A local meshless method for numerical simulation of dendritic crystal growth

Solidification processes are present in a wide range of manufacturing methods and applications, from metallurgy to food processing. In recent years, Phase-Field models have been increasingly used to simulate and predict the formation and evolution of material microstructure and phase change interfacial kinetics. In this article, we study the phase-field model of solidification for numerical simulation of dendritic crystal growth that occurs during the casting of metals and alloys based on the kobayashi model. At first, the kobayashi  phase-field model, which describes the solidification of a pure material from an undercooled melt, is introduced in detail. In discretization process of this model, the time derivatives are approximated via finite difference method.   Then the local meshless moving Kriging method is applied for discretization of the model in space direction. The moving Kriging method is a truly meshless method in which the unknown function can be approximated locally, and this leads to the sparsity of the coefficient matrix. As the shape functions possess the Kronecker delta function property, boundary conditions can be implemented without any difficulties. The model is simulated for various values of it’s parameters.  Numerical simulations illustrate the applicability and effectiveness of the proposed method. As a consequence, it is found that the method is very efficient and accurate for phase-field models compared with those of other conventional methods. Therefore, this method can be considered as an attractive alternative to existing mesh-based methods in solving phase-field models.