Isotropic finite-difference approximations for phase-field simulations of polycrystalline alloy solidification

TL;DR

This paper develops isotropic finite-difference schemes for phase-field models of alloy solidification, significantly reducing lattice anisotropy artifacts in simulations on parallel architectures.

Contribution

It introduces new isotropic finite-difference approximations for differential operators in phase-field models, improving accuracy in polycrystalline alloy solidification simulations.

Findings

Isotropic schemes reduce lattice anisotropy effects.

Simulations show improved interface dynamics accuracy.

Method applicable in 2D and 3D models.

Abstract

Phase-field models of microstructural pattern formation during alloy solidification are commonly solved numerically using the finite-difference method, which is ideally suited to carry out computationally efficient simulations on massively parallel computer architectures such as Graphic Processing Units. However, one known drawback of this method is that the discretization of differential terms involving spatial derivatives introduces a spurious lattice anisotropy that can influence the solid-liquid interface dynamics. We find that this influence is significant for the case of polycrystalline dendritic solidification, where the crystal axes of different grains do not generally coincide with the reference axes of the finite-difference lattice. In particular, we find that with the commonly used finite-difference implementation of the quantitative phase-field model of binary alloy solidification, both the operating state of the dendrite tip and the dendrite growth orientation are strongly affected by the lattice anisotropy. To circumvent this problem, we use known methods in both real and Fourier space to derive finite-difference approximations of leading differential terms in 2D and 3D that are isotropic at order $h^2$ of the lattice spacing $h$. Importantly, those terms include the divergence of the anti-trapping current that is found to have a critical influence on pattern selection. The 2D and 3D discretizations use an approximated form of the anti-trapping current that facilitates the Fourier-space derivation of the associated isotropic differential operator at $O(h^2)$, but we also derive a 2D discretization of the standard form of this current. Finally, we present 2D and 3D phase-field simulations of alloy solidification, showing that the isotropic finite-difference implementations can dramatically reduce spurious lattice anisotropy effects.