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Stochastic finite elements as a bridge between random material microstructure and global response
Abstract
This paper presents a finite element method aimed at the introduction of microstructural material randomness below the level of a single finite element. A consideration of dependence of effective moduli on scale and on the defining boundary conditions leads to an identification of a finite element as a mesoscale window (or, a mesoscale finite element) in a stochastic finite element method (SFE). An estimation of the global response can be obtained through bounds stemming from minimum potential energy and complementary energy principles, which involve Dirichlet and Neumann boundary conditions on all the mesoscale finite elements, respectively. While in the classical case of a homogeneous material, these two bounds converge to each other as the finite element mesh becomes sufficiently fine, an optimal mesoscale with respect to the difference between both bounds may exist in the case of a heterogeneous material. The proposed SFE method is illustrated with a numerical example of a sample two-phase Voronoi composite (with some 26 000 grains), where a reference solution taking into account the entire microstructure without any smearing out, is shown to fall between both energy bounds. A generalization to an ensemble response is straightforward.
... These random fields would have to be obtained from a large number of experiments in order to obtain accurate results, which is not always possible. In order to avoid this costly step, these random fields can be defined from micromechanical information ( 2 , 3 ) that contains the statistic properties of the uncertainties present on the microstructure of the material, being then possible to generate realistic virtual microstructures that contain the same stochastic properties as the real material. This has been done for multiple microstructures as shown in the review by L. Noels 4 . ...
... where vol ( ) = ( ) 3 . This normal is used in the computation of the plastic evolution, which is determined through the plastic strain rate tensor as:̇p l = pl ⋅ pl . ...
This paper presents the construction of a mean‐field homogenization (MFH) surrogate for nonlinear stochastic multiscale analyses of two‐phase composites that allows the material response to be studied up to its failure. The homogenized stochastic behavior of the studied unidirectional composite material is first characterized through full‐field simulations on stochastic volume elements (SVEs) of the material microstructure, permitting to capture the effect of the microstructural geometric uncertainties on the material response. Then, in order to conduct the stochastic nonlinear multiscale simulations, the microscale problem is substituted by a pressure‐dependent MFH reduced order micromechanical model, that is, a MF‐ROM, whose properties are identified by an inverse process from the full‐field SVE realizations. Homogenized stress‐strain curves can be used for the identification process of the nonlinear range, however, a loss of size objectivity is encountered once the strain softening onset is reached. This work addresses this problematic introducing a calibration of the energy release rate obtained with a nonlocal MFH micromechanical model, allowing to scale the variability found on each SVE failure characteristics to the macroscale. The obtained random effective properties are then used as input of a data‐driven stochastic model to generate the complete random fields used to feed the stochastic MF‐ROM. To show the consistency of the methodology, two MF‐ROM constructed from SVEs of two different sizes are successively considered. The performance of the MF‐ROM is then verified against an experimental transverse‐compression test and against full‐field simulations through nonlocal Stochastic Finite Element Method (SFEM) simulations. The implementation of the energy release rate calibration allows to conduct stochastic studies on the failure characteristics of material samples without the need for costly experimental campaigns, paving the way for more complete and affordable virtual testing.
... In order for the multi-scale process to be accurate, the RVE geometry should be representative of the reality. One method is to construct the RVE from image acquisition [116], or to build the RVE using micro-structure generation tools [117]. In both cases, the recourse to level-set functions is a key ingredient. ...
... In both cases, the recourse to level-set functions is a key ingredient. Because of the resulting complex micro-structure, a resolution of the meso-scale BVP using the X-FEM method, avoiding the need for conforming mesh, can be envisioned [116]. ...
... for the notation ω(w). Following Alzebdeh and Ostoja-Starzewski (1996); Ostoja-Starzewski and Wang (1999), ω(X; w) is called a Stochastic Volume Element (SVE), see also the review by Ostoja-Starzewski et al. (2016). ...
... In these equations we assume that the uncertainties result from the micro-scale information only. The uncertainties of the material properties can then be propagated to the macro-scale structural response through a stochastic finite element discretisation Ω ∼ ∪ e Ω e of the boundary value problem problem (313) as suggested by Alzebdeh and Ostoja-Starzewski (1996);Ostoja-Starzewski and Wang (1999); Ghanem and Spanos (1991) ;Le Maitre and Knio (2010). To this end, the problem is completed by the stochastic constitutive behaviour (251) which is formulated under the form of a random field, see Section 5.3 in the linear range and Section 5.4 in the non-linear range. ...
The scientific community has realized that nondeterminism is a major issue that affects structural and material performance and reliability. Because experimental characterization alone cannot reliably sample the tails of distributions, virtual stochastic testing has thus become a research field of growing interest. Since the uncertainties at the structural level also result from the variability of the microstructure, there is a need to develop computationally efficient stochastic multiscale methods. The purpose of this work is to provide a summary of the different methods that have been developed in the context of microstructure characterization and reconstruction, of stochastic homogenization and of uncertainties upscaling.
