# Statistical learning Techniques for Microstructure Classification

Document Sample

```					      STATISTICAL LEARNING TECHNIQUES
FOR MICROSTRUCTURE
CLASSIFICATION AND
REPRESENTATION WITH APPLICATIONS
IN MATERIALS DESIGN
Prof. Nicholas Zabaras
Materials Process Design and Control Laboratory
Sibley School of Mechanical and Aerospace Engineering
188 Frank H. T. Rhodes Hall
Cornell University
Ithaca, NY 14853-3801

Email: zabaras@cornell.edu
URL: http://www.mae.cornell.edu/zabaras/

Army Research Office/GATECH workshop on Inverse Techniques in Materials Design, Atlanta, April 26-28, 2004

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
MOTIVATION FOR MICROSTRUCTURE-SENSITIVE DESIGN

 Critical hardware components in the
aerospace, naval & automotive industry,
armor/projectile design, require
improved material response,
performance robustness, increased
utility & lifing, etc.
Design LIGHT armor for SUPERIOR           Introduce complex materials and
PERFORMANCE                     process design solutions using
physically-based mathematical and
statistical techniques

STATISTICAL LEARNING
TOOLBOX

Aircraft engines
PROGNOSIS - Life & response prediction
Understanding and
blocking fracture by   REAL TIME CONTROL – Real time property
MSD
analysis and control (during processing)
Microstructure-sensitive design of:
 fracture toughness, corrosion, elastic,     SIMULATION MATCHING DESIGN –
creep, fatigue & other properties             Predictive modeling of processes (AIM)

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
STATISTICAL LEARNING TOOLBOX

Training
samples                                                         NUMERICAL
SIMULATION OF
MATERIAL
RESPONSE
1.   Multi-length
Image                   STATISTICAL                                scale analysis
LEARNING                             2.   Polycrystalline
plasticity
TOOLBOX
Functions:
1. Classification
methods                             PROCESS
2. Identify new                         DESIGN
ODF                        classes
ALGORITHMS

1.   Exact methods
(Sensitvities)
2.   Heuristic
methods
Process
Pole figures          controller

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
DEFORMATION PROCESS DESIGN SIMULATOR

Research Objectives:                             Current capabilities:
To develop a mathematically and                  • Development of a general purpose
computationally rigorous gradient-based          continuum sensitivity method for the
optimization methodology for virtual materials   design of multi-stage industrial
process design that is based on quantified       deformation processes
product quality and accounts for process         • Deformation process design for
targets and constraints.                         porous materials
• Design of 3D realistic preforms and
dies
•Extension to
polycrystal
plasticity based
constitutive
models with
evolution of
crystallographic
texture

Initial guess            Optimal preform

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
PROCESS DESIGN CAPABILITIES
DESIGN OBJECTIVE                                                                                                                                                                                                       DESIGN OBJECTIVE
Design the preforming die for a fixed volume                                                                                                                                           Design the extrusion die for a fixed
of the workpiece (1100-Al) such that the                                                                                                                                        reduction such that the variation in state in
variation in state in the product is a minimum                                                                                                                                       the product at the exit is a minimum
Preforming                                                                                                                                                                                                                                                                        Initial die                  Optimal die
Finishing Stage
Stage                                                                    0.25                                                                                                                          0. 94

State Var (MPa)              State Var (MPa)

Objective
0.2                                                                                                                          0. 93
n                                                                                                                                                                                                                            37.2737                      37.6207
o
State variable ( MPa )
i
t
c
n                                                                                                                                                                                                                            36.7569                      37.0812
u
36.5418

Objective
f        0.15                                                                                                                                                                                                                36.2402
54.431                      e
v
0. 92
i
t
c                                                                                                                                                                                                                            35.7234                      36.0023
51.729                      e
j
b
O                                                                                                                                                                                                                            35.2066                      35.4628
49.028                                0.1                                                                                                                          0. 91                                                                                                              34.9233
34.6899
46.326                                                                                                                                                                                                                                                                                34.3839
34.1731
43.625                               0.05                                                                                                                                                                                                                33.6563                      33.8444
.
