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Oscillation-free numerical schemes for Biot's model and their iterative coupling solution

Álvaro Pé de la Riva, Francisco J. Gaspar, Xiaozhe Hu, James Adler, Carmen Rodrigo, Ludmil Zikatanov

TL;DR

This work tackles non-physical pressure oscillations in Biot's poroelasticity models at low permeability or small time steps by introducing a novel stabilization that does not depend on the mesh size. The stabilization, combined with a fixed-stress–like iterative coupling, yields a convergent, parameter-robust sequential solver for FE discretizations (P1-P1 and MINI) of Biot's model, without requiring additional stabilization for convergence. A thorough one-dimensional analysis (Terzaghi's problem) identifies optimal stabilization and coupling parameters that guarantee two-iteration convergence, with near-optimal performance extending to two- and three-dimensional benchmarks (Barry & Mercer test and a 3D footing problem). The approach also generalizes to broader fluid regimes by introducing a two-parameter coupling scheme, maintaining robustness with respect to discretization and physical parameters. Practically, the method offers stable, efficient solvers for poroelastic simulations across a wide range of settings.

Abstract

In this work, we present a new stabilization method aimed at removing spurious oscillations in the pressure approximation of Biot's model for poroelasticity with low permeabilities and/or small time steps. We consider different finite-element discretizations and illustrate how not only does such a stabilized scheme provide numerical solutions that are free of non-physical oscillations, but it also allows one to iterate the fluid and mechanics problems in a fashion similar to the well-known fixed-stress split method. The resulting solution method is convergent without the necessity for additional terms to stabilize the iteration. Finally, we present numerical results illustrating the robust behavior of both the stabilization and iterative solver with respect to the physical and discretization parameters of the model.

Oscillation-free numerical schemes for Biot's model and their iterative coupling solution

TL;DR

This work tackles non-physical pressure oscillations in Biot's poroelasticity models at low permeability or small time steps by introducing a novel stabilization that does not depend on the mesh size. The stabilization, combined with a fixed-stress–like iterative coupling, yields a convergent, parameter-robust sequential solver for FE discretizations (P1-P1 and MINI) of Biot's model, without requiring additional stabilization for convergence. A thorough one-dimensional analysis (Terzaghi's problem) identifies optimal stabilization and coupling parameters that guarantee two-iteration convergence, with near-optimal performance extending to two- and three-dimensional benchmarks (Barry & Mercer test and a 3D footing problem). The approach also generalizes to broader fluid regimes by introducing a two-parameter coupling scheme, maintaining robustness with respect to discretization and physical parameters. Practically, the method offers stable, efficient solvers for poroelastic simulations across a wide range of settings.

Abstract

In this work, we present a new stabilization method aimed at removing spurious oscillations in the pressure approximation of Biot's model for poroelasticity with low permeabilities and/or small time steps. We consider different finite-element discretizations and illustrate how not only does such a stabilized scheme provide numerical solutions that are free of non-physical oscillations, but it also allows one to iterate the fluid and mechanics problems in a fashion similar to the well-known fixed-stress split method. The resulting solution method is convergent without the necessity for additional terms to stabilize the iteration. Finally, we present numerical results illustrating the robust behavior of both the stabilization and iterative solver with respect to the physical and discretization parameters of the model.
Paper Structure (14 sections, 7 theorems, 52 equations, 11 figures, 4 tables)

This paper contains 14 sections, 7 theorems, 52 equations, 11 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Biot's model: Mathematical model and discretization
  3. Mathematical model
  4. Discretization
  5. On the stability of the proposed method
  6. On the convergence of the iterative coupling method
  7. One-dimensional problem and choice of parameter $\gamma$
  8. Discretization with linear finite elements (P1-P1)
  9. Discretization with the MINI-element for displacement
  10. Numerical experiments
  11. Two-dimensional Barry & Mercer problem
  12. Three-dimensional footing problem
  13. More general fluid regimes
  14. Conclusions

Key Result

Theorem 1

If the weak inf-sup condition ine:weak-inf-sup holds, the iterative method given in it_pressure2-it_displacement2 is convergent for any parameters $\gamma \in (1/2,2]$ and $L$ such that $\gamma L = \omega \frac{\alpha^2}{ (\lambda+2\mu/d)} \geq \frac{\alpha^2}{(\lambda+2\mu/d)}$, i.e., $\omega \geq where $\theta^* \geq \frac{\epsilon^2C_2 \gamma}{4 \omega(\gamma - |1-\gamma|)}$ is a root of the f

Figures (11)

  • Figure 1: Comparison between the analytical solution (in blue) and the numerical solution (in red) for the pressure field obtained with (a) the P1-P1 finite-element method and (b) the stabilized P1-P1 finite-element method. The hydraulic conductivity is fixed at $K=10^{-6}$, and the grid size is $h=1/32$.
  • Figure 2: Number of iterations of the iterative coupling method for different values of $\gamma$, when the stabilized P1-P1 discretization of Terzaghi's problem is considered.
  • Figure 3: Comparison between the analytical solution (in blue) and the numerical solution (in red) for the pressure field obtained with (a) the MINI-element and (b) the stabilized MINI-element for displacement. The hydraulic conductivity is fixed as $K=10^{-6}$, and the grid size is $h=1/32$.
  • Figure 4: Number of iterations of the iterative coupling method for different values of $\gamma$, when the stabilized MINI-element discretization of Terzaghi's problem is considered.
  • Figure 5: Computational domain and boundary conditions for the Barry and Mercer's source problem.
  • ...and 6 more figures

Theorems & Definitions (12)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 2.1
  • proof
  • Corollary 2.2
  • proof
  • Theorem 3
  • proof
  • ...and 2 more