Nicholas Zabaras, Veera Sundararaghavan and Sethuraman Sankaran

1. An information-theoretic approach for obtaining property PDFs from macro specifications of microstructural variability Nicholas Zabaras, Veera Sundararaghavan and Sethuraman Sankaran Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering188 Frank H. T. Rhodes Hall Cornell University Ithaca, NY 14853-3801 Email: zabaras@cornell.edu, vs85@cornell.edu, ss524@cornell.edu URL: http://mpdc.mae.cornell.edu/

2. Idea Behind Information Theoretic Approach 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. 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 Information Theory Linkage? Information Theory Statistical Mechanics

3. 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.

4. The MAXENT principle E.T. Jaynes 1957 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. • MAXENT is a guiding principle to constructPDFs based onlimited 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.

5. 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. Commonly seen distributions Mean, variance given Mean provided No information provided (unconstrained optimiz.) 1-parameter exponential family (similar to Poisson distribution) Gaussian distribution The uniform distribution

6. 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.

7. e3 ^ e’3 ^ ^ e’2 ^ e’1 ^ e2 e1 ^ Microstructural feature : ODF Crystal/lattice reference frame • CRYSTAL SYMMETRIES? Same axis of rotation => planes Each symmetry reduces the space by a pair of planes Sample reference frame crystal RODRIGUES’ REPRESENTATION FCC FUNDAMENTAL REGION n • ORIENTATION SPACE Euler angles – symmetries Neo Eulerian representation Particular crystal orientation Rodrigues’ parametrization Cubic crystal

8. Partition Function MAXENT as an optimization problem Find feature constraints Subject to features of image I Lagrange Multiplier optimization Lagrange Multiplier optimization

9. Equivalent log-likelihood problem Find that minimizes Kuhn-Tucker theorem: The that minimizes the dual function L also maximizes the system entropy and satisfies the constraints posed by the problem Equivalent log-linear model A comparison

10. Gradient Evaluation • Objective function and its gradients: • 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)

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

12. 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. Optimization Schemes Convergence analysis with stabilization

13. Voronoi cell tessellation : Division of n-D space into non-overlapping regions such that the union of all regions fill the entire space. Division of into subdivisions so that for each point, pi there is an associated convex cell, Voronoi structure {p1,p2,…,pk}: generator points. Cell division of k-dimensional space : Voronoi tessellation of 3d space. Each cell is a microstructural grain.

14. 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. Volumes need to be hulled to obtain consistent representation with commercial packages • 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: • The face name, which is a string. • The boundary of the face, which is a list of faces of one lower dimension. • 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 Convex hulling to obtain a triangulation of surfaces/grain boundaries

15. 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

16. Preprocessing: stage 2 • 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 ANSYSTM 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.

17. Monte Carlo techniques for matching grain size distributions Problem : Generate voronoi-cell structures whose grain size distribution matches grain size PDF obtained using MAXENT • Generate a database of microstructures using Monte Carlo schemes (on the generator points) based on voronoi tessellation • Obtain correlation coefficient between the MAXENT and actual grain size measure. Accept microstructures whose correlation is above a cutoff. Assign random voronoi centers. Evaluate grain size distribution correlation coefficient > Rcut Accept microstructure

18. Meshing Tetrahedral element meshed. Grain boundaries conform with the mesh shapes. Pixel based meshing scheme. Boundary is distorted since element shapes and sizes are fixed.

19. Mesh refinement Tetrahedral mesh Hexahedral mesh Input to homogenization tool to obtain plastic property and eventually property statistics

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

21. 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 Materials Process Design and Control Laboratory

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

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

24. 2D random microstructures: evaluation of property statistics 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)

25. MAXENT distribution of grain sizes 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.

26. 0.14 0.12 0.1 0.08 Orientation distribution function 0.06 0.04 0.02 0 -2 -1 0 1 2 Orientation angle (in radians) 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 Input ODF (corresponds to a pure shear deformation, Zabaras et al. 2004) Ensemble properties of ODF from reconstructed distribution

27. 80 70 60 Bounding plastic curves over a set Equivalent Strain (MPa) of microstructural samples 50 40 30 0 0.05 0.1 0.15 0.2 Equivalent Stress 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

28. MAXENT tool 3D random microstructures – evaluation of property statistics Problem definition: Given lower moments of grain sizes and an input ODF, obtain 3D reconstructions of the microstructure and evaluate statistics of its homogenized properties Input constraints: macro grain size observable. First three grain size moments , expected value of the ODF are given as constraints. Output: Entire variability (PDF) of grain size and ODFs in the microstructure is obtained.

29. Reconstructing microstructures

30. Microstructure meshing Nodal degrees of freedom 500000 Nodal degrees of freedom 50000 Nodal degrees of freedom 300000 No. of hexahedral elements ~140000 No. of brick elements ~13500 No. of hexahedral elements ~95000

31. ODF reconstruction using MAXENT Input ODF Reconstructed samples using MAXENT

32. Ensemble properties Ensemble properties of reconstructed samples of microstructures Input ODF

33. Stress-contours obtained using conforming meshes Orientations are sampled from the PDF of ODF’s obtained using MAXENT and grains are assigned these orientations. These microstructures are interrogated for evaluating homogenized properties. Microstructure Conforming hexahedral mesh Sharp stress-contours are seen near the grain boundaries which is attributed to a comparatively higher heterogeneities near these boundaries.

34. 30 25 20 Equivalent stress (MPa) 15 10 5 0 0 0.2 0.4 0.6 0.8 1 Equivalent strain -3 x 10 Homogenized equivalent stress-strain curves

35. Applications (many …) Statistics of plastic properties

36. 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.