Incorporating the Min-Max Mesh Optimization Method within the Target-Matrix Paradigm
The Target-Matrix Optimization Paradigm (TMOP) is a method for improving mesh quality via node movement. Through the use of appropriate pre-defined quality metrics and Target-matrices, TMOP provides the ability to define mesh quality in terms of element shape, volume, and edge orientation as a function of position within the mesh, thus facilitating high-level goals such as size-adapted meshing. The Paradigm is developed further in this paper by showing how it can be applied to the problem of improving the mesh in locations where the quality is the worst. Improving the worst quality portions of a mesh was investigated by Freitag, Amenta, and others, who showed that the problem can be formulated as a min-max optimization problem. The Target-paradigm is used to extend the previous work in two important respects. First, by using quality metrics from the Target-paradigm, quality can be user-specified via Target-matrices instead of the quality definition being hard-wired into the quality metric. Second, by making use of the sample point concept in TMOP, one can extend min-max mesh optimization to non-simplicial element types such as quadrilaterals. Numerical experiments were performed to illustrate these extensions and to investigate their utility. A non-gradient solver known as the amoeba was used to numerically compute the optimal solutions. This derivative-free method, in conjunction with the use of barrier metrics from TMOP, enables the solution of the min-max problem without the use of explicit constraints to guarantee that the optimal mesh is non-inverted. The experiments showed that the use of Target-metrics in min-max with triangle (or tetrahedral) element meshes is able to effectively improve the worst quality elements in terms of shape or in terms of size+shape. The experiments also showed that the mesh min-max extension to quadrilateral element meshes can lead to acceptable meshes using barrier metrics from the Target-paradigm, provided that inter-element mesh smoothness is not important. Poor behavior was observed when using non-barrier Target-metrics in min-max with quadrilateral element meshes. For both simplicial and quadrilateral meshes, the min-sum method can be competitive with min-max, however.