1. Materials Process Design and Control Laboratory An Information-Theoretic Approach to Multiscale Modeling and Design of Materials 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.edu URL: http://www.mae.cornell.edu/zabaras /

2. Materials Process Design and Control Laboratory Filtering and two way flow of statistical information 110 2 4 6 9 Engineering Length Scales ( ) Physics Chemistry Materials 0 A Information flow Statistical filter Electronic Nanoscale Microscale Mesoscale Continuum INFORMATION FLOW ACROSS SCALES

3. Materials Process Design and Control Laboratory DEFORMATION PROCESS DESIGN (Minimal barreling) Initial guess Optimal preform Optimal preform shape Final optimal forged product Final forged product Initial preform shape

4. Materials Process Design and Control Laboratory ROBUST DESIGN OF DEFORMATION PROCESSES Metal forming Forging velocity Lubrication – friction at die-workpiece interface Intermediate material state variation over a multistage sequence –residual-stresses, temperature, change in microstructure, expansion/contraction of the workpiece Die shape – is it constant over repeated forgings ? Damage evolution through processing stages Preform shapes (tolerances) Composites – fiber orientation, fiber spacing, constitutive model Biomechanics – material properties, constitutive model, fibers in tissues Material heterogeneity

5. Materials Process Design and Control Laboratory UNCERTAINTY REPRESENTATION TECHNIQUES Sample space Real interval Reinterpret random variables as functions Any stochastic process is a spatially and temporally varying random variable We can use following function approximation techniques Spectral expansion Finite elements Wavelet expansion Spectral expansion Finite element – support-space method Mean Higher order statistics Karhunen-Loeve expansion Generalized polynomial chaos Techniques Support-space is region where joint PDF of the uncertain quantity is not zero Mesh the support-space Refine the mesh where PDF has large values Use piecewise polynomials to represent any function of the uncertain quantity

6. Materials Process Design and Control Laboratory UNCERTAINTY DUE TO MATERIAL HETEROGENEITY 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.

7. 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

8. Materials Process Design and Control Laboratory INFORMATION THEORETIC FRAMEWORK Wavelet basis ( ) a,b are scale and space parameters Wavelet coefficients at macro scale Wavelet coefficients at meso scale Correlation kernels at macro scale Correlation kernels based on intrascale mutual information criterion Information filtering based on Renyi’s entropy and Linsker’s maximum mutual information KLE – effective method to model material heterogeneity using correlation kernels. From phenomenology to explicit derivation of kernels using multiscale information Information transfer and filtering between scales based on maximum entropy criterion and wavelet parameters.

9. Materials Process Design and Control Laboratory IDEA BEHIND AN INFORMATION THEORETIC APPROACH Statistical Mechanics Information Theory Rigorously quantifying and modeling uncertainty, linking scales using criteria derived from information theory. Use information theoretic tools to predict parameters in the face of incomplete information, etc. Linkage? Information Theory

10. Materials Process Design and Control Laboratory Information-Theoretic Methods for Multilength Scale Modeling Source information micro scale macro scale Wavelet based coding of parameters Decoding of wavelet parameters Information Theoretic upscaling of wavelet coefficients Information Upscaling Channel Received information Wavelet Basis at higher scale Information lost here  How much information is required at each scale and what is the acceptable loss of information during upscaling to answer performance related questions at the macro scale ?  Maximum entropy methods for extracting higher order information from lower order statistics (in microstructures) by maximizing entropy across unconstrained dimensions.  Use of wavelets as a tool to project the multiscale parameters across various scales. Wavelets are tools to represent signals hierarchically at different resolutions.  Information theoretic measures to quantify the process of upscaling and homogenization and study the scale-coupling problem rigorously from a mathematical stand-point. Wavelet Basis at lower scale

11. Materials Process Design and Control Laboratory NEED FOR WAVELETS  A very useful tool in areas where a multiscale analysis is important. quantify information  Could be used as a tool to quantify information of physical parameters of interest.  Very useful for such analyses because it is mathematically compact and consistent FoFo F1F1 Q1FoQ1Fo Q2FoQ2Fo F2F2 Q3FoQ3Fo FnFn Micro scale Meso scale  a,b : wavelet coefficients at scale a and spatial location b. Wavelets as a multiscale tool Compound Wavelet Matrix Method: Independent simulations done at two different scales and solutions obtained mapped onto wavelet domain. Use the above-mentioned to bridge the scales between atomic and continuum, both spatially as well as temporally Frantziskonis, Deymier (2000,2003) Information Lost Schematic of wavelet representation Information across all scales

