A study on different implementations of Neumann boundary conditions in the meshless RBF-FD method for the phase-field modelling of dendrite growth

Abstract

This paper studies and assesses different Neumann boundary conditions (BC) implementations in the radial basis function generated finite difference (RBF-FD) method. We analyse four BC implementations by solving a phase-field model for single dendrite growth in supercooled pure melts. In the first BC implementation, the BC are satisfied when constructing interpolation problems in the local support domains near the boundary. In the second one, the BC are satisfied by solving an additional system of linear equations for the field values in the boundary nodes. In the third one, we add a layer of ghost nodes to the boundary nodes; the BC are satisfied by solving an additional system of linear equations for the field values in the ghost nodes. The fourth BC implementation uses the same node distribution as the third one; since we are dealing with the symmetric BC, we set the values in the ghost nodes by direct mirroring. We analyse the influence of the size of a local support domain and the type of node distribution (regular/scattered) on the accuracy. We show that using ghost nodes is recommended to consider Neumann BC in the RBF-FD method accurately when solving phase-field models for dendritic growth.