Mesh Shape and Anisotropic Elements

Apel, Thomas; Berzins, Martin; Jimack, Peter K.; Kunert, Gerd; Plaks, A.; Tsukerman, Igor; Walkley, Mark A.

doi:10.1016/b978-008043568-8/50024-9

Cited by 12 publications

(21 citation statements)

References 11 publications

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“…This error bound extends the optimal interpolation error estimates for linear elements in [2,10,11,16] to higher order elements in R 2 . The above conclusions also agree with those based on the exact error formulas in the model problems of linear interpolation of a quadratic function (k = 1) and quadratic interpolation of a cubic function (k = 2) presented in [8] and [9], respectively.…”

Section: Introductionmentioning

confidence: 90%

“…There are only a few papers considering the anisotropic error estimates and mesh refinement for higher order elements. For instance, denote by Π k u the interpolation of u by polynomials of degree k. Apel derived in [2] the following estimate for the interpolation error u − Π k u over an anisotropic element τ :…”

Section: Introductionmentioning

confidence: 99%

“…Then p * 2 can be factored into two linear functions as in (2), and the magnitude, orientation, and anisotropic ratio of ∇ 2 u can be determined equivalently in terms of the level-0 lines of p * 2 (ξ).…”

Section: Weiming Caomentioning

confidence: 99%

“…Numerous examples have shown that long and thin elements are useful in computation of problems with boundary or internal layers [1,2,14,15,19,22]. A practical question is in what direction an element should be long and how long and thin it should be.…”

Section: Introductionmentioning

confidence: 99%

“…Accurately understanding the anisotropic behavior of ∇ k+1 u is crucial for the anisotropic error estimates and mesh refinements. As pointed out in [2], page 7, a tight error estimate for k ≥ 2 relies on a "sufficiently fine description of the properties of the solution u". A major motivation for our work in this paper is to find a way to measure quantitatively the anisotropic behavior of ∇ k+1 u.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

Section: Weiming Caomentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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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Anisotropic Level Set Adaptation for Accurate Interface Capturing

Ducrot

Frey

2008

Proceedings of the 17th International Meshing Roundtable

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In numerical simulations, interface capturing and tracking is considered as a very challenging problem and has a strong impact on industrial application. We describe an adaptive scheme based on the definition of an anisotropic metric tensor to control the generation of highly streched elements near an interface described with a levelset function. The accuracy of the method is verified and various numerical experiments are presented to show its efficiency.

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Anisotropic Delaunay Mesh Adaptation for Unsteady Simulations

Dobrzynski

Frey

2008

Proceedings of the 17th International Meshing Roundtable

119

123

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Summary. Anisotropic mesh adaptation is a key feature in many numerical simulations to capture the physical behavior of a complex phenomenon at a reasonable computational cost. It is a challenging problem, especially when dealing with time dependent and interface capturing or tracking problems. Here, we describe an extension of the Delaunay kernel for creating anisotropic mesh elements based on adequate metric tensors. The accuracy and efficiency of the method is assessed on various numerical examples of complex three-dimensional simulations.

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Mesh Shape and Anisotropic Elements

Cited by 12 publications

References 11 publications

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

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

Anisotropic Level Set Adaptation for Accurate Interface Capturing

Anisotropic Delaunay Mesh Adaptation for Unsteady Simulations

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