a data driven approach for generating reduced-order stochastic models of

8/14/2019 A Data Driven Approach for Generating Reduced-Order Stochastic Models Of

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Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory CCOORRNNEELLLLU N I V E R S I T YCCOORRNNEELLLLU N I V E R S I T Y

A DATA DRIVEN APPROACH FOR

GENERATING REDUCED-ORDER

STOCHASTIC MODELS OF RANDOM

HETEROGENEOUS MEDIA

Nicholas Zabaras and Baskar Ganapathysubramanian

Materials Process Design and Control Laboratory

Sibley School of Mechanical and Aerospace Engineering

Cornell UniversityIthaca, NY 14853-3801

[email protected]

http://mpdc.mae.cornell.edu/


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MOTIVATIONPROVIDE LOW-DIMENSIONALITY

REPRESENTATION OF THE

MICROSTRUCTURE, PROPERTY

AND PROCESS SPACES

Applications:

(i) Identify microstructures that have

extremal properties.

(ii) Identify processing sequences

that lead to desired microstructures

and properties.

3.05 3.06 3.07 3.08 3.09 3.1 3.11 3.123.03

3.04

3.05

3.06

3.07

3.08

3.09

3.1

3.11

3.12

Taylor factor along RD

Taylo

rfactoralo

ngTD

b

2. Rolling

3. Rolling followed by drawing

1. Drawing

R

C

a

d

fe

Process

Property-process

space

-1

-0.5

0

0.5

-1

-0.50

0.51

-1.8

-1.7

-1.6

-1.5

-1.4

-1.3

Microstructurerepresentations

Process-structure

space

Process paths

A100

A1000

A80

Property-structure

space


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STRATEGY

Given limited experimental information

(microstructural features):

Represent several plausible microstructures

Encode this information into a low-dimensional

parameterization ofall such possible

microstructures

WHY?

Can incorporate effects of limited information of

macro behavior

Low-dimensional embedding significantly aidssearching and contouring of high dimensional

microstructural space


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OUTLINE

Linear reduced-order modeling framework of

microstructures

Non-linear reduced order modeling framework for

microstructures

Applications of classification and reduced models of

structure-property-process maps for tailored

materials


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Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

CCOORRNNEELLLLU N I V E R S I T YCCOORRNNEELLLLU N I V E R S I T Y

LINEAR REDUCED-ORDER

MODELING FRAMEWORK


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DEVELOPING LINEAR TOPOLOGICAL MODELSData driven techniques for encoding the variability in properties into a viable, finite dimensional

stochastic model.

Advances in using Bayesian modeling, Random domain decomposition

Aim is to create a seamless technique that utilizes the tools of the mature field of property/

microstructure reconstruction

First investigations into constructing data-driven reduced order representation of topological/

material/ property distributions utilized a Principal Component Analysis (PCA/POD/KLE) based

approach.

Generate 3D samples from the microstructure space and apply PCA to them

= a1 a2 +..+ an+

1. B. Ganapathysubramanian, N. Zabaras, Modelling diffusion in random heterogeneous media: Data-driven models, stochastic

collocation and the variational multi-scale method, J Comp Physics 226 (2007) 326-353.

Convert variability of property/microstructure to variability of coefficients.

Not all combinations allowed. Developed subspace reducing methodology1 to find the space of

allowable coefficients that reconstruct plausible microstructures


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OVERVIEW OF METHODOLOGY

Extract properties P1, P2, .. Pn, that

the structure satisfies.

These properties are usually

statistical: Volume fraction, 2-point

correlation, auto correlation

Reconstruct realizations of the

structure satisfying the properties.

Monte Carlo, Gaussian Random

Fields, Stochastic optimization etc.

Construct a reduced-orderstochastic model from the data. This

model must be able to approximate

the class of structures.

KL expansions, FFT and other

transforms, Autoregressive models,

ARMA models

Extract structure-property-process

relations

Link with microstructure

classification and statistical learning

algorithms

1. Property extraction 2. Microstructure reconstruction

3. Reduced model4. Applications


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DATA TO CONSTRUCT INPUT MODELS

2D microstructure

characterization

tomographic characterization

Only have characterization of property variation in

finite number of regions or finite realizations

Consider the property variation and/or microstructureto be a stochastic process.

Identify this stochastic process using the experimental

information available

volume fraction, 2-point correlation, 3-point

correlations .