12. Materials Process Design and Control Laboratory HOMOGENIZATION IN WAVELET SPACES Full microstructure information Homogenized properties at next scale Complete homogenization Wavelet Basis

13. Materials Process Design and Control Laboratory Decreasing resolution of microstructure using Daubechies- 1 wavelets. Choose a scale with truncated wavelet basis functions so that only parameters above that scale could be resolved. Information lost when approximated to fourth scale Choose level of analysis so that computational time is significantly reduced (at lower resolutions) while ensuring that information loss of the omitted wavelets is tenable Tradeoff WAVELET BASED REDUCED ORDER STUDY Completely averaged scale. Chosen wavelet basis elements

14. Materials Process Design and Control Laboratory Mutual Information Comparison Across Scales Daubechies Family Biorthogonal Family Mutual Information: The information that parameters in a scale are able to convey about parameters in another scale. A higher information loss occurs when we try to reduce the dimensionality of the solution when the physics involves lower order scales. Hence a hierarchical wavelet based method to be employed while ensuring that information lost in the truncated wavelet bases is minimized.

15. Materials Process Design and Control Laboratory INFORMATION AND WAVELET MEASURES Entropy Measures (Shannon) (Renyi) (Tsallis) Renyi’s and Shannon’s Entropy have the same minima Renyi’s quadratic entropy is computationally very efficient and fast Mean square error criterion for training is a very special case of Renyi’s mutual information maximization criterion Renyi vs Shannon (a : Scale parameter,b : space parameter, w : wavelet coefficients) Wavelet Maps Map parameters at lower scale onto a wavelet basis Upscale these coefficients by maximizing mutual information between multiscale wavelet coefficients Obtain the macro scale information maximized parameters Wavelet Families Haar Daubechies Biorthogonal Morlet

16. Materials Process Design and Control Laboratory INFORMATION THEORETIC DOWNSCALING Averaged velocity gradient Variations across averaged values as seen from micro scale Constant velocity gradient applied at the macro scale to the specimen Micro scale parameters would be distributed across this macro value. Hence a stochastic simulation needed at the micro. MAXENT (Jaynes): The entropy of variables must be maximized over the parameter space to obtain micro parameters subjected to macro averages

17. Materials Process Design and Control Laboratory MAXENT AS A DOWNSCALING TOOL Microstructure Reconstruction via MAXENT MAXENT provides means to obtain the entire microstructural variability of entities whose average and certain moments are available at higher scales (Sobczyk, 2003) A deterministic simulation at higher scale is equivalent to a stochastic simulation at lower scales where the stochastic parameters are obtained using MAXENT and higher scale parameters Experimental simulations when microstructure approximated as PV tessellations using MC analysis (Kumar et al, 1992)

18. Materials Process Design and Control Laboratory MAXENT AS A RECONSTRUCTION TOOL Most of the simulations at the microscale use deterministic samples/microstructures as input to their simulations. Actual samples, on the other hand could only be characterized stochastically From a set of statistical samples, maximize the uncertainty over the unspecified informational direction (MAXENT) to obtain the best estimates of the stochastic description of the microstructure Use the stochastic description of the microstructure to simulate the evolution processes at the micro scale Upscale the outputs from these simulations in a wavelet based information-theoretic framework. Obtain bounds on properties and serve as an input to stochastic simulations (SSFEM) at the macro.

