Gradient Augmented Reinitialization Scheme for the Level Set Method

A hybrid scheme for reinitializing the level set function and its gradient within the frame work of the augmented level set method is presented. It is based on first dividing the domain into an interfacial region (i.e. nodes close to the interface) and its complement.  Within the interfacial region,  the level set and its gradient are updated explicitly through a modified version of Newton’s method (Chopp, 2001, SIAM J. Sci. Comput. 230-244) and is implemented here within the context of Hermite  polynomials. In the region away from the interface, the solution pertains to a semi-Lagrangian implementation of the reinitialization equations, which are solved based on Hermite polynomials and are time marched with a single step and a multipoint scheme. It is shown that for various exercises, the present method predicts the signed distance function and its gradient to 4th & 3rd order (in space), respectively with regards to the L1, L2, and L? norms, provided the level set field is sufficiently smooth.  A range of test cases are also considered from the literature, where the present method is compared to existing methods and shown to be generally more accurate. Moreover, the well known issue of volume loss due to reinitialization is addressed successfully with the current implementation, even for objects that are of the size of one grid cell, and whose local radius of curvature falls below the local grid size.  For both time marching schemes, it is shown that the L2 and L? errors decay to negligible levels, are smooth in space, and do not exhibit temporal oscillations. Finally the performance of the hybrid scheme is evaluated by applying it on various kinematic test cases. For solid body rotation problems (zero deformation flow field) the benefit stemming from hybrid reinitialization is marginal. When applied to kinematic cases involving severe deformation, such as the standard vortex flow, the reinitialization strategy helps maintain a smooth level set field, which prevents serious numerical errors from developing.


Figure showing reinitialized profiles for a poorly resolved object in 2D centered at (1.8,2), (red) initial profile, (blue) reinitialized profile. Erroneous interfacial shift can be seen for the reinitialized profiles in (a), (b) and (c), where the reinitialization was performed using the CIR implementation in the entire domain. Reinitialized profiles shown in (d), (e) and (f) are obtained using hybrid reinitialization strategy.