Convert this representation into a computationally

useful form: Finite dimensional representation

Process data for statistical invariance of the structure

All realizations of the stochastic process satisfy the

experimental statistical relations

These microstructures belong to a very large(possibly) infinite dimensional space.


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FINITE DIMENSIONAL REPRESENTATION

The data extraction/reconstruction procedure gives a set of 3D microstructures.

These are samples from the microstructural space.Need a qualitative, functional representation of the topological variation.

Must be finite dimensional for this description to be useful

Necessity of model reduction arises

I = Iavg + I1a1 + I2a2+ I3a3 + + Inan

Represent any microstructure as a linear combination of the microstructures or

some eigenimages

= a1 a2 an+ + ..+

Move randomness from image to coefficients


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REDUCED MODEL OF TOPOLOGICAL VARIATION

Construct descriptor from sample images.

Use POD

Microstructure images (nxnxn pixels) represented as vectors Ii i=1,..,M

The eigenvectors of the covariance matrix are computed

The first N eigenimages are chosen to represent the microstructures

= a1 a2 an+ + ..+

Represent any microstructure as a linear combination of the microstructures or

some eigenimagesMove randomness from image to coefficients


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PROPER ORTHOGONAL DECOMPOSITION

Suppose we had a collection of data

(from experiments or simulations) for

the some variable/process/parameter

1 1{ ( )} , { ( , )}N N

i iS A x A x t = ==

Is it possible to identify a basis such

that this data can be represented in

the smallest possible space. I.e find

1{ ( )} , M NM

ix = =

Such that it is optimal for the data to

be represented as

1( ) ( ),

M

i i

i A x a x

==

1

( , ) ( ) ( ),M

i i

i

A x t a t x=

=

Proper Orthogonal Decomposition(POD), Principal Component Analysis

(PCA), Karhunen LoeveDecomposition (KLE), Sirovich,

Lumley, Ravindran, Ito

PCA is mathematically defined as an

orthogonal linear transformation that

transforms the data to a new

coordinate system such that the

greatest variance by any projection of

the data comes to lie on the first

coordinate (called the first principal

component),

PCA is theoretically the most optimum

transform for a given data in least

square terms.


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Data usually collated in terms of a

matrix XT.

XT is shifted to a mean zero value

The covariance matrix of this data is

computed.

TY V=

Solve the optimization problem

Method of snapshots

where

Eigen-value problem

where

Requires the computation of the

covariance matrix of the data and

subsequent eigen decomposition. Canbecome computationally demanding as

N increases. A computationally simpler

POD technique is the method of

snapshots

1 TXXN

=C

Compute the eigen values and eigen

vectors of the covariance matrixTC V V= The reduced description is given by

,

1 Ti jC X X

N=

PROPER ORTHOGONAL DECOMPOSITION


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REDUCED MODEL : CONSTRAINTS

Let I be an arbitrary microstructure satisfying the experimental statistical

correlations

The PCA method provides a unique representation of the image

That is, the PCA provides a function

The function is injective but nor surjective

Every image has a unique mapping

But every point need not define an image

in

Construct the subspace ofallowable n-tuples


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Image I belongs to the class of structures?

It must satisfy certain conditions

a) Its volume fraction must equal the specified volume fraction

b) Volume fraction at every pixel must be between 0 and 1

c) It should satisfy the given two point correlation

Thus the n tuple (a1,a2,..,an) must further satisfy some constraints.

Enforce these constraints sequentially

CONSTRUCTING THE REDUCED SUBSPACE H

1. Pixel based constraints

Microstructures represented as discrete images. Pixels have bounds

This results in 2n3 inequality constraints


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2. First order constraints

The Microstructure must satisfy the experimental volume fraction

This results in one linear equality constraint on the n-tuple

3. Second order constraints

The Microstructure must satisfy the experimental two point correlation.

This results in a set of quadratic equality constraints

This can be written as

CONSTRUCTING THE REDUCED SUBSPACE H


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SEQUENTIAL CONSTRUCTION OF THE SUBSPACE

Computational complexity

Pixel based constraints + first order constraints result in a simple convex hull problem

Enforcing second order constraints becomes a problem in quadratic programming

Sequential construction of the subspace

First enforce first order statistics,

On this reduced subspace, enforce second order statistics

Example for a three dimensional space: 3 eigen images

15

20-15 -10

-5 05

10 1


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THE REDUCED MODEL

The sequential contraction procedure a subspace H, such that all n-tuples

from this space result in acceptable microstructures

H represents the space of coefficients that map to allowable

microstructures.

