Computational Physics Group

Karel Matous










Three-dimensional reconstruction of statistically optimal unit
cells of polydisperse particulate composites from

H. Lee2, M. Brandyberry2, A. Tudor2 and K. Matous1

1Department of Aerospace and Mechanical Engineering
University of Notre Dame
Notre Dame, IN 46556, USA.

2Computational Science and Engineering
University of Illinois at Urbana-Champaign
Urbana, IL 61801, USA.


    In this paper, we present a systematic approach for characterization and reconstruction of statistically optimal representative unit cells of polydisperse particulate composites. Microtomography is used to gather rich three-dimensional data of a packed glass beads system. First-, second- and third-order probability functions are used to characterize the morphology of the material, and the parallel augmented simulated annealing algorithm is employed for reconstruction of the statistically equivalent medium. Both the fully resolved probability spectrum and the geometrically exact particle shapes are considered in this study, rendering the optimization problem multidimensional with a highly complex objective function. A ten-phase particulate composite composed of packed glass beads in a cylindrical specimen is investigated, and a unit cell is reconstructed on massively parallel computers. Further, rigorous error analysis of the statistical descriptors (probability functions) is presented and a detailed comparison between statistics of the voxel-derived pack and the representative cell is made.


    The paper describes a reconstruction procedure for statistically optimal representative unit cells from rich three-dimensional tomographic data. The particulate composite under investigation consists of glass beads packed in a cylindrical container. High resolution microtomography is employed to gather the material data, and the image recognition software Amira is used for data processing. The first-, second- and third-order probability functions are used to characterize a polydisperse particulate medium. Error measures are established to assess the quality of the statistical characterization. A fully represented probability spectrum is optimized without distortion of the particle shape and with a constraint on the particle overlap, furnishing the resulting minimization problem highly complex with several local minima. Therefore, the parallel augmented simulated annealing technique is employed to solve the optimization problem on massively parallel computers. Presented results show good repeatability of the reconstruction procedure. Excellent agreement is obtained for statistics of the voxel based pack and statistics of the reconstructed unit cell.
    Investigation of the higher order probability functions reveals disagreement in the third-order probabilities between the pack and the cell, even though the first- and second-order functions are well optimized. Thus, potential extension of this work is in expansion of the fitness function for the third-order statistics. Also, optimization of polydisperse composites with different inclusion shapes, such as ellipsoids, rhombi, etc., is of interest.
    It is important to note that the reconstructed unit cells are only representative from a geometrical statistics point of view and that the representativity of the unit cell must also account for the physical processes of interest, such as mechanical or transport properties. However, the construction of a geometrically equivalent representative unit cell is an important first step in describing behavior of complex heterogeneous materials, and both computational and experimental evidence suggests that a statistical approach adopted in this work accounts for the most important interactions [27, 42]. Moreover, advances in parallel computing are making fully resolved simulations of complex physical phenomena, such as combustion [29], on cells presented in this work a reality.


    The authors would like to gratefully acknowledge the support from the Center for Simulation of Advanced Rockets (CSAR) at the University of Illinois under the contract number B523819 by the U.S. Department of Energy as a part of its Advanced Simulation and Computing (ASC) program. K. Matous and H. Lee would like to also acknowledge the support from Buckmaster Research - DoD STTR program, AFOSR: Dr. J. Buckmaster (Buckmaster Research) and Dr. A. Nachman (AFOSR) program managers. Moreover, the authors thank Michael Campbell for running the reconstruction code on Red Storm computer located at Sandia National Laboratories, NM, and to Sergei Poliakov for running the statistics code on the Turing cluster. Finally, the authors gratefully acknowledge the use of the Turing cluster maintained and operated by the Computational Science and Engineering Program at the University of Illinois.

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© 2009 Notre Dame and Dr. Karel Matous