I discuss how to implement efficiently phase field models in 2 and 3 dimensions, using adaptive mesh refinement. The accuracy of the computations is compared to solvability theory. I also present level set methods, which attain accuracy comparable to the phase field calculations, and discuss the pros and cons of both methods. Lastly, I discuss our recent work in which we have been able to use phase field methods to compute 3 dimensional solidification patterns in the presence of fluid flow.
This work was performed in collaboration with Jon Dantzig, Nik Provatas, Tae Kim and Jun-Ho Jeong, and was supported by NASA through grants NAG8-1657 and NAG 8-1249.