Since H is a plane in N dimensional space, we call this the material

plane

Since each of the microstructures in the material plane satisfies all required

statistical properties, they are equally probable. This observation provides a way

to construct the stochastic model for the allowable microstructures:

Define such that

This is our reduced stochastic model of the random topology of the microstructure

class


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Reconstruction of well characterized material

Tungsten-Silver composite1

Produced by infiltrating porous tungsten solid with molten

silver

1. S. Umekawa, R. Kotfila, O. D. Sherby, Elastic properties of a tungsten-silver composite above and below the melting

point of silver, J. Mech. Phys. Solids 13 (1965) 229-230

640x640 pixels = 198 m x 198 m

PROPERTY EXTRACTION


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r (um)

g(r)

0 5 10 15 20

0

0.1

0.20.3

0.4

0.5

0.6

0.7

0.80.9

1

Digitized two phase microstructure

image

White phase- W

Black phase- Ag

Simple matrix operations to extract

image statistics

First order statistics: Volume fraction: 0.2

Second order statistics: 2 pt correlation

PROPERTY EXTRACTION


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MICROSTRUCTURE RECONSTRUCTION

Statistical information available- First and second order statistics

Reconstruct Three dimensional microstructures that satisfy these experimental

statistical relations

GAUSSIAN RANDOM FIELDS

GRF- model interfaces as level cuts of a function

Build a function y(r). Model microstructure is given by level cuts of this function.

y(r) has a field-field correlation given by g(r)

If this function is known, y(r) can be constructed as

Uniformly distributed over the unit sphere

Uniformly distributed over [0, 2)

Distributed according to where


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Relate experimental properties to

1. Two phase microstructure, impose level cuts on y(r). Phase 1 if

2. Relate to statistics

first order statistics

where

second order statistics

Set , and

For the Gaussian Random Field to match experimental statistics

MICROSTRUCTURE RECONSTRUCTION


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FITTING THE GRF PARAMETERS

Assume a simplified form for the far field correlation function

Three parameters, is the correlation length, d is the domain

length and rc is the cutoff length

Use least square minimization to find optimal fit

r (um)

g(r)

0 5 10 15 200

0.20.4

0.6

0.8

1


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20 m x 20 m x 20 m

40 m x 40 m x 40 m

200 m x 200 m

3D MICROSTRUCTURE RECONSTRUCTION

128x128x128pixel

64x64x64 pixel


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Eigen number

Normalized

eigenvalue

5 10 15 20 25 300

0.05

0.1

0.15

0.2

0.25

MODEL REDUCTION

Principal component analysis

First 9 eigen values from the spectrum

chosen

Constructing the reduced subspace

and the stochastic model

- Enforcing the pixel based bounds and the linear

equality constraint (of volume fraction) was

developed as a convex hull problem. A primal-dual

polytope method was employed to construct the set

of vertices.- Enforcing the second order constraints was

performed through the quadratic programming toolsin the optimization toolbox in Matlab.

- Two separate cases are considered in this

example. In the first case, only the first-order

constraints (volume fraction) are used to reconstruct

the subspace H. In the second case, both first-order

as well as second-order constraints (volume fraction

and two-point correlation) are used to construct thesubspace H.


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PHYSICAL PROBLEM

T= -0.5 T= 0.5

Structure size 40x40x40 m

Tungsten Silver Matrix

Heterogeneous property is the

thermal diffusivity.

Tungsten: 19250 kg/m3k 174 W/mK

c 130 J/kgK

Silver: 10490 kg/m3

k 430 W/mK

c 235 J/kgK

Diffusivity ratio Ag /W = 2.5

Left wall maintained at -0.5

Right wall maintained at +0.5

All other surfaces insulated

Investigate effects of limited topological information on diffusion in

heterogeneous random media


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Computational domain of each deterministic problem: 128x128x128 pixels

COMPUTATIONAL DETAILS

The construction of the stochastic solution: through sparse grid collocation

level 5 interpolation scheme used

Number of deterministic problems solved: 15713

Each deterministic problem solution: solved ona 8 8 8 coarse element grid (uniform hexahedral

elements) with each coarse element having 16 16

16 fine-scale elements.

