Materials Process Design and Control Laboratory Nicholas Zabaras and Sethuraman Sankaran Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 188 Frank H. T. Rhodes Hall Cornell University Ithaca, NY URL: Computing property variability of polycrystals induced by grain size and orientation uncertainties

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

Materials Process Design and Control Laboratory Why do we need a statistical model? A macro specimen has thousands of microsructures and in practice the microstructure at different material points will be different. How do we compute microstructures at all points in the specimen?

Materials Process Design and Control Laboratory Microstructure as a class We compute a microstructural class where there is variability in microstructural features within a class but certain experimentally obtained information is incorporated within the class. MICROSTRUCTURE CLASS TOPOLOGICAL FEATURES TEXTURAL FEATURES A class is defined based on statistics – mean grain size, variance of grain sizes, mean texture etc. A microstructure in the entire specimen is considered to be a sample from this class where a probability is assigned to each microstructure in the class

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

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

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 At any point in the domain of the microstructure, only one of the variables take a value of 1 and others are 0. Hence, any point is indexed by the field or simply the q for which takes the value 1 Define free energy function (Allen/Cahn 1979, Fan/Chen 1997) : Non-zero only near grain boundaries

Materials Process Design and Control Laboratory Phase field model (contd…) 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

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.

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

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

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

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

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

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

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 and lower moments 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.

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.

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 (Poisson distribution) Gaussian distribution Mean, variance given No information provided (unconstrained optimiz.) The uniform distribution Commonly seen distributions

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

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)

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

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.

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

Materials Process Design and Control Laboratory Stochastic modeling of microstructures Topological uncertainties within the microstructure is utilized as microstructural feature. Stochastic modeling refers to computing microstructures whose grain size distribution is computed using MaxEnt principle Mean Grain size Probability Samples whose topological uncertainty matches the given grain size distribution

Materials Process Design and Control Laboratory Applications Diffusion in heterogeneous random media driven by topological uncertainties Region of large mean grain size Region of small mean grain size Heterogeneties in microstructure Directional properties of the microstructure Diffusion properties of the microstructures strongly dictated by variabilities of grain size

Materials Process Design and Control Laboratory 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 > R cut Accept microstructure

Materials Process Design and Control Laboratory Heuristic algorithm for generating voronoi centers Generate sample points on a uniform grid and each point is associated with a grain size drawn from the given distribution, d Mean Grain size Probability Approximate shapes using spheres and define a forcing function to ensure non- overlap. 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

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

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

Materials Process Design and Control Laboratory Numerical Example: 2D microstructure reconstruction

Materials Process Design and Control Laboratory 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

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

Materials Process Design and Control Laboratory Reconstructed microstructures Reconstruction of microstructures based on correlation with the MAXENT grain size distribution. All voronoi tessellations which lead to a size distribution that has correlation coefficient more than 0.9 are accepted.

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 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

Materials Process Design and Control Laboratory 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

Materials Process Design and Control Laboratory Numerical Example: 3D microstructure reconstruction

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 samples consistent with the sampled distributions.

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

Materials Process Design and Control Laboratory Grain volume (voxels) Probability mass function R corr = KL= Reconstructing microstructures Computing microstructures using the Sobel sequence method

Materials Process Design and Control Laboratory Grain volume (voxels) Probability mass function R corr = KL=0.05 Reconstructing microstructures (contd..) Computing microstructures using the Sobel sequence method

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

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

Materials Process Design and Control Laboratory How many samples are required to predict the mean/standard deviation of the stress-strain curves effectively? Convergence analysis Almost sure convergence: Convergence in probability: Convergence in moments: What we are interested We say that converges to X if Estimates Increase the number of microstructure samples and test if the mean and standard deviation values converge at different locations in the stress-strain curve

Materials Process Design and Control Laboratory Convergence analysis A B C D x Equivalent strain Equivalent stress (MPa) A Number of samples Standard deviation of stress (MPa) Number of samples Standard deviation of stress (MPa) A B

Materials Process Design and Control Laboratory Number of samples Standard deviation of stress (MPa) Number of samples Standard deviation of stress (MPa) C D A B C D x Equivalent strain Equivalent stress (MPa) Convergence analysis – contd..

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.

Materials Process Design and Control Laboratory Data-driven in-situ estimation of microstructure classes

Materials Process Design and Control Laboratory Motivation Continuous stream of data about the microstructure MaxEnt optimization algorithm Class of microstructures Steepest descent Time inefficient INFORMATION LEARNING

Materials Process Design and Control Laboratory Grain size moments (like mean, std, etc) Input Output A class of microstructures satisfying the information Statistical relation Problem statement: We want to predict the class of microstructures which satisfies a given statistic of grain size moment. Input is a 2-tuple and output is a microstructural class. The data is generated using MaxEnt. Moment no.Moment value Method Use MAXENT for certain microstructure classes. Generate MAXENT distribution for certain inputs TRAIN A NON- LINEAR STATISTICAL MAPPER

Materials Process Design and Control Laboratory Distribution that is given from the database Distribution from training weights of the network Optimization T: number of microstructure classes that have been pre- computed and stored in a database Kullback-Leibler divergence Non-linear relationship between input moments and parameters of the MaxEnt PDF Input moment Parameters of the MaxEnt PDF

Materials Process Design and Control Laboratory Technique-Backpropagation Initialize weights and biases Objective function Gradients Update weights/biases If less than tolerance, terminate

Materials Process Design and Control Laboratory Iterations Objective function Grain size (  10 3 voxels) Probability mass function output predicted using information learning output predicted using MaxEnt voxels) Grain size (  10 3 Probability mass function Information learning – convergence for 2 input moments

Materials Process Design and Control Laboratory Iterations Objective function A B C Grain size (  10 3 voxels) Probability mass function Distribution at point A Distribution at point B Distribution at point C MaxEnt distribution for the input  1 = voxels  2 =343.2e3 voxel 2  3 = e6 voxel 3 Information learning – convergence plot Distributions at various iterations Information learning – convergence for 3 input moments

Materials Process Design and Control Laboratory Probability mass function Distribution computed using L  1 =391 voxels  2 =265e3 voxel 2  3 =218e6 voxel 3  4 =199e9 voxel 4 Grain size (  10 3 voxels) Database containing information about two moments Database containing information about three moments Database containing information about four moments Recomputed MaxEnt distributions using information learning

Materials Process Design and Control Laboratory Conclusions  We generated a class of microstructures based on insufficient data using maximum entropy method using (a) grain size features and (b) textural features  Microstructures were reconstructed using the inverse voronoi tessellation technique  Samples were interrogated using multiscale homogenization methods and we were able to compute statistics of plastic properties  A learning algorithm to accelerate the computation of MaxEnt classes was found to be efficient in computation of the microstructure class

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 polycrystals induced by topological uncertainty Diffusion coefficient Probability Variability of effective diffusion coefficient of microstructure

Materials Process Design and Control Laboratory Information RELEVANT PUBLICATIONS S.Sankaran and N. Zabaras, Maximum entropy method for statistical modeling of microstructures, Acta Materialia, 2007 Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 188 Frank H. T. Rhodes Hall Cornell University Ithaca, NY URL: Prof. Nicholas Zabaras CONTACT INFORMATION N. Zabaras andS.Sankaran, An information theoretic approach to stochastic materials modeling, IEEE Computing in Science and Engineering, 2007 N. Zabaras and S.Sankaran, An information theoretic approach to stochastic materials modeling, IEEE Computing in Science and Engineering, 2007