Research Papers

Stochastic Reassembly Strategy for Managing Information Complexity in Heterogeneous Materials Analysis and Design

[+] Author and Article Information
Hongyi Xu, Hua Deng

Department of Mechanical Engineering,
Northwestern University,
Evanston, IL 60208

M. Steven Greene

Theoretical & Applied Mechanics,
Northwestern University,
Evanston, IL 60208

Dmitriy Dikin

Department of Mechanical Engineering,
Northwestern University,
Evanston, IL 60208;
Department of Physics and Astronomy,
Northwestern University,
Evanston, IL 60208

Catherine Brinson

Department of Mechanical Engineering,
Northwestern University,
Evanston, IL 60208;
Department of Material Science and Engineering,
Northwestern University,
Evanston, IL 60208

Wing Kam Liu

Department of Mechanical Engineering,
Northwestern University,
Evanston, IL 60208

George Papakonstantopoulos

Global Materials Science Division,
The Goodyear Tire & Rubber Company,
Akron, OH 44305

Mike Poldneff

External Science & Technology Division,
The Goodyear Tire & Rubber Company,
Akron, OH 44309

Wei Chen

Department of Mechanical Engineering,
Northwestern University,
Evanston, IL 60208
e-mail: weichen@northwestern.edu

A heterogeneous material, by definition, contains multiple phases. For instance, filled elastomers contain polymer-like synthetic rubber and filler-like silica, nanoclay, or carbon black. These constituents have individual properties. However, designers using the composite material are interested in how the multiple phases behave together (for instance the tensile strength). The properties of the whole material system are labeled “apparent” and are often quite different than the sum of the constituents' properties.

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received August 22, 2012; final manuscript received June 19, 2013; published online September 11, 2013. Assoc. Editor: Michael Kokkolaras.

J. Mech. Des 135(10), 101010 (Sep 11, 2013) (12 pages) Paper No: MD-12-1427; doi: 10.1115/1.4025117 History: Received August 22, 2012; Revised June 19, 2013

Design of high performance materials system requires highly efficient methods for assessing microstructure–property relations of heterogeneous materials. Toward this end, a domain decomposition, affordable analysis, and subsequent stochastic reassembly approach is proposed in this paper. The approach hierarchically decomposes the statistically representative cell (representative volume element (RVE)) into computationally tractable unrepresentative ones (statistical volume element (SVE)) at the cost of introducing uncertainty into subdomain property predictions. Random property predictions at the subscale are modeled with a random field that is subsequently reassembled into a coarse representation of the RVE. The infinite dimension of microstructure is reduced by clustering SVEs into bins defined by common microstructure attributes, with each bin containing a different apparent property random field. We additionally mitigate the computational burden in this strategy by presenting an algorithm that minimizes the number of SVEs required for convergent random field characterization. In the proposed method, the RVE thus becomes a coarse representation, or mosaic, of itself. The mosaic approach maintains sufficient microstructure detail to accurately predict the macroproperty but becomes far cheaper from a computational standpoint. A nice feature of the approach is that the stochastic reassembly process naturally creates an apparent-SVE property database whose elements may be used as mosaic building blocks. This feature enables material design because SVE-apparent properties become the building blocks of new, albeit conceptual, material mosaics. Some simple examples of possible designs are shown. The approach is demonstrated on polymer nanocomposites.

Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.