... in which the external forces include a perturbation vector R r , characterized as the white noise V t , formal derivative of a Wiener process W t [Ostoja-Starzewsk and Wang (1999)]. Let y =ẋ, we can express the governing equation into a typical dynamical system form ...
... where for the i th mode the drift term is represented by λ i ξ i and the diffusive term increment c i dW t yields from φ −1 BV t dt along with the definition of the white noise as the formal derivative of a Wiener process W t [Ostoja-Starzewsk and Wang (1999)]. ...
In this work, we employ fluid-structure interaction (FSI) systems with immersed flexible structures with or without free surfaces to explore both Singular Value Decomposition (SVD)-based model reduction methods and mode superposition methods. For acoustoelastic FSI systems, we adopt a three-field mixed finite element formulation with displacement, pressure, and vorticity moment unknowns to effectively enforce the irrotationality constraint. We also propose in this paper a new Inf-Sup test based on the lowest non-zero singular value of the coupling matrix for the selection of reliable sets of finite element discretizations for displacement and pressure as well as vorticity moment. Our numerical examples demonstrate that mixed finite element formulations can be effectively used to predict resonance frequencies of fully coupled FSI systems within different ranges of respective physical motions, namely, acoustic, structural, and slosh motions, without the contamination of spurious (non-physical) modes with nonzero frequencies. Our numerical results also confirm that SVD-based model reduction methods can be effectively used to reconstruct from a few snapshots of transient solutions the dominant principal components with moderate level of signal to noise ratio, which may eventually open doors for simulation of long-term behaviors of both linear and nonlinear FSI systems.
... These random fields have to be properly defined to describe the material property definition [2]. For modeling the stochastic response of composite materials, stochastic multi-scale approaches were developed, where the random response is extracted from the material microstructure [5,6]. By experimentally characteriz-ing and statistically representing the material microstructure based on microscopy images, synthetic microstructures, which exhibit the same stochastic features, can be generated [2]. ...
Heterogeneous materials exhibit considerable spatial variations in properties, im-pacting structural performance and local stress and strain fields. Recent research has focused on considering material behaviour uncertainties and quantifying the impact of uncertainties on the structural response, requiring the definition of random fields for describing the material variability. In this paper, a stochastic modeling methodology for additively manufactured structures is implemented (forward model). The stochastic parameters are determined from experimental Digital Image Correlation (DIC) images (inverse problem) using a Convolutional Neural Network (CNN), and the CNN is trained using the forward model. Validation of the CNN estimates using previously unseen data shows adequate performance of the network, and consistent predictions are found when estimating the stochastic parameters from experimental results. The proposed methodology allows examination of the stochastic response and uncertainty quantification of additively manufactured structures, while requiring only minor experimental efforts to fully define the random fields. Once the CNN is trained, computational expense for predicting stochastic parameters is minimal.
Conditions under which samples of continuous stochastic processes X(t) on bounded time intervals [0,τ] can be represented by samples of finite dimensional (FD) processes Xd(t) are augmented such that samples of Slepian models Sd,a(t) of Xd(t) can be used as surrogates for samples of Slepian models Sa(t) of X(t). FD processes are deterministic functions of time and d<∞ random variables. The discrepancy between target and FD samples is quantified by the metric of the space C[0,τ] of continuous functions. The numerical illustrations, which include Gaussian/non-Gaussian FD processes and solutions of linear/nonlinear random vibration problems, are consistent with the theoretical findings in the paper.
Fibre reinforced polymer (FRP) composites have been increasingly applied in engineering structures especially for achieving high demands on structural performance, but they are susceptible to variations in material properties, geometry parameters, etc. mainly arising from manufacturing processes. Due to these uncertainties, FRP structures are usually adopted conservative designs. In order to fully explore the merits of FRP composites, it is therefore of paramount importance to understand and quantify uncertainties in FRP composite structures. Although there are mainly studies on uncertainty quantification for FRP structures, there is a lack of a systematic review. The present study intends to provide a comprehensive review on multiscale uncertainty quantification techniques from the following three aspects: (1) the source of uncertainty: uncertainties induced by manufacturing defects are presented and their influences on the mechanical properties of composite materials are analysed; (2) the prediction of the effective material properties with uncertainties: probabilistic homogenization methods used to propagate uncertainties from microscale to macroscale are described, and influences of microscale uncertainties on the effective material properties are discussed; (3) the probabilistic structural analysis: methods for static analysis, dynamic analysis, buckling stability analysis and reliability analysis with considerations of multiscale uncertainties are presented.
