A computational statistics and stochastic modeling approach to materials-by-design 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/ Materials Process Design and Control Laboratory

INFORMATION FLOW ACROSS SCALES Continuum Engineering Mesoscale Information flow Microscale Materials Statistical filter Nanoscale Filtering and two way flow of statistical information Chemistry Electronic Physics Length Scales ( ) A 1 10 2 10 4 10 6 10 9

OVERVIEW OF STOCHASTIC FRAMEWORK Adaptive spectral and support methods Multiscale information theoretic material heterogeneity models Existing deterministic application software Stochastic analysis framework Geometric, boundary, material uncertainties Information theoretic correlation kernels Complete statistical response of outputs Accurate risk and reliability estimates Explicit uncertainty quantification

UNCERTAINTY IN FINITE DEFORMATION PROBLEMS Material heterogeneity 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) Material heterogeneity Composites – fiber orientation, fiber spacing, constitutive model Biomechanics – material properties, constitutive model, fibers in tissues Materials Process Design and Control Laboratory

UNCERTAINTY ANALYSIS USING SSFEM xn+1(θ) X Bn+1(θ) B0 xn+1(θ)=x(X,tn+1, θ,) Key features Total Lagrangian formulation – (assumed deterministic initial configuration) Spectral decomposition of the current configuration leading to a stochastic deformation gradient Materials Process Design and Control Laboratory

Two independent random variables with order 4 PCE (Legendre Chaos) EFFECT OF UNCERTAIN FIBER ORIENTATION Aircraft nozzle flap – composite material, subjected to pressure on the free end Orthotropic hyperelastic material model with uncertain angle of orthotropy modeled using KL expansion with exponential covariance Two independent random variables with order 4 PCE (Legendre Chaos)

MODELING INITIAL CONFIGURATION UNCERTAINTY xn+1(θ)=x(XR,tn+1, θ,) F(θ) X(θ) xn+1(θ) FR(θ) Bn+1(θ) B0 XR F*(θ) BR Introduce a deterministic reference configuration BR which maps onto a stochastic initial configuration by a stochastic reference deformation gradient FR(θ). The deformation problem is then solved in this reference configuration.

Strain Localization due to Uncertainty in Initial Configuration Deterministic simulation- Uniform bar under tension with effective plastic strain of 0.7 . Power law constitutive model. Initial configuration assumed to vary uniformly between two extremes with strain maxima in different regions in the stochastic simulation. Plastic strain 0.7

Strain Localization due to Uncertainty in Initial Configuration Stochastic simulation Results plotted in mean deformed configuration Plastic strain 0.7

Strain Localization due to Uncertainty in Initial Configuration Stochastic simulation Plastic strain 0.7

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. Initial and mean deformed config. Eigenvectors

Dominant effect of material heterogeneity on response statistics UNCERTAINTY DUE TO MATERIAL HETEROGENEITY Dominant effect of material heterogeneity on response statistics Load vs Displacement SD Load vs Displacement

CORRELATION KERNELS Model Based Correlation Kernels Information Theoretic transfer based on Wavelets Process based microstructure models Raw Material Manufacture Process Multiscale parameters of stochastic deformation problem Correlation Kernels at various scales in a SSFEM framework Correlation in wavelet domain A multiscale analysis

INFORMATION THEORETIC FRAMEWORK Correlation kernels based on intrascale mutual information criterion Wavelet basis ( ) a,b are scale and space parameters Correlation kernels at macro scale Information filtering based on Renyi’s entropy and Linsker’s maximum mutual information Wavelet coefficients at macro scale Wavelet coefficients at meso scale 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.

INFORMATION THEORY AND MULTISCALE MODELING? Field of mathematics founded by Shannon in 1948 Revolutionized the outlook towards communication of information (rigorous mathematical standpoint) Information Theory used to link simulations at various scales in multiscale simulations Try to transfer as much information as possible about parameters of interest (displacements, stresses, strains etc) Linkage? Information Theory

AN INFORMATION THEORETIC VIEWPOINT Informational entropy of parameter ensembles during upscaling reduces due to averaging out of fine details present at micro scale Micro scale Meso scale 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 ?

