1. Materials Process Design and Control Laboratory Nicholas Zabaras 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.eduzabaras@cornell.edu URL: http://mpdc.mae.cornell.edu/ An information-theoretic approach for property prediction of random microstructures

2. Materials Process Design and Control Laboratory NEED FOR UNCERTAINTY ANALYSIS  Variation in properties, constitutive relations  Imprecise knowledge of governing physics, surroundings  Simulation based uncertainties (irreducible)  Uncertainty is everywhere Porous media Silicon wafer Aircraft engines Material process From DOE From Intel website From NIST From GE-AE website

3. Materials Process Design and Control Laboratory UNCERTAINTY AND MULTISCALING Macro Meso Micro  Uncertainties introduced across various length scales have a non-trivial interaction  Current sophistications – resolve macro uncertainties  Use micro averaged models for resolving physical scales  Imprecise boundary conditions  Initial perturbations  Physical properties, structure follow a statistical description

4. Materials Process Design and Control Laboratory Initial preform shape Material properties/models Forging velocity Texture, grain sizes Die/workpiece friction Die shape Small change in preform shape could lead to underfill Material Model Forging rate Die/Billet shape Friction Cooling rate Stroke length Billet temperature Stereology/Grain texture Dynamic recrystallization Phase transformation Phase separation Internal fracture Other heterogeneities Yield surface changes Isotropic/Kinematic hardening Softening laws Rate sensitivity Internal state variables Dependance Nature and degree of correlation Process UNCERTAINTY IN METAL FORMING PROCESSES

5. Materials Process Design and Control Laboratory RANDOM VARIABLES = FUNCTIONS ?  Math: Probability space ( , F, P ) Sample space Sigma-algebra Probability measure  : Random variable  Random variable  A stochastic process is a random field with variations across space and time

6. Materials Process Design and Control Laboratory SPECTRAL STOCHASTIC REPRESENTATION  A stochastic process = spatially, temporally varying random function CHOOSE APPROPRIATE BASIS FOR THE PROBABILITY SPACE HYPERGEOMETRIC ASKEY POLYNOMIALS PIECEWISE POLYNOMIALS (FE TYPE) SPECTRAL DECOMPOSITION COLLOCATION, MC (DELTA FUNCTIONS) GENERALIZED POLYNOMIAL CHAOS EXPANSION SUPPORT-SPACE REPRESENTATION KARHUNEN-LOÈVE EXPANSION SMOLYAK QUADRATURE, CUBATURE, LH

7. Materials Process Design and Control Laboratory KARHUNEN-LOEVE EXPANSION Stochastic process Mean function ON random variables Deterministic functions  Deterministic functions ~ eigen-values, eigenvectors of the covariance function  Orthonormal random variables ~ type of stochastic process  In practice, we truncate (KL) to first N terms

8. Materials Process Design and Control Laboratory GENERALIZED POLYNOMIAL CHAOS  Generalized polynomial chaos expansion is used to represent the stochastic output in terms of the input Stochastic output Askey polynomials in input Deterministic functions Stochastic input  Askey polynomials ~ type of input stochastic process  Usually, Hermite, Legendre, Jacobi etc.

9. Materials Process Design and Control Laboratory SUPPORT-SPACE REPRESENTATION  Any function of the inputs, thus can be represented as a function defined over the support-space JOINT PDF OF A TWO RANDOM VARIABLE INPUT FINITE ELEMENT GRID REFINED IN HIGH-DENSITY REGIONS OUTPUT REPRESENTED ALONG SPECIAL COLLOCATION POINTS – SMOLYAK QUADRATURE – IMPORTANCE MONTE CARLO

10. Materials Process Design and Control Laboratory State variable based power law model. State variable – Measure of deformation resistance- mesoscale property Material heterogeneity in the state variable assumed to be a second order random process with an exponential covariance kernel. Eigen decomposition of the kernel using KLE. Eigenvectors Initial and mean deformed config. UNCERTAINTY DUE TO MATERIAL HETEROGENEITY

11. Materials Process Design and Control Laboratory Load vs Displacement SD Load vs Displacement Dominant effect of material heterogeneity on response statistics UNCERTAINTY DUE TO MATERIAL HETEROGENEITY