Many problems in solid and geomechanics require the concept of a meso-continuum, which allows a resolution of stress and other dependent fields over scales not infinitely larger than the typical microscale. Passage from the microstructure to such a meso-continuum is based on a scale depen-dent window playing the role of a Representative Volume Element (RVE). It turns out that the material properties at the mesoscale cannot be uniquely approximated by a random field of stiffness with continuous realizations, but, rather, two random continuum fields, corresponding to essential and natural boundary conditions on RVE, need to be introduced to bound the material response from above and from below. In this paper Monte Carlo simulations are used to obtain the first-and second-order one-and two-point characteristics of these two random fields for random chessboards and matrix-inclusion composites. Special focus is on the correlation functions describing the auto-covariances and crosscovariances of effective random meso-scale conductivity tensor Q; and its dual Sjj. Following issues are investigated: i) scale-dependence of noise-to-signal ratios of various components of Cy and Sjj, ii) spatial structure of the correlation function, iii) uniform strain versus exact calculations in determination of the correlation function, iv) correlation structure of compos-ites with inclusions without and with overlap.
The physical properties of polycrystalline two-phase alloys depend on the properties and the amounts of the constituent phases and on the geometrical arrangement of the grains in the two-phase microstucture. Establishing microstructure-property relationships for two-phase materials requires the correct quantitative characterization of all topological features of the microstructure. Stereology and quantitative metallography provide the means to analyse both real and idealized model microstructures with this respect. The two most important quantitative parameters involved in the formulation of microstructure-property relationships are the contiguity C and the fraction of clusters r which quantify the continuity of the phases and to which extent the phases are present as matrix or as inclusion, respectively. Idealized random model microstructures closely approximating real microstructures are generated and joined with continuum and micromechanical models. The essence of the micromechanical model is the unit cell approach combined with finite element calculations. Properties computed for the unit cell are then representative for the overall microstructure. With this method four important physical properties of a ferritic-austenitic stainless duplex steel are modeled successfully: the magnetic permeability, the diffusion of hydrogen, the thermal expansion behavior and the mechanical properties at elevated temperatures. From these examples the relevance of the parameters C and r is evident. Furthermore, the linear rule of mixture is not appropriate to describe both experimentally obtained properties and the results from the numerical analyses for the respective entity.
A generalization of conventional deterministic finite element and difference methods to deal with spatial material fluctuations hinges on the problem of determination of stochastic constitutive laws. This problem is analyzed here through a paradigm of micromechanics of elastic polycrystals and matrix-inclusion composites. Passage to a sought-for random meso-continuum is based on a scale dependent window playing the role of a Representative Volume Element (RVE). It turns out that the microstructure cannot be uniquely approximated by a random field of stiffness with continuous realizations, but, rather, two random continuum fields may be introduced to bound the material response from above and from below. Since the RVE corresponds to a single finite element, or finite difference cell, not infinitely larger than the crystal size, these two random fields are to be used to bound the solution of a given boundary value problem at a given scale of resolution. The window-based random continuum formulation is also employed in analysis of rigid perfectly-plastic materials , whereby the classical method of slip-lines is generalized to a stochastic finite difference scheme. The present paper is complemented by a comparison of this methodology to other existing stochastic solution methods.
We review some recent advances in modeling of elastic composites and polycrystals by spring networks. In the first part, spring network models of anti-plane elasticity, planar classical elasticity, and planar micropolar elasticity are developed. In the second part, applications to progressive breakdown in elastic-brittle matrix-inclusion composites and aluminum sheets are discussed.
Spatial randomness, as opposed to periodic geometries, may have a significant effect on damage formation in composite materials. This issue was studied extensively over the last few years [1, 2, 3, 4], and in this paper we report new results on effects of scale and boundary conditions in the determination of meso-scale continuum-type models for elasticity and fracture. These models are formulated on scales larger than the single inclusion, yet smaller than the conventional continuum limit. The latter corresponds to the classical concept of aRepresentative Volume Element (RVE) which presupposes the presence representation of the microstructure with all the typical microheterogeneities, and thus calls for relatively large volumes. Indeed, according to Hill [5], an RVE should be such that the relations between volume average stress and strain should be the same regardless of whether kinematic or stress boundary conditions have been used.
The present stochastic finite element (SFE) study amplifies a recently developed micromechanically based approach in which two estimates (upper and lower) of the finite element stiffness matrix and of the global response need first to be calculated. These two estimates correspond, respectively, to the principles of stationary potential and complementary energy on which the SFE is based. Both estimates of the stiffness matrix are anisotropic and tend to converge towards one another only in the finite scale limit; this points to the fact that an approximating meso-scale continuum random field is neither unique nor isotropic. The SFE methodology based on this approach is implemented in a Monte Carlo sense for a conductivity (equivalently, out-of-plane elasticity) problem of a matrix-inclusion composite under mixed boundary conditions. Two versions are developed: in one an exact calculation of all the elements' stiffness matrices from the microstructures over the entire finite element mesh is carried out, while in the second one a second-order statistical characterization of the mesoscale continuum random field is used to generate these matrices.
A stochastic finite element method for analysis of effects of spatial variability of material properties is developed with the help of a micromechanics approach. The method is illustrated by evaluating the first and second moments of the global response of a membrane with microstructure of a spatially random inclusion-matrix composite under a deterministic uniformly distributed load. It is shown that two mesoscale random continuum fields have to be introduced to bound the material properties and, in turn, the global response from above and from below. The intrinsic scale dependence of these two random fields is dictated by the choice of the finite element mesh.