Simulation of 3-Dimensional Cell Population Growth Processes in Polyhedral Cellular Systems


Article Preview

In order to simulate the polyhedral grain nucleation in alloys, 3-D cell population growth processes are studied in space-filling periodic cellular systems. We discussed two different methods by which space-filling polyhedral cellular systems can be constructed by topological transformations performed on “stable” 3-D cellular systems. It has been demonstrated that an infinite sequence of stable periodic space-filling polyhedral systems can be generated by means of a simple recursion procedure based on a vertex based tetrahedron insertion. On the basis of computed results it is conjectured that in a 3-D periodic, topologically stable cellular system the minimum value of the average face number 〈f〉 of polyhedral cells is larger than eight (i.e. 〈f〉 > 8). The outlined algorithms (which are based on cell decomposition and/or cell nucleation) provide a new perspective to simulate grain population growth processes in materials with polyhedral microstructure.



Materials Science Forum (Volumes 537-538)

Edited by:

J. Gyulai and P.J. Szabó




T. Réti and I. Zsoldos, "Simulation of 3-Dimensional Cell Population Growth Processes in Polyhedral Cellular Systems ", Materials Science Forum, Vols. 537-538, pp. 579-590, 2007

Online since:

February 2007




[1] D. Weaire and M. Rivier: Soap, Cells and Statistics - Random Pattern in Two Dimensions, Contemp. Phys. Vol. 25 (1984) pp.59-99.

[2] J. Baracs: Juxtapositions, Structural Topology, Vol. 1 (1979) pp.59-71.

[3] M. A. Fortes and A. C. Ferro: Trivalent polyhedra: Properties, representation and enumeration, Acta Metall. Vol. 33 (1985) pp.1683-1696.


[4] M. A. Fortes and A. C. Ferro: Topology and transformations in cellular structures, Acta Metall. Vol. 33 (1985) pp.1697-1708.


[5] T. Aste, D. Boosé and N. Rivier: From on cell to the whole froth: A dynamical map, Phys. Review. E, Vol. 53 (1996) pp.6181-6191.


[6] T Aste: Dynamical partitions of space in any dimension, J. Phys. A: Math. Gen. Vol. 31 (1998) pp.8577-8593.


[7] H. Honda, M. Tanemura and T. Nagai: A three-dimensional vertex dynamic model of spacefilling polyhedra simulating cell behavior in a cell aggregate, Journal of Theoretical Biology, Vol. 226, (2004) pp.439-453.


[8] N. Rivier: Combinatorics in Glass, Math. Comput. Modelling, Vol. 26 (1997) pp.255-267.

[9] H.S.M. Coxeter and A.V. Kharchenko: Frieze patterns for regular star polytopes and statistical honeycombs, Periodica Mathematica Hungarica, Vol. 39 (1999) pp.51-63.

[10] J. Ohser and F. Müchlich: Statistical Analysis of Microstructures in Materials, John Wiley and Sons, New York, (2000).

[11] I. Saxl, P. Ponizil and K. Sülleiova : Stereology and simulation of heterogeneous crystalline media, Int. J. of Materials and Product Technology, Vol. 18, (2003) pp.1-25. ffff and.


[12] B. Grünbaum and G.C. Shephard: Incidence symbols and their applications. Proc. Symp. Pure Math. Vol. 34, (1979) p. l99-224.

[13] B. Grünbaum and G.C. Shephard: Some Comments on Juxtapositions, Structural Topology, Vol. 3 (1979) pp.58-61.

[14] R. Williams: The Geometrical Foundation of Natural Structure: A Source of Book of Design, New York, Dove, (1979).

[15] B. Grünbaum and G.C. Shephard: Tilings with congruent tiles, Bull. Amer. Math. Soc. Vol. 3 (1980) pp.951-973.


[16] F.W. Smith: The structure of aggregates - a class of 20-faceted space-filling polyhedra, Canadian Journal of Physics, Vol. 43 (1965) p.2052-(2055).


[17] E. Koch and W. Fischer: Wirkungsbereichstypen einer verzerrten Diamantkonfiguration mit Kugelpackungscharacter, Zeitschrift für Kristallographie, Vol. 135 (1972) pp.73-92.


[18] P. Engel: Über Wirkungsbereiche von kubischer Symmetrie, Zeitschrift für Kristallographie, Vol 154, (1981) pp.199-215.


[19] G.O. Brunner and F. Laves: How many faces has the largest space-filling polyhedron? Zeitschrift für Kristallographie, Vol 147, (1978) pp.39-43.


[20] E. Schulte: Tiling three-space by combinatorially equivalent convex polytopes, Proc. London Math. Soc. Vol. 49 (1984) pp.128-140.


[21] F. Aurenhammer: Voronoi Digarams - A Survey of a Fundamental Geometric Data Structure, ACM Computing Survey, Vol. 23 (1991) pp.345-405.


[22] P. Engel: Geometric Crystallography: An Axiomatic Introduction to Crystallography, D. Reidel Publishing Company, Dordrecht, (1986).

[23] D. Bochis and F. Santos: On the number of facets of Dirichlet stereohedra I: Groups with reflections, Discrete Comput. Geom., Vol. 25 (2001) pp.419-444.


[24] N. Dolbilin and E. Schulte: The Local Theorem for Monotypic Tilings, The Electronic Journal of Combinatorics, Vol. 11, (2004) pp.73-91.

[25] M. O'Keeffe: On a space-filling polyhedron of Aste et al., Philosophical Magazine Letters, Vol. 76. (1997) pp.423-426.