On Bhattacharyya relative entropy in a homogenization of composite materials

The main idea of this work is an application of relative entropy in the numerical analysis of probabilistic divergence between original material tensors of the composite constituents and its effective tensor in the presence of material uncertainties. The homogenization method is based upon the deformation energy of the representative volume elements for the fiber-reinforced and particulate composites and uncertainty propagation begins with elastic moduli of the fibers, particles, and composite matrices. Relative entropy follows a mathematical model originating from Bhattacharyya probabilistic divergence and has been applied here for Gaussian distributions. The semi-analytical probabilistic method based on analytical integration of polynomial bases obtained via the least squares method fittings enables for determination of the basic probabilistic characteristics of the effective tensor and the relative entropies. The methodology invented in this work may be extended toward other probability distributions and relative entropies, for homogenization of nonlinear composites and also accounting for some structural interface defects.     

On the dual iterative stochastic perturbation‐based finite element method in solid mechanics with Gaussian uncertainties

The main idea is a dual mathematical formulation and computational implementation of the iterative stochastic perturbation‐based finite element method for both linear and nonlinear problems in solid mechanics. A general‐order Taylor expansion with random coefficients serves here for the iterative determination of the basic probabilistic characteristics, where linearization procedure widely applicable in stochastic perturbation technique is replaced with the iterative one. The expected values and, in turn, the variances are derived first, and then, they are substituted into the equations for higher central probabilistic moments and additional probabilistic characteristics. The additional formulas for up to the fourth‐order probabilistic characteristics are derived thanks to the 10th‐order Taylor expansion. Computational implementation of this idea in the stochastic finite element method is provided by using the direct differentiation method and, independently, the response function method with polynomial basis. Numerical experiments include the simple tension of the elastic bar, nonlinear elasto‐plastic analysis of the aluminum 2D truss, and solution to the homogenization problem of periodic fiber‐reinforced composite with random elastic properties. The expected values, coefficients of variation, skewness, and kurtosis of the structural response determined via this iterative scheme are contrasted with these estimated by the Monte Carlo simulation as well as with the results of the semi‐analytical probabilistic technique following the response function method itself. Although the entire methodology is illustrated here by using the Gaussian variables where all odd‐order terms simply vanish, it can be extended towards non‐Gaussian processes as well and completed with all the perturbation orders. Copyright © 2015 John Wiley & Sons, Ltd.     

On semi‐analytical probabilistic finite element method for homogenization of the periodic fiber‐reinforced composites

The main aim of this paper is a development of the semi‐analytical probabilistic version of the finite element method (FEM) related to the homogenization problem. This approach is based on the global version of the response function method and symbolic integral calculation of basic probabilistic moments of the homogenized tensor and is applied in conjunction with the effective modules method. It originates from the generalized stochastic perturbation‐based FEM, where Taylor expansion with random parameters is not necessary now and is simply replaced with the integration of the response functions. The hybrid computational implementation of the system MAPLE with homogenization‐oriented FEM code MCCEFF is invented to provide probabilistic analysis of the homogenized elasticity tensor for the periodic fiber‐reinforced composites. Although numerical illustration deals with a homogenization of a composite with material properties defined as Gaussian random variables, other composite parameters as well as other probabilistic distributions may be taken into account. The methodology is independent of the boundary value problem considered and may be useful for general numerical solutions using finite or boundary elements, finite differences or volumes as well as for meshless numerical strategies. Copyright © 2011 John Wiley & Sons, Ltd.     

On probabilistic fatigue models for composite materials

The general purpose of this article is to review the main ideas in fatigue analysis of composites in the context of the application of probabilistic methods, both theoretical and computational. That is why most deterministic concepts of composite materials fatigue are summarized together with stochastic approaches. The application of the perturbation based Stochastic Finite Element Method (SFEM) to fatigue analysis of homogeneous and heterogeneous media is shown. Further, homogenization method in its effective modules approach is proposed below for application in fatigue processes modeling of linear elastic periodic random composites. Considering stochastic character of the analysis, the reliability tools appropriate to multicomponent materials are presented together with the specially adopted brittle and ductile fracture criteria.     

