Enumeration of Polyhedra for Grain Growth Analysis

  • Trevor Keller
  • Barb Cutler
  • Martin Glicksman
  • Dan Lewis


The advent of three dimensional data collection, grain reconstruction, and subsequent materials analysis has created opportunities to revisit problems in grain growth and polycrystalline structure. In this paper we review the relevant literature concerning the total number of polyhedral grains of a given number of faces and take the additional first steps at enumerating the topologies of the members within each set. Analysis of the dispersion in topology relative to the idealized N-hedra is presented. The relevance to grain growth simulations and experiments is discussed.


Polyhedral graphs Grain growth Monte Carlo methods 


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  1. [1]
    P. J. Federico, “The number of polyhedra,” Philips Research Reports, 30 (1975) 220–231.Google Scholar
  2. [2]
    A. J. W. Duijvestijn and P. J. Federico, “The number of polyhedral (3-connected planar) graphs,” Mathematics of Computation, 37 (1981) 523–532.Google Scholar
  3. [3]
    P. Engel, “On the enumeration of the simple 3-polyhedra,” Acta Crystallographica Section A, 59 (2003) 14–17.Google Scholar
  4. [4]
    C. S. Smith, “Grain shapes and other metallurgical applications of topology,” in “Metal Interfaces,” pages 65–118, 1952.Google Scholar
  5. [5]
    W. M. Williams and C. S. Smith, “A study of grain shape in an aluminum alloy and other applications of stereoscopic microradiography,” Transactions of the American Institute of Mining and Metallurgical Engineers, 194 (1952) 755–765.Google Scholar
  6. [6]
    W. Thomson, “On the division of space with minimum partitional area,” Philosophical Magazine, 24 (1887) 503.CrossRefGoogle Scholar
  7. [7]
    M. E. Glicksman, “Analysis of 3-D network structures,” Philosophical Magazine, 85 (2005) 3–31.CrossRefGoogle Scholar
  8. [8]
    D. J. Rowenhorst, A. C. Lewis, and G. Spanos, “Three-dimensional analysis of grain topology and interface curvature in a β-titanium alloy,” Acta Materialia, 58 (2010) 5511–5519.CrossRefGoogle Scholar
  9. [9]
    Y. Suwa, Y. Saito, and H. Onodera, “Parallel computer simulation of three-dimensional grain growth using the multi-phase-field model,” Materials Transactions, 49 (2008) 704–709.CrossRefGoogle Scholar
  10. [10]
    M. Elsey, S. Esedoḡlu, and P. Smereka, “Large-scale simulation of normal grain growth via diffusion-generated motion,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, 467 (2011) 381–401.CrossRefGoogle Scholar
  11. [11]
    E. A. Lazar, J. K. Mason, R. D. MacPherson, and D. J. Srolovitz, “A more accurate three-dimensional grain growth algorithm,” Acta Materialia, 59 (2011) 6837–6847.CrossRefGoogle Scholar
  12. [12]
    A. J. W. Duijvestijn, “List of 3-connected planar graphs with 6 to 22 edges,” 1979, unpublished computer tape, Twente University of Technology, Enschede, The Netherlands.Google Scholar
  13. [13]
    Y. L. Voytekhovsky and D. G. Stepenshchikov, “On the symmetry of simple 14-and 15-hedra,” Acta Crystallographica Section A, 59 (2003) 367–370.CrossRefGoogle Scholar
  14. [14]
    Y. L. Voytekhovsky and D. G. Stepenshchikov, “On the symmetry of simple 16-hedra,” Acta Crystallographica Section A, 62 (2006) 230–232.CrossRefGoogle Scholar
  15. [15]
    J. A. F. Plateau, Statique Expérimentale et Théorique des Liquides Soumis aux Seules Forces Mol´ecularies, Paris: Gauthier-Villars, 1873.Google Scholar
  16. [16]
    E. A. Lazar, The Evolution of Cellular Structures via Curvature Flow, Ph.D. thesis, Princeton University, 2011.Google Scholar

Copyright information

© TMS (The Minerals, Metals & Materials Society) 2012

Authors and Affiliations

  • Trevor Keller
    • 1
  • Barb Cutler
    • 2
  • Martin Glicksman
    • 3
  • Dan Lewis
    • 1
  1. 1.Materials Science and Engineering DepartmentRensselaer Polytechnic InstituteTroyUSA
  2. 2.Computer Science DepartmentRensselaer Polytechnic InstituteTroyUSA
  3. 3.Mechanical & Aerospace EngineeringFlorida Institute of TechnologyMelbourneUSA

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