COMMUNICATION AND MULTISCALE ANALYSIS? Channel capacity is the maximum rate at which information could be transmitted with negligible loss of information In a communication channel, data is transferred over a physical channel where unknown noise may act on the system Shannon’s source coding theorem states that there exists data encoding systems to achieve rates of the order of channel capacity There is a limit on the maximum information transfer of parameters of interest that could take place between two given scales Data is transferred between lower to upper scales over an hypothetical “homogenization” channel There exists an ideal transfer process which maximizes the information transfer across scales??

INFORMATION TRANSFER ACROSS CHANNELS Received messages Sent messages symbols Source coding encoder decoder Channel coding Channel decoding Source decoding channel source channel receiver Compression Error Correction Decompression Source Entropy Channel Capacity Rate vs Distortion Capacity vs Efficiency

AN INTERESTING ANALOGY Wavelet Basis at lower scale Information Upscaling Channel Wavelet Basis at higher scale Wavelet based coding of parameters Information Theoretic upscaling of wavelet coefficients Decoding of wavelet parameters micro scale macro scale Source information Received information Information lost here

a,b: wavelet coefficients at scale a and spatial location b. NEED FOR WAVELETS Schematic of wavelet representation A very useful tool in areas where a multiscale analysis is important. 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 Information across all scales Fo Micro scale Q1Fo F1 F2 Q2Fo Frantziskonis, Deymier (2000,2003) Q3Fo Information Lost 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 Meso scale Fn a,b: wavelet coefficients at scale a and spatial location b.

HOMOGENIZATION IN WAVELET SPACES Full Microstructure Information Homogenized Properties at next scale Wavelet Basis Complete Homogenization

Wavelet based Reduced Order study Information lost when approximated to fourth scale 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. Tradeoff 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 Chosen wavelet basis elements Completely averaged scale.

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.

INFORMATION AND WAVELET MEASURES Entropy Measures Renyi vs Shannon 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 (Shannon) (Tsallis) (Renyi) (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

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

Microstructure Reconstruction via MAXENT MAXENT as an upscaling tool Experimental Simulations when microstructure approximated as PV tessellations using MC analysis (Kumar et al, 1992) 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

Linsker’s maximum Mutual Information INFORMATION LEARNING Linsker’s maximum Mutual Information Mutual information between desired signal and output signal should be maximized Desired Macroscale entities Information Potential Information Force Normalized Information Potential Basis Microstructures

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

MICROSTRUCTURE BASED MODELS Model chosen based on microstructure Lineal analysis of microstructure photograph Orientation distribution function model Poly-phase material Dendritic Pure metal Spatial Correlation Structure of Models are known

Process captured by large scale computational grid IDEA BEHIND STOCHASTIC VMS Subgrid solution obtained semi-analytically using element-wise stochastic Fourier transform approach Coarse element Compatibility of stochastic subgrid solution with respect to large scale solution uncertainty is examined The subgrid scale approximate stochastic solution is substituted to large scale weak form yielding a stabilized formulation Process captured by large scale computational grid Variational multiscale decomposition – Stochastic solution considered as a sum of two scale components viz. large scale and subgrid scale Large scale can be captured by the computational grid Subgrid scale has to be modeled The approximate subgrid effects are considered in the large scale variational formulation to yield a stochastic finite element method with stabilized properties

Representation techniques for a Space-time stochastic process STOCHASTIC INPUT-OUTPUT MODELING Uncertainty in inputs and outputs modeled by considering them as stochastic processes Representation techniques for a Space-time stochastic process STOCHASTIC INPUT REPRESENTATION Stochastic inputs modeled using Karhunen-Loeve expansion based on spectral decomposition of the covariance kernel are independent random variables that form a basis for the input probability space For output representation schemes, we assume that input can be represented in terms of a few independent random variables GLOBAL OUTPUT REPRESENTATION The output is represented in the above Wiener-Askey generalized polynomial chaos expansion Specific polynomials are chosen based on the input joint probability density function. Examples are: Gaussian—Hermite, Uniform—Legendre and Beta—Jacobi. LOCAL OUTPUT REPRESENTATION Stochastic output is represented as a piecewise polynomial on the input stochastic support space We use a finite element mesh on the support space that is refined towards regions of high input PDF – importance based gridding approach

