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We develop a projection method to treat the motion of multiple junctions (such as contact lines) in the level set formulation. Multiple junctions are relevant to many fields including fluid dynamics, foams, and semiconductor manufacture. In the level set method an interface is defined as the zero level set of a smooth function. For an N-phase system the location of all interfaces can be specified by N - 1 functions (hence only one level set function is needed for a two-phase system). For N > 2 we describe a symmetric projection of the N level set functions onto an N - 1 dimensional manifold. This reduction in phase space eliminates unacceptable values of the level set functions (such as cases where more than one is positive at a given point.) This prevents the formation of vacuums or overlaps at multiple junctions during interface evolution. Further, this method can be applied to any number of phases and spatial dimensions. We present two- and three-dimensional results showing that the method gives correct equilibrium contact angles and produces accurate dynamics in multi-phase fluids.
Smith et al. (Mon,) studied this question.