Nicholas Zabaras and Sethuraman Sankaran

1
Computing property variability of polycrystals
induced by grain size and orientation
uncertainties
Nicholas Zabaras and Sethuraman Sankaran
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,
ss524_at_cornell.edu URL http//mpdc.mae.cornell.edu
/
2
Research Sponsors
U.S. AIR FORCE PARTNERS Materials Process
Design Branch, AFRL Computational
Mathematics Program, AFOSR
ARMY RESEARCH OFFICE
Mechanical Behavior of Materials Program
NATIONAL SCIENCE FOUNDATION (NSF) Design
and Integration Engineering Program
CORNELL THEORY CENTER
3
Why do we need a statistical model?
A macro specimen has thousands of microsructures
and in practice the microstructure at different
material points will be different. How do we
compute microstructures at all points in the
specimen?
4
Microstructure as a class
We compute a microstructural class where there is
variability in microstructural features within a
class but certain experimentally obtained
information is incorporated within the class.
TOPOLOGICAL FEATURES
A class is defined based on statistics mean
grain size, variance of grain sizes, mean texture
etc.
MICROSTRUCTURE CLASS
A microstructure in the entire specimen is
considered to be a sample from this class where a
probability is assigned to each microstructure in
the class
TEXTURAL FEATURES
5
Development of a mathematical model
Compute a PDF of microstructures
Orientation Distribution functions
Grain size features
ODF (a function of 145 random parameters)
Grain size
Assign microstructures to the macro specimen
after sampling from the PDF
6
The main idea
Experimental microstructures
Phase field simulations
Extract features of the microstructure
Geometrical grain size Texture ODFs
Compute a PDF of microstructures
MAXENT
Compute bounds on macro properties
7
Generating input microstructures The phase field
model
Define order parameters
where Q is the total number of orientations
possible
Non-zero only near grain boundaries
Define free energy function (Allen/Cahn 1979,
Fan/Chen 1997)
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Phase field model (contd)

• Driving force for grain growth
• Reduction in free energy thermodynamic driving
  force to eliminate grain boundary area
  (Ginzburg-Landau equations)

kinetic rate coefficients related to the mobility
of grain boundaries
Assumption Grain boundary mobilties are constant
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Phase Field Problem parameters

• Isotropic mobility (L1)
• Discretization
• problem size 75x75x75
• Order parameters
• Q20
• Timesteps 1000
• First nearest neighbor approx.

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Input microstructural samples
2D microstructural samples
3D microstructural samples
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The main idea
Experimental microstructures
Phase field simulations
Extract features of the microstructure Geometrical
grain size Texture ODFs
Compute a PDF of microstructures
MAXENT
Compute bounds on macro properties
12
Microstructural feature Grain sizes
Grain size obtained by using a series 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
Grain size is computed from the volumes of
individual grains
3D microstructures
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Microstructural feature ODF
Crystal/lattice reference frame

• CRYSTAL SYMMETRIES?

Same axis of rotation gt planes Each symmetry
reduces the space by a pair of planes
Sample reference frame
crystal
RODRIGUES REPRESENTATION FCC FUNDAMENTAL REGION
n

• ORIENTATION SPACE
• Euler angles symmetries
• Neo Eulerian representation

Particular crystal orientation
Rodrigues parametrization
Cubic crystal
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The main idea
Experimental microstructures
Phase field simulations
Extract features of the microstructure Geometrical
grain size Texture ODFs
Tool for microstructure modeling
Compute a PDF of microstructures
MAXENT
Compute bounds on macro properties
15
Distribution of microstructures
ODF (a function of 145 random parameters)
Given Microstructures at some points Obtain
PDF of microstructures
Grain size
Know microstructures at some points
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MAXENT as a tool for microstructure reconstruction
Input Given average (and lower moments) of grain
sizes and ODFs Obtain microstructures that
satisfy the given properties

• Constraints are viewed as expectations and lower
  moments of features over a random field. Problem
  is viewed as finding that distribution whose
  ensemble properties match those that are given.
• Since, problem is ill-posed, we choose the
  distribution that has the maximum entropy.
• Microstructures are considered as realizations
  of a random field which comprises of randomness
  in grain sizes and orientation distribution
  functions.

