Materials Process Design and Control Laboratory Veera Sundararaghavan and Prof. Nicholas Zabaras...

Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

CCOORRNNEELLLL U N I V E R S I T Y


Veera Sundararaghavan and Prof. 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], [email protected] URL: http://mpdc.mae.cornell.edu/

STATISTICAL LEARNING METHODS FOR MICROSTRUCTURES

mailto:[email protected]

mailto:[email protected]




WHAT IS STATISTICAL LEARNING

Statistical learning is all about automating the process of searching for patterns from large Statistical learning is all about automating the process of searching for patterns from large scale statistics.scale statistics.

Which patterns are interesting?Which patterns are interesting?

Mathematical techniques for associating input data with desired attributes and identifying Mathematical techniques for associating input data with desired attributes and identifying correlationscorrelations

A powerful tool for designing materialsA powerful tool for designing materials




FOR MICROSTRUCTURES?FOR MICROSTRUCTURES?

Properties of a material are affected by the underlying microstructure

• Microstructural attributes related to specific properties

•Examples: Correlation functions -> Elastic moduli

Orientation distribution ->Yield stress in polycrystals

• Attributes evolve during processing (thermo mechanical, chemical, solidification etc.)

• Can we identify specific patterns in these relationships?

• Is it possible to probabilistically predict the best microstructure and the best processing paths for optimizing properties based on available structural attributes?




TERMINOLOGYTERMINOLOGY

Microstructure can be represented in terms of typical attributesMicrostructure can be represented in terms of typical attributesExamples are volume fractions, probability functions, shape/size attributes, Examples are volume fractions, probability functions, shape/size attributes,

orientation of grains, cluster functions, lineal measures and so onorientation of grains, cluster functions, lineal measures and so on

All these attributes affect physical propertiesAll these attributes affect physical properties

Attributes evolve during processing of a microstructureAttributes evolve during processing of a microstructure

Attributes are represented in a discrete (vector) form as Attributes are represented in a discrete (vector) form as ‘features’‘features’

‘‘features’ are represented as a vector xfeatures’ are represented as a vector xkk, k = 1,…,n, k = 1,…,n where n is where n is

the dimensionality of the featurethe dimensionality of the feature

Every different feature is represented as Every different feature is represented as xxkk(i)(i) where superscript where superscript

denotes the denotes the iithth feature feature that we are interested in that we are interested in




TERMINOLOGYTERMINOLOGY

Given a data set of computational or experimental Given a data set of computational or experimental microstructures, can we learn the functional differences microstructures, can we learn the functional differences

between them based on features?between them based on features?

We denote microstructures that are similar in attributes in We denote microstructures that are similar in attributes in terms of a class representation terms of a class representation ‘y’, y = 1..k‘y’, y = 1..k where k is where k is

number of classes. number of classes.

Classes are formed into hierarchies: Each level Classes are formed into hierarchies: Each level represented by feature represented by feature xx(i)(i)..

Structure based classes are affiliated with process and Structure based classes are affiliated with process and properties:properties: powerful tool for exploring complex powerful tool for exploring complex

microstructure design spacemicrostructure design space




APPLICATIONSAPPLICATIONS




quantification and mining associations

Input microstructure

Classifier

Feature Detection

MICROSTRUCTURE LIBRARIES FOR REPRESENTATION

Identify and add new classes

Employ lower-order features

Pre-processing

Sundararaghavan & Zabaras, Acta Materialia, 2004




MICROSTRUCTURE RECONSTRUCTION

vision

Database

2D Imaging techniques

MicrostructureAnalysis

(FEM/Bounding theory)

Feature extraction

Pattern recognition Microstructure

evolution models

Process

Reverse engineerprocess parameters

3D realizations

Sundararaghavan and Zabaras, Computational Materials Sci, 2005




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




DESIGNING PROCESSES FOR MICROSTRUCTURES

Process sequence-1

Process parameters

ODF history

Reduced basis

Process sequence-2

New process parameters

ODF history

Reduced basis

Classifier Adaptive basis selection

Optimization

Reduced basisProcess

Probable Process

sequences & Initial parameters

Desired texture/propert

y

Stage - 1 Stage - 2

New dataset added

DATABASE

Optimum parameters

Sundararaghavan and Zabaras, Acta Materialia, 2005




THIS LECTURE WILL COVER….THIS LECTURE WILL COVER….