Tsallis entropy in dual homogenization of random composites using the stochastic finite element method

This work concerns an application of the Tsallis entropy to homogenization problem of the fiber‐reinforced and also of the particle‐filled composites with random material and geometrical characteristics. Calculation of the effective material parameters is done with two alternative homogenization methods—the first is based upon the deformation energy of the Representative Volume Element (RVE) subjected to the few specific deformations, while the second uses explicitly the so‐called homogenization functions determined under periodic boundary conditions imposed on this RVE. Probabilistic homogenization is made with the use of three concurrent non‐deterministic methods, namely Monte‐Carlo simulation, iterative generalized stochastic perturbation technique as well as the semi‐analytical approach. The last two approaches are based on the Least Squares Method with polynomial basis of the statistically optimized order— this basis serves for further differentiation in the 10th‐order stochastic perturbation technique, while semi‐analytical method uses it in probabilistic integrals. These three approaches are implemented all as the extensions of the traditional Finite Element Method (FEM) with contrastively different mesh sizes, and they serve in computations of Tsallis entropies of the homogenized tensor components as the functions of input coefficient of variation.     

A study on the prediction of the mechanical properties of nanoparticulate composites using the homogenization method with the effective interface concept

A sequential multi‐scale homogenization method combined with molecular dynamics (MD) simulation is developed for the mechanical characterization of nanoparticulate composites. In order to characterize the particle‐size effect of nanocomposites, the effective interface, which has been adopted in continuum micromechanics approaches, is considered as the characteristic phase. Owing to the existence of the interface and the size‐dependent elastic modulus that is observed from MD simulations, an analysis of the mechanical properties of nanocomposites with continuum micromechanics requires careful consideration of the particle‐concentration effect. Therefore, this study focuses on hierarchical information transfer from the molecular model to the continuum model through the homogenization method in lieu of an analytical micromechanics bridging method. Using the present multi‐scale homogenization method, the elastic properties of the effective interface are numerically evaluated and compared with the analytically obtained micromechanics solutions. In addition, the overall elastic modulus of nanocomposites is obtained from the present model and compared with the results of MD simulation, the micromechanics bridging model, and finite‐element analysis (FEA). Copyright © 2010 John Wiley & Sons, Ltd.     

Homogenization of metallic fiber-reinforced composites under stochastic ageing

The main aim of this paper is to present an application of the generalized stochastic perturbation technique to model stochastic ageing processes of the metallic fibre-reinforced periodic composite materials in terms of their effective properties. Those ageing processes are modelled here as two-parametric time series having Gaussian random initial values and time rate, both defined uniquely by their expectations and standard deviations. Computational homogenization procedure is discrete and based on the Finite Element Method program MCCEFF as well as the computer algebra system MAPLE, where the Response Function Method and the stochastic analysis are entirely implemented. This numerical strategy is used to analyze probabilistic moments of the effective elastic tensor of the few metal matrix composites as well as to simulate stochastic ageing of two representative composites - MoSio₂-SiC and Ti-SiC. The approach proposed and results of computations may be further applied in the reliability analysis of metallic or the other composites.     

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.     

A spectral stochastic element free Galerkin method for the problems with random material parameter

This paper presents a spectral stochastic element free Galerkin method (SSEFGM) for the problems involving a random material property. The random material property and resulting system response quantity are represented by a probabilistic spectral expansion techniques (Karhunen–Loeve expansion and Polynomical Chaos series, respectively) and implemented into the element free Galerkin (EFG) analysis. Numerical solutions in 1D linear elastic problem with random elastic modulus are introduced, and compared with those of Monte Carlo simulation (MCS) so as to provide the validation of the proposed approach. The present SSEFGM approach can produce a probabilistic density distribution as well as a first‐ and second‐order statistical moments (mean and variance) of response quantity by a single calculation, which is distinguished from an iterative MCS. Moreover, the method is based on an element free analysis so that there is no need of nodal connectivities, which usually require more time and labourious task than main calculations. Thus the proposed SSEFGM approach can provide an alternative analysis tool for the problems contains a stochastic material property, and demands complex mesh structures. Copyright © 2004 John Wiley & Sons, Ltd.     

