skip to main content

Algorithm 964: An Efficient Algorithm to Compute the Genus of Discrete Surfaces and Applications to Turbulent Flows

Published:14 June 2016Publication History
Skip Abstract Section

Abstract

A simple and efficient algorithm to numerically compute the genus of surfaces of three-dimensional objects using the Euler characteristic formula is presented. The algorithm applies to objects obtained by thresholding a scalar field in a structured-collocated grid and does not require any triangulation of the data. This makes the algorithm fast, memory efficient, and suitable for large datasets. Applications to the characterization of complex surfaces in turbulent flows are presented to illustrate the method.

Skip Supplemental Material Section

Supplemental Material

References

  1. Dolors Ayala, Eduard Vergés, and Irving Cruz. 2012. A polyhedral approach to compute the genus of a volume dataset. In Proceedings of the International Conference on Computer Graphic Theory and Applications (GRAPP'12). 38--47.Google ScholarGoogle Scholar
  2. D. K. Bisset, J. C. R. Hunt, and M. M. Rogers. 2002. The turbulent/non-turbulent interface bounding a far wake. Journal of Fluid Mechanics 451, 383--410. DOI:http://dx.doi.org/10.1017/S0022112001006759Google ScholarGoogle ScholarCross RefCross Ref
  3. G. Borrell and J. Jiménez. 2013. Geometrical properties and scaling of the turbulent-nonturbulent interface in boundary layers. In Proceedings of the 66th Annual Meeting of the APS Division of Fluid Dynamics. http://adsabs.harvard.edu/abs/2013APS..DFDR31002B.Google ScholarGoogle Scholar
  4. Kapil Chauhan, Jimmy Philip, Charitha M. de Silva, Nicholas Hutchins, and Ivan Marusic. 2014. The turbulent/non-turbulent interface and entrainment in a boundary layer. Journal of Fluid Mechanics 742, 119--151. DOI:http://dx.doi.org/10.1017/jfm.2013.641Google ScholarGoogle ScholarCross RefCross Ref
  5. Isaac Chavel. 2006. Riemannian Geometry (2nd ed.). Cambridge University Press, Cambridge, UK.Google ScholarGoogle Scholar
  6. Li Chen and Yongwu Rong. 2010. Digital topological method for computing genus and the Betti numbers. Topology and Its Applications 157, 12, 1931--1936. DOI:http://dx.doi.org/10.1016/j.topol.2010.04.006Google ScholarGoogle ScholarCross RefCross Ref
  7. Computational Fluid Mechanics Lab. 2015. Index of /Genus. Retrieved May 20, 2016, from http://torroja.dmt. upm.es/genus/.Google ScholarGoogle Scholar
  8. Irving Cruz and Dolors Ayala. 2013. An efficient alternative to compute the genus of binary volume models. In Proceedings of the International Conference on Computer Graphics Theory and Applications (GRAPP'13). 18--26.Google ScholarGoogle Scholar
  9. Brian Curless and Marc Levoy. 1996. A volumetric method for building complex models from range images. In Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH'96). ACM, New York, NY, 303--312. DOI:http://dx.doi.org/10.1145/237170.237269 Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. Carlos B. da Silva, Julian C. R. Hunt, Ian Eames, and Jerry Westerweel. 2014a. Interfacial layers between regions of different turbulence intensity. Annual Review of Fluid Mechanics 46, 1, 567--590. DOI:http://dx.doi.org/10.1146/annurev-fluid-010313-141357Google ScholarGoogle ScholarCross RefCross Ref
  11. Carlos B. da Silva and Rodrigo R. Taveira. 2010. The thickness of the turbulent/nonturbulent interface is equal to the radius of the large vorticity structures near the edge of the shear layer. Physics of Fluids 22, 121702.Google ScholarGoogle ScholarCross RefCross Ref
