A Generalized Multiscale Finite Element Method for Poroelasticity Problems II: Nonlinear Coupling

TL;DR

This paper develops a generalized multiscale finite element method for efficiently solving nonlinear poroelasticity problems with heterogenous media, by constructing multiscale basis functions that handle nonlinearities locally and reduce computational complexity.

Contribution

It introduces a novel GMsFEM approach for nonlinear Biot-type poroelasticity problems, enabling coarse-grid solutions with local nonlinear handling and reduced basis dimension.

Findings

Efficient multiscale solution for nonlinear poroelasticity problems.

Accurate approximation of fine-scale solutions with reduced computational cost.

Effective handling of heterogenous media with nonlinear coupling.

Abstract

In this paper, we consider the numerical solution of some nonlinear poroelasticity problems that are of Biot type and develop a general algorithm for solving nonlinear coupled systems. We discuss the difficulties associated with flow and mechanics in heterogenous media with nonlinear coupling. The central issue being how to handle the nonlinearities and the multiscale scale nature of the media. To compute an efficient numerical solution we develop and implement a Generalized Multiscale Finite Element Method (GMsFEM) that solves nonlinear problems on a coarse grid by constructing local multiscale basis functions and treating part of the nonlinearity locally as a parametric value. After linearization with a Picard Iteration, the procedure begins with construction of multiscale bases for both displacement and pressure in each coarse block by treating the staggered nonlinearity as a parametric value. Using a snapshot space and local spectral problems, we construct an offline basis of reduced dimension. From here an online, parametric dependent, space is constructed. Finally, after multiplying by a multiscale partitions of unity, the multiscale basis is constructed and the coarse grid problem then can be solved for arbitrary forcing and boundary conditions. We implement this algorithm on a geometry with a linear and nonlinear pressure dependent permeability field and compute error between the multiscale solution with the fine-scale solutions.