McDowell, D. L., and Olson, G. B., 2008, “Concurrent Design of Hierarchical Materials and Structures,” Sci. Model. Simul., 15, pp. 1–34. [CrossRef]
Panchal, J. H., Choi, H. J., Allen, J. K., McDowell, D. L., and Mistree, F., 2007, “A Systems-Based Approach for Integrated Design of Materials, Products and Design Process Chains,” J. Comput.-Aided Mater. Des., 14, pp. 265–293. [CrossRef]
Chen, W., Yin, X. L., Lee, S., and Liu, W. K., 2010, “A Multiscale Design Methodology for Hierarchical Systems With Random Field Uncertainty,” ASME J. Mech. Des., 132(4), p. 041006. [CrossRef]
Yin, X. L., Lee, S., Chen, W., Liu, W. K., and Horstemeyer, M. F., 2009, “Efficient Random Field Uncertainty Propagation in Design Using Multiscale Analysis” ASME J. Mech. Des., 131(2), p. 021006. [CrossRef]
Allen, J. K., Seepersad, C., Choi, H. J., and Mistree, F., 2006, “Robust Design for Multiscale and Multidisciplinary Applications,” ASME J. Mech. Des., 128(4), pp. 832–843. [CrossRef]
Tian, R., Chan, S., Tang, S., Kopacz, A. M., Wang, J. S., Jou, H. J., Siad, L., Lindgren, L. E., Olson, G. B., and Liu, W. K., 2010, “A Multiresolution Continuum Simulation of the Ductile Fracture Process,” J. Mech. Phys. Solids, 58(10), pp. 1681–1700. [CrossRef]
Babuska, I., 1975, Homogenization and Its Application—Mathematical and Computational Problems (Partial Differential Equation Solutions for Diffusion and Composite Material Analysis) in Numerical Solution of Partial Differential Equations—III, University of Maryland, College Park, MD.
Guedes, J. M., and Kikuchi, N., 1990, “Preprocessing and Postprocessing for Materials Based on the Homogenization Method With Adaptive Finite-Element Methods,” Comput. Methods Appl. Mech. Eng., 83(2), pp. 143–198. [CrossRef]
Miehe, C., Schotte, J., and Lambrecht, M., 2002, “Homogenization of Inelastic Solid Materials at Finite Strains Based on Incremental Minimization Principles. Application to the Texture Analysis of Polycrystals,” J. Mech. Phys. Solids, 50(10), pp. 2123–2167. [CrossRef]
Fish, J., and Shek, K., 2000, “Multiscale Analysis of Composite Materials and Structures,” Compos. Sci. Technol., 60(12–13), pp. 2547–2556. [CrossRef]
Chung, P. W., Tamma, K. K., and Namburu, R. R., 1999, “Asymptotic Expansion Homogenization for Heterogeneous Media: Computational Issues and Applications,” Composites, Part A, 32(9), pp. 1291–1301. [CrossRef]
Oden, J. T., and Zohdi, T. I., 1997, “Analysis and Adaptive Modeling of Highly Heterogeneous Elastic Structures,” Comput. Methods Appl. Mech. Eng., 148(3–4), pp. 367–391. [CrossRef]
Raghavan, P., and Ghosh, S., 2004, “Concurrent Multi-Scale Analysis of Elastic Composites by a Multi-Level Computational Model,” Comput. Methods Appl. Mech. Eng., 193(6–8), pp. 497–538. [CrossRef]
Hill, R., 1963, “Elastic Properties of Reinforced Solids—Some Theoretical Principles,” J. Mech. Phys. Solids, 11(5), pp. 357–372. [CrossRef]
Ostoja-Starzewski, M., 2006, “Material Spatial Randomness: From Statistical to Representative Volume Element,” Probab. Eng. Mech., 21(2), pp. 112–132. [CrossRef]
Yin, X. L., Chen, W., To, A., McVeigh, C., and Liu, W. K., 2008, “Statistical Volume Element Method for Predicting Micro Structure-Constitutive Property Relations,” Comput. Methods Appl. Mech. Eng., 197(43–44), pp. 3516–3529. [CrossRef]
Greene, M. S., Liu, Y., Chen, W., and Liu, W. K., 2011, “Computational Uncertainty Analysis in Multiresolution Materials via Stochastic Constitutive Theory,” Comput. Methods Appl. Mech. Eng., 200(1–4), pp. 309–325. [CrossRef]
Ostoja-Starzewski, M., 1998, “Random Field Models of Heterogeneous Materials,” Int. J. Solids Struct., 35(19), pp. 2429–2455. [CrossRef]
Torquato, S., 2002, Random Heterogeneous Materials: Microstructure and Macroscopic Properties, Springer-Verlag, New York.
Jiang, Z., Chen, W., and Burkhart, C., 2012, “A Hybrid Optimization Approach to 3D Porous Microstructure Reconstruction via Gaussian Random Field,” ASME 2012 International Design Engineering Technical Conferences (IDETC) and Computers and Information in Engineering Conference (CIE), Chicago, IL.
Tewari, A., and Gokhale, A. M., 2004, “Nearest-Neighbor Distances Between Particles of Finite Size in Three-Dimensional Uniform Random Microstructures,” Mater. Sci. Eng., A, 385(1–2), pp. 332–341.
Holotescu, S., and Stoian, F. D., 2011, “Prediction of Particle Size Distribution Effects on Thermal Conductivity of Particulate Composites,” Materialwiss. Werkstofftech., 42(5), pp. 379–385. [CrossRef]
Al-Ostaz, A., Diwakar, A., and Alzebdeh, K. I., 2007, “Statistical Model for Characterizing Random Microstructure of Inclusion-Matrix Composites,” J. Mater. Sci., 42(16), pp. 7016–7030. [CrossRef]
