Isotropic discretization methods of Laplacian and generalized divergence operators in phase field models

Abstract

Phase field models have been extensively used to address various physical phenomena. However, discretization-induced anisotropy remains a longstanding challenge for phase field models. By using a hexagonal mesh in 2D, we describe isotropic discretization methods for the computation of Laplacian and generalized divergence operators. Quantitative analyses derived from discrete Fourier analysis prove that our methods are more isotropic than commonly used approaches, including isotropic methods for square mesh. To compare the performance of conventional and our discretization methods, a specific phase field model of alloy solidification was selected to perform benchmark simulations. Various 2D simulations using different discretization methods were carried out to verify the accuracy and efficiency of the improved numerical methods in a hexagonal mesh. We emphasize that the improved numerical methods using a hexagonal mesh are general and may be equally applied to other physical models that include the same operators.