Meshfitting for the Level-set Method

Abstract

In applying the level-set method in the context of a finite-element method, errors can be minimized by adjusting the mesh to the shape of the level-set curve. The size of the different types of errors that occur depend on the goodness of fit to the zero levelset curve, the skewness of the triangles and the size of the triangles of the mesh. To fit the mesh basic methods where designed by Javierre, Den Ouden and Verbeek. These methods adjust the mesh only locally and add many edges in the process. The aim of this research was to find better ways in which to adjust the mesh, without having to add new edges. First quality measures were set up: the maximum skewness, average skewness, standard deviation of skewnesses, maximum size, minimum size, standard deviation of sizes, where the sizes and skewnesses are taken over all the triangles in the mesh. Then the existing ‘cut’ method was analysed in depth and an extension called the ‘flip’ method was added. Afterwards a new approach was used, using shortest path algorithms to decide which points to move, and orthogonal projection to the level-set curve to move them. A second method was designed using a physical model, modelling every non-fitted non-boundary edge as a spring, in order to improve the rest of the mesh. These last two methods combined improved the quality of the created meshes greatly, with its only downside being its more limited applicability.