09
40.923                                                                                                                                                                                                                                                   33.1395                      33.3049
32.6228                      32.7655
0                    1           2       3           4       5   6   7    8
Iteration numberIteration index                                                         0. 89                                                                                 32.106                       32.226

State variable ( MPa )
0. 88
55.210
53.487                                                                                                                                                                      0    1       2   3       4   5       6   7       8   9

51.764                                                                                                                                                                                   Iteration index
50.040
48.317
46.594
Material
Process
DESIGN OBJECTIVE                                                                                                                          Design                                                                                                DESIGN OBJECTIVE
Evaluate preform of a porous material                                                                                                                  Simulator                                  Design the preforming die for a fixed
(2024- T351Al) for a given die shape                                                                                                                                                           volume of the workpiece (1100-Al)
such that the die cavity is completely filled                                                                                                                                             such that the finishing die is completely filled
8
Preforming Stage                 Finishing Stage
Objective function (x1e-03)

Sh e ar m od ul u s ( M P a)
Unfilled
Nondimensionalized

8       2 .5 8 E+ 0 4
7
6
2 .5 5 E+ 0 4
2 .5 1 E+ 0 4
7                                                                                                                                                                                                      cavity
5       2 .4 8 E+ 0 4
4       2 .4 5 E+ 0 4
3       2 .4 2 E+ 0 4
8.0
6
3

2       2 .3 8 E+ 0 4
Objective function

1       2 .3 5 E+ 0 4
6.0
(x1.0E-05)

5
Fully
4.0                                                                                                                          filled
S h e ar m o d u l u s ( M P a)
4                                                                                                                                                                                                       cavity
8        2.59E+04
7        2.55E+04                                                                                                                          2.0
6        2.52E+04
5        2.49E+04                                               3
4        2.45E+04                                                   0       2           4         6     8   10   12
0.0
3        2.42E+04                                                           Iteration index                                                      0         1                   2           3           4           5           6
2        2.39E+04
1        2.36E+04
Iteration number

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
MULTI-LENGTH SCALE DESIGN APPROACH
Phenomenology                                                      Polycrystal plasticity
Initial configuration           Deformed configuration     Initial configuration                                     Deformed configuration
F
Bo                               B                     s0                                       F                  s n
n0

F                           e             Bo                            p                                          B
F                                           F              s0                F*
n0
Intermediate
thermal
Fp                Stress free
(relaxed)                                         Stress free (relaxed)
configuration                              configuration                                        configuration

(1) Continuum framework                                                (1) Single crystal plasticity
(2) State variable evolution laws                                      (2) State evolves for each crystal
(3) Desired effectiveness in terms                                     (3) Ability to tune microstructure for
of state variables                                                            desired properties

Need for polycrystalline analysis                              Multi length scale design approach
The effectiveness of design for desired product                                                                      The inverse
properties is limited by the ability of                                                                    problem “of tuning
phenomenological state-variables to capture the
dynamics of the underlying microstructural
the microstructure
mechanisms                                                                                     for desired material
response”
Polycrystal plasticity provides us with the ability
ODF: 1 2 3 4 5 6 7

is difficult due to the large number of
to capture material properties in terms of the
crystal properties. This approach is essential for              microstructural degrees of freedom
realistic design leading to desired microstructure-
sensitive properties                          Solution: Microstructure model reduction

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
SYNERGY BETWEEN MICROSTRUCTURES & PROCESSING
Real time microstructure                                       Materials design – optimization
analysis – metrology tools      Reduced order models             in reduced microstructural
for efficient                           space

microstructure                ODF: 1 2 3 4 5 6 7

representation

● Averaging principles
● Reduced models
Process design simulator                                 ● Digital library

Materials testing   Sensing across
driven by design      length-scales
robustness limits and quantification
of uncertainty

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
POLYCRYSTAL PLASTICITY BASED APPROACH

Macro problem driven by the macro-design variable β           Micro problem driven by the

Bn+1
Fn+1

B0    X

L = L (X, t; β)
x = x(X, t; β)
ODF: 1 2 3 4 5 6 7

Polycrystal                     ~
L = velocity gradient          plasticity      Ω = Ω (r, t; L)

Design variables (β) are macro Design objectives are micro-scale
design variables                  averaged material/process
1. Die shapes                             properties
2. Preform shapes
3. Processing conditions
Etc.

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
MICROSTRUCTURE REDUCED MODEL REPRESENTATION

Suppose we had a collection of          Method of snapshots
data (from experiments or             Solve the optimization problem
simulations) for the ODF:

Is it possible to identify a basis          where

such that it is optimal for the
data represented as                          Eigenvalue problem

where

POD technique – Proper Orthogonal
Decomposition

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory

Uniaxial                                                        Plane strain
tension                                                         compression

Shear

Design vector α = {α1, α2, α3, α4, α5}
Design problem: Determine α so as to obtain desired
properties in the final product.

