## Three-dimensional
reconstruction of statistically optimal unit

cells of polydisperse particulate composites from

microtomography

H. Lee^{2}, M. Brandyberry^{2}, A. Tudor^{2}
and K. Matous^{1} ^{}

^{1}Department of Aerospace and Mechanical Engineering

University of Notre Dame

Notre Dame, IN 46556, USA.

^{2}Computational Science and Engineering

University of Illinois at Urbana-Champaign

Urbana, IL 61801, USA.

^{
}
#### Abstract

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.

#### Conclusions

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.

#### Acknowledgment

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.

Download the paper here
© 2009 Notre Dame and Dr.
Karel
Matous