[Paper Review] Multiphysics Finite Element Methods for a Poroelasticity Model

Xiaobing Feng, Zhihao Ge+1 more|arXiv (Cornell University)|Nov 27, 2014

Advanced Numerical Methods in Computational Mathematics22 references6 citations

TL;DR

This paper proposes a novel multiphysics finite element method for a quasi-static poroelasticity model by introducing two pseudo-pressures to decouple the system into a generalized Stokes problem and a diffusion equation. The approach ensures discrete energy stability, optimal convergence, and eliminates the locking phenomenon in numerical simulations, even under rapid initial pressure changes or vanishing constrained specific storage.

ABSTRACT

This paper concerns with finite element approximations of a quasi-static poroelasticity model in displacement-pressure formulation which describes the dynamics of poro-elastic materials under an applied mechanical force on the boundary. To better describe the multiphysics process of deformation and diffusion for poro-elastic materials, we first present a reformulation of the original model by introducing two pseudo-pressures, one of them is shown to satisfy a diffusion equation, we then propose a time-stepping algorithm which decouples (or couples) the reformulated PDE problem at each time step into two sub-problems, one of which is a generalized Stokes problem for the displacement vector field (of the solid network of the poro-elastic material) along with one pseudo-pressure field and the other is a diffusion problem for the other pseudo-pressure field (of the solvent of the material). In the paper, the Taylor-Hood mixed finite element method combined with the $P_1$-conforming finite element method is used as an example to demonstrate the viability of the proposed multiphysics approach. It is proved that the solutions of the fully discrete finite element methods fulfill a discrete energy law which mimics the differential energy law satisfied by the PDE solution and converges optimally in the energy norm. Moreover, it is showed that the proposed formulation also has a built-in mechanism to overcome so-called "locking phenomenon" associated with the numerical approximations of the poroelasticity model. Numerical experiments are presented to show the performance of the proposed approach and methods and to demonstrate the absence of "locking phenomenon" in our numerical experiments.

Motivation & Objective

• To address the numerical instability known as the 'locking phenomenon' in finite element approximations of poroelasticity models, especially under rapid initial pressure changes.
• To develop a time-stepping algorithm that decouples the poroelasticity system into two solvable sub-problems: a generalized Stokes problem for displacement and one pseudo-pressure, and a diffusion problem for the other pseudo-pressure.
• To ensure unique solvability of both sub-problems by deriving appropriate boundary conditions from conserved quantities and coupling via the generalized Stokes solution.
• To demonstrate that the fully discrete finite element scheme satisfies a discrete energy law and converges optimally in the energy norm.
• To show robustness of the method under the limit process where the constrained specific storage coefficient tends to zero, recovering Biot’s consolidation model.

Proposed method

• Reformulate the original poroelasticity model by introducing two pseudo-pressures, one of which satisfies a diffusion equation, enabling decoupling of the system.
• Decouple the time-discretized system into two sub-problems: a generalized Stokes problem for displacement and one pseudo-pressure, and a diffusion problem for the other pseudo-pressure.
• Use conserved quantities of the PDE solution to construct a boundary condition that ensures unique solvability of the generalized Stokes problem.
• Generate a boundary condition for the diffusion problem from the solution of the generalized Stokes problem, ensuring consistency and uniqueness.
• Employ the Taylor-Hood mixed finite element method for velocity and pressure, and the P1-conforming finite element method for the pseudo-pressures, enabling use of existing solvers for each sub-problem.
• Prove that the fully discrete scheme satisfies a discrete energy law mimicking the continuous energy law and achieves optimal convergence in the energy norm.

Experimental results

Research questions

• RQ1 Can a multiphysics finite element approach be designed to decouple the poroelasticity system into two solvable sub-problems while preserving stability and accuracy?
• RQ2 How can unique solvability be ensured for both the generalized Stokes and diffusion sub-problems in the time-stepping algorithm?
• RQ3 Does the proposed method eliminate the locking phenomenon observed in previous mixed finite element methods under rapid initial pressure changes?
• RQ4 How does the method behave in the limit as the constrained specific storage coefficient tends to zero, and does it recover Biot’s consolidation model?
• RQ5 Can the method be implemented using existing solvers for Stokes and diffusion problems without modifying their code?

Key findings

• The proposed method ensures unique solvability of both sub-problems by deriving boundary conditions from conserved quantities and coupling between the sub-problems.
• The fully discrete finite element scheme satisfies a discrete energy law that mirrors the continuous energy law of the PDE solution.
• Optimal convergence rates are achieved in the energy norm, as proven by rigorous error estimates.
• Numerical experiments confirm the absence of pressure oscillations and the elimination of the locking phenomenon, even at very small time steps with rapid initial pressure changes.
• The method remains robust in the limit as the constrained specific storage coefficient tends to zero, correctly recovering Biot’s consolidation model.
• The method is insensitive to the regularity of the pressure field because pressure is no longer a primary variable but a derived quantity from the pseudo-pressures.