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A Parametric Finite Element Approach for an Anisotropic Multi-Phase Mullins-Sekerka Problem with Kinetic Undercooling

Tokuhiro Eto, Harald Garcke, Robert Nürnberg

TL;DR

The study develops a sharp-interface, anisotropic multi-phase Mullins–Sekerka model with kinetic undercooling and boundary effects, and introduces a fully discrete unfitted parametric finite element method that is unconditionally stable. A variational weak formulation is combined with an anisotropy decomposition to yield a computable curvature operator, enabling robust simulation of complex interface networks with triple junctions. Numerical experiments in 2D and 3D demonstrate energy dissipation, stability, and the method’s ability to capture intricate morphologies such as ice-crystal–like structures under varying boundary and kinetic parameters. This provides a topology-flexible, stable computational tool for diffusion-controlled crystal growth with anisotropic surface energies.

Abstract

We consider a sharp interface formulation for an anisotropic multi-phase Mullins-Sekerka problem with kinetic undercooling. The flow is characterized by a cluster of surfaces evolving such that the total surface energy plus a weighted sum of the volumes of the enclosed phases decreases in time. Upon deriving a suitable variational formulation, we introduce a fully discrete unfitted finite element method. In this approach, the approximations of the moving interfaces are independent of the triangulations used for the equations in the bulk. Our method can be shown to be unconditionally stable. Several numerical examples demonstrate the capabilities of the introduced method. In particular, it is demonstrated that the evolution of multiple ice crystals with junctions can be modeled using the proposed approach.

A Parametric Finite Element Approach for an Anisotropic Multi-Phase Mullins-Sekerka Problem with Kinetic Undercooling

TL;DR

The study develops a sharp-interface, anisotropic multi-phase Mullins–Sekerka model with kinetic undercooling and boundary effects, and introduces a fully discrete unfitted parametric finite element method that is unconditionally stable. A variational weak formulation is combined with an anisotropy decomposition to yield a computable curvature operator, enabling robust simulation of complex interface networks with triple junctions. Numerical experiments in 2D and 3D demonstrate energy dissipation, stability, and the method’s ability to capture intricate morphologies such as ice-crystal–like structures under varying boundary and kinetic parameters. This provides a topology-flexible, stable computational tool for diffusion-controlled crystal growth with anisotropic surface energies.

Abstract

We consider a sharp interface formulation for an anisotropic multi-phase Mullins-Sekerka problem with kinetic undercooling. The flow is characterized by a cluster of surfaces evolving such that the total surface energy plus a weighted sum of the volumes of the enclosed phases decreases in time. Upon deriving a suitable variational formulation, we introduce a fully discrete unfitted finite element method. In this approach, the approximations of the moving interfaces are independent of the triangulations used for the equations in the bulk. Our method can be shown to be unconditionally stable. Several numerical examples demonstrate the capabilities of the introduced method. In particular, it is demonstrated that the evolution of multiple ice crystals with junctions can be modeled using the proposed approach.
Paper Structure (9 sections, 3 theorems, 72 equations, 16 figures)

This paper contains 9 sections, 3 theorems, 72 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Mathematical formulation
  3. Weak formulation
  4. Finite element approximation
  5. Solution methods
  6. Numerical results
  7. Simulations in 2D with $\partial\Omega_N = \partial\Omega$
  8. Simulations in 2D with undercooling on $\partial\Omega$
  9. Numerical results in 3D

Key Result

Proposition 2.1

Assume that $(\hbox{\boldmath{$w$}}(\cdot, t),\,\Gamma^{}_{}(t))_{t\in[0,T]}$ is a classical solution to eq:AMMS. Then, we have that recall eq:en. In addition, it holds that

Figures (16)

  • Figure 1: A surface cluster in 2D made up of three open curves and two triple junctions $(d = 2,\, I_S = 3,\, I_R = 3,\, I_T = 2)$.
  • Figure 2: The solution at times $t=0, 0.2, 0.4, 1$, and a plot of the discrete energy over time. Below we show the adaptive bulk mesh at times $t=0$ and $t=1$.
  • Figure 3: The solution at times $t=0, 1, 2, 4$, and a plot of the discrete energy over time.
  • Figure 4: The solution at times $t=0, 0.5, 1, 2, 3, 4, 5, 6, 7, 8$, and a plot of the discrete energy over time.
  • Figure 5: The solution at times $t=0, 0.5, 1, 2$, and a plot of the discrete energy over time.
  • ...and 11 more figures

Theorems & Definitions (9)

  • Proposition 2.1
  • proof
  • Remark 2.2
  • Theorem 4.1
  • proof
  • Theorem 4.2: Discrete energy dissipation
  • proof
  • Remark 4.3
  • Remark 5.1