Journal of Statistical Physics

, Volume 154, Issue 4, pp 913–928 | Cite as

Parametric Representation of 3D Grain Ensembles in Polycrystalline Microstructures

  • Aaron Spettl
  • Thomas Werz
  • Carl E. KrillIII
  • Volker Schmidt


As a straightforward generalization of the well-known Voronoi construction, Laguerre tessellations have long found application in the modelling, analysis and simulation of polycrystalline microstructures. The application of Laguerre tessellations to real (as opposed to computed) microstructures—such as those obtained by modern 3D characterization techniques like X-ray microtomography or focused-ion-beam serial sectioning—is hindered by the mathematical difficulty of determining the correct seed location and weighting factor for each of the grains in the measured volume. In this paper, we propose an alternative to the Laguerre approach, representing grain ensembles with convex cells parametrized by orthogonal regression with respect to 3D image data. Applying our algorithm to artificial microstructures and to microtomographic data sets of an Al-5 wt% Cu alloy, we demonstrate that the new approach represents statistical features of the underlying data—like distributions of grain sizes and coordination numbers—as well as or better than a recently introduced approximation method based on the Laguerre tessellation; furthermore, our method reproduces the local arrangement of grains (i.e., grain shapes and connectivities) much more accurately. The additional computational cost associated with orthogonal regression is marginal.


Polycrystalline microstructure Laguerre tessellation Orthogonal regression Convex cells 



The authors would like to thank D. Molodov of the Institute of Physical Metallurgy and Metal Physics, RWTH Aachen, for sample preparation; the Institute of Orthopaedic Research and Biomechanics, Ulm University, for X-ray microtomography beamtime; and especially Uwe Wolfram for assistance with the tomography measurements and numerous discussions. Furthermore, the authors are grateful to the Deutsche Forschungsgemeinschaft for funding through NSF/DFG Materials World Network Project KR 1658/4-1.


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Aaron Spettl
    • 1
  • Thomas Werz
    • 2
  • Carl E. KrillIII
    • 2
  • Volker Schmidt
    • 1
  1. 1.Institute of StochasticsUlm UniversityUlmGermany
  2. 2.Institute of Micro and NanomaterialsUlm UniversityUlmGermany

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