  12. Carlos B. da Silva, Rodrigo R. Taveira, and Guillem Borrell. 2014b. Characteristics of the turbulent/nonturbulent interface in boundary layers, jets and shear-free turbulence. Journal of Physics: Conference Series 506, Article No. 15.Google ScholarGoogle ScholarCross RefCross Ref
  13. Juan C. del Álamo, Javier Jiménez, Paulo Zandonade, and Robert D. Moser. 2006. Self-similar vortex clusters in the turbulent logarithmic region. Journal of Fluid Mechanics, 329--358. DOI:http://dx.doi.org/ 10.1017/S0022112006000814Google ScholarGoogle Scholar
  14. J. Einasto, M. Einasto, E. Tago, E. Saar, G. Htsi, M. Jeveer, L. J. Liivamgi, et al. 2007. Superclusters of galaxies from the 2D Redshift survey. Astronomy and Astrophysics 462, 2, 811--825. DOI:http://dx.doi.org/ 10.1051/0004-6361:20065296Google ScholarGoogle ScholarCross RefCross Ref
  15. Markus Gampert, Jonas Boschung, Fabian Hennig, Michael Gauding, and Norbert Peters. 2014. The vorticity versus the scalar criterion for the detection of the turbulent/non-turbulent interface. Journal of Fluid Mechanics 750, 578--596.Google ScholarGoogle ScholarCross RefCross Ref
  16. J. R. Gott III, M. Dickinson, and A. L. Melott. 1986. The sponge-like topology of large-scale structure in the universe. Astrophysical Journal 306, 341--357. DOI:http://dx.doi.org/10.1086/164347Google ScholarGoogle ScholarCross RefCross Ref
  17. J. R. Gott III, J. Miller, T. X. Thuan, S. E. Schneider, D. H. Weinberg, C. Gammie, K. Polk, et al. 1989. The topology of large-scale structure. III—Analysis of observations. Astrophysical Journal 340, 625--646. DOI:http://dx.doi.org/10.1086/167425Google ScholarGoogle ScholarCross RefCross Ref
  18. J. R. Gott III, D. H. Weinberg, and A. L. Melott. 1987. A quantitative approach to the topology of large-scale structure. Astrophysical Journal 319, 1--8. DOI:http://dx.doi.org/10.1086/165427Google ScholarGoogle ScholarCross RefCross Ref
  19. Alan H. Guth. 1981. Inflationary universe: A possible solution to the horizon and flatness problems. Physical Review D 23, 2, 347--356. DOI:http://dx.doi.org/10.1103/PhysRevD.23.347Google ScholarGoogle ScholarCross RefCross Ref
  20. A. J. S. Hamilton, J. R. Gott III, and D. Weinberg. 1986. The topology of the large-scale structure of the universe. Astrophysical Journal 309, 1--12. DOI:http://dx.doi.org/10.1086/164571Google ScholarGoogle ScholarCross RefCross Ref
  21. J. Hoshen and R. Kopelman. 1976. Percolation and cluster distribution. I. Cluster multiple labeling technique and critical concentration algorithm. Physical Review B 14, 8, 3438--3445. DOI:http://dx.doi.org/ 10.1103/PhysRevB.14.3438Google ScholarGoogle ScholarCross RefCross Ref
  22. Sergio Hoyas and Javier Jiménez. 2008. Reynolds number effects on the Reynolds-stress budgets in turbulent channels. Physics of Fluids 20, 10, 101511. DOI:http://dx.doi.org/10.1063/1.3005862Google ScholarGoogle ScholarCross RefCross Ref
  23. Yukio Kaneda, Takashi Ishihara, Mitsuo Yokokawa, Kenichi Itakura, and Atsuya Uno. 2003. Energy dissipation rate and energy spectrum in high resolution direct numerical simulations of turbulence in a periodic box. Physics of Fluids 15, 2, L21--L24. DOI:http://dx.doi.org/10.1063/1.1539855Google ScholarGoogle ScholarCross RefCross Ref
