## 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.

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here
© 2009 Notre Dame and Dr.
Karel Matous