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SUMMARYDendritic solidiÿcation with forced convection and free convection driven by contraction and thermosolutal buoyancy is simulated in two-dimensional space using a sharp-interface model. Both pure substances and alloys are considered. The model is formulated using the ÿnite element method and works directly with primitive variables. The coupled energy-and solutal concentration-equations, along with the Navier-Stokes equations for incompressible ow, are solved using di erent meshes. Temperature is solved in a ÿxed mesh that covers the whole domain (solid + liquid) where the solid-liquid interface is explicitly tracked using marker points. The concentration and momentum equations are solved in the liquid region using an adaptive mesh of triangular elements that conforms to the interface. The velocity boundary conditions are applied directly on the interface. The model is validated using a series of problems that have analytical, experimental and numerical results. Four simulations are presented: (1) crystal growth of succinonitrile with thermal convection under two small undercoolings; (2) dendritic growth into an undercooled pure melt with a uniform forced ow; (3) equiaxial dendritic growth of a pure substance and an alloy with contraction-induced convection; and (4) directional solidiÿcation of Pb-0:2 wt% Sb alloy with convection driven by the combined action of contraction, thermal and solutal buoyancy. Some of the simulation results are compared to those reported using other methods including the phase-ÿeld method; others are new. In each case, the e ects of convection on dendritic solidiÿcation are analysed.

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SUMMARYA two-dimensional ÿnite element model for dendritic solidiÿcation has been developed that is based on the direct solution of the energy equation over a ÿxed 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 solidiÿcation (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 solidiÿcation problems (the Stefan problem), it was shown that the method converges with second-order accuracy. When applied to the unstable solidiÿcation 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.