  24. S. E. Konkle, P. Moran, B. Hamann, and K. Joy. 2003. Fast Methods for Computing Isosurface Topology with Betti Numbers. Kluwer Academic, Norwell, MA, 363--375.Google ScholarGoogle Scholar
  25. T. Leung, N. Swaminathan, and P. A. Davidson. 2012. Geometry and interaction of structures in homogeneous isotropic turbulence. Journal of Fluid Mechanics 710, 453--481. DOI:http://dx.doi.org/ 10.1017/jfm.2012.373Google ScholarGoogle ScholarCross RefCross Ref
  26. A. D. Linde. 1983. Chaotic inflation. Physical Letters 129, 34, 177--181. DOI:http://dx.doi.org/10.1016/ 0370-2693(83)90837-7Google ScholarGoogle ScholarCross RefCross Ref
  27. Adrián Lozano-Durán, Oscar Flores, and Javier Jiménez. 2012. The three-dimensional structure of momentum transfer in turbulent channels. Journal of Fluid Mechanics 694, 100--130. DOI:http://dx.doi.org/10.1017/ jfm.2011.524Google ScholarGoogle ScholarCross RefCross Ref
  28. Adrián Lozano-Durán and Javier Jiménez. 2014. Effect of the computational domain on direct simulations of turbulent channels up to Reτ = 4200. Physics of Fluids 26, 1, 7. DOI:http://dx.doi.org/10.1063/1.4862918Google ScholarGoogle ScholarCross RefCross Ref
  29. E. Martin-Badosa, A. Elmoutaouakkil, S. Nuzzo, D. Amblard, L. Vico, and F. Peyrin. 2003. A method for the automatic characterization of bone architecture in 3D mice microtomographic images. Computerized Medical Imaging and Graphics 27, 6, 447--458.Google ScholarGoogle ScholarCross RefCross Ref
  30. K. R. Mecke, T. Buchert, and H. Wagner. 1994. Robust morphological measures for large-scale structure in the universe. Astronomy and Astrophysics 288, 697--704.Google ScholarGoogle Scholar
  31. Patrick Min. 2015. Binvox: Mesh Voxelizer. Retrieved May 20, 2016, from http://www.cs.princeton.edu/∼min/binvox/.Google ScholarGoogle Scholar
  32. F. S. Nooruddin and G. Turk. 2003. Simplification and repair of polygonal models using volumetric techniques. IEEE Transactions on Visualization and Computer Graphics 9, 2, 191--205. DOI:http://dx.doi.org/10.1109/TVCG.2003.1196006 Google ScholarGoogle ScholarDigital LibraryDigital Library
  33. A. Odgaard and H. J. G. Gundersen. 1993. Quantification of connectivity in cancellous bone, with special emphasis on 3-D reconstructions. Bone 14, 2, 173--182. DOI:http://dx.doi.org/10.1016/8756-3282(93)90245-6Google ScholarGoogle ScholarCross RefCross Ref
  34. C. Park, Y.-Y. Choi, M. S. Vogeley, J. R. Gott III, J. Kim, C. Hikage, T. Matsubara, et al. 2005a. Topology analysis of the Sloan Digital Sky Survey. I. Scale and luminosity dependence. Astrophysical Journal 633, 11--22. DOI:http://dx.doi.org/10.1086/452625Google ScholarGoogle ScholarCross RefCross Ref
  35. C. Park, J. Kim, and J. R. Gott III. 2005b. Effects of gravitational evolution, biasing, and Redshift space distortion on topology. Astrophysical Journal 633, 1--10. DOI:http://dx.doi.org/10.1086/452621Google ScholarGoogle ScholarCross RefCross Ref
  36. S. B. Pope. 2000. Turbulent Flows. Cambridge University Press, Cambridge, UK.Google ScholarGoogle Scholar
  37. V. Sahni, B. S. Sathyaprakash, and S. F. Shandarin. 1998. Shapefinders: A new shape diagnostic for large-scale structure. Astrophysical Journal 495, 1, L5.Google ScholarGoogle ScholarCross RefCross Ref
  38. Juan A. Sillero, Javier Jiménez, and Robert D. Moser. 2013. One-point statistics for turbulent wall-bounded flows at Reynolds numbers up to Δ+2000. Physics of Fluids 25, 10, 105102.Google ScholarGoogle ScholarCross RefCross Ref
