Sethuraman Sankaran and Nicholas Zabaras

1. Maximum entropy approach for statistical modeling of three-dimensional polycrystal microstructures Sethuraman Sankaran and Nicholas Zabaras 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: ss524@cornell.edu, zabaras@cornell.edu URL: http://mpdc.mae.cornell.edu/

2. Research Sponsors • U.S. AIR FORCE PARTNERS • Materials Process Design Branch, AFRL • Computational Mathematics Program, AFOSR ARMY RESEARCH OFFICE Mechanical Behavior of Materials Program NATIONAL SCIENCE FOUNDATION (NSF) Design and Integration Engineering Program • CORNELL THEORY CENTER

3. Why do we need a statistical model? Different statistical samples of the manufactured specimen When a specimen is manufactured, the microstructures at a sample point will not be the same always. How do we compute the class of microstructures based on some limited information?

4. Development of a mathematical model Random variable 2: High dimensions Random variable 1 (scalar or vector) Compute a PDF of microstructures Orientation Distribution functions Grain size features ODF (a function of 145 random parameters) Grain size Assign microstructures to the macro specimen after sampling from the PDF

5. The main idea Experimental microstructures Phase field simulations Extract features of the microstructure Geometrical: grain size Texture: ODFs Compute a PDF of microstructures MAXENT Compute bounds on macro properties

6. Generating input microstructures: The phase field model Define order parameters: where Q is the total number of orientations possible Non-zero only near grain boundaries Define free energy function (Allen/Cahn 1979, Fan/Chen 1997) :

7. Physics of phase field method Driving force for grain growth: • Reduction in free energy: thermodynamic driving force to eliminate grain boundary area (Ginzburg-Landau equations) kinetic rate coefficients related to the mobility of grain boundaries Assumption: Grain boundary mobilties are constant

8. Phase Field – Problem parameters • Isotropic mobility (L=1) • Discretization : problem size : 75x75x75 Order parameters: Q=20 • Timesteps = 1000 • First nearest neighbor approx.

9. Input microstructural samples 2D microstructural samples 3D microstructural samples

10. The main idea Experimental microstructures Phase field simulations Extract features of the microstructure Geometrical: grain size Texture: ODFs Compute a PDF of microstructures MAXENT Compute bounds on macro properties

11. 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. 2D microstructures Grain size is computed from the volumes of individual grains 3D microstructures

12. e3 ^ e’3 ^ ^ e’2 ^ e’1 ^ e2 e1 ^ Microstructural feature : ODF Crystal/lattice reference frame • Orientation Distribution Function Sample reference frame Volume fraction of crystals with a specific orientation crystal RODRIGUES’ REPRESENTATION FCC FUNDAMENTAL REGION n • ORIENTATION SPACE Euler angles – symmetries Neo Eulerian representation Particular crystal orientation Rodrigues’ parametrization Cubic crystal

13. The main idea Experimental microstructures Phase field simulations Extract features of the microstructure Geometrical: grain size Texture: ODFs Tool for microstructure modeling Compute a PDF of microstructures MAXENT Compute bounds on macro properties

14. Review Given: Microstructures at some points Obtain: PDF of microstructures ODF (a function of 145 random parameters) Grain size Know microstructures at some points

15. 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.

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

17. 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)

18. The main idea Experimental microstructures Phase field simulations Extract features of the microstructure Geometrical: grain size Texture: ODFs Tool for microstructure modeling Compute a PDF of microstructures MAXENT Compute bounds on macroscopic properties

19. 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, Microstructure modeling : the Voronoi structure {p1,p2,…,pk}: generator points. Cell division of k-dimensional space : Voronoi tessellation of 3d space. Each cell is a microstructural grain.

20. 0.2 0.2 0.18 0.18 0.16 0.16 0.14 0.14 Probability 0.12 0.12 Probability 0.1 0.1 0.08 0.08 0.06 0.06 0.04 0.04 0.02 0.02 0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Grain size Mean Grain size Stochastic modeling of microstructures Sampling using grain size distribution Sampling using mean grain size Each microstructure is referred to by its mean value. Match the PDF of a microstructure with PDF of grain sizes computed from MaxEnt Strongly consistent scheme Weakly consistent scheme

21. 0.2 0.18 0.16 0.14 Probability 0.12 0.1 0.08 0.06 0.04 0.02 0 0 5 10 15 20 25 30 Mean Grain size Heuristic algorithm for generating voronoi centers Given: grain size distribution Construct: a microstructure which matches the given distribution Generate sample points on a uniform grid from Sobel sequence No Forcing function Yes stop Rcorr(y,d)>0.95? Objective is to minimize norm (F). Update the voronoi centers based on F Construct a voronoi diagram based on these centers. Let the grain size distribution be y.