12. Materials Process Design and Control Laboratory Parameters of interest in stochastic analysis are the moment information (mean, standard deviation, kurtosis etc.) and the PDF. For a stochastic process Definition of moments NISG - Random space discretized using finite elements to Output PDF computed using local least squares interpolation from function evaluations at integration points. Deterministic evaluations at fixed points NISG - FORMULATION

13. Materials Process Design and Control Laboratory Finite element representation of the support space. Inherits properties of FEM – piece wise representations, allows discontinuous functions, quadrature based integration rules, local support. Provides complete response statistics. Decoupled function evaluations at element integration points. True PDF Interpolant FE Grid Convergence rate identical to usual finite elements, depends on order of interpolation, mesh size (h, p versions). NISG - DETAILS

14. Materials Process Design and Control Laboratory Mean InitialFinal Using 6x6 uniform support space grid Uniform 0.02 Material heterogeneity induced by random distribution of micro-voids modeled using KLE and an exponential kernel. Gurson type model for damage evolution EFFECT OF RANDOM VOIDS ON MATERIAL BEHAVIOR

15. Materials Process Design and Control Laboratory Load displacement curves EFFECT OF RANDOM VOIDS ON MATERIAL BEHAVIOR

16. Materials Process Design and Control Laboratory Axisymmetric cylinder upsetting – 60% height reduction (Initial height 1.5 mm) Random initial radius – 10% variation about mean (1 mm)– uniformly distributed Random die workpiece friction U[0.1,0.5] Power law constitutive model Using 10x10 support space grid Random ? Shape Random ? friction PROCESS UNCERTAINTY

17. Materials Process Design and Control Laboratory Force SD Force PROCESS STATISTICS

18. Materials Process Design and Control Laboratory ParameterMonte Carlo (20000 LHS samples) Support space 10x10 Mean2.2859e32.2863e6 SD297.912299.59 m3-8.156e6 - -9.545e6 m41.850e101.979e10 Final force statisticsConvergence study PROCESS STATISTICS Relative Error

19. Materials Process Design and Control Laboratory As the number of random variables increases, problem size rises exponentially. (assume 10 evaluations per random dimension) CURSE OF DIMENSIONALITY

20. Materials Process Design and Control Laboratory ADAPTIVE DISCRETIZATION BASED ON OUTPUT STOCHASTIC FIELD Refine/Coarsen input support space grid based on output defined control parameter (Gradients, standard deviations etc.) Applicable using standard h,p adaptive schemes. Support-space of input Importance spaced grid PROPOSED SOLUTIONS

21. Materials Process Design and Control Laboratory DIMENSION ADAPTIVE QUADRATURE (Gerstner et. al. 2003) Full grid SchemeSparse grid SchemeDimension adaptive Scheme Very popular in computational finance applications. Has been used in as high as 256 dimensions. PROPOSED SOLUTIONS

22. Materials Process Design and Control Laboratory Idea Behind Information Theoretic Approach Statistical Mechanics Information Theory Rigorously quantifying and modeling uncertainty, linking scales using criterion derived from information theory, and use information theoretic tools to predict parameters in the face of incomplete Information etc Linkage? Information Theory Basic Questions: 1. Microstructures are realizations of a random field. Is there a principle by which the underlying pdf itself can be obtained. 2. If so, how can the known information about microstructure be incorporated in the solution. 3. How do we obtain actual statistics of properties of the microstructure characterized at macro scale.

23. Materials Process Design and Control Laboratory MAXENT as a tool for microstructure reconstruction Input: Given average and lower moments of grain sizes and ODFs Obtain: microstructures that satisfy the given properties Constraints are viewed as expectations of features over a random field. Problem is viewed as finding that distribution whose ensemble properties match those that are given. Since, problem is ill-posed, we choose the distribution that has the maximum entropy. Microstructures are considered as realizations of a random field which comprises of randomness in grain sizes and orientation distribution functions.

24. Materials Process Design and Control Laboratory The MAXENT principle The principle of maximum entropy (MAXENT) states that amongst the probability distributions that satisfy our incomplete information about the system, the probability distribution that maximizes entropy is the least-biased estimate that can be made. It agrees with everything that is known but carefully avoids anything that is unknown. E.T. Jaynes 1957 MAXENT is a guiding principle to construct PDFs based on limited information There is no proof behind the MAXENT principle. The intuition for choosing distribution with maximum entropy is derived from several diverse natural phenomenon and it works in practice. The missing information in the input data is fit into a probabilistic model such that randomness induced by the missing data is maximized. This step minimizes assumptions about unknown information about the system.

