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## Article electronically published on January 8, 2003 ANALYSIS OF TRIANGLE QUALITY MEASURES

### Citations

393 |
Introduction to geometry.
- Coxeter
- 1969
(Show Context)
Citation Context ...tations: |t|0 |t|2 = = min(a, b, c), � a2 + b2 + c2 , |t|∞ = max(a, b, c), θ0 = min(α, β, γ), θ∞ = max(α, β, γ). We shall assume without proof a number of results from elementary geometry (see, e.g., =-=[6]-=- for proofs and details). In particular, we will make use of the well-known relations (1.1) 2R = abc a b c = = = 2A sin α sin β sin γ , where A is given by (1.2) A = rp, as well as by Heron’s formula ... |

153 |
On the angle condition in the finite element method
- Babuska, Aziz
- 1976
(Show Context)
Citation Context ...iangle quality measures. Introduction Assessment of mesh quality is an important requirement in both the selection of a finite element mesh and the evaluation of meshes that have undergone adaptation =-=[2, 5]-=-. Several measures of element quality have been proposed [1, 8, 11] based on the dimensionless ratios of various geometric parameters. Apart from the work of [11], there appears to be almost no discus... |

67 | Mesh Smoothing Using A Posteriori Error Estimates.
- Bank, K
- 1997
(Show Context)
Citation Context ...sin θ∞ sin(θ0 + θ∞) The Frobenius norm has been investigated in [7, 9]. Interestingly, this quality measure, more specifically its inverse called shape regularity, has been independently suggested by =-=[4]-=- in the context of mesh adaptation based on a posteriori error estimates. For p =1,weget (7.15) κ1 = (σ1 + σ2) 2 =2+κ2, and for p →∞we obtain (7.16) (7.17) κ∞ = σ2 σ1σ2 = σ1 σ2 2 σ1σ2 = 1 2κ2 + 1 2 = ... |

58 | Algebraic Mesh Quality Metrics
- Knupp
- 2001
(Show Context)
Citation Context ...e work of [11], there appears to be almost no discussion in the literature on the relative merits of these particular quality measures. More recently, alternative quality measures have been suggested =-=[3, 7, 9, 10]-=-. These alternative measures are derived from the singular values of a matrix whose columns are formed by the edge vectors of the mesh element. An element is said to be degenerate if its volume is zer... |

45 | Tetrahedral element shape optimization via the jacobian determinate and condition number
- Freitag, Knupp
- 1999
(Show Context)
Citation Context ...e work of [11], there appears to be almost no discussion in the literature on the relative merits of these particular quality measures. More recently, alternative quality measures have been suggested =-=[3, 7, 9, 10]-=-. These alternative measures are derived from the singular values of a matrix whose columns are formed by the edge vectors of the mesh element. An element is said to be degenerate if its volume is zer... |

45 |
On the angle condition in the finite element method
- Babuˇska, Aziz
- 1976
(Show Context)
Citation Context ...iangle quality measures. Introduction Assessment of mesh quality is an important requirement in both the selection of a finite element mesh and the evaluation of meshes that have undergone adaptation =-=[2, 5]-=-. Several measures of element quality have been proposed [1, 8, 11] based on the dimensionless ratios of various geometric parameters. Apart from the work of [11], there appears to be almost no discus... |

41 | On the shape of tetrahedra from bisection,
- Liu, Joe
- 1994
(Show Context)
Citation Context ...e work of [11], there appears to be almost no discussion in the literature on the relative merits of these particular quality measures. More recently, alternative quality measures have been suggested =-=[3, 7, 9, 10]-=-. These alternative measures are derived from the singular values of a matrix whose columns are formed by the edge vectors of the mesh element. An element is said to be degenerate if its volume is zer... |

31 |
General lagrange and hermit interpolation in R n with applications to the finite element method
- Ciarlet, Raviart
- 1972
(Show Context)
Citation Context ...iangle quality measures. Introduction Assessment of mesh quality is an important requirement in both the selection of a finite element mesh and the evaluation of meshes that have undergone adaptation =-=[2, 5]-=-. Several measures of element quality have been proposed [1, 8, 11] based on the dimensionless ratios of various geometric parameters. Apart from the work of [11], there appears to be almost no discus... |

15 |
Element Quality in Tetrahedral Meshes
- Baker
(Show Context)
Citation Context ...y is an important requirement in both the selection of a finite element mesh and the evaluation of meshes that have undergone adaptation [2, 5]. Several measures of element quality have been proposed =-=[1, 8, 11]-=- based on the dimensionless ratios of various geometric parameters. Apart from the work of [11], there appears to be almost no discussion in the literature on the relative merits of these particular q... |

9 |
Optimality of the Delaunay Triangulation in R d ,” Discrete Comput
- Rajan
- 1994
(Show Context)
Citation Context ...inθ0 sin θ∞ sin (θ0 + θ∞) . It is therefore interesting to examine how sensitive the radius-ratio is to the information provided by an extremal angle measurement. In particular, it is well known (cf. =-=[12]-=-) that among all triangulations of the convex hull of a given set of points in R2 , any Delaunay triangulation maximizes the minimal vertex angle; this property is generally considered to be a guarant... |

5 |
Deformation and quality measures for tetrahedral meshes
- Baker
- 2000
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Citation Context |

2 |
A comparison of tetrahedral quality measures
- Parthasarathy, Graichen, et al.
- 1993
(Show Context)
Citation Context ...y is an important requirement in both the selection of a finite element mesh and the evaluation of meshes that have undergone adaptation [2, 5]. Several measures of element quality have been proposed =-=[1, 8, 11]-=- based on the dimensionless ratios of various geometric parameters. Apart from the work of [11], there appears to be almost no discussion in the literature on the relative merits of these particular q... |