A smoothed particle hydrodynamics-phase field method with radial basis functions and moving least squares for meshfree simulation of dendritic solidification

We present a robust and efficient approach to meshfree phase-field (PF) simulation of dendritic solidification on arbitrary domain geometries using smoothed particle hydrodynamics (SPH). We use radial basis functions (RBFs) and moving least squares (MLS) as alternative approaches for constructing kernel approximation functions exhibiting a higher order of consistency than traditional kernel functions used in SPH. In the proposed smoothed particle hydrodynamics-phase field method (SPH–PFM), proper discretization of the PF order parameter at the diffuse interface region can be easily accomplished independently from the particle spacing resolution used for computing the thermal field distribution. We use an implicit geometry construction approach to automatically generate virtual boundary particles to impose Neumann-type boundary conditions at the domain boundaries. We solve the Allen–Cahn equation locally at particles constructed at a narrow band around the interface region. Additionally, only first-order derivatives of the meshfree approximation functions are needed in our implementation to solve the governing equations. Mathematical formulation and detailed analysis will be presented and discussed where we investigate the effect of the meshfree approximation scheme on the final morphology of the grown dendrite.