The solution of each deterministic VMS

problem: about 34 minutes, In comparison, afully-resolved fine scale FEM solution took nearly 40

hours.

Computational platform: 40 nodes on local Linux

clusterTotal time: 56 hours

Number of collocation points

Error

100

101

102

103

104

10

-2

10-1

100


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f

g

e

b c d

a

FIRST ORDER STATISTICS: MEAN TEMPERATURE


28/73



Temperature

Probabilitydistributio

nfunction

-0.4 -0.2 0 0.2 0.40

0.5

1

1.5

2

2.5

3

3.5

4

Temperature

Probabilitydistributi

onfunction

0 0.2 0.40

1

2

3

4

5

6

7

f

e

b c

d

a

FIRST ORDER STATISTICS: HIGHER MOMENTS


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f

g

e

b c d

a

SECOND ORDER STATISTICS: MEAN TEMPERATURE


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f

e

d

a

Temperature

Probabilitydistributi

onfunction

-0.4 -0.2 0 0.2 0.40

1

2

3

4

5

6

7

Temperature

Probabilitydistributionfunction

-0.2 0 0.2 0.40

1

2

3

4

5

6

7

8

9

10

b c

SECOND ORDER STATISTICS: HIGHER MOMENTS


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Temperature

Prob

abilitydistributionfunction

-0.4 -0.2 0 0.2 0.40

1

2

3

4

5

6

7

Comparison of temperature PDFs at a

point due to the application of first andsecond order constraints

A general methodology was presented for

constructing a reduced-order microstructuremodel

Using more sophisticated model reduction

techniques to build the reduced-order

microstructure model,

Extending the methodology to arbitrary

types of microstructures

INCORPORATING MORE INFORMATION

As more information is incorporated into the

analysis, the subspace of allowable

microstructures shrinksThis corresponds to tighter probability

distributions


32/73



NON-LINEAR REDUCED ORDER

MODELING FRAMEWORK


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INPUT STOCHASTIC MODELS: LINEAR APPROACH

Further related issues:- How to generalize it to other properties/structures? Can

PCA be applied to other classes of microstructures, say,

polycrystals?- How does convergence change as the amount of

information increases? Computationally?

- PCA based approaches find the smallest coordinate representation of the data .

but assumes that the data lies in a linear vector space

What is the result when the data lies in a nonlinear space?

As the number of input samples increases, PCA based approaches tend to overestimate the

dimensionality of the reduced representation.

Becomes computationally challenging

# of samples

#

of

eigen

vecto

rs

Only guaranteed to discover the true structure of data lying on a linear subspace of the highdimensional input space

NONLINEAR APPROACHES TO MODEL REDUCTION:

IDEAS FROM IMAGE PROCESSING, PSYCOLOGY


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NONLINEAR REDUCTION: THE KEY IDEASet of images. Each image = 64x64 = 4096 pixels

Each image is a point in 4096 dimensional space.

But each and every image is related (they arepictures of the same object). Same object but

different poses.

That is, all these images lie on a unique curve

(manifold) in 4096 .Can we get a parametric representation of this

curve?

Problem: Can the parameters that define this

manifold be extracted, ONLY given these images

(points in4096 )

Solution: Each image can be uniquely

represented as a point in 2D space (UD, LR).

Strategy: based on the manifold learning

problem

Different images of the same

object: changes in up-down (UD)

and left-right (LR) poses


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NONLINEAR REDUCTION: EXTENSION TO INPUT MODELS

Different microstructure realizations

satisfying some experimental

correlations

Given some experimental correlation that the

microstructure/property variation satisfies.

Construct several plausible images of the

microstructure/property.

Each of these images consists of , say, n pixels.

Each image is a point in n dimensional space.

But each and every image is related.

That is, all these images lie on a unique curve

(manifold) inn.

Can a low dimensional parameterization of this

curve be computed?

Strategy: based on a variant of the manifold

learning problem.

A FORMAL DEFINITION OF THE PROBLEM


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A FORMAL DEFINITION OF THE PROBLEM

State the problem as a parameterization problem (also called the manifold learning

problem)

Given a set of N unordered points belonging to a manifold embedded in ahigh dimensional space n, find a low dimensional region d that

parameterizes , where d


37/73



AN INTUITIVE PICTURE OF THE STRATEGY- Attempt to reduce dimensionality while preserving the geometry at all scales.

