A data mining approach for the design of optimized polycrystalline materials

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A data mining approach for the design
ofoptimized polycrystalline materials
Veera Sundararaghavan and Prof. Nicholas Zabaras
Materials Process Design and Control
Laboratory Sibley School of Mechanical and
Aerospace Engineering188 Frank H. T. Rhodes
Hall Cornell University Ithaca, NY
14853-3801 Email zabaras_at_cornell.edu URL
http//www.mae.cornell.edu/zabaras/
Materials Process Design and Control Laboratory
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MOTIVATION FOR MICROSTRUCTURE BASED DESIGN
Materials Process Design and Control Laboratory
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MICROSTRUCTURE-PROPERTY-PROCESSING
Microstructure
Identify processes for desired microstructure
for optimal properties.
TEM
3 FOLD DESIGN

• Texture
• Defects
• Composition

Processing
Realistic process sequence selection
Materials and process modeling
Materials Process Design and Control Laboratory
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MATERIALS DESIGN FRAMEWORK
Machine learning schemes
Computational process design simulator
Microstructure Information library
Virtual materials design framework
Virtual process simulations to evaluate alternate
designs
Accelerated Insertion of new materials
Optimization of existing materials
Tailored application specific material properties
Materials Process Design and Control Laboratory
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DATABASE FOR POLYCRYSTAL MATERIALS
Multi-scale microstructure evolution models
Statistical Learning
Database
Feature Extraction
Divisive Clustering
Class hierarchies
Class Prediction
Database
Reduced order basis generation
Tension process basis
Materials Process Design and Control Laboratory
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DESIGNING MATERIALS WITH TAILORED PROPERTIES
Macro problem driven by the macro-design variable
ß
Multi-scale Computation
Micro problem driven by the velocity gradient L
Bn1
Fn1
B0
L L (X, t ß)
Polycrystal plasticity
L velocity gradient
Data mining techniques
Reduced Order Modes
Database
Design variables (ß) are macro design variables
Processing sequence/parameters
Design objectives are micro-scale averaged
material/process properties
Materials Process Design and Control Laboratory
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FEATURES OF AN ODF ORIENTATION FIBERS
Fibers h1,2,3, y 1,0,1
Sample Axis y
For a particular (h), the pole figure takes
values P(h,y) at locations y on a unit sphere.
angle
Point y (1,0,1)
1,2,3 Pole Figure
Crystal Axis h
Integrated over all fibers corresponding to
crystal direction h and sample direction y
Points (r) of a (h,y) fiber in the fundamental
region
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SIGNIFICANCE OF ORIENTATION FIBERS
Lower order features in the form of pole density
functions over orientation fibers are good
features for classification due to their close
affiliation with processes
Important fiber families lt110gt uniaxial
compression, plane strain compression and simple
shear. lt111gt Torsion, lt100gt,lt411gt fibers
Tension a fiber (ND lt110gt ) b fiber FCC
metals under plane strain compression
z-axis lt110gt fiber BB
Uniaxial (z-axis) Compression Texture
z-axis lt111gt fiber CC
During deformation, Transport of crystals is
structured relative to orientation fiber families
z-axis lt100gt fiber AA
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LIBRARY FOR TEXTURES
Uni-axial (z-axis) Compression Texture
110 fiber family
Feature
q fiber path corresponding to crystal direction
h and sample direction y
z-axis lt110gt fiber (BB)
Materials Process Design and Control Laboratory
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SUPERVISED CLASSIFICATION USING SUPPORT VECTOR
MACHINES
Multi-stage classification with each class
affiliated with a unique process
Tension (T)
Stage 1
Stage 2
Stage 3
Identifies a unique processing sequence Fails to
capture the non-uniqueness in the solution
Given ODF/texture
Materials Process Design and Control Laboratory
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UNSUPERVISED CLASSIFICATION
Find the cluster centers C1,C2,,Ck such that
the sum of the 2-norm distance squared between
each feature xi , i 1,..,n and its nearest
cluster center Ch is minimized.
Each class is affiliated with multiple processes
Cost function
Feature Space
DATABASE OF ODFs
Clusters
Identify clusters
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ODF CLASSIFICATION

• Automatic class-discovery without class labels.
• Hierarchical Classification model
• Association of classes with processes, to
  facilitate data-mining
• Can be used to identify multiple process routes
  for obtaining a desired ODF