MORE ON SUPPORT-SPACE METHODOLOGY We assume that the stochastic input has been represented in the KL expansion represents the stochastic input vector represents the joint PDF of inputs is called the stochastic support space characterized by positive input joint PDF Any stochastic output can be represented as a function defined on this input space We consider a finite element piecewise representation Each nodal point on the spatial mesh is linked to a support-space grid Statistics of relevance or complete PDF passed on to the spatial mesh Spatial domain with finite element grid Grid on support-space Support-space of input Importance spaced grid Two-level grid approach

BIG PICTURE – STOCHASTIC VMS FRAMEWORK Subgrid mesh Localized subgrid problem solved using FEM/wavelets in regions where multiscale physics is important. GPCE for smooth solutions, support-space for discontinuous solutions Using feedback control and posteriori error estimates separate domain into multiscale and non-multiscale regions Multiscale physics important Semi analytical solutions to reduced operator problems dictate the subgrid boundary conditions Multiscale physics not important Renyi Shannon entropy, Linsker’s information maximization theorem Information theoretic filters based on entropy criterion to ensure that only the relevant statistical information is transferred from subgrid to large scales Modify large scale residual based on subgrid solutions Fully resolved solution

EXPLICIT SUBGRID MODELING TECHNIQUES Macro-residual passed to subgrid Case with high subgrid stochasticity Fully stochastic VMS decomposition For GPCE this means, the subgrid basis has higher dimensional Wiener-Askey polynomials For support-space, the subgrid support space is represented in terms of hierarchical support-space basis functions Macro component discretized using finite element coarse mesh Each coarse element further contains a link to the subgrid mesh This linkage is null if the coarse element does not possess any multiscale behavior Subgrid BCs calculated Macro domain Explicit subgrid model Combining concept + different subgrid equation Volume-averaging concept valid far from mold and at larger times Generate subgrid basis functions forces by partition of unity or wavelets Melt Near wall regions using higher-order transport equations derived from ab-initio principles Mold Compatibility of boundary conditions, statistical description and linking hypothesis developed in a consistent mathematical framework Ability to capture discontinuities in stochastic response, mixed-random variables with both discrete and continuous behavior Modify large scale weak form GPCE for smooth subgrid variations, support-space for discontinuous subgrid solutions

MULTISCALE REGION IDENTIFICATION Multiscale regions Transition regions As shown in figure, all points in macro component do not possess multiscale characteristic Multiscale identification toolbox consists of following algorithms with implementation Stochastic dual-problem based posterior error estimates Feedback control of posterior error Adaptive tree-cased algorithms for distribution of multiscale elements among different processors Mathematical techniques for parallel solution reconstruction Multiscale identification toolbox based on feedback control and posterior error estimates Feedback/ adaptive toolbox + Reconstructed solution after parallel computations Multiscale and transition regions distributed separately among parallel processors for subgrid calculations Mathematical posterior error estimates

FRAMEWORK APPLIED TO MICROSTRUCTURES Large scale information captured 3a Variational consistent upscaling 3b 3c Meso-to-macro upscaling Macro-state variable statistics obtained by upscaling the subgrid statistics using a consistent VMS approach Wavelet homogenized properties, state variable solutions passed from microstructure to subgrid Fine scale detail filtered using entropy based information filter, updation of wavelet coefficients 2a Spectral stochastic FEM, RFB, Greens function 2b 2c Spectral stochastic FEM at macro level Macro-to-meso downscaling Statistics of large scale solutions for macro-state variables represented in wavelet expansions Evolution of subgrid statistics driven by macro residuals Evolution of microstructure driven by micromechanical models Component at start of processing stage – Approximate regions with multiscale physics identified 1a Subgrid scale model equations 1b Subgrid element mapped to microstructure clouds 1c Three-scale VMS Three scale stochastic variational multiscale framework Upscaling and downscaling of statistical information is based on information theoretic concepts Wavelets are used instead of a finite element representation for the support-space output representation Stochastic homogenization applied to upscale from microstructure to subgrid scale

