2. Babak Kouchmeshky Uncertainty propagation in deformation processes Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 169 Frank H. T. Rhodes Hall Cornell University Ithaca, NY 14853-3801 Email: bk84@cornell.edu Presentation for the B exam

3. Modeling uncertainty propagation in deformation processes Materials Process Design and Control Laboratory

4. Variation of macro-scale properties due to multi-scale sources of uncertainty Macro-scale properties depend on the underlying micro-structure Represent the micro-structure in a continuous framework using Orientation distribution function (ODF) MaxEnt All possible macro-scale properties due to the effect of multiple sources of uncertainties on multi-scales Error bars in stress-strain response due to the effect of multiple sources of uncertainties on multi-scales. Continuum representation of texture in Rodrigues space Underlying Microstructure Obtain the variability of macro-scale properties due to uncertainties on multi- scales. Problem definition Uncertainty in process parameters Uncertainty in underlying micro-structure Approach Using Karhunen-Loeve (KL) expansion the uncertainty on micro-structure is represented by a set of random variables. In absence of sufficient information, Maximum Entropy (MaxEnt) should be used to obtain the joint probability of these random variables. KL Approach (cont.) The effect of various multi-scale uncertainties on macro-scale properties can be quantified by solving the governing stochastic partial differential equations (SPDE). Use sparse grid collocation to solve the SPDE’s. All possible macro-scale properties due to uncertainty can be represented by a convex hull. The convex hull can be used in calculating the risk associated with obtaining macro-scale properties less than specific critical values.

5. Materials Process Design and Control Laboratory Problem definition Sources of uncertainty: - Process parameters - Micro-structural texture Obtain the variability of macro-scale properties due to multiple sources of uncertainty in absence of sufficient information that can completely characterizes them.

6. Materials Process Design and Control Laboratory Sources of uncertainty (process parameters) Since incompressibility is assumed only eight components of L are independent. The coefficients correspond to tension/compression,plain strain compression, shear and rotation.

7. Materials Process Design and Control Laboratory Underlying Microstructure Continuum representation of texture in Rodrigues space Fundamental part of Rodrigues space Variation of final micro-structure due to various sources of uncertainty Sources of uncertainty (Micro-structural texture)

8. Materials Process Design and Control Laboratory Variation of macro-scale properties due to multiple sources of uncertainty on different scales Uncertain initial microstructure use Frank- Rodrigues space for continuous representation Limited snap shots of a random field Use Karhunen-Loeve expansion to reduce this random filed to few random variables Considering the limited information Maximum Entropy principle should be used to obtain pdf for these random variables Use Rosenblatt transformation to map these random variables to hypercube Use Stochastic collocation to obtain the effect of these random initial texture on final macro-scale properties.

9. Materials Process Design and Control Laboratory Evolution of texture Any macroscale property can be expressed as an expectation value if the corresponding single crystal property χ (r,t) is known. Determines the volume fraction of crystals within a region R' of the fundamental region R Probability of finding a crystal orientation within a region R' of the fundamental region Characterizes texture evolution ORIENTATION DISTRIBUTION FUNCTION – A(s,t) ODF EVOLUTION EQUATION – LAGRANGIAN DESCRIPTION

10. Materials Process Design and Control Laboratory Constitutive theory D = Macroscopic stretch = Schmid tensor = Lattice spin W = Macroscopic spin = Lattice spin vector Reorientation velocity Symmetric and spin components Velocity gradient Divergence of reorientation velocity Polycrystal plasticity Initial configuration B o B F * F p F Deformed configuration Stress free (relaxed) configuration n0n0 s0s0 n0n0 s0s0 n s (2) Ability to capture material properties in terms of the crystal properties (1) State evolves for each crystal

11. Materials Process Design and Control Laboratory Karhunen-Loeve Expansion: and is a set of uncorrelated random variables whose distribution depends on the type of stochastic process. Then its KLE approximation is defined as where and are eigenvalues and eigenvectors of Representing the uncertain micro-structure Let be a second-order stochastic process defined on a closed spatial domain D and a closed time interval T. If are row vectors representing realizations of then the unbiased estimate of the covariance matrix is

