Computational Physics Group

Karel Matous










Three-dimensional reconstruction of statistically optimal unit
cells of multimodal particulate composites

B.C. Collins2, K. Matous1 and D. Rypl3

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.

3Department of Mechanics
Czech Technical University in Prague
Prague, 160 00, Czech Republic.


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.


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.


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