0
Research Papers

A Level Set Method for the Construction of Boundary Conforming Voronoi Regions and Delaunay Triangulations Governed by a Spatial Distribution of Metrics

[+] Author and Article Information
Ü. Keskin

Department of Aeronautics,
Imperial College London,
South Kensington Campus,
London SW7 2AZ, UK
e-mail: u.keskin09@imperial.ac.uk

J. Peiró

Department of Aeronautics,
Imperial College London,
South Kensington Campus,
London SW7 2AZ, UK
e-mail: j.peiro@imperial.ac.uk

An isolated region is a Voronoi region interior to another Voronoi region.

Orphan regions are regions with no seeds on them.

Contributed by the Computers and Information Division of ASME for publication in the JOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING. Manuscript received August 8, 2012; final manuscript received May 7, 2014; published online June 2, 2014. Assoc. Editor: Vijay Srinivasan.

J. Comput. Inf. Sci. Eng 14(3), 031007 (Jun 02, 2014) (7 pages) Paper No: JCISE-12-1130; doi: 10.1115/1.4027698 History: Received August 08, 2012; Revised May 07, 2014

A Voronoi region can be interpreted as the shape achieved by a crystal that grows from a seed and stops growing when it reaches either the domain boundary or another crystal. This analogy is exploited here to devise a method for the generation of anisotropic boundary-conforming Voronoi regions for a set of points. This is achieved by simulating the propagation of crystals as evolving fronts modeled by a level set method. The techniques to detect the collision of fronts (crystals), formation of interfaces between seeds, and treatment of boundaries as additional (inner or outer) restricting seeds are described in detail. The generation of anisotropic Voronoi regions consistent with a user-prescribed Riemannian metric is achieved by re-interpreting the metric tensor in terms of the speed of propagation normal to the boundary of the crystal. This re-interpretation offers a better means of restricting metric fields for mesh generation.

FIGURES IN THIS ARTICLE
<>
Copyright © 2014 by ASME
Your Session has timed out. Please sign back in to continue.

References

Okabe, A., Boots, B., Sugihara, K., and Chiu, S., 2000, Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, 2nd ed., Wiley, Chichester, UK.
Aurenhammer, F., 1991, “Voronoi Diagrams–A Survey of a Fundamental Geometric Data Structure,” ACM Comput. Surv., 23(3), pp. 345–405. [CrossRef]
Klein, R., 1989, Concrete and Abstract Voronoi Diagrams, Springer, Berlin, Germany.
George, P.-L., and Borouchaki, H., 1998, Delaunay Triangulation and Meshing: Application to Finite Elements, Hermes, Paris, France.
Du, Q., and Wang, D., 2005, “Anisotropic Centroidal Voronoi Tessellations and Their Applications,” SIAM J. Sci. Comput., 26(3), pp. 737–761. [CrossRef]
Leibon, G., and Letscher, D., 2000, “Delaunay Triangulations and Voronoi Diagrams for Riemannian Manifolds,” 16th Annual Symposium on Computational Geometry, SCG’00, ACM, New York, pp. 341–349. [CrossRef]
Labelle, F., and Shewchuk, J., 2003, “Anisotropic Voronoi Diagrams and Guaranteed-Quality Anisotropic Mesh Generation,” 19th Annual Symposium on Computational GeometrySCG’03, ACM, New York, pp. 191–200. [CrossRef]
Yokosuka, Y., and Imai, K., 2009, “Guaranteed-Quality Anisotropic Mesh Generation for Domains With Curved Boundaries,” Comput.-Aided Des., 41(5), pp. 385–393. [CrossRef]
Boissonnat, J.-D., Wormser, C., and Yvinec, M., 2008, “Anisotropic Diagrams: Labelle Shewchuk Approach Revisited,” Theor. Comput. Sci., 408(2–3), pp. 163–173. [CrossRef]
Sethian, J. A., 1999, “Level Set Methods and Fast Marching Methods. Evolving Interfaces in Computational Geometry,” Fluid Mechanics, Computer Vision, and Materials Science (Cambridge Monograph on Applied and Computational Mathematics), 2nd ed., Cambridge University Press, Cambridge, UK.
Russo, G., and Smereka, P., 2000, “A Level Set Method for the Evolution of Faceted Crystals,” SIAM J. Sci. Comput., 21(6), pp. 2073–2095. [CrossRef]
Sethian, J., and Strain, J., 1992, “Crystal Growth and Dendritic Solidification,” J. Comput. Phys., 98(2), pp. 231–253. [CrossRef]
Shepard, D., 1968, “A Two-Dimensional Interpolation Function for Irregularly-Spaced Data,” Proceedings of the 23rd ACM National Conference, ACM, New York, pp. 517–524.
Barth, T., 1994, “Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations,” (VKI Lecture Notes 1994-05), Von Karman Institute, Sint-Genesius-Rode, Belgium.
Peyre, G., and Cohen, L., 2009, “Geodesic Methods for Shape and Surface Processing,” Advances in Computational Vision and Medical Image Processing: Methods and Applications (No. 13 in Computational Methods in Applied Sciences), J.Tavares and R.Jorge, eds., Springer, Berlin, Germany, pp. 29–56.

Figures

Grahic Jump Location
Fig. 1

Growth of crystals governed by a metric field that specifies a spatially varying distribution of growth speeds: (a) ellipses are used to represent the metric field; and (b) growth of the crystal boundaries

Grahic Jump Location
Fig. 2

Voronoi regions generated by crystal growth in the absence of boundaries. The figures on the left represent the metric field by ellipses and those on the right are the shapes of the crystals, i.e., the Voronoi regions: (a) and (b) constant isotropic metric field; (c) and (d) constant anisotropic metric field; and (e) and (f) variable metric field.

Grahic Jump Location
Fig. 3

Boundary conditions for crystal growth: (a) wet contact; and (b) dry contact

Grahic Jump Location
Fig. 4

Labelle–Shewchuk example re-interpreted: (a) the metric distribution which is assumed to be constant within the Voronoi region of a seed; (b) the Voronoi regions obtained using the method proposed here. It can be observed that no “orphan” regions are generated, but some “isolated” Voronoi regions have only one neighbor region.

Grahic Jump Location
Fig. 5

Labelle–Shewchuk example re-interpreted: (a) the spatial distribution of metric tensors (thin lines) has been interpolated from the previous metric tensors (thick lines) to obtain a continuous metric field that does not produce “isolated” regions; (b) the corresponding spatial partition into Voronoi regions

Grahic Jump Location
Fig. 6

A complex domain: (a) prescribed anisotropic metric field and (b) boundary-conforming Voronoi partition

Grahic Jump Location
Fig. 7

Comparison of the distribution of the isotropic (a) and anisotropic (b) metric fields and corresponding Voronoi partitions (c) and (d), respectively, obtained for a domain around a symmetric aerofoil

Grahic Jump Location
Fig. 8

Comparison of the Delaunay triangulations obtained for a domain around a symmetric aerofoil using isotropic (left) and anisotropic (right) metric fields: (a) and (b) whole mesh; and (c) and (d) enlargement near the wake of the aerofoil. Notice how the introduction of anisotropy in the metric field aligns the triangles with the wake as required.

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In