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
VALIDATION OF REDUCED-ORDER REPRESENTATION
Uniaxial tension test α = {1,0,0,0,0}              t = 0.1 s

Reduced model                          Full model                           Reduced model
(Basis I)                                                                  (Basis II)

ODF:   2   2.625 3.25 3.875 4.5

•Basis – I    3 POD modes from a uniaxial tension test with strain rate of 1s-1 for t=0.2 s
•Basis – II   9 POD modes from five tests ( αi= 1s-1 for the ith test, αj= 0, i ≠ j ) for t = 0.2 s

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
MULTI-LENGTH SCALE SENSITIVITY ANALYSIS

Bn+1
Sensitivity
Fn+1                                                                               kinematic
sub-problem
x = x(X, t; β)
L = L (X, t; β)                                                         Sensitivity
B0                          L = velocity gradient                                                      thermal
I + (Ls)n+1
ODF: 1 2 3 4 5 6 7       sub-problem
~
Ls = design            Ω = Ω (r, t; L)
constitutive
o
Bn+1                                                                 sub-problem
Fn+1 + Fn+1
Sensitivity
o                                                                         contact & friction
x + x = x(X, t; β+Δ                                  ODF: 1 2 3 4 5 6 7
sub-problem
o                                          0      ~
L β) L = L (X, t; β+Δ β)
+                                       Ω + Ω = Ω (r, t; L+ΔL)
r – orientation parameter

The velocity gradient – depends on a macro                                A micro-field – depends on a macro design
design parameter                                              parameter (and) the velocity gradient as

Sensitivity of the velocity gradient – driven by                            Sensitivity of this micro-field driven by the
perturbation to the macro design parameter                                               velocity gradient

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
MATERIAL POINT DESIGN SIMULATOR

Research Objectives:                                                                                                          Current capabilities:
To develop a mathematically and                                                                                               • Multi-length scale analysis of
optimization methodology for material                                                                                         • Development of a general purpose
properties dependent on the underlying                                                                                        multi-length scale continuum
microstructure of the material.                                                                                               sensitivity method for the design of
microstructures
• Polycrystal

Normalized hysteresis loss
Crystal <100>                                                                                          1.035                                       Initial
direction.                                                                                            1.03
1.025
Intermediate
Optimal
plasticity based
Easy direction                                                                                          1.02                                       Desired      constitutive
1.015
of
magnetization
1.01
1.005
models with
– zero power                                                                                                 1
evolution of
0.995
loss                                                                                               0.99
crystallographic
0.985

h
0     10    20   30   40   50   60
Angle from rolling axis
70   80
texture
90

External magnetization
• Reduced-order
Initial

direction                                                                                         Intermediate
Optimal
representation of
1.032
Desired
microstructures
• Classification of
R value

y                                                                                                                           Control of R-value,
1.012

Lankford coefficient
microstructures
variation on a plane                for better process
(sample plane)                    and property
θ
0.992
x                        0   20       40      60
Angle from rolling axis
80
selection

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
MICROSTRUCTURE RESPONSE MODELING CAPABILITIES
Modeling BCC material response – Tantalum                                                                                                                                                                                 Modeling FCC material response – FCC Al
400

0.4                                                                                                                                                                                                         0.4

1000                                                                                                                                                                                                                                350
0.2
Numerical predictions                                 0.2

Experimental results
0
900                                                                             0
Z

Z
-0.2                                                                                                                                                                                                         -0.2
300                                                               20K
-0.4                                                                                                                             800                                                                         -0.4

-0.4                                                                           -0.4                                                                                                                         -0.4                                                                                         -0.4

Equivalent Stress (MPa)
-0.2                                                     -0.2                                                                                                                                               -0.2                                                                   -0.2

Y 0                                      0                                                                                                                                                                        0                                             0
X
700                                                                                                     Y                                                     X
250
Equivalent stress (MPa)

0.2                   0.2                                                                                                                                                                                   0.2                           0.2

0.4   0.4                                                                                                                                                                                                       0.4   0.4

Z

600
X          Y

200
500                                                                                    0.4

0.2
0.4

0.2
400                                                                                     0
150

Z
-0.2

0
140K
Z

-0.4

-0.2
300                                                                                 -0.6

-0.4
-0.8                                                                                                                              100                                                               195K
-0.4
200                                                                                          -0.4
-0.2                                                     -0.2
-0.4

-0.2                                                            -0.2
-0.4
Y 0
0.2                           0.2
0
X                                                                                                                             300K
Y 0                                            0                                                                                                                                                                                    0.4               0.4
X
0.2

50
0.2
0.4           0.4                                                                       100

0
0               0.1          0.2        0.3            0.4   0.5
Equivalent strain
Microstructure                                                                                                                                     0
0                     0.1
Equivalent strain
0.2                    0.3

DESIGN OBJECTIVE
Design
Simulator                                                                                                                                                    DESIGN OBJECTIVE
Evaluate the microstructure and                                                                                                                                                                                                                                                                                  Evaluate the microstructure and
process for a desired Yield stress                                                                                                                                                                                                                                                                                process for a desired hysteretic
distribution                                                                                                                                                                                                                                                                                                     loss pattern
Initial
1.002                                                                                                                                                Intermediate                                                                                                                                                              1.035                                                            Initial
Optimal
1.001                                                                                                                                                Desired                                                                                                                                                                       1.03                                                         Intermediate
1                                                                                                                                                Optimal (full)
1.025                                                            Optimal
0.999                                                                                                                                                                                                                                                                                                                              1.02                                                         Desired
Iteration 3
0.998                                                                                                                                                                                                                                                                                                                          1.015
0.997                                                                                                                                                                                                                                                                                                                              1.01
Iteration 6                                                                                                                                                                                                1.005
0.996
1
0.995
0.995
0.994
0.99
0.993
0.985
0                                               20                                      40                 60             80                                                                                                                                                                                    0   10   20         30      40        50       60   70     80       90
Angle from rolling direction                                                                                                                                                                                                                                                    Angle from rolling direction

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
MULTI-STAGE PROCESS DESIGN FOR ODF

Identify & design stage 1 so that a                                                                  A multistage approach towards
desired ODF is obtained                                                                         CLASSIFICATION & QUANTIFICATION
at the end of stage 2 (tension)                                                                   of Lagrangian ODF’s
 NO EXISTING framework for
comparing Lagrangian ODF’s
DESIRED ODF AT THE END
 SYNERGY between process design &
OF STAGE 2
classification

Identify process

Displacement
TL ODF
and parameters
0.4

3.79
3.54
0.2
3.29                                                 ODF at the end of
3.04
2.79
stage 2 is specified
by the desired ODF
0
Z

2.54
2.29
-0.2

Stage 1?
