# Using data to account for lack of data: Linking material informatics with stochastic analysis

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## Abstract

Many material systems of fundamental as well as industrial importance are significantly affected by underlying fluctuations and variations in boundary conditions, initial conditions, material property, as well as variabilities in operating and surrounding conditions. There has been increasing interest in analyzing, quantifying, and controlling the effects of such uncertain inputs on complex systems. This has resulted in the development of techniques in stochastic analysis and predictive modeling. A general application of stochastic techniques to many significant problems in engineering has been limited due to the lack of realistic, viable models of the input variability. Ideas of material informatics can be used to utilize available data about the input to construct a data-driven, computationally viable model of the input variability. This promising coupling of material informatics and stochastic analysis is investigated and some results are showcased using a simple example of analyzing diffusion in heterogeneous random media. Limited microstructural data is utilized to construct a realistic model of the thermal conductivity variability in a system. This reduced-order model then serves as the input to the stochastic partial differential equation describing thermal diffusion through random heterogeneous media.

## Keywords

Stochastic Analysis Sparse Grid Stochastic Partial Differential Equation Input Uncertainty Material Informatics## Preview

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## References

- 1.Major Aircraft Disasters, dnausers.d-n-a.net/dnet-GOJG/Disasters.htm.Google Scholar
- 2.Accelerated Insertion of Materials (AIM) Program (Arlington, VA: DARPA Defense Sciences Office), www.darpa.mil/dso/thrusts/materials/novelmat/aimd3d/index.htm.Google Scholar
- 3.Prognosis, Program (Arlington, VA: DARPA Defense Sciences Office), www.darpa.mil/dso/thrusts/materials/novelmat/prognosis/index.htm.Google Scholar
- 4.National Security Assessment of the U.S. Forging Industry (Washington, D.C.: Bureau of Industry and Security, U.S. Department of Commerce), www.bis.doc.gov/defenseindustrialbaseprograms/osies/.Google Scholar
- 5.R. Martin and D.L. Evans “Reducing Costs in Aircraft: The Metals Affordability Initiative Consortium,”
*JOM*, 52(3) (200), pp. 24–28.Google Scholar - 6.N. Wiener, “The Homogeneous Chaos,”
*Amer. J. Math.*, 60 (1938), pp. 897–936.CrossRefMathSciNetGoogle Scholar - 7.R.H. Cameron and W.T. Martin, “The Orthogonal Development of Non-linear Functionals in Series of Fourier-Hermite Functionals,”
*Ann. Math.*, 48 (1947), pp. 385–392.CrossRefMathSciNetGoogle Scholar - 8.R.G. Ghanem and P.D. Spanos,
*Stochastic Finite Elements: A Spectral Approach*(Mineola, NY: Dover Publications, 1991).zbMATHGoogle Scholar - 9.R. Ghanem, “Probabilistic Characterization of Transport in Heterogeneous Porous Media,”
*Comput. Methods Appl. Mech. Engrg.*, 158 (1998), pp. 199–220.zbMATHCrossRefGoogle Scholar - 10.D. Xiu and G.E. Karniadakis, “Modeling Uncertainty in Steady State Diffusion Problems via Generalized Polynomial Chaos,”
*Comput. Methods Appl. Mech. Engrg.*, 191 (2002), pp. 4927–4948.zbMATHCrossRefMathSciNetGoogle Scholar - 11.B. Velamur Asokan and N. Zabaras, “Variational Multiscale Stabilized FEM Formulations for Transport Equations: Stochastic Advection-Diffusion and Incompressible Stochastic Navier-Stokes Equations,”
*J. Comp. Physics*, 202 (2005), pp. 94–133.zbMATHCrossRefMathSciNetGoogle Scholar - 12.M.K. Deb, I.K. Babuska, and J.T. Oden, “Solution of Stochastic Partial Differential Equations using the Galerkin Finite Element Techniques,”
*Comput. Methods Appl. Mech. Engrg.*, 190 (2001), pp. 6359–6372.zbMATHCrossRefMathSciNetGoogle Scholar - 13.I. Babuska, R. Tempone, and G.E. Zouraris, “Solving Elliptic Boundary Value Problems with Uncertain Coefficients by the Finite Element Method: The Stochastic Formulation,”
*Comput. Methods Appl. Mech. Engrg.*, 194 (2005), pp. 1251–1294.zbMATHCrossRefMathSciNetGoogle Scholar - 14.R. Tempone, “Numerical Complexity Analysis of Weak Approximation of Stochastic Differential Equations” (Ph.D. thesis, Royal Institute of Technology, Sweden, 2002).Google Scholar
- 15.D.M. Tartakovsky and D. Xiu, “Stochastic Analysis of Transport in Tubes with Rough Walls,”
*J. Comput. Phys.*, 217 (2006), pp. 248–259.zbMATHCrossRefADSMathSciNetGoogle Scholar - 16.B. Velamur Asokan and N. Zabaras, “Using Stochastic Analysis to Capture Unstable Equilibrium in Natural Convection,”
*J. Comp. Phys.*, 208 (2005), pp. 134–153.zbMATHCrossRefADSMathSciNetGoogle Scholar - 17.I. Babuska, F. Nobile, and R. Tempone,
*A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data*, ICES Report 05-47 (Austin, TX: Institute for computational Engineering and Science, 2005).Google Scholar - 18.D. Xiu and J.S. Hesthaven, “High-order Collocation Methods for the Differential Equation with Random Inputs,”
*SIAM J. Sci. Comput.*, 27 (2005), pp. 1118–1139.zbMATHCrossRefMathSciNetGoogle Scholar - 19.B. Ganapathysubramanian and N. Zabaras, “Sparse Grid Collocation Schemes for Stochastic Natural Convection Problems,”
*J. Comp. Phys.*, 225 (2007), pp. 652–685.zbMATHCrossRefADSMathSciNetGoogle Scholar - 20.C. Desceliers, R. Ghanem, and C. Soize, “Maximum Likelihood Estimation of Stochastic Chaos Representations from Experimental Data,”
*Int. J. Numer. Meth. Engng.*, 66 (2006), pp. 978–1001.zbMATHCrossRefMathSciNetGoogle Scholar - 21.L. Guadagnini, A. Guadagnini, and D.M. Tartakovsky, “Probabilistic Reconstruction of Geologic Facies,”
*J. Hydrol.*, 294 (2004), pp. 57–67.CrossRefADSGoogle Scholar - 22.B. Ganapathysubramanian and N. Zabaras, “Modelling Diffusion in Random Heterogeneous Media: Data-Driven Models, Stochastic Collocation and the Variational Multi-Scale Method,”
*J. Comput. Phys.*, 226 (2007), pp. 326–353.zbMATHCrossRefADSMathSciNetGoogle Scholar - 23.B. Ganapathysubramanian and N. Zabaras, “A Non-linear Dimension Reduction Methodology for Generating Data-Driven Stochastic Input Models,”
*J. Comput. Phys.*, 227 (2008), pp. 6612–6637.zbMATHCrossRefADSMathSciNetGoogle Scholar - 24.S. Umekawa, R. Kotfila, and O.D. Sherby, “Elastic Properties of a Tungsten-Silver Composite above and below the Melting Point of Silver,”
*J. Mech. Phys. Solids*, 13 (1965), pp. 229–230.CrossRefADSGoogle Scholar