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...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 ...

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...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...

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...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 = ...

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...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...

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...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...

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...

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...

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...

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...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...

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...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...

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...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...

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...