ODF 2,12,32,97
One ODF, several process paths
Data-mining for Process information with ODF
Classification
Materials Process Design and Control Laboratory
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DATABASE STRUCTURE
DATABASE
Process sequence-2 New process parameters ODF
history Reduced basis
Process sequence-1 Process parameters ODF
history Reduced basis
New dataset added
Desired texture/property
Classifier
Adaptive basis selection
Process
Reduced basis
Optimization
Probable Process sequences Initial parameters
Stage - 1
Stage - 2
Optimum parameters
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PROCESS PARAMETERS LEADING TO DESIRED PROPERTIES
ODF Classification
Database for ODFs
Property Extraction
Identify multiple solutions
Velocity Gradient
Different processes, Similar properties
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K-MEANS ALGORITHM FOR UNSUPERVISED CLASSIFICATION

• Lloyds Algorithm
• Start with k randomly initialized centers
• Change encoding so that xi is owned by its
  nearest center.
• Reset each center to the centroid of the points
  it owns.
• Alternate steps 1 and 2 until converged.

• User needs to provide k, the number of
  clusters.

But, No. of clusters is unknown for the texture
classification problem
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INFORMATION CRITERION FOR IDENTIFYING NO. OF
CLUSTERS
Maximum likelihood of the variance assuming
Gaussian data distribution
Probability of a point in cluster i
Log-likelihood of the data in a cluster
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CENTROID SPLIT TESTS

• X-MEANS algorithm
• Start with k clusters found through k-means
  algorithm
• Split each centroid into two centroids, and move
  the new centroids along a distance proportional
  to the cluster size in an arbitrarily chosen
  direction
• Run local k-means (k 2) in each cluster
• Accept split cluster in each region if BIC(k
  1) lt BIC(k 2)
• Test for various initial values of k and
  select the k with maximum overall BIC

Materials Process Design and Control Laboratory
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COMPARISON OF K-MEANS AND X-MEANS
Local Optimum produced by the kmeans algorithm
with k 4
Cluster configuration produced by k-means with k
6 Over-estimates the natural number of clusters
Configuration produced by the x-means algorithm
Input range of k 2 to 15. x-means found 4
clusters from the data-set based on the Bayesian
Information Criterion
Materials Process Design and Control Laboratory
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MATERIAL POINT SIMULATOR
ORIENTATION DISTRIBUTION FUNCTION A(r,t)

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

ODF EVOLUTION EQUATION EULERIAN DESCRIPTION
reorientation velocity
Any macroscale property lt ? gt can be expressed
as an expectation value if the corresponding
single crystal property ? ( ,t) is known.
Materials Process Design and Control Laboratory
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CRYSTAL CONSTITUTIVE RELATIONSHIPS
SALIENT FEATURES Viscoplastic rate dependent
model no hardening Extended Taylor hypothesis
assumed macro scale velocity gradient identical
to the crystal velocity gradient
Shear rate
Velocity gradient
Symmetric and spin components
D Macroscopic stretch Schmid tensor
Lattice spin W Macroscopic
spin Lattice spin vector
Reorientation velocity
Solve for the reorientation velocity and its
divergence given the velocity gradient
Divergence of reorientation velocity
Materials Process Design and Control Laboratory
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METHOD OF SNAPSHOTS
Method of snapshots
Suppose we had a collection of data (from
experiments or simulations) for the ODF
Solve the optimization problem
Is it possible to identify a basis
where
such that it is optimal for the data represented
as
Eigen-value problem
where
POD technique Proper Orthogonal Decomposition
Materials Process Design and Control Laboratory
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REDUCED ORDER MODEL DIRECT PROBLEM
Represent the ODF as

• Benefits
• Reduction in number of
  degrees of freedom
• PDE converted to ODE
• No stabilization required in the FE formulation