- Ensure that nearby points on the manifold map to nearby points in the low-

dimensional space and faraway points map to faraway points in the low dimensional

space.

3D dataPCA

-

-10

-5

0

5

10

15

-10

-5

0

5

10

15

20

40

Linear approach Non-linear approach: unraveling the curve

KEY CONCEPT


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KEY CONCEPT

Pt A Pt B

Euclidian dist

Geodesic dist

1) Geometry can be preserved if the distances between the points are

preserved Isometric mapping.

2) The geometry of the manifold is reflected in the geodesic distance between

points

3) First step towards reduced representation is to construct the geodesic

distances between all the sample points

THE NONLINEAR MODEL REDUCTION ALGORITHM


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THE NONLINEAR MODEL REDUCTION ALGORITHM

Given N unordered

samplesCompute pairwise

geodesic distance

1) Given the N unordered sample points ( microstructures, property maps )

2) Compute the geodesic distance between each pair of samples (i,j) .3) Given the pairwise distance matrix between N objects, compute the location

of N points, {i} in dsuch that the distance between these points isarbitrarily close to the given distance matrix

. Basic premise of group of

statistical methods called Multi Dimensional Scaling1 (MDS)

Perform MDS on thisdistance matrix

N points in a low

dimensional space

1. T.F.Cox, M.A.A.Cox, Multidimensional scaling, 1994, Chapman and Hall

MATHEMATICAL DETAILS


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How to compute geodesic distance?

Sum over short hops.

Need the notion of distance between

samples

Flexibility in defining the distance

measure..


The distance measure defines the properties of the manifold that the samples lie on1. Properties of the manifold n.

The distance measure,, based on how much the microstructures vary. Defined as

the difference in statistical correlation between two microstructures. ( , ) | ( ) ( ) |i j S i S j= D

The key to a reasonable dimension reduction is a good choice of the distance measure

1. B. Ganapathysubramanian and N. Zabaras, "A non-linear dimension reduction methodology for

generating data-driven stochastic input models", Journal of Computational Physics, in press.

Any choice of functions are allowable as long as they satisfy the metric

properties



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1. Properties of the manifold n.

1. B. Ganapathysubramanian and N. Zabaras, "A non-linear dimension reduction methodology

for generating data-driven stochastic input models", submitted to Journal of Computational

Physics

2. J. R. Munkres, Topology, Second edition, Prentice-Hall, 2000.

( , ) | ( ) ( ) |i j S i S j= D

a) (, ) is a metric space.

Ensure that satisfies the properties of non-negativity,

symmetry and the triangle inequality

Equivalence between microstructures: Two microstructures are equivalent if

they share the same higher order statistical correlation

b) (,) is a bounded.

c) (,) is dense.

e) (

,

) is a compact metric space1,2 .

d) (,) is complete.



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2. Mapping a compact manifold to a low-dimensional set

Have no notion of the geometry of the manifold to start with. Hence cannot constructtrue geodesic distances!

( , ) inf{length( )}i j

=Mm

D

Approximate the geodesic distance using the concept of graph distance G(i,j) : thedistance of points far away is computed as a sequence of small hops.

This approximation,G, asymptotically matches the actual geodesic distance . Inthe limit of large number of samples1,2 . (Theorem 4.5 in 1)

1 2(1 ) ( , ) ( , ) (1 ) ( , )i j i j i j +GM Mm m m

D D D


generating data-driven stochastic input models", submitted to Journal of Computational Physics.

2. M.Bernstein, V. deSilva, J.C.Langford, J.B.Tenenbaum, Graph approximations to geodesics on

embedded manifolds, Dec 2000

Based on results on graph approximations to geodesics2.



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Perform MDS on the geodesic matrix. i.e perform an eigenvalue decomposition of the

squared geodesic matrix.The largest d eigenvalues are the coordinates of the N points.

MATHEMATICAL DETAILS3. MDS and choosing the dimensionality of the reduced space

Estimate the dimensionality of the manifold based on a novel geometrical probability

approach (developed by A. Hero et. al.)

The manifold has an intrinsic dimensionality. How to choose the correct value of d?

(related with issues of accuracy and computational effort)

Based on ideas from graph theory. The rate of convergence of the length functional, L

of the minimal spanning tree of the geodesic distance matrix is related to the

dimensionality1,2, d.


generating data-driven stochastic input models", submitted to Journal of Computational Physics.

