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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. Copyright © 2006 John Wiley & Sons, Ltd.