Phase-field Simulation of Dendritic Solidification

We consider the solidi cation of a pure material from its undercooled melt, taking into account e ects of surface tension, kinetic undercooling and crystalline anisotropy. The classical mathematical formulation consists of two heat equations coupled with boundary conditions prescribed at the moving solid-liquid interface. Analytic solutions are known only for a few special cases. Since the problem has important industrial applications one is interested in numerical methods for accurate simulation. The classical formulation is not well suited for computation since it requires explicit tracking of geometrically complex free boundaries. Phaseeld methods have emerged as a popular alternative. They are based on the notion that there exists a smooth approximation of an indicator function, or phaseeld, uniquely identifying the phase at a given location and time. Phaseeld models are derived from thermodynamics and are formulated as reaction-di usion equations valid in the entire domain. Free boundaries are replaced by di use interfaces of width corresponding to transitions in the phaseeld variable. There is no need to explicitly track interfaces since they are implicitly de ned by the computed solution. The main purpose of this work is to nd computationally eÆcient methods for phaseeld simulation of dendritic solidi cation. To this end nite di erence approximations and their implementations are considered. A comparison between a rst order accurate semi-explicit time-stepping scheme and an implicit scheme based on BDF-discretization is presented. Numerical experiments in two space dimensions show that for suÆciently low error tolerances the BDF-scheme gives a solution in less simulation time. To further reduce simulation times for the implicit scheme an implementation has been written for parallel distributed memory architectures. Performance measurements on an IBM SP2 show that a parallel eÆciency exceeding 70% can be obtained for grids as small as 200 200 points on 16 processors. Agreement between phaseeld model and classical formulation is investigated in the limit of vanishing di use interface thickness. It is shown how the accuracy in the interface conditions can be increased from rst to second order in by modifying a single scalar parameter. Simulation can be performed with larger , and hence smaller computational grids, without loss of accuracy. An alternative limit where the undercooling tends to zero much faster than the di use interface thickness is considered. The analysis shows that the interfacial temperature becomes independent of to leading order only when the modi cation is applied. At low undercoolings the smallest geometric scale in the dendrite increases, and solidi cation proceeds slower. The correction predicted by analysis allows simulation for smaller undercoolings with the same computational resources. ISBN 91-7283-343-2 TRITA-NA-0219 ISSN 0348-2952 ISRN KTH/NA/R-02/19-SE