22. The main idea Experimental microstructures Phase field simulations Extract features of the microstructure Geometrical: grain size Texture: ODFs Tool for microstructure modeling Compute a PDF of microstructures MAXENT Compute bounds on macroscopic properties

23. (First order) homogenization scheme • 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

24. Numerical Example: Strong sampling

25. MAXENT tool 3D random microstructures – evaluation of property statistics Problem definition: Given microstructures generated using phase field technique, compute grain size distributions using MaxEnt technique as well as compute bounds in properties. Input constraints: macro grain size observable. First four 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.

26. Grain size distribution computed using MaxEnt 0.25 Grain volume distribution using phase field simulations pmf reconstructed using MaxEnt 0.2 0.15 Probability mass function Comparison of MaxEnt grain size distribution with the distribution of a phase field microstructure 0.1 K.L.Divergence=0.0672 nats 0.05 0 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 Grain volume (voxels)

27. 0.35 0.3 0.25 0.2 Probability mass function 0.15 R =0.9644 corr KL=0.0383 0.1 0.05 0 0 5000 10000 15000 20000 25000 Grain volume (voxels) Reconstructing strongly consistent microstructures Computing microstructures using the Sobel sequence method

28. 0.35 0.3 0.25 0.2 Probability mass function R =0.9830 0.15 corr KL=0.05 0.1 0.05 0 0 5000 10000 15000 20000 25000 Grain volume (voxels) Reconstructing strongly consistent microstructures (contd..) Computing microstructures using the Sobel sequence method

29. ODF reconstruction using MAXENT Representation in Frank-Rodrigues space Input ODF Reconstructed samples using MAXENT

30. Ensemble properties Expected property of reconstructed samples of microstructures Input ODF

31. 60 Mean ± std 50 Mean stress-strain 40 curve Equivalent stress (MPa) 30 20 10 0 0 1 2 3 Equivalent strain -4 x 10 Statistical variation of properties Aluminium polycrystal with rate-independent strain hardening. Pure tensile test. Statistical variation of homogenized stress-strain curves.

32. Numerical Example: Weak sampling

33. 3D microstructures: Grain boundary topology network Distribution of microstructures computed using MaxEnt technique using mean grain size as a microstructural feature A grain boundary network of one microstructural sample

34. Samples of microstructures computed at different points of the PDF Microstructures computed using the mean grain sizes, which are sampled from the PDF

35. Randomness in texture Samples of the reconstructed ODF function Expected ODF distribution that is given as a constraint to the MaxEnt algorithm Each grain is attributed an orientation that is sampled from a MaxEnt distribution of ODFs. Some of the samples of textures that are constructed are shown in the figure above.

36. Meshing microstructure samples using hexahedral elements (Cubit TM)

37. 60 60 50 50 40 40 Bounds on plastic Bounds on plastic properties properties 30 30 20 20 Equivalent stress (MPa) Equivalent stress (MPa) Equivalent stress (MPa) 10 10 0 0 0 0 0.5 0.5 1 1 1.5 1.5 2 2 2.5 2.5 3 3 3.5 3.5 Equivalent strain Equivalent strain Equivalent strain - - 3 3 x 10 x 10 Extremal bounds of homogenized stress-strain properties Aluminium polycrystal with rate-independent strain hardening. Pure tensile test. Statistical variation of homogenized stress-strain curves.

38. Probability Diffusion coefficient Future work: Diffusion in microstructures induced by topological uncertainty Diffusivity properties in a statistical class of microstructures Statistical samples of microstructure at certain collocation points computed using maximum entropy technique Limited set of input microstructures computed using phase field technique Variability of effective diffusion coefficient of microstructure

39. Information RELEVANT PUBLICATIONS S.Sankaran and N. Zabaras, Maximum entropy method for statistical modeling of microstructures, Acta Materialia, 2006 CONTACT INFORMATION Prof. Nicholas Zabaras Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering188 Frank H. T. Rhodes Hall Cornell University Ithaca, NY 14853-3801Email: zabaras@cornell.edu URL: http://mpdc.mae.cornell.edu/ Materials Process Design and Control Laboratory