25. Materials Process Design and Control Laboratory MAXENT : a statistical viewpoint MAXENT solution to any problem with set of features is Parameters of the distribution Input features of the microstructure Fit an exponential family with N parameters (N is the number of features given), MAXENT reduces to a parameter estimation problem. Mean provided 1-parameter exponential family (similar to Poisson distribution) Gaussian distribution Mean, variance given No information provided (unconstrained optimiz.) The uniform distribution Commonly seen distributions

26. Materials Process Design and Control Laboratory Microstructural feature: Grain sizes Grain size obtained by using a series of equidistant, parallel lines on a given microstructure at different angles. In 3D, the size of a grain is chosen as the number of voxels (proportional to volume) inside a particular grain.

27. Materials Process Design and Control Laboratory Cubic crystal Microstructural feature : ODF RODRIGUES’ REPRESENTATION FCC FUNDAMENTAL REGION Crystal/lattice reference frame e2e2 ^ Sample reference frame e1e1 ^ e’ 1 ^ e’ 2 ^crystal e’ 3 ^ e3e3 ^  ORIENTATION SPACE Euler angles – symmetries Neo Eulerian representation n Rodrigues’ parametrization  CRYSTAL SYMMETRIES? Same axis of rotation => planes Each symmetry reduces the space by a pair of planes Particular crystal orientation

28. Materials Process Design and Control Laboratory Subject to Lagrange Multiplier optimization feature constraints features of image I MAXENT as an optimization problem Partition Function Find

29. Materials Process Design and Control Laboratory Equivalent log-linear model Find that minimizes Equivalent log-likelihood problem Kuhn-Tucker theorem: The that minimizes the dual function L also maximizes the system entropy and satisfies the constraints posed by the problem Direct models Log-linear models ConcaveConcave Constrained (simplex) Unconstrained “Count and normalize” (closed form solution) Gradient based methods Acomparison

30. Materials Process Design and Control Laboratory Gradient Evaluation Objective function and its gradients: Objective function and its gradients: Infeasible to compute at all points in one conjugate gradient iteration Infeasible to compute at all points in one conjugate gradient iteration Use sampling techniques to sample from the distribution evaluated at the previous point. (Gibbs Sampler) Use sampling techniques to sample from the distribution evaluated at the previous point. (Gibbs Sampler)

31. Materials Process Design and Control Laboratory Improper pdf (function of lagrange multipliers) Start from a random microstructure. Go through each grain of the microstructure and sample an ODF according to the conditional probability distribution (conditioned on the other grains) continue till the samples converge to the distribution Processor 1 Processor r … Each processor goes through only a subset of the grains. Parallel Gibbs sampler algorithm

32. Materials Process Design and Control Laboratory Optimization Schemes Convergence analysis with stabilization Convergence analysis w/o stabilization Noise in function evaluation increases as step size for the next minima increases. This ensures that the impact on the next evaluation is reduced.

33. Materials Process Design and Control Laboratory Voronoi structure Division of n-D space into non-overlapping regions such that the union of all regions fill the entire space. Voronoi cell tessellation : {p 1,p 2,…,p k } : generator points. Division of into subdivisions so that for each point, p i there is an associated convex cell, Cell division of k-dimensional space : Voronoi tessellation of 3d space. Each cell is a microstructural grain.

34. Materials Process Design and Control Laboratory Mathematical representation OFF file representation (used by Qhull package) Initial lines consists of keywords (OFF), number of vertices and volumes. Next n lines consists of the coordinates of each vertex. The remaining lines consists of vertices that are contained in each volume. Brep (used by qmg, mesh generator) Dimension of the problem. A table of control points (vertices). Its faces listed in increasing order of dimension (i.e., vertices first, etc) each associated with it the following: 1.The face name, which is a string. 2.The boundary of the face, which is a list of faces of one lower dimension. 3.The geometric entities making up the face. its type (vertex, curve, triangle, or quadrilateral), its degree (for a curve or triangle) or degree-pair (for a quad), and its list of control points Volumes need to be hulled to obtain consistent representation with commercial packages Convex hulling to obtain a triangulation of surfaces/grain boundaries

35. Materials Process Design and Control Laboratory Preprocessing: stage 1 Growth of big grains to accommodate small grains entrenched in-between Compute volumes of all grains Adjust vertices of neighboring grains so that the new voronoi tessellation fills the volume of initial grain Recompute surfaces and planes of the new geometry

