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Sequential solution strategies for the Cahn-Hilliard-Biot model

Erlend Storvik, Cedric Riethmüller, Jakub Wiktor Both, Florin Adrian Radu

TL;DR

The paper addresses the challenge of solving the coupled nonlinear CH-Biot equations that model flow in deformable porous media with phase changes. It compares two solution strategies: a monolithic Newton-based solver for the fully coupled system and a decoupled splitting scheme that iterates the CH, elasticity, and flow subsystems, both using a semi-implicit time discretization with a convex-concave split of the double-well potential. The study finds that the splitting approach offers superior robustness and often lower total CPU time, especially under stronger coupling governed by parameters such as surface tension $\gamma$ and swelling $\xi$, though it is not immune to convergence issues at high coupling. These findings inform practical choices for numerical solvers in poromechanics with phase-field dynamics and highlight directions for further stabilizing and refining splitting methods. The work advances numerical strategies for multi-physics phase-field poromechanics and supports more reliable simulations in engineering and materials science contexts.

Abstract

This paper presents a study of solution strategies for the Cahn-Hilliard-Biot equations, a complex mathematical model for understanding flow in deformable porous media with changing solid phases. Solving the Cahn-Hilliard-Biot system poses significant challenges due to its coupled, nonlinear and non-convex nature. We explore various solution algorithms, comparing monolithic and splitting strategies, focusing on both their computational efficiency and robustness.

Sequential solution strategies for the Cahn-Hilliard-Biot model

TL;DR

The paper addresses the challenge of solving the coupled nonlinear CH-Biot equations that model flow in deformable porous media with phase changes. It compares two solution strategies: a monolithic Newton-based solver for the fully coupled system and a decoupled splitting scheme that iterates the CH, elasticity, and flow subsystems, both using a semi-implicit time discretization with a convex-concave split of the double-well potential. The study finds that the splitting approach offers superior robustness and often lower total CPU time, especially under stronger coupling governed by parameters such as surface tension and swelling , though it is not immune to convergence issues at high coupling. These findings inform practical choices for numerical solvers in poromechanics with phase-field dynamics and highlight directions for further stabilizing and refining splitting methods. The work advances numerical strategies for multi-physics phase-field poromechanics and supports more reliable simulations in engineering and materials science contexts.

Abstract

This paper presents a study of solution strategies for the Cahn-Hilliard-Biot equations, a complex mathematical model for understanding flow in deformable porous media with changing solid phases. Solving the Cahn-Hilliard-Biot system poses significant challenges due to its coupled, nonlinear and non-convex nature. We explore various solution algorithms, comparing monolithic and splitting strategies, focusing on both their computational efficiency and robustness.
Paper Structure (12 sections, 7 equations, 3 figures, 1 table)

This paper contains 12 sections, 7 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Cahn-Hilliard-Biot model
  3. Numerical solution strategies
  4. Time-discretization: Implicit with partly convex-concave split
  5. Spatial discretization
  6. Monolithic solution strategy
  7. Splitting schemes
  8. Numerical experiments
  9. Implementation
  10. Dependence on surface tension parameter
  11. Dependence on swelling parameter
  12. Conclusion

Figures (3)

  • Figure 1: Plot of phase-field solutions $\varphi$ from numerical study. The phase-field was initialized with the distribution from figure (a) for all cases. The other plots (b)--(e) show the phase-field at the final time $T_\mathrm{f} = 0.003$ for different values of the swelling parameter $\xi$.
  • Figure 2: The total number of iterations and CPU time to complete the simulation for different values of the surface tension parameter $\gamma$. The material parameters from Table \ref{['tab:1']} are applied with swelling parameter $\xi = 0.5$.
  • Figure 3: The total number of iterations and CPU time to complete the simulation for different values of the swelling parameter $\xi$. The material parameters from Table \ref{['tab:1']} are applied with surface tension parameter $\gamma = 5$.