Constraint energy minimizing generalized multiscale finite element method for nonlinear 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.

arXiv comment32 pages, 7 figures, 6 tables, submitted to Journal of Computational Physics

References in corpus (15)

• Generalized Multiscale Finite Element Methods (GMsFEM)
• Adaptive multiscale model reduction with Generalized Multiscale Finite Element Methods
• Constraint Energy Minimizing Generalized Multiscale Finite Element Method
• Finite strain FFT-based non-linear solvers made simple
• An adaptive GMsFEM for high-contrast flow problems
• Non-local Multi-continua Upscaling for Flows in Heterogeneous Fractured Media
• Fast Online Generalized Multiscale Finite Element Method using Constraint Energy Minimization
• Generalized Multiscale Finite Element Methods. Nonlinear Elliptic Equations
• The Heterogeneous Multiscale Finite Element Method for the Homogenization of Linear Elastic Solids and a Comparison with the FE${}^2$ Method
• On the existence of integrable solutions to nonlinear elliptic systems and variational problems with linear growth
• Generalized multiscale finite element method for a strain-limiting nonlinear elasticity model
• Computational Multiscale Methods for Linear Poroelasticity with High Contrast
• Multiscale simulations for upscaled multi-continuum flows
• Nonlinear nonlocal multicontinua upscaling framework and its applications
• Constraint Energy Minimizing Generalized Multiscale Finite Element Method for high-contrast linear elasticity problem

Cited by in corpus (11)

• Theory of functional connections applied to quadratic and nonlinear programming under equality constraints
• Prediction of numerical homogenization using deep learning for the Richards equation
• Multiscale simulations for multi-continuum Richards equations
• Local multiscale model reduction using discontinuous Galerkin coupling for elasticity problems
• Constraint Energy Minimizing Generalized Multiscale Finite Element Method for multi-continuum Richards equations
• CEM-GMsFEM for Poisson equations in heterogeneous perforated domains
• New twofold saddle-point formulations for Biot poroelasticity with porosity-dependent permeability
• Nonlinear Strain-limiting Elasticity for Fracture Propagation with Phase-Field Approach
• Constraint energy minimization generalized multiscale finite element method in mixed formulation for parabolic equations
• A finite element model for a coupled thermo-mechanical system: nonlinear strain-limiting thermoelastic body
• A semi-implicit stochastic multiscale method for radiative heat transfer problem