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The MAXENT principle
E.T. Jaynes 1957
The principle of maximum entropy (MAXENT) states
that amongst the probability distributions that
satisfy our incomplete information about the
system, the probability distribution that
maximizes entropy is the least-biased estimate
that can be made. It agrees with everything that
is known but carefully avoids anything that is
unknown.

• MAXENT is a guiding principle to construct PDFs
  based on limited information
• There is no proof behind the MAXENT principle.
  The intuition for choosing distribution with
  maximum entropy is derived from several diverse
  natural phenomenon and it works in practice.
• The missing information in the input data is
  fit into a probabilistic model such that
  randomness induced by the missing data is
  maximized. This step minimizes assumptions about
  unknown information about the system.

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MAXENT a statistical viewpoint
MAXENT solution to any problem with set of
features is
Parameters of the distribution
Input features of the microstructure
Fit an exponential family with N parameters (N is
the number of features given), MAXENT reduces to
a parameter estimation problem.
Commonly seen distributions
Mean, variance given
Mean provided
No information provided (unconstrained optimiz.)
1-parameter exponential family (Poisson
distribution)
Gaussian distribution
The uniform distribution
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MAXENT as an optimization problem
Find
feature constraints
Subject to
features of image I
Lagrange Multiplier optimization
Lagrange Multiplier optimization
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Gradient Evaluation

• Objective function and its gradients
• Infeasible to compute at all points in one
  conjugate gradient iteration
• Use sampling techniques to sample from the
  distribution evaluated at the previous point.
  (Gibbs Sampler)

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Parallel Gibbs sampler algorithm
Improper pdf (function of lagrange multipliers)
continue till the samples converge to the
distribution
Go through each grain of the microstructure and
sample an ODF according to the conditional
probability distribution (conditioned on the
other grains)
Start from a random microstructure.
Each processor goes through only a subset of the
grains.

Processor r
Processor 1
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Microstructure modeling the Voronoi structure
p1,p2,,pk generator points.
Cell division of k-dimensional space
Voronoi tessellation of 3d space. Each cell is a
microstructural grain.
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Mathematical representation

• OFF file representation (used by Qhull package)
• Initial lines consists of keywords (OFF), number
  of vertices and volumes.
• Next n lines consists of the coordinates of each
  vertex.
• The remaining lines consists of vertices that
  are contained in each volume.

Volumes need to be hulled to obtain consistent
representation with commercial packages

• Brep (used by qmg, mesh generator)
• Dimension of the problem.
• A table of control points (vertices).
• Its faces listed in increasing order of dimension
  (i.e., vertices first, etc) each associated with
  it the following
• The face name, which is a string.
• The boundary of the face, which is a list of
  faces of one lower dimension.
• The geometric entities making up the face. its
  type (vertex, curve, triangle, or quadrilateral),
  
• its degree (for a curve or triangle) or
  degree-pair (for a quad), and
• its list of control points

Convex hulling to obtain a triangulation of
surfaces/grain boundaries
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Stochastic modeling of microstructures
Topological uncertainties within the
microstructure is utilized as microstructural
feature. Stochastic modeling refers to computing
microstructures whose grain size distribution is
computed using MaxEnt principle
Samples whose topological uncertainty matches the
given grain size distribution
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Applications
Diffusion in heterogeneous random media driven by
topological uncertainties
Region of small mean grain size
Heterogeneties in microstructure
Region of large mean grain size
Directional properties of the microstructure Diff
usion properties of the microstructures strongly
dictated by variabilities of grain size
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Monte Carlo techniques for matching grain size
distributions
Problem Generate voronoi-cell structures whose
grain size distribution matches grain size PDF
obtained using MAXENT

• Generate a database of microstructures using
  Monte Carlo schemes (on the generator points)
  based on voronoi tessellation
• Obtain correlation coefficient between the
  MAXENT and actual grain size measure. Accept
  microstructures whose correlation is above a
  cutoff.

