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

JOURNAL OF COMPUTATIONAL PHYSICS

卷:417

Article

Fu, Shubin¹  Chung, Eric²  Mai, Tina^3,4 

[1] Univ Wisconsin, Dept Math, Madison, WI 53706 USA

[2] Chinese Univ Hong Kong, Dept Math, Shatin, Hong Kong, Peoples R China

[3] Duy Tan Univ, Inst Res & Dev, Da Nang 550000, Vietnam

[4] Duy Tan Univ, Fac Nat Sci, Da Nang 550000, Vietnam

关键词: Constraint energy minimizing;    Generalized multiscale finite element method;    Strain-limiting;    Nonlinear poroelasticity;    Nonlinear elasticity;    Residual based online multiscale basis functions;   

DOI  :  10.1016/j.jcp.2020.109569

来源: Elsevier

【 摘 要 】

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. (c) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free