability space and the Fourier transform of the convolution equations, an efficient numerical scheme is implemented using an iterative algorithm that takes advantage of fast Fourier transform. Two algorithms are provided for Gaussian and non-Gaussian random media, respectively, with corresponding sufficient conditions for convergence derived. Finally, numerical experiments are conducted for a Gaussian and a non-Gaussian random medium, respectively, with verifications obtained from Monte Carlo simulation.

Comments on rarity and chaos, Thomas D. Rogers, math. biosci. 72:13-18 (1984)

By using effective properties derived from homogenization techniques, a heterogeneous medium is transformed into a homogenous one, and the number of the degrees of freedom (DOF) can therefore be significantly reduced. This dimension reduction approach has been a most widely used computational approach for efficient simulation of heterogeneous materials. The success of such a deterministic Representative Volume Element (RVE) practice critically relies on satisfaction of the assumption on scale decoupling, i.e. the size of heterogeneity is significantly smaller than the size of finite elements or RVE. The condition however has been incautiously or unknowingly ignored in quite a number of engineering practices. One typical example is finite element modeling of concrete materials where the element size is often chosen to be comparable to or even smaller than the size of aggregates. In computational modeling of small scale systems such as NEMS and MEMS, the scale decoupling assumption is commonly invalid. For such circumstances, the use of deterministic homogenization to reduce DOF results in large errors and new methodology using stochastic homogenization and non-deterministic material constants in finite elements should be considered. This paper is aimed to quantify the size effect of RVE and therefore provide estimates for the threshold of scale separation. The generalized variational principles developed in Xu [Xu, X.F., submitted for publication. Generalized variational principles for uncertainty quantification of boundary value problems of random heterogeneous materials] are adapted to stochastic homogenization problems, which results in size-dependent energy bounds and Hashin–Shtrikman (HS) bounds. The difference between the size dependent stochastic bounds and the classical deterministic bounds demonstrates the size effect of RVE. Numerical characterization suggests that for three-dimensional problems the size threshold of scale-decoupling, or equivalently the minimum size of deterministic RVE, is approximately between 10 and 20 times of “correlation diameter” based on a range of accuracy criteria between 1% and 0.1%. The threshold for two-dimensional problems is approximately between 15 and 50 times of correlation diameter for the same criteria, which shows stronger size effect than the 3-D cases.