An analysis of simplex shape measures for anisotropic meshes

This paper analyses and generalises several simplex shape measures documented in the literature and currently used for mesh adaptation and mesh optimisation. It first summarises important properties of simplices and their degeneration in Euclidean space. Different simplex shape measures are then defined and validated according to a validity criterion. The shape measures are generalised to Riemannian spaces in order to extend their use to anisotropic meshes. They are then analysed and compared using complementary approaches: a visualisation method helping to show their regularity, some theoretical relations establishing their equivalence, and a discussion on the evaluation of the global quality of a mesh. Conclusions are drawn on the choice of a simplex shape measure to guide mesh optimisation.