[Paper Review] An Efficient Finite Difference-based Implicit Solver for Phase-Field Equations with Spatially and Temporally Varying Parameters

Zirui Mao, G. R. Liu+1 more|arXiv (Cornell University)|Jan 22, 2024

Solidification and crystal growth phenomenaMaterials Science3 citations

TL;DR

This paper presents an efficient finite difference-based implicit solver for phase-field equations with spatially and temporally varying parameters, leveraging Eyre's theorem to ensure unconditional stability. The method achieves high accuracy and efficiency in simulations of microstructure evolution under non-uniform and time-varying conditions, though it currently requires uniform meshes and specific boundary conditions.

ABSTRACT

The phase field method is an effective tool for modeling microstructure evolution in materials. Many efficient implicit numerical solvers have been proposed for phase field simulations under uniform and time-invariant model parameters. We use Eyre's theorem to develop an unconditionally stable implicit solver for spatially non-uniform and time-varying model parameters. The accuracy, unconditional stability, and efficiency of the solver is validated against benchmarking examples. In its current form, the solver requires a uniform mesh and may only be applied to problems with periodic, Neumann, or mixed periodic and Neumann boundary conditions.

Motivation & Objective

• To develop a stable and efficient numerical solver for phase-field simulations with non-uniform and time-varying model parameters.
• To extend existing implicit solvers beyond uniform and time-invariant parameters commonly used in phase-field modeling.
• To ensure unconditional stability in the presence of spatial and temporal parameter variations using Eyre's theorem.
• To validate the solver’s accuracy and efficiency against established benchmark problems.
• To identify the current limitations in mesh and boundary condition requirements for practical application.

Proposed method

• The solver is based on a finite difference discretization of phase-field equations with spatially and temporally varying parameters.
• Eyre's theorem is applied to construct a convex splitting scheme that guarantees unconditional energy stability.
• The resulting nonlinear system is solved iteratively using a Newton-Raphson method for robust convergence.
• The method assumes a uniform computational mesh to maintain stability and efficiency.
• Boundary conditions are restricted to periodic, Neumann, or mixed periodic-Neumann types.
• The solver is implemented in a semi-implicit fashion to balance accuracy and computational cost.

Experimental results

Research questions

• RQ1 Can an implicit finite difference solver maintain unconditional stability when model parameters vary spatially and temporally?
• RQ2 How does the proposed solver compare in accuracy and efficiency to existing solvers under non-uniform parameter conditions?
• RQ3 What are the limitations in mesh type and boundary condition applicability for the proposed method?
• RQ4 To what extent does the use of Eyre's theorem ensure energy stability in complex phase-field simulations?
• RQ5 How well does the solver perform on standard benchmark problems with varying parameters?

Key findings

• The proposed solver demonstrates unconditional stability for phase-field equations with spatially and temporally varying parameters, as guaranteed by Eyre's theorem.
• The solver achieves high accuracy in benchmark simulations, closely matching known reference solutions.
• Computational efficiency is maintained through a semi-implicit, Newton-based iterative solution strategy.
• The method is restricted to uniform meshes and specific boundary conditions, limiting its immediate applicability to complex geometries.
• Validation against standard benchmarks confirms the solver’s reliability for microstructure evolution simulations.
• The approach enables stable simulations under non-uniform and time-varying parameter regimes, which are common in real materials processes.