19. Materials Process Design and Control Laboratory RECONSTRUCTION PROBLEM Higher order information in the form of the expected lineal path functions are specified (corresponding to lineal path functions of circular shaped phase two embedded inside another phase). Such microstructures cannot be deterministically characterized with only lower order correlation functions. It is desired to produce samples of this microstructure whose statistical properties match the given information. Another set of microstructures correspond to square checked phase structure are also specified. Here the correlation functions are not uniform in all directions. Circular phase embedded in a larger phase Checked microstructure with anisotropic correlation functions

20. Materials Process Design and Control Laboratory RECONSTRUCTION SCHEMES Uses the given information and starts from a random configuration. An equivalent energy function is defined and the final microstructure is obtained so that the energy function is minimized. However, only microstructures compatible with the expected averages of given functions are obtained in contrast to a probabilistic representation by MAXENT (Torquato) Stochastic Optimization Scheme for ill posed problems where the amount of information given is incommensurate with the total information required to characterize the material. Here the optimization problem is to maximize entropy over the entire probabilistic space. Methods such as Conjugate Gradient may not be necessarily suited as the evaluation of function requires a sampling method whose probability could only be approximated using the previous distribution. This noise represents one of the major drawbacks in using this scheme. Another possibility is to define an Information Functional and ensure that the Information Norm in the constrained dimensions is close to unity (Information Learning) MAXENT

21. Materials Process Design and Control Laboratory ALGORITHMS USED FOR MAXENT  The probability distribution corresponding to the Maximum entropy is given by while satisfying the constraints  Hence, the original problem is now posed as an Equivalent optimization problem for the Lagrange Multipliers.  This could be done using gradient based algorithms but the inherent noise in the sampling algorithms may impede exact convergence.  The algorithm for computing the values and gradients are explained. Optimization Algorithm Sampling Algorithm Z is to be computed using sampling algorithms. Start from an initial value of equal to 0 so that all distributions are equally probable. Samples can be developed from this. For i>0, can be found from by importance sampling estimates as: The gradient of the L’s could also be found out from these importance sampling methods Noise at the estimates at these set of points would hinder the accuracy and convergence of the estimates

22. Materials Process Design and Control Laboratory MAXENT Vs STOCHASTIC OPTIMIZATION PROCEDURES Uses lower order information to simulate microstructures compatible with the given inputs. However, the stochastic field over the probabilistic microstructures is not rigorously formed which is a necessity for doing a stochastic simulation. Stochastic Optimization Reconstructed from correlation functions corresponding to checked microstructures Comparison of path functions for a simulated microstructure with circular inclusions of the second phase MAXENT The entropy is maximized over the whole space of random fields while satisfying the constraints posed by the given information. The probability distributions follow the Gibbs path (Jaynes ’57). Optimization is performed either using standard gradient-based algorithms or maximizing mutual information (Information Learning Schemes). Stochastic samples are generated by asymptotically sampling through the exponential distribution using MCMC techniques Comparison of path functions for a simulated microstructure with circular inclusions of the second phase

23. Materials Process Design and Control Laboratory INFORMATION THEORY AND STATISTICAL UPSCALING Another crucial application of Information Theory is that it could serve as input to upscaling methods in a statistical framework. This is optimally done by coupling with MAXENT method to generate maximally distributed samples satisfying known information. This ensures that no unknown information is neglected. The analysis involves analyzing the problem using methods such as Finite Elements and/or Green’s Function and utility of ensemble averaging/wavelet tools for the upscaling. Limited Information Space Sampling Set: Experimental Images MAXENT Maximized over Information Space. Stochastic Samples FEM/Green’s function simulation for evolution Upscaled Material Properties using Statistical Averaging/wavelet tools

24. Materials Process Design and Control Laboratory Applications of information theory with multiscale methods Informatio n Theoretic Framework Obtaining Property Bounds at the Macro from micro Information (upscaling) Serve as an Input to Stochastic Simulations at macro A rationale to use with Multiscale tools such as wavelets Generation of samples from limited Information Information Theoretic Correlation Kernels Information Learning (neural networks) for upscaling data dynamically Used in conjunction with frameworks such as OOF An useful tool for linking scales in a Variational Multiscale Framework Some currently ongoing and envisaged applications of Information Theory in a Multiscale Framework

25. Materials Process Design and Control Laboratory INFORMATION LEARNING Information Force Normalized Information Potential Information Potential Basis Microstructures Desired Macroscale entities Linsker’s maximum Mutual Information Mutual information between desired signal and output signal should be maximized

