Delaunay mesh generation governed by metric specifications Part II. Applications

Borouchaki, Houman; George, Paul‐Louis; Mohammadi, Bijan

doi:10.1016/s0168-874x(96)00065-0

Cited by 103 publications

(67 citation statements)

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“…For the description of our algorithm, see our recent work [26,24]. More sophisticated algorithms for a Delaunay-type mesh generation algorithm governed by Riemannian metrics can be found in [17,18]. .…”

Section: Numerical Experimentsmentioning

confidence: 99%

Optimal anisotropic meshes for minimizing interpolation errors in $L^p$-norm

2007

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Abstract. In this paper, we present a new optimal interpolation error estimate in L p norm (1 ≤ p ≤ ∞) for finite element simplicial meshes in any spatial dimension. A sufficient condition for a mesh to be nearly optimal is that it is quasi-uniform under a new metric defined by a modified Hessian matrix of the function to be interpolated. We also give new functionals for the global moving mesh method and obtain optimal monitor functions from the viewpoint of minimizing interpolation error in the L p norm. Some numerical examples are also given to support the theoretical estimates.

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Section: Numerical Experimentsmentioning

confidence: 99%

Optimal anisotropic meshes for minimizing interpolation errors in $L^p$-norm

2007

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“…We define the magnitude of ∇ m u as the coefficient D m in (6). It is an upper bound for all the m-th order directional derivatives of u at x.…”

Section: Higher Order Derivativementioning

confidence: 99%

“…Since we wish to have a more precise control of the geometric features of each element, we choose to "manually" create the anisotropic triangulations. For more general domain and applications, readers may resort to some robust algorithms, e.g., [5,6], to generate the anisotropic mesh under a given Riemannian metric. To start our process, we note that u is a function of the radial variable only.…”

Section: An Interpolation Error Estimate 281mentioning

confidence: 99%

An interpolation error estimate in $\mathcal{R}^2$ based on the anisotropic measures of higher order derivatives

Cao

2008

Math. Comp.

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Abstract. In this paper, we introduce the magnitude, orientation, and anisotropic ratio for the higher order derivative ∇ k+1 u (with k ≥ 1) of a function u to characterize its anisotropic behavior. The magnitude is equivalent to its usual Euclidean norm. The orientation is the direction along which the absolute value of the k + 1-th directional derivative is about the smallest, while along its perpendicular direction it is about the largest. The anisotropic ratio measures the strength of the anisotropic behavior of ∇ k+1 u. These quantities are invariant under translation and rotation of the independent variables. They correspond to the area, orientation, and aspect ratio for triangular elements. Based on these measures, we derive an anisotropic error estimate for the piecewise polynomial interpolation over a family of triangulations that are quasi-uniform under a given Riemannian metric. Among the meshes of a fixed number of elements it is identified that the interpolation error is nearly the minimum on the one in which all the elements are aligned with the orientation of ∇ k+1 u, their aspect ratios are about the anisotropic ratio of ∇ k+1 u, and their areas make the error evenly distributed over every element.

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“…Methods that apply isotropic techniques in a metric mapped space have been successful in two dimensions (2D). 19,[21][22][23][24] The metric-based adaptation process has two principle components: determining an improved resolution request and creating an improved grid that satisfies that request. The improved resolution request is commonly based on local error estimates 17,19,25,26 and can include the effect of local errors on a global output quantity.…”

Section: -14mentioning

confidence: 99%

Parallel Anisotropic Tetrahedral Adaptation

Park¹,

Darmofal

2008

46th AIAA Aerospace Sciences Meeting and Exhibit

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An adaptive method that robustly produces high aspect ratio tetrahedra to a general 3D metric specification without introducing hybrid semi-structured regions is presented. The elemental operators and higher-level logic is described with their respective domaindecomposed parallelizations. An anisotropic tetrahedral grid adaptation scheme is demonstrated for 1000-1 stretching for a simple cube geometry. This form of adaptation is applicable to more complex domain boundaries via a cut-cell approach as demonstrated by a parallel 3D supersonic simulation of a complex fighter aircraft. To avoid the assumptions and approximations required to form a metric to specify adaptation, an approach is introduced that directly evaluates interpolation error. The grid is adapted to reduce and equidistribute this interpolation error calculation without the use of an intervening anisotropic metric. Direct interpolation error adaptation is illustrated for 1D and 3D domains.

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Delaunay mesh generation governed by metric specifications Part II. Applications

Cited by 103 publications

References 1 publication

Optimal anisotropic meshes for minimizing interpolation errors in $L^p$-norm

Optimal anisotropic meshes for minimizing interpolation errors in $L^p$-norm

An interpolation error estimate in $\mathcal{R}^2$ based on the anisotropic measures of higher order derivatives

Parallel Anisotropic Tetrahedral Adaptation

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