Numerical approximation of a thermally driven interface using finite elements

Abstract

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.

A level set simulation of dendritic solidification with combined features of front-tracking and fixed-domain methods

“…As pointed out in [33], the formed secondary dendritic arms are different for coarse and fine grid simulations (see Figs. 9 (a) and (b)), while the primary dendrite tips are growing with the same velocity for both grids (they all reach the computational boundary at time about 0.036).…”

“…The morphologies obtained in [1] using a front-tracking technique appear to have a much higher mesh-dependency than the results reported here. The difference in published results [1,3,16,33] for this problem suggest that its solution is highly sensitive to perturbations during the solution process and that the problem is indeed a non-trivial one.…”

“…We also computed the solution to a crystal growth problem under anisotropy with six-fold symmetry examined previously in [33]. The problem definition is similar to the earlier example.…”

“…9 (a) and (b)), while the primary dendrite tips are growing with the same velocity for both grids (they all reach the computational boundary at time about 0.036). However, note that in the work of [33] using a front tracking method with markers, the total time for the crystal to reach the computational boundary varied from 0.035 with grid 800 × 800 to 0.045 with grid 400 × 400. The apparent improvement provided by the present methodology may be due to its energy conserving nature.…”

Characterization of the Al-3wt.%Si alloy in unsteady-state horizontal directional solidification

“…The insertion of analytical expressions for V L and T R into experimental equations has been proposed in order to establish experimental formulae that relate cellular 19,22 and dendritic 20,23 spacings with the unsteady-state solidification variables. On the other hand, various numerical methods including cellular automata [26][27][28] , front-tracking methods [29][30][31] , phase field techniques [32][33][34] and level set methods [35][36] have been developed to study the growth of dendrites.…”