Improved finite difference method for phase-field modelling of dendritic solidification

This paper introduces a novel numerical approach for solving phase-field models of dendritic solidification in 3-D. Traditional approaches utilising finite difference or finite element methods often introduce discretisation-induced anisotropy in the phase-field modelling of dendrite growth, particularly noticeable at low surface energy anisotropy strengths, necessitating adjustments to phase-field model parameters. Our study demonstrates an effective reduction of discretisation-induced anisotropy without altering phase-field model parameters, achieved by employing the generalised finite difference method (GFDM) with sufficiently large local support domains (stencils). We show that the GFDM can employ larger node spacings than the finite difference method and, therefore, larger time steps in the explicit time marching schemes, mitigating the increased computational cost due to the requirement for larger stencils. The GFDM is based on the polynomial weighted least squares approximation in the local support domains. Although the GFDM is usually applied to scattered node distributions, we apply the standard uniform regular distribution of nodes; hence, the insights of the current study can be straightforwardly applied to any phase-field modelling utilising the finite difference method by simply increasing the stencil size and updating the finite difference coefficients. The efficacy of the novel numerical procedure is assessed through simulations of dendrite growth in a supercooled pure melt, varying the strength of surface energy anisotropy. Additionally, we test how the nonlinear preconditioning of the phase-field model enhances computational efficiency. We mitigate the high computational cost of phase-field simulations by employing an octree-based space-time adaptive algorithm.