Algebraic Mesh Quality Metrics
- 1 January 2001
- journal article
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Scientific Computing
- Vol. 23 (1), 193-218
- https://doi.org/10.1137/s1064827500371499
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.This publication has 13 references indexed in Scilit:
- Achieving finite element mesh quality via optimization of the Jacobian matrix norm and associated quantities. Part II?A framework for volume mesh optimization and the condition number of the Jacobian matrixInternational Journal for Numerical Methods in Engineering, 2000
- Achieving finite element mesh quality via optimization of the Jacobian matrix norm and associated quantities. Part I?a framework for surface mesh optimizationInternational Journal for Numerical Methods in Engineering, 2000
- Qualitative measures for initial meshesInternational Journal for Numerical Methods in Engineering, 2000
- A Framework for Variational Grid Generation: Conditioning the Jacobian Matrix with Matrix NormsSIAM Journal on Scientific Computing, 2000
- Applied Numerical Linear AlgebraPublished by Society for Industrial & Applied Mathematics (SIAM) ,1997
- On the shape of tetrahedra from bisectionMathematics of Computation, 1994
- Relationship between tetrahedron shape measuresBIT Numerical Mathematics, 1994
- Variational approach to grid generationNumerical Methods for Partial Differential Equations, 1992
- A DISTORTION METRIC FOR ISOPARAMETRIC FINITE ELEMENTSTransactions of the Canadian Society for Mechanical Engineering, 1988
- Adaptive zoning for singular problems in two dimensionsJournal of Computational Physics, 1982