INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2015; 104:1061–1084 Published online 1 June 2015 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.4953 Computational homogenization of nonlinear elastic materials using neural networks B. A. Le, J. Yvonnet * ,† and Q.-C. He Université Paris-Est, Laboratoire Modélisation et Simulation Multi Echelle, MSME UMR 8208 CNRS, 5 Bd Descartes, 77454 Marne-la-Vallée Cedex 2, France SUMMARY In this work, a decoupled computational homogenization method for nonlinear elastic materials is proposed using neural networks. In this method, the effective potential is represented as a response surface parame- terized by the macroscopic strains and some microstructural parameters. The discrete values of the effective potential are computed by finite element method through random sampling in the parameter space, and neural networks are used to approximate the surface response and to derive the macroscopic stress and tan- gent tensor components. We show through several numerical convergence analyses that smooth functions can be efficiently evaluated in parameter spaces with dimension up to 10, allowing to consider three-dimensional representative volume elements and an explicit dependence of the effective behavior on microstructural parameters like volume fraction. We present several applications of this technique to the homogenization of nonlinear elastic composites, involving a two-scale example of heterogeneous structure with graded nonlinear properties. Copyright © 2015 John Wiley & Sons, Ltd. Received 6 November 2014; Revised 11 February 2015; Accepted 7 May 2015 KEY WORDS: neural networks; high-dimensional approximation; computational homogenization; nonlin- ear homogenization; multiscale methods 1. INTRODUCTION Computational homogenization is a powerful tool to estimate the response of structures made of heterogeneous, nonlinear materials. Starting from the knowledge of the microstructure (phase morphology, nonlinear constitutive behavior of each phase), the objective is to estimate numerically the effective behavior of the equivalent homogeneous material, so as to obtain the strain–stress relationship at the macroscopic scale. As compared with the linear case, where the effective behav- ior can be estimated from a limited number of elementary computations on a representative volume element (RVE), nonlinear homogenization is a much tougher problem as no general analytical form of the effective behavior can be assumed in the general case, except for some limited problems where specific assumptions are made on the geometry of the microstructure and the behavior of the different phases. Several classes of computational homogenization methods have been proposed in the past for nonlinear materials. The first class includes the so-called ‘multilevel’ computational homogeniza- tion, also referred to in the literature as ‘FE 2 ’ method [1–4]. In such techniques, the macroscopic stress is simply obtained at every point of a structure by solving a nonlinear problem on the corresponding RVE, for the boundary conditions given by the strain state at the same point of the structure. Iterations are then needed to balance equations at both microscale and macroscale. The advantages of these approaches can be summarized as follows: (1) the constitutive behavior of local phases can be arbitrary; (2) history effects can be considered; (3) microstructure evolution can *Correspondence to: Julien Yvonnet, Université Paris-Est, 5 Bd Descartes, 77454 Marne-la-Vallée Cedex, France. † E-mail: Julien.yvonnet@univ-paris-est.fr Copyright © 2015 John Wiley & Sons, Ltd.