•This lecture we will try to go into the math behind statistical learning and learn two really useful techniques – Support Vector Machines and Bayesian Clustering.

•Applications to microstructure representation, reconstruction and process design will be shown

• We will skim over the physics and some important computational tools behind these problems




STATISTICAL LEARNING TECHNIQUESSTATISTICAL LEARNING TECHNIQUES

Regressor Prediction ofreal-valued output

InputAttributes

DensityEstimator

ProbabilityInput

Attributes

Classifier Prediction ofcategorical output

InputAttributes





InputAttributes

DensityEstimator

ProbabilityInput

Attributes


InputAttributes

This lecture






InputAttributes

DensityEstimator

ProbabilityInput

Attributes


InputAttributes

This lecture

Function approximation: Useful for prediction in regions that are computationally unreachable (not covered in this lecture)





PRELIMINARIES OF SUPERVISED CLASSIFIERSPRELIMINARIES OF SUPERVISED CLASSIFIERS

Classifiery

x

Decision

Function

y = w.f(x)+bMicrostructure classes eg. based on a property

Microstructure features

denotes +1

denotes -1Two class problem: The classes for the test specimens are known apriori

Aim: To predict the strength of a new microstructureVolume fraction

Po

re d

ensi

ty

High strength

Low strength




SUPPORT VECTOR MACHINESSUPPORT VECTOR MACHINES

denotes +1

denotes -1

f(x,w,b) = sign(w. x - b)

How would you classify this data?




OCCAM’S RAZOROCCAM’S RAZOR

plurality should not be assumed without necessity William of Ockham, Surrey (England) 1285-1347 AD, theologian

•Simpler models are more likely to be correct than complex ones•Nature prefers simplicity. •principle of uncertainty maximization





denotes +1

denotes -1


How would you classify this data?





denotes +1

denotes -1


Any of these would be fine..

..but which is best?





denotes +1

denotes -1


Define the margin of a linear classifier as the width that the boundary could be increased by before hitting a datapoint.





denotes +1

denotes -1


The maximum margin linear classifier is the linear classifier with the, um, maximum margin.

This is the simplest kind of SVM (Called an LSVM)Linear SVM

Support Vectors are those datapoints that the margin pushes up against





• Plus-plane = Plus-plane = { { xx : : ww . . xx + b = +1 } + b = +1 }

• Minus-plane = Minus-plane = { { xx : : ww . . xx + b = -1 } + b = -1 }

• The vector The vector ww is perpendicular to the Plus Plane. Why? is perpendicular to the Plus Plane. Why?

“Predict Class

= +1”

zone

“Predict Class

= -1”

zonewx+b=1

wx+b=0

wx+b=-

1

M = Margin Width

How do we compute M in terms of w and b?

Let u and v be two vectors on the Plus Plane. What is w . ( u – v ) ?

And so of course the vector w is also perpendicular to the Minus Plane

Claim: Claim: xx++ = = xx-- + + ww for for some value of some value of . Why?. Why?

x+

x-





• What we know:What we know:

• ww . . x+x+ + b = +1 + b = +1

• ww . . x-x- + b = -1 + b = -1

• x+x+ = = x-x- + + ww

• ||x+x+ - - x-x- | | = M= M

• It’s now easy to get It’s now easy to get MM in terms of in terms of ww and and bb

“Predict Class

= +1”

zone

“Predict Class

= -1”

zonewx+b=1

wx+b=0

wx+b=-

1

M = Margin Width

w . (x - + w) + b = 1

=>

w . x - + b + w .w = 1

=>

-1 + w .w = 1

=>

x-

x+Computing the margin widthComputing the margin width





Learning the Maximum Margin Learning the Maximum Margin ClassifierClassifier

“Predict Class

= +1”

zone

“Predict Class

= -1”

zonewx+b=1

wx+b=0

wx+b=-1

M =

MinimizeMinimize w.ww.wWhat are the constraints?What are the constraints?

ww . . xxkk + b >= 1 if y + b >= 1 if ykk = 1 = 1

ww . . xxkk + b <= -1 if y + b <= -1 if ykk = -1 = -1





denotes +1

denotes -1

This is going to be a problem!

What should we do?