12. Materials Process Design and Control Laboratory Karhunen-Loeve Expansion can be obtained by Realization of random variables where denotes the scalar product in. The random variables have the following two properties

13. Materials Process Design and Control Laboratory Obtaining the probability distribution of the random variables using limited information In absence of enough information, Maximum Entropy principle is used to obtain the probability distribution of random variables. Maximize the entropy of information considering the available information as set of constraints

14. Materials Process Design and Control Laboratory Maximum Entropy Principle Constraints at the final iteration

15. Materials Process Design and Control Laboratory Inverse Rosenblatt transformation (i)Inverse Rosenblatt transformation has been used to map these random variables to 3 independent identically distributed uniform random variables in a hypercube [0,1]^3. (ii) Adaptive sparse collocation of this hypercube is used to propagate the uncertainty through material processing incorporating the polycrystal plasticity.

16. STOCHASTIC COLLOCATION STRATEGY Use Adaptive Sparse Grid Collocation (ASGC) to construct the complete stochastic solution by sampling the stochastic space at M distinct points Two issues with constructing accurate interpolating functions: 1)What is the choice of optimal points to sample at? 2) How can one construct multidimensional polynomial functions? Materials Process Design and Control Laboratory 1.X. Ma, N. Zabaras, A stabilized stochastic finite element second order projection methodology for modeling natural convection in random porous media, JCPA stabilized stochastic finite element second order projection methodology for modeling natural convection in random porous media 2.D. Xiu and G. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comp. 24 (2002) 619-644The Wiener-Askey polynomial chaos for stochastic differential equations 3.X. Wan and G.E. Karniadakis, Beyond Wiener-Askey expansions: Handling arbitrary PDFs, SIAM J Sci Comp 28(3) (2006) 455-464Beyond Wiener-Askey expansions: Handling arbitrary PDFs Assuming a finite dimension for the stochastic space, one can represent a function in random space as

17. From one-dimension to higher dimensions  Denote the one dimensional interpolation formula as with the set of support nodes:  In higher dimension, a simple case is the tensor product formula For instance, if M=10 dimensions and we use k points in each direction Number of points in each direction, k Total number of sampling points 21024 359049 4 1.05x10 6 5 9.76x10 6 10 1x10 10 This quickly becomes impossible to use. One idea is only to pick the most important points from the tensor product grid. Materials Process Design and Control Laboratory

18. FROM ONE-DIMENSION to MULTI-DIMENSIONS Extension to multiple dimensions: Use simple tensor product strategy Number of points in each direction, k Total number of sampling points 21024 359049 4 1.05x10 6 5 9.76x10 6 10 1x10 10 Results in combinatorial explosion One idea is only to pick the more important points from the tensor product grid 1.S. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions, Soviet Math. Dokl., 4 (1963) 240--243 Smolyak (1963) came up with a set of rules to construct such products 1 Materials Process Design and Control Laboratory where, is the multi-index representation of the support nodes. N is the dimension of the function f and q is an integer (q>N). k = q-N is called the depth of the interpolation. As q is increased more and more points are sampled.

19. ADAPTIVE SPARSE GRID COLLOCATION Anisotropic sampling for interpolating functions with steep gradients and other localized phenomena. Have to detect it on-the-fly. Utilize piecewise linear interpolating functions: local support Utilize hierarchical form of basis function: provides natural stopping criterion Define a threshold value. If magnitude of the hierarchical surplus is greater than this threshold, refine around this point. Add 2N neighbor points. Scales linearly instead of O(2 N ) Circumvent the curse-of-dimensionality? 1.B. Ganapathysubramanian and N. Zabaras, Sparse grid collocation schemes for stochastic natural convection problems, J. Comp. Phys 225 (2007) 652-685Sparse grid collocation schemes for stochastic natural convection problems 2.X. Ma and N. Zabaras, An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations, submittedAn adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations Materials Process Design and Control Laboratory

20. ADAPTIVE SPARSE GRID COLLOCATION Given a user-defined threshold, ε>0. For points where w > ε, refine the grid to include 2N daughters. Compute the hierarchical surpluses at these new points. Refine until all w< ε or maximum depth of interpolation is reached Implementation:  Keep track of uniqueness of new points  Efficient searching and inserting  Parallelizability Error estimate of the adaptive interpolant Materials Process Design and Control Laboratory

