A self-adaptive oriented particles Level-Set method for tracking interfaces

A new method for tracking evolving interfaces by lagrangian particles in conjunction with a Level-Set approach is introduced. This numerical technique is based on the use of time evo- lution equations for fundamental vector and tensor quantities defined on the front and rep- resents a new and convenient way to couple the advantages of the Eulerian description given by a Level-Set function / to the use of Lagrangian massless particles. The term ori- ented points out that the information advected by the particles not only concern the spatial location, but also the local (outward) normal vector n to the interface C and the second fundamental tensor (the shape operator) rn. The particles are exactly located upon C and provide all the requested information for tracking the interface on their own. In addi- tion, a self-adaptive mechanism suitably modifies, at each time step, the markers distribu- tion in the numerical domain: each particle behaves both as a potential seeder of new markers on C (so as to guarantee an accurate reconstruction of the interface) and a de-see- der (to avoid any useless gathering of markers and to limit the computational effort). The algorithm is conceived to avoid any transport equation for / and to confine the Level-Set function to the role of a mere post-processing tool; thus, all the numerical diffusion prob- lems usually affecting the Level-Set methodology are removed. The method has been tested both on 2D and 3D configurations; it carries out a fast reconstruction of the interface and its accuracy is only limited by the spatial resolution of the mesh.