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.
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Ungar, L.H., Ramprasad, N. & Brown, R.A. Finite element methods for unsteady solidification problems arising in prediction of morphological structure. J Sci Comput 3, 77–108 (1988). https://doi.org/10.1007/BF01066483
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DOI: https://doi.org/10.1007/BF01066483