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

## TL;DR

This paper develops a multiscale finite element method for nonlinear elasticity with strain-limiting behavior, efficiently capturing multiscale features and nonlinearities through offline and online basis functions.

## Contribution

It introduces a GMsFEM framework tailored for strain-limiting nonlinear elasticity, incorporating adaptive basis function updates based on material property changes.

## Key findings

- Accurate solutions with few basis functions per coarse region.
- Adaptive basis updates improve solution accuracy.
- Method effectively handles multiscale nonlinear elasticity problems.

## 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.