2D Centroidal Voronoi Tessellations with Constraints

Tournois, Jane

doi:10.4208/nmtma.2010.32s.6

Cited by 22 publications

(9 citation statements)

References 16 publications

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“…We do this by dividing the Voronoi cell into a set of triangles and using a quadrature rule on each of them. 15,93 The density field can be set manually, but here we use an automatic procedure that forces the triangles in the interior of the domain to be large, the triangles near the boundary to be small, all while ensuring a smooth gradation in mesh size. 85,[93][94][95] Figure 10(d) shows the resulting Voronoi tessellation using the same number of n = 1468 points.…”

Section: E Numerical Analysismentioning

confidence: 99%

Voronoi cell analysis: The shapes of particle systems

Lazar,

Lu,

Rycroft

2022

Preprint

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Many physical systems can be studied as collections of particles embedded in space, evolving through deterministic evolution equations. Natural questions arise concerning how to characterize these arrangements -are they ordered or disordered? If they are ordered, how are they ordered and what kinds of defects do they possess? Originally introduced to study problems in pure mathematics, Voronoi tessellations have become a powerful and versatile tool for analyzing countless problems in pure and applied physics. In this paper we explain the basics of Voronoi tessellations and the shapes they produce, and describe how they can be used to study many physical systems.

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Section: E Numerical Analysismentioning

confidence: 99%

Voronoi cell analysis: The shapes of particle systems

Lazar,

Lu,

Rycroft

2022

Preprint

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“…Voronoi tesselations can be constructed by computing a DT of the generators. This can be regarded as the typical approach for finding VTs (Tournois et al 2010;Hateley et al 2015). Given points in ℝ 2 , a DT is a triangulation, such that no point lies within the circumcircle of any triangle (Preparata and Shamos 1985).…”

Section: Geometric Approachmentioning

confidence: 99%

“…With the geometric approach, this case requires more advanced implementations, see e.g. Tournois et al (2010). With the PDE-based approach, non-convex shapes are significantly easier to handle.…”

Section: Comparison Of Geometric and Pde-based Approachmentioning

confidence: 99%

Stable honeycomb structures and temperature based trajectory optimization for wire-arc additive manufacturing

et al. 2020

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We consider two mathematical problems that are connected and occur in the layer-wise production process of a workpiece using wire-arc additive manufacturing. As the first task, we consider the automatic construction of a honeycomb structure, given the boundary of a shape of interest. In doing this, we employ Lloyd’s algorithm in two different realizations. For computing the incorporated Voronoi tesselation we consider the use of a Delaunay triangulation or alternatively, the eikonal equation. We compare and modify these approaches with the aim of combining their respective advantages. Then in the second task, to find an optimal tool path guaranteeing minimal production time and high quality of the workpiece, a mixed-integer linear programming problem is derived. The model takes thermal conduction and radiation during the process into account and aims to minimize temperature gradients inside the material. Its solvability for standard mixed-integer solvers is demonstrated on several test-instances. The results are compared with manufactured workpieces.

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“…To distribute the samples evenly on a triangle f i , we first generate n(f i ) samples randomly, and then perform a Lloyd relaxation process onto a bounded Voronoi Diagram (BVD) [42] (Fig. 5(b)).…”

Section: Approximating D H With Stratified Samplingmentioning

confidence: 99%

Error-Bounded and Feature Preserving Surface Remeshing with Minimal Angle Improvement

Yan

Bommes

et al. 2017

IEEE Trans. Visual. Comput. Graphics

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Abstract-The typical goal of surface remeshing consists in finding a mesh that is (1) geometrically faithful to the original geometry, (2) as coarse as possible to obtain a low-complexity representation and (3) free of bad elements that would hamper the desired application. In this paper, we design an algorithm to address all three optimization goals simultaneously. The user specifies desired bounds on approximation error δ, minimal interior angle θ and maximum mesh complexity N (number of vertices). Since such a desired mesh might not even exist, our optimization framework treats only the approximation error bound δ as a hard constraint and the other two criteria as optimization goals. More specifically, we iteratively perform carefully prioritized local operators, whenever they do not violate the approximation error bound and improve the mesh otherwise. Our optimization framework greedily searches for the coarsest mesh with minimal interior angle above θ and approximation error bounded by δ. Fast runtime is enabled by a local approximation error estimation, while implicit feature preservation is obtained by specifically designed vertex relocation operators. Experiments show that our approach delivers high-quality meshes with implicitly preserved features and better balances between geometric fidelity, mesh complexity and element quality than the state-of-the-art.Index Terms-surface remeshing, error-bounded, feature preserving, minimal angle improvement, feature intensity.

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2D Centroidal Voronoi Tessellations with Constraints

Cited by 22 publications

References 16 publications

Voronoi cell analysis: The shapes of particle systems

Voronoi cell analysis: The shapes of particle systems

Stable honeycomb structures and temperature based trajectory optimization for wire-arc additive manufacturing

Error-Bounded and Feature Preserving Surface Remeshing with Minimal Angle Improvement

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