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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
An Information-Theoretic Approach to Multiscale Modeling and Design
of Materials
Nicholas Zabaras
Materials Process Design and Control LaboratorySibley School of Mechanical and Aerospace Engineering
188 Frank H. T. Rhodes HallCornell University
Ithaca, NY 14853-3801
Email: [email protected]: http://www.mae.cornell.edu/zabaras/
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Filte
ring
and
two
way fl
ow o
f sta
tistic
al in
form
atio
n
1 102 104 106 109
Eng
inee
ring
Length Scales ( )
Phy
sics
Che
mis
try
Mat
eria
ls
0
A
Info
rmati
on fl
ow
Statistical filter
Electronic
Nanoscale
Microscale
Mesoscale
Continuum
INFORMATION FLOW ACROSS SCALES
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
DEFORMATION PROCESS DESIGN
(Minimal barreling)Initial guess Optimal preform
Optimal preform shape
Final optimal forged productFinal forged product
Initial preform shape
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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
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UNCERTAINTY REPRESENTATION
TECHNIQUESSample 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
01
( , , ) ( , ) ( , ) ( )N
n nn
W x t W x t W x t
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
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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.
0
n
fs
21 1( ,0, ,0) exp
r
bp pR
2
01
( ) (1 ( )
i n ii
s s vp p
V20.3398190.2390330.1382470.0374605
-0.0633257-0.164112-0.264898-0.365684-0.466471-0.567257
V10.4093960.3958130.382230.3686460.3550630.3414790.3278960.3143130.3007290.287146
Eigenvectors Initial and mean deformed config.
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CCOORRNNEELLLL U N I V E R S I T Y
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Displacement (mm)
SD
Loa
d (N
)
Homogeneous materialHeterogeneous material
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80
2
4
6
8
10
12
14
Displacement (mm)
Load
(N)
Mean
Load vs Displacement SD Load vs Displacement
Dominant effect of material heterogeneity on response statistics
UNCERTAINTY DUE TO MATERIAL HETEROGENEITY
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INFORMATION THEORETIC FRAMEWORK
Wavelet basis ( )ba,a,b are scale and space parameters
Wavelet coefficientsat macro scale
Wavelet coefficientsat meso scale
Correlation kernelsat macro scale
Correlation kernels based on intrascale mutual
information criterion
Information filteringbased on Renyi’s entropyand 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.
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IDEA BEHIND AN INFORMATION THEORETIC APPROACHIDEA BEHIND AN INFORMATION THEORETIC APPROACH
Statistical Mechanics
InformationTheory
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
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Information-Theoretic Methods for Multilength Scale Modeling
Source information
micro scale
macro scale
Wavelet basedcoding
of parameters
Decoding of
waveletparameters
InformationTheoretic upscalingof waveletcoefficients
InformationUpscaling 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
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NEED FOR WAVELETS
A very useful tool in areas where a multiscale analysis is important. Could be used as a tool to quantify quantify informationinformation of physical parameters of interest. Very useful for such analyses because it is mathematically compact and consistent
Fo
F1Q1Fo
Q2FoF2
Q3Fo
Fn
Micro scale
Meso scale
a,b: wavelet coefficients at scale a and
spatial location b.
Wavelets as a multiscale toolWavelets 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
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HOMOGENIZATION IN WAVELET SPACES
Full microstructure information
Homogenized properties at next scale
Complete homogenization
WaveletBasis
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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
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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.
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INFORMATION AND WAVELET MEASURES
, ,
,
,
( ) log( ( ))
( , ) (1 ( ))1
1( , ) log( ( ))
1
a a b a bb
qa a b
b
a a bb
h p w p w
kh k q p w
q
h R p w
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
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INFORMATION THEORETIC DOWNSCALING
Averaged velocity gradient
Variations acrossaveraged 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
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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)
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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.