Minimize w.w + C (distance of error points to their correct place)





wx+b=1

wx+b=0

wx+b=-1

M =

1

1.

2

R

kk

C ε

w w

7

11 2

Constraints?Constraints?w . xk + b >= 1-k if yk = 1

w . xk + b <= -1+k if yk = -1

k >= 0 for all k

MinimizeMinimize





Harder 1-dimensional datasetHarder 1-dimensional dataset

What can be done about this?

x=0





x=0

Quadratic Basis Quadratic Basis FunctionsFunctions




SUPPORT VECTOR MACHINES WITH KERNELSSUPPORT VECTOR MACHINES WITH KERNELS

1

1.

2

R

kk

C ε

w w

Constraints?Constraints?w . (xk)+ b >= 1-k if yk = 1

w . (xk)+ b <= -1+k if yk = -1

k >= 0 for all k

MinimizeMinimize

Φ: x → φ(x)




SUPPORT VECTOR MACHINES: QUADRATIC PROGRAMMINGSUPPORT VECTOR MACHINES: QUADRATIC PROGRAMMING

Datapoints with k > 0 will be the support vectors

Maximize

where

Subject to these constraints:

Then define:

Then classify with:

f(x,w,b) = sign(w. (x) - b)Φ




MULTIPLE CLASSESMULTIPLE CLASSES

Class-AClass-B

Class-CA

CB

AB

C

Given a new microstructure with its ‘s’ features given by

find the class of 3D microstructure (y ) to which it is most likely to belong.

[1,2,3,..., ]p1 2

1 1 1 2 2 21 1 2 2 1 2 1 2{ , ,...., }, { , ,...., },..., { , ,...., }

s

T T T s s sm m s mx x x x x x x x x x x x

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




MULTIPLE FEATURESMULTIPLE FEATURES

Class - 1

3D Microstructures3D Microstructures

Class - 2

FEATURE – 1: GRAIN SHAPE

FEATURE – 2 GRAIN SIZES

Class - 1

Class - 2

Class - 3

Class - 4

Rose of intersectionsHeyn int. Histogram

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HIERARCHICAL LIBRARIES – (a.k.a) DIVISIVE CLUSTERING




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




QUANTIFICATION OF DIVERSE MICROSTRUCTURE

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




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.




PRINCIPAL COMPONENT ANALYSISPRINCIPAL 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 microstructures (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




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)




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




EIGEN VALUES AND RECONSTRUCTION OVER THE BASISEIGEN VALUES AND RECONSTRUCTION OVER THE BASIS

1.Reconstruction with 100% basis

2. Reconstruction with 80% basis



4 23 1

Reconstruction of microstructures over fractions of the basis

Significant eigen values capture most of the image features




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




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




GRAIN SHAPE FEATURE: EXAMPLESGRAIN SHAPE FEATURE: EXAMPLES




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







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




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




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




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




MICROSTRUCTURE REPRESENTATION USING SVM & PCAMICROSTRUCTURE REPRESENTATION USING SVM & PCA

COMMON-BASIS FOR MICROSTRUCTURE REPRESENTATION

Does not decay to zero

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




PCA MICROSTRUCTURE RECONSTRUCTIONPCA MICROSTRUCTURE RECONSTRUCTION

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)




MOTIVATIONMOTIVATION

1. Creation of 3D microstructure models from 2D images

2. 3D imaging requires time and effort. Need to address real–time methodologies for generating 3D realizations.

3. Make intelligent use of available information from computational models and experiments.

vision

Database

Pattern recognition

MicrostructureAnalysis

2D Imaging techniques




LITERATURE: STOCHASTIC MICROSTRUCTURE RECONSTRUCTION

Methods available are optimization based: Features of 2D image are matched to that of a 3D microstructure by posing an optimization problem.

1) Does not make use of available information (experimental/simulated data)

2) Cannot perform reconstructions in real-time.

Need to take into account the processes that create these microstructure (Oren and Bakke, 2003) for correctly modeling the geometric connectivity.

Key assumptions employed for 3D image reconstruction from a single 2D image

Randomness Assumption (Ohser and Mucklich – 2000).