26. Materials Process Design and Control Laboratory INFORMATION THEORETIC LEARNING Information Learning Used to reduce the computational time when the parameters needs to be transferred continuously at each time step. Train a neural network with Information criterion, that is mutual information between actual and nn based outputs is maximized A convergence study of neural network based single level upscaling process employing information theoretic criterion Information potential of one implies that the nn based output can predict exactly the result of upscaling process

27. Materials Process Design and Control Laboratory Microstructure Based Models Model chosen based on microstructure Poly-phase material Pure metal Lineal analysis of microstructure photograph Orientation distribution function model Dendritic Spatial correlation structure of models are known

28. Materials Process Design and Control Laboratory Training samples ODF Image Pole figures STATISTICAL LEARNING TOOLBOX Functions: 1.Classification methods 2.Identify new classes NUMERICAL SIMULATION OF MATERIAL RESPONSE 1.Multi-length scale analysis 2.Polycrystalline plasticity PROCESS DESIGN ALGORITHMS 1. Exact methods (Sensitvities) 2.Heuristic methods Update data In the library Associate data with a class; update classes Process controller STATISTICAL LEARNING TOOLBOX

29. Materials Process Design and Control Laboratory APPLICATION: MICROSTRUCTURE RECONSTRUCTION vision Database 2D Imaging techniques Microstructure Analysis (FEM/Bounding theory) Feature extraction Pattern recognition Microstructure evolution models Process Reverse engineer process parameters 3D realizations

30. Materials Process Design and Control Laboratory THE PROBLEM STATEMENT A Common Framework for Quantification of Diverse Microstructure Representation space of all possible polyhedral microstructures Equiaxial grain microstructure space Qualitative representation Lower order descriptor approach Equiax grains Grain size: small Grain size distribution Grain size number No. of grains Quantitative approach1.42.64.00.9….. Microstructure represented by a set of numbers

31. Materials Process Design and Control Laboratory LOWER ORDER DESCRIPTOR BASED RECONSTRUCTION (Yeong & Torquato, 1998) Descriptor: Two-point probability function and lineal measure 1.Non-uniqueness 2.Computationally expensive 3.Incomplete How many descriptors? Under constrained Descriptor-1: P (2) ( r ) Reconstructed Actual New Descriptor: P (3) ( r,s,t ) (plotted as a vector) Reconstructed Actual An under constrained case

32. Materials Process Design and Control Laboratory REQUIREMENTS OF A REPRESENTATION SCHEME REPRESENTATION SPACE OF A PARTICULAR MICROSTRUCTURE Need for a technique that is autonomous, applicable to a variety of microstructures, computationally feasible and provides complete representation A set of numbers which completely represents a microstructure within its class2.73.61.20.1….. 8.42.15.71.9….. Must differentiate other cases: (must be statistically representative)

33. Materials Process Design and Control Laboratory Microstructure Representation: PRINCIPAL COMPONENT ANALYSIS Let be n images. 1. Vectorize input images 2. Create an average image 3. Generate training images 4. Create correlation matrix (L mn ) 5. Find eigen basis (v i ) of the correlation matrix 6. Eigen faces (u i ) are generated from the basis (v i ) as 7. Any new face image ( ) can be transformed to eigen face components through ‘n’ coefficients (w k ) as, Representation coefficients Reduced basis Data Points

34. Materials Process Design and Control Laboratory PCA REPRESENTATION OF MICROSTRUCTURE – AN EXAMPLE Eigen-microstructures Input Microstructures Representation coefficients (x 0.001) Image-1 quantified by 5 coefficients over the eigen- microstructures 0.01251.3142-4.234.5429-1.6396 -0.84060.8463-3.02320.34242.6752 3.943-4.2162-0.6817-97181.9268 1.1796-1.3354-2.84016.2064-3.2106 5.82945.2287-3.7972-3.6095-3.6515 Basis 5 Basis 1

35. Materials Process Design and Control Laboratory EIGEN VALUES AND RECONSTRUCTION OVER THE BASIS 1.Reconstruction with 100% basis 2. Reconstruction with 80% basis 3. Reconstruction with 60% basis 4. Reconstruction with 40% basis 4231 Reconstruction of microstructures over fractions of the basis Significant eigen values capture most of the image features

