1. 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
URL:
Materials Process Design and Control Laboratory

2. 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

3. 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

4. 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

5. 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

6. 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)

7. 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.

8. Strain Localization due to Uncertainty in Initial Configuration
Deterministic simulation- Uniform bar under tension with effective plastic strain of 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

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

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

11. 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

12. 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

13. 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

14. 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.

15. 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

16. 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 ?

17. 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??

18. 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

19. 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

20. 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.

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

22. 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.

23. 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.

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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

36. 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

37. 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
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38. 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

39. 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

40. 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

41. ADAPTIVE REDUCED ORDER MULTI-STAGE PROCESS DESIGN
Process – Plane strain compression a =
Process – Tension a =
Initial Conditions: Stage 1
DATABASE
Reduced Basis
f(1)
f(2)
Initial Conditions- stage 2
Sensitivity of material property
Direct problem a
Sensitivity problem
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42. 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

43. 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
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44. EXAMPLE: DESIGN FOR DESIRED MAGNETIC PROPERTY
Crystal <100> direction. Easy direction of magnetization – zero power loss h
External magnetization direction
Stage: 1 Shear – (a1 = )
TWO STAGE PROCESS
Stage: 2 Tension (a2 = )
Materials Process Design and Control Laboratory

45. 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

46. MULTI-SCALE DATA MINING –MICRO/MESO SCALE
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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

47. ATOMISTIC STATISTICAL LEARNING
Base alloy: Al-Li
Property statistics: Micro-alloying elements Si La
Bulk Modulus Mn Os
Atomic % Al Co Ni Zr
GPa per atomic% -1
D(Bulk Modulus)
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

48. 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
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49. 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
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50. 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

51. 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 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