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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
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RECONSTRUCTION SCHEMESRECONSTRUCTION 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 OptimizationScheme 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
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ALGORITHMS USED FOR MAXENTALGORITHMS 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
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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 tochecked microstructures
Comparisonof path
functions for a simulated
microstructure with circular
inclusions of the
second phase
MAXENTThe 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 Comparisonof path
functions for a simulated
microstructure with circular
inclusions of the second
phase
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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
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Applications of information theory with multiscale methods
Information 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
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INFORMATION LEARNING
1( ) ( ) ( ( ))R e eV e f e E f e
( ) Rj
j
VF e
w
Information Force
( )
(0)R
normR
V eV
V
Normalized Information Potential
Information Potential
1R
k kj
Vw w
w
Learning System
Y q X W =Input Signal Output Signal
X Y
Desired Signal D
OptimizationInformation Measure
I Y D
Basis Microstructures
Desired Macroscale entitiesLinsker’s maximum Mutual Information
Mutual information between desired signal and output signal should be maximized
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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
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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
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CCOORRNNEELLLL U N I V E R S I T Y
Training samples
ODF
Image
Pole figures
STATISTICALLEARNING TOOLBOX
Functions:1. Classification
methods2. 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 datawith a class;
update classesProcesscontroller
STATISTICAL LEARNING TOOLBOXSTATISTICAL LEARNING TOOLBOX
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APPLICATION: MICROSTRUCTURE RECONSTRUCTION
vision
Database
2D Imaging techniques
MicrostructureAnalysis
(FEM/Bounding theory)
Feature extraction
Pattern recognition Microstructure
evolution models
Process
Reverse engineerprocess parameters
3D realizations
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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
grai
ns
Quantitative approach
1.41.4 2.62.6 4.04.0 0.90.9 ……....
Microstructure represented by a set of numbers
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LOWER ORDER DESCRIPTOR BASED RECONSTRUCTIONLOWER 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
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REQUIREMENTS OF A REPRESENTATION SCHEMEREQUIREMENTS 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 class
2.72.7 3.63.6 1.21.2 0.10.1 ……....
8.48.4 2.12.1 5.75.7 1.91.9 ……....
Must differentiate other cases: (must be statistically representative)
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Microstructure Representation: PRINCIPAL COMPONENT ANALYSISMicrostructure Representation: PRINCIPAL COMPONENT ANALYSIS
Let be n images.
1. Vectorize input images2. Create an average image
3. Generate training images
1 2 n, ,.....
1
1=
n
iin
i i 4. Create correlation matrix (Lmn)
5. Find eigen basis (vi) of the correlation matrix
6. Eigen faces (ui) are generated from the basis (vi) as
7. Any new face image ( ) can be transformed to eigen face components through ‘n’ coefficients (wk) as,
Tmn m nL
i i iLv v
i ij ju v
( )Tk ku
Representation coefficients
Reduced basis
Data Points
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PCA REPRESENTATION OF MICROSTRUCTURE – AN EXAMPLEPCA REPRESENTATION OF MICROSTRUCTURE – AN EXAMPLE
Eigen-microstructures
Input Microstructures
Representation coefficients (x 0.001)
Image-1 quantified by 5 coefficients over the eigen-microstructures
0.0125 1.3142 -4.23 4.5429 -1.6396
-0.8406 0.8463 -3.0232 0.3424 2.6752
3.943 -4.2162 -0.6817 -9718 1.9268
1.17961.1796 -1.3354-1.3354 -2.8401-2.8401 6.20646.2064 -3.2106-3.2106
5.82945.8294 5.22875.2287 -3.7972-3.7972 -3.6095-3.6095 -3.6515-3.6515Basis 5
Basis 1
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EIGEN VALUES AND RECONSTRUCTION OVER THE BASISEIGEN 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
4 23 1
Reconstruction of microstructures over fractions of the basis
Significant eigen values capture most of the image features
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INCREMENTAL PCA METHODINCREMENTAL 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 10th microstructure is based on a PCA of 10 x 1 coefficient vectors instead of a 16384 x 1 size microstructures.
