Mathematical Problems in Engineering

Volume 2018 (2018), Article ID 3081078, 11 pages

https://doi.org/10.1155/2018/3081078

## Solution of the Nonlinear High-Fidelity Generalized Method of Cells Micromechanics Relations via Order-Reduction Techniques

^{1}Department of Aerospace Engineering, Mississippi State University, P.O. Box A, Mississippi State, MS 39762, USA^{2}Multiscale and Multiphysics Modeling Branch, Materials and Structures Division, NASA Glenn Research Center, 21000 Brookpark Rd., MS 49-7, Cleveland, OH 44135, USA^{3}BAM Federal Institute for Material Research and Testing, Unter den Eichen 87, 12205 Berlin, Germany

Correspondence should be addressed to Trenton M. Ricks

Received 17 November 2017; Accepted 29 January 2018; Published 28 February 2018

Academic Editor: Jose Merodio

Copyright © 2018 Trenton M. Ricks et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The High-Fidelity Generalized Method of Cells (HFGMC) is one technique, distinct from traditional finite-element approaches, for accurately simulating nonlinear composite material behavior. In this work, the HFGMC global system of equations for doubly periodic repeating unit cells with nonlinear constituents has been reduced in size through the novel application of a Petrov-Galerkin Proper Orthogonal Decomposition order-reduction scheme in order to improve its computational efficiency. Order-reduced models of an E-glass/Nylon 12 composite led to a 4.8–6.3*x* speedup in the equation assembly/solution runtime while maintaining model accuracy. This corresponded to a 21–38% reduction in total runtime. The significant difference in assembly/solution and total runtimes was attributed to the evaluation of integration point inelastic field quantities; this step was identical between the unreduced and order-reduced models. Nonetheless, order-reduced techniques offer the potential to significantly improve the computational efficiency of multiscale calculations.

#### 1. Introduction

The High-Fidelity Generalized Method of Cells (HFGMC) is a micromechanics technique that can be used to simulate nonlinear composite materials [1]. The core computational effort of this method involves repeatedly finding the solution to sets of simultaneous linear algebraic equations in order to determine local/global field quantities and effective properties for heterogeneous materials with a periodic microstructure. However, when material nonlinearity is admitted, the computational runtimes can become excessive, particularly as the problem size is increased due to a more detailed microstructural representation. Nonlinear analyses of such detailed, high-fidelity repeating unit cells (RUCs) are needed to accurately simulate realistic composite microstructures necessary for process modeling, prediction of residual stress states, progressive failure analysis, and other computational predictions that depend heavily on subscale features. The use of order-reduction techniques is one possibility of improving the computational efficiency of high-fidelity analyses. Furthermore, although HFGMC is fundamentally distinct and more computationally efficient than traditional finite-element (FE) approaches [1], both methods are relatively inefficient for multiscale simulations of realistic composite microstructures.

Proper Orthogonal Decomposition (POD) [2, 3] and Proper Generalized Decomposition (PGD) [4, 5] are two commonly used order-reduction approaches. In order to generate an order-reduced model using POD, the full solution to a particular problem (often found by solving a set of simultaneous equations) must be known a priori. If this solution cannot be practically obtained due to model size or computational limits, PGD can be used to generate an order-reduced model. However, for most solid mechanics problems of interest, a priori solutions can be easily obtained. More detailed information on PGD can be found in the review article by Chinesta et al. [4]. In this study, a POD approach was used due to its wide use in the literature and ease of implementation.

A significant number of FE studies have employed POD to generate order-reduced models that reduce the dimensionality of the ensuing large set of simultaneous equations. The goal of POD is to generate a set of basis vectors capable of capturing the dominant components of a system, optimally represent a full set of equations, and provide a mapping relationship between the unreduced and order-reduced domains. In this context, an order-reduced POD approach has two main components: (i) approximation of the solution to a set of equations and (ii) projection to the order-reduced domain. In general, FE-based POD techniques employ Galerkin projection (i.e., the projection is performed with the same set of basis vectors used for approximation). Carlberg et al. [6] noted that Galerkin projection may not be optimal in the presence of nonlinearity and can lead to computational instabilities. A more complex Petrov-Galerkin POD method was developed to overcome these limitations by modifying the form of the projection at the cost of some added calculations [6].

While POD-based order-reduction techniques have been commonly used to solve problems in computational fluid dynamics [7–9], these techniques have also been extended to include nonlinear solid mechanics problems [6, 10–14]. For instance, Radermacher et al. [10] were able to demonstrate improvements of the computational speed by a factor of 60–260 by employing a POD-based order-reduction technique in the analysis of an inelastic metal matrix composite. POD techniques have also been implemented within a multiscale framework. Multiscale methods are often based on an FE^{2} [15, 16] modeling approach, wherein a microscale FE model is called at each integration point within a macroscale FE model. Yvonnet and He [13] were able to achieve significant computational and memory savings for multiscale simulations of hyperelastic media. Radermacher et al. [10] demonstrated two orders of magnitude speedup in the computational time of nonlinear multiscale simulations by implementing POD at the microscale. Similarly, Ricks et al. [17] obtained significant computational savings by imbedding HFGMC within a macroscale linearly elastic FE model.

Several authors have also proposed methods to modify/update the original set of basis vectors in order to achieve better computational performance. Hernández et al. [12] formed a set of basis vectors by accounting for all elastic modes and only the essential inelastic modes. Ryckelynck [18] developed a procedure to adaptively update the subspace spanned by the original set of basis vectors during an analysis. Additional computational savings were achieved by using only a subset of the FEs to control the adaptive process [18]. This “hyperreduction” approach is similar in concept to the discrete empirical interpolation method [19] and gappy POD technique [20, 21]. Kerfriden et al. [14] proposed updating the original set of basis vectors using appropriately normalized unconverged/converged iterative solutions.

In the present work, the HFGMC global system of equations for doubly periodic RUCs with nonlinear constituents is reduced in size through the use of POD. This approach was previously shown to yield significant computational savings when applied to the HFGMC equations for linearly elastic materials only [17]. The order-reduced HFGMC models are then compared to the traditional HFGMC approach for multiple RUC discretizations in order to assess their accuracy and computational efficiency.

#### 2. High-Fidelity Generalized Method of Cells (HFGMC)

The HFGMC is a micromechanics technique used for modeling heterogeneous materials [1] and is an adaptation of classical homogenization theory [22–24]. The HFGMC has been extensively validated and shown to accurately calculate both effective properties and thermoinelastic material behavior for a wide range of composite material systems (see [1] for a partial summary). In contrast to the generalized method of cells [1], the HFGMC gives a higher accuracy in the subcell stress/strain fields, at an increased computational cost, by employing a higher-order subcell displacement field. Using the HFGMC, a doubly periodic RUC is discretized into an arbitrary number of subcells (see Figure 1). A doubly periodic RUC may be defined in the plane and is discretized into and subcells along the -direction (height) and the -direction (width), respectively, while any inhomogeneities/inclusions (e.g., fibers) extend infinitely in the -direction (length). A local coordinate system may be defined relatively to the centroid of each subcell. The height and length of each subcell are given by and , respectively. The discussion that follows presents key aspects of the HFGMC formulation that are relevant to this study. An exhaustive derivation of the HFGMC can be found in [1].