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A data mining approach for the design of optimized polycrystalline materials

A data mining approach for the design of optimized polycrystalline materials. Veera Sundararaghavan and Prof. Nicholas Zabaras. Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 188 Frank H. T. Rhodes Hall Cornell University

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A data mining approach for the design of optimized polycrystalline materials

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  1. A data mining approach for the design ofoptimized polycrystalline materials Veera Sundararaghavan and 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 14853-3801 Email: zabaras@cornell.edu URL: http://www.mae.cornell.edu/zabaras/ Materials Process Design and Control Laboratory

  2. Design processes to produce highly optimized microstructure with directional material properties Design LIGHT armor for SUPERIOR PERFORMANCE • Especially important for critical hardware components in the aerospace, naval and automotive industry to reduce material utilization for reduced process cost, fuel consumption and higher mobility Aircraft engines • Reduce dependence on other expensive methods such as alloying Understanding and blocking fracture by MSD MOTIVATION FOR MICROSTRUCTURE BASED DESIGN Materials Process Design and Control Laboratory

  3. Properties • Averaging principles • Pole figures => ODF • Model validation • Friction Tool wear • Geometric parameters • Process parameters • Material interaction • Material composition MICROSTRUCTURE-PROPERTY-PROCESSING Microstructure Identify processes for desired microstructure for optimal properties. TEM 3 FOLD DESIGN • Texture • Defects • Composition Processing Realistic process sequence selection Materials and process modeling Materials Process Design and Control Laboratory

  4. MATERIALS DESIGN FRAMEWORK Machine learning schemes Computational process design simulator Microstructure Information library Virtual materials design framework Virtual process simulations to evaluate alternate designs Accelerated Insertion of new materials Optimization of existing materials Tailored application specific material properties Materials Process Design and Control Laboratory

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

  6. DESIGNING MATERIALS WITH TAILORED PROPERTIES x = x(X, t; β) ~ Ω =Ω (r, t; L) Macro problem driven by the macro-design variable β Multi-scale Computation Micro problem driven by the velocity gradient L Bn+1 Fn+1 B0 L = L (X, t; β) Polycrystal plasticity L = velocity gradient Data mining techniques Reduced Order Modes Database Design variables (β) are macro design variables Processing sequence/parameters Design objectives are micro-scale averaged material/process properties Materials Process Design and Control Laboratory

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

  8. SIGNIFICANCE OF ORIENTATION FIBERS 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 Important fiber families: <110> : uniaxial compression, plane strain compression and simple shear. <111>: Torsion, <100>,<411> fibers: Tension a fiber (ND <110> ) & b fiber: FCC metals under plane strain compression z-axis <110> fiber BB’ Uniaxial (z-axis) Compression Texture z-axis <111> fiber CC’ During deformation, Transport of crystals is structured relative to orientation fiber families z-axis <100> fiber AA’ Materials Process Design and Control Laboratory

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

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

  11. UNSUPERVISED CLASSIFICATION Find the cluster centers {C1,C2,…,Ck} such that the sum of the 2-norm distance squared between each feature xi, i = 1,..,n and its nearest cluster center Ch is minimized. Each class is affiliated with multiple processes Cost function Feature Space DATABASE OF ODFs Clusters Identify clusters Materials Process Design and Control Laboratory

  12. 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 ODF 2,12,32,97 One ODF, several process paths Data-mining for Process information with ODF Classification Materials Process Design and Control Laboratory

  13. DATABASE STRUCTURE DATABASE Process sequence-2 New process parameters ODF history Reduced basis Process sequence-1 Process parameters ODF history Reduced basis New dataset added Desired texture/property Classifier Adaptive basis selection Process Reduced basis Optimization Probable Process sequences & Initial parameters Stage - 1 Stage - 2 Optimum parameters Materials Process Design and Control Laboratory

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

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

  16. INFORMATION CRITERION FOR IDENTIFYING NO. OF CLUSTERS The number of clusters chosen maximizes the Bayesian information criterion given by: Where is the is the log-likelihood of the data taken at the maximum likelihood point, p is the number of free parameters in the model Maximum likelihood of the variance assuming Gaussian data distribution Probability of a point in cluster i Log-likelihood of the data in a cluster Materials Process Design and Control Laboratory

  17. CENTROID SPLIT TESTS New clusters based on BIC Split centers Run local k-means (k = 2) in each cluster • X-MEANS algorithm: • Start with k clusters found through k-means algorithm • Split each centroid into two centroids, and move the new centroids along a distance proportional to the cluster size in an arbitrarily chosen direction • Run local k-means (k = 2) in each cluster • Accept split cluster in each region if BIC(k = 1) < BIC(k = 2) • Test for various initial values of ‘k’ and select the ‘k’ with maximum overall BIC Materials Process Design and Control Laboratory

  18. COMPARISON OF K-MEANS AND X-MEANS Local Optimum produced by the kmeans algorithm with k = 4 Cluster configuration produced by k-means with k = 6: Over-estimates the natural number of clusters Configuration produced by the x-means algorithm: Input range of k = 2 to 15. x-means found 4 clusters from the data-set based on the Bayesian Information Criterion Materials Process Design and Control Laboratory