36. Materials Process Design and Control Laboratory Steps Obtain input voronoi representation in OFF format. Obtain the convex hull of the volumes/grains so that each surface is a triangle (triangulation of surfaces). Use ANSYS TM to convert this representation to the universal IGES (Initial Graphics Exchange specification) format. Surface database : To ensure non-duplication of surfaces, a database consisting of previously encountered hyper-planes is searched. When a new surface is created, if it is already in the database and if all the vertices of the surface were not present in a previous grain, no new surface is made. Domain smoothing: The regions of the microstructure inside the region [0 1] 3 is chosen. Edges are smoothed so that the boundaries represent edges of a k-dimensional cube of unit side. Preprocessing: stage 2

37. Materials Process Design and Control Laboratory Meshing Pixel based meshing scheme. Boundary is distorted since element shapes and sizes are fixed. Tetrahedral element meshed. Grain boundaries conform with the mesh shapes.

38. Materials Process Design and Control Laboratory Mesh refinement Tetrahedral mesh Hexahedral mesh Input to homogenization tool to obtain plastic property and eventually property statistics

39. Materials Process Design and Control Laboratory (First order) homogenization scheme How does macro loading affect the microstructure 1.Microstructure is a representation of a material point at a smaller scale 2.Deformation at a macro-scale point can be represented by the motion of the exterior boundary of the microstructure. (Hill, R., 1972)Hill

40. Materials Process Design and Control Laboratory Homogenization of deformation gradient How does macro loading affect the microstructure Microstructure without cracks Use BC: = 0 on the boundary Note w = 0 on the volume is the Taylor assumption, which is the upper bound

41. Materials Process Design and Control Laboratory Implementation Largedef formulation for macro scale Update macro displacements Boundary value problem for microstructure Solve for deformation field Consistent tangent formulation (macro) Integration of constitutive equations Continuum slip theory Consistent tangent formulation (meso) Macro-deformation gradient Homogenized (macro) stress, Consistent tangent Mesoscale stress, consistent tangent meso deformation gradient Macro Meso Micro

42. Homogenized properties X Y Z (a) (c) (b) (d) Materials Process Design and Control Laboratory

43. Problem definition: Given an experimental image of an aluminium alloy (AA3302), properties of individual components and given the expected orientation properties of grains, it is desired to obtain the entire variability of the class of microstructures that satisfy these given constraints. Polarized light micrograph of aluminium alloy AA3302 (source Wittridge NJ et al. Mat.Sci.Eng. A, 1999) 2D random microstructures: evaluation of property statistics

44. Materials Process Design and Control Laboratory Grain sizes: Heyn’s intercept method. An equidistant network of parallel lines drawn on a microstructure and intersections with grain boundaries are computed. Input constraints in the form of first two moments. The corresponding MAXENT distribution is shown on the right. MAXENT distribution of grain sizes

45. Materials Process Design and Control Laboratory Assigning orientation to grains Given: Expected value of the orientation distribution function. To obtain: Samples of orientation distribution function that satisfies the given ensemble properties -2012 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Orientation angle (in radians) Orientation distribution function Input ODF (corresponds to a pure shear deformation, Zabaras et al. 2004) Ensemble properties of ODF from reconstructed distribution

46. Materials Process Design and Control Laboratory 00.050.10.150.2 30 40 50 60 70 80 Equivalent Stress Equivalent Strain (MPa) Bounding plastic curves over a set of microstructural samples Evaluation of plastic property bounds Orientations assigned to individual grains from the ODF samples obtained using MAXENT. Bounds on plastic properties obtained from the samples of the microstructure

47. Materials Process Design and Control Laboratory Motivation Uncertainties induced due to non- uniformities in grain growth patterns

48. Materials Process Design and Control Laboratory Input uncertainties Problem inputs: Microstructures obtained using monte-carlo grain growth model at different stages of the growth. Sources of uncertainty: Anything that changes the driving force for grain growth (curvature driven, reduction in surface energy) (e.g) ambient conditions not exactly same in microstructures near surface and in the bulk. Problem parameters: 1.10 input microstructures used that constraint the input information 2.Time lag of ~50 MC steps between each sample. 3.Simulated on a 9261 point grid

