## Three-dimensional
reconstruction of statistically optimal unit

cells of multimodal particulate composites

B.C. Collins^{2}, K. Matous^{1}
and D. Rypl^{3}

^{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.

^{3}Department of Mechanics

Czech Technical University in Prague

Prague, 160 00, Czech Republic.

^{
}
#### Abstract

In the current digital age, it is befitting that complex
heterogeneous materials, such as solid propellants, are
characterized by digital computational and/or experimental
techniques. Of those, micro computer tomography (micro-CT)
and advanced packing algorithms are the most popular for
identifying the statistics of multimodal, random,
particulate composites. In this work, we develop a
procedure for the characterization and reconstruction of
periodic unit cells of highly filled, multimodal,
particulate composites from a packing algorithm.

Rocpack, a particle
packing software, is used to generate the solid propellant
microstructures and one-, two- and three-point probability
functions are used to describe their statistical
morphology. However, both the experimentally scanned or
computationally designed packs are usually non optimal in
size and likely too big to be fully numerically resolved
when complex nonlinear processes such as combustion,
decohesion, matrix tearing, etc. are modeled. Thus, domain
reduction techniques, which can reconstruct the optimal
periodic unit cell, are important to narrow the problem
size while preserving the statistics. The three
dimensional reconstruction is carried out using a parallel
Augmented Simulated Annealing algorithm. Then, the
resulting cell geometries are discretized, taking into
consideration the periodic layout using our master/slave
approach implemented into a sophisticated meshing
generator

T3D.
Final discretized geometries show only a small loss of
volume fraction. Particulate systems composed of 40% and
70% volume fractions are investigated, and the unit cells
are reconstructed such that the statistical correspondence
to the original packs is maintained.

#### Conclusions

An efficient and highly parallel procedure to reconstruct
a statistically optimal periodic unit cell from a computer
generated microstructure has been developed. This
procedure consists of finding the statistical descriptors
of a random composite, then using stochastic optimization
methods to create a PUC with statistics that are similar
to those of the original microstructure. It is important
to note that the reconstructed periodic unit cells are
only representative from a geometrical statistics point of
view and that the representativity of the PUC must also
account for the physical processes of interest. However,
the construction of a geometrically equivalent periodic
unit cell is an important first step in describing
behavior of complex particulate materials, such as solid
propellants.

For the present study, one-, two- and
three-point probability functions have been identified as
the suitable statistical descriptors. Higher order
statistics will allow for more accurate material
description once the nonlinear processes are investigated.
Such processes are highly influenced, for example, by
small particles acting like stress concentrators in
between two big particles. Such occurrences can be
statistically measured by the third-order probability
functions and this information can be used in advanced
homogenization schemes.

Computer generated, highly filled,
particulate composites have been statistically
characterized and optimal unit cells have been
reconstructed with a high accuracy. For a highly packed
system, unit cell dimensions obtained from our analysis
are consistent with those experimentally observed. Novel
periodic meshing, based on the master/slave approach, has
been extended to three-dimensions and the reconstructed
cells of 40% and 70% volume fraction have been
successfully discretized for subsequent analysis. The
linear scalability of the optimization scheme has been
demonstrated.

A natural direction of further research
is to extend this procedure to include optimization of
three-point probability functions. Another possible future
research direction involves extending the genetic
algorithm to include optimization of other geometric
objects, such as ellipsoids, rhombi, etc. Also, direct
reconstruction from tomographic data is of interest.

#### Acknowledgment

K.Matous and B.C. Collins would like to gratefully
acknowledge the support from ARK/Thiokol (ATK-21316), with
J. Thompson and Dr. I. L. Davis serving as program
monitors and 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. D. Rypl would like to gratefully acknowledge the
Ministry of Education of Czech Republic in the framework
of the project No. MSM 6840770003. The authors also thank
Michael Campbell for running the reconstruction code on
Red Storm computer located at Sandia National
Laboratories, NM. Moreover, 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