1. Cornell University- Zabaras, FA9550-07-1-0139 An information-theoretic multiscale framework with applications to polycrystal materials Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 188 Frank H. T. Rhodes Hall Cornell University Ithaca, NY 14853-3801 Email: zabaras@cornell.edu URL: http://mpdc.mae.cornell.eduhttp://mpdc.mae.cornell.edu Prof. Nicholas Zabaras

2. 2 Cornell University- Zabaras, FA9550-07-1-0139 Overview Background – Robust control of microstructure-sensitive properties in polycrystalline materials – Stochastic optimization, stochastic multiscale design and control algorithms Technical progress to date – Adaptive stochastic analysis – Stochastic input reduced-model generation – Stochastic multiscale analysis framework – Stochastic optimization algorithms Impact Future plans Transition/collaboration opportunities

3. 3 Cornell University- Zabaras, FA9550-07-1-0139 Background Forging velocity Die/workpiece friction Die shape Initial preform shape Uncertainty in constitutive relation Uncertainty in Texture Uncertainty in grain sizes Design critical components with extremal properties Microstructure-sensitivity properties require a multiscale nature of analysis and design Operational variabilities and process/parameter/material uncertainties necessitate stochastic analysis. Approach: Adaptive collocation, data-driven models, stochastic multi-scaling and stochastic optimization

4. 4 Cornell University- Zabaras, FA9550-07-1-0139 Technical progress: Adaptive stochastic collocation Collocation based approach: - Approximate stochastic solution as an interpolation function. - Completely decoupled. Require deterministic evaluation at finite number of stochastic points - Error bounds and convergence proofs available - Further reduction possible through detection of solution structure - The conventional sparse grid method treats every dimension equally. Dimension adaptive collocation: - Functions may have widely varying characteristics in different directions (discontinuities, steep gradients) or the function may have some special structure (additive, nearly-additive, multiplicative). - Have to come up with a way to make the Smolyak algorithm treat different dimensions differently. - Convention sparse grids imposes a strict admissibility condition on the indices. Relaxing this restriction results in Generalized Sparse Grids. - Provides significant reduction in the number of function evaluations T = 0 u=v=0 T = 1 u=v=0 Porous medium Free fluid u=v=0

5. 5 Cornell University- Zabaras, FA9550-07-1-0139 Technical progress: Data-driven stochastic input model generation Realistic stochastic analysis: Must construct viable input uncertainty models for property variation. In our analysis, property variation is related to topology variation. Given some data about the media, construct a viable stochastic model of its topology. PCA based approach: Represent the space of allowable variations as linear combinations of eigen-images. Assumes that the space is a linear vector space. Non-linear approach: Assume that the space is a manifold. Given a finite set of points belonging to this manifold, construct a parameterization of the manifold. Convert to a problem in ‘manifold learning’. Utilize ideas from differential geometry and image processing to construct a linear vector space isometric to the nonlinear manifold.

6. 6 Cornell University- Zabaras, FA9550-07-1-0139 Technical progress: Multiscale stochastic modeling framework Seamlessly couple stochastic analysis with multiscale analysis. Multiscale framework (large deformation/thermal evolution) + Adaptive stochastic collocation framework Provides roadmap to efficiently link any validated multiscale framework Coupled with a data-driven input model strategy to analyze realistic stochastic multiscale problems. T= -0.5 T= 0.5 Limited data Statistics extraction + model generation Stochastic multiscale framework Mean statistics Higher-order statistics

7. 7 Cornell University- Zabaras, FA9550-07-1-0139 Technical progress: Stochastic design/optimization framework Inverse problems posed as optimization problems meanvariance Actual Compu ted meanvariance t=0.1 t=0.8 Contaminant source reconstruction problem Design problems posed as optimization problems Design thermal flux variability on one wall to maintain temperature (PDF) at another wall for all possible material realizations Design fluxMaintain temp Mean temperature Temperature std Designed mean flux Std of designed flux Coupled stochastic analysis with an optimization framework. Novel, highly-efficient way to compute stochastic sensitivities based on parallel sparse grid collocation schemes. Major advance in non-intrusive stochastic optimization – any stochastic optimization problem can now be posed as a deterministic optimization problem in a large dimensional space. Classical gradient and gradient-free optimization methods are directly applicable. Random heterogeneous material

8. 8 Cornell University- Zabaras, FA9550-07-1-0139 Impact: 3D Multiscale Design of Deformation Processes - A two-length scale continuum sensitivity framework has been developed that allows the efficient and accurate computation of the effects of perturbations of macroscopic design variables (e.g. dies and preforms) on microscopic variables (slip resistances, strength, etc.) - Using this sensitivity framework, the first 3D multiscale deformation process design simulator has been developed for the control of texture-dependent material properties and applied to complex processes.

9. 9 Cornell University- Zabaras, FA9550-07-1-0139 Impact: Application to designing critical components with extremal properties Forging velocity Die/workpiece friction Die shape Initial preform shape Uncertainty in constitutive relation Uncertainty in Texture Uncertainty in grain sizes - Significant developments towards the design of processing paths and materials for manufacturing critical components that have extremal properties. - Dimension reduction strategy has significant applications to problems where working in high dimensional spaces is computationally intractable: visualizing property evolution, process- property maps, searching and contouring. - Stochastic multiscale framework provides an approach to incorporate in the multiscale design framework shown earlier operational variabilities across multiple scales. Provide rigorous failure criteria and the associated probabilities.

