Die Berechnung dreidimensionaler Dendriten mit Finiten Elementen
A. Schmidt
Institut für Angewandte Mathematik,
Hermann-Herder-Str. 10, 79104 Freiburg, Germany
Abstract:
Starting from an initial seed crystal inside of an undercooled liquid,
the solid phase begins to grow rapidly and develops instable growth
patterns. Some growth directions are preferred because of
anisotropic parameters in the physical model. This results
in the development of dendrites.
The physical model includes the heat equation for both the liquid and
solid phases. The Gibbs--Thomson law couples the velocity of the
interphase, its curvature and the temperature.
We describe a numerical method that enables us to compute
dendritic growth of crystals in two and three space dimensions.
The method consists of two coupled finite element algorithms. The
first one solves the heat equation in the container; the other one
operates on a discretization of the free boundary and computes the
evolution of this moving interface. The two methods work with totally
independent grids. By using timedependent, locally refined and
coarsened adaptive meshes in both methods, we are able to reach a
spatial resolution necessary to compute dendritic growth in two and
three space dimensions.
AMS-Classification: 65C20, 65M50, 65M60, 65N30
Keywords:
crystal growth, dendritic growth, finite element method,
adaptive mesh refinement, moving grid
Dissertation (125 pages), Freiburg 1993, and
SFB 256 Bonn, Preprint no. 332, 1993 (125 pages)
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