Karasek, L., and Sumita, M., 1996, “Characterization of Dispersion State of Filler and Polymer-Filler Interactions in Rubber Carbon Black Composites,” J. Mater. Sci., 31(2), pp. 281–289. [CrossRef]
Belytschko, T., Liu, W. K., and Moran, B., 2000, Nonlinear Finite Elements for Continua and Structures, Wiley, Chichester, New York.
Greene, M. S., Xu, H., Tang, S., Chen, W., and Liu, W. K., 2012, “A Generalized Uncertainty Propagation Criterioark Studies of Microstructured Material Systems,” Comput. Methods Appl. Mech. Eng., 254, pp. 271–291. [CrossRef]
Liu, Y., Greene, M. S., Chen, W., Dikin, D., and Liu, W. K., 2011, “Computational Microstructure Characterization and Reconstruction to Enable Stochastic Multiscale Design,” CAD, 45, pp. 65–76.
Sundararaghavan, V., and Zabaras, N., 2005, “Classification and Reconstruction of Three-Dimensional Microstructures Using Support Vector Machines,” Comput. Mater. Sci., 32(2), pp. 223–239. [CrossRef]
Morisita, M., 1962, “Iσ-Index, a Measure of Dispersion of Individuals,” Res. Popul. Ecol., 4, pp. 1–7. [CrossRef]
Morozov, I. A., Lauke, B., and Heinrich, G., 2011, “A Novel Method of Quantitative Characterization of Filled Rubber Structures by AFM,” Kautsch. Gummi Kunstst., 64(1–2), pp. 24–27.
MacQueen, J. B., 1967, “Some Methods of Classification and Analysis of Multivariate Observations,” Proceedings of 5th Berkeley Symposium on Mathematical Statistics and Probability, University of California, Berkeley, pp. 281–297.
Lloyd, S. P., 1982, “Least-Squares Quantization in Pcm,” IEEE Trans. Inf. Theory, 28(2), pp. 129–137. [CrossRef]
Sibson, R., 1973, “Slink—Optimally Efficient Algorithm for Single-Link Cluster Method,” Comput. J., 16(1), pp. 30–34. [CrossRef]
Ester, M., Kriegel, H. P., Sander, J., and Xu, X.1996, “Density-Based Algorithm for Discovering Clusters in Large Spatial Databases With Noise,” Second International Conference on Knowledge Discovery and Data Mining, Portland, OR.
Ketchen, D. J., and Shook, C. L., 1996, “The Application of Cluster Analysis in Strategic Management Research: An Analysis and Critique,” Strategic Manage. J., 17(6), pp. 441–458. [CrossRef]
Goutte, C., Toft, P., Rostrup, E., Nielsen, F. A., and Hansen, L. K., 1999, “On Clustering fMRI Time Series,” Neuroimage, 9(3), pp. 298–310. [CrossRef]
Turk, M., and Pentland, A., 1991, “Eigenfaces for Recognition,” J. Cogn Neurosci., 3(1), pp. 71–86. [CrossRef]
Sudret, B., and Der Kiureghian, A., 2000, Stochastic Finite Element Methods and Reliability: A State-Of-The-Art Report, Department of Civil and Environmental Engineering, University of California, Berkeley.
Xiu, D., 2010, Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton University, Princeton, NJ.
Adcock, C. J., 1997, “Sample Size Determination: A Review,” Statistician, 46(2), pp. 261–283.
Stein, C., 1945, “A Two-Sample Test for a Linear Hypothesis Where Power Is Independent of Variance,” Ann. Math. Statist., 16, pp. 243–258. [CrossRef]
Cohen, J., 1977, Statistical Power Analysis for the Behavior Science, Revised ed., Academic, New York.
Ramanathan, T., Abdala, A. A., Stankovich, S., Dikin, D. A., Herrera-Alonso, M., Piner, R. D., Adamson, D. H., Schniepp, H. C., Chen, X., Ruoff, R. S., Nguyen, S. T., Aksay, I. A., Prud'homme, R. K., and Brinson, L. C., 2008, “Functionalized Graphene Sheets for Polymer Nanocomposites,” Nat. Nanotechnol., 3(6), pp. 327–331. [CrossRef]
Zheng, W., and Wong, S. C., 2003, “Electrical Conductivity and Dielectric Properties of PMMA/Expanded Graphite Composites,” Compos. Sci. Technol., 63(2), pp. 225–235. [CrossRef]
Putz, K. W., Mitchell, C. A., Krishnamoorti, R., and Green, P. F., 2004, “Elastic Modulus of Single-Walled Carbon Nanotube/Poly(Methyl Methacrylate) Nanocomposites,” J. Polym. Sci., Part B: Polym. Phys., 42(12), pp. 2286–2293. [CrossRef]
Brinson, H. F., and Brinson, L. C., 2007, Polymer Engineering Science and Viscoelasticity: An Introduction., Springer, New York.
Deng, H., Liu, Y., Gai, D., DikinD. A., Putz, K. W., Chen, W., and Brinson, L. C., 2011, “Utilizing Real and Statistically Reconstructed Microstructures for the Viscoelastic Modeling of Polymer Nanocomposites,” Compos. Sci. Technol., 72, pp. 1725–1732. [CrossRef]
Qiao, R., Deng, H., Putz, K. W., and Brinson, L. C., 2011, “Effect of Particle Agglomeration and Interphase on the Glass Transition Temperature of Polymer Nanocomposites,” J. Polym. Sci., Part B: Polym. Phys., 49(10), pp. 740–748. [CrossRef]
Coomans, D., and Massart, D. L., 1982, “Alternative K-Nearest Neighbor Rules in Supervised Pattern-Recognition. 2. Probabilistic Classification on the Basis of the Knn Method Modified for Direct Density-Estimation,” Anal. Chim. Acta, 138, pp. 153–165. [CrossRef]