36. Materials Process Design and Control Laboratory INCREMENTAL PCA METHOD For updating the representation basis when new microstructures are added in real-time. Basis update is based on an error measure of the reconstructed microstructure over the existing basis and the original microstructure IPCA : Given the Eigen basis for 9 microstructures, the update in the basis for the 10 th microstructure is based on a PCA of 10 x 1 coefficient vectors instead of a 16384 x 1 size microstructures. Updated Basis Newly added data point

37. Materials Process Design and Control Laboratory DYNAMIC MICROSTRUCTURE LIBRARY: CONCEPTS Space of all possible microstructures New class New class: partition Expandable class partitions (retraining) Hierarchical sub- classes (eg. medium grains) A class of microstructures (eg. equiaxial grains) Dynamic Representation: Axis for representation New microstructure added Updated representation distance measures

38. Materials Process Design and Control Laboratory BENEFITS 1.A data-abstraction layer for describing microstructural information. 2.An unbiased representation for comparing simulations and experiments AND for evaluating correlation between microstructure and properties. 3.A self-organizing database of valuable microstructural information which can be associated with processes and properties. Data mining: Process sequence selection for obtaining desired properties Identification of multiple process paths leading to the same microstructure Adaptive selection of basis for reduced order microstructural simulations. Hierarchical libraries for 3D microstructure reconstruction in real-time by matching multiple lower order features. Quality control: Allows machine inspection and unambiguous quantitative specification of microstructures.

39. Materials Process Design and Control Laboratory DIGITIZATION Conversion of RGB format of *.bmp file to a 2D image matrix PREPROCESSING Brings the image to the library format (RD : x-axis, TD : y-axis) – Rotate and scale image – Image enhancement steps – Boundary detection for feature extraction Inputs:Microstructure Image (*.bmp Format), Magnification, Rotation (With respect to rolling direction) Preprocessing based on user inputs of magnification and rotation PREPROCESSING

40. Materials Process Design and Control Laboratory ROSE OF INTERSECTIONS FEATURE – ALGORITHM (Saltykov, 1974) Identify intercepts of lines with grain boundaries plotted within a circular domain Count the number of intercepts over several lines placed at various angles. Total number of intercepts of lines at each angle is given as a polar plot called rose of intersections

41. Materials Process Design and Control Laboratory GRAIN SHAPE FEATURE: EXAMPLES

42. Materials Process Design and Control Laboratory GRAIN SIZE PARAMETER Several lines are superimposed on the microstructure and the intercept length of the lines with the grain boundaries are recorded (Vander Voort, 1993) The intercept length (x-axis) versus number of lines (y-axis) histogram is used as the measure of grain size.

43. GRAIN SIZE FEATURE: EXAMPLES Materials Process Design and Control Laboratory

44. SUPPORT VECTOR MACHINES: A BINARY CLASSIFIER Find w and b such that is maximized and for all ( x i,y i ) w. x i + b ≥ 1 if y i =1; w. x i + b ≤ -1 if y i = -1 Support Vectors Margin ( ) w.x i + b > 1 w.x i + b < -1 Class – I feature (y = 1) Class – II feature (y = -1) Class Labels (Supervised classifier)

45. Materials Process Design and Control Laboratory Map the non- separable data set to a higher dimensional space (using kernel functions) where it becomes linearly separable. Φ: x → φ(x) Non-separable case Minimize Relax constraints w. x i + b ≥ 1- if y i =1; w. x i + b ≤ -1+ if y i = -1 BETTER CLASSIFIERS

46. Materials Process Design and Control Laboratory SVM MULTI-CLASS CLASSIFICATION Class-A Class-B Class-C A C B A B C p = 3 One against one method: Step 1: Pair-wise classification, for a p class problem Step 2: Given a data point, select class with maximum votes out of

47. Materials Process Design and Control Laboratory SVM TRAINING FORMAT CLASSIFICATION SUCCESS % Total images Number of classes Number of Training images Highest success rate Average success rate 375114095.8292.53 3751110098.5495.80 Class Feature number Feature value Feature number Feature value 1123.32221.52 2124.12231.52 Data point GRAIN FEATURES: GIVEN AS INPUT TO SVM TRAINING ALGORITHM

