Algebraic Mesh Quality Metrics

Abstract

Quality metrics for structured and unstructured mesh generation are placed within an algebraic framework to form a mathematical theory of mesh quality metrics. The theory, based on the Jacobian and related matrices, provides a means of constructing, classifying, and evaluating mesh quality metrics. The Jacobian matrix is factored into geometrically meaningful parts. A nodally invariant Jacobian matrix can be defined for simplicial elements using a weight matrix derived from the Jacobian matrix of an ideal reference element. Scale and orientation-invariant algebraic mesh quality metrics are defined. The singular value decomposition is used to study relationships between metrics. Equivalence of the element condition number and mean ratio metrics is proved. The condition number is shown to measure the distance of an element to the set of degenerate elements. Algebraic measures for skew, length ratio, shape, volume, and orientation are defined abstractly, with specific examples given. Two combined metrics, shape-volume and shape-volume orientation, are algebraically defined and examples of such metrics are given. Algebraic mesh quality metrics are extended to nonsimplicial elements. A series of numerical tests verifies the theoretical properties of the metrics defined.