Abstract
A stochastic data-driven multilevel finite-element (FE) method is introduced for random nonlinear multiscale calculations. A hybrid neural-network–interpolation (NN–I) scheme is proposed to construct a surrogate model of the macroscopic nonlinear constitutive law from representative-volume-element calculations, whose results are used as input data. Then, a FE method replacing the nonlinear multiscale calculations by the NN–I is developed. The NN–I scheme improved the accuracy of the neural-network surrogate model when insufficient data were available. Due to the achieved reduction in computational time, which was several orders of magnitude less than that to direct FE, the use of such a machine-learning method is demonstrated for performing Monte Carlo simulations in nonlinear heterogeneous structures and propagating uncertainties in this context, and the identification of probabilistic models at the macroscale on some quantities of interest. Applications to nonlinear electric conduction in graphene–polymer composites are presented.
Highlights
Predicting the nonlinear behavior of materials from knowledge of their microstructure is a critical topic in engineering
For one realization of the volume-fraction distribution generated by Equation (30), the cost of one two-scale nonlinear structural problem is drastically reduced with the present NN surrogate model, allowing for performing a large number of macrocalculations at a low cost to conduct statistics on quantities of interest in a structure
A hybrid neural-network–interpolation (NN–I) scheme was developed to improve the accuracy of NN surrogate models, allowing for the use of a lower number of representative volume element (RVE) nonlinear calculations, which serve as a database to train the neural networks
Summary
Predicting the nonlinear behavior of materials from knowledge of their microstructure is a critical topic in engineering. To more accurately describe the behavior of general nonlinear materials, the so-called multilevel finite-element (FE2) method [9,10,11,12,13,14,15,16] or computational homogenization has been developed in recent years In this approach, an RVE is associated to each Gaussian point of a finiteelement macrostructure, and a nonlinear problem must be solved in each integration point and for each iteration of the macrosolving procedure. A new hybrid neural-network–interpolation (NN–I) surrogate model is proposed to provide an accurate response with a limited number of realizations Once constructed, this model can be used within stochastic analysis of two-scale nonlinear structure calculations. FE2 calculations can be reduced by several orders of magnitude, allowing for Monte Carlo simulation on stochastic nonlinear multiscale structures It is demonstrated for the first time that uncertainties can be propagated in this context, and probabilistic models can be identified.
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