2. J.A.Costa, A.O.Hero, Geodesic Entropic Graphs for Dimension and Entropy Estimation in Manifold

Learning, IEEE Trans. on Signal Processing, 52 (2004) 2210--2221.

log( ) log( ) L a N = +1d

ad

=with

THE REDUCED ORDER TOPOLOGICAL MODEL


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Given N unordered

samples

N points in a low

dimensional space

n. d

The procedure results in N points in a low-dimensional space. The geodesic distance

+ MDS step (Isomap algorithm1

) results in a low-dimensional convex, connectedspace2, d.

1. J. B. Tenenbaum, V. De Silva, J. C. Langford, A global geometric framework for nonlinear dimension reduction Science 290

(2000), 2319-2323.

2. B. Ganapathysubramanian and N. Zabaras, "A non-linear dimension reduction methodology for generating data-driven

stochastic input models", submitted to Journal of Computational Physics.

Using the N samples, the reduced space is given asiconvex hull({ })=A =

serves as the surrogate space for .Access variability in by sampling over.

BUT have only come up withmap . Need map too



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Only have N pairs to construct map. Various possibilities based on specific

problem at hand. But have to be conscious about computational effort and efficiency.

Illustrate 3 such possibilities below. Error bounds can be computed1

.

n d n d

n

d

1. Nearest neighbor map 2. Local linear interpolation

3. Local linear interpolation with projection

1. B. Ganapathysubramanian and N. Zabaras, "A non-linear dimension reduction methodology for generating data-driven

stochastic input models", submitted to Journal of Computational Physics.




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Given N

unordered

samples

Compute pairwise

geodesic distancePerform MDS on

this distance matrix

N points in a

low dimensional

space

Algorithm consists of two parts.

1) Compute the low dimensional representation of a set of N unordered sample points

belonging to a high dimensional space

For using this model in a stochastic collocation framework, must sample points in

2) For an arbitrary point must fins the corresponding point x . Compute themapping from

n. d


NON LINEAR DIMENSION REDUCTION


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NON LINEAR DIMENSION REDUCTIONThe developments detailed before are applied

to find a low dimensional representation of

these 1000 microstructure samples.

The optimal representation of these points was

a 9 dimensional region

Able to theoretically show that these points

in 9D space form a convex region in 9.This convex region now represents the lowdimensional stochastic input space

Use sparse grid collocation strategies to

sample this space.

10

15

20-15 -10

-5 05 10

1

-10

-5

0

5

10

15

Log(Samples)

Log(lengthofMST

)

100 300 500

10000

20000

30000

40000



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# of collocation points

Erro

r

100

101

102

103

10410

-6

10-5

10-4

10-3

10-2

10-1

100

101

Computational domain of each

deterministic problem: 65x65x65pixels


The construction of the stochastic

solution: through sparse grid

collocation level 5 interpolation

scheme used

Number of deterministic problems

solved: 26017

Computational platform: 50 nodes

on local Linux cluster (x2 3.2 GHz)

Total time: 210 minutes

Total number of dofs: 653x26017 ~

7x109

MEAN TEMPERATURE PROFILE


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MEAN TEMPERATURE PROFILE

a

b c d e

f

g

(a) Temp contour

(b-d) Temp isocontours

(e-g) Temp slices

HIGHER ORDER STATISTICS


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Temperature

Probabilitydistributionfunction

-0.2 0 0.20

1

2

3

4

5

6

HIGHER ORDER STATISTICS

a

(a) Temp contour

(b) Temp isocontours

(c) PDF of temp

(d-f) Temp slices

b d

e

f

c

MODELS OF POLYCRYSTALLINE MATERIALS


51/73



Microstructural variations affect macro-scale

properties

It is lot more difficult to analyze than two-phase

or multi-phase materials

Multiple layers of representation (a) grain

distribution (b) orientation distribution

Continuum distribution of orientation. Solvable

problem

Limit analysis to grain distribution

This is a tricky problem: Have to faithfully

encode grain distribution features andshould quickly reconstruct approximate

grains

MODELS OF POLYCRYSTALLINE MATERIALS

MICROSTRUCTURAL FEATURE: GRAIN SIZE


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Grain size obtained by using aseries of equidistant, parallel

lines on a given microstructure at

different angles. In 3D, the size

of a grain is chosen as the

number of voxels (proportional to

volume) inside a particular grain.

