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International Journal for Multiscale Computational Engineering
IF: 1.016 5-Year IF: 1.194 SJR: 0.452 SNIP: 0.68 CiteScore™: 1.18

ISSN Print: 1543-1649
ISSN Online: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.v8.i5.50
pages 489-507

Three-Dimensional Reconstruction of Statistically Optimal Unit Cells of Multimodal Particulate Composites

B. C. Collins
Computational Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL61801, USA
Karel Matous
Department of Aerospace and Mechanical Engineering, Center for Shock Wave-processing of Advanced Reactive Materials, University of Notre Dame, Notre Dame, Indiana 46556, USA
D. Rypl
Department 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, microcomputer 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 nonoptimal 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.


  1. Beran, M. J., Statistical Continuum Theories.

  2. Bittnar, Z. and Rypl, D., Direct triangulation of 3d surfaces using advancing front technique.

  3. Bochenek, B. and Pyrz, R., Reconstruction of random microstructures–A stochastic optimization problem. DOI: 10.1016/j.commatsci.2004.01.038

  4. Cignoni, P., Montani, C., and Scopigno, R., DeWall: A fast divide and conquer Delaunay triangulation algorithm in Ed. DOI: 10.1016/S0010-4485(97)00082-1

  5. Collins, B. C., Reconstruction of statistically optimal periodic unit cells of multimodal particulate composites.

  6. Collins, B., Maggi, F., Matous, K., Jackson, T. L., and Buckmaster, J., Using tomography to characterize heterogeneous propellants. DOI: 10.2514/6.2008-941

  7. Corson, P. B., Correlation functions for predicting properties of heterogeneous materials. i. experimental measurement of spatial correlation functions in multiphase solids. DOI: 10.1063/1.1663741

  8. Demmel, J. W., Eisenstat, S. C., Gilbert, J. R., Li, X. S., and Liu, J. W. H., A supernodal approach to sparse partial pivoting. DOI: 10.1137/S0895479895291765

  9. Fish, J. and Chen, W., RVE based multilevel method for periodic heterogeneous media with strong scale mixing. DOI: 10.1023/A:1022828324366

  10. Goldberg, D., Genetic Algorithms in Search, Optimization and Machine Learning.

  11. Hashin, Z. and Shtrikman, S., On some variational principles in anisotropic and non-homogeneous elasticity. DOI: 10.1016/0022-5096(62)90004-2

  12. Hill, R., Elastic properties of reinforced solids–some theoretical principles. DOI: 10.1016/0022-5096(63)90036-X

  13. Hughes, T. J. R., The Finite Element Method: Linear Static and Dynamic Finite Element Analysis.

  14. Knott, G. M., Jackson, T. L., and Buckmaster, J., The random packing of heterogeneous propellants. DOI: 10.2514/2.1361

  15. Kochevets, S. V., Buckmaster, J., Jackson, T. L., and Hegab, A., Random packs and their use in the modeling of heterogeneous solid propellant combustion. DOI: 10.2514/2.5820

  16. Kumar, N. C., Matouˇs, K., and Geubelle, P. H., Reconstruction of periodic unit cells of multimodal random particulate composites using genetic algorithms. DOI: 10.1016/j.commatsci.2007.07.043

  17. Lions, J. L., Some methode in the mathematical analysis of systems and their control.

  18. Liu, C., On the minimum size of representative volume element: An experimental investigation. DOI: 10.1007/BF02427947

  19. Matous, K., Leps, M., Zeman, J., and Sejnoha, M., Applying genetic algorithms to selected topics commonly encountered in engineering practice. DOI: 10.1016/S0045-7825(00)00192-4

  20. Matous, K. and Dvorak, J. G., Optimization of electromagnetic absorption in laminated composite plates. DOI: 10.1109/TMAG.2003.809861

  21. Matous, K. and Geubelle, P. H., Multiscale modelling of particle debonding in reinforced elastomers subjected to finite deformations. DOI: 10.1002/nme.1446

  22. Matous, K., Inglis, H. M., Gu, X., Rypl, D., Jackson, T. L., and Geubelle, P. H., Multiscale modeling of solid propellants: From particle packing to failure. DOI: 10.1016/j.compscitech.2006.06.017

  23. Povirk, G. L., Incorporation of microstructural information into materials.

  24. Rypl, D., Sequential and parallel generation of unstructured 3D meshes.

  25. Smith, P. and Torquato, S., Computer simulation results for the two-point probability function of composite media. DOI: 10.1016/0021-9991(88)90136-2

  26. Sundararaghavan, V. and Zabaras, N., Classification and reconstruction of three-dimensional microstructures using support vector machines. DOI: 10.1016/j.commatsci.2004.07.004

  27. Swaminathan, S. and Ghosh, S., Statistically equivalent representative volume elements for unidirectional composite microstructures: Part II.With interfacial debonding. DOI: 10.1177/0021998305055274

  28. Talbot, D. R. S. and Willis, J. R., Variational principles for inhomogeneous nonlinear media. DOI: 10.1093/imamat/35.1.39

  29. Torquato, S. and Stell, G., Microstructure of two-phase random media. I. The N-point probability functions. DOI: 10.1063/1.444011

  30. Torquato, S., Random Heterogeneous Media.

  31. Wentorf, R., Collar, R., Shephard, M. S., and Fish, J., Automated modeling for complex woven mesostructures. DOI: 10.1016/S0045-7825(98)00232-1

  32. Yeong, C. L. Y. and Torquato, S., Reconstructing random media. DOI: 10.1103/physreve.57.495

  33. Zeman, J. and Sejnoha, M., Numerical evaluation of effective elastic properties of graphite fiber tow impregnated by polymer matrix. DOI: 10.1016/S0022-5096(00)00027-2