48. Materials Process Design and Control Laboratory CLASS HIERARCHY Class –2 Class –1 Class 1(a)Class 1(b)Class 1(c)Class 2(a)Class 2(b)Class 2(c) Level 1 : Grain shapes Level 2 : Subclasses based on grain sizes New classes: Distance of image feature from the average feature vector of a class

49. IPCA QUANTIFICATION WITHIN CLASSES Materials Process Design and Control Laboratory Class-j Microstructures (Equiaxial grains, medium grain size) Class-i Microstructures (Elongated 45 degrees, small grain size) Representation Matrix Image -1Image-2Image-3… Component in basis vector 1 1232338 29154-85 3-549012 Average Image 212324 … Eigen Basis 0.90.840.23.. 0.540.210.74.. The Library – Quantification and image representation

50. Materials Process Design and Control Laboratory REPRESENTATION FORMAT FOR MICROSTRUCTURE Improvement of microstructure representation due to classification Date: 1/12 02:23PM, Basis updated Shape Class: 3, (Oriented 40 degrees, elongated) Size Class : 1, (Large grains) Coefficients in the basis:[2.42, 12.35, -4.14, 1.95, 1.96, -1.25] Reconstruction with 6 coefficients (24% basis): A class with 25 images Improvement in reconstruction: 6 coefficients (10 % of basis) Class of 60 images Original image Reconstruction over 15 coefficients

51. Materials Process Design and Control Laboratory Reconstruction Of Polyhedral Microstructure Polarized light micrographs of Aluminum alloy AA3002 representing the rolling plane (Wittridge & Knutsen 1999) A reconstructed 3D image Comparison of the average feature of 3D class and the 2D image

52. Materials Process Design and Control Laboratory Stereological Distributions (Geometrical) 3D reconstruction 2D grain profile 3D grain 3D grain size distribution based on assumption that particles are randomly oriented cubes ( ) N a,F a (s) : density of grains and grain size distribution in 2D image N v,F v (u) : density of grains and grain size distribution in 3D microstructure

53. Materials Process Design and Control Laboratory Two Phase Microstructure: Class Hierarchy Class - 1 3D Microstructures Feature vector : Three point probability function 3D Microstructures Class - 2 Feature: Autocorrelation function LEVEL - 1 LEVEL - 2 r  m 

54. Materials Process Design and Control Laboratory Example: 3D Reconstruction Using SVMS Ag-W composite (Umekawa 1969) A reconstructed 3D microstructure 3 point probability function Autocorrelation function

55. Materials Process Design and Control Laboratory Microstructure Property Estimation 3D image derived through pattern recognition Experimental image

56. Materials Process Design and Control Laboratory Microstructure Representation Using SVM & PCA COMMON-BASIS FOR MICROSTRUCTURE REPRESENTATION A DYNAMIC LIBRARY APPROACH Classify microstructures based on lower order descriptors. Create a common basis for representing images in each class at the last level in the class hierarchy. Represent 3D microstructures as coefficients over a reduced basis in the base classes. Dynamically update the basis and the representation for new microstructures

57. Materials Process Design and Control Laboratory Quantification using incremental PCA Input Image Classifier Feature Detection Dynamic Microstructure Library Identify and add new classes Employ lower- order features Pre-processing

58. Materials Process Design and Control Laboratory PCA Microstructure Representation Pixel value round-off Basis Components X 5.89 X 14.86 + Project onto basis Reconstruct using two basis components Representation using just 2 coefficients (5.89,14.86)

59. Materials Process Design and Control Laboratory DATABASE FOR POLYCRYSTAL MATERIALS Statistical Learning Feature Extraction Reduced order basis generation Multi-scale microstructure evolution models Process design for desired properties RD R-value TD Meso-scale database COMPONENTS ODF TD Youngs Modulus RD Database Divisive Clustering Class hierarchies Class Prediction Database Tension process basis

60. Materials Process Design and Control Laboratory DESIGNING MATERIALS WITH TAILORED PROPERTIES Micro problem driven by the velocity gradient L Macro problem driven by the macro-design variable β B n+1 Ω = Ω (r, t; L) ~ Polycrystal plasticity x = x(X, t; β ) L = L (X, t; β ) L = velocity gradient F n+1 B0B0 Reduced Order Modes Data mining techniques Multi-scale Computation Design variables (β) are macro design variables Processing sequence/parameters Design objectives are micro-scale averaged material/process properties Database