2.04
1.79                                             Tension
-0.4       1.54
-0.4                                                                 -0.4

-0.2                                               -0.2

Time
Z
0                           0
Y                                   X
0.2               0.2

0.4   0.4
X                 Y                                 Random ODF

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
MULTI-STAGE PROCESS DESIGN FOR ODF

OBSERVATIONS
0.4       TL ODF                                                                                                                                  Evolution of ODF
0.2
3.79
3.54                    1.                Process re-engineering and
3.29

0
3.04
2.79                                      accelerated materials
Z
2.54
-0.2
2.29
2.04                                      insertion
1.79
-0.4       1.54
-0.4

-0.2                                                                       -0.2
Z
-0.4
2.                Microstructure tuning for
0                                                 0

arresting cracks, better
Y                                                         X
0.2                             0.2

0.4          0.4
X                         Y

armor and projectiles
Desired ODF

Texture at the end of stage 1

0.4
TL ODF
0.4
2.65
0.2
2.60
2.54    0.2

2.48
0
2.42
Z

0
2.36
Z

-0.2
2.30
2.25    -0.2

-0.4
-0.4

-0.4
-0.4
0.4                         Evolution of the
0.2

fundamental region
-0.2                                                                               -0.2   Z                                -0.25
0        Z
0                                                     0                                                                                            Y
Y                                                             X                                                           0
X                     -0.2                 Y
0.2                                 0.2
0.25
0.4          0.4                 X                 Y                                                -0.4
X

Lagrangian ODF                                                                                                           Fundamental region

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
CLASSIFICATION BASED ON SUPPORT VECTOR MACHINES

B
C            Class-B
Class-A

C

Margin (w)                                       Class-C
Training features                            A
B
A

Given N samples: (xi,yi)          One Against One Method:
where ‘x’ is a feature vector and ‘y’
is the class label for data point,     • Step 1: Pair-wise classification, K(K-
1)/2 for a K class problem
Find a classifier with the decision    • Step 2: Given a data point, select class
function, f(x) such that y = f(x),    with maximum votes out of K(K-1)/2
where y is the class label for x.

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
IDEA OF A BINARY CLASSIFIER

w.xi + b < 1
Maximal Margin Classifier –
The optimization problem

w.xi + b > 1

Class – I feature                           Margin
Class – II feature

Find w and b such that
2
s           is maximized and for all (xi ,yi)
w
w . xi + b ≥ 1 if yi=1; w . xi + b ≤ -1 if yi = -1

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
BINARY CLASSIFIER: THE OPTIMIZATION PROBLEM

Maximal Margin           Find w and b such that
Classifier –               s
2
w   is maximized and for all (xi ,yi)
wTxi + b ≥ 1 if yi=1; wTxi + b ≤ -1 if yi = -1
optimization problem

Let w be of the form, w =Σαiyixi and b= yk- w . xk , k = arg maxk αk

Find α1…αN such that
Q(α) =Σαi - ½ΣΣαiαjyiyjxiTxj is maximized and
(1) Σαiyi = 0
(2) αi ≥ 0 for all i

Kernel function
Decision function f(x) = sgn (ΣαiyixiTx + b)

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
BETTER CLASSIFIERS

Non-separable case
n
1
J ( w,  )  w  C   j
2
Minimize
2      j 1
j
Relax constraints
w . xi + b ≥ 1- i if yi=1; w . xi + b ≤ -1+i if yi = -1
i
Further improvement
Map the non-separable data set to a higher dimensional space (using kernel
functions) where it becomes linearly separable

Φ: x → φ(x)

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
MULTIPLE CLASSES, MULTIPLE FEATURES

Given a new planar microstructure with its ‘s’ features given by
x1  {x1 , x1 ,...., x1 1 }, x2  {x1 , x2 ,...., xm2 },..., xs  {x1s , x2 ,...., xms }
T     1
2         m
T     2    2         2          T           s         s

find the class of 3D microstructure (y [1, 2,3,..., p] ) to which it is most likely to
belong.
p=3
B
One Against One Method:                          Class-A            C            Class-B

• Step 1: Pair-wise                                                                      C
classification, p( p  1) for a p
class problem 2
• Step 2: Given a data point,
select class with maximum                             A
B
votes out of p( p  1)                                               A       Class-C
2

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
CLASSIFICATION, TEXTURE RECOGNITION & REPRESENTATION

Key fibers in the
fundamental region
Level 1   α fiber     β fiber                       θ fiber

Level 2        MEASURES: ODF along a line in orientation
space, Average variation in values of the ODF

Equation of a fiber
in the fundamental
Level n             ODF value at the nodes
region

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
STATISTICAL LEARNING TOOLBOX

Library of training samples from
numerical & experimental tests

I) Training steps
1. Identify classes
2. Develop feature measures
3. Train statistical models like
Support Vector Machines

II) Model testing and validation

III) Identify reduced bases using the
POD/PCA analysis. These bases will
be used in process simulation.

