Numerical Simulation of Dendritic Solidification on Non-Graded Adaptive Cartesian Grids

We present a new computational paradigm to simulate time dependent two-dimensional dendritic solidification of pure substances. The method is based on a finite difference approximation of the heat equation and implicit level set tracking of the liquid-solid interface. We use the quadtree data structure to discretize the computational domain and a simple recursive algorithm to automatically generate the non-graded adaptive Cartesian grids, i.e., grids for which the size ratio between neighboring cells is not constrained. In our adaptive grids, the finest cells are placed around the liquid-solid interface and cells far away from the interface have much coarser resolutions. We demonstrate the second-order accuracy for both the temperature and its gradients in terms of both the average and the maximum errors. Since the evolution velocity of the interface is related to the jump of the temperature gradient across the interface, we also achieve second-order accuracy in the locations of the evolving interface. Numerical results demonstrate that our method can simulate physical effects such as surface tension and crystalline anisotropy. We also demonstrate the tremendous saving in computational time with our choice of non-graded adaptive grids.