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

TL;DR

This paper develops a multiscale finite element method using constraint energy minimization to efficiently solve nonlinear poroelasticity and elasticity problems, demonstrating convergence and accuracy through numerical simulations.

Contribution

It introduces a novel CEM-GMsFEM approach for nonlinear poroelasticity and elasticity, with adaptive basis enrichment and proven convergence.

Findings

Accurate solutions with coarse meshes and few basis functions.

Convergence demonstrated through numerical simulations.

Effective handling of nonlinearities in poroelasticity and elasticity.

Abstract

In this paper, we apply the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) to first solving a nonlinear poroelasticity problem. The arising system consists of a nonlinear pressure equation and a nonlinear stress equation in strain-limiting setting, where strains keep bounded while stresses can grow arbitrarily large. After time discretization of the system, to tackle the nonlinearity, we linearize the resulting equations by Picard iteration. To handle the linearized equations, we employ the CEM-GMsFEM and obtain appropriate offline multiscale basis functions for the pressure and the displacement. More specifically, first, auxiliary multiscale basis functions are generated by solving local spectral problems, via the GMsFEM. Then, multiscale spaces are constructed in oversampled regions, by solving a constraint energy minimizing (CEM) problem. After that, this strategy (with the CEM-GMsFEM) is also applied to a static case of the above nonlinear poroelasticity problem, that is, elasticity problem, where the residual based online multiscale basis functions are generated by an adaptive enrichment procedure, to further reduce the error. Convergence of the two cases is demonstrated by several numerical simulations, which give accurate solutions, with converging coarse-mesh sizes as well as few basis functions (degrees of freedom) and oversampling layers.