Digital library

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
STATISTICAL LEARNING TOOLBOX

Can offline knowledge
be used to identify the
to the desired ODF?

Methodology:

1. Create a digital library
using experimental or
computational
snapshots of
microstructures.

2. Classification is based
on a combination of
process sequences.
Desired
ODF/texture                 3. Use multiple levels (or
hierarchies)
of classification.

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
STATISTICAL LEARNING TOOLBOX

CLASSIFICATION STEP 1
SVM Step 1 – Identify dominant process
Digital library

Tension (T)    Plane strain                Shear – 1
Compression (P)                (S1)

Given
ODF/texture                   Shear – 2         Shear – 3
(S2)              (S3)

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
STATISTICAL LEARNING TOOLBOX

CLASSIFICATION STEP 1
SVM Step – Identify dominant process

Tension (T)

Plane strain
Compression (P)

Shear – 1
(S1)

Given
Shear – 2
ODF/texture
(S2)

Shear – 3
(S3)

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
STATISTICAL LEARNING TOOLBOX

CLASSIFICATION STEP 1
Tension identified

Tension (T)

CLASSIFICATION STEP 2
SVM Step – Identify
secondary process

Given
ODF/texture

T+P          T+S1             T+S2                T+S3

Secondary modes – Secondary process identified
as plane strain compression

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
STATISTICAL LEARNING TOOLBOX

CLASSIFICATION STEP 1
Tension identified

Tension (T)

Tree structure
classification
Given
ODF/texture
T+P
CLASSIFICATION STEP 2
Plane strain compression

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
SYNERGY BETWEEN MICROSTRUCTURE & PROCESSING

CLASSIFICATION STEP 1
Actual                     Tension identified
process
T+P+S1                               Tension (T)
CLASSIFICATION
STEP 3
T + P + Shear-1

Given                                                 T+P+S1
ODF/texture
T+P
CLASSIFICATION STEP 2
Tension + Plane strain
compression

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
DEVELOPMENT OF MULTI-STAGE PROCESSES

Initial Product              Based on the statistical “designer knowledge”, evaluate
Node:       practicable stage number (n) and select a process sequence p
Intermediate   from all feasible paths (j=1 … m), such that:
1st Stage               preform

m                 n

J=1
Cost
min Function   =         Cost
of Dies
+ Consumption + Material
Energy
Usage
Arc:                         i=1
Processing
Stage      such that:
• Equipment constraint (press force, ram speed,
maximum stroke, etc)
ith Stage
• Process temperature constraint
• Other process constraints

Finishing
Stage(nth)                                     •   Number of stages - n
Final                 •   Force constraints for each stage
Product                •   Stroke allocation for each stage
Optimal Path (pth)                •   Stage temperature, etc.
Feasible Paths (jth)

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
SAMPLE PROBLEM OF PROCESS CONTROL

Identify the process sequence and process parameters to obtain the
following ODF for a given F.C.C material

Key assumptions:
1. 2 unknown process stages
2. Each stage acts for 10
seconds
3. Random initial (stage 1)
microstructure
Find:
1. The type of the two
unknown stages
each stage, i.e. find
α = {α1, α2, α3, α4, α5}

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
SAMPLE PROBLEM OF PROCESS CONTROL

Statistical   SVM CHARACTERIZATION RESULTS
Stage 1 – Tension
Methods        Stage 2 – Plane strain compression

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
SYNERGY BETWEEN MODEL REDUCTION & CLASSIFICATION

POD MODEL REDUCTION
Desired ODF

Process 2
CLASSIFICATION            T+P
TREE

COMBINED POD BASIS

Process 1            PROCESS TUNING BY OPTIMIZATION
Tension

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
SAMPLE PROBLEM OF PROCESS CONTROL

Statistical            SVM CHARACTERIZATION RESULTS
Stage 1 – Tension
Methods                Stage 2 – Plane strain compression

PROCESS SIMULATION
 polycrystalline plasticity
 Visco-plastic analysis

20
PROCESS DESIGN
 Design for process strain rates
Best objective function

 Heuristic design algorithm –
simulated annealing
10                                                        Initial guess {0.3, 0.0}

OPTIMUM SOLUTION
Stage 1: α = {0.68, 0, 0, 0, 0}
0
0    25            50                75
Simulated annealing iteration index
100      Stage 2: α = {0, 0.28, 0, 0, 0}

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
SAMPLE PROBLEM OF PROCESS CONTROL

Process identified: Stage 1 – Tension, Stage 2 – Plane
strain compression
Stage 1: α = {0.68, 0, 0, 0, 0}
Stage 2: α = {0, 0.28, 0, 0, 0}
1

0.9

0.8

0.7

0.6
Stage 2

Desired ODF                 0.5
ODF from optimum
0.4

0.3
solution
0.2

0.1

0
Global solution
0   0.1   0.2   0.3   0.4     0.5   0.6   0.7   0.8   0.9   1
Stage 1

Original process used to generate desired ODF:
Stage 1 – Tension, Stage 2 – Plane strain compression
Stage 1: α = {0.7, 0, 0, 0, 0}
Stage 2: α = {0, 0.3, 0, 0, 0}

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
MICROSTRUCTURE LIBRARY FOR CLASSIFICATION

Input Image

Feature Detection

Classifier

Quantification using incremental PCA within classes

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
MICROSTRUCTURE REPRESENTATION: PCA

Let 1 ,  2 ,.....  n be n images.         i  i  
4.   Create correlation matrix (Lmn)
1.     Vectorize input images
2.     Create an average image                 Lmn   mT  n
1 n
 =  i
n i 1                 5.   Find eigen basis (vi) of the
correlation matrix