2D microstructures

3D microstructures

Grain size is computed from the

volumes of individual grains

MICROSTRUCTURAL FEATURE: GRAIN SIZE

EXPERIMENTAL DATA


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Polarized light micrograph

of aluminium alloy AA3302

(source Wittridge NJ et al.Mat.Sci.Eng. A, 1999)

0 2 4 6 8 10 12 14 16 18 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

GrainSize( m)

probability

=10.97

=124.90

Extract grain size

distribution from image

EXPERIMENTAL DATA

RECONSTRUCTING PLAUSIBLE DATA SET


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Reconstruct N=200 microstructures that satisfy the experimental grain size

distribution

Utilize stochastic optimization to construct microstructures

RECONSTRUCTING PLAUSIBLE DATA SET

MODEL REDUCTION OF POLYCRYSTALS


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The key feature to encode is the grain boundary

But grain boundaries are sparsely distributed in the domainNeed a strategy to compress all the grain boundary information and

remove all the interior information

Look at different types of transforms

1) Transform and its inverse should be computationally efficient

2) Data size should be limited.

3) Should be able to process grain boundaries : monochromatic lines on a

monochromatic background

4) Translation and rotation invariant

Radon and Fourier transform.

MODEL REDUCTION OF POLYCRYSTALS

IMAGE TRANSFORMATION


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Radon transform:

The Radon transform of an image represented by the function f(x,y) can

be defined as a series of line integrals through f(x,y) at different offsetsfrom the origin.

( , ) ( , ) ( cos sin ) R f x y x y r dxdy

= +

Why Radon transform?

It collects line integral information.

In some sense it is similar to the

Heyns intercept method

Extensively utilized in CATscanning and medical imaging..

Mature applications




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Forward radon transform

Inverse radon transform- Filtered

back projection: Two steps, filteringand then projection

0

( , ) ( cos sin , )i f x y f x y d

= +The reconstructed image is heavily blurred. Use

a high pass filter to the sinogram data in thefrequency domain.

Apply a 1-D DFT to the sinogram data for each

angle, multiply by the filter, and then using the

inverse DFT to reconstruct the data.




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REDUCED ORDER MODEL


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The distance measure, , based on how much the microstructures vary.Since the Radon transform encodes (in a sense) the grain volume,

define the distance as the difference in Radon transformed images

( , ) || ( , ) ( , ) ||i ji j R R = D

Use filtered or un-filtered Radon transform?

Given N unordered

samplesCompute pairwise

geodesic distance

Perform MDS on thisdistance matrix

N points in a lowdimensional space

REDUCED ORDER MODEL

DIMENSIONALITY OF THE REDUCED MODEL


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Dimensionality of

the parametric

space computedfrom application of

the BHH theorem.

Which connects

the dimensionality

of the surface to

the length

functional of a

graph

d = 31

DIMENSIONALITY OF THE REDUCED MODEL

Log(Number of samples)

Log(Length

functional)

25 50 75 100 125

5E+10

1E+11

1.5E+11

2E+11

2.5E+11

3E+11

SAMPLING AND RECONSTRUCTION


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Consider random points in the hyper cube

Reconstruct polycrystals corresponding to this point based on data. Direct

interpolation of the radon transform followed by inversion

SAMPLING AND RECONSTRUCTION

SOME CHALLENGES


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Reconstruction improves as the number of

data point improves .. i.e. A is densely

populated

As the number of neighbors used in

reconstruction increases the reconstruction

degrades. Reason is averaging

No filtering or heuristics used so far. But

seems like a good idea. Could result inbetter microstructures (line merging, pre-

filtering)

Investigate other translation and rotation

invariant transforms: Hough transform,

Steerable pyramid

Extension to 3D is straightforward. Use 3D

Radon transform.

SOME CHALLENGES

Experimental information,

gappy data

10

15

20-15 -10

-5 05 10

1

-10

-5

0

5

10

15


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Applications of classification and

reduced models of structure-property-process maps for tailored

materials

PROCESSING PATH DESIGN


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Tailored microstructures so that desired

properties can be achieved: controlled

deformation or thermal treatment.

Processing path design to realize

microstructures with optimal properties.

Non-uniqueness in processing path

solution. This problem cannot beaddressed solely using conventional

optimization schemes.

