Computational nonlinear stochastic homogenization using a nonconcurrent multiscale approach for hyperelastic heterogeneous microstructures analysis

Abstract

This paper is devoted to the computational nonlinear stochastic homogenization of a hyperelastic heterogeneous microstructure using a nonconcurrent multiscale approach. The geometry of the microstructure is random. The nonconcurrent multiscale approach for micro-macro nonlinear mechanics is extended to the stochastic case. Because the nonconcurrent multiscale approach is based on the use of a tensorial decomposition, which is then submitted to the curse of dimensionality, we perform an analysis with respect to the stochastic dimension. The technique uses a database describing the strain energy density function (potential) in both the macroscopic Cauchy green strain space and the geometrical random parameters domain. Each value of the potential is numerically computed by means of the FEM on an elementary cell whose geometry is given by the random parameters and the corresponding macroscopic strains being prescribed as boundary conditions. An interpolation scheme is finally introduced to obtain a continuous explicit form of the potential, which, by derivation, allows to evaluate the macroscopic stress and elastic tangent tensors during the macroscopic structural computations. Two numerical examples are presented.A. CLÉMENT, C. SOIZE AND J. YVONNET concurrent multiscale methods were developed and aim at solving simultaneously the nonlinear problems at both scales (see for instance [6][7][8][9][10][11][12][13]). The macroscopic problem provides the strain states in all integration points and gives the boundary conditions used to solve each microscopic problem. This kind of techniques allows taking into account different types of nonlinearities but may lead to significant computational costs. Alternatively, the nonconcurrent multiscale methods aim at decoupling the numerical simulations at both scales. They consist in identifying parameters characterizing the macroscopic constitutive law from experimental or numerical test [14] or using a database formed of microscopic calculations to build a numerical constitutive law without any assumption on the constitutive equations [15][16][17][18][19]. In this work, we focus on the methodology proposed in [17][18][19].On the other hand, it may also be necessary to take into account the different sources of uncertainty at the microscopic level if one seeks to obtain a reliable model of the macroscopic behavior of the material. Thus, the general area of stochastic multiscale modeling of complex random heterogeneous microstructures has recently attracted a continuously growing attention in the scientific community. A simple way to solve this kind of problem consists in coupling numerical simulations of random microstructures (based on experimental identification or some geometrical characteristics such that the volume fraction or others parameters such that the spatial and size distributions [20,21]) with classical finite element computations. Then, one can perform a statistical study of average quantities of interest characterizing the macroscopic behavior of the material (e.g. s...

Uncertainty quantification in computational stochastic multiscale analysis of nonlinear elastic materials

“…In this section, we detail the nonlinear homogenization scheme applied to hyperelastic heterogeneous materials and we present the deterministic method of Numerical EXplicit Potentials [50,51,52,7] (NEXP) leading to a continuous explicit form of the strain energy density function which characterizes the effective constitutive equations. In the field of homogenization, knowledge on the separation of the scales is vital to perform an appropriate mechanical analysis.…”

“…A great challenge thus comes from the extension of the deterministic methods stated above to the stochastic framework with reasonable computational costs. Based on a novel efficient non-concurrent multiscale approach developed by Yvonnet and coworkers [50,51,52], we have extended this method to the stochastic case in [7]. The so-called Stochastic Numerical EXplicit Potentials method (S-NEXP) aims at numerically determine the apparent strain energy density function according to the large scale strain states and the random variables describing the uncertainties related to the microstructure (geometrical or material parameters).…”

“…A new methodology has been recently introduced to deal with the identification of polynomial chaos representations in high-dimension [39,41]. We propose to use this novel technique in order to obtain a representation of the stochastic nonlinear constitutive equations which can thus be seen as a stochastic non-intrusive technique as opposed to the Stochastic Numerical EXplicit Potentials method [7] which suffers from the classical tensor-product interpolation rules since it acts as an intrusive technique. Then, we based our approach on the same nonlinear homogenization scheme presented in [7] but the methodology proposed to characterize the stochastic apparent nonlinear constitutive equations is totally different.…”

“…We propose to use this novel technique in order to obtain a representation of the stochastic nonlinear constitutive equations which can thus be seen as a stochastic non-intrusive technique as opposed to the Stochastic Numerical EXplicit Potentials method [7] which suffers from the classical tensor-product interpolation rules since it acts as an intrusive technique. Then, we based our approach on the same nonlinear homogenization scheme presented in [7] but the methodology proposed to characterize the stochastic apparent nonlinear constitutive equations is totally different. Indeed, we use the NEXP approach [52] as a deterministic solver, which is not directly extended to the stochastic framework, and we reformulate the problem into the identification of polynomial chaos expansions in high-dimension.…”

A stochastic computational multiscale approach; Application to MEMS resonators

“…In [26], the simulations of SVEs are used to capture the stochastic properties of the parameters in the constitutive model with which the uncertainties are propagated to the structural scale using the stochastic finite element method [27]. In the case of finite elasticity, the resolution of composite material elementary cells is used to explicitly define a meso-scale potential with the aim of studying the uncertainties in the fibers geometry/distribution [28].…”