61. Materials Process Design and Control Laboratory FEATURES OF AN ODF: ORIENTATION FIBERS Points (r) of a (h,y) fiber in the fundamental region angle Crystal Axis = h Sample Axis = y Rotation (R) required to align h with y (invariant to, ) Fibers: h{1,2,3}, y || [1,0,1] {1,2,3} Pole Figure Point y (1,0,1) Integrated over all fibers corresponding to crystal direction h and sample direction y For a particular (h), the pole figure takes values P(h,y) at locations y on a unit sphere.

62. Materials Process Design and Control Laboratory SIGNIFICANCE OF ORIENTATION FIBERS Uniaxial (z-axis) Compression Texture z-axis fiber BB’ z-axis fiber AA’ z-axis fiber CC’ During deformation, Transport of crystals is structured relative to orientation fiber families Important fiber families: : uniaxial compression, plane strain compression and simple shear. : Torsion,, fibers: Tension  fiber (ND ) &  fiber: FCC metals under plane strain compression Lower order features in the form of pole density functions over orientation fibers are good features for classification due to their close affiliation with processes

63. Materials Process Design and Control Laboratory LIBRARY FOR TEXTURES [110] fiber family DATABASE OF ODFs Uni-axial (z-axis) Compression Texture z-axis fiber (BB’) Feature:  fiber path corresponding to crystal direction h and sample direction y

64. Materials Process Design and Control Laboratory SUPERVISED CLASSIFICATION USING SUPPORT VECTOR MACHINES Given ODF/texture Tension (T) Stage 1 LEVEL – 2 CLASSIFICATION Plane strain compression T+P LEVEL – I CLASSIFICATION Tension identified Stage 2 Stage 3 Multi-stage classification with each class affiliated with a unique process Identifies a unique processing sequence: Fails to capture the non-uniqueness in the solution

65. Materials Process Design and Control Laboratory UNSUPERVISED CLASSIFICATION Find the cluster centers {C 1,C 2,…,C k } such that the sum of the 2-norm distance squared between each feature x i, i = 1,..,n and its nearest cluster center C h is minimized. Identify clusters Clusters DATABASE OF ODFs Feature Space Cost function Each class is affiliated with multiple processes

66. Materials Process Design and Control Laboratory ODF CLASSIFICATION Desired ODF Search path Automatic class-discovery without class labels. Hierarchical Classification model Association of classes with processes, to facilitate data-mining Can be used to identify multiple process routes for obtaining a desired ODF Data-mining for Process information with ODF Classification ODF 2,12,32,97 One ODF, several process paths

67. Materials Process Design and Control Laboratory PROCESS PARAMETERS LEADING TO DESIRED PROPERTIES Young’s Modulus (GPa) Angle from rolling direction CLASSIFICATION BASED ON PROPERTIES Class - 1 Class - 2 Class - 3 Class - 4 Velocity Gradient Different processes, Similar properties Database for ODFs Property Extraction ODF Classification Identify multiple solutions

68. Materials Process Design and Control Laboratory K-MEANS ALGORITHM FOR UNSUPERVISED CLASSIFICATION User needs to provide ‘k’, the number of clusters. Lloyds Algorithm: 1.Start with ‘k’ randomly initialized centers 2.Change encoding so that x i is owned by its nearest center. 3.Reset each center to the centroid of the points it owns. Alternate steps 1 and 2 until converged. But, No. of clusters is unknown for the texture classification problem

69. Materials Process Design and Control Laboratory A TWO-STAGE PROBLEM Process – 2 Plane strain compression  = 0.3515 Process – 1 Tension  = 0.9539 Initial Conditions: Stage 1 Sensitivity of material property Initial Conditions- stage 2 DATABASE Reduced Basis  (1)  (2) Direct problem  Sensitivity problem