3.     Generate training images
Lvi  i vi
Reduced basis                         6.   Eigen faces (ui) are generated from
the basis (vi) as

ui  vij  j
Data               7.   Any new face image (  ) can be
Points                  transformed to eigen face
components through ‘n’ coefficients
(wk) as,

Representation coefficients
 k  u k T (   )

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
PREPROCESSING OPERATIONS

Inputs: Microstructure Image (*.bmp Format), Magnification , Rotation
(With respect to rolling direction)

DIGITIZATION
Conversion of RGB format of
*.bmp file to a 2D array which is
used for image operations

PREPROCESSING
Brings the image to the library
format
(RD : x-axis, TD : y-axis)
– Rotate and scale image
– Image enhancement
– Boundary detection for feature   Preprocessing based on user inputs of magnification
extraction                      and rotation

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
ROSE OF INTERSECTIONS FEATURE - ALGORITHM

Identify intercepts of lines with
grain boundaries plotted within
a circular domain

Total number of intercepts
of lines at each angle is
given as a polar plot
called rose of
intersections

Count the number of
intercepts over several lines
placed at various angles.

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
POLYHEDRAL MICROSTRUCTURES: GRAIN SIZE FEATURES

Intercept lengths of
parallel network of lines
with the grain boundaries
are recorded at several
angles

The intercept length (x-axis) versus
number of lines (y-axis) histogram
is used as the measure of grain size.

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
CONVERSION OF FEATURES TO SVM TRAINING FORMAT

GRAIN FEATURES: GIVEN AS INPUT TO SVM TRAINING ALGORITHM

Class       Feature      Feature        Feature         Feature
number       value          number          value
1           1            23.32          2               21.52

2           1            24.12          2               31.52

Data point

CLASSIFICATION SUCCESS %

Total  Number of Number of             Highest           Average
images classes   Training images       success rate      success rate
375      11          40                95.82             92.53
375      11          100               98.54             95.80

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
CLASS HIERARCHY

Grain shapes

Grain sizes

Distance measure to identify new classes
Distance of image feature
from the average feature
vector of a class

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
PCA REPRESENTATION OF MICROSTRUCTURE - EXAMPLE

Input Microstructures

Reduced basis

Representation coefficients

Image-2 quantified
by 5 coefficients

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
EIGENVALUES & RECONSTRUCTION OVER THE BASIS

Significant eigen values capture
most of the image features
4      3      2      1
Reconstruction of microstructures are
based on the eigen values of the class
of images.
A fraction of the basis is sufficient to
represent an image.

1.Reconstruction         2. Reconstruction   3. Reconstruction        4. Reconstruction
with 100% basis          with 80% basis      with 60% basis           with 40% basis

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
INCREMENTAL PCA (IPCA) METHOD

• For updating the representation basis within a class when new microstructures are
added to the classes in real-time.
• Basis update is based on a error measure of the reconstructed and original image over
the existing basis
point                Updated Basis

IPCA (Skocaj & Leonardis, 2002):
• Storage: Microstructure images          Noisy data
do not need to be stored and are          (outlier)
• Computational cost is
minimized.
• Robust extension possible that
would adjust for erroneous images                               Non-robust PCA
basis is affected by
noise

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
REPRESENTATION FORMAT FOR MICROSTRUCTURE

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]

Improvement of microstructure representation due to classification

Reconstruction          Improvement in
Original image                                                        Reconstruction with
(without basis          reconstruction
update): A class        (without basis            basis update
with 25 images          update): class of
60 images

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
MOTIVATION

1. Creation of 3D microstructure models for property analysis 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.
2D Imaging techniques                                           Database
Pattern recognition

vision
Microstructure
Analysis

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
PATTERN RECOGNITION STEPS

PATTERN RECOGNITION : A DATA-DRIVEN OPTIMIZATION TOOL
•Feature matching for reconstruction of 3D microstructures Real-time
•Texture(ODF) classification for process path selection
•Model reduction: Increases efficiency of techniques like PCA/POD
•Adaptive selection of reduced order (POD) modes for faster analysis

A library of microstructures from
DATABASE CREATION
experiments or physical models
Extraction of statistical           FEATURE EXTRACTION
features from the database
Creation of a microstructure
TRAINING
class hierarchy
Prediction of 3D reconstruction,
PREDICTION
process paths, etc,

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
DATABASE CREATION

3D Microstructure database from:
• Experiments – Serial section reconstruction, X-ray computed tomography,
scanning laser confocal microscopy
• Models of microstructure evolution – Phase field methods, Monte Carlo method,
cellular automata, etc.