Updated BasisNewly added data point
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DYNAMIC MICROSTRUCTURE LIBRARY: CONCEPTSDYNAMIC 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
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BENEFITSBENEFITS
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.
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DIGITIZATIONDIGITIZATION
Conversion of RGB format of Conversion of RGB format of *.bmp file to a 2D image matrix*.bmp file to a 2D image matrix
PREPROCESSINGPREPROCESSING
Brings the image to the library Brings the image to the library formatformat
(RD : x-axis, TD : y-axis)(RD : x-axis, TD : y-axis)– Rotate and scale imageRotate and scale image– Image enhancement stepsImage enhancement steps– Boundary detection for Boundary detection for
feature extractionfeature extraction
Inputs: Microstructure Image (*.bmp Format), Magnification , Rotation (With respect to rolling direction)
Preprocessing based on user inputs of magnification and rotation
PREPROCESSINGPREPROCESSING
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ROSE OF INTERSECTIONS FEATURE – ALGORITHM (Saltykov, 1974)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
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CCOORRNNEELLLL U N I V E R S I T Y
GRAIN SHAPE FEATURE: EXAMPLESGRAIN SHAPE FEATURE: EXAMPLES
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
GRAIN SIZE PARAMETERGRAIN 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.
GRAIN SIZE FEATURE: EXAMPLESGRAIN SIZE FEATURE: EXAMPLES
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
SUPPORT VECTOR MACHINES: A BINARY CLASSIFIERSUPPORT VECTOR MACHINES: A BINARY CLASSIFIER
Find w and b such that
is maximized and for all (xi ,yi)
w . xi + b ≥ 1 if yi=1; w . xi + b ≤ -1 if yi = -1
2s
w
Support Vectors
Margin ( )
w.xi + b > 1
w.xi + b < -1
Class – I feature (y = 1) Class – II feature (y = -1)
2
w
Class Labels (Supervised classifier)
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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 2
1
1( , )
2
n
jj
J w w C
Relax constraints
w . xi + b ≥ 1- if yi=1; w . xi + b ≤ -1+ if yi = -1i i i
j
BETTER CLASSIFIERSBETTER CLASSIFIERS
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
SVM MULTI-CLASS CLASSIFICATIONSVM MULTI-CLASS CLASSIFICATION
Class-AClass-B
Class-CA
CB
AB
C
p = 3One 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 ( 1)
2
p p
( 1)
2
p p
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CCOORRNNEELLLL U N I V E R S I T Y
SVM TRAINING FORMAT
CLASSIFICATION SUCCESS %
Total Total imagesimages
Number of Number of classesclasses
Number of Number of Training imagesTraining images
Highest Highest success ratesuccess rate
Average Average success ratesuccess rate
375375 1111 4040 95.8295.82 92.5392.53
375375 1111 100100 98.5498.54 95.8095.80
ClassClass Feature Feature numbernumber
Feature Feature valuevalue
Feature Feature numbernumber
Feature Feature valuevalue
11 11 23.3223.32 22 21.5221.52
22 11 24.1224.12 22 31.5231.52
Data point
GRAIN FEATURES: GIVEN AS INPUT TO SVM TRAINING ALGORITHM
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CCOORRNNEELLLL U N I V E R S I T Y
CLASS HIERARCHYCLASS HIERARCHY
Class –2Class –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
IPCA QUANTIFICATION WITHIN CLASSESIPCA QUANTIFICATION WITHIN CLASSES
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Class-j Microstructures (Equiaxial grains, medium grain size)
Class-i Microstructures (Elongated 45 degrees, small grain size)
Representation Matrix
Image -1 Image-2 Image-3…
Component in basis vector 1
123 23 38
2 91 54 -85
3 -54 90 12
Average Image
21 23 24…
Eigen Basis
0.9 0.84 0.23..
0.54 0.21 0.74..