1. Grains in a polyhedral microstructure are assumed to be of the similar shapes but of different sizes.

2. Two phase microstructures can be characterized using rotationally-invariant probability functions




PATTERN RECOGNITION (PR) STEPSPATTERN RECOGNITION (PR) STEPS

DATABASE CREATION

FEATURE EXTRACTION

TRAINING

PREDICTION

Datasets: microstructures from experiments or physical models

Extraction of statistical features from the database

Creation of a microstructure class hierarchy: Classification methods

Prediction of 3D reconstruction, process paths, etc,

PATTERN RECOGNITION : A DATA-DRIVEN OPTIMIZATION TOOL•Feature matching for reconstruction of 3D microstructures

Real-time




POLYHEDRAL MICROSTRUCTURES: MC MODELPOLYHEDRAL MICROSTRUCTURES: MC MODEL

1

( )

1

(1 )

s

n

i j

N

ii

N i

i s sj

H J H

H

Potts Hamiltonian (H)Algorithm (1 Monte Carlo Step): • Calculation of the free energy of a randomly selected node (Hi)• Random choice of a new crystallographic orientation for the node• New calculation of the free energy of the element (Hf)• The orientation that minimizes the energy (min(Hf,Hi)) is chosen.

Ns: Total No. of nodes

Nn(i) : No. of neighbors of node ‘i’

Microstructure Database

Classes of microstructures based on grain size feature




POLYHEDRAL MICROSTRUCTURES : GRAIN SIZE FEATUREPOLYHEDRAL MICROSTRUCTURES : GRAIN SIZE FEATURE

Intercept lengths of parallel network of lines with the grain boundaries are recorded at several anglesThe intercept length (x-axis) versus number of lines (y-axis) histogram is the measure of grain size (Heyn intercept histogram).

Slice




FEATURE BASED CLASSIFICATIONFEATURE BASED CLASSIFICATION

Class - 1

3D Microstructures3D Microstructures

Class - 2

LEVEL - 1 LEVEL - 2

Class - 1

Class - 2

Class - 3

Class - 4

Rose of intersectionsHeyn int. Histogram

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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




STEREOLOGICAL ESTIMATES OF 3D GRAIN SIZESSTEREOLOGICAL ESTIMATES OF 3D GRAIN SIZES

The stereological integral equation for estimating the 3D grain size distribution from a 2D image for polyhedral microstructures

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

: rotation average of the size of a particle with maximum size = 1

Gu(s): Size distribution function of the section profiles under the condition that a random size ‘U’ equals the 3D particle mean size (u).

Remark: Sizes are defined as the maximum calliper diameter of a grain

b

Numerical Scheme:

Let

Then

: Mean number/volume of grains with size ui

Let yk be the mean number per unit area of section profiles with size between [sk-1,sk]

And let ui = ai and sk = ak, then, y = Pwhere

P is a matrix formed from a set of coefficients ( based onthe shape assumption of grains

( , ) (1 ( ))up u s bu G s (1 ( )) ( , )a a i i iN F s p u s

i

1( ( ) ( ))k a a k a ky N F s F s

ii iba

0

[1 ( )] (1 ( )) ( )a a u v vN F s bu G s N dF u




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




STATISTICAL CORRELATION MEASURESSTATISTICAL CORRELATION MEASURES

MC Sampling: Computing the three point probability function of a 3D microstructure(40x40x40 mic)

S3(r,s,t), r = s = t = 2, 5000 initial points, 4 samples at each initial point.

Rotationally invariant probability functions (Si

N ) can be interpreted as the probability of finding the N vertices of a polyhedron separated by relative distances x1, x2,..,xN in phase i when tossed, without regard to orientation, in the microstructure.

When a voxel solidifies, liquid is expelled to its neighbors, creating solute concentration (ci,j,k) gradients. Movement of solute to minimize concentration gradients is modeled using fick’s law




MC MODEL FOR TWO-PHASE MICROSTRUCTURESMC MODEL FOR TWO-PHASE MICROSTRUCTURES

Weights (w) of neighbors

Face neighbors = 1

Edge neighbors = 1

2Solid voxels

Microstructure is represented using voxels.Probability of solidification (P) depends on1) Net weight (w) of the No. of neighbors of a solid voxel:• If w >= 8.6568: voxel solidifies (P = 1)• If 3.8284 < w < 8.6568, P = 0.1• If weight < 3.8284, the voxel remains liquid (P = 0)

2) The solute concentration: A linear probability distribution with P = 0 at critical concentration and P = 1 when concentration is 0.

