## 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.

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