Generalized multiscale finite element method for a strain-limiting nonlinear elasticity model
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AbstractIn this paper, we consider multiscale methods for nonlinear elasticity. In particular, we investigate the Generalized Multiscale Finite Element Method (GMsFEM) for a strain-limiting elasticity problem. Being a special case of the naturally implicit constitutive theory of nonlinear elasticity, strain-limiting relation has presented an interesting class of material bodies, for which strains remain bounded (even infinitesimal) while stresses can become arbitrarily large. The nonlinearity and material heterogeneities can create multiscale features in the solution, and multiscale methods are therefore necessary. To handle the resulting nonlinear monotone quasilinear elliptic equation, we use linearization based on the Picard iteration. We consider two types of basis functions, offline and online basis functions, following the general framework of GMsFEM. The offline basis functions depend nonlinearly on the solution. Thus, we design an indicator function and we will recompute the offline basis functions when the indicator function predicts that the material property has significant change during the iterations. On the other hand, we will use the residual based online basis functions to reduce the error substantially when updating basis functions is necessary. Our numerical results show that the above combination of offline and online basis functions is able to give accurate solutions with only a few basis functions per each coarse region and adaptive updating basis functions in selected iterations.
All Author(s) ListShubin Fu, Eric Chung, Tina Mai
Journal nameJournal of Computational and Applied Mathematics
Year2019
Month10
Day15
Volume Number359
Pages153 - 165
ISSN0377-0427
eISSN1879-1778
LanguagesEnglish-United States

Last updated on 2020-05-08 at 01:22