Defining Quality Measures for Mesh Optimization on Parameterized CAD Surfaces

We present a technique to extend any distortion (quality) measure for planar meshes to meshes on parameterized surfaces. The resulting distortion (quality) measure is expressed in terms of the parametric coordinates of the nodes. This extended distortion (quality) measure can be used to check the quality and validity of a surface mesh. We also apply it to simultaneously smooth and untangle surface meshes by minimizing the extended distortion measure. The minimization is performed in terms of the parametric coordinates of the nodes and therefore, the nodes always lie on the surface. In addition, we apply the extension technique to define quality measures for high-order meshes on parameterized surfaces. Finally, we include some examples to show the application of the proposed technique. Specifically, we extend several Jacobian based measures, and we smooth and untangle linear and high-order meshes on CAD surfaces.