[Paper Review] A Variational Lagrangian Scheme for a Phase Field Model: A Discrete Energetic Variational Approach

Chun Liu, Yiwei Wang|arXiv (Cornell University)|Mar 23, 2020

Solidification and crystal growth phenomena70 references4 citations

TL;DR

This paper proposes a variational Lagrangian scheme for a modified Allen-Cahn phase-field model that preserves the energy-dissipation structure via a discrete energetic variational approach. By formulating the dynamics through a Lagrangian map with a regularization term, the method achieves energy stability and accurately captures thin diffuse interfaces with minimal mesh points, enabling robust computation of equilibrium states.

ABSTRACT

In this paper, we propose a variational Lagrangian scheme for a modified phase-field model, which can compute the equilibrium states for the original Allen-Cahn type model. Our discretization is based on a prescribed energy-dissipation law in terms of the flow map. By employing a discrete energetic variational approach, this scheme preserves the variational structure of the original energy-dissipation law and is energy stable. Plentiful numerical tests show that, by choosing the initial value properly, our methods can produce the desired equilibrium and capture the thin diffuse interface with a small number of mesh points.

Motivation & Objective

• To develop a structure-preserving numerical scheme for computing equilibrium states in Allen-Cahn-type phase-field models.
• To overcome the limitations of Eulerian methods in resolving thin diffuse interfaces with minimal computational cost.
• To extend Lagrangian methods to $L^2$-gradient flows, which lack natural variational structures on Lagrangian maps.
• To ensure energy stability and mesh quality through a regularization term in the energy-dissipation law.
• To demonstrate the effectiveness of the scheme in capturing sharp interfaces and converging to correct equilibrium states with minimal mesh resolution.

Proposed method

• The method is based on a prescribed energy-dissipation law expressed in terms of the flow map, ensuring variational structure preservation.
• A discrete energetic variational approach is employed to derive the time-discrete scheme, maintaining the continuous energy-dissipation law at the discrete level.
• The scheme incorporates a regularization term $\nu|\nabla\mathbf{u}|^2$ in the dissipation to prevent mesh tangling and ensure stability.
• The system is formulated using a Lagrangian map $\mathbf{X}(\mathbf{X}_0, t)$, where the phase function $\varphi$ evolves via $\varphi_t + \nabla\varphi \cdot \mathbf{u} = 0$.
• The discrete system is solved using finite elements with a consistent mass matrix $\mathbf{M}^e$ and stiffness matrix $\mathbf{K}^e$, ensuring positive definiteness for $\nu > 0$.
• The method is self-adaptive, concentrating mesh points at the diffuse interface without requiring adaptive mesh refinement.

Experimental results

Research questions

• RQ1 Can a variational Lagrangian scheme preserve the energy-dissipation structure of Allen-Cahn-type phase-field models at the discrete level?
• RQ2 How can Lagrangian methods be adapted to $L^2$-gradient flows, which lack natural variational structure on Lagrangian maps?
• RQ3 Can the proposed scheme compute equilibrium states accurately with minimal mesh resolution and avoid mesh tangling?
• RQ4 What role does the regularization term $\nu|\nabla\mathbf{u}|^2$ play in ensuring numerical stability and mesh quality?
• RQ5 How does the choice of initial condition $\varphi_0$ affect the convergence to the correct equilibrium state in Lagrangian simulations?

Key findings

• The proposed scheme is energy stable and preserves the variational structure of the continuous energy-dissipation law through discrete energetic variational formulation.
• With properly chosen initial conditions, the method successfully computes the correct equilibrium state of the Allen-Cahn model.
• The scheme captures thin diffuse interfaces with a small number of mesh points, demonstrating high resolution efficiency.
• Numerical tests show that the method avoids mesh tangling and maintains good mesh quality even at late times, unlike naive Lagrangian approaches.
• The regularization term $\nu|\nabla\mathbf{u}|^2$ is essential for ensuring the positive definiteness of the discrete dissipation matrix $\mathbf{D}$, which prevents numerical breakdown.
• A failed example with an improper initial condition $\varphi_0$ shows that the Lagrangian method can fail to reach the correct equilibrium if $\varphi_0$ is not bounded in $[-1,1]$, highlighting the importance of initial data selection.