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

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

3. Materials Process Design and Control Laboratory Why do we need a statistical model? 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? Different statistical samples of the manufactured specimen

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

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

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

7. Materials Process Design and Control Laboratory 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. Materials Process Design and Control Laboratory Phase Field – Problem parameters Isotropic mobility (L=1) Isotropic mobility (L=1) Discretization : Discretization : problem size : 75x75x75 Order parameters: Q=20 Timesteps = 1000 Timesteps = 1000 First nearest neighbor approx. First nearest neighbor approx.

9. Materials Process Design and Control Laboratory Input microstructural samples 3D microstructural samples 2D microstructural samples

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

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

12. 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  Orientation Distribution Function Volume fraction of crystals with a specific orientation Particular crystal orientation

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

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

15. 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.

16. 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

17. 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)

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

19. Materials Process Design and Control Laboratory Microstructure modeling : the 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.

20. Materials Process Design and Control Laboratory Stochastic modeling of microstructures Sampling using grain size distribution Sampling using mean grain size Match the PDF of a microstructure with PDF of grain sizes computed from MaxEnt Each microstructure is referred to by its mean value. 051015202530 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Mean Grain size Probability Weakly consistent scheme 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 Strongly consistent scheme

21. Materials Process Design and Control Laboratory Heuristic algorithm for generating voronoi centers Generate sample points on a uniform grid from Sobel sequence 051015202530 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Mean Grain size Probability Forcing function 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. R corr (y,d)>0.95? No Yes stop Given: grain size distribution Construct: a microstructure which matches the given distribution

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

23. Materials Process Design and Control Laboratory (First order) homogenization scheme 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

24. Materials Process Design and Control Laboratory Numerical Example: Strong sampling

25. Materials Process Design and Control Laboratory 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. 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.

26. Materials Process Design and Control Laboratory 02000400060008000100001200014000160001800020000 0 0.05 0.1 0.15 0.2 0.25 Grain volume (voxels) Probability mass function Grain volume distribution using phase field simulations pmf reconstructed using MaxEnt K.L.Divergence=0.0672 nats Grain size distribution computed using MaxEnt Comparison of MaxEnt grain size distribution with the distribution of a phase field microstructure

27. Materials Process Design and Control Laboratory 0500010000150002000025000 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Grain volume (voxels) Probability mass function R corr =0.9644 KL=0.0383 Reconstructing strongly consistent microstructures Computing microstructures using the Sobel sequence method

28. Materials Process Design and Control Laboratory 0500010000150002000025000 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Grain volume (voxels) Probability mass function R corr =0.9830 KL=0.05 Reconstructing strongly consistent microstructures (contd..) Computing microstructures using the Sobel sequence method

29. Materials Process Design and Control Laboratory Input ODF Reconstructed samples using MAXENT ODF reconstruction using MAXENT Representation in Frank- Rodrigues space

30. Materials Process Design and Control Laboratory Input ODF Expected property of reconstructed samples of microstructures Ensemble properties

31. Materials Process Design and Control Laboratory Statistical variation of properties Statistical variation of homogenized stress- strain curves. Aluminium polycrystal with rate-independent strain hardening. Pure tensile test.

32. Materials Process Design and Control Laboratory Numerical Example: Weak sampling

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

34. Materials Process Design and Control Laboratory Samples of microstructures computed at different points of the PDF Microstructures computed using the mean grain sizes, which are sampled from the PDF

35. Materials Process Design and Control Laboratory Randomness in texture 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. Expected ODF distribution that is given as a constraint to the MaxEnt algorithm Samples of the reconstructed ODF function

36. Materials Process Design and Control Laboratory Meshing microstructure samples using hexahedral elements (Cubit TM )

37. Materials Process Design and Control Laboratory Equivalent strain Equivalent stress (MPa) 00.511.522.533.5 x 10 -3 0 10 20 30 40 50 60 Equivalent strain Equivalent stress (MPa) Bounds on plastic properties 00.511.522.533.5 x 10 -3 0 10 20 30 40 50 60 Equivalent strain Equivalent stress (MPa) Bounds on plastic properties Statistical variation of homogenized stress- strain curves. Aluminium polycrystal with rate-independent strain hardening. Pure tensile test. Extremal bounds of homogenized stress-strain properties

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

39. Materials Process Design and Control Laboratory Information RELEVANT PUBLICATIONS S.Sankaran and N. Zabaras, Maximum entropy method for statistical modeling of microstructures, Acta Materialia, 2006 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.edu/ Prof. Nicholas Zabaras CONTACT INFORMATION