L p Lloyd's Energy Minimization for Quadrilateral Surface Mesh Generation

Indirect methods recombine the elements of triangular meshes to produce quadrilaterals. The resulting quadrilaterals are usually randomly oriented, which is not desirable. However, by aligning the vertices of the initial triangular mesh, precisely oriented quads can be produced. Levy’s algorithm is a non-linear optimization procedure that can align points according to a locally defined metric. It minimizes an energy functional based on the Lp distance in the local metric. The triangulation of a set of vertices smoothed with Levy’s algorithm is mainly composed of right-angled triangles, which is ideal for quad recombination. An implementation of Levy’s algorithm for the purpose of finite element computation has been developed. The implementation can create quads of desired size and orientation. The algorithm has been tested on two-dimensional geometries as well as parametrized curved surfaces. The results show an improvement of the quads alignment.