Reduced model for the evolution of the ODF
Initial conditions
Materials Process Design and Control Laboratory
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REDUCED ORDER SENSITIVITY PROBLEM
Reduced model for the evolution of the
sensitivity of the ODF
The reduced basis for the ODF ensemble has been
evaluated as
Observing that the basis is independent of the
macro design parameters, we conclude that the
basis generated in the direct analysis can be
used for the sensitivity problem.
where
Using this basis, the sensitivity of the ODF is
represented as follows
Initial conditions
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DESIGN OF PROCESS SEQUENCE
Velocity gradient
Modes
Tension/ compression
Shear
Plane strain compression
Rotation
Shear
Design vector a a1, a2, ..., anT (n-stage
problem)
Design problem Determine the process sequence so
as to obtain desired properties in the final
product
Materials Process Design and Control Laboratory
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A TWO-STAGE PROBLEM
Reduced Basis
DATABASE
f(2)
f(1)
Process 2 Plane strain compression
a 0.3515
Process 1 Tension
a 0.9539
Initial Conditions Stage 1
Initial Conditions- stage 2
Sensitivity of material property
Direct problem a
Sensitivity problem
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PROCESS DESIGN WITH A FIXED BASIS
Initial basis based on Tension process
1,0,0,0,0
The basis functions used for the control problem
not only needs to represent the solution but also
the textures arising from intermediate iterates
of the design variable
Final process iterate 1 -0.5 -0.25 0 0
Actual ODF corresponding to the process identified
ODF reconstructed using the initial fixed basis
Materials Process Design and Control Laboratory
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ADAPTIVE REDUCED-ORDER MODELING
Stage 1 Compression a -0.8 Stage 2 PSC a
-1.0
Sensitivity problem
Direct problem
Stage 2 sensitivity Adaptive reduced order
model (Threshold e 0.05)
Stage 2 sensitivity finite differences (da
0.01)
Reduced-order model
Full-order model
Materials Process Design and Control Laboratory
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MULTIPLE PROCESS ROUTES
Desired Youngs Modulus distribution
Stage 1 Tension a 0.9495
Stage 1 Tension a 0.9699
Stage 2 Rotation-1 a -0.2408
Stage 2 Shear-1 a 0.3384
Classification
Magnetic hysteresis loss distribution
Stage 1 Shear-1 a 0.9580
Stage 1 Shear -1 a 0.9454
Stage 2 Plane strain compression (a
-0.1597 )
Stage 2 Rotation-1 (a -0.2748)
Materials Process Design and Control Laboratory
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DESIGN FOR DESIRED ODF A MULTI STAGE PROBLEM
Desired ODF
Optimal- Reduced order control
Stage 1 Plane strain compression (a1 0.9472)
Stage 2 Compression (a2 -0.2847)
Initial guess, a1 0.65, a2 -0.1
Full order ODF based on reduced order control
parameters
Materials Process Design and Control Laboratory
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DESIGN FOR DESIRED MAGNETIC PROPERTY
Crystal lt100gt direction. Easy direction of
magnetization zero power loss
h
External magnetization direction
Stage 1 Shear 1 (a1 0.9745)
Stage 2 Tension (a2 0.4821)
Materials Process Design and Control Laboratory
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DESIGN FOR DESIRED YOUNGS MODULUS
Stiffness of F.C.C Cu in crystal frame
Elastic modulus is found using the polycrystal
average ltCgt over the ODF as,
Stage 1 Shear (a1 -0.03579)
Stage 2 Tension (a2 0.17339)
Materials Process Design and Control Laboratory
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BETTER MECHANICS

1. ThermoElasto-ViscoPlastic analysis
2. Coupled length scale design methods
3. Slip Twinning in FCC, BCC and HCP materials

x(X, t)
F(X, t)
Texture map ?
Texture map ?t
r
s
Reorientation map r(s, t)

Reference fundamental region
Current fundamental region
CONTINUUM MICROSTRUCTURE DESCRIPTION
Materials Process Design and Control Laboratory
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Electron scale database
DFT
Alloy systems
Hyperplanes quantify correlation of local length
scale features with the objective and higher
length scale effects
Phase Field
DD
Hierarchical class structure at each length scale
Statistical features at the local length scale
Dynamic update of class structures with new data
Reduced models for higher length scales
Objective
Design decisions
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RELEVANT PUBLICATIONS
S. Acharjee and N. Zabaras, A proper orthogonal
decomposition approach to microstructure model
reduction in Rodrigues space with applications
to the control of material properties, Acta
Materialia 51, 56275646, 2003.
CONTACT INFORMATION
Prof. Nicholas Zabaras
Materials Process Design and Control
Laboratory Sibley School of Mechanical and
Aerospace Engineering188 Frank H. T. Rhodes
Hall Cornell University Ithaca, NY
14853-3801Email zabaras_at_cornell.edu URL
http//www.mae.cornell.edu/zabaras/
Materials Process Design and Control Laboratory