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Computation of three dimensional dendrites with finite elements. (English) Zbl 0844.65096
Summary: Starting from an initial seed crystal inside an undercooled liquid, the solid phase begins to grow rapidly and develops unstable 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 velocity and curvature of the interface 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; the other 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 time-dependent, 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.

MSC:
65Z05Applications of numerical analysis to physics
82C26Dynamic and nonequilibrium phase transitions (general)
35Q72Other PDE from mechanics (MSC2000)
65M60Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE)
82D25Crystals (statistical mechanics)
35R35Free boundary problems for PDE
65M50Mesh generation and refinement (IVP of PDE)
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