49. Materials Process Design and Control Laboratory Maximum-entropic distribution of grain sizes 0100200300400500600 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 Grain size (  m 3 ) Probability =383.4967 =41.4490

50. Materials Process Design and Control Laboratory 051015202530 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Grain size Probability Sampling technique employed Weakly consistent scheme

51. Materials Process Design and Control Laboratory Input ODF Some representative ODF samples from the MaxEnt distribution ODF reconstruction using MAXENT Problem inputs/algorithm parameters: 1.145 degrees of freedom 2.MaxEnt algorithm using Brent’s line search method 3.Eighty Gibbs iteration through each grain of the microstructure

52. Materials Process Design and Control Laboratory Input ODF Ensemble properties of reconstructed samples of microstructures Ensemble properties

53. Materials Process Design and Control Laboratory Final uncertainty representation pdf Grain size ODF (a function of 145 random parameters) Microstructures sampled as points from the joint pdf space

54. Materials Process Design and Control Laboratory Microstructure models & meshes Tetrahedral meshes

55. Materials Process Design and Control Laboratory Obtaining statistics of non-linear properties Different microstructural models of a polycrystal Aluminium microstructure is obtained by sampling the resultant distribution. Each of these specimens is subject to a pure axial tension along the x direction. Plots of the resultant stress-contour and the resulting homogenized stress-strain curves are plotted for different realizations

56. Materials Process Design and Control Laboratory Homogenized stress fields on the microstructure Pixel based meshing Hexahedral meshing

57. Materials Process Design and Control Laboratory Homogenized stress fields on the microstructure

58. Materials Process Design and Control Laboratory Comparison of pixel based versus hexahedral meshing schemes The pixel based meshing scheme distorts grain boundaries and not only increases their area but also twists their shape which leads to a higher degree of stress localization as viewed in previous plot.

59. Materials Process Design and Control Laboratory Plots of homogenized stress-strain curves A plot showing three different samples of the stress-strain plots obtained for different statistical models of the microstructure generated using the MaxEnt scheme.

60. Materials Process Design and Control Laboratory Stress contours across grain boundaries and triple junctions Orientation 0.4142 -0.2071 -0.0858 Orientation -0.2929 -0.4142 0.2929 Orientation 0.4142 0.0858 -0.2071 Orientation 0.2071 -0.4142 0.0858 Orientation 0.4142 0.0858 -0.2071 Extreme sharp variation in texture across the triple junction. Hence, leads to a large degree of stress localization

61. Materials Process Design and Control Laboratory Applications (many …) Statistics of plastic properties

62. Materials Process Design and Control Laboratory Discussion A statistical distributions of mictrostructure was obtained incorporating variability in grain sizes and grain orientations. A statistical distributions of mictrostructure was obtained incorporating variability in grain sizes and grain orientations. Stress field distributions show a significant difference between the pixel based mesh and the hexahedral mesh. One possible reason may be attributed to the fact that grain boundaries are distorted as a result of which the localized stresses near the grain boundaries are felt in some regions in the bulk of the grain. Also, for the hexahedral grid 21960 elements were used while for the pixel based grid, 13824 elements were used. We are currently performing convergence studies with respect to the mesh sizes but the number of elements used were roughly equivalent. Also, sharp changes in the field were noticed in the vicinity of the grain boundaries due to steep variations in texture. Stress field distributions show a significant difference between the pixel based mesh and the hexahedral mesh. One possible reason may be attributed to the fact that grain boundaries are distorted as a result of which the localized stresses near the grain boundaries are felt in some regions in the bulk of the grain. Also, for the hexahedral grid 21960 elements were used while for the pixel based grid, 13824 elements were used. We are currently performing convergence studies with respect to the mesh sizes but the number of elements used were roughly equivalent. Also, sharp changes in the field were noticed in the vicinity of the grain boundaries due to steep variations in texture. Statistical samples of microstructure model were used to obtained different samples of homogenized stress-strain curves. Statistical samples of microstructure model were used to obtained different samples of homogenized stress-strain curves.

63. Materials Process Design and Control Laboratory MODELING GRAIN BOUNDARY PHYSICS Equivalent stress contours–Include failure mechanisms –Grain boundary properties –Local stress concentrations develop to cause the emission of a few partial dislocations from grain boundaries, and these high stresses drive the partial dislocations across the grain interiors –MD studies indicate that this is the major mechanism of the limited inelastic deformation in the grain interiors of nanocrystalline materials.