Generalized multiscale finite element method for a strain-limiting nonlinear elasticity model

Abstract

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 updating basis functions in selected iterations.

Constraint energy minimizing generalized multiscale finite element method for nonlinear poroelasticity and elasticity

“…Thanks to [7], we derive the following results, which were also stated in [8] (p. 19) and proved in our recent GMsFEM paper [18].…”

“…For the sake of simplicity, the case d = 2 is considered here. We refer the readers to our previous paper [18] for more details about the strain-limiting nonlinear elasticity model. We now consider an arising nonlinear poroelasticity system, where the unknowns are displacement u : Ω × [0, T ] and pressure p : Ω × [0, T ] satisfying…”

“…Function spaces. We refer the readers to [14,18] for the preliminaries. Latin indices are in the set {1, 2}.…”

“…As an interesting case of the considered nonlinear poroelasticity, a static strain-limiting nonlinear elasticity model (as in [18]) is also investigated by similar strategy and via the CEM-GMsFEM. To take into consideration the influence of source and global information, as in the linear elasticity case ( [19]), we use more efficient residual based online basis functions (via adaptive enrichment procedure [10]), which are not used in the poroelasticity case (where only offline multiscale basis functions are applied).…”

Generalized multiscale finite element method for highly heterogeneous compressible flow

“…To further improve the performances of GMsFEM, residual driven online multiscale basis functions were proposed in [17], these multiscale basis contains local and global media information and source information, which significantly facilitate convergence of the multiscale method. It is observed for time dependent problems or nonlinear problems, one can reuse the residual driven multiscale basis functions computed at certain time step or iteration [14,25,24,41,42], which tremendously increase the accuracy of the GMsFEM solution compared with only using equal numbers of offline basis functions.…”