21. Numerical Examples A sequence of modes is considered in which a simple compression mode is followed by a shear mode hence the velocity gradient is considered as: where are uniformly distributed random variables between 0.2 and 0.6 (1/sec). Example 1 : The effect of uncertainty in process parameters on macro-scale material properties for FCC copper Number of random variables: 2

22. Materials Process Design and Control Laboratory 1.28e05 4.02e07 3.92e07 1.28e05 Adaptive Sparse grid (level 8) MC (10000 runs) Numerical Examples ( Example 1)

23. Materials Process Design and Control Laboratory A simple compression mode is assumed with an initial texture represented by a random field A The random field is approximated by Karhunen-Loeve approximation and truncated after three terms. The correlation matrix has been obtained from 500 samples. The samples are obtained from final texture of a point simulator subjected to a sequence of deformation modes with two random parameters uniformly distributed between 0.2 and 0.6 sec^-1 (example1) Numerical Examples ( Example 2) Example 2 : The effect of uncertainty in initial texture on macro-scale material properties for FCC copper

24. Materials Process Design and Control Laboratory Numerical Examples ( Example 2) Step1. Reduce the random field to a set of random variables (KL expansion)

25. Materials Process Design and Control Laboratory Numerical Examples ( Example 2) Enforce positiveness of texture Step2. In absence of sufficient information,use Maximum Entropy to obtain the joint probability of these random variables

26. Materials Process Design and Control Laboratory Numerical Examples ( Example 2) Rosenblatt M, Remarks on multivariate transformation, Ann. Math. Statist.,1952;23:470-472 Rosenblatt transformation Step3. Map the random variables to independent identically distributed uniform random variables on a hypercube [0 1]^3 are needed. The last one is obtained from the MaxEnt problem and the first 2 can be obtained by MC for integrating in the convex hull D.

27. Materials Process Design and Control Laboratory Numerical Examples ( Example 2) Step4. Use sparse grid collocation to obtain the stochastic characteristic of macro scale properties Mean of A at the end of deformation process Variance of A at the end of deformation process Variation of stress-strain response FCC copper 1.41e054.42e08 Adaptive Sparse grid (level 8) MC 10,000 runs 4.39e08 1.41e05

28. A model reduction of the multiscale input for uncertainty quantification in multiscale deformation processes Materials Process Design and Control Laboratory

29. Simple Karhunen-Loeve Expansion The realizations of random variables One can use methods from previous slides to construct the probability distributions of these random variables at each integration point j. Now if the random variables at different integration points are correlated to each other then the aforementioned methodology has no means of figuring that out in another words it can not see the correlation between the set of random variables from different integration points. Bi-orthogonal Karhunen-Loeve Expansion Quantifying the effect of uncertain initial texture

30. An eigenvalue problem in Rodrigues space

31. Materials Process Design and Control Laboratory Step3: Construct the Covariance using the snapshots Step4: Obtain the eigenvalues and eigenvectors: ; Step5: Obtain the spatial modes Step6: Decompose the spatial modes using the polynomial Chaos: are in a one to one correspondent to the Hermite polynomials. Construct the reduced order representation of the texture Use the reduced order model to reconstruct the texture Top left: Distribution of Bulk modulus, Top right: Distribution of Young modulus, Bottom left: Distribution of Shear modulus. For one point on macro- scale. The bars represent the distribution obtained using the realizations of the texture and the solid line is the distribution obtained using the reduced order model for the texture. The relative error with respect to the order of polynomial chaos. Step1: Start from realizations of the texture Step2: Transform the realization using

