Abstract
I present a method for numerical integration of time-dependent diffusion problems with dynamically moving boundaries such as solidification fronts. The method is meant to be used for problems in which supercooling (and possibly superheating) occurs, and in which the effects of nonequilibrium interface kinetics and surface tension at the front are important. Thus it is necessary to track the front; this is done by using the Stefan boundary condition to move the front explicitly at the beginning of each time step. The diffusion field is then updated by a Galerkin finite element method. In order to generalize the Galerkin discretization to these problems in a systematic way, I base the finite element calculation on space-time finite elements: The spatial grid is fixed, but grid cells which are crossed by the front are divided into solid and liquid portions, with Galerkin integrals calculated separately over each portion. Results will be presented for the growth of a dendrite in a narrow channel.
This material is based upon work supported by the U.S. National Science Foundation under Grant No. DMR83-11053.
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References
For a review, see J.S. Langer, Instabilities and pattern formation in crystal growth, Revs. Mod. Phys. 52 (1980), 1–28.
W.W. Mullins and R.F. Sekerka, Stability of a planar interface during solidification of a dilute binary alloy, J. Appl. Phys. 35 (1964), 444–451.
D.J. Wollkind and L.A. Segel, A nonlinear stability analysis of the freezing of a dilute binary alloy, Phil. Trans. R. Soc. (Lond.) 268 (1970), 351–380.
See, e.g., D.A. Kurtze, W. van Saarloos, and J.D. Weeks, Front propagation in self-sustained and laser-driven explosive crystal growth: stability analysis and morphological aspects, Phys. Rev. B 30 (1984), 1398–1415 and references therein.
D.A. Kurtze, Nonlinear development of morphological instabilities in explosive crystallization, Phys. Rev. B 40 (1989), 11104–11119.
For a recent review, see D.A. Kessler, J. Koplik, and H. Levine, Pattern selection in fingered growth phenomena, Adv. Phys. 37 (1988), 255–339.
J. Yoo and B. Rubinsky, Numerical computation using finite elements for the moving interface in heat transfer problems with phase transformation, Numer. Heat Transfer 6 (1983), 209–222.
R. Bonnerot and P. Jamet, Numerical computation of the free boundary for the two-dimensional Stefan problem by space-time finite elements, J. Computational Phys. 25 (1977), 163–181.
D.A. Kessler, J. Koplik, and H. Levine, Dendritic growth in a channel, Phys. Rev. A 34 (1986), 4980–4987.
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© 1991 Springer Basel AG
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Kurtze, D.A. (1991). A fixed-grid finite element method for solidification. In: Neittaanmäki, P. (eds) Numerical Methods for Free Boundary Problems. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale d’Analyse Numérique, vol 99. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5715-4_20
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DOI: https://doi.org/10.1007/978-3-0348-5715-4_20
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