Assign random voronoi centers.
Evaluate grain size distribution
correlation coefficient gt Rcut
Accept microstructure
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Heuristic algorithm for generating voronoi centers
Generate sample points on a uniform grid and each
point is associated with a grain size drawn from
the given distribution, d.
No
Yes
stop
Rcorr(y,d)gt0.95?
Objective is to minimize norm (F). Update the
voronoi centers based on F
Construct a voronoi diagram based on these
centers. Let the grain size distribution be y.
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The main idea
Experimental microstructures
Phase field simulations
Extract features of the microstructure Geometrical
grain size Texture ODFs
Tool for microstructure modeling
Compute a PDF of microstructures
MAXENT
Compute bounds on macroscopic properties
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(First order) homogenization scheme

1. Microstructure is a representation of a material
   point at a smaller scale
2. Deformation at a macro-scale point can be
   represented by the motion of the exterior
   boundary of the microstructure. (Hill, R., 1972)

Materials Process Design and Control Laboratory
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Numerical Example 2D microstructure
reconstruction
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2D random microstructures evaluation of property
statistics
Problem definition Given an experimental image
of an aluminium alloy (AA3302), properties of
individual components and given the expected
orientation properties of grains, it is desired
to obtain the entire variability of the class of
microstructures that satisfy these given
constraints.
Polarized light micrograph of aluminium alloy
AA3302 (source Wittridge NJ et al. Mat.Sci.Eng.
A, 1999)
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MAXENT distribution of grain sizes

Grain sizes Heyns intercept method. An
equidistant network of parallel lines drawn on a
microstructure and intersections with grain
boundaries are computed.
Input constraints in the form of first two
moments. The corresponding MAXENT distribution is
shown on the right.
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Reconstructed microstructures
Reconstruction of microstructures based on
correlation with the MAXENT grain size
distribution. All voronoi tessellations which
lead to a size distribution that has correlation
coefficient more than 0.9 are accepted.
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Assigning orientation to grains
Given Expected value of the orientation
distribution function. To obtain Samples of
orientation distribution function that satisfies
the given ensemble properties
Input ODF (corresponds to a pure shear
deformation, Zabaras et al. 2004)
Ensemble properties of ODF from reconstructed
distribution
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Evaluation of plastic property bounds
Orientations assigned to individual grains from
the ODF samples obtained using MAXENT.
Bounds on plastic properties obtained from the
samples of the microstructure
80
70
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Bounding plastic curves over a set
Equivalent Strain (MPa)
of microstructural samples
50
40
30
0
0.05
0.1
0.15
0.2
Equivalent Stress
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Numerical Example 3D microstructure
reconstruction
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3D random microstructures evaluation of
property statistics
Problem definition Given microstructures
generated using phase field technique, compute
grain size distributions using MaxEnt technique
as well as compute samples consistent with the
sampled distributions.
Input constraints macro grain size observable.
First four grain size moments , expected value of
the ODF are given as constraints.
Output Entire variability (PDF) of grain size
and ODFs in the microstructure is obtained.
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Grain size distribution computed using MaxEnt
0.25

Grain volume distribution
0.2
Comparison of MaxEnt grain size distribution with
the distribution of a phase field microstructure
0.15
Probability mass function
0.1
K.L.Divergence0.0672 nats
0.05
0