32. Materials Process Design and Control Laboratory Use Rosenblatt transform to transfer the coordinates of the points inside a hypercube to the realizations of the random variables. Propagation of uncertainty In this method the stochastic space is sampled using collocation points. The coordinates of each of these collocation points correspond to the realizations of the random variables driving the problem. So, for each point we have the realizations of the random variables. These realizations can be used to solve a deterministic problem of deformation process and obtain the final texture. Q: How to quantify the effect of uncertainty in initial texture on the final texture of a work-piece that has gone through a deformation process? A: Collocation strategy is used for quantifying the effect of uncertainty. Use the realizations of the random variables to construct the realizations of the initial texture The results at the collocation points are used to construct interpolants in the stochastic space. Q: What about using global polynomial chaos expansion method? A: Needs significant changes in the formulation of the corresponding deterministic problem. Q: But you said you are using global polynomial chaos expansion ? A: Yes. It was used in the bi-orthogonal framework to approximate the initial random field. It could also be used in the stochastic PDEs along a stochastic Galerkin projection to reduce the SPDEs to a set of deterministic PDEs. But the latter step in not necessary and once you have a model to obtain the realization of the initial random field you can propagate uncertainty using any stochastic tools.

33. Materials Process Design and Control Laboratory Problem definition -Obtain the effect of uncertainty in initial texture on macro- scale material properties Uncertain initial microstructure

34. Materials Process Design and Control Laboratory Deterministic multi-scale deformation process

35. Materials Process Design and Control Laboratory Implementation of the deterministic problem Meso Macro formulation for macro scale Update macro displacements Texture evolution update Polycrystal averaging for macro-quantities Integration of single crystal slip and twinning laws Macro-deformation gradient microscale stress Macro-deformation gradient Micro

36. Materials Process Design and Control Laboratory THE DIRECT CONTACT PROBLEM r n Inadmissible region Reference configuration Current configuration Admissible region Impenetrability Constraints Augmented Lagrangian approach to enforce impenetrability

37. Polycrystal average of orientation dependent property Continuous representation of texture Materials Process Design and Control Laboratory REORIENTATION & TEXTURING

38. Materials Process Design and Control Laboratory Convergence of the deterministic problem Bulk modulus

39. Materials Process Design and Control Laboratory Convergence of the deterministic problem Young modulus

40. Materials Process Design and Control Laboratory Convergence of the deterministic problem Shear modulus

41. Materials Process Design and Control Laboratory Convergence of the deterministic problem

42. Materials Process Design and Control Laboratory Convergence of ODF

43. Materials Process Design and Control Laboratory Convergence of ODF

44. Materials Process Design and Control Laboratory Stochastic multi-scale deformation process

45. Materials Process Design and Control Laboratory H Curved surface parametrization – Cross section can at most be an ellipse Model semi-major and semi-minor axes as 6 degree bezier curves Random parameters Deterministic parameters The effect of uncertainty in the initial geometry of the work- piece on the macro-scale properties

46. STOCHASTIC COLLOCATION STRATEGY Use Adaptive Sparse Grid Collocation (ASGC) to construct the complete stochastic solution by sampling the stochastic space at M distinct points Two issues with constructing accurate interpolating functions: 1)What is the choice of optimal points to sample at? 2) How can one construct multidimensional polynomial functions? Materials Process Design and Control Laboratory 1.X. Ma, N. Zabaras, A stabilized stochastic finite element second order projection methodology for modeling natural convection in random porous media, JCPA stabilized stochastic finite element second order projection methodology for modeling natural convection in random porous media 2.D. Xiu and G. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comp. 24 (2002) 619-644The Wiener-Askey polynomial chaos for stochastic differential equations 3.X. Wan and G.E. Karniadakis, Beyond Wiener-Askey expansions: Handling arbitrary PDFs, SIAM J Sci Comp 28(3) (2006) 455-464Beyond Wiener-Askey expansions: Handling arbitrary PDFs Assuming a finite dimension for the stochastic space: =

47. Materials Process Design and Control Laboratory Mean(G) Var(G) Mean(B) Var(B) Mean(E) Var(E) The effect of uncertainty in the initial geometry

48. Materials Process Design and Control Laboratory Error of Mean(B) Error of Var(B) Comparison with Monte-Carlo Error of Mean(E) Error of Var(E) Error of Mean(G) Error of Var(G)

49. Materials Process Design and Control Laboratory A MODEL REDUCTION of the MULTISCALE INPUT Current method 1- D. Venturi, X. Wan, G.E. Karniadakis, J. Fluid Mech. 2008, vol 606, pp 339-367 where are modes strongly orthogonal in Rodrigues space and are spatial modes weakly orthogonal in space

50. Materials Process Design and Control Laboratory Reconstructing a stochastic microstructure Step1: Construct the covariance using the snapshots Step2: Obtain the eigenvalues and eigenvectors: ;

51. Materials Process Design and Control Laboratory Reconstructing a stochastic microstructure Step3: Obtain the spatial modes Step4: Decompose the spatial modes using the polynomial Chaos: are in a one to one correspondent to the Hermite polynomials.

