A projection method for motion of triple junctions by level sets

Abstract

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.

A parametric finite element method for fourth order geometric evolution equations

“…A level set approach for mean curvature flow of curve networks has been considered in Merriman, Bence, and Osher (1994), Zhao, Merriman, Osher, and Wang (1998) and Smith, Solis, and Chopp (2002). A phase field approximation of the motion of surface diffusion of a closed curve was studied in Barrett, Nürnberg, and Styles (2004), and its extension to curve networks is given in Barrett, Garcke, and Nürnberg (2006a).…”

The Voronoi Implicit Interface Method for computing multiphase physics

“…Building robust numerical methods to tackle these problems is equally difficult, requiring numerical resolution of sharp corners and singularities, and recharacterization of domains when topologies change. A variety of methods have been proposed to handle these problems, including (i) front tracking methods, which explicitly track the interface, modeled as moving segments in two dimensions and moving triangles in three dimensions, (ii) volume of fluid methods, which use fixed Eulerian cells and assign a volume fraction for each phase within a cell, (iii) level set methods (1), which use an implicit formulation to represent the interface, and treat each region/phase separately, followed by a repair procedure which reattaches the evolving regions to each other (1)(2)(3), and (iv) diffusion generated motion which combine diffusion via convolution with reconstruction procedures to simulate multiphase mean curvature flow (4). Although there are advantages and disadvantages to each of these approaches, it has remained a challenge to robustly and accurately handle the wide range of possible motions of evolving, highly complex interconnected interfaces separating a large number of phases under time-resolved physics.…”

Comparison of Numerical Methods for Ternary Fluid Flows: Immersed Boundary, Level-Set, and Phase-Field Methods

“…tension coefficient σ k (see [19]). Recall that σ 3,1 would not be defined in our compound droplet case.…”

“…Merrian et al [18] represented each phase by using an individual level set function. The projection method, which uses only (n − 1)-level set functions to represent the interfaces of n-phases, was developed to resolve the triple junction problem by Smith et al [19].…”