Finite element methods for unsteady solidification problems arising in prediction of morphological structure

Abstract

Galerkin finite element methods are presented for calculation of the dynamic transitions between planar and deep two-dimensional cellular interface morphologies in directional solidification of a binary alloy from models that include solute transport, the phase diagram, and the interfacial free energy between melt and crystals. The unknown melt-solid interface shape is accounted for in the finite element formulation by mapping the equations to a fixed domain. Novel nonorthogonal transformations are introduced combining cylindrical and Cartesian coordinate interface representations for approximating the deep cellular interfaces that evolve from a planar solidification front. The algorithm for time integration combines a fully implicit Adams-Moulton algorithm with the Isotherm-Newton method for solving the nonlinear set of differential-algebraic equations that result from the spatial discretization of the moving-boundary problem. The fully implicit scheme is found to be more accurate and efficient than an explicit predictor-corrector algorithm. Sample calculations show the connectivity between families of shapes with resonant spatial wavelengths.