Numerical simulation of crystal growth in three dimensions using a sharp-interface finite element method

A sharp-interface numerical model is presented to simulate thermally driven crystal growth in three-dimensional space. The model is formulated using the finite element method and works directly with primitive variables. It solves the energy equation in a fixed volume mesh while explicitly tracking the motion of the solid-liquid interface. The three-dimensional interface is represented by connected planar triangles that form a surface mesh. To accurately capture the morphology of the growing dendrite, the surface mesh is updated every few time-steps so that the quality of the triangles is maintained and the size of the triangles is always kept in a range associated with the element size of the fixed volume mesh. The interface curvature is calculated by a least-squares paraboloid-fitting to neighbouring nodes. The model is validated through a comparison with an exact solution of a three-dimensional Stefan problem, a mesh refinement study, a mesh orientation test and a comparison with solvability theory. It is shown that the interface position is tracked to second-order accuracy. Simulations are performed under different combinations of the undercooling and surface energy. The effects of these parameters on the growth and morphology of the dendrites are studied.