Delaunay mesh generation governed by metric specifications. Part I. Algorithms

Borouchaki, Houman; George, Paul‐Louis; Hecht, Frédéric; Laug, Patrick; Saltel, Eric

doi:10.1016/s0168-874x(96)00057-1

Cited by 215 publications

(177 citation statements)

References 11 publications

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“…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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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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“…The difference between the two methodologies becomes more evident when dealing with anisotropic meshes. Although anisotropic versions of Delaunay Triangulations and Constrained Delaunay Triangulations have been used to handle anisotropic meshes [23], such techniques demand either tensor estimation to define the anisotropy or an explicit manipulation of constraints, which can become as complex as the connectivity management during merging operations. In fact, our approach can be seen as a first step toward a new framework for connectivity oblivious mesh representation using the idea of regular triangulations.…”

Section: Discussion and Limitationsmentioning

confidence: 99%

Connectivity Oblivious Merging of Triangulations

Silva

Scheidegger

Etiene

et al. 2012

2012 25th SIBGRAPI Conference on Graphics, Patterns and Images

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Fig. 1:Merging a buddha tetrahedral mesh with a background grid. Our technique is able to handle meshes with distinct levels of refinement -observe how the internal tetrahedra of the buddha have not been refined.Abstract-Simplicial meshes are extremely useful as discrete approximations of continuous spaces in numerical simulations. In some applications, however, meshes need to be modified over time. Mesh update operations are often expensive and brittle, which tends to make the numerical simulations unstable. In this paper we propose an alternative technique for updating simplicial meshes that undergo geometric and topological changes. We exploit the property that a Weighted Delaunay Triangulation (WDT) can be used to implicitly define the connectivity of a mesh. Instead of explicitly maintaining connectivity information, we simply keep a collection of weights associated with each vertex. This approach allows for a simple way to merge triangulations, which we illustrate with examples in 2D and 3D.

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“…Several versions of octree structures were also implemented following the typical approaches used in the area of meshing [5,12,18] as reference for comparison with the proposed kd-tree structures.…”

Section: Octree Structuresmentioning

confidence: 99%

“…In order to generate a mesh fulfilling the specified quality criteria, it is necessary to assign the desired size and element shape to each point of the modeled object. A common way to achieve such a result is to use the Riemannian metric, changing it locally in various sub-areas of a three-dimensional object [1,2,5,11]. The sources of the metric may be of a various nature, relating to the characteristics of the domain and specific application of the mesh.…”

Section: Introductionmentioning

confidence: 99%

Tree Structures for Adaptive Control Space in 3d Meshing

Jurczyk¹,

Głut

2016

csci

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The article presents a comparison of several octree-and kd-tree-based structures used for the construction of control space in the process of anisotropic mesh generation and adaptation. The adaptive control space utilized by the authors supervises the construction of meshes by providing the required metric information regarding the desired shape and size of elements of the mesh at each point of the modeled domain. Comparative tests of these auxiliary structures were carried out based on different versions of the tree structures with respect to computational and memory complexity as well as the quality of the generated mesh. Analysis of the results shows that kd-trees (not present in the meshing literature in this role) offer good performance and may become a reasonable alternative to octree structures.

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Delaunay mesh generation governed by metric specifications. Part I. Algorithms

Cited by 215 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

Connectivity Oblivious Merging of Triangulations

Tree Structures for Adaptive Control Space in 3d Meshing

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