The Library – Quantification and image representation
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CCOORRNNEELLLL U N I V E R S I T Y
REPRESENTATION FORMAT FOR MICROSTRUCTUREREPRESENTATION FORMAT FOR MICROSTRUCTURE
Improvement of microstructure representation due to classificationImprovement 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
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CCOORRNNEELLLL U N I V E R S I T Y
Reconstruction Of Polyhedral MicrostructureReconstruction 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
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CCOORRNNEELLLL U N I V E R S I T Y
Stereological Distributions (Geometrical)Stereological Distributions (Geometrical)
3D reconstruction2D grain profile
3D grain
3D grain size distribution based on assumption that particles are randomly oriented cubes ( )3 / 2b
0
[1 ( )] (1 ( )) ( )a a u v vN F s bu G s N dF u
Na,Fa(s) : density of grains and grain size distribution in 2D image
Nv,Fv(u) : density of grains and grain size distribution in 3D microstructure
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Two Phase Microstructure: Class HierarchyTwo 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
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CCOORRNNEELLLL U N I V E R S I T Y
Example: 3D Reconstruction Using SVMSExample: 3D Reconstruction Using SVMS
Ag-W composite (Umekawa 1969) A reconstructed 3D microstructure
3 point probability function
Autocorrelation function
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CCOORRNNEELLLL U N I V E R S I T Y
Microstructure Property EstimationMicrostructure Property Estimation
170
190
210
230
250
270
290
310
0 200 400 600 800 1000Temperature (deg-C)
You
ngs
Mod
ulus
(G
Pa)
HS boundsBMMP boundsExperimentalFEM
3D image derived through pattern recognition
Experimental image
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CCOORRNNEELLLL U N I V E R S I T Y
Microstructure Representation Using SVM & PCAMicrostructure 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
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CCOORRNNEELLLL U N I V E R S I T Y
Quantification using incremental PCA
Input Image
Classifier
Feature Detection
Dynamic Microstructure Library
Identify and add new classes
Employ lower-order features
Pre-processing
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
PCA Microstructure RepresentationPCA 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)
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CCOORRNNEELLLL U N I V E R S I T Y
DATABASE FOR POLYCRYSTAL MATERIALS
Statistical Learning
Feature Extraction
Reduced order basis generation
Multi-scale microstructure
evolution models
Process design for desired properties
RD
R-v
alue
0.985
0.99
0.995
1
1.005
1.01
1.015
1.02
1.025
1.03
1.035
0 10 20 30 40 50 60 70 80 90
Angle from rolling direction
InitialIntermediateOptimalDesired
TD
Process Process parameters Values ..Tension Strain rate, time 0.56Forging Forging velocity ,Initial Temperature 2.13
Meso-scale database COMPONENTS
ODF TD
You
ngs
Mod
ulus
RD0 20 40 60 80
144
144.1
144.2
144.3
144.4
144.5
144.6
144.7
Database
Divisive Clustering
Class hierarchies
Class PredictionDatabase
Tension process basis
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
DESIGNING MATERIALS WITH TAILORED PROPERTIES
Micro problem driven by the velocity gradient L
Macro problem driven by the macro-design
variable βBn+1
Ω = Ω (r, t; L)~Polycrystal
plasticityx = x(X, t; β)
L = L (X, t; β)ODF: 1234567
L = velocity gradient
Fn+1
B0
Reduced Order Modes
Data mining techniques
Multi-scale Computation
Design variables (β) are macrodesign variables Processing sequence/parameters
Design objectives are micro-scale
averaged material/processproperties
Database
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CCOORRNNEELLLL U N I V E R S I T Y
FEATURES OF AN ODF: ORIENTATION FIBERS
1(
1 .r h y+ (h+y))
h y
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 FigurePoint y (1,0,1)
0 0
h||y
R.h=h, h||y
1P(h,y) = (P (h,y)+P (-h,y))
21
P(h,y) = 2
Ad
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.