10. 10 Cornell University- Zabaras, FA9550-07-1-0139 Future Plans 1.Develop wavelet based collocation strategies for enhancing the stochastic framework. 2.Utilize model reduction strategy to compute realistic input stochastic models of polycrystalline materials based on limited information. Couple with Maximum Entropy methods to extract PDFs from limited information. 3.Couple stochastic framework with the multiscale framework for large deformation modeling and design. Investigate uncertainty propagation and interaction of uncertainties across multiple scales. Identify zones/regions where failure initiation possibly occurs, regions that cause sub-optimal properties, etc. 4.Extend stochastic optimization framework for designing: (a) processing paths, (b) processing parameters (c) material properties, of critical polycrystalline components that have extremal properties. 5.Utilize model reduction strategy as a technique for reducing problems in high- dimensional spaces (visualizing property evolution, process-property maps, searching and contouring) into more tractable low dimensional surrogate spaces

11. 11 Cornell University- Zabaras, FA9550-07-1-0139 Areas where collaborative effort and cross-application is anticipated: 1)Accurate and efficient solution techniques for stochastic PDEs– Adaptivity and data storage 2)Advances in solution strategies and fast visualization techniques – major impact on enhanced decision making 3)Analyzing component performance over extended time periods-- accurate long time integration 4)A framework that precludes the necessity of any expert knowledge- Validation and verification of complex systems using experimental data, gappy data 5)Wide applicability of the model reduction techniques to any data-driven problem. In the case of stochastic modeling, it will have significant applications in diverse areas as it eliminates the need for assumptions on input random models (thus leading to KLE-free approximations). Transition/Collaboration opportunities

12. 12 Cornell University- Zabaras, FA9550-07-1-0139 Publications & Awards SELECTED PUBLICATIONS [1] S. Acharjee and N. Zabaras, "A non-intrusive stochastic Galerkin approach for modeling uncertainty propagation in deformation processes", Computers and Structures Vol. 85, Issues 5-6, pp. 244-254, 2007. [2] S. Sankaran and N. Zabaras, "Computing property variability of polycrystals induced by grain size and orientation uncertainties", Acta Materialia, Vol. 55, Issue 7, pp. 2279-2290, 2007. [3] B. Ganapathysubramanian and N. Zabaras, "Sparse grid collocation methods for stochastic natural convection problems", Journal of Computational Physics, Vol. 225, pp. 652-685, 2007. [4] B. Ganapathysubramanian and N. Zabaras, "Modelling diffusion in random heterogeneous media: Data-driven models, stochastic collocation and the variational multi-scale method", Journal of Computational Physics, in press [5] N. Zabaras and S. Sankaran, "An information-theoretic approach to stochastic materials modeling", IEEE Computing in Science and Engineering (CiSE), special issue of "Stochastic Modeling of Complex Systems" (guest edts. D. M. Tartakovsky and D. Xiu), March/April issue, pp. 50-59, 2007. [6] B. Ganapathysubramanian and N. Zabaras, "A non-linear dimension reduction methodology for generating data-driven stochastic input models", Journal of Computational Physics, submitted. [7] N. Zabaras, S. Sankaran and B. Ganapathysubramanian, "An efficient approach to stochastic sensitivity analysis using a sparse grid collocation scheme", J. Computational Physics, submitted. [8] V. Sundararaghavan and N.Zabaras, "A multiscale design framework for the control of texture- dependent properties in deformation processes", Int J Plasticity, submitted

13. 13 Cornell University- Zabaras, FA9550-07-1-0139 Uncertainties at multiple scales Design in the presence of uncertainties LONG-TERM PAYOFF : Decrease processing costs and enhance properties of forged aerospace components. Robust design of critical components in the presence of multiple sources of uncertainty. OBJECTIVES Analyze performance of critical components in the presence of uncertainties across multiple length scales. Translate limited information about properties/topology into performance bounds and failure norms. FUNDING ($K) FY07 FY08 FY09 FY10 FY11 AFOSR Funds120K 120K 120K TRANSITIONS Numerous journal publications can be found in http://mpdc.mae.cornell.edu/ STUDENTS V Sundararaghavan, B. Ganapathysubrmanian, S. Sankaran, Xiang Ma LABORATORY POINT OF CONTACT Dr. Jeff P. Simmons, AFRL/MLLMP, WPAFB, OH APPROACH/TECHNICAL CHALLENGES A non intrusive multiscale framework for stochastic analysis. Uncertainty quantification and transfer: Ideas from information theory, stochastic homogenization and multiscale methods. Linear and non-linear techniques for generating input models. Nonlinear analysis based on the principle of ‘Manifold learning’ (provides solutions where linear techniques- PCA, KLE- fail) Stochastic optimization based on computing stochastic sensitivities using sparse grid collocation strategies AN INFORMATION THEORETIC MULTISCALE FRAMEWORK WITH APPLICATIONS TO POLYCRYSTAL MATERIALS Cornell University, Nicholas Zabaras

14. 14 Cornell University- Zabaras, FA9550-07-1-0139 A stochastic multiscale framework coupled with a data-driven input strategy Limited information Provide reliable bounds and statistics Multiple 3D reconstructions Reduced order stochastic model Statistics extraction Deterministic Multiscale Framework Stochastic sparse grid collocation framework