Grahic Jump Location
Fig. 3

Material property upscaling from SVE to RVE level using mosaic image of SVE clusters

Grahic Jump Location
Fig. 2

Comparison of proposed strategy and the typical homogenization approach. The method alters the representation of the RVE to a coarse version of itself through the hierarchical decomposition and reassembly processes. SVEs with similar morphology features are classified into different clusters marked by different color in the coarsened RVE.

Grahic Jump Location
Fig. 1

The concept of upscaling properties in multiscale methods with RVE and SVE. RVEs produce deterministic apparent material properties; SVEs introduce uncertainty. When comparing the RVE (top) to the SVE (bottom), the material is the same. The difference is only the window size lSVE<lRVE.

Grahic Jump Location
Fig. 4

Illustration of “elbow point” in optimal cluster number determination

Grahic Jump Location
Fig. 5

The flowchart of the sample size determination process

Grahic Jump Location
Fig. 6

Property for pure polymer (G′ and G″). G′ measures the stored energy, representing the elastic portion, and G″ measures the energy dissipated as heat, representing the viscous portion.

Grahic Jump Location
Fig. 7

(a) Double layer gradient interphase model in FEM; (b) viscoelastic properties of interphase

Grahic Jump Location
Fig. 8

Two examples of comparisons between SVE clusters generated by clustering method based solely on gray level values (a, b) and clustering method based on five key microstructure descriptors (c, d)

Grahic Jump Location
Fig. 9

Comparison of standard deviations of middle points' value on the G′ and G″ curves in each cluster. The standard deviations are sorted following the ascending trend instead of following the sequence of cluster number to facilitate visual comparison.

Grahic Jump Location
Fig. 10

The reduced sample sets' sizes compared with full sample sets' sizes of each cluster

Grahic Jump Location
Fig. 11

The distributions of values at five frequencies on the tan δ curve are compared with determine the sample size

Grahic Jump Location
Fig. 12

Comparison of RVE material damping property (G′ and G″). Meanings of titles in the legend: (1) DNS, direct numerical simulation; (2) real SVE properties: using properties of real SVE images divided for RVE image; (3) regenerated SVE properties, all SVE samples: regenerated SVE properties using random field model established by all SVE sample images; (4) regenerated SVE properties, α = X: regenerated SVE properties using random field model based on minimized sample set. X = 0.5 is used in t test.

Grahic Jump Location
Fig. 13

Simulation accuracy and computation time cost of reassembly approach using different SVE sizes

Grahic Jump Location
Fig. 14

Microstructure descriptors and correlation functions for two material designs

Grahic Jump Location
Fig. 15

Reconstructed microstructures, mosaic RVE, and the correspondent properties



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In