Final state

Where (i,j,k) is a voxel coordinate, n is the time step and D is the diffusion coefficient




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




EXAMPLE: 3D RECONSTRUCTION USING SVMSEXAMPLE: 3D RECONSTRUCTION USING SVMS

Ag-W composite (Umekawa 1969) A reconstructed 3D microstructure

3 point probability function

Autocorrelation function




MICROSTRUCTURE ELASTIC PROPERTIESMICROSTRUCTURE ELASTIC PROPERTIES

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




WHAT IS MICROSTRUCTURE DESIGN

Initial microstructure

processing sequence? Final microstructure/

property

Microstructure?Known operating conditions

Known property limits

Initial Microstructure

Known operating conditions

Property?

Direct problem

Design problems

Use finite elements, experiments etc.

Design for best microstructure

Design for best processes




SUPERVISED VS UNSUPERVISED LEARNING

Supervised classification for design:

1. Classify microstructures based on known process sequence classes

2. Given a desired microstructure, identify the processing stages required through classification

3. Drawback: Identifies a unique process sequence, but we that find many processing paths to lead to similar properties!

UNSUPERVISED CLASSIFICATION

1. Identify classes purely based on structural attributes

2. Associate processes and properties through databases

3. Explores the structural attribute space for similarities and unearths non-unique processing paths leading to similar microstructural properties




K MEANSK MEANS

Suppose the coordinates of points drawn randomly from this dataset are transmitted.

You can install decoding software at the receiver.

You’re only allowed to send two bits per point.

It’ll have to be a “lossy transmission”.

Loss = Sum Squared Error between decoded coords and original coords.

What encoder/decoder will lose the least information?




K MEANSK MEANS

Idea OneIdea One

00

1110

01

Break into a grid, decode each bit-pair as the middle of each grid-cell

QuestionsQuestions

• What are we trying to What are we trying to optimize?optimize?

• Are we sure it will find Are we sure it will find an optimal clustering?an optimal clustering?

Break into a grid, decode each bit-pair as the centroid of all data in that grid-cell




K MEANSK MEANS

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.

1 2 2( )

1

( , ,.., ) ( )i

Rk

i encodei

J c c c

xx cCost Function =

OwnedBy( )

1

| OwnedBy( ) |j

j iij

c

c xc

Cost function minimized by

transmitting centroids




THE EXPECTATION-MAXIMIZATION (EM) ALGORITHMTHE EXPECTATION-MAXIMIZATION (EM) ALGORITHM

What properties can be changed for centers c1 , c2 , … , ck have when distortion is not minimized?

Expectation step: Compute expected centers

(1) Change encoding so that xi is encoded by its nearest center

Maximization step: Compute maximum likelihood values of centers

(2) Set each Center to the centroid of points it owns.

There’s no point applying either operation twice in succession.

But it can be profitable to alternate.

…And that’s K-means!

2ENCODE( )

1

Distortion ( )i

R

ii

xx c

EM algorithm will be dealt with later




K-MEANSK-MEANS

1. Ask user how many clusters they’d like. (e.g. k=5)




K-MEANSK-MEANS


2. Randomly guess k cluster Center locations




K-MEANSK-MEANS



3. Each datapoint finds out which Center it’s closest to. (Thus each Center “owns” a set of datapoints)




K-MEANSK-MEANS



3. Each datapoint finds out which Center it’s closest to.

4. Each Center finds the centroid of the points it owns




K-MEANSK-MEANS



3. Each datapoint finds out which Center it’s closest to.

4. Each Center finds the centroid of the points it owns…

5. …and jumps there

6. …Repeat until terminated!

often unknown (is dependent on the features used for microstructure representation)




SHORTCOMINGS OF K-MEANS AND REMEDIESSHORTCOMINGS OF K-MEANS AND REMEDIES

1) K-MEANS gives hyper-spherical clusters: Not always the case with data

2) Number of classes must be known apriori: Beats the reasoning for unsupervised clusters – we do not know anything about the classes in the data

3) May converge to local optima – not so bad

We will discuss about new strategies to get improved clusters of microstructural features

1) Gaussian mixture models and Bayesian clustering

2) Later, an improved k-means algorithm called X-means which uses a Bayesian information criterion




PROBABILITY PRELIMINARIESPROBABILITY PRELIMINARIES

• A is a Boolean-valued random variable if A denotes an event, and there is some degree of uncertainty as to whether A occurs.

