Stochastic Model and Generator for Random Fields with Symmetry Properties: Application to the Mesoscopic Modeling of Elastic Random Media

Abstract

Abstract. This paper is concerned with the construction of a new class of generalized nonparametric probabilistic models for matrix-valued non-Gaussian random fields. More specifically, we consider the case where the random field may take its values in some subset of the set of real symmetric positive-definite matrices presenting sparsity and invariance with respect to given orthogonal transformations. Within the context of linear elasticity, this situation is typically faced in the multiscale analysis of heterogeneous microstructures, where the constitutive elasticity matrices may exhibit some material symmetry properties and may then belong to a given subset M sym n (R) of the set of symmetric positive-definite real matrices. First of all, we present an overall methodology relying on the framework of information theory and define a particular algebraic form for the random field. The representation involves two independent sources of uncertainties, namely one preserving almost surely the topological structure in M sym n (R) and the other one acting as a fully anisotropic stochastic germ. Such a parametrization does offer some flexibility for forward simulations and inverse identification by uncoupling the level of statistical fluctuations of the random field and the level of fluctuations associated with a stochastic measure of anisotropy. A novel numerical strategy for random generation is subsequently proposed and consists in solving a family of Itô stochastic differential equations. The algorithm turns out to be very efficient when the stochastic dimension increases and allows for the preservation of the statistical dependence between the components of the simulated random variables. A Störmer-Verlet algorithm is used for the discretization of the stochastic differential equation. The approach is finally exemplified by considering the class of almost isotropic random tensors.

Design optimization under uncertainties of a mesoscale implant in biological tissues using a probabilistic learning algorithm

“…This model has been used for modeling and identifying cortical bone in [61,63]. Note that extensions of this model can be found in [30,31,86] for some positive-definite lower and upper bounds introduced as constraints and also for random fields with any material symmetry property. Let [K] = E{[K(x)]} be the mean matrix in M + n , which corresponds to the positive-definite fourth-order tensor…”

Stochastic Representations and Statistical Inverse Identification for Uncertainty Quantification in Computational Mechanics

“…All the details of this development can be found in [48,49] concerning the stochastic model for the anisotropic statistical fluctuations. The case for which dominant statistical fluctuations belong to the isotropic class with anisotropic statistical fluctuations has been introduced in [65] and has been generalized to all the symmetry classes in [56] and included references. A very general theory is then proposed in [56,58,45,44] for the general case for which there are simultaneously some statistical fluctuations in a given symmetry class and statistical fluctuations in the anisotropic class.…”

“…A symmetry class, induced by a material symmetry, is defined as a subset M sym n (R) of M + n (R), whose algebraic dimension is n s ≤ n(n + 1)/2 (see its construction in [56] …”

“…in which {[M(x)], x ∈ Ω} is a non-Gaussian M sym (R)-valued random field that models the dominant statistical fluctuations in the symmetry class defined by M sym (R) (see its construction and its generator in [56]), where {[G 0 (x)], x ∈ Ω} is a non-Gaussian M + n (R)-valued random field that models the anisotropic statistical fluctuations around the symmetry class (see its construction and its generator in [48,49]). The random fields {[M(x)], x ∈ Ω} and {[G 0 (x)], x ∈ Ω} are statistically independent.…”

“…An important step in the methodology is the construction and the use of algebraic prior stochastic models (APSM) of such a nonGaussian random field for which some advanced generators have been developed. The APSM and the associated generators presented and used hereinafter are those that have been published in [44,45,48,49,50,51,52,53,54,55,40,56,57,58]. Three illustrations are presented: the stochastic modeling of track irregularities for high-speed trains and its experimental identification [59], the stochastic continuum modeling of random interphases from atomistic simulations for a polymer nanocomposite [60], and the multiscale identification of the random elasticity field at mesoscale of a heterogeneous microstructure using multiscale experimental observations [61,62,63].…”