Isotropic 2D Quadrangle Meshing with Size and Orientation Control

Pellenard, Bertrand; Alliez, Pierre; Morvan, Jean‐Marie

doi:10.1007/978-3-642-24734-7_5

Cited by 5 publications

(10 citation statements)

References 31 publications

(46 reference statements)

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“…It uses a feature-dependent direction field and thus also restricts the ability of element size control. Recently, a method [Pellenard et al 2011] has been proposed for controlling the size and orientation of isotropic 2D quadrangulation that addresses the requirement of size control before orientation alignment. Because the method relies on labeling the triangles of the background triangulation through local information, it is difficult to reduce mesh singularities; furthermore, it produces results that are sensitive to the initial tessellation.…”

Section: Related Workmentioning

confidence: 99%

Spectral Quadrangulation with Feature Curve Alignment and Element Size Control

et al. 2014

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Existing methods for surface quadrangulation cannot ensure accurate alignment with feature or boundary curves and tight control of local element size, which are important requirements in many numerical applications (e.g., FEA). Some methods rely on a prescribed direction field to guide quadrangulation for feature alignment, but such a direction field may conflict with a desired density field, thus making it difficult to control the element size. We propose a new spectral method that achieves both accurate feature curve alignment and tight control of local element size according to aThe work of W.given density field. Specifically, the following three technical contributions are made. First, to make the quadrangulation align accurately with feature curves or surface boundary curves, we introduce novel boundary conditions for wave-like functions that satisfy the Helmholtz equation approximately in the least squares sense. Such functions, called quasi-eigenfunctions, are computed efficiently as the solutions to a variational problem. Second, the mesh element size is effectively controlled by locally modulating the Laplace operator in the Helmholtz equation according to a given density field. Third, to improve robustness, we propose a novel scheme to minimize the vibration difference of the quasi-eigenfunction in two orthogonal directions. It is demonstrated by extensive experiments that our method outperforms previous methods in generating feature-aligned quadrilateral meshes with tight control of local elememt size. We further present some preliminary results to show that our method can be extended to generating hex-dominant volume meshes.

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Section: Related Workmentioning

confidence: 99%

Spectral Quadrangulation with Feature Curve Alignment and Element Size Control

et al. 2014

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“…A conforming relaxation method has been recently proposed [19] to improve conformity but it also suffers from partitioning defects when employed with strongly anisotropic metrics. The inset depicts a common defect that arises when a generator with a large metric competes with a nearby generator with a small metric.…”

Section: Previous Workmentioning

confidence: 99%

“…Remind that the general set-cover problem, shown to be N P -hard, consists in finding the minimum number of subsets of a set that cover the input domain [12]. Instead of guaranteeing partitioning by construction when using, e.g., L ∞ Voronoi diagrams [16,19], we favor during optimization the absolute matching of the metric while allowing a decomposition with both overlaps and orphans. In other words, the partitioning requirement now becomes an objective function to optimize through the notion of coverage.…”

Section: Decompositionmentioning

confidence: 99%

“…Similarly to previous work [19,7] we generate a partition by flooding the input mesh S one triangle at a time from the set of generators placed during decomposition. We use a global priority queue initialized with all feasible triangles incident to the generators.…”

Section: Meshingmentioning

confidence: 99%

“…A vertex is said of degree n (with n > 1) if it is adjacent to n subdomains. Following the terminology of Pellenard et al [19], we tag as meta-vertices the vertices of S with degree ≥ 3 in the interior, and ≥ 2 on the boundary of S. We then generate the polygon surface mesh M induced by the partition and its meta-vertices.…”

Section: Meshingmentioning

confidence: 99%

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Anisotropic Rectangular Metric for Polygonal Surface Remeshing

Pellenard

Morvan

Alliez

2013

Proceedings of the 21st International Meshing Roundtable

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Summary. We propose a new method for anisotropic polygonal surface remeshing. Our algorithm takes as input a surface triangle mesh. An anisotropic rectangular metric, defined at each triangle facet of the input mesh, is derived from both a user-specified normal-based tolerance error and the requirement to favor rectangleshaped polygons. Our algorithm uses a greedy optimization procedure that adds, deletes and relocates generators so as to match two criteria related to partitioning and conformity.

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A multiobjective mesh optimization framework for mesh quality improvement and mesh untangling

Kim

Panitanarak

Shontz

2013

Numerical Meth Engineering

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SUMMARYWe propose a multiobjective mesh optimization framework for mesh quality improvement and mesh untangling. Our framework combines two or more competing objective functions into a single objective function to be solved using one of various multiobjective optimization methods. Methods within our framework are able to optimize various aspects of the mesh such as the element shape, element size, associated PDE interpolation error, and number of inverted elements, but the improvement is not limited to these categories. The strength of our multiobjective mesh optimization framework lies in its ability to be extended to simultaneously optimize any aspects of the mesh and to optimize meshes with different element types. We propose the exponential sum, objective product, and equal sum multiobjective mesh optimization methods within our framework; these methods do not require articulation of preferences. However, the solutions obtained satisfy a sufficient condition of weak Pareto optimality. Experimental results show that our multiobjective mesh optimization methods are able to simultaneously optimize two or more aspects of the mesh and also are able to improve mesh qualities while eliminating inverted elements. We successfully apply our methods to real‐world applications such as hydrocephalus treatment and shape optimization. Copyright © 2013 John Wiley & Sons, Ltd.

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Isotropic 2D Quadrangle Meshing with Size and Orientation Control

Cited by 5 publications

References 31 publications

Spectral Quadrangulation with Feature Curve Alignment and Element Size Control

Spectral Quadrangulation with Feature Curve Alignment and Element Size Control

Anisotropic Rectangular Metric for Polygonal Surface Remeshing

A multiobjective mesh optimization framework for mesh quality improvement and mesh untangling

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