PROCESSING PATH DESIGN

Data mining strategies comes natural to such problems.

Development of a database that can accommodate unknown microstructures into

newly formed classes without user intervention

MODEL REDUCTION AND STATISTICAL LEARNING


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Model reduction results in low-complexity

and low-dimensional models of the

microstructural space

Interrogate sample microstructures to

construct corresponding process and

property spaces

Potentially results in huge reduction indimensions

Utilize classification and statistical learning

frameworks to construct relationships

between low dimensional models of

microstructure, process and property spaces

Unsupervised learning strategies for

automated design: X-means classifier

Materials Process Design and Control LaborMaterials Process Design and Control Labor

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

3.055

3.06

3.065

3.07

3.075

3.08

3.085

3.09

3.095

3.055 3.06 3.065 3.07 3.075 3.08 3.085 3.09 3.095

3.05

3.06

3.07

3.08

3.09

3.1

3.11

Taylor factor along RD

Taylor

factor

alongTD

R

C

Process plane

Process-property plane

CLASSIFICATION HIERARCHY: MICROSTRUCTURES


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Classify/tessellate reduced space based on features

CLASSIFICATION HIERARCHY: TEXTURE


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Classify/tessellate reduced space based on fiber orientations

K-MEANS CLUSTERING


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Find the cluster centers {C1,C2,,Ck} such that the sum of the 2-norm distance

squared between each featurexi, i = 1,..,n and its nearest cluster centerCh is

minimized.

21 2

21,..,1

1( , ,.., ) ( )

2min

nk h

ih ki

J c c c x C ==

=

Identify clusters

Clusters

DATABASE OF ODFs

Feature

Space

Cost functionEach class is affiliated with

multiple processes

ADAPTIVE REDUCED MODELS: ACCELERATED DESIGN


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Linking data-driven reduced order models for microstructure and texture

evolution potentially very significant

The classification technique is database-driven and the availability of

existing information can be further utilized to accelerate the texture

evolution models.

Adaptivity to account for the sensitivity of different features. This provides

addition information of significance: Which processes affect whichfeatures and which features affect different properties

Tangible input for further experimentation

= a1 a2 +..+ an+

THE DESIGN FRAMEWORK


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DESIGN FOR DESIRED ODF: A MULTI STAGE PROBLEM


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0 5 10 15 20 250

0.2

0.4

0.6

0.8

IterationIndex

Normalizedobjectivefunction

Initial guess, 1 =

0.65, 2 = -0.1

Desired ODF Optimal-

Reduced order

control

Stage: 1 Plane strain

compression ( 1 = 0.9472)

Stage: 2

Compression

( 2 = -0.2847)

20x faster than full

optimization.

Gradients are

obtained from

reduced order

sensitivity analysis.

DESIGN FOR DESIRED ODF: A MULTI STAGE PROBLEM


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IterationIndexNormalizedobjectiv

5 10 150

0.2

0.4

0.6

0.8

1

h

Crystal

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 fromthe rolling direction

Magnetichysteresisloss(W/Kg)

DesiredpropertydistributionOptimal (reduced)Initial

Stage: 1 Shear 1

( 1 = 0.9745)

Stage: 2Tension

( 2 = 0.4821)

CONCLUSIONS


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Data-driven non-linear reduced order models of microstructures

developed

Very significant when performing computationally demanding operations searching, contouring - in intrinsically high-dimensional property-

process-structure spaces

Naturally coupled with statistical learning and unsupervised classification

strategies to effectively estimate optimal processing routes for tailored

materials

1) B. Ganapathysubramanian and N. Zabaras, "Modelling diffusion in random heterogeneous

media: Data-driven models, stochastic collocation and the variational multi-scale method",

Journal of Computational Physics, Vol. 226, pp. 326-353, 2007

2) B. Ganapathysubramanian and N. Zabaras, "A non-linear dimension reduction methodology for

generating data-driven stochastic input models", Journal of Computational Physics, submitted.

3) V. Sundararaghavan and N. Zabaras, "A statistical learning approach for the design of

polycrystalline materials", Statistical Analysis and Data Mining, submitted

4) V. Sundararaghavan and N. Zabaras, "Linear analysis of texture-property relationships using

process-based representations of Rodrigues space", Acta Materialia, Vol. 55, Issue 5, pp.

1573-1587, 2007

a data driven approach for generating reduced-order stochastic models of

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