Numerical simulation of dendritic crystal growth in a channel

Abstract

A quasi-stationary mathematical model of dendritic growth from an undercooled melt in a channel is considered. This is based on the integral representation formula of the problem which assumes that the crystals grow steadily so that the temperature field satisfies a convection-diffusion equation in the moving frame of reference. Numerically, the above phenomenon is very unstable and any small perturbation in the interface shape may give rise to the growth of an artificial (numerical) branch in the dendrite. Although important advances have been made in the numerical simulation of this process in the last decade, more stable numerical schemes are required in order to understand, in detail, the dynamics of pattern formation. The proposed numerical approach is based on a BEM formulation where cubic B-splines are used to represent the contour geometry, as a result of the requirement of C 2-continuity for the evaluation of the surface curvature. A standard quadratic interpolation is used for the densities within each boundary element, with an appropriate smoothing scheme to avoid the zig-zag instability which develops at the interface after a large number of time steps. The results obtained with the proposed numerical formulation are compared with experimental, theoretical and other numerical results.