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CCOORRNNEELLLL U N I V E R S I T Y
SIGNIFICANCE OF ORIENTATION FIBERS
Uniaxial (z-axis) Compression Texture
z-axis <110> fiber BB’
z-axis <100> fiber AA’
z-axis <111> fiber CC’
During deformation, Transport of crystals is
structured relative to orientation fiber
families
Important fiber families: <110> : uniaxial compression, plane strain compression and simple shear.
<111>: Torsion, <100>,<411> fibers: Tension
fiber (ND <110> ) & 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
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
LIBRARY FOR TEXTURES
[110] fiber family
DATABASE OF ODFsUni-axial (z-axis) Compression Texture
z-axis <110> fiber (BB’)
Feature:
fiber path corresponding to crystal direction h and sample direction y
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
SUPERVISED CLASSIFICATION USING SUPPORT VECTOR MACHINES
Given ODF/texture
Tension (T)
Stage 1
LEVEL – 2 CLASSIFICATIONPlane strain compression
T+P
LEVEL – I CLASSIFICATIONTension identified
Sta
ge 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
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CCOORRNNEELLLL U N I V E R S I T Y
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.
21 2
21,..,1
1( , ,.., ) ( )
2minn
k hi
h ki
J c c c x C
Identify clusters
Clusters
DATABASE OF ODFs
Feature Space
Cost function Each class is affiliated with multiple processes
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
File index Process Description Number of parameters Process parameters Values ---------->1 Tension 2 (Strain rate, time, velocity gradient) 1 0.12 Plane Strain Compression 2 (Strain rate, time, velocity gradient) 1 0.43 Forging 7 (Forging velocity ,Time,Initial Temperature ) 1 -0.2
Data-mining for Process information with ODF Classification
ODF 2,12,32,97 One ODF, several process paths
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CCOORRNNEELLLL U N I V E R S I T Y
PROCESS PARAMETERS LEADING TO DESIRED PROPERTIESY
oung
’s M
odul
us (
GP
a)
Angle from rolling direction
CLASSIFICATION BASED ON PROPERTIES
Class - 1 Class - 2
Class - 3Class - 40.5 0.25 0
0.25 -1.25 00 0 0.75
0.5 0 00 0.75 00 0 -1.25
Velocity Gradient
Different processes, Similar properties
Database for ODFs
Property Extraction
ODF Classification
Identify multiple solutions
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CCOORRNNEELLLL U N I V E R S I T Y
K-MEANS ALGORITHM FOR UNSUPERVISED CLASSIFICATION
•User needs to provide ‘k’, the number of clusters.
( )
( )
1 2( ), 1,..,
( ) ( )( ) 1
2
( ) 0 (for a minimum)
Thus at a minimum, ( , ,.., )
x c
x c
x c
x c x cc
c c
x c
c mean x x x
i j
i j
i j
Ti j i j
clusterj
j j
Ti j
cluster
j ncluster i n
J
Lloyds Algorithm:
1. Start with ‘k’ randomly initialized centers
2. Change encoding so that xi 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
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CCOORRNNEELLLL U N I V E R S I T Y
A TWO-STAGE PROBLEMA 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
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CCOORRNNEELLLL U N I V E R S I T Y
PROCESS DESIGN WITH A FIXED BASISPROCESS 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
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CCOORRNNEELLLL U N I V E R S I T Y
ADAPTIVE REDUCED-ORDER MODELINGADAPTIVE 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
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CCOORRNNEELLLL U N I V E R S I T Y
MULTIPLE PROCESS ROUTESMULTIPLE PROCESS ROUTES
0 10 20 30 40 50 60 70 80 90144
144.5
145
145.5
Angle from the rolling direction
You
ngs
Mod
ulus
(G
Pa)
Desired Young’s Modulus distribution
Magnetic hysteresis loss distribution
0 10 20 30 40 50 60 70 80 901.205
1.21
1.215
1.22
1.225
1.23
1.235
1.24
Ma
gn
etic
hys
tere
sis
loss
(W
/kg
)
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
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CCOORRNNEELLLL U N I V E R S I T Y
DESIGN FOR DESIRED ODF: A MULTI STAGE PROBLEMDESIGN FOR DESIRED ODF: A MULTI STAGE PROBLEM
0 5 10 15 20 250
0.2
0.4
0.6
0.8
1
Iteration Index
No
rma
lize
d o
bje
ctiv
e fu
nct
ion
Initial guess, = 0.65, = -0.1
Desired ODF Optimal- Reduced order control
Full order ODF based on reduced order control parameters
Stage: 1 Plane strain compression ( = 0.9472)
Stage: 2 Compression ( = -0.2847)
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CCOORRNNEELLLL U N I V E R S I T Y
DESIGN FOR DESIRED MAGNETIC PROPERTYDESIGN FOR DESIRED MAGNETIC PROPERTY
Iteration Index
No
rma
lize
d o
bje
ctive
fu
nctio
n
5 10 150
0.2
0.4
0.6
0.8
1
h
Crystal <100> direction.
