Numerical Simulation of Dendritic Solidi cation Using a Phase Field Model

Solidi cation of a pure material from its undercooled melt is considered, taking into account e ects of surface tension, kinetic undercooling and anisotropy. The resulting crystals exhibit complicated dendritic shapes. To avoid explicit tracking of the solid-liquid interfaces a phase eld model is employed. A di use interface of arti cial width is introduced and the solidi cation is modeled by a set of reaction-di usion equations valid in the entire computational domain. The phase eld equations are discretized in space using second order nite di erences on uniform Cartesian grids. A semi-explicit rst order accurate time stepping scheme based on equation splitting is compared to a more accurate fully implicit scheme. Numerical methods for eÆcient computer implementation are discussed. Convergence rates are experimentally validated for decreasing time steps and it is shown that for some cases the rst order scheme may fail to capture the correct growth rate of the crystal. A formal asymptotic analysis of the phase eld model is presented in the limit of vanishing di use interface thickness. It was recently shown how an increased accuracy in the asymptotics can be used to obtain better agreement between the phase eld model and the corresponding sharp interface formulation. This has implications on computational eÆciency by allowing simulation with larger di use interface width which is veri ed by numerical experiments. An implementation of the fully implicit scheme has been written for parallel distributed memory architectures. Experiments conducted on an IBM SP2 and networks of Sun Ultra 5 workstations show that the code scales well with the number of processors. ISBN 91-7170-580-5 TRITA-NA-0013 ISSN 0348-2952 ISRN KTH/NA/R--00/13--SE