A variational level set approach to multiphase motion

A coupled level set method for the motion of multiple junctions (of, e.g., solid, liquid, and grain boundaries), which follows the gradient flow for an energy functional consisting of surface tension (proportional to length) and bulk energies (proportional to area), is developed. The approach combines the level set method of S. Osher and J. A. Sethian with a theoretical variational formulation of the motion by F. Reitich and H. M. Soner. The resulting method uses as many level set functions as there are regions and the energy functional is evaluated entirely in terms of level set functions. The gradient projection method leads to a coupled system of perturbed (by curvature terms) Hamilton-Jacobi equations. The coupling is enforced using a single Lagrange multiplier associated with a constraint which essentially prevents (a) regions from overlapping and (b) the development of a vacuum. The numerical implementation is relatively simple and the results agree with (and go beyond) the theory as given in [12]. Other applications of this methodology, including the decomposition of a domain into subregions with minimal interface length, are discussed. Finally, some new techniques and results in level set methodology are presented.