70. Materials Process Design and Control Laboratory PROCESS DESIGN WITH A FIXED BASIS Initial basis based on Tension process: [1,0,0,0,0] Final process iterate: [1 -0.5 -0.25 0 0] Actual ODF corresponding to the process identified ODF reconstructed using the initial fixed basis The basis functions used for the control problem not only needs to represent the solution but also the textures arising from intermediate iterates of the design variable

71. Materials Process Design and Control Laboratory ADAPTIVE REDUCED-ORDER MODELING Stage 1: Compression  -0.8 Stage 2: PSC  -1.0 Full-order model Reduced-order model Direct problem Stage –2 sensitivity: finite differences (  = 0.01) Stage –2 sensitivity: Adaptive reduced order model (Threshold  = 0.05) Sensitivity problem

72. Materials Process Design and Control Laboratory MULTIPLE PROCESS ROUTES Desired Young’s Modulus distribution Magnetic hysteresis loss distribution Stage: 1 Shear-1  = 0.9580 Stage: 2 Plane strain compression (  = -0.1597 ) Stage: 1 Shear -1  = 0.9454 Stage: 2 Rotation-1 (  = -0.2748) Stage 1: Tension  = 0.9495 Stage 2: Shear-1  = 0.3384 Stage 1: Tension  = 0.9699 Stage 2: Rotation-1  = -0.2408 Classification

73. Materials Process Design and Control Laboratory DESIGN FOR DESIRED ODF: A MULTI STAGE PROBLEM Initial guess,   = 0.65,   = -0.1 Desired ODFOptimal- Reduced order control Full order ODF based on reduced order control parameters Stage: 1 Plane strain compression (   = 0.9472) Stage: 2 Compression (   = -0.2847)

74. Materials Process Design and Control Laboratory DESIGN FOR DESIRED MAGNETIC PROPERTY h Crystal direction. Easy direction of magnetization – zero power loss External magnetization direction Stage: 1 Shear – 1 (   = 0.9745) Stage: 2 Tension (   = 0.4821)

75. Materials Process Design and Control Laboratory DESIGN FOR DESIRED YOUNGS MODULUS Stage: 1 Shear (   = - 0.03579) Stage: 2 Tension (   = 0.17339) Stiffness of F.C.C Cu in crystal frame Elastic modulus is found using the polycrystal average over the ODF as,

76. Materials Process Design and Control Laboratory MULTISCALE DATA MINING –MICRO/MESO SCALE Constitutive laws, Microstructure-dependent properties through bounding theories and FEM Phase field model Dislocation dynamics Microstructure Morphology Properties of individual phases and crystals LEVEL - 1 3 point probability Microstructure Class Hierarchy 3D Microstructures Meso-scale database Data- mining Expanded view of the meso-scale database Model reduction Autocorrelation Statistical learning To stochastic continuum models Data from DFT

77. Materials Process Design and Control Laboratory Electron scale database Alloy systems DFT Phase Field DD Meso-scale database Micro-scale database Statistical features at the local length scale Hierarchical class structure at each length scale Dynamic update of class structures with new data Reduced models for higher length scales ObjectiveDesign decisions Hyperplanes quantify correlation of local length scale features with the objective and higher length scale effects MATERIAL FEATURE REPRESENTATION AND DESIGN

78. Materials Process Design and Control Laboratory ATOMISTIC SCALE STATISTICAL LEARNING Divisive hierarchical learning Macro property design 0: Lattice type 1: Eqm volume 2: Cohesive energy DESCRIPTORS (Ab-initio) Lattice constants, Equilibrium volume Cohesive energy, Helmholtz free energy Structural energy difference between configurations (BCC/FCC) Bulk properties: bulk and shear moduli, Zener’s anisotropy constant CORRELATIONS WITH ENGINEERING PROPERTIES Material strength Phase stability Resistance to intergranular corrosion Resistance to pitting, stress corrosion cracking Hardness Ductility

79. Materials Process Design and Control Laboratory DESIGNING ALLOYS THROUGH STATISTICAL LEARNING Meshing and virtual experimentation (OOF) Property statistics Phase field model Thermodynamic variables (CALPHAD) Mobilities Interfacial energies Nucleation Models compositions that give optimum properties Design problems: 1) Determine the compositions that give optimum properties 2) Design process sequences to obtain desired properties Diffusion coefficients