Numerical approximation of a thermally driven interface using finite elements

A two-dimensional nite element model for dendritic solidi cation has been developed that is based on the direct solution of the energy equation over a xed mesh. The model tracks the position of the sharp solid–liquid interface using a set of marker points placed on the interface. The simulations require calculation of the temperature gradients on both sides of the interface in the direction normal to it; at the interface the heat ux is discontinuous due to the release of latent heat during the solidi cation (melting) process. Two ways to calculate the temperature gradients at the interface, evaluating their interpolants at Gauss points, were proposed. Using known one- and two-dimensional solutions to stable solidi cation problems (the Stefan problem), it was shown that the method converges with second-order accuracy. When applied to the unstable solidi cation of a crystal into an undercooled liquid, it was found that the numerical solution is extremely sensitive to the mesh size and the type of approximation used to calculate the temperature gradients at the interface, i.e. di erent approximations and di erent meshes can yield di erent solutions. The cause of these di culties is examined, the e ect of di erent types of interpolation on the simulations is investigated, and the necessary criteria to ensure converged solutions are established.