materials process design and control laboratory an information-theoretic approach to multiscale...

Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

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/




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




DEFORMATION PROCESS DESIGN

(Minimal barreling)Initial guess Optimal preform

Optimal preform shape

Final optimal forged productFinal forged product

Initial preform shape

Materials Process Design and Control Laboratory



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




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




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.




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




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.




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




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




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




HOMOGENIZATION IN WAVELET SPACES

Full microstructure information

Homogenized properties at next scale

Complete homogenization

WaveletBasis




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




Mutual Information Comparison Across Scales

Daubechies Family Biorthogonal Family

Mutual Information: The information that parameters in a scale are able to convey about parameters in another scale. A higher information loss

occurs when we try to reduce the dimensionality of the solution when the physics involves lower order scales. Hence a hierarchical wavelet based

method to be employed while ensuring that information lost in the truncated wavelet bases is minimized.




INFORMATION AND WAVELET MEASURES

, ,

,

,

( ) 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




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




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)




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.




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




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




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




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




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




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




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




INFORMATION THEORETIC LEARNING

Information Learning

Used to reduce the computational time when the

parameters needs to be transferred continuously at each

time step. Train a neural network with Information criterion, that is mutual

information between actual and nn based outputs is maximized

A convergence study of neural network based single level

upscaling process employing information theoretic criterion

Information potential of one implies that the nn based output can predict exactly the result of upscaling process




Microstructure Based Models

Model chosen based on

microstructure

Poly-phase material Pure metal

Lineal analysis of microstructure

photograph

Orientation distribution

function model

Dendritic

Spatial correlation structure of models are known




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




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




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




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




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)




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




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




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




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.




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




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







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)




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




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




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




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 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




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 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




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




Quantification using incremental PCA

Input Image

Classifier

Feature Detection

Dynamic Microstructure Library

Identify and add new classes

Employ lower-order features

Pre-processing




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)




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




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




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’

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




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)

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




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




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




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




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




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




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




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




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




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




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

materials process design and control laboratory an information-theoretic approach to multiscale...

Documents

order random process

information theory

information theoretic

control laboratoryidea

face of incomplete information

acceptable loss of information

macro scalecorrelation

macro scalewavelet