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Structure-preserving approximation of the Cahn-Hilliard-Biot system

Aaron Brunk, Marvin Fritz

TL;DR

The authors develop a structure-preserving, decoupled discretisation for the Cahn-Hilliard-Biot system, combining conforming finite elements with an IMEX Euler time stepping to maintain the gradient-flow structure. They prove discrete mass and energy-dissipation balances, establish existence of solutions, and provide a CFL-type uniqueness condition. The method enables efficient splitting into a linear poro-elastic step and a nonlinear Cahn-Hilliard step, while preserving key thermodynamic properties. Numerical experiments, including convergence tests and tumour-growth-inspired scenarios, validate the theoretical results and demonstrate the scheme’s stability and accuracy in multi-physics settings.

Abstract

In this work, we propose a structure-preserving discretisation for the recently studied Cahn-Hilliard-Biot system using conforming finite elements in space and problem-adapted explicit-implicit Euler time integration. We prove that the scheme preserves the thermodynamic structure, that is, the balance of mass and volumetric fluid content and the energy dissipation balance. The existence of discrete solutions is established under suitable growth conditions. Furthermore, it is shown that the algorithm can be realised as a splitting method, that is, decoupling the Cahn-Hilliard subsystem from the poro-elasticity subsystem, while the first one is nonlinear and the second subsystem is linear. The schemes are illustrated by numerical examples and a convergence test.

Structure-preserving approximation of the Cahn-Hilliard-Biot system

TL;DR

The authors develop a structure-preserving, decoupled discretisation for the Cahn-Hilliard-Biot system, combining conforming finite elements with an IMEX Euler time stepping to maintain the gradient-flow structure. They prove discrete mass and energy-dissipation balances, establish existence of solutions, and provide a CFL-type uniqueness condition. The method enables efficient splitting into a linear poro-elastic step and a nonlinear Cahn-Hilliard step, while preserving key thermodynamic properties. Numerical experiments, including convergence tests and tumour-growth-inspired scenarios, validate the theoretical results and demonstrate the scheme’s stability and accuracy in multi-physics settings.

Abstract

In this work, we propose a structure-preserving discretisation for the recently studied Cahn-Hilliard-Biot system using conforming finite elements in space and problem-adapted explicit-implicit Euler time integration. We prove that the scheme preserves the thermodynamic structure, that is, the balance of mass and volumetric fluid content and the energy dissipation balance. The existence of discrete solutions is established under suitable growth conditions. Furthermore, it is shown that the algorithm can be realised as a splitting method, that is, decoupling the Cahn-Hilliard subsystem from the poro-elasticity subsystem, while the first one is nonlinear and the second subsystem is linear. The schemes are illustrated by numerical examples and a convergence test.
Paper Structure (12 sections, 3 theorems, 63 equations, 5 figures, 2 tables)

This paper contains 12 sections, 3 theorems, 63 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notation and variational formulation
  3. Structure-preserving discretisation
  4. Proof of Theorem 1
  5. Balance laws
  6. A priori bounds
  7. Existence
  8. Uniqueness
  9. Numerical illustrations
  10. Convergence test
  11. Validation experiments
  12. Conclusion and Outlook

Key Result

lemma 2.1

Sufficiently regular solutions of the system eq:abs1--eq:abs3 satisfy the variational formulations eq:var1--eq:var5. Furthermore, the balance of mass and volumetric fluid content as well as the energy-dissipation balance hold, which are given by with dissipation rate and source rate

Figures (5)

  • Figure 1: Experiment \ref{['exp:lshape']}: Snapshots of the phase field $\phi$ at the times $t\in\{0,0.02,0.06,2\}$ with mesh size $h_{\max} \approx 10^{-2}$ and time-step $\tau=10^{-3}$ (from top left to bottom right).
  • Figure 2: Experiment \ref{['exp:tumor']} for the CHB system: Snapshots of the phase field $\phi$ at the times $t\in\{0,0.5,0.75,1\}$ with mesh size $h_{\max} \approx 10^{-2}$ and time-step $\tau=10^{-3}$ (from top left to bottom right).
  • Figure 3: Experiment \ref{['exp:tumor']} for the CHL system: Snapshots of the phase field $\phi$ at times $t\in\{0,0.5,0.75,1\}$ with mesh size $h_{\max} \approx 10^{-2}$ and time-step $\tau=10^{-3}$ (from top left to bottom right).
  • Figure 4: Experiment \ref{['exp:tumor']}: Difference between the phase-fields of the CHB and CHL solutions in the times $t\in\{0.01,0.5,0.75,1\}$ (from top left to bottom right).
  • Figure 5: Snapshots of the phase field $\phi$ at the times $t\in\{0.2,0.5,0.8,1\}$ with mesh size $h_{\max} \approx 2\cdot 10^{-2}$ and time-step $\tau=5\cdot 10^{-4}$. Visualisation of the tumour core ($\phi\geq 0.9$) in red and the isosurfaces $\phi=-0.25$ (green) and $\phi=0.15$ (yellow) (from top left to bottom right).

Theorems & Definitions (5)

  • lemma 2.1
  • proof
  • theorem 1
  • lemma 3.1
  • proof