Domain decomposition of stochastic PDEs: Theoretical formulations

We present a novel theoretical framework for the domain decomposition of uncertain systems defined by stochastic partial differential equations. The methodology involves a domain decomposition method in the geometric space and a functional decomposition in the probabilistic space. The probabilistic decomposition is based on a version of stochastic finite elements based on orthogonal decompositions and projections of stochastic processes. The spatial decomposition is achieved through a Schur‐complement‐based domain decomposition. The methodology aims to exploit the full potential of high‐performance computing platforms by reducing discretization errors with high‐resolution numerical model in conjunction to giving due regards to uncertainty in the system. The mathematical formulation is numerically validated with an example of waves in random media. Copyright © 2008 John Wiley & Sons, Ltd.     

On Stochastic Modeling of Interface Defects in Composite Materials

The article presents mathematical and computational research dealing with the problem of stochastic interface defects occurring in composite materials between their constituents. A mathematical model of the periodic composite with such defects is presented in detail, as well as probabilistic numerical methods enabling computational experiments which are shown in a further part of the text. The fiber-reinforced and laminated composite has been tested in numerical tests as well as the superconducting coil cable-four-component composite to verify how the structural defects considered, according to the model introduced, influence the static behavior of the composites analyzed. All the results obtained and discussed in the article are summarized in concluding remarks which show the directions of further model development, while numerous references enable the reader to study the problem further.     

Boundary element method homogenization of the periodic linear elastic fiber composites

The paper presented is devoted to the Boundary Element Method based homogenization of the periodic transversely isotropic linear elastic fiber-reinforced composites. The composite material under consideration has deterministically defined elastic properties while its components are perfectly bonded. To have a good comparison with the FEM-based computational techniques used previously, the additional Finite Element discretization is presented and compared numerically against BEM homogenization implementation on the example of engineering glass–epoxy composite. The homogenization method proposed has rather general characteristics and, as it is shown, can be easily extended on n-component composites. On the contrary, we can consider and homogenize the heterogeneous media with randomly defined material properties using Monte-Carlo simulation technique or second order perturbation second probabilistic moment approach.     

Computational homogenization of nonlinear elastic materials using neural networks

In this work, a decoupled computational homogenization method for nonlinear elastic materials is proposed using neural networks. In this method, the effective potential is represented as a response surface parameterized by the macroscopic strains and some microstructural parameters. The discrete values of the effective potential are computed by finite element method through random sampling in the parameter space, and neural networks are used to approximate the surface response and to derive the macroscopic stress and tangent tensor components. We show through several numerical convergence analyses that smooth functions can be efficiently evaluated in parameter spaces with dimension up to 10, allowing to consider three‐dimensional representative volume elements and an explicit dependence of the effective behavior on microstructural parameters like volume fraction. We present several applications of this technique to the homogenization of nonlinear elastic composites, involving a two‐scale example of heterogeneous structure with graded nonlinear properties. Copyright © 2015 John Wiley & Sons, Ltd.     

Two-scale homogenization of elastic layered composites with interfaces oscillating in two directions

In a variety of situations of practical interest, the interface between two phases in a composite cannot be reasonably assumed to be smooth but has to be taken as being rough at the microscopic scale. How to determine the effective properties of such a composite remains a largely open problem in micromechanics. The present work is concerned with layered composites in which the interface between two neighboring layers oscillates quickly and periodically along two directions in the plane normal to the layering direction. In this case, the classical homogenization theory of layered composites is no longer applicable, since the interfacial oscillations prevent the layered composite in question from being homogeneous in the plane perpendicular to the layering direction. To overcome this difficulty, a two-scale homogenization method is proposed in the present work. First, at the mesoscopic scale, each zone in which an interface oscillates is homogenized as an interphase by an asymptotic analysis. The effective elastic properties of this interphase are determined by using a numerical method based on the fast Fourier transform (FFT) or estimated by applying the generalized self-consistent scheme (GSCS). Then, at the macroscopic scale, the effective elastic moduli of the composite made of the resulting plane layers and interphases are calculated with the help of the classical homogenization theory of layered composites. Finally, numerical examples are provided to illustrate the results for the effective elastic moduli of a layered composite obtained by the two-scale homogenization method proposed and to compare them with the corresponding numerical results given by the finite element method (FEM).     

