Phase Field Instabilities and Adaptive Mesh Refinement

We show that the phase field model, used for numerical simulation of solidification, can possess unphysical stable intermediate states for certain values of the parameters. We demonstrate that these states can be seen in simple computations in one dimension; in two dimensions we use an adaptive mesh refinement algorithm to exhibit appearance of the intermediate states in realistic geometries of dendritic growth. The Phase-Field Model We study solutions of the one-dimensional phase-field model, which we write using the notation and scalings of [1]: ø OE t = f(OE; u) + ffl 2 OE xx e t = Du xx (1) e = u \Gamma 1 2 p(OE) Here OE(x; t) is a phase variable, taking the value 1 in solid and \Gamma1 in liquid, and u(x; t) is temperature scaled by the latent heat of fusion divided by the to appear in Modern Methods for Modeling Microstructure Evolution in Materials, TMS/SIAM, 1996 y The University of Chicago, Department of Mathematics, 5734 S. University Ave, Chicago, IL 60637; al..