  39. K. R. Sreenivasan, R. Ramshankar, and C. Meneveau. 1989. Mixing, entrainment and fractal dimensions of surfaces in turbulent flows. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 421, 1860, 79--108. DOI:http://dx.doi.org/10.1098/rspa.1989.0004Google ScholarGoogle Scholar
  40. Stanford. 2014. The Stanford 3D Scanning Repository. Retrieved May 20, 2016, from https://graphics.stanford.edu/data/3Dscanrep/.Google ScholarGoogle Scholar
  41. Anthony C. Thompson. 1996. Minkowski Geometry. Cambridge University Press, Cambridge, UK.Google ScholarGoogle Scholar
  42. J. Toriwaki and T. Yonekura. 2002. Euler number and connectivity indexes of a three dimensional digital picture. Forma 17, 183--209.Google ScholarGoogle Scholar
  43. Greg Turk and Marc Levoy. 1994. Zippered polygon meshes from range images. In Proceedings of the 21st Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH'94). ACM, New York, NY, 311--318. DOI:http://dx.doi.org/10.1145/192161.192241 Google ScholarGoogle ScholarDigital LibraryDigital Library
  44. Alberto Vela-Martín and Guillem Borrell. 2014. Computation of a Temporal Decaying Turbulent Jet with GPGPUS. Technical Report. Computational Fluid Mechanics Group, UPM.Google ScholarGoogle Scholar
  45. M. S. Vogeley, C. Park, M. J. Geller, J. P. Huchra, and J. R. Gott III. 1994. Topological analysis of the CfA Redshift survey. Astrophysical Journal 420, 525--544. DOI:http://dx.doi.org/10.1086/173583Google ScholarGoogle ScholarCross RefCross Ref
  46. D. H. Weinberg. 1988. Contour: a topological analysis program. Publications of the Astronomical Society of the Pacific 100, 1373--1385. DOI:http://dx.doi.org/10.1086/132337Google ScholarGoogle ScholarCross RefCross Ref
  47. J. Westerweel, C. Fukushima, J. M. Pedersen, and J. C. R. Hunt. 2009. Momentum and scalar transport at the turbulent/non-turbulent interface of a jet. Journal of Fluid Mechanics 631, 199--230. DOI:http://dx.doi.org/10.1017/S0022112009006600Google ScholarGoogle ScholarCross RefCross Ref

Index Terms

  1. Algorithm 964: An Efficient Algorithm to Compute the Genus of Discrete Surfaces and Applications to Turbulent Flows

        Recommendations

        Comments

        Login options

        Check if you have access through your login credentials or your institution to get full access on this article.

        Sign in

        Full Access

        • Published in

          cover image ACM Transactions on Mathematical Software
          ACM Transactions on Mathematical Software  Volume 42, Issue 4
          July 2016
          185 pages
          ISSN:0098-3500
          EISSN:1557-7295
          DOI:10.1145/2956571
          Issue’s Table of Contents

          Copyright © 2016 ACM

          Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

          Publisher

          Association for Computing Machinery

          New York, NY, United States

          Publication History

          • Published: 14 June 2016
          • Accepted: 1 November 2015
          • Revised: 1 October 2015
          • Received: 1 April 2015
          Published in toms Volume 42, Issue 4

          Permissions

          Request permissions about this article.

          Request Permissions

          Check for updates

          Qualifiers

          • research-article
          • Research
          • Refereed

        PDF Format

        View or Download as a PDF file.

        PDF

        eReader

        View online with eReader.

        eReader