Label-Invariant Mesh Quality Metrics

Mappings from a master element to the physical mesh element, in conjunction with local metrics such as those appearing in the Target-matrix paradigm, are used to measure quality at points within an element. The approach is applied to both linear and quadratic triangular elements; this enables, for example, one to measure quality within a quadratic finite element. Quality within an element may also be measured on a set of symmetry points, leading to so-called symmetry metrics. An important issue having to do with the labeling of the element vertices is relevant to mesh quality tools such as Verdict and Mesquite. Certain quality measures like area, volume, and shape should be label-invariant, while others such as aspect ratio and orientation should not. It is shown that local metrics whose Jacobian matrix is non-constant are label-invariant only at the center of the element, while symmetry metrics can be label-invariant anywhere within the element, provided the reference element is properly restricted. 1 Measuring Quality Within Mesh Elements Mesh quality is important for maintaining accuracy and efficiency of numerical simulations based on the solution of partial differential equations [6]. Mesh quality metrics are used to measure mesh quality and there is an extensive literature on the subject, particularly for finite element meshes [8], [14], [16], [17], [19]. For simplicial elements, ‘shape’ is an important measure [3], [15]. A shape measure based on condition number was proposed in [4], [5]. In [2] and [7] the notion of shape for simplicial elements was formalized. Few works discuss quality measures for quadratic elements [1], [18]; the latter reference being the only example that goes beyond detecting singular points. Significantly, the latter is limited to triangle elements. Engineers usually measure mesh quality by one of two basic approaches, depending on whether they are working with unstructured or structured meshes. The quality of an unstructured mesh is most often studied in terms of the individual elements within themesh. Elements aremost often polygons or polyhedra, with triangles, tetrahedra, quadrilaterals, hexahedra, prisms, and pyramids being the most commonly used types. A mesh element contains vertices and/or nodes, usually given in some canonical ordering. The vertices/nodes