Optimization-based Quadrilateral and Hexhedral Mesh

The accuracy and eeciency of the solution to numerical systems depends on the quality of the mesh used. Automatic mesh generation and adaptive reenement methods can, however, produce poor quality elements, even invalid elements, for quadrilateral or hexahedral mesh. We present an optimization-based approach for quadrilateral or hexahedral mesh untangling that maximizes the minimum area or volume of aaected simplicial of an element in a local submesh. We formulate the problem as a linear program that is solved by using the computationally inexpensive simplex method. By proving that the function level sets are convex regardless of the position of the free vertex, we show that the local subproblem is guaranteed to converge. We combine the mesh untangling technique with optimization-based mesh improvement techniques. Typical results showing the eeectiveness of the combined untangling and smoothing techniques are given for both quadrilateral and hexahedral meshes.