52. B E G Comparison between the original microstructure and the reduced order one Bars: obtained using the realizations of the texture that were used in constructing the covariance. Solid line: Obtained from the sampling the random variables and constructing the texture using the reduced order modeling.

53. Materials Process Design and Control Laboratory B E G Comparison between the original microstructure and the reduced order one

54. Materials Process Design and Control Laboratory B E G Comparison between the original microstructure and the reduced order one

55. Materials Process Design and Control Laboratory B E G Comparison between the original microstructure and the reduced order one

56. Materials Process Design and Control Laboratory B E G Comparison between the original microstructure and the reduced order one

57. Materials Process Design and Control Laboratory Mean(G) Mpa Mean(B) Mpa Mean(E) Mpa Original Reconstructed Mean(G) Mpa Mean(B) Mpa Mean(E) Mpa Comparison between the original microstructure and the reduced order one

58. Materials Process Design and Control Laboratory Var(G) Var(B) Var(E) Original Reconstructed Var(G) Var(B) Var(E) Comparison between the original microstructure and the reduced order one

59. Materials Process Design and Control Laboratory The effect of uncertainty in the initial texture of the work- piece on the macro-scale properties

60. Materials Process Design and Control Laboratory Mean(G) Var(G) Mean(B) Var(B) Mean(E) Var(E) The effect of uncertainty in the initial texture After the interpolants in the stochastic space for the texture have been obtained one can use them to obtain the realizations of the texture. Using these realizations statistics of the macro-scale properties can be obtained.

61. Materials Process Design and Control Laboratory The effect of uncertainty in the initial texture Comparison of Mean and variance of the macro-scale properties with MC, Top left: Bulk modulus, Top right: Young modulus, Bottom: Shear modulus Relative error for Mean(G) Relative error for Var(G) Relative error for Mean(B) Relative error for Var(B) Relative error for Mean(E) Relative error for Var(E)

62. Materials Process Design and Control Laboratory Conclusion The first half of the slides: The effect of uncertain process parameters and initial texture on the convex hull of the macro- scale properties is investigated. The second half of the slides: The previous method is extended to the multi- scale. A model reduction of the multi-scale input for the texture uncertainty in multi-scale deformation process has been provided.

63. Materials Process Design and Control Laboratory Publications and presentations since Aug. 2006 B. Kouchmeshky and N. Zabaras, A model reduction of the uncertain input for quantifying the effect of uncertainty in a multi-scale stochastic problem, In preparation. B. Kouchmeshky and N. Zabaras, The effect of multiple sources of uncertainty on the convex hull of material properties", Computational Materials Science, submitted. B. Kouchmeshky and N. Zabaras, Modeling the response of HCP polycrystals deforming by slip and twinning using a finite element representation of the orientation space, Computational Materials Science, Vol. 45, Issue 4, pp. 1043-1051, 2009 Publications: Presentations: B. Kouchmeshky and N. Zabaras, “Uncertainty quantification in multiscale deformation processes”, 2009, 10th U.S. National Congress on Computational Mechanics. B. Kouchmeshky and N. Zabaras, " Modeling uncertainty propagation in deformation processes", 2009 TMS Annual Meeting & Exhibition. B. Kouchmeshky and N. Zabaras, " Advances on multi-scale design of deformation processes for the control of material properties”, 2009 TMS Annual Meeting & Exhibition. B. Kouchmeshky and N. Zabaras, "A microstructure-sensitive design approach for controlling properties of HCP materials", 2008 TMS Annual Meeting & Exhibition. B. Kouchmeshky and N. Zabaras, "A simple non-hardening rate-independent constitutive model for HCP polycrystals deforming by slip and twinning”, 2008 TMS Annual Meeting & Exhibition.