Johann Guilleminot

Stochastic model and generator for random fields with symmetry properties: application to the mesoscopic modeling of elastic random media

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

Itô SDE-based generator for a class of non-Gaussian vector-valued random fields in uncertainty quantification

\mathbb{M}_n^{sym}(\mathbb{R})

Stochastic hyperelastic constitutive laws and identification procedure for soft biological tissues with intrinsic variability.

of the set of symmetric positive-definite real matrices. First, 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...

Approximate Solutions of Lagrange Multipliers for Information-Theoretic Random Field Models

This paper is concerned with the derivation of a generic sampling technique for a class of non-Gaussian vector-valued random fields. Such an issue typically arises in uncertainty quantification for complex systems, where the input coefficients associated with the elliptic operators must be identified by solving statistical inverse problems. Specifically, we consider the case of non-Gaussian random fields with values in some arbitrary bounded or semibounded subsets of

Multiscale prediction of acoustic properties for glass wools: Computational study and experimental validation

\mathbb{R}^n

Stochastic representations and statistical inverse identification for uncertainty quantification in computational mechanics

. The approach involves two main features. The first is the construction of a family of random fields converging, at a user-controlled rate, toward the target random field. Each of these auxialiary random fields can be subsequently simulated by solving a family of Ito stochastic differential equations. The second ingredient is the definition of an adaptive discretization algorithm. The latter allows refining the integration step on-the-fly and prevents the scheme from diverging. The proposed strategy is finally exemplified on th...

A Stochastic Model for Elasticity Tensors Exhibiting Uncertainties on Material Symmetries

In this work, we address the constitutive modeling, in a probabilistic framework, of the hyperelastic response of soft biological tissues. The aim is on the one hand to mimic the mean behavior and variability that are typically encountered in the experimental characterization of such materials, and on the other hand to derive mathematical models that are almost surely consistent with the theory of nonlinear elasticity. Towards this goal, we invoke information theory and discuss a stochastic model relying on a low-dimensional parametrization. We subsequently propose a two-step methodology allowing for the calibration of the model using standard data, such as mean and standard deviation values along a given loading path. The framework is finally applied and benchmarked on three experimental databases proposed elsewhere in the literature. It is shown that the stochastic model allows experiments to be accurately reproduced, regardless of the tissue under consideration.

A probabilistic model for bounded elasticity tensor random fields with application to polycrystalline microstructures

This work is concerned with the construction of approximate solutions for the Lagrange multipliers involved in information-theoretic non-Gaussian random field models. Specifically, representations of physical fields with invariance properties under some orthogonal transformations are considered. A methodology for solving the optimization problems raised by entropy maximization (for the family of first-order marginal probability distributions) is first presented and exemplified in the case of elasticity fields exhibiting fluctuations in a given symmetry class. Results for all classes ranging from isotropy to orthotropy are provided and discussed. The derivations are subsequently used for proving a few properties that are required in order to sample the above models by solving a family of stochastic differential equations---along the lines of the algorithm constructed in [J. Guilleminot and C. Soize, Multiscale Model. Simul., 11 (2013), pp. 840--870]. The results thus allow for forward simulations of the pr...

On the Statistical Dependence for the Components of Random Elasticity Tensors Exhibiting Material Symmetry Properties

This work is concerned with the multiscale prediction of the transport and sound absorption properties associated with industrial glass wool samples. In the first step, an experimental characterization is performed on various products using optical granulometry and porosity measurements. A morphological analysis, based on scanning electron imaging, is further conducted to identify the probability density functions associated with the fiber angular orientation. The key morphological characterization parameters of the microstructure, which serve as input parameters of the model, include the porosity, the weighted volume diameter accounting for both lengths and diameters of the analyzed fibers (and therefore the specific surface area of the random fibrous material), and the preferred out-of-plane fiber orientation generated by the manufacturing process. A computational framework is subsequently proposed and allows for the reconstruction of an equivalent fibrous network. A fully stochastic microstructural model, parameterized by the probability laws inferred from the database, is also proposed herein. Multiscale simulations are carried out to estimate transport properties and sound absorption. With no adjustable parameter, the results accounting for ten different samples obtained with various processing parameters are finally compared with the experimental data and used to assess the relevance of the reconstruction procedures and the multiscale computations.

Theoretical framework and experimental procedure for modelling mesoscopic volume fraction stochastic fluctuations in fiber reinforced composites

The paper deals with the statistical inverse problem for the identification of a non-Gaussian tensor-valued random field in high stochastic dimension. Such a random field can represent the parameter of a boundary value problem (BVP). The available experimental data, which correspond to observations, can be partial and limited. A general methodology and some algorithms are presented including some adapted stochastic representations for the non-Gaussian tensor-valued random fields and some ensembles of prior algebraic stochastic models for such random fields and the corresponding generators. Three illustrations are presented: (i) the stochastic modeling and the identification of track irregularities for dynamics of high-speed trains, (ii) a stochastic continuum modeling of random interphases from atomistic simulations for a polymer nanocomposite, and (iii) a multiscale experimental identification of the stochastic model of a heterogeneous random medium at mesoscale for mechanical characterization of a human cortical bone.