A multiscale method for periodic structures using domain decomposition and ECM-hyperreduction

Abstract

This paper presents a nonlinear multiscale approach for periodic structures in the quasi-static, small strain regime. The approach consists in combining a domain decomposition method in which interface conditions are established through a “fictitious” frame with reduced-order modeling (ROM). We propose to approximate interface displacements, subdomain displacements and Lagrange Multipliers as linear combinations of reduced sets of dominant modes, and to assign to the coefficients of such linear combinations the role of coarse-scale displacements, strains and stresses, respectively. We discuss and propose a method based on subspace rotations that ensures that these three modal approximations lead to stable formulations. The modes of such expansions are determined in an offline “training” stage by applying the truncated Singular Value Decomposition (SVD) to the solutions obtained from Finite Element (FE) analyses. In contrast to other multiscale approaches, such as FE-ROM homogenization schemes, which uses a single unit cell with periodic conditions, here the “training” structures are formed by several unit cells. This way, we allow the SVD to extract also dominant patterns of interaction between subdomains in terms of reactive forces and deformations. This original feature confers to the proposed approach three unique advantages over FE-ROM homogenization schemes, namely: (1) It can deal with unit cells of arbitrary size (no need for scale separation). (2) It can model, not only how forces are transmitted through the structure, but also local effects. (3) It can also handle domains which are only periodic along one or two directions (beam-like or shell structures, respectively). To deal with material nonlinearities, element-wise Gauss integration of reduced internal forces is replaced by an algorithmically improved version of the hyperreduction scheme recently proposed by the author elsewhere, called the Empirical Cubature Method(ECM). Furthermore, we demonstrate that the coarse-scale Degrees of Freedom (DOFs) of a subdomain can be expressed directly in terms of the (fine-scale) stresses and strains at the ECM integration points. This feature dispenses with the need of deploying a special algorithmic infrastructure of intertwined local/global problems, as it occurs in FE-ROM schemes, since the unit cell can be treated as a special type of finite element, in which the centroids of the interfaces and the ECM points play the role of nodes and Gauss points, respectively. To illustrate all these advantages, we present 4 distinct examples: two beam-like structures of rectangular-shape and I-shaped cross-sections, a 2D hexagonal cellular material, and a cylindrical shell made of a porous composite cell. In all 4 cases the proposed method is able to produce coarse-scale models reducing the number of DOFs and integration points by over two orders of magnitude, with errors below 5 %. Interestingly, in the case of the beam-like structures, we show that the method provides 2-node beam finite elements whose kinematics are identical to that predicted by “analytical” beam theories (6 DOFs per node in the case of the rectangular beam), and totally consistent with their 3D full-order counterparts.