An oriented particle level set method based on surface coordinates

In this work we present a new method for tracking evolving interfaces. Like the Adaptive Oriented Particle Level Set method (AOPLS, Ianniello and Mascio, 2010), it uses a combination of unconnected particles containing geometric information and a level set function to track the interface. In the present method, the geometric information, solved for in each particle, is a set of tangent vectors and the corresponding curvature tensor of the interface. Compared to the AOPLS method, the amount of geometric data stored in each particle is roughly halved, while still retaining a description of the interface that includes both orientation and curvature. A mechanism for ensuring a sufficiently dense distribution of particles on the interface is also included. From the information stored in each particle, a “continuous” representation of the interface is obtained by transferring the information to a level set function stored on an Eulerian grid. Unlike the AOPLS method, the present method takes the curvature into account when the zero level set is constructed. This results in significant improvements in regions of high curvature. Two different formulations of the new method have been explored. In the first, the tangent vectors deform purely according to the velocity field. In the second, the tangent vectors are forced to coincide with the principal directions of the surface. Both formulations are shown to be applicable.