A Finite Element Method for Evolving Multi-Phase Interfaces with Triple Junctions

Core Concepts

The authors introduce a structure-preserving finite element method for the numerical approximation of the multi-phase Mullins-Sekerka problem, which models the evolution of a network of curves driven by surface energy minimization and subject to area preservation constraints.

Abstract

The content presents a numerical method for the multi-phase Mullins-Sekerka problem, which describes the evolution of a network of curves that partition a domain into multiple phases. The key highlights and insights are:

The authors derive a weak formulation of the problem, which encodes the motion law, Gibbs-Thomson law, and the balance of forces at triple junctions.

They introduce a parametric finite element method that approximates the moving interfaces independently of the discretization used for the bulk equations. This scheme is shown to be unconditionally stable and to satisfy an exact volume conservation property.

The discretization features an inherent tangential velocity for the vertices on the discrete curves, leading to asymptotically equidistributed vertices without the need for remeshing.

The authors provide a detailed discussion on the solution of the linear systems arising from the discrete problem, including techniques to avoid complications due to the nonstandard finite element spaces.

Several numerical examples, including a convergence experiment for the three-phase Mullins-Sekerka flow, demonstrate the capabilities of the introduced method.

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Key Insights Distilled From

A structure-preserving finite element method for the multi-phase Mullins-Sekerka problem with triple junctions

by Toku... at arxiv.org 04-11-2024

https://arxiv.org/pdf/2309.11948.pdf

Deeper Inquiries

How can the proposed method be extended to handle more complex geometries or topological changes in the interface network

The proposed method can be extended to handle more complex geometries or topological changes in the interface network by incorporating adaptive mesh refinement techniques. By dynamically adjusting the mesh resolution based on the curvature and movement of the interface network, the method can effectively capture intricate geometries and topological changes. Additionally, the use of higher-order finite element methods can improve the accuracy of the solution in regions of high curvature or rapid changes. Furthermore, implementing advanced interface tracking algorithms, such as level set methods or phase field methods, can enhance the method's capability to handle complex interface dynamics and topological changes.

What are the potential applications of the multi-phase Mullins-Sekerka model beyond materials science, and how could the numerical scheme be adapted to those contexts

The multi-phase Mullins-Sekerka model has potential applications beyond materials science in various fields such as fluid dynamics, biological systems, and image processing. In fluid dynamics, the model can be used to simulate the behavior of multiple interacting fluids with different properties, such as in multiphase flow simulations. In biological systems, the model can be applied to study cell membrane dynamics, tissue growth, and morphogenesis. In image processing, the model can be utilized for image segmentation, shape analysis, and object tracking. To adapt the numerical scheme to these contexts, modifications may be needed to incorporate additional physical phenomena, boundary conditions, or constraints specific to each application domain.

The authors mention the connection between the sharp interface model and a diffuse interface Cahn-Hilliard formulation. Could the insights from this work be used to develop efficient numerical schemes for the latter

The insights from the sharp interface model and its connection to the diffuse interface Cahn-Hilliard formulation can be leveraged to develop efficient numerical schemes for the latter. By understanding the relationship between the two models and the underlying principles governing interface dynamics, one can design numerical algorithms that efficiently capture the diffuse interface behavior while maintaining stability and accuracy. Techniques such as implicit time integration, adaptive mesh refinement, and higher-order discretization can be employed to enhance the numerical scheme's performance for the diffuse interface Cahn-Hilliard formulation. Additionally, incorporating regularization techniques and energy minimization principles from the sharp interface model can improve the convergence and robustness of the numerical scheme for the diffuse interface formulation.