In 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.
https://www.sciencedirect.com/science/article/abs/pii/S0377042719301785?via%3Dihub
