Algebraic mesh quality metrics (Q2719276)

The author builds a mathematical theory of quality metrics for structured and unstructured meshes. A mesh has elements and nodes. An element quality metric is a scalar function of node positions that measures some geometric property of the element with the given nodes.NEWLINENEWLINENEWLINEThe author gives a list of desirable properties for a mesh quality metric and relates them to a host of mesh quality metrics defined over the years, such as maximum angle, ratio of quadrilateral diagonal lengths, aspect ratio and skew.NEWLINENEWLINENEWLINEThe general theory is based on the Jacobian matrix of a mesh element which in the case of a tetrahedron with vertices \(x_k\in \mathbb{R}^3\) \((k= 0,1,2,3)\) takes the form NEWLINE\[NEWLINEA_0= \begin{pmatrix} x_1- x_0 & x_2- x_0 & x_3- x_0\\ y_1- y_0 & y_2- y_0 & y_3- y_0\\ z_1- z_0 & z_2- z_0 & z_3- z_0\end{pmatrix}.NEWLINE\]NEWLINE This matrix is factorized in standard matrix factorizations. The theory provides a means of constructing, classifying and evaluating mesh quality metrics. Algebraic measures for skew, length ratio, shape, volume and orientation can be defined abstractly and then studied in specific examples. The paper contains extensive numerical examples.