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]