Efficient direct re-initialization approach of a level set method for unstructured meshes

A new direct re-initialization algorithm of a level set method is proposed for two- and three-dimensional unstructured meshes. In this method, the signed distances (level set values) to an interface are to be computed for all of the nodes inside a narrow band. In the 2D case, simple geometrical relations are employed for efficient calculation of the distance from a node inside a narrow band of an interface segment to the segment without finding the foot of the perpendicular drawn from the node onto the line including the interface segment. In the 3D case, an algorithm similar to the algorithm for the 2D case is introduced to determine nodes with feet far from the interface segment quickly. For the remaining nodes, a new technique to determine the existence of the foot on the interface segment is proposed in the re-initialization procedure. Benchmark problems with prescribed velocities are simulated to validate the proposed method for various mesh resolutions. The proposed method is not only demonstrated to be accurate but also highly efficient. Furthermore, assessment of the proposed method with simulations related to free surface flows governed by the incompressible Navier-Stokes equations is successfully performed in the present study.