  19. MATERIAL POINT SIMULATOR ORIENTATION DISTRIBUTION FUNCTION – A(r,t) • Determines the volume fraction of crystals within • a region R'of the fundamental region R • Probability of finding a crystal orientation within • a region R' of the fundamental region • Characterizes texture evolution ODFEVOLUTION EQUATION – EULERIAN DESCRIPTION – reorientation velocity Any macroscale property < χ > can be expressed as an expectation value if the corresponding single crystal property χ (,t) is known. Materials Process Design and Control Laboratory

  20. CRYSTAL CONSTITUTIVE RELATIONSHIPS SALIENT FEATURES Viscoplastic rate dependent model – no hardening Extended Taylor hypothesis assumed: macro scale velocity gradient identical to the crystal velocity gradient Shear rate Velocity gradient Symmetric and spin components D = Macroscopic stretch = Schmid tensor = Lattice spin W = Macroscopic spin = Lattice spin vector Reorientation velocity Solve for the reorientation velocity and its divergence given the velocity gradient Divergence of reorientation velocity Materials Process Design and Control Laboratory

  21. METHOD OF SNAPSHOTS Method of snapshots Suppose we had a collection of data (from experiments or simulations) for the ODF: Solve the optimization problem Is it possible to identify a basis where such that it is optimal for the data represented as Eigen-value problem where POD technique – Proper Orthogonal Decomposition Materials Process Design and Control Laboratory

  22. REDUCED ORDER MODEL: DIRECT PROBLEM Represent the ODF as • Benefits • Reduction in number of degrees of freedom • PDE converted to ODE • No stabilization required in the FE formulation Reduced model for the evolution of the ODF Initial conditions Materials Process Design and Control Laboratory

  23. REDUCED ORDER SENSITIVITY PROBLEM Reduced model for the evolution of the sensitivity of the ODF The reduced basis for the ODF ensemble has been evaluated as Observing that the basis is independent of the macro design parameters, we conclude that the basis generated in the direct analysis can be used for the sensitivity problem. where Using this basis, the sensitivity of the ODF is represented as follows: Initial conditions Materials Process Design and Control Laboratory

  24. DESIGN OF PROCESS SEQUENCE Velocity gradient Modes Tension/ compression Shear Plane strain compression Rotation Shear Design vectorα = {α1, α2, ..., αn}T (n-stage problem) Design problem: Determine the process sequence so as to obtain desired properties in the final product Materials Process Design and Control Laboratory

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

  26. PROCESS DESIGN WITH A FIXED BASIS Initial basis based on Tension process: [1,0,0,0,0] 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 Final process iterate: [1 -0.5 -0.25 0 0] Actual ODF corresponding to the process identified ODF reconstructed using the initial fixed basis Materials Process Design and Control Laboratory

  27. ADAPTIVE REDUCED-ORDER MODELING Stage 1: Compression a = -0.8 Stage 2: PSC a = -1.0 Sensitivity problem Direct problem Stage –2 sensitivity: Adaptive reduced order model (Threshold e = 0.05) Stage –2 sensitivity: finite differences (da = 0.01) Reduced-order model Full-order model Materials Process Design and Control Laboratory

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

  29. DESIGN FOR DESIRED ODF: A MULTI STAGE PROBLEM Desired ODF Optimal- Reduced order control Stage: 1 Plane strain compression (a1 = 0.9472) Stage: 2 Compression (a2 = -0.2847) Initial guess, a1 = 0.65, a2 = -0.1 Full order ODF based on reduced order control parameters Materials Process Design and Control Laboratory

  30. DESIGN FOR DESIRED MAGNETIC PROPERTY Crystal <100> direction. Easy direction of magnetization – zero power loss h External magnetization direction Stage: 1 Shear – 1 (a1 = 0.9745) Stage: 2 Tension (a2 = 0.4821) Materials Process Design and Control Laboratory

  31. DESIGN FOR DESIRED YOUNGS MODULUS Stiffness of F.C.C Cu in crystal frame Elastic modulus is found using the polycrystal average <C> over the ODF as, Stage: 1 Shear (a1 = -0.03579) Stage: 2 Tension (a2 = 0.17339) Materials Process Design and Control Laboratory

  32. Orientation space Ѕ BETTER MECHANICS • ThermoElasto-ViscoPlastic analysis • Coupled length scale design methods • Slip + Twinning in FCC, BCC and HCP materials x(X, t) F(X, t) Texture map χ Texture map χt r s Reorientation map r(s, t) ^ Reference fundamental region Current fundamental region CONTINUUM MICROSTRUCTURE DESCRIPTION Materials Process Design and Control Laboratory

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

  34. RELEVANT PUBLICATIONS V. Sundararaghavan and N. Zabaras, "On the synergy between classification of textures and deformation process sequence selection", Acta Materialia, Vol. 53/4, pp. 1015-1027, 2005 S. Ganapathysubramanian and N. Zabaras, "Design across length scales: A reduced-order model of polycrystal plasticity for the control of microstructure-sensitive material properties", Computer Methods in Applied Mechanics and Engineering, Vol. 193 (45-47), pp. 5017-5034, 2004 S. Acharjee and N. Zabaras, “A proper orthogonal decomposition approach to microstructure model reduction in Rodrigues’ space with applications to the control of material properties”, Acta Materialia 51, 5627—5646, 2003. 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://www.mae.cornell.edu/zabaras/ Materials Process Design and Control Laboratory

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