**1. Introduction**

Predicting the nonlinear behavior of materials from knowledge of their microstructure is a critical topic in engineering. For example, the development of 3D-printed micromaterials [1–3] or of nanomaterials [4,5] with nonlinear behaviors opens exciting opportunities for designing innovative functionalized and enhanced engineering systems. While linear effective properties of heterogeneous materials can be accurately estimated though either analytical [6,7] or numerical techniques [8], predicting the effective behavior of nonlinear materials requires more advanced techniques.

A direct but limited approach is the use of the representative volume element (RVE) to calibrate an empirical nonlinear model. A limitation of such techniques is the number of parameters to be calibrated for complex, nonlinear, or multiphysics problems. To more accurately describe the behavior of general nonlinear materials, the so-called multilevel finite-element (FE<sup>2</sup> ) method [9–16] 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. The drawback of this method, however, is that it induces unaffordable computational times in practical applications.

Several strategies were developed recently to alleviate FE<sup>2</sup> calculations. First, the strategy relies on reducing micro-RVE computations through efficient techniques such as model-order reduction [17,18], fast Fourier transform [19,20], wavelet transforms [21], NTFA [22], self-clustering analysis (SCA) [23,24], or GPU acceleration [25]. In [26], He et al. developed an adaptive strategy to reduce microcalculations by constructing the reduced basis on the fly during the macroscale calculation.

**Citation:** Lu, X.; Yvonnet, J.; Papadopoulos, L.; Kalogeris, I., Papadopoulos, V. A Stochastic FE<sup>2</sup> Data-Driven Method for Nonlinear Multiscale Modeling. *Materials* **2021**, *14*, 2875. https://doi.org/10.3390/ ma14112875

Received: 30 March 2021 Accepted: 21 May 2021 Published: 27 May 2021

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Another idea, initiated in [27,28], is the use of so-called data-driven approaches in which microscale calculations are performed offline, and are then used as data in an online stage to reconstruct the macroscopic (effective) behavior. For this purpose, several techniques were proposed, including interpolation methods [27,29], neural networks [28,30–35], Bayesian inference [36], Fourier series expansions [37], or Gaussian process regression [38]. In the related techniques, offline data collection is used in a regression process to construct an accurate surrogate model whose evaluation is several orders of magnitude lower than that performing one RVE nonlinear calculation. A critical comparison of several regression techniques used in data-driven multiscale approaches can be found in [39]. In [40], Avery et al. investigated and discussed several regression methods with ANN in homogenization problems of hyperelastic woven composites, and demonstrate its use in advanced dynamic or fluid structure applications. Recent advances of datadriven techniques, including handling history-dependent behaviors such as plasticity, can be found in [35,41,42]. On-the-fly construction of the surrogate model by probabilistic machine learning was proposed in [38]. Developments of neural-network techniques in FE<sup>2</sup> , including feed-forward and recurrent neural networks, can be found in [31,41]. In [43,44], a manifold-based nonlinear reduced-order model in tandem with a digital database was developed for the nonlinear multiscale analysis of hyperelastic structures involving neural networks, a kernel inverse/reconstruction map, and dimension reduction through an isomap.

Stochastic extensions of data-driven methods in multiscale applications are relatively new and unexplored. One of the first analyses in this context can be found in [45,46], where the NEXP method [27] was extended to stochastic problems. In these studies, stochastic parameters were introduced within the surrogate model using a separated representation-interpolation technique. Probability density functions related to the nonlinear macroscale problem were identified. In [47], a machine-learning strategy based on a three-dimensional convolutional neural network was introduced to evaluate the linear effective properties of random materials from geometrical descriptions of RVE. In [24], a framework for uncertainty quantification in a data-driven approach was proposed where self-consistent clustering analysis (SCA) [23,24] was used to reduce computational times in the learning step.

In this paper, the use of data-driven methods for heterogeneous nonlinear materials with uncertainties at both the micro- and the macroscale is addressed. Taking into account uncertainties in nonlinear multiscale methods implies (a) constructing a probabilistic surrogate macromodel from microcalculations, allowing for generating realizations of the macroresponse for a given macroloading; and (b) performing Monte Carlo simulations of the model at the macroscale to quantify uncertainties on the quantities of interest in the structure. In view of its immense computational requirements, direct use of FE<sup>2</sup> for stochastic nonlinear two-scale analysis is not possible. However, data-driven FE<sup>2</sup> approaches have comparable computational costs as compared to classical (one-scale) FEM calculations, and they open the route to developing stochastic two-scale nonlinear approaches. To the best of our knowledge, this problem remains relatively unexplored in the literature. A new stochastic data-driven approach based on RVE calculations was developed for taking into account random effects in nonlinear heterogeneous structures. First, preliminary RVE calculations were performed. These calculations include several microstructural features that varied, such as the distribution of heterogeneities and its volume fraction. Then, for each realization of the random microstructure, the space of macroscopic loading was sampled, and boundary conditions were prescribed on the RVE. Subsequently, the nonlinear problem was solved by FEM. This large database was used to construct a surrogate model whose inputs were the macroloading and the volume fraction, and its output was the macroscopic (homogenized) response. 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. At the macroscale,

the volume fraction of heterogeneities is considered to be random here, and it was modeled as a stochastic field with given probabilistic characteristics. Then, during the macro-nonlinear resolution, solving the full nonlinear RVE was replaced by the proposed fast surrogate model, which allowed for performing hundreds of macro-non-linear calculations at the cost of classical FEM problems. As a result, statistical postprocessing can be performed on the macroquantities of interest, and probabilistic models could be identified.

The novelties of this paper are twofold. The first is the proposed neural-network– Interpolation FE<sup>2</sup> method, which is an extension of our previous neural-network FE2 method, developed in [28,30]. The NN–I scheme allows for modeling the stochastic spatial variability of the volume fraction in the frame of the FE<sup>2</sup> procedure, leading to the improved accuracy of the surrogate model when limited data are available. The second novelty is the application of this machine-learning method to nonlinear multiscale stochastic problems. Using the proposed approach, FE<sup>2</sup> 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.

The paper is organized as follows. Section 3 presents the equations of the nonlinear RVE problem, and the definitions of the input (macroelectric load) and output (homogenized electric flux) in the nonlinear composite. Section 4 introduces the hybrid neural network/interpolation scheme, and its construction using offline data on RVE is described. In Section 5, the present stochastic data-driven strategy is proposed. Lastly, numerical examples are presented in Section 6.