0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
Grain volume (voxels)
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Reconstructing microstructures
Computing microstructures using the Sobel
sequence method
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Reconstructing microstructures (contd..)
Computing microstructures using the Sobel
sequence method
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ODF reconstruction using MAXENT
Representation in Frank-Rodrigues space
Input ODF
Reconstructed samples using MAXENT
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Ensemble properties
Expected property of reconstructed samples of
microstructures
Input ODF
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Convergence analysis
How many samples are required to predict the
mean/standard deviation of the stress-strain
curves effectively?
Estimates
We say that converges to X if
Convergence in moments
Almost sure convergence
Convergence in probability
What we are interested
Increase the number of microstructure samples and
test if the mean and standard deviation values
converge at different locations in the
stress-strain curve
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Convergence analysis
2
1.8
A
A
1.6
D
1.4
C
1.2
B
Standard deviation of stress (MPa)
1
0.8
A
0.6
0.4
0.2
0
0
5
10
15
20
25
30
35
40
45
Number of samples
2
B
1.8
1.6
1.4
1.2
Standard deviation of stress (MPa)
1
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
30
35
40
45
Number of samples
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Convergence analysis contd..
2
1.8
C
1.6
1.4
D
1.2
C
1
B
Standard deviation of stress (MPa)
0.8
A
0.6
0.4
0.2
0
0
5
10
15
20
25
30
35
40
45
Number of samples
2
D
1.8
1.6
1.4
1.2
Standard deviation of stress (MPa)
1
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
30
35
40
45
Number of samples
46
Statistical variation of properties
Aluminium polycrystal with rate-independent
strain hardening. Pure tensile test.
Statistical variation of homogenized
stress-strain curves.
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Data-driven in-situ estimation of microstructure
classes
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Motivation
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Method
Use MAXENT for certain microstructure classes.
Output
Input
Statistical relation
Grain size moments (like mean, std, etc)
A class of microstructures satisfying the
information
Moment no.
Moment value
Problem statement We want to predict the class
of microstructures which satisfies a given
statistic of grain size moment. Input is a
2-tuple and output is a microstructural class.
The data is generated using MaxEnt.
TRAIN A NON-LINEAR STATISTICAL MAPPER
Generate MAXENT distribution for certain inputs
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Optimization
T number of microstructure classes that have
been pre-computed and stored in a database
Kullback-Leibler divergence
Distribution that is given from the database
Non-linear relationship between input moments and
parameters of the MaxEnt PDF
Distribution from training weights of the network
Parameters of the MaxEnt PDF
Input moment
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Technique-Backpropagation
Initialize weights and biases
Objective function
If less than tolerance, terminate
Gradients
Update weights/biases
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Information learning convergence for 2 input
moments
1.8
0.2
1.6
1.4
0.15
1.2
Probability mass function
0.1
1
Objective function
0.8
0.05
0.6
0
0.4
0
0.5
1
3
Grain size (

10
0.2
0
-0.2
50
100
150
0
Iterations
output predicted using information learning
output predicted using MaxEnt
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Information learning convergence for 3 input
moments
0.7
Distributions at various iterations
A
0.6
0.5
0.4
Objective function
0.3
B
0.2
0.1
C
0
20
40
60
80
100
120
140
Iterations
Information learning convergence plot
54
Recomputed MaxEnt distributions using information
learning
Database containing information about two moments
Database containing information about three
moments
Database containing information about four moments
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Conclusions

• We generated a class of microstructures based on
  insufficient data using maximum entropy method
  using (a) grain size features and (b) textural
  features
• Microstructures were reconstructed using the
  inverse voronoi tessellation technique
• Samples were interrogated using multiscale
  homogenization methods and we were able to
  compute statistics of plastic properties
• A learning algorithm to accelerate the
  computation of MaxEnt classes was found to be
  efficient in computation of the microstructure
  class

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Future work Diffusion in polycrystals induced by
topological uncertainty
Diffusivity properties in a statistical class of
microstructures
Statistical samples of microstructure at certain
collocation points computed using maximum entropy
technique
Limited set of input microstructures computed
using phase field technique
Variability of effective diffusion coefficient of
microstructure
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Information
RELEVANT PUBLICATIONS
S.Sankaran and N. Zabaras, Maximum entropy method
for statistical modeling of microstructures, Acta
Materialia, 2007
N. Zabaras and S.Sankaran, An information
theoretic approach to stochastic materials
modeling, IEEE Computing in Science and
Engineering, 2007
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//mpdc.mae.cornell.edu/
Materials Process Design and Control Laboratory