Examples

• A = You win the toss

• A = Probability of failure of a structure

0 <= P(A) <= 1

P(True) = 1

P(False) = 0

P(A or B) = P(A) + P(B) - P(A and B)

P(~A) + P(A) = 1

P(B) = P(B ^ A) + P(B ^ ~A)

Discrete Random VariablesDiscrete Random Variables





P(A ^ B) P(A|B) = ----------- P(B)

P(A ^ B) = P(A|B) P(B)

P(A ^ B) P(A|B) P(B)P(A ^ B) P(A|B) P(B)

P(B|A) = ----------- = ---------------P(B|A) = ----------- = ---------------

P(A) P(A)P(A) P(A)

Corollary: The Chain Rule

Definition of Conditional ProbabilityDefinition of Conditional Probability

Bayes Rule





• MAP (Maximum A-Posteriori Estimator):predict

1 1argmax ( | )m mv

Y P Y v X u X u

predict1 1argmax ( | )m m

vY P X u X u Y v

What if Y = v itself is very unlikely?

• MLE (Maximum Likelihood Estimator):MLE (Maximum Likelihood Estimator):

Includes P(Y = v) information through Bayes rule (P(Y = v) is called as ‘prior’)

Class of data = argmaxi P(data | class = i)

Class of data = argmaxi P(class = i | data)





1 1

1 1

1 1

1 1

( | )

( | ) ( )

( )

1( | ) ( )

m m

m m

m m

m m

P Y v X u X u

P X u X u Y v P Y v

P X u X u

P X u X u Y v P Y vc

• MAP (Maximum A-Posteriori Estimator):predict

1 1argmax ( | )m mv

Y P Y v X u X u





Bayes Classifiers in a nutshellBayes Classifiers in a nutshell

predict1 1

1 1

argmax ( | )

argmax ( | ) ( )

m mv

m mv

Y P Y v X u X u

P X u X u Y v P Y v

1. Learn the distribution over inputs for each value Y.

2. This gives P(X1, X2, … Xm | Y=vi ).

3. Estimate P(Y=vi ). as fraction of records with Y=vi .

4. For a new prediction:




NAÏVE BAYES CLASSIFIERNAÏVE BAYES CLASSIFIER

predict1 1argmax ( | ) ( )m m

vY P X u X u Y v P Y v

In the case of the naive Bayes Classifier this can be simplified:

predict

1

argmax ( ) ( | )Yn

j jv j

Y P Y v P X u Y v

The independent features

assumption




Notation change:

The naïve Bayes classifier

0 ,( ) ( , ), ( | ) ( , )c i c i j c j i cP v p x v P x v p x v

New Bayes classifier

NAÏVE BAYES CLASSIFIER IS AN SVM?




Bayes classifier with feature weighting


wj = 1 (for naïve Bayes)

But, features may be correlated!

A two class classifier

Decision function given by the sign of fWBC given by

Class t

Class f





Class t

Class f

SVM classifier!

Feature space of a naïve Bayes classifier




INTRO TO BAYESIAN UNSUPERVISED CLASSIFICATIONINTRO TO BAYESIAN UNSUPERVISED CLASSIFICATION

( | ) ( )( | )

( )

p y i P y iP y i

p

xx

x

/ 2 1/ 2

1 1exp

(2 ) || || 2( | )

( )

T

k i i k i imi

p

P y ip

x μ Σ x μΣ

xx

Gaussian Mixture Models

Assume that each feature is generated as:

Pick a class at random. Choose class i with probability P(wi).

The feature is sampled from a Gaussian distribution : N(i, i )




GAUSSIAN MIXTURE MODELGAUSSIAN MIXTURE MODEL

1

3

• There are k components. The i’th component is called yi

• Component yi has an associated mean vector i

• Each component generates data from a Gaussian with mean i and covariance matrix i

2

1

Assuming features in each class can be modeled by a Gaussian distribution, identify the parameters (means,variances etc.) of the distributions

Probabilistic extension of K-MEANS





• We have x1 x2 … xn features of a microstructure

• We have P(y1) .. P(yk). We have σ.