Stochastic multiscale analysis in hydrodynamic lubrication

A stochastic multiscale analysis framework is developed for hydrodynamic lubrication problems with random surface roughness. The approach is based on a multi‐resolution computational strategy wherein the deterministic solution of the multiscale problem for each random surface realization is achieved through a coarse‐scale analysis with a local upscaling that is achieved through homogenization theory. The stochastic nature of this solution because of the underlying randomness is then characterized through local and global quantities of interest, accompanied by a detailed discussion regarding suitable choices of the numerical parameters in order to achieve a desired stochastic predictive capability while ensuring numerical efficiency. Finally, models of the stochastic interface response are constructed, and their performance is demonstrated for representative problem settings. Overall, the developed approach offers a computational framework, which can essentially predict the significant influence of interface heterogeneity in the absence of a strict scale separation. Copyright © 2017 John Wiley & Sons, Ltd.     

Tensor-valued random fields for meso-scale stochastic model of anisotropic elastic microstructure and probabilistic analysis of representative volume element size

The main objective of this paper is to present a generic meso-scale probability model for a large class of random anisotropic elastic microstructures in order to perform a parametric analysis of the Representative Volume Element (RVE) size. This new approach can be useful for a direct experimental identification of random anisotropic elastic microstructures when the standard method cannot easily be applied to anisotropic elastic microstructures. Such a RVE is used to construct the macroscopic properties in the context of stochastic homogenization. The probability analysis is not performed as usual for a given particular random microstructure defined in terms of its constituents. Instead, it is performed for a large class of random anisotropic elastic microstructures. For this class, the probability distribution of the random effective stiffness tensor is explicitly constructed. This allows a full probability analysis of the RVE size to be carried out and its convergence to be studied. The procedure of homogenization is based on a homogeneous Dirichlet condition on the boundary of the RVE. The probability model used for the stiffness tensor-valued random field of the random anisotropic elastic microstructure is an extension of the model recently introduced by the author for elliptic stochastic partial differential operators. The stochastic boundary value problem is numerically solved by using the stochastic finite element method. The probability analysis of the RVE size is performed by studying the probability distribution of the random operator norm of the random effective stiffness tensor with respect to the spatial correlation length of the random microstructure.     

Homogenization of transient heat transfer problems for some composite materials

The article presented is devoted to the homogenization of transient heat transfer problems in some composite materials. The mathematical model used in the FEM computation is based on the effective modules method introduced for periodic composites. The effective heat conductivity is calculated in the closed form; effective heat capacity and mass density for the composite are obtained by simple spatial averaging. Such a homogenization scheme makes it possible to significantly simplify the numerical analysis of transient heat phenomena in various types of composites. Computational experiments performed using symbolic mathematics show the variability of effective heat conductivity for 2D and 3D composites as a function of the reinforcement volume ratio, of composite components conductivity coefficients as well as of the probabilistic moments of material properties versus volume ratio. Finally, using the Finite Element Method program, the comparison of transient heat transfer problem for the real and homogenized composites models is carried out.     

Effective transverse elastic moduli of three-phase hybrid fiber-reinforced composites with randomly located and interacting aligned circular fibers of distinct elastic properties and sizes

A higher-order multi-scale structure for three-phase hybrid fiber-reinforced composites containing randomly located yet unidirectionally aligned circular fibers is proposed to predict effective transverse elastic moduli based on the probabilistic spatial distribution of circular fibers, the pairwise fiber interactions, and the ensemble-area homogenization method. Specifically, the two inhomogeneity phases feature distinct elastic properties and sizes. Two non-equivalent formulations are considered in detail to derive effective transverse elastic moduli of three-phase composites leading to new higher-order bounds. Numerical examples and comparisons among our theoretical predictions and other analytical predictions are rendered to illustrate the potential capability of the present method.     

A polynomial dimensional decomposition for stochastic computing

This article presents a new polynomial dimensional decomposition method for solving stochastic problems commonly encountered in engineering disciplines and applied sciences. The method involves a hierarchical decomposition of a multivariate response function in terms of variables with increasing dimensions, a broad range of orthonormal polynomial bases consistent with the probability measure for Fourier‐polynomial expansion of component functions, and an innovative dimension‐reduction integration for calculating the coefficients of the expansion. The new decomposition method does not require sample points as in the previous version; yet, it generates a convergent sequence of lower‐variate estimates of the probabilistic characteristics of a generic stochastic response. The results of five numerical examples indicate that the proposed decomposition method provides accurate, convergent, and computationally efficient estimates of the tail probability of random mathematical functions or the reliability of mechanical systems. Copyright © 2008 John Wiley & Sons, Ltd.