A grid based particle method for evolution of open curves and surfaces

We propose a new numerical method for modeling motion of open curves in two dimensions and open surfaces in three dimensions. Following the grid based particle method we proposed in [S. Leung, H.K. Zhao, A grid based particle method for moving interface problems. J. Comput. Phys. 228 (2009) 2993–3024], we represent the open curve or the open surface by meshless Lagrangian particles sampled according to an underlying fixed Eulerian mesh. The underlying grid is used to provide a quasi-uniform sampling and neighboring information for meshless particles. The key idea in the current paper is to represent and to track end-points of the open curve and boundary-points of the open surface explicitly and consistently with interior particles. We apply our algorithms to several applications including spiral crystal growth modeling and image segmentation using active contours.