MC grain growth model – polyhedral       MC solidification model – Two-phase
microstructures                          microstructures
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
3D MICROSTRUCTURE RECOGNITION: TWO-CLASS PROBLEM

Training Features
(using lower order
descriptors)

Class        Feature Vector (x) – single feature type (Grain size
(y)          feature)
Number        1            21.30             160.12                   20.01            9.52
of training   -1           24.02             52.15                    14.08            36.52
sets          1            20.10             158.20                   25.30            11.30
=5            1            23.32             154.12                   23.01            2.52
-1           24.12             52.65                    12.08            31.52
Feature dimensionality: 4

New Feature (From a 2D image) – To what 3D class does this belong?
21.45            153.14           24.10             2.31

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory

Heyn Intercept histogram data set (grain size
feature of polyhedral microstructures)

Data set dimensionality: 21
Total data sets: 120
Number of classes: 6

Cross validation sets   Probabilistic neural network   Linear discriminant Decision tree SVM (Accuracy %)
10                       81.6712                   81.3924         81.1538         85.9504
6                        81.4676                   81.4738         80.7619         85.9504
5                        81.1117                   80.5657          80.401          85.124
4                         80.701                   80.9036         79.9032         84.2975
3                        78.9123                   80.2818         79.0244         84.7769

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
RECONSTRUCTION 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
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
STEREOLOGICAL ESTIMATES OF 3D GRAIN SIZES

The stereological integral equation for estimating the 3D grain size
distribution from a 2D image

N a (1  Fa (s))
 b  u (1  Gu ( s))dFv (u), s  0
Nv            s

3D grain size distribution based on
assumption that particles are
randomly oriented cubes

2D grain profile

3D grain

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
TWO PHASE MICROSTRUCTURE: CLASS HIERARCHY
Feature vector :
Feature:                                 Three point
3D Microstructures                        3D Microstructures
Autocorrelation                                 probability
function                                       function

g                             Class - 1

r mm
Class - 2

LEVEL - 1                                       LEVEL - 2
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
TWO PHASE MICROSTRUCTURES

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

Autocorrelation                           3 point probability
function                                  function

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
ELASTIC PROPERTIES

310                                                        HS bounds
BMMP bounds
290                                                        Experimental
FEM
Youngs Modulus (GPa)

270

250

230

210
3D image Derived
190        through pattern
recognition       Experimental image
170
0          200           400         600            800             1000
Temperature (deg-C)

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
MICROSTRUCTURE REPRESENTATION USING PCA

A DYNAMIC LIBRARY
APPROACH
• Classify microstructures based
on lower order descriptors.                      Does not decay to zero
• 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.
• Dynamically update the basis
and the representation for new
microstructures
COMMON-BASIS FOR MICROSTRUCTURE
REPRESENTATION

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
PCA MICROSTRUCTURE RECONSTRUCTION

Basis Components
Reconstruct
using two
X 5.89              basis
components
+
X 14.86
Project
onto basis                                              Pixel value
round-off

Perfect representation using just 2 coefficients

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
INCORPORATING STEREOLOGY EFFECTS

•                                             Effect of stereology (Bunge 1998)                        Desired Property
ratio of max stereological
influence to max texture

0.3
0.25

Young's Modulus
Voigt upper bound
0.2
influence

0.15                 Cu          Cu-Zn
0.1            Fe
0.05                                                                                 Reuss lower
Al
0                                                                                    bound
W
0                    5             10
Crystal anisotropy                RD                                TD

Texture fixes the bounds, stereology specifies the actual values
• Figure above shows with which ‘weights’
texture and stereology must be controlled for a
desired property.
•Stereology effects can be incorporated using                                       Plates    Equiax     Fibres
higher order statistical functions.                                                    Reuss, Hill and Voigt
structures

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
THERMAL PROCESSING: RECOVERY & RECRYSTALLIZATION

• Rearrangement of dislocations (recovery) and nucleation of strain-free
grains (recrystallization) changes the texture and hence the properties of
materials.
•Current models explain but do not predict recrystallization textures, making
the process of controlling properties extremely challenging.

KEY QUESTIONS FOR MODELLING
• How many nuclei, what nucleation
rates?
• Orientation distribution of nuclei?
• Growth rate of nuclei?          Stored energies
• How to select the nucleation sites?
State Var (MPa)
37.6207
37.0812
36.5418
36.0023
35.4628

•Which models to use? (MC, continuum)                      34.9233
34.3839
33.8444
33.3049

•What assumptions can be made? (site
32.7655
32.226               ODF
saturation etc.)

