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Thermal Response Variability of Random Polycrystalline Microstructures

Published online by Cambridge University Press:  20 August 2015

Bin Wen
Materials Process Design and Control Laboratory, Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853-3801, USA
Zheng Li
Materials Process Design and Control Laboratory, Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853-3801, USA State Key Laboratory of Structural Analysis for Industrial Equipment, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian 116024, China
Nicholas Zabaras*
Materials Process Design and Control Laboratory, Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853-3801, USA
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A data-driven model reduction strategy is presented for the representation of random polycrystal microstructures. Given a set of microstructure snapshots that satisfy certain statistical constraints such as given low-order moments of the grain size distribution, using a non-linear manifold learning approach, we identify the intrinsic low-dimensionality of the microstructure manifold. In addition to grain size, a linear dimensionality reduction technique (Karhunun-Loéve Expansion) is used to reduce the texture representation. The space of viable microstructures is mapped to a low-dimensional region thus facilitating the analysis and design of polycrystal microstructures. This methodology allows us to sample microstructure features in the reduced-order space thus making it a highly efficient, low-dimensional surrogate for representing microstructures (grain size and texture). We demonstrate the model reduction approach by computing the variability of homogenized thermal properties using sparse grid collocation in the reduced-order space that describes the grain size and orientation variability.

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Copyright © Global Science Press Limited 2011

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[1]Sundararaghavan, V. and Zabaras, N., A statistical learning approach for the design of poly-crystalline materials, Statistical Analysis and Data Mining, 1 (2009), 306–321.Google Scholar
[2]Sundararaghavanand, V.Zabaras, N., Design of microstructure-sensitive properties in elasto-viscoplastic polycrystals using multi-scale homogenization, Int. J. Plast., 22 (2006), 1799–1824.Google Scholar
[3]Sankaran, S. and Zabaras, N., Computing property variability of polycrystals induced by grain size and orientation uncertainties, Acta. Mater., 55 (2007), 2279–2290.CrossRefGoogle Scholar
[4]Zabaras, N. and Sankaran, S., An information-theoretic approach to stochastic materials modeling, IEEE Computing in Science and Engineering (CiSE) (special issue of “Stochastic Modeling of Complex Systems”, Tartakovsky, D. M., and Xiu, D., edts), Mar/Apr (2007), 50–59.Google Scholar
[5]Ganapathysubramanian, B. and Zabaras, N., Modelling diffusion in random heterogeneous media: data-driven models, stochastic collocation and the variational multi-scale method, J. Comput. Phys., 226 (2007), 326–353.Google Scholar
[6]Ganapathysubramanian, B. and Zabaras, N., A non-linear dimension reduction methodology for generating data-driven stochastic input models, J. Comput. Phys., 227 (2008), 6612–6637.Google Scholar
[7]Frank, R. C., Orientation mapping, Met. Trans. A., 19A (1988), 403–408.Google Scholar
[8]Acharjee, S. and Zabaras, N., A proper orthogonal decomposition approach to microstructure model reduction in Rodrigues space with applications to optimal control of microstructure-sensitive properties, Acta. Mater., 51 (2003), 5627–5646.Google Scholar
[9]Ganapathysubramanian, S. and Zabaras, N., Design across length scales: a reduced-order model of polycrystal plasticity for the control of microstructure-sensitive material properties, Comput. Meth. App. Mech. Eng., 193 (2004), 5017–5034.Google Scholar
[10]Kouchmeshky, B. and Zabaras, N., The effectof multiple sources of uncertainty on the convex hull of material properties of polycrystals, Comput. Mater. Sci., 47 (2009), 342–352.Google Scholar
[11]Ma, X. and Zabaras, N., An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations, J. Comput. Phys., 228 (2009), 3084–3113.CrossRefGoogle Scholar
[12]Bowman, A.W. and Azzalini, A., Applied Smoothing Techniques for Data Analysis, Oxford University Press, 1997.Google Scholar
[13]Kent, J. T., Bibby, J. M., and Mardia, K. V., Multivariate Analysis, Probability and Mathematical Statistics, Elsevier, 2006.Google Scholar
[14]Roweis, S. T. and Saul, L.K., Nonlinear dimensionality reductionbylocally linear embedding, Sci., 290 (2000), 2323–2326.Google Scholar
[15]deSilva, V. and Tenenbaum, J. B., Global versus local methods in nonlinear dimensionality reduction, Adv. Neural. Inform. Pro. Syst., 15 (2003), 721–728.Google Scholar
[16]Tenenbaum, J., Silva, V. de, and Langford, J., A global geometric framework for nonlinear dimension reduction, Sci., 290 (2000), 2319–2323.Google Scholar
[17]A Global Geometric Framework for Nonlinear Dimensionality Reduction, freely downloadable software available at Scholar
[18]Krill, C. E. III and Chen, L.-Q., Computer simulation of 3-D grain growth using a phasefield model, Acta. Mater., 50 (2002), 3059–3075.Google Scholar
[19]Sankaran, S. and Zabaras, N., A maximum entropy approach for property prediction of random microstructures, Acta. Mater., 54 (2006), 2265–2276.Google Scholar
[20]Fan, D. and Chen, L.-Q., Computer simulation of grain growth using a continuum field model, Acta. Mater., 45 (1997), 611–622.Google Scholar
[21]Fan, D., Geng, C., and Chen, L.-Q., Computer simulation of topological evolution in 2-D grain growth using a continuum diffuse-interface field model, Acta. Mater., 45 (1997), 1115–1126.Google Scholar
[22]Vedantam, S. and Patnaik, B. S. V., Efficient numerical algorithm for multiphase field simulations, Phys. Rev. E., 73 (2006), 016703.Google Scholar
[23]Melnick, E. L. and Tenenbein, A., Misspecifications of the normal distribution, Am. Stat., 36(4) (1982), 372–373.Google Scholar
[24]Rosenblatt, M., Remarks on a multivariable transformation, Ann. Math. Stat., 23 (1952), 470–472.Google Scholar
[25]Yue, X. and W. E, , The local microscale problem in the multiscale modeling of strongly heterogeneous media: effects of boundary conditions and cell size, J. Comput. Phys., 222 (2007), 556–572.Google Scholar
[26]Kumar, S. and Singh, R. N., Thermal conductivity of polycrystalline materials, J. Am. Ceram. Soc., 78(3) (1995), 728–736.Google Scholar