Unconditionally stable method and numerical solution of the hyperbolic phase-field crystal equation

Area of Science:

• Computational Materials Science
• Mathematical Modeling
• Physics

Background:

• The phase-field crystal (PFC) model bridges atomistic and standard phase-field methods for atomic-scale simulations.
• A hyperbolic variant of the PFC model was developed to capture both fast propagative and slow diffusive dynamics.

Purpose of the Study:

• To present a novel finite-element method (FEM) for solving the hyperbolic PFC equation.
• To develop an unconditionally stable time integration algorithm for the hyperbolic PFC model.
• To validate the method's accuracy and applicability in simulating material microstructures.

Main Methods:

• A finite-element method employing C0-continuous Lagrange elements with quadratic shape functions for spatial discretization.
• An unconditionally stable, second-order accurate in time integration algorithm for the space-time discretization.
• Numerical simulations and benchmarks in two-dimensional space.

Main Results:

• The proposed FEM algorithm is analytically proven to be unconditionally stable.
• Numerical simulations demonstrate a monotonic decrease in free energy during transitions to striped patterns.
• Benchmarks confirm the algorithm's effectiveness in determining equilibrium states and modeling pattern formation.

Conclusions:

• The developed finite-element method provides a robust and accurate approach for simulating dynamics governed by the hyperbolic PFC equation.
• The algorithm is suitable for quantitative predictions, including lattice parameter and velocity selection in crystal growth scenarios.
• This work advances computational methods for materials science at atomic length and diffusive time scales.