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Mathematical Problems in Engineering
Volume 2018 (2018), Article ID 3081078, 11 pages
Research Article

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

1Department of Aerospace Engineering, Mississippi State University, P.O. Box A, Mississippi State, MS 39762, USA
2Multiscale and Multiphysics Modeling Branch, Materials and Structures Division, NASA Glenn Research Center, 21000 Brookpark Rd., MS 49-7, Cleveland, OH 44135, USA
3BAM 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.


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 4.8–6.3x 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.