A Variational Level Set Approach to Multiphase Motion

The point at which they meet (the triple junction) has prescribed angles which can be shown [12] to be defined by 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 sin u1 f23 5 sin u2 f31 5 sin u3 f12 , (1.2) 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 where ui is the angle between the two curves Gij and Gij9 , functional is evaluated entirely in terms of level set functions. The j ? j9; see Fig. 1. gradient projection method leads to a coupled system of perturbed This problem was defined and analyzed clearly in a paper (by curvature terms) Hamilton–Jacobi equations. The coupling is by Reitich and Soner [12], and we base our approach in enforced using a single Lagrange multiplier associated with a conpart on their theoretical framework. Their method does straint which essentially prevents (a) regions from overlapping and (b) the development of a vacuum. The numerical implementation not lend itself to a direct numerical treatment. is relatively simple and the results agree with (and go beyond) Our objective here is to develop and implement numerithe theory as given in [12]. Other applications of this methocal algorithms which ‘‘capture’’ rather than ‘‘track’’ the dology, including the decomposition of a domain into subregions interfaces, based on the level set method of Osher and with minimal interface length, are discussed. Finally, some new Sethian [9]. The usual advantages of the level set method techniques and results in level set methodology are presented. Q 1996 Academic Press, Inc. hold (see, e.g., [2, 8, 9, 16]). In the case of a single interface separating two phases the central idea is to follow the evolution of a function f, whose zero-level set corresponds