Easy direction of
magnetization – zero power
loss
External magnetization direction
0 20 40 60 80
1.21
1.215
1.22
1.225
1.23
1.235
Angle from the rolling direction
Ma
gn
etic
hys
tere
sis
loss
(W
/Kg
) Desired property distributionOptimal (reduced)Initial
Stage: 1 Shear – 1 ( = 0.9745)
Stage: 2 Tension ( = 0.4821)
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CCOORRNNEELLLL U N I V E R S I T Y
DESIGN FOR DESIRED YOUNGS MODULUSDESIGN 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 <C> over the ODF as,
0 10 20 30 40 50 60 70 80 90143.6
143.8
144
144.2
144.4
144.6
144.8
145
145.2
145.4
Angle from the rolling direction
Yo
un
gs
Mo
du
lus
(GP
a)
Desired property distributionInitialOptimal (reduced)
1 2 3 4 5 6 7 80.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Iteration Index
Nor
mal
ized
obj
ect
ive
func
tion
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CCOORRNNEELLLL U N I V E R S I T Y
MULTISCALE DATA MINING –MICRO/MESO SCALEMULTISCALE DATA MINING –MICRO/MESO SCALE
Constitutive laws, Microstructure-dependent properties through bounding theories and FEM
170
190
210
230
250
270
290
310
0 200 400 600 800 1000Temperature (deg-C)
Yo
un
gs
Mo
du
lus
HS boundsBMMP boundsExperimentalFEM
Phase field model Dislocation dynamics
Microstructure Morphology
Properties of individual phases and crystals
LEVEL - 1
0 5 100
0.5
1
0 10 20 30 400
0.05
0.1
3 point probability
Microstructure Class Hierarchy
3D Microstructures
Meso-scale database
Data-mining
Exp
ande
d vi
ew o
f the
mes
o-sc
ale
data
base Model reduction
Autocorrelation
Statistical learning
To stochastic continuum models
Data from DFT
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CCOORRNNEELLLL U N I V E R S I T Y
Electron scale databaseAlloy systems
DFT
Phase FieldDD
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
Objective Design decisions
Hyperplanes quantify correlation of local length scale features with the objective and higher length scale effects
MATERIAL FEATURE REPRESENTATION AND DESIGNMATERIAL FEATURE REPRESENTATION AND DESIGN
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CCOORRNNEELLLL U N I V E R S I T Y
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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CCOORRNNEELLLL U N I V E R S I T Y
DESIGNING ALLOYS THROUGH STATISTICAL LEARNING
Meshing and virtual experimentation (OOF)
Property statistics
Phase field modelThermodynamic
variables (CALPHAD) Mobilities
Interfacial energiesNucleation
Models
Design problems: 1) Determine the compositions that give optimum propertiescompositions that give optimum properties 2) Design process sequences to obtain desired properties
Diffusion coefficients