Variational approximation for a non-isothermal coupled phase-field system: Structure-preservation & Nonlinear stability

TL;DR

This paper tackles a non-isothermal, cross-diffusion phase-field system arising from a Cahn-Hilliard-Allen-Cahn model coupled to heat transfer, incorporating full non-diagonal mobility via $\mathbf{L}$. It reformulates the problem in terms of the inverse temperature $\theta=1/T$ to reveal a variational/gradient-structure and develops structure-preserving semi-discrete and fully discrete discretizations using conforming finite elements and backward Euler time stepping with convex-concave splitting. A central contribution is a nonlinear stability analysis based on a relative-entropy functional $\mathcal{W}_{\lambda}$ and a dissipation measure $\mathcal{D}_{\mathbf{L}}$, yielding Gronwall-type bounds for both semi-discrete and fully discrete schemes and supporting potential convergence and error analyses. Numerical experiments confirm mass and energy conservation, entropy production, and the scheme’s ability to capture temperature-driven differences in time scales and microstructural evolution, including applied scenarios with distinct initial temperature fields. Overall, the framework provides a thermodynamically consistent, robust foundation for analyzing non-isothermal phase-field systems and sets the stage for rigorous error estimates and generalized-solution concepts in this multi-physics context.

Abstract

A Cahn-Hilliard-Allen-Cahn phase-field model coupled with a heat transfer equation, particularly with full non-diagonal mobility matrices, is studied. After reformulating the problem w.r.t. the inverse of temperature, we proposed and analysed a structure-preserving approximation for the semi-discretisation in space and then a fully discrete approximation using conforming finite elements and time-stepping methods. We prove structure-preserving property and discrete stability using relative entropy methods for the semi-discrete and fully discrete case. The theoretical results are illustrated by numerical experiments.

Figures (4)

• Figure 1: Snapshots of the conserved phase-field $\rho$ at the times $t\in\{0.5,1.5,7.5,10\}$ with the three different test cases. (Upper row) Initial temperature profile A (Middle row) Initial temperature profile B (Lower row) Initial temperature profile C.
• Figure 2: Snapshots of the non-conserved phase-field $\eta$ at the times $t\in\{0.5,1.5,7.5,10\}$ with the three different test cases. (Upper row) Initial temperature profile A (Middle row) Initial temperature profile B (Lower row) Initial temperature profile C.
• Figure 3: Snapshots of $\rho(2\eta-1)$ at the times $t\in\{0.5,1.5,7.5,10\}$ with the three different test cases. (Upper row) Initial temperature profile A (Middle row) Initial temperature profile B (Lower row) Initial temperature profile C.
• Figure 4: Snapshots of the inverse temperature $\theta$ at the times $t\in\{1.5,7.5,10\}$ with the three different test cases. (Upper row) Initial temperature profile A (Middle row) Initial temperature profile B (Lower row) Initial temperature profile C.

Theorems & Definitions (17)

• Remark 1
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof
• Theorem 7
• proof
• Remark 8