Marching cubes: A high resolution 3D surface construction algorithm
SIGGRAPH '87 Proceedings of the 14th annual conference on Computer graphics and interactive techniques
Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations
Journal of Computational Physics
A continuum method for modeling surface tension
Journal of Computational Physics
Motion of multiple junctions: a level set approach
Journal of Computational Physics
A level set approach for computing solutions to incompressible two-phase flow
Journal of Computational Physics
Shape Modeling with Front Propagation: A Level Set Approach
IEEE Transactions on Pattern Analysis and Machine Intelligence
A fast level set method for propagating interfaces
Journal of Computational Physics
A numerical method for tracking curve networks moving with curvature motion
Journal of Computational Physics
A variational level set approach to multiphase motion
Journal of Computational Physics
Journal of Computational Physics
The fast construction of extension velocities in level set methods
Journal of Computational Physics
A multiphase field concept: numerical simulations of moving phase boundaries and multiple junctions
SIAM Journal on Applied Mathematics
Some Improvements of the Fast Marching Method
SIAM Journal on Scientific Computing
Ordered Upwind Methods for Static Hamilton--Jacobi Equations: Theory and Algorithms
SIAM Journal on Numerical Analysis
Diffusion generated motion for grain growth in two and three dimensions
Journal of Computational Physics
Numerical simulations of two-dimensional foam by the immersed boundary method
Journal of Computational Physics
Hi-index | 31.45 |
We analyze a new mathematical and numerical framework, the ''Voronoi Implicit Interface Method'' (''VIIM''), first introduced in Saye and Sethian (2011) [R.I. Saye, J.A. Sethian, The Voronoi Implicit Interface Method for computing multiphase physics, PNAS 108 (49) (2011) 19498-19503] for tracking multiple interacting and evolving regions (''phases'') whose motion is determined by complex physics (fluids, mechanics, elasticity, etc.). From a mathematical point of view, the method provides a theoretical framework for moving interface problems that involve multiple junctions, defining the motion as the formal limit of a sequence of related problems. Discretizing this theoretical framework provides a numerical methodolology which automatically handles multiple junctions, triple points and quadruple points in two dimensions, as well as triple lines, etc. in higher dimensions. Topological changes in the system occur naturally, with no surgery required. In this paper, we present the method in detail, and demonstrate several new extensions of the method to different physical phenomena, including curvature flow with surface energy densities defined on a per-phase basis, as well as multiphase fluid flow in which density, viscosity and surface tension can be defined on a per-phase basis. We test this method in a variety of ways. We perform rigorous analysis and demonstrate convergence in both two and three dimensions for a variety of evolving interface problems, including verification of von Neumann-Mullins' law in two dimensions (and its analog in three dimensions), as well as normal driven flow and curvature flow with and without constraints, demonstrating topological change and the effects of different boundary conditions. We couple the method to a second order projection method solver for incompressible fluid flow, and study the effects of membrane permeability and impermeability, large shearing torsional forces, and the effects of varying density, viscosity and surface tension on a per-phase basis. Finally, we demonstrate convergence in both space and time of a topological change in a multiphase foam.