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
ROBUST DEFORMATION PROCESS DESIGN (AFOSR Project)

Key question
How robust is the design solution to material and process variability and potential
deficiencies in modeling of a process, characterization of model parameters or
description of model boundary conditions that is solely due to lack of knowledge?

We are                                       Robust design environment
developing a
probabilistic       Required product with
Are levels
desired material                  Are PDFs
framework to        properties and shape            of design variables   Yes      of uncertainty (PDFs) in   Yes
other process conditions
analyze and            with specified                   technically
tolerable?
confidence (output                  feasible?
design               PDFs from SSFEM)                    No                               No
processes
Interface with digital library and expert               Can
that result in                                              advice to modify design objectives,       No        we obtain the
products with                                               material models, process models                    PDFs by existent
testing?
desired
Yes
properties
Reference                                                                                Update model PDFs
within limits of     input and                    High                                                          and database
variability (i.e.     process                 performance                                                       (digital library)
MATERIALS TESTING
conditions                computing
with given                                    environment
DRIVEN BY DESIGN
PDFs                                         ROBUSTNESS LIMITS
robustness).
BAYESIAN INFERENCE

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
SPECTRAL STOCHASTIC METHODS IN MATERIALS DESIGN

Uncertainty characterization by joint       A rigorous spectral framework for the
probability distribution functions         analysis and design of materials
 Ideal for formulation and solution of
inverse design problems using
stochastic optimization algorithms
 Methodology for using DATA ACROSS
LENGTH SCALES and for estimation of
properties (Stochastic VMS method)

• For each uncertainty input, a             Provides ability to identify which
property affects the optimal design to
probability distribution is attributed
what extent
• Statistical description of uncertainty
 Allows introduction of robustness
• Suitable for most engineering            limits on the desired properties in the
(continuum) systems                        product –declare a range of properties
that from performance point of view
 Represent uncertain quantities in        will be acceptable
a Fourier like expansion of random          Allows evaluation of optimal
variables - Karhunen Loève /               MATERIALS TESTING & METROLOGY
Polynomial Chaos expansion                 driven by product robustness limits

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
DATA-DRIVEN BAYESIAN INFERENCE FOR MATERIALS

 Framework for DECISION MAKING,                              Ideal for PROGNOSIS – Use Bayesian
ESTIMATION, CLASSIFICATION &                               inference and microstructure
ROBUST DESIGN                                              signatures to infer (in nearly real-time)
p ( Y | ) p( )                         component lifing and reliability
p ( | Y )                        p ( Y | ) p ( )
p (Y )                                Methodology for using DATA ACROSS
LENGTH SCALES and for estimation of
 MCMC – Markov Chain Monte Carlo                            properties
 Ideal for modeling the propagation of
UNCERTAINTY – from errors in
modeling to measurement uncertainty
 BAYESIAN techniques provide the
BEST INTERPRETATION for polluted
data in comparison with any other
method

 Spatial statistics – MRF                                    ROBUST design solutions
 Prior modeling, inherent                                    Allows evaluation of optimal
regularization: statistically-stated                        MATERIALS TESTING & METROLOGY
inverse problems are well-posed                             driven by product robustness limits
problems!

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
INFORMATION THEORETIC APPROACH TO MATERIALS MODELING

Physics Chemistr Material Engineering
Property averaging
Interfacial energies
Inter-Atomic
Potentials
s

Micro-             Continuum
y

structural
Atomistic
Electronic
Length Scales

1 nm                 1 mm                     1 mm                     1m
 Uncertainty propagation across length scales                                       What do you need to know, at what
 Information is lost in upscaling                                                  scales and to what accuracy level to
 Spatial statistics
performance related questions posed in
 Entropy in information theory                                                              the macro-scale ?

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
MULTISCALE BRIDGING: PHASE FIELD MODELS

Atomistic and                 Meso Scale                        Continuum Scale
dislocation length
scales                   Phase Field                       Continuum
Thermodynamic              (Evolution of Field              Deformation Problem
variables                   Variables)
Mobilities                                             Couple Field variables &
Interfacial energies                                           local free energy
Interstitial                                                   functions
/Substitutional Stress
Nucleation Models     Crystallographic Lattice
Parameters
• Libraries need to be developed at each length scale to provide data for upscaling
• Realistic analysis, design and control tools require development of consistent
model reduction techniques for phase field models: Adaptive proper orthogonal
decomposition (POD), centroidal Voronoi tessellations (CVT) & wavelet methods

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
REDUCED-ORDER METHODS FOR MICROSTRUCTURE CONTROL

PHASE FIELD MODEL
REDUCTION AND CONTROL
Control of solidification: POD
reduction (Volkwein - 2001)
updating basis libraries)
(Ravindran - 2002)
Uncontrolled case           Controlled case

Atom deposition: Control of
roughness in thin films using kinetic
MC models (Lou & Christofides
2003)
POD basis approach: Banks
2002
Uncontrolled case           Controlled case

CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory

```
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
 views: 18 posted: 10/4/2012 language: Unknown pages: 66