Simulation of dendritic growth on a spherical surface using a multi-component phase-field model

Abstract

We consider a numerical algorithm for a phase-field mathematical model of multiple dendritic growth on a spherical surface. One numerical method for curved surfaces is a triangular mesh-based computation method for surfaces. Calculating the governing equations with anisotropic properties using interface angles in a triangular grid is a significant challenge. To solve this issue, we compute the phase-field equation by rotating the triangular mesh relative to the vertex of the crystal seed and then projecting and interpolating it to Cartesian coordinates. When projecting the triangular mesh onto Cartesian coordinates, we apply an adaptive block region that embeds each dendritic phase. The growth simulations of multiple crystals present additional challenges. For multi-crystals, the criteria for rotation are ambiguous; hence, the criteria for rotation are clarified by applying the vector-valued phase-field equation to resolve this problem. Various numerical experiments are conducted on a spherical surface to verify the reliability and robustness of the proposed numerical algorithm to solve the phase-field equations of multiple dendritic growth. We comprehensively present the computational results, and show compelling evidence that validates the reliability and robustness of our computational method.