• We can define, for any x , P(x|yi , μ1, μ2 .. μk)

• Can we define P(x | μ1, μ2 .. μk) ?

• Can we define P(x1, x2, .. xn | μ1, μ2 .. μk) ?





Given a guess at Given a guess at μμ11, , μμ22 .... μμ k,k,

We can obtain the probability of the unlabeled data given We can obtain the probability of the unlabeled data given those those μμ‘s.‘s.

Inverse Problem: Find Inverse Problem: Find ’s given the points x’s given the points x1,1,xx22,…x,…xkk

The normal max likelihood trick:

Set d log Prob (….) = 0

d μi

and solve for μi‘s.

Using gradient descent, Slow but doable

Use a much faster and recently very popular method…




EM ALGORITHM REVISITEDEM ALGORITHM REVISITED

We have unlabeled microstructural features We have unlabeled microstructural features xx11 xx22 … … xxRR

• We know there are k classesWe know there are k classes

• We know P(yWe know P(y11), P(y), P(y22), P(y), P(y33), …, P(y), …, P(ykk))

• We We don’tdon’t know know μμ11 μμ22 .. .. μμkk

• We can write P( data | We can write P( data | μμ11…. …. μμkk) )

1 1

11

111

2

211

p ... μ ...μ

p μ ...μ

p ,μ ...μ P

1K exp μ P

2σ

R k

R

i ki

R k

i j k jji

R k

i j jji

x x

x

x y y

x y

Maximize this likelihood





1

11

11

For Max likelihood we know log Pr ob data μ ...μ 0μ

Some algebra turns this into: "For Max likelihood, for each j,

,μ ...μμ

,μ ...μ

ki

R

j i k ii

j R

j i ki

P y x x

P y x

This is n nonlinear equations in μj’s.”

If, for each xi we knew that for each yj the prob that μj was in class yj is P(yj|xi,μ1…μk) Then… we would easily compute μj.

If we knew each μj then we could easily compute P(yj|xi,μ1…μj) for each yj and xi.





Iterate. On the Iterate. On the tt’th iteration let our estimates be’th iteration let our estimates be

{ { μμ11(t), (t), μμ22(t) … (t) … μμcc(t) }(t) }

• E-stepE-step

Compute “expected” classes of all datapoints for each classCompute “expected” classes of all datapoints for each class

2

2

1

p , ( ), ( )p , PP ,

p p , ( ), ( )

k i i ik i t i ti k t c

k tk j j j

j

x y t p tx y yy x

x x y t p t

I

IM-step.

Compute Max. like μ given our data’s class membership distributions

P , μ 1

P ,

i k t kk

ii k t

k

y x xt

y x

Just evaluate a Gaussian at xk




GAUSSIAN MIXTURE MODEL: DENSITY ESTIMATIONGAUSSIAN MIXTURE MODEL: DENSITY ESTIMATION

Features in 2D

Complex PDF of the feature space

Classification + Probabilistic quantification of results

Ambiguity + Anomaly detection – Very popular in Genome mapping




DATABASE FOR POLYCRYSTAL MICROSTRUCTURES

Statistical Learning

Feature Extraction

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

TDProcess Process parameters Values ..Tension Strain rate, time 0.56Forging Forging velocity ,Initial Temperature 2.13

Meso-scale database COMPONENTS

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 Prediction

Driven by distance based (or) Probabilistic clustering






Feature Extraction


evolution models


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





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 Prediction

Cluster based on similar microstructural features






Feature Extraction


evolution models


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





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 Prediction

Associate process/ property info from database

Cluster based on similar microstructural features




ORIENTATION DISTRIBUTION FUNCTION

Any macroscale property < χ > can be expressed as an expectation value if the corresponding single crystal property χ ( ,t) is known.

• Determines the volume fraction of crystals within a region R' of the fundamental region R• Probability of finding a crystal orientation within a region R' of the fundamental region• Characterizes texture evolution

ORIENTATION DISTRIBUTION FUNCTION – A(r,t)

– reorientation velocity

ODF EVOLUTION EQUATION – EULERIAN DESCRIPTION




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.