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

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

TWO PHASE MICROSTRUCTURE: CLASS HIERARCHY Materials Process Design and Control Laboratory TWO PHASE MICROSTRUCTURE: CLASS HIERARCHY Feature vector : Three point probability function Feature: Autocorrelation function 3D Microstructures 3D Microstructures Class - 1 g r mm Class - 2 LEVEL - 1 LEVEL - 2

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

ADAPTIVE REDUCED ORDER MULTI-STAGE PROCESS DESIGN Process – 2 Plane strain compression a = 0.3515 Process – 1 Tension a = 0.9539 Initial Conditions: Stage 1 DATABASE Reduced Basis f(1) f(2) Initial Conditions- stage 2 Sensitivity of material property Direct problem a Sensitivity problem Materials Process Design and Control Laboratory

LIBRARY FOR TEXTURES Feature: 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

DATABASE DRIVEN COMPUTATIONAL PROCESS DESIGN Process sequence-1 Process parameters ODF history Reduced basis Process sequence-2 New process parameters Classifier Adaptive basis selection Optimization Process Probable Process Sequences & Initial Parameters Desired texture/property Stage - 1 Stage - 2 New dataset added DATABASE Optimum parameters Materials Process Design and Control Laboratory

EXAMPLE: 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) TWO STAGE PROCESS Stage: 2 Tension (a2 = 0.4821) Materials Process Design and Control Laboratory

MULTI-SCALE DATA MINING Materials Process Design and Control Laboratory MULTI-SCALE DATA MINING Information transferred to subsequent length scales: used for statistical learning Electronic scale: First principle Atomistics – DFT, Monte Carlo simulations Alloy systems: Newly discovered compositions Lattice energies, Interatomic potentials, Bulk free energies, Interfacial free energies, Bulk strength, crystal structure Knowledge discovery Electronic-scale database Statistical learning Information from other scales To higher length scales

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

ATOMISTIC STATISTICAL LEARNING Base alloy: Al-Li Property statistics: Micro-alloying elements Si La Bulk Modulus Mn 100 90 80 Os Atomic % Al Co Ni Zr GPa per atomic% -1 0 1 D(Bulk Modulus) 3 0 10 20 Macro-property correlations with atomistic properties Atomic % Li Alloy property maps Ductility Pitting corrosion Objective: Increase Bulk Moduli descriptor Desired property: Ductility, corrosion resistance Bulk Modulus Bulk Modulus 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 Materials Process Design and Control Laboratory

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

MATERIAL FEATURE REPRESENTATION AND DESIGN Materials Process Design and Control Laboratory MATERIAL FEATURE REPRESENTATION AND DESIGN Hyperplanes quantify correlation of local length scale features with the objective and higher length scale effects DFT Electron scale database Alloy: Al-Ni Composition: ?? (Atomistic) Process variables: ?? (Meso-Macro) Alloy systems Phase Field Meso-scale database Statistical features at the local length scales sYmax Processes & Microstructures (Phase field) Homogenized property (OOF) Desired strength distribution Objective Design decisions

MATERIAL DESIGN THROUGH STATISTICAL LEARNING Material properties Optimum process sequence Database update & retraining DATABASE compositions structures Desired property Descriptor generation through DFT Alloy composition Meso-scale evolution modeling Descriptor generation through image analysis Process sequence-2 New process parameters Structure history New properties Reduced order modes Process sequence-1 Process parameters Structure history New properties Atomistic Meso decoupled Machine learning Property correlations with descriptors Structure/ reduced order modes Processing sequence Optimization Multi-stage process design (FEM/CSM) Microstructure evolution & response modeling Process models with macro-meso linking Processing Stage - 1 Stage - 2 Optimal composition Property variation with composition Effect of micro-alloying From atomistic scale database Materials Process Design and Control Laboratory