A numerical study of different projection-based model reduction techniques applied to computational homogenisation

Soldner, D., Brands, B., Zabihyan, R., Steinmann, P. and Mergheim, J. (2017) A numerical study of different projection-based model reduction techniques applied to computational homogenisation. Computational Mechanics, 60(4), pp. 613-625. (doi: 10.1007/s00466-017-1428-x) (PMID:31258232) (PMCID:PMC6560488)

Abstract

Computing the macroscopic material response of a continuum body commonly involves the formulation of a phenomenological constitutive model. However, the response is mainly influenced by the heterogeneous microstructure. Computational homogenisation can be used to determine the constitutive behaviour on the macro-scale by solving a boundary value problem at the micro-scale for every so-called macroscopic material point within a nested solution scheme. Hence, this procedure requires the repeated solution of similar microscopic boundary value problems. To reduce the computational cost, model order reduction techniques can be applied. An important aspect thereby is the robustness of the obtained reduced model. Within this study reduced-order modelling (ROM) for the geometrically nonlinear case using hyperelastic materials is applied for the boundary value problem on the micro-scale. This involves the Proper Orthogonal Decomposition (POD) for the primary unknown and hyper-reduction methods for the arising nonlinearity. Therein three methods for hyper-reduction, differing in how the nonlinearity is approximated and the subsequent projection, are compared in terms of accuracy and robustness. Introducing interpolation or Gappy-POD based approximations may not preserve the symmetry of the system tangent, rendering the widely used Galerkin projection sub-optimal. Hence, a different projection related to a Gauss-Newton scheme (Gauss-Newton with Approximated Tensors- GNAT) is favoured to obtain an optimal projection and a robust reduced model.

• Item Type: Articles
• Additional Information: The first and fifth author gratefully acknowledge the financial support by the German Science Foundation (DFG) within the Collaborative Research Center 814: Additive Manufacturing (subproject C3). The second, third and fourth author gratefully acknowledge the financial support by the ERC Advanced Grant MOCOPOLY.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Steinmann, Professor Paul
• Authors: Soldner, D., Brands, B., Zabihyan, R., Steinmann, P., and Mergheim, J.
• College/School: College of Science and Engineering > School of Engineering > Infrastructure and Environment
• Journal Name: Computational Mechanics
• Publisher: Springer
• ISSN: 0178-7675
• ISSN (Online): 1432-0924
• Published Online: 08 June 2017
• Copyright Holders: Copyright © 2017 The Authors
• First Published: First published in Computational Mechanics 60:613-625
• Publisher Policy: Reproduced under a Creative Commons License