Sethuraman Sankaran and Nicholas Zabaras

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Maximum entropy approach for statistical modeling
of three-dimensional polycrystal microstructures
Sethuraman Sankaran and 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 ss524_at_cornell.edu,
zabaras_at_cornell.edu URL http//mpdc.mae.cornell.e
du/
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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
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Why do we need a statistical model?
Different statistical samples of the manufactured
specimen
When a specimen is manufactured, the
microstructures at a sample point will not be the
same always. How do we compute the class of
microstructures based on some limited information?
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Development of a mathematical model
Random variable 2 High dimensions
Random variable 1 (scalar or vector)
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
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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
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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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Physics of phase field method

• 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
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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

• Orientation Distribution Function

Sample reference frame
Volume fraction of crystals with a specific
orientation
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
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Review
Given Microstructures at some points Obtain PDF
of microstructures
ODF (a function of 145 random parameters)
Grain size
Know microstructures at some points
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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 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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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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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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Stochastic modeling of microstructures
Sampling using grain size distribution
Sampling using mean grain size
Match the PDF of a microstructure with PDF of
grain sizes computed from MaxEnt
Each microstructure is referred to by its mean
value.
Strongly consistent scheme
Weakly consistent scheme
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Heuristic algorithm for generating voronoi centers
Given grain size distribution Construct a
microstructure which matches the given
distribution
Generate sample points on a uniform grid from
Sobel sequence
No
Forcing function
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

• Microstructure is a representation of a material
  point at a smaller scale
• 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 Strong sampling
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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 bounds in properties.
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
using phase field simulations
pmf reconstructed using MaxEnt
0.2
0.15
Probability mass function
Comparison of MaxEnt grain size distribution with
the distribution of a phase field microstructure
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 strongly consistent microstructures
Computing microstructures using the Sobel
sequence method
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Reconstructing strongly consistent
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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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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Numerical Example Weak sampling
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3D microstructures Grain boundary topology
network
Distribution of microstructures computed using
MaxEnt technique using mean grain size as a
microstructural feature
A grain boundary network of one microstructural
sample
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Samples of microstructures computed at different
points of the PDF
Microstructures computed using the mean grain
sizes, which are sampled from the PDF
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Randomness in texture
Samples of the reconstructed ODF function
Expected ODF distribution that is given as a
constraint to the MaxEnt algorithm
Each grain is attributed an orientation that is
sampled from a MaxEnt distribution of ODFs. Some
of the samples of textures that are constructed
are shown in the figure above.
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Meshing microstructure samples using hexahedral
elements (Cubit TM)
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Extremal bounds of homogenized stress-strain
properties
Aluminium polycrystal with rate-independent
strain hardening. Pure tensile test.
Statistical variation of homogenized
stress-strain curves.
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Future work Diffusion in microstructures 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, 2006
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