SIGNIFICANCE OF ORIENTATION FIBERS

Uniaxial (z-axis) Compression Texture

z-axis <110> fiber BB’

z-axis <100> fiber AA’

z-axis <111> fiber CC’

Predictable fiber

development

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

close affiliation with processes




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




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




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




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




PROCESS PARAMETERS LEADING TO DESIRED PROPERTIESY

oung

’s M

odul

us (

GP

a)


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




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




SCHWARZ CRITERION FOR IDENTIFYING NUMBER OF CLUSTERS

The number of clusters chosen maximizes the Bayesian information criterion given by:

Where is the is the log-likelihood of the data taken at the maximum likelihood point, p is the number of free parameters in the model

Maximum likelihood of the variance assuming Gaussian data distribution

Probability of a point in cluster i

Log-likelihood of the data in a cluster




CENTROID SPLIT TESTS

X-MEANS algorithm:

• Start with k clusters found through k-means algorithm

• Split each centroid into two centroids, and move the new centroids along a distance proportional to the cluster size in an arbitrarily chosen direction

• Run local k-means (k = 2) in each cluster

•Accept split cluster in each region if BIC(k = 1) < BIC(k = 2)

• Test for various initial values of ‘k’ and select the ‘k’ with maximum overall BIC

Split centers Run local k-means (k = 2) in each cluster New clusters based on BIC




COMPARISON OF K-MEANS AND X-MEANS

Local Optimum produced by the kmeans algorithm with k = 4

Cluster configuration produced by k-means with k = 6: Over-estimates the natural number of clusters

Configuration produced by the x-means algorithm: Input range of k = 2 to 15. x-means found 4 clusters from the data-set based on the Bayesian Information Criterion




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




LIMITATIONS OF STATISTICAL LEARNING BASED DESIGN SOLUTIONSLIMITATIONS OF STATISTICAL LEARNING BASED DESIGN SOLUTIONS

Classification alone does not yield the final design solution

• Why? Since it is impossible to explore the infinite design space within a database of reasonable size.

• Use statistical learning for providing initial class of solutions

• Use local optimization schemes (details not given in this presentation) to identify the exact solutions

Response surface

Ob

ject

ive

to b

e m

inim

ized

Microstructure attributes

Stat Learning Design solutions




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)




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


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)




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


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




WHAT WE SHOULD KNOW

How to “learn” microstructure/process/property relationships given computational and experimental data

•Be happy with probabilistic tools: Bayesian analytics and Gaussian mixture models

•Understand simple tools like K-MEANS that can be readily used.

•Understand SVMs as a versatile statistical learning tool: For both feature selection and classification

Apply statistical learning to perform real-time decisions under high degrees of uncertainty

Appreciate the uses and understand the limitations of statistical learning applied to materials




USEFUL REFERENCESUSEFUL REFERENCES

• Andrew Moore’s Statistical learning course online:

http://www-2.cs.cmu.edu/~awm/tutorials/

• Books:

R.O. Duda, P.E. Hart and D.G. Stork, Pattern classification (2nd ed), John Wiley and Sons, New York (2001).

Example papers on microstructure/materials related applications for the tools presented in this talk:

• V. Sundararaghavan and N. Zabaras, "A dynamic material library for the representation of single phase polyhedral microstructures", Acta Materialia, Vol. 52/14, pp. 4111-4119, 2004

• V. Sundararaghavan and N. Zabaras, "Classification of three-dimensional microstructures using support vector machines", Computational Materials Science, Vol. 32, pp. 223-239, 2005

• V. Sundararaghavan and N. Zabaras, "On the synergy between classification of textures and deformation process sequence selection", Acta Materialia, Vol. 53/4, pp. 1015-1027, 2005

• T J Sabin, C A L Bailer-Jones and P J Withers, Accelerated learning using Gaussian process models to predict static recrystallization in an Al–Mg alloy, Modelling Simul. Mater. Sci. Eng. 8 (2000) 687–706

•C. A. L. Bailer-Jones, H. K. D. H. Bhadeshia and D. J. C. MacKay, Gaussian Process Modelling of Austenite Formation in Steel, Materials Science and Technology, Vol. 15, 1999, 287-294.




THANK YOUTHANK YOU

Materials Process Design and Control Laboratory Veera Sundararaghavan and Prof. Nicholas Zabaras...

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Transcript of Materials Process Design and Control Laboratory Veera Sundararaghavan and Prof. Nicholas Zabaras...