Quality Measures for Curvilinear Finite Elements

We present a method for computing robust shape quality measures defined for finite elements of any order and any type, including curved pyramids. The measures are heuristically defined as the minimum of the pointwise quality of curved elements. Three pointwise qualities are considered: the ICN that is related to the conditioning of the stiffness matrix for straight-sided simplicial elements, the scaled Jacobian that is defined for quadrangles and hexahedra, and a new shape quality that is defined for triangles and tetrahedra. Based on previous work presented by Johnen et al. (Journal of Computational Physics 233:359–372, 2013, [1]); Johnen and Geuzaine (Journal of Computational Physics 299:124–129, 2015, [2]), the computation of the minimum of the pointwise qualities is efficient. The key feature is to expand polynomial quantities into Bezier bases which allows to compute sharp bounds on the minimum of the pointwise quality measures.