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On the convergence and efficiency of splitting schemes for the Cahn-Hilliard-Biot model

Cedric Riethmüller, Erlend Storvik

Abstract

In this paper, we present a novel solution strategy for the Cahn-Hilliard-Biot model, a three-way coupled system that features the interplay of solid phase separation, fluid dynamics, and elastic deformations in porous media. It is a phase-field model that combines the Cahn-Hilliard regularized interface equation and Biot's equations of poroelasticity. Solving the system poses significant challenges due to its coupled, nonlinear, and non-convex nature. The main goal of this work is to provide a consistent and efficient solution strategy. With this in mind, we introduce a semi-implicit time discretization such that the resulting discrete system is equivalent to a convex minimization problem. Then, using abstract theory for convex problems, we prove the convergence of an alternating minimization method to the time-discrete system. The solution strategy is relatively flexible in terms of spatial discretization, although we require standard inverse inequalities for the guaranteed convergence of the alternating minimization method. Finally, we perform some numerical experiments that show the promise of the proposed solution strategy, both in terms of efficiency and robustness.

On the convergence and efficiency of splitting schemes for the Cahn-Hilliard-Biot model

Abstract

In this paper, we present a novel solution strategy for the Cahn-Hilliard-Biot model, a three-way coupled system that features the interplay of solid phase separation, fluid dynamics, and elastic deformations in porous media. It is a phase-field model that combines the Cahn-Hilliard regularized interface equation and Biot's equations of poroelasticity. Solving the system poses significant challenges due to its coupled, nonlinear, and non-convex nature. The main goal of this work is to provide a consistent and efficient solution strategy. With this in mind, we introduce a semi-implicit time discretization such that the resulting discrete system is equivalent to a convex minimization problem. Then, using abstract theory for convex problems, we prove the convergence of an alternating minimization method to the time-discrete system. The solution strategy is relatively flexible in terms of spatial discretization, although we require standard inverse inequalities for the guaranteed convergence of the alternating minimization method. Finally, we perform some numerical experiments that show the promise of the proposed solution strategy, both in terms of efficiency and robustness.
Paper Structure (21 sections, 7 theorems, 105 equations, 4 figures, 1 table)

This paper contains 21 sections, 7 theorems, 105 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model formulation
  3. Notation
  4. Assumptions on model parameters
  5. Numerical solution strategy
  6. Variational form
  7. Discrete function spaces and norms
  8. State-independent material parameters
  9. Discretization
  10. Alternating minimization
  11. State-dependent material parameters
  12. Discretization
  13. Alternating minimization
  14. Numerical experiments
  15. Model problem
  16. ...and 6 more sections

Key Result

Lemma 1

For all $q_h \in \mathcal{V}_{h,0}^{\mathrm{z}}$ (with $\|q_h\|_{h,y} \neq 0$) there exists a positive constant $C_\mathrm{inv}$ such that

Figures (4)

  • Figure 1: Evolution of the phase field $\varphi$ for the model problem. Presented above are four snap-shots at different times.
  • Figure 2: "Semi-impl. Mono" refers to a monolithic Newton solver applied to the discrete system \ref{['eq:ch1split_sd']}--\ref{['eq:pressuresplit_sd']}, "Semi-impl. Split" refers to the scheme \ref{['eq:ch1split_alter_sd']}--\ref{['eq:pressuresplit_alter_sd']}, and "Semi-impl. FS" refers to a three-way decoupling solver, sequentially solving the Cahn-Hilliard, the elasticity and the flow subsystems, for the semi-implicit time discretization \ref{['eq:ch1split_sd']}--\ref{['eq:pressuresplit_sd']}. On the other hand, "Mono", "Split" and "FS", refer to the same solution strategies but applied to the discrete system arising from discretizing with the implicit Euler method and the Eyre convex-concave split Eyre1998 of the double-well potential.
  • Figure 3: "Semi-impl. Mono" refers to a monolithic Newton solver applied to the discrete system \ref{['eq:ch1split_sd']}--\ref{['eq:pressuresplit_sd']}, "Semi-impl. Split" refers to the scheme \ref{['eq:ch1split_alter_sd']}--\ref{['eq:pressuresplit_alter_sd']}, and "Semi-impl. FS" refers to a three-way decoupling solver, sequentially solving the Cahn-Hilliard, the elasticity and the flow subsystems, for the semi-implicit time discretization \ref{['eq:ch1split_sd']}--\ref{['eq:pressuresplit_sd']}. On the other hand, "Mono", "Split" and "FS", refer to the same solution strategies but applied to the discrete system arising from discretizing with the implicit Euler method and the Eyre convex-concave split Eyre1998 of the double-well potential.
  • Figure 4: Total energy for each time step for varying surface tension (left) and swelling (right) parameters. Depicted are the energies calculated for the semi-implicit time discretization \ref{['eq:model_disc_sd']}, using the alternating minimization method \ref{['eq:model_disc_alter_sd']} to solve the systems in each time step.

Theorems & Definitions (19)

  • Remark 1: Assumption (A1)
  • Remark 2: Assumption (A4)
  • Remark 3
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 4
  • Lemma 3
  • proof
  • ...and 9 more